Identification in a binary choice panel data model with a predetermined covariate

We study identification in a binary choice panel data model with a single predetermined binary covariate (i.e., a covariate sequentially exogenous conditional on lagged outcomes and covariates). The choice model is indexed by a scalar parameter 0, whereas the distribution of unit-specific heterogeneity, as well as the feedback process that maps lagged outcomes into future covariate realizations, is left unrestricted. We provide a simple condition under which 0 is never point-identified, no matter the number of time periods available. This condition is satisfied in most models, including the logit one. We also characterize the identified set of 0 and show how to compute it using linear programming techniques. While 0 is not generally point-identified, its identified set is informative in the examples we analyze numerically, suggesting that meaningful learning about 0 may be possible even in short panels with feedback. As a complement, we report calculations of identified sets for an average partial effect and find informative sets in this case as well.