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Journal of the Indian Institute of Science

, Volume 99, Issue 4, pp 673–681 | Cite as

Discrete Choice Models with Alternate Kernel Error Distributions

  • Rajesh PaletiEmail author
Review Article
  • 54 Downloads

Abstract

The multinomial logit (MNL) and probit (MNP) models dominated the literature on consumer behavior analysis, particularly in the transportation planning context where the focus is on future travel demand prediction as an aggregated outcome of individual traveler choices. While Gumbel kernel errors in the MNL model are unbounded and positively skewed, normal kernel errors in the MNP model are symmetric and unbounded. However, choice models with alternative kernel errors (beyond Gumbel and normal distributions) have piqued the interest of choice modelers for behavioral and prediction accuracy reasons. In addition, researchers found evidence in support of these alternate kernel errors in a wide variety of empirical contexts. This paper compiles a synthesis of the past literature that developed choices models with flexible kernel errors, including both parametric and semi-parametric methods and concludes with possible avenues for further research.

Keywords

Choice models Utility functions Logit Probit Kernel errors Travel choices 

Notes

References

  1. 1.
    Alptekinoglu A, Semple JH (2016) The exponomial choice model: a new alternative for assortment and price optimization. Opera Res 64(1):79–93.  https://doi.org/10.2139/ssrn.2210478 CrossRefGoogle Scholar
  2. 2.
    Alptekinoglu A, Semple JH (2018) Heteroscedastic exponomial choice. SSRN Electron J.  https://doi.org/10.2139/ssrn.3232788 CrossRefGoogle Scholar
  3. 3.
    Beilner H, Jacobs F (1974) Probabilistic aspects of traffic assignment. In: Proceeedings of 5th International Symposium on the Theory of Traffic Flow and Transportation, Berkely, pp 183–194Google Scholar
  4. 4.
    Bhat CR (1995) A heteroscedastic extreme value model of intercity travel mode choice. Transp Res Part B.  https://doi.org/10.1016/0191-2615(95)00015-6 CrossRefGoogle Scholar
  5. 5.
    Bhat CR (2011) The maximum approximate composite marginal likelihood (MACML) estimation of multinomial probit-based unordered response choice models. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2011.04.005 CrossRefGoogle Scholar
  6. 6.
    Bhat CR (2018) New matrix-based methods for the analytic evaluation of the multivariate cumulative normal distribution function. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2018.01.011 CrossRefGoogle Scholar
  7. 7.
    Bhat CR, Astroza S, Hamdi AS (2017) A spatial generalized ordered-response model with skew normal kernel error terms with an application to bicycling frequency. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2016.10.014 CrossRefGoogle Scholar
  8. 8.
    Bhat CR, Dubey SK (2014) A new estimation approach to integrate latent psychological constructs in choice modeling. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2014.04.011 CrossRefGoogle Scholar
  9. 9.
    Bhat CR, Dubey SK, Nagel K (2015) Introducing non-normality of latent psychological constructs in choice modeling with an application to bicyclist route choice. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2015.04.005 CrossRefGoogle Scholar
  10. 10.
    Bhat CR, Sidharthan R (2012) A new approach to specify and estimate non-normally mixed multinomial probit models. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2012.02.007 CrossRefGoogle Scholar
  11. 11.
    Castillo E et al (2008) Closed form expressions for choice probabilities in the Weibull case. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2007.08.002 CrossRefGoogle Scholar
  12. 12.
    del Castillo J (2016) A class of RUM choice models that includes the model in which the utility has logistic distributed errors. Transp Res Part B Methodol 91:1–20.  https://doi.org/10.1016/j.trb.2016.04.022 CrossRefGoogle Scholar
  13. 13.
    Chikaraishi M, Nakayama S (2016) Discrete choice models with q-product random utilities. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2016.08.013 CrossRefGoogle Scholar
  14. 14.
    Chorus CG, Arentze TA, Timmermans HJP (2008) A random regret-minimization model of travel choice. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2007.05.004 CrossRefGoogle Scholar
  15. 15.
    Daganzo C (1979) Multinomial probit: the theory and its application to demand forecasting. Academic Press, Elsevier, Amsterdam.  https://doi.org/10.2307/2287751 CrossRefGoogle Scholar
  16. 16.
    Dubey S. et al. (2019) A generalized continuous-multinomial response model with a t-distributed error kernel, pp. 1–39. Available at: http://arxiv.org/abs/1904.08332. Accessed 15 June 2019
  17. 17.
    Eidsvik J et al (2014) Estimation and prediction in spatial models with block composite likelihoods. J Comput Graph Stat.  https://doi.org/10.1080/10618600.2012.760460 CrossRefGoogle Scholar
  18. 18.
    Eluru N, Bhat CR, Hensher DA (2008) A mixed generalized ordered response model for examining pedestrian and bicyclist injury severity level in traffic crashes. Accid Anal Prev.  https://doi.org/10.1016/j.aap.2007.11.010 CrossRefGoogle Scholar
  19. 19.
