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This paper introduces bias-corrected estimators for nonlinear panel data models with both time invariant and time varying heterogeneity. These include limited dependent variable models with both unobserved individual effects and endogenous explanatory variables, and sample selection models with unobserved individual effects. Our two-step approach first estimates the reduced form by fixed effects procedures to obtain estimates of the time variant heterogeneity underlying the endogeneity/selection bias. We then estimate the primary equation by fixed effects including an appropriately constructed control function from the reduced form estimates as an additional explanatory variable. The fixed effects approach in this second step captures the time invariant heterogeneity while the control function accounts for the time varying heterogeneity. Since either or both steps might employ nonlinear fixed effects procedures it is necessary to bias adjust the estimates due to the incidental parameters problem. This problem is exacerbated by the two step nature of the procedure. As these two step approaches are not covered in the existing literature we derive the appropriate correction thereby extending the use of large-T bias adjustments to an important class of models. Simulation evidence indicates our approach works well in finite samples and an empirical example illustrates the applicability of our estimator.

The relationship between democracy and economic growth is of long standing interest. We revisit the panel data analysis of this relationship by Acemoglu et al. (forthcoming) using state of the art econometric methods. We argue that this and lots of other panel data settings in economics are in fact high-dimensional, resulting in principal estimators - the fixed effects (FE) and Arellano-Bond (AB) estimators - to be biased to the degree that invalidates statistical inference. We can however remove these biases by using simple analytical and sample-splitting methods, and thereby restore valid statistical inference. We find that the debiased FE and AB estimators produce substantially higher esti-mates of the long-run effect of democracy on growth, providing even stronger support for the key hypothesis in Acemoglu et al. (forthcoming). Given the ubiquitous nature of panel data, we conclude that the use of debiased panel data estimators should substantially improve the quality of empirical inference in economics.