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Making History Count A Primer in Quantitative Methods for Historians
Making History Count introduces the main quantitative methods used in his-
torical research. The emphasis is on intuitive understanding and application of
the concepts, rather than formal statistics; no knowledge of mathematics
beyond simple arithmetic is required. The techniques are illustrated by appli-
cations in social, political, demographic, and economic history.
Students will learn to read and evaluate the application of the quantitative
methods used in many books and articles, and to assess the historical conclu-
sions drawn from them. They will see how quantitative techniques can open
up new aspects of an enquiry, and supplement and strengthen other methods
of research. This textbook will also encourage students to recognize the bene-
fits of using quantitative methods in their own research projects.
Making History Count is written by two senior economic historians with
considerable teaching experience on both sides of the Atlantic. It is designed
for use as the basic text for taught undergraduate and graduate courses, and is
also suitable for students working on their own. It is clearly illustrated with
numerous tables, graphs, and diagrams, leading the student through the
various key topics; the whole is supported by four specific historical data sets,
which can be downloaded from the Cambridge Unversity Press website at
http://uk.cambridge.org/resources/0521806631.
is Emeritus Professor of Economic History and
Emeritus Fellow of All Souls College, Oxford. His publications include
National Income, Expenditure and Output of the United Kingdom, 1855–1965
(1972), British Economic Growth, 1856–1973 (with R. C. O. Matthews and J.
Odling-Smee, 1982), and The European Economy between the Wars (with Peter
Temin and Gianni Toniolo, 1997).
is Associate Professor of History at the University of
Virginia. His publications include Capitalism in Context: Essays on Economic
Development and Cultural Change (ed., with John A. James, 1994),
Disintegration of the World Economy between the Wars (ed., 1996), and The
Economic Future in Historical Perspective (ed., with Paul David, 2002).
Making History Count A Primer in Quantitative Methods for
Historians
.
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press The Edinburgh Building, Cambridge , United Kingdom
First published in print format
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© Charles H. Feinstein and Mark Thomas 2002
2002
Information on this title: www.cambridge.org/9780521806633
This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.
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Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
hardback
paperback paperback
eBook (EBL) eBook (EBL)
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Contents
List of figures page xiii
List of tables xvii
List of panels xix
Preface xxi
Part I Elementary statistical analysis
Chapter 1 Introduction 3 1.1 Aims of the book 3
1.2 The four case studies and the data sets 6
1.3 Types of measurement 7
1.3.1 Cases, variables, and values 8
1.3.2 Cross-section and time-series variables 8
1.3.3 Levels of measurement 9
1.3.4 Populations and samples 10
1.3.5 Dummy variables 11
1.4 Simple notation 12
1.5 The equation of a straight line 13
1.6 Powers and logarithms 14
1.6.1 Powers 14
1.6.2 Logarithms 16
1.6.3 Logs as proportions and ratios 17
1.7 Index numbers 19
1.8 Trends and fluctuations 21
1.8.1 Measuring the trend by a moving average 22
1.8.2 Measuring the fluctuations by the ratio to the trend 25
Chapter 2 Descriptive statistics 33 2.1 Presentation of numerical data 33
2.1.1 Frequency distributions 33
v
2.1.2 Bar charts and histograms 36
2.1.3 Frequency curves 40
2.2 Measures of central tendency 42
2.2.1 The mean, the median, and the mode 43
2.2.2 Weighted averages and standardized rates 44
2.2.3 Percentiles, deciles, and quartiles 46
2.3 Measures of dispersion 47
2.3.1 The range and the quartile deviation 47
2.3.2 The mean deviation, the variance, and the standard deviation 47
2.3.3 The coefficient of variation 50
2.4 Measures of central tendency and dispersion with grouped data 51
2.4.1 The median and mean with grouped data 51
2.4.2 The variance and standard deviation with grouped data 52
2.5 The shape of the distribution 53
2.5.1 Symmetrical and skewed distributions 53
2.5.2 Measures of skewness 55
2.6 The normal distribution 56
2.6.1 Properties of the normal distribution 56
2.6.2 The standard normal distribution 61
2.7 Exercises for chapter 2 66
Chapter 3 Correlation 71 3.1 The concept of correlation 71
3.1.1 Correlation is not causation 72
3.1.2 Scatter diagrams and correlation 72
3.1.3 Outliers 74
3.2 The correlation coefficient 76
3.2.1 Measuring the strength of the association 76
3.2.2 Derivation of the correlation coefficient, r 77
3.2.3 Interpreting a correlation coefficient 82
3.2.4 An illustration of the use of correlation coefficients 83
3.3 Spearman’s rank correlation coefficient 86
3.4 Exercises for chapter 3 90
Chapter 4 Simple linear regression 93 4.1 The concept of regression 93
4.1.1 Explanatory and dependent variables 93
4.1.2 The questions addressed by regression 94
4.2 Fitting the regression line 95
4.2.1 How do we define a line? 95
4.2.2 What criterion should be adopted for fitting the line? 96
4.2.3 How do we find this best fitting line? 98
vi
4.2.4 The regression of Y on X versus the regression of X on Y 99
4.2.5 Fitting a linear trend to a time series 100
4.3 Measuring the goodness of fit 102
4.3.1 Explained variations and unexplained variations (residuals) 102
4.3.2 The coefficient of determination, r2 105
4.3.3 Some intuitive implications 106
4.4 Introducing the Ballantine 110
4.5 Exercises for chapter 4 112
Part II Samples and inductive statistics
Chapter 5 Standard errors and confidence intervals 117 5.1 Introduction 117
5.2 Sampling distributions 120
5.2.1 Sampling distributions and point estimates of the population
mean 120
5.3 Four fundamental statistical propositions 122
5.3.1 The Central Limit Theorem 122
5.4 Theoretical probability distributions 125
5.4.1 Empirical and theoretical frequency distributions 125
5.4.2 The Z- and t-distributions 131
5.5 Confidence intervals for a sample mean 136
5.5.1 Interval estimates and margins of error 137
5.5.2 Confidence interval for an estimate of a population mean 137
5.6 Confidence intervals for other sample statistics 140
5.6.1 Sample proportions 141
5.6.2 Correlation coefficients 141
5.6.3 Regression coefficients 143
5.7 Exercises for chapter 5 145
Chapter 6 Hypothesis testing 149 6.1 Testing hypotheses 149
6.1.1 A simple illustration 150
6.1.2 The five stages of hypothesis testing 151
6.2 The null hypothesis 151
6.2.1 Other assumptions of the model 152
6.3 How confident do we want to be about the results? 153
6.3.1 Two types of error and the null hypothesis 153
6.3.2 Significance levels and the critical region 155
6.3.3 One- and two-tailed tests 158
6.3.4 Statistical versus historical significance 160
vii
6.3.5 Conventional significance levels and prob-values 160
6.3.6 The critical region and the four cases of theft 162
6.4 Test statistics and their sampling distributions 163
6.4.1 The basic ideas 163
6.4.2 Z- and t-tests 164
6.5 Making a decision 167
6.6 Interpreting the results 170
6.7 Three illustrations of the use of t-tests 170
6.7.1 A t-test for a difference of means 170
6.7.2 A t-test for a correlation coefficient when the null hypothesis is
zero correlation 173
6.7.3 A t-test for a regression coefficient 175
6.8 Exercises for chapter 6 181
Chapter 7 Non-parametric tests 185 7.1 Introduction 185
7.2 Non-parametric tests for two independent samples 187
7.2.1 The Wald–Wolfowitz runs test 188
7.2.2 The Mann–Whitney U-test 193
7.2.3 The Kolmogorov–Smirnov test 198
7.2.4 Criteria for choosing between alternative tests 201
7.3 The �2-test 202
7.3.1 Contingency tables or cross-tabulations 203
7.3.2 The �2-test for a relationship between two variables 204
7.4 The one-sample runs test of randomness 210
7.5 One-sample non-parametric tests of goodness of fit 216
7.5.1 The one-sample �2-test 216
7.5.2 The one-sample Kolmogorov–Smirnov test 216
7.6 Non-parametric measures of strength of association 218
7.6.1 Four measures of association: �, Cramer’s V, the contingency
coefficient C, and Goodman and Kruskal’s tau 218
7.7 Exercises for chapter 7 224
Part III Multiple linear regression
Chapter 8 Multiple relationships 231 8.1 The inclusion of additional explanatory variables 231 8.2 Multiple regression 232
8.2.1 The coefficient of multiple determination, R2 232
8.2.2 Partial regression coefficients 234
8.2.3 Controlling for the influence of a second explanatory
variable 235
viii
8.2.4 Understanding the impact of additional explanatory
variables 238
8.2.5 The Ballantine for multiple regression 240
8.2.6 The least squares criterion in multiple regression 242
8.2.7 Standardized beta coefficients 243
8.2.8 The adjusted coefficient of multiple determination, R –2 245
8.2.9 Interpreting the intercept term 246
8.3 Partial and multiple correlation 248
8.3.1 The interpretation and order of partial correlation
coefficients 248
8.3.2 Removing the influence of the control variable 252
8.3.3 Calculation of partial correlation coefficients 253
8.3.4 The coefficient of multiple correlation, R 254
8.4 Exercises for chapter 8 255
Chapter 9 The classical linear regression model 258 9.1 Historical research and models of relationships between variables 258
9.1.1 Quantitative methods and historical judgement 260
9.2 Deviations of the dependent variable from the estimated regression
line 261
9.2.1 Errors of measurement and specification 261
9.2.2 Stochastic (random) factors and the error term, e 262
9.2.3 Sampling and the regression line 263
9.2.4 Assumptions about the error term 264
9.3 A significance test for the regression as a whole 268
9.3.1 The F-test and the F-distribution 269
9.3.2 F-tests of the statistical significance of the regression as a
whole 269
9.3.3 The relationship of F and R2 271
9.4 The standard error of the estimate, SEE 272
9.4.1 Confidence interval for a prediction of the mean value of a
dependent variable 273
9.4.2 Confidence interval for a prediction of a specific value of
a dependent variable 276
9.5 Exercises for chapter 9 277
Chapter 10 Dummy variables and lagged values 280 10.1 Dummy variables 280
10.1.1 Dummy variables with two or more categories 280
10.1.2 Dummy variables as a change in the intercept 283
10.1.3 Additional dummy variables 285
ix
10.1.4 Interaction between two dummy variables 286
10.1.5 Interaction between dummy and numerical variables 287
10.2 Lagged variables 289
10.2.1 Lagged dependent variables 291
10.2.2 Distributed lag models 292
10.3 Exercises for chapter 10 295
Chapter 11 Violating the assumptions of the classical linear regression model 300 11.1 The assumptions of the classical linear regression model 300
11.1.1 The model is correctly specified 301
11.1.2 The error term is appropriately specified 301
11.1.3 The variables are correctly measured and appropriately
specified 301
11.2 Problems of model specification 302
11.2.1 Non-linear models 302
11.2.2 Omitted and redundant variables 303
11.2.3 Unstable parameter values 305
11.3 Problems of error specification 308
11.3.1 Non-zero errors 308
11.3.2 Heteroscedasticity 309
11.3.3 Autocorrelation 311
11.3.4 Outliers 316
11.4 Problems of variable specification 319
11.4.1 Errors in variables 319
11.4.2 Multicollinearity 321
11.4.3 Simultaneity 324
11.5 Exercises for chapter 11 326
Part IV Further topics in regression analysis
Chapter 12 Non-linear models and functional forms 333 12.1 Different forms of non-linear relationships 334
12.2 Estimation of non-linear regression models 341
12.2.1 Two groups of non-linear models 342
12.2.2 Estimating the double logarithmic and semi-logarithmic
models 345
12.3 Interpretation of non-linear relationships 347
12.3.1 Interpretation of the regression coefficients for logarithmic
models 347
12.3.2 Elasticities 349
12.3.3 Long-run multipliers 351
x
12.4 Fitting a trend to a time series 351
12.4.1 Fitting a log-linear trend 351
12.4.2 Fitting non-linear trends 352
12.5 Modelling a relationship 359
12.5.1 Choice of functional form 360
12.5.2 Some general principles 362
12.5.3 Implementing the principles: some examples from economic
history 363
12.5.4 Implementing the principles: some examples from demographic,
social, and political history 367
12.6 Model specification 372
12.7 Exercises for chapter 12 380
Chapter 13 Logit, probit, and tobit models 384 13.1 Limited dependent variables 385
13.1.1 Three cases of limited dependent variables 385
13.1.2 Why linear models are not appropriate for limited dependent
variables 387
13.2 Estimating the logit model: coefficients, absolute effects, and
elasticities 390
13.2.1 The logit regression model 390
13.2.2 Interpreting logit coefficients 391
13.2.3 Absolute changes: marginal effects and impact
effects 399
13.2.4 Calculating the marginal and impact effects 402
13.2.5 Proportionate effects: logit elasticities 406
13.2.6 Non-linearity and the measurement of change 412
13.2.7 Goodness of fit 413
13.3 Estimating the logit model: an illustration 415
13.3.1 An illustration of the logit model 415
13.3.2 Estimating the logit model using grouped data 418
13.4 Estimating the probit model 419
13.4.1 The probit model 419
13.4.2 Interpreting the results from a probit model 420
13.5 Multi-response models 422
13.6 Censored dependent variables 423
13.6.1 A censored continuous dependent variable 423
13.6.2 The tobit model 424
13.6.3 Two-part censored models 427
13.7 Exercises for chapter 13 432
xi
Part V Specifying and interpreting models: four case studies
Chapter 14 Case studies 1 and 2: unemployment in Britain and emigration from Ireland 437
14.1 Inter-war benefits and unemployment in Great Britain 438
14.1.1 The Benjamin and Kochin model 438
14.1.2 Parameter stability and outliers in the Benjamin and
Kochin model 440
14.2 Emigration from Ireland 445
14.2.1 The time-series model of Irish emigration 446
14.2.2 The panel model of Irish emigration 451
14.2.3 The impact of the explanatory variables on changes in
emigration over time 458
Chapter 15 Case studies 3 and 4: the Old Poor Law in England and leaving home in the United States, 1850–1860 463
15.1 The effects of the Old Poor Law 463
15.1.1 Relief payments, wages, and unemployment 464
15.1.2 The Old Poor Law and the birth rate 476
15.2 Leaving home in the United States, 1850–1860 480
15.2.1 Influences on leaving home 482
15.2.2 The logit model 483
Appendix to §15.1.1 Derivation of reduced-form equations from a simultaneous-
equations model 491
Appendix A The four data sets 496 A.1 The iniquitous effects of the Old Poor Law 496
A.2 What explains the scale and timing of the massive emigration from
Ireland after the famine? 497
A.3 To what extent were generous unemployment benefits responsible
for the high level of unemployment in inter-war Britain? 501
A.4 What influenced children’s decision to leave home in the United
States in the mid-nineteenth century? 502
Appendix B Index numbers 507 B.1 Price and quantity relatives 507
B.2 Two types of index number 508
B.3 Laspeyres index numbers with fixed weights 510
B.4 The index number problem and chained indices 513
B.5 Paasche index numbers with changing (given year) weights 520
Bibliography 529
Index of subjects 539
Index of names 545
xii
Figures
1.1 Illustration of SPSS print-out of regression results page 4
1.2 Graph of a straight line
(a) With positive slope 15
(b) With negative slope 15
1.3 Miles of railway track added in the United States, 1831–1913, with
fitted trend 21
1.4 Miles of railway track added in the United States, 1831–1913
(a) Original series with 11-year moving average 26
(b) De-trended series 26
(c) Percentage deviation from trend 26
2.1 Bar chart of number of workhouses in selected counties in 1831 36
2.2 Histograms with different class intervals for per capita relief payments in
311 parishes in 1831
(a) With 8 equal class intervals, each 6 shillings wide 38
(b) With 23 equal class intervals, each 2 shillings wide 38
(c) With 6 class intervals of unequal width 39
2.3 Per capita relief payments in 311 parishes in 1831
(a) Histogram and frequency polygon 41
(b) Cumulative frequency distribution 41
2.4 Symmetrical and skewed frequency curves
(a) The normal curve 54
(b) Smooth curve skewed to the right (positive skewness) 54
(c) Smooth curve skewed to the left (negative skewness) 54
2.5 A family of normal curves
(a) Different means but the same standard deviation 57
(b) The same mean but different standard deviations 57
(c) Different means and standard deviations 57
2.6 Histogram of per capita relief payments in 311 parishes in 1831 with
normal curve superimposed 58
xiii
2.7 Areas under the normal curve
(a) 1.645 standard deviations either side of the mean covers 90 per cent
of the area 60
(b) 1.96 standard deviations either side of the mean covers 95 per cent
of the area 60
(c) 2.58 standard deviations either side of the mean covers 99 per cent
of the area 60
3.1 Scatter diagrams showing different strengths and directions of
relationships between X and Y 73
3.2 Scatter diagram of relief and unemployment, 24 Kent parishes in 1831 75
3.3 Deviations from the mean 78
4.1 Deviations from the regression line 97
4.2 Least squares deviations from the regression line 98
4.3 Scatter diagram of relief and unemployment with regression line,
24 Kent parishes in 1831 101
4.4 Net national product, United Kingdom, 1920–1938 and linear trend 103
4.5 Explained and unexplained deviations 104
4.6 The relationship between the correlation coefficient, r, the regression
coefficient, b, and the standard deviations of X and Y
(a) When sx is low relative to sy 109
(b) When sx is high relative to sy 109
4.7 Ballantine for a simple regression with one explanatory variable 110
5.1 Distribution of heads for 2, 4, 10, and 20 tosses of a coin 131
5.2 The t-distribution for selected degrees of freedom 134
6.1 The critical region
(a) A two-tailed test at the 5 per cent level of significance 157
(b) A one-tailed test at the 5 per cent level of significance 157
6.2 A one-tailed hypothesis test at the 5 per cent level of significance
(a) Rejection of the null hypothesis 169
(b) Failure to reject the null hypothesis 169
7.1 The chi-square distribution for selected degrees of freedom 208
8.1 Scatter diagrams for hypothetical data on WELFARE and HOMELESS
(a) Scatter diagram and regression line for all observations 236
(b) Scatter diagram identifying the effect of TEMP on the relationship
between WELFARE and HOMELESS 236
8.2 Ballantine for a regression with two explanatory variables 527
8.3 Ballantine for a regression with two explanatory variables
(a) With two independent explanatory variables 527
(b) With two explanatory variables which are not independent 527
xiv
8.4 Least squares deviation from the regression plane with two
explanatory variables
(a) Three-dimensional scatter diagram for a dependent variable and
two explanatory variables 244
(b) Three-dimensional diagram with regression plane 244
9.1 The population of Y for different values of X
(a) Hypothetical populations 265
(b) The form of the population of Y assumed in simple linear
regression 265
9.2 The true (population) regression line and the estimated (sample)
regression line 267
9.3 The F-distribution for selected degrees of freedom 272
9.4 95 per cent confidence interval for a prediction of the mean value of
UNEMP 275
10.1 An intercept dummy variable 285
10.2 A slope and an intercept dummy variable 289
11.1 The marginal cost of production of a statistical textbook 303
11.2 Omitted and redundant variables
(a) Ballantine to show the effect of a missing variable 527
(b) Ballantine to show the effect of a redundant variable 527
11.3 Illustration of the use of dummy variables to allow for structural
changes
(a) Introduction of an intercept dummy variable 306
(b) Introduction of a slope dummy variable 306
(c) Introduction of both an intercept and a slope dummy variable 306
11.4 Heteroscedastic error terms 310
11.5 Typical patterns of autocorrelated errors 313
11.6 Autocorrelation 315
11.7 Outliers
(a) An outlier as a leverage point 318
(b) A rogue outlier 318
12.1 A non-linear relationship between age and height 334
12.2 Geometric curves: Y�aXb 336
12.3 Exponential curves: Y�abX 337
12.4 Reciprocal curves: Y�a�b/X 338
12.5 Quadratic curves: Y�a�b1X�b2X 2
(a) Rising and falling without turning points 339
(b) Maximum and minimum turning points 339
12.6 Logistic curves: Y� 340 h
g � abx
xv
12.7 Double logarithmic transformation of a geometric model
(a) The original curve: Y�aXb 344
(b) The double logarithmic transformation: log Y�log a�b(log X) 344
12.8 Semi-logarithmic transformation of an exponential model
(a) The original curve: Y�abX 346
(b) The semi-logarithmic transformation: log Y�log a�X(log b) 346
13.1 Pattern of automobile ownership, Detroit, 1919 388
13.2 Cumulative distribution function of the standardized normal
distribution 389
13.3 Maximum-likelihood function 393
13.4 Logit curve of automobile ownership, Detroit, 1919 401
13.5 Flow chart for deriving logit results for the sample as a whole 410
13.6 OLS model with censored observations
(a) Censored from above 425
(b) Censored from below 425
15.1 Departures of children from home by age and sex, United States,
1850–1860 481
xvi
Tables
1.1 Calculation of 11-year moving average and deviations from trend,
miles of railway track added in the United States, 1831–1850 page 24
1.2 Calculation of a seasonally adjusted series, bankruptcy in England,
1780–1784 29
2.1 Per capita expenditure on relief in 24 Kent parishes in 1831 (shillings) 34
2.2 Frequency, relative frequency, and cumulative frequency of per capita
relief payments in Kent in 1831 34
2.3 Frequency of per capita relief payments in 311 parishes in 1831 37
2.4 Population and migration for four Irish counties in 1881 and 1911 45
2.5 Calculation of the sample variance for two sets of farm incomes 49
2.6 Heights of a sample of 40 male students and the standardized distribution 62
2.7 Areas under the standard normal curve for selected values of Z 64
3.1 Calculation of the coefficient of correlation for a sample of four
(imaginary) values of X and Y 81
3.2 Coefficients of correlation with the British business cycle, 1854–1913 85
3.3 Fifth-quintile male occupations rank ordered by completed fertility and
female age at marriage 88
4.1 Calculation of the regression coefficient for four values of X and Y 100
4.2 Calculation of the coefficient of determination for four values of X and Y 106
5.1 Distribution of t for selected degrees of freedom 135
6.1 Calculation of the regression coefficient and t-ratio for the regression of
UNEMP on BWRATIO 178
7.1 Rankings of two independent samples of students 189
7.2 The Wald–Wolfowitz runs test 192
7.3 The Mann–Whitney U-test and the Wilcoxon rank sums test 197
7.4 The Kolmogorov–Smirnov test 200
7.5 Cross-tabulation of patients admitted to the York Retreat, by
superintendent’s regime and by treatment outcomes 205
7.6 �2-test for statistical significance: calculations using data for York
Retreat, classified by superintendent’s regime and treatment outcomes 210
xvii
7.7 Cross-tabulation of 379 parishes in England by seasonal type in
1661–1700 and 1701–1740 221
8.1 Notation for simple and multiple correlation and regression 233
8.2 Average monthly data on welfare expenditure, homelessness,
temperature bands, and food costs 235
8.3 Correlation coefficients and partial correlation coefficients for five
variables
(a) Zero-order correlation coefficients 250
(b) Third-order partial correlation coefficients 250
8.4 Zero-order and partial correlations between church attendance,
population density, and church density in England in 1851 251
12.1 Absolute and proportionate regression coefficients 349
12.2 Calculation of change in fitted trend, miles of railway track added in the
United States, 1881, 1891, and 1901 359
13.1 Deriving the findings from a logit regression 411
13.2 Detroit automobile ownership, 1919: the full logit model 416
13.3 Detroit automobile ownership, 1919: censored models 428
14.1 Determinants of total Irish emigration rates, time-series data, 1877–1913 447
14.2 Determinants of Irish county emigration rates, panel data, 1881–1911 453
14.3 Decomposition of changes in Irish emigration rates, 1881–1911 459
14.4 Restricted model of Irish county emigration rates, panel data, 1881–1911 460
15.1 The agricultural labour market and the Old Poor Law, simultaneous-
equations and reduced-form models, cross-section data, c. 1831 472
15.2 Determinants of birth rates, cross-section data, c. 1831 478
15.3 Impact of child allowances and other factors on the increase in the
English birth rate, 1781–1821 480
15.4 The age at leaving home in the United States, 1850–1860, explaining the
probability of departure, boys and girls aged 5–11 and 12–18 486
A.1 Data set for investigation of relief payments in England in 1831 498
A.2 Data set for investigation of the birth rate in southern England,
c. 1826–1830 499
A.3 Data set for investigation of emigration from Ireland, 1877–1913 500
A.4 Data set for investigation of unemployment in inter-war Britain,
1920–1938 503
A.5 Data set for investigation of decisions to leave home in the United States,
1850–1860 504
B.1 Bituminous coal output in the United States, 1900–1913, original data
and two quantity relatives 508
B.2 Illustrative data for construction of price and quantity index numbers 510
B.3 Price and quantity indices for the data in table B.2 511
B.4 Chained index numbers with changes in the reference year 519
xviii
Panels
1.1 The log of a variable represents proportional changes in that
variable page 18
1.2 Decomposition of a time series into trend, seasonal factors, and
residual fluctuations 28
5.1 Sampling with and without replacement 126
5.2 Generating the normal theoretical probability curve 130
5.3 The concept of degrees of freedom 133
6.1 Calculation of the standard error of the difference between two
sample means 172
7.1 The chi-square distribution 207
7.2 The concept of degrees of freedom for a cross-tabulation 211
8.1 The degrees of freedom in a regression equation 246
9.1 Assumptions of the classical linear regression model 266
9.2 The theoretical F-distribution 270
10.1 Koyck’s transformation 294
12.1 Measurement of the compound rate of growth 354
12.2 Decomposition of a time series by means of a log-linear trend and
seasonal dummy variables 356
13.1 Expected values and probability theory 386
13.2 Maximum-likelihood models 392
14.1 Pooled or panel data 454
15.1 Simultaneous-equations models 468
15.2 Reduced-form models 469
15.3 Estimation by two-stage least squares 470
B.1 Laspeyres index numbers 514
B.2 Paasche index numbers 522
xix
Preface
We would like to express our deep appreciation to Daniel Benjamin, Levis
Kochin, Timothy Hatton, Jeffrey Williamson, George Boyer, and Richard
Steckel for agreeing to our use of their work for the case studies, which are a
central feature of the approach we have adopted for this book. We are par-
ticularly grateful to Boyer, Hatton, and Steckel for providing us with the
unpublished data that we have used throughout the text and exercises to
illustrate quantitative procedures, and for allowing us to reproduce their
data sets on our web site. We have also received very helpful comments
from a number of our colleagues, notably James Foreman-Peck, Jane
Humphries, Sharon Murphy, and Richard Smith. We owe a special debt of
gratitude to Joachim Voth, who read the entire text with great care and sent
us many thoughtful criticisms and suggestions. None of these scholars has
any responsibility for any errors of procedure or interpretation that
remain.
We particularly wish to acknowledge the contributions of generations
of students who took our courses in quantitative methods in Oxford and
Charlottesville. They showed that intuitive appreciation of the importance
and limitations of quantification in history could be developed by under-
graduates and graduates from a wide range of intellectual and disciplinary
backgrounds. The approach adopted in the present text was originally
designed primarily for students who had no prior knowledge of statistics,
and who were in many cases initially hostile to, or intimidated by, quantita-
tive procedures. It was, above all, their positive response, their success in
applying these methods in their own work, and their helpful comments
that motivated us to develop the text for a larger readership.
Our final debt is to our families, for their encouragement, forbearance,
and general support during the long period over which this book has
evolved.
xxi
Elementary statistical analysis
Introduction
1.1 Aims of the book
This text has three principal objectives. The first is to provide an elementary
and very informal introduction to the fundamental concepts and techniques
of modern quantitative methods. A primer cannot be comprehensive, but
we will cover many of the procedures most widely used in research in the his-
torical and social sciences. The book is deliberately written at a very basic
level. It does not include any statistical theory or mathematics, and there is
no attempt to prove any of the statistical propositions. It has been planned
on the assumption that those reading it have no retained knowledge of sta-
tistics, and very little of mathematics beyond simple arithmetic.
It is assumed that the material in the book will be taught in conjunction
with one of the several statistical packages now available for use with com-
puters, for example, SPSS for Windows, STATA, MINITAB, or SAS. By using
the computer to perform all the relevant statistical calculations and manip-
ulations it is possible to eliminate both the need to learn numerous formu-
lae, and also the tedious work of doing laborious calculations. However, if
the computer is going to provide the results, then it is absolutely essential
that the student should be able to understand and interpret the content and
terminology of the printouts, of which figure 1.1 is a typical specimen, and
the second objective of the book is to achieve this.
This leads naturally to the third and most important objective. The
book is throughout concerned to relate the quantitative techniques studied
to examples of their use by historians and social scientists, and by doing
this to promote the understanding and use of these methods. In the follow-
ing section we introduce four specific studies that will be deployed
throughout the book, but – at appropriate points in the text – we also refer
readers to other examples of the application of quantitative methods to
historical and other issues. A student who studies this text will not be able
Variables Entered/Removedb
FARMERS, CHILDALL, LONDON, WEALTH, GRAINa
Enter
Model 1
Variables Entered
Variables Removed Method
All requested variables entered.a.
Dependent Variable: RELIEFb.
Model Summary
.532a .283 .271 6.84746 Model 1
R R Square Adjusted R
Square
Std. Error of the
Estimate
Predictors: (Constant), FARMERS, CHILDALL, LONDON, WEALTH, GRAINa.
ANOVAb
5638.346 5 1127.669 24.050 .000a
14300.769 305 46.888 19939.116 310
Regression Residual Total
Model 1
Sum of Squares df
Mean Square F Sig.
Predictors: (Constant), FARMERS, CHILDALL, LONDON, WEALTH, GRAINa.
Dependent Variable: RELIEFb.
Coefficientsa
11.352 1.718 6.609 .000 .301 .089 .204 3.363 .001
5.609 1.030 .271 5.445 .000 -.276 .179 -.081 -1.544 .124
-4.156E-02 .010 -.234 -4.110 .000 5.873 2.131 .158 2.756 .006
(Constant) GRAIN CHILDALL WEALTH LONDON FARMERS
Model 1
B Std. Error
Unstandardized Coefficients
Beta
Standardized Coefficients
t Sig.
Dependent Variable: RELIEFa.
Figure 1.1 Illustration of SPSS print-out of regression results
to claim that she is a fully qualified statistician. However, she should have
the confidence to read chapters or articles which use quantitative methods,
to understand what the authors have done and why they have done it, and
to make her own critical evaluation of the procedures used and the histori-
cal conclusions drawn from the statistical results.
Students should also be able to see from the case studies and other
examples how the use of quantitative methods can open up new aspects of
an enquiry and can supplement and strengthen other methods of research.
We hope that they might then appreciate how their own research projects
might benefit from the application of these methods, and take their own
first steps in this direction.
The book is designed to be used both as the basic text for taught courses
and for students working on their own without an instructor. In planning
the content and sequence of the chapters, one of our primary considera-
tions has been to keep the material in the early chapters at a very elemen-
tary level. Many of those for whom this book is intended will naturally be
somewhat wary about taking a course in quantitative methods, but if they
find that they can make substantial progress in understanding some of the
basic statistical concepts they will gain confidence in their ability to handle
the slightly more difficult material in later chapters. There is a small price to
be paid for this approach, since it means that correlation and regression are
covered twice: first without any statistical theory (in chapters 3 and 4) and
then again in greater depth (in chapter 8). However, the text has been used
for a number of years to teach a class taken almost exclusively by statistical
novices who initially approached the course with deep suspicion, and
experience has shown that this strategy is very successful.
It is, of course, also possible for instructors to follow the material in a
different sequence, or – depending on the time available and the level it is
desired to achieve – to omit certain topics altogether. For example, chapter
7 on non-parametric methods has been included because these procedures
are often appropriate for the sort of problems faced by historians; and it has
been placed at this point in the text because it makes a useful complement
to the discussion of the standard principles of hypothesis testing in chapter
6. However, a course designed primarily to provide a very basic introduc-
tion to regression analysis in 10 sessions might skip this. It might, for
example, start with chapters 2–6 and 8–9, add material on dummy vari-
ables from chapter 10 and on the basic aspects of non-linear regression
from chapter 12, and then cover the appropriate applications in the case
studies in chapters 14 and 15.
This would be sufficient to give students a good grounding in some of
the main aspects of regression methods, and should enable them to cope
§.
with many examples of the use of these methods in the historical literature.
However, the omission of chapter 11 would mean that they would not have
acquired any knowledge of either the serious problems which can arise
when the assumptions underlying the standard regression model (dis-
cussed in chapter 9) are violated, or the associated procedures for diagnos-
ing and – where possible – correcting for these violations. This would also
be a substantial weakness for any students who wished to apply these
methods to their own research. One alternative, for students able to start
with some knowledge of very elementary statistics and wishing to aim a
little higher (while still limited to 10 sessions), would be to skip chapters
1–4 and 7, and work through chapters 5–6 and 8–15.
The normal text and tables are supplemented by material in boxes and
panels. Boxes are used to highlight the fundamental definitions and con-
cepts, and should be studied closely. Panels are used to provide explana-
tions or information at a slightly more advanced level than the rest of the
text. The panels should be helpful for some readers, but those who omit
them will not be at any disadvantage in understanding the remainder of the
text. In addition to footnotes (referred to by a letter), we have also used
endnotes (referred to by a number) where it seems desirable not to burden
the main text with lengthy annotations. Endnotes typically consist either of
lists of references to further examples of the applications of statistical
methods to historical topics, or of technical points which need not distract
all readers although they may be useful to some.
We have given a formula for all the basic concepts even though the book
is written on the assumption that the computers will provide what is
required for any particular statistical operation. It is usually possible to set
out these formulae in various ways, but we have always chosen the variant
that best explains the essential nature of the concept, rather than one that
facilitates computation (and we avoid the complication of alternative ver-
sions of the same formula). We recognize that many readers will not be
accustomed to working with symbols, and so may be uncomfortable ini-
tially with this method of setting out information. However, the multi-part
nature of many concepts means that a formula is the most concise and
effective ‘shorthand’ way of showing what is involved, and we would
strongly urge all readers to make the small initial effort required to learn to
read this language; the rewards for doing so are very substantial.
1.2 The four case studies and the data sets
Throughout the book we will make frequent reference to four data sets that
were compiled and used by historians to investigate specific historical
issues.a In order to illustrate and practise the various techniques discussed
in subsequent chapters we will normally draw on one of these data sets, and
the results will be reproduced and discussed in depth in chapters 14 and 15,
by which time readers will have encountered all the procedures used in
these studies. We are extremely grateful to the authors for their agreement
to the use of their statistical material and for making available their unpub-
lished data. These data sets are not reproduced in this book but can be
accessed without charge on the Cambridge University Press web site
<http://uk.cambridge.org/resources/0521806631>. Further instructions
for downloading the data are given on the web site.
A brief outline of the issues raised in these studies is given in appendix
A, together with a description of each of the individual statistical series that
make up the data sets. In all cases we refer to each series in a data set by a
short name given in capital letters, and since some computer programs
limit series titles to eight letters we have adopted that restriction. Even
when abbreviated these names should be reasonably self-explanatory (for
example, BRTHRATE or IRISHMIG) and they are much easier to remem-
ber and interpret than symbols. Where the authors we are quoting have
used a similar convention we have generally adopted their names; where
they have not done so we have created our own.
In addition to these primary data sets, most chapters include references
to other examples of the application of the given procedures by historians.
These illustrations are designed both to reinforce understanding of the
techniques and to show the range of issues that have been addressed by his-
torians with the aid of quantitative methods. The text also uses imaginary
data where it is desirable to have very simple numbers in order to illustrate
basic concepts or to make calculations without the need for a computer.
1.3 Types of measurement
Measurement is the assignment of numbers or codes to observations. The
measurements we work with can be classified in various ways. In this intro-
ductory section we note a number of basic terms and distinctions.
§.
a The four studies are George Boyer, An Economic History of the English Poor Law, Cambridge
University Press, 1990, chapters 4 and 5; Timothy J. Hatton and Jeffrey G. Williamson, ‘After the
famine: emigration from Ireland, 1850–1913’, Journal of Economic History, 53, 1993, pp.
575–600; Daniel K. Benjamin and Levis A. Kochin, ‘Searching for an explanation for unemploy-
ment in interwar Britain’, Journal of Political Economy, 87, 1979, pp. 441–78; and Richard H.
Steckel, ‘The age at leaving home in the United States, 1850–1860’, Social Science History, 20,
1996, pp. 507–32.
1.3.1 Cases, variables, and values A data set consists of a series of units or cases each of which has one or more characteristics, known as variables. For each variable there is a sequence of varying observations, each with its own particular value. The cases are the basic unit for which the measurements are available, and there are two
main types (see §1.3.2). They can be individuals, households, firms, towns,
churches, votes, or any other subject of analysis; or they can be weeks, quar-
ters, years, or any other period of time for which a data series is available.
In the English Poor Law data set:
Cases: The cases are 311 English parishes (identified by a number).
Variables: There are 15 variables for each parish. These include, for example,
the per capita relief expenditure of each parish (RELIEF), the annual
income of adult male agricultural labourers in the parish (INCOME), and
the proportion of unemployed labourers in the parish (UNEMP).
Values: For the RELIEF variable the values for the first five parishes are (in shil-
lings): 20.4, 29.1, 14.9, 24.1, 18.2, etc.b
In the data set for annual emigration from Ireland:
Cases: The cases are the 37 successive years from 1877 to 1913.
Variables: There are five basic variables for each year. These are the rates per
1,000 of the population emigrating from Ireland (IRISHMIG), the foreign
and domestic employment rates (EMPFOR and EMPDOM), the foreign
wage relative to the domestic wage (IRWRATIO), and the stock of previous
emigrants (MIGSTOCK).
Values: For the main emigration variable (IRISHMIG), the values of this rate
for the first five years are: 7.284, 7.786, 8.938, 18.358, and 15.238.
1.3.2 Cross-section and time-series variables A set of measurements that applies to a single case at different periods of
time is referred to as a time series. The series in the data sets dealing with migration from Ireland from 1877 to 1913, and with unemployment and
benefits in Great Britain each year from 1920 to 1938, are both examples of
annual time series.
A set of measurements that applies to different cases at a single point in
time is referred to as a cross-section. The data set for children leaving home is a cross-section, with values for each variable for each child in the sample
in 1850.
b The data on relief expenditure shown for the 24 Kent parishes in the data set are given to 3
decimal places (i.e. have 3 figures after the decimal point). In the text the values for the first five
parishes are rounded to 1 decimal place.
It is also possible to combine cross-section and time-series data to create
what is known as a panel or longitudinal data set. The second of the two data sets for Irish emigration (see part (b) of table A.3 in appendix A),
which has values for the variables for a cross-section of 32 counties for each
of the four census dates (1881, 1891, 1901, and 1911), is an example of
panel data. The process of combining the values for the cross-section over
time is referred to as pooling the data. In principle, time series are used to study changes over time, and cross-
sections to study differences between cases (for example, families, parishes,
or countries) at a particular date. However, it is not always possible to
obtain the necessary time series for periods in the past, and historians
sometimes attempt to infer how relationships might have changed over
time by looking at differences in a cross-section for a more recent date. For
example, the patterns of household expenditure of low-income families in
a modern cross-section might be used to infer something about overall
patterns of expenditure at an earlier date when all families had lower
incomes.c
1.3.3 Levels of measurement We can distinguish the following three levels or scales of measurement. Each has its own properties and the nature of the statistical exercises that
can be performed depends on the level of the data.
Nominal measurement This is the lowest level and conveys no information about the relations
between the values. Each value defines a distinct category but can give no
information other than the label or name (hence nominal level) of the cat- egory. They are sometimes also referred to as categorical variables.
For example, a study of migration to urban areas might include as one of
the variables the birthplace of the migrants. These towns or villages cannot
be ranked or placed in any order in terms of their value as place of birth
(though they could be by other criteria such as size, or distance from the
migrants’ final destination).
Ordinal measurement This applies when it is possible to order or rank all the categories according to some criterion without being able to specify the exact size of the interval
between any two categories.
§.
c A similar procedure is adopted by Boyer, Poor Law, p. 126 to explain the increase in relief expen-
diture over time on the basis of cross-section variations in expenditure across parishes.
This is a very common situation with historical data and can occur for
one of three reasons:
� The relevant variables are often ordinal by definition. For example, an
analysis of the labour force might classify workers as unskilled, semi-
skilled, skilled, and professional. We could agree that skilled workers
ranked above semi-skilled, and semi-skilled above unskilled, but could
not say anything about the distance between the categories.
� Lack of data might impose an ordinal scale even when a higher level
would, in principle, be possible. For example, in a study of education it
would in principle be possible to measure the number of years spent at
school by each member of the labour force, but the surviving records
may state only whether or not they completed five years’ primary edu-
cation, and if they did whether they then went on to secondary school.
We would thus have to treat ‘schooling’ as an ordinal variable with
three values: ‘less than five years’, ‘five years’, and ‘secondary or above’.
� The third reason for using ordinal scales might be doubts about the
accuracy of the data. In a study of wealth, for example, we might decide
that the actual values reported by property owners were too unreliable
to use, but that we could safely order them on the basis of, say, three
values: large, medium, and small.
Interval or ratio measurement1
These measurements have all the properties of an ordinal scale. In addition
it is now possible to measure the exact distance between any pair of values.
This level of measurement is thus truly quantitative, and any appropriate
statistical procedure can be applied to values on an interval or ratio scale.
Values measured on such a scale may be either continuous or discrete (discontinuous). A continuous variable, such as height or income, is meas- ured in units that can be reduced in size to a theoretically infinite degree,
limited only by the sensitivity of our measurement procedures. Discrete
variables, by contrast, can take only a limited number of pre-determined
values (most commonly whole numbers) and occur when we are counting
indivisible units such as cotton mills or people.
1.3.4 Populations and samples The term population (or universe) refers to all possible observations. If, for example, we were interested in the number of cotton mills in New England
in 1880, the ‘population’ would consist of all mills in existence at that date.2
In a few cases historians might have such information if, for example, a
complete census of textile mills had been taken at the relevant date.
Normally, however, we would have only a sample. The characteristics of the population variables are known as parame-
ters, those of the sample variables as statistics. Parameters are fixed values at any point in time and are normally unknown. Statistics, on the other
hand, are known from the sample, but may vary with each sample taken
from the population. The extent of such variation from sample to sample
will depend on the homogeneity (uniformity) of the population from
which it is drawn.d
A crucial feature of any sample is whether or not it is random. A random sample satisfies three basic conditions. First, every item in the population
(parishes in England and Wales, voters in an election, cards in a deck of
cards) has an equal chance of appearing in the sample. Secondly, every com-
bination of items has an equal chance of selection. Thirdly, there is indepen-
dence of selection: the fact that any one item in the population has been
selected has absolutely no influence on whether or not any other item will be
selected.
When the sample is drawn with replacement the same item can be selected more than once; for example, after a card is drawn it is put back in
the deck. If the sample is drawn without replacement the item can be selected only once. Almost all use of sampling in historical analysis is sam-
pling without replacement. Each parish, each voter, each cotton mill is
sampled only once when the data set is being compiled.
The proper procedures to be followed in constructing samples, whether
the sampling should be random or some other type, such as stratified or
cluster sampling, and the size of the sample required for any particular
project are complex subjects which go beyond the scope of this book and
specialized texts should be consulted.3
1.3.5 Dummy variables A variable that cannot be measured can still be used in quantitative work by
assigning values representing each of two (or more) categories. This is
known as a dummy variable. In the simplest case, where there are only two possible values (equivalent to yes or no) the dummy variable is formed by
setting the positive values equal to 1 and the negative values to 0. There are
a number of these in the Poor Law data set – for example, whether or not
the parish pays child allowances and whether or not it has a workhouse.
§.
d Harold Wilson, Britain’s only statistically informed prime minister, once remarked that he
needed to sip only one spoonful from a plate of soup to know whether it was too hot to drink.
They can also be used in time-series analysis – for example, years in which
there was a war might be given a value of 1, while peacetime years are
assigned a value of 0. In more complex cases there could be more than two
categories. For example, a study of voting behaviour might divide the
electorate into four categories according to whether in the previous elec-
tion they had voted Democrat, Republican, or Independent, or had not
voted.
This procedure effectively transfers the information into a numerical
form (albeit a limited one) and makes it possible to apply standard statisti-
cal tools to the variable. For fuller discussion of the use of dummy variables
see §10.1 and also chapter 13.
1.4 Simple notation
The use of symbols may deter some readers, but they are a very convenient
form of shorthand and if properly defined are not difficult to understand.
They are used sparingly in this book, mainly to supplement more cumber-
some verbal statements and to bring out the essential features of some of
the concepts. The following notes introduce the simple conventions
adopted.
It is customary to refer to variables by the letters X and Y. For each vari- able there would be a sequence of observations, each of which would have
its own particular value. For example, the values in a sequence of 5 obser-
vations for X could be listed as
X1, X2, X3, X4, and X5
More generally, if there were n observations this could be written as
X1, X2, ... , Xn
and if it is desired to indicate that we are referring to any one particular
observation without specifying which it is, we normally use a subscript i
Xi
It is sometimes necessary to refer to a constant rather than a variable, that is to a value which is fixed. Such constants are usually denoted by the
initial letters of the alphabet – for example, a, or its Greek equivalent, �
(alpha).
Summation The symbol used to refer to the total (or sum) of all the observations in the
sequence is the Greek letter capital sigma, �
Thus instead of writing X1�X2�X3�X4�X5 we could write
Xi
This is an instruction to add all the values in the series from i�1 to i�5.
If it is obvious which values are included this may be simplified to
�X
Differences The symbol used to refer to the difference between two series is the Greek
letter capital delta, �
Thus if instead of referring to the successive values of an annual variable
X, it is desired to refer to the change in these values from year to year, this is
done by writing
�X
Time and time lags If it is desired to indicate that the variable refers to a specific year (or other
period of time), the convention is to use the subscript t. Thus Xt would
indicate the current period, Xt 1 the previous period, and Xt�1 the follow-
ing period.e Xt 1 is known as a lagged variable.
Population and sample The characteristics of the population are usually denoted by Greek letters,
and those of samples from that population by Roman letters. For example,
the standard deviation of the population (this term will be defined in
§2.3.2) is designated by (lower case Greek sigma) and the standard devia-
tion of the sample by s.
1.5 The equation of a straight line
We shall refer at numerous points to linear, or straight-line, relationships. A straight line can be represented algebraically by an equation of the form
Y�a�b X
where X and Y are any two variables, and a and b are any two constants.
This equation tells us that each value of Y is equal to some constant, a,
� 5
i�1
§.
e The equation on p. 581 of Hatton and Williamson, ‘Emigration’, refers to current and previous
years in this way.
plus the value of X multiplied by another constant, b.f The precise position
of the straight line will then change depending on the specific values given
to a and b.
This is illustrated in figure 1.2 (a) for the particular case where a�2 and
b�3. The equation thus becomes
Y�2�3X
� a is the intercept. It determines the level at which the straight line crosses the vertical axis and is the value of Y when X�0. In figure 1.2
(a) the line intercepts (cuts) the vertical axis at 2.
� b is the slope of the line. It tells us by how much Y changes for every change of 1 unit in X. In figure 1.2 (a) Y rises by 3 units on the vertical
axis every time there is a change of 1 unit in X on the horizontal axis.
The sign of b is also important. In figure 1.2 (a), b is positive, so Y rises as X rises. By contrast, in the following equation, b is negative
Y�10 2X
so Y falls by 2 units for every increase of 1 unit in X , as shown in figure
1.2 (b).
1.6 Powers and logarithms
1.6.1 Powers (a) For any number, X, the statement ‘multiply X by itself n times’ is written
Xn. The technical term for this is: ‘raise the number X to the nth power.’ ‘n’ is called the exponent. Thus
X1�X; X2�X� X; X3�X�X�X, and so on
Note that X0�1.g
f Multiplication can be represented in various ways. Thus the statement ‘a multiplied by b’ can be
written as: a�b, as a . b, as a(b), or simply as ab. The order of multiplication makes no difference:
ab is exactly the same as ba.
If one of the terms to be multiplied has more than one component then these must be placed
in brackets; for example b(a�c) tells us to add a to c and then multiply the result by b. It is essen-
tial to complete the operation in brackets before the multiplication.
Similarly for division, the statement ‘a divided by b’ can be written as a ÷ b, or as a/b or as
, but here the order is critical. a/b is not the same as b/a. The term above the line is called
the numerator, and the term below the line the denominator. g If the reason for this is not immediately clear, consider the second example in paragraph (d)
below.
a
b
§.
X
543210
Y 12
10
8
6
4
2
0
Y = 10 – 2X 1
– 2
1
– 2
Figure 1.2 Graph of a straight line
X
543210
Y 20
16
12
8
4
0
Y = 2 + 3X
1
3
1
3
(a) With positive slope Y rises by 3 units when X increases by 1 unit
(b) With negative slope Y falls by 2 units when X increases by 1 unit
(b) If the exponent is negative, the result is obtained by dividing 1 by the
number raised to the same positive power. Thus
X 1�1/X1�1/X; X 2�1/X2, and so on
(c) We may also be interested in multiplication of powers. The product
of a number (X) raised to a power (say, 2) and of the same number raised to
another power (say 3) is obtained by adding the exponents. Thus
X 2�X 3�X 2�3�X5�X�X�X�X�X
(d) Just as multiplication of powers is done by addition, so division of
powers is done by subtraction
�X 3 2�X1�X
Also
�X 3 3�X0�1
(e) Finally, the exponent of a number may be a fraction, in which case
the result is obtained as follows
X � ; X �( )3; and so onh.
1.6.2 Logarithms For every positive number there is a corresponding logarithm (log). It is equal to the power to which a given base must be raised to equal that number.
Take, for example, the number 100 and the base 10. Then
the log of 100 to the base 10�2
because
102�100
i.e. the base 10 raised to the power 2�100
Logs can be calculated for any base greater than 1. Base 10 was originally
widely used, but with computers the base generally adopted is not 10 but a
constant with special mathematical properties known as e, equal to
2.71828 (to 5 decimal places).4
Logs to the base e are known as natural logs, and unless we explicitly state that we are using logs to base 10, all references in the remainder of this
book are to natural logs. The proper abbreviation for natural logs is ‘ln’,
but ‘log’ is also frequently used and since it is easier to recognize what it
refers to we have also adopted this term. Thus if you see a statement such as
4 �X
3 4
2 �X
1 2
X3
X3
X3
X2
h If the term to be raised to a power is given in brackets it is again essential to first complete the
operation in brackets. For example (a b)2 is calculated by deducting b from a and then squar-
ing the remainder; this is not the same as a2 b2 .
log INFTMORT, it refers to the natural log of the series in the Poor Law
data set for infant mortality.
When logarithms are used in a calculation, the result can always be con-
verted back to ordinary numbers by obtaining the corresponding expo- nential (of a natural log) or anti-log (of a log to base 10). As long as that is done consistently, the final result is unaffected by the choice of base.
Calculations using logs Logs were originally used extensively as a simpler way of doing certain cal-
culations, because it follows from their definition that:
(i) The product of two numbers (X and Y) can be found by adding their
logs, and then taking the exponential of the result
log (X�Y)�log X�log Y
(ii) Division of one number by another can be done by subtracting their
logs, and then taking the exponential of the result
log �log X log Y
(iii) A number can be raised to a power by multiplying the log of that number
by the exponent, and then taking the exponential of the result
log (X n)�n � log X
Since square roots and other radicals can be expressed as exponents in the
form of a fraction (see §1.6.1) they can also be calculated by multiplying
the log of the number by the appropriate fraction, and then taking the
exponential of the results
log( )�log (X )� log X
log( )�log (X )� log X
and so on.
1.6.3 Logs as proportions and ratios The use of logs for calculations has now been largely superseded by com-
puters, but they remain important because of their other properties; in
particular, the fact that they represent proportional rather than absolute
changes. The basis for this statement is shown in panel 1.1.
Logs can also be used to measure the ratio of one series to another. The
aim here is not to calculate the value of that ratio in ordinary numbers (as
described in §1.6.2) but to have a measure in logs of the ratio.
1
4
1 4
4 �X
1
2
1 2�X
�XY�
§.
Panel 1.1 The log of a variable represents proportional changes in that variable
A fundamental property of logarithms is that the absolute change in the log
of any variable corresponds to the proportionate change in that variable.
To illustrate this, consider the following simple example. In column (2),
Income is rising by 2 per cent in each period, so the proportionate change in
the series, given in column (3), is always 0.02. The natural log of Income is
shown in column (4), and the absolute change in this series in column (5).
The absolute change in the log of Income in column (5) is constant at
0.0198, corresponding to the constant proportionate change in Income
itself in column (3) of 0.02.*
(1) (2) (3) (4) (5)
Change in col. (2) as Absolute change
Period Income a proportion Log (Income) in col. (4)
1 100 ... 4.6052 ...
2 102 0.02 4.6250 0.0198
3 104.04 0.02 4.6448 0.0198
4 106.1208 0.02 4.6646 0.0198
5 108.24322 0.02 4.6844 0.0198
One very useful consequence of this property of logs is that it makes it pos-
sible to graph a time series for any variable on a proportionate basis by con-
verting it to logs. When a statistical program is used to plot such a graph it
typically offers the choice of making the vertical axis either linear or loga-
rithmic. If you select the former the program will plot the original values of
the variable; if you select the latter (usually referred to as a log scale) it will
convert the values to logs and plot these.
In the linear form, each absolute increment will be the same: the distance
on the vertical axis between, say, 100 and 200 units will be exactly the same
as the distance between 800 and 900. This can give a very misleading
impression. The change from 100 to 200 represents a doubling of the series;
the change from 800 to 900 is an increase of only 12.5 per cent.
* It may also be noted that the constant proportionate change in the series is approximately but
not exactly equal to the constant absolute change in the log of the series. If the constant abso-
lute change in the log of the series is converted back to ordinary numbers (by taking the
exponential) the result is exactly equal to 1�the proportionate change. Thus, in this
example, the unrounded change in column (5) is 0.0198026, and the exponential of this is
exactly 1.02.
There is an example of this procedure in the article by Benjamin and
Kochin on unemployment.i As indicated in table A.4 in appendix A, they
use the variable DEMAND to indicate the effects of aggregate demand on
unemployment. DEMAND is measured by the proportionate deviation of
output from its trend value (the higher the proportion, the stronger the
demand), where the variable for output is NNP and its trend value is
NNP*. We thus have
DEMAND�the ratio of NNP to NNP*�log NNP log NNP*
In the same way a multiplicative relationship between two series can be
expressed as the sum of their logs and, as we shall see in chapter 12, logs also
play an important part in regression analysis.
1.7 Index numbers
Many of the series used in quantitative analysis take the form of index
numbers of the changes in either prices or quantities. Common examples
of the former include indices of producer (or factory-gate) prices for
industrial products, of retail prices for consumer goods, of wage rates, and
of share prices such as the NASDAQ or the FTSE100. Among the latter are
indices of industrial or agricultural production, or of total gross national
product (GNP), indices of real wages, and indices of the quantity of goods
imported.j
These indices are constructed from time series on prices and quantities,
working either directly with these data or using them in the form of price
and quantity relatives. Such relatives look exactly like index numbers because they also have one item in the series shown as 100. However, they
are merely series that have been converted from an absolute to a relative
basis; they do not provide any additional information that is not already
contained in the original series on which they are based.
§.
i Benjamin and Kochin, ‘Unemployment’, pp. 452–3. j Quantity indices are also referred to as volume indices, or as indices ‘in real terms’. A national
accounts series such as GNP expressed in monetary units ‘at constant prices’ (or ‘in constant
dollars’) is also a form of quantity index.
By contrast, if the graph is plotted on a logarithmic vertical scale, each
proportionate increment will be the same. Thus the distance between 100
and 200 units will be equal to the distance between 800 and 1,600 units, and
so on. This conveys a much more accurate impression of the growth of the
series.
By contrast, a proper index number combines information on the changes in prices (or quantities) to create a series that cannot be directly
measured in any other way. For example, if an historian has collected infor-
mation on the changes over time in the prices of a representative selection
of goods and services purchased by working-class consumers, she could
employ an appropriate procedure to construct an index number of the
overall change in their cost of living. There is no way in which she could
measure the ‘cost of living’ directly.
Index numbers play a prominent role in many aspects of historical
research and debate. To give just three examples: the index of the prices of
consumables and of builders’ wage rates compiled by Phelps Brown and
Hopkins for the seven centuries from 1260 to 1954 has been used by
numerous scholars to indicate the influence of changes in prices and wages
during this period.5 Successive versions of an index of industrial produc-
tion have been at the heart of the long-running controversies about the
pace and pattern of economic growth in Britain during the industrial revo-
lution and in the later slow-down at the end of the century.6 Indices meas-
uring the relationship between the prices received by Britain, the United
States, and other advanced countries for the manufactured goods sold to
countries on the periphery, and the prices they paid for the food and raw
materials purchased from the periphery, lie at the centre of heated debates
about the nature of the relationship between the two groups of countries.7
Index numbers are commonly applied in historical research, not only in
the representation of the movement of aggregate prices, wages, or output,
but also as components of more complex models of historical processes. In
this role they have many of the attributes of a ‘black box’, in that it is
extremely hard to determine what is responsible for the behaviour of a par-
ticular index number over time, and the researcher is implicitly putting
faith in the person(s) who constructed the index. Unfortunately, not all
index numbers warrant this faith, either because of the low quality of their
ingredients, or because the recipe for their making has not been followed
properly.
Given the general significance of index numbers, and given our overall
strategy in this book of explaining, rather than assuming, it seems impor-
tant to provide guidance on how index numbers are constructed and why.
This may enable readers to gain some deeper understanding of their limita-
tions, and so provide some basis for scrutinizing the series used by histo-
rians. However, since none of what follows in this book requires a
knowledge of index numbers, we have postponed the discussion to appen-
dix B, at the end of the book.
1.8 Trends and fluctuations
Many of the time series that are of interest to historians display a combina-
tion of long-term growth or decline and short-term fluctuations. This
applies, for example, to economic series such as the tonnage of ships built
or index numbers of wholesale prices; to demographic series such as birth
or marriage rates; and to social series such as the number of indictable
offences reported to the police or the number of deaths from suicide. Both
features – the trend and the cycle – are well illustrated by the annual data on
the miles of railway track added in the United States from 1831 to 1913 (see
the solid line in figure 1.3).
The secular trend in a series represents the broad pattern of movement that is sustained in the same direction over long periods of time. It may be
interpreted as the general direction of movement ignoring all shorter-term
variations, and may indicate either long-term growth or decline, or it may
be broadly stable, with neither growth nor decline. In figure 1.3 the trend in
railway construction indicated by the broken line is initially upward for
some six decades, and then turns downward from the 1890s.
Superimposed on the long-run trend are varying types of fluctuation of shorter duration. If a series is available for periods shorter than a year it
may show more or less regular seasonal fluctuations. These may occur as a direct result of the weather (sales of fuel higher in the winter, building
activity higher in the summer) or of seasonal harvests, as with grain or
cotton. Or they may reflect the indirect impact of the seasons on social
§.
Figure 1.3 Miles of railway track added in the United States, 1831–1913 with fitted trend
191119011891188118711861185118411831
M ile
s of
r ai
lw ay
tr ac
k ad
de d
14,000
12,000
10,000
8,000
6,000
4,000
2,000
0
habits, as seen in the tendency for marriages in pre-industrial pastoral
communities to take place after the spring season of lambing and calving.
Holidays taken at Christmas and Easter also cause regular seasonal
fluctuations.
Annual series typically show less regular fluctuations. These are of
broadly two types. They may reflect some recurrent cyclical pattern such as
the business cycle experienced in many economies. This typically consisted
of a downswing in trade, employment, and other indicators of economic
activity from peak to trough, followed by an upswing to the next peak, the
whole cycle usually lasting for between seven and 10 years. These economic
cycles were in turn an important influence on a range of social, political,
and demographic variables such as marriage rates and elections.8
Fluctuations in economic activity of considerably longer duration – by
some accounts as long as 60 years – have also been detected by some schol-
ars, but are more controversial.9
Alternatively, the fluctuations may be completely irregular, occurring
either in response to exceptional events such as wars or floods, gold discov-
eries or dramatic new inventions, that affect either the country itself or its
major trading partners. Or they may simply be random variations in an
individual series as a consequence, for example, of idiosyncratic human
behaviour.
Some discussions of the analysis of time series suggest that it is possible
to split the series into these different components, producing separate
series for the trend, seasonal fluctuations (where applicable), regular
fluctuations, and a residual of irregular movements. However, some of the
procedures suggested for splitting year-to-year movements into regular
and irregular fluctuations are very unsatisfactory, and can actually gener-
ate spurious fluctuations which were not present in the original series,
while others are mathematically very complex and well beyond the level of
this text.10 For annual series we prefer, therefore, to make only a single divi-
sion into long-run trend and short-term fluctuations, and in the following
section we will describe a method for doing this.
1.8.1 Measuring the trend by a moving average There are two main procedures for determining the trend in a time series.
The simplest is to calculate a centred moving average. The other involves fitting a trend line to the data by regression analysis. This is a more reliable
but also a more complex procedure, and we must defer its introduction
until we have developed the necessary procedures in subsequent chapters,
though we may note that the trend line in figure 1.3 was obtained in this
way.11
The procedure for deriving a moving average is very straightforward. It
is calculated directly from the original time series by choosing an appropri-
ate number of years (for an annual series, or months for a monthly series)
and calculating the average of the series for those years. The selected period
of time is then moved forward one year, and the average calculated for that
new period, and so on until the series is completed.
It is conventional to choose an odd number of years for the moving
average, so that as each average is struck it can be allocated to the central
year of the time period to which it relates. The longer the period that is
chosen, the more the fluctuations will be smoothed out, but the choice
must depend to some extent on the length of the underlying time series.
If dealing with time series covering several centuries, as Wrigley and
Schofield were in their investigation of the long-term trends in English fer-
tility and mortality, an average as long as 25 years may be appropriate.12
For shorter series, averages of between five and 11 years are common,
and the more regular the periodicity in the underlying series, the more
effectively a moving average covering the same period will smooth out the
series. For example, if the series had alternating peaks and troughs, both
occurring every nine years with unfailing regularity, a nine-year moving
average would completely eliminate the cycle and leave only a smooth
trend. Unfortunately such regularity is seldom if ever encountered in his-
torical data.
We can illustrate the procedure by calculating an 11-year moving
average for the first 20 years of the data on railway construction in the
United States (see table 1.1). To obtain the 11-year moving average we start
with the original series for railway miles added in column (1) of table 1.1.
In column (2), the average of the first 11 years 1831–41 is calculated. This is
(72�134�...�516�717)/11�319.3, and so 319.3 is placed opposite the
middle of the 11 years, 1836, to give the first item in the moving average.
The average of the next 11 years, 1832–42 (�357.4) is then allocated to
1837, and so on until the column ends with the average for 1840–50 (�
610.8), placed opposite 1845.
We have already noted that one defect of moving averages is that they are
capable of creating spurious cyclical fluctuations. This is a very serious
weakness when the procedure is intended to identify shorter cycles in the
data, but matters less if the aim is merely to indicate the long-term trend. A
second weakness of the moving average is evident from the illustration in
table 1.1: the procedure leaves blank years at the beginning and end of the
sequence, and it is thus impossible to examine the precise relationship
between the original data and the trend for these missing years.
A further problem can be seen from a comparison of the 11-year
§.
Table 1.1 Calculation of 11-year moving average and deviations from trend, miles of railway track added in the United States, 1831–1850
(1) (2) (3) (4)
Original series Trend De-trended series
Centred Deviation of
Railway 11-year Original original data
miles moving data as from trend
added average % of trend as % of trend
1831 1,172
1832 1,134
1833 1,151
1834 1,253
1835 1,465
1836 1,175 319.3 54.8 45.2
1837 1,224 357.4 62.7 37.3
1838 1,416 359.6 115.7 15.7
1839 1,389 363.4 107.1 77.1
1840 1,516 363.6 141.9 41.9
1841 1,717 348.4 205.8 105.8
1842 1,491 393.2 124.9 24.9
1843 1,159 409.0 38.9 61.1
1844 1,192 495.6 38.7 61.3
1845 1,256 610.8 41.9 58.1
1846 1,297
1847 1,668
1848 1,398
1849 1,369
1850 1,656
Notes:
(1) Brinley Thomas, Migration and Economic Growth, Cambridge University Press, 1954,
p. 288.
(2) See text.
(3) �(1)/(2)�100.
(4) �[(1) – (2)]/(2)�100.
moving average shown in figure 1.4 (a) with the fitted trend in figure 1.3. It
is immediately evident that the former does not smooth out the
fluctuations as successfully as the latter. It would, of course, be possible to
take an even longer number of years, say 25, but only at the cost of still
longer periods at the beginning and end of the series for which there would
be no trend value. One additional disadvantage of the moving-average
procedure is that it has no mathematical form independent of the data
from which it is constructed, and so cannot be extrapolated by formula to
cover earlier or later periods. Despite these weaknesses it is very simple to
construct and has been widely used by historians.
We have so far referred to centred moving averages, but it is also pos-
sible to construct other forms of moving average. In particular, if the
focus of interest is the experience of the most recent years, it would be
appropriate to calculate an average of (say) the last five years in the series
at any given date, and to place the average against that date. This series
would then be moved forward one year at a time, so that at each date it
provided a measure of the movements in the underlying series over the
last five years. The ‘tracking polls’ used to provide a rolling measure of the
most recent poll results during the course of an election are one variant of
this. More elaborate variants might attach greater importance to the
experience of the most recent years by calculating a weighted average (see
§2.2.2), with progressively diminishing weights for the more distant
years.
1.8.2 Measuring the fluctuations by the ratio to the trend Having calculated the long-run trend in the series by means of a moving
average or a fitted trend, it is then possible to derive a de-trended series that measures the remaining fluctuations.13 This can be done by simply calcu-
lating the absolute deviation from trend, i.e. the absolute difference
between the original series and the trend. This is satisfactory if the underly-
ing series is broadly constant over time, but not if it is changing in value. In
the latter case, for example, if the series were increasing strongly, a devia-
tion of (say) 10 units that might seem relatively large at the beginning of the
series would be relatively small by the end. It is generally better, therefore,
to measure the ratio of the original series to the trend by expressing it as a percentage of the trend.
This is done in column (3) of table 1.1, by dividing the original series by
the centred 11-year moving average. Years such as 1838 and 1841, when
railway construction was booming and the series was above its trend, have
values greater than 100; years such as 1836 and 1843, when construction
§.
191119011891188118711861185118411831
M ile
s of
r ai
lw ay
tr ac
k ad
de d
14,000
12,000
10,000
8,000
6,000
4,000
2,000
0
Figure 1.4 Miles of railway track added in the United States, 1831–1913
191119011891188118711861185118411831
D ev
ia ti
on f
ro m
tr en
d as
p er
c en
t o f
tr en
d 150
100
50
0
–50
–100(c) Percentage deviation from trend
191119011891188118711861185118411831M il
es o
f ra
ilw ay
tr ac
k ad
de d
as p
er c
en t o
f tr
en d 250
200
150
100
50
0(b) De-trended series
(a) Original series with 11-year moving average
was depressed and below its trend, have values below 100. The trend has
been removed from the series, and what is left in column (3) is the remain-
ing component: the regular and irregular fluctuations. The corresponding
de-trended series calculated with the 11-year moving average is shown in
figure 1.4 (b).k
Much the same result can be obtained by calculating the percentage deviation from the trend. This is equal to the absolute difference between the trend and the original series, expressed as a percentage of the trend, as is
done in column (4) of table 1.1 and figure 1.4 (c). This is also a de-trended
series and will show exactly the same fluctuations as column (3). The only
difference is that the former fluctuates around a mean of 100 per cent and
the latter around a deviation of 0 per cent.
When dealing with monthly time series, or other periods of less than a
year, it is possible to make separate estimates of the seasonal variations and
to separate these from the residual cyclical and irregular fluctuations. Such
seasonal adjustment plays no part in any further work in this text, but it is a
useful illustration of the way in which historical data – for example a series
for birth rates or for agricultural production – can be decomposed into
these three elements: long-run trend, recurrent seasonal fluctuations, and
residual fluctuations, thus enabling the historian to analyse separately each
of these contributions to the overall change in the series. A brief account of
one simple seasonal adjustment procedure is set out in panel 1.2 for those
who wish to see how this three-way split is made.14
§.
k Since the ratio of two series can be calculated as the difference between their respective loga-
rithms, a de-trended series can also be derived in this way, as with the DEMAND variable meas-
ured by Benjamin and Kochin, ‘Unemployment’, pp. 452–3 (see also §1.6.3).
Panel 1.2 Decomposition of a time series into trend, seasonal factors, and residual fluctuations
This panel describes a simple procedure for splitting a time series into trend,
seasonal factors, and residual fluctuations, using quarterly data on bank-
ruptcy in England for 1780–84 to illustrate the calculations.* The original
data for the five years are set out with a column for each quarter in row (1) of
table 1.2. The trend is calculated as a 25-quarter moving average over the
longer period 1777–87 (so that there are trend values for all quarters from
1780 I to 1784 IV), and is given in row (2).
Row (3) gives the original series as a ratio to the trend for each quarter
and this reveals the seasonal pattern in the bankruptcy statistics. The main
seasonal feature is clearly the persistently low level in the third quarter of
the year. Hoppit attributes this primarily to the strong favourable effect of
the grain harvest on a wide range of activities and to the higher level of
building. Conversely, bankruptcies were at their highest in the winter
months, and this is also reflected in the ratios for the first and fourth quar-
ters in row (3).
The temptation at this point is to calculate the average of the quarterly
ratios for the five-year period, but a more accurate measure is obtained by
calculating the ratio of the five-year totals for the original series and the
trend. This is given in row (4). Row (5) then makes a minor correction to
these ratios; the average of the four quarterly ratios in row (4) is 0.9721, and
each quarter must be multiplied by 1/0.9721. As a result of doing this, the
average of the corrected adjustment factors in row (5), weighted by the five-
year total of the trend, is exactly equal to 1. This in turn ensures that the total
number of bankruptcies over the five years is not changed by the seasonal
adjustment to the component quarters.
The original series is then divided by the adjustment factors in row (5) to
give the seasonally adjusted series. As can be seen in row (6), the low
autumn levels are raised and the higher levels in the other three quarters are
all reduced. This series thus reflects both the trend in bankruptcy and the
remaining irregular fluctuations from quarter to quarter after the seasonal
variations have been removed.
* The data are taken from Julian Hoppit, Risk and Failure in English Business, 1700–1800,
Cambridge University Press, 1987, pp. 187–96. For his own de-seasonalized series Hoppit
used a 61-quarter moving average, so our results are not exactly the same as his, though the
differences are small. For his interesting discussion of the short-term fluctuations in bank-
ruptcy see ibid., pp. 104–21.
§.
The final adjustment, made in row (7), also eliminates the trend, thus
leaving only the irregular fluctuations in a de-trended and de-seasonalized
series. Since in this case there was little movement in the trend over these
years (see row (2)), the series in row (7) does not diverge much from the
series in row (6).
The decomposition of the original data into the three elements is now
easily checked. For example, for the first quarter of 1780, if we multiply the
trend (row (2)), by the recurrent seasonal factor (row (5)), and by the resid-
ual fluctuation (row (7)), and divide the result by 100 we get (133.32�
1.093�83.03)/100�121, which is the value of the original series in row
(1).
Table 1.2 Calculation of a seasonally adjusted series, bankruptcy in England, 1780–1784
QI QII QIII QIV
(1) Original series, number of bankruptcies (�B)
1780 121 117 74 123
1781 116 110 97 128
1782 151 144 110 144
1783 148 144 101 128
1784 148 145 94 150
(2) Trend: 25-quarter moving average (�T)
1780 133.3 133.2 131.8 132.7
1781 132.7 131.6 127.6 127.8
1782 126.2 125.5 123.6 125.0
1783 124.5 126.2 125.0 126.9
1784 127.1 127.7 127.4 129.0
(3) Ratio of original series to trend (�B/T)
1780 0.908 0.879 0.562 0.927
1781 0.874 0.836 0.760 1.002
1782 1.197 1.147 0.890 1.152
1783 1.189 1.141 0.808 1.009
1784 1.165 0.738 0.738 1.162
Table 1.2 (cont.)
QI QII QIII QIV
(4) Initial seasonal adjustment factors
Five-year totals, 1780–4
Original series 684 660 476 673
Trend 643.72 644.12 635.40 641.40
Ratio of original to trend 1.063 1.025 0.749 1.049
(5) Corrected seasonal adjustment factors (�S)
1.0931 1.0541 0.7707 1.0794
(6) Seasonally adjusted series (�B/S�T�R)
1780 110.7 111.0 96.0 113.9
1781 106.1 104.3 125.9 118.6
1782 138.1 136.6 142.7 133.4
1783 135.4 136.6 131.1 118.6
1784 135.4 137.6 122.0 139.0
(7) Residual fluctuations in de-trended and de-seasonalized series (�B/(T�S)�R)
1780 83.0 83.4 72.9 85.9
1781 80.0 79.3 98.6 92.8
1782 109.5 108.8 115.5 106.7
1783 108.8 108.3 104.8 93.4
1784 106.5 107.7 95.7 107.7
Notes:
(1) Hoppit, Risk and Failure, pp. 195–5.
(2) 25-quarter moving average calculated on quarterly series for 1777–87.
(3) �Row (1)/Row (2).
(4) The five-year total of row (1) divided by the five-year total of row (2).
(5) The quarterly correction factors in row (4) average 0.9721, and each is raised by
1/0.9721. This ensures that the five-year total for row (6) will be the same as for row
(1).
(6) �Row (1)/Row (5)�100.
(7) �Row (1)/(Row (2)�Row (5))�100.
Notes 1 Many statistical textbooks treat these as two separate levels of measurement, with the
ratio as the highest level. The distinction between them is that the interval level does
not have an inherently determined zero point. However, this issue is hardly ever
encountered in historical research; the example usually given relates to temperature
scales in which 0° is an arbitrarily defined point on the scale but does not represent an
absence of temperature. 2 In some contexts it might be appropriate to think of an infinite as opposed to a finite
population. In the case of the mills this would consist of all the mills which might
hypothetically ever have been constructed in the past or which will be constructed in
future; see further §9.2.3 below. 3 For a brief introduction to general sampling procedures, see H. M. Blalock, Social
Statistics, 2nd edn., McGraw-Hill, 1979. R. S. Schofield, ‘Sampling in historical
research’, in E. A. Wrigley (ed.), Nineteenth-Century Society, Essays in the Use of
Quantitative Methods for the Study of Social Data, Cambridge University Press, 1972,
pp. 146–90, is an excellent discussion of sampling procedures in an historical
context. 4 It is not necessary to know any more about the constant e, but for those who are
curious it is equal to the limiting value of the exponential expression
as n approaches infinity. For example, if n is taken as 10,000, e�2.71815, and taking
higher values of n will not change the first three decimal places. The value of 2.71828
was calculated by taking n as 1,000,000. 5 Henry Phelps Brown and Sheila V. Hopkins, ‘Seven centuries of the price of consum-
ables, compared with builders’ wage rates’, Economica, 23, 1956, reprinted in Henry
Phelps Brown and Sheila V. Hopkins, A Perspective of Wages and Prices, Methuen,
1981, pp. 13–59. For a consumer price index for the United States for 1774–1974, see
Paul A. David and Peter Solar, ‘A bicentenary contribution to the history of the cost
of living in America’, Research in Economic History, 2, 1977, pp. 1–80. 6 The first index of industrial production in Britain was compiled by Walther G.
Hoffmann, British Industry, 1700–1950, Basil Blackwell, 1955. There have been
several subsequent alternative indices, including Phyllis Deane and W. A. Cole,
British Economic Growth 1688–1959, Cambridge University Press, 1962; W. Arthur
Lewis, Growth and Fluctuations, 1870–1913, Allen & Unwin, 1978; N. F. R. Crafts,
British Economic Growth during the Industrial Revolution, Cambridge University
Press, 1985; and Robert V. Jackson, ‘Rates of industrial growth during the industrial
revolution’, Economic History Review, 45, 1992, pp. 1–23. The corresponding index
for the United States is Edwin Frickey, Production in the United States 1860–1914,
Harvard University Press, 1947. 7 For an excellent introduction to the debate about the terms of trade see John Spraos,
‘The statistical debate on the net barter terms of trade between primary commod-
ities and manufactures’, Economic Journal, 90, 1980, pp. 107–28.
�1 � 1n� n
8 Among the classic studies of business cycles are T. S. Ashton, Economic Fluctuations
in England, 1700–1800, Oxford University Press, 1959; A. D. Gayer, W. W. Rostow
and A. J. Schwartz, The Growth and Fluctuation of the British Economy, 1790–1850,
Oxford University Press, 1953; and Arthur F. Burns and W. C. Mitchell, Measuring
Business Cycles, NBER, 1947. 9 Solomos Solomou, Phases of Economic Growth, 1850–1973, Kondratieff Waves and
Kuznets Swings, Cambridge University Press, 1987, is a sceptical investigation of the
evidence for these longer fluctuations in Britain, Germany, France, and the United
States. 10 One such technique is known as spectral analysis; for a brief non-mathematical
introduction see C. W. J. Granger and C. M. Elliott, ‘A fresh look at wheat prices and
markets in the eighteenth century’, Economic History Review, 20, 1967, pp. 257–65. 11 The procedure for fitting a simple linear trend is described in §4.2.5. The more
widely used trend based on a logarithmic transformation is explained in panel 12.1
of chapter 12, and other non-linear trends (such as the one illustrated in figure 1.3)
are introduced in §12.4. 12 E. A. Wrigley and R. S. Schofield, The Population History of England 1541–1871,
Edward Arnold, 1981, pp. 402–53. 13 Examples of historical studies based heavily on patterns determined by such a dis-
tinction between the trend and the cycle include Brinley Thomas, Migration and
Economic Growth, Cambridge University Press, 1954; A. G. Ford, The Gold Standard
1880–1914, Britain and Argentina, Oxford University Press, 1962; and Jeffrey G.
Williamson, American Growth and the Balance of Payments 1820–1913, A Study of the
Long Swing, University of North Carolina Press, 1964. 14 There are alternative methods for calculating seasonally adjusted series using more
advanced statistical techniques; one such method is described in panel 12.2 in §12.4.
Descriptive statistics
This chapter presents an introduction to the key elements of descriptive statistics: the ways in which quantitative information can be presented and described. The first step in the analysis of quantitative data is its organiza-
tion and presentation in tables and graphs. The basic features of the data –
such as its central or most common values and the way the observations are
distributed around these central values – can then be summarized in
various ways.
2.1 Presentation of numerical data
2.1.1 Frequency distributions Quantitative data in their raw form consist of a series of numbers or cate-
gories. For example, the Poor Law data on per capita relief expenditure in
1831 by the 24 parishes in Kent could be set out in its original sequence as in
table 2.1, with the data rounded to one decimal place.
Even with 24 observations it is difficult to get much sense of the data in
this form. It would help a little if the values were arranged in ascending or
descending order of magnitude (known as an array). This would immedi- ately show the highest and lowest values and give some indication of the
most common level of payment, but with a large number of observations it
would still be hard to interpret the information in this form.
A much clearer picture can be obtained if the data are re-arranged as a
frequency distribution. This is the most common form in which data are summarized and presented. With a discrete variable (see §1.3.3) it makes
sense to measure the frequency of occurrence of each of the values. For
example, if a lecturer had set an examination with nine questions, she
might find it informative to compile a frequency distribution showing how
many candidates had answered each question.
However, this is less appropriate for a continuous variable such as the
payments of relief, since we would not normally be interested in minute
differences, e.g. in finding out how many parishes paid 18.1 shillings, and
how many paid 18.2 shillings, etc. Instead we choose appropriate class intervals and note the frequency of the occurrence of payments in each class.
For table 2.2 we have chosen seven class intervals, each with a width of 5
shillings; so that all the payments in the first interval are equal to or greater
than 5 shillings (abbreviated to �5) but less than 10 shillings ( 10); those
in the next are equal to or greater than 10 shillings but less than 15 shillings,
and so on.1 The frequency is shown in column (3) and the relative fre- quency in column (4). In order to show how the frequency is derived the underlying tally is shown in column (2), but this would not normally be
included in a frequency table. The relative frequency shows the share of the
Table 2.2 Frequency, relative frequency, and cumulative frequency of per capita relief payments in Kent in 1831
(1) (2) (3) (4) (5) (6)
Class Relative Cumulative
intervals Frequency frequency (%) Cumulative relative
(shillings) Tally (f ) (f /n�100) frequency frequency (%)
�5 but 10 1 1 4.2 1 4.2
�10 but 15 1 1 1 1 1 1 7 29.2 8 33.4
�15 but 20 1 1 1 1 4 16.6 12 50.0
�20 but 25 1 1 1 1 1 6 25.0 18 75.0
�25 but 30 1 1 1 1 4 16.6 22 91.6
�30 but 35 1 1 4.2 23 95.8
�35 but 40 1 1 4.2 24 100.0
24 100.0
Note:
In column (1) � stands for ‘equal to or greater than’, and for ‘less than’.
Table 2.1 Per capita expenditure on relief in 24 Kent parishes in 1831 (shillings)
20.4 29.1 14.9 24.1 18.2 20.7
8.1 14.0 18.4 34.5 16.1 24.6
25.4 12.6 13.3 27.3 29.6 13.6
11.4 21.5 20.9 11.6 18.2 37.9
observations in each class interval. In table 2.2 this is expressed as a per-
centage of the total (adding to 100), but it could equally well be a propor-
tion (adding to 1).
It is sometimes also useful to know the cumulative frequency of a vari- able. This gives the number of observations which are less than or equal to
any selected value or class interval. The cumulative relative frequency gives the corresponding proportions or percentages. For example, we can see
from column (5) of table 2.2 that per capita relief payments were less than
20 shillings in 12 Kent parishes, and from column (6) that this represented
50 per cent of the parishes.
By summarizing the data in the form of a frequency distribution some
information is lost, but it is now much easier to get a clear idea of the
pattern of relief payments in the Kent parishes.
Number and width of the class intervals The precise number and width of the class (or cell) intervals in a frequency
table is a matter of judgement. Too many narrow intervals and the pattern
is again obscured; too few and too much information is lost. For example,
in relation to table 2.2, covering a sample of 24 parishes, having (say) 2
classes would tell us very little, whereas 20 classes would be little improve-
ment on the presentation of all the data in table 2.1. A rough rule of thumb
is that it is usually best to have between five and 20 classes, depending on
the number of observations.
It is preferable if the width can be made uniform throughout. Classes of
unequal width (e.g. 10 – 14.9, 15 – 24.9, 25 – 29.9) are confusing to the
reader and a nuisance in computations. However, this is not always pos-
sible. For example, when there are a few extreme values (as in data on
wealth holdings) the intervals may have to be changed to accommodate the
very uneven distribution.
It is sometimes necessary to work with open-ended classes, e.g. ‘under
10 shillings’ or ‘50 shillings and over’. However, if the data were given in
this form in the original source this can cause problems in calculations that
require a figure for the mid-point of the class intervals (as in §2.4). At the
lower end it may not be plausible to assume that the values are (roughly)
evenly distributed all the way down to a lower bound of zero. At the upper
end, in the absence of information about the values in excess of the lower
limit given for the open-ended class, the best that can be done is to make an
informed guess about the likely values within this class.
§.
2.1.2 Bar charts and histograms It is often useful for both statistician and reader to present results in a form
that can be interpreted visually. Two common methods of doing this with
frequency distributions are bar charts and histograms.
Bar charts are commonly used to present data on the frequency distri- bution of qualitative data, or data which falls into discrete categories. Thus
the number of workhouses in the sample parishes in seven counties in the
south and south-east of England could be represented in a bar chart such as
figure 2.1, in which there is a bar for each of the selected counties, and the
height of each bar indicates the number (or frequency) of workhouses
reported in the Poor Law data set. There is no statistical convention relating
to the width of each bar, though it is visually more appealing to present bars
of common width for each category equally spaced.
Histograms are used where the frequencies to be charted are grouped in class intervals. Each bar (or rectangle) represents either the absolute
number or the proportion of cases with values in the class interval. To illus-
trate the construction of histograms we will use the Poor Law data on per
capita relief payments by all 311 parishes. The information is set out in
table 2.3 using eight class intervals, each with a width of 6 shillings. The
number of parishes in each class is given in column (2), the corresponding
Figure 2.1 Bar chart of number of workhouses in selected counties in 1831
KentSussexBedsCambsSuffolkNorfolkEssex
N um
be r
of w
or kh
ou se
s
40
30
20
10
0
relative frequency in column (3), and the cumulative relative frequency in
column (4). The relative frequencies in column (3) are shown with a total
of 100. It would make no difference if they were all divided by 100 and
summed to 1.
Since there is by construction a strictly proportional relationship
between the absolute frequency in column (2) and the relative frequency in
column (3) – the latter is always exactly a fraction of the former (in this
case, equal to 1�311) – it will make no difference whatsoever to the
appearance of the histogram whether we plot it using column (2) or
column (3).
If it is possible to keep all class intervals the same width, then the visual
impact of the histogram is effectively determined by the height of each of
the rectangles. Strictly speaking, however, it is the area of each bar that is
proportional to the absolute or relative frequency of the cases in the corre-
sponding class interval.
The first two versions of the histogram for the per capita relief payments
by all 311 parishes in figure 2.2 are of this type. In (a) there are eight class
intervals as in table 2.3, each with a width of 6 shillings; in (b) the number
of intervals is increased to 23, each with a width of only 2 shillings. For (b)
the horizontal axis is labelled by the mid-point of each range.
§.
Table 2.3 Frequency of per capita relief payments in 311 parishes in 1831
(1) (2) (3) (4)
Class Cumulative
intervals Relative relative
(shillings) Frequency frequency frequency
�0 but 6 6 1.93 1.93
�6 but 12 78 25.08 27.01
�12 but 18 93 29.90 56.91
�18 but 24 63 20.26 77.17
�24 but 30 46 14.79 91.96
�30 but 36 17 5.47 97.43
�36 but 42 5 1.61 99.04
�42 but 48 3 0.96 100.00
311 100.00
Note:
In column (1) � stands for ‘equal to or greater than’, and for ‘less than’.
N um
be r
of p
ar is
he s
100
80
60
40
20
0 0–6 6–12 12–18 18–24 24–30 30–36 36–42 42–48
Relief payments (shillings)
Figure 2.2 Histograms with different class intervals for per capita relief payments in 311 parishes in 1831
(a) With 8 equal class intervals, each 6 shillings wide
4844403632282420161284
N um
be r
of p
ar is
he s
50
40
30
20
10
0
Relief payments (shillings)
(b) With 23 equal class intervals, each 2 shillings wide
In this case we could select these (or other) class intervals with equal
widths because we have the data for every individual parish in the data set.
But what would we do if the data had already been grouped before it was
made available to us, and the class intervals were not of equal width?
Assume, for example, that the information on relief payments in the 311
parishes had been published with only six class intervals, the first four with
a width of 6 shillings as in table 2.3, and the last two with a width of 12 shil-
lings (i.e. with absolute frequencies of 63 and 8, and relative frequencies of
20.26 and 2.57).
Since the width of these last two intervals has been doubled, they must be
shown on the histogram with their height set at only half that of the 6 shil-
ling intervals. The areas (i.e. width � height) of the two sets of intervals will
then be proportional. In figure 2.2 (c) the original eight-class equal-width
histogram of figure 2.2 (a) is reproduced, with the alternative rectangles for
the two wider class intervals superimposed on the last four intervals.
The height of the first of these 12-shilling intervals is set at 31.5, equal to
half the 63 parishes in this class; the height of the second at 4, equal to half
of the eight parishes in this final class. It is easy to see that when done on
this basis the area of each of the new rectangles is exactly equal to that of the
two corresponding rectangles in the previous version.
§.
N um
be r
of p
ar is
he s
100
80
60
40
20
0 0–6 6–12 12–18 18–24 24–36 36–48
Relief payments (shillings)
Figure 2.2 (cont.)
(c) With 6 class intervals of unequal width
The advantage of the histogram in such situations is thus that despite
the change in width of the intervals it is still possible to represent either the
absolute or the relative frequency of the cases in each class interval by com-
parison of the area of their rectangles. Because it is the area rather than the
height that is proportional to the frequency, it is always wise when looking
at a histogram to refer first to the horizontal axis to ascertain the widths,
and not to be misled by the visual impact of the height of each of the
columns.
2.1.3 Frequency curves Imagine a line that is drawn so that it passes through the mid-point of each
of the class intervals of a frequency distribution for a continuous variable
such as per capita relief payments. This graph is called a frequency polygon. If the histogram is like the one in figure 2.2 (a), with relatively few cases and wide class intervals, the polygon will look very different to the
histogram. There will be numerous triangles that fall within the polygon
but lie outside the histogram, and others that are outside the polygon but
inside the histogram. However, if we look closely at figure 2.3 (a), in which
the two graphs are shown on the same diagram, we see that all these trian-
gles form exactly matching pairs, so that the area under the two graphs is
exactly the same.
Now assume that we can gradually reduce the width of the class intervals
to take full advantage of the whole scale over which a continuous variable is
measured.a As we increased the number of rectangles in the histogram and
narrowed their width, we would find that the shape of the polygons drawn
through the mid-points of the rectangles approximated more and more
closely to a smooth curve. We would ultimately get a perfectly smooth fre- quency curve.
It was noted in §2.1.2 that the relative frequencies in column (3) of table
2.3 could be treated as if they summed to 1, and also that the areas of the
rectangles making up a histogram are proportional to the relative frequen-
cies. Since together the rectangles cover all the class intervals, the total area
under the histogram can also be thought of as equal to 1. We have now seen
how we might in principle obtain a perfectly smooth frequency curve, and
that the area under such a curve is identical to the area under a histogram.
Thus the total area under the frequency curve can also be thought of as equal
a We also need to assume that we can increase the number of cases we are dealing with (i.e. that we
can work with very much larger numbers than the 311 parishes in our Poor Law data set). By
increasing the number we avoid the irregularities in the distribution that might occur with small
numbers of cases in narrow class intervals.
§.
Relief payments (shillings)
N um
be r
of p
ar is
he s
100
80
60
40
20
0 0–6 6–12 12–18 18–24 24–30 30–36 36–42 42–48
Figure 2.3 Per capita relief payments in 311 parishes in 1831
(a) Histogram and frequency polygon
Relief payments (shillings)
4842363024181260
C um
ul at
iv e
fr eq
ue nc
y di
st ri
bu ti
on
1.00
.80
.60
.40
.20
.00
62.2
124.4
186.6
248.8
311.0
(b) Cumulative frequency distribution
to 1. This simple but important idea of the area under a smooth frequency
curve will be extremely useful in relation to the normal distribution, a
concept which plays a major part in many of the statistical techniques we
employ later in this book.
A different form of frequency polygon can also be used to display the
cumulative frequency distribution defined in §2.1.1. This diagram (also known as an ogive) is illustrated in figure 2.3 (b) with the data from column (4) of table 2.3 for the per capita relief payments by the 311 par-
ishes. The scale shown on the horizontal axis indicates the upper limit of
the successive class intervals, and that on the vertical axis indicates both the
cumulative relative and cumulative absolute frequency. (The values for the
latter are obtained by cumulating column (2) of table 2.3.)
The curve of the cumulative frequency distribution traces either the
proportion or the number of cases that are less than the corresponding
value of the variable shown on the horizontal axis. Since there is an exact
correspondence between the absolute and relative frequencies, the two
graphs are identical. For example, the position marked off by the broken
lines in figure 2.3 (b) indicates that relief payments of less than 18 shillings
per person were made by 177 parishes, corresponding to a proportion of
0.569 (56.9 per cent). The s-shape is characteristic of such cumulative fre-
quency diagrams.
2.2 Measures of central tendency
If we refer again to the data set in table 2.1, there are 24 observations, each
giving the per capita expenditure on relief in one of the Kent parishes in
1831. Grouping the data in class intervals in a frequency table (such as table
2.2), and plotting this as a histogram, provides a more helpful indication of
some features of the data, but this is neither a convenient nor a precise way
in which to summarize the information. In order to achieve this we need to
determine three aspects of the data:
(a) Which are the central (i.e. the most common or typical) values within
the distribution?
(b) How is the distribution spread (dispersed) around those central
values?
(c) What is the shape of the distribution?
Each of these features can be described by one or more simple statistics.
These are the basic elements of descriptive statistics, and together they
provide a precise and comprehensive summary of a data set. We shall now
look at each in turn.
There are three principal measures that are used to locate the most
common or most typical cases. They are referred to collectively as meas- ures of central tendency.
2.2.1 The mean, the median, and the mode (a) The arithmetic mean, also called the arithmetic average, is obtained by adding up all the values and dividing this total by the number of observa-
tions.b It is usually denoted by X –
(X bar) when referring to a sample, and by
� (lower case Greek mu) for the corresponding population. Thus
X –
� (2.1)
where n is the number of observations in the data set for which there are
values for the variable, X.c
(b) The median is the value that has one-half of the number of observa- tions respectively above and below it, when the series is set out in an
ascending or descending array. Its value thus depends entirely on whatever
happens to be the value of the observation in the middle of the array; it is
not influenced by the values of any of the other observations in the series.
For example, if there are five observations, the median is the value of the
third observation, and there are two cases above this and two below. When
there is an even number of observations, the average of the two middle
observations is taken as the median.
(c) The mode is the value that occurs most frequently (i.e. is most fash- ionable). When data are grouped in class intervals the mode is taken as
the mid-point of the category with the highest frequency. In a frequency
distribution the mode is represented by the highest point on the curve. A
distribution may have more than one mode; one with two modes is
described as bimodal.
�X
n
§.
b This measure is frequently referred to simply as the mean. When no further qualification is given
the term ‘mean’ can be understood to refer to the arithmetic mean. However, it is also possible to
calculate other means, such as a geometric mean, and when there is any possibility of confusion
the full titles should be used. The geometric mean is calculated by multiplying all the values and
taking the appropriate root: thus if there are five terms in a series (X1, X2, ... , X5) the geometric
mean would be the fifth root of their product:
c We will follow the convention of using lower case n to refer to the number of observations in a
sample. The symbol for the corresponding number in the population is upper case N.
5 �X1�X2�X3� X4�X5
The three primary measures can be illustrated with the following simple
data set containing seven observations set out in ascending order:
3, 5, 5, 7, 9, 10 and 45
The arithmetic mean��X/n�84/7�12d
The median�7
The mode is 5
The most useful of these measures for most purposes are the mean and
the median. The mean is generally to be preferred because it is based on all
the observations in the series, but this can also be its weakness if there are
some extreme values. For example, in the series just given the mean is
raised by the final very large value, whereas this observation has no effect
on the median. It is a matter of judgement in any particular study whether
or not to give weight to the extreme values in reporting a measure of central
tendency. For the data on RELIEF in table 2.1 the mean is 20.3 and the
median 19.4.
The mode is the only one of these three measures that can be used with
either nominal or ordinal level measurements.
2.2.2 Weighted averages and standardized rates The simple mean defined in §2.2.1 effectively treats all n observations as
having equal importance, and is the measure of central tendency most
often referred to in statistical work. However, for certain purposes it may
not be appropriate to give the same weight to each observation when some
observations represent a larger share of the relevant total than others. For
example, if the aim were to measure the average change in prices of various
farm products between 1920 and 1938, it would not be appropriate to give
the same weight to minor products (such as pumpkins) as to major prod-
ucts (such as wheat).
To illustrate a very useful alternative procedure we will take the data for
CNTYMIG in the Irish emigration data set (for simplicity we restrict the
example to the first four counties and round the data to 1 decimal place).
To begin with, consider only the data in columns (1) and (2) of table 2.4 for
1881.
It is immediately evident that there are considerable differences in the
rate of migration per 1,000 of population from these four counties, and
that the one with the lowest rate – Dublin – also has the largest population.
d The geometric mean is 8.02, indicating that the extreme value (45) has less effect on this measure
than on the arithmetic mean.
Conversely Carlow has the highest rate of migration but the smallest popu-
lation. If we ignore these differences in the migration rates and size of the
counties and calculate the simple mean we get an average of 16.9 migrants
per 1,000 (as shown at the foot of table 2.4).
If we wish to take these differences into account, we must instead use a
weighted arithmetic mean. This requires that each county is given an appropriate weight, in this case its population size in 1881. The weighted
mean is then calculated by multiplying each rate of migration by its weight,
and dividing the total of these products by the sum of the weights. The cal-
culation is set out at the foot of table 2.4, and the result is 10.7 per 1,000.
This is considerably lower than the unweighted mean, and gives a much
more accurate measure of the overall rate of migration from these
counties.
More generally, the formula for a weighted mean, X – �, is
X – �� (2.1a)
where wi is the appropriate weight for each observation, Xi, in the series.
We will also introduce one particular form of weighted average that is
very widely used in demographic and medical history, the standardized rate.
�(wiXi)
�wi
§.
Table 2.4 Population and migration for four Irish counties in 1881 and 1911
(1) (2) (3) (4)
1881 1911
Population CNTYMIG Population CNTYMIG
County (000) (per 1,000) (000) (per 1,000)
Carlow 46.6 29.0 36.3 5.9
Dublin 418.9 6.8 477.2 2.3
Kildare 75.8 19.8 66.6 4.4
Kilkenny 99.5 11.9 75.0 4.9
Arithmetic mean of CNTYMIG in 1881� �16.9
Weighted arithmetic mean of CNTYMIG in 1881�
(29.0�46.6) � (6.8�418.9) � (19.8�75.8) � (11.9�99.5)
46.6 � 418.9 � 75.8 � 99.5 �
6884.7
640.8 � 10.7
29.0 � 6.8 � 19.8 � 11.9
4 �
67.5
4
If we look at the 1911 data in columns (3) and (4) of table 2.4 we see that the
emigration rates have fallen sharply in all four counties, and that the popula-
tion in Dublin has increased since 1881 while that in the other three counties
has fallen. The weighted average rate of CNTYMIG for 1911 is only 3.0.
However, this decline as compared to 10.7 in 1881 is partly a result of
changes in migration from the individual counties, and partly a result of
changes in the relative importance of the different counties. If we wish to
get a summary measure of the rate of emigration which is unaffected by the
population changes we can calculate a standardized rate. In this case we would ask, in effect: what would the 1911 rate have been if the distribution
of the population by county were fixed at the proportions of 1881? The
answer is obtained by weighting the four 1911 county rates by their respec-
tive 1881 populations, and is 3.2.
This procedure has very wide application, with the possibility of stan-
dardizing a variety of rates for one or more relevant factors. For example,
crude birth rates per 1,000 women are frequently standardized by age in
order to exclude the effects of changes in the age composition of the female
population. Crude death rates per 1,000 of the population might be stan-
dardized by both age and occupation; suicide rates by both geographical
region and gender; marriage rates by social class; and so on. In every case
the standardized rate is effectively a weighted average of a particular set of
specific rates such as the age-specific birth rates or the age- and occupa-
tion-specific death rates.
2.2.3 Percentiles, deciles, and quartiles It was noted in §2.2.1 that the median is calculated by taking the value in an
array above and below which there are one-half of the observations.
Exactly the same principle can be applied for other measures, but taking
other fractions.
(a) Quartiles: Instead of dividing the observations into two equal halves we can divide them into four equal quarters. The first quartile is then
equal to the value that has one-quarter of the values below it, and three-
quarters above it. Conversely the third quartile has three-quarters of the
valuesbelowit, andone-quarterabove.Thesecondquartilehas twoquar-
ters above and two quarters below, and is thus identical to the median.
(b) Percentiles and deciles: Other common divisions are percentiles, which divide the distribution into 100 portions of equal size, and
deciles, which divide it into 10 portions of equal size. So if you are told
at the end of your course on quantitative methods that your mark is at
the ninth decile, you will know that nine-tenths of the students had a
lower mark than you, and only one-tenth had a better mark.
The fifth decile and the fiftieth percentile are the same as the median; the
25th and 75th percentiles are the same as the first and third quartiles.
2.3 Measures of dispersion
Knowledge of the mean or some other measure of central tendency tells us
something important about a data set, but two sets of numbers can have
the same mean and still be markedly different. Consider, for example, the
following (fictitious) numbers relating to the profits (in dollars) earned by
two groups of five farmers in counties A and B:
County A: 14, 16, 18, 20, and 22
County B: 2, 8, 18, 29, and 33
Both these series sum to 90 and have the same mean of 18 dollars, but it is
immediately obvious that the spread of numbers around this shared
central value is very much greater in the second county. Our view of the
risks and rewards of farming in the two regions would be influenced
accordingly, and we would want to include some indication of this factor in
any summary statement about farm incomes.
To do this we need some measure of the dispersion or spread of the data around its central value, and there are a variety of different ways such a
measure can be constructed.
2.3.1 The range and the quartile deviation
(a) The range is a very crude measure of dispersion and is simply the difference between the highest and lowest value in the series. Its
obvious weakness is that it is based entirely on these two extreme
values, and takes no account whatsoever of all the other observations.
(b) The quartile deviation (or inter-quartile range) is a slightly better measure. Instead of representing the range as the difference between the
maximum and minimum values it measures the distance between the
first and third quartiles (see §2.2.3). It is sometimes divided by 2 and
called the semi-interquartile range. In either form it is thus a measure of the distance covered by the middle half of all the observations, and is less
sensitive to extreme values of the distribution than the range. However,
it still ignores a considerable part of the data, and it does not provide any
measure of the variability among the cases in the middle of the series.
The remaining measures of dispersion avoid both these weaknesses.
2.3.2 The mean deviation, the variance, and the standard deviation If the objective is to get a good measure of the spread of all the observations
in a series, one method which suggests itself is to find out how much each
§.
value differs from the mean (or some other central or typical value). If we
then simply added up all these differences, the result would obviously be
affected by the number of cases involved. So the final step must be to take
some kind of average of all the differences.
The three following measures all do this in varying ways.
(a) The mean deviation is obtained by calculating the difference between each observation (Xi) and the mean of the series (X
– ) and then
finding the average of those deviations.
In making this calculation the sign of the deviation (i.e. whether the
value of the observation is higher or lower than the mean) has to be
ignored. If not, the sum of the positive deviations would always be exactly
equal to the sum of the negative deviations. Since these two sums would
automatically cancel out, the result would always be zero.
The mean deviation does take account of all the observations in the
series and is easy to interpret. However, it is less satisfactory from a theoret-
ical statistical point of view than the next two measures.
(b) The variance uses an alternative way of getting rid of the negative signs: it does so by calculating the square of the deviations.e It is then cal-
culated by finding the mean of the squared deviations. This is equal to
the sum of the squared deviations from the mean, divided by the size of
the sample.f
THE VARIANCE
The variance, s2, is equal to
the arithmetic mean of the squared deviations from the mean.
The formula is thus
s2� (2.2) �(Xi X)2
n
e Taking squares solves the problem because one of the rules of elementary arithmetic is that when
a negative number is multiplied by another negative number the result is a positive number; e.g.
4 � 4��16. f When we get to the deeper waters of chapter 5 we will find that the statisticians recommend that
n should be replaced by n 1 when using sample data in the calculation of the variance and
certain other measures. We will ignore this distinction until we reach that point. Note, however,
that if you use a computer program or calculator to check some of the calculations done ‘by
hand’ in chapters 2–4 you will get different results if your program uses n 1 rather than n. The
distinction between the two definitions matters only when the sample size is small. If n�30 or
more there would be practically no difference between the two estimates.
The calculation of the variance for the simple data set given above for
the hypothetical farming incomes in the two counties is set out in table 2.5.
The variance of County A�40/5�8, while that of County B�702/5�
140.4, and is very much greater. Note that the result of squaring the devia-
tions from the mean is that the more extreme values (such as incomes of 2
dollars or 33 dollars) have a very large impact on the variance.
This example was deliberately kept very simple to illustrate how the var-
iance is calculated and to show how it captures the much greater spread of
incomes in County B. In a subsequent exercise we will take more realistic
(and larger) samples, and use a computer program to calculate the vari-
ance.
The variance has many valuable theoretical properties and is widely
used in statistical work. However, it has the disadvantage that it is expressed
in square units. In the farm income example we would have to say that the
variance in County A was 8 ‘squared dollars’, which is not very meaningful.
The obvious way to get rid of these awkward squared units is to take the
square root of the variance. This leads to our final measure of dispersion.
(c) The standard deviation is the square root of the variance. Thus:
The standard deviation in County A equals �2.8 dollars
The standard deviation in County B equals �11.8 dollars�140.4 �8
§.
Table 2.5 Calculation of the sample variance for two sets of farm incomes
(1) (2) (3) (4) (5) (6)
County A County B
Xi (Xi �X –) (Xi �X
–)2 Xi (Xi �X –) (Xi �X
–)2
14 4 16 2 16 256
16 2 4 8 10 100
18 0 0 18 0 0
20 2 4 29 11 121
22 4 16 33 15 225
Sum 90 0 40 90 0 702
n�5
Mean (X –
) 18 18
Variance (s2) 8 140.4
THE STANDARD DEVIATION
The standard deviation, s,
is equal to
the square root of the
arithmetic mean of the squared deviations from the mean.
The formula is thus
s� (2.3)
The standard deviation is the most useful and widely used measure of
dispersion. It is measured in the same units as the data series to which it
refers. Thus we can say that the mean income in County A is 18 dollars,
with a standard deviation (s.d.) of 2.8 dollars. Such results will often be
reported in the form: X –
�18, s.d.�2.8, or in even more summary form as:
X –
�18�2.8.
The standard deviation can be thought of as the average or typical
(hence standard) deviation from the mean. Thus it will be seen that in
County A the deviations from the mean (see column (2) of table 2.5) vary
between 0 and 4, and the standard deviation is 2.8. Similarly in County B
the spread of the deviations in column (5) is from 0 to 16, and the standard
deviation is 11.8. The standard deviation thus lies somewhere between the
smallest and largest deviation from the mean.
The variance and the standard deviation have several mathematical
properties that make them more useful than the mean deviation. Both play
an extremely important role in many aspects of statistics.
2.3.3 The coefficient of variation The standard deviation is calculated in the same units (e.g. of weight or
length or currency or time) as the series to which it refers. This makes it
specific to that series, and means that it is difficult or misleading to
compare the absolute standard deviations of two series measured in
different underlying units. This would also be true even with the same unit,
if that is used for values where there is a substantial change in level.
For example, we might discover that in 1914, when the mean weekly
wage of adult male workers in the United Kingdom was £1.60, the standard
deviation was £0.30; and that in 1975 when the mean was £75, the standard
��(Xi X) 2
n
deviation was £17.50 (these are very rough approximations to the actual
data). Because growth and inflation has completely altered the level of
wage payments, it is impossible to tell from this whether the dispersion of
wages was larger or smaller in the later period.
To do this we need a measure of relative rather than absolute variation.
This can be obtained by dividing the standard deviation by the mean. The
result is known as the coefficient of variation, abbreviated to CV (or cv)
CV�s/X –
(2.4)
The two estimates for CV are, therefore, 0.3/1.6�0.19 and 17.5/75�0.23
and it is evident that the variation of wages has increased slightly.g
2.4 Measures of central tendency and dispersion with grouped data
In the preceding description of the procedures for computation of the
measures of central tendency and dispersion it was taken for granted that
the full set of values was available for all the cases. It may be, however, that
for some projects the relevant information is available only in the form of
grouped data. Imagine, for example, that instead of the individual data for relief payments in each of the 311 parishes, the only information that sur-
vived in the historical record was the grouped data of table 2.3. Certain
additional assumptions would then be required in order to calculate the
measures of central tendency and dispersion.
The results obtained for grouped data on these assumptions will not be
exactly the same as those that we would get if we had all the individual data.
However the inaccuracies will generally be minor unless the class intervals
are very wide or there are serious errors in the values attributed to the
open-ended classes.h
2.4.1 The median and mean with grouped data In order to calculate the median, mean, and other measures with such
grouped data it is necessary to make some assumptions about the distri-
bution of the unknown individual cases within the class intervals. For
the median, quartiles, and similar measures the conventional assumption
is that the cases are all spread at equal distances within their respective
intervals.
§.
g CV is sometimes expressed as a percentage. In the above example multiplying by 100 would give
results of 19 per cent and 23 per cent, respectively. h The scale of the possible differences is illustrated by question 1 in the exercises at the end of this
chapter.
To illustrate this procedure, consider the data on per capita relief pay-
ments in table 2.3. Since there are 311 parishes, the median parish will be
the 156th. There are 84 parishes in the first two class intervals, so we need
another 72 to reach the median parish, and it will thus fall somewhere in
the third class interval, which has a lower limit of 12 shillings. There are a
total of 93 parishes in that class interval and, by assumption, they are
spread at equal distances along the interval of 6 shillings. The median
parish will therefore occur at the value equal to 12 shillings plus 72/93 of 6
shillings, or 16.6 shillings.i
For measures involving the mean, the corresponding assumption is that
all the cases within the group have a value equal to the mid-point of the
group. Thus we take a value of 15 shillings for all 93 parishes in table 2.3
with relief payments between 12 and 18 shillings, and similarly for the
other intervals.j
Let us denote each of these mid-points by Xi, and the frequency with
which the cases occur (as shown in column (2) of table 2.3) by f. These two
values can then be multiplied to get the product for each class interval, f Xi.
The sum of those products is thus �f Xi, and the formula for the mean with
grouped data is then this sum divided by the sum of the frequencies, �f
X –
� (2.5)
There are two points to note with regard to this formula. First, for the
data in table 2.3, the denominator in this formula, �f, is 311, which is pre-
cisely what n would be when calculating the mean with ungrouped data
using the formula in (2.1). Secondly, the procedure is exactly equivalent to
the weighted arithmetic mean introduced in §2.2.1, as can be readily seen
by comparing this formula with the one in (2.1a). The mid-points of the
class intervals correspond to the observations in the series, and the fre-
quencies become the weights.
2.4.2 The variance and standard deviation with grouped data The calculation of the variance with grouped data then involves the follow-
ing steps. Find the deviations of each of the mid-points from this mean value
(Xi – X –
); square each of these terms (Xi X –
)2; and multiply this by the corre-
sponding frequency, f, to obtain the product f(Xi X –
). The variance is then
equal to the sum of these products divided by the sum of the frequencies
�f Xi �f
i This result is rounded to 1 decimal place. Note that a result can never be more precise than the
data on which it is based. j If there are any open-ended classes (e.g. if the last class in table 2.3 had been reported only in the
form of a lower limit of ‘equal to or greater than 42 shillings’) it would be necessary to make an
intelligent guess as to a sensible value for the mid-point.
s2� (2.6)
The standard deviation can similarly be calculated as the square root of
the variance
s� (2.7)
2.5 The shape of the distribution
2.5.1 Symmetrical and skewed distributions The final aspect of the distribution that is of interest is its shape, i.e. the way
in which the data are distributed around the central point. The distribution
around the mean may be symmetrical or it may be skewed either to the left
or to the right.
In §2.1.3 we noted that if we have a sufficiently large number of cases for
a variable measured on a continuous scale, and if these cases are then
grouped in sufficiently narrow class intervals, a line drawn through the
mid-point of each of the class intervals ultimately becomes a perfectly
smooth frequency curve.
One subset of such smooth frequency curves is perfectly symmetric.
This applies to the rectangular distribution, in which case the proportion of frequencies in each of a succession of equally spaced intervals is identi-
cal, and also to the pyramidical distribution.2 The most important member of this family of perfectly smooth and symmetrical frequency
curves is also bell-shaped and is known as the normal curve, illustrated in figure 2.4 (a). Because the curve is symmetrical and unimodal, the mean,
median, and mode are all identical. The normal curve plays a very large
part in this book and will be discussed more fully in §2.6.1.
Other families of distribution are smooth but not symmetrical. For
example, a distribution may be skewed to the right, as in figure 2.4 (b),
where there are large numbers of small or medium values and a tail stretch-
ing to the right with a small number of very large values (positive skew- ness). Examples of such positive skewness are often found with data on income distribution or holdings of land and other property, and on the size
of firms. Data on the age of marriage or on the number of births per
married women would also be positively skewed.k
Alternatively, a distribution may be skewed to the left, as in figure 2.4 (c),
��f (Xi X) 2
�f
�f (Xi X)2
�f
§.
k Such distributions are sometimes loosely referred to as log-normal. A true log-normal distribu-
tion is one that is strongly positively skewed when the data are entered in ordinary numbers and
normally distributed when the data are converted to logs.
Figure 2.4 Symmetrical and skewed frequency curves
(a) The normal curve
Mean
Median
(c) Smooth curve skewed to the left (negative skewness)
Mean
Median
(b) Smooth curve skewed to the right (positive skewness)
with a tail made up of a small number of very low values (negative skew- ness). This would be characteristic of data on the age of death in Britain or the United States in any normal peacetime year. There would be a small
number of deaths at low ages as a result of accidents and illness, but the
great majority would occur at ages 60–90 and the right hand tail would
effectively end abruptly a little above age 100.
2.5.2 Measures of skewness Measures of skewness should be independent of the units in which the var-
iable is measured, and should be zero when the distribution is perfectly
symmetrical so that there is no skewness. One simple measure of the extent
to which a distribution is asymmetrical (skewed) is given by the relation-
ship between the mean and the median: the greater the skewness the larger
the difference between the mean and the median. If the distribution is pos-
itively skewed, the mean will be raised by the large values in the right-hand
tail of the distribution, whereas the median is unaffected by these extreme
values. The mean will thus be larger than the median. Conversely, if the dis-
tribution is negatively skewed, the mean will be smaller than the median. A
very rough indication of the presence of either positive or negative skew-
ness in any particular distribution can thus be obtained by a comparison of
the mean and median of a distribution.
A simple measure based on these features is a variant of the Pearson
coefficient of skewness
Skewness� (2.8)
This satisfies the two criteria mentioned above and would lie between –3
and �3, though these limits are seldom encountered in practice.
Statistical packages tend to use more precise measures based on the
deviations of the individual observations from the mean. In one such
measure, these deviations are cubed, and expressed in terms of the number
of standard deviations, also cubed. The arithmetic mean of this term is
then calculated. The formula is thus
(2.9) �(Xi X)3/s3
n
3(Mean Median) Standard deviation
§.
2.6 The normal distribution
2.6.1 Properties of the normal distribution Since the perfect normal curve is a theoretical distribution based on an
infinitely large number of observations it can be only approximated by
actual frequency distributions, but we will find many examples of random
variables that show a very good approximation to the theoretical normal
distribution. We will also find that the normal distribution has a number of
remarkable properties.
A variable is defined as a random variable if its numerical value is unknown until it is observed ; it is not perfectly predictable. The observation
can be the outcome of an experiment such as tossing a coin or planting
seeds, or it can be the measurement of a variable such as height, the price of
wheat, or the age of marriage. If it can take only a finite number of values
(for example, the number of births per married women is limited to posi-
tive whole numbers such as 0, 1, 2, or 3) it is called a discrete random vari- able. If it can take any value (within a relevant interval) it is called a
continuous random variable. The normal curve is defined by an equation involving two constants and
the mean and standard deviation of the distribution of a continuous
random variable, X. The equation gives the value of Y (the height of the
curve, shown on the vertical axis) for any value of X (measured along the
horizontal axis). Y thus measures the frequency with which each value of X
occurs. Although the form of this equation is considerably more compli-
cated than the one used in §1.5 for a straight line (Y�a�bX), the principle
is exactly the same. We select the values for the constants, in this case they
are � (lower case Greek pi, approximately 3.1416) and e (approximately
2.7183), plug in the changing values for the variable(s) in the equation (in
this case the values for X), and calculate the corresponding values for Y.
The normal curve is not a single, unique, curve. All normal curves will
have the same smooth and symmetrical bell-like shape, but there are very
many different normal curves, each defined by its particular mean and
standard deviation. Two normal curves may have the same spread (stan-
dard deviation) but different means, in which case the one with the larger
mean will be further to the right along the horizontal (X) axis, as in figure
2.5 (a). Or they may have the same mean but different standard deviations,
in which case the one with the larger standard deviation will be spread out
more widely on either side of the mean, as in figure 2.5 (b). Or both means
and standard deviations may be different, as in figure 2.5 (c).
When a statistical program is used to plot a histogram for any particu-
lar data set it can also be requested to add the particular normal curve
§.
Figure 2.5 A family of normal curves
(a) Different means but the same standard deviation
(b) The same mean but different standard deviations
(c) Different means and standard deviations
given by the mean and standard deviation of that data. It is thus possible to
see how far the data conform to the normal distribution. This is done, for
example, in figure 2.6 with the histogram of parish relief payments previ-
ously shown in figure 2.2 (b), and it can be seen that it is a reasonably good
approximation.
The fact that many actual distributions approximate the theoretical
normal distribution enables statisticians to make extensive use of the prop-
erties of the theoretical normal curve. One obvious property that we have
already noted implicitly is that the mean, median, and mode are all equal;
they coincide at the highest point of the curve and there is only one mode.
A second property of considerable importance relates to the area under
the normal curve. Irrespective of the particular mean or standard devia-
tion of the curve it will always be the case that a constant proportion of all the
cases will lie a given distance from the mean measured in terms of the standard
deviation. It is thus possible to calculate what the proportion is for any par-
ticular distance from the mean expressed in terms of standard deviations (std
devs).
Since the distribution is perfectly symmetrical we also know that exactly
one-half of the above proportions are to the right of (greater than) the
mean and one-half are to the left of (smaller than) the mean.
Figure 2.6 Histogram of per capita relief payments in 311 parishes in 1831 with normal curve superimposed
4844403632282420161284
N um
be r
of p
ar is
he s
50
40
30
20
10
0
Relief payments (shillings)
Three particular results, specified in the following box, are of special
interest.
SPREAD OF THE NORMAL CURVE
90 per cent of all cases are within 1.645 std devs either side of the mean, leaving 5 per cent in each of the two tails
95 per cent of all cases are within 1.96 std devs either side of the mean, leaving 2.5 per cent in each of the two tails
99 per cent of all cases are within 2.58 std devs either side of the mean, leaving 0.5 per cent in each of the two tails
These proportions are illustrated in the three panels of figure 2.7.
In the above description, particular areas were selected given in conven-
ient whole numbers. It is equally possible to select convenient whole
numbers for the number of standard deviations to be taken either side of
the mean, in particular:
� The distance of �1 std dev from the mean covers 68.26 per cent of all cases
� The distance of �2 std devs from the mean covers 95.46 per cent of all cases
� The distance of �3 std devs from the mean covers 99.73 per cent of all cases
To fix this extremely important property of the normal distribution in
your mind, consider an example of a roughly normal distribution that
should already be intuitively familiar. If you are told that the mean height
of a large sample of male students at the University of Virginia is 6 feet (�)
and the standard deviation (std dev) is 5 inches (�), you could apply the
characteristics stated above and would find that:
� Roughly two-thirds are between 5� 7� and 6� 5� tall (the mean �1 std dev)
� That only a small minority, less than 5 per cent, are shorter than 5� 2� or
taller than 6� 10� (the mean �2 std devs)
§.
–1.645 sd + 1.645 sd
5 per cent 5 per cent
45 per cent 45 per cent
Figure 2.7 Areas under the normal curve
(a) 1.645 standard deviations either side of the mean covers 90 per cent of the area. This leaves 5 per cent in each tail.
–2.58 sd + 2.58 sd0.5 per cent
0.5 per cent
49.5 per cent 49.5 per cent
(c) 2.58 standard deviations either side of the mean covers 99 per cent of the area. This leaves 0.5 per cent in each tail.
–1.96 sd + 1.96 sd 2.5 per cent 2.5 per cent
47.5 per cent 47.5 per cent (b) 1.96 standard
deviations either side of the mean covers 95 per cent of the area. This leaves 2.5 per cent in each tail.
� There are effectively none below 4� 9� or above 7� 3� (the mean �3 std
devs).l
These results should seem plausible, and should help you to understand
how to interpret information about the standard deviation for (approxi-
mately normal) distributions that are less familiar than heights.
This leads directly to the idea that areas under the normal curve can be
thought of in terms of the number of standard deviations.
2.6.2 The standard normal distribution We have seen (§2.6.1) that there is a large family of normal curves, each
defined by its particular mean and standard deviation. Because there are so
many different normal curves it is helpful to standardize them.
To do this, note that one way to measure the distance from the mean of
the distribution to any point above or below the mean is to ask how many
standard units that distance represents. Thus with the height example used
in the previous subsection, where the mean was 6� and the standard devia-
tion was 5�, we could say that if a student measured 6� 11� his height was 2.2
standard deviations above the mean.
This idea of the distance from the mean (X X –
) expressed in units of stan-
dard deviations can be generalized to standardize any normal distribution.
The result creates a new variable, designated Z, and the distribution of Z is
referred to as the standard normal distribution
Z� (2.10)
A very important feature of this standardized distribution is that the
mean must, by definition, always be zero, and the standard deviation must
always be equal to 1. This is illustrated in table 2.6, where a hypothetical
(and approximately normally distributed) series for the heights of a sample
of 40 male students is given in column (1), and the corresponding stan-
dardized distribution (Z) in column (4). (The distances exactly 1 or 2 stan-
dard deviations above and below the mean are highlighted.) Columns (2)
and (3) give the information required to calculate Z and the standard
(X X)
s
§.
l For those more accustomed to think in metric units, the equivalent units are a mean of 180 cm
and a standard deviation of 13 cm. The height of roughly two-thirds of the students is thus
between 167 and 193 cm, and fewer than 5 per cent are either shorter than 154 cm or taller than
206 cm.
