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Vector quantile regression: an optimal transport approach

Authors: Guillaume Carlier , Victor Chernozhukov and Alfred Galichon
Date: 22 September 2015
Type: cemmap Working Paper, CWP58/15
DOI: 10.1920/wp.cem.2015.5815

Abstract

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|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 QY|Z(U,z) has the distribution of Y conditional on Z=z. Moreover, we have a strong representation, Y =QY|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)T 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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Now published:
Guillaume Carlier, Victor Chernozhukov and Alfred Galichon June 2016, Vector quantile regression: an optimal transport approach, Journal article, Institute of Mathematical Statistics
Previous version:
Guillaume Carlier, Victor Chernozhukov and Alfred Galichon December 2014, Vector quantile regression, cemmap Working Paper, CWP48/14, Institute for Fiscal Studies

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