Open AccessOn the surjectivity of the mapping between utilities and choice probabilities
Author(s) -
Norets Andriy,
Takahashi Satoru
Publication year - 2013
Publication title -
quantitative economics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 4.062
H-Index - 27
eISSN - 1759-7331
pISSN - 1759-7323
DOI - 10.3982/qe252
Subject(s) - multinomial distribution , nonparametric statistics , estimator , econometrics , covariate , consistency (knowledge bases) , discrete choice , mathematics , surjective function , statistics , geometry , pure mathematics
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 deterministic 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 covariate dependent mixtures.