Table 2.6 Heights of a sample of 40 male students and the standardized distribution
(1) (2) (3) (4) (5)
Xi (Xi �X –)/s (Zi �Z
–)2
Height in inches (Xi �X –) (Xi �X
–)2 �Z �Z2
61.1 10.9 118.81 2.18 4.75 62.0 10.0 100.00 2.00 4.00 63.5 8.5 72.25 1.70 2.89 64.7 7.3 53.29 1.46 2.13 65.3 6.7 44.89 1.34 1.80 66.3 5.7 32.49 1.14. 1.30 67.0 5.0 25.00 1.00 1.00 68.0 4.0 16.00 0.80 0.64 68.2 3.8 14.44 0.76 0.58 68.3 3.7 13.69 0.74 0.55 68.5 3.5 12.25 0.70 0.49 69.2 2.8 7.84 0.56 0.31 69.3 2.7 7.29 0.54 0.29 70.0 2.0 4.00 0.40 0.16 70.3 1.7 2.89 0.34 0.12 70.7 1.3 1.69 0.26 0.07 71.4 0.6 0.36 0.12 0.01 71.7 0.3 0.09 0.06 0.00 72.0 0.0 0.00 0.00 0.00 72.4 0.4 0.16 0.08 0.01 72.6 0.6 0.36 0.12 0.01 72.8 0.8 0.64 0.16 0.03 72.9 0.9 0.81 0.18 0.03 73.4 1.4 1.96 0.28 0.08 73.7 1.7 2.89 0.34 0.12 73.9 1.9 3.61 0.38 0.14 74.1 2.1 4.41 0.42 0.18 74.3 2.3 5.29 0.46 0.21 74.5 2.5 6.25 0.50 0.25 75.4 3.4 11.56 0.68 0.46 75.6 3.6 12.96 0.72 0.52 76.3 4.3 18.49 0.86 0.74 76.5 4.5 20.25 0.90 0.81 77.0 5.0 25.00 1.00 1.00 77.3 5.3 28.09 1.06 1.12 77.5 5.5 30.25 1.10 1.21 78.8 6.8 46.24 1.36 1.85 80.3 8.3 68.89 1.66 2.76 81.2 9.2 84.64 1.84 3.39 82.0 10.0 100.00 2.00 4.00
� 2,880.0 0.00 1,000.00 0.00 40.00
n�40
deviation of Z is calculated in column (5). The mean of Z is seen to be 0 and
the standard deviation is 1.00.m
It will also be seen how the standardized distribution maps the initial
distribution, so that the mean value of the original series (�72 inches) cor-
responds to the mean value of the standardized distribution (�0).
Similarly, a distance of 1 standard deviation above or below the mean in the
original distribution (X –
�1 std dev�72�5�67 or 77 inches) corresponds
to values of Z�Z – �1 std dev�0�1�–1 or�1. In the same way a distance
of 2 standard deviations above or below the mean (62 or 82 inches) corre-
sponds to values of Z�–2 or�2.
Statisticians have compiled tables that show the proportion under the
standardized curve for all values of Z. The proportions in these tables apply
to the perfect theoretical distribution, but for any actual distribution that is
approximately normal the proportions will be roughly the same. An
extract from such a table of the standard normal distribution is reproduced
in table 2.7.n
In this particular version the area under the curve is subdivided in five
different ways, so that five different proportions of the total area are given
for each positive value of Z. Since the curve is perfectly symmetrical the
proportions for the corresponding negative values of Z would be exactly
the same.
The five different proportions in columns (2)–(6) are:
(a) The cumulative proportion up to Z.
(b) The proportion from the mean to the specific positive value of Z.
(Since the total proportion to the left of the mean�0.5, this is always
0.5 less than the value in (a).)
(c) The proportion in the right-hand tail beyond Z. (Since the total pro-
portion under the curve�1, this is always 1.0 minus the value in (a).)
(d) The proportion in both tails given by the positive and negative value of
Z. (This is always twice the proportion in (c).)
(e) The central proportion excluding the two tails defined by the positive
and negative values of Z. (Since the total proportion under the curve�
1, this is always 1.0 minus the value in (d).)
§.
m The calculations are as follows: The mean height for these students�X –
��X/n�2,880/40�
72.0. The variance of the heights�s2��(Xi X –
)2/n�1,000/40�25.00, and so the standard
deviation of these heights�s� �5.00. This standard deviation is then used in column (4)
to calculate Z. For Z, �Z/n�0.00/40�0.0, and this is used in column (5) to calculate the vari-
ance of Z��(Zi Z –
)2/n �40/40�1.00. The standard deviation is therefore �1.00. n A more complete table covering all values of Z can be consulted in D. V. Lindley and W. F. Scott,
New Cambridge Statistical Tables, Cambridge University Press, 2nd edn., 1995, p.34, and in most
general statistics books.
�1.00
�25
Table 2.7 Areas under the standard normal curve for selected values of Z
(1) (2) (3) (4) (5) (6)
Cumulative Area from Area in Area in
Selected area mean one tail both tails Central
values of Z up to Z up to Z beyond Z beyond Z area
0.00 0.5000 0.0000 0.5000 1.0000 0.0000
0.20 0.5793 0.0793 0.4207 0.8414 0.1586
0.40 0.6554 0.1554 0.3446 0.6892 0.3108
0.60 0.7257 0.2257 0.2743 0.5486 0.4514
0.80 0.7881 0.2881 0.2119 0.4238 0.5762
1.00 0.8413 0.3413 0.1587 0.3174 0.6826
1.20 0.8849 0.3849 0.1151 0.2302 0.7698
1.28 0.8997 0.3997 0.1003 0.2006 0.7994
1.40 0.9192 0.4192 0.0808 0.1616 0.8384
1.60 0.9452 0.4452 0.0548 0.1096 0.8904
1.645 0.9500 0.4500 0.0500 0.1000 0.9000
1.80 0.9641 0.4641 0.0359 0.0718 0.9282
1.96 0.9750 0.4750 0.0250 0.0500 0.9500
2.00 0.9772 0.4772 0.0228 0.0456 0.9544
2.20 0.9861 0.4861 0.0139 0.0278 0.9722
2.40 0.9918 0.4918 0.0082 0.0164 0.9836
2.58 0.9951 0.4951 0.0049 0.0098 0.9902
2.80 0.9974 0.4974 0.0026 0.0052 0.9948
3.00 0.9986 0.4986 0.0014 0.0028 0.9972
3.20 0.9993 0.4993 0.0007 0.0014 0.9986
3.40 0.9997 0.4997 0.0003 0.0006 0.9994
3.60 0.9999 0.4999 0.0001 0.0002 0.9998
3.80 0.9999 0.4999 0.0001 0.0002 0.9998
Note:
The two-tailed area in column (5) refers to both the positive and the negative values of Z.
Source: Lindley and Scott, Statistical Tables, p. 34 for column (2); other columns as explained
in the text.
You will find that authors usually give only one of these five possible
proportions, and different authors choose different proportions. This
divergence in the way the table is printed is confusing, and means that if
you want to use one of these tables you must first establish the form in
which the information is presented.
The proportions are normally given to 3 or 4 decimal places. It is often
more convenient to think in terms of percentages, and these are obtained
by multiplying the proportions by 100 (i.e. moving the decimal point 2
places to the right): for example, 0.500�50 per cent.
Once the form in which the data are presented in any particular table is
clarified, it is a simple matter to find the proportion up to or beyond any
given value of Z. Take, for instance, the student in table 2.6 with a height of
77 inches. This corresponds to a value for Z�1.0, and is thus 1.0 standard
deviations from the mean height of 72 inches. From column (4) of table 2.7
it can be seen that the proportion greater than this value of Z is 0.1587, or
15.9 per cent. Since there are 40 students this indicates that if the heights of
this sample are normally distributed there should be approximately six stu-
dents (15.9 per cent of 40) taller than this one. Table 2.6 in fact has six taller
students.
Notes 1 Note that the precise class intervals depend on the extent to which the underlying
data have been rounded. The data in table 2.1 were rounded to 1 decimal place, so the
true upper limit of the first class interval in table 2.2 would be 9.94 shillings. All
values from 9.90 to 9.94 would have been rounded down to 9.9 shillings, and so
would be included in the interval �5 but 10. All values from 9.95 to 9.99 would
have been rounded up to 10 shillings, and so would be included in the interval �10
but 15. 2 If a perfect die is thrown once the probability of obtaining a six is 1/6, and the prob-
ability of obtaining each of the other face values from 1 to 5 would be exactly the
same, so these outcomes would represent a rectangular distribution. A pyramid-
shaped distribution would be obtained for the probability of each value if the die
were thrown twice. With each successive throw beyond two the distribution moves
towards the bell-shape of the normal curve. For the corresponding successive distri-
butions when the probabilities relate to tossing increasing numbers of coins, see
figure 5.1.
2.7 Exercises for chapter 2
All exercises in this and subsequent chapters may be solved by computer, unless the question states specifically that they should be done by hand.
1. Extract the data on INCOME for Cambridgeshire (County 5) by parish
from the Boyer relief data set.
(i) By hand, use these data to construct:
� An array
� A frequency distribution using 5 class intervals of equal width of
£5, beginning at £20
� A frequency distribution using 10 class intervals of equal width of
£2, beginning at £20
� A frequency distribution using 16 class intervals of equal width of
£1, beginning at £20.
(ii) Which frequency distribution provides the information on
Cambridgeshire incomes in the most useful fashion?
(iii) What difference does it make if you round the income figures to 1
decimal place before constructing the frequency distributions?
(iv) What difference does it make if you round the income figures to the
nearest whole number before constructing the frequency distribu-
tions?
(v) Instruct the computer to construct a histogram with 10 intervals.
Are the resulting class intervals the same as for the relative frequency
distribution in (i) above? Explain your finding.
2. Take the WEALTH variable for every parish in the Boyer relief data set.
(i) Calculate the following statistics:
� Upper, lower, and middle quartiles
� 10th and 90th percentiles
� Range
� Mean, median, and mode
� Variance
� Standard deviation
� Coefficient of variation.
(ii) Plot a histogram of WEALTH with: (a) 31 intervals and (b) 11
intervals.
(iii) Plot a bar chart of mean WEALTH in each of 21 counties. (Hint : use
COUNTY for the category axis.)
Comment on the relationship between the different measures of central
tendency and dispersion, and discuss what they tell you about wealth in
these parishes.
3. Using Boyer’s birth rate data set, calculate the means and standard devi-
ations of INCOME, BRTHRATE, and DENSITY, counting each parish as a
single observation. Now calculate the means and standard deviations of
the same variables, with each parish weighted by its population (POP).
How would you explain the difference between the two measures? Is there
any reason to prefer one set of measures to the other?
4. The following table provides data on the percentage of women in
England and Wales, aged 10 and above, who worked outside the home in
1911 and 1991, classified by age and marital status. It also shows the per-
centage of women in each age group who were married in 1911 and 1991,
as well as the age structure of all women, single or married, in the two years.
1911 1991
% single % married % all % single % married % all
women women women % women women women women % women
Age working working married by age working working married by age
15 10.4 0.0 0.0 11.8 – – – –
20 69.5 13.4 1.2 11.3 54.7 47.2 2.0 6.2
25 77.6 12.9 24.2 11.3 75.8 62.3 22.9 9.1
35 73.7 10.6 63.2 21.0 75.7 62.2 62.8 18.4
45 65.3 10.6 75.3 16.9 75.4 72.4 78.5 16.8
55 53.5 10.5 70.9 12.3 73.0 70.5 79.9 13.9
65 36.6 8.8 58.4 8.2 35.9 38.0 72.5 12.6
75 18.7 5.7 37.5 5.1 4.9 5.1 53.6 11.9
≥75 6.3 2.3 16.1 2.1 1.1 1.5 23.2 11.2
ALL 50.4 10.3 44.6 100.0 46.0 53.0 56.1 100.0
Note:
Single includes single, widowed, and divorced women. Married refers to currently married
women.
Sources: Census of Population, England and Wales, 1911 and 1991.
(i) What proportion of ALL women were in paid employment in (a)
1911? (b) 1991?
(ii) How much of the overall change in the proportion of ALL women
working between 1911 and 1991 was due to:
§.
(a) The changing proportion at work among single and married
women. (Hint : evaluate the level of female market work if the
distribution of married and single women working had
remained the same as in 1911.)
(b) The changing age structure of the female population.
(c) The changing age at marriage among women.
Write a brief paragraph setting out what the table and your calculations
indicate about the changing patterns of women’s involvement in paid
employment.
5. Let us suppose that the data on WEALTH for the 311 parishes in the
Boyer relief data set have been aggregated in the original source to produce
the following table:
Average wealth Number of
per person (£) parishes Population
�0 but 2 14 26,023
�2 but 3 44 90,460
�3 but 4 83 123,595
�4 but 5 58 76,421
�5 but 6 48 48,008
�6 but 7 25 19,811
�7 but 8 16 11,086
�8 but 9 6 3,820
�9 but 10 9 5,225
�10 8 2,634
ALL 311 407,083
Note:
No other information is available.
(i) Calculate total wealth in each class interval using (a) the number of
parishes, and (b) the population, as the weighting system. What
assumptions did you make about the level of wealth per person in
each class interval and why?
(ii) Calculate the mean, median, mode, upper and lower quartiles, vari-
ance, standard deviation, and coefficient of variation for total
wealth.
(iii) Change your assumption about the level of wealth per person in the
top and bottom classes and recalibrate the measures of central ten-
dency and dispersion. Which measures are most sensitive to changes
in the measurement of average wealth? Explain your answers.
6. Using the WEALTH data from Boyer’s relief data set:
(i) Calculate the average level of wealth per person for each county.
Repeat the exercise of question 5, using both the number of parishes
and the number of people in each county to calculate total wealth.
(ii) Compare the measures of central value and dispersion with those
produced by question 5. Identify and account for any discrepancies.
How do your results in questions 5 and 6 compare to the results of
question 2?
Is the loss of information in these various permutations serious? How
might we evaluate any loss of information?
7. Instruct the computer to produce histograms of the following variables
from the Boyer data set on births:
BRTHRATE
INFTMORT
INCOME
DENSITY
(i) In each case, inspect the shapes of the histograms and assess whether
the data are normally distributed, negatively skewed, or positively
skewed. Record your findings.
(ii) Now ask the computer to calculate the degree of skewness in each of
these variables. Compare these results to your own findings.
8. The LONDON variable in the Boyer data set is measured by assigning to
each parish in a given county the same distance from the centre of London
as the mid-point of the county, in miles. This is likely to create a bias in the
measurement of distance for each parish. Why? How will the bias affect the
measured mean and standard deviation of the variable LONDON, relative
to its ‘true’ value? How might you test for the extent of any differences?
9. We are interested in determining which parishes are the most and least
generous providers of relief to the unemployed poor. How might we con-
struct such a variable from the data in the RELIEF file?
10. A random sample of 1,000 women was found to have a mean age of first
marriage of 23.6 years with a standard deviation of 3 years. The ages can be
assumed to be normally distributed. Use the table of the standard normal
distribution (table 2.7) to calculate:
§.
(i) How many of the women were first married between 20 and 29
years?
(ii) What was the minimum age of marriage of the oldest 5 per cent of
the sample?
(iii) What proportion of the women married at an age that differed from
the mean by more than 1.8 standard deviations?
Correlation
3.1 The concept of correlation
This chapter is devoted to one of the central issues in the quantitative study
of two variables: is there a relationship between them? Our aim is to explain
the basic concepts, and then to obtain a measure of the degree to which the
two variables are related. The statistical term for such a relationship or
association is correlation, and the measure of the strength of that relation- ship is called the correlation coefficient.
We will deal first with the relationship between ratio or interval level
(numerical) variables, and then look more briefly in §3.3 at the treatment
of nominal and ordinal level measurements.a In this initial discussion we
ignore the further matters that arise because the results are usually based
on data obtained from a sample. Treatment of this important aspect must
be deferred until the issues of confidence intervals and hypothesis testing
have been covered in chapters 5 and 6.
If there is a relationship between the two sets of paired variables (for
example, between the level of relief expenditure (RELIEF) and the propor-
tion of unemployed labourers (UNEMP) in each of the parishes, or
between EMPFOR, the annual series for foreign employment and IRMIG,
the number of immigrants from Ireland), it may be either positive or nega-
tive. When there is positive correlation, high values of the one variable are associated with high values of the other. When there is negative correla- tion, high values of the one variable are associated with low values of the other. In each case the closeness of the relationship may be strong or weak.
The third possibility is, of course, that there is no consistent relation-
ship: high values of one variable are sometimes associated with high values
of the other, and sometimes with low values.
a These different levels of measurement were explained in §1.3.3.
3.1.1 Correlation is not causation It must be stressed at the outset that correlation does not imply causation. It
is only some appropriate theory that can provide a hypothesis that might
lead us to believe that there is a causal relationship between two variables.
An empirical finding that there is an association does not imply that
changes in the one variable are causing the changes in the other, however
high the degree of correlation. Even if we have reason to believe that there is
a causal relationship, the fact of a high degree of correlation does not in
itself prove causality, although it may be regarded as another piece of evi-
dence in support of the underlying hypothesis.
It is very easy to get an association between two variables even when there
is no plausible hypothesis that might cause one to influence the other. This
can occur either coincidentally, because the two variables are independently
moving in the same directions, or – more commonly – because the move-
ments in both variables are influenced by some (unspecified) third variable.
Over a period of years there might, for example, be a strong positive cor-
relation between sales of umbrellas in New York and the number of slugs
counted in the city parks. But this would occur only because both are
influenced by how much rain has fallen; the umbrellas do not actually
cause the proliferation of slugs. The simultaneous decline of the birth rate
and of the number of storks in Sweden as a result of industrialization is a
more famous example.
Correlation of this sort is called spurious or nonsense correlation, but as noted by Wonnacott and Wonnacott, it would be more accurate to say that
the correlation is real enough, and that it is the naive inference of cause and
effect which is nonsense.b
3.1.2 Scatter diagrams and correlation The best way to explore the form and extent of any association between two
variables is to plot them on a scatter diagram (or scatter plot). This is a chart in which one variable is measured on the vertical axis, and the other
on the horizontal axis.
Each point on the diagram represents a pair of values of the two vari-
ables. In a time series, each pair relates to the same time period (for
example, UNEMP and BENEFITS, British unemployment rates and
benefit payments, in a given year, say 1925). In a cross-section, each pair
relates to the same case (for example, CNTYMIG and AGRIC, emigration
rates and the proportion of the male labour force in agriculture, in each
Irish county). Some illustrative scatter plots are shown in figure 3.1, with
the two variables referred to simply as X and Y.
b T. H. Wonnacott and R. J. Wonnacott, Introductory Statistics, 5th edn., John Wiley, 1990.
§.
X
Y
r = +1
(a)
(e)Y
X
r = 0
X
Y
r = 0
(f)
X
Y
r = +0.6
(c)
X
Y
r = –0.8
(d)
X
Y
r = –1
(b)
Figure 3.1 Scatter diagrams showing different strengths and directions of relationships between X and Y
Perfect correlation In the upper panel, plot (a) on the left has perfect positive correlation, and
plot (b) on the right shows perfect negative correlation. In each case all the
points fall on a perfectly straight line.
Such very strong association would seldom, if ever, be encountered in
the social sciences. It might, however, occur in a controlled physical experi-
ment; for example, X in plot (a) might be a measure of the flow of water
into a closed tank, and Y the height of the water in the tank.
Positive and negative correlation In the middle panel, plots (c) and (d) again illustrate positive and negative
correlation, but here the degree of association is somewhat weaker. In (c)
high values of X are generally associated with high values of Y, and low
values of X with low values of Y, though with some exceptions, giving a
modest positive correlation. In (d) there is a stronger negative correlation:
most high values of X are associated with low values of Y, and low values of
X with high values of Y.
No linear relationship The lower panel shows two examples where there is no linear relationship.
They are, however, very different. In (e) on the left there is no relationship
of any kind: the plots are all over the graph. In (f) on the right there very
clearly is a relationship, but it is not a linear (straight-line) one; it is U-
shaped.
Plot (f) is important as a reminder that the techniques for the measure-
ment of association that we are examining here (and until chapter 12)
relate only to linear relationships. Nevertheless, we will find in practice that there are a great many interesting relationships that are, at least approxi-
mately, linear. So even with this restriction, correlation is a very useful
technique.
As an illustration of an historical relationship, a scatter diagram based
on the data for 24 Kent parishes reproduced in table 2.1 is given in figure
3.2. UNEMP is shown on the (horizontal) X axis and RELIEF on the (verti-
cal) Y axis.
3.1.3 Outliers Note that one of the points in figure 3.2 (in the top right) is very different
from all the others. This is known as an outlier. One of the advantages of plotting a scatter diagram before proceeding to more elaborate procedures
is that it immediately reveals the presence of such outliers. The first step
should always be to check the data to ensure that the outlier is a genuine
observation and not an error in the original source or in the transcription
of the data. If it is genuine then its treatment in further analysis is a matter
of judgement and there is no simple rule.
If more is known about the circumstances creating the outlier it may be
decided that these represent such exceptional circumstances that the case
will distort the principal purposes of the study, and alternative procedures
should be invoked. This might apply, for example, when two time series
have been plotted on a scatter diagram and it is realized that the outlier rep-
resents a year in which there was some abnormal extraneous event such as a
war or a prolonged strike. In a regional cross-section it may be found that
the particular area includes a major city and that this is distorting the
results of what is intended to be primarily a study of rural areas.
Alternatively, it may be judged that despite the extreme values the
outlier is a fully representative case and should remain as one of the obser-
vations in the analysis and in the calculation of further statistical measures.
These and other issues relating to the treatment of outliers are consid-
ered in §11.3.4.
§.
Figure 3.2 Scatter diagram of relief and unemployment, 24 Kent parishes in 1831
30.020.010.00.0
R el
ie f
pa ym
en ts
( sh
ill in
gs )
40
30
20
10
0
Unemployment (per cent)
3.2 The correlation coefficient
A scatter diagram is an excellent way to get a good visual impression of the
presence or absence of a relationship between two variables. But if we want
to make a more precise statement about the degree of association we have
to use the numerical measure known as the correlation coefficient.c
3.2.1 Measuring the strength of the association This measurement of the strength of the association between two interval
level variables is designated by the symbol r. It is constructed so that it can
take any value in the range from 1.0 to �1.0:
r� 1.0 indicates perfect negative correlation
r��1.0 indicates perfect positive correlation
r�0 indicates the complete absence of any (linear) correlation.
In figure 3.1, r equals 0.6 for the modest positive relationship in plot (c),
and it equals 0.8 for the stronger negative relationship in plot (d). If we
calculate the correlation coefficient for UNEMP and RELIEF in the 24 Kent
parishes in figure 3.2 the result is found to be 0.52.
THE CORRELATION COEFFICIENT
The correlation coefficient, r,
is a measure of the
degree of linear relationship between two variables.
The value of r may be positive or negative,
and is independent of any specific units.
r will always be between 0 and 1.0 for negative correlation,
and between 0 and�1.0 for positive correlation.
The higher the value of r,
the stronger is the relationship
between the two variables.
c Or, more formally, as the Pearson product-moment correlation coefficient.
In calculating r, the variables X and Y appear symmetrically.
Interchanging the data for X and Y would not change the results in any way.
So the coefficient can tell us only the degree of association between the two
variables. An historian might have reason to believe that two variables that
show a strong relationship are also causally related, so that one of the vari-
ables is influenced by the other. But he cannot learn anything from the cor-
relation coefficient about which of the two might be exercising the
influence, and which responding to it.
3.2.2 Derivation of the correlation coefficient, r At this stage it would be possible simply to hand over to the computer and
leave an appropriate statistical program to apply the appropriate formula
and calculate r, the correlation coefficient. However, it is desirable to probe
a little further in order to develop a better grasp of the ideas lying behind
the calculation of the coefficient. These ideas are of great general impor-
tance to an understanding of the quantitative methods we shall be using,
both here and in many of the subsequent exercises involving regression
analysis.
The central issue to be addressed is the measurement of the association
between two variables, X and Y. As we have seen, the relationship can be
summarized as follows:
Values of X Values of Y
large large
Positive correlation: or or
small small
large small
Negative correlation: or or
small large
We have also seen in figure 3.1 that when there is perfect positive or neg-
ative correlation, the points cluster along the straight line (plots (a) or (b)).
The further the points deviate from this line the weaker is the correlation
between the two variables, until we reach the case in plot (e) where there is
no possible single line that would be a good fit through all the points.
To get a numerical measurement it is necessary to give a more precise
content to the scheme set out above, and to this sense of deviations from
the line of perfect correlation. What do we mean by ‘large’ or ‘small’ for any
particular pair of variables? Given the techniques that have already been
§.
explored in developing the measures of dispersion in §2.3, one obvious
possibility would be to think in terms of the size of the respective devia- tions from the mean.
The crucial first step in the calculation of the correlation coefficient is
thus to obtain the deviations of each value of X from the mean of the series
for the X variable (X –
), and the deviations of each value of Y from the mean
of that series (Y –
).
In order to be able to see exactly what is involved, consider a very simple
example, with only 4 pairs of observations:
Xi Yi 4 6
6 11
10 7
12 16
Mean: X –
�8 – – Y
– �10
–– ––
These four sets of points are plotted in figure 3.3, which is divided into four
quadrants by drawing a horizontal line through the mean value of Y
(Y –
�10) and a vertical line through the mean value of X (X –
�8).
The first pair of values (4, 6) appears in quadrant III, with both observa-
tions below their respective means. Thus both deviations from the mean
Figure 3.3 Deviations from the mean
X
14121086420
Y
18
16
14
12
10
8
6
4
2
0
IV
III
I
II
6, 11
4, 6 10, 7
12, 16
– 2 + 1
+ 4
+ 6
– 3
+ 2
– 4
– 4
Y
X
are negative. To get a measure of the combined value of the deviation of
both X and Y from their respective means we can either:
add the two deviations to get their sum : 4� 4� 8
or multiply the deviations to get their product : 4� 4��16.d
Multiplication has certain advantages, and this is the procedure adopted. It
then follows that the pairs of values in each quadrant can be classified as
follows:
� Any pair of values in quadrant I will have both values above their respec-
tive means, and both deviations from the mean will be positive. So the
product of the deviations for all such points will be positive.
� Any point in quadrant III will have both values below their respective
means, and both deviations from the mean will be negative. So the
product of the deviations for all such points will also be positive.
� Any pair of values in quadrants II and IV will have one of the pair above
its mean and one below, and so one deviation will be positive and the
other negative. So the product of the deviations for all such points will
be negative.
The basic element in the measure of association (r) is thus obtained by
adding up all the products of the deviations from the mean. Two further steps
are then required to get the final measure:
(a) We have obtained the total of the combined deviations from the mean
by adding up (summing) all the products of the deviations from the
mean. But this measure is obviously affected by the number of cases in
the data set: the larger the number the bigger the sum. To correct for
this we divide by the number of cases; this is equivalent to taking an
average of the product of the deviations. The resulting measure is
called the covariance of X and Y, or COV (XY). (b) The covariance is measured in the units in which X and Y are meas-
ured, and is thus awkward to use. If the units were changed (for
example from feet to metres) this would cause an inappropriate
change in the covariance. In order to correct for this, the covariance is
divided by both the standard deviation of X and the standard devia-
tion of Y, thus neutralizing the specific units in which X and Y happen
to be measured.
§.
d If one negative number is multiplied by another negative number the product is positive; if a
negative number is multiplied by a positive number the product is negative.
COVARIANCE
The covariance of two variables X and Y is equal to
the sum of the products
of their deviations from their respective means,
divided by
the number of cases.
COV(XY)� (3.1)
We thus get the final formula for the correlation coefficient.
THE CORRELATION COEFFICIENT, r
For two variables, X and Y,
the correlation coefficient, r, is equal to
the covariance of X and Y
divided by
the product of the
standard deviation of X and the standard deviation of Y
r� (3.2a)
This can also be written as1
r� (3.2b)
r is thus independent of any specific units of measure. Unlike the mean or
the standard deviation a change in the units in which the variables are
measured would have no effect on the value of r. The calculation of r is
illustrated for the simple four-case sample data set at the foot of table 3.1.
There should now be no difficulty in understanding intuitively the
properties of the correlation coefficient:
�(Xi X)(Yi Y)
��(Xi X)2���(Yi Y)2
COV(XY)
sX.sY
�(Xi X)(Yi Y)
n
� It will be low when there is no strong relationship between the vari-
ables, because in that case there will be a scatter of points in all the
quadrants. When the products are added together the positive values in
I and III will cancel out the negative values in II and IV, reducing the
size of the measure of association. The more evenly the points are scat-
tered over the positive and negative quadrants the closer the coefficient
will be to 0.
� The value of r will be large and positive (close to �1.00) when there is a
strong positive association between the two variables, because in that
case most of the pairs of values will fall in quadrants I and III, and there
will be little offsetting effect from points in the negative quadrants.2
� The value of r will be large and negative (close to 1.00) when there is a
strong negative association, because in that case most of the pairs of
values will fall in quadrants II and IV, and there will be little offsetting
effect from points in the positive quadrants.
§.
Table 3.1 Calculation of the coefficient of correlation for a sample of four (imaginary) values of X and Y
(1) (2) (3) (4) (5) (6) (7)
Deviations Deviations Product of
Variables from means squared deviations
Xi Yi (Xi –X –) (Yi –Y
–) (Xi –X –)2 (Yi –Y
–)2 (Xi –X –)�(Yi –Y
–)
4 6 4 4 16 16 �16
6 11 2 1 4 1 2
10 7 2 3 4 9 6
12 16 4 6 16 36 �24
Sum 32 40 0 0 40 62 �32
Mean 8 10
Standard deviation of X�sX� � 3.16
Standard deviation of Y� sY� � 3.94
COV(XY) � �32/4�8.0
Correlation coefficient�r �COV (XY)/sx sY �8/(3.16�3.94)�0.64
�(Xi X –
)(Yi Y –
)
n
��(Yi Y –
)2
n ��624
��(Xi X –
)2
n ��404
If it is possible to establish that there is a strong (positive or negative)
correlation between any two variables, and if there is some theoretical
reason to assume that one variable (Y) is influenced by the other (X), then
this suggests the possibility that if we know the values of X, we can say
something about the likely values of Y. This cannot be done by correlation
techniques, but requires the further stage of regression analysis, which we
take up in chapter 4.
3.2.3 Interpreting a correlation coefficient How should we interpret a correlation coefficient? We have already sug-
gested a fundamental rule of interpretation, namely that correlation must
not be confused with causation. When an r of 0.75 is recorded between two
variables, it indicates that they are related to each other, but it does not tell
us anything about the nature of the relationship. It could be that changes in
one variable were responsible for the behaviour of the other; however, it
could also be that both variables were influenced by the movement of a
further, unidentified, variable.
But even if we have decided this issue, there are other points to consider.
How, for example, should we judge a particular value of r? One pair of var-
iables may exhibit an r of 0.75, while r for another pair might be measured
as 0.25. Are these large or small values? It is standard to apply adjectives
indicating the strength of association, such as ‘weak’, ‘moderate’, or
‘strong’, but it should also be recognized that such terms are not absolute.
An r of 0.75 might be considered high or low, depending on what was
expected before the calculation was made.
Imagine, for example, that we have a sample of students from an histor-
ical statistics course. We record two pieces of information for each student,
height and weight, and analyse this together with indicators of their perfor-
mance in the statistics class. We would not expect that height and class per-
formance would be highly correlated; indeed, our expected value would be
close to zero. On the other hand, we would expect there to be a strong cor-
relation between height and weight, perhaps on the order of 0.8.
When we analyse correlation coefficients, we must consider the results
in the light of our expectations. Were we to find that height and class per-
formance were correlated with r of 0.3, while height and weight were corre-
lated with r of 0.50, we ought to consider the first figure to be high, and the
second low, relative to expectations. We should also try to understand why
such unexpected results occurred. The same principle applies to historical
data. We should try to establish criteria for judging the strength of an asso-
ciation that go beyond a mere mechanical reporting of the correlation
coefficients themselves.
A further point to bear in mind when interpreting correlation
coefficients is that differences between values of r are not fully symmetric.
A higher value of r clearly indicates a higher correlation between the two
variables, but the difference between an r of 0.3 and one of 0.5 is not neces-
sarily the same as the difference between 0.5 and 0.7. The relationship
between r and the strength of association is not an absolutely linear one.
Finally, we should note that the strength of association between two var-
iables may be affected by the way in which the information on the variables
is collected. It is often noted that time-series data tend to exhibit higher
values of r than do cross-section data on the same variables. It is, moreover,
often suggested that the primary reason for a higher correlation in the
time-series comparison is that it reflects the influence of a third variable,
namely time. But this is not quite right.
It is certainly true that were we to compare two series (such as popula-
tion and house-building in the United States in the nineteenth century) we
would anticipate a high correlation because both series are trending
upwards. It is also true that if we were to compare population levels and
house-building rates across regions in, say, 1850, we would expect a lower
correlation. Partly this reflects the removal of the influence of time, but the
difference is mostly due to the additional influence of a myriad of local
factors that affect the pattern of both variables. These would include
income levels, migration rates, city building and zoning regulations, the
responsiveness of local builders, and so on.
Most time-series data tend to focus on aggregate observations, such as
the national rate of house-building and population growth over time,
which expunges all the local variation from the statistics. With cross-
section analysis, the local variation is included, generating a greater degree
of deviation in both X and Y and resulting in a lower r between them.
3.2.4 An illustration of the use of correlation coefficients For an illustration of the use of correlation coefficients in historical
research we turn to the work of Dorothy Thomas, one of the pioneers in the
quantitative analysis of social statistics. In a study based on a London PhD
and published in 1925 she undertook a comprehensive investigation of the
relationship between the business cycle and social conditions in Britain
between 1854 and 1913.e
Her measure of the business cycle was based on seven series indicative of
§.
e Dorothy Swaine Thomas, Social Aspects of the Business Cycle, Routledge, 1925. The book
includes a reprint of part of a paper written jointly with W. F. Ogburn in 1922, when she was at
Columbia University, in which a similar analysis was made for the United States using data for
1870–1920.
business conditions, including the value of total exports, unemployment,
production of pig iron and of coal, and raw material prices. For each of
these she found the secular trend by fitting a suitable curve, calculated the
percentage deviation of the series from the long-run trend, and then
divided the deviations from trend by their standard deviation to obtain a
standardized series.f These standardized cycles in each series were then
averaged to obtain the overall measure of fluctuations in the ‘British
business cycle’. A similar procedure was followed for each of the series of
social statistics she wished to investigate, and correlation coefficients were
calculated.
She also considered the possibility of delayed responses, as a result of
which a peak or trough in the business cycle would not make its impact in
the same year, but might do so a year or more later. To allow for this she cal-
culated the correlation coefficient between the two series with both meas-
ured for the same year, and also with the social indicator lagged by one,
two, or three years. The study was further extended by making separate cal-
culations for subperiods to see whether any change in the relationship over
time could be detected.
Some of the main results are reported in table 3.2. The positive correla-
tion for, say, marriage rates, indicates that years in which there was a boom
in business activity were also years in which the number of marriages was
above trend. The negative correlation on suicide rates indicates an inverse
relationship: the higher the level of business activity the smaller the pro-
portion of population committing suicide relative to the trend.
The implication of this extended exercise was clearly that the
fluctuations in the social statistics were influenced by those of the business
cycle. However, Thomas was aware that this was not necessarily a direct
influence. She noted explicitly (p. 162) that ‘causality cannot be claimed
merely on the basis of the coefficients of correlation’ and discussed the pos-
sibility of indirect influences.
For example, her discussion of the correlation between the business
cycle and the birth rate recognized that this might in part be due to the
close connection between the cycle and the marriage rate, and between
the marriage rate and the birth rate. Similarly, the inverse correlation of the
cycle with the rate of illegitimate births was attributed to the restriction of
marriage during depressions and its expansion during booms; and she
suggested that the lagged positive correlation with infant mortality
might be the result of the increase in alcohol consumption during times of
f For the procedure for calculating the deviations from trend see §1.8. Her method of fitting a
trend is the one explained in §12.4.
§.
Table 3.2 Coefficients of correlation with the British business cycle, 1854–1913a
(1) (2)
Same year Social series lagged one yearb
Marriage rates �0.67 �0.45
Birth rates 0.07 �0.29
Illegitimate birth rates �0.07 0.26
Death ratesc �0.04 �0.31
Infant mortality ratesd �0.04 �0.28
Suicide rates 0.50 0.47
Divorce rates 0.02 0
Emigration rates
Total �0.48 �0.14
To United States �0.77 �0.59
Per capita consumption
Beer �0.30 �0.45
Spirits �0.36 �0.60
Prosecutions for drunkennesse �0.33 �0.54
Pauperismf
Indoor 0.32 0.52
Outdoor 0.11 0.32
All indictable crimese 0.25 �0.09
Crimes against propertye
Without violence 0.25 �0.02
With violence 0.44 0.37
Notes: a The series generally cover a period from the mid-1850s to 1913, but in a few cases the
initial date is later than this. The majority cover only England and Wales but those for con-
sumption of alcohol and for emigration relate to the whole of the United Kingdom. b Except for birth rates, which are lagged by 2 years. c Excluding deaths from epidemic diseases. d Excluding deaths from diarrhoea. e Per 100,000 population. f Number of paupers relieved indoors (i.e. in workhouses) or outdoors (i.e. in their own
homes) per 1,000 population.
Source: Dorothy Swaine Thomas, Social Aspects of the Business Cycle, Routledge, 1925
prosperity. Her comments on the size of the coefficients show that she was
also sensitive to the issues of interpretation discussed in the previous sub-
section.
Thomas included in her discussion a comparison with the correspond-
ing results obtained for the United States. Marriage rates were found to
follow business conditions even more closely (r��0.81) than in England
and Wales. The positive correlation between the cycle and the lagged death
rate was ‘contrary to expectation’ but was also found in the United States (r
��0.49); and both countries showed the same positive correlation
between the cycle and lagged infant mortality rates, with a somewhat
higher coefficient in the United States (�0.43).
One of the more interesting contrasts was the relationship of divorce
rates to the cycle. In England and Wales there was no relationship between
the two series (r� 0.02), whereas in the United States they were highly
correlated (r��0.70). The absence of correlation in the former case was
attributed to the fact that divorce was expensive and restricted to a small
class of wealthy people who were not much affected by the business cycle.
Thomas concluded from her investigation that there was good evidence
that the business cycle caused repercussions in various other spheres of
social activity, and that the interrelationship between these social phenom-
ena raised questions for social theorists as to the part played by group
influences on individual actions. She thought her study ‘brings to light ten-
dencies of great interest for the social reformer’and that, above all, it points
strongly to the need for further research.3
3.3 Spearman’s rank correlation coefficient
The measure of correlation which has been discussed is appropriate when
working with data that have interval or ratio levels of measurement. An
alternative measure, known as Spearman’s rank correlation coefficient, rs, is used:
(a) for data that can only be measured on an ordinal scale; or
(b) where ranks have been allotted to data which were originally measured
on an interval scale, but the point of interest lies in their rank; or
(c) where the margins of error in interval level data are so large that it is
advisable to confine attention to the ranking rather than the actual
values.
Like the correlation coefficient, r, the rank correlation coefficient, rs,
takes values ranging from �1.0, when there is perfect agreement between
the rankings, to 1.0, when there is perfect disagreement. It equals 0 when
there is no relationship whatsoever.
The method of calculation is very simple. Each of the two variables is
placed in rank order, starting with rank 1 for the highest (or lowest) value.
The difference in ranks for each pair, designated Di, is calculated. These
differences are then squared and the sum of the squared differences is used
in the following formula, where n equals the number of pairsg
rs�1 (3.3)
This measure of correlation was used by Simon Szreter in an examina-
tion of the fall in fertility which started in England and Wales in the late
nineteenth century.h Among his key variables is a measure of completed
fertility, classified by the occupation of the husband. This is defined as the
number of children born alive to the present marriage by 1911.i At the head
of the list (rank 1), with the lowest completed fertility, were barristers with
an average of 3.16 children born to their marriages; at the other end of the
list (rank 195) were coal miners working at the face, with an average of
7.60.
As one element in his very wide-ranging study of the relationship
between fertility and social class, occupation, and employment status,
Szreter looks at the extent to which this ranking of occupations according
to the limitation of fertility within marriage was correlated with a second
ranking, reflecting the alternative strategy of prudential marriage: limiting
fertility by delaying the age at which the female married. For this index the
lowest rank was scored by the occupation with the largest proportion mar-
rying at older ages.
For all occupations, the Spearman coefficient between completed fertil-
ity and prudential marriage shows a strong positive correlation of �0.799.
However, when Szreter calculated the coefficient for five separate quintile
groups (ordered by the completed fertility ranking) he found very interest-
ing differences. For example, for the 39 occupations in the first quintile
(those that had most effectively limited births within marriage), the rank
correlation coefficient was 0.06. By contrast, for the 39 occupations in
the fifth quintile (those with the highest fertility), it was �0.433.
For occupations in this quintile with the highest fertility there was thus a
reasonable degree of consistency in their behaviour: in general they were
6�D2i n(n2 1)
§. ’
g Note that precisely the same result could be produced by using formula (3.2b) but the Spearman
coefficient is much simpler to calculate when dealing with ordinal data. h Simon Szreter, Fertility, Class and Gender in Britain, 1860–1914, Cambridge University Press,
1996. The issues to which we refer are analysed in chapter 7, especially pp. 335–50. The main
source for the data is table 35 in the Fertility of Marriage Report in the 1911 Census of Population. i The main measure refers to couples married in 1881–5 when the wife was aged 20–24; Szreter
also uses data for families formed at earlier and later ages of marriage and dates.
Table 3.3 Fifth-quintile male occupations rank ordered by completed fertility and female age at marriage
(1) (2) (3) (4)
Rank by Rank by Square of
completed female age difference
Occupation fertility of marriage in rank
Boilermakers 1 21 400
Brass, bronze workers 2 5 9
China, pottery manufacture 3 12 81
Stone – miners, quarriers 4 2 4
Tanners, curriers 5 13 64
Plasterers 6 20 196
Refuse disposal 7 27 400
Ironfoundry labourers 8 10 4
Plaster, cement manufacture 9 22 169
French polishers 10 30 400
Steel – manufacture, smelting, founding 11 9 4
Bricklayers 12 24 144
Road labourers 13 16 9
Gas works service 14 8 36
Oil (vegetable) – millers, refiners 15 28 169
Navvies 16 14 4
Agricultural labourers i/c of horses 17 3 196
General labourers 18 15 9
Agricultural labourers i/c of cattle 19 1 324
Shipyard labourers 20 35 225
Shepherds 21 6 225
Agricultural labourers 22 18 16
Fishermen 23 34 121
Bargemen, lightermen, watermen 24 39 225
Brick, terracotta makers 25 19 36
Dock, wharf labourers 26 25 1
Coal – mineworkers above ground 27 23 16
Builders’ labourers 28 11 289
Glass manufacture 29 31 4
Masons’ labourers 30 7 529
Coalheavers, coal porters 31 29 4
Tinplate manufacture 32 4 784
Ship – platers, riveters 33 33 0
not controlling fertility either by birth control within marriage or by late
marriage. This was also true within the fourth quintile. By contrast, within
the three lower fertility quintile groups there was an almost complete
absence of a relationship between age at marriage and completed fertility.
A considerable number of these occupations were thus achieving low fer-
tility within marriage despite also marrying at a relatively young age.
The procedure for calculating the coefficient is illustrated with the data
for the fifth quintile in table 3.3. The two rankings are given in columns (2)
and (3), and the difference in ranking is obtained and squared in column
(4). �D2, the sum of the squared differences, is found to be 5,604. N is 39,
so N2(N 1) in the denominator is 59,280. Inserting these values in (3.3)
gives
rs�1 �1 0.567 � 0.433
Notes 1 The second expression is obtained by filling in the formulae for COV (XY) and the
two standard deviations. Note that when this is done all the terms in n cancel out.
The formula for COV (XY) above the line is divided by n; while those for Sx and Sy below the line are each divided by the square root of n, and the product of
�n. Thus, both numerator and denominator are divided by n, and so it is eliminated
from the formula. 2 It is beyond the scope of this text to prove that r can never be greater than 1, but it is
�n ��n
6�5,604
59,280
Table 3.3 (cont.)
(1) (2) (3) (4)
Rank by Rank by Square of
completed female age difference
Occupation fertility of marriage in rank
Iron – miners, quarries 34 38 16
Bricklayers’ labourers 35 17 324
Coal – mineworkers below ground 36 37 1
Puddlers – iron, steel rolling mills 37 26 121
Pig iron manufacture –(blast furnace) 38 32 36
Coal – mineworkers at the face 39 36 9
5604
Source: Szreter, Fertility, Appendix C, pp. 612–13. The rank order is given in Appendix C for
all 195 occupations and the occupations in the fifth quintile were re-ranked from 1 to 39 for
this table.
perhaps intuitively evident that if there is perfect correlation between X and Y so that
all the points lie on a straight line, then there will be a constant ratio between every
pair of values of X and Y. Let us call this ratio b (as in the equation for a straight line, Y
�a�bX, in §1.5). The ratio between the means of X and Y, and between their devia-
tions from their respective means, must then also be equal to b. All the terms in
(Y Y –
) in the formula for r can thus be replaced by (X X –
). If this is done, the numer-
ator and the denominator in the formula both become b�(X X –
)2, and so r�1. 3 Later work, also using correlation coefficients, has generally tended to confirm
Thomas’ findings. See, for example, Jay Winter, ‘Unemployment, nutrition and
infant mortality in Britain, 1920–50’, at pp. 240–45 and 255–6 in Jay Winter (ed.),
The Working Class in Modern British History, Cambridge University Press, 1983, pp.
232–56, on the inverse relationship between infant mortality and unemployment;
and Humphrey Southall and David Gilbert, ‘A good time to wed? Marriage and eco-
nomic distress in England and Wales, 1839–1914’, in Economic history Review, 49,
1996, pp. 35–57, on the negative correlation between marriage and unemployment
(with no lag) in a sample of towns. For a graphical analysis of the incidence of crimi-
nal behaviour in England and Wales between 1805 and 1892 see V. A. C. Gattrell and
T. B. Hadden, ‘Criminal statistics and their interpretation’, in E. A. Wrigley (ed.),
Nineteenth-Century Society, Essays in the Use of Quantitative Methods for the Study of
Social Data, Cambridge University Press, 1972, pp. 363–96. Their finding that ‘more
people stole in hard times than good’, while the rate of violent crime was stimulated
by ‘high wages and high employment’ and a consequent ‘higher consumption of
liquor’ also supports Thomas.
3.4 Exercises for chapter 3
1. Calculate by hand the correlation coefficient between the following pairs
of observations:
Regional grain prices in England, 1691–1692:
(shillings per quarter)
Wheat Oats
Thames Valley 43.4 15.1
Eastern 36.3 13.4
Midland 32.7 9.7
Southern 40.0 12.3
South-Western 39.9 10.9
Northern 29.7 8.8
Source: A. H. John, ‘The course of agricultural change,
1660-1760’, in L. Pressnell (ed.), Studies in the Industrial
Revolution, University of London, 1960, p. 136.
2. Plot scatter diagrams of the following pairs of variables in the Boyer relief
data set.
UNEMP and DENSITY
LONDON and INCOME
GRAIN and RELIEF
(i) In each case, use visual inspection to determine whether the data are
uncorrelated, positively correlated, or negatively correlated, and
indicate the probable strength of the association. Record your
results.
(ii) Now calculate the correlation coefficient. Compare these calcula-
tions with the results of your visual inspection.
3. Plot a scatter diagram of the data on UNEMP and BENEFIT from the
Benjamin–Kochin data set. Are there any outliers? Calculate the correla-
tion coefficient between these two variables, including and excluding the
outlier. What does this calculation tell us about the importance of this
unusual observation?
4. Plot a scatter diagram for the variables RELIEF and UNEMP for the 28
Essex parishes in the Boyer relief data set. Are there any outliers? What cri-
teria did you use to identify any unusual observations? Compare the corre-
lation coefficients for the data with and without any outliers. Interpret your
results.
5. The following data have been extracted from Angus Maddison’s compi-
lation of national income statistics for the nineteenth and twentieth centu-
ries (Monitoring the World Economy, 1820-1992, OECD, 1995, p. 23). What
are the correlation coefficients between the levels of income in 1820 and
1870; 1870 and 1913; and 1820 and 1913?
Income per person (constant
(1990) US dollars) in: 1820 1870 1913
Austria–Hungary 1,295 1,875 3,488
Australia 1,528 3,801 5,505
Canada 893 1,620 4,213
France 1,218 1,858 3,452
Germany 1,112 1,913 3,833
Netherlands 1,561 2,640 3,950
United Kingdom 1,756 3,263 5,032
United States 1,287 2,457 5,307
§.
A critic judges that the probable margins of error in the national income
data, especially for 1820 and 1870, make it ‘advisable to confine attention to
the ranking rather than the actual values’ of income per person. In
response, calculate the Spearman rank correlation coefficient for each pair
of years and compare the results to your previous calculations. Do the
comparisons validate the critic’s judgement?
6. Calculate the Spearman rank correlation coefficients for the following
pairs of variables in the Irish migration data set for 1881:
URBAN and ILLITRTE
CATHOLIC and RELIEF
URBAN and AGRIC
Calculate the Pearson correlation coefficients for the same pairs of vari-
ables. Interpret your results, being sure to indicate which measure is more
appropriate and why.
Simple linear regression
The aim in this chapter is to extend the analysis of the relationship between
two variables to cover the topic of regression. In this introductory discus-
sion we will deal only with linear (straight-line) relationships between two variables (bivariate regression). In chapter 8 this analysis will be extended to include more than two variables (multiple or multivariate regression), and non-linear relationships will be discussed in chapter 12. As with the discussion of correlation, problems arising from the use of sample data are
deferred until the issues of confidence intervals and hypothesis testing are
covered in chapters 5 and 6.
4.1 The concept of regression
In the discussion of correlation in chapter 3 we emphasized that no distinc-
tion was made between the two variables, X and Y, and that interchanging
them would have no effect on the correlation coefficient. In the present
chapter we change procedure and introduce a fundamental distinction
between the two variables.
4.1.1 Explanatory and dependent variables It will often be the case that we have some theoretical reason to think that
movements in one of the variables are influenced by movements in the other.
In that case, the convention is to use X for the variable that is having the
influence, and Y for the variable that is influenced.
We could write that as
Y is influenced by X
or, more formally,
Y�f(X)
which says that Y is some function of X.
Y is then referred to as the dependent variable and X as the independent or explanatory (Xplanatory!) variable.1 When these are plotted in a scatter diagram the dependent variable (Y) is always shown on the vertical axis
and the explanatory variable (X) on the horizontal axis.
We might, for example, have two time series, one showing the number
of crimes of theft committed each year in the nineteenth century, and the
other showing the average price of wheat in that year. The hypothesis
would be that people are more likely to steal when times are hard, and the
price of wheat is taken as an indicator of the prevailing economic condi-
tions.a In this case it is very clear that if there is a relationship the price of
wheat will be the explanatory variable (X) and the number of crimes the
dependent variable (Y); there is no reason to think that the frequency of
crime could influence the wheat price.
However, the decision as to which is the explanatory variable is not
always so simple. It will often happen that there is a mutual interaction between the two variables. A cross-section study of the relationship
between the level of national income per head in a sample of countries, and
the expenditure on education by those countries, might find a good corre-
lation. That relationship could occur either (a) because richer countries
were able to spend more on education than poor countries; or (b) because
high levels of education increased worker productivity and raised the level
of national income. More advanced regression techniques would then be
needed to investigate these possibilities.
In their study of the impact of benefits on unemployment in inter-war
Britain, Benjamin and Kochin consider the possibility that they may have
‘interpreted the direction of causation incorrectly, that is, the high level of
unemployment may be the cause of the high benefit-to-wage ratio rather
than the reverse’. This could happen if, say, the British parliament had been
influenced by the extent of unemployment to provide improved benefits.
However, they conclude that causation in this direction was not a major
problem for their analysis.b
Having established this terminology for the two variables we can now
take up the concept of regression.
4.1.2 The questions addressed by regression When we examined the relationship between two interval or ratio level var-
iables (X and Y) in the discussion of correlation, the two main questions
addressed were:
a For a quantitative historical analysis of a relationship of this type see Douglas Hay, ‘War, dearth
and theft in the eighteenth century’, Past and Present, 95, 1982, pp.117–60. b Daniel K. Benjamin and Levis A. Kochin, ‘Searching for an explanation for unemployment in
interwar Britain’, Journal of Political Economy, 87, 1979, pp. 453–4.
(a) Is there any relationship or association between X and Y?
(b) If so, how strong is that association, and is it positive or negative?c
Once we have established that there is a close relationship between them
(and have some reason for thinking that it is not a spurious relationship)
we are likely to want to explore two further questions. These are:
(c) What is the precise nature of the relationship between X and Y, and how
exactly will Y change when X changes by a specified amount?
(d) How much of the change (variation) in Y is explained by the specified
change in X?
With these questions we are moving beyond issues of mere association
to issues of causation, in which changes in one variable cause changes in the
other.
Regression cannot establish causality, but it can tackle the fundamental
issue of how the two variables are related. Question (c) is equivalent to
asking: What is the predicted value of Y for any given value of X? The answer
is obtained by fitting an appropriate regression line to the data on X and Y. The answer to question (d) is given by calculating the coefficient of deter- mination (r2). We examine each of these procedures in turn.
4.2 Fitting the regression line
The procedure for fitting a line poses three issues.
4.2.1 How do we define a line? Recall from §1.5 that a straight line is defined by a general equation of the
form
Y�a�bX (4.1)
where a and b are constants.
This equation tells us that for any value of X:
Y�the value of a plus b times the value of X
In fact we need to know only two values of X and Y to draw a straight line
since, by definition, all points given by the equation lie on the same straight
line.
As shown in figure 1.1 (see §1.5), b measures the slope of the line.
§.
c An initial visual answer to the first of these questions was obtained in §3.1.2 from a scatter
diagram. The correlation coefficient (r) was introduced in §3.2 as the measure of the strength
and direction of the association.
If b is positive, the line slopes upward : when X rises there is a rise in Y
when X falls there is a fall in Y
If b is negative, the line slopes downward : when X rises there is a fall in Y
when X falls there is a rise in Y
The larger b is, the steeper the slope, and thus the larger the magnitude of
the change in Y for any given change in X.
In the specific context of regression, b is known as the regression coefficient, and the basic purpose of a great deal of quantitative analysis is directed to the task of establishing the regression coefficients for particular
relationships. It is this coefficient that ideally quantifies the influence of X
on Y and enables us to ‘predict’ how Y will change when X changes.
A primary objective of the quantitative historian and social scientist is
thus to test the propositions (hypotheses) suggested by relevant theories
in order to discover whether predicted relationships between sets of
variables are supported by the data. And, if they are, to measure the
regression coefficients in order to give substance to the theory by specify-
ing exactly how the dependent variable is affected by the explanatory
variable.
4.2.2 What criterion should be adopted for fitting the line? We have a cluster or ‘swarm’ of points, each based on one pair of observa-
tions of the two variables. We want to find the line which is the ‘best fit’
through all these pairs of values, i.e. which best describes the linear rela-
tionship between X and Y.
If the two variables were perfectly correlated (as in plots (a) and (b) in
figure 3.1), it would be easy to do this, since all the points would lie on a
single straight line. But that hardly ever occurs. Instead we are more likely
to have a spread of points, such as that shown by the RELIEF and UNEMP
variables for the 24 Kent parishes in figure 3.2.
There are various possible ways in which the deviations of each point
from a line through the middle of the cluster could be measured. The
method normally adopted is to calculate the vertical distance from each
point to the regression line.2 This is illustrated in figure 4.1 for our simple
four-case example from §3.2.2.
What the regression line does is, in effect, to predict the values of Y that
will be given by the corresponding values of X. A new symbol, Ŷ(Y hat), is
used to designate this value of Y predicted by the regression line.
Because the correlation is not perfect, most – or all – of the actual Ys will
lie some distance from the line, and thus from their predicted value. These
deviations of the actual Ys from their predicted values are known as residu- als. The advantage of taking the vertical distance from the line as the meas- urement of this deviation or residual (Yi Ŷi) is that it can then be
compared directly with the corresponding vertical deviation of the point
from the mean – this was the distance adopted in §3.2.2 as the measure of
the total deviation.
However, if the residuals are calculated in this way the positive and neg-
ative values will automatically cancel out. The sum of all the deviations
below the best fitting regression line will equal the sum of all those above
the line. The position would thus be comparable to the one encountered in
measuring dispersion by the mean arithmetic deviation in §2.3.2. There is
a further difficulty: more than one line can be found which satisfies the
condition of equality of positive and negative vertical deviations from the
line.
The way to get round both these problems is to take the square of these
deviations, (Yi Ŷi) 2. Since we are interested in all the points in the data set,
we then take the sum of these squared deviations from the regression line,
�(Yi Ŷi) 2. There is only one line that minimizes this sum of squares of the
vertical deviations, and it is this line that is selected as the ‘best fit’.
To show this visually, figure 4.2 repeats figure 4.1, but this time with
the actual squares drawn on each of the deviations for the four pairs of
observations.
§.
Figure 4.1 Deviations from the regression line
1412108642
18
16
14
12
10
8
6
4
Ŷ
Yi
Yi
Ŷ
Ŷ
Yi
Yi
Ŷ
X
Y
Y
X
i
ii
i
THE REGRESSION LINE
For two variables, X and Y,
the ‘best fitting’ regression line is defined as the line that
minimizes
the sum of the squares of the vertical deviations
of all the pairs of values of X and Y
from the regression line.
Because of this criterion, the basic procedure is known as ordinary least squares regression (frequently abbreviated to OLS).
4.2.3 How do we find this best fitting line? The formal procedure for minimizing the sum of the squares requires the
use of differential calculus, and the details need not concern us. The proce-
dure yields the two formulae needed to calculate the regression line which
satisfies the least squares criterion: one for the regression coefficient or
slope, b, and the second for the intercept, a.
Figure 4.2 Least squares deviations from the regression line
X
1412108642
Y 18
16
14
12
10
8
6
4
X
Y
THE REGRESSION COEFFICIENT
The regression coefficient, b, is the
slope of the regression line
b� (4.2)
The components should be familiar, and in fact b equals the covariance
of X and Y, divided by the variance of X.d
Given b, the intercept, a, can be quickly found from the second formula:
a�Y –
bX –
(4.3)
To increase familiarity with this procedure it will be helpful to work
through the simple four-case data previously given in table 3.1. The calcu-
lation of the regression line for these pairs of values is set out in table 4.1.
All the necessary information was given in table 3.1, but is repeated here for
convenience.
With larger data sets one of the statistical computer packages can be
used to do the necessary work. For example, for the RELIEF and UNEMP
data for the 24 Kent parishes, it applies these formulae to derive the best
fitting regression line, giving a�14.9 and b�76.4. This regression line is
shown in figure 4.3.
4.2.4 The regression of Y on X versus the regression of X on Y When estimating the strength of the association between two variables it
was observed that the coefficient of correlation (r) is wholly symmetrical:
the correlation of X with Y is always identically equal to the correlation of Y
with X (see §3.2.1).
Such symmetry is not present when the regression coefficient, b, is esti-
mated. As shown above, the variance of X is part of the calculation of b, but
the variance of Y is not. It is thus essential to distinguish between:
� The regression of Y on X, in which X is the explanatory variable by refer-
ence to which Y can be predicted; and
� The regression of X on Y, in which Y is the explanatory variable by refer-
ence to which X can be predicted.
�(Xi X) (Yi Y) �(Xi X)2
§.
d The terms for n in the covariance and the variance cancel out and so do not appear in the formula
given for b in (4.2).
For any given data set these two regressions are not the same. If we have
reason to be interested in both these forms of a mutual inter-relationship
(as in some of the examples discussed in §4.1.1), two separate regression
equations have to be found.
The two regressions will have different intercepts and different slopes. If,
for example, the two variables are expenditure on education and national
income, the amount by which national income grows, when there is an
increase in expenditure on education, will not be the same as the amount
by which expenditure on education increases, when there is a rise in
national income.
4.2.5 Fitting a linear trend to a time series In §1.8 we discussed the distinction between the long-run trend and short-
term fluctuations in a time series. We described a simple method for meas-
uring the trend by means of a moving average, but noted that this
procedure suffered from a number of weaknesses. We can now improve on
this by using our regression technique to fit the trend. Essentially all that is involved in doing this is that the series for which we
wish to find the trend is taken as the dependent variable, and time is treated
Table 4.1 Calculation of the regression coefficient for four values of X and Y
(1) (2) (3) (4) (5) (6) (7)
Deviations Deviations Product of
Variables from means squared deviations
Xi Yi (Xi –X –) (Yi –Y
–) (Xi –X –)2 (Yi –Y
–)2 (Xi –X –)�(Yi –Y
–)
4 6 4 4 16 16 �16
6 11 2 1 4 1 2
10 7 2 3 4 9 6
12 16 4 6 16 36 �24
Sum 32 40 0 0 40 62 �32
Mean 8 10
b� �32/40�0.80
a�Y –
– bX –
�10 – (0.8)(8)�3.60
�(Xi X –
)(Yi Y –
)
�(Xi X –
)2
as the explanatory variable. The ‘names’ of the actual years covered by the
series (1920, 1921, and so on) have no significance in this context, and we
would in the end obtain the same results if we substituted any other
sequence of numbers that increased by 1 unit each year. The choice of units
for the measure of time will, however, affect the size of the regression
coefficients, as we show below.
Given X and Y, the intercept, a, and the regression coefficient, b, can be
derived by the standard procedure given in §4.2.3, and these two constants
define the trend line. To illustrate the technique we will use the series for
NNP (real net national product in £ million at 1938 factor cost) from the
Benjamin and Kochin data set.3 We do so in two forms: in the first we use
the actual dates (1920–38) as the measure of time (YEAR); in the second
we replace these by the numbers 1–19 (TIME).
When we use YEAR as the explanatory variable, the regression line fitted
by our computer package is
Y� 132,130.60�70.54 YEAR
When we use TIME as the explanatory variable, the regression is
Y�3,236.33�70.54 TIME
§.
Figure 4.3 Scatter diagram of relief and unemployment with regression line, 24 Kent parishes in 1831
30.020.010.00.0
R el
ie f
pa ym
en ts
( sh
ill in
gs )
40
30
20
10
0
Y = 14.9 + 76.4X
Unemployment (per cent)
We see that the two results have the same slope but very different inter-
cepts. In both forms the trend line will rise by £70.54 million every year (i.e.
for every increase of 1 unit in either measure of time). But we are trying to
obtain the trend in NNP and so need to know the position of the line in
relation to this series.
If we take the first year 1920, the equation with YEAR tells us that the pre-
dicted value of the line (Ŷ in the terminology of §4.2.2) will be 132,130.6
�(70.54�1920), which is £3,307 million. For the same year, the alternative
equation with TIME tells us that Ŷ will be 3,236.33�(70.54 �1), which is
again £3,307 million. We will similarly get the same result for every other
year, whichever form of the regression we use. The contrast is nevertheless a
useful reminder of the need to think about the units in which the dependent
and explanatory variables are measured before drawing any conclusions
from a regression. We return to this issue in §4.3.3.
The use of simple linear regression is in many respects an improvement
on the moving average procedure. There are no missing years at the begin-
ning and end of the series, the trend is perfectly smooth, and – if desired – it
could be extrapolated to earlier or later years. However, the method is itself
subject to the important limitation that it can generate only a linear trend,
and can thus be used only when inspection of the data indicates that a
straight line is appropriate. In this case, as can be seen in figure 4.4, the
linear trend does fit the data well. However, if there is clearly evidence of a
non-linear trend, then the calculation must be made with slightly more
advanced methods, notably the log-linear (semi-logarithmic) trend that
will be described in chapter 12 (see especially panel 12.1).
4.3 Measuring the goodness of fit
The procedure for obtaining the regression line for any pair of variables, X
and Y, has now been established. There will always be one regression line
which we can fit to a given data set by applying the ordinary least squares
(OLS) criterion. By definition it is the best linear fit that can be achieved on
that criterion.
However, that still leaves the crucial research question: How good a fit is
it? It may be the best fit there is, but is it a good fit?
4.3.1 Explained variations and unexplained variations (residuals) Asking how well the line fits the data points is equivalent to asking: How
much of the variation in Y is explained by this regression line? In the ideal case
of perfect correlation, all the points would lie exactly on the line, and the
regression line would ‘explain’100 per cent of the variation in Y. If we know
how X changes, we could say by exactly how much Y will change.
In history and the social sciences we always have to settle for something
less than this. However powerful the selected explanatory factor may be,
there will always be a variety of other factors – both systematic and random
– which will also affect the behaviour of the dependent variable (Y).
It is thus necessary to have a measure of how successful the regression
line is in explaining (in the limited sense of ‘accounting for’) the move-
ments in Y. This measure is known as the coefficient of determination (r2) and is equal to the square of the correlation coefficient. The value of r2 is
always between 0 and 1.e The closer it is to 1, the larger the proportion of
the variation in Y that has been ‘explained’ by the movements in the
explanatory variable, X.
The statistical package will quickly find r2 for us, but in order to see more
clearly what is happening it will be useful to look briefly at the underlying
logic of the calculation.
The objective is to find a measure of the success with which the move-
ments in Y are explained by the movements in X. We thus need some way to
specify these movements in Y, and we take for this purpose the deviation
from the mean for each value in the series. The justification for this is that if
we knew nothing about the factors determining the behaviour of Y, our
‘best guess’ for the value of any individual case, Yi would have to be that it
would be equal to the mean of the series, Y –
.
§.
e Since r cannot be greater than �1, r2 cannot be greater than 12, and �1��1� 1� 1�1.
Figure 4.4 Net national product, United Kingdom, 1920–1938 and linear trend
19361932192819241920
N et
n at
io na
l p ro
du ct
a t 1
93 8
fa ct
or c
os t (
£m ) 5,000
4,500
4,000
3,500
3,000
The regression line should enable us to improve on this estimate by pre-
dicting the value of Y for any given value of X. As noted earlier, the symbol
Ŷ is used to designate this predicted value. There are thus three possible
values of Y to be considered:
Yi�the value of any particular observation
Y –
�the mean value of all the observations, and
Ŷi�the value of Yi predicted by the regression line.
The relationship between them is illustrated in figure 4.5, using again
the data for our simple four-case example from table 4.1. The graph shows
the four points together with:
(a) a horizontal line drawn through the mean value Y –
; and
(b) the regression line, Ŷi�3.60�0.80 Xi.
The deviations (or variations) for each value of Y can then be analysed
as follows:
(Yi Y –
)�the total deviation
(Ŷi Y –
)�the part of the deviation predicted or ‘explained’ by the
regression line
(Yi Ŷi)�the residual or part left unexplained�(Yi Y –
) (Ŷi Y –
)
Figure 4.5 Explained and unexplained deviations
X
1412108642
Y 18
16
14
12
10
8
6
4
Ŷ
Yi
Yi
Explained
Unexplained
Unexplained
Explained
Ŷ
X
Y YY
i
i
X
This decomposition of the deviations or variation is illustrated in
figure 4.5 for two of the four cases: those where X�12 and Y�16, and
where X�6 and Y�11. In the former the regression line comes between
the mean and Yi, and the total deviation is a straightforward sum of the
explained and unexplained components. However, for the latter point, Yi is above both the mean and the predicted value, so that the value pre-
dicted by the regression line actually takes us even further from the value
of Yi, leaving an unexplained residual which is larger than the total devia-
tion.
On the basis of this form of decomposition of the deviations we can now
turn to the construction of a measurement of the goodness of fit.
4.3.2 The coefficient of determination, r2
It is – as usual – preferable to work with the squares of the deviations, and
since we are interested in the outcome over all pairs of values, it is again
necessary to sum all the cases. Fortunately, the decomposition given above
also holds when we take the sum of squares. We can thus write:
� (Yi Y –
) 2 � � (Ŷi Y –
) 2 � � (Yi Ŷi) 2 (4.4)
Total variation � Explained variation � Unexplained variation
Alternative terms are sometimes used. The total variation may be called
the total sum of squares. The explained variation may be called either the explained sum of squares or the regression sum of squares, and the unex- plained variation may be called either the residual sum of squares or the error sum of squares.
Equation (4.4) provides the means to measure the coefficient of determination.
THE COEFFICIENT OF DETERMINATION
The coefficient of determination, r2,
is the proportion of the total variation
that is explained by the regression line.
r2 is equal to the ratio of
the explained sum of squares
to the total sum of squares.
r2� (4.5) �(Ŷi Y)2
�(Yi Y)2
§.
We can now apply this measure to our basic parish example. For RELIEF
and UNEMP in Kent we get an r2 of 0.275. Unemployment is thus able to
account for only 27.5 per cent of the variation in payments by these par-
ishes, and the greater part of the variation has thus to be explained by other
factors. These could include the generosity of the ratepayers, the different
systems of child allowances, the level of income from wages and other
sources such as allotments, the extent of sickness, and the proportion of
old people in the parish.
The actual details of the calculation of r2 for the simple four-case data set
are shown in table 4.2.
4.3.3 Some intuitive implications We conclude this section by exploring the relationship between the regres-
sion coefficient, b, and the coefficient of determination, r2. When we
derived the best fitting line in §4.2.3, it was noted that the formula given for
the regression coefficient, b, in an equation of Y on X, was equal to the
covariance of X and Y, divided by the variance of X.
Similarly, in §3.2.2, when we derived the coefficient of correlation
between X and Y, it was noted that r could be written as the covariance of X
and Y, divided by the product of the standard deviation of X and the stan-
dard deviation of Y (see (3.2a)). It is worth writing both of these out as
equations
Table 4.2 Calculation of the coefficient of determination for four values of X and Y
(1) (2) (3) (4) (5) (6) (7)
Variables Predicted value of Y Explained Deviation Total Deviation
Xi Yi a � bXi � Ŷi (Ŷ–Y –) (Ŷ–Y–)2 (Yi –Y
–) (Yi –Y –)2
4 6 3.6�0.8 (4) � 6.8 3.2 10.24 4 16
6 11 3.6�0.8 (6) � 8.4 1.6 2.56 1 1
10 7 3.6�0.8 (10) � 11.6 �1.6 2.56 3 9
12 16 3.6�0.8 (12) � 13.2 �3.2 10.24 6 36
32 40 0.0 25.60 0 62
Note:
For column (3) a�3.60 and b�0.80 as calculated in table 4.1.
r2�Explained deviation/Total deviation
r2�25.60/62�0.41
b� (4.6)
r� (4.7)
The first of these two equations can easily be re-written as
b� (4.6a)
and if the first of these terms in (4.6a) is replaced by r we have
b� r � (4.8)
This states that the regression coefficient is equal to the correlation
coefficient (the square root of the coefficient of determination) multiplied
by the ratio of the standard errors of Y and X. Three intuitive results follow
immediately from (4.8).
First, if r�0, then b must also be 0.
Secondly, although r is independent of the units of measurement of X and
Y, b is not. To return to our earlier example, if RELIEF expenses were meas-
ured not in shillings but in pounds, the standard error of X would fall to one-
twentieth of its level (since there are 20 shillings to a pound). The correlation
coefficient, r, would remain unchanged. Thus, b would be 20 times larger in
this alternative regression. This is an example of a scalar effect, which arises when a variable is revalued consistently across all observations.f
The scalar effect of changing the denomination of a variable can be very
useful to researchers. Let us imagine, for example, that a PhD student is
interested in the demographic origins of homelessness in the United States
in the 1930s and considers urban size to be a major influence. She therefore
collects cross-section data on homelessness and total population for each
city. If she decided to regress the proportion of each city’s population who
were homeless in 1930 on the number of individuals in each city according
to the US Census of that year, she would find that the coefficient, b, would
be very small, on the order of 0.00000001.g
sY sX
COV(XY)
sXsY �
sY sX
COV(XY)
sXsY
COV(XY)
s2X
§.
f Note that the relationship between YEAR and TIME in the regression fitted to NNP in §4.2.5 is
not consistent in this sense: the ratio of 1920 to 1 is very different from the ratio of 1938 to 19. g Assume that there are five cities, with populations of 3,400,000, 5,000,000, 1,500,000, 500,000,
and 200,000. Proportions of homelessness are 0.03, 0.06, 0.02, 0.01, and 0.0085. From this, we
can calculate that sy�0.018814; sx�1,823,623; r�0.97; and b�0.00000001.
Presenting the coefficient in this way would be cumbersome. Were she
to express homelessness as a percentage rather than a proportion of the
population (e.g. 5.0 per cent, rather than 0.05), b would rise to 0.000001.
Alternatively, were she to express urban size in millions of residents, b
would rise to 0.01. If she combined both these scalar transformations of the
data, b would become 1.
This final result indicates that, according to the regression, a city that is
1 million people larger than another would experience a 1 per cent increase
in homelessness. This second result has exactly the same meaning as the
original version, in which an increase in population from one city to
another of one person increased the proportion of homelessness by one-
millionth of a per cent. However, the second version is clearly very much
easier to think about and work with.
It is thus important to look at the units of measurement of the depen-
dent and explanatory variables before drawing strong inferences about the
historical importance of an estimated regression coefficient.
Thirdly, the size of the coefficient, b, will be influenced by the relative
size of the variation of X and Y, even when the two variables are expressed
in the same units (e.g. both X and Y are proportions, or they are both
denominated in the same currency unit). As should be clear from (4.8), for
any given value of r, the larger the ratio of sY to sX, the larger will be the
regression coefficient, b.
To illustrate, assume that X and Y are highly correlated, say with r�0.9.
If the values of X are very highly clustered (so that sx is relatively low), while
those of Y are more spread out (so that sY is relatively high), the regression
coefficient, b, will be high, indicating a steep regression line of Y on X (see
figure (4.6(a)). Alternatively, if X varies a great deal around its mean, while
Y is more tightly distributed, the ratio of sY to sx will be lower than in the
previous case and b will be small, indicating a shallow regression line (see
figure (4.6(b)). At the extreme, if there is no variation in X (so that sx is
zero), the regression line will be vertical (b is infinite). If there is no varia-
tion in Y (so that sY is zero), the regression line will be horizontal (b is zero).
It is thus essential to pay attention to the variation of the regression var-
iables before evaluating the regression as a whole. If there is little or no vari-
ation in the explanatory variable (low sx), it is not going to explain much of
the behaviour of the dependent variable. The regression model will have
little explanatory power. On the other hand, if there is little or no variation
in the dependent variable (low sY), there is little to be explained, and apply-
ing a formal regression model may be a case of ‘using a sledgehammer to
crack a nut’.
§.
X
201612840
Y 25
20
15
10
5
0
Figure 4.6 The relationship between the correlation coefficient, r, the regression coefficient, b, and the standard deviations of X and Y
X
201612840
Y 25
20
15
10
5
0
(a) When sx is low relative to sy
(b) When sx is high relative to sy
X
X
4.4 Introducing the Ballantine
One way of presenting this somewhat technical material in a more intuitive
format is with the aid of a diagram known as the Ballantine.4 The Ballantine for the case of simple regression with one explanatory variable is
presented in figure 4.7. The two variables, X and Y, are each represented by
a circle, the areas of which represent their respective variances.
In figure 4.7, we have assumed that the variances of X and Y are equal
and have therefore drawn the circles to be the same size. The shaded area in
the diagram shows the overlap between the two variables. This overlap (or
union) of the two circles indicates the extent to which the variance of X
explains the variance of Y. It is equal to r2, the coefficient of determination.
In this case, the shaded area is 30 per cent of the total area of Y’s circle: the
coefficient of determination is thus 0.3.
That part of the variance that cannot be explained by the regression is
shown by the unshaded part of each circle, which in this case is equal to 0.7.
The greater the degree of overlap, the more of the variation of Y is
explained by the variation of X. If the two circles overlap completely, X
explains Y completely and the coefficient of determination is 1. Conversely,
if the two circles do not overlap at all, then the two variables are completely
independent and the coefficient of determination will be 0.
The Ballantine of figure 4.7 is a very stylized representation of regres-
sion analysis. In particular, it sets the variances of X and Y to be equal.
Normally, of course, this would not be the case: each variable will have a
Figure 4.7 Ballantine for a simple regression with one explanatory variable
Y X
different variance.5 How does this difference in the relative size of the vari-
ances affect our understanding of the Ballantine? In order to understand
this, let us delve into the derivation of the Ballantine a little further, with
the aid of our simple regression model from §4.2 based on the data given in
table 4.1.
The variances of X and Y can be calculated as 10 and 15.5, respectively
(from columns (5) and (6)); thus, in the Ballantine representation of this
regression, Y would have a larger circle than X. The covariance between X
and Y is 8.0 (from column (7)). The covariance is a measure of how far the
two variables move together, or covary; this is the statistical equivalent of
the shaded area in figure 4.7.
The size of the shaded area relative to the total area of each circle is equal
to the ratio of the covariance to the variance of X and Y, respectively. The
coefficient of determination is simply the product of these two ratios. Thus
r2� �0.41
This result is thus the same as the one calculated in a different way in table
4.2.
In the bivariate case, the Ballantine also incorporates all the information
necessary to calculate the regression coefficient. Recalling that r is simply
the square root of the coefficient of determination, and that the standard
deviations of X and Y are equal to the square roots of their respective vari-
ances, we can use (4.8) to calculate the regression coefficient, b, as
b� �0.64� �0.80
which is the same figure that is derived in table 4.1.
In later chapters, we shall use the Ballantine to explore the complexities
of regression analysis. In future, we shall always draw the circles with the
same area, and ignore the complications involved in having variables with
different variances. Therefore, when looking at the Ballantines in later
chapters, it should be remembered that they are an analytical device, rather
than a precise depiction of the structure of any regression and its compo-
nent variables.
Notes 1 The dependent variable may also be referred to as the regressand, or as the explained,
endogenous, or target variable. The explanatory variable can similarly be referred
to as the regressor or the predictor, or as the independent, exogenous, or control
variable.
3.94
3.16 �0.41 �
�15.5 �10
8
10 �
8
15.5
2 Alternative possibilities would be to measure either the horizontal distance or the
distance represented by a line drawn at right angles to the regression line. 3 Note though that this is not the form of trend actually used by Benjamin and Kochin.
They calculated a so-called log-linear trend by a procedure we will explain in panel
12.1 in chapter 12. 4 The Ballantine was originally introduced by Jacob and Patricia Cohen, Applied
Multiple Regression/Correlation Analysis for the Behavioral Sciences, Lawrence
Erlbaum Associates, 1975, to discuss correlation coefficients, and later mis-spelt and
extended by Peter Kennedy to the analysis of regression techniques, Peter E.
Kennedy, ‘The “Ballentine”: a graphical aid for econometrics’, Australian Economic
Papers, 20, 1981, pp. 414–16. 5 One way to conceptualize the Ballantine with equal circles is to assume that all vari-
ables have been standardized by the Z transformation, so that all variances are forced
to one.
4.5 Exercises for chapter 4
1. Which is the explanatory variable and which the dependent variable in
the following combinations?
Height and weight
Birth rate and marriage rate
Unemployment and poverty
Rainfall and crop yields
Rate of unemployment and level of relief expenditures in parishes
in Southern Britain
Government spending on welfare programmes and the voting
share of left-liberal parties
In which of these cases might you expect a mutual interaction between the
two variables?
2. Using the following values of X and Y:
X: 4, 8, 12, 16
Y: 10, 10, 3, 7
Calculate by hand :
The regression coefficient, b
The intercept, a
The total sum of squares
The explained sum of squares
The residual sum of squares
The coefficient of determination.
3. In the table below, (2) is an imaginary series. (3) is the official (London
Gazette) market price of wheat, rounded to the nearest 5 shillings. (4) is a
hypothetical series.
(1) (2) (3) (4)
Number Price of Number of popular wheat of
Year disturbances (shillings per qtr) shipwrecks
1810 20 105 12
1811 50 95 23
1812 60 125 32
1813 40 105 22
1814 20 75 8
1815 10 65 7
1816 30 75 17
1817 30 95 17
1818 20 85 8
1819 40 75 18
1820 20 65 8
1821 20 55 8
(i) By hand, calculate the correlation coefficient for wheat prices and the
number of disturbances.
(ii) By hand, find the equation for the regression line for wheat prices
and the number of disturbances. (Hint : think first about which is the
appropriate explanatory variable.)
(iii) Plot the corresponding scatter diagram and regression line.
(iv) By hand, calculate the correlation coefficient for number of ship-
wrecks and the number of disturbances.
(v) Compare the coefficients calculated in (i) and (iv).
Write a short note interpreting your results.
4. Using the Hatton–Williamson data set on Irish migration by county,
determine how much of the total variance in FAMSIZE in 1911 is explained
by the behaviour of:
CATHOLIC
AGE
ILLITRTE
Interpret your results.
§.
5. Run simple regressions on the Benjamin–Kochin data set of
UNEMP on BENEFITS, 1920–39
UNEMP on DEMAND, 1920–39
Record the coefficients and standard errors. Plot the residuals from each
regression. What do they tell you about the extent to which 1920 is an
outlier? Re-run these regressions dropping 1920. Record the coefficients
and standard errors and plot the residuals.
6. Critics of your analysis of disturbances and wheat prices in question 3
suggest the following errors in the statistics:
(i) The number of disturbances in 1819 was 60 not 40.
(ii) The price of wheat in 1819 was 70 not 75.
(iii) All the figures on the number of disturbances have a margin of error
of plus or minus 10 per cent.
(iv) The figures of wheat prices are too low, since they represent market
prices received by farmers, not retail prices paid by workers; retail
prices are 25 per cent higher in each year.
Draw up a statistical response to all of these criticisms. Generate new
results where applicable.
7. Use the methodology of §4.4 to calculate by hand the coefficient of deter-
mination and the regression coefficient from the data in question 2. What
do your results tell you about the limitations of this approach?
8. In a bivariate regression of Y on X, the variance of the dependent variable
is 100, while the variance of the explanatory variable is 4. The covariance of
X and Y is 2.
(i) Draw the Ballantine that illustrates the regression, being careful to
draw the variances of X and Y in due proportion. Calculate the
values of b and r2 in this model.
In a second regression, the variance of X is 100 and the variance of Y is 4.
(ii) Draw the Ballantine that illustrates the regression; calculate the
values of r2 and b.
(iii) How do you explain any differences between your answers for (i)
and (ii)?
Samples and inductive statistics
Standard errors and confidence intervals
5.1 Introduction
The previous chapters have been concerned with a review of basic descrip-
tive statistics. We now move to the much more important topic of induc- tive statistics. This chapter and chapter 6 will be devoted to an exploration of some of the implications of working with data from samples. The basic
purpose of inductive statistics is to make it possible to say something
about selected characteristics of a population on the basis of what can be inferred from one or more samples drawn from that population (see §1.3.4).
Samples can be obtained in various ways. Typically, the historian works
with information extracted from surviving documents such as parish reg-
isters, manuscript census schedules, household inventories, farm accounts,
private diaries, and records of legal proceedings. She will usually be
working with a sample of such records, either because that is all that was
ever available (only some people keep diaries), or because only some of the
original documents have survived, or because the records are so detailed
that it would be too expensive and time-consuming to extract all the infor-
mation they contain.
Whenever sample data are used it is necessary to pose the fundamental
question: How good are the results from the sample? How much can we
learn about the population as a whole from the data available to us, and
with what confidence? The overall answer to these questions involves three
separate issues that must be considered briefly before turning to the statis-
tical aspects.
(a) Is the sample of records that has survived representative of the full set
of records that was originally created?
(b) Is the sample drawn from the records representative of the information
in those records?
(c) Is the information in those records representative of a wider popula-
tion than that covered by the records?
We may consider two categories of source material. In the first, evi-
dence for the entire set of records is recoverable (as would normally be
true of census manuscripts, parliamentary debates, certain of the records
of legal proceedings, and certain parish registers), in which case the two
remaining issues are (b) and (c). Dealing with (b) is a matter of adopting
an appropriate sampling method. As long as each member of the popula-
tion is equally likely to be selected, so that the sample is random, (b) will be
achieved.1
The judgement with respect to (c) will depend on the nature of the
records concerned. Census schedules should cover the entire population,
though certain groups may be under-represented because they are, or
choose to make themselves, less visible to the enumerators. Surviving
records of various forms of taxation (for example, hearth and poll tax
returns, income taxes, and probate inventories) will often be complete and
are a valuable source for historians, but they typically omit a substantial
part of the population: usually the poor, often women, always those who
managed to evade assessment. Parish registers may provide data that are
representative of members of the established church, but not of dissen-
tients such as Methodists or Quakers; or the parishes for which registers
survive may not be representative of all parishes in the country. Court
records may not be representative of all the individuals falling within the
court’s jurisdiction if the costs of initiating legal proceedings were prohibi-
tive for some. This group of issues under (c) is thus a matter for careful
consideration in relation to each particular source.
The second category of sources consists of cases where the surviving
records are highly fragmentary (e.g. private diaries, business records,
household inventories) so that there are problems with respect to (a). In
that case care needs to be taken to establish that the source provides a reli-
able guide to the original population, before proceeding to address (b) and
(c). For some sources this can be done by comparing key characteristics of
the surviving records to some known attributes of the population. If these
characteristics are generally consistent, this may be enough to establish
representativeness.2
Sometimes the comparison indicates that records survive for some parts
of the population more frequently than for others (e.g. that business
records survive more often for bigger firms than for smaller ones). In such
cases, it may be possible to develop sampling strategies to counteract the
bias in the surviving data and create a more representative data set. Finally,
there may be cases where it is not possible to establish the bona fides of the
surviving data at all, either because there is no basis for comparison with
the population (e.g. household inventories) or because the data are clearly
not representative (e.g. private diaries in an era of limited literacy).
We do not suggest that statistical analysis of unrepresentative evidence
is uninteresting, but in what follows we are assuming that the sample is rep-
resentative of the population in all three of the dimensions mentioned,
such that we can apply the logic of inferential statistics to recover informa-
tion about the population attributes from the sample available to us.
Once representativeness has been established, we can focus on the
nature of the sample results. So, when we ask, ‘How good are the results
obtained from the sample?’, we are asking, in effect, whether the same
results would have been obtained if a different sample had been used.
Unfortunately, it is normally not possible to find out by drawing more
samples (either because the information is not available or because the
exercise would be too expensive). Instead we have to rely on statistical
theory to tell us about the probability that the results we have obtained can
be regarded as a true reflection of the corresponding characteristics of the
underlying population. Our task is thus to learn how we can make infer- ences about various parameters of the population, given what we know from the sample statistics (see §1.3.4).
In particular, in this chapter we will consider how to achieve a given
probability that a specified range or confidence interval around the sample statistic will cover the true (unknown) value we wish to estimate; and in
chapter 6 we will explore the closely related topic of the procedure for
testing statistical hypotheses. If we were considering only one variable, these statistics might be, for example, estimates of the mean and standard
deviation of the sample. If we were dealing with the relationship between
samples of data for two variables, we might have sample statistics for the
correlation and regression coefficients.
As noted before (§1.4), Greek letters are usually used to denote charac-
teristics of the population, while Roman letters are used for the corre-
sponding estimates from samples from that population. The Greek and
Roman symbols we will use in this and other chapters are set out below.
§.
SYMBOLS FOR CHARACTERISTICS OF THE POPULATION
AND ASSOCIATED SAMPLES
Characteristic Population Sample
Mean � X –
Variance 2 s2
Standard deviation s
Proportion � p
Correlation coefficient � r
Intercept for regression line � a
Regression coefficient � b
Number of cases N n
5.2 Sampling distributions
To introduce the subject of confidence intervals we will confine our atten-
tion to estimates of a sample mean. All the points that are made would
apply equally to any other measure derived from the sample, such as a pro-
portion or a correlation coefficient.
Take, for example, the information on BRTHRATE in 1826–30 in the
Poor Law data set consisting of a sample of 214 English parishes. The mean
rate was 15.636 births per 100 families. There were, however, over 15,000
parishes in England at that time. So the fundamental question that has to
be considered is: if the population mean is � (lower case Greek mu),
How good is this sample mean, X –
, as an estimator of �?
At this point probability becomes crucial to the procedure and would be
studied intensively in a formal statistics course. Although we are not going
to do this, we must take just a fleeting glance at what is involved in order to
get an intuitive understanding of some basic features of the procedures we
will use.
5.2.1 Sampling distributions and point estimates of the population mean To answer our fundamental question about the reliability of the sample
mean it is necessary to introduce the concept of a sampling distribution. In this case we are dealing with the sampling distribution of the sample mean,
but there would be a corresponding sampling distribution for every other
statistic derived from a sample, for example, the sampling distribution of the sample regression coefficient. To understand this notion of a sampling distribution it is essential to appreciate that it is one of three distinct –
though related – distributions.
� The first is the population distribution. The information required for this is not actually available in the case of BRTHRATE. However, it
would, in principle, have been possible to obtain and plot a distribu-
tion of BRTHRATE for all parishes in England. On the horizontal (X)
axis this would show the range of values recorded for births per 100
families. On the vertical (Y) axis it would show the total number (or
proportion) of parishes in England that had recorded each successive
rate.
Some birth rates would occur very infrequently; these would be
shown at the extremes of the distribution, either to the left if they were
very low rates, or to the right if they were exceptionally high. Other
rates would occur very frequently, particularly those close to the mean,
and these would be plotted near the centre of the distribution.
� The second is the distribution of the particular sample drawn from this population. For this we actually have the relevant data on the mean
BRTHRATE for the sample of 214 English parishes and could plot the
distribution. On the horizontal axis this would show the range of
values recorded for births per 100 families. On the vertical axis it would
show the number (or proportion) of parishes in the sample that had
recorded each successive birth rate.
� The third is the sampling distribution of the sample mean. This is rather different in character from the first two. Imagine that there is not
just one sample of 214 parishes, but a very large number of successive
random samples of that size, each drawn from the same population.a
For every separate sample the mean BRTHRATE shown by that sample
could be calculated.
We could then plot a distribution of these sample means. On the hor-
izontal axis this would show the range of values recorded for the mean
BRTHRATE. On the vertical axis it would show the number (or pro-
portion) of samples that had produced each value of the mean. This
hypothetical distribution is called the sampling distribution of the
sample mean.
§.
a Strictly, it should be all possible samples of the given size.
For a practical illustration of the concept of the sampling distribution,
see questions 7 and 8 in the exercises for this chapter.
More generally, it can be defined as shown in the following box.
SAMPLING DISTRIBUTION
A sampling distribution
is the distribution of a sample statistic
that would be obtained if
a large number of random samples of a given size
were drawn from a given population.
Note that this sampling distribution is not observed empirically. It is the
hypothetical distribution that would be obtained if a large number of
samples were taken from a given population. This notion of a sampling dis-
tribution can now be applied to some fundamental propositions in statisti-
cal theory.
5.3 Four fundamental statistical propositions
Four fundamental propositions relating to the sample results can be
derived from statistical theory. They are known collectively as the Central
Limit Theorem, and in this text these are simply accepted gratefully
without further enquiry into their mathematical foundations.
5.3.1 The Central Limit Theorem
1 (a) If the population has a normal distribution the sampling distribu- tion of a known sample mean (X
– ) will also be normal; and
1 (b) even if the population distribution is not normal, the sampling dis- tribution of X
– will nevertheless be approximately normal, provided
that the sample size is sufficiently large.b
2 (a) The mean of the sampling distribution of the sample means will be equal to �, the unknown population mean.
b A sample size of 10 or 20 will often be enough for this purpose, but anything less than 30 is
usually treated as a small sample. The reason for this is discussed in §5.4.2.
1 (b) On average, the known sample mean (X –
) will be equal to the mean of
the sampling distribution.
1 (c) It therefore follows that, on average, the known sample mean (X –
) will
be equal to �, the unknown population mean.
The value of the sample mean can thus be taken as a point estimate of the population mean.c Naturally any individual sample is likely to be a little
above or below the population mean. The second contribution made by
statistical theory enables us to calculate by how much any single sample is
likely to miss the target.
3 (a) The estimation error for any given point estimate, X –
, is determined
by the shape of the sampling distribution.
The flatter the sampling distribution, the more the possible values of X –
will be spread out, and the wider the distance between any given X –
and �.
Conversely, the greater the clustering of sample means, the less the likely
estimation error of any given X –
relative to �. Because the standard devia-
tion of the sampling distribution thus indicates the range of possible error
in estimating � owing to differences between samples, it is commonly
referred to as the standard error.
THE STANDARD ERROR
The estimated
standard deviation of the sampling distribution
is known as
the Standard Error.
This may be abbreviated to SE.
1 (b) The size of the standard error depends on the size of the sample being drawn from the population. As the sample size increases, the more
clustered will be the array of sampling means around the population
mean, �, and the smaller will be the estimation error attributable to
any given sample mean, X –
.
The ideal formula for the standard error of the sample mean is / ,
where (lower case Greek sigma) is the standard deviation of the
�n
§.
c A point estimate is a single value. The alternative is an interval estimate covering a range of
values and thus making a less precise claim to identify the single (unknown) population value.
population distribution, and n is the size of the sample. Unfortunately,
however, this formula requires the value of the population standard devia-
tion ( ), and we usually do not know this. Once again the theorem comes
to the rescue.
4 The standard deviation of the sample (s) can be taken as the best esti-
mate of the population standard deviation ( ). The standard error of the
sample mean can thus be calculated by substituting s for in the previ-
ous formula to produce:3
Standard error of the sample mean�s/ (5.1)
This equation enables us to calculate the error involved in adopting X –
as
an estimate of �, on the basis of information known only from the sample
itself.d
THE STANDARD ERROR OF A SAMPLE MEAN
The best estimate
of the standard error of a sample mean,
i.e. of the standard deviation
of the sampling distribution of a sample mean, is
(5.1)
where s is the standard deviation of the sample
and n is the size of the sample.
For the BRTHRATE variable, s�4.505 and n�214, so the standard error of the sample mean is 4.505/ �0.308.
You will often see these characteristics of a sample summarized in one
expression, namely that the value of X –
is equal to:
� (5.2) s
�n X
�214
s
�n
�n
d As noted in chapter 2, computer programs and calculators may calculate the standard error
using n 1 rather than n. The reason for doing this is explained in §5.4.2. The distinction
between the two definitions matters only when the sample size is small. If n�30 or more there
would be practically no difference between the two estimates.
This expression provides information on both the measured mean of a
sample and its standard error. In the case of BRTHRATE, the information
would be displayed as:
BRTHRATE�15.636�0.308
We have thus far derived the standard error of the sample mean.
Standard errors can also be calculated for other sample statistics; it is neces-
sary, therefore, to indicate which standard error is being reported in any
case. This is often accomplished by adding the symbol for the characteristic
in brackets after the abbreviation SE (for standard error); so for the sample
mean it would be SE(X –
). At a later point in this chapter (see §5.6) we will
give the formulae for the standard error for some other common sample
statistics.
On the basis of these four theoretical propositions, we are able to use
statistics derived from a sample to estimate the value of a population
parameter and to indicate how accurate we consider the estimate to be. In
§5.5, we will use these simple rules to construct intervals within which we
believe our estimate of the unknown population mean to fall, and to calcu-
late the probability that we will be correct in this belief.
However, we are not yet ready for this. We must first spend some time
developing the procedures for calculating the confidence interval, and we
do this in §5.4. It is also necessary to pay some more attention to the
meaning of the sample being drawn, and one important aspect of this is
discussed in panel 5.1.
5.4 Theoretical probability distributions
In §5.3.1, we used the Central Limit Theorem to show that X –
is on average
equal to the population mean, �, and that its sampling distribution is nor-
mally distributed with an estimation, or standard, error equal to s/ , where s is the standard deviation of the sample and n is the sample size.
Before we use that information to construct confidence intervals around
our sample estimate of X –
we should spend a little time reviewing and
extending our understanding of the normal distribution.
5.4.1 Empirical and theoretical frequency distributions It may be helpful to begin by recapitulating some of the material presented
in chapter 2. When we first encountered the normal curve and its standard-
ized version, the standard normal distribution, in §2.6 we were dealing
with frequency distributions, i.e. with the number of occurrences of par- ticular values of a continuous random variable.
�n
§.
Panel 5.1 Sampling with and without replacement
Almost all sampling theory is concerned with the process of sampling with
replacement. Each drawing of an observation to create the sample is made
over the entire population. Every observation has the same probability of
being sampled with each successive drawing. In the simple case of drawing a
card from a pack, sampling with replacement requires that a card once
drawn be placed back into the pack, available for a repeat draw.
In historical research, however, once an observation is drawn, it is nor-
mally excluded from successive drawings. Every observation has the chance
of being sampled once, but no observation can be sampled more than once.
Thus, when a historian draws a 10 per cent sample of households from the
US Census in 1860 in order to analyse the pattern of wealth holding, he will
probably draw every tenth observation from the list of census manuscript
schedules. Once a household is chosen, it cannot be chosen again. Likewise,
each parish in Boyer’s data set is chosen only once. These are examples of
sampling without replacement.
Some statistical texts imply that sampling without replacement
significantly distorts the application of inferential statistics. A standard
example is the probability of drawing aces from a deck of cards. The expec-
tation of drawing an ace from a virgin deck is 1 in 13; if cards are continually
replaced after selection, this probability will remain the same in successive
draws. This is the notion of sampling independence.
If, however, cards are not replaced, then the probability of selecting an ace
in any given draw will not be independent of previous draws. In the second
draw, for example, if an ace has already been chosen, the probability of
selecting another falls to 1 in 17 (3 aces remain in 51 cards); if a different
card is chosen in the first draw, the probability rises to 1 in 12.75 (4 in 51). In
this example, the smaller the population, the greater the distortion.
If we had 1,000 decks, the removal of an ace in the first draw would reduce
the chances of choosing an ace next time from 1 in 13 to 1 in 13.003, a fairly
trivial change. The larger the population (or more accurately, the larger the
ratio of the population to the sample), the less damaging is the use of sam-
pling without replacement. If the population were assumed to be infinite in
size, the problem would disappear completely.
So how detrimental is it to our use of statistical techniques that the his-
torical procedure does not replicate the ideal sampling method of the statis-
tics textbook? In practical terms, fortunately for the quantitative historian,
the answer is that it does not matter much as long as the population is reason-
§.
ably large compared to the sample. This is comforting; historians usually have
to make do with samples that are too small, so it is not likely that they will
often have to worry about samples which are too large.
For two of the most important statistics that we deal with in this chapter –
the mean and standard error of the sampling distribution – we can be
more precise. The following propositions may be stated without further
proof:*
(a) The mean of the sampling distribution of the sample mean is
unaffected by the shift to sampling without replacement, as long as the
drawing of the sample is truly random.
(b) The standard error of the sampling distribution of the sample mean (as
defined in (5.2)) will always be smaller when samples are drawn
without replacement. This can be corrected by dividing the result
obtained from the sample by
where N is the size of the population and n the size of the sample.
If the sample is not more than about one-fifth of the population the cor-
rection needed is only 2–3 per cent of the standard error that would be
obtained if the sample were drawn with replacement, and can generally be
ignored. However, if the sample drawn without replacement is a larger frac-
tion of the population than this, the correction should generally be made.
Correction is desirable in such cases because under-statement of the
standard error will lead to a corresponding under-statement of the sam-
pling errors and confidence intervals explained below (see §5.5), and may
thus create a misleading impression of the reliability of the inferences that
can be made on the basis of the evidence from the sample.
It is also important to take the correction factor into account when com-
paring the means from two different samples, especially when both are
small relative to the population as a whole. Thus, if we wish to determine
whether the mean birth rate in Kent parishes was higher than in Sussex par-
ishes and whether this difference is statistically meaningful, we should rec-
ognize that both samples were drawn without replacement and construct
the standard errors of the means accordingly. We shall return to this issue
more formally in §6.7.1.
* For illustrations of these propositions see exercises 2, 3, and 5 at the end of this chapter.
�N nN 1
We began with a normal curve, a purely theoretical curve specified
mathematically by an equation involving the constants � and e, and the
mean and standard deviation of the distribution. This was a perfectly
smooth and symmetrical curve with a bell-shape. Different normal curves
were illustrated in figure 2.5. It was noted that many real-life empirical
values for specific variables such as height had a distribution that was a rea-
sonable approximation to the normal distribution.
Then, in §2.6.2, we developed the idea that any normal curve could be
standardized. For any particular observation in the sample, Xi, we could
take the value of the sample mean (X –
) as a benchmark with which to
compare the observation. The distance of any observation from the mean
of the sample was measured in units of the sample standard deviation. The
result was called Z, defined as
Z � (5.3)
It was observed that the distribution of Z always has a mean of zero and a
standard deviation of 1. The proportion under the curve (i.e. the propor-
tion of the standardized distribution) up to or beyond any specific value of
Z can be obtained from a published table (an extract from which was
reproduced in table 2.7). This can be used to calculate the position of any
given value relative to the rest of the distribution.
We now move from such empirical distributions and introduce the idea
of a theoretical probability distribution of a random variable. We will first discuss such a distribution and then consider its standardized form. It is
easiest to begin with a simple example for a discrete random variable (one which can take only a finite number of values), such as the number of heads
that would be obtained in two tosses of a coin. A comprehensive list of all
the possible combinations is:
First coin Second coin
Tail Tail
Tail Head
Head Tail
Head Head
We can thus see that there is only a one in four chance of getting either
no heads or two heads, and two in four chances of getting one head. If we
think of these chances in terms of probabilities they can be summarized as
follows:
(Xi X)
s
Number of heads Probability
0 1/4
1 2/4
2 1/4
Notice that the total of the probabilities is 1, because the three results
exhaust all the possible outcomes. These results represent the probability distribution or probability density function for this particular discrete random variable: the number of heads obtained in two tosses of a coin.
Exactly the same ideas can be applied to a continuous random variable (which can take any value within a relevant range) such as GNP or age of
marriage, though the mathematics needed to determine the probability
distribution may be considerably more complicated. If such a probability
distribution is depicted graphically, the value of the random variable, X, is
measured along the horizontal axis, and the vertical axis shows the prob-
ability that X lies within some specified interval; for example, the probabil-
ity that the age of marriage of a woman lies between 16.0 and 18.0 years.4
The total area under the curve again exhausts all the possibilities and is
equal to 1.
One particular form of such a curve is the very important normal theo- retical probability curve. This looks exactly like the normal curve dis- cussed and illustrated in chapter 2. It is also a perfectly smooth and
symmetrical, bell-shaped, curve. The location and spread of the curve
along the horizontal axis are similarly determined by its mean and stan-
dard deviation. It too has a standardized equivalent, Z, with mean equal to
zero and standard deviation equal to 1. The way in which this shape is
derived from the probability distribution of a discrete variable is explained
in panel 5.2 and illustrated in figure 5.1.
It may, at first, seem rather confusing to regard a particular distribution
as both an empirical frequency distribution and a measure of probability.
However, this follows directly from the objective notion of probability as long-run relative frequency. This is, in effect, what is meant when it is said that the probability of getting a head when tossing a single coin is 1/2, or
that the probability of getting an ace when one card is drawn from a pack of
52 is 1/13. We would not expect to get one ace every time we drew 13 cards
from the pack, but if we drew a single card 13,000,000 times (from an ultra-
durable pack) then we would expect the number of aces to be very close to –
although not exactly equal to – 1,000,000.5
This standardized normal probability distribution has all the same
properties as the standard normal curve discussed in §2.6.2. We can thus
§.
Panel 5.2 Generating the normal theoretical probability curve
Why does the probability distribution of a random event take the shape of
the normal curve? Let us use the case of the proportion of heads in a series of
coin tosses to show the process.
The probability of producing a head in a coin toss is 0.5. Thus, after one
coin toss, we would expect to produce either 0 heads or 1 head, each with a
50:50 probability. After two tosses, the probability structure changes. There
are now four possible combinations of heads and tails: as noted in the text,
the outcomes are such that there is a 1 in 4 chance of producing either 0
heads or 2 heads, and a 2 in 4 chance of producing 1 head.
With three tosses, the number of combinations increases to 8, for which
the probabilities are:
0 heads: 1/8; 1 head: 3/8; 2 heads: 3/8; 3 heads: 1/8.
The average (or, in the language of probability theory, expected) number
of heads is 1.5.*
As we increase the number of tosses, the probable number of heads will
be distributed symmetrically around its mean of 0.5. But, with each addi-
tional coin toss, the tails of the distribution become more elongated. Thus,
after 5 throws, the chances of producing either no heads or all heads falls to
1 in 32; after 10, to 1 in 1,024; after 15, to 1 in 32,768.** Moreover, as the
number of tosses increases, so the distribution as a whole begins to spread
out, as more combinations become possible and the variance of the distri-
bution increases.
We show the process in figure 5.1, which reproduces the distribution of
heads for 2, 4, 10, and 20 coin tosses. The gradual transition towards the
familiar bell-shape of the normal distribution is clearly evident.
* This is in effect a weighted average (see §2.3.2) calculated as the sum of the number of heads
multiplied by their respective probabilities:
�[(0�0.125)�(1�0.375)�(2�0.375)�(3�0.125)]�1.5.
** Note that the number of possible permutations of heads and tails is equal to 2n, where n is
the number of separate coin tosses. No matter how many tosses, only one permutation in
each sequence will be either all tails or all heads.
use the information in published tables such as table 2.7 to find the propor-
tion of the distribution in the area defined by any value of Z. We might, for
example, ascertain the proportion of the distribution up to or beyond a
specific value of Z, and can then interpret that proportion as the probability
of getting some given result that is greater than (or, if Z is negative, less
than) that value of Z.
Similarly the proportion of the distribution lying between two specific
values of Z can be interpreted as the probability of getting some result in
the range defined by those two values. Each of these values of Z corre-
sponds to some value in the original distribution before it was standard-
ized, according to the definition of Z given by (5.3).
5.4.2 The Z- and t-distributions We now extend the idea of a standardized theoretical probability distribu-
tion to apply to the sampling distribution of a sample statistic ; for example,
the sampling distribution of the sample mean discussed in §5.2. In this
case the population mean, �, is taken as the benchmark with which to
compare the value of the mean derived from a sample, X –
. The distance of
the sample mean, X –
, from this benchmark can be expressed in units of
the standard deviation of the sampling distribution. As noted in §5.3, this
§.
Figure 5.1 Distribution of heads for 2, 4, 10, and 20 tosses of a coin
Proportion of heads
1.0.9.8.7.6.5.4.3.2.1.0
Pr ob
ab ili
ty
0.6
0.5
0.4
0.3
0.2
0.1
00.0
2 coins
4 coins
10 coins
20 coins
particular standard deviation is known as the standard error (SE), and for a
sample mean its value – as given in (5.2) – is s/ . Using this SE to stan-
dardize the deviation of the sample mean from the population mean, we
can define Z as
Z � (5.4)
We would thus obtain a separate value for Z for every sample we draw
from a given population and for which we calculate a mean. These values of
Z represent a theoretical probability distribution. If we depicted this on a graph the successive values of X
– would be measured along the horizontal
axis, and the vertical axis would record the probability of obtaining those
values. The total area under the curve again exhausts all the possibilities
and is equal to 1.
However, in order to obtain this formula for Z, it was necessary to
replace the unknown by the known sample standard deviation, s. Such
substitution will almost always be necessary in historical research, because
there is usually no way of knowing the population values. Unfortunately,
the statisticians have established that this procedure is not entirely satisfac-
tory. It turns out that s/ is a better estimate of the standard devia-
tion of the sampling distribution than s/ . The reason for this can be
explained intuitively in terms of the concept of degrees of freedom (df). This is discussed in panel 5.3.
As a consequence, an alternative standardized measure has been
devised, which is known as Student’s t, or simply t, where
t � (5.5)
Here, too, a separate value for t would be obtained for every sample
drawn from a given population for which a mean was calculated. These
values of t similarly represent a theoretical probability distribution of a sample statistic, known as the t-distribution.
The t-distribution is symmetric and bell-shaped, with mean and mode
equal to zero. It is thus similar in appearance to the standard normal prob-
ability distribution, but its precise shape depends solely on a parameter, k,
which is determined by the number of degrees of freedom (df) in the esti-
mate of the population standard deviation. This equals n 1, i.e. df is one
less than the size of the sample.
There is thus a different distribution – and so different probabilities –
for each df, and the distribution is flatter (less peaked) and more spread out
(X �)
s/�n 1
�n �n 1
(X �)
s/�n
�n
the lower the number of df (see figure 5.2). However, the difference
between successive t-distributions diminishes rapidly as the sample size
(and so the number of df) increases, and the distributions become less
spread out. After we reach about 30 df, the t-distribution approximates
increasingly closely to the standard normal probability distribution.6
In principle, the t-distribution is always superior to the standard normal
Z-distribution when is unknown, but its advantages are especially impor-
tant when the sample size is small (less than about 30). In such cases the
§.
Panel 5.3 The concept of degrees of freedom
The use of s/ rather than s/ in the formula for the t-distribution
(5.5), can be explained intuitively in terms of the concept of degrees of
freedom.
To introduce this concept, consider a simple example in which there is a
series consisting of only three numbers, with a mean value of 4. If you were
invited to choose the constituents of this series you would be able to choose
any number you liked for the first two values, but once you had done that the
final number would be determined for you. Say, for example, you chose first
1 and then 5. The third number would then be constrained to be 6; no other
value would be consistent with your first two choices and a mean of 4. Using
our standard notation we can say that if the number of items in a sample
series (n), and the sample mean of that series (X –
) are both known, then only
n 1 of the items in the series are ‘free’ to take any value; there are only n 1
degrees of freedom (df). One degree of freedom has been lost.
Since the sample mean is used in the formula for the sample variance, 1
degree of freedom is lost and there are only n 1 ‘free’ observations
involved in its calculation. This is what is meant by stating that there are
only n 1 degrees of freedom (df). The same principle applies to the calcu-
lation of the standard deviation or the covariance.
We shall encounter this concept again in other contexts in later chapters
(see §7.3 and §8.2). A general definition which applies to these and other
measures is:
The degrees of freedom of a statistic is the number of
quantities that enter into the calculation of the statistic, minus
the number of constraints connecting those quantities.*
* Peter E. Kennedy, A Guide to Econometrics, 3rd edn., Blackwell, 1992, p. 67.
�n�n 1
differences between the two distributions can have an important effect on
the decisions based on these distributions (such as those in §5.5 and §5.6 of
this chapter and in §6.4 and §6.5 of chapter 6). As the sample size increases
above this the differences between the two distributions cease to matter.
An extract from the published table for the theoretical t-distribution is
given in table 5.1, with the number of degrees of freedom specified in the
left-hand column. Note that the information is presented in a different form
from the table for the standard normal distribution. In table 2.7 we read
across from a specific value for Z to find the proportion of the total area
under the curve (i.e. the probability) relating to that value of Z. For example,
for Z�1.96 the proportion in one tail (column (4)) was 0.025 (2.5 per cent).
For the t-distribution the procedure is to read across from the number
of degrees of freedom in column (1) to find the value of t that leaves a
specified proportion in either tail of the distribution. For example, for 10 df
the value of t corresponding to the same proportion (probability) of 0.025
(2.5 per cent) in one tail is given in column (4) of table 5.1 and is found to
be t�2.228.
It can be seen from this example that, for small samples, the probability
of getting a result that is greater (or less) than some specific value is higher
with the t-distribution than with the standard normal distribution.e
Conversely, the probability of getting a result between any two values will be
lower with the t-distribution than with the standard normal distribution.
e With the standard normal distribution, if the probability of getting a result greater than Z�1.96
is 2.5 per cent, the probability of getting a result greater (further to the right) than that, such as Z
�2.228, would be less than 2.5 per cent, and thus less than the probability of getting that result
with the t-distribution.
Figure 5.2 The t-distribution for selected degrees of freedom
t
df = 30
df = 2
df = 4
§.
Table 5.1 Distribution of t for selected degrees of freedom
(1) (2) (3) (4) (5) (6)
Value of t for which area in one tail is:
df 0.10 0.05 0.025 0.01 0.005
1 3.078 6.314 12.706 31.821 63.657
2 1.886 2.920 4.303 6.965 9.925
3 1.638 2.353 3.182 4.541 5.841
4 1.533 2.132 2.776 3.747 4.604
5 1.476 2.015 2.571 3.365 4.032
6 1.440 1.943 2.447 3.143 3.707
7 1.415 1.895 2.365 2.998 3.499
8 1.397 1.860 2.306 2.896 3.355
9 1.383 1.833 2.262 2.821 3.250
10 1.372 1.812 2.228 2.764 3.169
11 1.363 1.796 2.201 2.718 3.106
12 1.356 1.782 2.179 2.681 3.055
13 1.350 1.771 2.160 2.650 3.012
14 1.345 1.761 2.145 2.624 2.977
15 1.341 1.753 2.131 2.602 2.947
16 1.337 1.746 2.120 2.583 2.921
17 1.333 1.740 2.110 2.567 2.898
18 1.330 1.734 2.101 2.552 2.878
19 1.328 1.729 2.093 2.539 2.861
20 1.325 1.725 2.086 2.528 2.845
21 1.323 1.721 2.080 2.518 2.831
22 1.321 1.717 2.074 2.508 2.819
23 1.319 1.714 2.069 2.500 2.807
24 1.318 1.711 2.064 2.492 2.797
25 1.316 1.708 2.060 2.485 2.787
26 1.315 1.706 2.056 2.479 2.779
27 1.314 1.703 2.052 2.473 2.771
28 1.313 1.701 2.048 2.467 2.763
29 1.311 1.699 2.045 2.462 2.756
30 1.310 1.697 2.042 2.457 2.750
32 1.309 1.694 2.037 2.449 2.738
The column headings in table 5.1 refer only to one tail, but because the
distribution is symmetrical the area in both tails marked off by any given
value of t is always exactly double the one-tailed area.f The value that marks
off a specified tail in this way is called the critical value or critical point, and will feature prominently in chapter 6 (see §6.3.2).
5.5 Confidence intervals for a sample mean
After this digression into two of the core ideas of mathematical statistics,
sampling distributions and probability distributions, we can now revert to
the problem posed at the start of §5.2. If we know the sample mean, X –
, how
good is this as an estimator of what we really want to know, which is the
population mean, �?
Our first step towards an answer to this question is to fix a confidence interval around the estimate of the mean obtained from the sample.
f Similarly, the corresponding cumulative and central areas, or the area to the right of the mean –
which were shown in table 2.7 – can easily be found, given that the total area�1 and the area to
the right (or left) of the mean�0.5.
Table 5.1 (cont.)
(1) (2) (3) (4) (5) (6)
Value of t for which area in one tail is:
df 0.10 0.05 0.025 0.01 0.005
34 1.307 1.691 2.032 2.441 2.728
36 1.306 1.688 2.028 2.434 2.719
38 1.304 1.686 2.024 2.429 2.712
40 1.303 1.684 2.021 2.423 2.704
50 1.299 1.676 2.009 2.403 2.678
60 1.296 1.671 2.000 2.390 2.660
120 1.289 1.658 1.980 2.358 2.617
� 1.282 1.645 1.960 2.326 2.576
Source: D. V. Lindley and W. F. Scott, New Cambridge Statistical Tables, 2nd edn., Cambridge
University Press, 1995, table 10, p. 45.
5.5.1 Interval estimates and margins of error The second of the fundamental propositions in §5.3.1 stated that the
sample mean, X –
, could be taken as an estimate of the unknown population
mean, �. It was noted that although this would be correct on average, each
individual sample would be likely to miss the target.
For this reason it is often preferable to replace the point estimate given by the single sample mean by an interval estimate. This interval is equal to the sampling error involved in relying on the sample mean. We can thus
write
X –
� the sampling error (5.6)
We can also decide how large we want this margin of error to be. The
interval can be made wider or narrower depending on how confident we
want to be that the interval estimate we adopt does contain the true �.
Given the sample size, if it is desired to be more confident – at the cost of
being less precise – a wider interval should be chosen, i.e. the sampling
error would be increased.
The usual practice is to opt for a 95 per cent confidence interval. This means that we expect to be correct 19 times out of 20 (or 95 times out of
100) that our interval estimate does include �. Equally, of course, it means
that we have to recognize that we will be wrong (in the sense that our inter-
val estimate will not actually include �) once in every 20 times. In other
words, by choosing a 95 per cent confidence interval we are satisfied with a
95 per cent probability that the estimate will be correct.
It is important to note that this is not the same as saying that we can be
95 per cent confident that � lies within the particular interval estimate
derived from a single sample. The value of � is fixed and cannot change;
what can change is the value of X –
obtained from successive samples, and
thus the range around X –
given by the confidence interval. With a 95 per
cent confidence interval it is expected that only one in every 20 intervals
derived from successive samples will not contain �.
It is thus a statement of confidence in the procedure we are adopting,
embracing all the (hypothetical) interval estimates we would obtain if we
were able to repeat the calculations with a large number of samples of a
given size.
5.5.2 Confidence interval for an estimate of a population mean What size interval do we need in order to have the required degree of
confidence that the range we are giving either side of the (known) sample
mean will contain the (unknown) population mean?
§.
More specifically, if we want a 95 per cent confidence interval (i.e. if we
want a 95 per cent probability) how many standard errors either side of the
sample mean do we have to add to achieve this degree of confidence?
We know from the Central Limit Theorem (see proposition 1 in §5.3.1)
that the sampling distribution of the sample mean will be (approximately)
normal. If it is necessary to use the sample standard deviation, s, in place of
the population standard deviation, , then the table for the t-distribution
(see table 5.1) can be consulted to find the value of t that leaves the specified
proportion in the two tails of the distribution.7 Thus to achieve 95 per cent
confidence we have to exclude an area with 5 per cent in the two tails, equal
to 2.5 per cent (usually given as a proportion, 0.025) in each of the tails.
95 PER CENT CONFIDENCE INTERVAL FOR A MEAN
An interval estimate of a population mean takes the form
X –
�sampling error. (5.6)
For the 95 per cent confidence interval this can be stated as
X –
� t 0.025 SE(X –
) (5.7)
where t 0.025 is the value that excludes 2.5 per cent
in each of the two tails of the t-distribution
for any given sample size,
and SE(X –
) is the Standard Error
of the sampling distribution of the mean.
In §5.3 the standard error for BRTHRATE for the sample of 214 English
parishes (�s/ ) was found to be 0.308 births around a mean rate of
15.636 births per 100 families. If we look up t 0.025 for the t-distribution for
213 degrees of freedom in table 5.1 we find (by interpolation) that the crit-
ical value is 1.971.8
The 95 per cent confidence interval for the mean birth rate (births per
100 families) of the population is thus
15.636�(1.971�0.308)
�15.636�0.607
�n
We are thus able to say that there is a 95 per cent probability that a range
from 15.029 to 16.243 (i.e. from 15.636 0.607 to 15.636�0.607) births
per 100 families will contain the (unknown) population mean.g
This is equivalent to saying that with the statistical procedure we are
using we can expect our sample to provide confidence intervals that will
include the population mean about 95 per cent of the time. If 20 samples
were taken, it is probable that the inference about the population mean, �,
that we draw from the sample mean, X –
, would be right 19 times and wrong
once. Of course, if we only have one sample there is no way of knowing
whether this is one of the 19 that gives the correct result, or the twentieth
one that gives the wrong result!
If the prospect of getting the wrong result 5 per cent of the time is alarm-
ing then it is always open to the investigator to adopt a higher standard.
Choosing a 99 per cent confidence interval would mean that the investiga-
tor could expect these (wider) intervals to include the unvarying mean
about 99 per cent of the time.
A familiar variant of this problem usually occurs during an election
when opinion polls are used to predict the proportion of the electorate that
will vote for a particular party or presidential candidate.h In a run of such
polls there are typically some that excite special interest by deviating sub-
stantially from the remainder in the series, but once it becomes apparent
that they have not detected a sudden shift in opinion the pollsters acknowl-
edge reluctantly that this deviant result was a ‘rogue poll’.
In other words, this is one of the incorrect results that any sampling pro-
cedure is bound to produce if it is repeated sufficiently often. The fre-
quency of rogue polls is systematically related to their sample size. The only
way for the pollsters to reduce the probability of errors would be to work
with larger samples, but that is often too expensive for their sponsors.
To see what the confidence interval would be for the mean BRTHRATE
if the sample size were smaller, imagine that instead of the 214 parishes the
sample had been confined to the 23 parishes for which information was
found for Kent (county 1).i The mean BRTHRATE obtained from this
sample was 18.547 with a standard deviation of 4.333. Since the sample size
is small it is important to replace by in the formula for the�n 1�n
§.
g If we used s/ rather than s/ , the standard error would change imperceptibly to 0.309
and the confidence interval would be 15.636�0.608. h In this case the relevant sampling statistic is a proportion rather than a mean, but the statistical
principle is the same. i For those who recall that we worked with 24 Kent parishes in chapters 2–4 it may be helpful to
note that those statistics related to the first Boyer data set for poor relief. The number of parishes
in his birth rate data set is smaller, and in the case of Kent is 23.
�n�n 1
standard error.j SE(X –
) is therefore 4.333/ �0.924. With a sample of
this size there are 22 df, and if we look up table 5.1 for the t-distribution
with 22 df we find a critical value for t0.025 of 2.074.
