Use of composite likelihood methods for the estimation of probit models

• Applicant: Professor Dr. Dietmar Bauer
• Subject Area: Statistics and Econometrics ; City Planning, Spatial Planning, Transportation and Infrastructure Planning, Landscape Planning
• Term: from 2017 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 356500581

• Final Report Year: 2023

Final Report Abstract

Multinomial probit (MNP) models explain the decision between a finite number of alternatives. The model assumes that the deciders obtain information on the characteristics of the alternatives to form utilities, which are subject to a random Gaussian distributed error also containing the effects of unobserved taste heterogeneity. The model thus links the observable characteristics to the choice probabilities of the alternatives. These probabilities are evaluated using a multivariate Gaussian cumulative distribution function (CDF) which – due to the lack of closed form expressions – takes much computing time. As a remedy Chandra Bhat (University of Texas at Austin) proposed the Maximum approximated Composite Marginal Likelihood (MACML) approach for the estimation. However, the statistical properties of this estimator were unknown before the project. Within the project we discovered that one way to think about the MaCMLapproach is that it provides an approximation of the MNP model rather than the criterion function. Using this viewpoint the estimator has the standard statistical properties like consistency and asymptotic normality. This interpretation, however, can only be used if the approximated choice probabilities are normalized to sum to one. For the Solow-Joe (SJ) approximation we found that the distribution of the sums of the choice probabilities typically is highly concentrated around 1 without normalization. Therefore this approach does not suffer strong effects without the numerically costly normalization. For the competing Mendell-Elston (ME) approximation this is not the case and hence normalization is strictly advised. Some approximation concepts use a strategic reordering of components in the Gaussian CDF involving the current parameter vector. Such a reordering generally leads to discontinuity of the criterion function. It should hence be avoided. The approximation concepts are not interchangeable. Using one concept for the generation of the data and another one for the estimation may lead to substantial biases. Therefore estimation and usage of the estimated model (e.g. for prediction) should always use the same approximation concept. The CML approach requires the choice of a CML function influencing the asymptotic variance. In the cases considered by us the full pairwise CML carries only a modest penalty in accuracy compared to maximum likelihood estimation. Other versions such as the adjacent pairwise CML reduce the numerical load but also result in a considerable decrease of accuracy. Higher order CMLs do not appear to be attractive alternatives. To further speed up the numerical estimation, a consistent (in special cases) regression based initialization procedure has been developed in the project. The methods have been implemented in publicly available R packages that significantly enhance the capabilities for the estimation of MNP models (including taste heterogeneity) as is demonstrated on a large panel data set.

Publications

• Model selection and model averaging in MaCML-estimated Multinomial Probit (MNP) models
  Batram, M. & Bauer, D.
  
• On consistency of the MACML approach to discrete choice modelling. Journal of Choice Modelling, 30, 1-16.
  Batram, Manuel & Bauer, Dietmar
  
• Using Motifs for Population Synthesis in Multi-agent Mobility Simulation Models. Springer Proceedings in Mathematics & Statistics, 335-349. Springer International Publishing.
  Büscher, Sebastian; Batram, Manuel & Bauer, Dietmar
  
• Bayes Estimation of Latent Class Mixed Multinomial Probit Models. Poster presented at the Annual Meeting of the TRB, 2021, Washington.
  Oelschlager, L. & Bauer, D.
  
• Comparison of Maximum—Approximate-Composite-Marginal-Likelihood and the MSL over simulations in R. Master thesis, Bielefeld University, 2022.
  Spies, E.
  
• Model Selection and Model Averaging in Computational Challenging. Econometric Models. Dissertation, Bielefeld University, July 2022.
  Batram, M.
  
• Non-parametric estimation of mixed discrete choice models
  Bauer, D., Buscher, S. & Batram, M.