# Econometric Analysis of Panel Data

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```					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   Fixed Effects
E(ui|Xi) = 0     E(ui|Xi) =/= 0
GLS or RE-OLS      Consistent and   Inconsistent
(Random Effects)   Efficient

LSDV or FE-OLS     Consistent       Consistent
(Fixed Effects)    Inefficient      Possibly Efficient
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
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

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