Vera Tabakova, East Carolina University Panel Data Models Modified JJ Vera Tabakova, East Carolina University

Chapter 15: Panel Data Models Grunfeld’s Investment Data Sets of Regression Equations Seemingly Unrelated Regressions The Fixed Effects Model The Random Effects Model Principles of Econometrics, 3rd Edition

Chapter 15: Panel Data Models The different types of panel data sets can be described as: “long and narrow,” with “long” describing the time dimension and “narrow” implying a relatively small number of cross sectional units; “short and wide,” indicating that there are many individuals observed over a relatively short period of time; “long and wide,” indicating that both N and T are relatively large. Principles of Econometrics, 3rd Edition

Grunfeld’s Investment Data The data consist of T = 20 years of data (1935-1954) for N = 10 large firms. Let yit = INVit and x2it = Vit and x3it = Kit 1 2 Principles of Econometrics, 3rd Edition

2 Firms: all coefficient equal 3b Principles of Econometrics, 3rd Edition

2 Firms: each has different coeffs (4b) Principles of Econometrics, 3rd Edition

Sets of Regression Equations Assumption (5) says that the errors in both investment functions (i) have zero mean, (ii) are homoskedastic with constant variance, and (iii) are not correlated over time; autocorrelation does not exist. The two equations do have different error variances (5) Principles of Econometrics, 3rd Edition

Separate OLS estimation of 2 Firms Principles of Econometrics, 3rd Edition

Joint OLS estimation using dummies Let Di be a dummy variable equal to 1 for the Westinghouse observations and 0 for the General Electric observations. This is one OLS regression of data for both firms using dummies. (6) Principles of Econometrics, 3rd Edition

15.2 Sets of Regression Equations Principles of Econometrics, 3rd Edition

Seemingly Unrelated Regressions This assumption says that the error terms in the two equations, at the same point in time, are correlated. This kind of correlation is called a contemporaneous correlation. (15.7) Principles of Econometrics, 3rd Edition

Seemingly Unrelated Regressions GLS Econometric software includes commands for SUR (or SURE) that carry out the following steps: Estimate the equations separately using least squares; Use the least squares residuals from step (i) to estimate ; Use the estimates from step (ii) to estimate the two equations jointly within a generalized least squares framework. GLS Principles of Econometrics, 3rd Edition

GLS estimation of SUR Principles of Econometrics, 3rd Edition

Separate OLS or Joint GLS Estimation? There are two situations where separate least squares estimation is just as good as the SUR technique : when the equation errors are not contemporaneously correlated; when the same explanatory variables appear in each equation. If the explanatory variables in each equation are different, then a test to see if the correlation between the errors is significantly different from zero is of interest. Principles of Econometrics, 3rd Edition

Separate OLS or Joint GLS Estimation? In this case Principles of Econometrics, 3rd Edition

Separate OLS or Joint GLS Estimation? Testing for correlated errors for two equations: LM = 10.628 > 3.84 Hence we reject the null hypothesis of no correlation between the errors and conclude that there are potential efficiency gains from estimating the two investment equations jointly using GLS Principles of Econometrics, 3rd Edition

Separate OLS or Joint GLS Estimation? If there are 3 instead of 2 firms Testing for correlated errors: Principles of Econometrics, 3rd Edition

Separate OLS or Joint SUR Estimation? If there are M firms: Testing for correlated errors in M equations: Under the null hypothesis that there are no contemporaneous correlations, this LM statistic has a χ2-distribution with M(M–1)/2 degrees of freedom, in large samples. Principles of Econometrics, 3rd Edition

Testing for Equality of Coefficients Most econometric software will perform an F-test and/or a Wald χ2–test; in the context of SUR equations both tests are large sample approximate tests. The F-statistic has J numerator degrees of freedom and (MTK) denominator degrees of freedom, where J is the number of hypotheses, M is the number of equations, and K is the total number of coefficients in the whole system, and T is the number of time series observations per equation. The χ2-statistic has J degrees of freedom. (8) Principles of Econometrics, 3rd Edition

The Fixed Effects Model We cannot consistently estimate the 3×N×T parameters in (9) with only NT total observations. (9) (10) Principles of Econometrics, 3rd Edition

Dummies to capture Fixed Effect All behavioral differences between individual firms and over time are captured by the intercept. Individual intercepts are the only coeffs allowed to vary and “control” for these firm specific differences. All other coefficients are assumed equal. (11) Principles of Econometrics, 3rd Edition

Dummies for fixed effect This specification is sometimes called the least squares dummy variable model, or the fixed effects model. (12) Principles of Econometrics, 3rd Edition

A Dummy Variable Model Principles of Econometrics, 3rd Edition

Test of equality of intercepts These N–1= 9 joint null hypotheses are tested using the usual F-test statistic. In the restricted model all the intercept parameters are equal. If we call their common value β1, then the restricted model is: (13) Principles of Econometrics, 3rd Edition

Pooled OLS regression Here all coefficients are assumed equal. This model is a pooled regression used as the restricted model for the test. It disregards the heteroscedasticity. Principles of Econometrics, 3rd Edition

Test of equality of coefficients We reject the null hypothesis that the intercept parameters for all firms are equal. We conclude that there are differences in firm intercepts, and that the data should not be pooled into a single model with a common intercept parameter. Principles of Econometrics, 3rd Edition

The Fixed Effects Estimator (14) (15) Principles of Econometrics, 3rd Edition

The Fixed Effects Estimator (16) (17) Principles of Econometrics, 3rd Edition

The Fixed Effects Estimator Principles of Econometrics, 3rd Edition

The Fixed Effects Estimator (18) Principles of Econometrics, 3rd Edition

The Fixed Effects Estimator Principles of Econometrics, 3rd Edition

The Fixed Effects Estimator (19) Principles of Econometrics, 3rd Edition

Fixed Effects Estimation Using a Microeconomic Panel Principles of Econometrics, 3rd Edition

The Random Effects Model (20) (21) (22) Principles of Econometrics, 3rd Edition

The Random Effects Model Because the random effects regression error in (24) has two components, one for the individual and one for the regression, the random effects model is often called an error components model. (23) (24) Principles of Econometrics, 3rd Edition

Error Term Assumptions (25) Principles of Econometrics, 3rd Edition

Error Term Assumptions There are several correlations that can be considered. The correlation between two individuals, i and j, at the same point in time, t. The covariance for this case is given by Principles of Econometrics, 3rd Edition

Error Term Assumptions The correlation between errors on the same individual (i) at different points in time, t and s. The covariance for this case is given by (26) Principles of Econometrics, 3rd Edition

Error Term Assumptions The correlation between errors for different individuals in different time periods. The covariance for this case is Principles of Econometrics, 3rd Edition

Error Term Assumptions (27) Principles of Econometrics, 3rd Edition

Testing for Random Effects (28) Principles of Econometrics, 3rd Edition

Estimation of the Random Effects Model (29) (30) (31) Principles of Econometrics, 3rd Edition

An Example Using the NLS Data Principles of Econometrics, 3rd Edition

Endogeneity in the Random Effects Model If the random error is correlated with any of the right- hand side explanatory variables in a random effects model then the least squares and GLS estimators of the parameters are biased and inconsistent. Principles of Econometrics, 3rd Edition

Chapter 15 Appendix Appendix 15A Estimation of Error Components Principles of Econometrics, 3rd Edition

Appendix 15A Estimation of Error Components Principles of Econometrics, 3rd Edition

Appendix 15A Estimation of Error Components Principles of Econometrics, 3rd Edition

Appendix 15A Estimation of Error Components Principles of Econometrics, 3rd Edition