Analytic approximations for computing probit choice probabilities

Abstract

The multinomial probit model has long been used in transport applications as the basis for mode- and route-choice in computing network flows, and in other choice contexts when estimating preference parameters. It is well known that computation of the probit choice probabilities presents a significant computational burden, since they are based on multivariate normal integrals. Various methods exist for computing these choice probabilities, though standard Monte Carlo is most commonly used. In this article we compare two analytical approximation methods (Mendell–Elston and Solow–Joe) with three Monte Carlo approaches for computing probit choice probabilities. We systematically investigate a wide range of parameter settings and report on the accuracy and computational efficiency of each method. The findings suggest that the accuracy and efficiency of an optimally ordered Mendell–Elston analytic approximation method offers great potential for wider use.

Keywords:

• multinomial probit
• multivariate normal integral
• analytical approximation
• choice probabilities

Acknowledgements

This work described in this article was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) through the project ‘Estimation of travel demand models from panel data’, grant EP/G033609/1. The second author also acknowledges the support of the Leverhulme Trust in the form of a Leverhulme Early Career Fellowship.

Notes

1. The CML approach works by replacing the complex multivariate integrals inherent to MVN likelihood functions by a combination of separate univariate or bivariate integrals. In the work by Bhat, this is applied to the case of probit structures, given the inherent suitability of normal distributions to a CML context.

2. Note that the ME algorithm in Kamakura (1989) has a slight error in equation 15a.

3. Uses QSIMVNV available from http://www.math.wsu.edu/faculty/genz/software/software.html