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SUMMARY
This paper considers mixed, or random coefficients, multinomial logit (MMNL) models for discrete response, and establishes the following results. Under mild regularity conditions, any discrete choice model derived from random utility maximization has choice probabilities that can be approximated as closely as one pleases by a MMNLmodel. Practical estimation of a parametric mixing family can be carried out by Maximum Simulated Likelihood Estimation or Method of Simulated Moments, and easily computed instruments are provided that make the latter procedure fairly efficient. The adequacy of a mixing specification can be tested simply as an omitted variable test with appropriately defined artificial variables.Anapplication to a problem of demand for alternative vehicles shows thatMMNLprovides a flexible and computationally practical approach to discrete response analysis. Copyright#2000 JohnWiley &Sons, Ltd.
1. INTRODUCTION
Define a mixed multinomial logit (MMNL) model as a MNL model with random coefficients a drawn from a cumulative distribution function G(a;y):
PCOi j x; yU a Z LCOi; x; aU-GOda; yU with LCOi; x; aU a exia=Xj2C exja O1U
In this setup, C a f1; . . . ; Jg is the choice set; the xi are 1 - K vectors of functions of observed attributes of alternative i and observed characteristics of the decision maker, with x a Ox1; . . . ; xJU; a is a K - 1 vector of random parameters; LCOi; x; aU is a MNL model for the choice set C; and y is a vector of deep parameters of the mixing distribution G. The random parameters a may be interpreted as arising from taste heterogeneity in a population of MNL decision makers. If the xi contain alternative-specific variables, then the corresponding components of a can be treated as alternative-specific random effects. Alternately, the model may simply be interpreted as a flexible approximation to choice probabilities generated by a random utility model. The mixing distribution G may come from a continuous parametric family, such as multivariate normal or log normal, or it may have a finite support. When G has finite support, MMNL models are also called latent class models. Equation (1) describes a single decision, but extension to dynamic choice models with multiple decisions is straightforward, by mixing over the parameters of a product of MNL models for...