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# Vector Quantile Regression: An Optimal Transport Approach

## Author

Listed:
• Guillaume Carlier

(Centre de recherches en mathématique de la décision (CNRS, Paris-Dauphine))

• Victor Chernozhukov

(Department of Economics (Massachusetts University of Technology))

• Alfred Galichon

(Département d'économie)

## 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 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.

## Suggested Citation

• Guillaume Carlier & Victor Chernozhukov & Alfred Galichon, 2016. "Vector Quantile Regression: An Optimal Transport Approach," Sciences Po publications info:hdl:2441/4c5431jp6o8, Sciences Po.
• Handle: RePEc:spo:wpmain:info:hdl:2441/4c5431jp6o888pdrcs0fuirl40
as

File URL: https://spire.sciencespo.fr/hdl:/2441/4c5431jp6o888pdrcs0fuirl40/resources/2016-carlier-chernozhukov-galichon-vector-quantile-regression-vauteur.pdf

## References listed on IDEAS

as
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.
Full references (including those not matched with items on IDEAS)

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
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Cited by:

1. 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).
2. 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.
3. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2017. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Sciences Po publications info:hdl:2441/64itsev5509, Sciences Po.
4. R H Spady & S Stouli, 2018. "Dual regression," Biometrika, Biometrika Trust, vol. 105(1), pages 1-18.
5. 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.
6. Daouia, Abdelaati & Paindaveine, Davy, 2019. "From Halfspace M-Depth to Multiple-output Expectile Regression," TSE Working Papers 19-1022, Toulouse School of Economics (TSE).
7. Marc Hallin, 2018. "From Mahalanobis to Bregman via Monge and Kantorovich," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 135-146, December.
8. Montes-Rojas, Gabriel, 2017. "Reduced form vector directional quantiles," Journal of Multivariate Analysis, Elsevier, vol. 158(C), pages 20-30.
9. Donald Lien & Zijun Wang, 2019. "Quantile information share," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 39(1), pages 38-55, January.
10. 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.

### Keywords

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

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