Thus with this sample the 95 per cent confidence interval for the mean
population birth rate is
18.547�(2.074�0.924)
�18.547�1.916
On the basis of this sample we are thus able to say that there is a 95 per
cent probability that a range from 16.631 to 20.463 births per 100 families
will contain the (unknown) population mean. With the smaller sample we
are able to place the population mean only within a fairly wide interval
(with 95 per cent confidence).
If a higher standard of confidence were required (say, 99 per cent) the
proportion to be excluded in each tail would be 0.005 (giving a total pro-
portion of 0.01 or 1 per cent in both tails). For 22 df, the critical value is
t0.005�2.819 (see column (6) of table 5.1) and the confidence interval for
the population BRTHRATE would be correspondingly wider: 15.942 to
21.152 births per 100 families.
It should also be noted that there is no overlap between the 95 per cent
confidence interval for the population mean birth rate obtained from the
sample of 23 Kent parishes and the corresponding interval obtained earlier
from the full sample of 214 parishes. The lower bound for the former
(16.631) is actually higher than the upper bound for the latter (16.243).
This is a sharp reminder that it would not be good sampling procedure to
attempt to derive an estimate of the mean birth rate in England as a whole
from a sample of parishes drawn exclusively from one county (see §1.3.4).
As should also be obvious, any confidence interval is directly affected by
the size of the sample on which it is based. The larger the sample, the
smaller will be the standard error, and hence the smaller the width of the
interval. The alternative to achieving a higher standard of confidence at the
expense of a wider interval is thus to increase the size of the sample where
this is not precluded by considerations of cost or availability of data.
5.6 Confidence intervals for other sample statistics
An analogous procedure can be used to determine the confidence interval
for any other result obtained from a sample, for example a proportion or a
regression coefficient. In each case:
�22
j See n. 3 (p. 144).
(a) There is a (hypothetical) sampling distribution of the sample statistic. (This important concept was discussed in §5.2.1 with reference to the
sampling distribution of the sample mean, but it was noted that there
would be a sampling distribution for every sample statistic.) The stan-
dard deviation from the mean of this sampling distribution is called
the Standard Error (SE).
(b) The shape of that sampling distribution and the size of the sample
determine how reliable the sample statistic is as an estimate of the cor-
responding population parameter.
(c) The required result can be expressed in terms of a confidence interval.
For any given level of confidence (say, 95 per cent) the size of the inter-
val is determined by
the selected probability of getting a result outside that range,
multiplied by
the relevant Standard Error.
5.6.1 Sample proportions For a sample proportion, p, the standard error is given by
SE(p) � (5.8)
where � (lower case Greek pi) is the (unknown) proportion in the popula-
tion. Because � is not known it is estimated by using the sample propor-
tion, p, and the formula becomes
SE(p) � (5.9)
The formula for the 95 per cent confidence interval for a sample size of
(say) 41 (so df�40) is thus
p� t0.025 SE (p)
p � 2.02 (5.10)
5.6.2 Correlation coefficients The correlation coefficient, r, was introduced in §3.2 as a measure of the
strength of the relationship between two variables. At that stage the result
was accepted without regard to the circumstances in which it was obtained.
�p(1 p)n 1
�p(1 p)n 1
��(1 �)n
§.
In the light of the concepts discussed in this chapter it should now be clear
that this is not satisfactory.
It is necessary to ask the same question about the correlation coefficient
as about any other result obtained from a sample: is the value obtained for r
simply the result of random (chance) features of the particular sample, or
does it reflect systematic and consistent features of the relationship
between the population of the two variables? In other words, if the correla-
tion coefficient for the population mean is � (lower case Greek rho), is this
sample correlation coefficient, r, a good estimator of �?
To answer this question in terms of confidence intervals we need to
know the standard error of the correlation coefficient, SE(r). This is
SE(r) � (5.11)
Unfortunately, this formula is of limited use, both because the value of �
is not usually known and because the sampling distribution of r is highly
skewed unless the sample size is quite large (a sample of at least 100 is rec-
ommended). To replace it statisticians have developed a transformation,
Fisher’s z, which is obtained from r as follows9
z � (5.12)
Fisher’s z should not be confused with the Z values introduced in earlier
chapters in relation to the standard normal distribution, and discussed
again in §5.4 of this chapter. The two are completely different.
The sampling distribution of Fisher’s z is approximately normal and its
standard error is given by
SE(z) � (5.13)
An illustration of Fisher’s z used for a confidence interval for a correlation coefficient To illustrate the procedure using Fisher’s z, we will take the example from
§3.2.1 where the coefficient of correlation between UNEMP and RELIEF
for the 24 Kent parishes was found to be 0.5244 (to 4 decimal places). From
(5.12)
z�1⁄2 loge (1.5244/0.4756)� 1⁄2 loge (3.2052)�
1⁄2 (1.1648)
�0.582
Then from (5.13)
1
�n 3
1
2 loge�1 � r1 r�
1 �2
�n 1
SE(z)�1/ �1/4.5826
�0.218
Because z is approximately normally distributed we can use the stan-
dard normal distribution (rather than the t-distribution) so the 95 per cent
confidence interval around z is given by
z�1.96 SE(z) �0.582�1.96 (0.218)
�0.582�0.427
The confidence interval around z thus ranges from �0.155 to�1.009. To
complete the exercise we then have to reverse the transformation and
convert back to the corresponding values of r. This can be worked out using
the formula in (5.12), and is also available in a published table, which gives
the value of r corresponding to the lower and upper limits of the interval as
0.154 and 0.765, respectively.k
On the basis of this rather small sample of 24 parishes we thus require a
quite wide range in order to be able to say that there is a 95 per cent prob-
ability that this confidence interval will contain the (unknown) population
correlation coefficient.
There are two further points about this confidence interval that should
be noted. First, it is not symmetrical about the correlation coefficient of
0.524: the upper limit of 0.765 is closer to r than the lower limit of 0.150.
The reason for this is that the upper limit of r is 1, and this in turn con-
strains the upper limit of the confidence interval. If we had taken an
example in which r was negative, the lower limit would be similarly con-
strained so that it could not exceed 1.
Secondly, the range given by the confidence interval around a positive
value for z could be larger than the value of z, so that the lower limit of the
range had a negative value. This would mean that the corresponding lower
limit of r would also be negative. Say, for example, that the correlation
coefficient obtained from a sample of 24 cases had a positive value of
�0.28. At the 95 per cent confidence level the interval that we could expect
to contain the true population correlation coefficient would stretch from
0.14 to �0.61, and would thus include the possibility of a negative
correlation.
5.6.3 Regression coefficients The estimated standard error for a regression coefficient involves some
very important concepts that we will investigate in chapter 9. For the
�21
§.
k Lindley and Scott, Statistical Tables, table 17, p. 59.
present, we can simply note that SE (b), the standard error of a sample
regression coefficient for the slope, b, of a regression line with only one explanatory variable, is
SE (b)� (5.14)
The formula for the 95 per cent confidence interval for the population
regression coefficient, ß, and a sample size of (say) 32 (df�30) is therefore
b�t0.025 SE (b) (5.15)
Substituting the formula for SE (b) from (5.14) in this equation gives
�b�2.04 (5.16)
In this chapter we have reviewed the formulae for the standard errors of
the sampling distributions of a mean, proportion, regression coefficient,
and correlation coefficient. These four statistics illustrate the basic princi-
ples. For other sample statistics your statistical program and computer will
provide the required formula for the standard error and do the necessary
calculations.
Notes 1 A full discussion of all the statistical (and financial) issues that arise in drawing an
appropriate sample cannot be provided here. A brief introduction to the topic is
given in the references given in §1.3.4. W. G. Cochran, Sampling Techniques, 3rd
edn., Wiley & Sons, 1977, is a good example of a more comprehensive text. 2 This is the procedure adopted by Boyer in his analysis of the English Poor Law Data
(An Economic History of the English Poor Law, Cambridge University Press, 1990, pp.
131, 149). Parishes that did not return relatively complete questionnaires to the Poor
Law Commission were excluded from his sample. However, every parish returned
some information and Boyer was able to compare these results with those for his
sample to indicate that incomplete respondents were not systematically different
from those that did reply in full. He thus decided that he could treat his data as
though they were drawn from a random sample. 3 The statisticians actually make a subtle distinction here. Although they tell us that
the sample standard deviation, s, is the best estimate of , they say that the best esti-
mate of the standard deviation of a sampling distribution is not s/ , but s/ .
If the sample size is large (more than 30) the difference made by using s/ rather
than s/ in the denominator hardly matters, as can be seen by reworking the
BRTHRATE example given below, but with smaller samples it becomes progressively
more important. We will say more about this distinction in §5.4.2 and offer an intui-
tive explanation for it in panel 5.3.
�n 1 �n
�n 1�n
� �(Yi Ŷ) 2
(n 2)�(Xi X)2
� �(Yi Ŷ) 2
(n 2)�(Xi X)2
4 We are told by the statisticians to refer in this way to the probability of a continuous
random variable having a value within some specified interval, because the probabil-
ity of any one specific value is always zero. 5 In this context, each card must of course be replaced immediately it has been drawn.
The number of aces would not be expected to be exactly 1,000,000 with replacement
because each draw is completely independent. Assume, for example, that at any point
in the sequence of drawing the cards, the proportion of aces had fallen 20 below 1 in
13. The cards have no memory of the previous draws, and so there is absolutely no
reason to expect that the next 20 draws will all be aces. At any stage, the probability
for each succeeding draw is exactly the same as it was at the outset of the experiment:
1 in 13. 6 The variance of the t-distribution is given by the formula k/(k 2), where k equals
the number of df (�n 1). This means that the larger the sample size (and hence k)
becomes, the closer the variance is to 1, and hence the more closely the distribution
approximates the standard normal probability distribution. As we know from
§2.6.2, the standard deviation (and hence also the variance) of this standard normal
distribution is also 1. 7 The t-distribution should be used in preference to the Z-distribution whenever is
not known, irrespective of the sample size. However, it should be intuitively obvious
that the difference between s and will be greater the smaller the size of the sample
on which s is based, and so the use of the t-distribution is again especially important
when dealing with small samples. 8 When the number of df is large it is recommended that interpolation should be har-
monic. For an explanation see Lindley and Scott, Statistical Tables, p. 96. 9 An alternative version of (5.12) is sometimes quoted, which is not calculated in
natural logs but in logs to the base 10 (see §1.6.2). This is
z � 1.1511log10 ,
and gives the identical result. The value of Fisher’s z for any given r can also be
obtained from published tables, for example, Lindley and Scott, Statistical Tables,
table 16, p. 58.
5.7 Exercises for chapter 5
1. If we have a perfect die the probability of getting each of the six faces on a
single throw is exactly 1/6. Since the probability of getting each of the
numbers from 1 to 6 is the same, the probability distribution for the throw
of a single die is rectangular. We can take such a rectangular distribution as
the population of all possible throws of a single die.
(i) Calculate the sampling distribution of the means of samples based
on throwing two dice.
(ii) Calculate the sampling distribution of the means of samples based
on throwing three dice.
�1 � r1 r�
§.
(iii) Plot the distributions obtained in (i) and (ii) and give an intuitive
explanation of why the distribution approximates more closely to
the normal distribution as the sample size increases.
2. A population consists of the five numbers: 2, 3, 6, 8, and 11.
(i) Calculate by hand the mean and standard deviation of the popula-
tion. (Note : this is the entire population; not a sample.)
(ii) Specify all possible samples of size 2 which can be drawn with replacement from this population, and calculate the means of each of these samples. (Hint : this means that you can have both 2, 3 and
3, 2 and also 2, 2, etc.)
(iii) Calculate the mean of the sampling distribution of means. Compare
your result with the population mean.
(iv) Calculate the standard error of the sampling distribution of means.
Compare your result with the value obtained from the formula SE�
/�n. (v) Comment on your results.
3. Take the same population as in question 2 and undertake the same exer-
cises, but using the approach of sampling without replacement. Comment on your results, especially with regard to any contrasts and similarities with
the answers to question 2.
4. Using the Boyer relief data set, calculate the mean and standard devia-
tion of INCOME for the 24 Kent parishes:
(i) Calculate by hand the 90 per cent and 99 per cent confidence inter-
vals for mean INCOME for the 24 Kent parishes. Compare your
result with the confidence intervals given by your statistical
program.
(ii) Calculate by hand the 99 per cent confidence interval for the propor-
tion of Kent parishes with a workhouse. Compare your result with
the confidence interval given by your statistical program. (Hint : treat
this a mean.)
(iii) If the 95 per cent confidence interval for INCOME is calculated for a
sample of 24 Kent parishes it is found to be 34.3625�2.129. How
many parishes would need to be included in the sample to reduce the
95 per cent confidence interval to �1? (You can take t 0.025�1.96.)
(iv) By how many times would the sample need to be increased in order
to reduce the confidence interval to one-eighth of the size of �1
adopted in question (iii); i.e. to 0.125? What general result can be
derived from this?
(v) Calculate the 95 per cent confidence interval for the regression
coefficient in the regression of RELIEF on UNEMP for all 311
parishes.
5. Calculate the standard error of parish population for Cambridge,
Oxford, and Sussex using the Boyer relief data set. Use the methodology of
panel 5.1 to correct the standard errors for the application of sampling
techniques without replacement, noting that the number of rural parishes
in each county in 1832 was 170, 295, and 155, respectively. Evaluate the sig-
nificance of the revisions for this data set. Would the same conclusions
apply if the total number of parishes in each county were (a) half as many
and (b) a third as many as the numbers presented?
6. Boyer has demonstrated that the selected parishes in his relief and birth
data sets are representative of all those who replied to the Poor Law
Commission’s questionnaires in 1832. But only a small proportion of all
parishes in Great Britain replied to the request for information. In the case
of Kent, 29 out of 404 rural parishes responded; Boyer used 24 of these
answers to construct his relief sample. You wish to determine whether
Boyer’s parishes are typical of all Kent parishes in 1831, including those
that did not respond to the questionnaires.
As a first step to testing for representativeness, you examine the relative
size of the parishes in the sample and the population, to determine whether
Boyer’s parishes are atypically large or small. The Census of 1831 indicates
that the average size of all rural Kent parishes is 965.75 persons, with a var-
iance of 1632.5. How would you use this information to determine the
validity of Boyer’s sample? What conclusions do you draw?
7. In a file on the web site, identified as LOCAL, we have compiled data on
the unemployment rate for adult males in July 1929; the data have been col-
lected for each of the 664 local Employment Exchange districts in Great
Britain. This data set may be considered the population for analysis of sampling procedures. Use this information to:
(i) Construct the population frequency distribution and measure the
population mean and variance. Note whether the distribution is
normal or skewed.
(ii) Draw 5 random samples of 5 local districts each and construct a
sampling distribution of the mean unemployment rate. (Hint : you
will need to use either a random number table, such as that given in
Lindley and Scott, Statistical Tables, p. 78 or many other statistics
texts; or a random number generator from your statistical program.)
(iii) How does the sampling distribution of the mean change as you
increase the number of samples of 5 districts each from 5 to 10 to 20?
What is the standard error of the sampling distribution in each case?
§.
Note whether the distribution is normal or skewed.
8. As a further test of the nature of sampling distributions, use the LOCAL
unemployment file on the web site to:
(i) Draw 5 random samples of 10 local districts each and construct a
sampling distribution of the mean unemployment rate. Increase the
number of samples (each of 10 districts) to 10 and then to 20,
recording the information on the mean and standard error of the
distribution. Note whether the distribution is normal or skewed.
(ii) Repeat this exercise using random samples of 20 local districts each,
and compare your results to those in (i).
What conclusions do you derive from these two exercises about the correct
approach to sampling?
Hypothesis testing
6.1 Testing hypotheses
In chapter 5 we established how to determine the confidence intervals for a
single result obtained from a sample. The next issue to be considered covers
the situation where we wish to investigate a specific hypothesis relating to a result obtained from one or more samples.
With confidence intervals the implicit hypothesis is that the specified
interval contains the required value. With hypothesis testing the hypothe-
sis is made explicit, and can be framed so as to refer to particular values of
the population statistic. It might, for example, be a hypothesis relating to a
comparison of the results of two different samples. Or it might be a
hypothesis about whether or not the result of a single sample can be con-
sidered as different from some specified value.
For example, RELIEF in the sample of Kent parishes has a mean value of
20.28 shillings with a standard deviation of 7.64 shillings. For Sussex the
corresponding values are a mean of 26.04 shillings and a standard devia-
tion of 8.04 shillings. The mean in Sussex was thus almost 6 shillings (28
per cent) higher than the mean in Kent.
This differential might have occurred, either
(a) because of chance factors and accidental irregularities reflected in the
samples we happened to get from the two counties; or
(b) because there were systematic and consistent differences in the underly-
ing structural features of the two counties; for example, in the sources
of income available to labourers, the generosity of relief payments, or
the incidence of poverty and sickness in the two areas.
The aim of inductive statistics is to enable us to decide which of these possibilities we should regard as more likely. If the answer is (a), the impli-
cation is that a similar differential would not be found if further samples
were drawn from additional parishes in each county. If so, the apparent
difference shown by the two existing samples of parishes is simply a
reflection of random factors that are of no real interest to historians.
If the answer is (b), then it is a potentially interesting finding and histo-
rians of the Poor Law should proceed further to consider why per capita
relief payments were some 28 per cent higher in Sussex than in Kent.
6.1.1 A simple illustration In order to illustrate some of the central ideas in a more familiar context,
consider the following simple story about four men, each of whom was
arrested and charged with theft. The four cases are quite separate. The
initial presumption is that each man is innocent until proved guilty. The
jury have to decide whether to uphold or reject this presumption on the
basis of the evidence presented to them.
The four cases, and the main evidence the prosecution have against the
men arrested, are as follows:
(a) Goods stolen from a house : A noticed loitering outside the house a
couple of days before the theft.
(b) A robbery at a drug store : B seen driving away from the store on the
night of the robbery.
(c) A break-in at an electrical goods warehouse : C seen in the vicinity of the
warehouse in the early evening and overheard boasting in a bar later
on the same night that he had done a good evening’s work and would
soon have loads of money to spend.
(d) A theft of jewellery and antiques : D reported trying to sell the stolen
goods to a second-hand dealer the following morning.
The jury rapidly acquitted A. They also acquitted B but only after a long
discussion. C was found guilty but on a majority verdict. D was quickly and
unanimously found to be guilty.
The jury’s decisions can be interpreted as follows. In each case the ques-
tion they had to ask themselves was: What is the chance that the man would
have been doing what the police reported if he was innocent? If they judged
the probability to be high, they would decide that he was innocent. If they
judged it to be low, they would reject the presumption of innocence and
find the man guilty. In making their judgement they have to recognize
that if they set their probability standard too high they run the risk of
acquitting a guilty man; if they set it too low they may convict someone
who is innocent.
In these four cases the jury evidently drew their dividing line between
high and low probability somewhere in the region between the chance of B
being seen driving away from the store if he was innocent, and the prob-
ability of C being seen near the warehouse and then being overheard boast-
ing in the pub if he was innocent. In A’s case the likelihood of innocence
was found to be very high, in D’s it was obviously very low.
We will return to this story in §6.3.6.
6.1.2 The five stages of hypothesis testing The essence of hypothesis testing is a decision about the plausibility of a
particular outcome given by the sample information. This decision is
reached by working through the following sequence:
(a) Decide the issue to be determined and specify a hypothesis in an appro-
priate form for testing (see §6.2).
(b) Select a level of probability on the basis of which it will be decided
whether or not the hypothesis should be rejected (see §6.3).
(c) Select the relevant test statistic and its related sampling distribution.
The test statistic expresses the specific result incorporated in the
hypothesis (e.g. a mean or a regression coefficient) in a form suitable
for comparison with the probability of all possible outcomes of that
particular statistic. The sampling distribution gives that theoretical
probability distribution for all the possible outcomes (see §6.4).
(d) Calculate the test statistic and compare this calculated value with the
tabulated theoretical probability distribution in order to reach a deci-
sion about the hypothesis (see §6.5).
(e) Interpret the results of the decision (see §6.6).
In the subsequent sections we will explain each of these stages, and then
in §6.7 we will illustrate three specific tests of hypotheses involving a
difference of sample means, a correlation coefficient, and a regression
coefficient.
6.2 The null hypothesis
It is a basic principle of inductive logic that a proposition can be refuted by
one observation but cannot be proved. To take a famous example, the
proposition that ‘all swans are white’ would have to be rejected if a single
black swan was observed, but would not be proved by the sighting of a
single white swan, or indeed even of a thousand.
Following from this principle, statisticians proceed by setting up a
hypothesis that can be rejected. This is known as the null hypothesis and is referred to as H0.
§.
THE NULL HYPOTHESIS
The null hypothesis (H0) specifies the negative form of the proposition to be tested, for example,
there is no difference between two means or proportions,
or no relationship between two variables,
or a new drug should not be introduced because it has adverse side effects.
The alternative or research hypothesis (H1) specifies the opposite:
that there is a difference,
or there is a relationship,
or the drug should be introduced.
Notice that the null hypothesis should normally be framed in terms of
the outcome that would be regarded by the researcher as undesirable : she
would normally prefer to find that there was a relationship between the two
variables or that the experiment was successful, or that the drug should be
introduced.
The aim of research is then to determine whether or not the null
hypothesis should be rejected. If the data used for this purpose are found
to be inconsistent with the hypothesis it should be rejected. If the data are
consistent with the hypothesis it should not be rejected. The language
used is important: because of the principle stated at the beginning of this
section the researcher must not claim that her experiment has established
that the null hypothesis is correct or that it should be accepted. The
appropriate response is that it is not rejected. The more frequently and
successfully a hypothesis resists rejection in successive independent
experiments, the better its status and the more it might be relied on, but it
would still remain a hypothesis that could be rejected in some future
experiment.
6.2.1 Other assumptions of the model It is important to bear in mind that in setting up a null hypothesis to be
tested, using data obtained from a sample, it is also necessary to make
certain assumptions about:
(a) the underlying population from which the sample was drawn; and
(b) the research procedures used to obtain the sample data.
It is only if one is certain about (a) and (b) that it is possible to isolate the
null hypothesis as the issue that is to be decided by the selected test.
One of the crucial questions with respect to (a) is whether or not the
population distribution is approximately normal (i.e. if it looks approxi-
mately like the theoretical frequency curve described in §2.6). If it is, then
one set of tests, in which this normality is assumed, is appropriate. If it is
not, then another set, known as non-parametric tests, which will be intro- duced in chapter 7, must be employed. These latter tests do not assume a
normal distribution and are also applicable to nominal and ordinal level
data.
The crucial question with respect to (b) is whether or not the sample is
truly random. (A brief indication of what is required to satisfy this condi- tion was given in §1.3.4.) If the sample is not random, then it is essential to
consider the extent to which there might be a bias in the data because of this. If there is a serious bias, the data are contaminated and the results
cannot be relied on.
It may, nevertheless, be possible to argue that even though the sample is
not random, the results are not invalidated. For example, as noted in
chapter 5 (see n. 2), Boyer (Poor Law, p. 129) claims this for his data, on the
grounds that the parishes that completed the returns to the Poor Law
Commission from which his samples are drawn were not systematically
different from those that did not.
6.3 How confident do we want to be about the results?
6.3.1 Two types of error and the null hypothesis The basic idea underlying this fundamental statistical procedure is that it is
not possible to achieve certainty in the decisions we make on the basis of
incomplete or imperfect information. To introduce the topic consider a
familiar non-statistical example: the everyday problem of a pedestrian
trying to cross a busy road:
The more carefully he waits until he is sure it is safe to cross, the less likely he is
to be run over. But if he tries to achieve maximum safety he might wait
forever and would never get to the other side of the road.
§. ?
There is thus an inescapable dilemma. If he minimizes the risk he maxi-
mizes delay; if he minimizes the delay he maximizes the risk. Both problems
cannot be avoided simultaneously.
In statistics this dilemma is expressed in terms of two types of error,
known respectively as Type I and Type II. There is always a trade-off between these: the more a statistician tries to eliminate one type, the more
likely he makes the other.
TWO TYPES OF ERROR
Type I: Rejecting a true hypothesis.
Type II: Failing to reject a false hypothesis.
Illustration: an experiment with a new drug The issue can be illustrated by the case of a trial for a new drug, where the
null hypothesis (H0) is that it should not be introduced because it may have
very adverse side effects for many users. The research hypothesis (H1) is
that there are no serious side effects and it should be introduced.
There are then four possibilities:
(a) If the scientists do not reject H0 when it is true (i.e. the drug does have
adverse side effects) they will make the correct decision. The drug will
not be introduced and no patients will suffer from the adverse
effects.
(b) If they reject H0 when it is true (and instead fail to reject H1) they will
make a Type I error: the drug will be introduced and some unfortu- nate patients will suffer the adverse effects.
(c) If they do not reject H0 when it is false the drug will not be introduced.
This will represent a Type II error, and the beneficial treatment the drug could have provided will be lost.
(d) If they reject H0 when it is false (and instead fail to reject H1) they will
again make the correct decision : the drug can be introduced and
patients will be treated without the risk of adverse side effects.
These possible outcomes can be summarized in the box below.
IMPLICATIONS OF REJECTING OR NOT REJECTING THE
NULL HYPOTHESIS
Decision:
Reject H0 Do not reject H0
H0: True Type I error Correct
H0: False Correct Type II error
If it was known whether H0 was true or false it would, of course, be easy
to make the correct decision: H0 would never be rejected when it was true
and would always be rejected if it was false. But since this cannot be known
it is necessary to rely on statistical theory for guidance on the probability
that H0 is true, given the outcome of the research, and should not be
rejected; or the probability that it is false, and should be rejected.
This in turn means that it is not possible to achieve complete certainty
with respect to the validity of any particular research hypothesis (H1). All
that can be done is to establish whether or not the hypothesis is probably
true in the light of the evidence. There is thus always some possibility that
an error will be made in relation to this decision.
The crucial point is thus that this procedure is designed to assist the
investigator to make a decision about the null hypothesis in the light of a
specific event or result, while recognizing that this decision will be subject
to the possibility of error.
The question the historian has to ask is: What is the probability that I
would have obtained that result if the null hypothesis is correct? If there is a
low probability then H0 should be rejected. If there is a high probability
then H0 should not be rejected.
This then leads us to the further issue: What should be regarded as a
high or low probability? This depends on the margin of error the investiga-
tor regards as appropriate in reaching a conclusion about the null hypothe-
sis. We discuss this in the following subsection.
6.3.2 Significance levels and the critical region In order to decide about this margin of error, the question the researcher
has to consider is: What risk is she willing to take of making Type I and Type
II errors? One crucial component of her answer will be the significance level, conventionally designated by �.
§. ?
SIGNIFICANCE LEVEL
The significance level, �, of a decision
is the probability of making
a Type I error.
The more there is at stake in rejecting a true null hypothesis, the greater
the need to select a low probability of making a Type I error, even though,
as we have seen, this will increase the prospect of making a Type II error.
The probability of making a Type II error is designated as �. The prob-
ability of not committing a Type II error (1 �) is known as the power (or efficiency) of a test of the null hypothesis.
THE POWER OF A TEST
The power (or efficiency) of a test (1 �)
is its ability to avoid a Type II error.
For any given sample size,
the greater the ability of a test to eliminate false hypotheses,
the greater is its relative power.
The critical region The concept of the critical region is very closely related to that of the level of significance. The lower the level of significance, the smaller the critical
region, the less the chance of making a Type I error, i.e. the less the probabil-
ity of rejecting H0 when it is true. The reason for introducing yet another
unfamiliar term is that it illustrates the procedure, and so helps to clarify
the rules to be followed in making the final decision.
THE CRITICAL REGION
The critical region constitutes
the set of unlikely (low-probability) outcomes
the occurrence of which would require
rejection of the null hypothesis.
A 5 per cent level of significance can thus be translated into a critical region covering 5 per cent of the area under the curve (shown in figure 6.1 (a) as the two shaded areas of 0.025 (2.5 per cent) in each of the two tails of
the distribution), and a non-rejection region, covering the remaining 95 per cent. The critical value marks the border between the two regions.
§. ?
Do not reject
in this region the null hypothesis
Critical region:
the null hypothesis reject
95 per cent 5 per cent
Figure 6.1 The critical region
Do not reject
in this region the null hypothesis
Critical region:
the null hypothesis reject
Critical region: reject
the null hypothesis
2.5 per cent 95 per cent 2.5 per cent
(a) A two-tailed test at the 5 per cent level of significance (a) At this significance level the critical region covers 2.5 per cent of the area
in each tail
(b) A one-tailed test at the 5 per cent level of significance (b) At this significance level the critical region covers 5 per cent of the area
in one tail
As good social scientists our procedure should always be to make it
difficult to reject the null hypothesis, H0, and thus also difficult to fail to
reject the research hypothesis, H1. It is essential to proceed in this way even
though (indeed, precisely because) we would actually like to reject H0 and
not reject H1 because the alternative would be more intellectually interest-
ing (or financially profitable). This good practice is achieved by selecting a
small critical region.
The trade-off There is, however, a price to be paid for this. Consider, for example, a com-
parison of the 1 per cent and 5 per cent significance levels. At the 1 per cent
significance level the critical (rejection) region is smaller, and the non-
rejection region is bigger, than at the 5 per cent level. Therefore, at the 1 per
cent level, with a smaller critical region, there is less likelihood of a Type I
error (rejecting H0 when it is correct). But then there will automatically be a
larger non-rejection region, which means there is more likelihood of a
Type II error (not rejecting H0 when it is false).
There is thus an inescapable trade-off: the less the chance of a Type I
error, the greater the chance of a Type II error. The standard approach to
hypothesis testing is to assume that Type I errors are less desirable than Type
II errors and to proceed accordingly. For this reason, the power of a test is
rarely invoked in empirical analysis. The conventional methodology
instead focuses on setting the significance level and then choosing a
method that minimizes the chances of committing a Type I error.
Thus in the example of the drug trial, the consequences of introducing a
drug with adverse side effects are usually viewed as far more serious than
the consequences of not introducing a drug with potential life-saving or
life-enhancing properties.1 This logic is reinforced by the recognition that,
in general, it is very hard to identify the sampling distribution of the
research hypothesis, and one cannot therefore easily observe the probabil-
ity of committing a Type II error. As a rule, therefore, researchers are
content to recognize the trade-off by not reducing � to extremely low
levels, but beyond this little more is done or said.
6.3.3 One- and two-tailed tests There is one further aspect of the critical region that must be clarified. In
figure 6.1 (a) the critical region was divided equally between the left- and
right-hand tails. But in principle the whole critical region could be shown
either in the right-hand tail, as in figure 6.1 (b), or in the left-hand tail. The
first refers to what is called a two-tailed test, the second (whether to the left or the right) to one-tailed tests.
Which of these should be used? The answer normally depends on
whether or not the hypothesis being investigated tells the researcher some-
thing about the direction of the result. For example, the null hypothesis
might simply state that there is no difference between two means. If so,
there is nothing in the hypothesis about direction and a two-tailed test
must be used.
Alternatively, the null hypothesis might state that the mean from the
first sample is larger than the mean from the second. If so, it is only
differences in that direction that are of interest, and a one-tailed test must
be used. In this case the relevant tail naturally depends on the direction
predicted by the hypothesis. If it is positive (larger than), it is the right-
hand tail; if negative (smaller than), the left-hand tail.
H1 is often expressed using the symbols � for ‘greater than’ and for
‘smaller than’. For example a researcher might state:
H0: �1��2 H1: �1��2
The direction of the arrowhead conveniently points to the direction of the
tail to be selected.
To illustrate this procedure consider two (hypothetical) samples from
which a demographer has derived figures for the mean age of marriage in
England in 1780 and 1820. If she has reason to think there may have been a
fall in the age of marriage over this period she is able to predict the direc-
tion of the change, and a one-tailed test is appropriate. H0 would be that
there is no difference between the two mean ages, and H1 would be that
X –
1820 was lower than X –
1780 (H1: X –
1820 X –
1780).
If the mean age in 1820 is then found to be actually higher than in 1780
the result is clearly inconsistent with the idea of a fall in the age of marriage,
and a test of the hypothesis is hardly necessary. But if it is lower, then the
demographer has to decide whether or not to reject the null hypothesis of
no difference and must concentrate attention on the left-hand tail.
Notice that the effect of selecting a one-tailed test is to increase the size
of the relevant critical region for any given level of significance. This
increases the probability of a result in the critical region and thus makes it
easier to reject the null hypothesis. For this reason, one-tailed tests should
be used only when there is a good a priori reason, independent of the data
being tested, for investigating a directional hypothesis. And if a one-tailed
test is adopted, the choice must be made before the results of the test are
calculated.
§. ?
6.3.4 Statistical versus historical significance Despite the importance that is attached to this concept of statistical
significance it is absolutely crucial to understand that it does not imply that
the result is necessarily meaningful or significant – in the everyday sense of
the word – for the researcher.
A typical null hypothesis for many statistical measures is that there is no
relationship between two variables, or that the value of a coefficient is zero.
If a measure is found to be statistically significant this establishes that the
null hypothesis can be rejected, but not that the relationship matters or
that the value of the coefficient is important in the context of the research.
Statistical significance is determined partly by the strength of the relation-
ship, or the extent of the difference between the coefficient and zero shown
by the sample, and partly by the size of the sample. If the sample size is large
enough it is almost always possible to obtain a ‘statistically significant’ result.
Thus, if it is established that a result is statistically significant, this indi-
cates that it merits further consideration, not that it is automatically to be
regarded as an important or interesting finding. That is a totally different
question, and statistical significance must never be confused with histori- cal significance, i.e. with the issue of what is substantial or important from the point of view of the historian.
Take, for example, the initial illustration in §6.1 of the difference
between the mean values of RELIEF in Kent and Sussex, respectively 20.3
shillings and 26.0 shillings. A difference of this magnitude that was shown
to be statistically significant at the 5 per cent level would be of considerable
interest to historians of poverty and the Poor Law. If, however, the results
had been 21.0 shillings in Kent and 21.1 shillings in Sussex the difference
would be of little historical significance, even if it could be shown to be sta-
tistically significant at the 1 per cent level, or lower.
Similarly a regression coefficient may be found to be statistically
significant, but be so small as to demonstrate that the explanatory variable
has no practical effect on the dependent variable.a
6.3.5 Conventional significance levels and prob-values
The traditional procedure The conventional levels of significance are 1 per cent, 5 per cent, and 10 per
cent. For most research in history and the social sciences it has become the
norm to adopt the 5 per cent level as appropriate. If a researcher has stated
a An important aspect of the analysis of any quantitative exercise is thus to examine the magni-
tudes of the regression coefficients and other results. This is discussed with several illustrations
from the literature in chapters 14 and 15.
that she will adopt the 5 per cent level and then finds that her results are
significant at the 1 per cent level, she will normally note this with special
satisfaction, and the results might then be described as ‘highly significant’.
If, however, the research related to something like the experiment to test
a new drug for adverse side effects, it would be advisable for the drug
company to require a higher degree of certainty that the drug was safe. It
would thus be necessary to choose a lower level of significance in order to
have an extremely low probability of making a Type I error (namely, reject-
ing the null hypothesis of adverse effects when it was true, i.e. when the
drug actually did have such effects). In such a crucial context a level even
lower than 1 per cent might be appropriate.
It is an important principle of good research practice that the appropri-
ate level should be decided in advance ; it should not be influenced by the
results once they have been obtained.
Significance levels can also be regarded as the complement of the
confidence intervals discussed in chapter 5. If a null hypothesis is rejected
at the 5 per cent level, i.e. with a 5 per cent probability of error in coming to
this conclusion, this is exactly equivalent to saying that it lies outside the 95
per cent confidence interval. Conversely, if it is not rejected at the 5 per cent
level, this is equivalent to saying that it falls within the 95 per cent
confidence interval.
Significance levels and prob-values The traditional procedure outlined above involved the selection of a
specific significance level, in the social sciences typically 5 per cent. This
can, however, be criticized as a rather arbitrary procedure. If the result falls
just within the critical region corresponding to the 5 per cent level H0 is
rejected; if it falls just outside the region H0 is not rejected, no matter how
small the difference in the probability of these two outcomes. There is no
valid justification for making such a distinction between significance levels
of, say, 5.1 per cent and 4.9 per cent.
The origins of this procedure lay in a pre-computing age when the rele-
vant probability distribution tables from which the levels were determined
were expensive to reproduce and had to be kept simple. For example, a
complete set of tables for the t-distribution (which was shown in abridged
form in table 5.1), with all the information for every individual degree of
freedom, would run to many pages.
With the advent of modern computers it is no longer necessary to use
published tables. Modern statistical packages like SPSS and STATA can
swiftly calculate all the necessary information, and their default procedure
is to report the actual probability of the outcome, i.e. the precise critical
§. ?
value equal to the calculated test statistic. Hypothesis testing procedures
have changed accordingly. Instead of indicating whether or not the test sta-
tistics fall within the critical region defined by one of the conventional
significance levels, many researchers now report these actual probabilities.
Researchers should still indicate, at least broadly, the level they regard as
appropriate for their project, but their readers are given the fullest possible
information and can make their own judgements about how the
significance tests should be interpreted.
This actual probability is given under various headings, including prob- value or p-value or Sig. Thus a prob-value of 0.003 means that the prob- ability of obtaining that result – if H0 is true – is only 0.3 per cent. The
smaller the probability, the stronger the case for rejecting H0, so for most
purposes this would mean that the null hypothesis could be rejected. A
prob-value of 0.053 would mean that the probability of obtaining that
result if H0 is true is 5.3 per cent. If the conventional 5 per cent significance
level were being strictly followed, this would lead to a decision that H0 should not be rejected, but a different judgement might be made depend-
ing on the context and the size of the sample.
PROB-VALUE OR p-VALUE
Prob-value or p-value
(or sig or statistical significance)
is the probability
that the outcome observed would be present
if
the null hypothesis were true.
6.3.6 The critical region and the four cases of theft It may help to consolidate understanding of the concepts introduced in the
preceding sections if we revert now to the story in §6.1.1 of the four men
charged with theft. The presumption of innocence is equivalent to the null
hypothesis. The four types of evidence represent the sample statistical
results in the light of which the null hypothesis is to be rejected or not
rejected. The dividing line between the non-rejection and critical regions is
implicit rather than explicit, but the majority of the jury clearly located it
somewhere between the higher probability of B being seen driving away
from the drug store if he was innocent and the lower probability of C being
seen near the warehouse and then being overheard boasting in the bar if he
was innocent.
A’s case falls well into the non-rejection region: there is a very good
chance (high probability) that he could have been noticed outside the
house a couple of days before the theft even if he was innocent. Conversely,
D’s case is well into the critical region: the jury obviously thought that there
was very little chance (very low probability) that he would have been trying
to sell the stolen goods to a dealer the morning after the theft if the pre-
sumption of his innocence were true. A low probability thus leads to rejec-
tion of the null hypothesis.
Note that Type II errors (failing to reject the null hypothesis when it is
false) do not come into this story at all. The emphasis in the Anglo-
American legal system on the notion of reasonable doubt is consistent with
setting a low significance level, � (but not too low, since it is reasonable
rather than absolute doubt that is at issue). In other words, these judicial
systems are primarily concerned to avoid making Type I errors: finding
accused persons guilty when they are innocent.
However, in jurisdictions in which the rights of the accused are less sac-
rosanct, or in a situation where it is seen as more important to avoid cases
where the guilty are acquitted, the judicial system might be more con-
cerned to avoid Type II errors: finding the accused innocent when they are
actually guilty. In these circumstances it is considered more important to
lock up potential wrongdoers without too much concern for any infringe-
ment of their civil rights if they are indeed innocent. A prime example of
this in the case of the United Kingdom was the policy of internment of sus-
pected terrorists during the 1980s. The movement towards parole denial in
certain categories of crime in the United States (especially involving vio-
lence against children and sexual deviancy) may be viewed similarly.
6.4 Test statistics and their sampling distributions
We have now stated a null hypothesis, and selected an appropriate
significance level and critical region (or used a statistical program to deter-
mine its probability equivalent). The next stage in the procedure is to select
the relevant test statistic and its related sampling distribution.
6.4.1 The basic ideas The null hypothesis refers to one of the basic statistical measures derived
from one or more samples. This sample measure (or sample statistic)
might be, for example, a mean or median, a proportion, a variance, a
§.
correlation or regression coefficient, or a comparison of observed and
expected frequencies.
For every such sample statistic it is necessary (a) to calculate a test statis-
tic and (b) to ascertain the sampling distribution for that statistic.
� The test statistic expresses the specific result calculated from the sample data in a form suitable for comparison with the sampling distri-
bution.
� The sampling distribution is associated with a theoretical probability distribution for all possible outcomes of that particular test statistic.
The concept of a sampling distribution of a sample statistic was intro-
duced in §5.2. In that instance the concept was illustrated by a discussion of
the sampling distribution of the sample mean, but exactly the same ideas
apply to any other statistic derived from a sample. For any such sample sta-
tistic or measure there is a corresponding sampling distribution, i.e. the dis-
tribution of the value of that statistic which would be obtained if the statistic
was calculated repeatedly on the basis of a large number of random samples.
The statisticians calculate these sampling distributions on the basis of
mathematical reasoning that we do not need to consider. Each of these
sampling distributions has specific properties (for example, its mean and
standard deviation).
These will generally agree more or less closely with the corresponding
properties of one of a number of theoretical probability distributions, for
example, the standard normal distribution. We can also rely on the statisti-
cians to tell us which sampling distributions correspond to which theoreti-
cal distributions.
In fact, several different sampling distributions follow the same theoret-
ical distribution. So we do not use a different theoretical distribution for
each individual test statistic. Two of the theoretical probability distribu-
tions and associated tests that are most widely used in research in history
and the social sciences are those for Z and t, both of which were introduced
in §5.4.2.
In the following sections we will first look briefly at the test statistics
associated with these two distributions, and then complete our discussion
of hypothesis testing.
6.4.2 Z- and t-tests The first two theoretical probability distributions we encountered were the
standard normal (or Z-) distribution and the closely related t-distribution
(see §5.4.2).
In the case of a mean, for example, the essence of the theoretical Z-
distribution was that the value of the population mean, � (mu), was taken
as the benchmark, and the deviation (or distance) from that benchmark of
a particular sample mean, X –
, was expressed in terms of the standard devia-
tion of the sampling distribution of the sample mean.
This standard deviation of a sampling distribution was given the special
name of standard error. It is commonly designated by the abbreviation SE followed by the statistic to which it relates. Thus for a sample mean it would
be SE(X –
), for a sample regression coefficient it would be SE(r), and so on.
The question thus posed is: How many standard errors from the bench-
mark is the value of the mean obtained from the given sample?
It was stated in §5.3 that SE(X –
) is equal to / . So in order to answer
that question, it would be necessary to know the population standard devi-
ation ( ), but that information is hardly ever available. However, as
explained in §5.4.2, if the sample size is large enough (generally taken to be
more than 30), we can use the standard deviation of the sample, s, as an
estimate of . The standardized deviation of the sample mean from the
population benchmark can then be measured as
Z � � (6.1)
The table for the proportionate areas under the curve of the standard
normal distribution can then be used to assess the probability of getting a
particular value greater (or less) than Z, or the probability of getting a par-
ticular value within some range, for example from the mean to Z.
The alternative t-distribution replaces s/ , by a better estimate of the
standard error, s/ , to give
t� � (6.2)
It was further noted that although the t-distribution is, in principle,
always superior to the Z-distribution when is unknown, its advantages
are especially important when the sample size is less than about 30.
We can now apply these distributions to our primary theme of hypothe-
sis tests. In what follows we refer, for simplicity, to the t-tests, since histo-
rians often have to work with small samples, but the same fundamental
ideas apply to both Z- and t-tests.
The basic idea behind the main test statistics is exactly the same as in the
above review of the theoretical distributions, except that in the context of
hypothesis tests it is the value specified in the null hypothesis that is taken as
the benchmark. In a test of a sample mean this would typically be the value
of the population mean as in (6.2).2 In a test of a sample regression
(X �)
s/�n 1 (X �)
SE(X)
�n 1 �n
(X �) s/�n
(X �)
SE(X)
�n
§.
coefficient the null hypothesis might be that there is no association
between the variables, so the benchmark is r equals zero; and so on.
The result of this standardized deviation from the benchmark is often
referred to as tcalc. b So (6.2) might be written as
tcalc� � (6.2a)
For other sample statistics there would be a corresponding calculation,
with an appropriate estimate for the standard error of the sampling distri-
bution of the given statistic. A number of examples of such alternative test
statistics will be given in §6.7. The table for the areas under the curve of the
t-distribution (see table 5.1) can then be used to assess the probability of
getting a particular value greater (or less) than that test statistic, tcalc.
We thus have two values that have to be compared. One is the test statis-
tic, tcalc, calculated by means of an appropriate formula such as (6.2a) from
the actual sample data available to the researcher. The other is the theoreti-
cal probability distribution of t (or Z) indicating the values that would be
obtained from an infinite number of random samples.
This fundamental principle for the calculation of test statistics can be
applied much more widely, i.e. it need not be limited to means or to a single
sample. The more general form of these test statistics can be given as
tcalc� (6.3)
STATISTICS FOR t-TESTS
The test statistic, tcalc, equals:
the deviation of the sample statistic from
the value for that statistic specified in the null hypothesis
divided by
the standard error (SE)
of the sampling distribution of that statistic:
tcalc� (6.3) Sample estimate Null hypothesis
SE
Sample estimate Null hypothesis
SE
(X �)
s/�n 1 (X �)
SE(X)
b The corresponding test statistic calculated for a Z-test would be Zcalc, and so on. An alternative
notation uses an asterisk to distinguish the test statistic; thus t*, Z*, etc.
The test statistic given by (6.3) is used if the null hypothesis relates to
some specific non-zero value. This would apply if relevant values are given
by some other source; for example, if it is known from past experience that
the proportion of students failing their statistics course is 15 per cent, and it
is desired to test whether the current failure rate of 17 per cent is indicative
of a deterioration in performance. Alternatively, the specific value might be
derived from a theoretical proposition regarding, say, the slope of the
regression coefficient; for an illustration of the case where the null hypoth-
esis is b�1 see §6.7.3.
However, if – as is often the case – the null hypothesis is that the value is
zero, then the second term in the numerator can be eliminated, and the
formula reduces to the simpler form known as the t-ratio
tcalc� (6.4)
t-RATIOS
Where the null hypothesis is that the value of the sample statistic�0
the test statistic, tcalc, becomes:
the ratio of
the sample estimate
to
the standard error (SE) of the sampling distribution of that statistic:
tcalc � (6.4)
In general we can rely on our statistical software to provide the required
formula for SE and thus to calculate the various test statistics, but we
will look at some examples of the procedure in relation to differences
between two sample means, and to correlation and regression coefficients,
in §6.7.
6.5 Making a decision
We are now ready to take the final step in the testing of a hypothesis. We
have selected an appropriate sampling distribution and test statistic, and
have decided on the significance level and whether it should be a one- or
two-tailed test. We proceed as follows:
Sample estimate
SE
Sample estimate
SE
§.
1. Calculate the test statistic (e.g. tcalc) from the sample data using the rel-
evant formula.
2. Compare this result with the corresponding tabulated sampling distri-
bution, the theoretical probability of getting a result greater than the
selected critical value given by the choice of significance level and crit-
ical region(s).
3a. If the calculated statistic is larger than the specified theoretical prob-
ability (and thus falls in the critical region) the null hypothesis is
rejected : see figure 6.2 (a).
3b. If it is smaller than the specified theoretical probability (and thus falls
in the non-rejection region) the null hypothesis cannot be rejected : see
figure 6.2 (b).
The logic of this procedure should be clear from the preceding discus-
sion. The fundamental principle is that H0 should be rejected only if the
results obtained from the sample are unlikely to have occurred if H0 is true.
What is meant by ‘unlikely’ has been decided in advance by the selection of
the significance level and critical region.
To recapitulate the full procedure, consider again the case of the drug
trial:
� H0 is that the drug should not be introduced because it has adverse
effects.
� The company should reject H0 only if the results obtained from the trial
are extremely unlikely to have occurred if H0 is true.
� The theoretical distribution defines the critical region of results that
are extremely unlikely to occur.
� If the test result falls in this region, H0 can be rejected.
� If it falls in the much larger non-rejection region, H0 cannot be rejected
and the drug should not be introduced.
More generally, the statistical packages calculate the probability that the
observed relationship found in the given data set would be present if H0 is
true. The lower the prob-value the less likely it is that such a result would be
obtained if H0 were true. So if a very low prob-value (or equivalent term) is
reported the null can be rejected.
This procedure is often found to be rather confusing. There are a
number of reasons for this, and it may help to remember the following
points:
� First, a low level of significance (probability) corresponds to a high value
for the test statistic which takes it towards or into the tails of the
distribution, and thus towards or into the critical region where the
probability is low.
� Secondly, when this occurs and the test statistic is in the critical region,
the procedure is to reject the null hypothesis, which sounds depressing.
But remember that H0 is usually specified as the undesirable outcome,
so rejection of the null is actually good news for the researcher.
§.
The test statistic
Non-rejection Critical regionregion
95 per cent 5 per cent
Figure 6.2 A one- tailed hypothesis test at the 5 per cent level of significance
The test statistic
Non-rejection Critical regionregion
95 per cent 5 per cent
(a) Rejection of the null hypothesis The test statistic is greater than the critical value: Ho is rejected
(b) Failure to reject the null hypothesis The test statistic is less than the critical value: Ho cannot be rejected
6.6 Interpreting the results
The final stage in the process is the interpretation of the results. Once the
statistical program has tested the hypothesis the important task is to con-
sider the implications of the result.
Assume, for example, that the test was designed to see whether or not
emigration from Ireland (IRISHMIG) was associated with the level of
employment in the United States and other foreign destinations
(EMPFOR). H0 would then be that there was no association. If H0 cannot
be rejected it follows that the research hypothesis, H1, must be rejected. The
conclusion would have to be that there is no association, at least so far as
this can be tested with the particular data set you have used. However, if H0 is rejected, then the researcher can proceed to consider the further implica-
tions of the research hypothesis, H1, that there is an association – i.e. that
emigration was influenced by employment opportunities in the receiving
countries.
It is worthwhile to stress again the point made in §6.3.4. A finding that a
result is ‘statistically significant’ does not mean that it is important to an
historian. All that has been established is that the result is unlikely to have
arisen by chance given the degree of relationship that exists between the
variables and the size of the particular sample that was tested.
However, if the magnitudes show that the result is of interest, the
research can proceed underpinned by statistical support for the view that
there is an association between IRISHMIG and EMPFOR. The researcher
must then apply her historical knowledge and understanding to consider
why this association applied, and what consequences follow from it for her
analysis of the determinants of Irish emigration.
6.7 Three illustrations of the use of t-tests
To complete this chapter on hypothesis tests we will take three common
sample statistics to illustrate the use of t-tests. The first is a test of the
difference between two sample means, the second a coefficient of correla-
tion between two variables, and the third a regression coefficient.
6.7.1 A t-test for a difference of means This first illustration would normally be done with a statistical computer
package that would report the various values derived in the remainder of
this section. The reason for working through an example in this more labo-
rious way is that the process should help to explain what the computer is
doing and give you a better feel for the procedure.
We use the data for RELIEF for the counties of Kent and Sussex referred
to at the beginning of this chapter, and the issue to be considered is whether
the difference of 5.76 shillings between the sample means is statistically
significant. If so, it is clearly large enough to be also historically interesting.
The relevant summary statistics are:
Kent Sussex
Sample size (n1 and n2) 24 42
Mean (shillings) (X –
1 and X –
2) 20.28 26.04
Standard deviation (s1 and s2) 7.641 8.041
SE of the mean (�s/�n) 1.560 1.241
The null hypothesis is that there is no difference between the two sample
means:
H0: �1��2
and the test statistic is formally:
tcalc � (6.5)
But if, as the null hypothesis states, �1��2 , then �1 �2�0, and (6.5)
reduces to:
tcalc � (6.6)
It is then necessary to apply an appropriate formula to obtain the value
of SE(X –
1 X –
2). This involves a further procedure that is explained in panel
6.1. The result in this instance is found to be 2.021. This value can then be
used in (6.6).
tcalc � � 2.850
We can now complete the procedure by comparing this value for tcalc with the theoretical t-distribution given in table 5.1. In this case
the number of degrees freedom (df) is (n1�n2 2)�64. For 64 df and a
two-tailed test at the 5 per cent significance level the critical value is just
under 2.000. tcalc is substantially greater than this and so falls well within
the critical region. Thus H0 can be rejected.
If a statistical computer program is used in place of table 5.1 it is possible
to be more precise. It reports that the prob-value is actually 0.006, thus
indicating that the probability of getting a t-statistic of 2.85 when H0 is
X1 X2 SE(X1 X2)
� 5.761
2.021
X1 X2 SE(X1 X2)
X1 X2 (�1 �2)
SE(X1 X2)
§. t-
Panel 6.1 Calculation of the standard error of the difference between two sample means
The calculation of SE(X –
1 X –
2) in the denominator of (6.6) depends on
whether or not the variances of the (unknown) underlying populations
from which the two samples are drawn, 1 2 and 22, can be assumed to be
equal. (A special test, known as the F-test, can be applied to test the null
hypothesis, H0: 1 2 � 22)*.
If this null hypothesis cannot be rejected, the population variances can be
assumed to be equal, and the information from the two samples can be
pooled. The formula for the pooled estimate of that common variance (sp 2)
uses information from both samples as follows.
(sp 2) � (6.7)
�
�
This pooled variance (sp 2) can then be used in the formula given in (6.8) to
calculate the SE of the difference between the two sample means:
SE(X –
1 X –
2) � (6.8)
�
What is done if the outcome of the F test is that the null hypothesis of no
difference between the variance of the populations can be rejected? In that
case it has to be assumed that the population variances are unequal and the
information from the two samples cannot be pooled. It is then necessary to
use each of the separate sample variances in the calculation of the SE. The
formula for the standard error is then:
* The F-test will be introduced in §9.3.1: it is applied to the testing of equal sample variances
in question 6 in the exercises for chapter 9. An alternative test for equality of sample vari-
ances, known as Levene’s test, is used by SPSS.
�62.40324 � 62.40342 � �2.600 � 1.486 � 2.021 � s
2 p
n1 �
sp2
n2
1342.8523 � 2650.965
64 � 62.403
(23)(7.641)2 � (41)(8.041)2
24 � 42 2
(n1 1)s21 � (n2 1)s22
n1 � n2 2
that there is no difference between the means is only 0.6 per cent. This is
very low and so the null hypothesis can be rejected at a markedly lower
significance level than 5 per cent.3
6.7.2 A t-test for a correlation coefficient when the null hypothesis is zero correlation Consider, for example, the Irish emigration data for the census year 1881. If
we use a statistical program to measure the correlation between migration
(CNTYMIG) and the degree of population pressure indicated by average
family size (FAMSIZE) the coefficient, r, is found to be �0.528. The null
hypothesis would be that there is no association between these two vari-
ables, H0: ��0, where � is the population correlation coefficient. We will
apply a 1 per cent significance level to test this null hypothesis.
As explained in §6.4.2, when the null hypothesis to be tested is that the
value of the population coefficient is zero, the numerator of the test statis-
tic reduces to r 0�r. This is divided by the standard error (SE) of r to give
the t-ratio or t-statistic
tcalc � (6.10) r
SE(r)
§. t-
SE(X –
1 X –
2) � � � 1.993 (6.9)
This is marginally smaller than the result with the pooled variance, and the
SE of the difference between the means will be marginally larger.
However, it must be noted that if the alternative correction for unequal
variances does have to be applied, it is also necessary to correct for a change
in the number of degrees of freedom to be used in testing the SE for statisti-
cal significance. The correction, which arises from the inefficiency of the
additive formula (6.9) as an estimate of x̄1 x̄2, may be quite large.
Thus, in the case of the comparison between Kent and Sussex, the
number of degrees of freedom for the case of unequal variances is 35, com-
pared to a figure of 64 under the assumption of equal variances.4 A compar-
ison of the critical t-values at the 5 per cent significance level in these two
cases (1.98 for 64 df; 2.04 for 35 df) indicates that the need to correct for
unequal variances tends to make the t-test more stringent. The smaller the
samples, the greater the effect of the correction.
�2.433 � 1.539� s 2 1
n1 �
s22 n2
c The Spearman rank correlation coefficient (rs) was introduced in §3.3 as an alternative measure
of correlation for use with ordinal or ranked data. For large samples (probably above 30 and
To work this out we need to know SE(r), the standard error of a correla-
tion coefficient (i.e. the standard deviation of the sampling distribution of
all the correlation coefficients that would be derived from an infinite
number of random samples). The statisticians have calculated that for the
special case of ��0
SE(r)� (6.11)
where r is the correlation coefficient and n is the size of the sample, i.e. the
number of pairs of values of X and Y for which the strength of the associa-
tion is being tested.5
Substituting this formula for SE(r) in the previous equation gives the
test statistic in the following box.
THE TEST STATISTIC FOR A CORRELATION COEFFICIENT
When ��0, the test statistic
for a correlation coefficient, r, is
tcalc �
� � (6.12)
with n 2 degrees of freedom.
tcalc is then compared with the theoretical probability of getting such a
result as indicated by the t-distribution (which was described in §5.4.2),
and a decision made about whether or not the null hypothesis should be
rejected at the chosen significance level.c
For the correlation of CNTYMIG with FAMSIZE in 1881 across a
sample of 32 counties this would give
r�n 2 �1 r2
r
�1 r 2
n 2
r �
SE(r) �
r 0
SE(r)
�1 r 2
n 2
tcalc �
� � 3.406
From table 5.1 we see that for n 2�30 degrees of freedom the critical
value for a two-tailed significance level of 1 per cent (0.005 in one tail) is
2.750. tcalc is considerably greater than this and thus lies well within the crit-
ical region, so the null hypothesis can be rejected at the 1 per cent level.6
6.7.3 A t-test for a regression coefficient One of the most important results of regression analysis is the sign and size
of the regression coefficient, b, for the explanatory variable. A major aspect
of the evaluation of a model is thus the test of the reliability of such
coefficients. This vital procedure is encountered on numerous occasions in
the literature of the social sciences.
The null hypothesis in relation to the regression coefficients would be
that there is no relationship between X and Y: any change in the explana-
tory variable, X, has no effect on the dependent variable, Y. In the case of
simple regression with only one explanatory variable this is equivalent to
saying that the regression coefficient for the population, �, is zero; the
regression line is absolutely horizontal, it has no slope.
Once again we have to decide how good our sample regression
coefficent, b, is as an estimator of �. Under the null hypothesis of ��0, the
numerator of the test statistic reduces to b 0�b. The value of b is then
divided by SE(b), the standard error of the coefficient (i.e. the standard
deviation of the sampling distribution of regression coefficients obtained
from an infinite number of random samples), to give the t-ratio
tcalc� (6.13)
SE (b) was given previously in (5.14) in §5.6.3, and is repeated here
SE (b)� (6.14)� �(Yi Ŷi) 2
(n k)�(Xi X)2
b
SE(b)
2.892
0.849
0.528��32 2 �1 (0.528)(0.528)
� 0.528�5.477
�0.721
§. t-
certainly above 50), the statistical significance of rs can be gauged by the statistic, z�rs ,
using the procedure introduced in §2.5.2. For smaller samples, a special table must be used, such
as that provided by Sidney Siegel and N. John Castellan, Jr., Nonparametric Statistics for the
Behavioral Sciences, 2nd edn., McGraw-Hill, 1988, pp. 360–1.
�n 1
where k is the number of parameters. In a simple regression k would be 2
(the intercept and one explanatory variable). This formula for SE(b) is
then substituted in (6.13) to derive the test statistic, tcalc.
THE TEST STATISTIC FOR A REGRESSION COEFFICIENT
If the null hypothesis is that
there is no relationship
between the dependent variable
and a specified explanatory variable,
the test statistic for the regression coefficient, b,
on that explanatory variable is the t-ratio:
tcalc � (6.15)
tcalc can then be compared with the theoretical value of the t-distribution in
the usual way and a decision reached with regard to the null hypothesis.
Before working through an illustration of this test, there are a few addi-
tional points that should be mentioned in relation to t-tests for regression
coefficients. The first relates to a widely used rule of thumb. Reference to
the table of the t-distribution shows that for a two-tailed test at the 5 per
cent level of significance the value of t is very close to 2 for sample sizes of
20 or more (see column (4) of table 5.1). (This is the small-sample equiva-
lent of the statement in §2.6.1 that 1.96 standard deviations cover 95 per
cent of a normal distribution, leaving 5 per cent in the two tails.)
This leads to a very convenient rule of thumb for establishing the statis-
tical significance of a regression coefficient without undertaking more
formal calculations. If the t-ratio (i.e. the ratio of the regression coefficient
to its standard error) is equal to 2 or more, the relationship between that
explanatory variable and the dependent variable is statistically significant
at the 5 per cent level.
Secondly, it is also straightforward to test other null hypotheses regard-
ing the relationship between the dependent and explanatory variables; for
example, that � has some specified value, say 1.7 This simply requires the
insertion of that value in the numerator in place of the zero, so the test sta-
tistic becomes
b
� �(Yi Ŷi) 2
(n k)�(Xi X)2
b 0
SE(b) �
tcalc� (6.16)
Thirdly, irrespective of the value attributed to � under the null hypothe-
sis, the evaluation of tcalc against the theoretical distribution is convention-
ally done using a two-tailed test. This implies that no a priori view has been
specified regarding the sign of the regression coefficient.
However, it is very often the case that the researcher’s specification of the
model does embody a view of what the sign of the coefficient will be; in
other words that he does have a theoretical expectation about whether a
change in the explanatory variable in question will cause the dependent
variable to rise or to fall. If so, he is interested in the direction of the result
(see §6.3.3) and a one-tailed test would be more appropriate.
It is important to remember the distinction between statistical and his-
torical significance when evaluating these t-statistics. All the points that we
have made thus far relate to the issue of statistical significance. It may well
be that even though the t-statistic is large enough to establish that the
coefficient is significantly different from zero, the result is nevertheless of
no interest to the historian. This could be because the regression
coefficient, while clearly non-zero, is too small to have historical relevance;
or because the range of the coefficient (as indicated by the confidence inter-
val) is so wide that it provides no helpful guidance as to the likely effect of
the variable on the historical event; or because the correlation between the
two variables is spurious and only accidentally significant.
Finally, note that the procedure for testing hypotheses about the inter-
cept of the regression line, a, is essentially the same, though there is a slight
difference in the calculation of its standard error, SE(a) which we need not
reproduce here. It is automatically given in the results calculated by most
statistical computer programs.
Illustration of the t-test for a regression coefficient To illustrate the application of this very important test we will derive the
regression coefficient for the regression of unemployment (UNEMP) on
the replacement rate given by the ratio of unemployment benefits to wages
(BWRATIO) in the data set on inter-war unemployment in Britain. This is
a simplified version of the model developed by Benjamin and Kochin that
is discussed in chapter 15 because we have only one explanatory variable.
We would normally leave the calculations to the computer, but in order
to illustrate the procedure we will work them out ‘by hand’. The values
required for this are shown in table 6.1, omitting the intervening stages
b 1
� �(Yi Ŷi) 2
(n k)�(Xi X)2
§. t-
(Yi Y –
), (Xi X –
), and (Yi Ŷ). The first step is to calculate the value of the
regression coefficient, b, and the intercept, a, using the formulae given in
(4.2) and (4.3) in §4.2.3. This gives
b � � 15.64
and
a�13.70 15.64(0.467)�6.40
3.472
0.222
Table 6.1 Calculation of the regression coefficient and t-ratio for the regression of UNEMP on BWRATIO
(1) (2) (3) (4) (5) (6)
Product of Deviation Deviation
deviations of of X from of Y from
X and Y from its mean Predicted its predicted
UNEMP BWRATIO their means squared value of Y value squared
Yi Xi (Xi – X –)(Yi – Y
–) (Xi – X –)2 Ŷ (Yi –Ŷ)
2
1920 3.90 0.15 3.105 0.100 8.740 23.430
1921 17.00 0.24 0.749 0.051 10.149 46.933
1922 14.30 0.37 0.058 0.009 12.184 4.477
1923 11.70 0.40 0.134 0.004 12.654 0.910
1924 10.30 0.42 0.159 0.002 12.967 7.112
1925 11.30 0.48 0.032 0.000 13.906 6.791
1926 12.50 0.48 0.016 0.000 13.906 1.977
1927 9.70 0.48 0.053 0.000 13.906 17.690
1928 10.80 0.50 0.096 0.001 14.219 11.690
1929 10.40 0.50 0.109 0.001 14.219 14.585
1930 16.10 0.53 0.152 0.004 14.689 1.992
1931 21.30 0.54 0.556 0.005 14.845 41.665
1932 22.10 0.50 0.279 0.001 14.219 62.110
1933 19.90 0.51 0.268 0.002 14.376 30.520
1934 16.70 0.53 0.189 0.004 14.689 4.046
1935 15.50 0.55 0.150 0.007 15.002 0.248
1936 13.10 0.57 0.062 0.011 15.315 4.905
1937 10.80 0.56 0.270 0.009 15.158 18.994
1938 12.90 0.56 0.075 0.009 15.158 5.099
Sum 3.472 0.222 305.173
Mean 13.70 0.467
Source: For (1) and (2) see the data set for inter-war unemployment.
The regression equation is therefore
UNEMP�6.40�15.64 BWRATIO
tcalc can then be calculated using the formula in (6.15)
tcalc �
�1.74
As the final step in the procedure this value for tcalc is compared with the
theoretical t-distribution (summarized in table 5.1). In this case the
number of degrees freedom (df) is (n 2)�17. For 17 df and a two-tailed
test at the 5 per cent significance level the critical value is 2.110. Since tcalc is
less than this it falls well within the non-rejection region, and H0 cannot be
rejected.
If a statistical computer program is used in place of table 5.1 it is possible
to be more precise. For this regression it reports that the prob-value for the
explanatory variable is actually 0.101. This indicates that the probability of
getting a t-statistic of 1.74 when H0 is that there is no relationship between
UNEMP and BWRATIO is as much as 10.1 per cent. The null hypothesis
would thus fail to be rejected even if the significance level had been set at 10
per cent.
Notes 1 A classic illustration of the consequences if such an approach is not adopted was the
case of Thalidomide. This anti-nausea drug was introduced in the 1950s for the
benefit of pregnant women, but was later demonstrated to have caused substantial
numbers of severe birth defects among the children of those who took the drug. Note
that in the wake of such cases, the legal rules of liability certainly intensify the prefer-
ence for significance over efficiency among drug companies. 2 Since we have laid stress on the fact that the population standard deviation, , is typ-
ically unknown, it might legitimately occur to you to ask how we know the value of
the population mean, �. The answer would be that we very seldom do know what �
is, and the test statistics historians calculate are normally variants of (6.2a) that do
not involve hypotheses about the value of a single mean, and thus do not involve �.
See, for example, §6.7. 3 For a number of applications of a difference of means test by an historian see Ann
Kussmaul, Servants in Husbandry in Early Modern England, Cambridge University
Press, 1981, pp. 57–9 and 64–5. For example, she has two samples giving information
on the distances travelled by servants between successive hirings in the late eight-
eenth century. The larger sample, based on the Statute Sessions held at Spalding (in
15.64
� 305.173(19 2)0.222 �
15.64
�80.860 �
15.640
8.992
Lincolnshire), indicates a mean distance for male servants of 12.32 km. The figure
calculated from the smaller sample, based on settlement examinations (held to
determine entitlement to poor relief) in Suffolk, was 8.24 km. The two estimates
were found to differ significantly (p�0.0031). Kussmaul also uses the corresponding
procedure to test for differences of proportions. 4 The corrected number of degrees of freedom will be calculated automatically by the
computer, using the formula:
df� 2
rounded to the nearest whole number. 5 The fact that the hypothesis being tested is that ��0 must be emphasized. It is only
when this is true that it is possible to apply the formula for the standard error given in
(6.11). In all other circumstances it would be necessary to use the alternative formula
for SE(r) and transformation of r into Fisher’s z outlined in §5.6.2. 6 For extensive use of tests of the significance of correlation coefficients see Dorothy
Swaine Thomas, Social Aspects of the Business Cycle, Routledge, 1925. As mentioned
in §3.2.4, Thomas measured the correlation between the business cycle and a
number of social variables such as births, marriages, and crime. For each of the
numerous coefficients she also calculated a standard error corresponding to (6.11)
(though with n rather than n 2 in the denominator), and stated that she would take
as ‘a test of significance that r should be as much as three times the value of this
expression’ (p. 178). This would be more than sufficient to cover the 1 per cent
significance level, and is thus a very conservative standard (compare the rule of
thumb mentioned in §6.7.3 for a ratio of two as an approximation to the 5 per cent
level). 7 This null hypothesis would be appropriate, for example, in a study of market inte-
gration. If a standardized commodity such as wheat is sold in two neighbouring
markets, say Liverpool and Manchester, and these markets are linked by trade, then
one would expect any move in price in one market to be closely followed in the other
market. In these circumstances, a regression of the price in Liverpool on the price in
Manchester should show a regression coefficient of unity. For an application of this
approach to international markets (for which it is also necessary to take account of
tariffs and transport costs) see Donald N. McCloskey and J. Richard Zecher, ‘How
the gold standard worked, 1880–1913’, pp.192–4 in Donald N. McCloskey,
Enterprise and Trade in Victorian Britain, Allen & Unwin, 1981, pp. 184–208.
See also the analysis of the prices quoted for three British stocks on the London
and Amsterdam stock exchanges in the eighteenth century in Larry Neal, The Rise of
Financial Capitalism, Cambridge University Press, 1990, pp.120–32. The rationality
of the market prices is established by showing that when the share price is regressed
on the present value of the expected future dividends the coefficient ‘is statistically
significant in each case and is very close to the theoretical value of 1.0’ (p. 131).
� s21n1 � s22 n2�
2
� s21n1� 2
� 1n1 �1� � � s22 n2�
2
� 1n2 �1�
6.8 Exercises for chapter 6
1. Identify the null hypothesis and research hypothesis for the following
cases:
(i) The death rate from tuberculosis was the same in Britain and the
United States in the 1920s.
(ii) Death rates from tuberculosis fell in Britain between 1920 and 1950.
(iii) Death rates from tuberculosis in the United States in 1930 were
systematically related to the level of urbanization.
(iv) Tuberculosis is caused by consumption of raw, untreated milk.
(v) The American Civil War generated a significant increase in total iron
production to make armaments.
(vi) The American Civil War caused no change in the level of iron pro-
duction in the United States.
Which of the hypotheses are directional and which are not?
2. Write a brief essay assessing the relevance of Type II errors in historical
research. Under what circumstances are Type II errors likely to be impor-
tant? Are there situations in which the possibility of a Type II error can
safely be ignored?
3. Test the following hypotheses using Hatton and Williamson’s cross-
section data on Irish emigration by county:
(i) The correlation between Catholicism and family size was statistically
significant in 1881, but not in 1911.
(ii) The correlation between housing density and degree of urbanization
was statistically significant in 1881, but not statistically different
from 0.5.
(iii) The chances of finding a county with an average family size of less
than four children is less than 5 per cent; however, the chances of
finding a county with an average family size of more than six chil-
dren is greater than 5 per cent.
(iv) The proportion of the Irish population who could not read or write
in 1911 was the same as in 1881.
(v) The decline in the Irish emigration rate between 1881 and 1911 was
dramatic. Every county in 1911 fell outside the 99 per cent confi-
dence interval for the migration rate in 1881.
4. Calculate by hand the value of tcalc for the regression coefficient of distur-
bances on wheat prices in chapter 4, question 3. Compare this with the
appropriate test statistic, being sure to identify the correct numbers of
degrees of freedom. Does this suggest that the regression coefficient is
statistically significant?
§.
5. The example in the text of the difference of means test of relief payments
in Kent and Sussex does not take into consideration the fact that these data
were collected by sampling without replacement. Using the procedures introduced in chapter 5, recalculate the example incorporating the finite
population correction. Report your results. Does the correction weaken or
strengthen the conclusions reached in the text? Explain your conclusions.
Can any general rules be learned from this example?
6. Use the Boyer relief data set to test the hypothesis that workhouses in
Southern England were more likely to be built in parishes with more aggre-
gate wealth at their disposal (you will need to construct a new variable that
measures total wealth per parish; this can be achieved by multiplying
average wealth by parish population).
(i) Apply a difference of means test of this hypothesis. Interpret your
results in the light of the maintained hypothesis.
A critic of your analysis of the relationship between wealth and the work-
house notes that the distribution of total wealth by parishes is not even
approximately normally distributed. Indeed, the critic observes that the
distribution is log-normal (see §2.4.1).
(ii) Generate bar charts of total wealth and the log of total wealth by
parish to assess these criticisms. Then repeat (i) with the log of total
wealth. Does this cause you to amend your evaluation of the main-
tained hypothesis that wealthier parishes were more likely to build
workhouses?
(iii) Is there a more straightforward answer to your critic, that does not
involve recalculation of the data? (Hint : remember the Central Limit
Theorem.)
7. Generate the means and standard deviations of BRTHRATE for parishes
in Bedfordshire, Buckinghamshire, Essex, and Sussex from Boyer’s birth
data set.
(i) Use the data to test for equality in the average level of births per 100
families in Bedfordshire and Essex, assuming equal variances in the
two counties. Note the relative size of the standard deviations of the
birth rate for the two counties. What difference does it make to your
conclusions if you treat the population variances in the two counties
as unequal rather than equal?
(ii) Use the data to test for equality in the average level of births per 100
families in Buckinghamshire and Sussex. Once again, there is a
notable difference in the standard deviation of the birth rate
between the two counties. What difference does it make to your con-
clusions if you treat the population variances in the two counties as
unequal rather than equal?
What lessons can be drawn from these two examples about the importance
of correcting for unequal variance when carrying out difference of means
tests?
8. Let us assume that more responses to the Poor Law questionnaires have
been discovered in the depths of the Public Record Office. Reassuringly, all
the means and standard deviations of the variables remain the same, both
for the entire sample, and also for the individual counties.
(i) With this new evidence in hand, you decide to reassess the results of
question 7. The number of observations has been increased, such
that there are now 35 parishes reported for Bedfordshire and Essex;
37 parishes for Buckinghamshire, and 45 for Sussex.
(ii) What do you find when you repeat the exercise? How would you
explain any change in the results?
(iii) What do the results of questions 7 and 8 tell us about the robustness
of the t-test for difference of means between samples?
9. In Steckel’s data set on the patterns of leaving home in the United States,
1850–60, there are two variables. YNGERFAM and OLDERFAM, that
record the number of younger and older children in each household.
(i) Calculate the mean and standard deviation of each variable.
(ii) You wish to record the total number of children in the household.
Create a new variable, TOTALFAM. Calculate the mean and stan-
dard deviation of the new variable.
(iii) Compare the mean and standard deviation of TOTALFAM with the
means and standard deviations of YNGERFAM and OLDERFAM.
What do these comparisons tell you about the additivity of these
measures of central tendency?
10. In a revision of Benjamin and Kochin’s analysis of inter-war unemploy-
ment in Britain, a researcher addresses the question of the separate effects
of benefits and wages on unemployment. He wishes to test whether the
impact on unemployment of a change in the benefit level (BENEFITS),
holding the wage rate constant, was the same as a change in the average
wage (WAGES), holding the benefit rate constant. He also wishes to test
whether the coefficient on the benefit rate is significantly different from
§.
unity (i.e. b1�1), and whether the coefficient on DEMAND is significantly
different in this specification from the one reported in (14.2).
The result of the revised regression is (t-statistics in parentheses):
UNEMP� 3.127�0.461 BENEFITS�0.081 WAGES 96.492 DEMAND
(0.29) (3.41) (0.60) (9.10)
What are the answers to the researcher’s hypotheses?
Non-parametric tests
7.1 Introduction
Historians cannot always work with problems for which all the relevant
data are based on quantitative measurements. Very often the only informa-
tion available for analysis relates to the number of cases falling into
different categories; the category itself cannot be quantified.a Thus house-
hold heads might be classified according to their sex, political affiliation, or
ethnicity. Wars might be grouped into epic, major, and minor conflicts.
Women might be subdivided by their religion, the forms of birth control
they practised, or the socio-economic status of their fathers.
Alternatively the historian might have data that can be ranked in order,
but the precise distance between the ranks either cannot be measured or is
unhelpful for the problem under consideration. One example might be a
ranking of all the universities in the country by a newspaper combining on
some arbitrary basis a medley of criteria such as the quality of students
admitted, library expenditure per student, and total grants received for
research. Another might be a ranking of the power of politicians in an
assembly on the basis of some measure of their influence on voting in the
assembly.1 Similarly, an historian of religion might construct a ranking of
the intensity of religious belief of the members of a community according
to the frequency of their church attendance in a given period.
In all such cases it is not possible to apply the techniques of hypothesis
testing described in chapter 6. There are two fundamental reasons why par- ametric tests such as Z and t cannot be used.b First, these statistical tests
a In the terminology of §1.3.3 the information is at the nominal level of measurement, as opposed
to the interval or ratio level of most economic data. b Statistical tests such as Z and t (or the F-test that will be introduced in §9.3) are known as para-
metric tests because they test hypotheses about specific parameters (characteristics of the popu-
lation) such as the mean or the regression coefficient.
cannot be made with nominal or ordinal levels of data, they require vari-
ables that are measured on an interval scale. With nominal level data it is
simply not possible to perform any of the basic operations of arithmetic
such as would be needed to calculate a mean. With some forms of ordinal
level data, for example, the ranking of politicians on the basis suggested
above, it might be possible to perform such operations, but the interpreta-
tion of the results would be problematic. The successive differences
between any two ‘scores’ are not true intervals with substantive meaning,
and consequently the interpretation of test results, such as the position of
the test statistic in relation to the critical region, may be misleading.
Secondly, the parametric tests assume that the sample observations
satisfy certain conditions, in particular, that they are drawn from popula-
tions that are approximately normally distributed (see §6.2.1).
To take the place of the parametric tests in situations where they are not
appropriate, a large battery of non-parametric tests has been developed. These can be applied to nominal or ordinal level data, and require the
minimum of assumptions about the nature of the underlying distribution.
It is thus appropriate to use a non-parametric test when there is reason to
think that the relevant population distribution is not normal (or its exact
form cannot be specified). This is particularly likely to be the case when
dealing with small samples.
The fundamental principles of hypothesis testing outlined in chapter 6
apply equally to the various non-parametric tests. It is again necessary:
� to specify a null hypothesis
� to select a significance level and critical region
� to calculate a test statistic
� to compare this with the theoretical probability given by the sampling
distribution for that statistic.
There are a number of non-parametric tests that are quite widely used,
and are available with several statistical computer programs. We will begin
in §7.2 by discussing three such tests that can be used in relation to com-
parisons between data from two small samples using ranked (ordinal level)
data. Inevitably, there is a price to be paid for the advantages of these non-
parametric tests, and this is considered briefly in §7.2.4.
In the following section, §7.3, we introduce another non-parametric
test of statistical significance known as �2 (lower case Greek chi, pro-
nounced ki-squared). This is a very widely used test for dealing with data
classified into two or more categories. §7.4 describes a very useful non-
parametric test than can be applied to establish whether or not events in a
sample occur in a random sequence. Two further tests that can be used to
determine the goodness of fit of an actual distribution in comparison with
some specified alternative are briefly noted in §7.5.
As with numerical (interval level) measures, the preceding tests relating
to statistical significance are only part of the story. If an association is found
to be statistically significant it is also necessary to establish the strength of
that association, and §7.6 discusses some of the non-parametric measures
which are available for this purpose.
All the non-parametric tests discussed in §7.2–§7.5 establish critical
values that determine whether or not the null hypothesis should be
rejected (see §6.3.2). For small samples these are usually based on exact
sampling distributions for the relevant test statistic, and there are pub-
lished tables of the critical values for a range of combinations of sample size
and desired significance level.c For larger samples the test may rely on an
approximation to a particular sampling distribution that becomes more
precise as the size of the samples increases.
The tables of critical values have to a large extent been superseded in
recent years by computer programs, with the benefits noted in relation to
parametric tests in §6.3.5. The programs are able to cover all sample sizes,
and in addition to calculating the required test statistic will also report the
corresponding prob-value. This will be either the exact prob-value or its
large-sample approximation. However, reference to the published tables
helps one to get a better feel for the way in which critical values change with
sample size or number of degrees of freedom, and we will generally discuss
both methods of reaching a decision.
The topics in this chapter build on ideas developed in previous chapters,
but are not themselves required for the topics covered in the later part of
this book. They could be skipped by students whose main interest is in the
subject of regression taken up in the subsequent chapters.
7.2 Non-parametric tests for two independent samples
We will look at each of these first three tests in the context of two indepen- dent samples.d The definition of independence has two components. As between the samples, it means that the units selected for the first sample are
completely independent of those selected for the second sample. (The
§. -
c Tables are given in D. V. Lindley and W. F. Scott, New Cambridge Statistical Tables, 2nd edn.,
Cambridge University Press, 1995, for all the tests referred to in this chapter. d In addition to these tests for two independent samples there are other non-parametric tests for
two related samples (i.e. those that are not independent), or for three or more samples, usually
referred to collectively as k samples. There are also tests for a single sample, and three examples of
these tests will be discussed in §7.4 and §7.5.
alternative would be two related samples, as in a medical trial when every person in one sample has a family relation in the other sample, or the same
person is used in ‘before and after’ samples. Such related samples hardly
ever occur in the data available to historians.) The second aspect of inde-
pendence applies within each sample, and requires that each unit is inde-
pendent of every other unit within that sample.
The essential principle behind all three of these tests for two indepen-
dent samples is quite straightforward. Each test consists of an intuitively
obvious way of comparing the results from the two samples in order to
decide whether or not they come from underlying populations with exactly
the same distribution, and hence can be treated as coming from the same
population. In some cases only the general nature of the distribution can
be considered; in others it is possible to compare more specific features
such as the mean or variance.
The null hypothesis in each case is that there is no difference between
the two samples: the two distributions are identical and come from the
same population. The research hypothesis (H1) might be simply that the
two distributions are not identical, leading to a two-tailed test. Or H1 might
specify something about the direction of the difference leading to a one-
tailed test in the specified direction. For example, this difference might
relate to the mean or some other measure of central tendency, in which case
H1 would be that the values of one sample are larger (smaller) than the
values of the other.
In order to highlight the principal features of the three tests we have
created four variants of an artificially simple data set, and will use this to
illustrate the comparison of two samples with each of the tests. The data set
relates to two independent samples of newly enrolled students, 10 from
Harvard and 10 from Yale. Each student was ranked by an impartial panel
from 1 (highest) to 20 (lowest) on the basis of subjective assessments of
their intellectual ability and application. The four variants (I, II, III, and
IV) of the rankings are given in table 7.1. The advantage of this imaginary
example is that it enables the tests to be illustrated in relation to a wider
range of possible distributions than would normally be encountered in
practice.
7.2.1 The Wald–Wolfowitz runs test This is the simplest of the three tests. The two samples are combined and
the values from each sample are ranked in either ascending or descending
order. Are Harvard and Yale admitting their students from the same under-
lying population, or is there a statistically significant difference in the aca-
demic quality of their intake? The procedure for answering this question by
means of a runs test would be to list the 20 observations in rank order. If the
symbols H and Y are used to designate the respective samples, the list for
variant III of table 7.1 – ranked in descending order – would look like this:
HHHH YYYY HHH Y HH YYYYY H
Each sequence of the same letter constitutes a run. In the above illustration
there are a total of seven runs: 4 in H and 3 in Y.
The basic idea behind the test is that if the underlying distributions are
identical, the letters representing the two samples will be randomly scat-
tered all through the list, creating large numbers of short runs. On the other
hand, if the underlying distributions are different, the ranks from one of the
samples are likely to predominate in some parts of the list, and those from
the other sample in other parts, giving rise to a small number of long runs.
The test cannot say anything about how the distributions might differ
§. -
Table 7.1 Rankings of two independent samples of students (imaginary data, four variants)
University I II III IV
Harvard 1 1 1 1
Harvard 2 2 2 3
Harvard 3 3 3 5
Harvard 4 4 4 7
Harvard 5 5 9 9
Harvard 6 11 10 11
Harvard 7 12 11 13
Harvard 8 13 13 15
Harvard 9 14 14 17
Harvard 10 20 20 19
Yale 11 6 5 2
Yale 12 7 6 4
Yale 13 8 7 6
Yale 14 9 8 8
Yale 15 10 12 10
Yale 16 15 15 12
Yale 17 16 16 14
Yale 18 17 17 16
Yale 19 18 18 18
Yale 20 19 19 20
– it might be, for example, with respect to either the location (mean or
median) or the shape (dispersion or skewness) of the distribution, or to
both. It is thus mainly useful as a quick and easy test when the focus of
interest is not on the specific features of the distribution but only on detect-
ing a general difference.
The null hypothesis is that there is no difference between the two
samples, i.e. they come from the same population. Even if no direction is
predicted, the runs test is effectively a one-tailed test because H0 can be
rejected only if there are a small number of long runs.e However a direction
may be predicted in advance; for example, the research hypothesis might
be that one sample would have a higher ranking than the other. This would
be a possible reason for selecting a lower significance level, such as 2.5 per
cent, thus forming a smaller critical region and so making it harder to reject
the null hypothesis.
The test statistic, r, is the number of runs and can be counted very easily.
Given the choice of significance level and the test statistic, the decision can
be made on whether or not to reject the null hypothesis. These final steps in
the test can be performed on the computer by your statistical software,
though for small samples it may be as quick to carry out the work by hand.
There is no need to memorize the formulae the computer will apply, but
they are discussed below so that those who wish to increase their under-
standing can see how the results are obtained. It is also useful to show expli-
citly that this non-parametric procedure belongs to the same family of test
procedures as those already discussed in chapter 6.
If the samples are small (neither more than 20), the exact sampling dis-
tribution for the number of runs, r, can be calculated for different combi-
nations of sample size, and is given in a published table.f
If the sample sizes are larger than this, the Central Limit Theorem (see
§5.3.1) applies and the sampling distribution of r is approximately normal.g
The standardized value of Z is calculated from the mean, �r, and standard
deviation, r, of the sampling distribution of the number of runs, r.
e This is an exception, therefore, to the general principle stated in §6.3.3 that a one-tailed test
should be used only when direction is predicted. f Lindley and Scott, Statistical Tables, table 18, pp. 60–2. The table is in two parts, the upper and
lower percentage points (corresponding to the left- and right-hand tails of the distribution), but
for the present test it is only the latter we are interested in. (The upper percentage points are
required when the runs test is used below for the one-sample test of randomness, for which both
tails may be relevant; see §7.4.) g Recall from §5.3.1 that it is possible for the distribution of a sampling distribution to be approxi-
mately normal even when the underlying population from which the samples are drawn is not
assumed to be normal.
THE WALD–WOLFOWITZ RUNS TEST
For large samples (either n1 or n2 greater than 20)
the sampling distribution of the test statistic, r,
is approximately normal,
with mean
�r � , (7.1)
and standard error, SE(r)
� r � (7.2)
The standardized value Zcalc can thus be derived in the usual way by cal-
culating the difference in units of standard errors between the actual
number of runs given by the particular sample, and the theoretical mean
given by the sampling distribution.h
THE STANDARDIZED TEST STATISTIC FOR THE
WALD–WOLFOWITZ RUNS TEST
This is
Zcalc � (7.3)
To make the test decision, the resulting Zcalc is compared with the theo-
retical sampling distribution. For small samples this is given by the special
table; for large samples the table for the standard normal distribution (par-
tially reproduced in table 2.7) is used.i
An illustration of the runs test To illustrate the nature of this test, the runs given by each of the four vari-
ants of the imaginary data are shown in the upper panel of table 7.2.
r �r r
� 2n1n2(2n1n2 n1 n2)(n1 � n2)2(n1 � n2 1)
2n1n2 n1 � n2
� 1
§. -
h Compare the initial definitions of Zcalc and tcalc in §6.4.2. i For the complete table of the normal distribution see Lindley and Scott, Statistical Tables, table 4,
pp. 34–5.
Table 7.2 The Wald–Wolfowitz runs test
1 Data: Four alternative series of runs for two independent samples
(students from Harvard (H) and Yale (Y), imaginary ranked data from table 7.1)
I II III IV
(2 runs) (5 runs) (7 runs) (20 runs)
H H H H
H H H Y
H H H H
H H H Y
H H Y H
H Y Y Y
H Y Y H
H Y Y Y
H Y H H
H Y H Y
Y H H H
Y H Y Y
Y H H H
Y H H Y
Y Y Y H
Y Y Y Y
Y Y Y H
Y Y Y Y
Y Y Y H
Y H H Y
2 Test statistics, significance values, and results
(1) (2) (3) (4) (5)
Test Standardized
statistic value Significance
Series r Z (1-tailed) * Decision
I 2 3.905 0.0000 (0.0%) Reject H0 II 5 2.527 0.0045 (0.45%) Reject H0 III 7 1.608 0.0513 (5.13%) Cannot reject H0 IV 20 4.365 1.0000 (100.0%) Cannot reject H0
Note:
* Probability of obtaining this value of r if the null hypothesis is correct. Both sample sizes are
less than 20, so the exact distribution of the test statistic can be used rather than the normal
approximation.
Variants I and IV illustrate two extreme possibilities: in I every observation
for Harvard is above any observation for Yale and there are thus only 2 long
runs; in IV the observations alternate perfectly creating 20 short runs.
Columns II and III cover intermediate possibilities with 5 and 7 runs,
respectively.
We look first at the procedure when this decision is made on the basis of
published tables and then at the results obtained from the computer
program. For the present test we will adopt the 5 per cent level with no
direction predicted. If we enter the table for the lower percentage points of
the distribution we find that for two samples, both of size 10, the critical
value for the 5 per cent level is 6. Any value of the test statistic, r, less than or
equal to 6 will fall in the critical region, and the null hypothesis should be
rejected. For any value of r greater than 6 the null hypothesis cannot be
rejected.
As should be intuitively obvious, the null hypothesis that there is no
difference between the samples is easily rejected in the case of I (2 runs) and
is not rejected in the case of IV (20 runs). The two remaining cases are less
obvious: the result of the test is that H0 is rejected for II (with 5 runs), but
cannot be rejected for III (with 7 runs).
The exercise is now repeated using a computer program and the stan-
dardized values of the test statistic, Zcalc. The test statistic, r, is reported for
each variant in column (2) of the lower panel of table 7.2, and the value of
Zcalc obtained by the computer in column (3). 2 The corresponding
significance level for a one-tailed test and the final decision are reported in
columns (4) and (5). Because we are working with small samples the
program is able to calculate the exact significance based on the sampling
distribution corresponding to the critical values in the published tables.
With larger samples it would report the approximate significance based on
the normal distribution.
The significance levels for I (2 runs) and IV (20 runs) are, respectively,
0.0000 (zero probability of getting this result if H0 is true) and 1.000 (100
per cent probability of getting this result if H0 is true), so H0 can easily be
rejected for the former but cannot be rejected for the latter. For II (5 runs)
the significance level is 0.45 per cent, indicating as before that H0 must be
rejected. By contrast, for III (7 runs) the significance level of 5.13 per cent
falls just outside the conventional 5 per cent critical region, and H0 cannot
be rejected at this level.
7.2.2 The Mann–Whitney U-test This test is similarly based on the ranking of the observations in the two
samples. It is a popular test of whether or not there are differences between
§. -
the sample means (or of some other measure of location), but can also be
used as a test for differences in dispersion or skewness. The basic idea is that
if the underlying distributions are identical, the ranking of the observa-
tions from the two samples will be broadly similar. If they are not identical,
the ranks of the one sample will be predominantly higher (or lower) than
those of the other. This difference might occur because, for example, the
mean of one sample was above the mean of the other, or because one
sample was more spread out than the other.
The null hypothesis is thus that the two distributions are identical. The
research hypothesis might either be simply that they are different (a two-
tailed test); or, more specifically, that the rank values from one of the distri-
butions are predominantly higher (lower) than those from the other (a
one-tailed test in the specified direction).
The hypothetical ranking of the 20 students from Harvard and Yale is
again used to illustrate the procedure, which is as follows:
(a) Combine both samples, and rank the students in ascending (or
descending) order from 1 to 20.j
(b) Take the lowest rank in one of the samples (if the samples differ in size
it is usual to take the smaller but this does not affect the result – we will
take the Harvard sample), and count the number of students with
higher ranks than this in the second sample (Yale). Repeat this for each
successive rank in the Harvard sample.
(c) Add the results of this counting exercise for all 10 Harvard students.
This sum is the test statistic, U.
(d) Compare U with the appropriate theoretical sampling distribution to
test the probability that the observed difference in ranks could have
come from the same distribution.
The Wilcoxon rank sum test is a variant of the Mann–Whitney test that simply counts the sum of the ranks of the smaller (or first) sample. This
sum, W, is the test statistic, and there is a fixed relationship between W and
U.3 Furthermore, the exact significance levels of W and U are the same, so
the two tests always give the same result.
The relevant calculations for the subsequent stages of this test are again
given in order to clarify what the computer does if asked to perform the
test. If the samples are small (neither more than 20) it is possible to use
j If the underlying scores result in a tie, the rank allocated is the mean of the ranks that would have
otherwise been allocated to the tied scores. If the tie is within one of the samples the calculations
are otherwise unaffected; if it is a tie between samples a correction to the formula for Zcalc is
required.
a special table that gives the exact sampling distribution for the
Mann–Whitney test statistic, U.k
With larger sample sizes the sampling distribution of U will be approxi-
mately normal. In this case the mean and standard deviation of this sam-
pling distribution can be calculated as in the following box:
THE MANN–WHITNEY U-TEST
For large samples (either n1 or n2 greater than 20)
the sampling distribution of the test statistic, U,
is approximately normal,
with mean
�u � , (7.4)
and standard error, SE(U)
� u � (7.5)
A standardized value for the test statistic Zcalc in units of standard errors
can be derived as before.
THE STANDARDIZED TEST STATISTIC FOR THE
MANN–WHITNEY U-TEST
This is
Zcalc � (7.6)
To make the test decision, the resulting Zcalc is compared with the theo-
retical sampling distribution. For small samples this is given by the special
table; for large samples the table for the standard normal distribution (par-
tially reproduced in table 2.7) is used.l
U �u u
�n1n2(n1 � n2 � 1)12
n1n2 2
§. -
k Lindley and Scott, Statistical Tables, table 21, pp. 66–7. The table gives the lower percentage point
of the distribution. l For the complete table of the normal distribution see Lindley and Scott, Statistical Tables, table 4,
pp. 34–5.
An illustration of the Mann–Whitney U-test The calculations are illustrated in the upper panel of table 7.3 using the
hypothetical data for series IV from table 7.1. The significance level is taken
as 5 per cent in a two-tailed test with no direction predicted. If we start with
the Harvard sample we get U�45 and W�110. If we had started instead
with the Yale sample the respective results would be 55 and 100.4
We start with the procedure for making the decision on the basis of the
published table for small samples. Because of the form in which the tables
are constructed it is conventional to take the lower of the two possible
values for the test; in this case, 45 for the Mann–Whitney test (or 100 for the
Wilcoxon rank sum test).5
If we consult the table for two samples each of 10, we see that the critical
value for the 5 per cent level is 27. This is the largest value that falls in the
critical region. For any value of the test statistic, U less than or equal to 27, the
null hypothesis should be rejected. For any value of U greater than 27 the null
hypothesis cannot be rejected. Thus for variant IV, U is 45; this is markedly
greater than 27 and so H0 cannot be rejected.
The values of U for all four variants are given in column (2) of the lower
panel of table 7.3. For the two extreme variants, I and IV, we find as
expected that H0 is easily rejected for I (U�0), but cannot be rejected for
IV (U�45). For the intermediate variants H0 cannot be rejected either for
III (U�32) or for II (U�30) since these test statistics are both greater than
27.
It will be seen that the difference between U and the critical value is quite
small in the case of II, and that it leads to a different result from the one
obtained in §7.2.1 with the runs test, according to which H0 should be
rejected for II. The issue of different test results is discussed briefly in §7.2.4
after the introduction of one further non-parametric test.
With larger samples we would not be able to use the tables and would
instead rely on the computer. To see how this procedure works we repeat
the exercise using a statistical program (SPSS). The standardized value
of the test statistic, Zcalc, calculated by the program is shown in column (4)
of the lower panel of table 7.3. A quick check shows that it is equal to the
results given by (7.6) based on the formulae for �u and u in (7.4) and
(7.5).6
The corresponding two-tailed significance levels are reported in column
(5) of the lower panel of table 7.3.7 The significance levels are 0.00 (zero
probability of getting this result if H0 is true) for I and 0.739 (73.9 per cent
probability of getting this result if H0 is true) for IV. The significance levels
for II and III are lower but still well outside the critical region at 0.143 and
0.190, respectively. We thus reach the same decisions as we made when
§. -
Table 7.3 The Mann–Whitney U-test and the Wilcoxon rank sums test
1 Calculation of test statistics for two independent samples:
(imaginary data, series IV, table 7.1)
(1) (2) (3) (4)
Rank of Number of Rank of Number of
Harvard Yale students Yale Harvard students
students with higher rank students with higher rank
W� 19 U� 9 W�� 20 U��10
W� 17 U� 8 W�� 18 U�� 9
W� 15 U� 7 W�� 16 U�� 8
W� 13 U� 6 W�� 14 U�� 7
W� 11 U� 5 W�� 12 U�� 6
W� 9 U� 4 W� �10 U�� 5
W� 7 U� 3 W�� 8 U�� 4
W� 5 U� 2 W�� 6 U�� 3
W� 3 U� 1 W�� 4 U�� 2
W� 1 U� 0 W�� 2 U�� 1
W�100 U�45 W��110 U��55
2 Test statistics, significance values, and results for all four series in table 7.1
(1) (2) (3) (4) (5) (6)
Test statistics
Mann– Standardized Exact
Whitney Wilcoxon value significance
Series U W Z (2-tailed)* Decision
I 0 55 3.780 0.000 (0.0%) Reject H0 II 30 85 1.512 0.143 (14.3%) Cannot reject H0 III 32 87 1.361 0.190 (19.0%) Cannot reject H0 IV 45 100 0.378 0.739 (73.9%) Cannot reject H0
Note:
* Probability of obtaining this value of U (or W) if the null hypothesis is correct. Both sample
sizes are less than 20, so the exact distribution of the test statistic can be used rather than the
normal approximation.
using the tables, but with the precise prob-values we have more
information.8
7.2.3 The Kolmogorov–Smirnov test The key idea underlying this third test is that if the two independent
samples have been drawn from the same population, their cumulative fre- quency distributions will be essentially similar because they should both show only random deviations from the common population distribu-
tion.m The test is sensitive to any kind of difference there might be between
the two distributions, for example differences in means or other measures
of central tendency, in dispersion, or in skewness.
The null hypothesis is thus that there is no difference between the two
samples. It is designed for use with continuous data and is more conserva-
tive (less likely to reject the null hypothesis) with discrete data. Therefore, if
H0 is rejected in an application of the Kolmogorov–Smirnov test with dis-
crete data the researcher can have more confidence in the decision.
The test statistic is based on D, the maximum difference between the two
cumulative frequency distributions, expressed as a proportion. If the
research hypothesis is simply that there is a difference between the two
samples, D is calculated without regard to sign for a two-tailed test. If H1 predicts the direction of the difference, D is the maximum positive or
maximum negative difference for a one-tailed test. The actual test statistic
is not D, but n1�n2�D, where n1 and n2 are the sample sizes.
The particular sampling distribution used for the Kolmogorov–
Smirnov test depends on whether the sample sizes are small or large, and
whether they are of equal size. Published tables are available for selected
sizes of small samples, indicating critical values for different levels of
significance.n The table gives the upper percentage point, (i.e. the right-
hand tail) and so differs in this respect from those referred to in the two
previous sections. The decision rule thus changes and for this test is that the
m A cumulative absolute frequency distribution measures the number of observations below – or
equal to – any selected value in the distribution; see §2.1.1 and §2.1.3. With ordinal data this
would refer to the number in or below each ordered category (e.g. a classification of workers by
social class). The corresponding cumulative relative frequency expresses this as a proportion
(0.1, 0.2, etc.). The cumulative relative frequency could also be expressed as a percentage as in
table 2.2 and table 2.3, or in fractions, but for the purpose of the test statistic it must be in the form of
a proportion. n Lindley and Scott, Statistical Tables, table 19, pp. 62–4 gives critical values for samples of n1�n2
up to n2�20, and for n1�n2 from 20 to 100. The figures given for each significance level are for
two-tailed tests and the corresponding value for a one-tailed test is obtained by moving to half
the significance level. For example, for samples of n1�n2�10, the critical value for a two-tailed
test at 10 per cent is 60, and this would be the critical value for a one-tailed test at 5 per cent.
null hypothesis should be rejected if the test statistic, n1n2D, is equal to or
greater than these values. Computer packages provide more exact prob-
values and also cover large samples.
An illustration of the Kolmogorov–Smirnov test The procedure for calculation of D is illustrated in the upper panel of table
7.4 using variant III of the imaginary data for the two groups of students. In
this instance the available information is maximized by treating each rank
(from 1 to 20) as a separate class interval, but the procedure can also be
applied when the data are grouped (see §2.1.1). Because there are 10 items
in each sample, the proportion of the total distribution represented by each
entry is 0.1, and because there is only one entry in each class interval, each
successive entry adds 0.1 to the cumulative relative frequency distributions
shown in columns (3) and (4). The final row for each sample must be 1.
The difference between the two cumulative relative frequency distribu-
tions, D, is then calculated in column (5). If sign is ignored, the largest
difference is 0.4, indicated by arrows in rows 6 and 16 (the fact that it occurs
twice is irrelevant); and this is also the largest negative difference. The
largest positive difference is 0.1 in the first row. The test statistic n1n2D is
then derived from this; in the present case this simply involves multiplying
by 100, since n1�n2�10�10.
The values of D for all four series, I to IV, are given in column (2) of the
lower panel of table 7.4, and the corresponding test statistics, n1n2D, in
column (3). We will deal first with the procedure for making a decision on
the basis of the exact sampling distribution in the published tables. The
significance level is taken as 5 per cent with no direction predicted.
The published table gives 70 as the critical value for samples of this size
at the 5 per cent level. It is easily seen that the results for all four cases are the
same as with the Mann–Whitney test. For the two extreme cases, the null
hypothesis is rejected for I (n1n2D�100) but cannot be rejected for D
(n1n2D�10). It also cannot be rejected for either of the intermediate cases,
II (n1n2D�50) or III (n1n2D�40).
It is obvious that the last three results are falling progressively further
from the critical region and thus that the probability of getting these results
if H0 is true is getting increasingly large. If we want to know the exact prob-
values the exercise can be repeated using a computer. SPSS calculates a sta-
tistic it calls Kolmogorov–Smirnov Z (hereafter KS Z), for which the
formula is
KS Z�D (7.7)� n1n2n1 � n2
§. -
Table 7.4 The Kolmogorov–Smirnov test
1 Calculation of test statistic for two independent samples
(imaginary data, series III, table 7.1)
(1) (2) (3) (4) (5)
Cumulative D�
Ranking relative frequency Difference
Harvard Yale Harvard Yale Harvard–Yale
20 0.1 0.0 0.1 �
19 0.1 0.1 0.0
18 0.1 0.2 0.1
17 0.1 0.3 0.2
16 0.1 0.4 0.3
15 0.1 0.5 0.4 �
14 0.2 0.5 0.3
13 0.3 0.5 0.2
12 0.3 0.6 0.3
11 0.4 0.6 0.2
10 0.5 0.6 0.1
9 0.6 0.6 0.0
8 0.6 0.7 0.1
7 0.6 0.8 0.2
6 0.6 0.9 0.3
5 0.6 1.0 0.4 �
4 0.7 1.0 0.3
3 0.8 1.0 0.2
2 0.9 1.0 0.1
1 1.0 1.0 0.0
2 Test statistics, significance values, and results for all four series in table 7.1
(1) (2) (3) (4) (5) (6)
Test statistic Kolmogorov– Significance
Series D n1n2D Smirnov Z (2-tailed)* Decision
I 1.0 100 2.236 0.000 (0.0%) Reject H0 II 0.5 50 1.118 0.164 (16.4%) Cannot reject H0 III 0.4 40 0.894 0.400 (40.0%) Cannot reject H0 IV 0.1 10 0.224 1.000 (100.0%) Cannot reject H0
Note:
* Probability of obtaining this value of D if the null hypothesis is correct. The significance
levels quoted are an approximation.
Thus for case III, where D�0.4,
KS Z�0.4 � � 0.4 � 2.2361 � 0.8944
The values of KS Z for all four cases are reported in column (4) of the lower
panel of table 7.4.9 These are then used to calculate the approximate
significance levels in column (5).
The prob-value for I is 0.00 (zero probability of getting this result if H0 is
true) and so H0 must clearly be rejected. The prob-values then rise from
0.164 for II to 0.40 for III and finally – for the extreme case of IV – to 1.00
(100 per cent probability of getting this result if H0 is true). The decisions
listed in column (6) are thus identical to those made on the basis of the
tables.10
7.2.4 Criteria for choosing between alternative tests We have seen in the preceding sections that different non-parametric tests
can be applied to the same data and lead to different decisions on whether
or not the null hypothesis should be rejected. There are also circumstances
where it is possible to use both parametric and non-parametric procedures
to test a null hypothesis, and this again can result in different decisions.
How should the choice between different tests be made?
The choice between alternative non-parametric tests can be made on
the basis of a number of different criteria. One set of considerations
includes the power of the tests (as defined in §6.3.2), the level of measure-
ment, and the size of the sample(s). A second is the precise hypothesis that
is to be tested. For example, some tests are more sensitive to differences in
central tendency, others to differences in dispersion, others to any
difference between the underlying populations (in central tendency or dis-
persion or skewness). For more detailed discussion of the relative merits
of the different tests a specialist text on non-parametric tests should be
consulted.o
With regard to the choice between parametric and non-parametric
tests, two possible reasons were given at the outset of this chapter for
choosing the latter. They involve fewer assumptions about the population
distributions, and they do not require interval or ratio measurements.
Against this, in cases where the underlying data are normally distributed
and the measurements are at interval level, non-parametric tests are less
powerful. This means that for a given sample size and a given level of
significance, there is more risk of a Type II error (not rejecting the null
�10 �1010 � 10
§. -
o We particularly recommend Sidney Siegel and N. John Castellan Jr., Nonparametric Statistics for
the Behavioral Sciences, 2nd edn., McGraw-Hill, New York, 1988.
hypothesis when it is false). Thus, non-parametric tests should be reserved
either for non-normal populations, or when very few data observations are
available, or where information is presented in nominal or ordinal form.
Comparison of a t-test and a Mann–Whitney U-test In chapter 6 (§6.7) a parametric difference of means t-test was used to
compare the mean value of RELIEF in the two counties of Kent and Sussex.
The conclusion drawn was that the null hypothesis of no difference
between the two sample means could be rejected: the difference was statis-
tically highly significant with a prob-value of only 0.6 per cent. If the same
comparison test is made using the non-parametric Mann–Whitney test,
the same decision is reached, with a statistically highly significant prob-
value of 0.7 per cent.
However, this will not always be the case. If these two tests are run on the
mean values in the same two counties of a different variable, INCOME, the
tests yield different decisions. In the t-test, corrected for unequal variances,
the prob-value is 15.6 per cent, and the null hypothesis of no difference
between the two sample means could not be rejected in a two-tailed test at
the 5 per cent level. In the Mann–Whitney test, by contrast, the prob-value
is 4.5 per cent and H0 would be rejected at the 5 per cent level. For this vari-
able the non-parametric test is to be preferred in this instance because it
cannot be established that the underlying populations from which the (rel-
atively small) samples are drawn are normally distributed. It is not pos-
sible, therefore, to be sure that the conditions required for the parametric
t-test are satisfied.p
7.3 The �2-test
In chapter 3 we considered the possibility of an association between the
numerical values of two variables, and described a measure of the strength
of that association, the correlation coefficient, r. We saw, for example, that
the 24 Kent parishes with a relatively high (or low) level of UNEMP also
had a relatively high (or low) level of RELIEF, with r�0.52 (§3.2.1). We
also reviewed an alternative measure, Spearman’s rs, that could be used to
assess the degree of correlation between ranked variables (§3.3).
We may, however, be interested in some features of a variable that are
p As a general principle, tests that throw away information, in this case by eliminating information
on the scale of differences between values of INCOME, are less powerful. However, this does not
always hold, as this feature may be over-ridden by other considerations. In difference of means
tests, applied to data sets that have both very different variances and very different sample sizes,
the t-test tends to be less powerful than non-parametric alternatives, such as Mann–Whitney.
not reflected in a numerical value or a rank order, but instead involve its
classification into two or more categories. For example, a study of voting
patterns in the United States in the presidential election of 1896 might clas-
sify voting both by degree of urbanization (large cities, other urban areas,
rural areas) and by political choice (Democrat, Republican, Populist). It
would then be possible to see whether there were any significant differences
between voters related to their location. Or a historical demographer
might classify women in a given locality on the basis of their fathers’ occu-
pation (merchant, farmer, craftsman, labourer), and of their own marital
status (single, married, widowed, remarried). The purpose would be to see
whether there was any relationship between their fathers’ occupation and
their own marital status.
Our aim in the remaining sections of this chapter is to explain some of
the procedures that can be used when dealing with nominal (categorical)
data of this sort. On some occasions a simple comparison of percentages
may reveal such clear differences that nothing more is needed. But if the
differences are small, or the relationship between different categories is
more complex, then it would be necessary to adopt an appropriate statisti-
cal procedure.
We begin in §7.3.1 with a brief digression to introduce the basic idea of
contingency tables or cross-tabulations used for such data. In §7.3.2 we
discuss an application of the �2-test (lower case Greek chi, pronounced ki-
squared) that can be used when it is desired to compare one or more
samples with respect to a variable that is classified into two or more catego-
ries. This is a test of whether or not the differences between the categories
are statistically significant. We also introduce a new theoretical probability
distribution that is required for this test.
In §7.5 we will look at two other procedures that can be used when there
is only one sample and the object is to test the observed distribution against
some theoretical alternative. Finally, some of the corresponding non-para-
metric measures of the strength of association between the categories are
discussed in §7.6.
7.3.1 Contingency tables or cross-tabulations The �2-test is used where data are cross-classified in contingency tables or cross-tabulations. In their most general form these tabulations show the frequency with which the characteristics of one variable are associated with
those for another variable for one or more samples of cases.
As an illustration of such a table we will use some data compiled from
the records of the Retreat, an asylum established by Quakers in York in
1796, and famous for its promotion of moral therapy in place of the
§. �-
harsher treatments of the insane practised by many other institutions.q
One of the points of interest is whether the success of the therapy changed
over time as Quaker patients were joined by non-Quakers, and lay thera-
pists were replaced as superintendents by qualified doctors. We can
compare the outcomes under the regime of the first superintendent,
George Jepson (1796–1823) with those of his successors Thomas Allis
(1823–41), John Thurnam (1841–9) and John Kitching (1849–74). The
first two were lay therapists, the latter two had medical qualifications.
A double classification of the patients can thus be made according to:
(a) the regime under which they were admitted, and
(b) which of three possible treatment outcomes was recorded in their
case-books: Recovered, Improved or relieved, or Not improved.
The cases are the individual patients; the two variables are the superinten-
dent’s regime and the outcome. Note that both these variables are on the
lowest level of data, the nominal level; in neither case is there any actual
measurement beyond a simple count of the number of patients in each of
the nominal categories.
In table 7.5 the information for this double classification is set out in the
rows and columns of a contingency table. The data for Thurnam and
Kitching have been combined to cover a longer period. The upper panel
gives the absolute numbers of patients, and in the lower panel these figures
are expressed as a percentage of the total number in the three samples.
Table 7.5 is described as a 3�3 table, because it has 3 rows and 3
columns. It has 9 cells, in which are entered the observed frequencies or counts. The dependent variable (the patient outcomes) is shown in the columns, and the possible explanatory variable (superintendent) in the
rows. At the bottom are the column totals and on the right the row totals. These totals are also referred to more generally as marginal totals. The grand total, which is also the total number of observations (n) on which the table is based, is in the bottom right-hand corner.
7.3.2 The �2-test for a relationship between two variables Cross-tabulations such as the one in table 7.5 show how the different
samples (of patients admitted by the successive superintendents) compare
with respect to the different categories of a variable (treatment outcomes).
The table specifies the frequency with which the observations in each
sample are associated with those for the treatment outcomes. The �2-test
q Anne Digby, Madness, Morality and Medicine, A Study of the York Retreat, 1796–1914, Cambridge
University Press, 1985.
can then be applied to establish whether the observed joint frequency dis- tribution in table 7.5 could have occurred by chance.11 The use of the test for this purpose is sometimes referred to as a test of independence (or, con- versely, of association) between the variables.
The �2-test involves a comparison of the observed frequencies (fo) with
those that would be expected if there were no relationship between the vari-
ables (fe). At the core of this measure is the difference between the observed
and the expected frequency in each cell.
The difference is squared (for familiar reasons – compare §2.3.2), and
the squares are then standardized by dividing by the expected frequency in
each cell.r Finally, this measure is summed over all cells. If fo and fe agree
exactly, �2�0. The greater the discrepancy between the observed and
expected frequencies, the larger the value of �2.
§. �-
r This ensures that the biggest contributions to �2 come from the biggest discrepancies between fo and fe , not from the cells with the largest number of cases.
Table 7.5 Cross-tabulation of patients admitted to the York Retreat, by superintendent’s regime and by treatment outcomes
Treatment outcomes
Recovered Improved Not improved Total
Numbers
Superintendent
Jepson (1796–1823) 117 31 11 159
Allis (1823–41) 96 29 22 147
Thurnam and Kitching
(1841–74) 158 68 30 256
Total 371 128 63 562
Percentage
Superintendent
Jepson (1796–1823) 20.8 5.5 2.0 28.3
Allis (1823–41) 17.1 5.2 3.9 26.2
Thurnam and Kitching
(1841–74) 28.1 12.1 5.3 45.5
Total 66.0 22.8 11.2 100.0
Source: Digby, Madness, p. 231. The classification excludes patients who died in the Retreat.
THE �2-TEST
�2 is a measure of the difference between
the observed frequencies, fo,
and
the frequencies that would be expected
if there were no relationship between the variables, fe,
It is defined as follows:
�2 � � (7.8)
The probability of getting a result greater than the calculated value of �2
can then be determined by reference to the appropriate theoretical prob-
ability distribution. This is one we have not yet encountered, the chi-square
distribution, and is described in panel 7.1 and illustrated in figure 7.1.
The sampling distribution of �2 approximates this theoretical chi-
square distribution very closely, provided the number of expected frequen-
cies is at least equal to 5. However, if the number of expected frequencies in
any of the cells is 5 or less (which can often happen if n, the total number of
observations, is small) the distribution of the test statistic is not a good
approximation of the theoretical chi-square distribution and the test has to
be modified.
For a 2 � 2 table, this can be done by an alternative procedure known as
Fisher’s exact test. The outcome of this test is the probability of getting the observed results by chance. This probability can be assessed in the usual
way; for example if the selected level of significance is 5 per cent, then a
probability of less than 5 per cent would lead to rejection of the null
hypothesis.
For larger cross-tabulations there is no easy correction that can be
made. If there are a reasonably large number of cells and only a few have
frequencies of 5 or less, it is possible to accept the results of the �2-test.
However, if there are several cells with small frequencies it would be better
to combine some of the categories in the original classification so as to
eliminate the small frequencies.
(fo fe)2
fe
§. �-
Panel 7.1 The chi-square distribution
The concept of a theoretical probability distribution was first introduced in
§5.4, where we discussed both the normal distribution and the t-distribution.
We now need to refer to a further theoretical distribution, the chi-square dis-
tribution (pronounced ki-square). Like the t-distribution, it is related to the
normal distribution, which can be thought of as its ‘parent’ distribution.
In formal terms the chi-square distribution is the sampling distribution
of a random variable created by squaring a single standard normal random
variable with mean zero and standard deviation of 1, and then summing all
the squared terms.
Its basis is thus a statistic with which we are already familiar: take a
random variable, X, which is normally distributed with a mean of � and a
standard deviation of . Then derive from this a standard normal variable
Z1�
This will have a mean of zero and a standard deviation of 1. Then square
this to obtain Z1 2. If a second independent variable is calculated in the same
way, this will be Z2 2, a third will be Z3
2, and so on.
The total number of such squared terms is given by the parameter k, and k
can have any value from 1 to infinity. If we designate the chi-square random
variable as V, then
V � Z21 � Z 2 2 � ... � Z
2 k (7.9)
The precise shape of the distribution, including its mean and standard
deviation, depends solely on k, which represents the number of degrees of
freedom (df) of the chi-square distribution. Just as the normal curve was
completely defined by two parameters, its mean and standard deviation, so
the chi-square distribution is completely defined by this one parameter, k. A
further property of this distribution is that the mean is also k, and the vari-
ance is 2k.
Because the chi-square distribution is formed by the process of squaring
a variable it can have only positive numbers, and when df is small the distri-
bution is heavily skewed to the right. However as df increases the distribu-
tion becomes more symmetric, and at about 30 df it becomes effectively the
same as a normal distribution. Some illustrative distributions are shown in
figure 7.1.*
* The percentage points of the chi-square distribution are given for selected df from 1 to 100 in
Lindley and Scott, Statistical Tables, table 8, p. 41.
(X �)
Illustration of the �2-test To illustrate the �2-test we will now apply it to the data for the Retreat in
table 7.5. In accordance with the standard test procedure the null hypothe-
sis is that there was no difference between the two outcomes with respect to
the regimes of the successive superintendents.
One research hypothesis might be that patients were less likely to
recover under the later superintendents than in Jepson’s regime because
the admission of non-Quakers made a uniform therapeutic regime less
effective. Another might be that they were more likely to recover under
Thurnam and Kitching than under the previous lay regimes because of
benefits gained from the specifically medical expertise of the later superin-
tendents. We adopt a 5 per cent significance level, and since direction
cannot be predicted it is effectively a two-tailed test.12
The test procedure always involves rejection of the null hypothesis if the
test statistic is larger than the critical value given by the sampling distribu-
tion for chi-square. Since �2 can never be negative there is no equivalent of
moving to the left-hand tail if the alternative to the null hypothesis points
in that direction; for example, if the research hypothesis is unequivocally
that later patients in the Retreat would show a lower recovery rate than
earlier ones.
However, if the researcher does feel able to predict direction, she can use
this to justify reporting a lower significance level than the one actually
achieved. Thus if the value obtained for �2 is large enough to indicate statis-
tical significance at the 5 per cent level with no direction predicted, she
could report it as being significant at the 2.5 per cent level, if direction had
been predicted in advance.
Figure 7.1 The chi- square distribution for selected degrees of freedom
Chi-square
Pr ob
ab ili
ty d
en si
ty
df = 4
df = 10
df = 30
The test statistic �2calc measures the difference between the observed fre-
quencies (shown in table 7.5) and the frequencies that would be expected if
there were no relationship between the two variables.
Out of a total of 562 patients in the three samples, 371 (66.0 per cent)
were classified as recovered. If there were no relationship between regime
and outcome, the same proportion would be expected in all patients. The
expected frequencies would thus be 66 per cent of the 159 patients admitted
by Jepson, 66 per cent of the 147 admitted by Allis, and 66 per cent of the
256 admitted by Thurnam and Kitching.
This gives expected frequencies for the patients classified as recovered of
104.9, 97.0, and 169.0 to compare with the observed frequencies of 117, 96,
and 158. Expected frequencies for the patients classified as improved (22.8
per cent) and not improved (11.2 per cent) can be derived in the same way.
The full calculation of �2 is carried out in table 7.6, and produces a result for
�2calc�9.64.
In order to compare this with the theoretical chi-square distribution so
as to make a decision regarding its significance, it is first necessary to deter-
mine the number of degrees of freedom associated with the cross-tabula- tion. This concept was first introduced in panel 5.3 in §5.4.2, and its
application in the present context is explained in panel 7.2. We see from
this that for the 3�3 tabulation of table 7.6 there are 4 degrees of
freedom (df).
If we enter the published tables for the theoretical probability distribu-
tion of chi-square with 4 degrees of freedom (df) we find that for a 5 per
cent significance level, the critical point is 9.488.s In other words, 5 per cent
of the area under the curve lies to the right of this value. Any calculated
value for the test statistic larger than that falls in the critical (rejection)
region; any statistic lower than that falls in the region in which the null
hypothesis cannot be rejected. Since the calculated test statistic is 9.64 it is
greater than the critical value and the null hypothesis of no difference can
be rejected.
It is thus established that there is a statistically significant association
between the regimes of the successive superintendents and the results of
treatment at the Retreat. As with all tests of statistical significance, the
result is partly determined by the sample size. It is important to be aware
that the value of �2calc is directly proportional to the size of the combined
samples used in the calculation. If the data in table 7.6 were all scaled down
by half, the value of �2 would also be halved. It is thus relatively easy to get a
large and statistically significant result with large samples.
§. �-
s Lindley and Scott, Statistical Tables, table 8, pp. 40–1.
The test has shown that the differences are statistically significant. It
could be, however, that it is only a very weak association, of no real histori-
cal interest for medical historians. It remains for historians of the institu-
tion to investigate the strength of the association, and to evaluate its
underlying causes and consequences. A number of such measures of the
strength of association will be considered in §7.6.
