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Generalized autoregressive errors in the multinomial probit model

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  • Bolduc, Denis

Abstract

In discrete choice analysis, the multinational probit (MNP) provides the most flexible framework to allow for general interdependencies among the alternatives. These interdependencies are usually modeled through the correlation structure of the error term. This framework suffers from two serious impediments, however. The first and major one is computational and is related to the evaluation of the response probabilities, which are multidimensional normal integrals. In the past, this has restricted its utilization to studies involving less than five alternatives where using numerical integration remains practical. A recent solution to the dimensionality problem consists in replacing the choice probabilities with easy to compute efficient simulators. The second impediment arises in models with large choice sets when a fully unconstrained error correlation structure is postulated. In that case, the large number of nuisance parameters to estimate in the error covariance matrix becomes a problematic issue that can exacerbate the estimation process. To tackle that problem, a first-order generalized autoregressive [GAR(1)] error approach is suggested. The approach enables one to approximate general correlation structures with parsimonious parametric specifications. The key feature of the approach is that the number of nuisance parameters grows linearly with the number of alternatives considered. The methodology is most useful in models with large choice sets where the estimation also requires to use probability simulators. The paper focuses on the GAR(1) solution to the error covariance matrix estimation problem. The issue of identification of the nuisance parameters is examined in detail and a rank condition is suggested. Some theoretical and numerical examples based on synthetic data are presented.

Suggested Citation

  • Bolduc, Denis, 1992. "Generalized autoregressive errors in the multinomial probit model," Transportation Research Part B: Methodological, Elsevier, vol. 26(2), pages 155-170, April.
  • Handle: RePEc:eee:transb:v:26:y:1992:i:2:p:155-170
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    Cited by:

    1. Denis Bolduc, "undated". "A Fast Maximum Simulated Likelihood Estimation Technique for NMP Models," Computing in Economics and Finance 1997 155, Society for Computational Economics.
    2. Hajivassiliou, Vassilis A. & Ruud, Paul A., 1986. "Classical estimation methods for LDV models using simulation," Handbook of Econometrics,in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 40, pages 2383-2441 Elsevier.
    3. Daniel McFadden, 2001. "Economic Choices," American Economic Review, American Economic Association, vol. 91(3), pages 351-378, June.
    4. Munizaga, Marcela A. & Heydecker, Benjamin G. & Ortúzar, Juan de Dios, 2000. "Representation of heteroskedasticity in discrete choice models," Transportation Research Part B: Methodological, Elsevier, vol. 34(3), pages 219-240, April.
    5. Wall, Melanie M. & Liu, Xuan, 2009. "Spatial latent class analysis model for spatially distributed multivariate binary data," Computational Statistics & Data Analysis, Elsevier, vol. 53(8), pages 3057-3069, June.
    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.
    7. Young, Gary & Valdez, Emiliano A. & Kohn, Robert, 2009. "Multivariate probit models for conditional claim-types," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 214-228, April.
    8. Yai, Tetsuo & Iwakura, Seiji & Morichi, Shigeru, 1997. "Multinomial probit with structured covariance for route choice behavior," Transportation Research Part B: Methodological, Elsevier, vol. 31(3), pages 195-207, June.
    9. Bhat, Chandra R. & Guo, Jessica, 2004. "A mixed spatially correlated logit model: formulation and application to residential choice modeling," Transportation Research Part B: Methodological, Elsevier, vol. 38(2), pages 147-168, February.
    10. Schmidheiny, Kurt, 2006. "Income segregation and local progressive taxation: Empirical evidence from Switzerland," Journal of Public Economics, Elsevier, vol. 90(3), pages 429-458, February.
    11. Bolduc, Denis & Kaci, Mustapha, 1993. "Estimation des modèles probit polytomiques : un survol des techniques," L'Actualité Economique, Société Canadienne de Science Economique, vol. 69(3), pages 161-191, septembre.
    12. Kenneth Train, "undated". "Simulation Methods for Probit and Related Models Based on Convenient Error Partitioning," Working Papers _009, University of California at Berkeley, Econometrics Laboratory Software Archive.
    13. Liu, Yu-Hsin, 2011. "Incorporating scatter search and threshold accepting in finding maximum likelihood estimates for the multinomial probit model," European Journal of Operational Research, Elsevier, vol. 211(1), pages 130-138, May.
    14. Bolduc, Denis & Khalaf, Lynda & Yélou, Clément, 2010. "Identification robust confidence set methods for inference on parameter ratios with application to discrete choice models," Journal of Econometrics, Elsevier, vol. 157(2), pages 317-327, August.
    15. GRAMMIG, Joachim & HUJER, Reinhard & SCHEIDLER, Michael, 2001. "The econometrics of airline network management," CORE Discussion Papers 2001055, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    16. Ziegler, Andreas, 2001. "Simulated z-tests in multinomial probit models," ZEW Discussion Papers 01-53, ZEW - Zentrum für Europäische Wirtschaftsforschung / Center for European Economic Research.
    17. Denis Bolduc & Lynda Khalaf & Clément Yélou, 2005. "Identification Robust Confidence Sets Methods for Inference on Parameter Ratios and their Application to Estimating Value-of-Time," Computing in Economics and Finance 2005 48, Society for Computational Economics.
    18. Bolduc, Denis, 1999. "A practical technique to estimate multinomial probit models in transportation," Transportation Research Part B: Methodological, Elsevier, vol. 33(1), pages 63-79, February.
    19. Daziano, Ricardo A., 2015. "Inference on mode preferences, vehicle purchases, and the energy paradox using a Bayesian structural choice model," Transportation Research Part B: Methodological, Elsevier, vol. 76(C), pages 1-26.
    20. Bolduc, Denis & Khalaf, Lynda & Moyneur, Érick, 2008. "Identification-robust simulation-based inference in joint discrete/continuous models for energy markets," Computational Statistics & Data Analysis, Elsevier, vol. 52(6), pages 3148-3161, February.

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