stats short essay week 6

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Multiple Linear Regression Equation for Economic Dimension of Standard of Living

Nicoleta Mihaela Florea

1 , Georgeta Mădălina Meghișan

2 , Cristina Nistor

3

1,2, 3 University of Craiova

2 Faculty of Economics and Business Administration, Craiova

Scientific Researcher III, Romanian Academy National Institute of Economic Research ”Costin C. Kiritescu”, Bucharest

[email protected];[email protected];[email protected]

Abstract. The purpose of this analysis is to determine the signification of the following factors: population, population density and inflation rate to the measurement of the standard of living. The analysis takes into consideration demographic and economic data for a number of 10 EU member states. After calculating the standard of living in these countries, we analysed the impact of the mentioned factors using a multiple linear regression equation. We concluded that a part of the variation of the standard of living depends on the evolution of the three variables taken into consideration in the analysis.The standard of living of a population assesses the economic dimension of a country and the quality of life for a population. Keywords: living standard, European Union, inflation, population density, population, multiple linear regression equation. JEL classification: C2, E2, E3.

1. Introduction

Many economists have long been concerned with evaluating the factors that may have an influence on the standard of living of a population. Some authors (Romer, 1986; Wolff, 1991; Mankiw et al., 1992) designed several models of convergence between the level per capita production at an initial point and the growth rate per capita production over a time period series. In their studies, Evans and Karras (1993) underlined the fact that there is a convergence between per capita consumption properties within rich countries. Ravallion’s study (1994) focuses on standard of living measurement errors for estimating an individual-specific poverty line, due to imperfect information that can be found on “various consumption needs”. Other authors (Cooper et al., 2015) evaluated the standard of living within an era of fast technological change, focusing on two consumer demand systems: AIDS (Almost Ideal Demand System) of Deaton and Muellbauer (1980) and QAIDS (Quadratic Almost Ideal Demand System), which were adapted in order to estimate the income effects. In our study, we focus on the measurement of the standard of living, using the following independent variables: population, population density and inflation rate, considered to be important measures of welfare. This analysis, connected with the studies made before by other researchers, can be part of a complex standard of living measurement model.

2. Method and results

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This article focuses on the analysis of the dimension of standard of living in 10 member states of the European Union, during the period 2004-2013. In order to determine the key factors that have a contribution to this, we used panel data, which was analyzed using multiple linear regression equation. The dependent variable, the standard of living, was determined as a ratio between annual GDP per capita and the annual final consumption expenditures per capita. The population consumption expenditures were taken into consideration, because they measure the objective economic dimension of the standard of living, while the other expenses (e.g. the access to IT, telecommunications services) emphasize more on the subjective dimension of the standard of living, meaning the quality of life. The multiple linear regression focuses on the explanation of a dependent variable, with the help of m independent variables, where m>1. According to the mathematical model, the following relations characterize our equation (1, 2):

y = f (x1, x2, …, xm) (1)

where, f (x) = α + β1x1 + β2x2 + … + βmxm (2) with c, c1, c2, …, cm representing the parameters of the equation.

We will suppose that this is a stochastic relationship, where the econometric

model is (3):

y = f (x1, x2, …, xm) + ε (3) According to multiple linear regression, in an analytical writing, the equation becomes (4): yi = α + β1x1i + β2x2i + … + βmxmi + εi , i= 1,n (4) where εi is the error of specification and it has a known probabilistic distribution (it is an random variable), while n represents the number of observations.

The forecast model is (5): SOLt = α + β1POPt + β2DENSPOPt + β3INFLt + εt (5)

All the data in the panel regression model is in annual frequency and takes into consideration the period 2004-2013. Where (Table 1):

Table 1. Variables definition

Variable Definition of the variable Time series

SOL Standard of living (GDP per capita/ total consumption expenditure per capita)

2004-2013

POP Population (measured in thousands) – logarithmic function was used for this

2004-2013

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data, for a better precision.

DENSPOP Density of population (number of persons/ squared kilometer)

2004-2013

INFL Inflation rate (expressed in percentage)

2004-2013

Source: Authors’ own encoding The following results were obtained, using the EViews informatics program package, in order to estimate the parameters of the model:

Figure 1. Estimation of the parameters of the model

Source: Data according to Eviews informatics program package

Based on the information from the Figure 1, obtained with the Eviews

informatics program package, the following statements on the regression equation can be made:

 the free element from the regression equation (c) is  = 3,545030 and it represents the point where all the explanatory variables (population, density of the population and inflation rate) are equal to zero and have a standard error of 0,509786;

 the value of the β1 coefficient is -0,332416. This value can be explained the following way: when the population undertakes a raise of a unity, the standard of living decreases with 0,332416 unities. Because the value of P for this parameter is

0,0002  0,05, we can affirm that the parameter is statistically significant;

 the value of β2 is 0,005348. Thus, at a raise of the population density with a unity, the standard of living would rise with 0,005348 unities. The positive coefficient associated with the population density was expected because entertainment and leisure related industries thrive in dense population countries. For example, one finds