7.4 The one-sample runs test of randomness
The Wald–Wolfowitz runs test was introduced in §7.2.1 in the context of
two independent samples. In this section we describe a different form of
runs test that can be used by a researcher who has only a single sample and
wishes to establish whether or not the observations in the sample occur
in a random sequence, so that there is no relationship between each
Table 7.6 �2-test for statistical significance: calculations using data for York Retreat, classified by superintendent’s regime and treatment outcomes
(1) (2) (3) (4) (5)
Observed Expected Square of Square of difference/
frequency frequency Difference difference expected frequency
Category fo fe (fo – fe) (fo – fe) 2 (fo – fe)
2/fe
Jepson
Recovered 117 104.9 12.1 145.3 1.38
Improved or relieved 31 36.2 5 5.2 27.5 0.76
Not improved 11 17.8 5 6.8 46.3 2.60
Allis
Recovered 96 97.0 5 1.0 1.0 0.01
Improved or relieved 29 33.5 5 4.5 20.4 0.61
Not improved 22 16.5 5 5.5 30.7 1.86
Thurnam and Kitching
Recovered 158 169.0 11.0 121.6 0.72
Improved or relieved 68 58.4 59.8 95.3 1.63
Not improved 30 28.7 5 1.3 1.6 0.06
Total 562 562.0 5 0.0 489.7 9.64
Note:
(1) See table 7.5.
(2) See text.
§. -
Panel 7.2 The concept of degrees of freedom for a cross-tabulation
Let us begin by thinking of the simplest 2�2 cross-tabulation in which 2
row and 2 column totals have been entered, but all the cells are blank. If you
were invited to choose a set of entries for the cells that would be consistent
with these row and column totals you would be able to choose a value for
only one cell. As soon as you had done this, all the others would be automat-
ically determined since they must sum to the row and column totals. There
are four cells, but 3 degrees of freedom have been lost and you have only 1
degree of freedom.
On the same basis if we are calculating the expected frequencies for a 2�
2 cross-tabulation, then as soon as fe has been calculated for any one cell, all
the others are determined since they must sum to the row and column
totals. There is thus only 1 degree of freedom in this case.
What happens if we have a larger table, for example, the 3�3 cross-
tabulation in table 7.5? The expected proportion that would be classified as
recovered if there were no association with the superintendent’s regime was
66 per cent. We can thus calculate that for Jepson’s 159 patients the expected
number who would recover would be 105, and for those of Allis it would be
97. The corresponding figures for the 22.8 per cent of patients classified as
improved would be 36 for Jepson and 34 for Allis.
Once these four expected frequencies have been calculated the remaining
five cells are all determined automatically. From the row totals, we know that
the expected numbers classified as not improved under Jepson must be
159 (105 �36)�18, and under Allis must be 147 (97�34)�16.
From the column totals we then know that the expected figures for the
three categories of Thurnam and Kitching’s patients must be recovered:
371 (105�97)�169; improved: 128 (36�34)�58; and not improved:
63 (18�16)�29. In this 3�3 table there are thus only 4 degrees of
freedom. We could start by filling some cells other than the four used in this
illustration, but as soon as any four expected values have been calculated
those of all the remaining cells would be predetermined.
More generally, the rule for cross-tabulations is that the number of
degrees of freedom is equal to one less than the number of rows, multiplied by
one less than the number of columns.
df�(r 1)(c 1) (7.10)
This is thus a further illustration of the principle stated in panel 5.3 in
§5.4.2 that the number of degrees of freedom equals the number of obser-
vations less the number of constraints linking them.
observation and all previous or subsequent observations. Like all non-par-
ametric tests, this one has the important advantage that it need not be
assumed that the observations are normally distributed.
The definition of a run is exactly the same as before, but in this case the
comparison is made between the number of runs shown by the sample
data, and the number that would be expected if the sample were truly
random. The test can be applied to any events that can be classified into two
categories (heads or tails when tossing a coin, married or single if selecting
from a group of women), and the order in which the two categories occur is
all that matters.
The null hypothesis is that the order is random. The research hypothesis
might simply be that the order is not random, in which case a two-tailed
test would be made. Alternatively, however, the researcher might be inter-
ested in a proposition suggesting that some specified factor was causing the
events to occur in clusters. In that case the research hypothesis would be
that the number of runs should be fewer than would be expected if the data
were random, and since direction is predicted a one-tailed test is required.
To illustrate the procedure, we take the imaginary case of an inspector
who comes to visit a school and obtains from the principal an alphabetical
list of surnames from which 25 pupils are to be selected for questioning.
The selection is supposed to be made at random without regard to the
intelligence of the pupils, although a measure of that is known to the school
from internal tests. The agency employing the inspector thinks she may be
colluding with the school, and so asks us to test whether the selection was
truly random. The suspicion is not that too many of the more intelligent
students are being selected at the expense of the less able, but rather that the
inspector is somehow being guided to particular students in both catego-
ries who are expected to perform well because, for example, they are more
articulate.
To test this we divide the 25 pupils into two categories (high and low)
according to their test scores, and count the number of runs (�r) made up
of pupils with high scores (H) and those with low scores (L) in the sample.
If we find that H and L pupils were drawn in strict rotation (giving 25 runs,
the maximum number possible with a sample of this size) we would
strongly suspect that the choice was not random, and that the principal had
influenced the selection in some way. We would be equally suspicious if the
inspector chose first 13 H pupils in a row, and then 12 L pupils, so that there
were only 2 runs.
In both these cases we would probably not need a statistical technique to
tell us that we should reject the null hypothesis of a random sample. Such a
procedure would be necessary, however, if the order is not at either of these
extremes, and a runs test would be appropriate for this. If the sample is
large, with either or both the number of events (H and L) greater than 20,
then the distribution of r is approximately normal, with mean, �r, and
standard deviation, r, exactly as given previously by (7.1) and (7.2). The
test statistic, Zcalc can thus be obtained as in 7.3. This can then be compared
with the normal distribution (as in table 2.7) to obtain the prob-value for
either a two-tailed or a one-tailed test.t
If the sample is small, with neither event (H or L) greater than 20, the
required result can be obtained directly from a published table. The statis-
ticians have worked out the sampling distribution of r that could be
expected from repeated random samples, and we can refer to the tables of
this distribution to establish what intermediate number of runs can be
treated as consistent with a random sample.u With small samples we can
either ask the computer to make the calculation or we can do it directly
from the table. For larger samples the computer is obviously quicker.v
To see how the procedure works for a small sample we will consider four
cases. I and II are those already mentioned, with r�25 and 2, respectively.
In III the sequence was
H LL HHH LLL HH LL HHH LLLL H LL HH
giving 11 runs, and in IV it was
LL HHH LLLL HHHH LLL HHH LLLL HH
giving 8 runs.w
Entering the table for the distribution of the number of runs for n1�12
and n2�13 and adopting the 5 per cent level, we find that the upper bound
is 18 and the lower is 9. The upper bound indicates that if r is 19 or higher it
is so large that the probability of obtaining it with a random sample is 5 per
cent or less. Similarly, the lower bound indicates that if r is 8 or lower the
probability of obtaining it with a random sample is 5 per cent or less.13
Accordingly, the null hypothesis cannot be rejected for values of r
between 9 and 18, but can be rejected for any samples with r outside this
range. Since three of our variants (I, II, and IV) fall outside this range it
§. -
t For the complete table of the normal distribution see Lindley and Scott, Statistical Tables, table 4,
pp. 34–5. u Lindley and Scott, Statistical Tables, table 18, pp. 60–2. v Note that in order to perform the test with a computer package it may be necessary to recode the
sequences from letters to numbers; for example by replacing all the Ls by a 1 and all the Hs by a 2. w To simplify the illustration we have kept the number of both events constant in each case, but if a
random selection had really been undertaken four times we would not expect that there would
always be 12 Hs and 13 Ls on the list.
leaves only one, III, with 11 runs, for which the null hypothesis of a random
sample cannot be rejected.
Note that the essence of this test is that it is based solely on the order in
which the events occur, not on their frequency. It is the pattern of the
ranking that informs us that the null hypothesis should be rejected for var-
iants I, II, and IV. There is nothing in the frequencies (in every case 12 Hs
and 13 Ls) to identify either the extreme lack of randomness in samples
I and II, or the narrower difference in respect of randomness between III
and IV.
It is also worth pointing out the difference between this test and the two-
sample test described in §7.2.1. In the present test, perfect alternation (as in
II) – and more generally a large number of runs – leads to rejection of the
null hypothesis of randomness. By contrast, in the Wald–Wolfowitz runs
test, perfect alternation (as in variant IV of table 7.1) – and more generally
a large number of runs – means that the null hypothesis of no difference
between the two samples cannot be rejected.
An illustration of the application of the runs test for randomness For many centuries, and for millions of people, the state of the grain
harvest each autumn was by far the most important element in determin-
ing their standard of living in the following year. Historians have naturally
been very interested in the annual grain yield, and because grain prices
were recorded much more frequently than grain yields, have often
attempted to infer information about harvest fluctuations from the data on
prices.
One particular issue that has claimed their attention is the very striking
tendency of prices to run in clusters, with years of above-average prices fol-
lowing each other in sequence before giving way to a run of below-average
prices. One possibility is that these are simply the result of random
fluctuations in weather conditions, but two more interesting explanations
have been proposed to account for the clusters.
The first is that runs of good and bad yields were caused by the need to
set aside a substantial proportion of each year’s crop for next year’s seed. In
a bad year the rural population would be forced to maintain their required
minimum standard by consuming part of next year’s seed corn; in a good
year more could be set aside, and seed sown more liberally. In this way each
year helped to generate a second year of similar quality until the sequence
was broken by some larger shock induced by abnormal weather. This
hypothesis in turn implies that the observed runs in prices simply reflect an
underlying pattern in the (unknown) yields, with runs of low prices
matching runs of above-average yields, and vice versa.14
The rival hypothesis is that the runs in prices are largely independent of
the sequence of yields because of the possibility of grain storage. An excep-
tionally abundant harvest in one year will enable grain to be carried over
not only for the next year, but also for subsequent years. Even if the follow-
ing year was one of below average yields, prices might remain below
average because grain was being sold from store. Thus there would be runs
in prices that would not reflect corresponding runs in yields.
In the light of this analysis, one element in the investigation of the data
on prices (and on yields where that exists) would be to establish whether or
not the clusters of above and below average observations are random. Tony
Wrigley applied a one-tailed runs test to French data on wheat yields and
prices over the period 1828–1900, and showed that the null hypothesis of
randomness could not be rejected for the yields (prob-value�0.36), but
was easily rejected for the prices (prob-value�0.0028).x
When the same one-tailed test was applied to a series on wheat prices at
Exeter over three subperiods between 1328 and 1789, the prob-values were
respectively 0.0436, 0.0021 and 0.0003. As Wrigley noted, the null hypoth-
esis can only just be rejected at the 5 per cent significance level for the first
subperiod, but in the two other subperiods ‘it is increasingly and ultimately
extremely improbable that the runs were random’ (p. 129).
On the basis of these findings Wrigley suggested that the presence of non-
random runs in the price data, but not – for France – the data on yields, sup-
ported the storage rather than the seed corn hypothesis; and further, that the
trend over time towards a stronger pattern of non-random fluctuations in
prices also suggested that storage was the relevant factor since the impact of
seed corn would have been most pronounced in medieval times.
This conclusion has been reinforced by the application of the runs test
to data on both wheat yields and prices at the Winchester manors in the
years 1283–1350. For yields the prob-value was 0.402, whereas for prices it
was 0.0017. This suggested a high probability that the yield sequence was
random but the price sequence was not. The contrast thus confirmed the
message of the French yield data. Further support for the storage hypothe-
sis was given by the application of this and other tests to data on butter
prices. For butter, unlike wheat, the null hypothesis of random fluctuations
in price could not be rejected, and this is what would be expected for a
commodity that could not be stored.y
§. -
x E. A. Wrigley, ‘Some reflections on corn yields and prices in pre-industrial economies’, in E. A.
Wrigley, People, Cities and Wealth, Blackwell, 1987, pp. 92–132. y Randall Nielsen, ‘Storage and English government in early modern grain markets’, Journal of
Economic History, 57, 1997, pp. 1–33. For further discussion of this issue see Karl Gunnar
Persson, Grain Markets in Europe 1500–1900, Cambridge University Press, 1999.
7.5 One-sample non-parametric tests of goodness of fit
7.5.1 The one-sample �2-test The �2-test introduced in §7.3.2 is applicable when the object is to test
whether a null hypothesis of no association between two or more vari-
ables should be accepted or rejected. An alternative problem that might
face an historian occurs when there is only one sample, and the point at
issue is whether the observed distribution of the sample across certain
categories is consistent with some alternative distribution specified by the
researcher.
The procedure for making the test is identical to that already described
in §7.3, except that the researcher has to specify the expected frequencies for
comparison with the observed frequencies given by the sample. This is
referred to as a test of goodness of fit, and is used to determine how well the observed observations fit the specified alternative. As with the previous �2-
test, the expected frequency in each category should be at least 5.
The number of degrees of freedom is the number of rows (categories) in
the distribution minus 1. Thus if there are 6 categories there will be 5
degrees of freedom since once five numbers have been entered in the table
the final one is determined by the previous five and the total.
There are several possible alternative distributions that might be
specified. One of the simplest is that the proportions are the same for all
categories. For example, for a sample of cases reporting the day on which
people were observed as being at work, the null hypothesis would be that
the proportions should be the same for all six working days.z Other pos-
sibilities involve one of the theoretical distributions, such as the normal or
the binomial. Alternatively the researcher might specify a particular empir-
ical distribution; for example, that the ethnic distribution of the inmates of
an institution is the same as their distribution in the general population, or
that certain characteristics of a sample are the same as those for the popula-
tion as a whole as given by a census.15
7.5.2 The one-sample Kolmogorov–Smirnov test The Kolmogorov–Smirnov procedure described in §7.2.3 for comparing
cumulative frequency distributions from two samples can also be adapted
to provide a one-sample test of goodness of fit. The single sample provides
z See for example, Voth, Time and Work, p. 86.
the observed distribution and, as in §7.5.1, the researcher specifies an alter-
native theoretical distribution. The procedure can thus be used to test the
null hypothesis that the sample observations are drawn from a population
with the specified alternative distribution.
This alternative might be, for instance, a theoretical distribution such as
the normal distribution. Or it might be that the distribution is uniform, so
that the proportion of observations in each class interval is the same. In
both these cases the computer will compile the alternative from the sample
data. For example, it will calculate the normal distribution of a series with
the same mean and standard deviation as the single sample, and then
convert this to a cumulative frequency distribution.
It is also possible, however, for the researcher to specify the alternative
by formulating a statistical model to represent some postulated theory of
how the observations might be distributed. The issue would then be to test
how well the observed data fit the predictions of the model.
The maximum difference between the observed and theoretical cumu-
lative relative frequencies, D, is calculated as before, and provides the test
statistic. Published tables of the sampling distribution of D can then be
consulted to decide whether or not to reject H0 at different levels of
significance. The tables effectively cover all samples, regardless of size. In
some versions, the critical values are tabulated for D ; in other – more com-
prehensive – tables, for D , where n is the size of the single sample.aa For
the latter, H0 should be rejected if the value of D is equal to or greater
than the critical value on a one-tailed test with direction predicted. For a
two-tailed test the critical value is applicable to half the given significance
level.
To illustrate, assume that the test is applied to a sample of 70 observa-
tions with mean 35.5 and standard deviation of 20. The specified alterna-
tive is the normal distribution. D is found to be 0.0627 and D is,
therefore, 0.5244. Entering the table for the upper percentage points of the
one-sample KS distribution, it will be seen that this is less than the value of
1.203 given for the 10 per cent significance level. It is thus not possible to
reject the null hypothesis on a one-tailed test.
This test is superior in certain respects to the �2-test of goodness of fit. It
can treat individual categories separately, whereas the �2-test may need to
combine categories if the sample sizes are small.
�n
�n �n
§. - -
aa Lindley and Scott, Statistical Tables, table 23, p. 70 has this form, and it is this value which SPSS
reports under the name of Kolmogorov–Smirnov Z.
7.6 Non-parametric measures of strength of association
In §3.2 the correlation coefficient, r, was introduced as a measure of the
strength of association between two interval level variables, X and Y. r has
the very convenient property that it takes a value of 0 when there is no
(linear) correlation between the two variables, and of �1 or –1 when there
is perfect positive or negative correlation. However, it has two limitations.
First, it can be used only with interval data. Secondly, inferences about the
corresponding estimate of the strength of the relationship between the var-
iables in the population, � (lower case Greek rho) require the assumption
that the population distribution is normal.
In §3.3 we introduced one alternative to r, Spearman’s rank correlation
coefficient, rs, for use with ranked data. This is a non-parametric measure
of strength of association, though the term was not mentioned at that
stage. We will now introduce, but not discuss in detail, four further non-
parametric measures of strength of association which are not subject to the
limitations affecting the correlation coefficient, and are appropriate for use
with nominal and ordered data arranged in contingency tables.
Unfortunately these non-parametric measures can be awkward to inter-
pret and compare. They do not all take a value of unity when there is
perfect correlation; a few do, but for some the upper limit is more than 1,
for others it is less than 1. Moreover, for some measures the upper limit
depends on the number of rows and columns in the contingency table.
There is also an untidy inconsistency with respect to notation: some of
these measures of strength of association are conventionally described by
Roman letters, whereas Greek letters are used for others; and some are cal-
culated as squared measures whereas others are not.
7.6.1 Four measures of association: �, Cramer’s V, the contingency coefficient C, and Goodman and Kruskal’s tau We will mention four of these measures, the first three of which are based
on �2. Their starting point is the fact (as noted in §7.3.2) that the �2-test for
the statistical significance of a relationship varies directly with the size of
the sample. For example, if the sample given in table 7.5 had covered 1,124
rather than 562 patients, the value calculated for �2 would also double,
from 9.64 to 19.28. A simple correction for this is to divide �2 by the size of
the sample. This leads to a measure denoted � (lower case Greek phi)
� � (7.11)�� 2
n
If the contingency table has only 2 rows (i.e. it is a 2�k table), the
maximum value � can take will be 1, but in any other case the upper limit
can be substantially greater than 1.
An alternative introduced by Cramer is derived by dividing �2 by the
product of the sample size and whichever of r 1 and c 1 is the smaller
V � (7.12)
where ‘whichever is the smaller’ is represented by min (for minimum).
Thus in a 4�3 table there are fewer columns than rows, so the lower of
(r 1) and (c 1) would be (3 1)�2, and V would be equal to ,
which is the same as �/2. The advantage of Cramer’s V is that the
maximum value it can take is always 1.
A third measure of association based on �2 is the contingency
coefficient, denoted by C, and calculated as
C � (7.13)
C is easily computed once �2 has been calculated, but suffers from the limi-
tation that its upper limit depends on the number of categories in the con-
tingency table, although it is always less than 1. For example, if it is a 2�2
table the upper limit is 0.707, if it is a 3�3 table the maximum value is
0.816, and so on.
The final measure we will refer to is known as Goodman and Kruskal’s
tau, where tau is the name for the Greek letter �. This measure of associa-
tion is calculated on a quite different basis, and involves a procedure
(which we need not consider here) for using values of the independent var-
iable to predict values of the dependent variable.
If there is no association between the variables, the explanatory variable
is no help in predicting the dependent variable and ��0. If there is perfect
association between the variables the explanatory variable would be a
perfect predictor and ��1. This version of the measure is denoted �b because A columns of the explanatory variable are used to predict B rows of
the dependent variable.
In some circumstances, however, the researcher may simply be inter-
ested in the strength of association between the two variables without
regarding either one as the possible explanatory variable. It is then possible
to calculate an alternate version, denoted �a, in which the procedure is
� � 2
�2 � n
�� 2
2n
� � 2
n�min(r 1, c 1)
§. -
reversed and the B rows are used to predict the A columns. Except for the
extreme cases of zero and perfect correlation, the numerical values of
�a,and �b will not be the same.
An illustration of the use of Cramer’s V Cramer’s V was used by Ann Kussmaul in her fascinating study of farming
patterns in early modern England as indicated by the seasonality of mar-
riage. The predominant form of economic activity in a sample of 379 non-
market parishes was determined by reference to the timing of marriages.
Very broadly, in those parishes in which the majority of marriages
occurred in the autumn after the harvest had been gathered in, economic
activity was classified as predominantly arable (A-type). In those in which
weddings were typically celebrated in spring and early summer following
the birth of lambs and calves in late winter and early spring, activity was
classified as predominantly pastoral (P-type).
Those parishes that showed little tendency to peaks in marriages in
either of the two great agricultural seasons were classified as places of rural
industry (X-type). Finally, a small number of parishes that displayed both
high spring and high autumn peaks were classified as heterogeneous (H-
type).bb
One of many features of rural England that Kussmaul was able to
examine on the basis of this classification of data drawn from the marriage
registers was the extent to which these rural patterns had changed over
time. What was the strength of the association between the types of eco-
nomic activity in successive periods? To establish this she uses Cramer’s V.
If the seasonal typing of the parishes in successive periods were completely
independent, V would equal 1.0; if activity in one period were com-
pletely determined by activity in the previous period, V would equal �1.0.
This measure was calculated for 12 successive periods from 1541–80 to
1801–40.
Table 7.7 reproduces Kussmaul’s data for one such pair of periods,
1661–1700 and 1701–40. Calculating the expected frequencies and apply-
ing (7.8), �2 is found to be 168.59. Cramer’s V can then be obtained by
means of (7.12) as
V �
��0.38 � 168.59379�(4 1) ��168.591137 � �0.1483
bb Ann Kussmaul, A General View of the Rural Economy of England 1538–1840, Cambridge
University Press, 1990. This brief description of the classification does not do justice to the depth
and subtlety of the work.
By the final pair of periods, 1741–80 and 1781–1820, the strength of the
association over time had risen to �0.60. The full set of calculations thus
shows the constancy of seasonal typing rising steadily from the late eight-
eenth century, as parishes settled into increasingly specialized patterns.cc
Notes 1 A number of such measures of political power are discussed in Allan C. Bogue,
‘Some dimensions of power in the thirty-seventh Senate’, in William O. Aydelotte,
Allan C. Bogue and Robert W. Fogel (eds.), The Dimensions of Quantitative Research
in History, Oxford University Press, 1972. For example, one measure attributed more
power to Senator A than Senator B, if the number of times A voted with the majority
on a set of proposals when B was in the minority was greater than the number of
times B voted with the majority when A was opposed. All senators could then be
ranked on the basis of the total number of their positive power relationships in all
possible pairings. 2 The value given for Zcalc in table 7.2 incorporates a small correction to the formula given
in (7.3). The computer program (in this instance SPSS) has added �0.5 to the numera-
tor when Zcalc is negative, and –0.5 when it is positive. The scale of the correction is thus
equal to 0.5/ r, and since r is determined solely by the size of the two samples (see
(7.2)) the correction diminishes as the sample sizes increase. In table 7.2 the sample
sizes are the same for each variant, and so the correction is constant at 0.230.
It is thus a correction that is effective when using the approximation of the
sampling distribution to the normal distribution with small samples. When Zcalc is
negative the correction makes it smaller; when it is positive the correction makes it
cc Kussmaul, General View, pp. 76–8. For other examples of the use of Cramer’s V by Kussmaul, see
pp. 68 and 134–6.
Table 7.7 Cross-tabulation of 379 parishes in England by seasonal type in 1661–1700 and 1701–1740
Seasonal type in 1701–40
X A P H Total
Seasonal type in 1661–70
X 57 31 16 5 109
A 17 92 3 15 127
P 21 17 50 7 95
H 6 25 7 10 48
Total 101 165 76 37 379
Source: Kussmaul, General View, p. 77. For the classification of parishes by seasonal type
see text.
larger. It thus moves the critical value further into the respective tails, making it more
difficult to reject the null hypothesis. 3 Given the lower of the two possible values for U, the corresponding minimum value
of W can be obtained as follows:
W � � U
where n1 is the size of the sample with which the calculation of U was started. 4 It is not actually necessary to repeat the procedure with the other sample as the start-
ing point since there is a simple relationship between U and U�, the sum obtained by
starting with the second sample: U��(n1n2) – U, where n1 and n2 are the numbers in
the respective samples. W� is then easily derived from U� by the formula for W and U
in n. 3. 5 We have specified a two-tailed test so the research hypothesis is simply H1: �u1��u2.
If, however we had selected a two-tailed test with direction predicted it might have
been H1: �u1 �u2, in which case we could have moved to the 2.5 per cent level. The
table gives a critical value for this level of 23. Since the decision rule would be the
same (reject H0, for any value of U equal to or less than this), it would be harder to
reject the null hypothesis. The other possibility is H1: �u1��u2. To test this alterna-
tive it is necessary to move to the higher of the two possible values for U obtained by
starting with the other sample (for example, to U��55 for series D). With this as the
research hypothesis the decision rule would be to reject the null hypothesis if U�
(rather than U) is equal to or less than the critical value. 6 As can easily be seen, the value of Z is the same (though the sign is different) whichever
sample one starts from. The fact that U and U� give the same absolute result for Z
(though with different signs) can be seen from the following example. For variant B
and two samples each of 10 observations, �u� 1⁄2 (n1n2)�
1⁄2 (100)�50. Since U�45,
the numerator of Z�U �u�45 50� 5. If we take the higher value for U of 55 the
numerator is 55 50��5. The sign of Z is relevant only if direction is predicted, i.e. if
H0 is that the mean of one of the samples is larger (smaller) than the other. 7 Because our examples are based on small samples SPSS is able to report two meas-
ures of significance. One is described as ‘exact significance’ and this is the only one
quoted in table 7.3. It is based on the exact sampling distribution of the test statistic
for small samples, and is thus consistent with the critical values given for these
samples in the published tables. The other is described as ‘asymptotic significance’
and is the approximation based on the normal distribution. For large samples this is
the only measure of significance available; it becomes progressively more accurate as
the sample size increases. 8 For an example of the use of the Mann–Whitney test by an historian see James H.
Stock, ‘Real estate mortgages, foreclosures, and Midwestern agrarian unrest,
1865–1920’, Journal of Economic History, 44, 1984, pp. 89–106. Stock used this non-
parametric test (see pp. 97–9) to explore the relationship of three qualitative levels of
agrarian protest (ranked across 12 Midwestern states) with the corresponding rank-
ings of various measures of the levels and volatility of farm debt in those states. The
null hypothesis of no relationship between debt and unrest was rejected at the 1 per
cent level.
n1(n1 � 1) 2
See also Hans-Joachim Voth, Time and Work in England, 1750–1830, Oxford
University Press, 2001, pp. 88–94, 227–8. Voth assembled evidence of people being at
work on different days of the week from statements by witnesses who appeared before
the Old Bailey in London between 1749 and 1763. He calculated the Mann–Whitney
U to test whether Monday was statistically different from other days of the week
(because workers were observing the practice of St Monday); and also to test whether
work was markedly less frequent on the 46 days of the year that were the old Catholic
holy days. In both cases the null hypothesis of no difference was rejected. 9 The test reported in table 7.4 is a two-tailed test. If a one-tailed test is required, it is
necessary only to divide the reported prob-value for KS Z by 2. There is, however, an
alternative presentation of the one-tailed test statistic, known as �2, calculated as
4D2
This is equal to four times the square of KS Z. The �2 test statistic is compared to the
chi-squared distribution with 2 degrees of freedom; this distribution will be
described in §7.3. The critical value of the �2 test statistic at the 5 per cent level is
1.36
and there are corresponding values for other levels of significance. 10 This test (referred to simply as the Smirnov test) was applied by Michael Anderson,
Family Structure in Nineteenth Century Lancashire, Cambridge University Press,
1971, pp. 56–61 and 206–7. Anderson worked with the �2 version mentioned in n. 9.
He used the procedure to compare how close family members lived to each other in
his study town of Preston. The actual pattern of residence of members of the same
family could be determined for a small subsample from the addresses in the census
enumerator’s books; the alternative was obtained by allocating people to numbered
areas on a map by means of a random number table. The propinquity of the Preston
families was shown to be statistically significant at the 0.01 per cent level when the
exercise was based on related pairs of kin; but it was not quite significant at the 5 per
cent level when it was done with a larger sample using individuals with the same
surname and birthplace.
For a slightly different application of the test see David Loschky and Ben D.
Childers, ‘Early English mortality’, Journal of Interdisciplinary History, 24, 1993, pp.
85–97. In this case the two-sample Kolmogorov–Smirnov test was used to compare
pairs of life tables for years before and after 1500. The authors’ primary aim was to
establish whether there were any statistically distinguishable differences in the crude
death rates implied by the pre-1500 and post-1500 tables. They were able to reject
with great confidence (prob-value 0.01) the hypothesis that the samples of pre-
and post-1500 data were drawn from the same underlying population. 11 A subtly different application of the �2-test, which uses precisely the same procedure,
is referred to as a test of homogeneity. The object of such a test is to establish whether
two or more samples have the same proportion of cases possessing some particular
characteristics. For example, a social historian might be interested in testing whether
three regions in antebellum Virginia are homogeneous with respect to the proportion
�n1 � n2n1n2
n1n2 n1 � n2
of households in each region in respect of four categories of use of house servants: no
servants, employed wage servants, owned slaves, and rented slaves. There is no sugges-
tion of any association between the respective regions and the type of servant used. 12 There is one issue that might be puzzling for some readers. The �2-test is intended for
two or more independent samples, but the data for the Retreat are not at first sight
samples: they are comprehensive records of all patients admitted to the institution. A
statistician’s response would be to say that each of the three sets of patients could
nevertheless be thought of as a sample drawn from an infinite population composed
of all the people who might have been admitted to the Retreat in the period of the
particular superintendent. On this basis it is possible to apply the theories of sam-
pling and sampling distributions introduced in chapter 5; see also §9.2.3. 13 If we want to establish the exact prob-values we can use the computer to obtain the
values of Zcalc (from (7.3)) and then establish either one-tailed or two-tailed
significance levels by reference to the table of the normal distribution. For cases I–IV
the values of Z are respectively, 4.510, –4.494, –0.810, and –2.038, giving prob-values
for a two-tailed test of 0.000 for both I and II, –0.810 for III, and 0.042 for IV. If a one-
tailed test is required these prob-values would be halved. 14 For the hypothesis on seed corn see W. G. Hoskins, ‘Harvest fluctuations and English
economic history, 1480–1619’, Agricultural History Review, 12, 1964, pp. 28–46 and
Hoskins, ‘Harvest fluctuations and English economic history, 1620–1759’,
Agricultural History Review, 16, 1968, pp. 15–31. 15 See for example, Alan Armstrong, Stability and Change in an English County Town, A
Social Study of York, 1801–51, Cambridge University Press, 1974, pp. 203–5.
Armstrong drew a 10 per cent sample from the Census enumerators’ books for York
for both the 1841 and 1851 Censuses, and used a �2-test to compare the sample dis-
tribution by age, gender, and birthplace with the distributions shown by the Census.
For the 1851 sample, the null hypothesis of no difference was accepted at the 5 per
cent level for age and sex structure but not for birthplace. There was a deficiency of
Irish-born because the sampling procedure had omitted the institutions and large
lodging-houses in which many of the Irish resided.
See also Gary Magee, Productivity and Performance in the Paper Industry,
Cambridge University Press, 1997, pp. 62–4. Magee compared the regional distribu-
tion of papermaking patents with the distribution that would be expected if patent-
ing activity were directly proportional to the number of paper mills in each region.
7.7 Exercises for chapter 7 1. Which non-parametric test is best suited to the following data sets? In
each case, write a sentence explaining your selection.
(i) Data are available on the ranking of two variables within a popula-
tion, but with no evidence on the scale of differences between them.
(ii) Data are available on the ranking of two variables within a popula-
tion, and the data set includes information on the scale of differences
between them.
(iii) Data are available on the ranking of two attributes for a particular
sample; you wish to determine whether the rankings are statistically
different.
(iv) Data are available on the distribution of attributes for two indepen-
dent samples; you wish to determine whether there is a significant
difference between the two subsets.
(v) Data are available on the frequency of an attribute for two indepen-
dent samples; you wish to determine whether the samples are drawn
from the same population.
2. In his testimony to a 1909 government inquiry into the effects of part-
time work on child health, Mr Robert Morley, President of the Halifax
District Trades and Labour Council, handed in data on the heights and
weights of 40 boys aged 12–13, 20 of whom worked in a woollen factory
half-time, and 20 of whom were in school full-time. In the following table,
we have combined these data into a single measure of physical develop-
ment, the body–mass index.
Schoolboys Factory boys
March October March October
1908 1908 1908 1908
A1 16.84 17.14 B1 16.86 15.86
A2 20.62 20.97 B2 17.77 16.69
A3 17.02 17.35 B3 17.93 16.50
A4 16.08 15.73 B4 19.77 19.17
A5 17.87 18.63 B5 18.29 18.02
A6 17.09 16.95 B6 17.00 17.29
A7 17.84 17.99 B7 16.89 16.58
A8 16.83 17.00 B8 18.15 16.32
A9 20.03 21.30 B9 16.11 16.45
A10 16.71 18.84 B10 17.99 18.08
A11 16.09 17.00 B11 20.54 20.52
A12 15.77 15.69 B12 19.06 18.42
A13 17.04 16.74 B13 19.50 18.49
A14 17.85 18.56 B14 16.33 16.51
A15 14.47 14.78 B15 17.23 16.71
A16 18.90 19.36 B16 17.44 16.83
A17 18.36 20.15 B17 14.91 14.41
A18 15.34 16.38 B18 16.74 15.67
A19 15.47 15.99 B19 16.01 14.71
A20 15.08 15.78 B20 16.96 15.84
Note:
The body–mass index is calculated as (703 � weight in lb)/(height in feet)2.
Source: British Parliamentary Papers, Report on Partial Exemption from School Attendance
(Cd. 4887, 1909), p. 281.
§.
The data were used to support Mr Morley’s argument that ‘the effect of
half-time [labour] is nearly universally bad’.
Apply the Wald–Wolfowitz runs test to determine whether the statistics
bear out this interpretation.
(i) Use the body–mass index for March 1908 to determine whether
these boys were drawn from the same population.
(ii) Then use the change in the body–mass index between March and
October to test whether there are any systematic differences in the
change in the body–mass index between the two groups after the
beginning of half-time work.
3. The sequence of US presidential administrations (R indicates
Republican, D indicates Democratic) since the Civil War has been as
follows:
1865–1932: R R R R R D R D R R R R D D R R R
1932–2000: D D D D D R R D D R R D R R R D D
(i) By hand, use the runs test for randomness to determine whether
there is evidence that the incumbent party has an advantage in suc-
cessive elections over the entire period, 1865–2000.
(ii) Are your results any different if you subdivide the period into two
parts: 1865–1932, and 1932–2000?
4. The density of retailers in different parts of Manchester in 1871 was as
follows (higher scores indicate a greater density; the average for the whole
of Manchester being 1):
Bakers: 2.04, 1.30, 1.50, 2.10, 1.68, 0.87, 1.80, 1.12, 1.22
Butchers: 1.40, 0.82, 1.01, 1.27, 0.95, 0.86, 1.11, 0.58, 0.95
Source : Roger Scola, Feeding the Victorian City: The Food Supply of Victorian
Manchester, 1770-1870, Manchester University Press, 1992, p. 299.
(i) Do these figures support the hypothesis that there was no difference
in the distribution of butchers and bakers in these regions? Choose
the appropriate non-parametric test and make all necessary calcula-
tions by hand.
(ii) Test the hypothesis that butchers were more concentrated in some
parts of Manchester than others.
5. Evaluate the hypothesis that average wealth holdings in the neighbour-
ing counties of Berkshire and Buckinghamshire are drawn from the same
population. Apply the parametric difference of means test as well as the
non-parametric Mann–Whitney and Kolmogorov–Smirnov tests.
What differences do you detect between these test results? Which is to be
preferred and why? Consider both historical and statistical reasons in your
answer.
6. A survey of households in Liverpool in 1927-9 recorded information on
the first occupation of children after leaving secondary school and on the
occupation of their fathers.
Child’s occupation
Occupations Professional Clerical and Manual All
of father and business commercial labour occupations
Professional and business 321 323 122 766
Clerical and commercial 98 229 52 379
Manual labour 131 323 115 569
All occupations 550 875 289 1,714
Source : D. Caradog Jones (ed.), The Social Survey of Merseyside, 3, Liverpool University Press,
1934, pp. 178–180.
(i) Do the data support the report’s conclusion that, ‘the occupational
grade of the parent has considerable weight in the determination of
the occupational grade of the child’?
(ii) Do the data support the inference that the children of ‘higher social
class’ (as measured by a higher occupational grade) are more likely to
enter higher occupations?
(iii) Would your responses to (i) and (ii) differ if you were asked to evalu-
ate whether children of each occupational class were more likely to
be selected from the same occupational class as their father?
(iv) Would your responses to (i) and (ii) differ if the survey were based
on a sample of 171 (i.e. one-tenth as large)?
Write a few sentences explaining what these figures indicate about the
extent and pattern of social mobility in inter-war Britain.
7. By hand, construct a contingency table of occupational structure by
region, using the Steckel data set. (Hint : the table will be 5 � 4.) Express the
cross-tabulations as both counts and relative frequencies.
(i) By hand, calculate the �2 test statistic. Compare to the critical point
at an appropriate level of significance, being careful to specify cor-
rectly the number of degrees of freedom.
(ii) Does the evidence support the hypothesis that the occupational
structure was identical in all US regions in 1850?
§.
(iii) How would you test the hypothesis that occupational structure was
identical in all regions outside the North-East? What do you find?
8. By hand, use the one-sample Kolmogorov–Smirnov test on the data for
Cambridge parishes in the Boyer relief data set to determine whether there
are statistically significant differences between their populations. (Hint :
order your data from lowest to highest.) Report your findings.
Why does it matter that you order the data properly before undertaking
the calculation?
9. Using the data constructed for question 7, calculate the four measures of
strength of association discussed in §7.6.1. Report your findings. Do the
four tests reach the same conclusion about the strength of association
between region and occupational structure? How do you explain any
differences?
10. Can you think of a historical case in which data are available for related
samples? (Hint : look at the exercises for this chapter.)
Multiple linear regression
Multiple relationships
In this chapter we once again take up the subject of regression, first intro-
duced in chapter 4, and this will now be our central theme for the remain-
der of this book. In chapter 4 we dealt only with simple regression, with one
dependent and one explanatory variable. In the present chapter we will
extend the model to see what happens when there is more than one explan-
atory variable. We introduce this idea in §8.1, and explain various aspects
of the concept of multiple regression in §8.2. The related concepts of partial and multiple correlation are covered in §8.3.
In chapter 9 we will examine some of the underlying ideas in more
depth, and will also deal with some of the issues arising from the fact that
the data underlying our regressions are typically drawn from samples and
so are subject to sampling error. Two further extensions of the basic linear
regression model, the use of dummy variables and of lagged values, are
then introduced in chapter 10.
8.1 The inclusion of additional explanatory variables
Extension of a regression to include several explanatory variables is fre-
quently desirable, because it is usually appropriate to formulate and
discuss relationships in which the behaviour of the dependent variable is
explained by more than one factor.
The basic idea is very simple. In the initial treatment of the relationship
between two variables (bivariate regression) in §4.2.1 there was one dependent variable (Y), which was influenced by one explanatory variable
(X). In multiple regression we can have more than one explanatory vari- able (X1, X2, ... , Xn), each of which may exercise an influence on the single
dependent variable.
Where previously the (linear) relationship was expressed as
Y�a�bX (8.1)
it might now be
Y�a�b1X1�b2X2 b3X3 (8.2)
Thus in an analysis of the level of per capita relief payments by English
parishes in 1831, Y would be RELIEF and X1 might still be UNEMP, as in
§4.2.3, but additional explanatory variables could also be incorporated in
the investigation to assess their effect on RELIEF. For example, X2 might be
FARMERS, the proportion of labour-hiring farmers in the total number of
parish taxpayers; and X3 might be LONDON, a measure of the distance of
the parish from London, used as a proxy for the cost of migration. The
signs before the explanatory variables indicate that Y is expected to rise as
X1 and X2 rise, but to fall as X3 rises.
The fundamental underlying theoretical principles and statistical pro-
cedures required for the estimation of multiple regression equations, and
for the evaluation of the coefficients, are essentially the same as those
already discussed in earlier chapters in relation to simple regression. The
formulae and calculations naturally become more laborious and compli-
cated with the inclusion of additional explanatory variables, and the com-
putations can be left to the statistical computer programs.
There are, however, a few points relating to the interpretation of the
results of multiple regression and correlation that it will be helpful to
discuss.
The notation and terminology used for simple regression and correla-
tion is adapted as shown in the right-hand column of table 8.1. As will be
seen, the introduction of additional explanatory variables produces four
new terms. In the following sections we will look at these terms and con-
sider some of the more important changes in interpretation involved when
dealing with multivariate relationships.
8.2 Multiple regression
8.2.1 The coefficient of multiple determination, R2
As with simple regression it is desirable to have a measure of the proportion
of the variation in the dependent variable that is explained by the several
explanatory variables included in a multiple regression model. This
measure is called the coefficient of multiple determination. The logic and definitions are exactly the same as with the coefficient of
determination (covered in §4.3.2), but it is referred to as R2 rather than r2.
Subscripts can be added to show which variables were included; for
example, R2Y.X1X2X3 would indicate that the coefficient had been calculated
for the regression of Y on three explanatory variables, X1, X2,and X3.
THE COEFFICIENT OF MULTIPLE DETERMINATION, R2
R2 is a measure of the proportion of the variation
in the dependent variable
explained by
the several explanatory variables
in a multiple regression.
The value of R2 always lies between 0 and 1, and the higher it is the more
the variation in Y that has been explained (sometimes referred to as an
indication of ‘the goodness of fit’). However, the total variation in Y (which
is the denominator in the calculation of R2) is unaffected by the number of
explanatory variables, whereas each additional explanatory variable will
add a term to the numerator. As a consequence of this, the addition of
another explanatory variable will almost always raise the value of R2, and
will certainly never reduce it. A small correction to R2 to take account of the
number of explanatory variables used is described in §8.2.8.
§.
Table 8.1 Notation for simple and multiple correlation and regression
One explanatory variable Two or more explanatory variables
(simple or bivariate) (multiple or multivariate)
r2 Coefficient of determination R2 Coefficient of multiple
determination (see §8.2.1 and
§8.2.8)
b Regression coefficient b1, b2, . . . Partial regression coefficients
(see §8.2.2–8.2.6)
rYX1 . X2 Partial correlation coefficients� (see §8.3.1–8.3.3)r Correlation coefficient � rYX2 . X1
RY . X1X2 Multiple correlation coefficient
(see §8.3.4)
8.2.2 Partial regression coefficients With multiple regression there is one partial regression coefficient (b1, b2, ... , bk) for each of the k explanatory variables. In (8.2) there were
three such coefficients.
When there was only one regression coefficient, it was interpreted (see
§4.2.1) as the measure of the influence of the explanatory variable (X) on
the dependent variable (Y). If the value obtained for b was, say, �2, this
meant that for every 1 unit that X increased, Y increased by 2 units. What is
the corresponding interpretation for each of the partial regression
coefficients? It is basically the same interpretation but with one additional
– and very important – element.
The use of partial regression coefficients is an extremely powerful ana-
lytical device that enables us to examine the influence of each of the explan-
atory variables in turn while controlling for the influence of the others.
PARTIAL REGRESSION COEFFICIENTS
The partial regression coefficient
for each of the explanatory variables
is a measure of
the influence of that particular variable
on the dependent variable
while the influence of all the other explanatory variables
is held constant.
In experimental sciences it is possible to achieve this effect directly. For
example, a plant scientist can compare the yield of a new variety of seed
with that obtained from an older variety, by planting them in adjoining
beds with identical soil quality, sunlight, and rainfall, and controlling for
all other relevant factors (such as weeding or application of fertilizer). He
can thus isolate the effects of the new variety of seed. A medical scientist
can test a new drug by administering it to a groups of patients while giving
a placebo to a second (control) group selected so as to be as similar as pos-
sible in terms of factors such as age, sex, and health status.
In historical studies and the social sciences this ability to assess the
influence of one particular variable while controlling for the influence of
others can normally be achieved only indirectly, by the application of
appropriate statistical procedures, of which multiple regression is the most
powerful.
8.2.3 Controlling for the influence of a second explanatory variable To illustrate this very important procedure consider the hypothetical data
given in table 8.2. A charity mission operating in Chicago has information
in its records on the amount spent each month, averaged over a period of
years in the 1930s (WELFARE). These outlays covered the costs of provid-
ing accommodation and clothing, operating a soup kitchen, and dispens-
ing medicines. A PhD student writing a dissertation on the history of the
mission is trying to explain the variation from month to month. She ini-
tially finds data for a possible explanatory variable: the number of people
who were sleeping rough expressed as a percentage of the city’s population
(HOMELESS).
She begins by plotting a scatter diagram of WELFARE and HOMELESS
as shown in figure 8.1(a), and finds what appears to be a strong positive
relationship. The regression line (with the t-ratios – see §6.4.2 – given in
brackets) isa
§.
a For simplicity of presentation the results of the regression are given to the nearest whole number
and the t-ratios are rounded to one decimal place.
Table 8.2 Average monthly data on welfare expenditure, homelessness, temperature bands, and food costs (imaginary data)
(1) (2) (3) (4) (5)
Y X1 X2 X3 MONTH WELFARE HOMELESS TEMP FOODCOST
(dollars) (%) (Band) (Jan�100)
Jan 6,500 5.0 3 100
Feb 6,500 5.5 3 105
Mar 3,000 2.5 2 96
Apr 2,500 1.5 2 99
May 2,000 0.5 1 84
Jun 1,500 3.5 1 83
July 1,000 1.5 1 80
Aug 1,500 1.0 1 77
Sept 3,000 4.5 2 77
Oct 3,500 5.0 2 70
Nov 4,500 5.5 2 75
Dec 5,000 4.5 3 79
HOMELESS
6.05.04.03.02.01.00.0
W E
L FA
R E
7,000
6,000
5,000
4,000
3,000
2,000
1,000
0
Figure 8.1 Scatter diagrams for hypothetical data on WELFARE and HOMELESS
HOMELESS
6.05.04.03.02.01.00.0
W E
L FA
R E
7,000
6,000
5,000
4,000
3,000
2,000
1,000
0
(a) Scatter diagram and regression line for all observations
(b) Scatter diagram identifying the effect of TEMP on the relationship between WELFARE and HOMELESS
WELFARE�431�848 HOMELESS (8.3)
(0.5) (4.2)
with r2�0.63. The regression coefficient on HOMELESS is highly
significant (with a t-ratio of 4.2 corresponding to a prob-value of 0.2 per
cent), and indicates that welfare disbursements responded quite strongly
to the proportion of the population sleeping rough, with a rise of 848
dollars for every 1 unit increase in homelessness. (Since the units for HOME-
LESS are percentages, a rise of 1 unit would be a rise of one percentage
point; for example, from 2.0 per cent to 3.0 per cent.)
However, comments by the director of the mission suggested to the
researcher that homeless people were more likely to seek refuge in the
shelter when the weather was bad. She therefore decided to add informa-
tion on weather to the regression. As a measure for this she is able to use a
record of monthly average temperatures (TEMP) grouped in three bands,
warm (coded as 1), cold (2), and very cold (3). When account is taken of
this second explanatory variable a very different picture emerges. The
multiple regression equation (with t-ratios in brackets) is
WELFARE�–1,245�260 HOMELESS�1,910 TEMP (8.4)
(2.3) (1.6) (5.1)
The coefficient of multiple determination, R2 is now 0.91. This indicates
that the new model can explain 91 per cent of the variation in welfare
expenses, a marked improvement on the 63 per cent explained by the
model with only one dependent variable. Other changes are even more
striking.
Compared to (8.3), the coefficient on HOMELESS has dropped from
848 to 260. Thus when the effect of temperature is held constant the partial
regression coefficient shows that benefit payments rise by only 260 dollars
for every 1 percentage point increase in homelessness. Furthermore, the t-
ratio on HOMELESS is now well under 2, and the coefficient is no longer
statistically significant even at the 10 per cent level (prob-value�13.7 per
cent). By contrast, the coefficient on the new variable, TEMP, has a t-ratio
of over 5 and is highly significant (prob-value�0.1 per cent).
To see why the estimate of the response to homelessness has changed so
sharply look at figure 8.1 (b). This has exactly the same data as figure 8.1
(a), but this time the points on the scatter diagram also distinguish the
three different levels of payment according to the temperature bands. It can
be seen immediately that:
� in summer months when the temperature was warm (band 1, marked
with a cross) the outlays on WELFARE were relatively low
§.
� in winter months when it was very cold (band 3, marked by circles)
outlays were relatively high, and
� in early spring and autumn (band 2, marked by squares) they were at an
intermediate level.
It is thus the additional factor of the amount spent in response to the
changes in temperature that is largely responsible for the steep slope of the
bivariate regression line of WELFARE on HOMELESS shown in both dia-
grams.
The distinction between the omission of a relevant explanatory variable
from a regression model and its inclusion is thus very striking:
� If it is omitted, its effect on the dependent variable is simply ignored.
� When it is included, its effect is held constant.
Figure 8.1 (b) also shows three dotted straight lines, one for each tem-
perature band. Each line has the same slope, determined by the partial
regression coefficient on HOMELESS of 260 given by the multiple regres-
sion equation.b These lines illustrate the essence of multiple regression. In
the present example, each of the dotted lines can be thought of as the rela-
tionship one would find between WELFARE and HOMELESS while con-
trolling for the temperature level.
The top dotted line with an intercept of 4,485 is thus in a rough sense the
simple regression line we would get if the only available observations on
WELFARE and HOMELESS were the three circles (i.e. the expenditures by
the mission when the temperature was in band 3).c Corresponding com-
ments apply to the two other dotted lines for the squares and crosses.
8.2.4 Understanding the impact of additional explanatory variables Not every additional explanatory variable will have as startling an impact
on the rest of the regression model as the addition of TEMP. To see this
point in practice, imagine that our researcher, before her attention turned
to weather as a potential explanator of welfare expenses, had collected
information on the cost of food over the year. She found a monthly index
number for food costs in Chicago (FOODCOST) and decided to enter this
as an additional explanatory variable.
The values of the index are presented as X3 in table 8.2. Costs fell
b The lines are also equidistant, but that is not a general feature; it applies in this example only
because the temperature bands move up in equal steps of 1. c The value of the intercept is obtained by substituting the level of TEMP in the multiple regres-
sion equation when HOMELESS�0. For example, when TEMP is in band 3 it becomes
1,245�(260�0)�(1,910�3)� 1,245�5,730�4,485
throughout the year. When she re-ran her initial regression with FOOD-
COST as a second explanatory variable, the results were:
WELFARE� 6,182�846 HOMELESS�77.5 FOODCOST (8.5)
(3.0) (5.9) (3.3)
With the addition of this variable the R2 has increased from 0.63 to 0.83.
The impact of adding the cost of food to the regression is revealing. The
overall explanatory power of the equation has, once again, risen consider-
ably. Clearly, the added variable is helping to explain more of the variation
in the level of WELFARE over the year. Moreover, the high t-statistic on
FOODCOST indicates that it is a statistically significant explanatory vari-
able (the prob-value is 0.9 per cent).
But what is really interesting is that the coefficient on HOMELESS has
barely changed at all from the simple regression model of (8.3). Moreover,
its t-statistic, far from collapsing with the addition of a new statistically
significant coefficient, has actually risen (from 4.2 to 5.9, or from a prob-
value of 0.2 per cent to one below 0.05 per cent). The addition of FOOD-
COST has sharpened the relationship between welfare payments and the
level of homelessness.
Why has this happened? Why is the impact of this alternative second
variable so very different from the addition of TEMP? The answer is to be
found in the relationships among the explanatory variables. In the case of
HOMELESS and TEMP, the reason why the addition of TEMP to the equa-
tion has such a startling effect on HOMELESS is that the two variables were
not only strongly related to WELFARE, but were also strongly related to
each other.
The correlation between HOMELESS and TEMP is 0.73, indicating a
strong degree of interdependence between them.d So when TEMP was
added to the equation, it essentially replicated much of the information
previously provided by HOMELESS. Once it is also recognized that TEMP
was more strongly correlated with WELFARE than was HOMELESS, it
becomes clear that TEMP has more power to explain the behaviour of the
dependent variable. In effect, the addition of TEMP swamps HOMELESS
as an explanatory variable.
In contrast, the movements of HOMELESS and FOODCOST were
almost completely independent of each other.e The correlation of the two
variables is only 0.0026. The addition of the new variable thus provided new
§.
d When two explanatory variables are highly correlated with each other, they are sometimes
referred to as being highly collinear. e When two explanatory variables exhibit independence, as measured by a very low correlation,
they are sometimes referred to as orthogonal to each other.
and independent information that the regression model could use to explain
WELFARE. This information did not replicate any of the information con-
tained in the movement of HOMELESS; its addition therefore did not alter
the relationship between HOMELESS and WELFARE contained in (8.3).
Two morals emerge from this simple example. First, the effect of con-
trolling for additional explanatory variables is a matter of empirical analy-
sis and cannot be determined a priori. Second, it is important for the
historian to understand the underlying structure of the model being tested.
The regression equation produces parameter values net of interdepen-
dences among the variables; but in many cases, it is these interdependen-
cies that are important for a full understanding of the historical process
being analysed.
A good start to a better understanding of the connections among the
explanatory variables is to ask the computer to construct a correlation
matrix, which sets out the simple correlation coefficients for each pair of
variables (including the dependent variable) in tabular form (see table 8.3
in §8.3.1 for an example). A full discussion of how these simple correla-
tions may be used to measure the impact of controlling for an additional
explanatory variable is presented in the analysis of partial and multiple
correlation in §8.3.
8.2.5 The Ballantine for multiple regression The case of multivariate regression can be illustrated using the Ballantine,
first introduced in chapter 4.1 Figure 8.2, on the pullout sheet, p. 527,
depicts the Ballantine for a regression in which two explanatory variables
(X1, X2) influence the behaviour of the dependent variable, Y. Each circle
represents the variation in the variables.f The variation in Y is indicated by
the yellow circle; the variation in X1 by the blue circle; the variation in X2 by
the red circle.
Note that there are significant overlaps between the three circles. These
represent the extent of common variation among variables. The green area
indicates the overlap between X1 and Y, independent of X2; the orange area,
the overlap between X2 and Y, independent of X1. The purple area indicates
the overlap between X1 and X2, independent of Y; the brown area indicates
where all three variables overlap.
The coefficient of multiple determination, R2, measures the extent to
which the variation in the dependent variable, Y, is explained by the varia-
tion in the explanatory variables, X1 and X2. The explained sum of squares
f For convenience, each of the three circles is drawn to the same size. This is not meant to imply
that the variance of each variable is the same. However, as noted in §4.4, having circles of equal
size makes the analytical interpretation of the Ballantine easier.
in this case is equal to the sum of the green, orange, and brown areas. The
yellow area shows the unexplained sum of squares, that part of Y that is not
explained by the behaviour of X1 or X2. The R 2 is thus equal to the ratio of
the green, orange, and brown areas to the total area of the yellow circle.
Although all three areas determine the value of R2, only that part of the
overlap that is unique to each variable is used to determine the partial regres-
sion coefficient. Thus, only the information contained in the green area is
used to determine the value of b1, the coefficient on X1. Similarly, only the
orange area is used to determine the value of b2, the coefficient on X2.
The regression analysis ignores the information contained in the brown
area, since it is impossible to allocate it between X1 and X2. This is the
reason why the regression coefficient on X1 changes with the addition of a
second explanatory variable. It is less that some part of the original explan-
atory power of X1 is displaced onto X2, but rather that the addition of X2 compromises the original structure of the relationship between X1 and Y.
One further effect of the inclusion of a second variable will be that the
relationship between Y and X1 is now estimated with less information (the
size of the overlap is smaller). This means that the accuracy with which we
can measure the value of the regression coefficient declines – the standard
error of b1 increases, and its t-statistic drops.
The Ballantine clearly indicates what is happening in the case of the
Chicago welfare mission. In the original regression formulation in §8.2.3 it
was assumed that only the proportion of homelessness among the popula-
tion influenced outlays on welfare. In terms of the Ballantine, the green and
brown areas in combination were used to determine b1 and R 2.
When TEMP was added to the regression, more information (in the
form of the orange area) was added to the overall model of welfare outlays.
A higher proportion of the yellow circle is thus overlapped, and R2 rises. At
the same time, less information was used to estimate the value of b1, which
now depends on the green area alone, producing both a different
coefficient and a different standard error. With the addition of TEMP, not
only does the coefficient on HOMELESS change (indicating a high degree
of correlation between the temperature on the street and the number of
people sleeping rough), but its statistical significance declines.g
The Ballantine further indicates that the impact on b1 of introducing an
additional explanatory variable will be greater, the higher the degree of
correlation between X1 and X2. In figure 8.3(a), on the pullout sheet, p. 527,
the two variables are independent (or orthogonal to each other). There is no overlap between them. Thus, the addition of X2 to a regression of Y on
§.
g The simple correlation of TEMP and HOMELESS is 0.73.
X1 will have no impact on the estimation of the regression coefficient, b1.
This is an extreme form of the situation that arose when we added FOOD-
COST to the simple regression of WELFARE on HOMELESS in the
example of the Chicago welfare mission.h
In contrast, figure 8.3(b) on the pullout sheet depicts a Ballantine in
which there is considerable overlap between the two variables. The brown
area is very large relative to the green area, indicating that the addition of X2 will reduce the contribution of X1 to Y significantly. A situation when two
(or more) explanatory variables are highly correlated (or collinear) with each other is known as multicollinearity; its implications for the assess- ment of the regression results are discussed in detail in §11.4.2.
8.2.6 The least squares criterion in multiple regression The actual derivation of the partial regression coefficient in multiple
regression models becomes more complex as the number of explanatory
variables increases. In a simple bivariate regression everything can be
understood in terms of two dimensions (as in chapter 4). For a model with
three variables (Y, X1, and X2) it is necessary (and possible) to think in
terms of a three-dimensional diagram. Think, for example, of the corner of
a room: the X1 axis is where one wall meets the floor; the X2 axis is where the
adjoining wall meets the floor, and the Y axis is where the two walls meet.
Then think of a collection of balloons floating above the floor at varying
heights and at varying distances from the two walls. For each balloon, the
distance from each wall represents the values of the two explanatory vari-
ables, and the height above the floor represents the value of the dependent
variable, a picture roughly illustrated in figure 8.4 (a).
Finally, imagine a flat plane such as a piece of cardboard that is to be
suspended in the middle of the space containing the balloons, as in figure
8.4 (b). Some of the balloons will be above the plane (shaded as before),
some below (unshaded). The statisticians’ least squares calculation then
finds the ‘best fitting’ position for the plane by minimizing the sum of
squares of the distance (deviations) of the balloons from the plane. The
analysis is exactly analogous to the procedure discussed in §4.2.2 in which
the least squares criterion was used to select the best fitting line in the
context of a two-dimensional scatter diagram.
The plane can slope up or down independently in two directions: either
from top to bottom or from front to back. The angle of the slope in one
direction (from front to back) measures the effect of X1 while X2 is held con-
h Complete orthogonality would require that the simple correlation of X1 and X2 be 0.00; in the
case of HOMELESS and FOODCOST, the correlation was 0.0026, implying a very small overlap
between the blue and red circles.
stant ; the angle of the slope in the other direction (from top to bottom)
measures the effect of X2 while X1 is held constant.
It is in this sense that multiple regression holds one explanatory variable
constant, while measuring the effect of the other. It was reflected in figure 8.1
(b) by the dotted parallel lines (each with the same slope) showing the effect
of homelessness while controlling for the influence of the temperature.
If there are more than two explanatory variables, all but one have to be
held constant in the calculation of the partial regression coefficient for the
remaining variable. This makes the calculations quite complex, and we can
be grateful that computers do them for us.
8.2.7 Standardized beta coefficients The influence of each explanatory variable in a linear multiple regression
model is measured by the partial regression coefficients in whatever units
the explanatory variable is itself measured. In the example of the Chicago
mission, one would say that WELFARE would increase by 260 dollars if
HOMELESS increased by 1 percentage point ; but by 1,910 dollars if TEMP
increased by 1 unit.
Each of these is an absolute measure of the effect on the dependent vari-
able, but it is not possible to form any impression from them of their rela-
tive importance as influences on expenditure on WELFARE. A suitable
relative measure can be obtained by standardizing the coefficients.2 Thus
for any explanatory variable, Xk, its partial regression coefficient bk is stan-
dardized by multiplying it by the ratio of the standard deviation of its
explanatory variable (sX) to the standard deviation of the dependent
variable (sY).
THE STANDARDIZED BETA COEFFICIENT
The beta coefficient
is a measure of the relative effect
of each of the partial regression coefficients
in comparable units.
For any explanatory variable, Xk,
betak � bk (8.6)
where sX is the standard deviation of the relevant explanatory variable,
Xk,and sY is the standard deviation of the dependent variable, Y.
sX sY
§.
X2 X1
Y
Figure 8.4 Least squares deviation from the regression plane with two explanatory variables
X2 X1
Y
(a) Three-dimensional scatter diagram for a dependent variable and two explanatory variables
(b) Three-dimensional diagram with regression plane
The beta coefficient thus measures how many standard deviations of
movement in Y are caused by a change of 1 standard deviation in each of
the explanatory variables. For the Chicago mission, for example, when the
two explanatory variables were HOMELESS and TEMP, the standardized
betas were 0.24 and 0.76, respectively. For HOMELESS and FOODCOST
they were 0.79 and 0.45.
8.2.8 The adjusted coefficient of multiple determination, R–2
When the coefficient of multiple determination, R2, was introduced in
§8.2.1 it was noted that the introduction of an additional explanatory vari-
able would always raise its value. It is appropriate to make a correction for
the number of explanatory variables used in the model. This is done in
terms of the degrees of freedom (see panel 8.1), given by the size of the
sample and the number of parameters to be estimated (excluding the inter-
cept term).
The resulting adjusted coefficient of multiple determination is called R–2
(R bar squared).
THE ADJUSTED COEFFICIENT OF MULTIPLE
DETERMINATION, R–2
R–2 is equal to R2 corrected for
the number of explanatory variables plus the intercept :
R–2 � 1 (8.7)
where k�the number of explanatory variables plus the intercept,
and the number of df�k 1.
In the simple Chicago mission example used in §8.2.3, the addition of
TEMP raised R2 from 0.63 to 0.91. Correcting the latter for the additional
explanatory variable leads to only a small reduction, with R–2�0.88.
In general, if the sample size is large, there will be very little difference
between R2 and R–2, but if the model is based on a small sample and has a
large number of explanatory variables, the difference can be substantial.
This is a warning that in such a case there may be too few degrees of
freedom to estimate a reliable relationship.
�(1 R2) n 1n k�
§.
8.2.9 Interpreting the intercept term The intercept, or constant term, indicates the intersection of the regression
line with the vertical axis. It shows the value of the dependent variable
when all of the explanatory variables are set equal to zero. But care needs to
Panel 8.1 The degrees of freedom in a regression equation
The concept of degrees of freedom (df) was first encountered in the context
of the t-distribution (§5.4.2 and panel 5.3) and again in connection with
cross-tabulations for �2-tests (§7.3.2 and panel 7.2). In the context of the
coefficient of a regression line, the number of degrees of freedom is deter-
mined by the size of the sample reduced by the number of items of informa-
tion used in specifying the required unknown values.
Consider, for example, a series for which there are only two observations.
It would then be possible to find a linear relationship between that series
and any independent variable (however bizarrely unrelated it might be),
and this would give a perfect fit. The equation for this line would have two
parameters (the intercept, and the slope or regression coefficient) and
would pass exactly through the two points; there would be no degrees of
freedom.
If a third observation was added the line would be unlikely to pass exactly
through all three points, but it might still be a reasonably good fit because
there is still effectively only one ‘free’ observation to explain; there is only
one degree of freedom. With each additional observation the number of
degrees of freedom would increase, and so would the opportunity for the
intercept and slope of the regression line to change. The number of degrees
of freedom thus indicates the number of ‘free’ observations used to calcu-
late the particular statistic.*
For example, in simple regression with one independent variable, using a
sample of n observations, it is necessary to estimate 1 constant and 1 regres-
sion coefficient, leaving n–2 degrees of freedom. In multiple regressions
with k independent variables it is necessary to estimate 1 constant and k
regression coefficients, leaving n (k�1) degrees of freedom. Thus with a
sample size of 20, and 4 independent variables, there would be 15 degrees of
freedom.
* This explanation is based on Peter E. Kennedy, A Guide to Econometrics, 3rd edn., Blackwell,
1992, pp. 66–7.
be exercised when drawing inferences from the intercept. In many cases,
the explanatory variables cannot take the value of zero by construction (the
temperature band in table 8.2, for example) or by observation (the home-
less rate in table 8.2). In many other cases the data over which the regres-
sion is run do not include the value of zero (or even a value close to zero)
for the dependent variable.
This suggests that the intercept is estimated by extrapolating from the
known values of the sample to values outside the sample. Such out of
sample extrapolation is often dangerous, since the underlying relationship
between the explanatory variables and the dependent variable may be very
different for a different range of data. When zero is an unrecorded value for
the dependent variable (Y), great caution needs to be used in interpreting
the predicted or fitted value of the dependent variable (Ŷ).
Even when the value of the intercept meets the requirements of relevance
and can therefore be ascribed meaning, it is still important to understand
what it does mean. The classic example of a meaningful intercept term
involves a regression of household food expenditure on family earnings and
the number of boarders (paying guests) in the household. People do not stop
eating even if income falls to zero; they continue to buy food for themselves,
even if they no longer have a boarder to feed. Thus, even if both explanatory
variables are zero, the dependent variable still takes on a positive value.
In this case, the intercept therefore makes economic sense. Moreover,
since the sample is likely to include some observations in which the house-
hold has no boarders, and in which unemployment reduces earnings to
zero or close to it, the intercept makes statistical sense and is not contami-
nated by out of sample bias.
But what is the positive intercept actually measuring? The family that
buys food is either paying for it out of past savings, or from money lent or
given by family or friends, or is buying it on credit, to be paid for out of a
future pay packet. These are factors that influence food expenditure, but
which are excluded from the list of explanatory variables. Some statisti-
cians generalize from this case to suggest that the value of the intercept
merely indicates the impact of all the variables excluded from the regression.
This is a rather dubious claim (especially since, as §8.2.3 shows,
excluded variables can also influence the coefficients on the included vari-
ables). However, it does point up the general proposition: that thought
needs to be given to the meaning behind an intercept. It should not be
interpreted automatically as the default value of the dependent variable
when all the explanatory variables have a value of zero.
§.
We can illustrate these issues by focusing on the intercepts in two of our
case studies. In Benjamin and Kochin’s analysis of inter-war Britain, unem-
ployment is regressed on two variables, the ratio of unemployment
benefits to wages, and the deviation of log output from its trend. Each of
these could in theory take on zero values. Indeed, the observed value of the
second variable was close to zero for some years but the benefit/wage ratio
never was. So, although the intercept of 5.19 might be interpreted as the
minimum level of unemployment in inter-war Britain (i.e. 5.19 per cent)
without an insurance scheme and with ‘full employment’, it is important
to recognize that such an interpretation requires that the model be extrap-
olated outside of its sample range to produce such a number.
In Hatton and Williamson’s time-series model of Irish emigration, the
intercept is –11.61, implying that if all the dependent variables were zero,
the model would predict Irish immigration at this rate (i.e. a negative inter-
cept would mean that migrants would be entering rather than departing
from Ireland). It is hard to make sense of this term. Not only is the concept
of Irish immigration to Ireland hard to accept, especially if the migrant
stock of Irish abroad is set equal to zero, but so too is the concept of univer-
sal zero unemployment and zero relative wages.
8.3 Partial and multiple correlation
8.3.1 The interpretation and order of partial correlation coefficients Just as the partial regression coefficient measures the effect of one variable
while holding the others constant, so the partial correlation coefficients control for the influence of the other variables specified in the relationship.
The basic principles are the same as in simple correlation, but the nota-
tion changes slightly to show which variables are included in the correla-
tion and which are held constant. Thus rYX1 . X2X3 indicates the correlation of
Y and X1, while controlling for the influence of X2 and X3.
PARTIAL CORRELATION COEFFICIENTS
The partial correlation coefficient
is a measure of
the strength of the linear relationship
between two variables,
while controlling for the influence of the other variables
included in the correlation.
The number of control variables on the right of the point determines what is
called the order of the partial correlation coefficient. Thus simple correlation, which ignores any other variables, is called a zero-order correlation. With two control variables, as in the example in the previous paragraph, it is a second- order correlation (or partial), with three it is a third-order, and so on.
Consider, for example, the inter-relationship between five variables
from the Poor Law data set for a sample of 311 parishes: RELIEF, UNEMP,
FARMERS, GRAIN, and LONDON. As a first step in the analysis of the
relationships between these five variables a statistical computer program
might be used to calculate the zero-order correlation between all possible
pairs; in this case there would be 10 pairs. The results are typically pre-
sented in a matrix of five rows and five columns, as in the upper panel of
table 8.3, together with the two-tailed significance levels of the coefficients.
With each of these simple (bivariate) correlations no account is taken of
any possible effect of the three remaining variables. In order to do this it is
necessary to calculate first-, second-, and third-order partials, taking
various combinations of two variables while controlling for one, two, or
three of the others.
The lower panel of table 8.3 shows two of the third-order partials. In the first, RELIEF is correlated with UNEMP while controlling for FARMERS,
GRAIN, and LONDON. In the second, RELIEF is correlated with
LONDON, while controlling for the three remaining variables.
These tables can then be scrutinized to see how the coefficients change
with the introduction of the various control variables. For example, the
simple (bivariate) correlation coefficient for RELIEF and LONDON was
–0.35; when the influence of the three other variables was controlled for,
the third-order coefficient dropped to –0.26.
This partial correlation procedure can thus help one to get a better
understanding of the possible causal links within a set of related variables.
It not only measures the strength of the various combinations of relation-
ships, but can also be a very effective way of detecting hidden relationships.
Whereas the OLS regression coefficients indicate the net effect of the rela-
tionships among the explanatory variables, partial correlation coefficients
enable the researcher to diagnose more completely the process by which
the regression coefficients are generated.
They can also help to uncover spurious relationships, by revealing when
an apparent relationship between X1 and Y is actually the result of two sep-
arate causal relationships between these two variables and a third variable,
X2. The fact that the actual causal relationship is not between X1 and Y
would emerge through the calculation of a first-order partial correlation
coefficient in which X1 was correlated with Y while controlling for X2.
§.
Table 8.3 Correlation coefficients and partial correlation coefficients for five variables (data relate to 311 parishes in England and Wales)
(a) Zero-order correlation coefficients
RELIEF UNEMP FARMER GRAIN LONDON
RELIEF r 1.000 0.439 0.198 0.365 0.347
Sig 0.000 0.000 0.000 0.000
UNEMP r 1.000 0.113 0.196 0.170
Sig 0.047 0.001 0.003
FARMER r 1.000 0.385 0.150
Sig 0.000 0.008
GRAIN r 1.000 0.372
Sig 0.000
LONDON r 1.000
Sig
(b) Third-order partial correlation coefficients
i Correlation of RELIEF and LONDON
controlling for FARMER, GRAIN, and UNEMP
RELIEF LONDON
RELIEF r 1.000 0.256
Sig 0.000
LONDON r 1.000
Sig
ii Correlation of RELIEF and UNEMP
controlling for FARMER, GRAIN, and LONDON
RELIEF UNEMP
RELIEF r 1.000 0.383
Sig 0.000
UNEMP r 1.000
Sig
An illustration of the use of partial correlation coefficients An excellent illustration of the way in which the inter-relationship between
variables can be illuminated by the use of partial correlation coefficients is
found in a recent study of attendance at church in Victorian England.i Using
the rich data available in the 1851 Religious Census, Alasdair Crockett exam-
ines the inter-relationship between three variables. These are an index of
churchgoing based on total attendances at all churches on Sunday 30 March
1850, expressed as a percentage of the total population (which we will call
ATTEND); population per square kilometre (POPDENS); and churches per
square kilometre (CHURCHES). His data cover a cross-section of 258 regis-
tration districts in rural England and 318 urban districts.
He presents the results of a series of zero-order and partial, first-order
correlations between these three variables across the rural and urban dis-
tricts, which we have reproduced in table 8.4.
The zero-order correlations in rows 1 and 2 of the table show both
POPDENS and CHURCHES positively correlated with ATTEND among
rural districts, and both negatively correlated with ATTEND among urban
districts. All these correlations are statistically significant, with prob-values
less than 0.001.
§.
i Alasdair Crockett, ‘Variations in churchgoing rates in England in 1851: supply-side deficiency or
demand-led decline?’, in Alasdair Crockett and Richard O’Leary (eds), Religion in Modernity:
Patterns and Processes in Europe and America, Cambridge University Press, forthcoming.
Table 8.4 Zero-order and partial correlations between church attendance, population density, and church density in England in 1851a
258 rural 318 urban
districts districts
Zero-order correlations
1 ATTEND and POPDENS 0.353** 0.536**
2 ATTEND and CHURCHES 0.572** 0.418**
First-order partial correlations
3 ATTEND and POPDENS
controlling for CHURCHES 0.099 0.617**
4 ATTEND and CHURCHES
controlling for POPDENS 0.379** 0.532**
Notes: a Note that log values are used for both POPDENS and CHURCHES in the urban analysis.
** indicates p 0.001.
Source: Crockett, ‘Variations in churchgoing rates’.
However, when we turn to the partial correlations it is evident that in
rural England there is no remaining statistically significant positive associ-
ation between ATTEND and POPDENS once the third variable,
CHURCHES, is controlled for (row 3). In contrast, the strong positive
association between ATTEND and CHURCHES persists when POPDENS
is controlled for (row 4). Crockett concludes from this that population
density appeared to be positively associated with churchgoing across rural
England only because of its underlying relationship with church density.
The story revealed by the first-order correlations for the urban districts
is also interesting. The strong negative association between ATTEND and
POPDENS (row 1) is still evident when CHURCHES is controlled for (row
3). However, when POPDENS is controlled for, the relationship between
ATTEND and CHURCHES is strongly positive (row 4). It only appeared to
be negatively correlated (row 2) because of the underlying relationship
with population density.
8.3.2 Removing the influence of the control variable How are the control variables actually held constant? The essential princi-
ple can be explained for the case of a first-order coefficient, i.e. where
there is one control variable in a total of three variables. To illustrate the
explanation let us continue with the churchgoing variables, with the aim of
calculating the partial correlation coefficient between ATTEND and
POPDENS while controlling for CHURCHES.
The basic technique starts with the calculation of two simple regression
lines, in each of which the control variable appears as the explanatory variable :
In the first POPDENS is regressed on CHURCHES
In the second ATTEND is regressed on CHURCHES
The first of these fitted regression lines explains as much of the variation
in POPDENS as can be attributed to CHURCHES. The remaining, unex-
plained, variation is reflected in the residuals. ATTEND will be one factor
contributing to these residuals.
The second regression line explains as much of the deviation in
ATTEND as can be attributed to CHURCHES. The remaining, unex-
plained, variation is again reflected in the residuals. POPDENS will be one
factor contributing to these residuals.
With these two regressions CHURCHES has been allowed to explain as
much of the variation in the two other variables as it can. The final step is
then to correlate the residuals from the two regressions. This gives the
desired measure of the relationship between POPDENS and ATTEND that
is independent of any influence from CHURCHES.
On the basis of these correlations, Crockett advances a spatial interpre-
tation of English churchgoing in the nineteenth century that focuses on the
different geographies of church density in rural and urban parishes. Rural
parishes were more spread out and many parishioners had long distances
to walk on a Sunday. This constrained church attendance and explains the
insignificant partial correlation between POPDENS and ATTEND after
correcting for CHURCHES. In contrast, geographic isolation played no
role in an urban environment replete with churches and chapels of many
denominations.
Crockett instead argues that the strong negative correlation of
POPDENS and ATTEND after correcting for CHURCHES in urban dis-
tricts is better explained by the effects of industrialization on the propensity
to attend church. In Crockett’s terminology, rural church attendance was
limited by supply factors, and urban church attendance by demand factors.
8.3.3 Calculation of partial correlation coefficients The derivation of the formula for the calculation of partial coefficients will
not be explained here. It is sufficient to note that the required measurement
of the correlation between the residuals is obtained from the simple corre-
lation coefficients among the several variables for which there is a multiple
relationship.
Assume, for example, that there are three variables, Y1, X2, and X3, for
which there are three simple (zero-order) correlation coefficients, rY1 X2,
rY1 X3, and rX2X3. (For simplicity the Xs and Ys will be omitted in the subse-
quent paragraphs.)
The first of the (first-order) partial correlation coefficients is r12 . 3. This
measures the strength of the relationship between Y1 and X2 when any
influence of X3 has been removed. The formula is
r12.3 � (8.8)
The second partial coefficient r13.2 similarly measures the relationship
between Y1 and X3 when any influence of X2 has been removed. It can be
derived in the same way by an appropriate re-arrangement of the simple
correlation coefficients.
This formula can be used to explain what happened in our example of
the Chicago welfare mission. The impact of adding a second explanatory
variable to the basic regression of WELFARE on HOMELESS differed,
depending on whether FOODCOST or TEMP was included. This impact is
essentially the difference between the zero-order partial correlation
between WELFARE and HOMELESS, and the two first-order partials, con-
trolling for FOODCOST and TEMP, respectively.
r12 (r13)(r23)
�(1 r132 )(1 r232 )
§.
The matrix of zero-order partial correlations among our variables looks
like this j
HOMELESS FOODCOST TEMP
WELFARE 0.7960 0.4495 0.9368
HOMELESS 0.0026 0.7271
FOODCOST 0.4521
If we plug these figures into (8.8) for the two cases, we find that the first-
order partial correlation between WELFARE and HOMELESS controlling
for FOODCOST is 0.8898, an increase compared to the zero-order correla-
tion of 0.7960. The calculation is
� � 0.8898
By contrast, the corresponding first-order partial correlation between
WELFARE and HOMELESS controlling for TEMP is only 0.4782, a much
lower value than the zero-order correlation.
This contrast explains why the strength of HOMELESS in the regression
in (8.3) fell dramatically with the addition of TEMP as an explanatory var-
iable, but rose slightly when FOODCOST was added.
In general, the explanatory power of an explanatory variable will increase
with the addition of a new variable that is weakly correlated to it ; and will fall
with the addition of a new variable that is strongly correlated to it.
8.3.4 The coefficient of multiple correlation, R Finally, we may want to know how much of the variation in the dependent
variable is provided by all the explanatory variables taken together, rather than
by each separately. This is given by the coefficient of multiple correlation.
THE COEFFICIENT OF MULTIPLE CORRELATION, R
R is a measure of the
closeness of the linear relationship
between all the variables
taken together.
The letter R is used for this measure, with subscripts to indicate which
variables have been included in the calculation. Thus RY · X1X2X3 would
indicate the correlation of Y with three other variables.
0.7948
�0.7979�0.9999 0.7960 (0.4495)(0.0026)
�[1 (0.4495)2]�[1 (0.0026)2]
j We have simplified the presentation by comparison with table 8.3 by omitting both the own cor-
relations (e.g. WELFARE with WELFARE) and the statistical significance.
Numerically, for any given set of variables, R is simply the square root of
the corresponding coefficient of multiple determination, R2 (see §8.2.1).
By convention R is always presented as positive, but the sign of the overall
correlation coefficient has no meaning, since some of the explanatory vari-
ables may be positively correlated and others negatively. R can vary
between 0 and 1. A value of 0 indicates no linear relationship, but – as with
the two-variable case – it is always possible that there is a strong non-linear
relationship.
Notes 1 The name Ballantine was originally given to the shape of interlocking circles because
of the similarity between figure 8.2 and the advertising logo of P. Ballantine & Co., a
producer of beer and ales in Newark, New Jersey in the nineteenth century. The
Ballantine logo consisted of three interlocking rings, each symbolizing the essential
characteristics of Ballantine ale. Before Prohibition, these characteristics were
identified as Purity, Strength, and Flavor; after the repeal of the 18th Amendment in
1933, the ring of Strength was renamed the ring of Body. 2 For examples of the use of beta coefficients to compare the relative magnitudes of the
effects on the dependent variable of different explanatory variables see William A.
Sundstrom and Paul A. David, ‘Old-age security motives, labor markets and farm
family fertility in antebellum America’, Explorations in Economic History, 25, 1988,
pp. 190–2; and Michael Edelstein, Overseas Investment in the Age of High
Imperialism, Columbia University Press, 1982, pp. 102–9.
8.4 Exercises for chapter 8
1. Write a paragraph interpreting the following results from a multiple
regression model explaining a government’s VOTE share in each General
Election since 1945 in terms of the current economic GROWTH rate (per
cent per annum), the current level of UNEMPloyment (per cent per
annum), the current level of INFLTION (per cent per annum), and the
number of STRIKES in the previous year. Be sure to discuss the meaning of
the regression coefficients and associated t-statistics, the intercept term,
and the coefficient of multiple determination.
VOTEt�45.5�5.4 GROWTHt 1.2 UNEMPt 2.6 INFLTIONt 5.1 STRIKESt 1 (8.3) (4.2) (3.6) (2.1) (1.4)
R2�0.65
2. You wish to build a model of family size in Irish counties in the early
twentieth century. Use the Hatton and Williamson cross-section data set to
test the hypotheses that family size is directly related to religion (CATHO-
LIC), age (AGE), and literacy (ILLITRTE).
§.
(i) Test the hypotheses independently, by calculating the simple (zero-
order partial) correlation coefficients for the variables across Irish
counties in 1911.
(ii) Test the hypotheses jointly by calculating the partial correlation
coefficients, holding the other variables constant.
(iii) Test the hypotheses by running a multiple regression of FAMSIZE
on the three variables.
Compare the results. What points of similarity and dissimilarity do you
find between the three approaches? Explain your findings.
3. Use the Ballantine methodology to interpret what happens in the
Benjamin and Kochin model of unemployment when the number of
explanatory variables is increased from one to two. Does it matter in which
order the variables are added? Use the Ballantine method to illustrate what
is happening when YEAR is added as a third variable in the equation.
(Hint : look at the correlation matrix of all three variables.)
4. How would you respond to a critic of regression methodology who sug-
gests that the analysis of partial correlations produces all the information
you need to test hypotheses and evaluate relationships between variables,
without any of the costs implicit in formal modelling?
5. Boyer’s preferred OLS equation to explain county-by-county variation
in RELIEF payments is reproduced in table 15.1 (‘reduced-form model’).
Generate beta coefficients for this regression. How does converting the
results in this way alter our perception of the importance of the variables in
explaining parish-by-parish variation in relief payments per capita?
6. What would you infer from a result in which the R2 is positive but small,
while the R–2 is small but negative?
7. You run two regressions. In the first, the R–2 is high, but the individual t-
statistics on each variable are low. In the second, the individual t-statistics
are high, but the overall R–2 is low. Write a paragraph interpreting these
results.
8. You run two variants of the same basic regression. In the first version, the
coefficient on variable X2 is 0.3 with a t-statistic of 1.41; in the second
version, the coefficient is �0.25 and the t-statistic is 1.02. How do you
interpret these differences? What do they tell you about the status of X2 in
the regression as a whole?
9. Are the following statements true or false? Explain your reasoning.
(i) In a regression, if the values of X3 are doubled, the coefficient on X3 will also be doubled.
(ii) In a regression, if the values of Y are halved, the coefficients on each
X will also be halved.
(iii) In a regression of Y on log X, the regression coefficient is b. If the
regression were re-estimated using Y on X, the regression coefficient
would be log b.
(iv) In a regression, if every value of Y is larger by 5 units, the value of the
intercept will be larger by 5 units.
(v) The addition of an orthogonal variable will raise R2 but lower R–2,
whereas the addition of a collinear variable will raise both.
10. Evaluate the following propositions concerning R2:
(i) ‘The correct way to test for the relevance of an additional explana-
tory variable is to compare the level of R2 before and after its inclu-
sion. So long as the R2 goes up, the additional variable should be
included in the regression.’
(ii) ‘The whole purpose of running regressions is to explain as much as
possible about the behaviour of the dependent variable. This means
getting the best fit from the data. The essence of the OLS model is to
minimize the least squares, i.e. to minimize the residual sum of
squares from a regression. Since the R2 statistic is simply the ratio of
the explained to the total sum of squares, it follows logically that the
best approach to regression estimation is to maximize the value of
R2.’
§.
The classical linear regression model
This chapter builds on our previous work and takes a succession of giant
strides towards realistic quantitative analysis of relationships between two
or more variables. After completing this material it will be possible to
examine a large range of potential historical problems, and to read criti-
cally many of the studies in history and the social sciences that make use of
regression and related quantitative techniques.
We will begin in §9.1 with a brief discussion of the concept of a ‘model’
and of the associated methodology of quantitative research into the rela-
tionships between two or more variables. The basic technique of linear
regression (discussed in chapters 4 and 8) is then extended in §9.2 to
examine the reasons why the observed values of the dependent variable
deviate from the regression line, and to consider the implications of these
deviations. In §9.3 a new test statistic, the F-test, is introduced and used to
test the significance of the multiple regression as a whole. Finally, §9.4 is
devoted to a further useful summary statistic, the standard error of the
estimate.
9.1 Historical research and models of relationships between variables
One of the principal features of quantitative research in the historical and
social sciences is the attempt to analyse and explain the behaviour of some
particular variable. The social historian may wish to explore changing pat-
terns of racial discrimination; the urban historian may study the growth of
cities. For the economic historian the subject may be the factors that
influenced the supply of – and demand for – wheat or cotton or hogs, or the
choice made by women between staying at home and going out to work.
The political historian may investigate the voting behaviour of the electo-
rate, or the effect of an incumbent president on congressional elections; for
§.
the demographer it may be the trends and fluctuations in fertility and mor-
tality or in international migration.
In each of these and many similar problems a quantitative analysis
begins with the specification of a model.a This is the researcher’s formula- tion of the relationship between the phenomenon that she wishes to
analyse, and other factors that she thinks will help to explain that phenom-
enon. In our earlier terminology she has one dependent variable, and
attempts to specify which are the relevant explanatory variables, and how
they are related to the dependent variable.
In most historical writing this is, of course, done implicitly. The non-
quantitative historian does not say in as many words: ‘here is the model I
have formulated to explain why the number of criminal offences first
increased and then declined’, but her discussion of the issue is, in effect, an
attempt to do this.1
The distinctive feature of the quantitative approach is first, that the rela-
tionships are made explicit; secondly, that they can be measured and tested.
Whether or not this is a better approach depends largely on how well the
relationships can be specified by the researcher, and how accurately the
major relevant variables can be measured. Even if the results are statisti-
cally poor, the exercise will probably help to clarify the crucial issues.
The specification of the model will thus determine:
(a) The variables to be included.
(b) The researcher’s a priori expectations about the sign of the regression
coefficients, i.e. whether a rise in an explanatory variable will be asso-
ciated with a rise or a fall in the dependent variable.
(c) The mathematical form of the relationship between the variables (for
example, whether it is linear or has some other particular non-linear
form).
Once a model has been specified, the subsequent stages in the analysis
are the estimation and the evaluation of the model. Estimation is the col- lection of the necessary data and the application of appropriate statistical
procedures to obtain estimates of the size and sign of the regression
coefficients (i.e. the values for b1, b2, etc.). Evaluation is then the applica- tion of the hypothesis testing procedures to decide whether the estimates
are reliable and statistically significant, and of historical or other criteria
(including common sense) to decide whether they are meaningful and
important.
a This is also referred to as the formulation of a maintained hypothesis.
9.1.1 Quantitative methods and historical judgement A great deal of the material covered in this text is precise and mechanical. If
one is required to calculate the mean and variance of a series, or a correla-
tion coefficient for two variables, there is only one correct answer.
When dealing with confidence intervals and hypothesis testing there
may be a little more scope for judgement – both when setting significance
levels and when evaluating minor divergences about the chosen level – but
the basic procedures are laid down and otherwise leave little room for dis-
agreement. However, when it comes to the formulation of quantitative
models to analyse and explain the behaviour of a particular variable, the
position changes dramatically. Elements of the procedures will still involve
precise mathematical or statistical calculations, but the fundamental
approach is entirely dependent on the judgement of the researcher, and
there is no uniquely correct answer.
It is helpful to stress this, because as soon as one starts to read the books
and articles that use these methods it becomes evident that many topics
remain highly controversial. Where authors disagree strongly, quantitative
methods seldom resolve disputes. This may seem surprising given the
apparent precision of quantitative measurements and techniques. But
no model of the relationship between variables can ever be precisely or
comprehensively specified. It is always possible to question the specifi-
cation of the model, and to suggest other formulations that might be more
satisfactory.
It then becomes necessary to judge the merits of the competing models,
and here, too, there is enormous room for discretion and disagreement.
There are likely to be differences in the choice of explanatory variables or in
the way they are measured. Furthermore, it is seldom the case that all the
relevant criteria point unambiguously in the same direction. One model
may have a high R2, but the regression coefficients may have large standard
errors so that they are not statistically significant at acceptable levels. The
rival model may have lower standard errors on the coefficients but per-
form less well on other tests, such as the ability of the model to predict the
movement in the variable outside the period for which the regression was
calculated.
A fundamental aim of this chapter (and of the further discussion in
§12.5 and §12.6, and in the case studies in chapters 14 and 15) is thus to help
readers understand (a) how these differences arise, (b) why the various cri-
teria for evaluating the merits of different models may give conflicting ver-
dicts, and (c) how the approach may be of considerable benefit to historical
understanding even when the results are not conclusive.
§.
9.2 Deviations of the dependent variable from the estimated regression line
In the preliminary consideration of the fitting of the regression line by the
ordinary least squares (OLS) procedure in §4.2 we were of course aware
that many of the observations of the dependent variable (Y) did not lie on
the fitted line.
The measurement of the goodness of fit (r2) given in §4.3 explicitly dis-
tinguished between:
(a) the part of the deviation of Yi from Y –
which was explained by the
regression line (�ŶI– Y –
), and
(b) the part which remained unexplained (�Yi Ŷi).
It is now necessary to consider more carefully the reasons for these devia-
tions from the regression line and to examine their implications. The
reasons why the observed values of Y do not all lie neatly on the regression
line can be broadly divided into two groups.
9.2.1 Errors of measurement and specification The first group of reasons consists of errors of measurement and
specification.b These can take several forms, including:
(a) The dependent variable, Y, may be measured incorrectly. This might, for
example, be the result of inaccuracies in the collection or processing of
the data, or of ambiguities about the precise definition of Y.
(b) Relevant explanatory variables may be omitted from the model. This
could happen because it did not occur to the researcher to include them;
or because the necessary data were not available; or because they consist
of aspects such as tastes or attitudes that are not readily measurable.
(c) There may be other forms of mis-specification of the model, including:
(i) Assuming a linear relationship between the variables when it is
actually non-linear.c
(ii) Including irrelevant explanatory variables, because the researcher
wrongly thought they were important.
(iii)Assuming that a time-series relationship is constant over time (so
that there is no change in the intercept or regression coefficients)
when it is actually changing.
b The technical aspects of ‘model specification’, i.e. of the formulation of a relationship between a
dependent variable and one or more independent variables, are examined in §11.2 and there is a
discussion of the more general issues in §12.5 and §12.6. c The specification of non-linear models will be discussed in chapter 12.
(iv) Assuming that a cross-section relationship is constant across all the
households, parishes, or other units of observation (so that the same
intercepts and regression coefficients apply to all units) when
there is actually a structural difference between two groups of
units (e.g. between states in the north-east of the United States
and those on the frontier).
(v) Assuming that there is a one-way relationship from the explanatory
to the dependent variable when there is actually a two-way rela-
tionship. For example, consumer demand for a product may be
influenced by its price, but the price may – at the same time – be
influenced by the volume of demand.
9.2.2 Stochastic (random) factors and the error term, e However accurately and fully the researcher specifies a relationship, there
will always be additional random factors that occur irregularly and unpre-
dictably, and will have an impact on the dependent variable. These are
referred to as stochastic factors and make up the second group of reasons why the observed values of Y do not all lie neatly on the regression line.
Stochastic factors may be relatively large events such as strikes, wars,
riots, fires, droughts, or epidemics. Or they may be factors which are indi-
vidually trivial and beyond analysis (for example, the individual attitudes
or erratic behaviour of a particular parish officer on a particular occasion),
but collectively can have an important effect on the actual value of the
dependent variable.
More generally, it is characteristic of the human behaviour with which
we are concerned in history and the social sciences that it is not absolutely
uniform and consistent. Individuals and groups do not always respond in
an identical way to any given stimulus.
These stochastic factors are thus equivalent to additional explanatory
variables that, even if they were measurable, could not be sufficiently
identified to be incorporated in the model of the relationship. We deal with
them by means of the concept of an error term. When a simple linear relationship between two variables was initially
specified, it was expressed as
Y�a�bX (9.1)
In order to recognize both the existence of possible errors of measurement
and specification, and the potential importance of the stochastic factors,
statisticians prefer to add to this an additional ‘error term’ designated by e.d
d Some authors refer to the error term as a disturbance or a stochastic term, and it is also desig-
nated in some texts by u or by the Greek letter � (epsilon).
§.
The effect of doing this is to transform the model from an exact or deterministic relationship to a stochastic one. This is an extremely impor- tant distinction, and represents the adaptation of a purely statistical
method to the stochastic nature of almost all social phenomena.
The estimated relationship thus becomes
Y�a�bX�e (9.2)
This adds an additional (random) variable that covers all the possible
reasons for the deviation of Yi from the value Ŷi indicated by the regression
line.
Since the value of this random variable, ei, cannot be actually observed
in the same way as the other explanatory variables, certain assumptions
have to be made about how it might behave. These assumptions are of crit-
ical importance to the evaluation of the model.
9.2.3 Sampling and the regression line Before considering these assumptions it will be helpful to emphasize that
we are essentially dealing with a characteristic sampling problem, created
because our data consist of a single sample drawn from a much larger pop-
ulation containing many more observations.
This is not always immediately obvious. Say, for example, that the issue
to be considered is the relationship between emigration from Ireland and
wages in the United States, to see whether wages are a possible explanatory
variable. The researcher may have available aggregate national data on
these two variables for the period 1877–1913. He would perhaps not
immediately think of this as a ‘sample’ in the same sense as the Boyer data
from a handful of parishes, or a poll of a small sample of adults across the
country to ask about their voting intentions in a forthcoming presidential
election.
But if he is interested in the way in which decisions about migration are
influenced by the prevailing level of wages, then the information generated
by the specific experience of the particular individuals who emigrated from
Ireland in the years 1877–1913 can be thought of conceptually as a sample
from the infinite population. This infinite population would consist of all the possible migration decisions made by all those who contemplated emi-
grating in the sample period or in other periods, or who might do so in the
future. It is thus necessary to determine how reliable this sample informa-
tion is as a measure of the underlying population.
As before, Greek letters are used to indicate this ‘true’ regression line
for the population, replacing a by � (alpha), b by � (beta), and e by �
(epsilon).
Y����X�� (9.3)
We do not know the values of � and � for the population; all we have are
the estimates of a and b in the regression line derived from the sample. It is
thus necessary to establish how reliable a is as an estimate of �, and how
reliable b is as an estimate of �. In order to do this we must revert to the
issue of the error term.
9.2.4 Assumptions about the error term Imagine that it is possible to draw a very large (theoretically infinite)
number of successive samples from a given population, each showing
the relationship between two variables, X and Y.e For example, Y could be
the quantity of meat supplied to urban markets by cattle farmers, and X the
price of beef. A possible set of hypothetical outcomes is illustrated in figure
9.1 (a). For every value of X on the horizontal axis, there are numerous
different values of Y.
Consider any single value of X, say X1. When the price of beef (X) is at
this particular level, some quantities supplied (Y) occur repeatedly in
different samples, others only very occasionally. Those values that occur
more than once rise vertically above each other, illustrated by the dashed
line drawn at the mean value, �1. When all the samples had been completed
we would be able to plot a distribution of all the possible values Y could
take when beef prices have a value of X1. The hypothetical result pictured in
figure 9.1 (a) shows this distribution as smooth and symmetrical, with a
narrow range between the left- and right-hand tails of the distribution, and
thus a low variance.
Similarly, when the beef price has the higher value X2, there would again
be a distribution of values for Y, but in this case it is assumed that it is
spread out over a much greater range, and thus has a higher variance. It has
a higher mean value (�2) and is strongly skewed to the right. And for X3,
there is yet another pattern of distribution, with an m-shape that is neither
smooth nor symmetrical, and with two modes. The mean, �3, is higher
than for X2, but the dispersion (variance) is narrower.
There could thus be a separate hypothetical distribution of the values of
Y for every value of X, each with its own shape and its own mean and vari-
ance. The preceding discussion of the error term explains why these
differences occur, so that there is not a single uniform value of Y for each
e This would be analogous to the exercise specified in question 7 of chapter 5, except that the infor-
mation assembled from the successive samples would be the values of the two variables which
are assumed to be related, not the sample means of a single variable.
§.
X3
Frequency of Y
X2
X1
Y
µ1 µ2 µ3
Figure 9.1 The population of Y for different values of X
X3
Frequency of Y
X2
X1
Y
µ1 µ2 µ3
(a) Hypothetical populations
(b) The form of the population of Y assumed in simple linear regression
value of X. The differences between the distribution of Y at X1, X2, and at X3 are thus determined by the behaviour of the corresponding error terms, e1,
e2, and e3.
In practice, of course, we do not have the luxury of the information
about the population which could be obtained from an infinite number of
samples; we have only a single sample, and thus a single value of Y for each
value of X. But it is nevertheless necessary to know something about the
possible distribution of the error terms in order to fit the best possible
regression line to the sample data. To get round this problem the statisti-
cians assume that the error term, e, is a random variable, and that its distri-
butions obey certain simple and plausible rules. These key assumptions
about the error term are illustrated in figure 9.1 (b), and are set out more
formally in panel 9.1.
Panel 9.1 Assumptions of the classical linear regression model
The four basic assumptions of classical linear regression (CLR) are that for
any particular model:
1. The mean value of the random error, e, is zero (��0) for all values of
X.
2. The variance of the random error, e, is the same for all values of X, and
is equal to the population variance, 2.
3. Values of e for different values of X are statistically independent so that
if, say, Y1 is large, there is no reason to think that this will have any
effect on Y2, making it more likely to be either large or small.
4. The mean values of Y all lie on a straight line. This is the population
regression line.
In addition, it may optionally be assumed that the values of Y are nor-
mally distributed about their mean for each value of Y, and thus that the
values of the random error, e, are also normally distributed about their
mean. This assumption is often made, (and is adopted in figures 9.1 (b) and
9.2) but it is not required for the estimation of the coefficients in the most
widely used form of regression analysis, the classical linear OLS model.
However, the normality assumption is required in order for hypotheses
about the values of the coefficients to be tested by standard procedures for
statistical inference, such as those to be discussed in §9.3.*
* The reason for this was explained in §6.2.1.
§.
If we had the information for all possible values of the relationship be-
tween X and Y we could fit the true population regression line, Y����X.
Since we have only the sample information the best that can be done is to
estimate the sample regression line, Ŷ�a�bX, assuming that the distribu-
tion of the error term is as specified in panel 9.1.
Since the true population regression line is not actually known, the error
terms representing the deviation of the actual values of Y from the true
regression line cannot be measured. We can, however, observe the devia-
tions of the estimated regression line from the sample values of Y.
The relationship between this estimated regression line and the true
(unknown) regression line that would be given by the population data is
shown in figure 9.2. The values of Y observed in the sample are Y1, Y2, and
Y3, and the estimated regression line passes through the values Ŷ1, Ŷ2, and
Ŷ3. Because of the random error terms, e1, e2, and e3, the values of Y pre-
dicted by the true regression line differ from the observed values of Y. These
error terms reflect the combined effect of the errors of measurement and
specification discussed in §9.2.1, and the stochastic factors considered in
§9.2.2.
The best model When the four basic assumptions of the classical linear regression model all
hold, the OLS model is the best approach to regression analysis. In the lan-
guage of the statistician, the OLS model is BLUE, which stands for the best linear unbiased estimator of the regression.
To see what is meant by BLUE, recall from §5.2 the discussion of the
Figure 9.2 The true (population) regression line and the estimated (sample) regression line
X3
Frequency of Y
X2 X1
Y2 Y3
Y1
Estimated regression line Ŷ = a + bX
True regression line Y = α + βX
Y1
sampling distribution of a sample statistic. In the case of a regression
coefficient, b, the mean of its sampling distribution is said to be unbiased if it is equal to the true (unknown) population coefficient, �. If the assump-
tions underlying an OLS regression model are correct, then the estimates of
its regression coefficients satisfy this condition.
This does not mean that the estimate formed from the particular sample
we happen to have will equal the true �; the only estimate that would do
this would be the average of all the estimates we would get if we could
repeat the sampling process an infinite number of times. If there is more
than one unbiased estimator, the one which is called the best, or the most efficient, is the one whose sampling distribution has the smallest variance, that is, is concentrated in a narrow range, not spread widely either side of
the mean.
In essence, the notion of the OLS estimator as BLUE is that the perfor-
mance of least squares regression could not be beaten by any other type of
estimation technique (such as maximum likelihood) fitted to the same
data. The OLS regression will provide the best and most appropriate fit to
the data, as long the conditions set out in panel 9.1 are met.f The
coefficients are measured accurately and have the lowest possible standard
errors (and thus the highest possible t-statistics).
The OLS model will therefore give the model its best possible showing as
an explanation for the behaviour of the dependent variable. But it is still
necessary to ask: How good is the model as a whole? Just as we need to eval-
uate the performance of a single explanatory variable by measuring its sta-
tistical significance, we should also evaluate the model as a whole by
measuring its statistical significance. This is the theme of the next section.
9.3 A significance test for the regression as a whole
The use of the t-test was introduced in chapters 5 and 6. In §6.7.3 it was
applied to an example of a simple linear regression to test whether the rela-
tionship between the dependent variable, Y, and the explanatory variable,
X, was statistically significant. The null hypothesis was that ��0, meaning
that the true (population) coefficient was zero so that there was no rela-
tionship between the two variables.
The principles of the test are exactly the same for a multiple regression
model with more than one explanatory variable. The standard error can be
calculated for each separate regression coefficient, and for each such
coefficient a separate decision can be made as to whether or not the
f What happens if any of these assumptions are violated is discussed in chapter 11.
§.
coefficient is statistically significant. It is also possible to test the overall
significance of the estimated regression line. This involves a new test, and
we introduce this in the following section. It is then applied to the regres-
sion model in §9.3.2 and 9.3.3.
9.3.1 The F-test and the F-distribution The basic F-test is concerned with two independent sample variances, s1
2
and s2 2, each drawn from a normal population, and the point at issue is
whether the variances of these two populations, 1 2 and 2
2, are the same.
The null hypothesis would be 1 2� 2
2, where the alternative hypothesis is
that 1 2is greater than 2
2. This involves a one-tailed test.
Fcalc is measured as the ratio of the two sample variances, and by conven-
tion the larger variance is taken as the numerator of the ratio, so that the
value of the test statistic must be equal to, or greater than 1. The greater the
difference between the two population variances, the greater Fcalc will be.
The sampling distribution for the probability of getting a result greater
than Fcalc follows the theoretical F-distribution outlined in panel 9.2 and
illustrated in figure 9.3. The distribution is skewed, not symmetrical: one
tail stretches out to the right to incorporate the small probability of getting
exceptionally large values. The skewness varies according to the degrees of
freedom of the two sample estimates, generating a family of curves for
different sample sizes.
F-TESTS
The F-test is the ratio of two sample variances, s1 2 and s2
2,
where s1 2�s2
2.
F � (9.4)
9.3.2 F-tests of the statistical significance of the regression as a whole One important and widely quoted use of the F-test is as a test of the overall
explanatory power of the regression as measured by the coefficient of
multiple determination, R2 (see §8.2.1). If R2 is not statistically significant
it means that none of the explanatory variables are linearly associated with
the dependent variable.g
s21 s22
g Another use of the F-test is in hypothesis testing of statistical differences between sample vari-
ances; see panel 6.1.
Panel 9.2 The theoretical F-distribution
This is the last of the four theoretical probability distributions we shall con-
sider and is derived from the chi-square distribution described in panel 7.1
of §7.3.2.* The theoretical F-distribution is the sampling distribution of the
ratio of two independent chi-square random variables, divided by their
respective degrees of freedom, k1 and k2.
Using the same notation as in (7.8) of panel 7.1, if the first chi-square
random variable is
V1 � Z 2 1 � Z
2 2 � ... � Z
2 k1 (9.5)
and the second independent chi-square random variable is
V2 � Z 2 1 � Z
2 2 � ... � Z
2 k2 (9.6)
then
F � (9.7)
The shape of the F-distribution depends solely on these two parame-
ters: k1 in the numerator and k2 in the denominator of the ratio. The dis-
tribution can take any value between 0 and infinity, and for small values
of k1 and k2 it is skewed, with a long tail to the right. As the two parame-
ters become large, however, the F-distribution approximates to the
normal distribution. It is illustrated for selected values of k1 and k2 in
figure 9.3.
Published tables for the F-distribution are given in many statistical
texts, though they are usually heavily abridged since it is necessary to cover a
range of combinations of the two degrees of freedom (k1 and k2 ) for each
proportion of the distribution (for example, the upper 5 per cent).**
* The other two are the Z- and t-distributions encountered in §5.4. ** D. V. Lindley and W. F. Scott, New Cambridge Statistical Tables, 2nd edn., Cambridge
University Press, 1995, pp.50–5 give the upper percentage points (right-hand tail) of the F-
distribution for proportions from 10 per cent to 0.1 per cent for various combinations of
df.
V1/k1 V2/k2
§.
As explained in §9.3.1, the F-test involves the test of a ratio of two
sample variances. In this case the two variances are the explained and the
unexplained variation obtained from the regression equation. When there
is only one explanatory variable in the model this reduces to the t-test
already described in §6.7.3.h When there are two or more, it is a single test
of the hypothesis that all the true regression coefficients (�s) are zero. The
test statistic is Fcalc.
THE F-TEST FOR THE STATISTICAL SIGNIFICANCE OF THE
REGRESSION AS A WHOLE
The F-test is a test of the null hypothesis
that all the true regression coefficients are zero,
so that there is no (linear) relationship
between the set of explanatory variables
and the dependent variable.
Fcalc� (9.8)
where k is the total number of explanatory variables plus the intercept,
and n is the sample size.
Fcalc can be compared in the usual way with the theoretical probability of
getting this particular result as determined by the theoretical F-distribution
(see panel 9.2) at the chosen level of significance, where the required
degrees of freedom are k 1 and n k. If Fcalc is greater than the theoretical
(tabulated) value of F the null hypothesis is rejected. In other words, we
reject the proposition that all the �s taken together are zero and have no
influence on the dependent variable.
9.3.3 The relationship of F and R2
When the coefficient of determination, r2, was introduced in §4.3 it was
defined as the ratio of the explained to the total (explained plus unex-
plained) variation. It will be seen that the first term in the numerator of the
formula given above for Fcalc is the explained variation, and the first term in
the denominator is the unexplained variation. It can be shown by simple
�(Ŷi Y– )2/(k 1) �(Yi Ŷi)2/(n k)
h It can be shown that for any individual regression coefficient the two tests are formally equiva-
lent and that F�t2.
manipulation of these terms that the relationship between Fcalc and the
equivalent coefficient of multiple determination, R2, is as follows
Fcalc� (9.9)
If the regression model explains very little of the variations in the depen-
dent variable, R2 will be very small and therefore, as the above formula
shows, the value of Fcalc will be very low. Conversely, the stronger the rela-
tionship specified in the model, the higher the value of Fcalc. Thus a high
value of Fcalc is generally also an indicator of strong overall relationships
between the dependent variable and the set of explanatory variables as a
whole.
9.4 The standard error of the estimate, SEE
The standard error of the estimate (SEE) is another useful summary statis- tic that is automatically calculated by most statistical programs. It meas-
ures the standard deviation of the residuals from the regression line and
provides the basis for a confidence interval for a prediction based on the regression line. It is formally equal to the square root of the sum of the
unexplained variation divided by the number of degrees of freedom.
R2/(k 1) (1 R2)/(n k)
Figure 9.3 The F-distribution for selected degrees of freedom
F
Pr ob
ab ili
ty de
ns ity
F (20,30)
F (8,4)
F (4,2)
§. ,
THE STANDARD ERROR OF THE ESTIMATE, SEE
SEE is a measure of the
standard deviation of the unexplained variation
SEE � (9.10)
It is used to calculate a confidence interval
around the regression line.
9.4.1 Confidence interval for a prediction of the mean value of a dependent variable The first value we might wish to predict on the basis of a regression equa-
tion is the mean value of the dependent variable. Take, for example, a
simple linear regression model relating the unemployment rate in each of a
series of years (UNEMP) to a single explanatory variable, the benefits avail-
able to those unemployed, measured as a ratio to the average wage
(BWRATIO):
UNEMP�a�b1 BWRATIO�e (9.11)
The coefficients from this regression could be used to calculate the mean
value of UNEMP that would be predicted for any given value of BWRATIO
(say, X0). i However, the regression line in (9.11) is estimated on the basis of
only one set of observations and thus represents only one sample from the
population of all possible relationships between the two variables. Other
samples would give different values for the constant, a, and the regression
coefficient, b1, and thus different answers to the calculation.
It is thus necessary to have a confidence interval for this prediction of
the mean value of UNEMP when BWRATIO�X0, and to specify the
desired level for this, say 95 per cent.
��(Yi Ŷi) 2
n 2
i Note that X0 is not one of the specific values taken by the explanatory variable. It is any value
chosen by the researcher, and may well lie outside the actual range of values for BWRATIO used
in the estimation of the regression line if the aim is to predict what UNEMP might be at some
other level.
CONFIDENCE INTERVAL FOR THE PREDICTION OF THE
MEAN VALUE OF Y
For the 95 per cent confidence interval
the prediction of the mean value
of Y is based on
the standard error of the estimate, SEE.
For any value, X0, the mean value of Y0 equals
(a�b1X0) � t0.025SEE (9.12)
Note two features of (9.12). First, if the mean value of UNEMP is calcu-
lated when the value of X0 chosen for BWRATIO is equal to the mean value
of that series, then X0 � X –
. The numerator in the second term under the
square root (X0 X –
) thus becomes 0, and the confidence interval simplifies
to SEE .
Second, when X0 is not equal to the sample mean value of X, then the
further it is from X –
in either direction, the larger that second term under the
square root becomes, and so the greater the width of the confidence interval
around the predicted mean value of UNEMP. This widening of the interval
as the value of the explanatory variable moves away from its mean value is
illustrated in figure 9.4 (where a broken vertical line is drawn through X –
).
Thus while an estimated regression line can be used to extrapolate to
periods (or cases) outside the sample, it is essential to be cautious in
attempting to predict the mean value of Y for a value of X that is any dis-
tance away from the sample mean of X.
Illustration of the calculation of a confidence interval The calculation underlying figure 9.4 can be used to illustrate the proce-
dure for obtaining the confidence intervals. The values of UNEMP and
BWRATIO are taken from the data set for unemployment in inter-war
Britain. The value of the standard error of the estimate, SEE, is calculated
according to (9.10). The unexplained sum of squares in the numerator of
the equation is 305.17.j The data cover the 19 years from 1920 to 1938, so
1
�n
�1n � (X0 X –
)2
�(Xi X –
)2
j This and other statistics quoted in this section were calculated in table 6.1 and §6.7.3, but would
also be given by any computer package used to estimate the regression line.
§. ,
n 2 is 17. The standard error of the estimate is thus
SEE � � 4.24
The next step is to obtain the values required for (9.12). This has three
components in addition to SEE. First, the regression line specified in (9.11)
must be estimated to obtain the values for the constant, a, and the regres-
sion coefficient, b1. These are found to be 6.40 and 15.64, respectively.
Secondly, the table for the theoretical t-distribution (see table 5.1)
shows that for 17 degrees of freedom, the two-tailed value of t at the 5 per
cent level is 2.110. Thirdly, the values in the final term of (9.12) must be
derived. Let us take X0 at a value of 0.10, a little below the lowest value of
BWRATIO observed in the sample. X –
, the mean value of BWRATIO in the
sample, is 0.467, and the sum of the deviations of X from its mean,
�(X–X –
)2, is 0.222. Since n is 19, the required term is
�
� 0.813.
� 119 � (0.10 0.467) 2
0.222 ��0.053 � 0.1350.222 � �0.660
�1n � (X0 X –
)2
�(Xi X –
)2
�305.1717
U ne
m pl
oy m
en t (
pe r
ce nt
)
Ratio of benefits to wages
Figure 9.4 95 per cent confidence interval for a prediction of the mean value of UNEMP
The full calculation of the 95 per cent confidence interval for X0�0.10
on the basis of (9.12) is thus
Y0 � a � b1X0 � t0.025SEE
� 6.40 � (15.64 � 0.10) � 2.110 � 4.24 � 0.813
� 7.96 � 7.27
� 0.69 to 15.23.
At this distance from the mean level of BWRATIO the width of the
confidence band around a forecast of the mean value of the unemployment
is thus rather large.
The same procedure was followed for a range of alternative values of X0 to derive the full set of confidence intervals drawn in figure 9.4.
9.4.2 Confidence interval for a prediction of a specific value of a dependent variable There is a second question we could ask on the basis of the regression equa-
tion. Instead of predicting the mean value of UNEMP we may wish to
predict a single value, given a particular value for BWRATIO (X0).
The answer to this question will be exactly the same, because the values
for a, b, and X0 are exactly the same. But the error around that answer will
be greater because we are trying to estimate only one value for Y, rather
then the mean of all possible Ys.
CONFIDENCE INTERVAL FOR THE PREDICTION OF A
SPECIFIC VALUE OF Y
For the 95 per cent confidence interval
the prediction of a specific value
of Y is based on
the standard error of the estimate, SEE.
For any value, X0, the value of Y0 equals
(a � b1X0) � t0.025SEE (9.13)�1n � (X0 X –
)2
�(Xi X –
)2 � 1
�1n � (X0 X –
)2
(Xi X –
)2
§.
Compared to (9.12), (9.13) has an additional term under the square
root,�1. Therefore, the confidence interval (or prediction interval) will
always be larger for this prediction of a specific value of the dependent var-
iable than for the prediction of the mean value.
As the sample size gets larger, the first two terms under the square root in
(9.13) approach zero, leaving just the final term, 1.k The confidence interval
thus approaches t0.025 SEE. It is for this reason that SEE is called the standard
error of the estimate.
Note 1 For a provocative statement of a leading quantitative economic historian’s views on
‘covert theory’, see Robert W. Fogel, ‘The specification problem in economic
history’, Journal of Economic History, 27, 1967, pp. 283–308.
9.5 Exercises for chapter 9
1. Is an estimate BLUE if:
(i) The dependent variable is always in error by �5 per cent?
(ii) The dependent variable is in error by �5 per cent for the bottom half
of the distribution, but by �15 per cent for the top half?
(iii) The dependent variable is under-measured for the lower half of the
distribution, but over-measured for the upper half?
(iv) The dependent variable is always in error by �5 units?
(v) The dependent variable is measured as only half of its true value for
all observations?
2. Which of the following series used as an explanatory variable are subject
to stochastic errors and which to systematic errors? In each case, write a
sentence explaining your reasoning.
(i) A series on incomes based on tax data, given tax evasion by the rich
(ii) A series on poverty based on reported income, given welfare fraud
by the poor
(iii) A series compiled by a census taker who leaves out every other house
(iv) A series compiled by census subjects who round their ages to the
nearest fifth year
(v) A series compiled by a government official who inverts every
number, e.g. writing 75 not 57
(vi) A series compiled by a government official who inverts every other
number
k The first term diminishes because as n gets larger, 1/n gets smaller. The second term diminishes
because the bigger the sample, the more cases of X there are to include in the sum of the devia-
tions of Xi from in the denominator. �(Xi X –
)2 thus gets larger and larger relative to (Xo X –
)2.
(vii) A series compiled by a researcher who misreads data as lb per inch,
rather than kg per cm
(viii) A series compiled by a researcher who misreads data as being
expressed in numbers, rather than in logs
(ix) A series that uses urban prices rather than rural prices to explain
farm consumption patterns
(x) A series that measures urban prices by collecting data only from
large retailers.
3. Write a brief paragraph discussing the following proposition:
Since the data on which the regression is being estimated are a sample drawn from
the population, it must therefore follow that the estimators (coefficients) for that
regression are themselves sample statistics – just one of many possible that could be
drawn from the sampling distribution. It is important to recognize, therefore, that
minimizing least squares gives the best fit to this particular set of numbers, but that
this is not necessarily the best fit for any other sample, let alone for the population
as a whole.
4. The error term is not only assumed to have zero mean and to be symmet-
rically distributed, it is also assumed to be independent of both the explan-
atory variables and the dependent variable. Why are the last two
assumptions necessary to maintain unbiasedness?
5. Calculate by hand the value of F for the regression coefficient of distur-
bances on wheat prices in chapter 4, question 3. Compare this with the
appropriate test statistic, being sure to identify the correct numbers of
degrees of freedom. Does this suggest that the regression is statistically sig-
nificant? Compare your findings with the answer to question 4 in chapter 6.
6. In chapter 6, question 7, a question was raised about whether the vari-
ances of birth rate were (a) the same for Bedfordshire and Essex, and (b)
the same for Buckinghamshire and Sussex. Set up an F-test of these two
hypotheses and report your finding (use the formula in equation 9.4).
In chapter 6, question 8, the number of parishes in each sample was
increased. Maintain this fiction here. What difference do the larger samples
make to (a) the value of the F-statistic, and (b) its statistical significance?
7. Run the following regression on the Benjamin–Kochin data set:
UNEMP�a�b BWRATIO�e
(i) Confirm that the reported value of the F-statistic is equal to the
square of the t-statistic and that the statistical significance of the two
tests are identical.
(ii) Construct a new variable, by adding 10 units to unemployment for
§.
each observation. Re-run the basic regression above and report the
results.
(a) What do you notice about the statistical significance of the
intercept?
(b) What do you notice about the relationship between the F- and t-
statistics?
(c) What do you notice about the overall statistical significance of
the equation? What does this tell you about the relationship
between the F-test and the intercept?
(iii) Now run a variant of the basic regression, separating out the compo-
nent parts of BWRATIO in the form
UNEMP�a�b1 BENEFIT�b2 WAGE�e
(a) Report the F-statistic and its significance.
(b) Use the formula (9.9) to explain why the value of the F-statistic
has fallen so much, despite the small increase in R2.
8. How would you use the F-test to determine the statistical significance of
the R2 statistic? (Hint : use (9.9).)
9. Calculate by hand the standard error of the estimate (SEE) for the regres-
sion coefficient of disturbances on wheat prices in chapter 4, question 3.
(i) What is the 95 per cent confidence interval around the predicted
number of disturbances at a wheat price of 100 shillings per qtr?
(ii) Will this be larger or smaller than the 95 per cent confidence interval
around the predicted number of disturbances at 85 shillings per qtr?
(Hint : this second question should not require a second calculation.)
10. Using the Benjamin–Kochin data set:
(i) Apply the procedure of §9.4.1 to calculate the confidence interval
around the predicted value of UNEMP for the full
Benjamin–Kochin model, namely
UNEMP�a�b1 BWRATIO�b2 DEMAND�e
at the mean of BWRATIO.
(ii) Calculate the 95 per cent confidence interval around the value of
BWRATIO in 1920.
(iii) How would your results differ if you were measuring the 95 per cent
confidence interval around the mean value of DEMAND and the
value of DEMAND in 1920, respectively?
(iv) How do your results influence your interpretation of the
Benjamin–Kochin explanation of inter-war unemployment?
Dummy variables and lagged values
In chapter 8 the simple linear regression model was extended to cover the
introduction of two or more explanatory variables, and in chapter 9 the
model was given its essential stochastic form.
In the present chapter two further extensions are described. First, §10.1
is devoted to the use of dummy variables. This is a procedure developed to enable us to include in a regression a variable that cannot be measured in
the same way as a continuous numerical value (for example, income, or age
at marriage) but is instead represented by two or more categories (for
example, single, married, or widowed). This special form of a nominal (or
categorical) scale is known as a dummy variable.
Secondly, in §10.2 we develop the implications of the idea that it may be
appropriate for one or more of the explanatory variables to refer to an
earlier period than the one to which the dependent variable relates. Such
lagged values recognize the fact that there may be a delay before the changes in the explanatory variable make their full impact.
10.1 Dummy variables
10.1.1 Dummy variables with two or more categories In historical research we often want to take account of factors in a regres-
sion that are not measurable in the usual way, but can be expressed as rep-
resenting one of two (or more) categories. This is a special form of a
nominal scale and is called a dummy variable. In this chapter we consider such variables only when they are introduced as explanatory variables. It is
also possible for the dependent variable to be a dummy variable, and this
will be discussed separately in chapter 13.
