Identification et Information in Monotone Binary Models

Let, y, a binary outcome, v a continuous explanatory variable and x someother explanatory variables. Assume that the population distribution of the random variablew = (y, v, x) satisfies Monotone (1) and Large Support (2) assumptions: (1) E(y | v, x) ismonotone in v and (2) E(y | v, x) varies from 0 to 1 when v varies over its support. Withinthis framework, this paper studies inference on the parameters of the semiparametric binaryregression model y = 1(xß + v + > 0). It shows that the moment restrictions that Lewbel(2000) proposed lead to exact identification of the parameter of interest, ß. In other words, anuncorrelated-error restriction (E(x) = 0) combined with a partial-independance assumption(F( | v, x) = F( | x)) are sufficient and necessary for identification. We also show thatLewbel’s moment estimator attains the semi-parametric efficiency bound in the set of latentmodels that he considers. Yet, uncorrelated-error and partial-independence assumptionsare not sufficient to identify ß when the support of v is not sufficiently rich. We proposeintuitive additional identifying assumptions under which ß remains just identified. Monte-Carlo experiments show that the estimation performs well in moderately small samples. Anextension to ordered choice models is also provided.