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 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.