    Fosgerau M (2006) Investigating the distribution of the value of travel time savings. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2005.09.007 CrossRefGoogle Scholar
  20. 20.
    Fosgerau M, Bierlaire M (2009) Discrete choice models with multiplicative error terms. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2008.10.004 CrossRefGoogle Scholar
  21. 21.
    Geweke J, Keane M, Runkle D (2006) Alternative computational approaches to inference in the multinomial probit model. Rev Econ Stat.  https://doi.org/10.2307/2109766 CrossRefGoogle Scholar
  22. 22.
    Greene W et al (2014) Heterogeneity in ordered choice models: a review with applications to self-assessed health. J Econ Surv.  https://doi.org/10.1111/joes.12002 CrossRefGoogle Scholar
  23. 23.
    Greene WH, Hensher DA (2010) Modeling ordered choices: a primer. Cambridge University Press, Cambridge.  https://doi.org/10.1017/CBO9780511845062 CrossRefGoogle Scholar
  24. 24.
    Hausman JA, Wise DA (1978) A conditional probit model for qualitative choice: discrete decisions recognizing interdependence and heterogeneous preferences. Econometrica 46(2):403–426.  https://doi.org/10.2307/1913909 CrossRefGoogle Scholar
  25. 25.
    Heagerty PJ, Lele SR (1998) A composite likelihood approach to binary spatial data. J Am Stat Assoc.  https://doi.org/10.1080/01621459.1998.10473771 CrossRefGoogle Scholar
  26. 26.
    Hess S, Daly A, Batley R (2018) Revisiting consistency with random utility maximisation: theory and implications for practical work. Theory Decis.  https://doi.org/10.1007/s11238-017-9651-7 CrossRefGoogle Scholar
  27. 27.
    Koppelman FS, Wen CH (2000) The paired combinatorial logit model: properties, estimation and application. Transp Res Part B Methodol.  https://doi.org/10.1016/s0191-2615(99)00012-0 CrossRefGoogle Scholar
  28. 28.
    Li B (2011) The multinomial logit model revisited: a semi-parametric approach in discrete choice analysis. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2010.09.007 CrossRefGoogle Scholar
  29. 29.
    McCulloch R, Polson NG, Rossi PE (2000) A Bayesian analysis of the multinomial probit model with fully identified parameters. J Econ 99(1):173–193.  https://doi.org/10.1016/S0304-4076(00)00034-8 CrossRefGoogle Scholar
  30. 30.
    McCulloch R, Rossi PE (1994) An exact likelihood analysis of the multinomial probit model. J Econ 64(1–2):207–240.  https://doi.org/10.1016/0304-4076(94)90064-7 CrossRefGoogle Scholar
  31. 31.
    McFadden D (1973) Conditional logit analysis of qualitative choice behavior. In: Zaremb P (ed) Frontiers in econometrics. Academic Press, New York.  https://doi.org/10.1108/eb028592 CrossRefGoogle Scholar
  32. 32.
    McFadden D, Train K (2002) Mixed MNL models for discrete response. J Appl Econo 15(5):447–470.  https://doi.org/10.1002/1099-1255(200009/10)15:5%3c447:aid-jae570%3e3.3.co;2-t CrossRefGoogle Scholar
  33. 33.
    Paleti R (2018) Generalized multinomial probit model: accommodating constrained random parameters. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2018.10.019 CrossRefGoogle Scholar
  34. 34.
    Paleti R (2019) Multinomial probit model with truncated normal kernel errors: analysis of airline itinerary choices. Technical Paper, The Pennsylvania State University, University ParkGoogle Scholar
  35. 35.
    Paleti R, Pinjari A (2019) A new class of ordered response models with stochastic thresholds. Technical Paper, The Pennsylvania State University, University ParkGoogle Scholar
  36. 36.
    Patil PN et al (2017) Simulation evaluation of emerging estimation techniques for multinomial probit models. J Choice Model.  https://doi.org/10.1016/j.jocm.2017.01.007 CrossRefGoogle Scholar
  37. 37.
    Small KA (1987) A discrete choice model for ordered alternatives. Econometrica 55(2):409.  https://doi.org/10.2307/1913243 CrossRefGoogle Scholar
  38. 38.
    Train K (2003) Discrete choice methods with simulation, discrete choice methods with simulation. Cambridge University Press, Cambridge.  https://doi.org/10.1017/CBO9780511753930 CrossRefGoogle Scholar
  39. 39.
    Vij A, Walker JL (2016) How, when and why integrated choice and latent variable models are latently useful. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2016.04.021 CrossRefGoogle Scholar
  40. 40.
    Vovsha P (2007) Application of cross-nested logit model to mode choice in Tel Aviv, Israel, Metropolitan area. Transp Res Rec J Transp Res Board.  https://doi.org/10.3141/1607-02 CrossRefGoogle Scholar
  41. 41.
    Wang K et al (2017) On the development of a semi-nonparametric generalized multinomial logit model for travel-related choices. PLoS One 12(10):e0186689CrossRefGoogle Scholar
  42. 42.
    Wen CH, Koppelman FS (2001) The generalized nested logit model. Transp Res Part B Methodol.  https://doi.org/10.1016/s0191-2615(00)00045-x CrossRefGoogle Scholar
  43. 43.
    Ye X et al (2017) A practical method to test the validity of the standard Gumbel distribution in logit-based multinomial choice models of travel behavior. Transp Res Part B Methodol.  https://doi.org/10.1016/j.trb.2017.10.009 CrossRefGoogle Scholar

Copyright information

© Indian Institute of Science 2019

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringThe Pennsylvania State UniversityUniversity ParkUSA

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