centre for microdata methods and practice

ESRC centre

cemmap is an ESRC research centre

ESRC

Keep in touch

Subscribe to cemmap news

Vector quantile regression: an optimal transport approach

Authors: Guillaume Carlier , Victor Chernozhukov and Alfred Galichon
Date: 01 June 2016
Type: Journal Article, Annals of Statistics, Vol. 44, No. 3, pp. 1165--1192

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">YY, taking values in Rd">RdRd given covariates Z=z">Z=zZ=z, taking values in Rk">RkRk, is a map u⟼QY|Z(u,z)">uQY|Z(u,z)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">UU follows a reference non-atomic distribution FU">FUFU, for instance uniform distribution on a unit cube in Rd">RdRd, the random vector QY|Z(U,z)">QY|Z(U,z)QY|Z(U,z) has the distribution of Y">YY conditional on Z=z">Z=zZ=z. Moreover, we have a strong representation, Y=QY|Z(U,Z)">Y=QY|Z(U,Z)Y=QY|Z(U,Z) almost surely, for some version of U">UU. The vector quantile regression (VQR) is a linear model for CVQF of Y">YY given Z">ZZ. Under correct specification, the notion produces strong representation, Y=β(U)⊤f(Z)">Y=β(U)f(Z)Y=β(U)⊤f(Z), for f(Z)">f(Z)f(Z) denoting a known set of transformations of Z">ZZ, where u⟼β(u)⊤f(Z)">uβ(u)f(Z)u⟼β(u)⊤f(Z) is a monotone map, the gradient of a convex function and the quantile regression coefficients u⟼β(u)">uβ(u)u⟼β(u) have the interpretations analogous to that of the standard scalar quantile regression. As f(Z)">f(Z)f(Z) becomes a richer class of transformations of Z">ZZ, 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">YY 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.

Previous version:
Guillaume Carlier, Victor Chernozhukov and Alfred Galichon September 2015, Vector quantile regression: an optimal transport approach, cemmap Working Paper, CWP58/15, Institute for Fiscal Studies

Publications feeds

Subscribe to cemmap working papers via RSS

Search cemmap

Search by title, topic or name.

Contact cemmap

Centre for Microdata Methods and Practice

How to find us

Tel: +44 (0)20 7291 4800

E-mail us