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Probit Model Estimation Revisited: Trinomial Models of Household Car Ownership

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  • Bunch, David S.
  • Kitamura, Ryuichi

Abstract

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.

Suggested Citation

  • Bunch, David S. & Kitamura, Ryuichi, 1991. "Probit Model Estimation Revisited: Trinomial Models of Household Car Ownership," University of California Transportation Center, Working Papers qt2hr8d4bs, University of California Transportation Center.
    • Handle: RePEc:cdl:uctcwp:qt2hr8d4bs
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    1. Randall Filer & Marjorie Honig, 2005. "Endogenous Pensions and Retirement Behavior," CESifo Working Paper Series 1547, CESifo.
    2. Yeshwanth Cherapanamjeri & Constantinos Daskalakis & Gabriele Farina & Sobhan Mohammadpour, 2025. "Learning Correlated Reward Models: Statistical Barriers and Opportunities," Papers 2510.15839, arXiv.org, revised May 2026.
    3. Matteo Ricciarelli, 2009. "Investment choice and asset allocation of Italian households: the discrete-continuous approach," Applied Economics, Taylor & Francis Journals, vol. 43(6), pages 651-662.
    4. David S. Bunch, 2014. "Numerical methods for optimization-based model estimation and inference," Chapters, in: Stephane Hess & Andrew Daly (ed.), Handbook of Choice Modelling, chapter 23, pages 565-598, Edward Elgar Publishing.
    5. Rinus Haaijer & Michel Wedel & Marco Vriens & Tom Wansbeek, 1998. "Utility Covariances and Context Effects in Conjoint MNP Models," Marketing Science, INFORMS, vol. 17(3), pages 236-252.
    6. Victoria Prowse, 2012. "Modeling Employment Dynamics With State Dependence and Unobserved Heterogeneity," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 30(3), pages 411-431, April.

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