To examine what influence individuals’ decisions on donations, the linear regression model

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School of Economics

Introductory Econometrics

ECON2206

Tutorial Program

Session 1, 2014

Assignment 1 is on page 5.

Assignment 2 is on page 12.

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Week 2 Tutorial Exercises

Readings • Read Chapter 1 thoroughly. • Make sure that you know the meanings of the Key Terms at the chapter end.

Problem Set (these will be discussed in tutorial classes) • Q1. Wooldridge 1.1 • Q2. Wooldridge 1.2 • Q3. Wooldridge C1.3 • Q4. Wooldridge C1.4 (These are selected from the end-of-chapter Problems and Computer Exercises.)

Computer Exercise and STATA Hints • All data files and data description files are in the course website, suffixed with “.raw” and

“.des” respectively. Sometimes, data files may have a “.txt” or “.csv” suffix. You must read the description files always!

• Example STATA do-files, are also posted in the course website, suffixed with “.do”. The solution do-files will be posted with one week delay.

• One of the course objectives is to learn to use STATA. • To complete Q3 and Q4, you should follow the steps below.

o Step 1. Create a work folder, say F:\ie, on your USB drive or your computer. o Step 2. Download data and description files to the work folder. Download

dataUpload.zip and unzip its contents into your USB ECON2206 folder, say F:\ie. o Step 3. Download do files. Download doFilesUpload.zip and unzip its contents into

your USB ECON2206 folder, say F:\ie. o Step 4. Follow the instructions in Slides01 or A Guide to Using STATA (both on the

website) to run bwght_1st.do and read the output carefully. o Step 5. Make sure that you understand the effect of each command in bwght_1st.do.

You can type each command in the lower panel (command panel) of STATA to see the corresponding result.

o Step 6. Modify bwght_1st.do (or find proper do files in your ECON2206 folder) to complete Q3 and Q4.

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Week 3 Tutorial Exercises

Readings • Read Chapter 2 thoroughly. • Make sure that you know the meanings of the Key Terms at the chapter end.

Review Questions (these may or may not be discussed in tutorial classes) • The minimum requirement for OLS to be carried out for the data set {(xi, yi), i=1,…,n} with

the sample size n > 2 is that the sample variance of x is positive. In what circumstances is the sample variance of x zero?

• The OLS estimation of the simple regression model has the following properties: a) the sum of the residuals is zero; b) the sample covariance of the residuals and x is zero.

Why? How would you relate them to the “least squares” principle? • Convince yourself that the point ),( yx , the sample means of x and y, is on the sample

regression function (SRF), which is a straight line. • How do you know that SST = SSE + SSR is true? • Which of the following models is (are) nonlinear model(s)?

a) sales = β0 /[1 + exp(-β1 ad_expenditure)] + u; b) sales = β0 + β1 log(ad_expenditure) + u; c) sales = β0 + β1 exp(ad_expenditure) + u; d) sales = exp(β0 + β1 ad_expenditure + u).

• Can you follow the proofs of Theorems 2.1-2.3?

Problem Set (these will be discussed in tutorial classes) • Q1. Wooldridge 2.4 • Q2. Wooldridge 2.7 • Q3. Wooldridge C2.6 • Q4. Wooldridge C2.7 (not in 3rd edition)

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Week 4 Tutorial Exercises

Readings • Read Chapter 3 thoroughly. • Make sure that you know the meanings of the Key Terms at the chapter end.

Review Questions (these may or may not be discussed in tutorial classes) • What do we mean when we say “regress wage on educ and expr”? • Why and under what circumstance do we need to “control for” expr in the regression model

in order to quantify the effect of educ on wage? • What is the bias of an estimator? • What is the “omitted variable bias”? • What is the consequence of adding an irrelevant variable to a regression model? • What is the requirement of the ZCM assumption, in your own words? • Why assuming E(u) = 0 is not restrictive when an intercept is included in the regression

model? • In terms of notation, why do we need two subscripts for independent variables? • Using OLS to estimate a multiple regression model with k independent variables and an

intercept, how many first-order conditions are there? • How do you know that the OLS estimators are linear combination of the observations on the

dependent variable? • What is Gauss-Markov theorem? • Try to explain why R2 never decreases (it is likely to increase) when additional explanatory

variables are added to the regression model. • What is an endogenous explanatory variable? Exogenous explanatory variable? • What is multicollinearity and its likely effect on the OLS estimators?

