Papers › Vector Quantile Regression: An Optimal Transport Approach
Vector Quantile Regression: An Optimal Transport Approach
Guillaume Carlier, Victor Chernozhukov, Alfred Galichon
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We propose a notion of conditional vector quantile function and a vector quantile regression. A \emph{conditional vector quantile function} (CVQF) of a random vector Y, taking values in ℝᵈ given covariates Z=z, taking values in ℝ, 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 ℝᵈ, 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 \emph{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) ^⊤ f(Z), for f(Z) denoting a known set of transformations of Z, where u ⟼β(u)^⊤ 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.
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