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Discrete Choice Data with Unobserved Heterogeneity: A Conditional Binary Quantile Model

  • Xiao Lu (a1)


In political science, data with heterogeneous units are used in many studies, such as those involving legislative proposals in different policy areas, electoral choices by different types of voters, and government formation in varying party systems. To disentangle decision-making mechanisms by units, traditional discrete choice models focus exclusively on the conditional mean and ignore the heterogeneous effects within a population. This paper proposes a conditional binary quantile model that goes beyond this limitation to analyze discrete response data with varying alternative-specific features. This model offers an in-depth understanding of the relationship between the explanatory and response variables. Compared to conditional mean-based models, the conditional binary quantile model relies on weak distributional assumptions and is more robust to distributional misspecification. The model also relaxes the assumption of the independence of irrelevant alternatives, which is often violated in practice. The method is applied to a range of political studies to show the heterogeneous effects of explanatory variables across the conditional distribution. Substantive interpretations from counterfactual scenarios are used to illustrate how the conditional binary quantile model captures unobserved heterogeneity, which extant models fail to do. The results point to the risk of averaging out the heterogeneous effects across units by conditional mean-based models.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Author’s note: This work was supported by the Collaborative Research Center “Political Economy of Reforms” (SFB 884 Project C1: Legislative Reforms and Party Competition), which is funded by the German Research Foundation. The author also acknowledges support by the state of Baden-Württemberg through bwHPC (high-performance computing cluster MISO Production) and the German Research Foundation through grant INST 35/1134-1 FUGG. An earlier version of this paper has been presented in the poster session of the 6th Asian Political Methodology Meeting in Kyoto. I thank Thomas König, James Lo, Jamie Monogan, Jong Hee Park, Richard Traumüller, the Editor-in-Chief, Jeff Gill, and three anonymous reviewers for their very helpful discussions and suggestions. Replication materials for this paper are available (Lu 2019).