Problem Set (these will be discussed in tutorial classes) • Q1. Wooldridge 3.2 • Q2. Wooldridge 3.10 • Q3. Wooldridge C3.2 • Q4. Wooldridge C3.6

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Assignment 1 To investigate the effects of the land size (lotsize)and house size (sqrft) on the house price (price) in a particular suburb, the following model log(𝑝𝑟𝑖𝑐𝑒) = 𝛽0 + 𝛽1 log(𝑙𝑜𝑡𝑠𝑖𝑧𝑒) + 𝛽2 log(𝑠𝑞𝑟𝑓𝑡) + 𝑢 is used.

(a) What is the meaning of 𝛽1in this model? (b) Explain why you would (or would not) expect that 𝛽1 > 0. (c) Explain the meaning of the zero-conditional-mean assumption for this model. (d) Download the description file “hprice1.des” and data file “hprice1.raw” from the course

website and read the descriptions about the data. Find the (i) the units of measurement; (ii) the sample means; and (iii) the sample standard deviations of price , lotsize , sqrft and bdrms.

(e) With the data in “hprice1.raw”, estimate the above model by OLS and report the results in equation form, including the sample size and 𝑅2. Interpret the estimated coefficient on log (𝑠𝑞𝑟𝑓𝑡). Would you say the coefficient is economically significant? Why?

(f) Suppose that two houses (A and B) have exactly the same characteristics, except that house A has lotsize = 9000 and house B has lotsize = 9900. According to the results in (e), what is the expected percentage difference of the prices of the two houses? Approximation can be used.

(g) Add the variable bdrms (as an explanatory variable) to the above model and estimate the extended model by OLS and report the results in equation form, including the sample size and 𝑅2. Interpret the estimated coefficient on bdrms. Find the correlation between bdrms and log(𝑠𝑞𝑟𝑓𝑡) (using the “cor” command in STATA). Does the correlation agree with your expectations?

(h) Suppose that the model in (g) is the true model that satisfies the Gauss-Markov assumptions MLR.1-5. Explain why the estimation results in (e) may not be desirable under this supposition. Comment on the differences in estimated coefficients on log (𝑙𝑜𝑡𝑠𝑖𝑧𝑒) and log (𝑠𝑞𝑟𝑓𝑡) in (e) and (g).

Notes on Assignment 1 Your answers to the above questions must be submitted to your tutor at (or before) the beginning of your Week 5 tutorial class. These will be marked for the course assessment (worth 10%). You must attach your STATA do-file and relevant output to your answers. Your submission, including STATA printout, should not exceed 4 pages. Make sure that your name, student number, course name, tutorial group number and tutor's name are included in your submission. Do not use plastic sheets or binders. Simply staple the pages together. Do not submit loose-leaf sheets of paper. All submissions will be checked for plagiarism, which will lead to heavy penalty. Each of Parts (a)-(g) is worth 1 mark. Part (h) is worth 2 marks. The overall presentation is worth 1 mark.

The University regards plagiarism as a form of academic misconduct, and has very strict rules regarding plagiarism. For UNSW’s policies, penalties, and information to help you avoid plagiarism see: http://www.lc.unsw.edu.au/plagiarism/index.html.

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Week 5 Tutorial Exercises

Readings • Read Chapter 4.1 to 4.3 thoroughly. • Make sure that you know the meanings of the Key Terms at the chapter end.

