This paper studies identification in multiple discrete choice models in which there may be endogenous explanatory variables, that is, explanatory variables that are not restricted to be distributed independently of the unobserved determinants of latent utilities. The model does not employ large support, special regressor, or control function restrictions; indeed, it is silent about the process that delivers values of endogenous explanatory variables, and in this respect it is incomplete. Instead, the model employs instrumental variable restrictions that require the existence of instrumental variables that are excluded from latent utilities and distributed independently of the unobserved components of utilities. We show that the model delivers set identification of latent utility functions and the distribution of unobserved heterogeneity, and we characterize sharp bounds on these objects. We develop easy‐to‐compute outer regions that, in parametric models, require little more calculation than what is involved in a conventional maximum likelihood analysis. The results are illustrated using a model that is essentially the conditional logit model of 41, but with potentially endogenous explanatory variables and instrumental variable restrictions. The method employed has wide applicability and for the first time brings instrumental variable methods to bear on structural models in which there are multiple unobservables in a structural equation.

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Abstract.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 Mathematica TM 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.

a b s t r a c tThis paper studies single equation instrumental variable models of ordered choice in which explanatory variables may be endogenous. The models are weakly restrictive, leaving unspecified the mechanism that generates endogenous variables. These incomplete models are set, not point, identifying for parametrically (e.g. ordered probit) or nonparametrically specified structural functions. The paper gives results on the properties of the identified set for the case in which potentially endogenous explanatory variables are discrete. The results are used as the basis for calculations showing the rate of shrinkage of identified sets as the number of classes in which the outcome is categorised increases.