Correlated Random Effects Models with Endogenous Explanatory Variables and Unbalanced Panels

This paper shows how the correlated random effects approach can be extended to linear panel data models when instrumental variables are needed and the panel is unbalanced. We obtain the algebraic equivalence between the fixed effects two stage least squares (FE2SLS) estimator and a pooled 2SLS (P2SLS) estimator on a transformed equation. This equivalence allows us to obtain fully robust Hausman tests comparing random effects 2SLS (RE2SLS) and FE2SLS. In addition, we obtain an equivalence result for control function estimates and FE2SLS estimates in an unbalanced panel. We use this result to obtain a robust variable addition Hausman test that effectively compares the FE and FE2SLS estimates. We illustrate the tests using an unbalanced panel on student performance and spending at the school level.