Inference for Fixed Effects Estimators when Panels are Unbalanced

We derive the asymptotic properties of two-way fixed effects M-estimators with missing observations in an asymptotic framework in which the numbers of cross-sectional units and time periods grow jointly. We allow the selection process to be deterministic (conditional on the unobserved effects and initial conditions), stochastic, or mixed, and we impose only a conditional mean restriction. The uncorrected estimators are asymptotically normal but not centered at zero, suffering from incidental parameter and feedback biases. Feedback bias can be induced by predetermined regressors in the outcome equation and by a predetermined selection process. We propose debiased estimators that handle both sources without requiring knowledge of which regressors or selection components are predetermined.