Econometric Analysis of Panel Data Panel Data Analysis

Econometric Analysis of Panel Data • Panel Data Analysis – Random Effects • • Assumptions GLS Estimator Panel-Robust Variance-Covariance Matrix ML Estimator – Hypothesis Testing • Test for Random Effects • Fixed Effects vs. Random Effects

Panel Data Analysis • Random Effects Model – ui is random, independent of eit and xit. – Define eit = ui + eit the error components.

Random Effects Model • Assumptions – Strict Exogeneity • X includes a constant term, otherwise E(ui|X)=u. – Homoschedasticity – Constant Auto-covariance (within panels)

Random Effects Model • Assumptions – Cross Section Independence

Random Effects Model • Extensions – Weak Exogeneity – Heteroscedasticity

Random Effects Model • Extensions – Serial Correlation – Spatial Correlation

Model Estimation: GLS • Model Representation

Model Estimation: GLS • GLS

Model Estimation: RE-OLS • Partial Group Mean Deviations

Model Estimation: RE-OLS • Model Assumptions • OLS

Model Estimation: RE-OLS • Need a consistent estimator of q: – Estimate the fixed effects model to obtain – Estimate the between model to obtain – Or, estimate the pooled model to obtain – Based on the estimated large sample variances, it is safe to obtain

Model Estimation: RE-OLS • Panel-Robust Variance-Covariance Matrix – Consistent statistical inference for general heteroscedasticity, time series and cross section correlation.

Model Estimation: ML • Log-Likelihood Function

Model Estimation: ML • ML Estimator

Hypothesis Testing To Pool or Not To Pool, Continued • Test for Var(ui) = 0, that is – If Ti=T for all i, the Lagrange-multiplier test statistic (Breusch-Pagan, 1980) is:

Hypothesis Testing To Pool or Not To Pool, Continued – For unbalanced panels, the modified Breusch. Pagan LM test for random effects (Baltagi-Li, 1990) is: – Alternative one-side test:

Hypothesis Testing To Pool or Not To Pool, Continued • References – Baltagi, B. H. , and Q. Li, A Langrange Multiplier Test for the Error Components Model with Incomplete Panels, Econometric Review, 9, 1990, 103 -107. – Breusch, T. and A. Pagan, “The LM Test and Its Applications to Model Specification in Econometrics, ” Review of Economic Studies, 47, 1980, 239 -254.

Hypothesis Testing Fixed Effects vs. Random Effects Estimator Random Effects E(ui|Xi) = 0 Fixed Effects E(ui|Xi) =/= 0 GLS or RE-OLS (Random Effects) Consistent and Efficient Inconsistent LSDV or FE-OLS (Fixed Effects) Consistent Inefficient Consistent Possibly Efficient

Hypothesis Testing Fixed Effects vs. Random Effects • Fixed effects estimator is consistent under H 0 and H 1; Random effects estimator is efficient under H 0, but it is inconsistent under H 1. • Hausman Test Statistic

Hypothesis Testing Fixed Effects vs. Random Effects • Alternative (Asym. Eq. ) Hausman Test – Estimate any of the random effects models – F Test that g = 0

Hypothesis Testing Fixed Effects vs. Random Effects • Ahn-Low Test (1996) – Based on the estimated errors (GLS residuals) of the random effects model, estimate the following regression:

Hypothesis Testing Fixed Effects vs. Random Effects • References – Ahn, S. C. , and S. Low, A Reformulation of the Hausman Test for Regression Models with Pooled Cross-Section Time-Series Data, Journal of Econometrics, 71, 1996, 309 -319. – Baltagi, B. H. , and L. Liu, Alternative Ways of Obtaining Hausman’s Test Using Artificial Regressions, Statistics and Probability Letters, 77, 2007, 1413 -1417. – Hausman, J. A. , Specification Tests in Econometrics, Econometrica, 46, 1978, 1251 -1271. – Hausman, J. A. and W. E. Taylor, Panel Data and Unobservable Individual Effects, Econometrics, 49, 1981, 1377 -1398. – Mundlak, Y. , On the Pooling of Time Series and Cross-Section Data, Econometrica, 46, 1978, 69 -85.

Example: Investment Demand • Grunfeld and Griliches [1960] – i = 10 firms: GM, CH, GE, WE, US, AF, DM, GY, UN, IBM; t = 20 years: 1935 -1954 – Iit = Gross investment – Fit = Market value – Cit = Value of the stock of plant and equipment