A simple approximation for the bivariate normal integral

Abstract

A simple approximation for the bivariate normal cumulative distribution function based on the error function is derived. We compare the accuracy of our approximation with those of several existing approximation methods and the command in MATLAB. Our approximation performs quite well in the central portion and extreme tails of the distribution. We also apply the proposed method to evaluate the likelihood function of the binary endogenous regressor probit (BERP) model. The simulations indicate that the numerical stability and the accuracy of the proposed approach is promising when conducting the maximum likelihood estimation of the BERP model.