106 Finance – Challenges of the Future

more movie theatres, bowling alleys, skate rinks, etc. in densely populated countries versus non-densely populated countries. These leisure related industries also tend to be located in more affluent areas, where the people can afford not only to pay for the goods and services with their money, but also with available time. This aspect can be mainly explained by the existing situation of two of the analyzed countries, Czech Republic and Poland, which have the highest density of the population and also the highest standard of living. The value of P (0,0000) is higher than 0,05, se we can affirm that the parameter is statistically significant;

 the β3 coefficient has the value of 0,021120. In other words, at a raise of the inflation rate with a unity, the standard of living will encounter a growth of 0,021120 unities. The fact that the probability (P) associated to this parameter is 0,0360 (lower

than 0,05) leads us to the conclusion that the parameter 3 is statistically significant;

 the coefficient of determination (R 2 ) is 0,2033, which indicates that only

20,3388% of the standard of living variation can be explained by the three independent variables taken into analysis. A higher percent could be encountered by adding other variables that were omitted in this model;

 log likelihood = -7,310934 represents the logarithm of the likelihood function (supposing that the errors have a normal distribution); this function is determined taking into consideration the estimated values of the parameters. The function for the calculation of this indicator, used by EViews informatics program package, is (6):

L = n 2( ) 1+ ln 2p( ) + ln ût 2

å n( )( ) (6) where: 

2ˆ t

u sum of squared errors;

k = number of exogenous variables;

n = number of observations.

This indicator is used for statistical tests that find omitted variables from an econometric model, together with some tests for finding out redundant variables from an econometric model, such as, for instance, LR test or the ratio of verisimilitudes (Likelihood Ratio).

 from the calculation of Fisher statistics (F-statistic = 8,170097), the hypothesis according to which the model is not valid is rejected and we can conclude that the regression model is statistically valid. However, the associated probability (F- statistic = 0,000067 < 0,05) strengthens the affirmation that the constructed multiple linear regression model is statistically valid;

 mean dependent var = 1,838165 represents the average of the dependent

or endogenous variable, having the following calculation relationship (7):

y = y i

i=1

n

å n (7)

 Akaike criterion is used for comparing two or more econometric models.

The calculation relationship for it, used by the EViews informatics program package, is

(8):

AIC = -2L n+ 2k n (8)

where L = Log likelihood;

The decision rule used for this test is the following: it will be chosen the

econometric model for which the lowest value for this indicator was obtained.

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 Schwartz criteria is also used in order to compare two or more econometric

models. The calculation relation for this, used by EViews informatics program package,

is (9):

SC = -2L n+klnn n (9)

In this case also, it will be chosen the econometric model for which the lowest

value for this indicator was obtained.

The low Durbin-Watson statistic indicates serial correlation. An AR (1) process

could offset the serial correlation. However, after taking into consideration the AR (1)

process (Figure 2), the model estimates become spurious. The presence of serial

correlation in the model does not take away its significance. The model remains non-

bias and consistent. Serial correlation only affects parameter estimates. If anything, the

presence of serial correlation downplays the significance of the model.

Dependent Variable: SOL Method: Panel Least Squares Date: 03/23/15 Time: 22:55 Sample (adjusted): 2005 2013

Periods included: 9 Cross-sections included: 10 Total panel (balanced) observations: 90 Convergence achieved after 4 iterations

Variable Coefficient Std. Error t-Statistic Prob. C 3.702752 0.712018 5.200361 0.0000

POP -0.382257 0.117256 -3.260016 0.0016 DENSPOP 0.005632 0.001441 3.909761 0.0002

INFL -0.007427 0.003543 -2.095973 0.0391 AR(1) 0.745756 0.035581 20.95912 0.0000

R-squared 0.876270 Mean dependent var 1.788292

Adjusted R-squared 0.870447 S.D. dependent var 0.252556 S.E. of regression 0.090904 Akaike info criterion -1.904081 Sum squared resid 0.702395 Schwarz criterion -1.765203 Log likelihood 90.68366 Hannan-Quinn criter. -1.848077 F-statistic 150.4943 Durbin-Watson stat 1.967292 Prob(F-statistic) 0.000000

Figure 2. Model significance Source: Data according to Eviews informatics program package

3. Conclusions

According to the analysis made, the evolution of the standard of living in EU member states is influenced by the three independent variables chosen: population, density of the population and inflation rate. However, the multiple linear regression equation can be improved by the inclusion of other variables such as: poverty rate, unemployment rate, personal bankruptcy and by the enlargement of the number of

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observations. These variables could not be included in the regression equation, because available data for the analyzed states could not be found. References Cooper, R.J., McLaren, K.R., Rehman, F., Szewczyk, W.A., 2015. Economic welfare evaluation in an era of rapid technological change. J. Economics Letters, in press- accepted manuscript. Deaton, A., Muellbauer, J., 1980. An almost ideal demand system. Am. Econ. Rev. 70 (3), 312–326. Evans, P., Karras, G., 1993. Do standards of living converge?: Some cross-country evidence. J. Economics Letters. 43 (2), 149-155. Mankiw, N.G., Romer, D., Weil, D.N., 1992. A contribution to the empirics of economic growth. Quarterly Journal of Economics, 107, 407–438. Ravallion, M., 1994. Poverty rankings using noisy data on living standards. J. Economics Letters. 45, 481-485. Romer, P.M., 1986. Increasing returns and long-run growth, Journal of Political Economy, 94, 1002–1037. Wolff, E.N., 1991. Capital formation and productivity convergence over the long term. American Economic Review, 81, 565–579.

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