Many panel data methods, while allowing for general dependence between covariates and time-invariant agent-specific heterogeneity, place strong a priori restrictions on feedback: how past outcomes, covariates, and heterogeneity map into future covariate levels. Ruling out feedback entirely, as often occurs in practice, is unattractive in many dynamic economic settings. We provide a general characterization of all feedback and heterogeneity robust (FHR) moment conditions for nonlinear panel data models and present constructive methods to derive feasible moment-based estimators for specific models. We characterize semiparametric efficiency bounds in this case, quantifying the information loss associated with accommodating feedback as well as providing insight into how to construct estimators with good efficiency properties in practice. When FHR moments point-identify the target parameter - the case we emphasize - we show they attain the efficiency bound associated with the full model. These results apply both to the finite dimensional parameter indexing the parametric part of the model as well as to estimands that involve averages over the distribution of unobserved heterogeneity or features of the feedback process. Lastly, we illustrate our methods by providing a complete characterization of all FHR moment functions in the multi-spell mixed proportional hazards model, and compute efficient moment functions for both model parameters and average effects in this setting.

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