A Comparison of Hierarchical Bayes and Maximum Simulated Likelihood for Mixed Logit

Mixed logit is a flexible discrete choice model that allows for random coefficients and/or error components that induce correlation over alternatives and time. Procedures for estimating mixed logits have been developed within both the classical (e.g., Revelt and Train, 1998) and Bayesian traditions (Sawtooth Software, 1999.) Asymptotically, the two procedures provide the same information, and Huber and Train (2001) found that the two methods provide very similar results on their typical sample. The relative convenience of the two methods, in terms of programming and computation time, depends on the specification of the model. The purpose of this paper is to elucidate these realms of relative convenience. Analogous concepts apply for probit models. The classical approach is usually implemented with the GHK probit simulator developed by Geweke (1989), Hajivassiliou (1990), and Keane (1990). A Bayesian approach has been developed by Albert and Chib (1993), McCulloch and Rossi (1994), and Allenby and Rossi (1999). A run-time comparison of the two procedures has been conducted by Bolduc et al. (1996), who found the Bayesian approach to be about twice as fast for their particular specification. Some of the comparisons that we make for mixed logit, such as correlated versus uncorrelated random coefficients and the inclusion of fixed as well as random coefficients, are applicable to probits. Other comparisons, such as the use of lognormal and triangular distributions, are not, since probit and the classical and Bayesian procedures to estimate it depend on the assumption that all random terms are normal. From both perspectives (though for different reasons) the Bayesian procedure has a theoretical advantage over the classical procedure, independent of convenience. The classical perspective focuses on the sampling distribution of an estimator. As described below, the classical and Bayesian estimators both require integration, though of a different integrand. If the integrals could be evaluated exactly, then the Bayesian and