In recent years simulation-based estimation of multinomial probit (MNP) models has attracted an increasing interest among econometricians. With the help of more powerful and cheaper computers, MNP models have become a well-accepted alternative to multinomial logit (MNL) models that impose the restrictive Independence of Irrelevant Alternatives (IIA) assumption. In contrast to the large volume of research in the estimation of MNP models, there is little evidence on the properties of specification tests when applied to these simulation-based estimates. Specification tests are very important for discrete choice models. Contrary to linear regression models, Yatchew and Griliches (1984) showed that errors in specification for logit and probit models, such as omitted variables and heteroskedasticity, can cause the estimators to be inconsistent. In this paper we propose some simulated Wald and LM tests for testing panel MNP model specification. Unlike LR (or distance) statistic, Wald and LM tests only require model estimation under the alternative hypothesis or the null hypothesis, but not both. Thus Wald and LM tests are less computationally demanding for simulation-based estimation of panel MNP models. LM tests have been traditionally used for testing model specifications for binary choice models and MNL models, for example, see Davidson and MacKinnon (1984) and McFadden (1987). For classical estimation methods such as SML and MSM, Hajivassiliou (2000) suggested that the trinity of tests using a very long simulation can be applied for testing MNP model specifications, and the test statistics are free of simulation noise. We will focus on four different aspects of a panel MNP model specification: omitted variables, independence of irrelative alternatives, unobserved heterogeneity, and intertemporal correlation. To make the simulation tests concrete and practical to use, we choose a specific parametric model form. B{\\"o}rsch-Supan, Hajivassiliou, Kotlikoff and Morris (1992) proposed this form and it has ever since been adopted by many researchers for estimating panel MNP models. B{\\"o}rsch-Supan, Hajivassiliou, Kotlikoff and Morris (1992) showed that this model nests eight different models, depending on if there is intratemporal correlation between alternatives, intertemporal correlation among the same alternatives, or random effects for modeling unobserved heterogeneity. Therefore to test for IIA, the existence of unobserved heterogeneity, and intertemporal correlation, we can construct hypotheses by simply imposing constraints on the corresponding model parameters. To test for omitted variables, we will follow McFadden (1987) and construct a similar LM test based on simulation. We will derive the forms of various test statistics for use with MSM, and discuss how to simulate these tests in practice. Although similar tests can be derived for SML, we choose to use MSM based on the Monte Carlo results in Geweke, Keane and Runkle (1994) and Geweke, Keane and Runkle (1996), who found that although both SML and MSM perform reasonably well for point estimation, there is a closer agreement between root mean squared errors and the mean of asymptotic standard errors for MSM than for SML. Depending on how to estimate asymptotic covariance matrix of model parameters, we can usually construct the same test in different forms that are asymptotically equivalent. However, the finite sample properties of these asymptotically equivalent tests are usually different. By using some extensive Monte Carlo experiments, in this paper we will study the finite sample properties of the proposed simulated tests for MSM, and provide some guidance for using these tests in practice. The simulation-based tests proposed in this paper are implemented for the object oriented S-PLUS environment, and can be routinely used by empirical researchers.
