Random and Fixed Effect Models in Panel Regression

When performing a panel regression analysis, the problem of choosing an appropriate model specification arises. A pooled OLS regression assumes that the differences in data between cross-sections and time periods are insignificant so that they can be ignored and treated using a simple OLS regression.

However, in real life, distinctions between cross-sections and between time periods can be substantial. In these cases, individual effects have to be introduced in the model. To account for these changes fixed effect (FE) and random effect (RE) models are used. The difference between FE and RE specifications is based on how exactly they treated the un-observable differences between cross-sections or periods.


FE and RE models

An FE model resolves the issue of heterogeneity in observed objects by introducing individual dummy variables. This greatly increases the number of parameters to be estimated in the model and consumes degrees of freedom. As a result, the estimation of FE models may not be the most efficient compared to alternatives. However, benefit of the FE model is that the coefficients it produces are always unbiased and consistent.

This cannot be said about the RE model, which suggests a different way to treat the issue of heterogeneity. Instead of adding many dummy variables, RE models add a single stochastic term which accounts for cross-sectional or period variations. This method does not consume as many degrees of freedom but it creates an additional error term associated with the new added term. This error term may be correlated with the regression residuals. In this case, one of the assumptions of the Gauss-Markov theorem would be violated and the coefficients will not be BLUE (Best Linear Unbiased Estimates). In particular, they may no longer remain consistent even though such specification would be more efficient compared to FE models.


Final Note

When conducting a panel regression analysis in Stata or other econometric software, it is necessary to run both the FE and RE specifications to decide which of them generates more appropriate results. After the outcomes of both models have been obtained, the Hausman specification test will help to determine which output to choose for interpretation. The null hypothesis of this test is that the coefficients of the RE model are consistent. If the test does not reject the null hypothesis, the RE model is accepted as more efficient. In the opposite case, when the null hypothesis is rejected, the FE specification is preferred in order to remain consistent with the Gauss-Markov theorem.


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