https://hal-sciencespo.archives-ouvertes.fr/hal-03567920Carlier, GuillaumeGuillaumeCarlierCEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris sciences et lettres - CNRS - Centre National de la Recherche ScientifiqueChernozhukov, VictorVictorChernozhukovGalichon, AlfredAlfredGalichonECON - Département d'économie (Sciences Po) - Sciences Po - Sciences Po - CNRS - Centre National de la Recherche ScientifiqueVector Quantile Regression: An Optimal Transport ApproachHAL CCSD2016Vector quantile regressionVector conditional quantile functionMonge-Kantorovich-Brenier[SHS.ECO] Humanities and Social Sciences/Economics and FinanceSciences Po Institutional Repository, Spire2022-02-12 08:50:062022-10-26 14:33:512022-03-24 13:55:23enJournal articleshttps://hal-sciencespo.archives-ouvertes.fr/hal-03567920/document10.1214/15-AOS1401application/pdf1We 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.