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This paper analyzes small sample properties of several versions of z-tests in multinomial probit models under simulated maximum likelihood estimation. Our Monte Carlo experiments show that z-tests on utility function coefficients provide more robust results than z-tests on variance covariance parameters. As expected, both the number of observations and the number of random draws in the incorporated Geweke-Hajivassiliou-Keane (GHK) simulator have on average a positive impact on the conformities between the shares of type I errors and the nominal significance levels. Furthermore, an increase of the number of observations leads to an expected decrease of the shares of type II errors, whereas the number of random draws in the GHK simulator surprisingly has no significant effect in this respect. One main result of our study is that the use of the robust version of the simulated z-test statistics is not systematically more favorable than the use of other versions. However, the application of the z-test statistics that exclusively include the Hessian matrix of the simulated loglikelihood function to estimate the information matrix often leads to substantial computational problems.

Z-Tests in Multinomial Probit Models under Simulated Maximum Likelihood Estimation: Some Small Sample Properties