A number of examples of dummy variables with two categories occur in
the Poor Law data set, such as whether or not there was a workhouse in the
parish (WORKHSE), or whether or not child allowances (CHILDALL)
§.
were paid. These are dealt with by assigning a value of 1 whenever the
answer is Yes, and a value of 0 whenever the answer is No. Other examples
of classifications into two nominal categories that could be represented
by a dummy variable include male/female, urban/rural, graduate/non-
graduate, and house owner/not house owner.
A corresponding procedure in the context of a time series would be to
assign a value of 1 to every year in the period when there was some excep-
tional event such as a war, a strike, or an earthquake, and a value of 0 to all
other years.
Dummy variables can also be applied to a nominal variable with more
than two categories. For example, in a study of earnings it might be appro-
priate to distinguish four categories of social class: professional, skilled,
semi-skilled, and unskilled. Similarly, in a political study of US policy in
the period since the Second World War it might be useful to include the
role of different presidents as one of the explanatory variables, assigning a
separate dummy variable for each of the periods in which successive presi-
dents had held office. Other multiple category variables for which dummy
variables might be used include religion, occupation, region, or time
period.
Whatever the number of categories, one of them must be selected as the
reference category (also referred to as the control category or benchmark), and this reference category must be omitted from the regression. An alterna-
tive, but generally less satisfactory procedure, is to omit the intercept rather
than the reference category. Failure to adopt one or other of these proce-
dures is known as the dummy variable trap. It is essential not to fall into this trap, because what is measured if neither the control category nor the
intercept is omitted is effectively an identity with a perfect linear relation-
ship between the dummies and the intercept. In this situation; the r2
becomes 1 and the standard errors on all the coefficients become 0. Clearly
such results are nonsensical.
To illustrate the proper application of dummy variables, consider the
data set for children leaving home in the United States in the decade
1850–60. One of the explanatory variables was the place of residence of the
family in 1850, subdivided into four areas, the Northeast, North-central,
South, and Frontier.a Northeast was selected as the reference category.
There are then three separate dummies (one fewer than the number of cate-
gories), each of which takes a value of either 1 or 0, as follows.
a The Frontier area was defined as Minnesota, Iowa, Kansas, Oklahoma, Texas, and states further
west: Richard H. Steckel, ‘The age at leaving home in the United States, 1850–1860’, Social
Science History, 20, 1996, p. 512.
NCENTRAL � 1 for families living in North-central states 0 for all other
families
SOUTH � 1 for families living in the South 0 for all other
families
FRONTIER � 1 for families living on the Frontier 0 for all other
families
The omitted reference category of residence in the Northeast then acts as
the benchmark against which the results for these three categories are com-
pared.
Students have been known to think that in a case like this the values
given to the dummy variables would be 3, 2, 1, and 0. So it must be stressed
that the values assigned to each dummy are always either 1 or 0, irrespective
of the number of categories for which there are dummy variables, and that
the reference category is omitted.
Once these dummy variables have been formed they can be included in
a regression in the same way as any other explanatory variable. For
example, in the time series analysis a wartime/peacetime dummy variable
would enable the researcher to include the occurrence of war as one of the
explanatory variables, and thus to control for its effects on the dependent
variable in the manner explained in §8.2.3 above.
The coefficients on the dummy variables can be tested for significance in
the same way as those on numerical variables, and all the other regression
procedures, such as the calculation of the coefficient of multiple determi-
nation, R2, or the F-test, are made and interpreted in the usual way.
An illustration of the use of dummy variables The following regression equation is taken from a paper on the influence of
London on wages paid in 1903 in a cross-section of districts in southern
England.b The districts were classified in five groups according to their dis-
tance from London. The first four categories were for intervals of 25 miles
(0–25 miles, 25–50 miles, and so on). The fifth category covered districts
more than 100 miles from London, and was used as the control group.
WAGE�a�b1LON 0–25�b2LON 25–50�b3LON 50–70�
b4LON 75–100�b5GRAIN�e (10.1)
where
WAGE � weekly summer cash wage in shillings for agricultural labourers in
southern England
b George R. Boyer, ‘The influence of London on labor markets in southern England, 1830–1914’,
Social Science History, 22, 1998, pp. 257–85.
§.
LON � dummy variables measuring distance from London in 25-mile
intervals
GRAIN� dummy variable�1 for districts located in counties specializing in
grain production.
The regression results (with their t-statistics in parentheses) were:
Constant a 13.71 (64.71)
LON 25 b1 3.11 (8.77)
LON 25–50 b2 1.17 (4.21)
LON 50–75 b3 0.74 (2.65)
LON 75–100 b4 0.48 (1.50)
GRAIN b5 1.40 (7.37)
R2 0.357
N 239
The author’s comment (italics added)c was:
in 1903, weekly [agricultural] wages in districts within 25 miles of London were
3.1s. higher than those in districts more than 100 miles from London. . . . weekly
wages were 1.2s. higher in districts 25–50 miles from London, and 0.7s. higher in
districts 50–75 miles from London, than in those districts more than 100 miles
from the metropolis. Agricultural wages in districts 75–100 miles from London
were not significantly different than wages in districts more than 100 miles from
London.
This illustrates two aspects of the use of dummy variables very clearly. First,
the four dummy variables enable districts to be subdivided into five catego-
ries, and the separate regression coefficients measure the effect on wages as
the districts get progressively further from London. Secondly, the districts
more than 100 miles from London are used as the reference category, and
are thus the basis for comparison with wages in districts nearer to the
capital.
10.1.2 Dummy variables as a change in the intercept In this section we will extend the discussion of dummy variables to show
how the coefficients on the dummy variables can be interpreted as a change
in the intercept of the regression equation; in this context they are some-
times referred to as intercept dummy variables. This will clarify the proce- dure and will also be helpful as a preliminary to the slightly more complex
aspects in the discussion of additional dummy variables below.
c Boyer, ‘The influence of London’, pp. 266 and 267.
Consider, for example, an enquiry into a sample of workers from which
information can be derived on INCOME (measured in $ per week);
EDUC, a continuous variable for the number of years of education each
worker had received beyond primary school; and a nominal category, race,
distinguishing whether the workers were BLACK or WHITE.
The model could specify a linear relationship with INCOME as the
dependent variable and two explanatory variables: EDUC and BLACK.
The latter is the intercept dummy variable for race, with white workers as
the reference category. We then have
INCOME�a�b1EDUC�b2 BLACK�e (10.2)
EDUC thus measures the effect on INCOME of number of years of edu-
cation, while controlling for race. BLACK measures the difference it makes
to INCOME being a black rather than a white worker, while holding EDUC
constant (i.e. effectively assuming both black and white workers have had
the same number of years of education).
If the regression equation is computed in the usual way there will be a
constant, a, and coefficients for EDUC and BLACK. Imagine that the latter
turns out to be –40. There are effectively two versions of the regression
equation. For all white workers, BLACK�0, so –40 � 0�0, and the equa-
tion reduces to
INCOME�a�b1EDUC�e (10.3)
For all black workers, however, BLACK�1, so –40 x 1� 40. Since this
is a constant it can be added to (or, rather, because in this case it is negative,
deducted from) the intercept, and the equation can be written as
INCOME�(a 40)�b1EDUC�e (10.4)
If these two regressions are plotted on a graph (as in figure 10.1, using
hypothetical figures) there are two parallel lines, one for white workers
(whose incomes are marked by a cross) and one for black workers (marked
by a square). The regression line for white workers has an intercept at
a�190, the one for black workers has an intercept at (a 40)�150. The
difference in intercept thus represents the effect of being a black worker
(controlling for EDUC). It indicates that a unit change in the dummy vari-
able (from 0 for white workers to 1 for black workers) reduces INCOME for
black workers as compared to white workers by $40 per week.
If the enquiry had subdivided the workers into four categories rather
than two (say, white, Asian–American, African–American, and Hispanic),
three dummy variables would be needed. There would then be four parallel
lines, one for the reference category, and three others above or below that,
§.
their position relative to the intercept depending on the sign and size of the
coefficients on the respective dummy variables.
10.1.3 Additional dummy variables It may occur to the researcher that gender may also have an influence on
INCOME. If information on this topic had been collected in the enquiry as
well, the linear regression model could be reformulated with a second
dummy variable for FEMALE, so that the reference category is now white
males
INCOME�a�b1EDUC�b2 BLACK�b3 FEMALE�e (10.5)
The workers can thus be subdivided into four categories, with the two
dummy variables given the values of 1 or 0 as appropriate, and there would
be a separate equation for each category. For example, the equation for
white female workers would be
INCOME�a�b1EDUC�(b2�0)�(b3�1)�e (10.6)
which reduces to
INCOME�a�b1EDUC�0�b3�e (10.7)
The intercept for white female workers would thus be (a�b3). Treating
the other three categories in the same way (and omitting the error term),
the right-hand side of the four regressions can be summarized as
Figure 10.1 An intercept dummy variable
EDUC (number of years education beyond primary school)
1086420
IN C
O M
E (
do ll
ar s
pe r w
ee k)
500
400
300
200
100
190 – 40) + 32 EDUC
Y = 190 + 32 EDUC (BLACK = 0)
(BLACK = 1) Y = (
40
Males Females
White (a�0�0)�b1EDUC (a�0�b3)�b1EDUC
Black (a�b2 �0)�b1EDUC (a�b2�b3)�b1EDUC
The intercept for white males is a and, as before, the coefficients b2 and
b3 are added to this intercept to determine the difference it makes to
INCOME if a worker is in one of the other categories (controlling for
EDUC). Thus for any level of EDUC, INCOME for male black workers
will differ from that for male white workers by b2; INCOME for female
white workers will differ by b3; and INCOME for female black workers
will differ by b2�b3. We might expect to find that both b2 and b3 are
negative.d
10.1.4 Interaction between two dummy variables It will be seen, however, that under this procedure INCOME for females is
constrained so that it always differs from that for males by the amount
of the single coefficient, b3, regardless of whether the workers are
black or white. An alternative procedure, which avoids this constraint,
involves the introduction of interaction terms. These are a special type of dummy variable formed by combining variables, in this case two dummy
variables.
In this example, three new dummy variables, combining the categories
of race and gender, would be formed in place of the two previous ones for
the separate categories. The regression model becomes
INCOME�a�b1EDUC�b2 BLACK–MALE�
b3 BLACK–FEMALE�
b4 WHITE–FEMALE�e (10.8)
The fourth category, white males, serves as the control.
In order to assign a value of 1 to these new compound variables, both
elements must be satisfied. For example, BLACK–MALE will take a value
of 1 only for workers who are both black and male; for all other workers it
will be 0. With this approach the right-hand side values for the four catego-
ries (omitting the error term) are now
Males Females
White (a�0�0�0)�b1EDUC (a�0�0�b4)�b1EDUC
Black (a�b2�0�0)�b1EDUC (a�0�b3�0)�b1EDUC
d For an application of this approach to the measure of wage discrimination against women in the
United States see Claudia Goldin, Understanding the Gender Gap. An Economic History of
American Women, Oxford University Press, 1990, pp. 84–7.
§.
The difference in income between males and females is no longer forced
to be the same regardless of race; it is now possible to determine by how
much INCOME for each of the gender/race combinations will differ from
that for white males (controlling for EDUC).
These interaction terms between intercept dummy variables need not
be restricted to combinations of only two variables. If, for example, the
enquiry had also assembled data on whether the workers were urban or
rural it would be possible to form a term for categories such as female,
black, rural, workers and to compare their income (controlling for EDUC)
with a reference category of white, male, urban, workers.
10.1.5 Interaction between dummy and numerical variables The previous subsection explained the formation of interaction terms
between two or more dummy variables. For the example used, it was
shown that this enables the researcher to determine (i) the effect on
INCOME of number of years of education, while controlling for combina-
tions of race and gender, and (ii) the difference it makes to INCOME being
in any one of the race/gender combinations compared to the income of a
white, male worker, while holding EDUC constant.
This is effectively a model of a single linear relationship between INCOME
and EDUC, with a different intercept and regression line for each race/gender
category. If the regression lines were plotted there would be three parallel
lines, i.e. they would all have the same slope, given in the example above by b1,
the common coefficient on EDUC for all the different categories.
It may be, however, that the researcher thinks that there should be a
different linear relationship between INCOME and EDUC for the different
categories. In this case it is not only the intercepts that would shift as the
categories changed, there would also be a change in the slope of the regres-
sion line.
This can be done by introducing a slope dummy variable, formed by interacting a dummy variable with a numerical or continuous variable. To
simplify the illustration we will omit the gender category and revert to the
initial linear relationship, repeating (10.2)
INCOME�a�b1EDUC�b2 BLACK�e (10.2)
We now modify this by introducing a further interaction term or inter- action variable, obtained by multiplying the numerical variable by the dummy variable.1 The modified (but still linear) relationship is then
INCOME�a�b1EDUC�b2 BLACK�
b3 EDUC�BLACK�e (10.9)
For all white workers, both BLACK and EDUC�BLACK are zero. So
the coefficients b2 and b3 are eliminated and the coefficient on EDUC
remains as b1. However, for all black workers BLACK�1, and so EDUC�
BLACK�EDUC. The intercept becomes (a�b2), and the coefficient on
EDUC becomes (b1�b3).
The regression for white workers thus reduces to
INCOME�a�b1EDUC�e (10.10a)
whereas that for black workers is
INCOME�(a�b2)�(b1�b3)EDUC�e (10.10b)
To illustrate this we use another hypothetical data set. For white workers
the details are exactly the same as for figure 10.1, but for black workers they
are different. Fitting the regression line corresponding to (10.9) we get the
following results. For white workers
INCOME�190�32 EDUC
and for black workers
INCOME �190�32 EDUC 65 BLACK 5 EDUC�BLACK
�(190 65)�27 EDUC
The corresponding regression lines for the separate groups are shown in
figure 10.2. As can be seen, the line for black workers has both a different
intercept and a different slope.
The degree of discrimination revealed by this second data set is thus
greater than that in the data underlying figure 10.1. In that case each addi-
tional year of post-primary education had the same effect on the incomes
of the two groups of workers: incomes rose by $32 for each extra year. In
figure 10.2 that is still true for white workers, but for black workers the
return to an extra year of education is only $27. In addition when we
control for the effects of education there is a bigger gap: the difference
between the two intercepts is now $65 rather than $40.
However, this formulation of the two regression lines immediately
raises an awkward question: has anything been achieved by working
through the procedure for interacting a dummy variable with a numerical
variable? We have simply ended with two separate regressions. Why could
we not simply have formed two separate regressions to begin with?
The answer is that in this case we would get exactly the same estimates
from the regression with the EDUC�BLACK interaction term as we
would from two separate regressions, one for white and a second for black
workers. In order for it to be worthwhile using the interaction between a
§.
numerical variable and a dummy variable, the regression model must
include at least one other numerical variable that is not interacted. To illus-
trate this we could add an additional explanatory variable to (10.2) to cover
experience (EXP), measured by the length of time each worker has held
his/her current job.
If EXP is not interacted with BLACK, the slope coefficient for EXP is
assumed to be the same for both groups of workers: INCOME for a black
worker is assumed to benefit in exactly the same way from an extra year of
experience as for a white worker. This restriction would not be imposed if
two separate regressions were estimated, and the parameters derived from
the separate models would be less efficient (as defined at the end of §9.2.4) than those produced by the single equation with an interaction term for the
other numerical variable (EDUC).
10.2 Lagged variables
In all our previous discussions of time-series regressions we have implicitly
assumed that all the observations relate to the same year. What do we do if
we think it appropriate to include among the explanatory variables a vari-
able referring to an earlier year, say 1949, when the dependent variable and
the other explanatory variables all refer to 1950?
For example, an economic historian might consider that the expendi-
ture on new machinery (INV) undertaken by firms in any year depends
partly on the cost of borrowing as measured by the current interest rate
Figure 10.2 A slope and an intercept dummy variable
EDUC (number of years education beyond primary school)
1086420
IN C
O M
E (
do lla
rs p
er w
ee k)
500
400
300
200
100
190 – 65) + 27 EDUC
Y = 190 + 32 EDUC (BLACK = 0)
(BLACK = 1) Y = (
65
(INT) and partly on the value of goods sold (SALES) in the previous year.
Using the subscripts t for the current year, and t 1 for the year one before
that, the regression would be specified as
INVt�a�b1INTt�b2SALESt 1�et (10.11)
SALESt 1 is an example of what is known as a lagged variable. Lagged variables are treated in exactly the same way as any other explanatory vari-
ables and in a simple model such as the above do not require any special
procedures. Their use is not restricted to annual time periods; the periods
might equally be months or quarters if the required data are available.
When the effects of a variable are spread over several time periods in this
way the resulting model is known as a distributed lag model. Such models are particularly common in economic analysis, where it is assumed that
firms, households, or other agents need time to assemble and react to the
information which influences their actions. However, these models might
also be used in other contexts.
For example, a political historian might think that voters’ political
choices are influenced by the economic conditions they experienced in the
year before an election and not just by the conditions prevailing in the year
of the election. She might then model this by including lagged values of rel-
evant economic variables, such as income or inflation.2 Other possible
explanatory variables reflecting successes or failures in foreign policy, or
major political scandals, might be similarly lagged.
One question that is immediately posed by such models is the length of
time over which this influence from the past is effective. One assumption,
which avoids the need to select a specific period, is that the effects last
forever: all previous periods are relevant. This is known as an infinite dis- tributed lag model. Alternatively, the researcher may assume that the effect
lasts for only a fixed period of time, say five years.
For example, the model for INV in (10.11) might be altered to specify
that INV is affected by SALES both in the current year and also in each of
the five preceding years
INVt�a�b1INTt�b2SALESt�b3SALESt 1�b4SALESt 2 �
b5SALESt 3�b6SALESt 4�b7SALESt 5�et (10.12)
However, the choice of five lagged values is rather arbitrary since the
theory of investment on which the model is based does not really provide
guidance on this issue. Furthermore, equations like this might be difficult
to estimate for two reasons. First, each time period involves a separate
regression coefficient, and thus one additional parameter that has to be
estimated; and each one uses up one further degree of freedom. In addi-
§.
tion, the use of lagged variables uses up observations at the beginning of
the period. For example, if the data set starts in 1850 but one of the explan-
atory variables is lagged by five years, the first value for the dependent vari-
able will be 1855, and the number of years in the sample is reduced by five.
If the distributed lag stretches back over many periods, or if the sample is
small, this could be a serious problem.
Secondly, it is likely that there will be a close linear relationship between
SALES in the current year and SALES in each of the previous years, so that
the model is subject to the technical problem known as multicollinearity.e
We will discuss one technique for circumventing these difficulties in
§10.2.2, but before we come to that it will be convenient to discuss first the
ways in which it is possible for the lagged value of the dependent variable to
appear as an explanatory variable on the right-hand side of the regression
equation.
10.2.1 Lagged dependent variables There are two main ways in which a lagged dependent variable can enter the regression model as one of the explanatory variables. The first and most
important is because the researcher considers that the level of the depen-
dent variable in any one period may be influenced by its own level in previ-
ous periods.f The justification for this is the belief that there is a strong
element of habit, persistence, or inertia underlying many decisions.
For example, a model might specify that household expenditure on
consumption (CONSUMP) is influenced by household income
(INCOME) in the current and previous periods and also by consumption
in the previous period. The case for this would be partly that households
seek to maintain the level and pattern of spending to which they are accus-
tomed, and partly that they need time to determine whether any shock to
their income is transitory or permanent. The model would thus have
lagged values of both the dependent and explanatory variables
CONSUMPt�a�b1CONSUMPt 1�b2INCOMEt�
b3INCOMEt 1�et (10.13)
Similarly, in models of emigration it is common to assume that previ-
ous emigration is a powerful influence, a process known as ‘chain migra-
tion’. Former migrants from one’s own country may encourage the next
e Multicollinearity will be discussed more fully in §11.4.2. f When a model assumes in this way that the dependent variable is influenced by its own past
values this is referred to as autoregression. Special care is necessary when interpreting the results
from such a model because the inclusion of a lagged dependent variable violates the assumptions
underlying the standard OLS regression model. This issue is further discussed in §11.3.3.
generation to emigrate in a variety of ways. They can reduce some of the
uncertainty involved in moving to a foreign country, they may send back
money for fares, and they can help with accommodation and the search for
employment when the new immigrants arrive.
In their study of Irish emigration to the United States, for example,
Hatton and Williamson suggest that this important influence can be cap-
tured either by the lagged dependent variable or by a variable that measures
the total stock of Irish emigrants already in the receiving countries.g In
their test of the model they formulate to explain Irish emigration, they
include both the lagged value of the dependent variable IRISHMIGt 1, and
the current value of the stock of migrants, MIGSTOCKt.
To simplify the presentation here we will combine all their other explan-
atory variables into a single variable that we call OTHER, and this version
of their model is thus
IRISHMIGt�a�b1IRISHMIGt 1�b2MIGSTOCKt�
b3OTHERt�et (10.14)
The second way in which the lagged value of the dependent variables
can appear as one of the explanatory variables is when an equation is trans-
formed by mathematical manipulation. This is typically done in order to
overcome some problem in the original regression equation. There are a
number of procedures that have this result, and in §10.2.2 we shall look at
one of these to illustrate how it leads to a role for the lagged value of the
dependent variable on the right-hand side of the equation.
10.2.2 Distributed lag models Recall in relation to (10.12) above that the introduction of many or all past
values of an explanatory variable in addition to the current value could give
rise to two problems: too few degrees of freedom, and multicollinearity
between the different terms for the same variable, in that example between
current SALES and SALES for each of the five previous years.
One way to overcome the first of these problems is to impose some partic-
ular pattern on the coefficients on the successive lagged terms for the explan-
atory variable. This pattern effectively incorporates the researcher’s view as
to how the impact of the successive values of the lagged term might change
over time, and means that it is no longer necessary for these coefficients to be
estimated by the regression equation; they are already given.
There are many ways in which this can be done. One of the most
common is to assume that the coefficients decline geometrically.3 Assume
g Timothy J. Hatton and Jeffrey G. Williamson, ‘After the famine: emigration from Ireland,
1850–1913’, Journal of Economic History, 53, 1993, pp. 581–3.
§.
that � (lower case Greek lambda) is the rate of decline of the distributed lag and is a positive number with a value greater than 0 but less than 1. We can
incorporate this in an infinite version of (10.12)
INVt�a�b1INTt�b2SALESt�b2�SALESt 1�
b2� 2SALESt 2�…�b2�
nSALESt n�et (10.15)
Assume, for example, that ��0.5. Then �2�0.25, �3�0.125, �4�0.0625,
and so on. The closer � is to 1, the slower the rate of decline in the
coefficients on the successive terms for the lagged explanatory variable; the
closer it is to zero, the more rapid the decline. The result of choosing this
pattern is thus that the effect of more distant terms becomes progressively
weaker, which in many contexts seems intuitively plausible. It would not
require many periods before the effect of the lagged explanatory variable
was so negligible that it could effectively be ignored.
Whereas (10.12) required the estimation of eight parameters: an inter-
cept and seven regression coefficients, b1 to b7, (10.15) requires only four.
These are the intercept, the coefficients b1 on INT and b2 on SALES, and �.
The problem of losing too many degrees of freedom has thus been reduced
or eliminated, and the precise number of lagged periods to include in the
model has effectively been finessed. However, it would still be an extremely
difficult model to estimate, and because the parameters b2 and � are multi-
plied together it could not be done by the standard OLS procedure.
Fortunately, there is an ingenious way out of this problem, known in
honour of its author as Koyck’s transformation. This is described in panel 10.1 and leads to the following version of the model
INVt�(1 �)a�b2 SALESt��INVt 1�(et �et 1) (10.19)
The transformation has thus achieved a simple linear relationship, and
solves all three of the problems encountered above:
� There are only three parameters to be estimated: the intercept, a; the
regression coefficient on current SALES, b2; and the rate of decline of
the distributed lag, �.
� The successive values of the lagged explanatory variable have been
eliminated from the model so there is no longer a problem of multicol-
linearity.
� It is not necessary for the researcher to specify the precise length of the
distributed lag.
A further notable feature of (10.19) is that, as indicated at the end of
§10.2.1, the manipulation or transformation of the equation has had the
result of entering the lagged value of the dependent variable (INVt 1) as
one of the explanatory variables. The interpretation of the regression
coefficients on this and other terms in the model is discussed in §12.3.1.
There is inevitably a price to be paid for the substantial advantages
gained from this transformation and the new model raises a number of
technical issues in relation to the assumptions underlying the standard
estimating procedures. Discussion of these technicalities would, however,
take us beyond the scope of this book.4
Panel 10.1 Koyck’s transformation
To see how Koyck’s transformation can be applied we will simplify equation
10.15 by omitting the other explanatory variable, INT. The model becomes
INVt�a�b2SALESt�b2�SALESt 1�b2� 2SALESt 2�…
�b2� n SALESt n�et
�a�b2(SALESt��SALESt 1 ��2SALESt 2�…��
n SALESt n)�et (10.16)
It follows, that there must be an exactly corresponding relationship for INV
in year t 1
INVt 1�a�b2(SALESt 1��SALESt 2�� 2SALESt 3�…
��nSALESt n)�et 1 (10.17)
We come now to the transformation or manipulation. If every term in
(10.17) is multiplied by �, and the resulting equation then deducted from
(10.16), we get
INVt �INVt 1�[a�b2(SALESt��SALESt 1�
�2SALESt 2�…�� n SALESt n)�et]–
[�a�b2(�SALESt 1�� 2SALESt 2�
�3SALESt 3�…�� n�1SALESt n �
�et 1)] (10.18)
All except two of the terms in SALES cancel out, because �SALESt 1
�SALESt 1�0, and similarly for all the other terms in SALES except the first
(SALESt) and last (SALESt n). The latter is so far in the distant past that its
coefficient, �n�1, can effectively be treated as zero, leaving only the current
year in the model.
If we now move the lagged term for INVt 1 from the left-hand side of
(10.18) to the right-hand side (with the appropriate change in sign from
to �), we have drastically reduced (10.16) to
INVt�(1 �)a�b2SALESt��INVt 1�(et �et 1)
(10.19)
§.
Notes 1 We may note here that it is also possible to form an interaction variable as the
product of two numerical variables. One way in which this can be done will be
covered in §12.1, where we consider the possibility of specifying a non-linear rela-
tionship (such as a parabola) that includes X2 as one of the explanatory variables.
Since X2 is obtained by multiplying a numerical variable, X, by itself, a squared term
such as this can equally be thought of as a further type of interaction variable. The
final type is then one in which one numerical variable is multiplied by another, as in
the example of AGRIC and LAND which is discussed in §14.2.2. In both these forms
of interaction variable the effect is that the slope of the regression relationship is
changing continuously. 2 See, for example, Howard S. Bloom and H. Douglas Price, ‘Voter response to short-
run economic conditions: the asymmetric effect of prosperity and recession’,
American Political Science Review, 69, 1975, pp. 1240–54. Their model of voting in
the United States from 1896 to 1970 included a measure of income designed to repre-
sent ‘the change in the “average” voter’s real spending power prior to each election’.
The best available measure of income as perceived by the voter was judged to be the
‘percentage change in real per capita personal income during the year preceding each
election’ (p. 1243, italics added). 3 This geometric pattern is also known as a Koyck distributed lag. This is an attractive
specification of the influence of past events at a constant rate over time. However,
this may not be the most appropriate form of lag. There are many other patterns
which have been suggested, including an arithmetic lag, in which the impacts
declines linearly, and a polynomial lag, which can take various shapes, for example
one in which the impact first rises for earlier values of the variable and then falls.
To illustrate this last pattern, consider a change in the rate of interest that is
thought to build up to its maximum effect on INV over two years and then to die
away. The appropriate polynomial for a model with INVt as the dependent variable
would make the effect larger for a change in period t 1 than for one in t, larger still
for a change in t 2 than for one in t 1, but then progressively smaller for a change
in t 3 and all earlier periods. 4 There is a further point that might be made. Although Koyck’s transformation solves
certain problems, it is purely a mathematical manipulation and can be criticized
because it does not have any theoretical justification. There are, however, a number
of alternative models that do have a basis in economic theory, including the adaptive
expectations model and the stock adjustment (or partial adjustment) model.
Economic historians can find further discussion of these models in most economet-
ric textbooks.
10.3 Exercises for chapter 10
1. You are setting up a model of the relationship between voter turnout by
US state and a series of state-specific characteristics (such as average
income per head; age distribution of the population; racial composition of
the population; share of union workers in the labour force; size of the
government sector in the economy). The data for the model are drawn
from five successive elections between 1968 and 1984; a number of the
explanatory variables are in the form of dummy variables. Specify the
dummy variables for the regression if:
(i) The age distribution of the population distinguishes 18–24-year-
olds; 24–45-year-olds; 45–65-year-olds; over 65-year-olds.
(ii) The racial composition distinguishes Caucasian; African–
American; and Asian–American.
(iii) The year of the vote is being tested as an independent cause of varia-
tion in voter turnout.
(iv) You wish to test whether turnout varies by age differently from
election to election.
(v) You wish to test whether unionization of the government sector
affects election outcomes.
2. Let us imagine that a historian believes that certain counties were more
generous in their provision of poor relief than others and that this is a signifi-
cant explanation of the variation in relief payments across Southern parishes.
She finds qualitative support for this hypothesis for Sussex in the archives.
Use the dummy variable technique to test the hypothesis that Sussex
parishes were more generous than all others, holding other parish charac-
teristics constant. What do you find? Does the researcher’s insight stand up
to statistical scrutiny? What is the impact on the coefficients and standard
errors on the other variables in the Boyer equation?
3. The researcher, emboldened by her findings, sets out to test the more
general hypothesis that county was an important determinant of variation
across all Southern parishes.
Once again, use the dummy variable procedure to test this hypothesis.
Be certain to exclude one county to avoid the dummy variable trap. Which
county did you choose as the default and why? Do the coefficients and stan-
dard errors on the additional variables bear out the researcher’s hypothe-
sis? How do you interpret the size and signs of the coefficients?
Examine the coefficients and standard errors on the other variables. Do
these new results cause you to re-evaluate Boyer’s interpretation?
4. In §10.1.1, we reported the results of Boyer’s analysis of the influence of
proximity to London on agricultural wage levels in Southern England in
1903. Use the dummy variable procedure to repeat the exercise for 1831,
using the information in Boyer’s relief data set on agricultural wages
(INCOME), distance from LONDON, and specialization in GRAIN
§.
production. In order to ensure consistency in variable definition over time,
you will need to construct the following variables:
WAGE, which can be calculated by dividing INCOME by 2.6 to convert annual
income in pounds to the weekly wage in shillings
GRAIND, which is equal to 1 if the value of the continuous variable GRAIN is
greater than 20, and 0 otherwise
LON 0–25, which is equal to 1 if the distance from London is less than 25 miles
and 0 otherwise
LON 25-50, which is equal to 1 if the distance from London is �25 miles
but 50 miles, and 0 otherwise
LON 50-75, which is equal to 1 if the distance from London is �50 miles but
75 miles, and 0 otherwise
LON 75–100, which is equal to 1 if the distance from London is �75 miles but
100 miles, and 0 otherwise.
Compare your results to those reproduced in the text. Was the influence of
London stronger or weaker in 1831 than in 1903? Is the wage gradient
steeper or shallower? Is the influence of grain specialization similar? How
might you account for any differences?
5. How might you set up the model in question 4 to determine whether the
influence of grain on agricultural wages varied by distance from London?
Generate the appropriate new variables and test the hypothesis by regression.
(Hint : re-read §10.1.4 on interactive dummy variables.) Report your results.
6. Benjamin and Kochin, in their analysis of British unemployment
between 1920 and 1938, estimated the following equation:
UNEMP�a�b1 BWRATIO�b2 DEMAND�e
A critic of this model suggests that changes in the benefit–wage ratio on
unemployment would not have been contemporaneous, as suggested in
their formulation, but would have affected the labour market only after a
lag of one year.
(i) Use the data on the web site to test the hypothesis. Report your find-
ings.
(ii) How might you extend the model to test whether the impact of both
the benefit–wage ratio and the DEMAND variable on unemploy-
ment was concentrated in the current rather than the previous year,
using only a single regression? Run the appropriate regression and
report and interpret your results.
7. What lag structure (including the number of lags, and for which
periods), if any, would you recommend adopting in the following models?
(i) A quarterly version of Benjamin and Kochin’s unemployment
model, in which data are available four times a year.
(ii) A model of US presidential elections, in which the aim is to under-
stand the vote share of a sitting President by county.
(iii) A model explaining the level of the Irish migrant stock in Australia,
1880-1913.
(iv) A model explaining individual earnings relative to each worker’s
level of schooling and work experience.
(v) A model explaining the impact of the Stock Market collapse of
October 1987 on later values of the Dow–Jones (or FTSE) index of
share prices.
8. Hatton and Williamson, in their time-series analysis of Irish emigration,
1877–1913, estimate the following equation:
IRISHMIGt�a�b1 log EMPFORt�b2 � log EMPFORt 1�
b3 log EMPDOMt 1�b4 log IRWRATIOt 1�
b5 MIGSTOCKt�b6 IRISHMIGt 1�e
(i) Using the data on the web site, estimate the regression, making sure
that you can retrieve the same results as reproduced in table 14.1.
(Hint : be careful to take logs of the variables where necessary and
also to use lagged variables where necessary.)
(ii) Now re-estimate the equation to determine the effect of assuming
that the impact of the migrant stock is also lagged by one year. How
much difference does it make to the value and statistical significance
of the coefficients on MIGSTOCK and the other variables?
(iii) The lag of the dependent variable is included in the estimating equa-
tion. Why? What would be the impact on the equation if this variable
were dropped? Report the impact on the coefficients and statistical
significance of the variables in the model. Also report the impact on
the regression diagnostics, such as the R2 and the standard error of
the equation.
What do these differences suggest about the importance of the lagged
dependent variable to Hatton and Williamson’s interpretation of Irish
emigration, 1877–1913?
9. A critic of Hatton and Williamson’s cross-section model of Irish migra-
tion by county suggests that the model fails to take chain migration into
account. The suggestion is that emigration from each county will tend to
be heavily influenced by the level of county emigration 10 years earlier.
Reconfigure the Hatton–Williamson model to test the hypothesis. How
does the addition of lagged values of CNTYMIG for 1891, 1901, and 1911
§.
affect our understanding of the process of migration? Is there evidence of a
‘chain’ at work? Is the impact of other variables substantially altered? Is the
revision persuasive? (Hint : to produce the value of CNTYMIG, lagged 10
years, instruct the computer to construct a new variable (e.g. LAGMIG)
which will be equal to the value of CNTYMIG lagged 32 observations.)
10. A researcher reports the results from a regression of expenditure on
new machinery on sales and the interest rate as follows:
INVt�155.3�15.6 INTt�20.3 SALESt 1
where the variables are as defined in §10.2. It is unclear from the discussion
of these results whether the relationship between investment and sales is to
be interpreted as a Koyck distributed lag model, in which this year’s invest-
ment is determined by the history of sales over many previous years, or as a
statement that this period’s investment depends on last year’s sales.
As a commentator on these results, how would you interpret the coeffi-
cients on both INT and SALES if the underlying relationship between INV
and SALES were (a) a one-period lag, or (b) an infinite distributed lag?
Violating the assumptions of the classical linear regression model
We have thus far been developing the methodology of classical linear
regression (CLR) using the ordinary least squares (OLS) system of estima-
tion. This is a very powerful technique for uncovering the relationships
among variables. Yet it has its limitations. It is clearly important to under-
stand what these limitations are, to see how they affect the outcomes of the
regression, and to suggest some procedures both to identify and to correct
for these effects. That is the purpose of this chapter.
11.1 The assumptions of the classical linear regression model
The best way to proceed is first to indicate the main criteria that a model
must satisfy in order to qualify as a good estimator, that is, to be what the
econometricians call the best linear unbiased estimator (BLUE), and then to state the conditions under which OLS methods meet these criteria.
We have previously defined BLUE in §9.2.4 but repeat it here for con-
venience. In the case of a regression coefficient, b, the mean of its sampling
distribution is said to be unbiased if it is equal to the true (unknown) pop- ulation coefficient, �. If the assumptions underlying an OLS regression
model are correct, then the estimates of its regression coefficients satisfy
this criterion.
This does not mean that the estimate formed from the particular sample
we happen to have will equal the true �; only that the average of all the esti-
mates we would get if we could repeat the sampling process an infinite
number of times would do so. If there is more than one unbiased estimator,
the one which is called the best, or the most efficient, is the one whose sampling distribution has the smallest variance, that is, is most closely
concentrated around its mean.
§.
The conditions required for OLS to satisfy the criteria for BLUE fall into
three categories.
11.1.1 The model is correctly specified The most fundamental assumption of the classical regression approach is
that the relationship between the dependent variable (Y) and the explana-
tory variables (X1, X2, ... Xn) is correctly modelled:
(a) The OLS approach assumes that the relationship between the depen-
dent and explanatory variables is linear. In a simple bivariate regres- sion, the regression line is a straight line; in a multivariate regression,
the relationship between the dependent variable and each of the
explanatory variables is a straight line (see §11.2.1).
(b) The list of explanatory variables is complete and none is redundant.
There are no omissions of relevant explanators, no inclusion of irrele-
vant variables (see §11.2.2).
(c) The relationship between the dependent and explanatory variables is
stable across all observations (whether across individuals or over time)
(see §11.2.3).
11.1.2 The error term is appropriately specified The relationship between dependent and explanatory variables is inexact.
To begin with, the regression model is not trying to measure identities or
formulas. Secondly, it is simply not possible to include all the variables that
may influence an outcome. Finally, some observations may be measured
imprecisely. The combination of all of these factors is that the observations
(and their relationship to the dependent variable) are measured with error.
In the OLS model, this error (or disturbance) term, e, is assumed to have certain properties:a
(a) The mean value of the error term across all observations is zero (see
§11.3.1)
(b) Each of the error terms has the same variance (the assumption of
homoscedasticity) (see §11.3.2) (c) The individual error terms are uncorrelated with each other (see
§11.3.3).
11.1.3 The variables are correctly measured and appropriately specified The OLS model also relies upon certain assumptions about the nature of
the dependent and explanatory variables. In particular, it assumes that:
a These three assumptions were previously stated in panel 9.1, together with the linearity assump-
tion in paragraph (a) of §11.1.1.
(a) The explanatory variables are correctly measured, i.e. that they are not
systematically distorted so as to be always too high or always too low
(see §11.4.1).
(b) The explanatory variables are not systematically correlated with each
other (they are not collinear) (see §11.4.2). (c) The explanatory variables are not systematically correlated with the
error term (see §11.4.3).
The rest of this chapter shows what happens when these various
assumptions are not maintained. We shall also suggest some simple tools
for detecting and correcting these violations of the classical linear regres-
sion model. However, there are two important points it is worth noting at
the outset. First, not all deviations from the list of fundamental assump-
tions are equally disastrous. Second, not all violations are easily detectable
or correctable. Some can be identified or corrected only with techniques far
beyond the scope of this simple introduction; some cannot be corrected at
all without finding new and better data.
11.2 Problems of model specification
11.2.1 Non-linear models The OLS model assumes linear relationships between X and Y. But what if
the relationship is non-linear? Figure 11.1 provides an extreme version of
the problem, using fictional data on the marginal (extra) cost of producing
an additional statistical textbook. The true relationship between cost (Y)
and output (X) has a U-shape (one of a family of curves known as parabo-
las), but the regression line that we have estimated from the data is a
straight line.
The results are evidently unsatisfactory. The coefficient on X is clearly
biased from its true value; it is very poorly defined (the standard error will be very high); moreover, it is very sensitive to changes in the sample. Thus,
if those observations beyond the vertical line ZZ were missing, the slope of
the linear regression line would become negative. Certainly, this regression
would be worse than useless for the textbook publisher, whose marketing
and pricing policies would be severely distorted if they were based on this
inappropriate model specification.
How does one detect non-linearities? When there are only two variables,
the best starting point is to plot the residuals from the regression (the difference between the observed and predicted values of the dependent
variable) and see if they show any obvious patterns. Thus, in our textbook
analysis, the residuals when plotted against output will follow an inverted
U-shape; against marginal cost, they will trace an inverted L-shape.
§.
When there is more than one explanatory variable, this may not be a
reliable methodology. In this case, the simplest method may be to split the
sample into subgroups according to the size of the explanatory variable
suspected of having a non-linear impact on Y, and run separate regressions
for each subgroup. In the absence of non-linearities, the coefficients should
not be significantly different from each other.
Not all non-linearities are as severe in form or consequence as the text-
book example in figure 11.1. But without correction, all will produce
biased and inefficient estimates of the ‘true’ relationships among the vari- ables. Fortunately, most are amenable to the solution of transforming the
variables. The procedures for doing this are explained and illustrated in
detail in chapter 12 (§12.2).
11.2.2 Omitted and redundant variables It is clearly not sensible to leave out variables that are important for under-
standing a historical development; it may not be so obvious why it matters
if we add irrelevant variables. We can explain both with the aid of the
Ballantine.
The Ballantine for examining the case of the missing variable is shown in figure 11.2 (a), on the pullout sheet, p. 527. The diagram indicates that
the best model for estimating Y includes both X1 and X2. If the model is
Figure 11.1 The marginal cost of production of a statistical textbook
201612840
C os
t pe
r bo
ok (
sh ill
in gs
)
100
90
80
70
60
50
40
30
20
10
0
Z
Z
Output of textbooks (000)
properly specified, the green area (and only the green area) will be used to
estimate the impact of X1 on Y; the orange area (and only the orange area),
the impact of X2 on Y; the brown area (for reasons given in §8.2.5) is
ignored. In this case, the estimators will be unbiased. But, if X2 is omitted,
X1 will use the information in both the green and brown areas to estimate
the relationship with Y, generating biased regression coefficients.b The
extent of the bias will be directly related to the extent to which the two
circles overlap (i.e. the extent to which X1 and X2 are collinear). c
The Ballantine for examining the case of the redundant variable is shown in figure 11.2 (b), on the pullout. In this case, the true model for
estimating Y uses only information about X1; the coefficient is estimated
using the information in the green and brown areas. However, an addi-
tional variable (or set of variables) X2 is added spuriously; it has no real
impact on Y, although because of its strong collinearity with X1, it will
overlap with Y in the brown (and orange) areas.
This overlap reduces the information available to the regression model
for the estimation of the coefficient on X1. The resulting coefficient is not
biased (since the green area uses information unique to X1), but it is
inefficient: it has a higher variance than would result in the correctly
specified model (the green area being smaller than the green–brown area,
the coefficient is measured with too little information).
Thus, omission is more hazardous than redundancy. But this sort of logic
may lead to ‘kitchen-sink’ regression analysis in which every measurable var-
iable is included in a regression on the off-chance that it will turn out to be
important. The problem, as may readily be imagined, is that the more redun-
dant variables are added, the more the efficiency of the relevant explanatory
variable will suffer. In the extreme, the coefficient on X1 will have such a large
standard error that it will appear to be statistically insignificant, and will be
ignored when explaining the evolution of the dependent variable.d
b At the same time, because it uses more information to estimate the coefficient, it will have a
smaller variance. This indicates the danger of comparing t-statistics from different regressions to
determine which is the superior specification. c Obviously, if X1 and X2 are orthogonal (see §8.2.5), the Ballantine circles will not overlap and
omission of one variable will not generate bias in the evaluation of the other. If the researcher is
interested only in a hypothesis test of the impact of X1 on Y, the omission of orthogonal variables
is acceptable. However, if the purpose is to explain the behaviour of Y, omission of X2 will consti-
tute a mis-specification since the model does not capture all the influences on Y. d You may wish to add a fourth circle to the Ballantine of figure 11.2 (b) that intersects points A
and B; this will reduce the total amount of information used to link X1 to Y to a thin sliver,
effectively reducing its explanatory power to almost nothing.
§.
The appropriate antidote is the use of common sense. Only include
those variables that can, for reasons of theory or history, be considered
likely to affect the dependent variable. But at the same time never leave out
something that is likely to be important. If needs be, use a proxy to capture
an explanatory variable for which no actual data exists.e
Thus, if the aim of the regression is to model the growth and
fluctuations of US imports over time, it would be essential to include a
measure of the relative prices of domestic and imported goods. If such a
term is excluded, the coefficients on the other variables (income, popula-
tion, etc.) will be biased. If no such series has ever been constructed, it is
vital that the researcher either undertakes the task of construction, or that
he finds a substitute that moves together in a systematic way with the true
relative price series (such as the relative price deflator of national incomes
in the United States and the rest of the world).f
11.2.3 Unstable parameter values Suppose that we are measuring the relationship between consumer
demand for a luxury good (e.g. DVD players) and average household
income. It is perfectly possible that there is a threshold level of income
below which no households can afford the luxury good. Above that thresh-
old, households will choose whether or not to purchase the good; the
higher the income, the higher the proportion of households who make the
choice. Thus, the demand curve will not be continuous across all income
levels; there will be a distinct kink at the threshold level. Running a linear
regression across all income groups will therefore produce a biased esti-
mate of the coefficient on income (see figure 11.3 (b)).
Similarly, in the case of the relationship between benefits and unem-
ployment in inter-war Britain, government policies setting the require-
ments for receiving the dole did not remain stable over time. In 1931, the
government introduced the so-called Anomalies Regulations, which made
it more difficult for certain categories of unemployed workers to claim
benefits. For example, the burden of proof that someone was genuinely
seeking work was shifted onto the claimants; single persons were expected
to travel to seek work; and married women wishing to receive benefits had
to satisfy substantially more demanding contributory requirements.
Clearly, we would not expect the same relationship between BWRATIO
and UNEMP to exist before and after the introduction of such restrictions.
e There are also formal statistical tests for omitted variables. These are discussed in advanced
econometrics textbooks. f More formally, the substitute should be collinear with the missing series; for further discussion
see §11.4.2.
X
1086420
Y 40
30
20
10
0
Z
Figure 11.3 Illustration of the use of dummy variables to allow for structural changes
(a) Introduction of an intercept dummy variable
X 1086420
Y 40
30
20
10
0
Z (c) Introduction of both an intercept and a slope dummy variable
X 1086420
Y 30
20
10
0
Z
(b) Introduction of a slope dummy variable
These are both examples of structural change in the parameter values of the system being modelled. Failure to incorporate evidence on structural
change is an important source of error in the estimation of empirical
models. In many cases (as in the example of British unemployment), a
knowledge of history will reduce such errors. The timing of structural
change will be known and can be factored into the basic regression. But not
in all cases: in time-series analysis, it may not be known when such a regime
change took place; in cross-section, a threshold effect may only be sus-
pected. Moreover, it may be that a suspected structural break in the regres-
sion model turns out not to be important at all in the specification of the
model.
How then does one determine whether there has been a structural break
and whether it matters? Most tests involve splitting the sample into two
(or more) subgroups: before and after the regime change in the case of
time-series data, above and below the threshold in the case of cross-section
data. The basis regression is then run on the subgroups and the results are
compared.
The procedure followed most frequently is the Chow test, which runs an F-test on the two sub-samples in comparison to the pooled sample to
determine if they are derived from the same underlying population. We
will not here elaborate on the mechanics of the test; most statistical pack-
ages can be commanded to perform a Chow test. The packages will deter-
mine whether the model ‘passes’ the Chow test or not; if the F-statistic is
deemed to be statistically significant, the regression fails the null hypothe-
sis of parameter stability. The conclusion is then that there has been a
structural break in the model, and that it matters.1 We will reintroduce the
Chow test in §14.1.2 when discussing parameter stability in relation to
Benjamin and Kochin’s model of inter-war unemployment.
What procedures are available for dealing with structural change? If
there is a discrete structural change in the system (as in the case of the
Anomalies Regulations) or a series of structural breaks over time, it is pos-
sible to use dummy variables to capture the change in the parameter values.
As explained in §10.1, dummy variables can be used to alter the intercept of
an equation, the slope of an equation, or both. Three simple cases are
shown in figure 11.3.
In figure 11.3 (a) there is a change in the level of the variable Y at point Z.
The slope of the relationship is the same before and after the structural
break. In this case, an intercept dummy variable, taking the value 0 before
the break and 1 thereafter, will be sufficient to capture the impact of the
structural break on the system.
§.
Thus, the model takes the following shape
Yi�a0�a1Di�b1Xi (11.1)
where Di �0 if Xi Xt �1 if Xi�Xt
In figure 11.3 (b) the slope changes after the break (this is equivalent to
the threshold effect of the DVD case). This is commonly referred to as a piecewise linear regression. In this case, it is necessary to add a slope dummy variable, obtained by the interaction of the relevant dummy vari-
able and the explanatory (numerical) variable thought to be affected by the
structural break. The dummy variable will have a value of 0 up to the
threshold, and of 1 at and above it. In this case, the model takes the follow-
ing shape
Yi�a0�b1Xi�b2(Xi Xt) Di (11.2)
where Di �0 if Xi Xt �1 if Xi�Xt
In figure 11.3 (c) the entire relationship changes after the break: both
slope and intercept are different. It is therefore necessary to incorporate
both an intercept dummy and a slope dummy. In this case, it is imperative
that the regression also includes the untransformed variables (the constant
and X). Thus, the model takes the following form
Yi�a0�a1Di�b1Xi�b2(Xi Xt) Di (11.3)
where Di �0 if Xi Xt �1 if Xi�Xt
The dummy variables should be evaluated in the same way as any other
variable in the regression. Clearly, if they are not statistically significant, the
source of the structural change has been misdiagnosed. Alternatively, it
may be that the Chow test is identifying some other problem in the con-
struction of the model.
11.3 Problems of error specification
11.3.1 Non-zero errors Although error terms with a mean value which is not zero are commonly
cited as a violation of the basic assumptions of the CLR model, the only
important consequence of a regression having a non-zero mean is that the
coefficient on the constant (intercept) term will be biased. The slope
coefficients are unaffected. As noted in §8.2.9, for many regression models
the constant term has little or no significance (in some cases, indeed, it has
no practical meaning at all). Clearly, where the constant does have a mean-
ingful interpretation, the researcher needs to exercise care in evaluating
whether the error term has a biased mean, but otherwise this violation can
be safely ignored.
It is sometimes suggested that the CLR model requires that errors should
also be normally distributed. This, however, is not the case. The OLS regres-
sion produces unbiased and efficient coefficient estimates (is BLUE)
whether the errors are distributed normally or not. The only loss associated
with non-normality is that the t- and F-statistics used for hypothesis testing
can be applied only to large samples (say, above 30 observations). Thus, the
problem arises only when a researcher is dealing with small samples (as, for
example, when running regressions over annual observations between 1921
and 1938). But she should be cautious in such a situation in any case; there is
more than one trap to jeopardize the unwary when data are scarce.
11.3.2 Heteroscedasticity Heteroscedasticity is a complicated term that describes a fairly simple
violation of the assumptions of classical linear regression methods. The
CLR approach assumes that the variance of the error term is equal across
all observations ; this is known as homoscedasticity (equal spread). Heteroscedasticity (unequal spread) arises when the variance of the error term differs across observations.
In figure 11.4 it can easily be seen that the variance of the error (the
breadth of the error distribution) in Y becomes progressively larger as X
increases. This case of heteroscedastic errors may be compared to figure 9.1
(b), where the error term is shown, by assumption, as homoscedastic. Note
that the violation of the CLR assumption occurs only when the error term
shows a clear, systematic, pattern of distortion; random differences in the
size of the error term across observations do not constitute heteroscedas-
ticity.
Heteroscedasticity may arise for a number of reasons.
(a) In time-series analysis, it may arise because of a process of learning by
experience, or, as econometricians commonly say, error-learning. If for
example, the numbers of errors your class makes on statistics exams
declines as you take more of them, the error term in a regression of
§.
average class mark over time will be heteroscedastic. Similarly, if data
collection methods in, for example, the recording of unemployment
statistics improve over time, the error term of an equation seeking to
explain the behaviour of unemployment will show heteroscedasticity.
The converse of this, decay in the quality of measurement over time,
will also produce heteroscedastic errors; as will the impact of fatigue
on productivity over time among individuals of varying robustness.
(b) In cross-section analysis, the most common source of heteroscedastic-
ity originates in differences in scale or behaviour across reporting
units. Thus, for example, the ratio of consumption expenditures to
income tends to become more variable at higher incomes, as spending
becomes more discretionary (the capacity of households to save rises
with income, but different households exhibit differential levels of
thrift).
Similarly, the dispersion of average local expenditures across par-
ishes will tend to be greater in larger than in smaller counties, since the
opportunity for divergent behaviour is greater, the larger the number
of separate decision-makers.
In both cases, cross-section regressions applied to such data will be
contaminated by heteroscedastic errors.
Figure 11.4 Heteroscedastic error terms
X3
Frequency of Y
X2
X1
Y
X0
Regression coefficients produced under conditions of heteroscedastic-
ity remain unbiased; however, they are inefficient, in that the variance asso-
ciated with the coefficients will be larger than if the errors were
homoscedastic. This will increase the confidence interval around any esti-
mated coefficient, creating problems for testing hypotheses. In the
extreme, variables that are statistically significant explanators of the
dependent variable may be dismissed as irrelevant because their standard
errors are too high, all because of the exaggerated impact of heteroscedas-
ticity on the error term.
Because of this it is important to test for heteroscedasticity before testing
for statistical significance. The first step is usually to examine the residuals
from the estimating equation. These can be plotted against the (rank-
ordered) level of whatever explanatory variable is thought to be respon-
sible for distorting the error term. If the residuals show a distinct (upward
or downward) trend on the graph, this is prima facie evidence of heterosce-
dasticity and needs closer scrutiny.
Formal tests are provided by most regression packages. The Goldfeld–
Quandt, Breusch–Pagan, and White tests are all commonly employed, both
to diagnose formally the presence of heteroscedasticity and, in the case of the
White procedure, to create a heteroscedastic-corrected error term which
permits the use of standard techniques for testing statistical significance. A
similar test is used by Hatton and Williamson in their analysis of Irish emi-
gration by county in the late nineteenth century (see §14.2.2).
11.3.3 Autocorrelation The CLR approach requires that the errors associated with each observa-
tion are independent. If there is a systematic relationship among the errors,
this is known as autocorrelation. The most common environment for autocorrelation to occur is in time-
series analysis. In this case, the error in one time period is influenced by the
error in a previous period, which is in turn influenced by the error in a
period before that, and so on. The structure of errors when mapped against
time will show a systematic pattern, rather than a random distribution as
required by the CLR model. Time-series autocorrelation is also referred to
as serial correlation. Serial correlation of errors happens if a shock to the system being
described by the regression model creates echo or ripple effects over more
than one period. A collapse of the stock market, for example, will reverber-
ate through the financial and economic system for some time after its
§.
initial shock, because it takes time for market expectations to recover.
Buyers and sellers of stocks may be more cautious in response to good eco-
nomic news for some time to come; they may also be more sensitive to bad
economic news in the future, making any after-shocks more devastating
than would normally have been the case.
This example indicates that serial correlation may arise because of
inertia (or path dependence), in which current behaviour is strongly
influenced by past events. It may also occur because the economic system is
slow to rebound from random shocks. The economy of the American
South, for example, took a very long time to recover from the shock of the
Civil War on the market-place. The plantation system of the cotton
economy was severely disrupted by the war (and subsequent slave emanci-
pation), which required a long period of institutional adjustment of
market institutions after 1865. So, too, was the market for American cotton
abroad, which faced competition from Brazilian and Egyptian suppliers
who had begun to supply Britain during the Union blockade of
Confederate exports.
Serial correlation may also arise for more mundane reasons. Data are
often manipulated before publication in ways that create systematic inter-
dependence among errors. When information is missing for certain
observations, the gaps are often filled by interpolation between reported
data points, or by extrapolation from known information. If a data series
shows sharp peaks and troughs, it may be smoothed by the use of moving
averages to reveal the underlying trend; similarly, if there are strong sea-
sonal fluctuations in a series, researchers may have employed seasonal cor-
rection before publishing the data. In each of these cases, the errors are
redistributed from their original (true) structure, and serial correlation
results.
Some typical patterns of autocorrelated errors are shown in figure 11.5.
Four types are displayed. In (a) and (b), each successive term is positively
correlated to the last. In (a), the errors oscillate around a zero mean – this
pattern might emerge when modelling a business cycle or some other
system in which there are regular, fluctuating errors. In (b), the degree of
correlation is higher and cumulating – the errors become consistently
larger over time.
The cumulative effect is also seen in (c), but in reverse. Here the process
starts with a large shock at the beginning of the period; the negatively corre-
lated errors cause a gradual decumulation in the error term. Finally, (d)
illustrates another pattern of negative serial correlation, consistent with a
situation in which the aftermath of an original, positive shock follows a
distinctly non-linear trajectory over time, diverging from the linear
specification implicit in OLS regression.
Although autocorrelation is normally associated with time-series data,
it may also crop up in cross-section analysis. In this case, the error asso-
ciated with one observation (one Irish county, or one Poor Law district) is
systematically related to the error in another observation (or observa-
tions). Such spatial autocorrelation could occur in migration analysis, for example, if there were information chains which linked one county (say,
Dublin) to another (say, Limerick). In this case, a shock to the level of emi-
gration of Dubliners would also increase emigration from Limerick.
Similarly, if the authorities in charge of setting Poor Law rates in one Kent
parish were influenced by rate-setting in other parishes in the county, it
§.
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Figure 11.5 Typical patterns of autocorrelated errors
would no longer be appropriate to assume that the error term in a regres-
sion analysis of Poor Law rates was free of autocorrelation.2
The problem of autocorrelation for OLS regression with time-series
data is illustrated in figure 11.6. In this case, we have assumed systematic
positive serial correlation, such that the errors grow over time (as in figure
11.5 (b)). We have shown two possible outcomes relative to the true rela-
tionship between X and Y. In the top line, we have begun from the assump-
tion that there is a positive shock to the relationship (i.e. the error term at
time zero is positive); since the error in the next period is positively corre-
lated with the original shock, it too will be positive, as will be the next, and
so on. Thus, the estimated OLS regression line deviates progressively
further from the true OLS line over time.
Because the error terms are systematically biased upwards, the meas-
ured observations are also larger than their true values (in the sense of the
values predicted by the true underlying relationship between X and Y, as
represented by the middle line). In the bottom line, we have assumed the
same structure of autocorrelated errors, but imposed them from a
different starting point in which the original shock was negative. Again,
the error term shows a systematic bias from its non-zero mean, generating
observations that are consistently and progressively too low relative to the
true OLS line.g
It is quite clear that serial correlation creates biased estimates of the
true relationship between X and Y. Moreover, for any given system of
autocorrelation, the direction of bias is entirely dependent on the size and
direction of the original shock to the system. Unless the serial correlation
is eradicated, the results of any such regression may be severely mislead-
ing.
Autocorrelation is by far the most harmful of the mis-specifications of
the error term; fortunately, it has proven to be among the easiest to diag-
nose and to take care of. The procedure for correcting autocorrelation is to
identify the structure of autocorrelated errors and remove it from the error
term, leaving only random errors behind.
As always, the first step is to diagnose the problem. This may be achieved
by visual inspection of the residuals to identify any obvious pattern. The
difficulty with visual inspection, of course, is less in determining whether a
pattern exists than in deciding whether it is significant enough to warrant
correction. However, most regression analyses provide a direct test, known
g The alternative regression lines gradually converge towards the true relationship because the
influence of shocks tends to diminish over time.
as the Durbin–Watson, which produces a test statistic (the DW or d statis- tic) that may be used as benchmark against which to judge the extent of
serial correlation.
If there is no autocorrelation, the value of d will be approximately 2.0.
The further d is from 2.0 the more probable it is that autocorrelation con-
taminates the data set. However, there is also an indeterminate range of d,
in which autocorrelation may or may not exist. Given the seriousness of the
outcome, it is probably best to be conservative and to correct for autocorre-
lation, even when the test statistic remains in the indeterminate range.
What if the Durbin–Watson test indicates autocorrelation? The first
response should be to ensure that the problem is not one of model or vari-
able mis-specification, as is frequently the case. Possible culprits include
non-linearities or missing explanatory variables. If the researcher is
satisfied that this is a bona fide case of autocorrelation, any of the standard
correction procedures provided by the computer package may be applied.
The most common are the Cochrane–Orcutt procedure, Durbin’s two-
stage method, and the Hildreth–Lu technique. If the true underlying error
term of the regression is normally distributed, each of these correcting pro-
cedures is equivalent. Each of the procedures generates both an estimate of
the degree of autocorrelation (�, lower case Greek rho) as well as error-
corrected coefficient estimates. In terms of figure 11.6, each procedure will
take steps to reconstruct the error term to its random component, amend
the observation estimates accordingly, and effectively rotate the regression
line away from its biased location towards the true relationship.3
§.
Figure 11.6 Autocorrelation
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True OLS line
Biased OLS line
Biased OLS line
The Durbin–Watson procedure is an inappropriate test of autocorrela-
tion in the case of autoregression (including autoregressive transformations
of data such as the Koyck lag model; see §10.2.1). When a regression
includes the lagged dependent variable as a regressor, the value of the
Durbin–Watson statistic is biased towards 2, so that the null hypothesis of
serial correlation would invariably be rejected. It is necessary therefore to
find an alternative procedure.
A number of other procedures have been developed, of which one of the
most frequently cited is the Breusch–Gordon Lagrange Multiplier test. We
discuss this briefly in our analysis of Hatton and Williamson’s work on
Irish emigration in chapter 14, which uses this procedure to test for the
extent of serial correlation in an autoregressive framework.h
11.3.4 Outliers A potential further problem results when one (or more) of the errors in the
error term is unusually large, relative to the rest of the distribution. While
this does not formally contravene any of the assumptions of the CLR
model it does have some negative consequences, especially for OLS estima-
tion, and should not be ignored.
A potential warning sign of this problem is the presence of outliers in a sample. These are observations in a data set that do not conform to
the patterns suggested by the remainder of the sample.i They are unusual
observations.
Two types of outlier may be identified. One occurs when the value of an
explanatory variable for an observation is very different from the remain-
ing values in the sample. The other occurs when the dependent variable for
an observation is very different from that predicted by the regression
model (it is measured with a large residual).
Both types of outlier may seriously affect regression results. An unusual
value of an explanatory variable relative to the rest of the sample may con-
stitute a leverage point, giving it unusual influence over the value of the regression coefficient. Thus, in the bivariate model of figure 11.7 (a), most
of the data points are clustered together in the centre of the distribution,
while one observation clearly diverges. The slope of the OLS regression line
in this example depends entirely on the position of that one data point.
h One test that was much used in previous econometric work to test for serial correlation in auto-
regression is Durbin’s h. However, recent analysis suggests that this is an inappropriate proce-
dure. i For a preliminary analysis of the problem, see §3.1.3.
The second type of outlier is depicted in figure 11.7 (b), in which one
observation is measured with a large error. The least squares method will
cause the regression to pivot towards that observation, as it searches to
minimize the aggregate value of the residuals (‘the sum of squares’), thus
distorting the results. Once again, a single observation has a large influence
on the regression results.
Thus, the key issue is not whether an observation is an outlier per se, but
whether it is an influential observation – i.e. whether it has unusual influence on the regression model.
Not all influential observations are equally harmful. Indeed, it is often
pointed out that the existence of explanatory variables far from the centre
of the sample is a good thing, because it gives the regression model more
information with which to discern patterns in the relationship among var-
iables. As a general rule, the more tightly clustered the values of a particular
explanatory variable, the more poorly defined its coefficient estimate.
The real problem is not with leverage points (unless it can be shown that
they are misleading), but with rogue observations. These are observations that do not belong with the rest of the data set, not because one or more of
the explanatory variables were unusual, but rather because the model
linking X and Y is not the same for that observation as for the rest of the
data. Rogue observations carry the potential to distort parameter estimates,
invalidate test statistics, and may lead to incorrect statistical inference.
How should we test for outliers and determine whether they are good or
bad influences?
Clearly, the first step is to determine whether there are any influential
observations in the data set. The crucial question is whether the regression
model is significantly changed by the presence of any single observation.
This can be addressed by comparing the model results with and without
each data point in sequence. Tests can determine whether the absence of an
observation causes any regression coefficient to change by a significant
amount (the dfBeta test), or whether it significantly affects the measure-
ment of the dependent variable (the dfFits test).
Let us say that a model has been tested and one or more observations are
revealed to be influential. What should be done about them? It is tempting
to discard the observations entirely. But, because of the potential value of
leverage points, this should be resisted. Rogue observations are more
clearly expendable. But how can they be distinguished?
Possible leverage points are often identified by graphing a scatter plot of
the data and searching for unusual observations.j But there are clear limits
§.
j This was done in figure 3.2, in our preliminary discussion of the problem.
X
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Figure 11.7 Outliers
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(a) An outlier as a leverage point
(b) A rogue outlier
to the usefulness of graphical aids, especially when dealing with more com-
plicated multivariate models, where it may be difficult to spot an outlier by
eyeballing. More helpful are tests (such as Hadi’s) that systematically
examine the distributional parameters of the data set and determine
whether any observations fall into the upper or lower tails.
There are also a number of tests for outliers due to error in the estimat-
ing equation.k One is to include observation-specific dummies in the
regression, and to test whether the coefficients on these are statistically
significant. Ideally, this test should be run over all observations, in order to
pick up any outlier – it is a test of the entire system, not just of one apparent
odd case.
Regardless of the outcome of these tests, if outliers are identified, the best
course is to abandon OLS estimation, and to utilize one of the various
regression methods that have been developed to accommodate outliers.
These are known as robust regressions, implying that their results are robust to problems of this sort. There are a variety of robust estimators, almost all of
which use weighting techniques to reduce the emphasis on high-error obser-
vations. All are superior to OLS methods in this situation. They may be run
over the entire data set and their results compared to the OLS specification.
If the robust regression method generates very different results, it suggests
some further problems with the data. For an illustration, see the robust
regression we run in §14.1.2 to test the Benjamin and Kochin model.
11.4 Problems of variable specification
11.4.1 Errors in variables Both the dependent and the explanatory variables may suffer from meas-
urement errors, arising from incomplete information, mistakes in report-
ing or recording of information, or flawed responses to survey questions.
Lord Stamp’s famous quotation about the accuracy of official Indian statis-
tics is worth quoting here, as it is broadly applicable to the historical statis-
tics of many other countries:l
The Government are very keen on amassing statistics – they collect them, add
them, raise them to the nth power, take the cube root and prepare wonderful dia-
grams. But what you must never forget is that every one of those figures comes in
the first instance from the chowty dar (village watchman), who just puts down what
he damn well pleases.
§.
k Visual inspection of the regression residuals is not one of them, since the pivoting of the OLS line
to minimize the total sum of squares will tend to mask individually large residuals (see figure
11.7 (b)). l Sir Josiah Stamp, Some Economic Factors in Modern Life, P. S. King & Son, 1929, p. 258.
This, fortunately, over-states the extent of the problem, but it is clearly
right to be wary about the implications of any such errors in measurement
on our regression results.
We can divide the problem between errors in the dependent variable
and errors in the explanatory variables. In both cases, it is important to dis-
tinguish stochastic from systematic errors. Stochastic errors, which originate in rounding of data, accidental inac-
curacies in reporting or recording, or other random variability, may be
thought of as increasing the size of the error term, while maintaining its
fundamental characteristics of being symmetric around a zero mean.
Systematic errors, which arise from consistent distortion in the report-
ing or recording of information, are more troublesome. They clearly add to
the size of the error term, but also destroy its fundamental characteristics.
Statistical analysis can provide no solutions to this type of error, nor any
guide to its possible impact on regression results, other than to say that
any regression working with data that is systematically distorted will pro-
duce results that are also systematically distorted, perhaps to the point of
uselessness.
It is therefore imperative that researchers should try to understand the
data and any possible sources of such systematic error, and should attempt
to make corrections to the data before running a regression. If, for
example, it is known that the measurement of US imports was biased
downwards between 1861 and 1865 because of blockade running during
the Civil War, it is important that a correction be made for this before
running a regression on (say) the relationship of imports to incomes and
relative prices over 1840–1900. This correction might be made by adding
20 per cent to the level of imports in that period (the best guess of the
extent of smuggling). Since the correction is a best guess, it will obviously
be imprecise. It might be thought that this is merely substituting one error
for another. However, there is a crucial difference: it is substituting a
stochastic for a systematic error, and this is clearly a move in the right
direction.
Similarly, if it is known that a vital explanatory variable is missing from
a data set, it is imperative that a proxy be introduced to capture its effect on
the dependent variable. If this is not done, all the other explanatory vari-
ables will be biased. The proxy, once again, cannot be measured with abso-
lute accuracy, but if it is properly chosen, the added error it introduces to
the regression will be stochastic, and its addition will correct for the
systematic error created by the absence of the vital explanatory variable.
In what follows, we assume that systematic errors have been expunged
and that the only remaining errors in variables are random or stochastic.
Error measurement in the dependent variable is not very important.
The error term will be larger, suggesting an increase in the imprecision of
the regression as a whole, which will show up as a fall in the R2. The stan-
dard errors of the regression coefficients will be higher, leading to smaller
t-statistics, but the coefficients themselves will be unbiased.
Error measurement in the explanatory variables is more problematical.
The error is added to the error term. This not only makes it larger and the
regression less precise, it also destroys the fundamental assumption that
the error term is independent of the explanatory variables. Clearly, if X is in
error and this error is added to e, then X and e must be correlated. When a
regression is run over Y, X, and an error term, e, the coefficient on X will be
biased; the greater the extent of measurement error, the greater the extent
of the bias.
The solutions to errors-in-variables problems are far from perfect. The
most common solution involves instrumental variables (IV) estimation.m
An instrumental (or proxy) variable is one that is highly correlated to the
explanatory variable in error, but which is not correlated with the error term
of the equation. This sounds good in theory, but is very hard to apply in prac-
tice. Not only are good proxies difficult to find, it is even harder to determine
that they are indeed independent of the error term, as required by theory. For
this reason, many researchers are disposed to ignore the problem, on the basis
that most measurement errors are small, and that attempting to correct them
may cause more problems that leaving them alone.
There are however a couple of approaches which are worth pursuing in
cases where the errors-in-variables problem is too large to ignore.
One possibility that is widely canvassed in time-series analysis is substi-
tuting a lagged value of the explanatory variable in question. Lagged and
current values are likely to be correlated with each other; as long as their
errors are not so correlated, the lagged variable will be an appropriate
instrumental variable.
The most efficient approach to errors-in-variables problems in cross-
section analysis is the Durbin method. This solution operates by ranking
the explanatory variable by size (from 1, the lowest, to n, the highest) and
using the rank order (1, 2, ... , n) as its proxy replacement.
11.4.2 Multicollinearity Multicollinearity is a term that was originally introduced to capture the
effect of a perfect, or exact, linear relationship between two (or more) of
the explanatory (right-hand) variables in a regression.
§.
m Kennedy discusses the instrumental variables approach in a Ballantine framework in Peter E.
Kennedy, A Guide to Econometrics, 3rd edn., Blackwell, 1992, pp. 143–5.
We have already met this problem in the form of the ‘dummy variable
trap’ (see §10.1.1). The problem could also arise if, in a regression on
county-level Irish migration, both AGRIC (the share of the labour force in
agriculture) and NONAG (the share of the labour force employed outside
agriculture) were included; or if a regression of aggregate relief payments
per county were to include both the number of persons receiving relief and
the average payment per recipient. But, as these examples make clear, the
chances of encountering perfect linear relationships outside of design flaws
in the model are rare.
Given the rarity of this situation, the term multicollinearity is now gen- erally applied more broadly to situations in which two or more explanatory
variables are highly, rather than perfectly, correlated with each other.
Within our data sets, there are numerous cases where we would expect
to find significant correlations among variables. In the cross-section analy-
sis of migration by Irish county, for example, we might well expect RELIEF
and HOUSING to be significantly related to each other, as well as URBAN
and AGRIC. Neither would be perfectly collinear, but statistical tests will
reveal high correlation coefficients across the data set.
What will happen as a result of such multicollinearity? In the extreme
case of perfect collinearity, the computer will generate coefficient estimates
that are indeterminate with standard errors that are infinite. Indeed, if any
regression produces such a result (see, for example, question 3), it is an
indication that the researcher has either fallen prey to multicollinearity, or
has fallen into the dummy variable trap. The solution is easy: the offending
supernumerary variable must be identified and dropped from the list of
right-hand variables.
What about the weaker, but more frequent, case of high but imperfect
collinearity? The standard errors will be very high relative to the size of the
coefficient, generating a low t-statistic, and suggesting that the explanatory
power of the affected variable will be weak. The reported confidence inter-
val around the measured coefficient will be large; frequently, it will include
zero, suggesting that the variable is without statistical significance. In cases
of true multi-collinearity (in which more than two explanatory variables
are highly correlated with each other), the statistical significance of all
affected variables will be pushed towards zero. At the same time, the overall
explanatory power of the equation, as measured by R2, may be very high
(e.g. 0.9 or higher).
But not all cases of insignificant t-statistics can be blamed on multicolli-
nearity. In many cases, it may simply be that there is no systematic relation-
ship between X and Y. How can one distinguish between these two cases?
One indication is the presence of a high R2 but few significant t-statistics.
If the regression packet also reveals that the F-test rejects the null hypothesis
that all coefficients are simultaneously zero, then that is added reason to be
suspicious.
Another simple test is to examine the correlation coefficients across
explanatory variables to see if they are jointly determined. An r of 0.75 or
above is a frequently recommended diagnostic measure. But, more obvi-
ously, one should think about the variables being included in a model and
try to avoid introducing explanatory variables that are clearly measuring
the same, or very similar aspects, of a problem, hypothesis, or model.
A third strategy is to run a series of stepwise regressions, in which additional explanatory variables are added one at a time, recording all pos-
sible combinations of regression results. The changes in the coefficients at
each permutation of the variables can be used to infer the structure of
collinearity.
It is not always possible to avoid including two or more variables that are
collinear. For example, in measuring the level of Irish emigration across
counties, it is highly relevant to determine whether counties dominated by
agriculture in the post-famine era were more likely to shed excess labour,
and also to discover whether counties with a higher level of urbanization
had better networks of communication and information that promoted
emigration. Agriculture and urbanization may be highly (negatively) cor-
related with each other, but they are measuring different, separable factors
that may have influenced migration, and the costs of including both are
lower than the costs of leaving either out.
If we exclude either one, we jump from the frying pan of multicollinearity
into the fire of specification bias. In the absence of one variable, the other
would take on the added burden of explanation. Moreover, the coefficient on
the variable left in would be biased (it would be a mix of the two ‘true’
coefficients). If both are left in, the coefficients remain unbiased, only their
statistical significance is adversely affected. Of course, if one variable is left
out and the other remains statistically insignificant, this might tell us that
neither variable was important to begin with, even if they were collinear.
In the end, multicollinearity has to be accepted as an occupational
hazard of running regressions. In some sense, its existence represents a
minor morality play, indicating the limitation of quantitative methods and
emphasizing that running regressions is no substitute for historical reason-
ing and appropriate, logical, model-building.
§.
11.4.3 Simultaneity If we run a regression of Y on X, we are asserting that Y is the dependent
variable and X is its explanator. There is a strict causal link from the right-
hand side of the equation to the left.
But (as we have already noted in §4.1.1) it is not difficult to think of
circumstances when the relationship between X and Y is not unidirec-
tional. In such cases, Y is influenced by X but X is also influenced by Y. The
classic example relates to the market for a particular commodity. Increases
in the supply of this commodity will tend to reduce its price (as its scarcity
declines, people have to pay less to purchase it). At the same time, however,
as the price of the commodity falls, its supply tends to drop (as producers
shift towards manufacturing other items that will generate more revenue).
Price is determined by supply (and demand); at the same time, supply (and
demand) are determined by price.
This is the case of simultaneous equations, in which there is no truly independent, or exogenous, variable determined outside the framework of the model. All variables are, to a greater or lesser extent, endogenous – that is, determined within the model. The explanatory variables are no longer
independent of the error term (since a shift in the error term directly
changes the dependent variable, which in turn changes the explanatory
variable because of the feed-back effect), thus violating one of the principal
assumptions of the classical linear regression.
Simultaneity generates biased estimates of the relationship between X
and Y. Moreover, the bias does not disappear as the sample size increases (it
is therefore also inconsistent). In the extreme case, this arises because it is
simply not possible to characterize the relationships between X and Y with
a single estimating equation. The extent of bias will depend on the degree
of correlation between the explanatory variables and the error term: the
higher the correlation, the greater the bias.
What can be done about simultaneity? If the extent of such bias is small,
it is tempting to do nothing. In other words, if the extent of feed-back from
Y to X is very limited, the extent of correlation of the X and e will be small,
and the bias in the estimated coefficient on X will also be small.
If, however, the problem is considered to be more significant, then one
of two solutions may be followed. The first is to model the system more
appropriately and then estimate the relationships as a system, rather than
as a series of separate regressions. Such an approach, however, falls far
beyond the limits of this book. A more limited response is to use single-
equation methods, but to do so as a two-part procedure. This is the tech-
nique known as two-stage least squares (2SLS). It is a special case of the
instrumental variables approach, which was introduced previously to indi-
cate a solution to the errors-in-variables problem.n
The 2SLS procedure operates by locating an instrumental variable that
can substitute for the biased explanatory variables in the simultaneous
equation. The logic of the 2SLS procedure is to correct the biased explana-
tory variable by expunging it of the feed-back impact from the dependent
variable. The best way to do this is to treat the contaminated variable (X) as
itself a dependent variable, and regress it against the other explanatory var-
iables in the system (since the formal dependent variable is not included in
this first stage of the estimating system, the feed-back effects are necessarily
excluded).4
The predicted values of X from this regression are then used either as
instrumental variables in the original estimating equation, or alternatively
(and more simply) substituted for the original values of X in the equation.
The results from this second stage of the procedure will be consistent (i.e.
unbiased over large samples); moreover, most econometricians consider it
to be the best solution to the problem of simultaneity in small samples as
well. In the final chapter (§15.1.1) we will examine the use of this technique
by Boyer to deal with the problem of simultaneity in his model of the Poor
Law and the agricultural labour market.
We have treated each of the violations of the CLR model separately. However,
although our analysis implies that each problem of mis-specification may be
treated in isolation, most practitioners recommend that more general strate-
gies be employed that investigate the model as a whole. We discuss this and
other aspects of best-practice model evaluation in §12.6.
Notes 1 An alternative methodology for identifying structural breaks is to run the regression
over one subgroup, and then apply the estimated coefficients to the known explana-
tory variables to forecast the value of the dependent variable in that part of the
sample thought to lie beyond the structural break or threshold. The measured values
are then compared to the forecast; if the measured values fall outside the predicted
confidence interval, this is prima facie evidence of parameter change. 2 Note that autocorrelation may occur because of specification problems in either
model or variable selection. We observed an example of model mis-specification in
our discussion of the U-shaped cost curve in §11.1.1, where we noted that the resid-
uals from figure 11.1 follow an inverted U-shape against firm size. This situation is
n Both errors-in-variables and simultaneous equations raise the same problem of correlation
between X and e, although the causal connection is significantly different in each case.
similar to that depicted in figure 11.5 (d) and represents a clear case of autocorrela-
tion. A similar outcome may result from excluding a relevant explanator from a
regression, especially if the missing variable follows a systematic pattern over time.
In these cases, the correct procedure is to correct for the mis-specification; if this is
done properly, it should also take care of the autocorrelation. 3 The analysis in this chapter assumes first-order autocorrelation, i.e. that the correla-
tion across errors is from this period to the last, and so on. In quarterly data, it may be
that the error in this quarter’s data is correlated not to the last quarter’s error, but to
the error from one year (four quarters) ago. If this is the case, the DW statistic will
not diagnose the problem. More advanced textbooks discuss the procedures appro-
priate to this situation. 4 Simultaneity is not to be confused with the situation of reverse causation in which a
model is compromised because the causation really runs in the opposite direction
from that specified by the researcher; see, for example, the discussion of such a possi-
bility in relation to Benjamin and Kochin’s model in §14.1.1. Reverse causation is a
problem of model mis-specification that arises not from contraventions of the statis-
tical properties of the CLR model, but rather from a fundamental misreading of the
historical evidence. It is another instance of the principle that correlation need not
imply causation.
11.5 Exercises for chapter 11
1. Take the Boyer birth rate data set and run a regression of
BRTHRATE�a�b1 INCOME�b2 DENSITY�b3 INFTMORT�
�b4 CHILDAL3�b5 CHILDAL4�b6 CHILDAL5�
b7 HOUSING �b8 ALLOTMNT �b9 COTTIND�e.
(i) Check that your results are the same as those reprinted in table 15.2.
(ii) Instruct the computer to generate residuals (i.e. Ŷi Yi) from the
estimating equation. Plot these residuals against the value of (a)
BRTHRATE; (b) INCOME; and (c) POP.
Describe and interpret your findings. Are there any indications of model
mis-specification in these results?
2. With the aid of the combined Boyer data set, construct a model that tests
the importance of omitted and redundant variables for evaluating regres-
sions. For example, in a model explaining INCOME, you might choose to
exclude LONDON; alternatively, you might choose to include COUNTY
(as a continuous, rather than as a dummy, variable). Be sure to include the
impact of both problems on t-statistics and R2. Is it correct to assert that the
impact of omitted and redundant variables is larger, the smaller the
number of explanatory variables included in the regression?
3. Examine the correlation matrix of variables in the Hatton–Williamson
cross-section data set.
(i) Are there indications of possible multicollinearity in the matrix?
Which variables strike you as especially of concern? What difference
does it make if you remove one of the pair of variables from the
equation? (Hint :you will need to remove each in turn.) Is the impact
felt solely on that one variable or are others affected? Explain your
findings.
(ii) Instruct the computer to construct a new variable, RURAL, which is
equal to (1 – URBAN). Now run an equation of CNTYMIG on the
appropriate variables, including both RURAL and URBAN. What is
the result?
4. In your own words, write a paragraph making clear the distinction
between the concepts of autocorrelation and autoregression.
5. Write a brief essay, identifying what you consider to be the most likely
sources of error in historical statistics, being sure to indicate which errors
are most serious for statistical analysis and how you might try to evaluate
their significance.
6. A researcher uses the Boyer relief data set to construct a model explain-
ing the cross-section pattern of wealth per person across early nineteenth-
century English parishes. The model is simply a linear regression of
WEALTH on three variables, LONDON, DENSITY, and GRAIN. Is there
any reason to suspect that the data may be contaminated by heteroscedas-
ticity?
As a preliminary test, you examine the variance of wealth across
parishes in the data set. What do you find and how do your findings
compare with your prior expectations? (Hint : instruct the computer to
group the parish data by size, rather than trying to analyse the problem
using parishes singly.)
As the next step, you re-examine the researcher’s model for signs of
heteroscedastic disturbances. Run the regression model and examine the
residuals to determine whether there are signs of heteroscedasticity, and if
so whether:
(a) they conform with the preliminary analysis of the variance of wealth
(b) they appear to be serious.
Write a brief report to explain your findings.
7. Test the Benjamin–Kochin data set for signs of serial correlation in:
§.
(i) the behaviour of the benefit–wage ratio over time.
(ii) the behaviour of unemployment over time.
(iii) a regression of unemployment on the benefit–wage ratio.
Explain your method and interpret your findings.
8. Generate a scatter-plot of the Irish emigration rate for 1876–1913. Are
there any years that appear idiosyncratic?
Develop a methodology for testing whether these are leverage points,
rogue observations, or are not influential observations at all. Use the basic
Hatton–Williamson time-series regression model as your basis for evaluat-
ing these observations.
What do you find and how, if at all, do your findings modify our under-
standing of the history of Irish migration in this period?
9. What is the appropriate dummy variable model for the following exam-
ples of structural change?
(i) Clerical workers earned the same income as artisans in London in
1913, but chose to live in more expensive housing.
(ii) Clerical workers earned the same income as artisans in London in
1913, but chose to spend more of each extra pound on housing.
(iii) The number of overseas holidays is higher for business executives
than for clerical staff; fortunately for the executives, since they earn
more, they can afford it.
(iv) The number of overseas holidays has increased significantly now
that ordinary families can afford to travel abroad.
(v) Once ordinary families found they could afford to travel abroad,
they gained a taste for foreign culture and the growth of overseas
holidays has accelerated.
10. A critic of Hatton and Williamson’s time-series analysis of Irish emi-
gration, 1877–1913, notes that there was a structural break in the relation-
ship between relative wages and emigration around 1896. Before 1896,
relative wages were falling, whereas after 1896 they rose. However, Irish
emigration rates continued to decline.
You wish to evaluate this critique. Take the following steps:
(i) Graph the pattern of relative wages to determine whether there is a
change in behaviour after 1896.
(ii) Use a dummy variable test to determine whether there is a significant
difference in the relationship between the relative wage and Irish
emigration after 1896 (see chapter 10, question 8 for the basic regres-
sion equation).
The dummy variable test focuses on the stability of only one explanatory
variable. You wish to evaluate the stability of the entire model over the
period. In order to do so, you choose the Chow test. A working description
of the test may be found in §14.1.2.
(iii) Use the Chow test to evaluate whether there is a structural break in
the emigration model in 1896. What do you find?
(iv) Compare the coefficient estimates from the two subsamples with the
regression run over the entire period.
What is your overall response to the critic? How, if at all, does this alter our
understanding of Irish emigration in this period?
§.
Further topics in regression analysis
Non-linear models and functional forms
In all the preceding discussion of regression and correlation, both simple
and multiple, we have always assumed and tested for linear relationships.
This means that in a simple bivariate regression, the regression line is
always a straight line; in a multivariate regression, the relationship
between the dependent variable and each of the explanatory variables is a
straight line. However, we were warned in §11.1.1 that the results
obtained by applying the classical linear regression (CLR) model might
be seriously distorted if the actual relationship between the variables was
not linear.
One illustration of such a non-linear association was given for cost data in figure 11.1. As another, we can take the familiar relationship between age
and height. Data for males in Britain from birth to age 34 are plotted as a
scatter diagram in figure 12.1.1 We see a changing pattern with an initial
period of rapid increase between birth and 2 years, then steady growth with
a slight acceleration from about 12 to 16 years, and after that a period of
stability from an age of about 16 or 17. If we extended the chart to later ages
we would find a slight decrease in height from about 40 years of age, and
this would become more pronounced as osteoporosis takes its toll in old
age.
It is, of course, possible to fit a straight line to these data, but as figure
12.1 shows, this is clearly a very poor way to model this relationship over
the whole of the life cycle, and it becomes increasingly less appropriate after
age 16. Can we do better?
In §12.1 we discuss various forms that might be adopted to model non-
linear relationships. Then in §12.2 we explain the procedures available for
transforming some of these non-linear models so that they can be esti-
mated by familiar OLS procedures. §12.3 is devoted to the interpretation
of the regression coefficients when the basic relationship between the
explanatory and dependent variables is non-linear, and §12.4 looks at the
procedures for fitting log-linear and non-linear trends to time series.
§12.5 reviews some of the principles that might guide a researcher in
choosing to specify a particular non-linear relationship, and illustrates
these by examples from the historical literature. Finally, §12.6 discusses
how to judge whether the model specified represents a suitable way of
interpreting the data and conforms to the empirical realities.
12.1 Different forms of non-linear relationships
Our study of linear relationships was built on models such as
Y�a�b1X1 b2X2�e (12.1)
When we turn to non-linear models there are a number of possible
alternatives, each of which specifies a different type of relationship. Some
of the most common relationships are listed below; in each case a and b
(and for (12.7), g and h) are constants, and X and Y are variables.a
a If any of these relationships were used to model data in regression analysis, the model would also
have to include an error term.
Figure 12.1 A non- linear relationship between age and height
Age (years)
36322824201612840
H ei
gh t
(c m
)
220
200
180
160
140
120
100
80
60
40
� A quadratic curve or parabola Y�a�b1X�b2X 2 (12.2)b
� A reciprocal or inverse curve Y�a�b (12.3)
� A geometric curve Y�aXb (12.4)
� A multivariate version of (12.4) Y�aXb11 X b2 2 (12.5)
2
� An exponential curve Y�abX (12.6)
� A logistic curve Y� (12.7)
Some illustrative non-linear relationships are shown in figures
12.2–12.6. The shape and position of many of these curves depend primar-
ily on the value taken by the parameter, b. For example, the geometric curve in figure 12.2 slopes upwards and to the right if b is positive, but downward and to the left if b is negative. Similarly, the exponential curve in figure 12.3 slopes upward and to the right if b is greater than 1, but
downward and to the left if b is positive but less than 1.
All four reciprocal curves in figure 12.4 are drawn for values of a equal �2 or 2, and for b equal �3 or 3. As X increases the curves fall if b is
positive, and rise if it is negative, and they get progressively closer to the
value of the a, though they never quite reach it.c
The crucial parameters governing the shape of the four quadratic curves shown in figure 12.5 are b1 and b2. The curves depicted in figure 12.5 (a) slope upwards if both parameters are positive, and slope downwards if
both are negative. In figure 12.5(b), we have drawn two curves for which
the parameters have opposite signs. The left-hand curve has a positive
value of b1 and a negative value of b2; the right-hand curve has a negative
value of b1 and a positive value of b2. The intercept is determined by the
value of a, in the same manner as with a linear regression.d
The slope of each curve is determined by the relative value of b1 and b2,
in the form of (b1 2 b2X). For the two curves in figure 12.5 (a), the higher
the value of this expression, the steeper the slope. The fact that X is
h
g � abx
1
X
§. -
b Equation (12.2) has a single explanatory variable, X, which appears as X and also squared. It is
one of a family of curves (known as polynomials). A quadratic might have additional explana-
tory variables (X2, X3, and so on). A higher-order polynomial will have higher powers than X 2;
for example, the cubic equation will have a term in X3, and the model might be: Y�a�b1X1�
b2X2 3. Any polynomial will be non-linear.
c The mathematical name for a curve such as this, that approaches progressively closer to a line
without ever reaching it, is an asymptote. d In each case, we have set a�400; b1�45; and b2�1.5. Only the signs have been changed.
X
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Y 2.0
1.5
1.0
.5
0.0
Negative values of b
b = –3
b = –2
b = –1
0
Figure 12.2 Geometric curves: Y�aXb
X
11975
Y 9,000
8,000
7,000
6,000
5,000
4,000
3,000
2,000
1,000
0
Positive values of b
b = 3.5
b = 3
b = 2.5
§. -
X
18.514.09.55.00.5
Y 2.0
1.5
1.0
0.5
0.0
b less than 1
b = 0.9
b = 0.7
b = 0.5
Figure 12.3 Exponential curves: Y�abX
X
107.55
Y 20,000
15,000
10,000
5,000
0
b greater than 1
b = 2.5
b = 2.2 b = 2
included in this expression ensures that the relation between X and Y is
non-linear; as X rises, the value of the expression and, therefore, the slope
of the curve will change.
In the case of the lower panel, figure 12.5 (b), the location of the turning
point, at which the slope of the curve changes (either from positive to neg-
ative, or vice versa), is when X equals the ratio of (b1/2b2). In the left-hand
curve, b1 equals 45 and b2 is 1.5; thus, the curve reaches its maximum
when X equals 15. In the right-hand curve, b1 is 45 and b2 equals 1.5; the
curve reaches its minimum when X equals 15.e
X
302520151050
Y 5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
a and b both positive
X 302520151050
Y 2.5
2.0
1.5
1.0
0.5
0.0
–0.5
–1.0
a is positive, b is negative
X
302520151050
Y –1.5
–2.0
–2.5
–3.0
–3.5
–4.0
–4.5
–5.0
a and b both negative
X
302520151050
Y 1.0
0.5
0.0
–0.5
–1.0
–1.5
–2.0
–2.5
a is negative, b is positive
Figure 12.4 Reciprocal curves: Y�a�b/X
e Note the similarity between the top curve in figure 12.5 (a) and the geometric curve with b�0;
and also the similarity between the lower curve in figure 12.5 (b) below the turning point and the
exponential curve with b 1.
The final non-linear relationship to be considered is the logistic. The
most general form of the logistic curve is given in (12.7). With four con- stants and the variable, X, there are numerous possible permutations, and
we will mention only a selection of the various shapes the curve can take. If
the constants g and h are both positive, b is greater than 1, and the power x
in the denominator is negative, then the logistic curve slopes upward from
left to right in an elongated S-shape (also known as a sigmoid curve). If the
§. -
X
302520151050
Y 4,000
3,000
2,000
1,000
0
b and b both positive1 2
X
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Y 800
700
600
500
400
300
b is positive, b is negative1 2
X
302520151050
Y 500
400
300
200
100
0
b is negative, b is positive21
X
302520151050
Y 1,000
0
–1,000
–2,000
–3,000
b and b both negative1 2
Figure 12.5 Quadratic curves: Y�a�b1X�b2X
2
(a) Rising and falling without turning points
(b) Maximum and minimum turning points
value of b, or the sign of x or h changes – for example, if b is less than 1; or if
all three of them are changed – for example, h negative, b less than 1 and x
positive – the logistic curve slopes downward from left to right in the shape
of a backward-elongated S.
The constants g and h determine the upper and lower limits of the arms
of the S-shape at each end of the curve. For the upper limit, four special
cases are of interest; in each case the lower bound is zero. The four cases and
the resulting upper limits are:
Value of g Value of h Upper limit of curve
0 �0 �
1 �0 h
�1 �0 h/g
1 1 1
The constant, a, determines the point at which the curve begins to rise;
the greater a is, the further to the right will be the point at which the curve
begins to bend upwards. The final constant, b, determines the slope of the
curve: the greater it is, the steeper the slope.
Figure 12.6 shows two logistic curves, for both of which g and h�1, so
that the upper limit of both curves is 1. The two curves have the same value
of a (�2), but different slopes: for one b�1.5 and for the other b�e (the
exponential constant introduced in §1.6.2, with a value to four decimal
places of 2.7183). The reason for choosing this value for b is explained
below.
Figure 12.6 Logistic
curves: Y� h
g � abx
10.05.00–5.0–10.0
b = 2.7183
b = 1.5
1.0
0.0
One particular variant of the logistic curve is of special importance. In
this form g, h, and a are all set equal to 1, x is negative, and b is taken as
equal to e. The equation is thus
Y � (12.7a)
It can also be shown by manipulation that this is identical to3
Y � (12.7b)
This form of the logistic curve, with a lower limit of 0 and an upper limit
equal to 1, plays a central role in the procedures we will discuss in chapter
13.
All these relationships look very much more difficult than the simple
linear regressions, and of course the underlying mathematics is more com-
plicated. However, for our purposes the basic ideas essentially follow
through from the linear to the non-linear models. The procedures and
measurements for non-linear regression are thus very similar to those
already explained in chapters 4 and 8.
There are, however, two important additional aspects of the procedures
for non-linear relationships that must be considered. The first relates to the
method of estimation, the second to the interpretation of the regression
coefficients.
12.2 Estimation of non-linear regression models
For the purposes of undertaking regression analysis the non-linear models
can be divided into three groups: the first group require only a simple sub-
stitution, and then can be fitted by the same ordinary least squares (OLS)
procedure that we have been using for linear relationships. The second
group can also be estimated by the OLS procedure, but only after any non-
linear explanatory variables have been transformed into logarithms.
The third group are intractably non-linear, i.e. there is no suitable trans-
formation that can be applied. These include the logistic curve, (12.7), and
models in this group may also be created by certain features of the error term.f
As a result the underlying regressions are too complex to be solved in the
usual way and require more elaborate procedures, one of which (maximum
likelihood) is discussed briefly in panel 13.1 in the next chapter.
ex
1 � ex
1
1 � e x
§. -
f For example, if (12.5) were modelled with an additive rather than a multiplicative error, so that
the model was Y�aX1 bX2
b�e rather than Y�aX1 bX2
be, then it would be intractably non-linear.
In the remainder of this section we will review examples of non-linear
models in the first two groups.
12.2.1 Two groups of non-linear models
The first group of non-linear models The first group covers models such as those shown in (12.2) and (12.3).
These can easily be fitted by the same OLS procedure that is used for
linear relationships (see §4.2.2). All that is required is to define a new var-
iable in place of any terms in X which are non-linear, for example, X2 or
1/X.g If the non-linear term is X2 we simply square all the observations in
the series for X and give the resulting series a new name, say Z. Similarly, if
the non-linear term is 1/X we simply work out the reciprocal of X, and call
that Z.
If this new series, Z, is substituted in the original equation there is then a
standard linear relationship, and we can proceed in the usual way, with all
the advantageous properties of the OLS regression. For example, the quad-
ratic in (12.2),
Y�a�b1X�b2X 2�e, (12.2)
becomes
Y�a�b1X1�b2Z�e (12.2a)
in which there are no non-linear terms. This procedure is adopted in §12.4
to fit a non-linear trend to a time series. Similarly, the reciprocal model in
(12.3) becomes
Y�a�bZ�e (12.3a)
This simple procedure can be followed for all models in which the non-
linearity occurs only in the X variables, not in the regression coefficients.
The second group of non-linear models The second group covers models such as those shown in (12.4). These can
also be estimated by the regular OLS procedure but it is first necessary to
transform the explanatory variables into logs, and these logarithms are then
substituted for the original variables. Recall from §1.6 that
(i) two numbers can be multiplied by taking the sum of their logs, e.g.
log (a � b)�log a�log b;
g Terms such as X2 or 1/X are described as ‘non-linear’ because if any one of them is included in an
equation it automatically means that the regression relationship is no longer a straight line (or
the multivariate equivalent such as a plane) but has some form of curve.
and
(ii) a number raised to a power equals the exponent � the log of the number,
e.g.
log X3�3 � log X.
By applying both these transformations the geometric curve (12.4)h can
be expressed in logs as follows
log Y�log a�b(log X)�e (12.4a)
This is called a double logarithmic (or log-log) model, because both X and Y are in logs .i An example of a geometric curve (with b� 3) is shown
in figure 12.7 (a) and its double logarithmic transformation is given in
figure 12.7 (b). The procedure to be followed for estimating a model of this
type is set out in §12.2.2.
The alternative to this is a semi-logarithmic model. This can take several forms. In one, X is in logs but Y is not
Y�a�b log X�e (12.8)
Lin-log models of this form are frequently used to model relationships.
Hatton and Williamson specified a multivariate model of this form in
their study of migration, and Benjamin and Kochin also used this form
in their analysis of inter-war unemployment.j We will look more closely at
their models in chapter 14.
In another, sometimes described as an inverse semi-log model, Y is in
logs but X is not
log Y�a�b X�e (12.9)
§. -
h The multivariate version in (12.5) is essentially the same but has one extra term in log X2. The
transformation is thus: log Y�log a�b1 (log X1)�b2 (log X2). i Some writers also refer to this as a log-linear model because – as can be seen in figure 12.6 – it is
linear when plotted in logs. However, this usage may cause confusion, and we prefer to reserve
the term log-linear for semi-logarithmic models such as (12.6a) and (12.9), where only the
explanatory variable is in logs. By extension, the semi-log model (12.8), where only the depen-
dent variable is in logs, is sometimes called a lin-log model. Using the terms in this fashion has
the advantage of indicating immediately whether the logged variables are on the left-hand side
of the equation (log-linear), the right-hand side (lin-log), or both (log-log). j See Timothy J. Hatton and Jeffrey G. Williamson, ‘After the famine: emigration from Ireland,
1850–1913’, Journal of Economic History, 53, 1993, p. 581; and Daniel K. Benjamin and Lewis A.
Kochin, ‘Searching for an explanation for unemployment in interwar Britain’, Journal of Political
Economy, 87, 1979, p. 453.
1085421
Y –
lo ga
ri th
m ic
s ca
le
2
1 0.9 0.8
0.7
0.6
0.5
0.4
0.3
0.2
a = 2, b = –1
X – logarithmic scale
Figure 12.7 Double logarithmic transformation of a geometric model
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Y –
ar ith
m et
ic s
ca le
2.0
1.5
1.0
0.5
0.0
a = 2, b = –1
X – arithmetic scale
(a) The original curve: Y�aXb
(b) The double logarithmic transformation: log Y� log a�b(log X)
Models of this form are less common, but are used, for example, in
studies of earnings, with the log of the wage taken as the dependent vari-
able and factors such as age, education, and a dummy variable for union
membership as the explanatory variables.k
A third version of the semi-log (or log-linear) model is obtained by
transformation of the exponential curve (12.6), giving an equation in
which both Y and the coefficient b are in logs, but X is not
log Y�log a�(log b)X�e (12.6a)
An exponential curve (with b�2.5) and its semi-log transformation are
shown in figure 12.8 (a) and (b).
This transformation of the exponential curve is not often used to model
relationships in the social sciences. Its importance derives from the fact
that it is the model used to fit a long-run trend to a time series. The trend
obtained in this way will be linear in the logs (though not when converted
back to ordinary numbers), and is generally recommended in preference to
either the moving average described in §1.8 or the simple linear trend
introduced in §4.2.5.
The procedure for fitting this and other non-linear trends is described in
§12.4. Fitting this particular log-linear trend has the further advantage that
the annual compound rate of growth of the series can be obtained very
simply from the regression coefficient, and this is explained later in this
chapter (see panel 12.1).
12.2.2 Estimating the double logarithmic and semi-logarithmic models The first step in the procedure for fitting a regression based on the double
logarithmic model (12.4a) is to calculate two new variables by taking the
logarithms of the original variables. We will refer to each of these new vari-
ables by an asterisk (*) to indicate its relationship to the original variable:
Y*� log Y and
X*�log X
By substituting these three new variables in (12.4a) we get a standard linear
regression
Y*�a*�bX*�e (12.4b)
where a* is the logarithm of the intercept, a.
§. -
k This is the form used, for example, by Barry Eichengreen and Henry A. Gemery, ‘The earnings of
skilled and unskilled immigrants at the end of the nineteenth century’, Journal of Economic
History, 46, 1986, pp. 441–54.
20,000
10,000
5,000 4,000
3,000
2,000
1,000
500 400
300
200
100 5 6.25 7.5 8.75 10
a = 2, b = 2.5
Y –
lo ga
ri th
m ic
s ca
le
X – arithmetic scale
Figure 12.8 Semi- logarithmic transformation of an exponential model
10.008.757.506.255.00
20,000
15,000
10,000
5,000
0
a = 2, b = 2.5
Y –
a ri
th m
et ic
s ca
le
Y – arithmetic scale
(a) The original curve: Y�abX
(b) The semi-logarithmic transformation: log Y� log a�X(log b)
Suppose, for the purpose of illustration, that the linear regression in
(12.4b) has been computed by the standard OLS procedure; the intercept,
a*, is found to be 0.69315 and the regression coefficient, b, is 1.75. These
parameters can be substituted in the original transformation (12.4a) to get
log Y�0.69315�1.75 log X (12.4c)
In practice, the relationship would usually be reported in this logarith-
mic form. However, it is possible to get back to ordinary numbers by
finding the exponential of 0.69315 (which�2) and reversing both the log-
arithmic transformations. This gives
Y � 2X1.75 (12.4d)
Essentially the same procedure is followed for the estimation of the
semi-log models, except that there is only one new variable to be created in
each case: X*�log X for the model of (12.8), and Y*�log Y for the model of (12.9). In neither case is the intercept a in log form.
12.3 Interpretation of non-linear relationships
The second essential feature of non-linear models that must be considered
relates to the interpretation of the regression coefficients. In the initial dis-
cussion of linear relationships in §4.2 it was noted that the regression
coefficient measured the slope of the line. In a simple linear model such as
Y�a�bX�e, the coefficient b on the explanatory variable informs us that
when X changes by 1 unit, Y changes by b units; the units being whatever X
and Y are measured in.l
In slightly more formal terms, if we use the symbol � (capital Greek
delta) to represent ‘a small change’, then we can say that in a linear relation-
ship, �Y�b�X, and if �X is 1 unit then �Y is b units. What is the corre-
sponding interpretation of the regression coefficient when any of the
variables is in log form?
12.3.1 Interpretation of the regression coefficients for logarithmic models It follows from the properties of logs summarized in §1.6.2 that a change in
the log of a variable is a proportionate rather than an absolute change. Using
the same symbol, �, then such a proportionate change in Y would be �Y/Y,
and similarly for X. We then have three cases to consider.
§. -
l As one particular application of this rule, note that if the regression coefficient, b, is expressed as
a logarithm – as in (12.6a) – then its value will depend on whether the author has taken a natural
logarithm to base e, as is usually done, or a log to base 10 (see §1.6.2).
(a) The first case is the double logarithmic (log-log) model in which both
the dependent and the explanatory variable are in logs. In this form the
regression coefficient informs us that when there is a small proportion-
ate change in X there will be a small proportionate change in Y equal to
b. So
and if the proportionate change in X is 1 per cent, the proportionate
change in Y will be b per cent.
Economists will recognize that if this relationship is re-written
b � (12.10)
it shows that b defines an elasticity. Thus if this double logarithmic model was applied to the demand for a commodity, in which Y was
some measure of quantity and X was the price, the regression
coefficient, b, would measure the price elasticity (and would have a
negative sign).
In other contexts this expression for the regression coefficient, b,
can also be interpreted as the constant rate of growth of Y associated
with a constant rate of growth of X. In the example used in §12.2.2 in
relation to (12.4c) above, b was 1.75. This means that when X grows at
1 per cent p.a., Y grows at 1.75 per cent p.a.
(b) The second case is the semi-log (lin-log) model such as (12.8) in which
the explanatory variable, X, is in log form but not the dependent vari-
able, Y. In this case the interpretation of the regression coefficient, b, is
that when there is a small proportionate change in X, there will be an
absolute change of b units in Y. So
�Y � b
If, for example, the proportionate change in X is 0.01 (or 1 per cent)
then the absolute change in Y will be 0.01b; this is the same as b/100.
(c) Thirdly, we can consider the alternative form of semi-log (log-lin)
model in which the dependent variable, Y, is in log form but not the
explanatory variable, X; for example (12.9). In this case the regression
coefficient tells us that when there is an absolute change in X, there will
be a proportionate change in Y of b. So
� b�X �Y
Y
�X
X
�Y
Y �
�X
X
�Y
Y � b
�X
X
and if �X is 1 unit the proportionate change in Y will be b units, corre-
sponding to a percentage change of 100b per cent.
The possible combinations of units for X and Y are summarized in table
12.1. Columns (1) and (2) indicate the form in which X and Y are specified
by the model. Columns (3) and (4) show how this affects the meaning of
the change in X and the resulting change in Y. Note that absolute units are
measured in the same units as the relevant variable.
12.3.2 Elasticities It is often interesting to establish the elasticity implied by a particular
regression model. This is the ratio of the proportional change in the depen-
dent variable to the proportional change in the explanatory variable, or
As noted above (see (12.10)) when the model is specified as a double loga-
rithmic relationship, the elasticity can be read directly from the regression
coefficient, b. However, it is only for this model that b is identical to the
elasticity; it does not hold for any other model.
In every other case, if the researcher wants to determine the elasticity
she can always do so, but it requires an additional manipulation of the
regression coefficient. For example, in the case of the lin-log model consid-
ered in paragraph (b) above, where it is the explanatory variable X that is in
�Y
Y �
�X
X
§. -
Table 12.1 Absolute and proportionate regression coefficients
(1) (2) (3) (4)
X Y Change Resulting
Type of model measured in measured in in X change in Y
1 Linear models, e.g.
Y�a�b1X1 – b2X2 numbers numbers 1 unit b units
2 Semi-logarithmic models, e.g.
Y�a�b log X logs numbers 1% b/100 units
OR
log Y�a�b X numbers logs 1 unit (b�100)%
3 Double logarithmic models, e.g.
log Y�log a�b log X logs logs 1% b%
logs, a small proportionate change in X produces a small absolute change
in Y, and
b � �Y � (12.11)
This can be converted to an elasticity by dividing both sides by Y to get
(12.12)
Similarly, in the alternative form of log-lin model considered in para-
graph (c) above, where it is the dependent variable Y that is in logs, a small
absolute change in X (��X) produces a small proportionate change in Y
and so
b � ��X (12.13)
In order to convert this into an elasticity it is necessary to multiply both
sides by X, giving4
bX � (12.14)
We have thus established that for a semi-log model in which the explan-
atory variable is in log form, the elasticity is measured by b/Y. For the alter-
native semi-log model in which the dependent variable is in log form, the
proportionate change in Y caused by a proportionate change in X, or elas-
ticity, is measured by bX.
In the same way it can be shown that for a standard linear model the
elasticity is
and for the reciprocal model it is
b
It is important to note, however, that all these measures of elasticity
differ in one vital respect from the measure associated with the double log-
arithmic model. In that model the elasticity is constant irrespective of the
values of X and Y. In all the other models the elasticity is variable and its
precise value will depend on the value of X and Y. So when these elasticities
1
XY
b X
Y
�Y
Y �
�X
X
�Y
Y
� � �YY �
b
Y �
�Y
Y �
�X
X
�X
X
are reported, the single result given is typically calculated at the mean
values of the variables.
12.3.3 Long-run multipliers There is one final point to be made regarding the interpretation of regres-
sion coefficients. It arises in relation to the distributed lag models discussed
in §10.2.2. It was noted that a model could be specified that allowed all or
many past values of an explanatory variable to influence the dependent
variable. This was illustrated by (10.15) in §10.2.2, in which all past values
of SALES appeared as separate explanatory variables. It was shown that
such a model was difficult to estimate but that the problems could be over-
come by a transformation such as (10.19) (see panel 10.1). For convenience
this equation is reproduced here
INVt�(1 �)a�bSALESt��INVt 1�(et �et 1) (12.15)
In this form of the model only the current value of SALES is shown on
the right hand side. In this case the regression coefficient, b, measures the
change in INV caused by a change in SALES in the current period, and is
referred to as an impact multiplier. The total effect of the change in SALES in all previous periods, or long-run multiplier (which we will refer to as b̃ to distinguish it from the short-run coefficients), is not given directly by
(12.15). However, it can easily be derived as
where b is the coefficient on the current SALES, and � is the rate of decline
of the distributed lag and is given by the coefficient on the lagged depen-
dent variable, INVt 1. 5
12.4 Fitting a trend to a time series
In this section we continue the discussion of the measurement of long-run
trends first introduced in §1.8. We begin by looking at the log-linear trend
and then turn briefly to other forms of non-linear trend.
12.4.1 Fitting a log-linear trend The semi-log transformation of the exponential model (12.6a) is widely
used in the study of time series to fit a trend to the series. A log-linear trend of this nature grows at a constant rate, and will appear as a straight line when
plotted in logs. It thus differs from the linear trend described in §4.2.5; this
was also linear but in the original numbers, not in logs.
b
1 �
§.
The log-linear (or semi-logarithmic) trend is fitted to the data by treat-
ing the log of the relevant series (Y) as the dependent variable and time as the
explanatory variable (X). As explained in §4.2.5, it makes no difference to
the resulting trend whether time is measured by successive dates (1920,
1921, …) or by any other series that increases by 1 unit each year
(1, 2, 3, …), though the coefficients will be different.
To illustrate the procedure we can fit a log-linear trend to the series for
NNP (real net national product in £ million) for 1920–38 used by
Benjamin and Kochin in their analysis of inter-war unemployment in the
UK.m We do so using the actual years (YEAR) as the explanatory variable.
The model is thus
log NNP�log a�log b YEAR�e (12.16)
The computer estimates this regression as
log NNP�–26.084�0.01781 YEAR
This measure of the trend in NNP would normally be left in logs and
plotted in that form. If, however, it were desired to convert back to ordinary
numbers, the procedure for 1920 would be
log NNP�–26.084�(0.01781 � 1920)�8.1135
so NNP�exp (8.1135)�£3,339.2 m
and similarly for other years.
This procedure for fitting a log-linear trend has the further important
advantage that the regression coefficient provides a direct measure of the
annual compound rate of growth of the series. This method of calculating the rate of growth of a time series is described in panel 12.1, and compared
with alternative methods.
Where a time series relates to periods of less than a year, regression anal-
ysis can be used not only to fit a long-run trend, but also to eliminate the
seasonal variations in the series by means of dummy variables. This makes
it possible to analyse the three separate components of the series: long-run
trend, seasonal variation, and residual cyclical and random fluctuations.
This procedure is explained in panel 12.2.
12.4.2 Fitting non-linear trends The log-linear trend described in the previous subsection is appropriate
only if it provides a good fit to the data. This may be judged by inspection
or by suitable statistical tests such as the value of R2 and of the standard
m This is the series given in the data set and used previously when fitting a linear trend in §4.2.5.
§.
errors. There will, however, be many series for which a linear trend is not a
good fit, and it is then necessary to fit a more complex model.
To illustrate one such case, we revert to the series used in §1.8 for miles
of railway track added in the United States from 1831 to 1913 (RAILWAY);
this was shown together with its fitted trend in figure 1.3. To model this we
use a quadratic in which a term in X2 is added to the model used previously
(see (12.16)). For this version we take TIME as the explanatory variable,
using a sequence of numbers from 1 to 83. The model is thus
log RAILWAY�log a�log b1 TIME�log b2 (TIME) 2�e
(12.22)
This belongs to the first group of non-linear models described in §12.2,
and so can be converted to a linear form by simply replacing the term in
(TIME)2 by the corresponding squared values. Thus for 1832, (2)2 becomes
4, for 1833 (3)2 becomes 9, and so on all the way to the final year, where
(83)2 is replaced by 6,889. We now have a standard linear model with two
explanatory variables that can be estimated by OLS. When the computer
does this, the results (with t-statistics in brackets) are
log RAILWAY�4.7799�0.1185 TIME – 0.00095 (TIME)2
(25.23) (11.39) (7.88)
The trend in actual miles can then be derived for each year by applying
these results and finding the exponential.n For example, for 1881 (which is
51 in the series for TIME) the trend value is found from
logRAILWAY�4.7799�(0.1185�51) (0.00095�(51)2)�8.3653
so RAILWAY�exp(8.3653)�4,295 miles
and similarly for all other years. This is the long-term trend in construction
of railway track that was fitted to the data in figure 1.3.
The key to understanding why this model produces a trend which first
rises and then falls, is to notice the effect of the term for (TIME)2. Its
coefficient, log b2, is smaller than the coefficient on TIME, log b1, but is neg-
ative. In the early years of the series, the positive impact of b1 outweighs the
negative effect of b2, even though TIME is much smaller than (TIME) 2. But
in later years, (TIME)2 becomes so large relative to TIME that eventually its
negative coefficient dominates, and the trend starts to decline.
To illustrate this effect, we can compare the above calculation for 1881
with comparable calculations for 1891, when the trend value was close to
n These results are rounded to simplify the presentation, but the calculations are very sensitive to
the precise coefficients, and the illustrations for 1881 and other years were done with figures to 7
decimal places: b1�0.1185348 and b2� 0.0009457.
* Note that when the rate of growth is small, a close approximation to the compound growth
rate can be obtained more simply from the regression coefficient itself (multiplied by 100).
In the present example b�0.01781 so the approximation would be 1.78 per cent p. a. ** The linear trend fitted in §4.2.5 was also based on all points in the series, but the growth rate
cannot be obtained directly from the regression, and because the trend is not linear in the
logs, the slope of the line does not correspond to a constant rate of growth. The growth rate
can be calculated by applying the compound interest formula to two selected years, but it will
vary according to the years selected.
Panel 12.1 Measurement of the compound rate of growth
A great advantage of the procedure for fitting a log-linear trend described in
§12.4.1 is that the regression coefficient provides a direct measure of the
annual compound rate of growth of a series.
To illustrate, it should be evident from (12.16), and the calculation
made on the basis of the regression results for 1920, that the value of log
NNP will rise each year by a fixed absolute amount equal to the regression
coefficient, b (since the intercept does not change and the value of YEAR by
which the coefficient is multiplied always increases by exactly 1). In the
example used the fixed amount (in logs) was 0.01781.
Recall from §1.6 and panel 1.1 that an equal absolute change in the loga-
rithm of a variable represents a constant proportional change in the variable.
It follows, therefore, that if log NNP is growing each year by a constant
amount of 0.01781, then NNP must be growing at a constant rate equal to
the exponential of that amount, which is 1.01797. Thus the annual com-
pound rate of growth can be obtained very simply from the exponential of
the coefficient by subtracting 1 and multiplying by 100. The growth rate of
NNP is therefore 1.797 per cent p.a.*
It is important not to confuse this use of the coefficient on the log-linear
model with the coefficient on the log-log model interpreted in §12.3.1 as the
constant rate of growth of Y associated with a constant rate of growth of X.
The difference in this case is that X represents time and is not growing at a
constant rate, but by a constant absolute amount of one year each year.
The strength of this measurement of the rate of growth is that it is based
on the log-linear trend that has been fitted to all years in the series.** An
alternative procedure often adopted to measure the compound rate of
growth is to select two years, say the first and the last in the series, and calcu-
late the growth between these two points by means of the compound inter-
est formula
Xt � X0 (12.17a)�1 � r100� t
§.
where X0 and Xt are the values of the series in the two selected years, r is the
rate of growth, and t the number of years between X0 and Xt over which the
growth is to be calculated.
In order to solve (12.17a) it must first be converted to logarithms, giving
logXt � log X0 � (12.17b)
so
log
For NNP in 1920 and 1938, X0 is £3,426m, Xt is £4,807m and t�18.
Substituting these values in (12.17b) gives log (1�r/100)�0.01882. We
then take the exponential of this to get (1�r /100)�1.01899. By subtract-
ing 1 and multiplying by 100 we obtain the compound rate of growth of
1.90 per cent p.a.
This procedure amounts, in effect, to simply drawing a straight line
through the two selected dates and ignoring what happens to the series in all
other years. In this case the two methods give broadly similar results.
However, the use of the compound interest formula can give misleading
results, particularly if the two selected dates occur at different points in the
business cycle – for example, if the first is a year of depression and the
second at or close to a peak year.
Calculation of the growth rate from the coefficient in the log-linear
trend model is, in general, more reliable, and is recommended as the best-
practice procedure. This is, of course, subject to the proviso that a log-linear
trend is a good fit for the particular series. If the long-run growth of the
series is not constant, then a single estimate of the growth rate is not appro-
priate, and other procedures must be adopted to find the relevant growth
rate for different segments of the curve.
To conclude this discussion of growth rates we report a very helpful
rule of thumb. This states that the product of (i) the annual compound rate
of growth of a series and (ii) the time it will take for that series to double if it
is growing at that rate, is approximately equal to 70. Thus, if we know that,
say, the population of a country is growing at 5 per cent p.a., we can imme-
diately say that the population will double in 14 years. Conversely, if we
want to know what rate of economic growth would be required in order to
achieve a doubling of the average standard of living within a period of 20
years, we can immediately calculate that it would need to be 3.5 per cent p.a.
�1 � r100� � log Xt log X0t
�t �log �1 � r100��
Panel 12.2 Decomposition of a time series by means of a log-linear trend and seasonal dummy variables
In §1.8 we described a procedure for decomposing a time series for periods
of less than a year into a trend, a pattern of regular seasonal variation, and a
residual cyclical or irregular fluctuation. We can now introduce an alterna-
tive – and in some respects more satisfactory procedure – using regression
analysis, first to fit the trend and then, with the aid of seasonal dummy vari-
ables, to ascertain the seasonal adjustment factors.
To illustrate the method we will again take the quarterly data on bank-
ruptcy in eighteenth-century England (BANKRUPT) compiled by Hoppit
and used in table 1.2. In the first stage, the long-run trend in the series is esti-
mated by fitting a semi-log model with TIME as the explanatory variable.
The model is
log BANKRUPT�log a�log b TIME�e (12.18)
If this is estimated for the period from 1777QI to 1787QIV, using suc-
cessive numbers 1, 2, 3, … for each quarter to represent TIME, the com-
puter obtains
log BANKRUPT�4.9237 0.0032 TIME
The trend value for 1777QI (in ordinary numbers) would thus be
exp[4.9237 (0.0032�1)]�exp(4.9205)�137.07; for 1777QII it would be
exp[4.9237 (0.0032�2)]�exp(4.9173)�136.63, and so on.
Having calculated the trend, in the second stage the log of BANKRUPT
is regressed on its trend (TRENDB) and the seasonal dummy variables. The
trend is converted back to ordinary numbers because it is again a semi-log
model (see (12.6a)), with only the dependent variable in logs.
Dummies are created for three of the four quarters, leaving the fourth
as the control quarter. Thus, if QIV is taken as the control quarter, the
dummy variables are
DQI which has a value of 1 for every 1st quarter, 0 otherwise
DQII which has a value of 1 for every 2nd quarter, 0 otherwise,
and
DQIII which has a value of 1 for every 3rd quarter, 0 otherwise.
Each of the coefficients on the dummies captures the effect of its own
seasonal variations on the pattern of bankruptcy, relative to the pattern in
the control quarter.
§.
The model is thus
log BANKRUPT�log a�log b1TRENDB�log b2DQI
�log b3DQII�log b4DQIII�e (12.19)
Estimation of the model gives the following results, with t-statistics in
brackets.
log BANKRUPT� 4.00�0.0072 TRENDB� 0.0057 DQI
(7.99) (1.85) (0.10)
�0.0273 DQII 0.3170 DQIII
(0.474) (5.52)
The t-statistics show that the seasonal dummy for the third quarter is
highly significant, though those for the first and second quarters are not.
Together with the F-test (for which the prob value is 0.000) there is clear evi-
dence of a strong seasonal pattern in bankruptcies in this period.
Since we have used a semi-log model the seasonal coefficients are in
logs, and are multiplicative rather than additive. The first step in calculating
the quarterly adjustment factors is thus to obtain their exponentials (and
their relationship then becomes additive). The second step is to convert
these from adjustments relative to the control quarter, QIV, into the actual
adjustments required.
