1. 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

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

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

4. Random Effects Model
Assumptions
Cross Section Independence

5. Random Effects Model
Extensions
Weak Exogeneity
Heteroscedasticity

6. Random Effects Model
Extensions
Serial Correlation
Spatial Correlation

7. Model Estimation: GLS
Model Representation

8. Model Estimation: GLS
GLS

9. Model Estimation: RE-OLS
Partial Group Mean Deviations

10. Model Estimation: RE-OLS
Model Assumptions
OLS

11. 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

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

13. Model Estimation: ML
Log-Likelihood Function

14. Model Estimation: ML
ML Estimator

15. 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:

16. 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:

17. 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,
Breusch, T. and A. Pagan, “The LM Test and Its Applications to Model Specification in Econometrics,” Review of Economic Studies, 47, 1980,

18. 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
Possibly Efficient

19. Hypothesis Testing Fixed Effects vs. Random Effects
Fixed effects estimator is consistent under H0 and H1; Random effects estimator is efficient under H0, but it is inconsistent under H1.
Hausman Test Statistic

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

21. 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:

22. 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,
Baltagi, B.H., and L. Liu, Alternative Ways of Obtaining Hausman’s Test Using Artificial Regressions, Statistics and Probability Letters, 77, 2007,
Hausman, J.A., Specification Tests in Econometrics, Econometrica, 46, 1978,
Hausman, J.A. and W.E. Taylor, Panel Data and Unobservable Individual Effects, Econometrics, 49, 1981,
Mundlak, Y., On the Pooling of Time Series and Cross-Section Data, Econometrica, 46, 1978,

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