Identification of Semiparametric Discrete Choice Models

Abstract

The question of model identification is analyzed for the semiparametric random utility model of discrete choice. Attention is focused on settings where agents face a common choice between a set of J+l alternatives, but where actual choices are only partially observed. Necessary conditions are derived for the setting where the only data on actual choices consists of a binary indicator for one of the alternatives. Sufficient conditions are developed in this setting for a linear in parameters specification of indirect utility. It is found that relative to the parametric case, only a mild continuity restriction on the distribution of regressors is needed in the semiparametric model. Under these circumstances all of the choice probabilities are identified, even though actual choices are only partially observed. It is shown that estimators that rely only on the index structure of the model require substantially stronger prior restrictions on the parameters for identification when the number of alternatives is large. Finally, results on the model with partial observability of choices are used to analyze the special case of full observability.