Single equation endogenous binary response models

This paper studies single equation models for binary outcomes incorporating instrumental variable restrictions. The models are incomplete in the sense that they place no restriction on the way in which values of endogenous variables are generated. The models are set, not point, identifying. The paper explores the nature of set identi…cation in single equation IV models in which the binary outcome is determined by a threshold crossing condition. There is special attention to models which require the threshold crossing function to be a monotone function of a linear index involving observable endogenous and exogenous explanatory variables. Identi…ed sets can be large unless instrumental variables have substantial predictive power. A generic feature of the identi…ed sets is that they are not connected when instruments are weak. The results suggest that the strong point identifying power of triangular “control function” models restricted versions of the IV models considered here is fragile, the wide expanses of the IV model’s identi…ed set awaiting in the event of failure of the triangular model’s restrictions. Keywords: Binary Response, Control functions, Endogeneity, Incomplete models, Index Restrictions, Instrumental variables, Partial Identi…cation, Threshold Crossing Models, Triangular Models. 1. Introduction This paper explores the identifying power of single equation models for binary responses. The models allow explanatory variables to be endogenous and embody instrumental variable (IV) exclusion and independence restrictions. Probit and logit models with endogenous explanatory variables are familiar examples of parametric models to which the results of this paper apply. The analysis is essentially nonparametric but parametric restrictions are very easy to incorporate as will be demonstrated. These IV models place no restrictions on the genesis of endogenous explanatory variables. In this respect they are incomplete. One consequence of this is that the models are set not point identifying for deep structural features. One of the contributions of the paper is to characterize tight identi…ed sets in nonparametric and parametric versions of the IV binary response model. I thank Sokbae Lee, Lars Nesheim, Adam Rosen and Richard Spady for stimulating comments and discussions and Konrad Smolinski for excellent research assistance. Some of the results given here were presented at seminars at Caltech, UCLA and USC in November 2007 and subsequently at the Malinvaud Seminar in Paris in December 2007, and at seminars at EUI, University of Amsterdam, CeMMAP, Cowles Foundation, Georgetown University and Johns Hopkins University. I gratefully acknowledge the …nancial support of the Economic and Social Research Council through a grant (RES-589-28-0001) to the ESRC Centre for Microdata Methods and Practice (CeMMAP).