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Vector quantile regression: an optimal transport approach

Author

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  • Guillaume Carlier

    (Institute for Fiscal Studies)

  • Victor Chernozhukov

    () (Institute for Fiscal Studies and MIT)

  • Alfred Galichon

    () (Institute for Fiscal Studies and NYU)

Abstract

We propose a notion of conditional vector quantile function and a vector quantile regression. A conditional vector quantile function (CVQF) of a random vector Y, taking values in Rd given covariates Z=z, taking values in Rk, is a map u --> QY|Z(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 non-atomic distribution FU, for instance uniform distribution on a unit cube in Rd, the random vector QY|Z(U,z) has the distribution of Y conditional on Z=z. Moreover, we have a strong representation, Y =QY|Z(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=ß(U)Tf(Z),for f(Z) denoting a known set of transformations of Z, where u --> ß(u)T f(Z) is a monotone map, the gradient of a convex function, and the quantile regression coefficients u --> ß(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.

Suggested Citation

  • Guillaume Carlier & Victor Chernozhukov & Alfred Galichon, 2015. "Vector quantile regression: an optimal transport approach," CeMMAP working papers CWP58/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:58/15
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    References listed on IDEAS

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    1. Flavio Cunha & James J. Heckman & Susanne M. Schennach, 2010. "Estimating the Technology of Cognitive and Noncognitive Skill Formation," Econometrica, Econometric Society, vol. 78(3), pages 883-931, May.
    2. Wei, Ying, 2008. "An Approach to Multivariate Covariate-Dependent Quantile Contours With Application to Bivariate Conditional Growth Charts," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 397-409, March.
    3. Marc Hallin & Davy Paindaveine & Miroslav Siman, 2008. "Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth," Working Papers ECARES 2008_042, ULB -- Universite Libre de Bruxelles.
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    Cited by:

    1. R H Spady & S Stouli, 2018. "Dual regression," Biometrika, Biometrika Trust, vol. 105(1), pages 1-18.
    2. Marc Hallin, 2018. "From Mahalanobis to Bregman via Monge and Kantorovich towards a “General Generalised Distance”," Working Papers ECARES 2018-12, ULB -- Universite Libre de Bruxelles.
    3. Daouia, Abdelaati & Paindaveine, Davy, 2019. "From Halfspace M-Depth to Multiple-output Expectile Regression," TSE Working Papers 19-1022, Toulouse School of Economics (TSE).
    4. repec:eee:jmvana:v:158:y:2017:i:c:p:20-30 is not listed on IDEAS
    5. repec:wly:jfutmk:v:39:y:2019:i:1:p:38-55 is not listed on IDEAS
    6. repec:eee:jmvana:v:161:y:2017:i:c:p:96-102 is not listed on IDEAS

    More about this item

    Keywords

    Vector quantile regression; vector conditional quantile function; Monge-Kantorovich-Brenier;

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