To do this we need some simple algebra, based on the following four
equations:
DQI �1.0057 DQIV (12.20a)
DQII �1.0277 DQIV (12.20b)
DQIII�0.7283 DQIV (12.20c)
DQIV�4 (DQI�DQII�DQIII) (12.21)
The first three of these equations are simply statements of the value of
the seasonal dummies relative to QIV, based on the exponentials of the
regression coefficients. Equation (12.21) incorporates the rule that the four
adjustment factors must sum to 4; in other words, their average must be
exactly 1 so that the annual totals will not be changed by the removal of the
seasonal pattern.
Since all four equations relate to DQIV we can substitute the first three
in (12.21) to get
DQIV�4 (1.0057 DQIV�1.0277 DQIV�0.7283 DQIV)
its peak level; and with 1901, by when it had declined. Since the constant
term is always the same we ignore that and concentrate on the two remain-
ing terms. Since TIME increases in each case by 10 years, the change in the
term log b1 (TIME) will always be 10�0.11853�1.1853. It remains to
work out the change in the second term and to calculate the net effect of the
two terms. The former is done in columns (2) and (3) of table 12.2, the
latter in column (4).
This calculation shows that between 1881 and 1891 the combined effect
was an increase of 0.1261; between 1891 and 1901 it was a fall of 0.063.
The corresponding exponentials are 1.1344 and 0.9389. This is a (multivar-
iate) semi-log model with only the dependent variable in logs, so these
exponentials represent the ratios of the dependent variable. The final result
is thus that the ratio of 1891 to 1881 is 1.1344, corresponding to an increase
in the trend value by 13.44 per cent, from 4,295 to 4,873 miles. By contrast
the ratio of 1901 to 1891 is 0.9389, corresponding to a decrease over the
second period by 6.11 per cent, from 4,873 to 4,575 miles.
The exercise also underlines the fact that although the net change in the
two explanatory variables can be used to calculate the percentage change in
We then take all the terms in DQIV to the left-hand side
DQIV(1�1.0057�1.0277�0.7283)�4
so
DQIV�4/(3.7617)�1.0633
We now know the seasonal adjustment factor for QIV. Therefore the
factor for QI is 1.0057 � 1.0633�1.0694, and similar calculations for QII
and QIII complete the exercise. The full set of seasonal adjustments is set
out below, with those previously calculated in chapter 1 given for compari-
son. The differences are not large and lie within the 95 per cent confidence
intervals derived from the t-statistics.
Seasonal dummies 1.0694 1.0928 0.7745 1.0633
Previous method 1.0939 1.0548 0.7712 1.0802
As before, the original series is divided by these factors to obtain a series
with the seasonal variations removed, and that in turn can be divided by the
trend to reveal the remaining fluctuations without either trend or seasonal
variation.
the trend over any period, this will not be a constant rate of growth ; it will
vary according to the position of the selected dates on the fitted trend.
Inspection of the t-statistics and other tests suggests that this quadratic
provides a reasonably good fit for this series. If it did not, we could explore
higher-order polynomials such as cubics or quartics (i.e. models with terms
in X3 or X4). These additional terms would act in a similar fashion to the term
in (TIME)2, and would eventually produce additional turning points in the
trend.6
Alternatively we could try other curves. In his classic study of the trends in
a large number of production series for the United Kingdom, United States,
and other countries Simon Kuznets fitted a variety of non-linear functional
forms including parabolas and logistic curves.o The logistic is appropriate
for many production series that start at a low level, rise rapidly for a time, and
then level off as a result of factors such as declining prices and profitability as
a consequence of the entry of other producers, saturation of the market, and
competition from new products. The upper and lower bounds of this
process are captured by the coefficients g and h discussed in §12.1.
12.5 Modelling a relationship
The process of fitting a suitable trend to a time series can be thought of as
a special case of the more general problem of finding an appropriate
§.
Table 12.2 Calculation of change in fitted trend, miles of railway track added in the United States, 1881, 1891, and 1901
(1) (2) (3) (4)
Change in Change in Net change in
log b1 (TIME) log b2 (TIME) 2 log b2 (TIME)
2 log RAILWAY
1881 ... ( 0.00095�2601)� 2.4599 ... ...
1891 1.1853 ( 0.00095�3721)� 3.5191 1.0592 0.1261
1901 1.1853 ( 0.00095�5041)� 4.7674 1.2483 0.0630
Notes:
(1) and (2) calculated from results of trend fitted by (12.22)
log RAILWAY�4.7799�0.1185 TIME – 0.00095 (TIME)2
with TIME measured by a series running from 1 (for 1831) to 83 (for 1913).
(4)�(1)�(3).
o Simon Kuznets, Secular Movements in Production and Prices, Houghton Mifflin, 1930.
functional form with which to model the relationship between a depen- dent variable and one or more explanatory variables. How do we know
which of the possible non-linear relationships is the best one to choose
once we have decided that the most appropriate relationship is not linear?
More generally, when should we choose a non-linear rather than a linear
relationship?
We begin with a discussion of some general principles that might help to
inform this decision, and then look at how these have been implemented in
practice by authors investigating a wide variety of topics. We conclude the
chapter by returning to the general question of how to evaluate the
specification of a model, a conceptual issue that embraces not only matters
of non-linearity, but other issues, such as the selection of variables and how
best to evaluate the success or failure of the model when confronted with
historical data.
12.5.1 Choice of functional form The range of possible functional forms that might be selected is quite large,
and it should be emphasized at the outset that it is not confined to the
precise forms specified earlier in this chapter. For example, if a multivariate
double logarithmic model is derived by transformation of the geometric
curve given in (12.5), the result is
log Y�a�b1 log X1�b2 log X2�e (12.23)
This could obviously be extended by the inclusion of more explanatory
variables, but it is not necessary that all of these should be in logs. In exactly
the same way, a researcher might work with a variant of the lin-log model specified in (12.8) in which some of the right-hand-side variables are in
logs and others are not.
There are a number of reasons why logs might be used for some vari-
ables in these models but not for others. Logs would not normally be used
for dummy variables; and there is the technical consideration that it is
possible to express only positive values in logs, so a variable with a range
that includes values of 0 or less must necessarily be left in ordinary
numbers.p A more substantial reason would be the view that some partic-
ular variable should be entered in the model in ordinary numbers
because its relationship to the dependent variable was absolute, not pro-
portional.
p Some researchers who wish to use logs, but who are confronted with data points equal to 0, have
been known to add a small positive number (e.g. 0.001) to all the values in the series to permit a
logarithmic transformation.
There is an example of this in Hatton and Williamson’s time-series
model of Irish emigration, where all but one of the explanatory variables is
in logs (see §14.2.1). The exception is the measure of the proportion of the
total Irish population already living abroad (MIGSTOCK). The reason
why this is not converted to logs is that it is the level of this variable that is
seen as relevant to the flow of immigrants, not its rate of change. Another
case occurs in a model of wage determination in post-war Britain, where
the authors included both unemployment and the log of unemployment as
explanatory variables in an otherwise log-log model, because they believed
that its level as well as its rate of change would have an important influence
on the rate of change in wages.7
The issue of when it is appropriate to include linear and non-linear vari-
ables in the same regression equation is part of the general debate over func-
tional form.q How do we know which of the many possible functional forms
is the right one for a particular application? Is the log-linear model appro-
priate, or is the double log model superior? The problem is compounded by
the likelihood that a number of different non-linear forms could fit the same
data without clear evidence of statistical superiority, as indicated by tradi-
tional goodness of fit measures such as R2 and t-statistics.r
The decision on whether or not to take logs of explanatory and/or
dependent variables is only one of the issues to be considered in deciding
on the form in which a model is to be specified. A second is the use of
lagged variables, a procedure that might be adopted for the reasons dis-
cussed in §10.2. Yet another is the procedure that should be adopted to deal
with the effects of common trends in time series. This problem frequently
arises because of the risk of spurious correlation when both dependent and
explanatory variables are moving together over time.8 In a growing
economy, for example, the process of growth will be experienced by many
economic and social factors, and a regression that does not account separ-
ately for this is likely to obtain a high but misleading R2.
There are several methods available for dealing with this problem of
trendedness. One is to fit a suitable trend to every affected variable, and then work with the deviations from the trend (as discussed in §1.8.2)
rather than the original series. Another is to include time as one of the
explanatory variables so that this variable picks up the effects of the
changes over time. However, neither procedure is generally recommended.
§.
q There are advanced procedures, such as RESET, discussed in most econometrics textbooks, that
test whether the most appropriate functional form for the data is linear or non-linear. r As can be seen, for example, by comparison of figure 12.2 when b�0 with figure 12.5 (a), when
both b1 and b2�0; or of figure 12.2 when b� 2, with figure 12.3 when b�0.7.
A better solution involves working with the differences of successive
values of the time series rather than the actual values.9 For example, if the
value of NNP in 1924 was £3,622m and in 1925 it was £3,840m, then the
first difference is £218m. This would be designated as �NNP, where � (capital Greek delta) is the symbol used earlier in this chapter for a small
change. The second difference is then the first difference of the first difference, denoted by �2NNP. In general, first differencing of a time series
is sufficient to eliminate a linear long-run trend, and second differencing
would eliminate a non-linear trend with a quadratic term such as X2.
12.5.2 Some general principles Given this range of choices, what should determine our selection of vari-
ables and the functional form in which the model is specified?
The essential answer, the golden rule, is that the choice should be based on
some consideration of general principles combined with knowledge of the data
and of the behaviour of the relevant variables. An appropriate analytical
framework should always be the starting point of any investigation. But
this need not be a grand or formal theory. It may be, as with models of the
economy derived from the great structure of Keynesian macro-economics,
or rival monetarist theories, which lead to a complete set of equations for
variables such as consumption, investment, and prices.10 But more often
what is required is much less sophisticated than this.
Economic principles regarding the relationship between price and
quantity in supply and demand functions, or between profit maximization
and industry performance, have been fertile sources of ideas for quantita-
tive analysis. Other disciplines similarly provide broad principles and gen-
eralizations. Thus, some writers have suggested Durkheim’s theory of
anomie as a promising basis for a quantitative investigation of suicide; a
study of infant mortality might start from the analysis and reasoning of
medical scientists and epidemiologists; a proposition regarding relative
deprivation might be applied in an investigation of social unrest. In other
cases, the basis for a regression model may be found even more simply in
ideas put forward as no more than common sense.
The analytical framework underlying the choice of functional form and
the selection of variables should be set out explicitly in advance of the esti-
mation, and the expected signs – and if possible the size – of the coefficients
should be indicated. As will be seen, this is the procedure followed in all
four of the case studies discussed in chapters 14 and 15.
What should never be done is to take a bunch of possible relationships,
grab as many variables as possible, and set the computer to run regressions
until it comes up with a few that might be judged satisfactory. Results
obtained in this way can have no credibility; neither low standard errors
nor high R2s are worth anything in such circumstances. If a model has no
underlying principle there is absolutely no reason to believe that any rela-
tionship obtained by the regression reflects any sort of causal relationship
between the dependent and explanatory variables.
The implementation of these general ideas can now be illustrated by a
few examples from recent literature. In each case we have space to identify
only one or two principal features of the models, and our brief comments
do not do justice to the richness of the analytical frameworks developed in
these studies. In each case, we are focusing on the specific issue of including
and specifying a non-linear relationship in the model, although it should
be clear that the examples chosen have been selected for their general sensi-
tivity to the imperatives of model specification.
12.5.3 Implementing the principles: some examples from economic history We start with a very specific issue. An historian has found a set of appren-
ticeship contracts in the local archives. They contain information on
various matters, including the length of the contract, the date it was signed,
the occupation and location of the master, the age of the apprentice, and
certain details about issues such as probation, pay, and schooling. How
should this information be analysed?
A basic economic principle, known as the diminishing marginal produc-
tivity of labour, states that under competitive conditions a firm should con-
tinue to hire labour up to the point where the cost of an extra worker is
equal to the additional revenue he will produce. If the same principles are
applied to the demand for apprentices on the part of the master – under a
system with variable contract lengths – it might be expected that the dura-
tion of the contract would be related positively to the cost of the arrange-
ment to the master (in wages paid, board, working time lost while training
the apprentice, and so on), and negatively to the value of the output pro-
duced by the apprentice.
It is also necessary to consider the contract from the point of view of the
apprentice. Here the analysis can build on ideas derived from human
capital theory. The essence of this theory is that investments are made in
human resources in order to raise their productivity, and thus in time their
earnings. The forms of such investment in human capital include formal
schooling and on-the-job training, as well as broader factors such as good
health and nutrition. The implication of this theory is that apprentices will
be willing to sign on for longer contracts, the more they expect their future
earnings to be increased by the training received during the apprenticeship.
§.
Working from these ideas Gillian Hamilton was able to develop a suc-
cessful model of the duration of apprenticeship contracts in late eight-
eenth- and early nineteenth-century Montreal.s Most factors in her
model entered the regression as linear explanators. But theoretical con-
siderations recommended the addition of age-squared as a variable to test
for non-linear effects in age. Younger boys were both less mature and less
experienced, so that employers would require longer service contracts to
ensure full recompense for the expense of training. Moreover, the risks
that the apprentice would under-perform were highest for the very
young, about whom the employer had little or no prior knowledge. The
reported coefficient on age is negative while that on the square of age is
positive, indicating that contracts became increasingly short, the older
the apprentice.t
One way for employers to find out something about the potential risk
was to require a probationary period before negotiating a contract.
Hamilton found that contract lengths were much shorter for probationers,
as theory would suggest. Moreover, boys who were sponsored by a guar-
dian served longer terms, consistent with the hypothesis that information
was a key ingredient in contract bargaining. The signs and significance of
other variables in the regression equation further support the underlying
theory, with lengthier contracts for apprentices who negotiated money at
the end of the apprenticeship (a form of head-right), or access to schooling
or church during it.11
A further illustration of the value of human capital theory in historical
applications may be found in the analysis of the earnings of immigrants
into the United States compared to those of native-born workers in the
same occupations. The theory tells us that earnings will be systematically
related to variables that affect individual productivity, such as education,
work experience, and age. Moreover, it indicates that the relationships will
be non-linear, albeit in different ways. Thus, there are diminishing returns
to each additional year of experience; whereas the returns to added years of
education are likely first to rise (as literacy and learning skills are devel-
oped), and then to fall (as knowledge becomes less functional). For manual
workers, the relationship between age and earnings certainly has a strong
s Gillian Hamilton, ‘The market for Montreal apprentices: contract length and information’,
Explorations in Economic History, 33, 1996, pp. 496–523. t The respective coefficients are 2.033 and �0.043. Thus at age 10 their combined effect on the
length of contract is (10� 2.033)�(100�0.043)� 16.03. At age 20 it is (20� 2.033)�
(400�0.043)� 23.46, a reduction in contract length of only 7.4, not 10, years. The implied
shape of the age–duration relationship is similar to that of an exponential curve (figure 12.3)
with a value of b of about 0.9.
quadratic component, with productivity potential rising during early
adulthood and falling after middle age, because of factors such as declining
physical strength.12
Accordingly, models of earnings have been specified with non-linear
functional forms, typically by taking the log of earnings as the dependent
variable, and by squared terms for age, years of work experience, and – for
immigrants – years in the United States, among the explanatory variables.
This log-lin formulation thus measures the proportional effect on earnings
of these and other variables.13 One advantage of using age and age squared
terms in the human capital variables is that it permits a range of possible
functional forms, depending on the sign and size of the coefficients.
Human capital theory, complemented by the pioneering work of soci-
ological investigators such as Seebohm Rowntree in the late nineteenth
century, also provides strong foundations for life-cycle theories of income,
consumption, and wealth accumulation. A simple model of the path of
income and consumption over the life cycle leads to the hypothesis that
there will be a strong but changing association between age and accumu-
lated savings (or wealth), leading to hump-shaped wealth–age profiles.
While income first rises and then falls over an individual’s lifetime, con-
sumption needs tend to remain relatively stable. Elderly individuals, whose
human capital has depreciated with time, are likely to find that their earn-
ings are insufficient to cover living expenses. Workers are therefore moti-
vated to save for retirement during the peak earning years.
In any such life-cycle model, a non-linear specification with terms in age
and age squared would be required to produce the postulated turning point
in the relationship between age and wealth (or savings). When such models
are estimated, the regression coefficients on age are typically found to be
statistically significant, and the coefficient on age is large relative to the one
on age squared, but it is positive whereas the latter is negative. The result of
this specification is that the positive term with the large coefficient initially
outweighs the negative term, and predicted wealth rises with age. But even-
tually (as with the exercise fitting non-linear trends in §12.4.2), age squared
becomes so much larger than age that despite its smaller coefficient the neg-
ative term dominates, and predicted wealth starts to decline.14
Normally the turning point in historical studies is found somewhere in
late middle-age. At this stage in the life cycle, ageing begins to lower pro-
ductivity and reduces earning capacity. Consumption, however, continues
and households run down their stock of assets to pay for it. The speed of
decumulation is likely to increase as the worker moves from middle to old
age, either because of formal retirement, or because of reduced opportu-
nities for finding work at good pay.
§.
Life-cycle models focus primarily on the path of total consumption over
time. But historians are also interested in understanding the patterns of
spending, across income groups and across products. One of the funda-
mental relationships in the study of demand analysis is the way in which
expenditure on particular commodities varies with the income level of the
household. This is known as the Engel curve, after a nineteenth-century sta-
tistician who established that as the standard of living of the household
rises, the proportion of expenditure devoted to food decreases, while the
proportion devoted to luxuries increases. Engel curves do not necessarily
maintain the same shape for different commodities. This point was made by
Prais and Houthakker in a pioneering econometric study of spending pat-
terns of British working- and middle-class households in the late 1930s.u
Prais and Houthakker found that spending on some commodities was
linearly related to income – a unit change in income produced the same
increase in spending for the poor and the comfortably off. For other goods
and services, the relationship between expenditure and income was clearly
non-linear. However, even within the general category of non-linear rela-
tionships, there were a variety of responses, demanding a range of different
non-linear specifications. Thus, expenditure on household appliances rose
more rapidly than income, suggesting an exponential curve with b greater
than 1. By contrast, expenditure on food rose more slowly than income,
suggesting a geometric curve with b less than 0. Here is a classic case in
which theory tempered by common sense and practical familiarity guided
the choice among functional forms.
Micro-economic theories useful to historians are not limited to studies
of the individual. There is also a large literature devoted to the behaviour of
firms and industries. One very fertile area has been the study of production
functions and cost functions, which provide a theoretical specification of
the determinants of production or of costs suggesting the use of the double
logarithmic form in empirical work. These models have been used to
understand the changing structure of American industry in the nineteenth
century and its implications for the rate of productivity growth; to
compare the productivity and profit performance of state-owned and
private utility companies in nineteenth-century Britain; and as the basis
for a study of possible explanations for the higher level of productivity
attained by the United States, among other topics.15
For our final example in this section, we take a subject where a very
different form of non-linear relationship applies. A combination of theo-
u S. J. Prais and H. S. Houthakker, The Analysis of Family Budgets, Cambridge University Press,
1955, pp. 87–100.
retical insights and empirical observation, and of ideas drawn from epi-
demiological work on the spread of disease, stimulated modelling of the
diffusion of new technologies in terms of an S-shaped logistic curve, with
the dependent variable measuring the proportion of firms adopting the
innovative technology.
The general proposition is that there is an initial period of slow growth
when the product or process is first introduced by a small number of more
innovative firms; more rapid diffusion (comparable to contagion) as the
advantages of the technology become more widely known; and a final
phase in which diffusion slows down once the majority of potential users
have installed the technology. Factors underlying this pattern include the
initial lack of information about the advantages of the innovation and the
costs of acquiring this other than by observing the experience of early
adopters, and the gradual reduction of uncertainty as a common percep-
tion of the value of the innovation is formed.
For innovations to which these factors apply, these trends are effectively
modelled by a logistic curve, where the parameter, b, measures the speed of
diffusion, and is itself related positively to the profitability of the innova-
tion and negatively to the capital investment required to adopt it.16 A very
similar model is used in relation to demand for household appliances. In
this context, it is suggested that the speed of diffusion is determined by
whether the innovation merely increases the amount of discretionary time,
as with time-saving appliances such as vacuum cleaners and washing
machines – in which case diffusion is slow; or actually enhances its quality,
as with time-using appliances such as radios and TV sets – in which case
the appliances spread much more rapidly.17
Not all theories emanate from economics, of course. It is a strength of
economic history that it is able to draw upon a well-established array of
testable theories. But theory need not be formal to establish hypotheses
against which to judge the empirical evidence. Indeed, some critics of
model-building in economic history have suggested that its theoretical
base is too dependent on a narrow range of assumptions, arguing that the
less formal and more inductive theorizing of other disciplines has its
advantages.
12.5.4 Implementing the principles: some examples from demographic, social, and political history In the study of past demography, a keen understanding of empirical real-
ities, much of it shaped by graphical analysis, and much originating before
the spread of the computer, has influenced the application of more sophis-
ticated quantitative modelling. Two of the central issues in population
§.
history are the decline in birth and death rates, and the association of these
trends with the processes of industrialization and modernization. We
mention only a few studies here out of the wide array of quantitative analy-
ses of the demographic transition.
The pioneer of quantitative demography was William Farr, the first
Statistical Superintendent of the General Registry Office in London who,
during his tenure from 1838 to 1880, wrote a series of reports evaluating
the so-called ‘laws of mortality’. Among these was Farr’s Law, which
posited a mathematical relationship between the crude death rate (the
ratio of total deaths to population) and urban density. The best fit to the
data for registration districts outside London in the 1860s was a geometric
model, with b equal to 0.122. Farr’s Law has been criticized by more recent
scholars, but it represented an important first step towards understanding
mortality patterns in mid-Victorian Britain.18
A more elaborate model of late-nineteenth-century mortality in Britain
has been estimated on data for a sample of 36 towns. As one component of
this study, a double logarithmic specification was adopted in which the log
of age-specific mortality was regressed on the three sets of explanatory var-
iables, all in logs. These covered measures of density, of public health indi-
cators such as expenditure on sanitation and water, and a measure of
nutritional status proxied by the volume of food affordable with the level of
real income in the different towns.v
A comparative analysis of child mortality in Britain and America at the
end of the nineteenth century analysed differences across occupations,
focusing primarily on the influence of environment and income.
Environment was proxied by the proportion of workers in each occupation
who lived in urban centres; this was entered in the regression as a linear
variable. Income, however, was entered in log form, ‘since we expect to
observe diminishing returns in the mortality effects of income gains’.w
An earlier study of fertility analysed the number of surviving children
under the age of four among a survey of American workers in the late nine-
teenth century. Among the variables included in the model were measures
of household income, the industry and occupation of the (male) house-
hold head, whether the wife was working, the ethnicity of the husband, the
age of the wife, and the square of wife’s age. The last two variables were
included to capture ‘the curvilinear relationship of fertility to age of
woman’. The analysis found that the age variables were large and highly
v Robert Millward and Frances N. Bell, ‘Economic factors in the decline of mortality in late nine-
teenth century Britain’, European Review of Economic History, 2, 1998, pp. 263–88. w Samuel H. Preston and Michael R. Haines, Fatal Years: Child Mortality in Late Nineteenth-
Century America, Princeton University Press, 1991, pp. 194–8.
significant, sketching a parabolic relationship between age and number of
children.x
Another area in which general principles have been developed on the
basis of less formal theorizing is social and political history. Among the
many manifestations of social and political disorder are large-scale social
movements such as protest meetings, riots, boycotts, and lynchings, as well
as more individual actions such as crimes against property, drug addiction,
and suicide. On the basis of official records and contemporary newspaper
accounts it is generally possible to find information that can be used for
quantitative analysis, though the nature of such evidence always requires
very careful consideration.19 Given suitable data, what general principles
might inform such analysis?
Sociologists and political theorists have suggested a variety of explana-
tions for such actions. The most straightforward views attribute disorders
directly to hardships associated with factors such as high food prices or
unemployment. A more general category of explanations includes propo-
sitions about the breakdown of existing social norms and constraints, and
the creation of discontinuities as a result of developments such as urban-
ization and industrialization; the impact of rising well-being in encourag-
ing the growth of previously limited expectations; and awareness of
relative deprivation between rich and poor or between black and white.
Other explanations draw on more general notions of the pursuit of collec-
tive self-interest and class solidarity, notably those deriving from Marxist
ideas of class-consciousness.
When any of these broad ideas are applied to a specific issue they have, of
course, to be supplemented by additional, historically specific, explanatory
variables. Thus a study of protests and demonstrations in rural England in
the early nineteenth century will need to consider factors such as the effect
of enclosures or the introduction of new machinery. Similarly, a study of
strikes in a particular period will need to take into account the prevailing
labour legislation, the strength and experience of the trade unions, the eco-
nomic circumstances of the firm or industry in that period, and so on.
There is a voluminous theoretical and empirical literature in this field,
and though many studies give careful consideration to the available quan-
titative data, there are fewer that take the further step of applying a formal
statistical technique to explore the suggested hypotheses. We will discuss
just one contemporary study to illustrate what can be achieved when
regression analysis is used.20 The issue for consideration is the nature of
§.
x Michael Haines, Fertility and Occupation: Population Patterns in Industrialization, Academic
Press, 1979, pp. 212–23.
the relationship between crime and economic conditions.21 Most theories
in the literature of criminology and the related social sciences predict a
strong positive relationship for crimes against property, as did Dorothy
Thomas in the work reported in §3.2.4. However, in recent years several
quantitative studies have found a relationship that was weak or even nega-
tive, especially when unemployment was taken as the explanatory vari-
able.y In an attempt to explain this, current research has emphasized the
existence of two diverse channels relating the state of the economy to
crime rates.
An increase in prosperity makes more goods available and so creates
more opportunity for crime. However, a fall in prosperity leads to an
increase in dissatisfaction (or hunger or alienation or sense of deprivation)
and so creates a greater motivation for crime. Thus if the former is the dom-
inant factor, crime and prosperity will be positively related; if the latter pre-
dominates, there will be an inverse relationship.
An analysis made in the late 1980s of crime in post-war England and
Wales used regression analysis to explore these connections in depth,
focusing especially on short-run fluctuations.z The problem of spurious
correlation in time series (mentioned in §12.5.1) was overcome by specify-
ing the regression model entirely in first differences of the log of the vari-
ables; this is equivalent to measuring the annual growth rates of both the
dependent and the explanatory variables (see panel 12.1). Real personal
consumption per capita was taken as the main measure of economic condi-
tions, and an interesting subsidiary result of the study was that once the
effect of personal consumption is taken into account, unemployment fails
to add anything extra to the explanation of any type of crime. Other
explanatory variables included the proportion of young males in the popu-
lation, the weather, police strength, and beer consumption.
The primary finding was that economic factors did have a major
influence on trends in both property crime and crimes against the person.
For crimes such as burglary, robbery, and theft there was a strong tendency
for the crime rate to move inversely with the cycles of prosperity and
depression, and a similar association was found for the United States,
Japan, and France. By contrast, crimes against the person – such as violence
and sexual offences – tended to move in the opposite direction, increasing
as economic conditions improved.
y See the review of the issue in D. Cantor and K. C. Land, ‘Unemployment and crime rates in the
post-World War II United States: a theoretical and empirical analysis’, American Sociological
Review, 44, 1979, pp. 588–608. z Simon Field, Trends in Crime and their Interpretation: A Study of Recorded Crime in England and
Wales, HMSO, 1990.
The implication of this finding is thus that for property crime the moti-
vation effect was the most powerful, while for personal crime the opportu-
nity effect dominated. The study explores some of the reasons why this
might be so, and also looks at lagged effects to explain the difference
between short-term and long-term patterns.
Turning next to the politics of democracy, a variety of issues have been
the subject of quantitative analysis. These include voter participation, the
effects of incumbency, and – in the United States – the presidential ‘coat-
tails’ effect; but perhaps the most interesting is the famous proposition
associated with President Clinton: ‘It’s the economy, stupid’. The underly-
ing notion is that a vote represents a rational choice between alternatives,
and that one of the fundamental factors which influences that choice is the
state of the economy and its impact on the individual voter.22
One of the more persuasive of several attempts to investigate this rela-
tionship is the multivariate regression model proposed by Bloom and
Price.aa Their model postulates that voting behaviour is determined by
both long- and short-run forces, and is tested on data for US elections for
the House of Representatives from 1896 to 1970. It is a linear model, with
the dependent variable represented by the deviation of the Republican
share of the two-party vote in those elections from the share that would be
expected on the basis of a measure of long-run party identification. The
explanatory variable is simply the percentage change in real per capita
income during the year preceding each election, but two separate regres-
sions are estimated, for periods with Republican and Democratic presi-
dents respectively.
The model does not assume that economic conditions are the sole deter-
minants of voting behaviour. Other factors, such as foreign policy and can-
didates’ personalities, are clearly relevant and must be regarded as omitted
variables in the terminology of §11.2.2. They are thus represented by the
error term as sources of the unexplained variation in the dependent vari-
able, but as long as they are not correlated with the changes in the explana-
tory variable their omission will not bias resulting estimates of the impact
of economic factors on the vote.
In addition to the test over the full set of 37 elections, the regression was
run separately for elections preceded by declining income and those pre-
ceded by rising income. One of the principal findings is that the relation-
ship between voting and economic conditions is asymmetric. Politicians
§.
aa Howard S. Bloom and H. Douglas Price, ‘Voter response to short-run economic conditions: the
asymmetric effect of prosperity and recession’, American Political Science Review, 69, 1975, pp.
1240–54.
are punished by the voters for economic downturns, but are not rewarded
for economic upturns.
If the economy can influence the outcome of elections it is natural to
consider the possibility that politicians may seek to influence the economy
for political reasons. This aspect of political debate has stimulated numer-
ous quantitative studies of the ‘political cycle’. What are the motivations of
political parties when formulating their economic policies? Were periods
with Republican (or Labour) administrations more or less likely to be asso-
ciated with low inflation, full employment, rapid growth, and small budget
deficits than periods when Democrats (or Conservatives) were in office?
According to the ‘opportunistic’ theory, political parties have no policy
preferences of their own, and simply choose economic policies that max-
imize their chances of election. The rival ‘partisan’ theory postulates that
left-wing parties are more strongly motivated to promote growth and
reduce unemployment than to curb inflation, and that these priorities are
reversed for right-wing parties. In recent developments these models have
been further refined by the incorporation of rational expectations.
A comprehensive study of this issue by Alesina et al. has formulated a
series of predictions based on variants of these models, and tested them in a
large set of multivariate regressions for the United States, United Kingdom,
and other industrial democracies. For target variables such as output,
inflation, and the money supply their concern is with rates of growth and
so the dependent variable is formulated in logs, but for other targets such as
unemployment and the budget deficit it is the level that is relevant and so
the relationship is linear. The key explanatory variables are lagged values of
the dependent variable, and dummy variables for the periods when the
various political parties were in office. The major conclusion, for both the
United States and other countries, is that the post-war data generally
support the rational partisan theory, particularly with respect to growth
and unemployment.bb
12.6 Model specification
The models that we have analysed in the previous section have all demon-
strated the golden rule of empirical analysis – the establishment of an
appropriate analytical framework, derived from a close understanding of
the relevant theory, tempered with appreciation of any historical realities
bb Alberto Alesina, Nouriel Roubini and Gerald D. Cohen, Political Cycles and the Macroeconomy,
MIT Press, 1997. For a quantitative analysis of the same issues of policy formation in the United
Kingdom see Paul Mosley, The Making of Economic Policy, Harvester Press, 1984.
that warrant modifications to the framework before it is confronted with
the data.
Model specification is not simply about choosing and fine-tuning a
model to determine which explanatory variables should be chosen and in
what form they should appear in the regression. It is also about determin-
ing whether the model so chosen is appropriate, whether it represents a
suitable way of interpreting the data, and whether or not it conforms to the
empirical realities. After all, there are often competing theories of a partic-
ular process or event and we need to establish the appropriate criteria for
determining which is the best model.
Controversy exists among quantitative historians, as it does among
those working in the cultural and literary end of the discipline. There is no
universal consensus, even about theory, and certainly not about imple-
mentation. Debates have raged among economic historians over the pace
of industrialization in eighteenth- and nineteenth-century Britain, over
the economic impact of slavery in the antebellum United States, over the
causes of the global Great Depression in the 1930s. Quantitative analysis
has often clarified the terms of the debate. Initial disagreement has fre-
quently stimulated improvements in the data base and in model
specification, and these have helped to bring about a convergence among
critics. Nevertheless, some disputes do remain unresolved.
There are, however, even among disputants, some generally accepted
rules of engagement – some agreement about what constitutes best prac-
tice. Much of the common ground actually concerns worst practice – what
ought never to be done when testing a hypothesis or model. We have
already indicated that ‘kitchen-sink’ regression procedures, in which
everything is thrown into the model and variants of the basic regression are
run until an acceptable outcome is reached, are to be avoided. About this,
there is consensus.23
Nonetheless, the use of specification searches to identify the best repre-
sentation of the model remains common. When a model is run, the
researcher pays particular attention to the R2 of the equation as a whole, to
the signs of the regression coefficients and their standard errors, and to
diagnostic statistics such as the Durbin–Watson statistic for serial correla-
tion. Often, the researcher is unsure whether a particular variable should
be entered in logs or not, or whether lags of the variable should be
included. It seems straightforward just to add more terms to determine
whether they are statistically significant or not, or to correct for some
warning of serial correlation in the DW statistic by running a Cochrane–
Orcutt regression. The criticism of this approach is, however, that contin-
ual reassessment of the regression equation in this fashion compromises
§.
the very statistics used to evaluate the success of the model. If the
specification search is motivated by a desire to maximize the value of R2
and t-statistics, then it is inappropriate to employ them as neutral arbiters
of the success or failure of a model.
How, then, should a conscientious reader approach an article to deter-
mine whether the researcher has resisted the temptations of specification
searches in his work? How should a reader approach empirical controver-
sies? Clearly, we have suggested that traditional arbiters of success, such as
R2 and t-statistics, may mislead. This is not to say that authors who only
report such matters are obviously guilty of worst practice, nor that we
should automatically throw out their results. But readers would be wise to
look for evidence that the research has been undertaken in a less proble-
matical way.
Best-practice methodology focuses on mis-specification searches, i.e.
the use of the various instruments discussed in chapter 11 to determine
whether the model, or its empirical implementation, suffers from fatal
errors of omission or commission. Almost all such tests focus on the prop-
erties of the residuals from the regression; the best work indicates that these
residuals have been evaluated for problems, such as serial correlation,
heteroscedasticity, omitted variables, structural change, etc. Ideally, these
tests should be carried out simultaneously, in order to minimize the pos-
sibility that the test statistics for each procedure are themselves contami-
nated by the order in which they are carried out.
If any of the tests do reveal clear evidence of a problem, the researcher
should show evidence of having considered the possibility that the model is
mis-specified and needs revision, rather than having immediately turned
to one of the many tools at her disposal for ‘correcting’ for the problem.
Obviously, the menu of possible specification errors depends on the
type of regression exercise – cross-section models should be tested for
some problems, and time-series models for others. But such tests are an
essential ingredient of good model-building and should be evaluated by
readers.
Any model is, of course, only as good as the data on which it is based.
Problems can arise in many ways, including errors of measurement or of
omission in the underlying source, errors in drawing samples from the
data, and errors in the procedures subsequently adopted to record or
process the data. Even if a series is reliable for one purpose, it may be
unsuitable if used as a measure of a different concept.cc Careful scrutiny of
cc For example, unwillingness of victims to report crime may mean that an accurate measure of
recorded crime is a very poor measure of the crimes actually committed; evasion of tax may
all data before they are used for regression analysis is thus something to
which historians must give the highest priority, and visual inspection of the
data can be particularly valuable.
It is also very unwise to run regression models borrowed from other
studies on a data set that might well have very different properties. Thus,
the tendency among some economists to employ historical data in order to
run regressions derived from contemporary economic theory as a test of
the general validity of the theory, without any attempt to examine the
empirical and institutional differences between past and present, is to be
deplored.dd As a general conclusion, the more information about the pro-
cedures employed in evaluating the data before and after the regression is
run, the sounder are likely to be the results and the more convinced the
reader should be of their value.
Assume, however, that two researchers have both shown exemplary
commitment to the principles of best-practice modelling, but have none-
theless produced divergent results. Are there any other clues that might
assist the reader in choosing between them?
One procedure that is often invoked as a good way to test the strength of a
model is out-of-sample prediction. This involves reserving some of the data
points from the sample, running the model over the rest, and using the
regression results to predict the value of the dependent variable outside the
period over which the regression was estimated. If the model is a good one –
and if the underlying structural conditions have not changed – the predictions
should compare well with the actual values over this additional period.
Clearly, if one model has better predictive powers than another over the
same omitted sample, this is indicative of model superiority.
Another comparative strategy employed by some researchers is the
development of a general model that incorporates both rival interpreta-
tions of the event or process being studied. The general model is then grad-
ually pared down as empirically marginal features are discarded, or as rival
elements are tested for mis-specification and found wanting.
§.
mean that an accurate measure of income assessed for taxation is a very poor measure of the
income actually received by taxpayers. ddThe point might be illustrated by studies of the labour market, where theories of wage determi-
nation formulated in the late twentieth century might be totally inappropriate in relation to
earlier periods in which both the role of trade unions and the size of firms were very different.
More fundamentally, human behaviour itself is not immutable. Modern mores and codes of
conduct that underpin current theories of individual action in economics, politics, and other
disciplines are likely not to have applied in the distant past. See also the more detailed discussion
of quantitative studies of historical changes in Larry W. Isaac and Larry J. Griffin, ‘Ahistoricism
in time-series analysis of historical process: critique, redirection and illustrations from US labor
history’, American Sociological Review, 54, 1989, pp. 873–90.
Both of these initiatives require that the researcher follow a particular
empirical strategy; they cannot easily be invoked by readers ex post. The
reader is left with a less definitive, and possibly more subjective set of com-
parisons – does the underlying theory in one model seem more plausible
than another? Does one empirical application seem more relevant in a
specific historical setting than another? Do the regression coefficients in one
model make more sense than those in another? The procedure of looking at
coefficients and using them to examine the purported relationships between
variables can be most illuminating and is often ignored by researchers. Often,
the best question to ask of any model is simply: ‘Do I believe the results?’
In chapters 14 and 15, we shall focus on four case studies that use regres-
sion techniques. Part of what we will be looking for is evidence of the use of
diagnostic tools, both formal and informal, to evaluate the overall sound-
ness of the model specification being used.
Notes 1 We are indebted to Roderick Floud for these data. They are taken from J. M. Tanner,
R. H. Whitehouse and M. Takaishi, ‘Standards from birth to maturity for height,
weight, height velocity and weight velocity: British children, 1965 (Part I)’, Archives
of Disease in Childhood, 41, 1966, pp. 454–71; and I. Knight and J. Eldridge, The
Heights and Weights of Adults in Great Britain, HMSO, 1984. 2 If the symbols are changed, economists may recognise (12.5) as the familiar
Cobb–Douglas production function, in which output (Q) is determined by inputs of
labour (L) and capital (K): Q�A L� K�. The two features of this specification are
(i) the equation is non-linear in variables but linear in parameters, and (ii) the effect
of the explanatory variables is multiplicative, not additive. 3 The manipulation is based on the rules for powers given in §1.6.1, in particular, the
rule that any number raised to the power 0 is equal to 1. Thus ex�e x�ex x�e0�1,
and we can substitute ex�e x for 1 in both the numerator and the denominator of
(12.7a). We therefore get
Y �
The terms in e x in the numerator and the denominator then cancel out, to leave the
version in 12.7b. 4 The following points may help some students to see more easily why it is necessary to
multiply by X. First, the definition of an elasticity, which was initially given as
may be re-written as
�Y
Y �
X
�X
�Y
Y �
�X
X
ex �e x
ex �e x � e x �
ex �e x
e x(ex � 1) �
ex �e x
e x(1 � ex)
which in turn is the same as
Secondly, the definition of b in this semi-log model, which was initially given as
can similarly be re-written as
Multiplication of the second term in this revised definition by X then gives the
required elasticity. 5 b/1 � gives the value for the long-run multiplier because the sum of all the
coefficients on current and past values for SALES in the untransformed (10.15) in
§10.2.2, can be written as b(1����2�…��n) and the mathematicians tell us that
the expression in brackets is an example of a geometric progression which sums to
1/1 �. 6 See, for example, Robert Woods, The Demography of Victorian England and Wales,
Cambridge University Press, 2000, pp. 377–8, where a third-order polynomial is
used to plot the relationship between population density and distance from centre of
London. The trend shows an initial rise in density as distance increases and then a
decline. In quadratics and higher-order polynomials, the number of turning points
is always one less than the size of the highest power. 7 Richard Layard and Stephen Nickell, ‘The labour market’, in Rudiger Dornbusch
and Richard Layard, The Performance of the British Economy, Oxford University
Press, 1987, pp.131–79; see especially p.139. 8 When the problem is confined to two or more of the explanatory variables it repre-
sents a form of multicollinearity, a problem briefly discussed in §11.4.2. 9 Recent developments in time-series statistics have produced even better procedures,
such as error-correction mechanisms, but these are too advanced for this text. 10 For an example of a relatively simple Keynesian model see T. Thomas, ‘Aggregate
demand in the United Kingdom 1918–1945’, in Roderick Floud and Donald
McCloskey (eds.), The Economic History of Britain since 1700, II, 1860 to the 1970s, 1st
edn., Cambridge University Press, 1981, pp. 332–46. A much more advanced and
comprehensive model covering the same period is developed in Nicholas H.
Dimsdale and Nicholas Horsewood, ‘Fiscal policy and employment in inter-war
Britain; some evidence from a new model’, Oxford Economic Papers, 47, 1995,
pp. 369–96. 11 An earlier investigation of servant indentures in colonial America also found some
evidence of non-linearity. In this case, age was entered as a series of dummy variables
for each year, giving flexibility to the measured relationship between age and length
of indenture. The coefficients on the dummies fell from 2.75 for those aged less than
�Y
Y �
1
�X �
�Y
�X �
1
Y
�Y
Y ��X
�Y
�X �
X
Y
15 to 0.17 for age 19, showing that age and length of indenture were negatively
related, while the change in the coefficient for each successive year showed that the
relationship was not linear. David Galenson, White Servitude in Colonial America,
Cambridge University Press, 1981, pp. 97–113. 12 For a stylized representation of the age–earnings profile derived from data on US
households in 1889–90, see Hartmut Kaelble and Mark Thomas, ‘Introduction’, in Y.
S. Brenner, H. Kaelble and M. Thomas (eds.), Income Distribution in Historical
Perspective, Cambridge University Press, 1991, pp. 19–23. 13 These points are illustrated by Timothy J. Hatton, ‘The immigration assimilation
puzzle in late nineteenth-century America’, Journal of Economic History, 57, 1997,
pp. 34–63; and Christopher Hanes, ‘Immigrants’ relative rate of wage growth in the
late 19th century’, Explorations in Economic History, 33, 1996, pp. 35–64. 14 Severals models of asset accumulation have found this relationship with age. See, for
example, J. R. Kearl, Clayne L. Pope and Larry T. Wimmer, ‘Household wealth in a
settlement economy: Utah, 1850–1870’, Journal of Economic History, 40, 1980, pp.
477–96; and Livio Di Matteo, ‘Wealth accumulation and the life-cycle in economic
history: implications of alternative approaches to data’, Explorations in Economic
History, 35, 1998, pp. 296–324. 15 On American industry see Jeremy Atack, ‘Economies of scale and efficiency gains in
the rise of the factory in America, 1820–1900’, in Peter Kilby (ed.), From Quantity to
Quiddity, Wesleyan University Press, 1987, pp. 286–335; and John A. James,
‘Structural change in American manufacturing, 1850–1890’, Journal of Economic
History, 43, 1983, pp. 433–59. James Foreman-Peck and Robert Millward, Public and
Private Ownership of British Industry 1820–1990, Oxford University Press, 1994, pp.
197–239, is a masterly study of the performance of public and private industry. The
Anglo-American productivity gap is analysed in S. N. Broadberry and N. F. R. Crafts,
‘Britain’s productivity gap in the 1930s: some neglected factors’, Journal of Economic
History, 52, 1992, pp. 531–58. 16 The classic study is Z. Griliches, ‘Hybrid corn: an exploration in the economics of
technical change’, Econometrica, 25, 1957, pp. 501–22. See also the application to the
diffusion of the stationary high-pressure engine in Cornish mining in the early nine-
teenth century in G. N. von Tunzelmann, Steam Power and British Industrialization,
Oxford University Press, 1978, pp. 252–64. 17 For the application of the logistic curve to expenditure on household durables see
Sue Bowden and Paul Turner, ‘The demand for consumer durables in the United
Kingdom in the interwar period’, Journal of Economic History, 53, 1993, pp.244–58.
The suggested explanation for the logistic pattern is given by Sue Bowden and Avner
Offer, ‘Household appliances and the use of time: the United States and Britain since
the 1920s’, Economic History Review, 47, 1994, pp. 725–48. 18 For a critical evaluation of Farr’s Law see Woods, Demography of Victorian England
and Wales, pp. 190–202. Woods also analyses the evidence for a relationship between
the incidence of specific diseases, such as measles and tuberculosis, and population
density using log-linear models, see pp. 317–25.
19 See, for example, the exemplary studies of the surviving sixteenth-century judicial
records of suicide in S. J. Stevenson, ‘The rise of suicide verdicts in south-east
England, 1530–1590: the legal process’, Continuity and Change, 2, 1987, pp. 37–75;
of violent crime in J. S. Cockburn, ‘Patterns of violence in English society: homicide
in Kent, 1560–1985’, Past and Present, 130, 1991, pp. 70–106; and of the concept of
‘crowds’ in Mark Harrison, Crowds and History, Mass Phenomena in English Towns,
1790–1835, Cambridge University Press, 1988. 20 Some other examples include Edward Shorter and Charles Tilly, Strikes in France
1830–1968, Cambridge University Press, 1974; Neil Sheflin, Leo Troy and C.
Timothy Koeller, ‘Structural stability in models of American trade union growth’,
Quarterly Journal of Economics, 96, 1981, pp. 77–88; A. R. Gillis, ‘Crime and state sur-
veillance in nineteenth-century France’, American Journal of Sociology, 95, 1989, pp.
307–41; and E. M. Beck and Stewart E. Tolnay, ‘The killing fields of the deep South,
the market for cotton and the lynching of blacks, 1882–1930’, American Sociological
Review, 55, 1990, pp. 526–39. 21 For an application of a linear regression model of the relationship between crimes
against property and fluctuations in prices in the seventeenth and eighteenth centu-
ries, see the major study by J. M. Beattie, Crime and the Courts in England,
1660–1800, Oxford University Press, 1986, pp. 199–237. Prices are measured by the
Schumpeter–Gilboy index; the regressions are run separately for urban and for rural
parishes in Sussex, and for Surrey, and also distinguish wartime and peacetime
periods. The general finding is that there was a clear positive relationship between
the fluctuations in prosecutions and in prices (with an R2 of 0.41 for Surrey and of
0.52 for Sussex), but there were some variations over time and between urban and
rural parishes. 22 The early history of research on this topic is reviewed in one of the seminal modern
studies, Gerald H. Kramer, ‘Short-term fluctuations in US voting behavior,
1896–1964’, American Political Science Review, 65, 1971, pp. 131–43. Subsequent
studies in the United States include Edward R. Tufte, Political Control of the
Economy, Princeton University Press, 1978; Gregory B. Markus, ‘The impact of
personal and national economic conditions on the presidential vote: a pooled
cross-sectional analysis’, American Journal of Political Science, 32, 1988, pp. 137–54;
and Robert S. Erikson, ‘Economic conditions and the Congressional vote: a review
of the macrolevel evidence’, American Journal of Political Science, 34, 1990, pp.
373–99. For some of the corresponding studies of the United Kingdom see James
E. Alt, The Politics of Economic Decline, Cambridge University Press, 1979,
pp. 113–38. 23 No doubt part of the reason for this approach lies in the preference among research-
ers and readers for providing positive results, i.e. finding that a hypothesis is consis-
tent with the data. Were the criteria for judging intellectual quality to include
analyses that demonstrate that a model is inconsistent with the data, researchers
would have less incentive to employ specification searches until ‘good’ R2 and t-
statistics were produced.
12.7 Exercises for chapter 12
1. Plot the following time-series relationships from the Benjamin–Kochin
data set:
(i) The level of benefit.
(ii) The wage rate.
(iii) The unemployment rate.
In each case, identify the most appropriate functional form to fit these
shapes. (Hint : choose from the list in §12.1, but note that (iii) involves a
polynomial form.)
For each of these three series, estimate a regression model using your
chosen functional form. Report your results.
Now predict the residuals from the fitted regression line. If your intui-
tion about the right shape is correct, there should be no obvious non-
linear relationship in the residuals when graphed against time. What do
you find?
2. In a model of earnings by age for household heads in the United States in
1889–90, the fitted regression line is
EARNINGS�425.1�12.6 WORKAGE 0.3 WORKAGE2
where WORKAGE is measured as the actual age in years – 15.
(i) Draw the curve of this relationship freehand (and without using the
computer), indicating the direction of slope at younger and older
ages.
(ii) Calculate the age at which earnings begin to decline. What are the
average earnings at this age?
(iii) Can the average worker expect to earn more on retirement (at age
65) than when he started work at age 15?
3. In a model of death rates by age for the Massachusetts population in
1865, the fitted regression line is
MORTALITY�57.8 3.78 AGE�0.06 AGE2
where MORTALITY measures the death rate per 1,000 people.
(i) Draw the curve of this relationship freehand (and without using the
computer), indicating the direction of slope at younger and older
ages.
(ii) Calculate the age at which the death rate is (a) at its peak; and (b) at
its lowest level.
(iii) What is the expected death rate for this population: At the age of 10?
At the age of 70?
4. Evaluate the following propositions about non-linear models, indica-
ting whether they are true or false and explaining why:
(i) ‘Logarithmic transformation is appropriate for multiplicative non-
linear relationships; substitution is appropriate for additive non-
linear relationships.’
(ii) ‘The advantage of the linear trend model is that, since it is a straight
line, it produces a constant growth rate to the data.’
(iii) ‘The linear growth rate is measured as the elasticity of the dependent
variable against time.’
(iv) ‘The number of turning points in a polynomial series is always one
less than the size of the power; thus there is 1 turning point in a
quadratic equation, 2 in a cubic, etc.’
(v) ‘Equation (12.15) indicates that the long-run multiplier will always
be larger than the impact multiplier.’
5. You review a regression result of the following form:
Y�5�2X 1 X2
What is the predicted value of Y, given that X 1�2, X2�3 if:
(i) The model is linear in all terms?
(ii) The model is double-logarithmic?
(iii) The model is semi-logarithmic of the lin-log form?
(iv) The model is semi-logarithmic of the log-lin form?
(v) The model is semi-logarithmic of the exponential form?
What is the impact of a unit change in X1 in each of these cases?
6. Using the Hatton–Williamson time-series data set on Irish emigration:
(i) Draw a graph of the total emigration rate by year for 1880–1913.
(ii) Fit a linear trend to the data on total emigration by year, 1880–1913.
(iii) Fit a log-linear trend to the data on total emigration by year,
1880–1913.
Compare the results of the two regressions. Which provides the superior fit?
7. Using the log-linear trend regression estimated for question 6:
(i) Work out the implied growth rate for the emigration rate from 1880-
1913. How does this compare to the growth rate calculated using the
compound growth rate formula (12.17b)?
§.
Would you consider that a great deal of information has been lost by apply-
ing the short-hand formula in this case?
(ii) Re-run the log-linear regression for IRMIG over the years,
1877–1913. Report your results and work out the implied growth
rate of the emigration rate. How does this rate compare to that cal-
culated from the compound growth formula, fitted to the end-
points alone?
Would you consider that a great deal of information has been lost by apply-
ing the short-hand formula, in this case?
Are there any general lessons about the applicability of the compound
growth-rate formula to be learned from a comparison of your findings for
the two periods?
8. Use the basic Benjamin–Kochin (OLS) regression to calculate the elastic-
ity of unemployment with respect to the benefit–wage ratio.
(i) Re-run the Benjamin–Kochin equation with the logarithm of
UNEMP as the dependent variable. What is the elasticity of unem-
ployment with respect to the benefit–wage ratio at the mean?
(ii) Re-run the Benjamin–Kochin equation with the logarithm of
BWRATIO as the explanatory variable. What is the elasticity of
unemployment with respect to the benefit–wage ratio at the mean?
(iii) Re-run the Benjamin–Kochin equation with the logarithm of
UNEMP as the dependent variable and the logarithm of BWRATIO
as the explanatory variable. What is the elasticity of unemployment
with respect to the benefit–wage ratio at the mean?
(iv) Re-run the Benjamin–Kochin model as an OLS equation, with
lagged UNEMP as an additional right-hand-side variable. How
much different is the long-run multiplier of the benefit–wage ratio
on unemployment from the impact multiplier? What is the long-run
elasticity?
9. As a test of the Benjamin–Kochin model of inter-war British unemploy-
ment, you decide to use out-of-sample prediction. The strategy you decide
to pursue is to run the basic model over the period between 1920 and 1935
and evaluate how well it predicts the value of unemployment in 1936–8.
What do you find? (Hint : you will need to instruct the computer to predict
the fitted values from the regression for the additional years; many pro-
grams will do this automatically, but do check.) How do the results
compare both to actual unemployment and to the predicted value of
unemployment from the regression for 1920–38?
You also wish to use out-of-sample prediction to determine whether an
alternative model is superior. This model is:
UNEMP�a�b1 BENEFIT�b2 WAGE�b3 DEMAND�e
Once again, you wish to compare the fitted values for 1936–8 from a regres-
sion run over 1920–35 to the actual unemployment rates in those years and
to the fitted values from a regression run over the entire sample. Report
your results and draw any appropriate inferences.
What does this exercise suggest about the relative strengths of the two
models for explaining inter-war unemployment in Britain?
10. Set up a model explaining the time-series behaviour since the Second
World War of any ONE of (a) US imports of oil; (b) the divorce rate in the
United Kingdom; (c) the level of foreign aid from the G7 countries to
African states. Be sure to specify the explanatory variables to be used in the
model; the form of the relationship between each explanatory variable and
the dependent variable; and what you would look for when analysing the
results.
§.
Logit, probit, and tobit models
In our analysis of regression models thus far, we have employed models
that implicitly assume that the dependent variable is continuous, rather
than discrete, and complete, rather than restricted. But many variables that
historians are interested in studying do not share these characteristics. If
the historian wishes to analyse why a state voted Republican or Democrat
in a presidential election, or why some households decided to invest in the
stock market in 1929 and others did not, or which country a family decided
to migrate to, she is dealing in every case with responses that can take on
only a small number of possible values. Similarly, other variables are
limited in the range of values they can take – a labour historian investigat-
ing individual variation in the length of the working year in weeks will find
many more values in her data set, but none above 52 or less than 0.
Such variables are known as limited dependent variables. Dependent variables that can take on only a small number of discrete values (such as 0,
1, 2) are known as qualitative dependent variables; those that are continu- ous, but restricted in their range (e.g. never becoming negative), are known
as censored variables. Because of the restriction on the value of the depen- dent variable, these and other such questions involve different estimating
strategies and require different regression techniques from the standard
OLS approach.
In §13.1, we discuss the nature of limited dependent variables in more
detail and indicate why standard OLS models are inappropriate for this class
of data. In §13.2, §13.3, and §13.4, we introduce two models (known as logit
and probit) that are more suited to limited dependent variables and apply
them to the simple case of a binary dependent variable. In §13.5, we extend
these models to cases in which the dependent variable can take on more than
two discrete values. Finally, in §13.6, we focus on the problems of censored
dependent variables and briefly discuss two models (known as the tobit and
two-stage models) that have been developed expressly for this case.
13.1 Limited dependent variables
13.1.1 Three cases of limited dependent variables The first case is when the dependent variable is limited by the fact that it is a
categorical or qualitative dummy variable. As with the dummy explanatory
variables discussed in §10.1, dummy dependent variables may be restricted
so that they can take only the values of 0 or 1 (referred to as binary or dichotomous); or there may be a larger but still finite number of possible values.
Examples in historical research of a binary dependent variable include:
political variables, such as whether an elected representative voted ‘yes’ or
‘no’ in a parliamentary vote; economic variables, such as whether or not a
worker experienced unemployment last year; and social variables, such as
whether or not a widow has remarried. In each case, the response would be
coded as 0 (for the negative outcome) or 1 (for the positive).a
Alternatively, there may be more than two possible outcomes. The
dependent variable may, for example, take the values 0, 1, 2, and 3 (or
more).b This would be appropriate in a model of occupational status of
urban workers, distinguishing between white collar, blue collar, unskilled,
and professional employment; or of mode of transport to work (walk,
bicycle, automobile, bus, train, etc.).
The second case occurs when the dependent variable is limited because,
although it is continuous, it can take values only within the range of 0 to 1.
For example, the proportion of all debates in a parliamentary session in
which an elected representative votes together with her party of affiliation
can range from 0 (never) to 1 (always); the proportion of the previous year
that a worker was unemployed can range from 0 (not at all) to 1 (all the
time); the proportion of households that owns an automobile can range
from 0 (nobody) to 1 (everybody). In each case, these variables are calcu-
lated as the ratio of positive outcomes to total outcomes and may be
expressed as the average or expected value of the outcome for the group or individual (see panel 13.1).
The third case arises when a continuous dependent variable is limited
because it is bounded in one direction or the other. For example, some house-
holds may own more than one car, but no household can own fewer than
no cars. Thus, the lower limit of automobile ownership is zero. Variables
§.
a It would be possible to reverse the order and code negative responses as 1, but the standard con-
vention is as described here. However, students are advised to make sure that the historian has
followed convention before interpreting regression results. b These are sometimes referred to as either polychotomous or polyotomous variables, but we
eschew such linguistic barbarisms.
whose values are bounded in this way are known as censored. Most such variables are censored from below, that is they cannot fall below a certain
level (usually, they cannot take on negative values). Less commonly, vari-
ables are censored from above. One example is the measurement of house-
hold savings in a given period, which cannot be greater than the total
income received.c
c A related problem occurs if a sample excludes observations according to some criterion, such as
a minimum height requirement in a sample of army recruits, or maximum family size in a
survey of household expenditures. Such cases are known as truncated samples. Their solution
requires more complex econometric theory than is possible in this text. Unhappily, truncated
samples do occur with some frequency in historical research and researchers should always be
careful to consider the problem.
Panel 13.1 Expected values and probability theory
The term expected value is derived from probability theory. It refers to the
value that would be expected if a random observation were chosen from the
population. It is equal to the sum of all the possible values, weighted by their
individual probabilities.
Take a game of coin tossing in which you are paid $1 for every head and
nothing for a tail. The expected value of your gain from a single coin toss
(assuming a fair coin) will be the sum of each value, multiplied by its chance
of occurring. In this simple case, it is $1�0.5�$0�0.5, which equals $0.50.
Clearly, you would receive either $1 or nothing for any given toss, but if a
very large number of tosses were made, this would be the most likely average
of all the values (as in our discussion of sampling distributions in §5.2.1).
This probabilistic exercise can be extended to non-random occur-
rences. Thus, if half the population owned cars and half did not, the
expected value of automobile ownership would be 0.5, an impossible value
for any single individual, but the average value across the population as a
whole. It is clear that the expected value in this case could range from 0 (no-
one owns a car) to 1 (everyone does), and includes any value within these
boundaries.
The reason for introducing the term expected value in this chapter is
that logit and probit models are framed in terms of probability theory. The
dependent variable in the regression formulations is identified as the prob-
ability of an event, e.g. the probability of a household owning a car given
their income, family size, wealth, etc. The regression predicts the expected
value of car ownership for households sharing these characteristics.
All these situations complicate regression analysis. OLS methods are
generally inappropriate for qualitative and limited dependent variables,
and alternative procedures should be employed. The most common
approaches used in quantitative analysis of such cases are the so-called
logit, probit, and tobit models. These were developed to deal with some of
the complexities of modelling limited dependent variables. Twenty years
ago they were rarely applied in historical analysis. However, with the explo-
sion in computing power in recent years they are now included in all major
computer packages, and there has been an upsurge in their popularity.
They have been used to analyse such a wide array of issues in social, politi-
cal, demographic, and economic history it is hardly necessary to provide
examples, but a few references are given to illustrate the range of subjects to
which these models have been fruitfully applied.1
13.1.2 Why linear models are not appropriate for limited dependent variables As an illustration of the method of logit and probit models and their super-
iority over OLS procedures, let us develop a simple example. Figure 13.1
shows the pattern of automobile ownership for a fictitious sample of house-
holds in Detroit in 1919. Families either own a car or they do not.
Automobile ownership is thus a binary variable, taking the value of 1 (own)
or 0 (not own). The explanatory variable in this simple bivariate model is
income. The data points (marked by circles) thus lie on one of two horizon-
tal lines.
It is evident from the graph that the frequency of automobile ownership
rises with income. The best-fitting OLS regression is superimposed on the
data. The slope of the straight line indicates the increased likelihood of a
household owning a car as income rises. But the linear model has two
serious problems.
First, it produces awkward results. For households with an income
below $1,540, it predicts negative ownership; for households with income
above $8,815, it predicts a probability of owning a car in excess of one.
Neither outcome is possible. Moreover, the standard errors around the
OLS coefficients are likely to be high, given the inability of a linear model to
fit the pattern of a dependent variable strung along two parallel lines.
Secondly, the OLS regression suffers from heteroscedasticity: the size of
the error term will be different for those who own automobiles and those
who do not.2 Since automobile ownership is related to the level of income,
it therefore follows that the variance of the error term depends on the
value of the explanatory variable. This contradicts the assumption of
§.
homoscedasticity, one of the fundamental conditions of the classical
linear regression model (see panel 9.1 and §11.3.2).
The linear model is clearly not ideal. What should we use instead? The
first step is to drop the assumption of linearity and instead apply a func-
tional form for the regression line that is more consistent with the data. We
have already employed this tactic in chapter 12 when dealing with non-
linear variables. In essence, the solution to the problem of dichotomous
variables merely extends the logic of curve-fitting to cover the special case
of limited dependent variables.
Figure 13.1 clearly shows that the best way to fit a curve to the observa-
tions on Detroit automobile ownership is to use a sigmoid shape – an elon-
gated S-shaped curve that approaches zero at its lowest values and one at its
highest. A large number of functional forms take on this shape. However,
most analysis is based on the cumulative distribution function of one of
two theoretical distributions.
A cumulative distribution function (cdf) is an ogive (see §2.1.3) fitted to the density function of a particular distribution; for example, the
density function of the standard normal distribution explained in §5.4.1.
The cdf is generated by adding up (cumulating) the successive values of the
density function from its lowest to its highest value.
One such cdf is the logistic curve.3 We have already met this in chapter 12 and seen in figure 12.6 that it demonstrates the required S-shape. It is
this curve that is fitted to the data in figure 13.1.
Figure 13.1 Pattern of automobile ownership, Detroit, 1919
1.2
1.0
0.8
0.6
0.4
0.2
0.0
–0.2 0 2 4 6 8 10
Household income ($000)
OLS regression line
S-shaped curve Pr
ob ab
ili ty
o f
au to
o w
ne rs
hi p
12
o o o o o o
o o o o o o o o
The standard normal distribution also takes on the S-shape when it is
changed from a density function to a cumulative distribution. To see why
this is so, recall from §2.6.2 that in the standardized normal distribution,
the area under the curve sums to one. The speed with which the cdf
approaches one will be slow at the start of the distribution (the extreme fur-
thest left from the mean) and at the end (furthest right from the mean),
where the slope of the normal curve is low and thus the successive incre-
ments in the cdf are small. The speed will reach its maximum in the middle
of the distribution where the slope is greater, and the increments are large.
This is reflected in the shape of the cdf of the normal curve – shallow at the
extremes and steepest in the middle – as depicted in figure 13.2.
The regression strategy is to take one of these forms and fit it to the data.
Regression models that assume a logistic distribution function are known
as logit models. Those that assume a normal distribution function are known as probit models.d Every major computer program allows the researcher to estimate either probit or logit models on any given data set.
We focus on each of these models in turn, beginning with the logit model.
§.
d It would in some respects be more appropriate to refer to the probit model as the normit,
reflecting its origin in the normal curve, just as the logit is derived from the logistic curve.
Figure 13.2 Cumulative distribution function of the standardized normal distribution
1.2
1.0
0.8
0.6
0.4
0.2
0.0
–0.2 –4 –3 –2 –1 0 1 2 3 4
Standard deviations from the mean
V al
ue o
f cu
m ul
at iv
e di
st ri
bu ti
on f
un ct
io n
13.2 Estimating the logit model: coefficients, absolute effects, and elasticities
13.2.1 The logit regression model Consider an analysis of household ownership of automobiles, in which the
dependent variable is binary (either owning a car or not). The relevant
explanatory variables might include family income, the size of the family,
location, etc.
Were the researcher to choose a linear regression model, this would be
formulated as
Yi�a�b1X1�b2X2�b3X3�e (13.1)
where
Yi�1 if the household owned an automobile
and
Yi�0 if the household did not own an automobile
The dependent variable may be interpreted as the probability of owning a
car. However, as we have already seen, the linear model is inappropriate to
this case; and so she adopts the logit model instead.
This model assumes that the probability of owning a car is distributed
according to the logistic function. This is a variant of the logistic equation, the particular derivation of which was explained in §12.1
Y � (13.2)
The term eX stands for ‘e raised to the power X’, where e is the exponential
constant�2.7183 and the variable X is referred to as the exponent (see §1.6).e
The special feature of the logistic function is that the X in (13.2) is
replaced by the constant and all of the explanatory variables in the relevant
regression equation. For example, for the model specified in (13.1), the
logistic function would be
Y � (13.3a)
This is a very cumbersome expression to work with, and in order to sim-
plify it we will replace the intercept, all the explanatory variables and their
ea�b1X1�b2X2�b3X3
1 � ea�b1X1�b2X2�b3X3
ex
1 � ex
e An alternative way of writing ‘e to the power X’ is exp(X), and this variant is occasionally seen in
articles and textbooks. In this chapter, we will retain the use of ex.
coefficients by the single letter, w. w is not a conventional symbol, but is
simply used here as a convenient shorthand expression. Equation (13.3a)
thus becomes
Y � (13.3b)
We can now apply this procedure to the specific form of the logit model.
The logit model relates to each individual unit in the sample, in this case
the households that either own or do not own a car. It assumes that the
probability of owning a car is distributed according to the logistic function,
which takes the value
P(Yi � 1) � (13.4)
where P(Yi�1) indicates the probability of the ith household owning an
automobile.
The probability of the ith household not owning a car is similarly
P(Yi�0), which equals 1 P(Yi�1). Substituting this in (13.4) gives
P(Yi � 0) � 1 (13.5a)
By a process of manipulation, this can be simplified either to4
P(Yi � 0) � (13.5b)
or to
P(Yi � 0) � (13.5c)
The method by which this set of equations, (13.4) and (13.5c), is solved
is known as maximum likelihood. Essentially, the procedure chooses the value of the model parameters, a and bi, to provide the best fit to the distri-
bution of the dependent variable. An intuitive explanation is provided in
panel 13.2.
The parameters so chosen are the logit coefficients.
13.2.2 Interpreting logit coefficients The computer reports the logit results in standard format – a constant term
plus coefficients on each explanatory variable. The computer also reports
the standard errors (and/or t-statistics) for each term. In many cases histo-
rians are happy simply to reproduce the logit coefficients and t-statistics,
e w
1 � e w
1
1 � ew
ew
1 � ew
ew
1 � ew
ew
1 � ew
§.
Panel 13.2 Maximum-likelihood models
In previous chapters, our regression analysis has been based on the applica-
tion of classical linear regression models, such as OLS. Not all applications
are amenable to CLR models, however, and econometricians have devel-
oped alternative methodologies. Among the most important of these is a
class of models based on maximum-likelihood techniques.
Maximum-likelihood models are mathematically very complex,
requiring sophisticated calculus tools well beyond the level of this text.
Since they are now part and parcel of every statistical computer package,
there is no need to work through their formal mechanics. Nonetheless, it is
helpful to understand the general methodology involved, and we can
explain the procedure intuitively.
The basic approach of maximum-likelihood models is to fit a specified
model to a specific set of data. The researcher chooses the model and then
asks the computer to select the parameters of the model that best replicate
the data set. The model selection is taken from the range of probability dis-
tributions, including the normal distribution, the lognormal, the logistic,
etc. Each of these has specific attributes (mean, variance, skewness, etc.)
that govern its shape and position. The model may be chosen on theoretical
grounds (it reflects the process by which the events recorded in the data
were produced), empirical grounds (it looks like the distribution of the
data), or practical grounds (it is easy to estimate).
Note, however, that each distribution has an infinite number of pos-
sible outcomes, depending on the specific mix of attributes (parameters).
Every normal curve, for example, has the same basic bell-shape, but
different means and variances will produce different variants of the basic
shape (as illustrated in figure 2.5).
The maximum-likelihood model is designed to find out which combi-
nation of parameters best fits the sample data; in other words, to find which
combination maximizes the likelihood of reproducing the data. The model
searches among the possible parameter values until the optimal combina-
tion is found. The chosen parameters are the maximum-likelihood esti-
mates for this particular model fitted to this particular data set.
Let us use a simple example. We randomly draw ten individuals from
the entire US population to assist a sports manufacturer decide how many
left-handed golf clubs to produce. We signify right-handed people by 0 and
left-handed as 1. The results of the survey are (0, 0, 1, 1, 0, 0, 0, 0, 1, 0). What
distribution of attributes is most likely to produce this outcome? We assume
§.
that there is only one parameter that matters, namely the proportion of left-
handed people in the population. We symbolize this proportion by p. What
proportion is likely to produce this result?
Clearly, it cannot be that all Americans are either left- or right-
handed; thus the likelihood of p�0 and p�1 are both zero. If the propor-
tion were 0.1 (1 in 10), each occurrence of a right-handed observation
would occur with a probability of 0.9, and each occurrence of a left-
handed with a probability of 0.1. Since the individuals were chosen ran-
domly from a very large population, each observation is independent,
such that the probability of drawing this specific sample is the product of
each separate probability.
Thus, if p�0.1, the likelihood of generating our sample is
0.9�0.9�0.1�0.1�0.9�0.9�0.9�0.9�0.1�0.9�0.0004783
Similarly, if p�0.2, the likelihood would be:
0.8�0.8�0.2�0.2�0.8�0.8�0.8�0.8�0.2�0.8�0.0016777
which is a little higher than with p�0.1.
The likelihood can be applied to every other value of p between 0 and 1.
The distribution of probabilities from such a process is depicted in figure
13.3. The distribution reaches its peak at p�0.3, as we would anticipate.
Figure 13.3 Maximum- likelihood function
0.003
0.002
0.001
0.000 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
Proportion of left-handers
Va lu
e of
li ke
li ho
od f
un ct
io n
This picture sketches the likelihood function. The formula for the like-
lihood function in this case is:
L(p) � p � (1 p) (13.6)
where yi�1 indicates the product of all the positive terms and
yi�0 the product
of all the negative terms. Dealing with product terms can be complicated,
especially when working with more complicated distributional forms.
Generally, therefore, researchers prefer to use the logarithm of the likeli-
hood function, or the log-likelihood function, which is additive rather than
multiplicative in the terms. This is defined for our simple example as:
logL(p) � log p� log(1 p) (13.7)
which may be re-written as:
logL(p) � n1 log p � (n n1)log(1 p) (13.8)
where n and n1 indicate the total number of observations and the number of
positive observations, respectively.
Both the likelihood and the log-likelihood functions reach their highest
values when p�0.3. This is the maximum-likelihood estimate of p for this
class of distribution and this particular sample.
The example establishes the methodology commonly applied in
maximum-likelihood models:
(a) Select the distribution most appropriate to the data being investigated.
(b) Establish the likelihood function for given parameters of the distribu-
tion.
(c) Determine the value of the parameters that maximize the value of the
likelihood function, given the data.
The parallels between our simple model and the general problem
involving dichotomous dependent variables should be clear. A given
sample produces a series of observations that are binary, taking the
value of 0 or 1. We interpret these values as probabilities and try to
determine what factors influence the particular distribution of 0s and
1s among the sample.
In the case of the logit model, the assumed distribution of the data is
the logistic. The likelihood function is constructed by assigning (13.4)
� yi�0
� yi�1
� yi�0
� yi�1
and to limit inferences to the sign and significance of each coefficient.f The
signs and the standard errors of logit coefficients may be interpreted in
exactly the same way as for OLS regressions. For example, if the sign of a
coefficient is negative, it indicates that the dependent variable falls as the
explanatory variable increases in value. If the t-statistic falls below a critical
level, it indicates that the logit coefficient is not statistically significantly
different from zero at a certain level of �. Similarly, the larger the t-statistic,
the smaller the confidence interval around the reported coefficient.g
However, in other cases, an historian may not consider it sufficient to
establish the sign and significance of the logit coefficients. She may also be
interested in making comparisons across coefficients, to determine which
explanatory variable has a greater impact on a particular outcome.
Alternatively, she may wish to discover how big an impact a certain change
in the explanatory variable (say, income) has on the value of the dependent
variable (say, the probability of owning a car). In order to accomplish any
of these tasks, it is necessary to understand exactly how the logit coefficient
is to be interpreted.
It may be obvious from the way this has been stated, that interpreting
logit coefficients is not as straightforward as it is with an OLS regression.
Unlike the OLS formulation, the coefficient, bk, is not a direct measure of
the change in the dependent variable, P(Yi�1) for a unit change in explan-
atory variable, Xk. Moreover, we cannot simply substitute the value of Xk
§.
f Note, however, that no empirical meaning can be ascribed to the constant term in a logit func-
tion. g Some programs also produce prob-values for the logit coefficients.
and (13.5c) to the positive and negative outcomes respectively. Thus,
the log-likelihood function may be written as
log L� log � log (13.9)
The computer iterates around the possible values of the parameters
embedded in w until it finds that combination that maximizes the
value of the function, i.e. comes closest to mimicking the actual distri-
bution of outcomes in the sample. This produces the maximum-
likelihood estimates of a, the constant, and bi, the coefficients, as
reported in column (1) of table 13.2.
e w
1 � e w�yi�0 ew
1 � ew�yi�1
into the equation and read off the value of the dependent variable directly,
as with OLS formulations. The information can be abstracted, but it
requires some manipulation of the coefficients.
To understand what is at issue, let us return to our sample of Detroit
households in 1919. We have estimated a logit model for the pattern of
automobile ownership with income as the sole explanatory variable, X1.
The coefficient on income is found to be 0.6111; the constant term is
3.6921. We wish to know the predicted level of automobile ownership at
a certain income, say, $2 thousand. The value of the logit function at that income level can be calculated by substituting the coefficients and the value
of the explanatory variable into the regression equation, a�b1X1. This
produces
3.6921�(0.6111�2)� 2.4699
Clearly, this figure cannot be the expected probability of owning a car, since
that must lie between 0 and 1. How can we move from what we see (the
value of the logit function), to what we want to know (the probability of car
ownership at X1�2)?
The first step in answering this question is to note that if we divide (13.4)
by (13.5b) we get
ew � (13.10a)
where ew is the expected value of the dependent variable from the regres-
sion.5 The ratio of the two probabilities on the right-hand side of the equa-
tion is known as the odds ratio.
THE ODDS RATIO
This is the ratio of the
probability of a positive outcome, P(Yi�1),
to the
probability of a negative outcome, P(Yi�0).
It is thus equal to
(13.10b) P(Yi � 1)
P(Yi � 0)
P(Yi � 1)
P(Yi � 0)
If we take the logarithm of both sides of (13.10a), the left-hand side
becomes w, and the right-hand side equals the natural logarithm of the
odds ratio, usually referred to as the log odds ratio.6 Thus,
w � loge (13.11)
To complete the derivation, recall that w is our shorthand term for the com-
plete set of explanatory variables, a�bk Xk. Thus, the log odds ratio is equal to
the estimated value of the logit function at a given set of explanatory variables.
THE LOG ODDS RATIO
This is the
logarithm of the odds ratio.
It is equal to
loge (13.11)
and since this equals w,
the log odds ratio is thus also equal to
the value of the logit function
at a given set of Xk.
In our hypothetical Detroit sample, the measured value of the logit
function, w, at X1�$2 thousand was 2.4699. In order to transform this
into the predicted probability of owning an automobile, P(Yi�1), which
we may represent by P̂Y, we have to undertake the following steps:
(i) Take the exponent of 2.4699; this equals 0.0846. This step trans-
forms the log odds ratio into the odds ratio proper (ew, as in (13.10a)).
(ii) Divide 0.0846 by 1.0846; this produces 0.078. This step extracts the
value of P(Yi�1) from the odds ratio (using e w/(1�ew)), as previously
shown in (13.4).
Thus, P̂Y at an income level of $2 thousand is 0.078, or 7.8 per cent. We
can use the same procedure to calculate P̂Y for any other income level in the
sample.h
�P(Yi � 1)P(Yi � 0)�
�P(Yi � 1)P(Yi � 0)�
§.
h The procedure is the same, no matter how many explanatory variables there are in the regression.
The procedure may also be used to calculate the average level of auto-
mobile ownership for the sample as a whole from the logit regression.
This could be accomplished by first estimating P̂Y for each observation
and then calculating its average. However, such a laborious process is
unnecessary, since the average value of P̂Y is precisely equal to the mean of
P(Yi�1) for the sample as whole (which we designate as P –
Y). Thus,
�P̂Y/n�P –
Y. This result is the maximum-likelihood equivalent of the stan-
dard OLS finding that the sum of all the deviations from the regression
line is equal to zero; the logit regression provides the best fit to the data on
average.7
THE AVERAGE PREDICTED VALUE OF PY
The average predicted value
of the dependent variable
for all the observations from the logit regression is
P̂Yi
This is
precisely equal to
the sample mean of the dependent variable, P–Y.
The procedure outlined in this subsection shows how to calculate the
probability of a given outcome for a certain level of an explanatory variable.
But it does not permit us to measure the impact of a change in the explana-
tory variable on the outcome. Thus, we can work out what the chances of
owning a car are for any given level of income, but we do not yet know how
to calculate how much the chances will change as income rises. In the next
three subsections, we will focus on this second issue.
Our discussion of the effects of a change in an explanatory variable will
distinguish between the impact on the dependent variable, P(Yi�1) of an
absolute change in an explanatory variable and of a proportionate change.
The former would be most appropriate if the aim is to discover which
explanatory factor had the largest influence on a particular decision. The
latter would be most appropriate if the aim is to understand how sensitive a
decision is to a given change in one or more explanatory factors. We will
1
n �i
discuss the effects of an absolute change in §13.2.3 and §13.2.4, and will
deal with the proportionate changes in §13.2.5.
13.2.3 Absolute changes: marginal effects and impact effects The impact on the dependent variable, P(Yi�1), of an absolute change in
an explanatory variable, Xk, can be measured in two different ways. The
marginal effect measures the impact of an infinitesimal change in Xk. This impact is commonly indicated by the expression, ∂PY/∂Xk.
i
THE MARGINAL EFFECT
This is the
impact on the dependent variable, P(Yi�1),
of
an infinitesimal change in an explanatory variable, Xk.
It is indicated by the expression
The alternative impact effect measures the change in P(Yi�1) relative to a (larger) finite change in Xk. The impact effect of an absolute change in Xk is
commonly indicated by the expression, �PY/�Xk, where � (capital Greek
delta) is the mathematical symbol for a finite change. It is usually calcu-
lated for a change of one unit in Xk, although the impact of other finite
changes may also be estimated. Since �PY/�Xk is a ratio (unlike ∂PY/∂Xk), the scale of the impact effect differs as �Xk changes. If the effect is measured
for a change of one unit then �Xk�1, and the impact effect is simply equal
to �PY.
�PY �Xk
§.
i ∂PY/∂Xk is the mathematical symbol representing a very small change in P(Yi�1) caused by a very small change in one explanatory variable, Xk, while any other explanatory variables are held
constant. (For the mathematically inclined, it is the partial derivative of the logistic function
with regard to Xk at the point of measurement.) Note that it must be thought of as a single
expression; unlike �PY/�Xk, it is not a ratio of a numerator to a denominator, in which the value
of the denominator can change from one calculation to another.
THE IMPACT EFFECT
This is the
change in the dependent variable, P(Yi�1),
relative to
a finite change in an explanatory variable, Xk.
It is indicated by the expression
The difference between these two measurements can be illustrated with
reference to a diagram of the logistic curve. Figure 13.4 reproduces the fitted
values of the logit regression of automobile ownership on income for our
Detroit sample. It is constructed by calculating the logit value, w, for each
observation, using the coefficient values of a and b1 ( 3.6921 and 0.6111),
and then converting these to predicted values of automobile ownership for
each household, using the procedure laid down in steps (i) and (ii) in §13.2.2.
We have identified two points, A and B, on the logit curve. These show
the predicted level of automobile ownership (P̂Y) when income (X1) equals
$2 and $3 thousand respectively. The line joining the points indicates the
change in P̂Y for a one-unit change in X1 (in this case, �X1�$1 thousand).
The slope of this line indicates the impact effect on PY of a unit change in X1. In
terms of the expression for the impact effect, the size of �X1 is one unit;
�PY will be equal to the change in P̂Y between A and B.
The other way to measure the slope is to fit a line that is tangential to the
curve at a particular point, say A. The slope of this line is exactly equal to
the slope of the curve at that point. It is indicated on figure 13.4 as a dotted
line. The slope of this line indicates the marginal effect on PY of an
infinitesimal change in X1. Both ∂X1 and ∂PY are very small (too small for the eye to see, perhaps). However, the comparative changes of X1 and PY may be evaluated by extrapolating from the infinitesimal to a finite change
by moving along the slope of the dotted line.
What the slopes tell us There are two points to note regarding the slopes. First, the slope of the
dotted line and the solid line in figure 13.4 are different. Our two different
methods of measuring the change in PY resulting from an absolute change
�PY �Xk
in Xk thus produce two different estimates (this can be seen by comparing
the size of the increase in PY occasioned by a one-unit change in Xk along
the two lines). The marginal effect and the impact effect will be different for
the same logistic curve, when measured at the same level of Xk.
It is therefore important to determine when to use one measure rather
than the other. The general rule of thumb is that the researcher is interested
in measuring the effect of the smallest meaningful change in the explana-
tory variable. If we are dealing with a continuous variable, such as income,
this suggests calculation of the effect of an infinitesimal movement in Xk (the marginal effect). On the other hand, if we are dealing with a discrete
explanatory variable, whose values change by one unit at a time (a dummy
variable, for example, whose value changes from 0 to 1), it is more appro-
priate to measure the impact of a one-unit change in Xk (the impact effect).
The marginal effect can be calculated for a discrete variable, but the rele-
vance of measuring the impact of an infinitesimal change in a variable that
can move only by one unit at a time is clearly limited.
Second, the size of both the marginal effect and the impact effect depend on
the level of Xk at which they are estimated. In the linear OLS model, the slope
of the regression line remains the same throughout the sample. This is
clearly not true of the logit function. The slope of both the solid and the
dotted lines will vary depending on where they are measured on the logit
curve. If we measured the impact effect of a change from $4 to $5 thousand
dollars, we would find that the slope of the line joining the two points on
the logit curve would be steeper than the solid line joining A and B.
§.
Figure 13.4 Logit curve of automobile ownership, Detroit, 1919
1.0
0.8
0.6
0.4
0.2
0.0 0 2 4 6 8 10
Household income ($ 000)
Impact effect
Marginal effect
A
BP ro
ba bi
li ty
o f
au to
-o w
ne rs
hi p
12
Likewise, the slope of the dotted line also varies along the curve. This
can be seen by comparing the tangency lines at A and B in figure 13.4. It
happens that the slope (and therefore the size of the marginal effect) is
largest at the middle of the logit curve (i.e. when the predicted value of
P(Yi�1)�0.50). The further we move from the middle of the curve, the
more the slope will diminish (and the marginal effect will decline in
value). The varying slopes are due to the inherent non-linearity of the
logit curve.
It is therefore important not to place too much emphasis on the meas-
urement of either the marginal or the impact effect of a change in an
explanatory variable at a single point on the curve. But on the other hand,
researchers have no wish to report a whole battery of results for each indi-
vidual, or for each possible permutation of the explanatory variables. They
invariably do provide one indicator of the effect of an absolute change, and
this is usually the average marginal, or average impact, effect for all obser-
vations. This collapses the multiplicity of possible outcomes into a single
index, which gives equal weight to the effect of a change in, say, income on
car ownership for every household in the sample.
13.2.4 Calculating the marginal and impact effects Having explained these two terms, we must now show how the change in
P(Yi�1) is calculated, both for an individual case (in the Detroit example,
a particular household with a certain income), and for the sample as a
whole (all households). We begin with an explanation of the procedure for
calculating the marginal effect of a change in an explanatory variable.
The marginal effect for an individual case The first form of impact defined in §13.2.3 was the marginal effect. For the ith individual household this is equal to �PYi/�Xk. It can be shown that
8
� P(Yi�1) � P(Yi�0)�bk (13.12)
where P(Yi�1) and P(Yi�0) are the predicted values of the i th household
owning and not owning a car at the chosen value of Xk, and bk is the logit
coefficient.
In order to apply this formula it is necessary to calculate P(Yi�1) by the
procedure laid down in §13.2.2, and P(Yi�0) can be obtained directly
from this. For the Detroit sample the required sequence of steps when
income is $2 thousand is as follows:
�PYi �Xk
(i) Log odds ratio �Value of logit function at X�$2 thousand
� 3.6921�(0. 6111 � 2)
� 2.4699
(ii) Odds ratio �exp( 2.4699)
�0.0846
(iii) Value of P(Yi�1) �Odds ratio/(1�Odds ratio)
�0.0846/1.0846
�0.078
(iv) Value of P(Yi�0) �1 P(Yi�1)
�1 0.078
�0.922
The marginal effect on automobile ownership of a change in income for
a household whose income is $2 thousand can thus be calculated by substi-
tuting these results in (13.12) to get 0.078 � 0.922 � 0.6111�0.044.
The marginal effect for the sample as a whole This completes the calculation of the marginal effect for an individual case,
but it is also necessary to obtain the effect for the sample as a whole. The
marginal effect of a change in Xk for the whole sample is equal to the
average of all the individual marginal effects. Thus,
(13.13a)
On the basis of (13.12), this can be re-written as
(13.13b)
where P̂Yi indicates the predicted probability of owning an automobile for
the ith household, and (1 P̂Yi) is the predicted probability of the house-
hold not owning an automobile. The second term in the equation is less
daunting than it may appear; it is simply the average across all the observa-
tions of the product of the predicted probabilities of owning and not
owning a car.
In the specific case of the Detroit sample, the logit coefficient on income
is 0.6111; the sample mean of the product of the individual probabilities is
0.1542. So the marginal effect of an infinitesimal change in income on
automobile ownership is calculated as 0.6111 � 0.1542 �0.0942.
�PY �Xk
� bk �� i
P̂Yi �(1 P̂Yi)
n
�PY �Xk
� 1
n �i �PYi �Xk
§.
There remains the question of how the reported value of the marginal
effect should be interpreted. Is 0.0942 a large number or a small one?
Without a point of comparison, it is hard to tell. In a bivariate model,
therefore, with only one explanatory variable, the marginal effect is of
limited interest. In a multivariate model, however, it provides a means to
compare the impact on the dependent variable of very small changes in
each of the explanatory variables. One variant of this approach that is
often to be found in logit (and probit) exercises is the publication of the
ratio of the marginal effects, indicating the relative strength of the
explanatory variables in influencing the behaviour of the dependent
variable.
The impact effect for an individual case As an illustration of how to calculate the alternative measure of the impact effect, we measure the impact of a change in Xk of precisely one unit. The procedure to be followed is an extension of the procedure for calculating
the value of P̂Y at any given level of Xk. It therefore requires further manipu-
lation of the log odds ratio, introduced in (13.11).
Recall that the log odds ratio is equal to the value of the logit function
predicted by the regression model at any specified value of Xk. Our task
now is to see how the log odds ratio responds to a change in Xk, and how
we should translate this change into the impact on the ultimate depen-
dent variable, the probability of P(Yi�1). One way to work this out
would be to recalculate the value of the logit function replacing Xk with
its new value and then go through the procedure for extracting the value
of P(Yi�1) as in §13.2.2 or the calculation just made for the marginal
effect.
However, there is a slightly quicker route to a solution. The crucial
element in this revised procedure is to recognize that the coefficient bk measures the change in the log odds ratio for a unit change in Xk, while
holding all other variables constant. Thus,
bk � �loge (13.14)
The new value of the log odds ratio after the change is thus equal to the origi-
nal value plus the regression coefficient.
We can illustrate the procedure with reference to our Detroit sample,
and we again take the individual household with an income of $2 thou-
sand.9 The predicted value of P̂Y for this household has already been calcu-
lated (in §13.2.2) at 0.078. The sequence of steps to calculate the new
�P(Yi � 1)P(Yi � 0)�
predicted value of P̂Y for an increase in income of one unit ($1 thousand) is
as follows:
(i) New value of log odds ratio �original value�logit coefficient on
income
� 2.4699�0. 6111
� 1.8588
(ii) New value of odds ratio �exp( 1.8588)
�0.1559
(iii) New value of P(Yi�1) �new odds ratio/(1�new odds ratio)
�0.1559/1.1559
�0.1349
(iv) Change in P(Yi�1) �new value of P(Yi�1) old value
�0.1349 0.0780
�0.0569.j
The impact effect of a unit change in the explanatory variable is thus
0.0569. This might also be represented as a percentage, by multiplying by
100 to produce 5.69 per cent.
The impact effect for the sample as a whole The final version required is the calculation of the impact effect for the
sample as a whole. The impact effect of a unit change in Xk for all house-
holds in the sample is measured by the average of the individual impact
effects for each sample observation. Thus,
(13.15a)
In the unit case, where �Xk�1 for all observations, this simplifies to
�P̂Yi (13.15b)
On the face of it, this requires the tedious process of going through
steps (i) through (iv) above for every household in the sample.
�PY �Xk
� 1
n �i
�PY �Xk
� 1
n �i �P̂Yi �Xk
§.
j Researchers sometimes report odds-ratio coefficients rather than logit coefficients when pre-
senting their results. These are exactly equivalent: the odds-ratio coefficient is simply the expo-
nential of the logit coefficient. But, if the results are reported in this way, it becomes possible to
reduce the number of steps to calculate the change in P̂Y for a unit change in Xk. The new value of
the odds ratio (step (ii)) is equal to the odds-ratio coefficient multiplied by the original odds
ratio. The remaining steps are the same.
Fortunately, however, both theory and the computer come to the
researcher’s rescue. Another way of expressing (13.15b) is as the difference
between PY before and after a unit change in Xk, averaged over all observa-
tions. Theory tells us that the average value of PY before the change is pre-
cisely equal to the sample mean of PY in the original data set (P –
Y). The
difficulty comes from the need to measure the average value of PY after the
unit change in Xk.
This is where the computer comes to the rescue. It can be directed to cal-
culate the logit value of [a�bk � (Xk�1)] for every observation; it can
then convert these log odds ratios into the new value of P(Yi�1) for each
observation by following steps (ii) and (iii). The simple average of these
individual probabilities provides the new value for the sample as a whole
which, when compared to the original sample mean of P(Yi�1) produces
the logit estimate of the impact effect of a unit change in Xk for the sample
as a whole.
When this calculation is undertaken for the fictitious Detroit sample,
the average predicted value of P̂Y after a unit change in income is found to
be 0.249. The sample mean of automobile ownership (P–Y) is 0.175, so the
change is 0.074. This indicates that the rate of automobile ownership in the
sample as a whole increases by 0.074 (or by 7.4 per cent) for an increase in
household income of $1 thousand.
Some researchers are content to report the marginal effect alone (some-
times using the term, ‘marginal probability’), or the impact effect alone.
Others go beyond this calculation to produce estimates of the proportionate
impact on the dependent variable of a proportionate change in the explana-
tory variable. We have already met these proportionate changes in chapter
12 in the form of elasticities, and these represent a very helpful source of
additional information which can be derived from the logit model.
The next section explains the procedure for extracting elasticities from
the logit regression results.
13.2.5 Proportionate effects: logit elasticities In §12.3, an elasticity was defined as the ratio of the proportionate change
in the dependent variable, Y, to a proportionate change in the explanatory
variable, Xk. In this case, the dependent variable is PY, so that the formula
may be written as
k � (13.16a)
where k (lower case Greek eta) is the symbol for the elasticity. The terms
may be reordered to produce an alternative formula
�PY PY
� �Xk Xk
k � (13.16b)
Equation (13.16b) shows that the calculation of the elasticity involves
two separate elements: the ratio of the changes in PY and Xk, and the ratio of
the values of Xk and PY. The first element measures the slope of the relation-
ship between PY and Xk; the second element measures the location at which
this slope is to be calculated.
Since there are two ways to measure the slope of the logit curve at a given
point, it follows that there are two ways to calculate the elasticity.k For the
point elasticity, the first element is the marginal effect,
calculated in the previous section.
For the unit elasticity, the first element is the impact effect, �PY, calcu- lated for a unit change in the explanatory variable (�Xk � 1).
The second element of (13.16b), Xk/PY, is the same for both the unit and
the point elasticity. For an individual observation, it is the ratio of the value
of the explanatory variable, Xk, and the predicted value, P̂Y, at that point.
For the sample as a whole, we once again take advantage of the finding that
the average predicted value of P̂Y is equal to P –
Y, the sample mean of
P(Yi�1). The average value of each explanatory variable is also equal to its
sample mean, X –
k.
We will begin with the calculation of the point elasticity for the entire
sample (i.e. the average elasticity for all households). The formula is
k � (13.17)
The formula for the first term was previously given in (13.13b) as
bk� . If we substitute this into (13.17), we get the
formula for the point elasticity
k�bk� (13.18) � i
P̂Yi �(1 P̂Yi)
n �
X –
k
P –
Y
� i
P̂Yi �(1 P̂Yi)
n
�PY �Xk
� X –
k
P –
Y
�PY �Xk
�PY �Xk
� Xk PY
§.
k We have adopted the terms ‘point’ and ‘unit’ to distinguish the two elasticities, but it should be
noted that some authors simply refer to ‘the elasticity’ without specifying exactly which they
have measured.
THE POINT ELASTICITY
This measures the ratio of
a proportionate change in the dependent variable, PY,
to
a proportionate change in the explanatory variable, Xk,
when the change in Xk is infinitesimal.
For the sample as a whole
it is measured at the means by
k�bk� (13.18)
The corresponding calculation for the unit elasticity starts from the
formula
k� (13.19)
However, we saw in (13.15b) that for the sample as a whole, the first
term on the right-hand side was equal to
�P̂Yi
This term can be substituted in (13.19) to give the formula for the unit elas-
ticity for the sample as a whole
k� �P̂Yi � (13.20)
We may return once again to our Detroit example to illustrate the appli-
cation of these two formulae. In §13.2.4 we measured the marginal and
impact effects of a change in income for the sample as a whole at 0.0942 and
0.074, respectively. These may be converted to point and unit elasticities by
multiplying by the ratio of the sample means of income (X –
k�$2.95 thou-
sand) and car ownership (P–Y�0.175). The point elasticity is thus measured
at 1.5885; the unit elasticity at 1.2474.
X –
k
P –
Y
1
n �i
1
n �i
�PY �Xk
� X –
k
P –
Y
� i
P̂Yi �(1 P̂Yi)
n �
X –
k
P –
Y
THE UNIT ELASTICITY
This measures the ratio of
a proportionate change in the dependent variable, PY,
to
a proportionate change in the explanatory variable, Xk,
when Xk changes by one unit.
For the sample as a whole
it is measured at the means by
k� �P̂Yi � (13.20)
However, the point elasticity may mislead if the change in the explanatory
variable is not small. This would always be the case, for example, if we were
estimating the effect of a change in a dummy explanatory variable from 0
to 1. Under these circumstances, it is clearly better to calculate the unit elas-
ticity, even if its derivation is more complicated.
At this stage it may be helpful to summarize the rather complex set of
procedures for interpreting logit coefficients. We do so in two ways. First,
with the aid of figure 13.5, which provides a simple flow diagram incorpo-
rating all the relevant formulae for measuring the effect of the absolute and
proportionate changes for the sample as a whole.
Secondly, to show how these formulae and procedures work out in prac-
tice, we have also given in table 13.1 some stylized results from the simple
bivariate model of car ownership in Detroit.
Column (1) in table 13.1 indicates the level of income at thousand-
dollar intervals of the income distribution, from $1 thousand to $10 thou-
sand. The remaining entries follow logically from this information by the
formulae adopted above. The final row in the table gives the results for the
sample as a whole; it is a weighted average of the figures reported in the
income specific rows.l This row summarizes the average response of auto-
mobile ownership to changing income across all observations.
The non-linearity of the logit function is clearly revealed in the table.
The measurement of the impact of both the absolute and proportionate
X –
k
P –
Y
1
n �i
§.
l It is a weighted, rather than a simple, average because the income is not distributed evenly across
all income levels in the original data set.
Figure 13.5 Flow chart for deriving logit results for the sample as a whole
� �PY �Xk
� bk �� i
P̂Yi �(1 P̂Yi)
n
�bk�� i
P̂Yi �(1 P̂Yi)
n �
X –
k
P –
Y
changes clearly varies within the sample; the point elasticity, for example, is
0.58 at an income level of $1 thousand, rising to almost 2.0 at $6 thousand,
and declining steadily to 0.50 at $10 thousand.
For this reason, although our calculations thus far have focused primarily
on the average effect of changing income for the sample as a whole, it is clear
that such a summary calculation hides a great deal of information about the
behaviour of automobile ownership in our data set. The averages are clearly
useful, but it will often be desirable for a researcher to go behind them to
investigate changes at specific points of the logistic curve. For example, she
may be interested in the responsiveness of automobile ownership to changes
§.
Table 13.1 Deriving the findings from a logit regression (hypothetical data)
(1) (2) (3) (4) (5) (6) (7) (8)
Income Logit Odds Predicted Marginal Impact Point Unit
($000) value ratio value of Y effect effect elasticity elasticity
1 3.081 0.046 0.044 0.026 0.034 0.584 0.777
2 2.470 0.085 0.078 0.044 0.057 1.127 1.458
3 1.859 0.156 0.135 0.071 0.088 1.586 1.964
4 1.248 0.287 0.223 0.106 0.123 1.899 2.204
5 0.637 0.529 0.346 0.138 0.148 1.998 2.133
6 0.026 0.975 0.494 0.153 0.149 1.857 1.808
7 0.586 1.796 0.642 0.140 0.126 1.530 1.369
8 1.197 3.309 0.768 0.109 0.091 1.135 0.950
9 1.808 6.097 0.859 0.074 0.059 0.775 0.620
10 2.419 11.233 0.918 0.046 0.036 0.500 0.388
Average for the sample as a whole
2.95 1.889 0.212 0.175 0.094 0.074 1.588 1.247
Notes:
(2) is the predicted value of the logit function from the regression, a�bX, where X is
income, a�–3.6921, and b�0.6111.
(3) is the exponent of (2).
(4) is calculated using the formula OR/(1�OR), where OR is the value of the odds ratio in
(3).
(5) is (4)�[1 – (4)]�the regression coefficient, b [�0.6111].
(6) is the difference between successive values of (4) (the value in row 10 is the difference
between 0.918 and the value of (4) if X�11).
(7) is (5)�(1)�(4).
(8) is (6)�(1)�(4).
All columns are subject to rounding error.
in income for the median household, or she may wish to compare the beha-
viour of ownership for lower- and upper-income families. The non-linearity
of the logit function ensures that these valuations will diverge, as table 13.1
makes clear.
13.2.6 Non-linearity and the measurement of change Because of the non-linearity of the logit curve, it is worth discussing briefly
the mechanics of evaluating changes in Xk on the predicted value of P(Yi�1)
at particular points of the curve. It happens that the procedures deviate from
those laid down in the previous sections and summarized in table 13.1 only
in one dimension – the evaluation of P̂Y in the various formulae.
When dealing with the average behaviour across the entire sample, we
can apply the sample mean, P–Y, in the calculation of marginal and impact
effects and of point and unit elasticities. This is, however, no longer per-
missible when dealing with local, rather than average, estimates. If we wish
to calculate the point elasticity for the median income level, we must sub-
stitute the predicted value of P–Y at that particular value of Xk, using the
regression coefficients to generate the logit value, w, and working from
there to calculate P̂Y, according to the steps set out in §13.2.5. The other for-
mulae in §13.2.4 and §13.2.5 must similarly be revised, by substituting
local estimates of P̂Y for P –
Y in each case.
The calculation of the impact effects and the unit elasticities is set out
below for households at various points of the income distribution. We have
chosen the median ($2,420) and the upper and lower quartiles of the
Detroit sample ($1,500 and $3,750, respectively). Readers are encouraged
to follow the calculations through, using the procedures set out in table
13.1, and also to calculate the marginal effects and point elasticities on their
own. In both cases, the underlying proposition that motivated this exercise
is borne out – the impact effects and elasticities show considerable varia-
tion within the sample.
�PY/�X1 � X1/P̂Y � 1 Lower quartile: 0.044 1.50/0.059 1.119
Median income: 0.069 2.42/0.099 1.688
Upper quartile: 0.114 3.75/0.198 2.159
Such variability in predicted outcomes is likely to be true for all logit
models, because of the inherent non-linearity of the logistic curve. The
greater the degree of variation in the outcome being modelled over the
sample, the greater the effect of the non-linearity. If we have an outcome
that does not vary very much within the sample, however, the potential for
divergent results within the sample will be correspondingly small.
Once we have moved from average to local measurements of change, it
is also possible to introduce new levels of change in the explanatory vari-
able, other than the unit effect. One such measure that is sometimes cited is
the change in Y for a change of 1 standard deviation in the explanatory vari-
able. The s.d. of income in the Detroit sample is 2.05, so the measured
change in P̂Y would be 0.2124, or 21.24 per cent. m
13.2.7 Goodness of fit Researchers are often interested in the goodness of fit for a regression equa-
tion. Since the logit model does not use least squares methods, it is not pos-
sible to estimate the R2 directly. Instead, it is necessary to construct other
measures of the goodness of fit, which have become known as pseudo R2
measures.
As explained in panel 13.1, the optimal evaluation of the parameters of
the logit model is determined by maximizing the value of the log- likelihood function. It is this function that is the basis for constructing the pseudo R2 measures. Since there is no ideal value of the log-likelihood
function against which the model estimate can be compared (corre-
sponding to the explanation of 100 per cent of the variance in an OLS
model), the goodness-of-fit measure compares the value of the model
estimate to the value of the log-likelihood function if the model explained
nothing of the variation in PY. This estimate is known as the base likeli- hood.
Let the maximum log likelihood be symbolized as log L1 and the log
value of the base likelihood as log L0.
The first goodness-of-fit measure is defined as
pseudo R2�1 (13.21)
where
logL0 � n1 log � n0 log (13.22)
for a sample of size n with n1 positive outcomes and (n n1)�n0 negative
outcomes. The minimum value that pseudo R2 can take will be zero, but the
maximum is always less than 1. The precise upper bound varies with the
ratio n1/n, and this limits its usefulness as a measure of the goodness of fit
of a logit model.
�1 n1n ��n1n �
1
1 � [2(log L1 log L0)/n]
§.
m The crucial change is in step (i). The new value of the log odds ratio will be 1.8894�(0.6111 �
2.05)� 0.6366. The rest of the sequence proceeds accordingly.
This has led to the introduction of a number of alternative measures of
goodness of fit, one of which is McFadden’s R2. This is calculated as
McFadden’s R2�1 (13.23)
This will always lie between 0 and 1 and can theoretically attain a value
of 1.n However, it is not likely to do so in practice because of the usual prob-
lems with cross-section regressions.o
The comparison of likelihood estimates can help us further understand
the results of the logit model. One use to which the comparison of L1 and L0 has been put is to determine whether the model adds explanatory power
over a simple prediction that the same probability of a positive outcome
applies to all observations, regardless of the values of the related explana-
tory variables. The log-likelihood ratio, calculated as 2(log L1 log L0), is the logit equivalent of the F-test for the joint statistical significance of
coefficients in the OLS model (see §9.3.2). The name of the test is often
abbreviated to –2 log � (lower case Greek lambda). The test statistic of
the log-likelihood ratio is distributed as a chi-square with (k 1) degrees
of freedom, where k is the total number of explanatory variables in the
regression.
THE LOG-LIKELIHOOD RATIO, –2 LOG �
This ratio is a test of the
overall statistical significance of the coefficients
of the logit model.
It is a test of the null hypothesis that
the set of explanatory variables
have no effect on the
probability of a positive outcome.
log L1 log L0
n A perfect fit would require that the model precisely match the distribution of (Yi�0) and (Yi�
1) for all observations, in which case the value of log L1 would be zero. o Another method of measuring goodness of fit in logit models, which begins from a similar
premise, is to calculate the number of correct predictions of positive outcomes. Great care needs
to be taken when interpreting such measures, for technical reasons that are beyond the scope of
this text.
13.3 Estimating the logit model: an illustration
13.3.1 An illustration of the logit model To illustrate the procedures described in §13.2, let us now consider a fuller
version of our fictitious sample of Detroit households in 1919, with two
additional explanatory variables, family size and wealth. The data have
been coded and entered into the computer. We wish to analyse why some
households own cars (AUTO�1) and others do not (AUTO�0). Our
hypothesis is that ownership will be influenced by three factors: household
income (INCOME), family size (FAMSIZE), and, as a measure of accumu-
lated wealth, home ownership (HOME).
INCOME is a continuous variable, measured in thousands of dollars;
HOME is a binary dummy variable, taking the value of 1 if the household
owns its home outright and 0 otherwise; FAMSIZE is a categorical variable,
whose (discrete) values range from 2 to 8. The regression is estimated over
1,000 observations. The average level of car ownership in the sample is
again 17.5 per cent.
The results of the logit model are shown in column (1) of table 13.2. The
coefficients are clearly all statistically significant; the very high value of the
log-likelihood ratio, 2 log ��420.02, confirms that the equation is a
powerful device for explaining the variation of automobile ownership.p
This is also reflected in the high value of the two goodness-of-fit measures,
the pseudo R2 and McFadden’s R2. Each of these values is derived from
the comparison of the values of the base log-likelihood function (log
L0� 463.7264) and the maximum log-likelihood function (log
L1� 253.717).
Statistics aside, the regression makes intuitive sense as well. The
coefficients are all of the appropriate sign: the probability of owning a car
increases with income and home ownership, but is negatively related to
family size (reflecting the greater claim on household income of a large
family).
The impact of the sign on the dependent variable is similarly reflected in
column (2) of table 13.2, which translates the logit coefficients into odds- ratio coefficients. The odds-ratio coefficient is equal to the exponential of the logit coefficient (for example, for INCOME, exp(0.6476)�1.9109). As
noted in footnote j on p. 405, the odds-ratio coefficients can be used to cal-
culate the revised odds ratio resulting from a unit change in the explana-
tory variable.
§. :
p The critical value of the chi-squared distribution with 3 degrees of freedom at the 1 per cent level
is 6.635.
If the logit coefficient is negative, the odds-ratio coefficient will be less
than 1, indicating that the odds ratio declines as the explanatory variable
rises. This is the case with FAMSIZE, showing that the probability of a
household owning a car diminishes as the size of the family increases. By
contrast, the odds-ratio coefficients on INCOME and HOME are both
above 1, indicating that the odds ratio moves in the same direction as the
explanatory variable, i.e. that the probability of owning a car rises with
income and wealth. It should be noted that the t-statistics on these two var-
iants of the logit model are identical, as are the other statistical indicators,
such as –2 log �.
The sign is an important part of any model, but we also wish to know
how much of a difference the explanatory variables make to the outcome of
the model. How big a change in income is necessary to increase car owner-
Table 13.2 Detroit automobile ownership, 1919: the full logit model (hypothetical data)
Dependent variable: AUTO
(1) (2) (3) (4) (5) (6)
Logit Odds-ratio Marginal Impact Point Unit
model coefficients* effect** effect** Elasticity** Elasticity**
INCOME 0.6476 (11.9109 0.0935 0.0542 1.576 0.914
(11.353) (11.353)
FAMSIZE 0.9601 (10.3828 0.1386 0.0610 4.619 2.034
(8.011) 1(8.011)
HOME 2.2405 (19.3980 0.3235 0.2358 0.342 0.223***
(9.092) 1(9.092)
CONSTANT 0.7285
(1.137)
Log L0� 463.726 pseudo R 2�0.2958
Log L1� 253.717 McFadden’s R 2�0.4529
–2 log�� 420.02
Notes:
*** No constant term is reported for the odds-ratio specification, since the expected value of
the dependent variable is completely independent of the intercept term.
*** These effects and elasticities are reported for the sample averages.
*** The unit elasticity for HOME is evaluated at the sample mean of the dummy variable
(0.185).
Figures in parentheses are t-statistics.
ship by a certain amount? Is this a large or a small amount, relative to our
expectations? To address these questions, table 13.2 displays the relevant
marginal and impact effects, and the point and unit elasticities, for the
sample as a whole. It is worth making a few points about these results.
First, the equation has three different types of explanatory variable, for
each of which one or more of the various measures of the effect of absolute
and proportional changes is most appropriate. For INCOME, which is a
continuous variable, the impact of a change in income is best measured by
the marginal effect and the point elasticity, although application of the unit
variants is also feasible.
For HOME, which is a binary dummy variable, the impact effect is most
appropriate. A comparison between the predicted value of automobile
ownership when HOME�0 and when HOME�1 will indicate the sensi-
tivity of P(Yi�1) to changes in home ownership. In this case, it is not pos-
sible to work with sample means, but with predicted values before and after
the change in HOME.q Also, because the change in HOME is from a base of
0, it is not possible to calculate the unit elasticity at that point (since Xk�0,
the formula for the unit elasticity must also be zero).10
FAMSIZE is a discrete, non-binary variable. In this case it is again best to
measure the effect of changes in FAMSIZE on automobile ownership using
the impact effect. In this case, however, since it is not a binary variable, it is
possible to calculate the impact of a unit change in FAMSIZE for all house-
holds from the reported level of FAMSIZE in the data set. Thus, for house-
holds with two family members, we can evaluate the impact of adding a
further person to the household on the expected probability of owning a
car, and can do the same for all other observations. Since we are beginning
from the sample mean of Xk, we can use P –
Y in the formulae. Moreover,
since FAMSIZE is non-zero, it is feasible to calculate the unit elasticity.
Second, it is important to recognize that all such calculations are made
for a change in one explanatory variable, holding all other variables con-
stant. Thus, the point elasticity for an infinitesimal change in income
incorporates only direct effects of income change on car ownership and
pays no attention to indirect effects created by the impact of changing
income on HOME or FAMSIZE.
It is also possible to extend the results of table 13.2 by calculating the
effects and elasticities for households at particular points on the logit func-
tion. This would be especially relevant for understanding the dynamics of
automobile ownership at different income levels, but it is also possible to
§. :
q This is, of course, because the change in HOME is not being measured at its sample mean, which
is neither 0 nor 1.
evaluate the impact of changing family size. The procedure for calculating
these local effects is the same as outlined in §13.2.6.11
13.3.2 Estimating the logit model using grouped data Although the logit model is usually run using information on individual
units, it can also be run over aggregated data, in which the individual
observations are combined into groups. If specified properly, the group
model will produce broadly the same results as the same equation run over
all the individual data. However, if group estimation is used it is essential
that the grouping of individual observations is appropriately structured.
The rules for proper aggregation are: for continuous variables, any
grouping is appropriate, so long as it is consistent; for discrete variables,
there must be as many groups as there are values (thus, two for HOME and
seven for FAMSIZE in the Detroit data set). Were we to designate nine
groups for INCOME, the total number of groups (or cells) would be
(2�7�9)�126.
The dependent variable in group estimation is the ratio of positive to
total outcomes in each cell. It is the odds ratio for the group of observations
that share this particular combination of attributes (home ownership,
family size, and income, in the case of the Detroit survey). If all the house-
holds in that cell own a car, its value will be 1; if none do, it will be 0; if
some proportion does, it will fall between 0 and 1. The group logit regres-
sion is thus an example of estimation using a dependent variable that is
continuous within the bounds of 0 and 1. The logic of the approach can
thus be generalized to other similar data, such as the proportion of an indi-
vidual’s decision or experiences over a given period that falls into one of
two categories.
As an illustration of the group logit approach, we have run the basic
automobile regression across nine income classes, subdivided by home
ownership and family size. There were 126 possible data cells altogether, of
which 36 were empty (no observations met this combination of character-
istics). The grouped logit model was run over the 90 inhabited cells, using
the weighted least squares methodology included in the STATA statistical
program. The results are as follows
AUTO�0.7031�0.6496 INCOME 0.9570 FAMSIZE�2.2377 HOME
(1.099) (11.453) (7.994) (9.078)
log L1� 253.725; –2 log ��420.00; McFadden’s R 2�0.4529.
It will be seen that there is very little difference from the results of table
13.2, run over the original 1,000 observations, indicating the power of the
grouped methodology when properly applied.12
13.4 Estimating the probit model
13.4.1 The probit model We introduced this chapter by discussing the logit and probit models
together. There is very little to choose between them in terms of model
specification and parameter selection. The probit, being based on the
normal distribution, has a slightly more elongated S-shape, but for most
purposes the difference is too small to be a basis for choosing one model
over the other. The models are very close substitutes. For this reason,
neither dominates in empirical work, and historians are just as likely to
report probit results as logit results.
When an historian uses the probit model, most of the reported results
will be identical or nearly identical to the results of the logit specification.
This is true for the value of the likelihood function at its maximum, as well
as for any tests of goodness of fit or model specification, such as the log-like-
lihood ratio or the various pseudo R2 measures. Similarly, the t-statistics
will be the same in the two models. However, one set of results will look very
different. These are the coefficient estimates themselves.
The probit coefficients are systematically smaller than those produced by the logit model. The divergence is due to the different shape of the
normal and logistic distributions and is invariant to the data set being ana-
lysed. The ratio of the coefficients on the probit and logit models is equal to
the ratio of the standard deviation of their underlying distributions.
The probit, being based on the standard normal distribution, has zero
mean and unit variance, and thus its standard deviation is also 1. The logis-
tic curve also has zero mean, but its variance is equal to �2/3, and so its
standard deviation is equal to , which is approximately 1.8138. The
s.d. ratio of probit : logit is thus 1:1.8138, which is approximately equal to
0.5513. The coefficients on the probit model are, therefore, on average 0.55
times the value of those on the logit model.13
As an illustration of these propositions, we have re-estimated our esti-
mating equation for automobile ownership in our fictitious Detroit survey
(see §13.3.1) using a probit regression. The probit results are as follows
AUTO�0.1942�0.3575 INCOME 0.4957 FAMSIZE�1.2589 HOME
(0.599) (12.428) (8.045) (9.274)
log L1� 253.708; –2 log ��420.04; McFadden’s R 2�0.4529.
The value of the log-likelihood function, the log-likelihood ratio, and
the McFadden R2 measure are very similar to the results reported in table
13.2. The ratio of the probit to the logit coefficients ranges from 0.52 to
0.56; the t-statistics are identical in the two formulations.14
��2/3
§.
13.4.2 Interpreting the results from a probit model The most difficult aspect of working with probit models is interpreting the
meaning of the coefficients. The source of the difficulty is that the probit is
based on the cumulative standardized normal distribution, the interpreta-
tion of which is far from straightforward. Moreover, as with logits, the
probit model does not directly predict the expected value of the dependent
variable. The logit model generates an estimate of the log of the odds ratio;
the probit model produces another, and more complex, variable, usually
known as the probit index. It is indicated by the symbol ! (upper case
Greek phi).
The probit index is derived from the constant and the coefficients of the probit model, evaluated at the mean of all the explanatory variables. For the
model of automobile ownership in §13.4.1, the mean for INCOME is $2.95
thousand, for FAMSIZE it is 5.831 persons, and for HOME it is 0.185. So
the value of the probit index is
0.1942�(0.3575�2.95)�( 0.4957�5.831)�(1.2589�0.185)� 1.4084
How do we transform this result into something more familiar and useful?
As with the logit model, we can calculate the impact of two forms of abso-
lute change in an explanatory variable, and two forms of proportionate
change.
We begin with the calculation of the impact effect of a unit change in Xk,
the explanatory variable, INCOME. The coefficient on INCOME in the
probit regression, given in §13.4.1, is 0.3575. This probit coefficient has a
special interpretation: it indicates that a unit increase in income raises the
probit index by 0.3575 standard deviations. The impact of such a change on
the expected value of automobile ownership will vary according to where
on the standardized normal distribution this change of 0.3575 s.d. takes
place.
To measure this we need to translate the value of the probit index into
the expected value of AUTO, and this is done by means of the published
tables of the area under the standardized normal distribution, an excerpt
from which was presented in table 2.7. The value of the cumulative area to
Z�1.4084 is 0.9205.r In order to calculate AUTO we need to find the value
in the tail beyond Z. This is equal to 1 0.9205�0.0795, which gives us P̂Y,
the average expected level of car ownership at the means of the explanatory
variables.
r D. V. Lindley and W. F. Scott, New Cambridge Statistical Tables, 2nd edn., Cambridge University
Press, 1995, table 4, p. 34 by interpolation between the values for Z of 1.40 (0.9192) and 1.41
(0.9207) to get 0.9205. Because the table is symmetrical, the area in the lower tail beyond
1.4084 is identical to the area in the upper tail beyond �1.4084.
The value of the probit index after a unit change in INCOME is equal to
the original value of the probit index plus the regression coefficient on
INCOME, which equals 1.4084�0.3575� 1.0509. In the same way as
before, this can be translated into an expected value of AUTO using the
table for the standardized normal distribution. This gives an area to Z of
0.8533, so the new value is (1 0.8533)�0.1467. The original value was
0.0795, so the absolute increase in the level of car ownership is 0.1467
0.0795�0.0672. The estimate of the impact effect of a unit change in
INCOME, calculated at the means of all the explanatory variables, is thus
0.0672.
The alternative measure, the marginal effect (�PY/�Xk), is derived by
multiplying the probit coefficient for the relevant explanatory variable (in
this case 0.3575) by a tabulated statistic we have not previously encoun-
tered: the height of the normal density function, valued at the probit
index.15 For a probit index of 1.4084 the figure is 0.1480, so the marginal
effect is equal to 0.3575�0.1480�0.0529.
We turn next to the two elasticities. As with the logit elasticities in
§13.2.5 the calculations involve two elements: the ratio of the proportion-
ate changes in Py and Xk, and the ratios of the values of Py and Xk. For the
sample as a whole, the latter are again taken at their mean values, i.e. at X –
k
and P̂Y.
The unit elasticity can thus be calculated in the usual way (see (13.19)) as
the ratio of the change in the dependent variable (�PY) to the change in the
explanatory variable (�Xk), multiplied by the ratio of the two variables at
their mean values (X –
k/P̂Y). We have already calculated �PY�0.0672 and
P̂Y�0.0795. We know that the mean level of the explanatory variable, X –
k, is
$2.95 thousand, and for a unit change �Xk is 1. So the unit elasticity is
(0.0672/1) � (2.95/0.0795)�2.4936.
The alternative point elasticity is similarly calculated by multiplying the
marginal effect (�PY/�Xk) by the ratio of the two variables at their mean
values. The first term was calculated above as 0.0529 so the point elasticity
is 0.0529�(2.95/0.0795)�1.9633.
The difference in scale between the point and unit elasticity measure-
ments is the probit equivalent of the difference between the slope of the
marginal and impact effects in the logit function, as shown in figure 13.4.
These probit calculations are indicative of the procedure. However, they
are based on only one point on the normal distribution function. Since this
function is no less non-linear than the logistic curve, we have to be careful
about generalizing from local results to the sample as a whole. As with the
logit estimates of the average effects and elasticities, if we wish to know the
effect of a change in Xk for the overall sample, the correct procedure is to
§.
take the average effect of a given (infinitesimal or unit) change for each
observation. And, as with the logit model, theory and the computer make
this a relatively straightforward exercise. Once again, the average value of
P̂Y for the observations as a whole is P –
Y. And once again, the computer will
calculate the effect of a change in the explanatory variable on the probit
index for each observation, from which the impact on P(Yi�1) for the
sample as a whole can be calculated.
13.5 Multi-response models
At the start of the chapter, we stated that categorical dependent variables
need not be binary. Survey questions may elicit a more complicated array
of responses than a simple ‘yes’ or ‘no’, indicating the strength of support
for a proposition (e.g. perhaps, probably, rarely). Consumers may face a
greater selection of choices than simply ‘buy’ or ‘not buy’, such as deciding
which brand of automobile to favour. Voters may have to choose between
more than two parties at the ballot box (e.g., Liberal, Conservative, Labour
in the British election of 1910; Democrat, Republican, Progressive in the
US election of 1912). The class of regression techniques that has been
developed to analyse cases such as these, where there are more than two
possible outcomes, is known as multi-response models.s
Multi-response models may be divided into two types, depending on
whether a ranking of the dependent variable is ordered or random. Ordered response models are applied to situations in which the categories reflect a natural ordering of alternatives. A survey of household spending habits
may generate responses such as ‘never’, ‘sometimes’, ‘often’, ‘always’. Infant
mortality rates at the county level may be classified as ‘low’, ‘average’, ‘high’,
relative to the national norm. Workers may choose between working ‘full-
time’, ‘part-time’, or ‘not at all’. In each case, the value of the dependent var-
iable reflects a ranking of alternatives; it is not random, as it would be if the
categories were (say) ‘blue’, ‘red’, ‘green’, and ‘yellow’.
