An instrumental variable model of multiple discrete choice

Abstract

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 de- terminants of latent utilities. The model does not employ large support, spe- cial regressor, or control function restrictions; indeed, it is silent about the pro- cess 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 utili- ties. 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 paramet- ric models, require little more calculation than what is involved in a conven- tional maximum likelihood analysis. The results are illustrated using a model that is essentially the conditional logit model of McFadden (1974), but with po- tentially endogenous explanatory variables and instrumental variable restric- tions. The method employed has wide applicability and for the first time brings in- strumental variable methods to bear on structural models in which there are mul- tiple unobservables in a structural equation. Keywords. Partial identification, random sets, multiple discrete choice, endo- geneity, instrumental variables, incomplete models. JEL classification. C25, C26.