Vector quantile regression

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(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=β(U)^Tf(Z),for f(Z) denoting a known set of transformations of Z, where u --> β(u)^Tf(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. Several applications to diverse problems such as multiple Engel curve estimation, and measurement of financial risk, are considered.