Probit Model Estimation Revisited: Trinomial Models of Household Car Ownership

In this paper we revisit various important issues relating to practical estimation of the multinomial probit model, using an empirical analysis of car ownership as a test case. To provide context, a brief literature review of empirical probit studies is included. Estimates are obtained for a full range of model specifications, including models with random (uncorrelated and correlated) taste variation and/or a general random error structure, and issues of estimability, specification testing, and alternative normalizations for probit models are addressed. Three model trust region algorithms for finding maximum likelihood estimates are compared, and the superiority of a structured quasi-Newton method employing "model switching" over more traditional approaches (Broyden-Fletcher-Goldfarb-Shanno secant update, Berndt-Hall-Hall-Hausman) is demonstrated. The trust region algorithms have reliable convergence properties and provide useful diagnostic information. Finally, a comparison of some probit integral approximation schemes (Clark, and two variants of Mendell-Elston) versus numerical integration is included. There is additional evidence against using Clark's approximation, but a variant of the Mendell-Elston approach appears promising. Numerical problems with variable-ordering techniques (such as separated-split) are demonstrated.