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 ℝ% k, is a map u longmapsto QYmid 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 FU, for instance uniform distribution on a unit cube in ℝd, the random vector QYmid Z(U,z) has the distribution of Y conditional on Z=z. Moreover, we have a strong representation, Y = QYmid 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 left(Uright) ^top f(Z), for f(Z) denoting a known set of transformations of Z, where u longmapsto β(u)^top f(Z) is a monotone map, the gradient of a convex function, and the quantile regression coefficients u longmapsto 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.