Although ordered variables are ranked, they cannot be analysed by OLS
methods, because the numbers assigned to the different ranks are essen-
tially arbitrary. Thus, the distance between ‘never’ and ‘sometimes’ is not
necessarily the same as that between ‘often’ and ‘always’. Instead, the
ordered logit or ordered probit models are used, depending on whether the underlying distribution is assumed to be logistic or standard normal.16
s In what follows, we leave the technical details to econometrics texts and the method of solution
to the computer package.
The second type of response model applies where the assigning of values
to states is random rather than ordered. In such cases the appropriate
multi-response model is either the multinomial logit or the multinomial probit model. A model investigating emigration from Ireland might analyse the pattern of migrant destinations according to a set of individual
and locational characteristics. A dichotomous model of Irish emigration
that distinguished only between those who did not emigrate and those who
did, could be extended by subdividing the latter by assigning 1 to those who
went to the United Kingdom, and 2, 3, and 4 to emigrants to the United
States, Canada, and the Antipodes, respectively. Similarly, a study of auto-
mobile ownership in Detroit in 1919 might try to model brand selection,
by assigning 1 to Ford, 2 to Chevrolet, 3 to Buick, etc. In both cases, the
order by which the values are assigned is random, so clearly the ordered
probit model would be inappropriate.17
In the case of multinomial models, there is a significant difference
between the logit and probit alternatives, in contrast to their use in binary
choice models. The logit model, although less computationally awkward,
has one significant drawback. This is known as the independence of irrele- vant alternatives. It arises because of the assumption in the logit model that all options are equally likely.
Let us assume that the model of Irish emigration is recast by separating
the Antipodes into Australia and New Zealand. The logical expectation
would be that this change would have no impact on the measured probabil-
ities of emigrating to the other destinations. Unfortunately, in the multino-
mial logit model, this is not the case; all the probabilities will decline by the
same proportion. Thus, unless all categories of the dependent variable are
equally likely, the multinomial logit may be inappropriate.
The alternative is to use multinomial probit, which does not assume
independence of the error term across alternative states. Its disadvantage is
its high computational complexity and expense, limiting analysis to fewer
than five categories.
13.6 Censored dependent variables
13.6.1 A censored continuous dependent variable Finally, we discuss the situation in which the density of the dependent vari-
able, while continuous, is censored or incomplete. Censoring takes place either from above (a variable cannot take on a value more than, say, 10) or
from below (it cannot take a value below, say, 5). The most common form
of censoring is when the minimum value of a variable is 0.
§.
The effect of censoring on OLS estimation is shown in figure 13.6. In
13.6 (a), the dependent variable is censored from above. The maximum
value that the worker can choose to work is 50 hours. They may wish to
work longer hours, but are constrained by union, workplace, or govern-
ment rules. In the scatter diagram the hours desired by the worker are
represented by the hollow circles; whereas, for hours desired in excess of
50, the actual hours worked are given by the filled circles. Because the
OLS regression line omits the censored values in excess of 50 hours it
clearly provides a biased estimate of the impact of wages on hours of
work demanded. The true regression line would omit these ‘imposed’
values but would include the values in excess of 50 hours desired by the
workers.
Similarly, the result of censoring from below is shown in figure 13.6 (b).
The scatter diagram shows the results of a survey of consumer spending on
life insurance at different levels of household income. It is evident that
expenditure on insurance is positively related to income for those who
could afford to purchase a policy. But many households in the survey
bought no life insurance at all. They cannot buy negative amounts of life
insurance, so their expenditures are censored from below at zero. Because
the OLS regression line includes the censored values it is once again clearly
biased. The true line for the regression of expenditure on life insurance on
income would omit these zero values.
The extent of the bias in this case will depend on the frequency of the
uninsured and their distribution by income (and similarly for the censored
hours of work). If the proportion of censored observations is small, the
bias will be minor and OLS regression techniques may still be used.
Likewise, if the relationship between income and being uninsured is sto-
chastic, the bias from using OLS methods will be small.
It is tempting to consider running an OLS regression on the uncensored
observations alone. But temptation should be resisted. The coefficient esti-
mates will be biased, as inspection of figure 13.6 makes clear.
13.6.2 The tobit model The issue of censoring was first analysed systematically by James Tobin in
his study of the purchase of automobiles, refrigerators, and other consu-
mer durables by US households in the 1950s.t Tobin observed that many
households bought no durables during the survey year, indicating that a
linear regression model would be inappropriate.
t James Tobin, ‘Estimation of relationships for limited dependent variables’, Econometrica, 1958,
26, pp. 24–36.
§.
60
50
40
30
20
10
0
–10 0 100 200 300 400 500
Income
True line
Biased OLS line
E xp
en di
tu re
o n
li fe
in su
ra nc
e
Figure 13.6 OLS model with censored observations
80
70
60
50
40
30
20
10 0 2 4 6 8 10 12 14
Hourly wage
True line
Biased OLS line
H ou
rs o
f w
or k
(a) Censored from above
(b) Censored from below
The model that Tobin developed to deal with the problem of censored
dependent variables has become known as the tobit. Very simply the tobit model sets up the regression problem in the following form:
Yi�w�e (13.24)
where w again stands for the constant and all the explanatory variables
with their coefficients in the regression equation, and
Yi�Yi if expenditure is positive
and
Yi�0 if expenditure is zero
The tobit approach thus distinguishes the decision to buy from the deci-
sion of how much to buy. Note that the structure of (13.24) is similar to
that of (13.1), simply substituting a continuous variable for the binary
choice in the first restriction. The tobit model may thus be seen as an exten-
sion of the logit-probit method.u
The model is solved simultaneously using maximum-likelihood
methods, by setting up a likelihood function with two elements, one relat-
ing to the zero observations, the other to the positive outcomes. Each part
of the likelihood function is based on the standard normal distribution.
The tobit coefficients have a dual interpretation. They measure the impact of a change in the explanatory variables both on the probability of
making a purchase and on the size of the purchase made. By solving as a
single system, the tobit does not allow for differential effects of the set of
explanatory variables on the two components of the household decision.
Rather it asserts that the variables that determine whether to buy operate in
the same way to determine how much to buy. So, in the case of our Detroit
households, the sensitivity of expenditure on automobiles is the same
throughout the income distribution; but those families below a certain
income level, although desirous of a car, cannot buy one because it is not
available at their offer price.
Since Tobin’s pioneering work, deeper understanding of the underlying
theory behind censored regressions has unearthed some problems with the
tobit solution. In particular, it produces biased coefficient estimates if the
explanatory variables are not normally distributed. This is a serious
u Tobit models may be applied to censoring from above as well as below. They may also be adapted
to cases in which the dependent variable is constrained from above and below (e.g. when hours
of work cannot fall below 20 or rise above 50).
problem if income, a notoriously non-normally distributed variable, is a
primary explanator, as it often is in models of consumption patterns.v
Moreover, it is now understood that even under ideal conditions, the
coefficient estimates produced by the tobit model do not measure the
expected change in the dependent variable for a given change in an explan-
atory variable. Rather, the coefficient on income is a mixture of two separ-
ate (and separable) elements on automobile ownership: the raising of some
households above the income level at which purchase of a car is possible,
and the desire to spend more money on a car for those who could already
afford one.
13.6.3 Two-part censored models An alternative to the tobit specification, which uses only one regression to
deal with censoring, is to subdivide the problem into two parts. Thus, in
the case of consumer durable expenditure, the problem is separated into
(a) the decision to buy, and (b) the decision on how much to buy for those
who choose to. Similarly, if we were investigating workers’ participation in
strikes, we could separate out the decision to join a strike from the duration
of the strike.
The first stage of the two-part model is a model of binary choice, in
which the dependent variable is dichotomous, taking the value of 0 or 1. It
can be estimated by the logit or probit model (most empirical work uses
the latter). The second stage uses interval level quantitative information
(how much was spent, the length of the period on strike) only for those
observations that took the value of 1 in the first part. For this stage a straight-
forward OLS regression is run on those observations.
Thus, the coefficients on, say, the decision to purchase an item are
allowed to differ from the coefficients on how much to buy. This is espe-
cially useful if it is thought that the mix of explanatory variables that
influence the decision to buy are systematically different from those that
determine how much to spend. It is also an advantage over the standard
tobit specification, that neither the probit nor the OLS require that all
explanatory variables be normally distributed.
To indicate the contrasts between the tobit and the two-part specifi-
cation, we have extended our fictitious Detroit survey to incorporate infor-
mation on the value of cars owned. The results of the two regressions are
§.
v If income is distributed log-normally, it is possible to rescue the tobit specification by using the
logarithm of income instead, but other variables in the regression model may also be non-
normal.
shown in table 13.3. The most striking difference between the single-equa-
tion and the two-stage model is in the interpretation of the two secondary
explanatory variables, HOME and FAMSIZE. Whereas the tobit regression
suggests that the value of the automobile dropped as family size increased
and rose as wealth increased, the two-stage model permits a more sophisti-
cated interpretation.
A comparison of the probit and the second-stage OLS equations in
columns (2) and (3) of the table indicates that HOME was primarily a
determinant of whether a family purchased a car or not, but was not a sta-
tistically significant factor in explaining the value of the car owned. A
bigger family size reduced the probability of owning a car for any given
level of income or wealth; but for those with incomes above the threshold, a
larger family translated into a more expensive (perhaps larger?) car. It
seems clear that the two-part specification permits a richer interpretation
of the data than the tobit.
Table 13.3 Detroit automobile ownership, 1919: censored models (hypothetical data)
(1) (2) (3)
Two-stage model
Tobit model Probit equation OLS equation
Dependent variable: CARCOST AUTO CARCOST
(all) (if AUTO�1)
INCOME 235.6741 0.3575 116.3986
(14.857) (12.428) (10.663)
FAMSIZE 230.4858 0.4957 71.3988
(6.673) (8.045) (2.660)
HOME 650.3871 1.2589 60.3503
(8.425) (9.274) (1.116)
CONSTANT 326.3432 0.1942 344.8536
(1.672) (0.559) (–2.251)
pseudo R2 2 log � R2
0.1272 420.02 0.4165
Note:
Figures in parentheses are t-statistics.
Notes 1 The following are a few recent examples of studies that use logit or probit procedure
for the analysis of binary dependent variables. The effects of early industrialization
on the employment of children, in Sara Horrell and Jane Humphries, ‘“The exploita-
tion of little children”: child labor and the family economy in the industrial revolu-
tion’, Explorations in Economic History, 32, 1995, pp. 485–516. The retirement and
related decisions made by a sample of Union army veterans in 1900 and 1910, in
Dora L. Costa, The Evolution of Retirement, University of Chicago Press, 1998. The
probability of dying from or contracting disease for a sample of Civil War recruits
during their military service, in Chulhee Lee, ‘Socio-economic background, disease
and mortality among Union army recruits: implications for economic and demo-
graphic history’, Explorations in Economic History, 34, 1997, pp. 27–55. The prob-
ability of voting for or against tariff repeal in the British Parliament in the 1840s, in
Cheryl Schonhardt-Bailey, ‘Linking constituency interests to legislative voting beha-
viour: the role of district economic and electoral composition in the repeal of the
Corn Laws’, Parliamentary History, 13, 1994, pp. 86–118. 2 This can be easily shown by rewriting the OLS regression, Yi�a�bXi�ei in the
form ei�Yi a bXi. When Yi�1, the error term is equal to (1 a bXi); when
Yi�0, it is equal to (– a bXi). The error term will be larger for households for whom
Yi�1 than for those for whom Yi�0. 3 The logistic curve is the cdf of the hyperbolic-secant-square (sech2) distribution. 4 We can derive (13.5b) by re-writing (13.5a)
P(Yi � 0) � 1
as follows
P(Yi � 0) �
to produce (13.5b).
Similarly, we may derive (13.5c) as follows, recalling from §1.6.1 that the multipli-
cation of powers is achieved by addition of the exponents, so that ew � e w � ew w,
then
P(Yi � 0) �
�
�
�
which produces (13.5c).
e w
1 � e w
ewe w
ewe w � ew �
ewe w
ew(1 � e w)
1 � ew ew
1 � ew �
ewe w � ew ew
1 � ew
1 � ew
1 � ew
ew
1 � ew
1 � ew
1 � ew
ew
1 � ew �
1 � ew ew
1 � ew �
1
1 � ew
ew
1 � ew
5 Dividing (13.4) by (13.5b) is equivalent to multiplying (13.4) by the reciprocal
(inverse) of (13.5b), i.e.
�ew
to produce (13.10). 6 To see why the logarithmic transformation of ew produces w, we apply the rules for
the log of a number raised to a power (see §1.6.2) to the expression loge(e w) to
produce w � logee. Since log e to the base e�1, this gives w. 7 However, the individual predicted values of car ownership (P̂Yi) from the regression
almost never take on the exact value of the original data, although they are con-
strained to the range (0,1). In the Detroit data set, the fitted values range from 0.04 to
0.95. The logit function, although clearly superior to a linear specification, nonethe-
less produces only an approximation to the actual distribution of data points. 8 Equation (13.12) can be obtained by calculating the derivative of the logit function
with respect to Xk. For an individual household, this gives
� �bk (13.12a)
From (13.4) the first term is equal to P(Yi�1), and from (13.5b) the second term is
equal to P(Yi�0). Hence ∂PYi/∂Xk�P(Yi�1) � P(Yi�0) � bk. 9 It was suggested previously that the impact effect is best calculated for discrete vari-
ables, such as a dummy explanatory variable. In our simple bivariate model, we have
no dummies; we therefore calculate the impact effect for a unit change in income.
The procedure set out in the text may be generalized to the case of a discrete variable.
The one significant difference in the procedure for binary dummy variables is that
the absolute effect is measured by comparing P̂Y for the cases when the dummy
equals one and zero. When calculating the impact effect of a change in a dummy var-
iable for the sample as a whole, it is necessary to set the dummy variable to zero for all
observations to provide the base-line for comparison, and then to compare this
result to the case when the dummy variable is set to one for all observations. 10 The unit elasticity may either be measured by comparing the impact effect of moving
from 0 to 1 relative to the average value of the dummy in the sample as a whole, or by
using the average value of the dummy as the starting point for estimating the impact
effect, e.g. moving from 0.185 to 1.185 and using 1/0.185 as the second term in the
elasticity formula. 11 One cautionary note. It is important to ensure that the selection of the values of Xk
are internally consistent. Thus, if the researcher is working through the effects of
changing income at the lower quartile of INCOME in the sample, it is essential that
she use the average values of FAMSIZE and HOME at this income level, rather than
the lower quartiles of each variable, which would be inconsistent and, therefore,
inappropriate. 12 For an interesting illustration of the logit model applied to group data see Konrad H.
Jarausch and Gerhard Arminger, ‘The German teaching profession and Nazi party
ew
(1 � ew)2 � bk �
ew
(1 � ew) �
1
(1 � ew)
�PYi �Xk
ew
1 � ew �
1 � ew
1
membership: a demographic logit model’, Journal of Interdisciplinary History, 20
(Autumn 1989), pp. 197–225. The authors analyse the proportion of teachers who
joined the Nazi party by various demographic and social characteristics, including
religion, sex, and birth cohort. Since all of their variables are categorical, there is
absolutely no difference between running such a regression on the various cells of the
cross-tabulations, weighted by the number of observations in each, and running a
regression that counts each individual separately. 13 In practice, because the distributions do not share precisely the same shape and
because observations are not drawn equally from all parts of the distribution, the
actual ratio of the two coefficient estimates will vary around 0.55, normally falling in
the range, 0.475–0.625. The larger the sample, the more likely it is that all parts of the
distribution will be sampled, and the closer the ratio of coefficients will be to 0.55.
The smaller the sample variance, the greater is the distortion. The other factor that
influences the ratio of the coefficients is the proportion of positive outcomes in the
sample. If P(Yi�1) is either very large or very small, the distortion is likely to be
larger. 14 We note in passing that the probit approach run over properly structured grouped
data produces very similar results to those obtained using the full data set. When we
ran a grouped probit regression using the same 126 groups as in §13.3.2, the follow-
ing results were obtained:
AUTO�0.1596�0.3625 INCOME 0.4919 FAMSIZE�1.2559 HOME
(0.458) (12.400) (7.995) (9.241)
log L1� 253.615; –2 log��420.22;] McFadden’s R 2�0.4531.
15 The height of the normal density function is calculated from the formula:
where � is the constant pi and z is the probit index. The table is reproduced in G. S.
Maddala, Introduction to Econometrics, 2nd edn., Prentice-Hall, 1992, pp. 610–11. 16 An insightful application of an ordered response model to analyse successive votes in
the US Senate in 1929–1930 on the introduction of the Smoot–Hawley tariff is found
in Colleen M. Callahan, Judith A. McDonald and Anthony Patrick O’Brien, ‘Who
voted for Smoot–Hawley?’ Journal of Economic History, 54, 1994, pp. 683–90. The
variable VOTE has a value of 2 for a representative who voted yes for both the initial
passage of the bill in May 1929 and the final passage in June 1930; 1 if he or she voted
in favour of initial passage but against final passage; and 0 if the representative voted
no on both occasions. 17 One example of a model that uses multinomial logit regressions is Robert A. Margo,
‘The labor force participation of older Americans in 1900: Further Results,’
Explorations in Economic History, 30, 1993, pp. 409–23, in which the procedure was
used to distinguish between not working, being currently employed, and being in
long-term unemployment. Another is Daniel A. Ackerberg and Maristella Botticini,
1
�2� exp � 12 z2�
‘The choice of agrarian contracts in early Renaissance Tuscany: risk sharing, moral
hazard, or capital market imperfections?’, Explorations in Economic History, 37,
2000, pp. 241–57, in which the distinctions were between owner-occupation of land,
fixed-rent contracts, and share contracts.
13.7 Exercises for chapter 13
1. Which model (logit, tobit, or neither) is more appropriate for regression
analysis of the following variables?
(i) Whether a household owns or rents their dwelling
(ii) How much a household pays in rent for their dwelling
(iii) How much debt a household still owes on its (owned) dwelling
(iv) Whether a person voted in the last election
(v) Which party a person voted for in the last election
(vi) What proportion of all eligible voters voted in the last 10 elections
(vii) What proportion of all votes were cast for a particular party in the
last 10 elections
(viii) How many votes were cast for the same party in the last 10 elections.
2. Using the AUTO data set on the web page, run a logit regression of
HOME, with INCOME and FAMSIZE as the regressors. The regression
result (with t-statistics in parentheses) should be
HOME�0.027�0.2170 INCOME 0.4196 FAMSIZE
(0.03) (2.99) ( 2.45)
Report the values of the pseudo R2 and the two log-likelihood values.
You wish to work out the level of HOME when INCOME is $2 thousand
and $5 thousand. Report your answer, showing the requisite steps.
3. Use the regression analysis of HOME ownership to:
(i) Calculate the impact effect of an increase in INCOME on HOME at
$2 thousand and $5 thousand, respectively. Report your answer,
showing the requisite steps. Express this result as an elasticity.
(ii) Calculate the marginal effect of an increase in INCOME on HOME
at $2 thousand and $5 thousand, respectively. Report your answer,
showing the requisite steps. Express this result as an elasticity.
4. You wish to use the Steckel data set to test certain hypotheses about the
causes of migration in antebellum America. The hypotheses include:
(a) Literacy is a positive influence on migration
(b) Migration tends to fall as households age
(c) Larger families are less able to migrate, because of the increased
expense
(d) Immigrants, who have already migrated once, are more likely to
migrate again
(e) Urban households are more likely to migrate than rural
(f) The absence of a wife increases the probability of moving
(g) Migration tends to be more frequent in some regions than others
(h) Unskilled workers tend to move less often than other occupations
(i) Migration is less frequent for households without property
(j) For those with property, the wealthier the family, the less likely they
are to move.
(i) Set up a model to test the hypotheses. (Note : you will need to instruct
the computer to generate new dummy variables in order to test
hypotheses (c), (d), (e), and (i)).
Use the logit procedure to test these hypotheses. Record your results
and write a brief essay interpreting them. How effective is the model at
explaining household migration? Which hypotheses are supported and
which are not? Be sure not only to examine the signs and significance of
the coefficients, but pay attention to the historical significance of the
results. (Hint : think about the elasticities.) Is this a complete model of
household migration?
(ii) Run a probit regression and compare the results to the logit formula-
tion. Analyse any differences.
5. Take the probit results from question 4 and use them to interpret the
value of the probit index at the mean of the variables. Apply the method
introduced in §13.6 to calculate the impact effect of a unit change in the
value of real estate on the probability of migration at the mean. Compare
the result of this exercise to the impact effect of a unit change in PROPER-
TY using the logit coefficients.
6. A researcher wishes to analyse why some Southern parishes built work-
houses while others did not. Her maintained hypothesis is that workhouses
tended to be built in large, rich, densely settled parishes. These parish char-
acteristics are captured by POP, WEALTH, INCOME, and DENSITY.
(i) Set up a logit model to test the hypotheses and report the results.
(ii) Re-organize the data into groups and re-run the logit model. Set up
your groups carefully, using the criteria established in §13.3.2.
Report your results, being sure to identify any differences from the
ungrouped results.
§.
7. Set up a model using the Steckel data set to investigate the relationship
between the value of the household’s real property and the characteristics
of the family. Motivate your selection of explanatory variables.
(i) Evaluate your model using standard (OLS) regression methods.
Report your results.
(ii) Evaluate your model using a tobit regression. Report your results.
(iii) Evaluate your model using the two-part procedure, applying a logit
or probit equation to establish the determinants of holding any real
estate, and an OLS regression to evaluate the determinants of the
level of real estate for propertied households. (Hint : generate a
dummy variable for positive wealth holding.) Report your results.
Interpret the differences between these results. Are there empirical reasons
for preferring one formulation over the others? What do these various for-
mulations tell us about the determinants of wealth holding in antebellum
America?
8. A researcher argues that the provision of child allowances varied from
parish to parish according to the affluence of the parish (WEALTH,
INCOME), its economic structure (GRAIN, COTTIND), and whether or
not there was a local workhouse.
(i) Set up a probit model to test the hypotheses and report the results.
Does the regression support the argument?
The researcher further argues that the number of children at which allow-
ance is paid is influenced by the same variables. In order to test this hypoth-
esis, it is necessary to generate a multinomial variable, which takes the value
of 0 if no child allowance is paid; 1 if allowance begins with 3 children; 2 if it
begins with 4 children; and 3 if it begins with 5 or more children.
(ii) Run an ordered probit regression to analyse this second hypothesis.
What do the results tell us? Are the results consistent with the
simpler model? Does the new version add any information for our
understanding of the provision of child allowances in Southern
parishes?
9. Is the mean household or the median household a more relevant orga-
nizing entity for limited dependent variable analysis? Justify your choice.
10. Instruct the computer to set up a multinomial of occupation in the
Steckel data set. Run a multinomial logit equation of occupation on region,
degree of urbanization, and ethnic origin. What do you find? How would
you interpret the regression results?
Specifying and interpreting models: four case studies
Case studies 1 and 2: unemployment in Britain and emigration from Ireland
We noted in chapter 1 that the most important of our three aims in this text
was to enable our readers to read, understand, and evaluate articles or
books that make use of modern quantitative methods to support their
analyses of historical questions. We selected four specific case studies that
we have referred to and used as illustrations at various points in preceding
chapters. In this and the following chapter we want to look more closely at
each of these studies, and to see in particular how their models were
specified, how any technical problems such as autocorrelation or simultan-
eity were handled, and how the regression coefficients, standard errors
(and t-ratios or prob-values), and other statistical results were interpreted
and related to the historical issues addressed in the articles.
The present chapter covers two studies investigating the causes of
unemployment in inter-war Britain and of the nineteenth-century emigra-
tion from Ireland to the United States and other countries. Chapter 15 is
devoted to the impact of the Old Poor Law on relief payments, earnings,
and employment in England in the 1830s, and to the factors that
influenced children’s decisions to leave the family home in the United
States in the mid-nineteenth century. We assume that students will by now
have carefully read the case studies and thought about the institutional and
historical questions with which they deal. The comments that follow are
thus concerned primarily with their statistical aspects, and we will refer to
the historical background only where this is necessary to give a context for
the statistical issues.
Each of these studies is briefly introduced in appendix A and the vari-
ables used are described and listed in tables A.1–A.5. Each series is given
an abbreviated name written in capital letters, and we shall refer here to
the series by these names (as we have done throughout the text) rather
than by the symbols or alternative references used in the original studies.
In specifying the models in the following sections we shall also continue
our previous practice of referring to the constant as a, and to all other
coefficients as b1, b2, and so on, even if this differs from the notation
adopted by the authors. And we shall continue to use the term ‘log’ rather
than ‘ln’ to indicate a natural logarithm.
14.1 Inter-war benefits and unemployment in Great Britain
We begin with the earliest and simplest of the models in our four articles,
the one used by Benjamin and Kochin in their 1979 paper to study the
effect of the unemployment benefits paid to the millions who were out of
work in inter-war Britain (see §A.1 in appendix A for their basic hypothesis
and table A.4 for details of the data set).a We look first at their procedure
and then, in §14.1.2, at some of the criticisms that their paper evoked.
14.1.1 The Benjamin and Kochin model Benjamin and Kochin argued that the dominant explanation for the high
level of unemployment in inter-war Britain was not – as often suggested by
other writers – the depressed level of overall demand. Their central
hypothesis was that it was the generous payment of benefits to those on the
dole (compared to the amount they would have received in wages if in
work) that was primarily responsible.
Benjamin and Kochin’s basic model is a single equation specifying a
linear relationship between the dependent variable, UNEMP, and two
explanatory variables: BWRATIO (the replacement rate, measured as the
ratio of BENEFITS to WAGES), and DEMAND (measured by the
difference between log output and the trend of log output, and thus equiv-
alent to the ratio of output to its trend)
UNEMP�a�b1BWRATIO�b2 DEMAND�e (14.1)
They estimate this model using ordinary least squares (OLS), and report
the following results (p. 453), with t-statistics shown in parentheses1
UNEMP�5.19�18.3 BWRATIO 90.0 DEMAND (14.2)2
(2.64) (4.46) ( 8.30)
R2�0.84, R–2�0.82, DW�2.18, SEE�1.90
a Unless otherwise noted all page references in §14.1 are to Daniel K. Benjamin and Lewis A.
Kochin, ‘Searching for an explanation of unemployment in interwar Britain’, Journal of Political
Economy, 87, 1979, pp. 441–78.
We can see immediately that the t-ratios are all well in excess of 2 and thus
all the coefficients are statistically significant (see §6.7.3 and (6.15)). The
coefficients are also correctly signed: positive for BWRATIO (a rise in the rel-
ative level of benefits increases UNEMP) and negative for DEMAND (the
higher output is relative to its trend value, the lower the level of UNEMP).
The size of the coefficients appears to give greater predictive importance
to DEMAND than to BWRATIO. However, since coefficient size depends
crucially on the size of the explanatory variable, it makes sense to scale the
two coefficients by calculating standardized beta coefficients (see §8.2.7).
The betas are 0.4533 for BWRATIO and 0.8422 for DEMAND, indicating
that a change of one standard deviation in DEMAND has a much greater
impact on UNEMP than a change of one standard deviation in BWRATIO.
The coefficient of multiple determination, R2, is very high at 0.84, and
because there are only two explanatory variables the adjusted R–2 is only
marginally lower than this. The model thus accounts for 82 per cent of the
variation in inter-war unemployment. However, we should not attach
too much importance to this since a high R2 is not uncommon with time-
series data in which many variables may fluctuate together even if not
causally related. The value of DW, the Durbin–Watson statistic, is close to
2, thus indicating that there is no problem of autocorrelation (see
§11.3.3). SEE is the standard error of the estimate, as defined in (9.10)
and §9.4.
Benjamin and Kochin (pp. 453–5) also consider the possibility that
there might be a problem of mis-specification if the direction of causation
runs in the opposite direction from that assumed by their model. They
discuss two mechanisms that might lead to such reverse causation, in
which case a high level of unemployment would be the cause of a high
benefit–wage ratio rather than an effect. However, they conclude that
neither was operating in inter-war Britain.
The first mechanism might have occurred if high levels of unemploy-
ment caused low levels of wages and thus raised the BWRATIO. They
dismiss this possibility on the grounds that most of the variation in the
inter-war BWRATIO was caused by movements in BENEFITS rather than
in WAGES; and also show that if BENEFITS and WAGES are entered separ-
ately in an alternative specification of the model, the coefficient on the
former remains strongly significant (with a t-ratio of 4.04) whereas the
latter is insignificant (with a t-ratio of –0.82).
The second mechanism might have occurred if the level of benefits had
been influenced by the level of unemployment. Benjamin and Kochin argue
that there is no evidence in the parliamentary debates to suggest that this
was a relevant consideration, and that the most important determinant of
§. -
the scale of benefits was the current and prospective financial condition of
the Fund from which they were paid.3
In addition to the primary model in (14.1), they also experiment with a
variety of other specifications (p. 453, n. 16). These include one in which
the relationship between UNEMP and BWRATIO is modelled as non-
linear, thus allowing for the possibility that the ratio of benefits to wages
might have a very different impact on UNEMP at different levels of
BWRATIO. In another a time trend (a series which simply increases by 1 unit each year) was added to (14.1);b and in others the primary model was
estimated for different time periods, omitting successively 1920, 1920–1,
1920–2 and 1920–3. They find, however, that the results of all these alterna-
tive functional forms are statistically very similar to those given by (14.1).
Benjamin and Kochin (pp. 464–70) use the regression coefficients on
BWRATIO and DEMAND from (14.1) to estimate the effect of the unem-
ployment benefit scheme. Unfortunately, their model involves a complica-
tion which means that the coefficient on BWRATIO cannot be taken in the
usual way to measure its impact on the dependent variable, and the manner
in which they deal with this involves some rather complicated economic
and mathematical reasoning that goes well beyond the scope of this text. We
simply note that their final conclusion is that over the period as a whole ‘the
insurance system raised the average unemployment rate by about five to
eight percentage points’ (p. 468), and thus had a very substantial effect.
14.1.2 Parameter stability and outliers in the Benjamin and Kochin model The publication of this paper unleashed a flurry of criticisms. Many of
these concerned either historical and institutional questions, such as the
administration and actual working of the unemployment insurance
scheme, or aspects of the data, such as the appropriate measure of benefits
relative to wages. There were, however, a number of interesting points con-
cerned with the statistical procedures, and we will look briefly at the most
important of these.
One pair of critics, Ormerod and Worswick, argue that the estimated
coefficient on BWRATIO (and hence the estimate of its effect on UNEMP)
is ‘very sensitive to small changes in the sample period’ and thus ‘casts
serious doubts on the validity of their results’.c They show a scatter diagram
b A time trend can be regarded as a proxy for any unmeasured (or unmeasurable) variables that
might affect the dependent variable and have a similar upward or downward trend. c P. A. Ormerod and G. D. N. Worswick, ‘Unemployment in inter-war Britain’, Journal of Political
Economy, 90, 1982, pp. 400–9, quotation at p. 402.
for UNEMP and BWRATIO in which the observation for 1920 appears to
be a clear outlier, and the remaining years fall into two groups. The obser-
vations for 1921–9 lie on one straight line with a negative slope, and those
for 1930–8 on another line, again with a negative slope, though for these
years a very slight one.d A negative slope is, of course, the opposite of what
Benjamin and Kochin find, since their regression coefficient on BWRATIO
is positive.
In order to demonstrate the sensitivity of the BWRATIO coefficient to
the sample of years for which the equation is estimated, Ormerod and
Worswick re-estimate the equation successively omitting the years from
1920. Their first alternative is thus for 1921–38, the second for 1922–38,
and so on for 10 equations ending with 1930–8.
As one test of sensitivity they take a 95 per cent confidence interval
around the original regression coefficient on BWRATIO of 18.3, giving
them a range from 10.6 to 26.1.e They then show that six of the 10
coefficients on BWRATIO estimated for their 10 alternative sample periods
fall outside this range.f
As a further test Ormerod and Worswick examine the robustness of the
Benjamin and Kochin results when a time series is added to their 10 alter-
native time periods.4 The 95 per cent confidence interval is considerably
larger in this case (because the standard error on the regression coefficient,
BWRATIO, for the period 1920–38 is much larger) and the range when the
time trend is included is from 4.3 to 33.6. None of the coefficients for
periods starting in 1923 or later falls within this range, and only one is sta-
tistically significant.
In their vigorous reply, Benjamin and Kochin reject these tests of the
stability of their coefficients as being inappropriate.g In their place, they
apply two other tests for parameter stability. The first is a dummy variable
test, based on the equation for a piecewise linear regression, introduced in
§11.2.3. This test takes the basic regression and adds a slope dummy vari-
able, equal to the interaction of the explanatory variable whose stability is
being tested (in this case, the benefit–wage ratio) and a categorical variable
§. -
d You have already plotted this scatter diagram in question 3 of chapter 3. e The 95 per cent confidence interval corresponds broadly to 2 standard errors (SE) either side of
the estimate; it can be calculated as shown in §5.6.3, using (5.16). Ormerod and Worswick give
the SE as 3.87, and their confidence interval is thus 18.36�2�(3.87). f Ormerod and Worswick, ‘Unemployment’, table 3, p. 405. g Daniel K. Benjamin and Levis A. Kochin, ‘Unemployment and unemployment benefits in twen-
tieth-century Britain: a reply to our critics’, Journal of Political Economy, 90, 1982, pp. 410–36;
see especially pp. 412–15.
set at 0 below the threshold of change and 1 above. The revised equation is
thus
UNEMP�a�b1BWRATIO�b2DEMAND�
b3DUMMY*BWRATIO�e (14.3)
The test statistic is the t-statistic on b3, evaluated at standard levels of
significance. If the t-statistic falls below the critical level, then the model
does not indicate structural change and the null hypothesis of stability
should not be rejected. Since Benjamin and Kochin are testing for general
parameter stability, rather than the existence of a single structural break,
they ran the dummy variable test over a range of subperiods, beginning
with 1921–38 and changing the beginning period by one year up to 1930–8.
In no case did the t-statistic on b3 pass at the 5 per cent level of significance.
The dummy variable test asks whether a single parameter is stable over
the regression sample. The Chow test asks whether the entire model is stable, i.e. whether all the parameters are stable. The Chow procedure
divides the sample into two (or more) subperiods, running the same
regression over each, and then comparing their results to those from a
regression run over the entire sample. The comparison is based not on
parameter values, but on the overall explanatory power of the equations, as
measured by the unexplained variation, or residual sum of squares (RSS),
first introduced in §4.3.2.h
Let us identify the RSS for the entire sample as RSS1; for the two sub-
samples, we use RSS2 and RSS3. The test statistic for the Chow test is:
F � (14.4)
where n2 and n3 indicate the number of observations in the subsamples and
k is the number of explanatory variables in the original (and by extension,
the subsample) regression.
This calculated statistic may be compared to the critical value of the F-
statistic from any published F-table. The critical value (Fmn) is reported for
two degrees of freedom, one for the denominator and one for the numera-
tor. The df in the Chow test are equal to k and (n1�n2 2k), respectively. If
the calculated F is larger than the critical value at the appropriate level of
significance, then the null hypothesis of model stability is rejected; if F falls
below the critical value, the null hypothesis is not rejected.
Benjamin and Kochin once again ran the Chow test over a range of pos-
sible subsamples in order to test for general model stability. In no case do
(RSS1 RSS2 RSS3)/k (RSS2 � RSS3)/(n2 � n3 2k)
h The unexplained variation is also referred to as the error sum of squares (ESS).
they find the calculated F-statistic to be greater than the critical value at the
5 per cent level of significance, indicating that the null hypothesis of
parameter stability should not be rejected.
The starting point for Ormerod and Worswick’s criticism was that 1920
appeared to be a clear outlier. This suggestion has also been made by other
historians critical of Benjamin and Kochin’s results. To our knowledge, no
one has formally evaluated whether 1920 really is an outlier.5 In §11.3.4, we
discussed the general problem of outliers and influential observations. In
that section, we distinguished ‘good’ outliers from ‘bad’ outliers, suggest-
ing that leverage points, in which the values of explanatory variables are
significantly different from the rest of the sample, may provide valuable
information about the model as a whole. In contrast, rogue observations
marred by large prediction errors should be discarded.
Is 1920 a good or a bad outlier? A scatter-plot of the data on UNEMP
and BWRATIO certainly marks it out as a leverage point, and this is
confirmed by a more formal statistical test, due to Hadi, which determines
whether the value of the observation falls into the tail of the sample distri-
bution as a whole. The values of BWRATIO for 1920 and 1921 are
significantly different from the rest of the sample at the 5 per cent level (the
Hadi test detects no outliers in either UNEMP or DEMAND).
To determine whether any observations in the Benjamin–Kochin data
constituted outliers due to estimation error, we ran two tests. A dummy
variable test, which evaluated the statistical significance of observation
dummies in the estimating equation, identified 1921 and 1930, but not
1920, as outliers at the 5 per cent level. A second test involves running the
regression for as many times as there are data points, dropping each
observation in turn, and predicting the value of the dependent variable
for the missing variable. This is a variant of the method of out-of-sample
prediction. An observation is identified as an outlier if the difference
between the predicted and actual values of the dependent variable is sta-
tistically large.6 The null hypothesis of no difference between the reported
and predicted unemployment rates was rejected for a larger number of
observations, including 1921, but excluding both 1920 and 1930.i On this
basis, 1920 does not appear to have been an estimation outlier, while 1921
clearly was.
Thus it appears that 1920 was a leverage point, but not a rogue observa-
tion. 1921, on the other hand, may have been both. But how did these data
points influence the overall equation? The substance of Ormerod and
Worswick’s critique is that the high and significant value of the coefficient
§. -
i The other outliers by this criteria were 1924–6 and 1936–8.
on BWRATIO was largely determined by the influence of 1920 on the rela-
tionship between the benefit–wage ratio and unemployment. Is this criti-
cism appropriate?
The two standard tests (dfBeta and dfFits, described in §11.3.4) reveal
1920 and 1921 to have been influential observations.j The dfBeta statistics
indicate that the coefficient on BWRATIO was significantly different from
the rest of the sample for both years; the dfFit value confirms that this
difference in coefficients was substantial enough to cause a statistically
significant difference in the predicted dependent variable. Thus, the dfFit
values are consistent with the suspicion that unemployment in 1921 was
generated by a process different from the rest of the sample, while 1920
was not.
What should be done about such outliers? The temptation is to throw
them away, but this would clearly be inappropriate in the case of 1920 – the
reason why that year is influential is not that it is a rogue observation, but
rather that one of its explanatory variables falls into the tail of the distribu-
tion. Dropping 1920 would jettison valuable information about the model
as a whole. The judgement on 1921 is more mixed, however, since it shows
up as an outlier in both categories. There may be a case for excluding 1921 –
curiously, since it was not identified by Benjamin and Kochin’s critics as a
rogue observation.k
The alternative to discarding either or both observations is to apply
robust regression techniques to the data (§11.3.4). Robust regressions use weighting procedures to reduce the power of influential observations.
Purely for illustration, we have run one over the entire Benjamin and
Kochin data set, with the following results
UNEMP�4.87�18.83 BWRATIO 91.35 DEMAND
(2.49) (4.62) ( 8.37)
which is essentially identical to the result in (14.2).l
j As with all such tests, it is imperative that they be run over the entire sample; ‘cherry-picking’
certain observations, such as 1920–1, for special evaluation is inappropriate. k If 1921 is excluded, the OLS regression result becomes:
UNEMP�2.61�23.21 BWRATIO 84.77 DEMAND
(1.38) (5.99) ( 9.08)
R2�0.89, R –2�0.88 SEE�1.56
It is not possible to generate a Durbin–Watson statistic since the data are no longer continuous. l Or rather, to our results as given in n. 2. Note that robust regression techniques do not product
standard goodness-of-fit measures.
Benjamin and Kochin’s response to their critics was not the last word on
the origins of unemployment in the inter-war period, as might be pre-
dicted from the tone of the debate.m More sophisticated econometric mod-
elling of the inter-war labour market has been undertaken, both by
advocates of a benefit story and by those who find it unpersuasive. Much of
the debate has continued to work with annual observations, although
some scholars have attempted to increase the degrees of freedom of such
models by introducing quarterly data.7
However, much the most intriguing developments in inter-war labour
history have emerged from micro-economic analysis of survey data for the
1920s and 1930s. The pioneering work was undertaken by Barry
Eichengreen, who sampled the records of the New Survey of London Life
and Labour that was undertaken between 1928 and 1932. With this data,
Eichengreen was able to test Benjamin and Kochin’s hypothesis about the
influence of rising benefits on labour supply decisions for individual
workers, rather than generalizing from market aggregates.
Eichengreen’s conclusions offered a substantial revision of the benefit
argument. For household heads, who dominated employment in this period,
the ratio of benefits to wages played almost no role in determining whether
they were in work or out of work. But for non-household heads the replace-
ment rate did matter, perhaps causing their unemployment rate to rise to as
much as double what it would have been had replacement rates stayed at 1913
levels. However, since these secondary workers accounted for only a very
small part of the London labour force, the weighted average of the two effects
was small, leading Eichengreen to conclude that, ‘it does not appear that the
generosity of the dole had much effect on the overall unemployment rate’.n
14.2 Emigration from Ireland
Our next and more elaborate case study is the 1993 article by Hatton and
Williamson examining the massive emigration from Ireland to the United
States and other destinations after the famine (see §A.2 in appendix A for a
brief introduction and table A.3 for details of the data set).o The overall rate
§.
m There are clearly several respects in which their modelling procedure would not conform with
what is today regarded as best practice in the sense of §12.6, but the article reflects standard prac-
tice of the time it was written. n Barry Eichengreen, ‘Unemployment in inter-war Britain: dole or doldrums?’, in N.F.R. Crafts,
N. H. Dimsdale and S. Engerman (eds.), Quantitative Economic History, Oxford University
Press, 1991, pp.1–27, quotation at p. 22. o All page references in this section are to Timothy J. Hatton and Jeffrey G. Williamson, ‘After the
famine: emigration from Ireland, 1850–1913’, Journal of Economic History, 53, 1993, pp. 575–600.
of emigration fluctuated sharply from year to year, and also displayed a
very clear long-term trend. After the immediate post-famine flood had
abated, the trend rate remained very high at around 15 per 1,000 of the
total Irish population in the 1860s and 1870s, and then declined to about 8
per 1,000 in the years immediately before the First World War.
Hatton and Williamson explore the causes of that emigration,
attempting to explain both the decline over time in the long-term trend,
and also the annual fluctuations. Following the tradition of earlier studies
of migration they look for the principal causes in the economic condi-
tions in Ireland, notably wages and employment, and assume that poten-
tial migrants compare these with the prospects offered to them in the
United States and other possible destinations. These forces are supple-
mented by other economic and social conditions such as age, urbaniza-
tion and access to land, poverty, religion, and literacy. The basis of the
modelling strategy adopted by Hatton and Williamson is to specify com-
prehensive models covering as many as possible of these factors, and then
to use the results of the regression analysis to establish their relative
importance.
For some of the relevant variables this can best be done in a time-series
model; for other variables more information is available in selected years in
which the data can be analysed by a model that combines cross-sectional
information for Irish counties over time. We look at each of these in turn.
14.2.1 The time-series model of Irish emigration The first model, based on annual time-series data for the years 1877–1913,
is set out below, using the symbol � (capital Greek delta) to indicate
‘change in’, and the subscripts t and t 1 to indicate the current and previ-
ous year
IRISHMIGt�a�b1�log EMPFORt�b2�log EMPDOMt�
b3�log IRWRATIOt �b4log EMPFORt 1�
b5log EMPDOM t 1�b6log IRWRATIO t 1 �
b7MIGSTOCKt�b8IRISHMIG t 1�e (14.5)
The model thus specifies that IRISHMIG, the rate of emigration per
1,000 of the population, is partly determined by changes between the
current and previous periods in three variables, foreign employment,
domestic employment, and the relative wage, and partly by the lagged levels
of those same variables.8 A further explanatory variable is the proportion
of the total Irish population already living abroad; this is intended to
capture the effects of chain migration, in which one generation of emi-
grants assists the movement of the next generation by the provision of
information, and financial and psychological support.
The model also includes the lagged value of the dependent variable, a
procedure explained in §10.2.1. The dependent variable is not in log form,
but six of the eight explanatory variables on the right-hand side are
expressed as logs, so the model is semi-logarithmic (lin-log).
They estimated their model for total emigration, and also for emigra-
tion to particular regions, and for males and females. We focus here on the
total. In reporting their results Hatton and Williamson omit two of the var-
iables, �IRWRATIO and �EMPDOM, because the coefficients on these
were never statistically significant. The remaining results are reproduced in
table 14.1.
We see that all the t-ratios are well in excess of 2, and thus the regression
coefficients on all these terms are statistically significant at the 5 per cent
level or better. The high R2 enables Hatton and Williamson to claim that
‘Irish emigration can largely be explained by the variables included in
[table 14.1], which account for four-fifths of the variance in the emigration
rate’ (p. 584).
The value of the Durbin–Watson statistic is close to 2, implying an
absence of first-order autocorrelation. However, since it is known that the
§.
Table 14.1 Determinants of total Irish emigration rates, time-series data, 1877–1913
(1) (2)
Coefficient t-statistics
Constant 11.61 2.56
� log EMPFOR 108.95 4.52
log EMPFORt 1 64.57 2.63
log EMPDOM t 1 –17.24 2.65
log IRWRATIO t 1 13.16 3.88
MIGSTOCKt 22.87 2.29
IRISHMIG t 1 0.44 3.46
R2 0.86
Residual sum of squares 80.05
Durbin–Watson 1.90
LM(1) 0.15
Source: Hatton and Williamson, ‘Emigration’, p. 584.
Durbin–Watson test is biased against finding autocorrelation if a lagged
dependent variable is included on the right-hand side, Hatton and
Williamson also include the results from a second test for serial correlation,
the Lagrange Multiplier (LM(1)) test, that is appropriate for autoregressive
models.
This test (also known as the Breusch–Gordon test) is run by regressing
the residuals from (14.5) against their lagged values and the other explana-
tory variables from the original equation. The R2 from this second equa-
tion, multiplied by the degrees of freedom of the equation (n k) produces
a test statistic – known as LM(1) – that has a chi-squared distribution with
(n k) degrees of freedom. The value of LM(1) in this case is 0.15, well
below the critical value at any reasonable level of significance, indicating
that the null hypothesis of serial correlation can be rejected with
confidence. This indicates that the R2 of the test regression is very low,
which implies that lagged residuals are not explaining current residuals.
There is therefore no evidence of serial correlation.p
Hatton and Williamson conclude from the regression results that the
effect of the overseas employment rate is very powerful. ‘In the short run, a
10 per cent change in the employment rate (for example, a change in the
unemployment rate from 10 to 1 per cent) would raise the emigration rate
by more than 11 per 1,000’ (p. 584).q To see how they derive this result from
(14.5) we need to recall that this relationship in their model is semi-
logarithmic: �EMPFOR is in logs, IRISHMIG is not.
Reference to our discussion of non-linear regression coefficients in
§12.3 and table 12.1 indicates that in the case of a relationship of this form
a change of 1 per cent in X causes a change in Y of b/100 units. The relevant
value of b in this case is the regression coefficient on �EMPFOR in table
14.1, which is �108.95, so b/100 is 1.0895. However, Hatton and
Williamson have chosen to refer to a change in X (in this case, EMPFOR) of
10 per cent rather than 1 per cent, so we must make a corresponding
adjustment to the change in Y, and multiply b/100 by 10. The result is,
therefore, that a 10 per cent change in EMPFOR would raise the emigra-
tion rate by 10.895 per 1,000 population or, in round numbers, by 11 per
1,000, controlling for the effects of the other variables in the model.9
p The other statistic reported in table 14.1 is the residual sum of squares (RSS), which indicates
how much of the variance in the dependent variable is not explained by the regression. It is used
by Hatton and Williamson to enable comparisons between (14.5) run on total emigration, and
other versions run on slightly different data sets. q Remember that EMPFOR is measured as 1 minus the proportion of the labour force unem-
ployed each year. A fall in unemployment from 10 to 1 per cent is equivalent to a rise in employ-
ment from 90 per cent to 99 per cent, i.e. by 10 per cent.
Since Hatton and Williamson are interested in the long-term trend as well
as the annual fluctuations, they also formulate an alternative long-run version
of the model. In this the long run is characterized by the elimination of any
impact from short-term changes in wage and employment rates, or in the
emigration rate. This is achieved by setting �EMPFOR, �EMPDOM, and
�IRWRATIO equal to zero, and by regarding IRISHMIGt and IRISHMIGt 1 as equal. Making these changes (after which the time scripts are no longer
needed), and moving the second term for IRISHMIG to the left-hand side,
the model becomesr
(1 b8) IRISHMIG�a�b4log EMPFOR�
b5log EMPDOM�b6log IRWRATIO�
b7MIGSTOCK�e (14.6a)
Dividing all terms on both sides by (1 b8) we get
IRISHMIG� � log EMPFOR�
log EMPDOM�
log IRWRATIO�
MIGSTOCK�e (14.6b)
Recalling the symbol b̃ adopted to distinguish the long-run coefficient
(see §12.3.3) we can thus say that the long-run effect on IRISHMIG of a
change in the foreign employment prospects is measured by
b̃4�
where b8 is the coefficient on the lagged dependent variable estimated from
the full model in (14.5), and similarly for the other variables in (14.6b).s
We thus have the long-run multiplier,
b̃4� b4
1 b8
b4 1 b8
b7 1 b8
b6 1 b8
b5 1 b8
b4 1 b8
a
1 b8
§.
r The left-hand side is written as (1 b8) IRISHMIG. This is the same as IRISHMIG–b8IRISHMIG. s This procedure for deriving long-run multipliers was also described in §12.3.3.
where b4 is the coefficient on EMPFOR, given in table 14.1 as 64.57; and b8 is the coefficient on IRISHMIG, given as 0.44.t So
b̃4� �115.3
and
is 1.153
As in the previous case we are determining the effect of a 10 per cent rather
than a 1 per cent change, so must multiply this by 10. The result is, there-
fore, 11.5, or roughly the same magnitude as the short-run effect, as noted
by Hatton and Williamson (p. 584).
Hatton and Williamson then turn to the effect of a sustained rise of 10
per cent in IRWRATIO, the foreign-to-domestic wage ratio, and find that
this ‘would lead ultimately to an increase in the emigration rate of 2.35 per
1,000’ (p. 584). Since the form of this relationship is exactly the same as the
previous one, the calculation follows the same lines. Replacing b4 by b6 we get
b̃6� �23.5
and so
is 0.235
Multiplying this by 10 for a 10 per cent change gives 2.35.
Hatton and Williamson also run a separate regression (not reproduced
in the text) to determine whether migration rates are affected by relative
wages, or by Irish wages alone. The regression also determines the relative
weights of the home and foreign wage levels on the decision to migrate (see
p. 585). In this alternative regression, the logs of the Irish and overseas
wages are entered as separate explanatory variables. Hatton and
Williamson report that the test results easily pass the chi-squared test at the
5 per cent level, indicating that the coefficients are indeed of equal value
and opposite signs.10
The necessary data for direct measurement of EMPDOM were not
available, and it was proxied by a series for the percentage deviation of Irish
farm output from trend. The calculation of the long-run effect of a 10 per
b̃6 100
13.16
1 0.44
b̃4 100
64.57
1 0.44
t The t 1 time subscripts on both variables in table 14.1 are irrelevant when considering the
long-term model.
cent change in this variable follows the same lines as those for EMPFOR
and IRWRATIO, and is about –3 per 1,000 (p. 585).
For the final variable, the migrant stock, Hatton and Williamson calcu-
late that the long-run impact was that ‘for every 1,000 previous migrants
an additional 41 were attracted overseas each year’ (p. 585). In this case the
MIGSTOCK variable was not entered in the model in logarithmic form so
we are dealing with a simple linear relationship (as in the first row of table
12.1). A change of 1 unit in MIGSTOCK will cause a change of b units in
IRISHMIG (both measured per 1,000 population) and the long-run
impact is derived as
�40.8
Hatton and Williamson conclude from this part of their analysis that the
principal cause of the long-run downward trend in Irish emigration was the
increase in Irish real wages relative to those available to potential migrants
in the United States and other receiving countries. They measure the actual
fall in IRWRATIO between 1852–6 and 1909–13 at about 43 per cent, which
would have ‘lowered the long-run emigration rate by as much as 10 per
1,000, accounting for much of the secular decline in the emigration rate that
we observe in the data’ (p. 586).u They further conclude, on the basis of the
value of R2 in (14.5), that over 80 per cent of the variation in the short-run
fluctuations in emigration is explained by the combination of variables
(other than �EMPDOM and �IRWRATIO) specified in their model.
14.2.2 The panel model of Irish emigration In order to throw further light on the composition of the emigrants,
Hatton and Williamson compiled a second data set based mainly on
Census of Population data for each of 32 Irish counties for each of the
census years, 1881, 1891, 1901, and 1911. A few variables that they initially
thought might be relevant were excluded because they were found to have
very little significance, and the final data set covers the dependent variable
and 10 explanatory variables. These are listed and defined in the lower
panel of table A.3.
An important advantage of their approach is that it covers a large range of
variables and is thus able to control for the full set of characteristics simulta-
neously (p. 591). Some previous studies have drawn conclusions from
22.87
1 0.44
§.
u Their wage data cover only the period from 1876 and they estimate the decline from 1852–6 by
extrapolation of the later rate. The long-run effect on IRISHMIG of a change in IRWRATIO of
43 per cent would be 43�0.0235�10.1 per 1,000.
simple relationships between emigration and one potentially relevant char-
acteristic, but this effectively ignores the possible contribution of other char-
acteristics.v For example, it has been observed that emigration rates varied
greatly across different regions, and it has been suggested that emigration
was associated with religion and republicanism. However, without the con-
trols provided by a full regression model it is impossible to say how far
poverty, illiteracy, and differences in religion contributed to higher rates of
emigration, since these variables were themselves correlated across regions
and counties. Similarly, the flow abroad was lower from the more industrial-
ized areas of Ireland, but without a comprehensive regression model we
cannot say whether it was the presence of industry, the level of wages, or just
the extent of urbanization that acted as a deterrent to emigration.
The dependent variable is CNTYMIG, the proportion per 1,000 of the
county population that emigrated. The explanatory variables cover a range
of economic, demographic, religious, and social factors suggested by the
extensive literature on Irish emigration. The emigration rate is expected to
be higher in counties that have a larger proportion of their population who
were in the prime emigration age group of 15–34 (AGE), or were living in
towns of 2,000 or more (URBAN), or who were CATHOLIC. It is expected
to be lower in counties that have a larger proportion who were illiterate
(ILLITRTE). Poverty is measured by two variables: the proportion living in
the lowest grades of HOUSING, and the percentage receiving poor
RELIEF. Both these are expected to be positively associated with emigra-
tion. High rates of emigration are also expected to be associated with large
families, and the average size is measured by FAMSIZE.
Among the economic variables, the economic composition of the labour
force is measured by AGRIC, the proportion of the county labour force
working in agriculture. Access to land is proxied by LANDHLDG, the pro-
portion of county agricultural holdings of less than 5 acres, but this variable
is not entered in the model on its own. Instead an interaction variable is
created by multiplying LANDHLDG and AGRIC. This was done ‘to give it a
greater weight in the more agricultural counties’ (p. 591). The potential
income gain from migration is measured by CYWRATIO, the ratio of the
foreign real wage to the agricultural real wage in the county. This variable is
entered in the cross-sectional model in logarithmic form to facilitate com-
parison with the time-series results. None of the other variables are in logs.
Hatton and Williamson use a pooled or panel model to analyse these
data. Models of this type are discussed in more detail in panel 14.1. Their
v The critical distinction between controlling for other variables and simply ignoring them was
discussed in §8.2.3.
model uses data for 32 counties for each of four census years, and is thus a
pooled data set of 128 observations. They initially included year dummies
to allow for the possibility that the coefficients would not be stable across
all four dates, but this was found not to be the case and the year dummies
were omitted.11
The results of this pooled model are given in table 14.2, again focusing
exclusively on total emigration. R2 is 0.70 and the model thus explains 70
per cent of the variation in CNTYMIG, a good performance by the stan-
dard usually attained in cross-sectional regressions.
HETERO reports a test statistic for heteroscedasticity. The test employed
by Hatton and Williamson is a variant of the Lagrange Multiplier tests (such
as the Breusch–Pagan and White tests) introduced in §11.3.2. The test statis-
tic is derived by running the square of the residuals from the fitted regression
equation against a constant and the fitted values of the dependent variable. If
there is heteroscedasticity, this regression should show a systematic relation-
ship between the size of the residual and the size of the fitted value.
The test statistic of the null hypothesis of homoscedasticity is nR2,
where n is the number of observations and R2 is the coefficient of
§.
Table 14.2 Determinants of Irish county emigration rates, panel data, 1881–1911
(1) (2)
Coefficient t-statistics
Constant 45.92 4.20
AGE 13.60 0.88
URBAN 3.02 0.50
AGRIC 16.39 2.38
LANDHLDG–AGRIC –41.98 3.26
log CYWRATIO 6.43 2.12
RELIEF 1.96 4.40
HOUSING 7.69 1.45
FAMSIZE 7.93 5.24
CATHOLIC 4.97 1.79
ILLITRTE 1.53 0.19
R2 0.70
Residual sum of squares 1420.9
HETERO 0.73
Source: Hatton and Williamson, ‘Emigration’, p. 593.
Panel 14.1 Pooled or panel data
Hatton and Williamson’s use of a data set that combines a series of cross-
sections of information on individual units over time is known as either a
pooled or a panel data set. Panel data may include information on a group
of individual observations (individuals, households, firms) over time (these
are sometimes known as longitudinal data), or on aggregates, including
administrative units such as parishes, counties, states, or countries.
The virtue of panel data is that they provide more degrees of freedom
within which to explore relationships among variables; they also allow for
more sophisticated integrated treatment of variability across space and
time. The challenge of panel data is that the assumptions of independence
among explanatory variables that underpin much regression analysis are
less likely to apply, resulting in the need for more complex econometric pro-
cedures.
Obviously, such sophisticated econometric procedures are beyond
the scope of this simple text. However, since panel data are becoming
increasingly common in historical work, some simple observations are
worthwhile.
The linear regression approach to a panel data set suggests the follow-
ing model:
Yit�bitXit�eit (14.7)
where the subscript indicates unit i in time period t. However, such an
open-ended formulation is likely to be too general.
Two alternative specifications are commonly employed in panel data
analysis – the fixed effects model and the random effects model.
The fixed effects model posits that the b coefficients are constant for all
units and time periods. Sometimes, these models allow the intercept to vary
across either time or across units (i.e. the effects are fixed for units at any one
time, but may vary by unit from one cross-section to the next; or, they are
fixed in any given time period, but may vary by units within the panel).
Such models employ dummy variables to allow the intercept to change
accordingly.
Hatton and Williamson apply an unrestricted fixed effects model in
their pooled regression analysis of county level Irish emigration,
1881–1911. They included year dummies for 1881, 1891, 1901, and 1911 to
test whether emigration patterns were different in each year, irrespective of
the behaviour of the explanatory variables. In other words, they were testing
for the existence of some undefined, but time-specific parameters. Hatton
determination. This test statistic is distributed as a chi-square, with 1 d.f.
The value of HETERO reported in table 14.2 is 0.73. The tabulated critical
value of the chi-square at the 5 per cent level of significance is 3.841, indi-
cating that the null hypothesis should not be rejected.
We can now look briefly at the results of this model. For the relative
wage we can interpret the regression coefficient in the same way as we did
with short-run semi-logarithmic time-series relationships in the previous
section. A 10 per cent change in CYWRATIO will cause a change in
CNTYMIG of 10�b/100 units. Since b is given in table 14.2 as 6.43 (with a
§.
and Williamson report that these year dummies were statistically insigni-
ficant. They therefore employ the unrestricted pooled regression, with a
single intercept term.
The random effects model rejects the notion that shifts in the regres-
sion line arising from unknown influences can be captured by dummy vari-
ables and posits instead that these unknown factors should be incorporated
in the error term. Thus the regression model takes the form:
Yit�ai�bXit�ei�uit (14.8)
where ui represents a second error term representing the individual variance
of experience within the sample.
The random effects model is especially relevant to cases in which the
raw data represent a sample from the population, where the purpose of the
regression is to infer behaviour patterns for the population as a whole from
the sample, and when the sample observations may suffer from stochastic
error. However, random effects models may be hard to reconcile with the
assumption of independence of errors across observations that is funda-
mental to the classical linear regression approach. For this reason, more
complex econometrics are often applied to random effects models.
In cases (such as Hatton and Williamson’s) where the observations
include information on the entire population under study, the fixed effects
model is generally considered to be more appropriate. This can be estimated
using OLS techniques.
Panel data techniques exist not only for basic regression, such as in
Hatton and Williamson’s use of OLS procedures, but also for probit, logit,
and tobit analysis. Tests for violations of the CLR model (such as heterosce-
dasticity, autocorrelation, etc.) are generally available for panel analysis.
This is an approach that is becoming more routine in regression packages
and more frequent in social science and historical research.
t-ratio of 2.12) it can be said ‘that a 10 per cent rise in the relative wage
would raise the emigration rate by 0.64 per 1,000’ (p. 592.) As Hatton and
Williamson note, this estimate is noticeably smaller than the correspond-
ing short-run coefficient estimated on time series, and much lower than
the long-run time-series coefficient. The coefficient on the former is given
in table 14.1 as 13.16 so the effect of a 10 per cent rise would be 13.16�
100�by 10 � 1.32; the latter was found in the preceding section to be 2.35.
Since the model specifies a simple linear relationship between the
dependent variable and all the remaining explanatory variables, the inter-
pretation of the regression coefficients on each of these variables is
straightforward. A change in an explanatory variable of 1 unit will cause a
change in CNTYMIG of b units; that is of b per 1,000 of the population,
because this is the unit in which the dependent variable is measured. Since
all except one of the explanatory variables are measured as proportions (or
in the case of RELIEF as a percentage), we can thus say that a change of 1
per cent (equivalent to a change in a proportion of 0.01) will have an effect
on CNTYMIG of b units.
The exception is FAMSIZE, where the unit is average number of persons
per family. This variable has a large and highly significant coefficient which
‘implies that a one-person reduction in average family size would lower the
emigration rate by 7.93 per 1,000. This would support those who have
argued that demographic forces continued to influence emigration in the
late nineteenth century. High birth rates led to large families which precip-
itated a large flow of emigration’ (p. 593).
If we look next at the other economic factors, we can see that the
coefficients on AGRIC, and on the interaction variable created by the
product of LANDHLDG and AGRIC, are both large and statistically
significant. For the former alone a 1 per cent rise in the proportion of the
labour force in agriculture would raise the emigration rate by 16.4 per
1,000, for the latter a rise of 1 per cent would lower the rate by as much as 42
per 1,000. The interaction variable can be interpreted as indicating that the
higher the proportion of smallholdings, the smaller was emigration, con-
trolling for the share of the labour force in agriculture. As Hatton and
Williamson conclude: ‘these results strongly suggest that lack of opportu-
nities in agriculture (including opportunities to obtain or inherit a small
farm) was an important cause of emigration’ (p. 592).12
Turning now to the remaining variables we find from the coefficient on
AGE in table 14.2 that a 1 per cent rise in the proportion of the population
in the prime emigration age group would raise the rate by 13.6 per 1,000,
but that this coefficient is not statistically significant. However, Hatton and
Williamson report this result in a somewhat different form from usual.
What they say is: ‘the point estimate for the proportion of the population
between 15 and 34 indicates, plausibly, that emigration would be 13.6 per
1,000 higher for this group than for the population as a whole’ (p. 592, italics ours). How do they get to this?
Note that the total migration rate may be expressed as the weighted
average of two migration rates – that for 15–34-year-olds and that for the
rest of the population. Let p be the proportion of the population aged
15–34; since the proportion must sum to one, the rest of the population is
1 p. If the migration rates of the two groups are M1 and M2, we can
express total migration (MIG) as:
MIG�M1(p)�M2(1 p) (14.9a)
This may be re-written as
MIG�M1(p)�M2 M2(p)�M2�(M1 M2)p (14.9b)
If we were to run a regression with MIG as the dependent variable and p
as the explanatory variable, we would have
MIG�a�b(p)�e (14.10)
By comparing (14.9b) and (14.10) it is easily seen that b, the coefficient
on p in the regression, is an estimate of (M1 M2), i.e. of the difference
between the migration rate of those aged 15–34 and that of the remainder
of the population.w Hence we could report the regression coefficient, b, as
the amount by which the migration rate for 15–34-year-olds exceeds the
rate for the rest of the population: 13.6 per 1,000. (This is not quite what
Hatton and Williamson suggested, which was the rate by which migration
for 15–34 year-olds exceeded the overall migration rate.) Alternatively, the
coefficient could be reported in the usual way by noting that an increase in
the proportion of the population aged 15–34 by 1 unit (for example, from
0.33 to 0.34) would raise the migration rate by 13.6 per cent (for example,
from 11 per 1,000 to 12.5 per 1,000).
It follows from the principle captured by (14.9) and (14.10) that the
coefficients on any other explanatory variable that relates to a proportion of
the relevant population (for example, share of the labour force in manufac-
turing, share of the housing stock which is owner-occupied) can also be
interpreted in this way. Thus, the coefficient on URBAN in table 14.2 indi-
cates that the migration rate of urban inhabitants was 3.02 per 1,000 higher
than that for non-urbanites; while the migration rate for Catholics was on
average 4.97 per 1,000 lower than for non-Catholics. All such coefficients
are, of course, estimated holding other characteristics constant.
§.
w Also note that the intercept term is an estimate of M2.
URBAN has the expected positive sign but the effect of urbanization is
only 3.0 per 1,000 and the coefficient is again not significant. Of the two
poverty variables, RELIEF is highly significant, but the effect on emigration
is relatively small: a rise of 1 per cent in the proportion of the population
receiving poor relief would raise the emigration rate by 1.96 per 1,000.
HOUSING has a smaller impact: a 1 per cent increase in the proportion of
families living in the worst types of housing would raise the emigration rate
by 0.08 per 1,000.x
Finally we see that the sign of the coefficient on CATHOLIC is negative,
and that on ILLITRTE is positive. In both cases this is the opposite of what
was expected, though the coefficient is insignificant in the latter case (p. 594).
14.2.3 The impact of the explanatory variables on changes in emigration over time In order to assess the importance of the different explanatory variables it is
necessary to relate the size of the regression coefficients to the actual change
in the variables that occurred during the late nineteenth century. The
regression model may find that a variable has a very large and statistically
significant coefficient, as was the case with the interaction variable
AGRIC–LANDHLDG. But if the value of that variable remains broadly
constant over the period, then it is of little importance as an explanation
for the changes in emigration.
In order to assess the combined impact of the actual changes in their 10
explanatory variables and the indications of the effect of those changes
given by the regression coefficients, Hatton and Williamson measure the
contribution of each variable to the decline in county emigration rates over
the period 1881–1911 (p. 595). They do this by calculating the
(unweighted) mean value across the 32 counties for each of the variables,
and then obtaining from this the change in the mean value. They then
multiply this change by its corresponding regression coefficient. The effect
of doing this is to replace the previous indications of the effect of an arbi-
trary 1 per cent or 10 per cent change by an approximation to the actual
change that occurred over time.
We repeat their procedure in table 14.3 for the change from 1881 to
1911; they also report their results for the shorter period 1881–1901. The
results indicate that the main contribution to the decline in county emigra-
tion of 9.5 per 1,000 over the four decades was made by FAMSIZE, and by
x RELIEF is expressed in percentage terms. Hence a 1 per cent rise in RELIEF is a unit rise in its
value; the impact is calculated as 1.96 � 1. HOUSING, in contrast, is expressed as a proportion.
A 1 per cent rise in HOUSING is a 0.01 unit increase in its value; the impact is calculated as
7.69� 0.01, which equals 0.0769, or 0.08 when rounded to the second decimal place.
the three variables reflecting living standards: CYWRATIO, RELIEF, and
HOUSING. The decline in average family size reduced the emigration rate
by 3.6 per 1,000, and the other three variables taken together by 3.8 per
1,000. By contrast, the two agricultural variables together contributed only
a modest 1.4 per 1,000.
Hatton and Williamson observe that there may seem to be a discrepancy
between these panel findings and the time-series results, ‘which laid stress
on the importance of relative wage rates in driving Irish emigration’ (p.
595). However they argue that wage rates can be viewed as a summary sta-
tistic representing various aspects of living standards. In the time-series
model these other aspects are absent, and so the full impact of living stan-
dards is attributed to the relative wage term. In the cross-sectional model
§.
Table 14.3 Decomposition of changes in Irish emigration rates, 1881–1911
(1) (2) (3) (4) (5)
Mean value Change Regression Effect on
1881 1911 1881–1911 coefficient emigration rate
CNTYMIG 16.552 7.027 9.525
AGE 0.320 0.318 0.002 13.60 0.03
URBAN 0.177 0.214 0.037 43.02 0.11
AGRIC 0.663 0.646 0.017 16.39 0.28
AGRIC–LANDHLDG 0.078 0.105 0.027 41.98 1.13
log CYWRATIO 0.629 0.501 0.128 46.43 0.82
RELIEF 2.509 1.801 0.708 41.96 1.39
HOUSING 0.458 0.246 0.212 47.69 1.63
FAMSIZE 5.098 4.641 0.458 47.93 3.63
CATHOLIC 0.811 0.814 0.003 7 4.97 0.01
ILLITRTE 0.252 0.111 0.141 41.53 0.22
Total change in county emigration rates estimated from regression coefficients 9.03
Actual change in CNTYMIG as above 9.52
Notes:
(1) and (2) from the original Hatton and Williamson data set, not the version rounded to 2
decimal places in ‘Emigration’, p. 598.
(3)�(2) – (1)
(4) from table 14.2, column (1)
(5)�(3)�(4). These results differ very slightly from those reported by Hatton and
Williamson, p. 595 because of rounding.
these aspects are directly represented and so the effect of relative wages is
diminished.
To test this assertion, Hatton and Williamson run a revised version of
their panel model. In this they omit those aspects of living standards other
than the relative wage, and also the minor variables that were not statisti-
cally significant or were found to have little or no effect. This alternative
model thus contains only two of the original explanatory variables: CYW-
RATIO and FAMSIZE.
We report their results for total emigration in table 14.4. They show how
with this very restricted specification the relative wage term captures some
of the effects previously attributed to the other measures of living stan-
dards. The impact of a 10 per cent increase almost doubles from 0.64 to
1.19 per 1,000, about the same as the short-run coefficient in the time-
series equation of 1.32, and about half the long-run coefficient of 2.35.
They conclude from this that the results of the time-series and cross-
section models ‘can be largely reconciled’ (p. 596).
Notes 1 At the time this paper was published it was not yet standard practice to give actual
prob-values. The significance of regression coefficients was indicated by reporting
either the standard errors or the t-statistics (i.e. the ratios of the regression
coefficients to their standard errors) or, occasionally, both. Note also that the t-ratios
in this paper are expressed with the same sign as the coefficient; in Hatton and
Williamson’s paper discussed below, the t-ratios are expressed as absolute values.
While it is more accurate to report the sign, it is no longer standard practice, largely
because the sign is irrelevant to the interpretation of statistical significance.
Table 14.4 Restricted model of Irish county emigration rates, panel data, 1881–1911
(1) (2)
Coefficient t-statistics
Constant 54.42 10.07
CYWRATIO 11.91 4.15
FAMSIZE 12.17 10.58
R2 0.57
Residual sum of squares 2145.9
HETERO 0.66
Source : Hatton and Williamson, ‘Emigration’, p. 596.
2 We regret that we have not been able to reproduce Benjamin and Kochin’s results
precisely. When (14.2) is run on the data on the web site (with the two explanatory
variables taken to 5 decimal places) the result is
UNEMP�5.13�18.36 BWRATIO 91.24 DEMAND
(2.27) (4.75) (–8.82)
R2�0.86, R–2�0.84, D–W�2.22, SEE�1.80
Note that the results reported in Ormerod and Worswick are slightly different again.
None of the discrepancies is large enough to affect the interpretation of the model,
and in what follows we report the original results from Benjamin and Kochin. 3 Benjamin and Kochin also reject the possibility of simultaneity (see §11.4.3),
whereby unemployment influences the replacement rate at the same time as the
replacement rate causes unemployment, though they do so without the benefit of a
formal test, such as the two-stage least squares model introduced in §15.1.1 and
panel 15.3. 4 Ormerod and Worswick recognize that Benjamin and Kochin had experimented
with the addition of a time period for the full period 1920–38, but had found that the
coefficient on the time trend was itself insignificant and that there was little impact
on the estimated value of the other coefficients. However, they say that this conclu-
sion is correct only for the full period, not for the shorter subperiods. 5 Strictly, tests for parameter stability are not the same as tests for outliers, although
Benjamin and Kochin’s variant of the Chow test does go some way to providing a
formal statistical test of the status of 1920. 6 The test statistic in this case is whether the reported unemployment rate falls within
2.131 standard errors of predicted unemployment, where 2.131 is the t-statistic at
the 5 per cent level with 15 df. 7 Timothy J. Hatton, ‘A quarterly model of the labour market in interwar Britain’,
Oxford Bulletin of Economics and Statistics, 50, February 1988, pp. 1–25; Nicholas H.
Dimsdale, S. Nickell, and N. J. Horsewood, ‘Real wages and unemployment in
Britain during the 1930s’, Economic Journal, 99, June 1989, pp. 271–92. 8 For a separate paper showing how this model is formally derived from appropriate
economic principles, see Timothy J. Hatton, ‘A model of UK emigration,
1870–1913’, Review of Economics and Statistics, 77, 1995, pp. 407–15. 9 Note that Hatton and Williamson’s procedure for calculating the impact of a 10 per
cent change in each explanatory variable is an extrapolation from the impact of an
infinitesimal change. An alternative method for calculating the effect of a given per
cent change in EMPFOR would be to multiply the coefficient b4 by the change in the
value of log EMPFOR. Thus, a 10 per cent increase in EMPFOR (e.g. from 90 to 99
per cent) would generate a 10.38 per cent increase in IRISHMIG (�log EMPFOR�
0.09531 � 108.95�10.384). 10 In contrast, Hatton and Williamson enter the home and foreign employment rate as
separate variables in the main regression. They do this because they believe that
potential migrants interpret information about the labour market in Ireland and
abroad in different ways, so that it would be inappropriate to combine the two ele-
ments into one ratio (see p. 582). 11 To include year dummies one of the four census years would be chosen as the control
year, say 1881, and three separate year dummies would be created, which we can call
1891, 1901, and 1911. The year dummy 1891 would take a value of 1 for all observa-
tions relating to 1891, and 0 otherwise; 1901 would be 1 for all observations relating
to 1901, 0 otherwise; 1911 would be 1 for all observations relating to 1911, 0 other-
wise. 12 The interaction variable would normally be constructed as the product of the two
component variables. Unfortunately, at a late stage of their work, Hatton and
Williamson changed the value of their AGRIC variable without changing the inter-
action term. The unchanged values are included in the data set on the web page to
enable readers to reproduce the published regression results. As can be seen by con-
structing the correct interaction variable, this discrepancy makes little difference to
the results.
Case studies 3 and 4: the Old Poor Law in England and leaving home in the United States, 1850–1860
15.1 The effects of the Old Poor Law
This case study is concerned with two cross-sectional models formulated
to investigate the operation of the Old Poor Law in the south of England in
the early nineteenth century. As noted in the brief introduction of this
subject in §A.1 in appendix A, the rapid increase in the cost of relief during
and after the Napoleonic Wars provoked fierce criticism of the system
adopted to assist the poor in these decades. In his 1990 book Boyer
explored several aspects of this topic, and we review two of these. The first
is his analysis of the effect of the system of poor relief on the working of the
agricultural labour market; specifically, the relationship between relief
payments, the wages paid to farm labourers, and the rate of unemployment
(pp. 122–49).a Details of the data set that he created for this purpose are set
out in table A.1 in appendix A.
The second theme is Boyer’s investigation of the proposition advanced –
most famously – by Malthus, that relief payments related to the number of
children in the labourer’s family were a major cause of a high birth rate and
a rapid growth of population (pp. 150–72). The data set used for the inves-
tigation of this topic is listed in table A.2.
The institutional framework for this debate is, of course, specific to the
conditions in the south of England in the early nineteenth century, but the
underlying issues are of enduring relevance. Many of the points at issue
will be recognized by anyone familiar with recent discussions in the United
States and other countries about the adverse effects of benefit payments
and the policies adopted to get people ‘off welfare’.
a Unless otherwise noted all page references in this section are to George R. Boyer, An Economic
History of the English Poor Law, Cambridge University Press, 1990.
15.1.1 Relief payments, wages, and unemployment
The hypotheses and the two models The questions that Boyer wishes to address relate essentially to the steep
increase in total and per capita relief expenditure over time. The Royal Poor
Law Commission appointed to investigate the problem reported in 1834
that the principal cause of the increase was the system of outdoor relief that
enabled able-bodied labourers to obtain relief without having to endure
the rigours of life in a workhouse. This and other criticisms of the Old Poor
Law were later challenged by Blaug, Digby, and other historians.1
Ideally this debate should be investigated by a time-series approach that
models the rise in relief payments in relation to the behaviour of various
competing explanatory factors. The data necessary for this do not exist, but
Boyer argues that most of the relevant hypotheses can be tested indirectly
by a cross-sectional regression to explain variations in relief across parishes
(p. 126).
For example, one hypothesis advanced by those who challenged the
Commission’s analysis is that the rise in relief was due to falling rural
incomes, caused by a decline in employment opportunities for women and
children in cottage industries because of the competition from the rapidly
developing factory sector. Another is that it was a response by rural par-
ishes to the increased demand for labour in urban areas and the conse-
quent need to pay higher rural wages: farmers used the Poor Law to
supplement wages, effectively passing some of the burden of their labour
costs on to other ratepayers.
If these and other changes increased per capita relief expenditures over
time, then we should also find that at any point in time parishes with fewer
opportunities for cottage industry, or those with more competition from
urban industry, had higher per capita relief expenditures than parishes in a
more favourable position with respect to these factors, other things being
equal.
Boyer compiled a large cross-sectional data set covering a sample of 311
parishes from 20 counties in southern England. In addition to the measure
of RELIEF there are seven numerical variables and a further seven dummy
variables. With one exception the observations all relate either to the
census year 1831 or to the following year, when rural parishes completed an
elaborate questionnaire distributed by the Royal Poor Law Commission.b
b The sample parishes were selected on the basis of the completeness of their returns to the
Commission, but – for the reasons mentioned in n. 2 of chapter 5 – Boyer believes that there is no
obvious bias as a result of this (pp. 129, 149).
The first problem that Boyer faced in specifying his model is created by
the interdependence of several of the key hypotheses advanced by contem-
poraries and later historians (p. 127). It is widely held that both wage rates
for agricultural labourers (INCOME) and unemployment among labour-
ers (UNEMP) were determinants of RELIEF, but at the same time it is also
claimed that RELIEF led to lower INCOME and higher UNEMP.
Moreover, Boyer’s approach assumes that labour-hiring farmers were able
to choose what they considered would be for them the most profitable
combination of the three variables: the wages they paid, the employment
they offered during non-peak seasons, and the level of benefits paid
through the Poor Law to unemployed labourers.
These three variables, INCOME, UNEMP, and RELIEF are referred to as
endogenous, meaning that they are determined within the system of rela- tionships Boyer is modelling. By contrast, the hypothesized explanatory
variables are referred to as exogenous or pre-determined. Exogenous vari- ables are determined outside the model in question, and cannot be affected
by any changes in the other variables within the system of relationships.
Given this mutual interaction between the three endogenous variables
there is clearly a problem of simultaneity, and thus a violation of one of the assumptions of the classical linear regression (CLR) model (see §11.4.3).
In these circumstances it is not appropriate to use a single-equation model
to explain cross-parish variations in relief expenditures, and Boyer adopts
two alternative procedures; one known as a reduced-form model, the second as a simultaneous-equations or a structural model (p. 133). A full explanation of these two types of model is not possible at the level of expo-
sition appropriate to this book, but a very elementary introduction to what
is involved is given in panels 15.1 and 15.2 for those who wish to know a
little more. The panels should be read after completing this section.
Ordinary least squares (OLS) can be used to estimate the reduced-form
equations, but because of the effects of the simultaneity in the relationships
this procedure should not be employed to estimate a simultaneous-equa-
tions model. Instead Boyer adopts the alternative two-stage least squares (2SLS) procedure introduced in §11.4.3. Panel 15.3 gives a brief explana-
tion of the procedure. Once the model has been estimated by 2SLS, the
interpretation of the regression results is exactly the same as for models
estimated by OLS.
The reduced-form model Strictly speaking, the reduced-form model is derived from the structural
model in the manner explained in panel 15.2 and illustrated (with a
different example) in the appendix to this section. However, in the present
§.
context it is convenient to follow Boyer in beginning with the simpler
reduced-form equations. He specifies three separate reduced-form equa-
tions. In each of these, one of the three endogenous variables is the depen-
dent variable, while the remaining two simultaneously determined
endogenous variables are omitted from the equation.
For example, in (15.1) we have a simple linear relationship with RELIEF
as the dependent variable and twelve explanatory variables. INCOME and
UNEMP do not appear in this reduced-form equation.
RELIEF�a�b1COTTIND�b2ALLOTMNT�
b3LONDON �b4FARMERS�b5DENSITY�
b6CHILDALL �b7SUBSIDY�b8WORKHSE�
b9ROUNDSMN �b10LABRATE�b11GRAIN�
b12WEALTH�e (15.1)
The identical list of explanatory variables appears in the corresponding
reduced-form equations for INCOME and UNEMP. These equations thus
escape the problem of simultaneity by successively taking one of the
endogenous variables as the dependent variable and excluding the two
remaining endogenous variables from any role in the equation.
This solves the simultaneity problem, but it does so at the cost of making
it impossible to test the hypothesis that RELIEF, INCOME, and UNEMP
were mutually determined, and that this interaction was a major cause of
the long-term increase in expenditure on relief. In order to do this, it would
be necessary to specify a model with three equations in which all three
endogenous variables are simultaneously determined. The crucial rule for
such models is that there must be as many independent equations as there
are endogenous variables.
Unfortunately, Boyer is compelled to eliminate the equation for
UNEMP, and he therefore specifies a simultaneous-equations model with
only two endogenous variables and two equations.2 The effect of excluding
UNEMP in this way is that it ceases to be endogenous to the model and can
be treated as another exogenous variable.
The simultaneous-equations model The two equations in this simultaneous-equations model are given (p. 134)
as
RELIEF�a�b1COTTIND�b2ALLOTMNT�
b3LONDON �b4FARMERS�b5WEALTH�
b6WORKHSE �b7UNEMP�b8INCOME�e (15.2)
and
INCOME�a�b1COTTIND�b2ALLOTMNT�
b3LONDON �b4CHILDALL�b5SUBSIDY�
b6LABRATE�b7ROUNDSMN�b8DENSITY�
b9RELIEF�e (15.3)
Two features of this model should be noted. First, when one of the endog-
enous variables is the dependent variable, the other is not omitted (as was the
case in the reduced-form model) but appears on the right-hand side as an
explanatory variable. We thus have INCOME in (15.2) and RELIEF in (15.3).
Because INCOME is included on the right-hand side of (15.2) it is not
necessary to include also those explanatory variables that help to explain
INCOME but do not directly affect RELIEF. This explains why the four
variables in (15.1) that represent specific forms of outdoor relief (CHIL-
DALL, SUBSIDY, LABRATE, and ROUNDSMN) are omitted from (15.2).
They are expected to have a negative effect on INCOME, but are not a
direct determinant of the overall level of RELIEF.
Secondly, unlike the reduced-form model, the equations in the simulta-
neous-equations model for RELIEF and INCOME ((15.2) and (15.3)) do
not share the same explanatory variables. FARMERS is included in (15.2)
to test whether parishes in which labour-hiring farmers had greater politi-
cal power were able to pass on to other ratepayers more of the cost of main-
taining their workers, thus leading to higher RELIEF. However, this power
is not regarded as a determinant of INCOME, and so this variable does not
appear in (15.3). GRAIN is designed to be a proxy for the extent of season-
ality in the demand for labour and is omitted from (15.2) because it should
affect RELIEF only through its effect on UNEMP, which is already entered
in the equation. And as noted in the previous paragraph, the four variables
representing forms of outdoor relief are included as determinants of
INCOME but not of RELIEF.
We can now turn to the discussion of the results of the regression
procedures.
The regression results Boyer gives a very full discussion of the role of the different explanatory
variables (pp. 134–8). He also includes two useful tables (p. 135) compar-
ing the direction of impact that each variable is predicted to have according
to the various hypotheses in the Poor Law literature (these are known as the
expected signs of the coefficients) with the actual effect shown by the results
of the regressions. The expected results are shown as positive (�), negative
§.
Panel 15.1 Simultaneous-equations models
A simultaneous-equations (or structural) model relates each of the endog-
enous variables to some specified combination of exogenous and endoge-
nous variables. The crucial rule is that there must be as many independent
equations as there are endogenous variables.
Boyer’s model supposes that labour-hiring farmers were able to jointly
determine RELIEF and INCOME so as to maximize their profits. We thus
have two linear equations and two endogenous variables, RELIEF and
INCOME, which are determined by the model (p. 134). (As noted in
§15.1.1 it was not possible to include UNEMP in the model; if it could have
been included, as Boyer initially proposed, there would have been three
dependent variables and a set of three equations.)
There are a large number of exogenous or predetermined variables (for
example, COTTIND, ALLOTMNT, and WEALTH) which are taken as given
outside the model. To simplify the presentation at this point we shall refer to
all of the combinations of exogenous variables that Boyer enters in any
specific relationship as though they were 1 single variable, called EXOG 1,
EXOG 2, and so on. Thus EXOG 1 covers the seven explanatory variables set
out in full in (15.2) of §15.1.1, and similarly for EXOG 2 and (15.3). We then
have the following equations
RELIEF�a1�b1EXOG 1�b2INCOME�e (15.2a)
INCOME�a2�b3EXOG 2�b4RELIEF�e (15.3a)
There are three important features to note about this model.
(a) Unlike all the other models we have considered hitherto, this model
consists of a set of two equations that must be considered together and
contains two dependent variables.
(b) The relationships are interdependent, as indicated by the model:
changes in INCOME will lead to changes in RELIEF, and at the same
time changes in RELIEF will cause changes in INCOME.
(c) Because this interdependence or simultaneity is present in the rela-
tionships, one of the assumptions of classical linear regression (CLR)
is violated (see §11.4.3). For this reason the simultaneous-equations
model cannot be reliably estimated by the ordinary least squares
(OLS) procedure. Boyer adopts an alternative technique known as
two-stage least squares (2SLS), explained in panel 15.3.
§.
Panel 15.2 Reduced-form models
The equations in a reduced-form model are obtained by solving the simul-
taneous-equations model so that the endogenous variables are expressed as
functions of the exogenous variables. An illustration of what is meant by
‘solving the simultaneous-equations model’ is given in the appendix to this
section on pp. 491–5.
In the reduced form there is a single equation for each endogenous var-
iable, with only the exogenous variables on the right-hand side. None of the
endogenous variables appear simultaneously as explanatory variables.
Using the same summary term, EXOG, for a set of explanatory variables as
in panel 15.1, the reduced-form equations would emerge as
RELIEF�a1�b1EXOG 3�e (15.4)
INCOME�a2�b2EXOG 3�e (15.5)
Equation 15.1 and the two corresponding equations for INCOME and
UNEMP are of this type though they were not derived in precisely this way
from the corresponding structural model. All these reduced-form equa-
tions can be estimated in the standard way by ordinary least squares (OLS).
The change from the simultaneous-equations to the reduced-form
model not only changes the right-hand side to exclude the endogenous var-
iables, it also changes the value of the parameters, i.e. of the constant and the
regression coefficients. This happens because in the reduced-form model
the coefficients measure both
(a) the direct effect of the respective exogenous variables on the endoge-
nous variables, and
(b) their indirect effects through the changes in the endogenous variables
which react simultaneously on each other.
This change in the parameters is further discussed and illustrated in the
appendix to this section.
Panel 15.3 Estimation by two-stage least squares
As the name implies, this procedure involves estimation in two stages, each
of which is done by ordinary least squares (OLS). We will give here a brief
description of the 2SLS procedure, as it would be used to estimate the
simultaneous set of equations, (15.2) and (15.3).
The first stage
In the first stage an equation is estimated for each of the endogenous vari-
ables in the model, in this case for RELIEF and INCOME. These equations
contain all the exogenous variables in the model, that is, both the seven on
the right-hand side of (15.2), and the additional five from (15.3) that are not
already entered in (15.2). The endogenous variables on the right-hand side
of these equations are not included at this stage.
What we are doing, in effect, is to estimate a reduced-form equation for
RELIEF in which the influence of its own exogenous explanatory variables
is combined with the influence of INCOME as represented indirectly by its
exogenous variables.*
The second stage
In the second stage the original equations (in this case, (15.2) and (15.3))
are estimated separately, by OLS. The endogenous variables on the right
hand side are now included in the estimation, but they are measured not by
their actual values, as given in the relevant data set, but by their predicted
values as given by the first stage of estimation. The predicted values are
obtained in exactly the same way as those that were calculated for Ŷ in the
very simple regression model in table 4.2 of §4.3, though of course there are
many more variables and regression coefficients in (15.2) and (15.3).
Thus (15.2) is estimated with its list of seven explanatory variables and
the predicted value of INCOME; (15.3) is estimated with its list of eight
explanatory variables and the predicted value of RELIEF. The statistical
computer package will calculate these predicted values as part of the first-
stage procedure, and the program can then be instructed to use these pre-
dicted values to estimate the second-stage equations by OLS in the standard
way. The computer package will also correct the standard errors in the
second stage, which would otherwise be biased.
* This can be thought of as a special case of the instrumental variable technique (see §11.4.1)
in the sense that all the exogenous variables are taken together to create an instrumental
variable.
( ), or uncertain (?); the actual results are either positive (�) or negative
( ) or not statistically significant (0).
We reproduce in table 15.1 the results Boyer reports for both models for
RELIEF and INCOME. R2 is 0.30 for the reduced-form equation for
RELIEF and 0.36 for INCOME. This is quite satisfactory for cross-sectional
models, which typically yield lower coefficients of multiple determination
than time-series models. In the latter, aspects of the change through time
are typically common to many variables, creating a broadly similar pattern
of variation for dependent and explanatory variables. By contrast, with
cross-sectional models, there are likely to be many other factors at work in
the individual cases (in this instance parishes) that will influence the
dependent variable (see §3.2.3).3
Boyer reports both the t-ratios and the prob-values.c The two are, of
course, different ways of assessing the same issue. For example,
ALLOTMNT (in the reduced-form equation for RELIEF) has a very low
t-statistic (0.13) and a correspondingly high prob-value 0.899; in other
words there is a probability of 89.9 per cent of getting a value greater than t
if the null hypothesis of no association between RELIEF and ALLOTMNT
is correct, and the null hypothesis therefore cannot be rejected.
By contrast the t-ratio for LONDON is very high (3.85) and there is a
correspondingly low prob-value of 0.0001; in other words there is only a
0.01 per cent probability of getting a value greater than t if there is no asso-
ciation, and the null hypothesis can be rejected.
More generally, as would be expected from the discussion of this matter
in §6.3.5, we see that coefficients with a t-ratio of 2 or higher have a corre-
sponding prob-value of 0.05 or lower and are thus significant at the 5 per
cent level or better. Those (such as COTTIND) with a t-statistic a little
under 2 are significant at the 10 per cent level though not at the 5 per cent
level. As a more precise indicator, the prob-value allows more discretion in
the assessment of regression results.
We turn now to the values obtained for the regression coefficients. All
the relationships are linear so the results should all take the form ‘a small
absolute change in X will cause a small absolute change in Y of b units’. It is
necessary to bear in mind, however, that many of the explanatory variables
are dummy variables, and that some of the numerical variables are meas-
ured in proportions or percentages.
The first quantitative result given by Boyer is that per capita relief expen-
ditures were between 1.7 shillings and 3.4 shillings lower in parishes with
§.
c The two parallel lines either side of t in table 15.1 are known as a modulus. This simply indicates
that the author is stating the prob-value without regard to the sign of t.
Table 15.1 The agricultural labour market and the Old Poor Law, simultaneous- equations and reduced-form models, cross-section data, c. 1831
(1) (2) (3) (4) (5) (6)
Reduced-form model Simultaneous-equations model
Coefficient t-statistic Prob�|t| Coefficient t-statistic Prob�|t|
1 Dependent variable RELIEF (in shillings)
Constant 11.20 4.92 0.0001 32.75 1.51 0.133
COTTIND 1.70 1.75 0.082 3.40 1.79 0.075
ALLOTMNT 0.10 0.13 0.899 0.85 0.95 0.342
LONDON 0.04 3.85 0.0001 0.08 2.12 0.035
FARMERS 5.30 2.46 0.015 8.39 3.29 0.001
WEALTH 0.30 1.64 0.103 0.002 0.01 0.993
DENSITY 0.55 1.11 0.268
CHILDALL 5.52 5.31 0.0001
SUBSIDY 0.11 0.11 0.912
GRAIN 0.29 3.11 0.002
WORKHSE 1.24 1.41 0.158 0.75 0.70 0.484
ROUNDSMN 1.10 0.88 0.381
LABRATE 1.24 1.30 0.195
UNEMP 0.34 3.48 0.001
INCOME 0.49 0.75 0.454
R2 0.304
2 Dependent variable INCOME (in £)
Constant 35.67 26.54 0.0001 38.91 24.14 0.0001
COTTIND 2.83 4.93 0.0001 3.30 5.36 0.0001
ALLOTMNT 0.85 1.82 0.070 0.88 1.78 0.076
LONDON 0.06 9.57 0.0001 0.07 9.28 0.0001
FARMER 1.99 1.56 0.119
WEALTH 0.07 0.67 0.503
DENSITY 0.65 2.21 0.028 0.86 2.80 0.006
CHILDALL 1.36 2.21 0.028 0.19 0.24 0.810
SUBSIDY 0.40 0.67 0.503 0.39 0.64 0.523
GRAIN 0.09 1.67 0.097
WORKHSE 0.70 1.36 0.175
ROUNDSMN 0.49 0.66 0.508 0.91 1.17 0.244
LABRATE 0.004 0.01 0.994 0.16 0.26 0.793
RELIEF 0.19 2.33 0.020
R2 0.363
Source: Boyer, Poor Law, pp. 139 and 140.
cottage industry than in those without (p. 138). This follows directly from
the coefficients on the dummy variable for COTTIND. The range of values
is given by the point estimates for COTTIND in the reduced-form and
simultaneous-equation models, and the sign indicates that the effect on
RELIEF was negative. As with all the other dummy variables in the model
there are only two categories, and so there is a straight comparison between
those parishes that have cottage industry and the control category that do
not. The subsequent statements about the effects of COTTIND and
ALLOTMNT on INCOME (p. 140), and of CHILDALL on INCOME
(p. 142) are obtained in precisely the same way.
Boyer qualifies the results based on dummy variables, particularly for
ALLOTMNT. These are included in the model to test the hypothesis that a
further cause of the rise in per capita relief expenditure was the need to
compensate labourers for the loss of land caused by enclosures and other
forms of engrossment. However, because ‘dummy variables measure the
occurrence of a phenomenon rather than its magnitude, one cannot always
make meaningful time-series inferences from their cross-sectional
coefficients’ (p. 127). In this case, the typical amount of land lost by labour-
ers was much larger than the size of the average allotment, and therefore
‘the coefficient from the cross-sectional analysis understates the long-term
effect of labourers’ loss of land’ (p. 143).
The next set of quantitative estimates is a little more complicated. Boyer
observes that distance from LONDON (a proxy for the cost of migration)
had a negative effect on both RELIEF and INCOME. A 10 per cent increase
in LONDON resulted in a reduction in RELIEF of between 1.5 and 2.9 per
cent, and in INCOME of between 1.3 and 1.4 per cent (p. 141). How does
he get these results?
We know directly from the regression coefficients that a change in
LONDON of 1 mile will lead to an absolute fall in RELIEF of between 0.04
and 0.08 shillings and in INCOME of between £0.06 and £0.07.4 The values
come from the two models reported in table 15.1 and all the coefficients are
negative. However, Boyer has chosen to convert these absolute changes to
relative ones, and he does this in relation to the mean values of the three
variables.
The mean distance of all parishes from London is 65.1 miles, the mean
of RELIEF is 18.0 shillings, and the mean of INCOME is £29.6.d A 10 per
cent change in LONDON would thus be 6.51 miles, and the effect of this
§.
d The mean values of RELIEF and INCOME (and of most of the other variables) are given by
Boyer on p. 149. The mean distance from London is not given but can easily be calculated from
the data set.
would be 6.51 times larger than the effects of a change of 1 mile.
Multiplying the original coefficients by 6.51 gives a fall of between 0.26 and
0.52 shillings in RELIEF and between £0.39 and £0.42 in INCOME.
These absolute changes must then be related to the mean values to get
the proportionate changes quoted by Boyer. 0.26 shillings is 1.5 per cent of
the mean RELIEF expenditure of 18.0 shillings, and 0.52 shillings is 2.9 per
cent. Similarly £0.39 is 1.3 per cent of the mean INCOME of £29.6, and
£0.42 is 1.4 per cent. Similar roundabout procedures are used to derive
proportionate effects for a number of the other coefficients discussed by
Boyer, including those relating to specialization in grain and the political
power of labour-hiring farmers (p. 141).
A different route must be followed to reach another type of result: the
statement that ‘the elasticity of relief expenditures with respect to the
unemployment rate is 0.14’ (p. 141). The starting point here is the positive
coefficient on unemployment of 0.34 in the simultaneous-equations
model for RELIEF. Since the model specifies a simple linear relationship
between RELIEF and UNEMP, this means that a rise in UNEMP of 0.01
would increase RELIEF by 0.34 shillings. Using the symbol � (capital
Greek delta) to represent a ‘change in’, we could express this as
�0.34
However, as explained in §12.3.2 this is not the same as the elasticity, which
is defined as the ratio of the proportionate changes in the two variables:
If we re-arrange the terms we see that this definition can also be written as
To obtain the elasticity it is thus necessary to multiply the coefficient of
0.34 by
This can be done in relation to the mean values of the variables, respec-
tively, 7.4 per cent for UNEMP and 18.0 shillings for RELIEF. We thus have
0.34� �0.14 7.4
18.0
X
Y
�Y
�X �
X
Y
�Y
Y �
�X
X
�Y
�X
Since this is not a constant elasticity (see again §12.3.2) its value would
be different at other levels of the variables, but it is usual to evaluate the
elasticity ‘at the mean’.
Conclusions from the regressions Boyer draws several interesting conclusions from his regressions. Those
explanatory variables that proved not to be statistically significant are dis-
missed as being irrelevant to an explanation for the trends in relief expen-
ditures. He finds, for example, that payment of allowances in aid of wages
(measured by the dummy variables for LABRATE, ROUNDSMN, and
SUBSIDY) did not increase unemployment rates or per capita relief expen-
ditures, or reduce labourers’ earnings; the existence of workhouses did not
reduce unemployment rates (p. 143). The INCOME variable was also not
statistically significant in the simultaneous-equations model for RELIEF
and he notes: ‘Surprisingly, the hypothesis that poor relief was used to sup-
plement “substandard”wages was not supported by the data’ (p. 142).
By contrast, the coefficients on GRAIN, COTTIND, and FARMER were
all statistically significant and Boyer concludes: ‘Crop mix, income from
cottage industry, and the political power of labour-hiring farmers were
important determinants of per capita relief expenditures’ (p. 142). The
negative coefficients on LONDON were statistically highly significant in
the equations for both RELIEF and INCOME, supporting the view that the
need to pay more in both wages and relief diminished as the relative pull of
the capital weakened with distance.
Unlike Hatton and Williamson, Boyer does not have data that would
enable him to quantify the changes in the explanatory variables over time,
and thus make a formal quantitative assessment of the relative importance
of the different forces comparable to table 14.3. Instead he draws on the
historical literature to show that there were marked changes in three of the
factors proposed as explanations for the rise in RELIEF (pp. 143–5). First,
there was increased specialization in grain in the southeast of England in
response to the rise in grain prices from 1760 to 1815. Second, the political
power of labour-hiring farmers increased in southern parishes after 1760
as a result of changes in the economic and legal environment.
The third, and probably the most important, change was the combina-
tion of (a) the decline in employment opportunities and wage rates for
women and children in cottage industry, and (b) the rapid increase in
London wage rates. In order to secure an adequate labour force for the peak
harvest season, farmers were compelled to respond to these two trends by
offering improved incomes to their labourers in the form of an appropriate
combination of wages and relief.
§.
15.1.2 The Old Poor Law and the birth rate One of the most frequent of the contemporary criticisms of the Old Poor
Law was that the granting of outdoor relief to able-bodied labourers – and
particularly the payment of child allowances to labourers with large fami-
lies – was a powerful factor in promoting earlier marriage and higher birth
rates. Malthus argued that this undermined one of the essential preventive
checks to population growth, and would lead to subsistence crises. The
Royal Poor Law Commission made this a further part of its condemnation
of the system of outdoor relief.
A number of scholars have attempted to test the Malthusian hypothesis
empirically. The most substantial of these studies, by Huzel, claimed to
show that Malthus was wrong because the payment of child allowances was
actually associated with relatively low birth and marriage rates.5 However
Boyer argues that there are problems with each of the tests applied by
Huzel. His major criticism is that a test confined to comparisons across
parishes of relief payments and birth rates fails to control for other possible
determinants of differences in birth rates, and similarly for marriage rates
and infant mortality (p. 152). A simple bivariate comparison simply
ignores all the other potential causes of variation in these demographic var-
iables across parishes. There is thus an omitted-variables problem in
Huzel’s formulation of the sort analysed in §11.2.2. To be satisfactory as a
test of the Malthusian hypothesis, these other factors must be included,
and controlled for, in a more comprehensive model.
The double log model Boyer accordingly discusses the socio-economic determinants of fertility
(pp. 155–61). The model is initially specified as a linear relationship (p. 159),
but it is subsequently estimated as a double logarithmic specification,e and is
set out below in this form. All the numerical variables are in logs, the variable
names are as given in table A.2 in appendix A, and the sign on the coefficients
indicates the expected impact of each of the explanatory variables. Only two
variables are expected to have a negative effect: population pressure
(DENSITY) and lack of accommodation (HOUSING).
log BRTHRATE�a�b1log INCOME b2log DENSITY
b3log HOUSING�b4CHILDAL3�
b5CHILDAL4�b6CHILDAL5 �
b7ALLOTMNT�b8COTTIND�
b9log INFTMORT�e (15.6)
e The log-log transformation of the OLS regression is designed to assist comparison of the
coefficients with the non-linear model of fertility that is introduced later in this section. The
coefficients are reported in columns (1) and (4) of table 15.2.
The data set covers 214 parishes in 12 counties in southeast England,
c. 1831. Equation (15.6) can be estimated by the ordinary least squares
(OLS) procedure after substituting the logs of the numerical variables for
the original values (see §12.2.2).
Data constraints mean that BRTHRATE has to be measured per 100
families residing in the parish, not per 100 women aged 15–64, and thus
cannot take account of differences between parishes in the number of
women of childbearing age. This creates two problems that Boyer can only
note but not remedy (p. 158). First, variations across parishes in the ratio of
births to families could be due simply to differences in the age distribution
of married females.
Secondly, differential rates of migration away from the parishes might
also be a cause of variation in measured BRTHRATE. If out-migration
rates were higher from poor ‘unpromising’ parishes, then such parishes
should have contained a relatively smaller share of young unmarried adults
and married couples than more prosperous parishes. This would lead to a
lower BRTHRATE for these parishes, other things being equal. ‘Thus there
might be a spurious positive relationship between birth rates and measures
of parish prosperity, such as wage rates.’ Furthermore, if generous child
allowances were associated with poor parishes, there might also be a spuri-
ous negative relationship between birth rates and child allowances.
Boyer then confronts a further awkward problem (p. 161). This one is
created by the fact that the denominator of INFTMORT (infant deaths per
100 live births) is also the numerator of the dependent variable (number of
births per 100 families). Any errors in the measurement of births will lead
to a spurious negative relationship between BRTHRATE and INFTMORT.
He is able to find a way out of this dilemma, but it involves specifying a
model in a form that falls into the intractably non-linear category men-
tioned in §12.2.f It can thus be estimated only by non-linear least squares, a
procedure that falls outside the scope of this text.
Results from the regressions Fortunately, however, the results of this procedure are interpreted in
exactly the same way as those from OLS, and so there is no difficulty in
evaluating them. They are given, together with those from (15.6), in table
15.2, again reporting both t-ratios and prob-values. As Boyer observes, the
results of the two models are ‘qualitatively similar’ (p. 163) for all the vari-
ables except INFTMORT, and his discussion of the results is based on the
non-linear model.6 The t-statistics for this model show that all the variables
§.
f The feature that makes the model non-linear occurs in the error term in Boyer’s equation (2), p.
162.
except COTTIND and ALLOTMNT were statistically significant. R2 in the
OLS specification is only 0.124, indicating that there was a great deal of
random variation across parishes in birth rates and perhaps also that other
factors – not represented in Boyer’s model – were also important.
The important conclusion from this study is that the provision of child
allowances did have a positive effect on birth rates, other things being
equal, and that the quantitative impact was large. The regression
coefficient for CHILDAL3 is �0.25. Since the dependent variable is in log
form, this indicates that parishes that began allowances at three children
had birth rates 25 per cent higher than those of parishes without allow-
ances, controlling for INCOME and other factors. For parishes that began
relief at four children, birth rates were 17 per cent higher; and the same
differential was found for parishes that began at five or more children.
Three other Malthusian hypotheses are also supported by the regression
results. Since these are all double logarithmic relationships the elasticities
can be read directly from the coefficients, indicating that there would be a
change of b per cent in BRTHRATE for every 1 per cent change in the
explanatory variable (see §12.3.1 and table 12.1). Farm labourers’ annual
income had a positive effect on fertility, with an increase in INCOME of 10
Table 15.2 Determinants of birth rates, cross-section data, c. 1831
(1) (2) (3) (4) (5) (6)
Double logarithmic model Non-linear model
Coefficient t-statistic Prob�|t| Coefficient t-statistic Prob�|t|
Constant 1.12 2.06 0.041 2.81 4.28 0.0001
INCOME 0.45 2.95 0.004 0.44 2.40 0.017
DENSITY 0.09 2.70 0.007 0.10 2.52 0.013
INFTMORT 0.06 1.17 0.242 0.38 5.82 0.0001
CHILDAL3 0.13 2.24 0.026 0.25 3.67 0.0003
CHILDAL4 0.11 2.15 0.033 0.17 2.64 0.009
CHILDAL5 0.09 1.50 0.124 0.17 2.25 0.025
HOUSING 0.19 1.62 0.106 0.28 1.94 0.054
ALLOTMNT 0.001 0.03 0.981 0.01 0.21 0.838
COTTIND 0.04 0.76 0.446 0.06 0.91 0.364
R2 0.124
Source: Boyer, Poor Law, p. 163.
per cent resulting in a rise in BRTHRATE of 4.4 per cent. Lack of accom-
modation and high population density both had a negative effect. The
coefficient on HOUSING of 0.28 indicates that an increase of 10 per cent
in the ratio of families to the number of houses would account for a fall in
BRTHRATE of 2.8 per cent; and a 10 per cent increase in DENSITY would
result in a reduction of 1.0 per cent.
These cross-section results show that child allowances, INCOME,
HOUSING, and DENSITY all had a statistically significant impact on the
birth rate, but – as always – it is also necessary to consider whether these
factors were historically important. In the present context that can be done
by ascertaining what their effect was in relation to the rise in the crude birth
rate during the late eighteenth and early nineteenth century (pp. 167–71).
For this purpose Boyer assumes that there were no child allowances at
the beginning of this period. To obtain an estimate of their effect at the end
of the period he constructs an average across the different regions in
England in 1824. The parishes in each region were subdivided into four
categories: those not giving child allowances, and those giving allowances
beginning at three, four, or five or more children. The impact of child
allowances for each category is assumed to be the same in each region as in
the southern parishes covered by the regression (zero for those not paying
allowances, and 0.25, 0.17, and 0.17, for the other three categories). The
regions were then weighted according to their population in 1821. The
final result is a weighted average of 0.142. In other words, for England as a
whole, the payment of child allowances caused birth rates to be 14.2 per
cent higher in 1824, other things being equal, than they would have been in
the absence of this form of relief (p. 168).
For the other explanatory variables Boyer obtains estimates of the per-
centage change over the four decades from 1781 to 1821, and multiplies
these by the elasticities from the regression results for (15.6). The results
are summarized in table 15.3. The table indicates that the five variables
would have caused birth rate to rise by between 5.0 and 7.8 per cent. If child
allowances had not been adopted the birth rate would have declined by
between 6.4 and 9.2 per cent. According to Wrigley and Schofield the actual
increase over this period was 14.4 per cent.g
As Boyer concludes, ‘the early-nineteenth-century increase in birth
rates cannot be understood without taking child allowance policies into
account’ (p. 171). Similarly, the abolition of child allowances following the
introduction of the New Poor Law in 1834 was a major reason why birth
§.
g E. A. Wrigley and R. S. Schofield, The Population History of England, 1541–1871, Edward Arnold,
1981, p. 529.
rates remained broadly stable between 1831 and 1851, despite a more rapid
increase in workers’ real incomes.
15.2 Leaving home in the United States, 1850–1860 Our final case study analyses an important aspect of the social evolution of
the family and its relation to broader socio-cultural dynamics. The nine-
teenth century is generally recognized as a period in which the status of
children underwent considerable change. Broader economic opportu-
nities afforded by industrialization challenged the customary structure of
the family economy, as children began more frequently to work outside the
home, and as they found themselves able to marry and set up independent
households at an earlier age.
Richard Steckel addresses a key part of this story by analysing the age at
which youths left the family home. Steckel drew a sample of households
from the manuscript schedules of the US Census for 1860; he then matched
as many of these as he could back to the 1850 census manuscript sched-
ules.7 His sample incorporated information on almost 1,600 male-headed
households, including many who had migrated en famille between 1850
and 1860. He then compared the names of the members of the household
Table 15.3 Impact of child allowances and other factors on the increase in the English birth rate, 1781–1821
(1) (2) (3)
Percentage change Regression Effect on birth rate
1781–1821 coefficient (%)
INCOME �14 0.44 �6.16
DENSITY �63 0.10 6.30
HOUSING �10 to �20 0.28 2.80 to 5.60
CHILDALL ... 0.14 �14.20
INFTMORT 9 0.38 3.42
Total change in birth rate
as a result of above factors �5.0 to �7.8
Actual change in crude birth rate 14.4
Notes:
(1) From Boyer, Poor Law, p. 170;
(2) From table 15.2, column (4);
(3)�(1) multiplied by (2).
in the two years to determine which of the children who were listed in the
census in 1850 were not present in the household 10 years later. This
allowed Steckel to determine the probability that a child of a given age
would leave the family home within the next 10 years.
The complexity of American migration patterns made it impossible to
match all households from 1860 to 1850. Therefore, Steckel’s data set is not
a random sample of all US families, but rather focuses on households that
were already established in 1850 and for which the household head sur-
vived until 1860. It excludes all households established between 1850 and
1860, as well as those in which all the children left before 1860.
Nonetheless, consideration of other evidence leads Steckel to conclude that
his results are representative of the circumstances governing the decision to
leave the family home in this period.
The calculated departure rates are depicted in figure 15.1.8 The graph
shows the proportion of children of a given age in 1850 who had left the
household by 1860. The horizontal axis is the age of the children in 1850; the
vertical axis shows the departure rate over the next 10 years. It is important
to emphasize that Steckel’s data do not mean that 15.7 per cent of 5–6-year-
old boys were leaving the family home in the 1850s, but rather that 15.7 per
cent of those aged 5–6 in 1850 left home sometime over the next 10 years. The
graph shows the departure rates of children aged from 5 to 22 in 1850.9
The graph shows a broadly sigmoid (or logistic) pattern of departures
for both sexes. The leaving rate is low for 5–6-year-olds, rises steeply
through the early teenage years, before reaching a high plateau for children
over 17. It is evident from the graph that younger girls tended to leave the
§. , –
Figure 15.1 Departures of children from home by age and sex, United States, 1850–1860
100
80
60
40
20
0 5–6 7–8 9–10 11–12 13–14 15–16 17–18 19–20 21–22
Age in 1850
P er
ce nt
ag e
de pa
rt ed
b y
18 60
Boys
Girls
family home faster than boys, and also that women who remained in the
household after 18 were less likely to leave than men of the same age.
15.2.1 Influences on leaving home Steckel’s analysis is revealing of the pace at which children left home at
given stages of their life cycle. But it does not tell us what determined the
decision to leave home. There is a large literature that discusses decisions to
migrate and to leave home. Steckel examines this literature and establishes
a series of hypotheses about the dynamics of early departure. These
hypotheses focus on the relative costs and benefits of leaving home and
staying within the family. He finds six major variables that may have
influenced these costs and benefits:
� Household wealth : Wealthier families had more resources, which would
allow children to stay at home longer, rather than leave in search of
employment or a fortune. Moreover, greater household wealth enabled
children to stay at home and gain more skills and education; it also
raised the chances that the family had a business that could employ the
child.
� Gender : Girls tended to leave home at a younger age than boys. This
reflected a number of factors: a lower age at marriage for females;
the greater opportunity for factory employment of young girls (and the
greater opportunity for boys to find a job in the family business);
the earlier maturation of girls; and the greater likelihood that boys
would be offered additional schooling.
� Location : Since rents and other costs were higher in urban areas, the
costs of leaving home were also higher, suggesting that children would
stay at home longer in more urban settings. Also, migration tended to
be more frequent on the frontier and it is probable that this extends to
the age of leaving home as well.
� Family structure : The more children in a family, the greater the compe-
tition for household resources. It is therefore plausible to suggest that
children of larger families who could find work outside the home
would be encouraged to leave. Birth order might matter as well. In
some cultures, the youngest (female) child is expected to stay at home
to support ageing parents; in other cultures, it is the oldest (female)
child that fulfils this function. Finally, the death of a parent may affect
the resources available to the household, perhaps increasing the need of
older (male and female) children to stay at home to maintain family
status and wealth.
� Ethnicity : The children of foreign-born family heads may have retained
the culture of home-leaving of their parents, giving rise to different
patterns compared to children of American parents.
� Migration : If households migrated with the employment of their chil-
dren in mind, this would reduce the chances of children leaving the
home at a later stage.
� Literacy : Migration studies often indicate that illiterate individuals are
less likely to move, perhaps because they are less aware of the opportu-
nities available elsewhere, perhaps because they lack the confidence to
take risks, perhaps because their services are less in demand in new
regions than those of skilled workers.
15.2.2 The logit model In order to assess the importance of these various influences on the deci-
sion to leave home, Steckel models a regression in which the probability of
leaving home within the period 1850–60 is the dependent variable. The
independent variables take the values recorded in answers to questions
asked at the 1850 Census and therefore relate to the situation of the house-
hold and its individual members in 1850.
The regression has the following form:
LEAVE�a�b1 PROPERTY�b2 LITRACYF�
b3 MIGRATED �b4 LITRACYM�b5 DEATHM�
b6 AGEM�b7 AGECHILD �b8 YNGERFAM�
b9 OLDERFAM�b10 BORNENG �b11 BORNIRE�
b12 BORNOTHR�b13 WHITECLR �
b14 BLUECLR �b15 UNSKILD �b16 OTHROCC �
b17 NCENTRAL�b18 SOUTH�b19 FRONTIER �
b20 LARGEURB�b21 SMALLURB�e (15.7)
The dependent variable, LEAVE, is dichotomous, taking the value of 1 if
a child who lived with the family in 1850 was not reported as present in
1860. The regression model used by Steckel to estimate the equation is a
logit, as introduced in §13.2.1. The regression procedure effectively chooses the coefficient values to reproduce as closely as possible the
observed distribution of binary values in the sample. Thus, it generates
values of a and bk which when positive will fit the distribution of leavers in
the sample, and when negative will fit the distribution of stayers in the
sample.
The explanatory variables were chosen to reflect the various hypotheses
set out in §15.2.1. The importance of household wealth is tested by includ-
ing the value of real estate owned in 1850 (PROPERTY). Steckel also
§. , –
includes information on the father’s occupation (WHITECLR, BLUECLR,
UNSKILD, OTHROCC) as further evidence of the household’s economic
standing. The significance of household structure and the intensity of
competing claims on family resources is reflected in two variables that
measure family composition – the number of children under the age of 10
(YNGERFAM), and the number of children who were 10 years and older
(OLDERFAM) in 1850.
Other household characteristics include the literacy of each parent
(LITRACYF, LITRACYM), whether or not the mother was still living
(DEATHM), whether the family had migrated between 1850 and 1860
(MIGRATED), and the ethnic origins of the household head (BORNENG,
BORNIRE, BORNOTHR). Steckel incorporated information on location,
by region (NCENTRAL, SOUTH, FRONTIER), and by urban status
(LARGEURB, SMALLURB). Finally, he includes the age of the child in
1850 (AGECHILD) and, as an indicator of the birth order of the child, the
age of the mother in 1850 (AGEM).
None of the variables is transformed from their original value. There are
thus no log values and no rates of change. The majority of the 21 explana-
tory variables are dummy variables. The only five which are continuous
are: AGECHILD, AGEM, YNGERFAM, OLDERFAM, and PROPERTY. All
of these are measured in actual numbers, except for PROPERTY, which is
measured in thousands of dollars.
The dummy variables fall into two types:
One set represent simple binary observations – households either
migrated (MIGRATED�1) or they did not (MIGRATED�0). The
coefficient on MIGRATED therefore indicates the difference in the value of
LEAVE between a household that did and did not migrate between 1850
and 1860. LITRACYM, LITRACYF, and DEATHM also take this form.
A second set represent more complex patterns. Thus, Steckel distin-
guished five categories of father’s occupation. For reasons explained in
§10.1.1, one of these categories (in this case, farmer) must be excluded
from the regression to avoid the dummy variable trap. There are thus four
occupational dummies in (15.7), each taking the value of (0,1). For
example, a child whose father was a farmer would have 0 values for each of
the four occupational dummies. If the father was a white collar worker,
these dummies would take the values: WHITECLR�1, BLUECLR�0,
UNSKILD�0, OTHROCC�0; and similarly for fathers in each of the
three remaining occupations. The omitted occupation (farmer) is
included in the constant term. Thus, the coefficient on WHITECLR indi-
cates the difference in the value of LEAVE if the father were a white collar
worker, relative to the value of LEAVE if the father were a farmer.
Similarly, there are three dummy variables for father’s birthplace, for
which there are four categories (the three listed above and the omitted cat-
egory, born in the United States). A child with a US-born father would have
0 values for each ethnic dummy; a child with an English-born father would
record 1 for BORNENG and 0 for the others; and similarly for the two
remaining categories. There are four regional variables and three dummies
(with Northeast being omitted); and there are three urban variables and
two dummies (with rural as the omitted category). In each case, the
omitted variables are in the constant term, and in each case, the value of the
coefficient on the dummy variable is relative to the value of LEAVE for the rel-
evant omitted variable.
Steckel divided his sample by sex into groups aged 5–11 and 12–18 in
1850.10 There are over 1,000 observations for the younger age categories
and 579 observations for the older. The sample was divided evenly between
girls and boys. The results for each of these groups are shown in table 15.4.
Interpreting the coefficients The logit model is a more complex regression model than the OLS (or even
2SLS) versions that have been used in the other case studies. The signs and
the t-statistics may be interpreted in the standard way, but the coefficients
do not predict the values of LEAVE directly and must be transformed into
something more meaningful. In chapter 13, we discussed the transforma-
tion from logit coefficients into both marginal and impact effects and elas-
ticities. Steckel’s analysis is framed in terms of elasticities.
Two concepts of elasticity were introduced in §13.2.5: the point elastic-
ity and the unit elasticity. The latter measures the ratio of a proportionate
change in the dependent variable to a proportionate change in the explana-
tory variable, when the latter changes by one unit. It is the unit elasticity
that Steckel has calculated, and that is reproduced in columns (3) and (6) in
table 15.4. In what follows, we focus on the derivation of the unit elasticity
on AGECHILD in column (6); that is, the proportionate response of
LEAVE for a unit change in AGECHILD, calculated for the data set of boys
aged 12–18.
As explained in the discussion of logit coefficients in §13.2.2, the basis
for the calculation is the odds ratio, and its logarithm, the log odds ratio. In
this case the mean value of LEAVE in the data set of boys aged 12–18 is
0.6511, and the proportion of stayers in the sample is 1 0.6511�0.3489,
so the odds ratio is 0.6511/0.3489�1.8662.h The log of the odds ratio is
§. , –
h The mean value of LEAVE corresponds to the mean value of P(Yi�1) in the terminology
employed in chapter 13, and the mean value of stayers corresponds to P(Yi�0).
