A Simple Approximation for Bivariate Normal Integral Based on Error Function and its Application on Probit Model with Binary Endogenous Regressor

A simple approximation for the bivariate normal cumulative distribution function (BNCDF) based on the error function is derived. The worst error of our method is found to four decimal places under various configurations considered in this paper's Table 1. This finding is much better than that in Table 1 of Cox and Wermuth (1991) and in Table 1 of Lin (1995) where the worst error of both tables is up to 3 decimal places. We also apply the proposed method to approximate the likelihood function of the probit model with binary endogenous regressor. The simulations indicate that the bias and mean-squared-error (MSE) of the maximum likelihood estimator based on our method are very much similar to those obtained from using the exact method in the GAUSS package.