Estimation of time-invariant effects in static panel data models

This article proposes the Fixed Effects Filtered (FEF) and Fixed Effects Filtered instrumental variable (FEF-IV) estimators for estimation and inference in the case of time-invariant effects in static panel data models when N is large and T is fixed. The FEF-IV allows for endogenous time-invariant regressors but assumes that there exists a sufficient number of instruments for such regressors. It is shown that the FEF and FEF-IV estimators are -consistent and asymptotically normally distributed. The FEF estimator is compared with the Fixed Effects Vector Decomposition (FEVD) estimator proposed by Plumper and Troeger (2007 Plumper, T., Troeger, V. E. (2007). Efficient estimation of time-invariant and rarely changing variables in finite sample panel analyses with unit fixed effects. Political Analysis 15:124–139.Crossref, Web of Science ® , Google Scholar) and conditions under which the two estimators are equivalent are established. It is also shown that the variance estimator proposed for FEVD estimator is inconsistent and its use could lead to misleading inference. Alternative variance estimators are proposed for both FEF and FEF-IV estimators which are shown to be consistent under fairly general conditions. The small sample properties of the FEF and FEF-IV estimators are investigated by Monte Carlo experiments, and it is shown that FEF has smaller bias and RMSE, unless an intercept is included in the second stage of the FEVD procedure which renders the FEF and FEVD estimators identical. The FEVD procedure, however, results in substantial size distortions since it uses incorrect standard errors. In the case where some of the time-invariant regressors are endogenous, the FEF-IV procedure is compared with a modified version of Hausman and Taylor (1981 Hausman, B. Y. J. A., Taylor, W. (1981). Panel data and unobservable individual effects. Econometrica 49:1377–1398.Crossref, Web of Science ® , Google Scholar) (HT) estimator. It is shown that both estimators perform well and have similar small sample properties. But the application of standard HT procedure, that incorrectly assumes a subset of time-varying regressors are uncorrelated with the individual effects, will yield biased estimates and significant size distortions.