Inference for Fixed Effects Estimators when Panels are Unbalanced

Abstract

We develop the asymptotic theory for two-way fixed effects M-estimators in unbalanced panels, within a framework where both panel dimensions grow large at proportional rates. The selection process may be deterministic, stochastic, or a combination of the two. We require neither a missing-at-random condition nor a selection equation, only a conditional mean restriction on the outcome. The uncorrected estimators are asymptotically normal but not correctly centered due to incidental parameter bias and feedback bias. The latter arises when regressors or the selection indicator respond to past outcomes. We propose debiased estimators that remove both biases without requiring knowledge of which regressors or selection components are predetermined. Simulations show that the corrections remove most of the bias and restore coverage close to nominal levels. Revisiting a study on capital inflow surges and banking crises, we find that the corrections leave qualitative conclusions unchanged but substantially shift the estimated magnitudes.