Panel Binary Variables and Sufficiency: Generalizing Conditional Logit

This paper extends the conditional logit approach used in panel data models of binary variables with correlated fixed effects and strictly exogenous regressors. In a two-period two-state model, necessary and sufficient conditions on the joint distribution function of the individual-and-period specific shocks are given such that the sum of individual binary variables across time is a sufficient statistic for the individual effect. Under these conditions, root-n consistent conditional likelihood estimators exist. Moreover, it is shown by extending Chamberlain (1992) that root-n consistent regular estimators can be constructed in panel binary models if and only if the property of sufficiency holds. Imposing sufficiency is shown to reduce the dimensionality of the bivariate distribution function of the individual-and-period specific shocks. This setting is much less restrictive than the conditional logit approach (Rasch, Andersen, Chamberlain). In applied work, it amounts to quasi-difference the binary variables as if they were continuous variables and to transform a panel data model into a cross-section model. Semiparametric approaches can then be readily applied.