Contributing Editor: Jeff Gill



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Alhamzawi, R., and Yu, K.. 2013. “Conjugate Priors and Variable Selection for Bayesian Quantile Regression.” Computational Statistics and Data Analysis 64:209219.
Alhusseini, F. H. H. 2017. “Bayesian Quantile Regression with Scale Mixture of Uniform Prior Distributions.” International Journal of Pure and Applied Mathematics 115(1):7791.
Alvarez, R. M., and Glasgow, G.. 1999. “Two-stage Estimation of Nonrecursive Choice Models.” Political Analysis 8(2):147165.
Alvarez, R. M., and Nagler, J.. 1995. “Economics, Issues and the Perot Candidacy: Voter Choice in the 1992 Presidential Election.” American Journal of Political Science 39(3):714744.
Benoit, D. F., Alhamzawi, R., and Yu, K.. 2013. “Bayesian Lasso Binary Quantile Regression.” Computational Statistics 28:28612873.
Benoit, D. F., and Poel, D. V.. 2012. “Binary Quantile Regression: A Bayesian Approach based on the Asymmetric Laplace Distribution.” Journal of Applied Economics 27:11741188.
Betz, T. 2017. “Trading Interests: Domestic Institutions, International Negotiations, and the Politics of Trade.” The Journal of Politics 79(4):12371252.
Cantú, F. 2014. “Identifying Irregularities in Mexican Local Elections.” American Journal of Political Science 58(4):936951.
Carpenter, B., Gelman, A., Hoffman, M. D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P., and Riddell, A.. 2017. “Stan: A Probabilistic Programming Language.” Journal of Statistical Software 76(1):132.
Carter, D. B., and Signorino, C. S.. 2010. “Back to the Future: Modeling Time Dependence in Binary Data.” Political Analysis 18(3):271292.
Croissant, Y. et al. . 2012 “Estimation of Multinomial Logit Models in R: The mlogit Packages.” R package version 0.2-2. URL:
Delgado, M. A., Rodrıguez-Poo, J. M., and Wolf, M.. 2001. “Subsampling Inference in Cube Root Asymptotics with An Application to Manski’s Maximum Score Estimator.” Economics Letters 73(2):241250.
Florios, K., and Skouras, S.. 2008. “Exact Computation of Max Weighted Score Estimators.” Journal of Econometrics 146:8691.
Glasgow, G. 2011. “Introduction to the Virtual Issue: Recent Advances in Discrete Choice Methods in Political Science.” Political Analysis 19(1):13.
Glasgow, G., Golder, M., and Golder, S. N.. 2012. “New Empirical Strategies for the Study of Parliamentary Government Formation.” Political Analysis 20:248270.
Hausman, J. A., and Wise, D. A.. 1978. “A Conditional Probit Model for Qualitative Choice: Discrete Decisions Recognizing Interdependence and Heterogeneous Preferences.” Econometrica 46(2):403426.
Hensher, D. A., and Greene, W. H.. 2003. “The Mixed Logit Model: the State of Practice.” Transportation 30(2):133176.
Hensher, D. A., Rose, J. M., and Greene, W. H.. 2015. Applied Choice Analysis . 2nd edn Cambridge: Cambridge University Press.
Hoffman, M. D., and Gelman, A.. 2014. “The No-U-turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo.” Journal of Machine Learning Research 15(1):15931623.
Horowitz, J. L. 1992. “A Smoothed Maximum Score Estimator for the Binary Response Model.” Econometrica 60(3):505531.
Kim, J., and Pollard, D.. 1990. “Cube Root Asymptotics.” The Annals of Statistics 18(1):191219.
Koenker, R. 2005. Quantile Regression . Cambridge: Cambridge University Press.
Koenker, R., and Bassett, G.. 1978. “Regression Quantiles.” Econometrica 46(1):3350.
Koenker, R., and Hallock, K. F.. 2001. “Quantile Regression.” Journal of Economic Perspectives 15(4):143156.
Koenker, R., and Machado, J. A. F.. 1999. “Goodness of Fit and Related Process for Quantile Regression.” Journal of American Statistical Association 94(448):12961310.
Kordas, G. 2006. “Smoothed Binary Regression Quantiles.” Journal of Applied Economics 21(3):387407.
Kotlyarova, Y.2005 “Kernel Estimators: Testing and Bandwidth Selection in Models of Unknown Smoothness.” Ph. D. thesis, McGill University.
Kotz, S., Kozubowski, T. J., and Podgorski, K.. 2001. The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance . New York: Springer Science+Business Media, LLC.
Kotz, S., Kozubowski, T. J., and Podgorski, K.. 2003. “An Asymmetric Multivariate Laplace Distribution.” Working Paper, 1–26.
Kozumi, H., and Kobayashi, G.. 2011. “Gibbs Sampling Methods for Bayesian Quantile Regression.” Journal of Statistical Computation and Simulation 81(11):15651578.
Lu, X.2019. “Replication Data for: Discrete Choice Data with Unobserved Heterogeneity: A Conditional Binary Quantile Model,”, Harvard Dataverse, V1, UNF:6:IBiII3WhUYbd+L6CUgbnrA== [fileUNF].
Manski, C. F. 1975. “Maximum Score Estimation of the Stochastic Utility Model of Choice.” Journal of Econometrics 3:205228.
Manski, C. F. 1985. “Semiparametric Analysis of Discrete Response: Asymptotic Properties of the Maximum Score Estimator.” Journal of Econometrics 27(3):313333.
Martin, L. W., and Stevenson, R. T.. 2001. “Government Formation in Parliamentary Democracies.” American Journal of Political Science 45(1):3350.
McFadden, D. 1974. “Conditional Logit Analysis of Qualitative Choice Behavior.” In Frontiers in Econometrics , edited by Zarembka, P., 105142. New York: Academic Press.
McFadden, D. 1986. “The Choice Theory Approach to Market Research.” Marketing Science 5(4):275297.
McGrath, L. F. 2015. “Estimating Onsets of Binary Events in Panel Data.” Political Analysis 23(4):534549.
Miller, W. E. 1991. “Party Identification, Realignment, and Party Voting: Back to the Basics.” American Political Science Review 85(2):557568.
Miller, W. E., Kinder, D. R., and Rosenstone, S. J.. 1993. The National Election Studies. 1993. American National Election Study, 1992: Pre-and Post-election Survey CPS Early Release Version, computer file . Ann Arbor: University of Michigan, Center for Political Studies, and Inter-University Consortium for Political and Social Research.
Mosteller, F., and Tukey, J. W.. 1977. Data Analysis and Regression: A Second Course in Statistics . Addison-Wesley Series in Behavioral Science: Quantitative Methods. Reading, MA: Addison-Wesley.
Oh, M.-s., Park, E. S., and So, B.-S.. 2016. “Bayesian Variable Selection in Binary Quantile Regression.” Statistics and Probability Letters 118:177181.
Poole, K. T. 2001. “Nonparametric Unfolding of Binary Choice Data.” Political Analysis 8(3):211237.
Rainey, C. 2016. “Dealing with Separation in Logistic Regression Models.” Political Analysis 24(3):339355.
Rasmussen, A. 2010. “Early Conclusion in Bicameral Bargaining: Evidence from the Co-decision Legislative Procedure of the European Union.” European Union Politics 12(1):4164.
Ratkovic, M., and Tingley, D.. 2017. “Sparse Estimation and Uncertainty with Application to Subgroup Analysis.” Political Analysis 25(1):140.
Reed, C., and Yu, K.. 2009 “A Partially Collapsed Gibbs Sampler for Bayesian Quantile Regression.” Technical report, Department of Mathematical Sciences, Brunel University London.
Rosenberg, A. S., Knuppe, A. J., and Braumoeller, B. F.. 2017. “Unifying the Study of Asymmetric Hypotheses.” Political Analysis 25(3):381401.
Sartori, A. E. 2003. “An Estimator for Some Binary-Outcome Selection Models Without Exclusion Restrictions.” Political Analysis 11(2):111138.
Sriram, K., Ramamoorthi, R., and Ghosh, P. et al. . 2013. “Posterior Consistency of Bayesian Quantile Regression based on the Misspecified Asymmetric Laplace Density.” Bayesian Analysis 8(2):479504.
Traunmüller, R., Murr, A., and Gill, J.. 2014. “Modeling Latent Information in Voting Data with Dirichlet Process Priors.” Political Analysis 23(1):120.
Yu, K., and Moyeed, R. A.. 2001. “Bayesian Quantile Regression.” Statistics & Probability Letters 54:437447.
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