Identification & Information in Monotone Binary Models

Let, y, a binary outcome, v a continuous explanatory variable and x some other explanatory variables. We study inference on the parameter b of the semiparametric binary regression model y=1(xb+v+e>0). We show that the set-up introduced by Lewbel (2000) that is, an uncorrelated-error restriction (E(x'e)=0) combined with a partial-independance assumption (F(e/v,x)=F(e/x)) and a large support assumption (Supp(-xb-e) c Supp(v)) provides exact identification of b and F(e/x). The two restrictions that the population distribution of the random variable w=(y,v,x) should satisfy are Monotone (1) and Large Support (2) conditions: (1) E(y/v,x) is monotone in v and (2) E(y/v,x) varies from 0 to 1 when v varies over its support. Moreover, we show that Lewbel's moment estimator attains the semi-parametric efficiency bound in the set of latent models that he considers. Yet, the uncorrelated-error and partial-independence assumptions are not sufficient to identify b when the support of v is not sufficiently rich. We propose intuitive additional restrictions on the tails of the conditional distribution of e under which b remains exactly identified even when condition(2) is not satisfied. In such a case, Monte-Carlo experiments show that the estimation performs well in moderately small samples. An extension to ordered choice models is provided.