A sufficient and necessary condition for identification of binary choice models with fixed effects

We study the identification of binary choice models with fixed effects. We provide a condition called sign saturation and show that this condition is sufficient for the identification of the model. In particular, we can guarantee identification even with bounded regressors. We also show that without this condition, the model is never identified unless the error distribution belongs to a small class. A test is provided to check the sign saturation condition and can be implemented using existing algorithms for the maximum score estimator. We also discuss the practical implication of our results.