Contents 12 THE PANEL DATA MODEL2 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 12.1.1 Individual Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 12.1.2 Fixed Effects and Random Effects . . . . . . . . . . . . . . . . . . . . . .5 12.1.3 Some Algebraic Results Related to the G Matrix . . . . . . . . . . . . . .5 12.1.4 Partition of the Sum of Squares. . . . . . . . . . . . . . . . . . . . . . .6 12.1.5 The Within and Between Estimators . . . . . . . . . . . . . . . . . . . . .9 12.2 The Fixed-Effects Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 12.2.1 The Identifi cation Problem . . . . . . . . . . . . . . . . . . . . . . . . . .10 12.2.2 The Least Square Estimation . . . . . . . . . . . . . . . . . . . . . . . . .11 12.2.3 Properties of the Least Square Estimators . . . . . . . . . . . . . . . . . .13 12.2.4 The Between Estimator in the Fixed-Effects Model . . . . . . . . . . . . .14 12.2.5 The Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . .16 12.3 The Random-Effects Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 12.3.1 The Generalized Least Square Estimation . . . . . . . . . . . . . . . . . .18 12.3.2 Properties of the GLS Estimators. . . . . . . . . . . . . . . . . . . . . .19 12.3.3 Variance Estimators. . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 12.3.4 Maximum Likelihood Estimation. . . . . . . . . . . . . . . . . . . . . .22 12.3.5 Time-Invariant Regressors . . . . . . . . . . . . . . . . . . . . . . . . . .24 12.3.6 The Correlation between Regressors and the Effect . . . . . . . . . . . . .27 12.4 Some Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30 1 Chapter12 THE PANEL DATA MODEL 12.1Introduction If data on each individual in an economy, such as a household, a fi rm, a region, a country, etc., can be collected repeatedly over time, and there are many such individuals, then we have a sample that is called panel data, or longitudinal data. At each point of time the many individuals consists of a cross-section data set, while the repeated observations on each individual over time represent time series. Panel data are not simply a collection of several cross-section data sets over time, since in the latter case we do not necessarily have repeated time series data on the same individuals. Because of the “two-dimensional” characteristic of the panel data, we need to change the subscript notation for various variables. For example, the notation for the dependent variable yi changes to yit , where the fi rst subscript i indicates a particular individual in a cross-section while the second subscript t indicates the point of the time when individual i is observed. Given the double subscripts of each variable, the linear regression model for panel data is expressed as1 yit= o+ x0 it + it, (12.1) i = 1,.,n, and t = 1,.,T. The double subscripts also reminds us of the sample size of the panel data, as a product of the size n of the cross-section and the number T of time periods, is generally quite large. However, the usefulness of panel data is more than its large sample size. The very nature of repeated observations on the same individual offers the opportunity to infer behavioral changes of an individual over time while controlling the differences across individuals. To see this, consider the standard regression equation for the estimation of income elasticity of 1We separate the constant term from all other k 1 non-constant explanatory variables in the (k 1)-dimensional vector xit. So 0is the intercept and is the vector of k 1 slopes. The reason for such separate treatments will become clear shortly. 2 CHAPTER 12. THE PANEL DATA MODEL3 health care: lnci= 0+ 1lnmi+ i where ciand miare the ith observations on health care expenditure and income, respectively, while the coeffi cient 1is the income elasticity to be estimated. The estimation can be conducted using cross-sectional micro data in which case i represents an individual consumer or household. Time series data can also be employed, in which case i represents a particular time and the ci and miusually are aggregate data (for the entire country, for instance) at time i.2No matter which type of data are used, the above regression equation is obviously too simple since there are many factors other than income that can infl uence health care consumption decision. To control those other factors, we may want to include as many explanatory variables as possible into the regression equation. This consideration is particularly important in the cross sectional case where the data are collected at the micro level. Numerous personal traits like age, gender, marital status, etc., may all exert substantial infl uence on the health care consumption decision. Moreover, a number of individual characteristics, such as current health condition, attitude toward exercises,