We are grateful to Bo Honore for helpful discussions. We thank anonymous referees for comments that helped to improve the manuscript.
On the surjectivity of the mapping between utilities and choice probabilities
Article first published online: 12 MAR 2013
Copyright © 2013 Andriy Norets and Satoru Takahashi
Volume 4, Issue 1, pages 149–155, March 2013
How to Cite
Norets, A. and Takahashi, S. (2013), On the surjectivity of the mapping between utilities and choice probabilities. Quantitative Economics, 4: 149–155. doi: 10.3982/QE252
- Issue published online: 12 MAR 2013
- Article first published online: 12 MAR 2013
- Submitted October, 2012. Final version accepted October, 2012.
- Multinomial choice models;
- Hotz and Miller estimator;
- covariate dependent mixtures
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.