DRAFT Identification of the Logit Kernel ( or 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 (and more) 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 been included in popular statistical software packages as well as a recent edition of a widely used econometrics textbook and two texts specializing on discrete choice. While the basic structure of logit kernel models is well understood, there are important identification issues that are often overlooked. Misunderstanding of these issues can lead to biased estimates as well as a significant loss of fit. This paper presents a general framework for identifying the logit kernel model. Many of the special cases of the logit kernel model are discussed in detail, including heteroscedasticity, error components, nesting structures, random coefficients, auto correlation, and application to panel data. Specification and identification issues related to each special case are identified. Finally the findings are demonstrated 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. Introduction The logit kernel model is a straightforward concept: it is a discrete choice model in which the disturbances (of the utilities) consist of both a probit-like portion and an additive i.i.d. Gumbel portion (i.e., a multinomial logit disturbance). Multinomial logit (MNL) has its well-known blessing of tractability and its equally well-known curse of a rigid error structure leading to the IIA property. The nested logit model relaxes the rigidity of the MNL error structure and has the advantage of retaining a probability function in closed form. Nonetheless, nested logit is still limited and cannot capture many forms of unobserved heterogeneity, including, for example, random taste heterogeneity. The logit kernel model with its probit-like (but even more general) disturbances completely opens up the specification of the disturbances so that almost any desirable error structure can be represented in