Abstract (EN)

We propose a notion of conditional vector quantile function and a vectorquantile regression.A conditional vector quantile function (CVQF) of a random vectorY, taking valuesinRdgiven covariatesZ=z, taking values inRp, is a mapu7!QYjZ(u;z), which ismonotone, in the sense of being a gradient of a convex function, and such that given thatvectorUfollows a reference non-atomic distributionFU, for instance uniform distributionon a unit cube inRd, the random vectorQYjZ(U;z) has the conditional distribution ofYconditional onZ=z. Moreover, we have a strong representation,Y=QYjZ(U;Z) almostsurely, for some version ofU.The vector quantile regression (VQR) is a linear model for CVQF ofYgivenZ. Undercorrect speci cation, the notion produces strong representation,Y=(U)>f(Z), forf(Z) denoting a known set of transformations ofZ, whereu7!(u)>f(Z) is a monotonemap, the gradient of a convex function, and the quantile regression coe cientsu7!(u)have the interpretations analogous to that of the standard scalar quantile regression. Asf(Z) becomes a richer class of transformations ofZ, 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, whereYis scalar, VQR reduces to a version of the classical QR,and CVQF reduces to the scalar conditional quantile function. Several applications todiverse problems such as multiple Engel curve estimation, and measurement of nancialrisk, are considered.