Vector Quantile Regression: An Optimal Transport Approach

Guillaume Carlier, Victor Chernozhukov and Alfred Galichon ()
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Guillaume Carlier: CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique
Alfred Galichon: ECON - Département d'économie (Sciences Po) - Sciences Po - Sciences Po - CNRS - Centre National de la Recherche Scientifique

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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.

Keywords: Vector quantile regression; Vector conditional quantile function; Monge-Kantorovich-Brenier (search for similar items in EconPapers)
Date: 2016-06-01
Note: View the original document on HAL open archive server: https://sciencespo.hal.science/hal-03567920v1
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Citations: View citations in EconPapers (33)

Published in Annals of Statistics, 2016, 44 (3), pp.1165-1192. ⟨10.1214/15-AOS1401⟩

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Related works:
Working Paper: Vector Quantile Regression: An Optimal Transport Approach (2016)
Working Paper: Vector quantile regression: an optimal transport approach (2015)
Working Paper: Vector quantile regression: an optimal transport approach (2015)
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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-03567920

DOI: 10.1214/15-AOS1401

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