Review Questions (these may or may not be discussed in tutorial classes) • What are the CLM assumptions? • What is the sampling distribution of the OLS estimators under the CLM assumptions? • What are the standard errors of the OLS estimators? • What is the null hypothesis about a parameter? • What is a one-tailed (two-tailed) alternative hypothesis? • In testing hypotheses, what is a Type 1 (Type 2) error? What is the level of significance? • The decision rule we use can be stated as “reject the null if the t-statistic exceeds the critical

value”. How is the critical value determined? • Justify the statement “Given the observed test statistic, the p-value is the smallest significant

level at which the null hypothesis would be rejected.” • What is the 90% confidence interval for a parameter? • In constructing a confidence interval for a parameter, what is the level of confidence? • When the level of confidence increases, how would the width of the confidence interval

change (holding other things fixed)? • Try to convince yourself that the event “the 90% confidence interval covers a hypothesised

value of the parameter” is the same as the event “the null of the parameter being the hypothesised value cannot be rejected in favour of the two-tailed alternative at the 10% level of significance.”

Problem Set (these will be discussed in tutorial classes) • Q1. Wooldridge 4.1 • Q2. Wooldridge 4.2 • Q3. Wooldridge 4.5 • Q4. Wooldridge C4.8

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Week 6 Tutorial Exercises

Readings • Read Chapter 4.4-4.6 and Chapter 5 thoroughly. • Make sure that you know the meanings of the Key Terms at the chapter end.

Review Questions (these may or may not be discussed in tutorial classes) • How would you test hypotheses about a single linear combination of parameters? • What are exclusion restrictions for a regression model? • What are restricted and unrestricted models? • How do you compute the F-statistic, given that you have SSRs? • What are general linear restrictions on parameters? • What is the test for the overall significance of a regression? • How would you report your regression results? • Why would you care about the asymptotic properties of the OLS estimators? • Comparing the inference procedures in Chapter 5 with those in Chapter 4, can you list the

similarities and differences? • Under MLR.1-MLR.5, the OLS estimators are consistent, asymptotically normal, and

asymptotically efficient. Try to explain these properties in your own words.

Problem Set (these will be discussed in tutorial classes) • Q1. Wooldridge 4.6 • Q2. Wooldridge 4.8 • Q3. Wooldridge 4.10 • Q4. Wooldridge C4.9 • Q5. Wooldridge 5.2 • Q6. Wooldridge C5.1

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Week 7 Tutorial Exercises

Readings • Read Chapter 6 thoroughly. • Make sure that you know the meanings of the Key Terms at the chapter end.

Review Questions (these may or may not be discussed in tutorial classes) • What are the advantages of using the log of a variable in regression? Find the “rules of

thumb” for taking logs. • Be careful when you interpret the coefficients of explanatory variables in a model where

some variables are in logarithm. Do you remember Table 2.3? • How do you compute the change in y caused by Δx when the model is built for log(y)? • Why do we need the “interaction” terms in regression models? • What is the adjusted R-squared? What is the difference between it and the R-squared? • How do you construct interval prediction for given x-values? • How do you predict y for given x-values when the model is built for log(y)? • What is involved in “residual analysis”?

Problem Set (these will be discussed in tutorial classes) • Q1. Wooldridge 6.4 • Q2. Wooldridge 6.5 • Q3. Wooldridge C6.3 (C6.2 for the 3rd Edition) • Q4. Wooldridge C6.8

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Week 8 Tutorial Exercises

Readings • Read Chapter 7 thoroughly. • Make sure that you know the meanings of the Key Terms at the chapter end.

Review Questions (these may or may not be discussed in tutorial classes) • What is a qualitative factor? Give some examples. • How do you use dummy variables to represent qualitative factors? • How do you use dummy variables to represent an ordinal variable? • How do you test for differences in regression functions across different groups? • What can you achieve by interacting group dummy variables with other regressors? • What is “program evaluation”? • What are the interpretations of the PRF and SRF when the dependent variable is binary? • What are the shortcomings of the LPM? • Does MLR5 hold for the LPM?

Problem Set (these will be discussed in tutorial classes) • Q1. Wooldridge 7.4 • Q2. Wooldridge 7.6 • Q3. Wooldridge 7.9 • Q4. Wooldridge C7.13

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Week 9 Tutorial Exercises

Readings • Read Chapter 8 thoroughly. • Make sure that you know the meanings of the Key Terms at the chapter end.

