Quantitative Economics: Mar, 2013, Volume 4, Issue 1

On the surjectivity of the mapping between utilities and choice probabilities

Andriy Norets, Satoru Takahashi

This note considers a standard multinomial choice model. It is shown that if the
distribution of additive utility shocks has a density, then the mapping from de-
terministic components of utilities to choice probabilities is surjective. In other
words, any vector of choice probabilities can be obtained by selecting suitable
utilities for alternatives. This result has implications for at least three areas of
interest to econometricians: the Hotz and Miller (1993) estimator for structural
dynamic discrete choice models, nonparametric identification of multinomial
choice models, and consistency of conditional density estimators based on co-
variate dependent mixtures.
Keywords. Multinomial choice models, identification, Hotz and Miller estimator,
covariate dependent mixtures.
JEL classification. C35, C61.

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