Statistical inference in the multinomial multiperiod probit model

Abstract Statistical inference in multinomial multiperiod probit models has been hindered in the past by the high dimensional numerical integrations necessary to form the likelihood functions, posterior distributions, or moment conditions in these models. We describe three alternative estimators, implemented using simulation-based approaches to inference, that circumvent the integration problem: posterior means computed using Gibbs sampling and data augmentation (GIBBS), simulated maximum likelihood (SML) estimation using the GHK probability simulator, and method of simulated moment (MSM) estimation using GHK. We perform a set of Monte-Carlo experiments to compare the sampling distributions of these estimators. Although all three estimators perform reasonably well, some important differences emerge. Our most important finding is that, holding simulation size fixed, the relative and absolute performance of the classical methods, especially SML, gets worse when serial correlation in disturbances is strong. In data sets with an AR(1) parameter of 0.50, the RMSEs for SML and MSM based on GHK with 20 draws exceed those of GIBBS by 9% and 0%, respectively. But when the AR(1) parameter is 0.80, the RMSEs for SML and MSM based on 20 draws exceed those of GIBBS by 79% and 37%, respectively, and the number of draws needed to reduce the RMSEs to within 10% of GIBBS are 160 and 80 respectively. Also, the SML estimates of serial correlation parameters exhibit significant downward bias. Thus, while conventional wisdom suggests that 20 draws of GHK is ‘enough’ to render the bias and noise induced by simulation negligible, our results suggest that much larger simulation sizes are needed when serial correlation in disturbances is strong.