QY jZ(u; z), which is monotone, in the sense of being a gradient of a convex function, and such that given that vector U follows a reference nonatomic distribution FU, for instance uniform distribution on a unit cube in Rd, the random vector QY jZ(U; z) has the distribution of Y conditional on Z = z. Moreover, we have a strong representation, Y = QY jZ(U;Z) almost surely, for some version of U. The vector quantile regression (VQR) is a linear model for CVQF of Y given Z. Under correct specification, the notion produces strong representation, Y = (U)> f(Z), for f(Z) denoting a known set of transformations of Z, where u --> (u)>f(Z) is a monotone map, the gradient of a convex function, and the quantile regression coefficients u --> (u) have the interpretations analogous to that of the standard scalar quantile regression. As f(Z) becomes a richer class of transformations of Z, the model becomes nonparametric, as in series modelling. A key property of VQR is the embedding of the classical Monge-Kantorovich's optimal transportation problem at its core as a special case. In the classical case, where Y is scalar, VQR reduces to a version of the classical QR, and CVQF reduces to the scalar conditional quantile function. An application to multiple Engel curve estimation is considered.">
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Vector Quantile Regression: An Optimal Transport Approach

Citations

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Cited by:

  1. Dmitry Arkhangelsky, 2019. "Dealing with a Technological Bias: The Difference-in-Difference Approach," Working Papers wp2019_1903, CEMFI.
  2. Alfred Galichon, 2021. "The Unreasonable Effectiveness of Optimal Transport in Economics," SciencePo Working papers Main hal-03936221, HAL.
  3. Gabriel Montes-Rojas & Nicolás Bertholet, 2023. "When are devaluations more contractionary? A quantile VAR estimation for Argentina," Chapters, in: Fernando Toledo & Louis-Philippe Rochon (ed.), Monetary Policy Challenges in Latin America, chapter 8, pages 132-149, Edward Elgar Publishing.
  4. Keisuke Hirano & Jack R. Porter, 2023. "Asymptotic Representations for Sequential Decisions, Adaptive Experiments, and Batched Bandits," Papers 2302.03117, arXiv.org, revised Feb 2025.
  5. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
  6. Nadja Klein & Thomas Kneib, 2020. "Directional bivariate quantiles: a robust approach based on the cumulative distribution function," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 104(2), pages 225-260, June.
  7. Marc Hallin, 2021. "Measure Transportation and Statistical Decision Theory," Working Papers ECARES 2021-04, ULB -- Universite Libre de Bruxelles.
  8. Tongseok Lim, 2023. "Replication of financial derivatives under extreme market models given marginals," Papers 2307.00807, arXiv.org.
  9. repec:spo:wpmain:info:hdl:2441/64itsev5509q8aa5mrbhi0g0b6 is not listed on IDEAS
  10. Schennach, Susanne M., 2020. "Mismeasured and unobserved variables," Handbook of Econometrics, in: Steven N. Durlauf & Lars Peter Hansen & James J. Heckman & Rosa L. Matzkin (ed.), Handbook of Econometrics, edition 1, volume 7, chapter 0, pages 487-565, Elsevier.
  11. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
  12. R H Spady & S Stouli, 2018. "Dual regression," Biometrika, Biometrika Trust, vol. 105(1), pages 1-18.
  13. Nadja Klein & Torsten Hothorn & Luisa Barbanti & Thomas Kneib, 2022. "Multivariate conditional transformation models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 116-142, March.
  14. Daouia, Abdelaati & Paindaveine, Davy, 2019. "Multivariate Expectiles, Expectile Depth and Multiple-Output Expectile Regression," TSE Working Papers 19-1022, Toulouse School of Economics (TSE), revised Feb 2023.
  15. repec:hal:spmain:info:hdl:2441/64itsev5509q8aa5mrbhi0g0b6 is not listed on IDEAS
  16. Hongjian Shi & Mathias Drton & Marc Hallin & Fang Han, 2023. "Semiparametrically Efficient Tests of Multivariate Independence Using Center-Outward Quadrant, Spearman, and Kendall Statistics," Working Papers ECARES 2023-03, ULB -- Universite Libre de Bruxelles.
  17. Marc Hallin & Gilles Mordant, 2021. "On the Finite-Sample Performance of Measure Transportation-Based Multivariate Rank Tests," Working Papers ECARES 2021-24, ULB -- Universite Libre de Bruxelles.
  18. Ruodu Wang & Zhenyuan Zhang, 2022. "Simultaneous Optimal Transport," Papers 2201.03483, arXiv.org, revised Dec 2024.
  19. Alfred Galichon, 2021. "The Unreasonable Effectiveness of Optimal Transport in Economics," Working Papers hal-03936221, HAL.
  20. Eustasio del Barrio & Alberto González Sanz & Marc Hallin, 2025. "Nonparametric Multiple-Output Center-Outward Quantile Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 120(550), pages 818-832, April.
  21. Hubner, Stefan, 2023. "Identification of unobserved distribution factors and preferences in the collective household model," Journal of Econometrics, Elsevier, vol. 234(1), pages 301-326.
  22. Florian Gunsilius & Susanne M. Schennach, 2017. "A nonlinear principal component decomposition," CeMMAP working papers 16/17, Institute for Fiscal Studies.
  23. Christis Katsouris, 2023. "Quantile Time Series Regression Models Revisited," Papers 2308.06617, arXiv.org, revised Aug 2023.
  24. Balcilar, Mehmet & Ozdemir, Zeynel Abidin & Ozdemir, Huseyin & Wohar, Mark E., 2020. "Transmission of US and EU Economic Policy Uncertainty Shock to Asian Economies in Bad and Good Times," IZA Discussion Papers 13274, Institute of Labor Economics (IZA).
  25. Richard Blundell & Dennis Kristensen & Rosa Matzkin, 2017. "Individual counterfactuals with multidimensional unobserved heterogeneity," CeMMAP working papers CWP60/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  26. Florian Gunsilius & Susanne Schennach, 2023. "Independent Nonlinear Component Analysis," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 118(542), pages 1305-1318, April.
  27. Roger Koenker, 2017. "Quantile regression 40 years on," CeMMAP working papers CWP36/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  28. Walter W. Zhang & Sanjog Misra, 2022. "Coarse Personalization," Papers 2204.05793, arXiv.org, revised Jun 2025.
  29. Agarwal, Gaurav & Tu, Wei & Sun, Ying & Kong, Linglong, 2022. "Flexible quantile contour estimation for multivariate functional data: Beyond convexity," Computational Statistics & Data Analysis, Elsevier, vol. 168(C).
  30. Timo Dimitriadis & Yannick Hoga, 2022. "Dynamic CoVaR Modeling and Estimation," Papers 2206.14275, arXiv.org, revised Jan 2025.
  31. Daniel Hlubinka & Lukáš Kotík & Miroslav Šiman, 2022. "Multivariate quantiles with both overall and directional probability interpretation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(4), pages 1586-1604, December.
  32. Carlier, Guillaume & Chernozhukov, Victor & Galichon, Alfred, 2017. "Vector quantile regression beyond the specified case," Journal of Multivariate Analysis, Elsevier, vol. 161(C), pages 96-102.
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