Fast Simulated Maximum Likelihood Estimation of the Spatial Probit Model Capable of Handling Large Samples

We show how to quickly estimate spatial probit models for large data sets using maximum likelihood. Like Beron and Vijverberg (2004), we use the GHK (Geweke-Hajivassiliou-Keene) algorithm to perform maximum simulated likelihood estimation. However, using the GHK for large sample sizes has been viewed as extremely difficult (Wang, Iglesias, and Wooldridge, 2009). Nonetheless, for sparse covariance and precision matrices often encountered in spatial settings, the GHK can be applied to almost any sample size as its operation counts and memory requirements increase almost linearly with n when using sparse matrix techniques. We show the utility of the technique by conducting a Monte Carlo study involving 100,000 observation spatial probit regressions where each regression takes between two to four minutes. To place this in context, Beron and Vijverberg (2004, p. 181) found that it required $30.5$ minutes for a sample size of n=200 when estimating spatial probit using the GHK, and that doubling the sample size increased computing time by a factor of 3.75, which is close to order n-squared