Table 15.4 The age at leaving home in the United States, 1850–1860, explaining the probability of departure, boys and girls aged 5–11 and 12–18
(1) (2) (3) (4) (5) (6)
Boys aged 5–11 Boys aged 12–18
Coefficient t-statistic Elasticity Coefficient t-statistic Elasticity
Constant 2.971 4.36 2.412 2.065 2.05 0.676
PROPERTY 0.005 0.21 0.006 0.013 0.57 0.010
LITRACYF 0.219 0.73 0.017 0.263 0.62 0.007
MIGRATED 0.120 0.51 0.013 0.401 1.23 0.016
Wife
LITRACYM 0.156 0.64 0.019 0.043 0.12 0.002
DEATHM 0.607 2.93 0.066 0.075 0.26 0.003
AGEM 0.045 2.85 1.249 0.073 3.73 0.952
Child
AGECHILD 0.254 6.16 1.582 0.330 6.21 1.582
Family composition
YNGERFAM 0.124 1.95 0.335 0.068 0.93 0.059
OLDERFAM 0.099 1.87 0.165 0.105 1.52 0.133
Foreign-born
BORNENG 0.383 0.66 0.005 1.340 1.60 0.005
BORNIRE 0.138 0.28 0.003 0.384 0.53 0.002
BORNOTHR 0.099 1.87 0.165 0.105 1.52 0.133
Occupation
WHITECLR 0.499 1.84 0.042 0.263 0.66 0.007
BLUECLR 0.754 3.63 0.108 0.088 0.29 0.004
UNSKILD 0.589 1.69 0.024 0.177 0.39 0.003
OTHROCC 0.926 2.22 0.020 0.621 1.20 0.008
Place of residence
NCENTRAL 0.468 1.94 0.071 0.291 0.95 0.015
SOUTH 0.162 0.76 0.060 0.393 1.56 0.063
FRONTIER 0.778 2.72 0.066 0.006 0.02 0.000
LARGEURB 0.718 1.71 0.027 0.439 0.96 0.006
SMALLURB 0.653 1.14 0.014 1.222 1.61 0.009
–2 log � 89.08 82.05
df 21 21
n 1135 579
§. , –
Table 15.4 (cont.)
(1) (2) (3) (4) (5) (6)
Girls aged 5–11 Girls aged 12–18
Coefficient t-statistic Elasticity Coefficient t-statistic Elasticity
Constant 3.424 5.15 2.401 2.722 2.39 0.614
PROPERTY 0.033 1.17 0.035 0.010 0.32 0.004
LITRACYF 0.338 1.02 0.019 0.216 0.51 0.004
MIGRATED 0.035 0.15 0.003 0.205 0.53 0.005
Wife
LITRACYM 0.158 0.63 0.016 0.227 0.65 0.008
DEATHM 0.052 0.24 0.005 0.551 1.45 0.013
AGEM 0.041 2.68 0.981 0.023 1.17 0.200
Child
AGECHILD 0.439 10.53 2.370 0.300 5.05 0.973
Family composition
YNGERFAM 0.133 3.21 0.306 0.205 2.52 0.128
OLDERFAM 0.065 1.21 0.091 0.112 1.57 0.093
Foreign-born
BORNENG 0.120 0.20 0.002 0.661 0.78 0.003
BORNIRE 0.378 1.00 0.011 0.617 0.74 0.003
BORNOTHR 0.419 0.76 0.006 0.024 0.03 0.000
Occupation
WHITECLR 0.246 0.91 0.018 0.152 0.42 0.003
BLUECLR 0.104 0.48 0.011 0.104 0.35 0.004
UNSKILD 0.605 1.97 0.028 0.061 0.10 0.001
OTHROCC 1.114 2.75 0.024 0.737 0.98 0.003
Place of residence
NCENTRAL 0.178 0.80 0.025 0.477 1.52 0.022
SOUTH 0.100 0.51 0.030 0.362 1.38 0.038
FRONTIER 0.522 1.86 0.036 1.123 2.35 0.023
LARGEURB 0.554 1.52 0.019 1.272 2.56 0.011
SMALLURB 0.521 1.18 0.014 0.152 0.22 0.001
–2 log � 167.40 62.99
df 21 21
n 1038 579
Source: Richard H. Steckel, ‘The age at leaving home in the United States, 1850–1860’, Social
Science History, 20, 1996, pp. 521–2.
thus 0.6239. The coefficient on AGECHILD in column (4) of table 15.4 is
0.330, which represents the change in the log odds ratio for an increase of 1
year in AGECHILD. Adding the coefficient to the mean value of the log
odds ratio produces 0.6239�0.330�0.9539.
In order to work out what has happened to the value of LEAVE, we take
the exponent of 0.9539 (to move from log to natural value), which equals
2.5958. This is the value of the odds ratio itself. Since the odds ratio is
LEAVE/(1 LEAVE), the value of LEAVE can be extracted from the odds
ratio by calculating (odds ratio/1�odds ratio) or, in this case,
(2.5958/3.5958). This produces a revised value of LEAVE of 0.7219.
Thus, a unit change in AGECHILD changes LEAVE from 0.6511 to
0.7219, a proportionate increase of 0.0708/0.6511�0.1087. Since the
mean value of AGECHILD in this sample is 14.63, a unit increase of 1 year
represents a proportionate increase of 0.0684. The unit elasticity is calcu-
lated as the proportionate change in LEAVE divided by the proportionate
change in AGECHILD (see (13.10a) in §13.2.2); this is 0.1087/0.0684�
1.589.i
A complication arises when evaluating the elasticity of the dependent
variable with respect to a change in the dummy variable, namely that a unit
change in the dummy (from 0 to 1) cannot be expressed as a proportion of
its base value. Steckel therefore measures the elasticity relative to the mean
value of the dummy variable in the sample, i.e. the proportion of all house-
holds or individuals exhibiting this characteristic. In the case of LITRA-
CYF, for example, the responsiveness of LEAVE to a unit change can be
calculated by taking the exponent of (0.6239�0.263) the log value of the
odds ratio at the sample mean plus the coefficient on LITRACYF – to gen-
erate the odds ratio, and then proceeding as above.
The new value of LEAVE is 0.7083, an increase of 8.78 per cent. The
sample mean of LITRACYF is 0.08636; a unit increase in the dummy vari-
able, expressed relative to its sample mean, is thus (1.08635/0.08635), or
1,258 per cent. The ratio of 8.78 to 1,258 produces the elasticity estimate of
0.007, as reported by Steckel. The relatively large coefficient on LITRACYF
translates into only a modest effect on the dependent variable.
Conclusions from the logit model What conclusions may be drawn from the logit analysis? Table 15.4 shows
the importance of focusing on the elasticities rather than the actual
coefficients when evaluating the results of a logit regression. As the table
i Steckel’s estimate of the elasticity is 1.582, rather than the 1.589 cited here. The difference is very
slight and is almost certainly the result of some rounding on either his part or ours.
shows, most of the elasticities associated with a unit change in the explana-
tory variables are small. As Steckel notes (p. 520) the t-statistics and other
tests indicate that socio-economic variables were more powerful influences
on the younger children than on those in the older age group.
The most important single influence on the timing and frequency of
leaving home was clearly the age of the child. The large elasticity of AGE-
CHILD suggests that a difference in the child’s age of 1 year in 1850
increased the probability of leaving the home by almost 1.6 per cent for
boys, and by as much as 2.4 per cent for young girls.
Yet, after holding this influence constant, some of the hypotheses set out
in §15.2.1 received consistent support from the model. Thus, the
coefficient on the number of young children in the household (YNGER-
FAM) was large and significant for all cases except the older boys, suggest-
ing that younger children tended to push older ones out of crowded homes.
The age of the mother (AGEM) was also consistently significant, and
inversely related to the decision to leave home: an additional year of age of
the mother reduced the probability of leaving by about 1 per cent for both
age-groups of boys and for young girls. If this is a proxy for birth order, the
result is consistent with evidence from other societies that later-born chil-
dren tended to remain with the family longer.
Older children might be forced out by younger siblings, but the oldest
tended to stay behind, perhaps in their capacity as heirs apparent to the
family estate. Similarly, father’s occupation clearly had a strong influence
on departure from the home, especially for younger children. The
coefficients for all four occupations are positive, indicating that farm chil-
dren (the control category) were generally least likely to leave home. For
young boys the chances of leaving were markedly higher for the sons of
blue collar workers, and lowest for those in unskilled and ‘other’ occupa-
tions; while the young daughters of unskilled workers were among the least
likely to stay. For older children, the t-statistics on the occupation dummies
for the father’s occupation are uniformly low, indicating that this was not a
significant influence on their decision to leave home.
Households resident on the American frontier were more likely to see
their children leave, reflecting perhaps a combination of greater opportu-
nities and a greater awareness of the advantages of going further west.
Some part of the frontier effect can be attributed to other features of that
area. As Steckel observes (p. 523) the frontier was less urbanized, and fami-
lies living there had younger women as mothers, and had more children
under 10 per family, than those in the Northeast. When these characteris-
tics are controlled for, the expected chances of departure were higher for
young boys on the frontier than in the Northeast (the control category) or
§. , –
the South, and were highest in the north-central states. For young girls the
rate of departure was highest on the frontier, while for older girls it was
higher there than in the northeast and north-central states, but even higher
in the South.
Some of the other hypotheses received at best weak or inconsistent
support from the logit results. Thus, living in an urban community clearly
reduced the rate of leaving home relative to being in a rural setting for all
categories, although the elasticities were small and only one coefficient was
statistically significant. Similarly, the impact of the mother’s absence from
the household had a measurable effect only on the departure rate of young
boys. The wealth of the household, as proxied by the value of its real estate,
had little systematic impact on the decision to leave home; parental literacy
and ethnicity were likewise unimportant. Previous relocation decisions
also counted for little.
With so many variables being insignificant (in the regression on older
boys, only two variables plus the constant are significant at the 10 per cent
critical value), it might be asked whether the model as a whole has statisti-
cal power. The value of –2 log � given at the base of table 15.4 tests the prop-
osition, by providing a test of whether the logit regression is a better
predictor of the odds ratio than a model in which the proportion of leavers
is constant across all households (see §13.2.7). It is thus a joint test of the
statistical significance of all the explanatory variables. The null hypothesis
is that the explanatory variables have no effect on the odds ratio.
This statistic is distributed as a chi-square distribution with (k 1)
degrees of freedom, where k is the total number of regressors, including the
constant. Steckel observes that the df for this test is 21.
For the sample of boys, 12–18, the baseline log-likelihood (log L0) is
374.468; the log-likelihood of the logit model is (log L1) 333.442. The
test statistic is thus 2 ( 374.468�333.442)�82.05. The 1 per cent criti-
cal value of chi-square with 21 df is 38.93, indicating that we may safely
reject the null hypothesis that the logit specification produced no added
value. It is easily seen from table 15.4 that the logit specifications for the
other samples also pass muster.
In chapter 13, we discussed the relative merits of the logit and probit
specifications of models with dichotomous dependent variables. As a
comparative exercise, we ran the probit regression using the Steckel data
set for boys aged 12–18. The two models produce almost identical results,
allowing for the systematic difference in the size of the coefficients
explained in §13.4.1. The coefficient values in the logit formulation are
about 1.81 times as large as in the probit regression (the ratio is not precise
because of the slight difference in the shape of the two distributions and
the non-linearity of the specifications); the t-statistics are almost identi-
cal. The value of the 2 log � test is very similar in both specifications, as
are the log-likelihood value and the two goodness-of-fit tests introduced
in §13.2.7: the pseudo R2 and McFadden’s R2. All in all, it appears that
there is little to distinguish between the probit and logit specifications of
this particular model.
Appendix to §15.1.1 Derivation of reduced-form equations from a simultaneous-equations model This appendix is presented as an illustration of the procedure for deriving
the reduced-form equations from a simultaneous-equations model. Each
of the reduced-form equations has only the exogenous variables on the
right-hand side. We also show how the regression coefficients of the
reduced-form equations differ from those of the simultaneous-equations
model, and how these two sets of parameters are related.
Consider a simple three-equation economic model in which there are
three endogenous variables: consumption (CON), investment (INV), and
national product (NNP), and two exogenous variables: national product in
the previous period (NNPt 1) and government expenditure (GOV). In
terms of the model these variables are exogenous because last year’s NNP is
already determined and GOV is fixed by the public authorities without
regard to the other variables in this model.
The first equation in the model specifies that the level of consumption
depends on the level of national product. The second specifies that the level
of investment depends partly on the current level of NNP and partly on its
lagged value. To complete the model the third equation is an identity that
says that NNP is equal to the sum of CON, INV, and GOV. The model thus
says that changes in NNP will cause changes in CON, but also that changes
in CON will cause changes in NNP, and there is a similar simultaneous
relationship between INV and NNP.
To keep the algebra simple we omit the constants and the error terms.
The simultaneous-equations model is then:
CON �b1NNP (15.8)
INV �b2NNP�b3NNPt 1 (15.9)
NNP �CON�INV�GOV (15.10)
To derive the reduced-form equations from the structural equations we
proceed as follows:
§..
a Substitute (15.8) and (15.9) in (15.10) and collect all the terms in NNP on the left-hand side
NNP �[b1NNP]�[b2NNP�b3NNP t 1]
�GOV
NNP b1NNP b2NNP �b3NNPt 1�GOV
(1 b1 b2)NNP �b3NNPt 1�GOV
NNP � NNPt 1�
� GOV (15.11)
b Substitute (15.11) in (15.8)
CON �b1 NNPt 1� GOV
� NNPt 1� GOV (15.12)
c Substitute (15.12) in (15.9) and collect the terms in NNPt 1
INV �b2 NNPt 1� GOV � b3NNPt 1
� NNPt 1� GOV
� NNPt 1� GOV (15.13)
We now have three reduced-form equations, (15.11)–(15.13), each of
which has only the exogenous variables GOV and NNP t 1 as explanatory
variables. The equations all have very complicated regression coefficients.
Of course if we estimate the equations we will not see these complicated
coefficients, with separate results for b1, b2, and b3. What we will get, for
example if we estimate (15.13), is simply a regression coefficient for each
explanatory variable; we will call them respectively c1 and c2:
INV �c1NNP t 1�c2GOV (15.13a)
where c1 is measuring both the direct effect of NNP t 1 on INV (equivalent to b3 in (15.11) in the full simultaneous-equations model) and also the combination of all the indirect effects involving b1 and b2. There are corre-
sponding equations for (15.11) and (15.12):
b2 1 b1 b2
b3(1 b1)
1 b1 b2
b2 1 b1 b2
b2b3 � b3(1 b1 b2)
1 b1 b2
�11 b1 b2� b3
1 b1 b2
b1 1 b1 b2
b1b3 1 b1 b2
�11 b1 b2� b3
1 b1 b2
1
1 b1 b2
b3 1 b1 b2
NNP �c3NNPt 1�c4GOV (15.11a)
and
CON �c5NNPt 1�c6GOV (15.12a)
In the case of (15.13) these indirect effects on INV arise because:
(a) changes in INV lead to changes in NNP (as specified in (15.10))
(b) the resulting changes in NNP in turn lead to changes in INV (meas-
ured by b2 in (15.9)), and
(c) also lead to changes in CON (measured by b1 in (15.8));
(d) the resulting changes in CON in turn lead to further changes in NNP
(as specified in (15.10)), and
(e) these changes in NNP have a further effect on INV (measured by b2 in
(15.9)).
Although the values of the parameters of the simultaneous-equations
model (b1, b2, and b3) are not directly revealed by the reduced-form equa-
tions, it may be possible to extract them from the values of the coefficients
(c1, c2, c3, etc.) in the reduced-form equations.
In the present case, for example, we can see from a comparison of the
coefficient on GOV in (15.13a) with the corresponding coefficient in
(15.13) that
c2 �
and similarly, from a comparison of the coefficients on GOV in (15.11) and
(15.11a), we can obtain
c4 �
It follows, therefore, that c2�c4�b2, and thus that
b2�
By a similar procedure using (15.12a) and (15.12) with (15.11) and
(15.11a) we can obtain
b1�
To complete the exercise, we can see from the comparison of the
coefficients on NNPt 1 in (15.11) and (15.11a) that
c6 c4
c2 c4
1
1 b1 b2
b2 1 b1 b2
§..
c3�
and since we now know c3, b1, and b2, it is possible to work out what b3 must
be.
Notes 1 Mark Blaug, ‘The myth of the old poor law and the making of the new’, Journal of
Economic History, 23, 1963, pp. 151–84; Anne Digby, Pauper Palaces, Routledge &
Kegan Paul, 1978. 2 It was necessary for Boyer to eliminate UNEMP because per capita expenditure on
relief is equal to the product of (a) the rate at which relief is paid, and (b) the fre-
quency with which it is paid, and the latter factor is obviously correlated with the
unemployment rate. This correlation makes it impossible to model the independent
effect of the generosity of relief payments on unemployment. See further, Boyer,
Poor Law, p. 133. 3 Although Boyer does not report the R2 from the 2SLS regressions, they are broadly
similar to the OLS estimates (0.325 for RELIEF, 0.362 for INCOME). 4 These are the values as reported after rounding to 2 decimal places. To reproduce the
results as published by Boyer we need to work to 3 decimal places for the coefficient
in the simultaneous-equation model for INCOME, i.e. with a coefficient of £0.065
rather than £0.07, obtained by replicating the regression. The other three regression
coefficients are not sensitive to the additional decimal place. 5 J. P. Huzel, ‘Malthus, the Poor Law, and population in early nineteenth century
England’, Economic History Review, 22, 1969, pp. 430–52; and Huzel, ‘The demo-
graphic impact of the old poor law: more reflexions on Malthus’, Economic History
Review, 33, 1980, pp. 367–81. 6 In the first model the coefficient on INFTMORT is negative though not statistically
significant, while in the second it is positive and highly significant. Boyer regards this
as confirmation of the existence of a spurious negative relationship between
BRTHRATE and INFTMORT in (15.6). 7 In order to facilitate matching, Steckel sampled only households with at least one
native-born child over 10 years old in 1860. Steckel’s technique for matching was to
look for households by the name of the household head in the two years. In cases
where families stayed in the same state, this would be relatively simple, since there are
indexes of the names of household heads for each state in 1850 and 1860. For those
households that could not be located in the same state in 1850, Steckel was able to use
the census manuscript information on the birthplace of all family members. The
state in which any child aged 10 or above was born would be a good clue to the house-
hold’s place of residence in 1850; Steckel then consulted the name index of house-
hold heads for that state in 1850 to effect a match. 8 Steckel had to make one final correction in his calculation of average departure rates.
The absence of a child in the household in 1860 could be caused by mortality as well
as migration. Steckel used information on age-specific mortality rates to estimate the
b3 1 b1 b2
number of absences due to death at each age. The expected death toll was then sub-
tracted from the overall number of absences by age to generate departure rates. The
figures shown in figure 15.1 are for departures without correction for mortality. 9 Steckel excluded children under 5 in 1850, since their departure was probably domi-
nated by premature mortality; he also excluded children over 22, because there were
too few of them to generate statistically reliable data. Married children were also
excluded from the sample. 10 The separation by gender follows from the hypotheses in §15.2.1. The separation
into age groups is designed to minimize the inconvenience of being able to separate
out mortality from voluntarily leaving home in the dependent variable. Age-specific
mortality rates are stable between the ages of 12 and 18, suggesting that the bias will
be small. For younger children, however, the problem may be more severe. If we treat
this as a stochastic error in the measurement of the dependent variable, this will have
no effect on the coefficients in table 15.4 (see §11.4.1). The implicit assumption in
Steckel’s model is that child mortality is not systematically related to the explanatory
variables such that it disguises the true relationship between voluntary departure
and independent variables. Steckel receives support for this assumption from the rel-
ative stability of the coefficients and elasticities in the two age-groups.
The four data sets
The following sections provide a brief description of the four data sets that
are referred to throughout the book and form the basis for the case studies
discussed in chapters 14 and 15. The original works should be consulted for
furtherdiscussionof thehistoricalaspectsandformoredetailedinformation
about the sources. The data sets are not reproduced in this book but can be
easilyaccessedwithoutchargeviaaspecialpageontheCambridgeUniversity
Press web site: <http://uk.cambridge.org/resources/0521806631/>
Some of the series are entered in the regression models in the form of
logarithms or as first differences, but all series are given in the data sets in
their original form, and if any manipulation or transformation is required
this must be done by users of the data set.
A.1 The iniquitous effects of the Old Poor Law
During the late eighteenth and early nineteenth centuries there was a rapid
increase in expenditure on poor relief in England. Fierce criticisms of the
system led eventually to the establishment of a Royal Commission and to
the introduction of a New Poor Law in 1834. Many contemporary critics of
the Old Poor Law attributed the remorseless growth of expenditure to the
widespread adoption of systems of outdoor relief whereby those in need
could receive financial assistance at home without being forced into the
demeaning conditions of life in a workhouse.
A number of historians have proposed other explanations for the rising
level of relief payments. Some see the problem as arising from increasing
poverty among farm workers, due to the falling real value of wages paid to
agricultural labourers; the removal by enclosure of allotments on which
labourers could produce milk, potatoes, and other foods to supplement
their wages; and the decline in home employment opportunities for
women and children as the rise of factory-based textile industry wiped out
cottage industry. Others have emphasized the ability of arable farmers to
exploit their local political influence to get the parish to pay relief to
labourers in the winter months, thereby shifting part of the cost of keeping
seasonal workers in the parish on to non-farming ratepayers.
Apart from its cost to the taxpayer, the Old Poor Law was also alleged to
have had many iniquitous effects, chief of which was the widely-held belief
that the granting to able-bodied labourers of outdoor relief related to the
number of their children encouraged a high birth rate, and was thus
responsible for an excessively rapid growth of population. The most
famous exponent of this view was Thomas Malthus, who argued that relief
payments artificially reduced the cost of having children and so under-
mined the ‘preventive check’ to population growth in the form of late mar-
riage or abstention. This Malthusian argument was adopted by the Royal
Commission as one of its grounds for the reform of the Old Poor Law.
In chapters 4 and 5 of a study published in 1990, George Boyer addresses
these two issues with specific reference to conditions in southern England.a
He compiled two data sets covering a sample of parishes in the south of
England in 1831. To investigate the factors that influenced the level of relief
expenditure he used 15 series for a sample of 311 parishes; these are
described in table A.1.1 We added a further series, POP.
For the principal investigation of the birth rate he was able to obtain infor-
mation for a sample of 214 parishes for 10 series. Four of these are already
listed in table A.1 (INCOME, DENSITY, ALLOTMNT, and COTTIND); the
remaining six are set out in table A.2. The first column of the data set indi-
cates the county in which the parish is located, using a numerical code for the
name of the county: 1 is Kent, 2 is Sussex, 3 is Essex, and so on. The full list of
24 county code numbers is reproduced with the data set. The next column
identifies the parish, again using a numerical code for the name; the actual
names are of little interest and this code is not reproduced.
The main source for both these data sets was the Rural Queries (the
replies to 58 questions distributed among rural parishes in the summer of
1832 by the Royal Poor Law Commission) and the enumeration of popula-
tion, occupations, and other data in the 1831 Census of Population.2
A.2 What explains the scale and timing of the massive emigration from Ireland after the famine?
In the period between the famine and the First World War more than 4.5
million men and women emigrated from Ireland. As a proportion of the
§.
a George R. Boyer, An Economic History of the English Poor Law, Cambridge University Press, 1990.
Table A.1 Data set for investigation of relief payments in England in 1831
(a) Measured series for values c. 1831 unless otherwise noted
RELIEF Relief expenditure of each parish per head of the population (in
shillings)
INCOME Annual income of adult male agricultural labourers in each parish
(in £)
UNEMP The ratio of the average number of unemployed labourers in each
parish to the total number of wage labourers
LONDON The distance to London from the centre of the county in which
each parish was located (in miles)
FARMERS The ratio of the number of labour-hiring farmers in each parish
to the total number of parish ratepayers
GRAIN The estimated percentage of adult males in each parish employed
in grain production
DENSITY The ratio of the total population of each parish to the acreage
WEALTH The value of real property (land and buildings) in each parish in
1815 per head of the population (in £)
POP The population of each parish
(b) Series for which the answer for each parish is either ‘yes’ (recorded as 1) or
‘no’(recorded as 0)
COTTIND Does cottage industry exist in the parish?
ALLOTMNT Do labourers have allotments of farm land?
WORKHSE Is there a workhouse in the parish?
CHILDALL Does the parish pay child allowances?
SUBSIDY Does the parish subsidize the wage rates of privately employed
labourers?
LABRATE Does the parish use a labour rate?*
ROUNDSMN Does the parish use the roundsman system?**
Notes:
** Under the labour rate unemployed labourers were apportioned among all occupiers of
property according to ‘the extent of occupation, acreage rent or number of horses
employed’.
** Under this system unemployed labourers were sent round the ratepayers in the parish
with the expectation that they would be offered work.
population this was more than double the rate of emigration from any
other European country. The overwhelming majority went to the United
States, but smaller numbers moved to Great Britain and a few also went to
Canada, Australia, and New Zealand. In an article published in 1993
Timothy Hatton and Jeffrey Williamson explored several features of this
emigration.b They investigated the trend over time, the annual fluctua-
tions, and the possible characteristics within each county of Ireland that
influenced the outflow, for example, religion, age structure, dependence
on agriculture, and living standards.3
The authors compiled two basic data sets for this purpose. One consists
of annual series for 1877–1913, the period for which the emigration and
other data are thought to be reasonably reliable. The five basic series are
listed in the top panel of table A.3; for some purposes these series are also
subdivided by the country which received the emigrants or by gender; or
they are used with various modifications, for example by taking the change
in a series rather than its actual level, or by taking logs.
§.
Table A.2 Data set for investigation of the birth rate in southern England, c. 1826–1830*
(a) Measured series
BRTHRATE The ratio of births per 100 families in each parish in 1826–30
INFTMORT The ratio of deaths of infants aged 0–4 per 100 live births in each
parish in 1826–30
HOUSING The ratio of families per inhabited house in each parish in 1831
(b) Series for which the answer for each parish is either ‘yes’ (recorded as 1) or
‘no’(recorded as 0)
CHILDAL3 Does the parish begin payment of child allowances at three
children?
CHILDAL4 Does the parish begin payment of child allowances at four
children?
CHILDAL5 Does the parish begin payment of child allowances at five or more
children?
Notes:
* In addition the following four series listed in table A.1 are also used in the study of birth
rates: INCOME, DENSITY, ALLOTMNT, and COTTIND.
b Timothy J. Hatton and Jeffrey G. Williamson, ‘After the famine: emigration from Ireland,
1850–1913’, Journal of Economic History, 53, 1993, pp. 576–600.
Table A.3 Data set for investigation of emigration from Ireland, 1877–1913
(a) Annual series, 1877–1913
IRISHMIG Number emigrating from Ireland each year per 1,000 of
population. (In addition to total emigration, separate series are
given in the data set for adjusted total emigration,* and also for
total intercontinental emigration, emigration to the United States,
and total male and total female emigration.)
EMPFOR The foreign employment rate measured as 1 – the proportion
unemployed each year. (For total emigration the foreign rate is
calculated as a geometric weighted average of the rates for the
United States (weight 0.7), Great Britain (0.2), Canada (0.05), and
Australia (0.05), with the weights proportionate to the emigration
shares.)
EMPDOM The domestic (Irish) employment rate. This should in principle
be measured in the same way as the foreign rate, but in the
absence of a series for Irish unemployment a proxy was created by
measuring the proportionate deviation of Irish agricultural
output each year from its trend.
IRWRATIO The ratio of the foreign real wage of builders’ labourers to the
corresponding domestic real wage for Ireland as a whole for each
year. (For total emigration the foreign real wage is a weighted
average calculated in the same way as for foreign employment.)
MIGSTOCK The stock of previous emigrants already living abroad each year in
the United States, Great Britain, Canada, Australia, and New
Zealand, as a proportion of the total Irish population.
(b) Data for each county for each of the four census dates, 1881–1911
CNTYMIG The proportion per 1,000 of the county population that
emigrated
AGE The proportion of the county population aged 15–34
URBAN The proportion of the county population living in towns of 2,000
or more
AGRIC The proportion of the county male labour force in agriculture
LANDHLDG The proportion of the county agricultural holdings less than 5
acres
CYWRATIO The ratio of the foreign real wage (defined and weighted as above)
to the domestic real wage in agriculture in the county
The second set provides data for each of the 32 counties at four census
dates (1881, 1891, 1901, and 1911) for 11 items, listed in the lower panel of
the table.4
A.3 To what extent were generous unemployment benefits responsible for the high level of unemployment in inter-war Britain?
From 1921 until the outbreak of the Second World War in 1939 unemploy-
ment in Great Britain was at record levels: it averaged 14 per cent – far
higher than it had been before 1913 or than it would be for any sustained
period after 1948 – and it never fell below 9.5 per cent. Among various
explanations which have been suggested for this high and persistent inabil-
ity to provide work for such a large proportion of the labour force were the
role of the trade unions and the excessive level of real wages; the over-
valuation of sterling as a result of the decision to return to the gold stan-
dard with the pound valued at its pre-war rate against other currencies;
structural problems arising from over-commitment to the old staple
export industries such as coal, textiles, and shipbuilding; and a general lack
of demand.
Then in 1979, in a highly controversial article, Daniel Benjamin and
Levis Kochin injected a new factor into the debate.c Their suggestion was
§.
c Daniel K. Benjamin and Levis A. Kochin, ‘Searching for an explanation for unemployment in
interwar Britain’, Journal of Political Economy, 87, 1979, pp.441–78. They were not the first to
argue that unemployment benefits contributed to a willingness to remain unemployed for a
longer period, but the hypothesis had been displaced by alternative explanations and it was only
with their article that it returned to prominence in the literature on the inter-war period.
Table A.3 (cont.)
RELIEF The percentage of the county population receiving poor relief
HOUSING The proportion of county families living in third- and fourth-
class housing
FAMSIZE The average family size in the county
CATHOLIC The proportion of the county population who were Catholic
ILLITRTE The proportion of the county population who were illiterate
Notes:
* Emigration from Ireland to Britain is believed to be seriously under-stated, so Hatton and
Williamson construct an adjusted total using double the recorded flow to Britain.
that the persistently high level of unemployment ‘was due in large part . . .
to high unemployment benefits relative to wages’. Apart from the severe
crises of 1921 and 1930–2, the high unemployment of other inter-war
years ‘was the consequence almost solely of the dole. The army of the
unemployed standing watch in Britain at the publication of the General
Theory was largely a volunteer army’.d
The insurance scheme that they highlighted had first been introduced in
Britain in 1911, but only for a very small proportion of the workforce. An
Act of 1920 extended coverage to almost all manual workers over the age of
16 (the two main exceptions were agricultural workers and domestic ser-
vants – both relatively stable sectors with low unemployment) and pro-
vided much more generous levels of benefit. There were numerous
subsequent changes, but Benjamin and Kochin argue that the level of the
benefits – together with various institutional features of the system – made
the inter-war scheme ‘more generous relative to wages than ever before or
since’. In their view it was primarily this generosity that explains why
unemployment was so much higher than in earlier or later periods.
Their paper provoked a flurry of historical and statistical criticisms, but
most historians would now recognize that the benefit system did make
some contribution to the severity of inter-war unemployment, though not
by as much as Benjamin and Kochin had originally claimed.5
Their data set consisted of four annual primary series for the years
1920–38: UNEMP, WAGES, BENEFITS, and NNP.6 They are defined in table
A.4; all are taken from published sources.7 From these they constructed two
further series. BWRATIO is the replacement rate, and is a measure of the gen-
erosity of BENEFITS relative to WAGES (i.e. of the extent to which they are a
replacement for wages). DEMAND is a measure of the effects of changes in
aggregate demand calculated as the ratio of NNP to NNP*, its trend value.e
A.4 What influenced children’s decision to leave home in the United States in the mid-nineteenth century?
Most children eventually leave home, but what are the factors that
influence when and whether they do this? In order to find some of the his-
d The phrases quoted are from Benjamin and Kochin, p. 474; the reference to the General Theory is
to the book which the famous economist, John Maynard Keynes, published in 1936 to explain
his theory of the nature and causes of unemployment. e The ratio of output to its trend is included in the regression model as the difference between the
log of actual output (log NNP) and the log of its trend value (log NNP*). The use of logs in this
way as a means of measuring proportions was explained in §1.6.3. The log-linear trend value for
NNP* was estimated by using the procedure explained in §12.4.
torical answers to these questions Richard Steckel extracted data from the
manuscript schedules for the censuses of the United States for 1850 and
1860.f His analysis dealt with two issues: the average age at which the chil-
dren left home and the factors that influenced them to make this decision,
but our concern here is only with the data and procedures relating to the
latter issue.
To obtain his data set Steckel initially drew from the manuscript sched-
ules for the 1860 Census a random sample of households with at least one
native-born child aged 10 or above still living at home. Those same house-
holds were then identified in the 1850 Census, using the state in which the
child was born as a pointer to locate where the family had been enumerated
in 1850. This information was obtained for 1600 male-headed households,
and these yielded a national sample of unmarried children aged 5–29 who
were recorded as living at home in 1850.
It was thus possible to determine from these matched samples of house-
holds which of the children living at home in 1850 were still there a decade
later. To investigate what might have influenced the decisions of those who
had departed, Steckel assembled information on a range of socio-
economic and demographic characteristics of the father, the mother, the
§. ’
f Richard H. Steckel, ‘The age at leaving home in the United States, 1850–1860’, Social Science
History, 20, 1996, pp. 507–32. The manuscript schedules are the original household returns from
which the published census reports were compiled.
Table A.4 Data set for investigation of unemployment in inter-war Britain, 1920–1938
(a) Annual series, 1920–1938
UNEMP Total unemployment as a percentage of all insured employees
WAGES Average weekly earnings (wages and salaries) of all full-time
employees (in shillings)
BENEFITS Weekly benefit entitlement of an adult male with one adult
dependent and two dependent children (in shillings)
BWRATIO The replacement rate, measured as the ratio of BENEFITS to
WAGES
NNP Real output measured by net national product (in £ million at 1938
factor cost)
DEMAND The ratio of NNP to its trend value, NNP*
Table A.5 Data set for investigation of decisions to leave home in the United States, 1850–1860
PROPERTY The value of the father’s real estate (in $1,000)
LITRACYF Dummy variable�1 if the father was illiterate, 0 otherwise
MIGRATED Dummy variable�1 if the household had migrated between 1850
and 1860, 0 otherwise
LITRACYM Dummy variable�1 if the mother was illiterate, 0 otherwise
DEATHM Dummy variable�1 if the mother had died before 1850, 0
otherwise
AGEM The mother’s age
AGECHILD The child’s age
YNGERFAM The number of children in the family under the age of 10
OLDERFAM The number of children in the family aged 10 and over
BORNENG One of a set of four categories for ethnicity. It is a dummy variable
that takes the value of 1 if the father was born in England, 0
otherwise. There are similar dummy variables for BORNIRE
(born in Ireland) and BORNOTHR (born elsewhere). The control
category is Native Born.
WHITECLR One of a set of five categories for the father’s occupation. It is a
dummy variable that takes the value of 1 if the father was in a
white collar occupation, 0 otherwise. There are similar dummy
variables for BLUECLR (blue collar), UNSKILD (unskilled), and
OTHROCC (other or occupation not given). The control category
is Farmer.
NCENTRAL One of a set of four categories for place of residence. It is a dummy
variable that takes the value of 1 if the residence of the family in
1850 was in a North-central state, 0 otherwise. There are similar
dummy variables for the SOUTH and FRONTIER. The control
category is residence in a state in the Northeast.
LARGEURB One of a set of 2 dummy variables for urbanization. It is a dummy
variable that takes the value of 1 if the residence of the family in
1850 was in an urban area with a population of more than 25,000,
0 otherwise. There is a similar dummy variable for SMALLURB
(with a population of less than 25,000). The control category is
residence in a Rural area.
Note:
The terms dummy variable and control category are explained in §10.1.1.
child, and the household. All this information relates to the position in
1850 and was extracted from the Census schedules. The full list of charac-
teristics is set out in table A.5.
The results are reported for 1,714 boys and 1,617 girls in two age groups,
5–11, and 12–18. In order to keep the working data set for this textbook to a
manageable size we have reproduced only the information relating to 579
boys aged 12–18, but the identical procedures are used for the younger boys
and for the girls, and the results for the whole sample are discussed in
chapter 15.
It should be noted that the children are subdivided according to their
age in 1850, so that there is some overlap between the two groups in respect
of the age at which they eventually left home. The children aged 5–11 could
have departed at any age from 6 (a 5-year-old who left in 1851) to 21 (an
11-year-old who left shortly before the census was taken in 1860). The cor-
responding range of age at departure for those in the group aged 12–18
would be 13–28. Steckel estimates that the mean age at which children left
home for the sample as a whole was 26.5 years for males and 22 years for
females.g
Notes 1 For other quantitative studies of English poor relief see Mark Blaug, ‘The myth of the
old poor law and the making of the new’, Journal of Economic History, 23, 1963, pp.
151–84; J. P. Huzel, ‘Malthus, the poor law, and population in early nineteenth
century England’, Economic History Review, 22, 1969, pp. 430–52 and ‘The demo-
graphic impact of the poor law: more reflexions on Malthus’, Economic History
Review, 33, 1980, pp. 364–81; and G. S. L. Tucker, ‘The old poor law revisited’,
Explorations in Economic History, 12, 1975, pp. 233–52. 2 The detailed replies from parishes were printed in four large volumes as Appendix B
of the H. M. Commissioners for Inquiry into the Administration and Practical
Operation of the Poor Laws, Report, Appendix B1, Answers to Rural Questions (H.C.
44), Parliamentary Papers, 1834, XXX–XXXIII; for the 1831 Census of Population of
England and Wales see Parliamentary Papers, 1833 (H.C. 149), XXXVI–XXXVIII.
The data on the number of births and infant deaths listed in table A.2 were extracted
by Boyer from unpublished parish returns for the 1831 Census located in the Public
Record Office (PRO, HO 71). 3 For good general discussions of the issues posed in analysis of emigration see J. D.
Gould, ‘European inter-continental emigration, 1815–1914: patterns and causes’,
Journal of European Economic History, 8, 1979, pp. 593–679; Gould, ‘European inter-
continental emigration: the role of diffusion and feedback’, Journal of European
Economic History, 8, 1980, pp. 267–315; and Dudley Baines, ‘European emigration,
1815–1930: looking at the emigration decision again’, Economic History Review, 47,
g Steckel, ‘Leaving home’, p. 517.
1994, pp. 525–44. For some more recent quantitative studies see Timothy J. Hatton
and Jeffrey G. Williamson (eds.), Migration and the International Labour Market,
1850–1939, Routledge, 1994. 4 The main source for the annual migration series was Imre Ferenczi and Walter F.
Willcox, International Migrations: vol I, Statistics, NBER, 1929. The annual home
and foreign real wage series were compiled by Jeffrey G. Williamson, ‘The evolution
of global labor markets in the first and second world since 1830: background and evi-
dence’, NBER Working Papers on Historical Factors in Long-Run Growth, 36, NBER,
1992. The unemployment series are from J. R. Vernon, ‘Unemployment rates in
post-bellum America, 1869–1899’, University of Florida, 1991, manuscript; Charles
H. Feinstein, National Income, Expenditure and Output of the United Kingdom,
1855–1965, Cambridge University Press, 1972; Wray Vamplew (ed.), Australian
Historical Statistics, Fairfax, Syme & Weldon, 1987, p. 153; and a series specially con-
structed by Hatton and Williamson for Canada. The county data were taken mainly
from various volumes of the decennial Census of Ireland, Parliamentary Papers, or
the Emigration Statistics of Ireland, Parliamentary Papers, for the four dates. 5 The main initial contributions to the debate were Michael Collins, ‘Unemployment
in interwar Britain: still searching for an explanation’, Journal of Political Economy,
90, 1982, pp. 369–79; David Metcalfe, Stephen J. Nickell and Nicos Floros, ‘Still
searching for an explanation of unemployment in interwar Britain’, Journal of
Political Economy, 90, 1982, pp. 386–99; P. A. Ormerod and G.D.N. Worswick,
‘Unemployment in inter-war Britain’, Journal of Political Economy, 90, 1982, pp.
400–9; and Daniel K. Benjamin and Levis A. Kochin, ‘Unemployment and unem-
ployment benefits in twentieth-century Britain: a reply to our critics’, Journal of
Political Economy, 90, 1982, pp. 410–436. For later quantitative work on this topic see
also Timothy J. Hatton, ‘Unemployment benefits and the macroeconomics of the
interwar labour market’, Oxford Economic Papers, 35 (Supplement), 1983, pp.
486–505; and Barry Eichengreen, ‘Unemployment in interwar Britain: dole or dol-
drums?’, in N. F. R. Crafts, N. H. D. Dimsdale and S. Engerman (eds.), Quantitative
Economic History, Oxford University Press, 1991, pp. 1–27. 6 Benjamin and Kochin also tested their hypothesis with additional series for juvenile
unemployment (for 1924–35) and for male and female unemployment (for
1923–37), but these series are not reproduced in the data set. 7 Wages are from Agatha L. Chapman, Wages and Salaries in the United Kingdom,
1920–1938, Cambridge University Press, 1953; benefits from Eveline Burns, British
Unemployment Programs, 1920–1938, Social Science Research Council, 1941; unem-
ployment from official Ministry of Labour statistics reproduced in Board of Trade,
Statistical Abstract of the United Kingdom, HMSO, 1936 and 1939, and net national
product from Feinstein, National Income.
Index numbers
B.1 Price and quantity relatives
In §1.7 we noted that many of the series used in quantitative analysis of his-
torical trends and fluctuations take the form of index numbers of the
changes in either prices or quantities, and gave a few examples of controver-
sies in which they have figured prominently. The aim of the present appen-
dix is to outline the main principles involved in the construction of index
numbers. Slightly more formal presentation of the principal concepts and
definitions is also given in panels B.1 and B.2 for those comfortable with the
algebra.
It may help to clarify the nature of a proper index number if we refer first
to price and quantity relatives. These are not index numbers, although they look exactly like them because they also have one item in the series
shown as 100. However these relatives are merely series that have been con-
verted from an absolute to a relative basis. All that this involves is choosing
some particular year for which the value of the series is taken as 100, and
then expressing all the other items in the series as a ratio to the value in the
chosen year. The advantage of doing this is that it makes it easier to see the
relationship between values; fewer digits are required and the comparative
dimension is more readily grasped.
To illustrate this simple procedure, data on the quantity of bituminous
coal produced in the United States each year from 1900 to 1913 are given in
column (1) of table B.1. The corresponding quantity relative is calculated in column (2) with 1900 as 100 and in column (3) with 1913 as 100. Both
relatives are rounded to one decimal point. In the same way a price relative could be constructed from data on (say) the annual average price of a
specific grade of raw cotton at New Orleans in cents per lb.
The important point to note about such relatives is that the relationship
between any pair of years is completely unaffected by either the switch to
the relative form or the choice of year to take as 100. For example, the
output of coal in 1910 is always 60.3 per cent higher than the output in
1902, whichever column is used for the computation. Relatives are useful,
but they do not provide any additional information that is not already con-
tained in the original series. The distinguishing feature of a proper index
number is that it does exactly that.
B.2 Two types of index number
Proper index numbers differ from such quantity or price relatives because they combine data that would otherwise have to be considered separately,
and thus provide additional information that could not be accurately
measured in any other way. Assume for example, that we wish to construct
a measure of the overall industrial production of the United Kingdom
from 1770 to 1830. We would need to combine data on the output of cotton
goods, woollen goods, iron, coal, beer, leather, and other industrial prod-
Table B.1 Bituminous coal output in the United States, 1900–1913, original data and two quantity relatives
(1) (2) (3)
Output Relative Relative
Year (000 tons) (1900�100) (1913�100)
1900 212,316 100.0 44.4
1901 225,828 106.4 47.2
1902 260,217 122.6 54.4
1903 282,749 133.2 59.1
1904 278,660 131.2 58.2
1905 315,063 148.4 65.9
1906 342,875 161.5 71.7
1907 394,759 185.9 82.5
1908 332,574 156.6 69.5
1909 379,744 178.9 79.4
1910 417,111 196.5 87.2
1911 405,907 191.2 84.8
1912 450,105 212.0 94.1
1913 478,435 225.3 100.0
Source: US Department of Commerce, Historical Statistics of the United States, series G13,
Washington, DC, 1949.
ucts. In principle, a separate series should be included for every good pro-
duced in the United Kingdom in these years. However, that is not normally
practicable, and instead a representative sample of products would be
selected.
How then should the output data for these products be combined? It
would clearly not be sensible to simply add together a mishmash of figures
for tons of coal, yards of cotton cloth, gallons of beer, and so on. It is neces-
sary to find some way of combining the data that circumvents the difficulty
created by the heterogeneity of the units in which the quantities are meas-
ured. Furthermore, there were enormous differences in the rates at which
individual industries increased production over this period. To take an
extreme example, the increase in the output of cotton yarn was a staggering
6,500 per cent, whereas that for beer was only a paltry 43 per cent. It is
therefore essential to combine the series in such a way that the overall index
reflects the relative importance of the various industries, and this requires
some definition of ‘relative importance’ and some system of weighting the
components.
There are a number of ways in which these requirements can be
satisfied, but in practice only two main types of index number are widely
used, and we will concentrate on these. One type uses the prices of a fixed
year to convert the component quantities to comparable values and simul-
taneously to weight them. This is known as a Laspeyres quantity index. The year used for the weights is referred to as the base year.
The other type uses a changing set of prices to value the quantities and is
known as a Paasche quantity index. In both cases a weighted arithmetic mean (see §2.2.2) is used to combine the different quantities. The year
shown as 100 for the index is called the reference year and, as we shall see, this may not be the same as the base year.
Each of these quantity indices has its corresponding price index. In the
Laspeyres price index the quantities of a fixed base year are used to weight the component prices. In the Paasche price index a changing set of quan- tities is used to weight the prices. The price and quantity indices are thus
perfectly symmetrical.
In order to illustrate some of the key features of these two types of
index number we have invented a very simple data set in which there are
only two commodities, cotton cloth and beer. Table B.2 sets out all the rel-
evant information on the prices and quantities of these two goods for
the three years, 1900, 1901, and 1902. The quantities refer identically to
the amounts produced in each year and the amounts consumed, and the
product of the prices and quantities in columns (3) and (6) can be inter-
preted as the value at current prices of the expenditure on the individual
§.
commodities.1 The corresponding aggregate expenditure is given in
column (7).
We will look first at the Laspeyres type of index with fixed weights, and
then in §B.5 at the alternative Paasche index with changing weights. To
illustrate the procedures without excessive repetition we will refer from
now on only to an index of prices, but everything said about the construc-
tion and properties of price indices in the remainder of this appendix is
equally applicable to the corresponding indices of quantities. The only
difference is that instead of using quantities to combine prices, a quantity
index is formed by using prices to combine quantities. A number of pos-
sible price and quantity indices for the two goods are given in table B.3. The
working for the price indices in (1)–(5) in the upper panel is given below;
the construction of the corresponding quantity indices in the lower panel
should be undertaken as an exercise.
B.3 Laspeyres index numbers with fixed weights
To begin with we will take 1900 as the base year. There are then two possible
methods that can be used to construct a Laspeyres price index. The first is a
ratio of aggregates. We can think of the quantities purchased in the base
year as a fixed basket of goods, and then calculate what it would cost to buy
this basket of goods at the prices prevailing each year. We thus work directly
with the actual prices of each year (from columns (2) and (5) of table B.2),
and the quantities of 1900 (from columns (1) and (4)).
Table B.2 Illustrative data for construction of price and quantity index numbers
(1) (2) (3) (4) (5) (6) (7)
Cotton cloth Beer Total
Quantity Price Expenditure Quantity Price Expenditure expenditure
Year (m units) ($/unit) (m $) (m units) ($/unit) (m $) (m $)
1900 1 10 10 2 15 30 40
1901 5 11 55 3 18 54 109
1902 10 12 120 4 20 80 200
Notes:
(3)�(1)�(2);
(6)�(4)�(5);
(7)�(3)�(6).
An index with 1900�100 is formed by dividing the resulting aggregate
cost for each year by the cost in the base year and multiplying by 100. The
calculations are:
1900 � 100 � � 100 � � 100 � 100.0
1901 � 100 � � 100 � � 100 � 117.5
1902 � 100 � � 100 � � 100 � 130.0
In each case, the first term in the brackets is the price of the commodity; the
second is the base-year quantity. It will be seen that this quantity (1 unit for
cloth, 2 units for beer) does not change.
This is the index given in column (1) of table B.3. It has a very straight-
forward interpretation: the cost of purchasing the 1900 basket of goods
would be 17.5 per cent more at 1901 prices than it was at the actual prices
prevailing in 1900, and it would be 30 per cent more at 1902 prices.
This aggregate method reveals most clearly the fundamental nature of an
index number and should be carefully studied. However, it involves working
with all the detail of the original prices and quantities, and in a real-life cal-
culation there would of course be many more than two commodities.
It is usually simpler, therefore, to use a second method, in which the
52
40
12 � 40
40
(12 �1) � (20 �2)
(10 �1) � (15 �2)
47
40
11 � 36
40
(11 �1) � (18 �2)
(10 �1) � (15 �2)
40
40
10 � 30
10 � 30
(10 �1) � (15 �2)
(10 �1) � (15 �2)
§.
Table B.3 Price and quantity indices for the data in table B.2 (1900�100)
(1) (2) (3) (4) (5)
Laspeyres Laspeyres Paasche Fisher
(1900 (1902 (Given year Ideal Divisia
weights) weights) weights) index index
Price indices
1900 100.0 100.00 100.00 100.00 100.00
1901 117.5 113.75 114.74 116.11 116.12
1902 130.0 125.00 125.00 127.48 127.73
Quantity indices
1900 100.0 100.00 100.00 100.00 100.00
1901 237.5 230.77 231.92 234.69 236.25
1902 400.0 384.62 384.62 392.23 394.06
prices are first converted to price relatives with the fixed base year (in this
case 1900) as 100. These relatives are then multiplied by an appropriate
measure of the relative importance of the item to obtain a weighted average
of the price relatives. For the present example the appropriate weights are
the share of each product in total expenditure in the base year, as given by
the data for 1900 in columns (3), (6), and (7) of table B.2.
In our example, the successive relatives are 100, 110, and 120 for cloth;
100, 120, and 133.33 for beer. Because the weights do not change in the
Laspeyres index it is convenient to express them as a proportion, and then
use these fixed proportions in the calculations for each year.a The 1900 pro-
portion for cloth is thus 10/40�0.25, and for beer it is 30/40�0.75. The
required index with 1900�100 is then formed and the results obtained are
identical to those given by the aggregate method:
1900 (100�0.25)�(100�0.75) �(25�75) �100.0
1901 (110�0.25)�(120�0.75) �(27.5�90) �117.5
1902 (120�0.25)�(133.33�0.75) �(30�100) �130.0
The aggregate method and the weighted average of the price relatives
appear to follow different routes but the underlying arithmetic operations
are actually identical, and the two methods will always give the same result
provided that the year taken as 100 for the relatives is the same as the base
year. If it is not, the outcome is not an alternative measure of the change in
prices, it is simply wrong. The way in which failure to observe this crucial
condition will generate a meaningless result is demonstrated in the alge-
braic presentation in panel B.1.
With realistic data sets, the second method – using relatives – is gener-
ally easier to use, and is the method most commonly adopted for the con-
struction of index numbers. Because the two methods give identical
results, users of index numbers frequently refer to the shares in expendi-
ture as though these were the true weights, but this is not correct. As is
demonstrated in panel B.1, the true weights of the Laspeyres price index,
regardless of whether it is compiled by the first or the second method, are
the actual quantities in the base year.
This distinction matters for historians and other users of index
numbers, because changes in expenditure shares reflect changes in both
prices and quantities, whereas it is only the changes in quantities that are rel-
a Expressing the weights as proportions (rather than percentages) has the further advantage that
the sum of the weights is 1, and it is thus unnecessary to divide the sum of the weighted relatives
by the sum of the weights; compare (2.1a) in §2.2.2.
evant when analysing what effect different sets of weights might have on
the movement in a price index.b
The appropriate quantities to use for the weights are determined by the
nature of the price index. For example, for an index of farm prices it would
be the quantities of the different farm products sold in the base year; for an
index of the cost of living for pensioners it would be the quantities pur-
chased by pensioners in the base year; for an index of share prices the
number of shares issued by the company in the base year; and so on.
For index numbers, unlike relatives, the chosen base year is potentially a
crucial determinant of the measurement. It is thus important that the year
selected as the base for a price index should not be one in which the quan-
tities were seriously distorted by wars, strikes, or other abnormal events.
Indeed, it is quite common to base an index on an average of several years
rather than a single year, in the expectation that this will minimize any dis-
tortions in the relative quantities.
However, even when the selection is confined to broadly ‘normal’ years,
the results obtained with different base years may differ markedly because
the relative quantities used as weights will change in response to the under-
lying economic processes.
B.4 The index number problem and chained indices
This possibility of obtaining different results with different base years
brings us to the famous index number problem. When the increase in prices between 1900 and 1902 was measured using the Laspeyres formula
with 1900 quantities as weights we found a rise in prices from 100 to 130,
an increase of 30 per cent.
What happens if we repeat the calculation using 1902 quantities as
weights? If we choose the second method, the price relatives for 1900 and
1901 must be recalculated with 1902�100. For 1900, they are 83.33 for cloth
and 75 for beer; for 1901 they are 91.67 and 90. The respective 1902 expendi-
ture shares are 120/200�0.6 and 80/200�0.4. The calculations are thus:
1900 (83.33�0.6)�(75�0.4) �(50�30) � 80
1901 (91.67�0.6)�(90�0.4) �(55�36) � 91
1902 (100.0�0.6)�(100�0.4) �(60�40) �100
§.
b See, for example, the discussion of alternative weights in Charles H. Feinstein, ‘Pessimism per-
petuated, real wages and the standard of living in Britain during and after the industrial revolu-
tion’, Journal of Economic History, 58, 1998, pp. 640–1.
Panel B.1 Laspeyres index numbers
The underlying structure of index numbers can be seen more clearly if we
use some simple notation. Consider a bundle of commodities, such as those
in table B.2, each of which has a price, P, and a quantity, Q. Subscripts (P0,
P1, ...and Q0, Q1, …) indicate the year to which the information relates. The
value of the expenditure on any one commodity in year 0 at the prices of
year 0 would be the product, P0Q0, and for the value of the expenditure on
all commodities in that year we use the summation sign, �, and show this as
�P0Q0.
What will a Laspeyres price index look like? If we take year 0 as base
year, the aggregate method involves weighting the prices of each year (P0, P1,
P2 ...) by the quantities of the base year (Q0). This is summed over all com-
modities. It is then expressed as an index – with year 0 as 100 – by dividing
each year by �P0Q0 and multiplying by 100. This price index is set out in
column (1) below. The corresponding Laspeyres quantity index is given in
column (2).
(1) (2)
Laspeyres Laspeyres
Year price index quantity index
0
1
2
By running one’s eye down the left-hand side of the successive formu-
lae in column (1) it can be seen that for each year the prices of that year are
being compared with prices in the base year, always using the quantities of
the base year as weights. Similarly, on the right-hand side of the quantity
index in column (2), the quantities for each year are being compared with
quantities in the base year, using the prices of the base year as fixed
weights.
The second method of constructing the same price index involves
weighting the price relatives for each commodity by the base year share of
�P0Q2 �P0Q0
� 100 �P2Q0 �P0Q0
� 100
�P0Q1 �P0Q0
� 100 �P1Q0 �P0Q0
� 100
�P0Q0 �P0Q0
� 100 �P0Q0 �P0Q0
� 100
This index has 1902�100. In order to make it more easily comparable
with the previous index calculated with 1900 weights it is helpful to express
these results relative to 1900. If 1900 is scaled up from 80 to 100, then 1901
increases proportionately from 91 to 113.75, and 1902 from 100 to 125 (as
in column (2) of table B.3). Note that this scaling procedure changes the
reference year to 1900 but the base year is not affected by this: it remains
§.
that commodity in total expenditure. For example, for the price indices in
years 1 and 2 (year 0�100) this would be
� 100 and � 100
Since the P0 in the denominator of the price relative for each separate
commodity would cancel out with the P0 in the expenditure share for that
commodity, this will give a formula identical to the one in column (2)
above. Similarly, in the corresponding formula for a quantity relative, the
Q0 in the denominator of the relative would cancel out with the Q0 in the
expenditure share, leaving exactly the same formula as the one in column
(2) above.
The formula for the use of relatives demonstrates two important points
about index numbers. First, the fact that the formula for the second method
reduces in this way to the one used for the first method shows that the
weights for the index are actually the base year quantities, and are not the
base year expenditure shares. Secondly, it is easy to see that if the year taken
as 100 for the relatives is not the same as the base year, then the terms would
not cancel out in the formula above, and the result would be a meaningless
expression.
For example, if the index is calculated using year 0 expenditure shares
while the relatives have year 2 as 100, the result for year 1 would be
� 100
� � 100
and it would be impossible to give this a sensible interpretation.
� P1P0Q0
P2 �P0Q0
��P1P2�P0Q0 �P0Q0
��P2P0�P0Q0 �P0Q0
��P1P0�P0Q0 �P0Q0
1902. We thus find that with 1902 weights the change in prices between
1900 and 1902 is not a rise of 30 per cent, but of only 25 per cent.
The fact that the increase in prices in our simple example is greater with
1900 weights than with 1902 weights is not accidental. The data used for our
simple two-good economy exhibit one of the fundamental characteristics
of a market economy: an inverse relationship between relative movements
in prices and quantities. Cotton cloth shows a smaller increase in price than
beer, and a larger increase in output. Inverse relationships of this type are an
essential feature of index numbers, and one of which historians should
always be acutely aware.
They are driven by powerful forces on both the supply and the demand
sides of the economy. On the demand side, consumers generally tend to
switch their consumption in favour of those goods that are becoming rela-
tively cheaper. On the supply side, it is usually the case that the greater the
output, the lower the cost, and so products with the largest expansion in
output tend to have the smallest increase in price (the process known to
economists as economies of scale). However the process is initiated, these
two forces interact cumulatively: greater demand stimulates increased
output, and higher output reduces prices and increases demand.
For any economy that experiences this process, a price index con-
structed with a base year early in the period under review will always show a
greater increase in prices than one constructed with a late base year. This
occurs because the early-year index gives a relatively larger weight to those
products that show a relatively large increase in prices.
By contrast, by the end of the period, the production of those relatively
more expensive goods will have declined in relative importance, and the
quantity weights attached to them will be correspondingly lower than they
were at the beginning of the period. In our simple example 2 units of beer
were consumed for every 1 unit of cloth in 1900, but by 1902 the ratio was
only 0.4 units of beer for every unit of cloth.
Exactly the same tendency applies to quantity indices. A quantity index
constructed with a base year early in the period will always show a greater
increase in quantities than one constructed with a base year late in the
period. The more rapid the rate of structural change in an economy, the
more important this phenomenon will be.
One of the most striking historical examples of this effect occurred in
the period of exceptionally rapid economic change in the USSR between
1928 and 1937. At the beginning of this period the economy had relatively
few machines and other capital goods, and consequently their price was
relatively very high. By the end of the period of Stalin’s programme of
forced industrialization and collectivization, it was food and consumer
goods that were scarce and expensive relative to machinery.2 As a result of
this structural shift there were marked discrepancies in the results accord-
ing to whether index numbers of output used early or late years for their
weights, and similarly for prices.
For example, a Laspeyres index of industrial production with early-year
weights gave a relatively high weight to the initially relatively expensive –
but fast-growing – machinery, and a low weight to the initially relatively
cheap – but slow-growing – consumer goods. Conversely, late-year weights
were relatively low for the machinery and relatively high for the consumer
goods. One of the most reliable estimates of the GNP of the USSR for 1928
and 1937 showed an increase of 175 per cent when measured at the prices
of 1928, compared to only 62 per cent when 1937 was taken as the base
year.3
Given the possibility of more than one answer, what should the histo-
rian do in practice? In statistical terms, both early-year and late-year
answers are equally valid and there is no justification for choosing one
rather than the other.4 Furthermore, the fact of a marked divergence
between the early-year and late-year measures is a valuable signal to the
historian that there has been significant structural change. Notwith-
standing this, it may be desirable for some purposes to have a single
measure of the change in prices or quantities, and there are a number of
possible ways in which this can be accomplished.
Fisher Ideal and Divisia indices One procedure that can be adopted in cases where the early-year and late-
year indices show markedly different results, is to calculate what is known
as the Fisher Ideal index. This is simply a geometric mean (see §2.2.1) of the results derived from the two alternative indices. For example, for the
two price indices calculated with the data of table B.2, the Fisher Ideal price
index would be 116.110 for 1901 and 127.475 for 1902.
This index also has a number of attractive statistical and theoretical
properties. One of the former is that the product of a Fisher price index and
the corresponding Fisher quantity index is exactly equal to the index of the
value of total expenditure. The change in value can thus be perfectly
decomposed into these two measures of price and quantity change.
Another procedure sometimes recommended is a Divisia index. Unlike the other indices we have considered, the Divisia is based on the growth of
prices (or quantities) calculated over intervals as short as possible – with
historical data this is usually a year.5 The growth of each item in the index is
calculated as the difference of the logs, and this difference is then weighted
by the arithmetic average of its share in total expenditure at the beginning
and at the end of each period over which growth is calculated. The growth
of the index is then obtained as the exponential of this weighted average,
§.
and the series can then be reassembled as an index by starting at 100 and
multiplying by the successive exponentials of the weighted averages.
The procedure may seem somewhat complicated, and we can illustrate
it with a Divisia price index based on our simple example. From 1900 to
1901 the growth in prices is (log 11 log 10)�(2.3979 2.3026)�0.0953
for cloth, and (log 18 log 15)�(2.8904 2.7081)�0.1823 for beer.
The arithmetic mean of the weights for the beginning and end of this
year are 0.5�(10/40�55/109)�0.3773 for cloth and 0.5�(30/40�
54/109)�0.6227 for beer. So the weighted average growth for this year is
(0.3773�0.0953)�(0.6227�0.1823)�0.1495. The exponential of this is
1.16125, or a rate of growth of 16.125 per cent.
The corresponding calculation for the growth rate from 1901 to 1902
would be (0.5523�0.0870)�(0.4477�0.1054)�0.0952 and the expo-
nential of this is 1.0999, or a growth rate of 9.99 per cent.
Taking 1900 as 100, the Divisia price index would thus show an increase
of 16.125 per cent on 100 to equal 116.12 for 1901; and a further increase of
9.99 per cent on this, to reach 127.73 for 1902. This gives an index (see
column (5) of table B.3) that is fractionally higher in each year (relative to
1900) than the corresponding Fisher Ideal index. As can be seen from
columns (1) and (2) of table B.3, it also falls between the indices with early-
year and late-year weights.c For the corresponding quantity indices, given
in the lower panel of table B.3, the differences are marginally larger
(because the growth of the quantities is more rapid than the growth of the
prices) but the overall pattern relative to the other indices is the same.
The Divisia indices should also provide a perfect decomposition of the
change in value into changes in price and in quantity, but because growth is
calculated over discrete intervals (rather than continuously), the decom-
position is not exact.
Chained indices The second way of dealing with the index number problem is to construct a
chained index which links (or splices) a succession of shorter price indices in order to cover a longer span of time than would be appropriate with a
single base year. These successive indices form a single continuous series,
with a single reference year but a number of underlying base years. The
actual procedure for doing this is usually no more than an arithmetic
adjustment based on one or more overlapping years. In this way the new
c For the use of Fisher Ideal and Divisia indices with historical data, see N. F. R. Crafts, British
Economic Growth during the Industrial Revolution, Cambridge University Press, 1985, p. 26.
subindex is brought up to the same level as the old subindex over those
years, and the subsequent years are scaled proportionately, as illustrated in
column (3) of table B.4.
There are two main reasons for adopting this procedure. The first is that
over time it usually becomes impossible to continue the components of a
single index. Some goods either cease to be produced (horse carriages) or
their quality changes so radically (computers) that those in the market at
the end of the period are no longer comparable with those produced at the
beginning. It is then simply not possible to find market prices in the year
2000 which could be used to value directly either a carriage of the type con-
structed in 1800, or a computer of the type produced in 1950.
The second is that even when this extreme situation does not arise, a
chained index provides a means of circumventing the index number
problem by constantly revising the base year so that the quantities used are
always up to date and thus fully relevant as weights for the constituent
prices.d Each subindex is thus appropriate to its specific time period, and
the gap between the early-year and late-year weights is too short to matter.
For contemporary index numbers generated by official agencies this is now
standard practice. Researchers working with historical series may not be
§.
d The Divisia index can be interpreted as a chained index which is rebased in every time period; in
our example every year.
Table B.4 Chained index numbers with changes in the reference year (imaginary data)
(1) (2) (3) (4)
First Second Chained Same index with a
subindex subindex index new reference year
(1900�100) (1902�100) (1900�100) (1905�100)
1900 100 100 20
1901 150 150 30
1902 250 100 250 50
1903 120 300 60
1904 160 400 80
1905 200 500 100
able to do this if the data required are not available, but the chaining proce-
dure should be considered whenever possible.e
It is worth repeating that the year shown as 100 in a chained index will
not be the base year except for – at most – one of the subperiods. All other
subperiods in the index will have their own base year. Since, as we have
already demonstrated, the actual base year can have a substantial impact on
any measurement derived from index numbers, it is highly desirable to
identify the true base year in any long-run historical series.
The way in which chaining can conceal the true base year is illustrated
by the example given in table B.4. One imaginary index is given in column
(1) for 1900–2 with 1900 as the base year, and a second in column (2) for
1902–5 with 1902 as base year. The two indices could then be spliced in
1902 as shown in column (3). The resulting spliced index is then converted
in column (4) to an index with 1905 as the reference year.
This final procedure has not altered the pattern of price change in any
way, but it has effectively suppressed the actual base year. The good histo-
rian should always take pains to uncover the truth, so that her evaluation of
the change in the indices is not distorted by misunderstanding of the true
base year used to weight the index.
B.5 Paasche index numbers with changing (given year) weights
The Paasche type of index number follows the same underlying principles as the Laspeyres index, but instead of using the fixed quantities of a single
base year to weight the component prices it uses the changing quantities of
successive years. Thus each successive year in a time series (also referred to
as the given year) is compared with the selected reference year. So if 1900 is chosen as the reference year for the data set of table B.2, the first item in the
Paasche index compares 1901 prices with 1900 prices, using 1901 quan-
tities as the weights. The next item compares 1902 prices with 1900 prices,
using 1902 quantities as the weights, and so on.
If we use the aggregate method to make the calculation, we have first the
comparison of 1900 and 1901 with 1901 weights
1900 � 100 � � 100 � � 100 � 87.15 95
109
50 � 45
55 � 54
(10 �5) � (15 �3)
(11 �5) � (18 �3)
e For an example of such linking for a nineteenth-century series see Charles H. Feinstein,
‘Pessimism perpetuated, real wages and the standard of living in Britain during and after the
Industrial Revolution’, Journal of Economic History, 58, 1998, p. 634; and for linking of constitu-
ent series over a much longer period, Henry Phelps Brown and Sheila V. Hopkins, A Perspective
of Wages and Prices, Methuen, 1981, p. 40.
and then the comparison of 1900 and 1902 with 1902 weights
1900 � 100 � � 100 � � 100 � 80.00
Compared to the corresponding Laspeyres calculation in §B.3, the first
term in the brackets is still the price of the commodity in the current year,
but the second term for the quantity weights is no longer fixed: it changes
from year to year. As result, we have two separate comparisons, one with
1901 as 100 and as base year, the second with 1902 as 100 and as base year.
The interpretation is again straightforward. The basket of goods pur-
chased in 1901 would have cost 12.85 per cent less if it had been purchased
at the prices prevailing in 1900. The basket of goods purchased in 1902
would have cost 20 per cent less if it had been purchased at 1900 prices.
In order to obtain a single continuous index we can select 1900 as the
reference year (though it is not the base year) and then invert the previous
results. This says, in effect, if 1900 is 87.15 when 1901�100, then if 1900 is
raised to 100, 1901 becomes 114.74. Similarly, if 1900 is 80 when 1902�
100, then if 1900 is raised to 100, 1902 becomes 125. This result is shown as
column (3) of table B.3.
Alternatively, exactly the same result can be obtained by multiplying the
price relatives by the expenditure shares of the given year. The first step is to
calculate the required proportions and relatives. When 1901 is the given
year, the respective shares of cotton cloth and beer are 55/109�0.5046 and
54/109�0.4954. It is again essential that the reference year for the relatives
must be the same as the base year. As demonstrated in panel B.2, if this rule is
not observed the result is completely meaningless. When 1901 is the given
year, the price relatives are thus 10/11�100�90.91 for cloth and 15/18�
100�83.33 for beer.
Secondly, as we have already seen with the aggregate method, the initial
form of the Paasche index is not a single continuous index but a succession
of binary comparisons. Thus if we simply take the first step of calculating a
weighted average of the price relatives as given above – in the same way as
we did for the Laspeyres index in §B.3 – the results we would get for 1900
and 1901 with 1901�100 are
19001(90.91�0.5046)�1(83.33�0.4954)�(45.87�41.28)�187.15
1901(100.0�0.5046)�(100.0�0.4954)�(50.46�49.54)�100.01
In order to convert this into an index with 1900�100 it is again neces-
sary to invert these results, raising 87.15 to 100 and 100 to 114.74. This
is equivalent to taking the reciprocal of the weighted average of the price
160
200
100 � 60
120 � 80
(10 �10) � (15 �4)
(12 �10) � (20 �4)
§.
Panel B.2 Paasche index numbers
The formula for the Paasche price index is set out in column (1) below, with
year 0 taken as the reference year�100. The corresponding Paasche quan-
tity index is given in column (2).
(1) (2)
Paasche Paasche
Year price index quantity index
0
1
2
Running one’s eye down the successive Paasche formula shows how for
this index the weights change every year. Thus for each year in the price
index the prices of that year are compared with the reference year prices,
using the quantities of the given year as weights: Q1 for year 1, Q2 for year 2,
and so on.
An important feature of the Paasche indices is that, strictly speaking,
each given year should only be compared directly with the reference year. If,
instead, a Paasche index is used to measure the change in prices between
years 1 and 2 what we actually have is
�
In our numerical example this corresponds to 125/114.74�108.94 (see
column (3) of table B.3). This change between years 1 and 2 is thus meas-
ured as the product of the change from year 2 to year 0 measured with year 2
weights, and the change from year 0 to year 1 measured with year 1 weights.
Apart from its complexity this is awkward because the measured change is a
combination of the ‘pure’ change in prices and the change in weights from
year 2 quantities to year 1 quantities, and may be distorted by the latter.
The alternative method of calculating a Paasche price index is to take
the reciprocal of the price relatives multiplied by the given year expenditure
shares. For example, for the index for year 1 this would be
�P2Q2 �P0Q2
� �P0Q1 �P1Q1
�100 �P2Q2 �P0Q2
� �P1Q1 �P0Q1
�100
�P2Q2 �P2Q0
� 100 �P2Q2 �P0Q2
� 100
�P1Q1 �P1Q0
� 100 �P1Q1 �P0Q1
� 100
�P0Q0 �P0Q0
� 100 �P0Q0 �P0Q0
� 100
§.
As long as the prices are expressed relative to the given year, the P1s in
the denominator of the relative and the numerator of the expenditure share
will cancel out to give
� 100
and when this is inverted the formula is identical to the one for year 1 in
column (1) above.
Students (and others) often omit these two critical elements in the pro-
cedure for a Paasche index based on relatives. If the reciprocal is omitted,
the index measures the prices of year 0 relative to those of year 1, instead of
the reverse. If the price relative is not recalculated every year to have the
given year as 100 the result is quite meaningless. For example, if the price
relative for year 1 had been taken with year 0 as 100 the calculation would be
�
a complex expression for which there is no intelligible interpretation.
It is because of possible pitfalls of this type that it is so important to
think of index numbers in terms of a notation such as the one adopted in
this panel. It is always possible to make an arithmetic calculation which
looks correct on the surface, but it is only by scrutiny of its underlying struc-
ture that one can decide whether or not the index is sensible.
�P1Q1 P12Q1
P0
1/ ��P1P0�P1Q1
�P1Q1
�1 /�P0Q1�P1Q1�
�1/��P0P1�P1Q1 �P1Q1
��100
relatives and then multiplying by 100, and this is the form in which a
Paasche index is often calculated. (It is presented in this form in panel B.2.)
We can go directly to this version of the procedure to calculate the index
for 1902 with 1900 as the reference year. When 1902 is the given year, the
respective shares in expenditure are 120/200�0.6 and 80/200�0.4, and
the price relatives (now with 1902 as 100) are 10/12�100�83.33 for cloth,
and 15/20�100�75 for beer. The full calculation is thus
[(1 � 100)/(83.33�0.6�75�0.4)]�100�[100/(0.50�0.30)]�100�125
This is an essential but rather convoluted procedure, and Paasche
indices can be derived by an indirect route; indeed this is sometimes done
without the researcher realizing that what he has calculated is in fact a
Paasche index. The indirect route exploits the fact that the change in any
series at current values is made up of a combination of either a change in a
Laspeyres quantity index and a Paasche price index, or a change in a
Laspeyres price index and a Paasche quantity index.
Thus, for example, if the current value series represented by the expen-
diture series in column (7) of table B.2 is expressed as an index with 1900�
100, and then divided by the Laspeyres quantity index with 1900 weights
(as in column (4) of table B.3), the result will be the Paasche price index
exactly as calculated above for column (3) of that table.
There are thus three combinations of price and quantity indices by
which we can obtain an exact decomposition of an increase in total expen-
diture. For our two-commodity example, the various possibilities for the
fivefold increase between 1900 and 1902 are
� a Paasche price index (125) and a Laspeyres quantity index (400)
� a Laspeyres price index (130) and a Paasche quantity index (384.62)
� a Fisher Ideal price index (127.47) and a Fisher Ideal quantity index
(392.23)
On the basis of the first of these decompositions one could say, there-
fore, that the increase in the value of total expenditure from 100 to 500 con-
sisted of an increase in prices of 25 per cent and an increase in quantities (a
rise ‘in real terms’) of 300 per cent. The fact that the other two methods
would give a different partitioning between price and quantity is simply
another manifestation of the index number problem.
Finally, it should be noted that in the perspective of a comparison of
1900 and 1902, the calculation of the Paasche price index for 1902 (with
1902 quantities as weights) is identically equal at 125 to the result that
would be obtained using a Laspeyres price index with 1902 as the base year,
because the latter also has 1902 quantities as weights. This can be seen in
the figures for 1902 in the upper panel of table B.3 (and for the correspond-
ing quantity indices in the lower panel, for which both are equal to 384.62).
For this reason the index number problem discussed in §B.4 is some-
times presented as the difference between a Laspeyres and a Paasche
measure, but this arbitrarily assumes that the base of the former is an early
year and the base of the latter a late year, neither of which is necessarily
correct.
Notes 1 The product of the prices and quantities might also be interpreted as the value of the
output of beer and cloth produced. However, the measure of output relevant for the
construction of a quantity index of output, such as an index of industrial produc-
tion, would be the value added, i.e. for each constituent industry, the difference
between the value at which its products are sold and the value of the raw materials,
fuel, and other inputs which it purchased. The corresponding ‘value added prices’
are thus not the same as the prices at which the goods were purchased by consumers.
The prices in table B.2 are assumed to be consumer prices and so we refer to the series
in columns (3), (6), and (7) as the value of expenditure. It is measured at ‘current
prices’, i.e. at the prices prevailing in each year. 2 For a very clear and full discussion of these issues see Abram Bergson, The Real
National Income of Soviet Russia since 1928, Harvard University Press, 1961, pp.
25–41; and Janet G. Chapman, Trends in Consumption in the Soviet Union, Rand
Corporation, 1964, pp. 27–44. 3 Bergson, Real National Income, p. 217. For illustrations of a substantial index
number problem in relation to measures of industrial production in the United
Kingdom between 1924 and 1948 and in the United States between 1909 and 1937,
see W. E. G. Salter, Productivity and Technical Change, Cambridge University Press,
1960, pp. 151–2, 170. 4 There are, however, substantial differences in the underlying economic interpretation
of the alternative index number formulae. For an introduction to these principles see
Robin Marris, Economic Arithmetic, Macmillan, 1958, pp. 227–83 or Dan Usher, The
Measurement of Economic Growth, Basil Blackwell, 1980, pp. 12–64. The economic
principles are discussed briefly in an historical context in N. F. R. Crafts, British
Economic Growth during the Industrial Revolution, Cambridge University Press,
1985, pp. 25–8. 5 The form of this index used for empirical work is an approximation to a theoretical
version which cannot be applied in practice because it treats time as a continuous
variable, whereas empirical work can be undertaken only with discrete periods of
time such as a month or a year.
Figure 8.2 Ballantine for a regression with two explanatory variables
Y
X1 X2
Figure 8.3 Ballantine for a regression with two explanatory variables
Y
X1 X2
Y
X1 X2
(a) With two independent explanatory variables
(b) With two explanatory variables which are not independent
Figure 11.2 Omitted and redundant variables
X1
X2
Y
X1
X2
Y
(a) Ballantine to show the effect of a missing variable
(b) Ballantine to show the effect of a redundant variable
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arithmetic mean, see mean asymptotic significance, 222(n.7) autocorrelation, 312–6, 325(n.2), 326(n.3),
439 in cross-section data, 313–4 see also Breusch–Gordon test,
Cochrane–Orcutt correction, Durbin–Watson test, Hildreth–Liu correction
autoregression, 291(n.), 316, 447–8
Ballantine, 110–111, 112(n.4,5), 240–2, 255(n.1), 303–5, see also pullout, p.527
bar charts, 36 Benjamin–Kochin (unemployment) data set,
101, 177, 248, 274, 307, 319, 343, 437, 438–45, 501–3
beta coefficients, see standardized beta coefficients
BLUE (best linear unbiased estimator), 267–8, 300–1, 309
Boyer (Poor Law) data sets, 8, 11, 33, 36, 40(n.), 120, 126, 139(n.), 249, 263, 280, 313–4, 463–80, 496–7, 498–9
Breusch–Gordon test for autocorrelation, 316, 448
Breusch–Pagan test for heteroscedasticity, 311, 453
causation, 72, 82, 93–5, see also reverse causation
censoring, 384, 385, 423–28 Central Limit Theorem, 122–5, 138, 190 central tendency, measures of, 42–6 chi-square distribution, 207–8, 223(n.9), 416,
415(n.), 448, 450, 455, 490 �2-test,
of goodness of fit, 216, 217, 224(n.15) of homogeneity, 223(n.11) of statistical significance, 186, 206, 202–10,
218
�2calc, 209 Chow test, 307–8, 442–5, 461(n.5) class intervals, 34–40, 65(n.1) classical linear regression (CLR)
models, 266, 300–26, assumptions of, 300–2, see also error term
Cochrane–Orcutt correction for autocorrelation, 315, 373
coefficient of correlation (r), 76, 76–86, 106–9 partial coefficients, 248, 248–54 test statistic for, 174 see also rank correlation
coefficient of determination (r2), 105, 103–7, 110–1
coefficient of multiple correlation (R), 254 coefficient of multiple determination (R2),
233, 232–3, 373–4, 439, 447, 453, 471, 478
effect of multicollinearity, 322 relationship to Fcalc, 269, 271
coefficient of multiple determination, adjusted (R–2), 233, 245, 439
coefficient of variation (CV), 50–51 collinearity, 239(n.), 242, 307(n.), see also
multicollinearity compound rate of growth, see growth rates confidence intervals, 136–43, 149
for a mean, 138, 136–40 for other sample statistics, 140–4, 441 for a prediction of the mean value of Y, 274,
273–6 for a prediction of a specific value of Y, 276,
276–7 for logit coefficients, 395
contingency coefficient (C), 218–9 contingency tables, see cross-tabulations control category, for dummy variables, 281 correlation, 71–4
and causation, 72, 326(n.4) see also coefficient of correlation
(Page references for boxes with definitions of the main terms and concepts are shown in bold type.)
Subject index
covariance (COV(XY)), 80, 79–81, 89(n.1), 111 Cramer’s V, 218–21 critical point, 136, 209 critical region, 156, 156–9, 162–3, 168–9, 193,
199 critical value, 136, 157, 187, 193, 199, 208–9,
217, 442 cross-section models, 464, 471–3, 476–80
and correlation, 83 and error specification, 310, 313 and model specification, 262, 307 and variable specification, 322 see also panel data sets
cross-section variables, 8–9 cross-tabulations, 203–5, 221 cumulative distribution functions (cdf),
388–9, 429(n.3)
d, 315, see also Durbin–Watson test deciles, 46–7 degrees of freedom (df), 132, 133, 180(n.4),
187, 207, 209, 211, 216, 246, 442 dependent and explanatory variables, 93–4,
108, 111(n.1), 321, 324–5 see also dependent variables, explanatory
variables dependent variables,
binary, 385, 387, 390, 429(n.1), 430(n.9) censored, 384, 385–6, 423–28 continuous, 384, 385 dummy, 385, 430(n.9) limited, 384–9 multi-response, 422–3 qualitative, 384
deviations from the regression line, see residuals
dfBeta test for outliers, 317, 444 dfFits test for outliers, 317, 444 dispersion, measures of, 47–51 distributed lag models, 290–1, 292–4 disturbance term, see error term Divisia index, 517–8, 519(n.) double logarithmic models, 343–5, 347–50,
353, 368, 476 dummy variables, 11, 280–9, 319, 372,
377(n.11) and regression coefficients, 473 as a change in the intercept, 283–5, 286,
306–8 as a change in the slope, 286–89, 306–8,
441–2, 443 as dependent variables, 385 use in fixed effects models, 454–5, 462(n.11) use in seasonal adjustment, 356–8 with two categories, 280–1, 484 with more than two categories, 280–3, 484–5
dummy variable trap, 281, 322, 484 Durbin–Watson (DW) test for
autocorrelation, 315–6, 326(n.3), 373, 439, 444(n.), 447–8
Durbin’s two-stage correction for autocorrelation, 315
DW statistic, see Durbin–Watson test
efficiency of a test, 156 elasticities and regression coefficients, in OLS
models, 349–51, in logit models, 406–11, in probit models, 419–21
endogenous and exogenous variables, 324, 465–70, 491–4
error-correction mechanisms, 377(n.9) error in hypothesis tests, two types of, 154,
153–6, 158, 161 error sum of squares (ESS), 442(n.), see also
residual sum of squares error term (e),
and non-linearity, 341(n.) assumptions of classical linear regression
(CLR), 264–8, 300 errors of measurement and specification,
261–2, 320–2 in random effects models, 455 non-zero means, 308–9 stochastic errors, 262–3 violation of CLR assumptions, 300, 308–19,
371–8, 429(n.2), 477 expected frequencies, 205, 209–10, 216 expected values, 385, 386, 396 explanatory variables, 93–4, 99, 108, 111(n.1)
and CLR models, 261–3, 301–2 effects in logit and probit models, 399–413,
415–8, 420–2 impact of additional variables, 235–45 lagged values, 289–94 multiple relationships, 231–5, 248–52 non-linear relationships, 347–51, 360–72 t-tests for regression coefficients, 175–9 use of dummy variables, 280–9
exponential constant (e), 16, 31(n.4), 56, 390–1
exponential curves, see models, exponential exponents, 14, 16, 17
F-distribution, 269–72 F-test, 185(n.), 269, 271, 307, 309, 442
effect of multicollinearity, 323 Fcalc, 269, 271–2, 273
first differences, 362, 370 first-order partial correlations, 249, 251–3 Fisher Ideal index, 517–8, 524 Fisher’s exact test, 206 Fisher’s z, 142–3, 145(n.9) fitting trends, see trends fixed effects model, 454–5, see also random
effects model fluctuations, 21–2, 25–7, see also trends frequency curves, 40–2 frequency distributions, 33–42, 125
cumulative, 34–5, 37, 42, 198–200, 217 relative, 34–5, 40–2, 198–200
frequency polygons, 40–2
geometric curves, see models, geometric geometric mean, 43(n.), 44(n.) Goldfeld–Quandt test for heteroscedasticity,
311 Goodman and Kruskal’s tau, 219–20
goodness of fit, of a regression line, 102–6, 261 of logit models, 413–4, 414(n.), 415 one-sample non-parametric tests, 216–7
grouped data, 51–3, 418, 430(n.12) growth rates, 352, 354–5, 359
Hadi’s test for outliers, 319, 443 Hatton–Williamson (emigration) data set, 44,
173, 248, 292, 313, 316, 322, 343, 361, 437, 445–60, 497, 499–501
HETERO, test for heteroscedasticity, 453, 455 heteroscedasticity, 309–11, 387, 431(n.2)
see also Breusch–Pagan test, Goldfeld–Quandt test, HETERO, Lagrange multiplier test, White test
Hildreth–Lu correction for autocorrelation, 315
histograms, 36–40, 58 homogeneity, test of, 223(n.11) homoscedasticity, 301, 309, see also
heteroscedasticity hypothesis testing, 149–79, 186–7
impact effect, in logit models, 399, 400, 400–2, 404–6, 410,
411, 412, 430(n.9) in probit models, 420–2
impact multiplier, 351 independence of irrelevant alternatives, 423 independent variables, see explanatory
variables index number problem, 513, 515–20 index numbers, 19–20, 507–25
chained index numbers, 518–9 instrumental variables, 321, 325, 470(n.) interaction variables, 286–9, 295(n.1), 441–2,
456, 462(n.12) intercept term (a), 14, 246–8, 283–6, 306–9
in a logit model, 395(n.) interval estimate, 123(n.), 137, see also
confidence intervals interval level data, 10, 31(n.1) Irish emigration data set, see
Hatton–Williamson data set
Kolmogorov–Smirnov test (KS Z), for two independent samples, 198–201, 223(n.9,10)
of goodness of fit, 216–7, Koyck distributed lag, 295(n.3), 316 Koyck’s transformation, 293–4, 295(n.4)
lagged dependent variables, 291–2, 372, 447 lagged variables, 13, 289–94, 321, 361, 446–9 Lagrange Multiplier (LM) test for
heteroscedasticity, 448 453 Laspeyres (price and quantity) index numbers,
510–3, 514–5, 524 levels of measeurement, 9–10 Levene’s test for equality of sample variances,
172(n.) leverage points, 316, 318, 443, see also outliers
lin-log models, see semi-logarithmic models logarithms, 16–9
use in non-linear models, 342–51, 360–1 see also double logarithmic models, semi-
logarithmic models logistic curves, see models, logistic logistic distribution functions, 388–9 logit coefficients, 391, 395–6, 403, 404–6, 410,
411 comparison with probit coefficients, 419,
431(n.13), 490–1 in model of decision to leave home, 485–8
logit functions, value of, 396–7, 403, 405, 411 logit models, 387–9, 390–418
derivation, 390–1 with grouped data, 418, 430(n.12)
logit model of decision to leave home, 483–91 log-likelihood ratio ( 2log�), 414, 415, 419,
490, 491 log-lin models, see semi-logarithmic models log-log models, see double logarithmic models log odds ratios, 397, 403–5, 410–1, 485 longitudinal data, 9, see also panel data long-run multipliers, 351, 377(n.5), 449–50
Mann–Whitney U-test, 195, 193–8, 199, 202, 222(n.4,5,6,8)
standardized test statistic for, 195 marginal effect,
in logit models, 399, 399–404, 406, 410, 411, 412
in probit models, 421 maximum-likelihood models, 392–5 McFadden’s R2, 414, 415, 419, 491 mean (X
– ), 43–4
with grouped data, 51–2 weighted, 44–6
mean deviation, 47–8 measurement errors, 261–2, 374–5
systematic, 320–1 stochastic, 320–1
measures of association, non-parametric, 86–9, 218–21, see also coefficient of correlation
median, 43–4 with grouped data, 51–2
mode, 43–4 models,
exponential, 335, 337, 345, 351 geometric, 335, 336, 343, 368 logistic, 335, 339–41, 359, 367, 378(n.17),
388, 390–1 quadratic (parabola), 335, 339, 342, 359,
369 reciprocal (inverse), 335, 338, 342 see also non-linear models
model specification, 259, 301, 302–8, 325(n.2), 359–72
best-practice procedures, 372–6 modelling relationships
choice of functional form, 360–2 general principles, 362–3 historical illustrations, 363–72
moving average, 22–25 multicollinearity, 242, 291, 321–3, 377(n.8) multinomial logit or probit models, 423,
431(n.17) multiple regression, see regression multi-response models
ordered, 422, 431(n.16) random, 422–3
natural logs, 16, 347(n.) nominal level data, 9, 186 non-linear models, 302–3, 333–76, 388–9,
440 choice of functional form, 360–3 estimation by OLS, 341–7 see also RESET test
non-parametric tests, 153, 185–224 criteria for choice of, 201–2 for two independent samples, 187–202 when to use, 185–6
non-rejection region, 157, 168–9 normal distribution,
properties, 56–61 and theoretical probability distributions,
128–31 and probit models, 388–9, 419
null hypothesis (H0), 152, 151–63, in t and Z-tests, 165–70 in non-parametric tests, 188, 190, 193, 194,
196–7, 198–201, 208–9, 212, 214, 215, 216, 217
implications of rejecting, 155
observed frequencies, 205, 209–10, 216 odds ratios, 396, 403, 405, 410–1, 415–6, 418,
485 odds-ratio coefficients, 415–6 ogive, 42, 388 omitted and redundant variables, 303–5, 371 one-tailed tests, 158–9, 190(n.), 198, 212, 215,
217 ordinal level data, 9–10, 86, 186 ordinary least squares regression (OLS),
comparison with logit coefficients, 395–6, 401
effect of outliers, 316, 319 in linear regression, 98–9, 102, 242–3, 244,
266, 267–8 in non-linear regression, 341–2, 477–8 when not appropriate, 384, 387–8, 422,
424–5, 429(n.2), 466, 468 orthogonality, 239(n.), 241, 242(n.), 304(n.) out-of-sample prediction, 375, 443 outliers, 74–75, 316–9, 443–4, 461(n.5),
see also dfBeta, dfFits, Hadi’s test
Paasche (price and quantity) index numbers, 509–11, 520–5
panel data sets, 9, 451–3, 454–60 parameter stability, 305–8, 325(n.1), 440–3,
461(n.5), see also Chow test parametric tests, 185–7,
criteria for choice of, 201–2
partial correlation coefficients, see correlation coefficients
partial regression coefficients, see regression coefficients
Pearson product-moment correlation coefficient, 76
percentiles, 46–7 � (phi), measure of association, 218–9 piecewise linear regression, 308, 441 point elasticity,
in logit models, 406–9, 408, 410, 411, 412 in probit models, 421–2
point estimate, 123(n.), 137 polynomials, 335(n.), 359, 377(n.6) pooled data, 9, 453, 454–5, see also panel data Poor Law data set, see Boyer data sets population, 10–11, 117–9
characteristics, symbols for, 120 finite and infinite, 224(n.12), 263 parameters, 11, 119–20, 122–5, 263–4
power of a test (1–�), 156, 201, 202(n.) powers, 14, 16 probability density function, 129 probability distributions, 125, 128–36, 164,
207–8, 269, 270, 272 probability of an event, in logit models, 386,
390–1 average predicted value (P̂Y), 398
probit index, ! (upper case phi), 420–2, 431(n.15)
probit models, 387–9, 419–22, 428, 490–1 with grouped data, 431(n.14)
prob-value, 161, 162, 168, 187, 201, 215, 460(n.1), 471, see also significance levels
pseudo R2, 413, 415, 491 p-value, 162, see also significance levels
quadratic curves, see models, quadratic quantity indices, 19 quantity relatives, 19, 507–8 quartiles, 46–7 quartile deviation, 47
random effects model, 454–5, see also fixed effects model
randomness, one sample runs test, 210–5 range, 47 rank correlation coefficient (rs), 86–9, 174(n.),
202, 218 reduced form models, 465–6, 469, 471–2,
491–4 reference category, for dummy variables, 281 regression
bivariate (simple) linear, 93–111 fitting the least square line, 96–100 fitting the least squares plane, 242–3, 244 multiple (multivariate) linear, 231–55 non-linear, 333–59, 477–81
regression coefficients (b), 99 interpretation in linear and non-linear
models, 347–9, 448–51, 455–60, 461(n.9), 471, 473–5, 478–80
partial, 234, 234–45
relationship to correlation coefficients, 106–9, 111
test statistic for, 176 research hypothesis (H1), 152, 154–5, 158, 170,
198, 208, 212 RESET test of non-linearity, 361(n.) residual sum of squares (RSS) 105, 442,
448(n.) see also variation
residuals, 96–8, 102–6, 272, 302, 374 and heteroscedasticity, 311 and outliers, 319(n.)
reverse causation, 94, 326(n.4), 439–40 robust regressions, 319, 444 rogue observations, 317–9, 443, see also
outliers runs tests,
Wald–Wolfowitz, 188–93, 214 of randomness, 210, 212–5
s-shaped (sigmoid) curves, 388–9, 419, 481, see also models, logistic
sample characteristics, symbols for, 120 sample size, 48(n.), 122, 124, 133–4, 187, 190,
195–6, 198–9, 201–2, 209, 217, 218, 221(n.2)
sample statistics, 11, 119–20 and confidence intervals, 136–44
sample variances, 269 samples, 11, 117–9
independent, 187–8 random, 11, 153 related, 187(n.), 188 truncated, 386(n.)
sampling, 126–7, 144(n.1) and replacement, 11, 126–7, 145(n.5)
sampling distributions, 120–5, 122, 151, 163–4, 190–1, 194–5, 198, 269–71
sampling error, see confidence intervals sampling independence, 126, 145(n.5) scatter diagrams, 72–3 seasonal adjustment, 21–2, 27–30, 356–8 semi-logarithmic models, 343, 345–51, 353,
358–9, 365, 378(n.18), 447 serial correlation, 311–6, see also
autocorrelation sign of the regression coefficient, 96, 232, 439,
467, 471, 476 in logit and probit models, 395, 415, 416,
428 significance level (�), 155–8, 156, 160–62,
168–9, 190, 192–3, 194–5, 196–7, 198–201, 217
significance tests, see F-test, t-test, Z-test simultaneity, 324–5, 326(n.4), 461(n.3), 465 simultaneous-equation models, 465, 466–7,
468, 471–3, 491–4 skewed distributions, 53–5 skewness, measures of, 55 Smirnov test, see Kolmogorov–Smirnov test Spearman, see rank correlation coefficient specification, see model specification spurious correlation, 72, 249
in cross sections, 477, 494(n.6) in time series, 361, 370
standard deviation (s), 49–50, 50 with grouped data, 52–3
standard error (SE), 123, 123–5, 138–44, 165 of a correlation coefficient, 142–3 of a regression coefficient, 143–4 of a sample mean, 124, 124–5, 138–40 of a sample proportion, 141 of logit coefficients, 391, 395 of the difference between sample means,
172–3 standard error of the estimate (SEE), 272–7,
273 standard normal distribution, 61–5, 128,
and probit models, 388–9, 419 standardized beta coefficients, 243, 243,
255(n.2) standardized rates, 45–6 statistical significance, 160, 170, 210, see also
significance levels statistical significance of the regression as a
whole, see F-tests Steckel’s (leaving home) data set, 281, 480–91,
502–5 stepwise regressions, 323 stochastic factors, 262–3 strength of association,
non-parametric measures, 218–21 see also correlation
structural change, 307–8, 325(n.1) see also parameter stability, Chow test
structural models, 465, see also simultaneous- equation models
Student’s t, 132, see also t-tests
t-distributions, 132–6, 138, 145(n.6,7), 164–5 t-ratios, 167, see also t-tests t-statistics, see t-tests t-tests, 164–79, 185–6, 309, 374
effect of multicollinearity, 322 for a correlation coefficient, 174, 173–5,
180(n.5, 6) for a difference of means, 170–3, 179(n.3),
202 for a regression coefficient, 175–9, 176,
180(n.7), 442, 447, 460(n.1), 471 for logit and probit coefficients, 391, 395,
416, 419 tcalc, 166, 167
test statistics, 166, 163–9, 190, 193, 194, 196, 198–9, 209
time, as an explanatory variable, 100–2, 352–355, 356–8, 361–440
time-series models, 446–51, 458–60 and correlation, 83 and error specification, 309–10, 312, 314 and model specification, 261, 307 and variable specification, 322
time-series variables, 8–9 tobit coefficients, 426–7 tobit models, 387, 424, 425–8 trends, 21–30, 440
linear, 100–2, 352(n.) log-linear, 351–2, 354–5, 356–8, 502(n.) non-linear, 352–3, 358–9
two-part censored models, 427–8 two-stage least squares technique (2SLS),
324–5, 465, 468, 470 two-tailed tests, 158–9, 208, 217, 222(n.5) Type I and Type II errors, 153–6, 154, 155, 158,
161
unemployment data set, see Benjamin–Kochin data set
unit elasticity, in logit models, 406–9, 409, 410, 411, 412,
430(n.10), 485–9 in probit models 421–2
variables categorical, 9, 385, 422 continuous, 10, 56,
discrete, 10, 56, 128 lagged, see lagged variables pre-determined, 465 random, 56, 125, 128, 129
variance (s2), 48, 48–9 with grouped data, 52–3
variation, explained and unexplained, 102–6, 232–3, 442
volume indices, 19(n.)
Wald–Wolfowitz runs test, 188–93, 191, 210, 214
standardized test statistic for, 191 White test for heteroscedasticity, 311, 453 Wilcoxon rank sum test, 194, 196, 197
Z-distributions, 61–5, 131–4, 165 Z-tests, 164–5, 185–6
Zcalc, 166(n.), 191, 193, 195–7, 221(n.2) zero-order correlation, 249, 250, 251, 253
Ackerberg, Daniel A., 431(n.17) Alesina, Alberto, 372 Alt, James E., 379(n.22) Anderson, Michael, 223(n.10) Arminger, Gerhard, 430(n.12) Armstrong, Alan, 224(n.15) Ashton, T. S., 32(n.8) Atack, Jeremy, 378(n.15) Aydelotte, William O., 221(n.1)
Baines, Dudley, 505(n.3) Beattie, J. M., 379(n.21) Beck, E. M., 379(n.20) Bell, Frances N., 368(n.) Benjamin, Daniel K., 7(n.), 19, 27(n.), 94,
112(n.3), 177, 183, 343(n.), 380, 439(n.), 501, 506(n.5,6)
Bergson, Abram, 525(n.2) Blalock, H. M., 31(n.3) Blaug, Mark, 464, 494,(n.1), 505(n.1) Bloom, Howard S., 295(n.2), 371 Bogue, Allan C., 221(n.1) Botticini, Maristella, 431(n.17) Bowden, Sue, 378(n.17) Boyer, George R., 7(n.), 9(n.), 144(n.2), 153,
256, 282(n.), 283(n.), 326, 463, 494(n.2), 497, 505(n.2)
Brenner, Y. S., 378(n.12) Broadberry, S. N., 378(n.15) Burns, Arthur F., 32(n.8) Burns, Eveline, 506(n.7)
Callahan, Colleen M., 431(n.16) Cantor, D., 370(n.) Caradog Jones, D., 227 Castellan Jr., N. John, 175, 201(n.) Chapman, Agatha L., 506(n.7) Chapman, Janet G., 525(n.2) Childers, Ben D., 223(n.10) Cochran, W. G., 144(n.1) Cockburn, J. S., 379(n.19) Cohen, Gerald D., 372(n.) Cohen, Jacob, 112(n.4) Cohen, Patricia, 112(n.4) Cole, W. A., 31(n.6)
Collins, Michael, 506(n.5) Costa, Dora L., 429(n.1) Crafts, N. F. R., 31(n.6), 378(n.15), 445(n.),
506(n.5), 518(n.), 525(n.4) Crockett, Alasdair, 251–3
David, Paul A., 31(n.5), 255(n.2) Deane, Phyllis, 31(n.6) Digby, Anne, 204(n.), 205, 464, 494(n.1) Di Matteo, Livio, 378(n.14) Dimsdale, Nicholas H., 377(n.10), 445(n.),
461(n.7), 506(n.5) Dornbusch, Rudiger, 377(n.7) Durkheim, Émile, 362
Edelstein, Michael, 255(n.2) Eichengreen, Barry, 345(n.), 445, 506(n.5) Eldridge, J., 376(n.1) Elliott, C. M., 32(n.10) Engerman, S., 445(n.), 506(n.5) Erikson, Robert S., 379(n.22)
Farr, William, 368 Feinstein, Charles H., 506(n.4), 513(n.),
520(n.) Ferenczi, Imre, 506(n.4) Field, Simon, 370(n.) Floros, Nicos, 506(n.5) Floud, Roderick, 376(n.1), 377(n.10) Fogel, Robert W., 221(n.1), 278(n.1) Ford, A. G., 32(n.13) Foreman-Peck, James, 378(n.15) Frickey, Edwin, 31(n.6)
Galenson, David, 378(n.11) Gattrell, V. A. C., 90(n.3) Gayer, A. D., 32(n.8) Gemery, Henry A., 345(n.) Gilbert, David, 90(n.3) Gillis, A. R., 379(n.20) Goldin, Claudia, 286(n.) Gould, J. D., 505(n.3) Granger, C. W. J., 32(n.10) Griffin, Larry J., 375(n.) Griliches, Z., 378(n.16)
Name Index
Hadden, T. B., 90(n.3) Haines, Michael R., 368(n.), 369(n.) Hamilton, Gillian, 364 Hanes, Christopher, 378(n.13) Harrison, Mark, 379(n.19) Hatton, Timothy J., 7(n.), 343(n.), 378(n.13),
461(n.7,8), 475, 499, 506(n.3,5) Hay, Douglas, 94 Hoffmann, Walther G., 31(n.6) Hopkins, Sheila V., 20, 31(n.5), 520(n.) Hoppit, Julian, 28, 30, 356 Horrell, Sara, 429(n.1) Horsewood, Nicholas J., 377(n.10),
461(n.7) Hoskins, W. G., 224(n.14) Houthakker, H. S., 366 Humphries, Jane, 429(n.1) Huzel, J. P., 476, 494(n.5), 505(n.1)
Isaac, Larry W., 375(n.)
Jackson, Robert V., 31(n.6) James, John A., 378(n.15) Jarausch, Konrad H., 430(n.12) John, A. H., 90
Kaelble, Hartmut, 378(n.12) Kearl, J. R., 378(n.14) Kennedy, Peter E., 112(n.4), 133, 246(n.),
321(n.) Keynes, John Maynard, 362, 377(n.10),
502(n.) Kilby, Peter, 378(n.15) Knight, I., 376(n.1) Kochin, Levis A., 7(n.), 19, 27(n.), 94,
112(n.3), 177, 183, 343(n.), 438(n.), 501, 506(n.5,6)
Koeller, C. Timothy, 379(n.20) Kramer, Gerald H., 379(n.22) Kussmaul, Ann, 179(n.3), 220–1 Kuznets, Simon, 359
Land, K. C., 370(n.) Layard, Richard, 377(n.7) Lee, Chulhee, 429(n.1) Lewis, W. Arthur, 31(n.6) Loschky, David, 223(n.10)
Maddala, G. S., 431(n.15) Maddison, Angus, 91 Magee, Gary, 224(n.15) Malthus, Thomas, 463, 476, 478, 497 Margo, Robert A., 431(n.17) Markus, Gregory B., 379(n.22) Marris, Robin, 525(n.4) McCloskey, Donald N., 180(n.7),
377(n.10) McDonald, Judith A., 431(n.16) Metcalfe, David, 506(n.5) Millward, Robert, 368(n.), 378(n.15) Mitchell, W. C., 32(n.8) Mosley, Paul, 372(n.)
Neal, Larry, 180(n.7) Nickell, Stephen J., 377(n.7), 461(n.7),
506(n.5) Nielsen, Randall, 215(n.)
O’Brien, Anthony Patrick, 431(n.16) Offer, Avner, 378(n.17) Ogburn, W. F., 83(n.) O’Leary, Richard, 251(n.) Ormerod, P. A., 440–1, 443, 461(n.2,4)
506(n.5)
Persson, Karl Gunnar, 215(n.) Phelps Brown, Henry, 20, 31(n.5), 520(n.) Pope, Clayne L., 378(n.14) Prais, S. J., 366 Pressnell, L., 90(n.) Preston, Samuel H., 368(n.) Price, H. Douglas, 295(n.2), 371
Rostow, W. W., 32(n.8) Roubini, Nouriel, 372(n.) Rowntree, Seebohm, 365
Salter, W. E. G., 525(n.3) Schofield, R. S., 23, 31(n.3), 32(n.12), 479 Schonhardt-Bailey, Cheryl, 429(n.1) Schwartz, A. J., 32(n.8) Scola, Roger, 226(n.) Sheflin, Neil, 379(n.20) Shorter, Edward, 379(n.20) Siegel, Sidney, 175, 201(n.) Solar, Peter, 31(n.5) Solomou, Solomos, 32(n.9) Southall, Humphrey, 90(n.3) Spraos, John, 31(n.7) Stamp, Sir Josiah, 319 Steckel, Richard H., 7(n.), 281(n.), 494(n.7,8),
495(n.9,10), 503, 505 Stevenson, S. J., 379(n.19) Stock, James H., 222(n.8) Sundstrom, William A., 255(n.2) Szreter, Simon, 87
Takaishi, M., 376(n.1) Tanner, J. M., 376(n.1) Thomas, Brinley, 32(n.13) Thomas, Dorothy Swaine, 83–6, 90(n.3),
180(n.6), 370 Thomas, Mark, 378(n.12) Thomas, T., 377(n.10) Tilly, Charles, 379(n.20) Tobin, James, 424, 426 Tolnay, Stewart E., 379(n.20) Troy, Leo, 379(n.20) Tucker, G. S. L., 505(n.1) Tufte, Edward R., 379(n.22) Turner, Paul, 378(n.17)
Usher, Dan, 525(n.4)
Vamplew, Wray, 506(n.4)
Vernon, J. R., 506(n.4) von Tunzelmann, G. N., 378(n.16) Voth, Hans-Joachim, 216(n.), 223(n.8)
Whitehouse, R. H., 376(n.1) Willcox, Walter F., 506(n.4) Williamson, Jeffrey G., 7(n.), 32(n.13),
343(n.), 475, 499, 506(n.3,4) Wimmer, Larry T., 378(n.14) Winter, Jay, 90(n.3)
Wonnacott, R. J., 72 Wonnacott, T. H., 72 Woods, Robert, 377(n.6), 378(n.18) Worswick, G. D. N., 440–1, 443, 461(n.2,4),
506(n.5) Wrigley, E. A., 23, 31(n.3), 32(n.12), 90(n.2),
215, 479
Zecher, J. Richard, 180(n.7)
- Cover
- Half-title
- Title
- Copyright
- Contents
- Figures
- Tables
- Panels
- Preface
- PART 1 Elementary statistical analysis
- CHAPTER 1 Introduction
- 1.1 Aims of the book
- 1.2 The four case studies and the data sets
- 1.3 Types of measurement
- 1.3.1 Cases, variables, and values
- 1.3.2 Cross-section and time-series variables
- 1.3.3 Levels of measurement
- Nominal measurement
- Ordinal measurement
- Interval or ratio measurement
- 1.3.4 Populations and samples
- 1.3.5 Dummy variables
- 1.4 Simple notation
- 1.5 The equation of a straight line
- 1.6 Powers and logarithms
- 1.6.1 Powers
- 1.6.2 Logarithms
- Calculations using logs
- 1.6.3 Logs as proportions and ratios
- 1.7 Index numbers
- 1.8 Trends and fluctuations
- 1.8.1 Measuring the trend by a moving average
- 1.8.2 Measuring the fluctuations by the ratio to the trend
- Notes
- CHAPTER 2 Descriptive statistics
- 2.1 Presentation of numerical data
- 2.1.1 Frequency distributions
- Number and width of the class intervals
- 2.1.2 Bar charts and histograms
- 2.1.3 Frequency curves
- 2.2 Measures of central tendency
- 2.2.1 The mean, the median, and the mode
- 2.2.2 Weighted averages and standardized rates
- 2.2.3 Percentiles, deciles, and quartiles
- 2.3 Measures of dispersion
- 2.3.1 The range and the quartile deviation
- 2.3.2 The mean deviation, the variance, and the standard deviation
- 2.3.3 The coefficient of variation
- 2.4 Measures of central tendency and dispersion with grouped data
- 2.4.1 The median and mean with grouped data
- 2.4.2 The variance and standard deviation with grouped data
- 2.5 The shape of the distribution
- 2.5.1 Symmetrical and skewed distributions
- 2.5.2 Measures of skewness
- 2.6 The normal distribution
- 2.6.1 Properties of the normal distribution
- 2.6.2 The standard normal distribution
- Notes
- 2.7 Exercises for chapter 2
- CHAPTER 3 Correlation
- 3.1 The concept of correlation
- 3.1.1 Correlation is not causation
- 3.1.2 Scatter diagrams and correlation
- Perfect correlation
- Positive and negative correlation
- No linear relationship
- 3.1.3 Outliers
- 3.2 The correlation coefficient
- 3.2.1 Measuring the strength of the association
- 3.2.2 Derivation of the correlation coefficient, r
- 3.2.3 Interpreting a correlation coefficient
- 3.2.4 An illustration of the use of correlation coefficients
- 3.3 Spearman’s rank correlation coefficient
- Notes
- 3.4 Exercises for chapter 3
- CHAPTER 4 Simple linear regression
- 4.1 The concept of regression
- 4.1.1 Explanatory and dependent variables
- 4.1.2 The questions addressed by regression
- 4.2 Fitting the regression line
- 4.2.1 How do we define a line?
- 4.2.2 What criterion should be adopted for fitting the line?
- 4.2.3 How do we find this best fitting line?
- 4.2.4 The regression of Y on X versus the regression of X on Y
- 4.2.5 Fitting a linear trend to a time series
- 4.3 Measuring the goodness of fit
- 4.3.1 Explained variations and unexplained variations (residuals)
- 4.3.2 The coefficient of determination, r
- 4.3.3 Some intuitive implications
- 4.4 Introducing the Ballantine
- Notes
- 4.5 Exercises for chapter 4
- PART II Samples and inductive statistics
- CHAPTER 5 Standard errors and confidence intervals
- 5.1 Introduction
- 5.2 Sampling distributions
- 5.2.1 Sampling distributions and point estimates of the population mean
- 5.3 Four fundamental statistical propositions
- 5.3.1 The Central Limit Theorem
- 5.4 Theoretical probability distributions
- 5.4.1 Empirical and theoretical frequency distributions
- 5.4.2 The Z-and t-distributions
- 5.5 Confidence intervals for a sample mean
- 5.5.1 Interval estimates and margins of error
- 5.5.2 Confidence interval for an estimate of a population mean
- 5.6 Confidence intervals for other sample statistics
- 5.6.1 Sample proportions
- 5.6.2 Correlation coefficients
- 5.6.3 Regression coefficients
- Notes
- 5.7 Exercises for chapter 5
- CHAPTER 6 Hypothesis testing
- 6.1 Testing hypotheses
- 6.1.1 A simple illustration
- 6.1.2 The five stages of hypothesis testing
- 6.2 The null hypothesis
- 6.2.1 Other assumptions of the model
- 6.3 How confident do we want to be about the results?
- 6.3.1 Two types of error and the null hypothesis
- 6.3.2 Significance levels and the critical region
- The critical region
- The trade-off
- 6.3.3 One-and two-tailed tests
- 6.3.4 Statistical versus historical significance
- 6.3.5 Conventional significance levels and prob-values
- The traditional procedure
- Significance levels and prob-values
- 6.3.6 The critical region and the four cases of theft
- 6.4 Test statistics and their sampling distributions
- 6.4.1 The basic ideas
- 6.4.2 Z-and t-tests
- 6.5 Making a decision
- 6.6 Interpreting the results
- 6.7 Three illustrations of the use of t-tests
- 6.7.1 A t-test for a difference of means
- 6.7.2 A t-test for a correlation coefficient when the null hypothesis is zero correlation
- 6.7.3 A t-test for a regression coefficient
- Illustration of the t-test for a regression coefficient
- Notes
- 6.8 Exercises for chapter 6
- CHAPTER 7 Non-parametric tests
- 7.1 Introduction
- 7.2 Non-parametric tests for two independent samples
- 7.2.1 The Wald–Wolfowitz runs test
- An illustration of the runs test
- 7.2.2 The Mann–Whitney U-test
- An illustration of the Mann–Whitney U-test
- 7.2.3 The Kolmogorov–Smirnov test
- An illustration of the Kolmogorov–Smirnov test
- 7.2.4 Criteria for choosing between alternative tests
- Comparison of a t-test and a Mann–Whitney U-test
- 7.3 The Chi-test
- 7.3.1 Contingency tables or cross-tabulations
- 7.3.2 The Chi-test for a relationship between two variables
- Illustration of the Chi-test
- 7.4 The one-sample runs test of randomness
- An illustration of the application of the runs test for randomness
- 7.5 One-sample non-parametric tests of goodness of fit
- 7.5.1 The one-sample Chi-test
- 7.5.2 The one-sample Kolmogorov–Smirnov test
- 7.6 Non-parametric measures of strength of association
- 7.6.1 Four measures of association: Phi, Cramer’s V, the contingency coefficient C, and Goodman and Kruskal’s tau
- An illustration of the use of Cramer’s V
- Notes
- 7.7 Exercises for chapter 7
- PART III Multiple linear regression
- CHAPTER 8 Multiple relationships
- 8.1 The inclusion of additional explanatory variables
- 8.2 Multiple regression
- 8.2.1 The coefficient of multiple determination, R
- 8.2.2 Partial regression coefficients
- 8.2.3 Controlling for the influence of a second explanatory variable
- 8.2.4 Understanding the impact of additional explanatory variables
- 8.2.5 The Ballantine for multiple regression
- 8.2.6 The least squares criterion in multiple regression
- 8.2.7 Standardized beta coefficients
- 8.2.8 The adjusted coefficient of multiple determination, R
- 8.2.9 Interpreting the intercept term
- 8.3 Partial and multiple correlation
- 8.3.1 The interpretation and order of partial correlation coefficients
- An illustration of the use of partial correlation coefficients
- 8.3.2 Removing the influence of the control variable
- 8.3.3 Calculation of partial correlation coefficients
- 8.3.4 The coefficient of multiple correlation, R
- Notes
- 8.4 Exercises for chapter 8
- CHAPTER 9 The classical linear regression model
- 9.1 Historical research and models of relationships between variables
- 9.1.1 Quantitative methods and historical judgement
- 9.2 Deviations of the dependent variable from the estimated regression line
- 9.2.1 Errors of measurement and specification
- 9.2.2 Stochastic (random) factors and the error term, e
- 9.2.3 Sampling and the regression line
- 9.2.4 Assumptions about the error term
- The best model
- 9.3 A significance test for the regression as a whole
- 9.3.1 The F-test and the F-distribution
- 9.3.2 F-tests of the statistical significance of the regression as a whole
- 9.3.3 The relationship of Fand R
- 9.4 The standard error of the estimate, SEE
- 9.4.1 Confidence interval for a prediction of the mean value of a dependent variable
- Illustration of the calculation of a confidence interval
- 9.4.2 Confidence interval for a prediction of a specific value of a dependent variable
- 9.5 Exercises for chapter 9
- CHAPTER 10 Dummy variables and lagged values
- 10.1 Dummy variables
- 10.1.1 Dummy variables with two or more categories
- An illustration of the use of dummy variables
- 10.1.2 Dummy variables as a change in the intercept
- 10.1.3 Additional dummy variables
- 10.1.4 Interaction between two dummy variables
- 10.1.5 Interaction between dummy and numerical variables
- 10.2 Lagged variables
- 10.2.1 Lagged dependent variables
- 10.2.2 Distributed lag models
- Notes
- 10.3 Exercises for chapter 10
- CHAPTER 11 Violating the assumptions of the classical linear regression model
- 11.1 The assumptions of the classical linear regression model
- 11.1.1 The model is correctly specified
- 11.1.2 The error term is appropriately specified
- 11.1.3 The variables are correctly measured and appropriately specified
- 11.2 Problems of model specification
- 11.2.1 Non-linear models
- 11.2.2 Omitted and redundant variables
- 11.2.3 Unstable parameter values
- 11.3 Problems of error specification
- 11.3.1 Non-zero errors
- 11.3.2 Heteroscedasticity
- 11.3.3 Autocorrelation
- 11.3.4 Outliers
- 11.4 Problems of variable specification
- 11.4.1 Errors in variables
- 11.4.2 Multicollinearity
- 11.4.3 Simultaneity
- Notes
- 11.5 Exercises for chapter 11
- PART IV Further topics in regression analysis
- CHAPTER 12 Non-linear models and functional forms
- 12.1 Different forms of non-linear relationships
- 12.2 Estimation of non-linear regression models
- 12.2.1 Two groups of non-linear models
- The first group of non-linear models
- The second group of non-linear models
- 12.2.2 Estimating the double logarithmic and semi-logarithmic models
- 12.3 Interpretation of non-linear relationships
- 12.3.1 Interpretation of the regression coefficients for logarithmic models
- 12.3.2 Elasticities
- 12.3.3 Long-run multipliers
- 12.4 Fitting a trend to a time series
- 12.4.1 Fitting a log-linear trend
- 12.4.2 Fitting non-linear trends
- 12.5 Modelling a relationship
- 12.5.1 Choice of functional form
- 12.5.2 Some general principles
- 12.5.3 Implementing the principles: some examples from economic history
- 12.5.4 Implementing the principles: some examples from demographic, social, and political history
- 12.6 Model specification
- Notes
- 12.7 Exercises for chapter 12
- CHAPTER 13 Logit, probit, and tobit models
- 13.1 Limited dependent variables
- 13.1.1 Three cases of limited dependent variables
- 13.1.2 Why linear models are not appropriate for limited dependent variables
- 13.2 Estimating the logit model: coefficients, absolute effects, and elasticities
- 13.2.1 The logit regression model
- 13.2.2 Interpreting logit coefficients
- 13.2.3 Absolute changes: marginal effects and impact effects
- What the slopes tell us
- 13.2.4 Calculating the marginal and impact effects
- The marginal effect for an individual case
- The marginal effect for the sample as a whole
- The impact efiect for an individual case
- The impact effect for the sample as a whole
- 13.2.5 Proportionate effects: logit elasticities
- 13.2.6 Non-linearity and the measurement of change
- 13.2.7 Goodness of fit
- 13.3 Estimating the logit model: an illustration
- 13.3.1 An illustration of the logit model
- 13.3.2 Estimating the logit model using grouped data
- 13.4 Estimating the probit model
- 13.4.1 The probit model
- 13.4.2 Interpreting the results from a probit model
- 13.5 Multi-response models
- 13.6 Censored dependent variables
- 13.6.1 A censored continuous dependent variable
- 13.6.2 The tobit model
- 13.6.3 Two-part censored models
- Notes
- 13.7 Exercises for chapter 13
- PART V Specifying and interpreting models: four case studies
- CHAPTER 14 Case studies 1 and 2: unemployment in Britain and emigration from Ireland
- 14.1 Inter-war benefits and unemployment in Great Britain
- 14.1.1 The Benjamin and Kochin model
- 14.1.2 Parameter stability and outliers in the Benjamin and Kochin model
- 14.2 Emigration from Ireland
- 14.2.1 The time-series model of Irish emigration
- 14.2.2 The panel model of Irish emigration
- 14.2.3 The impact of the explanatory variables on changes in emigration over time
- Notes
- CHAPTER 15 Case studies 3 and 4: the Old Poor Law in England and leaving home in the United States, 1850–1860
- 15.1 The effects of the Old Poor Law
- 15.1.1 Relief payments, wages, and unemployment
- The hypotheses and the two models
- The reduced-form model
- The simultaneous-equations model
- The regression results
- Conclusions from the regressions
- 15.1.2 The Old Poor Law and the birth rate
- The double log model
- Results from the regressions
- 15.2 Leaving home in the United States, 1850–1860
- 15.2.1 Influences on leaving home
- 15.2.2 The logit model
- Interpreting the coefficients
- Conclusions from the logit model
- Appendix to…15.1.1 Derivation of reduced-form equations from a simultaneous-equations model
- Notes
- APPENDIX A The four data sets
- A.1 The iniquitous effects of the Old Poor Law
- A.2 What explains the scale and timing of the massive emigration from Ireland after the famine?
- A.3 To what extent were generous unemployment benefits responsible for the high level of unemployment in inter-war Britain?
- A.4 What influenced children’s decision to leave home in the United States in the mid-nineteenth century?
- Notes
- APPENDIX B Index numbers
- B.1 Price and quantity relatives
- B.2 Two types of index number
- B.3 Laspeyres index numbers with fixed weights
- B.4 The index number problem and chained indices
- Fisher Ideal and Divisia indices
- Chained indices
- B.5 Paasche index numbers with changing (given year) weights
- Notes
- Bibliography
- Subject index
- Name Index
