Semiparametric Estimation and Inference in Multinomial Choice Models

The purpose of this paper is to incorporate semiparametric alternatives to maximum likelihood estimation and inference in the context of unordered multinomial response data when in practice there is often insufficient information to specify the parametric form of the function linking the observables to the unknown probabilities. We specify the function linking the observables to the unknown probabilities using a very general flexible class of functions belonging to the Pearson system of cumulative distribution equations. In this setting we consider the observations as arising from a multinomial distribution characterized by one of the CDFs in the Pearson system. Given this situation, it is possible to utilize the concept of unbiased estimating functions (EFs), combined with the concept of empirical likelihood (EL) to define an (empirical) likelihood function for the parameter vector based on a nonparametric representation of the sample's PDF. This leads to the concept of maximum empirical likelihood (MEL) estimation and inference, which is analogous to parametric maximum likelihood methods in many respects.