Alternative Computational Approaches to Inference in the Multinomial Probit Model

This research compares several approaches to inference in the multinomial probit model, based on two Monte Carlo experiments for a seven choice model.The methods compared are the simulated maximum likelihood estimator using the GHK recursive probability,simulator, the method of simulated moments estimator using the GHK recursive simulator and kernel-smoothed frequency simulators, and posterior means using a Gibbs sampling-data augmentation algorithm.Overall, the Gibbs sampling algorithm has a slight edge, with the relative performance of MSM and SML based on the GHK simulator being difficult to evaluate.The MSM estimator with the kernel-smoothed frequency simulator is clearly inferior.I. Introduction T HE multinomial probit is an appealing model of choice behavior because it allows a flexible pattern of conditional covariance among the latent utilities of alternatives.Nevertheless, multinomial probit applications have been limited because the required integrations of the multivariate normal density over subsets of Euclidean space are computationally burdensome.The computational simplicity of the multinomial logit has made it the model of choice for applied work.However, because the multinomial probit model relaxes the assumption of independence of irrelevant alternatives, it is generally preferred in principle to the multinomial logit model (McFadden, 1984, pp.1395-1458).Recently the method of simulated moments (McFadden, 1989; Pakes and Pollard, 1989) and Gibbs sampling with data augmentation (Albert and Chib, 1993; McCulloch and Rossi, 1994) have shown promise of making the required computations in the multinomial probit model practical.The development of the highly accurate GHK probability simulator (see Geweke, 1991; Hajivassiliou and McFadden, 1990; and Keane, 1990, 1994a) has also led to renewed interest in simulated maximum likelihood (Albright, Lerman, and Manski, 1977) as a method for estimating multinomial probit models.The objective of the research reported here is to provide a systematic comparison of the numerical properties of different simulation-based methods of inference in the multinomial probit model.Rather than considering the performance of these methods on a single model for a single data set, we attempt to control for a number of features of the inference problem, such as the number and nature of the unknown parameters of interest and the information content of the data on which inference is based.Also, we investigate for the first time how the performance of MSM estimation is affected by the type of probability simulator employed (i.e., GHK vs. kernel smoothing).While some investigators have examined the performance of particular estimators and computational techniques, Borsch-Supan and Hajivassiliou (1993), Hajivassiliou (1992), and Hajivassiliou, McFadden, and Ruud (1992) have made systematic comparison of alternative probability simulators, we are aware of only one systematic comparison of different estimators per se: Keane (1994a) compares method of simulated moments and simulated maximum likelihood estimators for an eight period binomial probit model in a Monte Carlo study.This paper is the first to compare performance of alternative methods of inference for the multinomial probit model and the first to examine how the relative performance of alternative methods differs across model specifications and across different data sets.In addition to addressing this main objective, this work introduces a new factor structure for the disturbances that may help to alleviate the proliferation of covariance matrix parameter problems in MNP models.We also illustrate Bayesian inference in a multinomial probit model with a factor structure for the first time.(See Elrod and Keane (forthcoming) for a discussion of factor structures for probit models.)