Vector quantile regression: An optimal transport approach

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 ℝ^d given covariates Z = z, taking values in ℝ^k, is a map u→→Q_Y|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 F_U for instance uniform distribution on a unit cube in ℝ^d the random vector Q_Y|Z(U z) has the distribution of Y conditional on Z = z. Moreover we have a strong representation Y =Q_Y|Z(UZ) 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 = β(U)Τ f (Z) for f (Z) denoting a known set of transformations of Z where u →β(u)Τ f (Z) is a monotone map the gradient of a convex function and the quantile regression coefficients u →β(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.