Multinomial choice and nonparametric average derivatives

Well known parametric estimators for multinomial choice (multinomial logit, nested logit, multinomial probit, mixed logit) have the disadvantage of dealing with nuisance parameters in the error term distribution: multinomial logit assumes them away, nested logit imposes priori restrictions on them, multinomial probit usually fails to converge with unrestricted error term covariances, and mixed logit needs to specify the mixing distribution. This paper shows that, under a multiple index assumption for the choice probabilities, certain restrictions on the multinomial choice regression parameters are identified with average derivatives of the choice probabilities, and that ratios of coefficients for alternative-variant regressors are identified by the corresponding ratios of the average derivatives. Using nonparametric average derivative estimators (ADE), we avoid the nuisance parameters in the error term distribution. Also, differently from the parametric estimators, ADE does not require any optimization, nor any analytic or numerical integration. A simulation study is conducted to compare the parametric estimators other than mixed logit with the ADE approach: the parametric estimators are particularly vulnerable to regression function misspecifications, whereas the parametric estimators perform better than the ADE approach and the differences among the parametric estimators are small when the regression functions are correctly specified. Also, an empirical illustration is provided.