Review Questions (these may or may not be discussed in tutorial classes) • What is heteroskedasticity in a regression model? • In the presence of heteroskedasticity, are the t-stat and F-stat from the usual OLS still valid?

Why? Are there any other problems with the OLS under heteroskedasticity? • What are the heteroskedasticity-robust standard errors? How do you use them in STATA? • How do you detect if there is heteroskedasticity? • If heteroskedasticity is present in a known form, how would you estimate the model? • If heteroskedasticity is present in an unknown form, how would you estimate the model? • What are the steps in the FGLS estimation? • How would you handle the heteroskedasticity of the LPM?

Problem Set (these will be discussed in tutorial classes) • Q1. Wooldridge 8.1 • Q2. Wooldridge 8.2 • Q3. Wooldridge 8.3 • Q4. Wooldridge 8.5 • Q5. Wooldridge C8.10

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Week 10 Tutorial Exercises

Readings • Read Chapter 9 thoroughly. • Make sure that you know the meanings of the Key Terms at the chapter end.

Review Questions (these may or may not be discussed in tutorial classes) • What is functional form misspecification? What is its major consequence in regression

analysis? • How would you test for functional form misspecification? • What are nested models? And, nonnested models? • What is the purpose of testing one model against another? • How would you test for two nonnested models? • What is a proxy variable? What are the conditions for a proxy variable to be valid in

regression analysis? • Can you analyse the consequences of measurement errors? • In what circumstances missing observations will cause major concerns in regression

analysis? • What is exogenous sample selection? What is endogenous sample selection? • What are outliers and influential observations? • Consider the simple regression model yi = α + (β + bi)xi + ui for i = 1, 2,…, n, where the “slop

parameter” contains a random variable bi; α and β are constant parameters. Assume that the usual MLR1-5 hold for yi = α + βxi + ui, ; E(bi|xi) = 0 and Var(bi|xi) = ω2. If we regress y on x with a intercept, will the estimator of the “slope parameter” be biased from β? Will the usual OLS standard errors be valid for statistical inference?

Problem Set (these will be discussed in tutorial classes) • Q1. Wooldridge 9.1 • Q2. Wooldridge 9.3 • Q3. Wooldridge 9.5 • Q4. Wooldridge C9.3

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Assignment 2 This assignment is based on the data set in “CHARITY.RAW”, see also tutorial exercises C2.7 and C7.14. The definitions of variables are in the file “CHARITY.DES”. To examine what influence individuals’ decisions on donations, the linear regression model

respond = β0 + β1 resplast + β2 avggift + β3 propresp + β4 mailsyear +u

is used. For hypothesis-testing questions, please always present hypotheses, test statistic and its distribution under the null, decision rule and conclusion.

a) Why the model is known as a linear probability model (LPM)? What is the meaning of β1? b) Suppose that MLR.1-4 hold for the model when all variables are correctly measured.

Further suppose that respond is measured with error: respond = respond* + e (i.e., observed = truth + error). Would the OLS estimators of 𝛽’s still be unbiased and consistent and why?

c) Suppose that MLR.1-4 hold for the model when all variables are correctly measured. Further suppose that one regressor, mailsyear, is measured with an additive error and the error is uncorrelated with the truth mailsyear*. How would the OLS estimator of, say, β1 be affected by the measurement error and why?

d) Estimate the model, using OLS, and interpret the estimate of β4. Find “het.-robust” standard errors and compare these with the usual OLS standard errors.

e) Test for the presence of heteroskedasticity, using the Breusch-Pagan LM test. f) Suppose that MLR.1-4 hold for the model. Find the expression for Var(u|all regressors). g) Given your answer to f), how would you use weighted least squares (WLS) to estimate

the model? What would be the weights for the WLS? Practically, what may prevent you from implementing the WLS?

h) Re-estimate the model, using FGLS with the textbook equation (8.32). Compare the estimates with those in d).

i) Test for possible model misspecifications, using the RESET. What does your conclusion suggest?

