DRAFT Specification , Identification , & Estimation of the Logit Kernel ( or Continuous Mixed Logit ) Model

Logit kernel is a discrete choice model that has both probit-like disturbances as well as an additive i.i.d. extreme value (or Gumbel) disturbance à la multinomial logit. The result is an intuitive, practical, and powerful model that combines the flexibility of probit with the tractability of logit. For this reason, logit kernel has been deemed the “model of the future” and is becoming extremely popular in the literature. It has already been included in a recent edition of a widely used econometrics textbook. While the basic structure of logit kernel models is well understood, there are important formulation and practical issues that are critical for estimation and yet are often overlooked. We aim to highlight some of these issues in the paper. One key point is that the addition of the Gumbel term is not necessarily innocuous, and thus the normalization required for logit kernel can be different than for an analogous pure probit model. Another point is that there are interesting and non-intuitive identification rules regarding nested structures and random coefficient models. Misunderstanding of these issues can lead to biased estimates as well as a significant loss of fit. A clear understanding of identification becomes even more critical given the fact that simulation, which is often used to estimate these models due to the high dimensionality of the integrals, has a tendency to cover up identification problems. In the paper we present a general framework for specification, identification, and estimation of the logit kernel model. We specify the model using a general factor analytic error structure. We show that the factor analytic form includes all known (additive) error structures as special cases, including heteroscedasticity, error components, nesting structures, random coefficients, and auto correlation. We discuss in detail many of the special cases of the logit kernel model and highlight specification and identification issues related to each. Finally we demonstrate our findings with empirical examples using both simulated and real data. The objectives of the paper are to present our specific findings, as well as highlight the broader themes and provide tools for uncovering identification issues pertaining to logit kernel models. * This work was partially supported by the Social Sciences and Humanities Research Council of Canada, Le Fonds FCAR, and a UPS fellowship. This paper is a major revision of the working paper by Ben-Akiva and Bolduc (1996), "Multinomial Probit with a Logit Kernel and a General Parametric Specification of the Covariance Structure" based on recent work by Walker (2001). 4 Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, U.S.A. 5 Université Laval, Québec, Canada, G1K 7P4.