New matrix-based methods for the analytic evaluation of the multivariate cumulative normal distribution function

Abstract

In this paper, we develop a new matrix-based implementation of the Mendell and Elston (ME) analytic approximation to evaluate the multivariate normal cumulative distribution (MVNCD) function, using an LDLT decomposition method followed by a rank 1 update of the LDLT factorization. Our implementation is easy to code for individuals familiar with matrix-based coding. Further, our new matrix-based implementation for the ME algorithm allows us to efficiently write the analytic matrix-based gradients of the approximated MVNCD function with respect to the abscissae and correlation parameters, an issue that is important in econometric model estimation. In addition, we propose four new analytic methods for approximating the MVNCD function. The paper then evaluates the ability of the multiple approximations for individual MVNCD evaluations as well as multinomial probit model estimation. As expected, in our tests for evaluating individual MVNCD functions, we found that the traditional GHK approach degrades rapidly as the dimensionality of integration increases. Concomitant with this degradation in accuracy is a rapid increase in computational time. The analytic approximation methods are also much more stable across different numbers of dimensions of integration, and even the simplest of these methods is superior to the GHK-500 beyond seven dimensions of integration. Based on all the evaluation results in this paper, we recommend the new Two-Variate Bivariate Screening (TVBS) method proposed in this paper as the evaluation approach for MVNCD function evaluation.