Vector quantile regression: an optimal transport approach

Guillaume Carlier, Victor Chernozhukov and Alfred Galichon
Additional contact information
Guillaume Carlier: Institute for Fiscal Studies

No CWP58/15, CeMMAP working papers from Centre for Microdata Methods and Practice, Institute for Fiscal Studies

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=ß(U)Tf(Z),for f(Z) denoting a known set of transformations of Z, where u --> ß(u)T f(Z) is a monotone map, the gradient of a convex function, and the quantile regression coefficients u --> ß(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.

Keywords: Vector quantile regression; vector conditional quantile function; Monge-Kantorovich-Brenier (search for similar items in EconPapers)
Date: 2015-09-22
New Economics Papers: this item is included in nep-ecm
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.ifs.org.uk/uploads/cemmap/wps/cwp581515.pdf (application/pdf)
Our link check indicates that this URL is bad, the error code is: 404 Not Found (https://www.ifs.org.uk/uploads/cemmap/wps/cwp581515.pdf [302 Found]--> https://ifs.org.uk/uploads/cemmap/wps/cwp581515.pdf)

Related works:
Working Paper: Vector Quantile Regression: An Optimal Transport Approach (2016)
Working Paper: Vector Quantile Regression: An Optimal Transport Approach (2016)
Working Paper: Vector quantile regression: an optimal transport approach (2015)
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:ifs:cemmap:58/15

Ordering information: This working paper can be ordered from
The Institute for Fiscal Studies 7 Ridgmount Street LONDON WC1E 7AE

Access Statistics for this paper

More papers in CeMMAP working papers from Centre for Microdata Methods and Practice, Institute for Fiscal Studies The Institute for Fiscal Studies 7 Ridgmount Street LONDON WC1E 7AE. Contact information at EDIRC.
Bibliographic data for series maintained by Emma Hyman ().