Fixed-effects binary choice models with three or more periods

We consider fixed-effects binary choice models with a fixed number of periods T and regressors without a large support. If the time-varying unobserved terms are i.i.d. with known distribution F, Chamberlain (2010) shows that the common slope parameter is point identified if and only if F is logistic. However, he only considers in his proof T = 2. We show that the result does not generalize to T ≥ 3: the common slope parameter can be identified when F belongs to a family including the logit distribution. Identification is based on a conditional moment restriction. Under restrictions on the covariates, these moment conditions lead to point identification of relative effects. If T = 3 and mild conditions hold, GMM estimators based on these conditional moment restrictions reach the semiparametric efficiency bound. Finally, we illustrate our method by revisiting Brender and Drazen (2008).