Sharp identified sets for discrete variable IV models

Instrumental variable models for discrete outcomes are set, not point, identifying. The paper characterises identi…ed sets of structural functions when endogenous variables are discrete. Identi…ed sets are unions of large numbers of convex sets and may not be convex nor even connected. Each of the component sets is a projection of a convex set that resides in a much higher dimensional space onto the space in which a structural function resides. The paper develops a symbolic expression for this projection and gives a constructive demonstration that it is indeed the identi…ed set. We provide a MathematicaTM notebook which computes the set symbolically. We derive properties of the set, suggest how the set can be used in practical econometric analysis when outcomes and endogenous variables are discrete and propose a method for estimating identi…ed sets under parametric or shape restrictions. We develop an expression for a set of structural functions for the case in which endogenous variables are continuous or mixed discrete-continuous and show that this set contains all structural functions in the identi…ed set in the non-discrete case. Keywords: Discrete endogenous variables, Discrete outcomes, Endogeneity, Fourier-Motzkin Elimination, Incomplete models, Instrumental variables, Set identi…cation, Threshold Crossing Models. JEL Codes: C10, C14, C50, C51. 1. Introduction This paper gives new results on the identifying power of single equation instrumental variable (SEIV) models in which both the outcome of interest and potentially endogenous explanatory variables are discrete. These models generally set rather than point identify structural functions.1 The paper derives the sharp identi…ed set for the general case in which there is an M -valued outcome and there are endogenous variables with K points of support. The discrete outcome, discrete endogenous variable case studied here arises frequently in applied econometrics practice. Examples of settings in which the results of the paper are useful include situations in which a binary or ordered probit, or a logit or a count data analysis or some semiparametric or nonparametric alternative would be considered and explanatory variables are endogenous. We study nonparametric models but, as we show, characterizations of identi…ed sets for nonparametric models are very useful in constructing identi…ed sets in parametric cases. This is a revised and extended version of the paper “Set identifying models with discrete outcomes and endogenous variables”February 5th 2010. We 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) and the intellectual support of our colleagues at CeMMAP with whom we have had many useful discussions of this work. See Chesher (2010).