Semiparametric structural models of binary response: Shape restrictions and partial identification

The paper studies the partial identifying power of structural single equation threshold crossing models for binary responses when explanatory variables may be endogenous. The paper derives the sharp identi…ed set of threshold functions for the case in which explanatory variables are discrete and provides a constructive proof of sharpness. There is special attention to a widely employed semiparametric shape restriction which requires the threshold crossing function to be a monotone function of a linear index involving the observable explanatory variables. It is shown that the restriction brings great computational bene…ts, allowing direct calculation of the identi…ed set of index coe¢ cients without calculating the nonparametrically speci…ed threshold function. With the restriction in place the methods of the paper can be applied to produce identi…ed sets in a class of binary response models with mis-measured explanatory variables. Keywords: Binary Response, Endogeneity, Incomplete models, Index Restrictions, Instrumental variables, Measurement Error Models, Partial Identi…cation, Probit Models, Shape Restrictions, Threshold Crossing Models. JEL Codes: C10, C14, C50, C51. 1. Introduction This paper gives new results on the identifying power of a single equation, limited information, instrumental variable (IV) model for a binary response in a structural econometric model that admits the possibility that explanatory variables are endogenous. The model involves a scalar continuously distributed latent variable U and a threshold crossing function p(X) which depends on a vector random variable X some of whose elements may be jointly dependently distributed with U . In this threshold crossing model realizations of U exceeding p(X) lead to the outcome Y = 1; realizations less than or equal to p(X) lead to Y = 0. The marginal distribution of U is normalized uniform on the unit interval. Realizations of (Y;X;Z) are observed where Z is a list of instrumental variables excluded from the threshold crossing function. The structural latent variable U is independent of Z in the sense that Pr[U ujZ = z] is independent of z for all values u in the unit interval and for all values z in some support set Z. I thank Martin Cripps, Lars Nesheim, Adam Rosen and Richard Spady for stimulating comments and discussions and Konrad Smolinski for excellent research assistance. Some of the results of this paper were presented at seminars at Caltech, UCLA and USC in November 2007, at the Malinvaud Seminar in Paris in December 2007 and at the inaugural Asian Econometric Theory Lecture at the SETA Conference, Kyoto, Japan, July 30th 2009. I thank participants for comments. I gratefully acknowledge the …nancial support of the UK Economic and Social Research Council through a grant (RES-589-28-0001) to the ESRC Centre for Microdata Methods and Practice (CeMMAP).