Analyzing Probit Bayes Estimator for Flexible Covariance Structures in Discrete Choice Modeling

Research in discrete choice modeling in recent decades has devoted an enormous effort to generalizing the distribution of the error term and to developing estimation methods that account for more flexible structures of error heterogeneity. Whereas the multinomial probit model offers a fully flexible covariance matrix, the maximum simulated likelihood estimator is extremely involved. However, Bayesian techniques have the potential to break down the complexity of the estimator. By using a Monte Carlo study, this paper tests the ability of a probit Bayes estimator based on Gibbs sampling to recover different substitution patterns. The results show that it is possible to use the Bayes estimator of a full covariance matrix to recover different covariance structures, even when small samples are used. Thus, the model can identify the true substitution patterns, by avoiding misspecification, even if these patterns are the result of multiple restrictions over the covariance matrix. In fact, the recovery of simpler covariance structures, such as that of the independent and identically distributed and heteroskedastic covariance without correlation, is more accurate than the recovery of more complicated structures, including fully unrestricted substitution patterns.