Notes on Assignment 2 Your answers to the above questions must be submitted to your tutor at (or before) the beginning of your Week 12 tutorial class. These will be marked for the course assessment (worth 10%). You must attach your STATA do-file and relevant output to your answers. Your submission, including STATA printout, should not exceed 5 pages. Make sure that your name, student number, course name, tutorial group number and tutor's name are included in your submission. Do not use plastic sheets or binders. Simply staple the pages together. Do not submit loose-leaf sheets of paper. All submissions will be checked for plagiarism, which will lead to heavy penalty. Each of Parts (a)-(i) is worth 1 mark. The overall presentation is worth 1 mark.

The University regards plagiarism as a form of academic misconduct, and has very strict rules regarding plagiarism. For UNSW’s policies, penalties, and information to help you avoid plagiarism see: http://www.lc.unsw.edu.au/plagiarism/index.html.

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Week 11 Tutorial Exercises

Readings • Read Chapter 10 thoroughly. • Make sure that you know the meanings of the Key Terms at the chapter end.

Review Questions (these may or may not be discussed in tutorial classes) • What are the main features of time series data? How do time series data differ from cross-

sectional data? • What is a stochastic process and its realisation? • What is serial correlation or autocorrelation? • What is a finite distributed lag model? What is the long-run propensity (LRP)? How would

you estimate the LRP and the associated standard error (say in SHAZAM)? • What are TS1-6 (assumptions about time series regression)? How do they differ from the

assumptions in MLR1-6? • What are “strictly exogenous” regressors and “contemporaneously exogenous” regressors? • What is a trending time series? What is a time trend? • Why may a regression with trending time series produce “spurious” results? • Why would you include a time trend in regressions with trending variables? • What is seasonality in a time series? Give an example of time series variable with seasonality. • For quarterly data, how would you define seasonal dummy variables for a regression model?

Problem Set • Q1. Wooldridge 10.1 • Q2. Wooldridge 10.2 • Q3. Wooldridge 10.7 • Q4. Wooldridge C10.10

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Week 12 Tutorial Exercises

Readings • Read Chapter 11 thoroughly. Also read sections 18.2, 18.3 and 18.4. • Make sure that you know the meanings of the Key Terms at the chapter end.

Review Questions (these may or may not be discussed in tutorial classes) • What is a strictly stationary stochastic process (SP)? What is a covariance stationary SP? • In this course, what do we mean by “weakly dependent (WD) time series”? • What is a dynamically complete (DC) model? • How would you test for the serial correlation in the disturbance of a regression model? • Will the usual OLS standard errors be valid when the disturbance has autocorrelation? • What is a random walk? What are the main properties of the random walk? What are the

properties of the difference of the random walk? • What are I(1) and I(0) time series? How do we decide whether a time series is I(1) or I(0)? • What is a “stochastic trend”? What is a spurious regression? • Can you explain the notion of “cointegration”?

Problem Set (these will be discussed in tutorial classes) • Q1. Wooldridge 11.1 • Q2. Wooldridge 11.4 • Q3. Wooldridge 11.6 • Q4. Wooldridge C11.7

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Week 13 Tutorial Exercises

Readings • Read Chapter 13 thoroughly. • Make sure that you know the meanings of the Key Terms at the chapter end.

Review Questions (these may or may not be discussed in tutorial classes) • What are pooled cross sections? How do they compare to a single cross sectional sample? • What is policy analysis? Give an example. • How are pooled cross sections used for policy analysis? What is the “difference-in-difference”

estimator? Estimating what? • What is a set of panel data? How does it differ from a set of pooled cross sections? • What is an unobserved (or fixed) effect model? What does the unobserved (or fixed) effect

represent? Is there a fixed effect in the usual cross sectional regression? • What is the first-differenced estimation for a two-period panel? Dose the validity of the first-

differenced estimation rely on the assumption that the “unobserved or fixed effect” is uncorrelated with the regressors?

• Can we estimate the unobserved or fixed effect with a two-period panel? • Comment on the advantage of using panel data for policy analysis.

Problem Set (these will be discussed in tutorial classes) • Q1. Wooldridge 13.1 • Q2. Wooldridge 13.2 • Q3. Wooldridge 13.7 • Q4. Wooldridge C13.5