An instrumental variable model of multiple discrete choice

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 McFadden (1974), 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.