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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

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**Panel Data Analysis Random Effects Model**

ui is random, independent of eit and xit. Define eit = ui + eit the error components.

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**Random Effects Model Assumptions Strict Exogeneity Homoschedasticity**

X includes a constant term, otherwise E(ui|X)=u. Homoschedasticity Constant Auto-covariance (within panels)

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Random Effects Model Assumptions Cross Section Independence

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Random Effects Model Extensions Weak Exogeneity Heteroscedasticity

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Random Effects Model Extensions Serial Correlation Spatial Correlation

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Model Estimation: GLS Model Representation

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Model Estimation: GLS GLS

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**Model Estimation: RE-OLS**

Partial Group Mean Deviations

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**Model Estimation: RE-OLS**

Model Assumptions OLS

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**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

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**Model Estimation: RE-OLS**

Panel-Robust Variance-Covariance Matrix Consistent statistical inference for general heteroscedasticity, time series and cross section correlation.

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Model Estimation: ML Log-Likelihood Function

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Model Estimation: ML ML Estimator

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**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:

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**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:

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**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,

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**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

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**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

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**Hypothesis Testing Fixed Effects vs. Random Effects**

Alternative (Asym. Eq.) Hausman Test Estimate any of the random effects models F Test that g = 0

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**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:

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**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,

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**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

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