  This paper studies vector quantile regression (VQR), which is a way to model
the dependence of a random vector of interest with respect to a vector of
explanatory variables so to capture the whole conditional distribution, and not
only the conditional mean. The problem of vector quantile regression is
formulated as an optimal transport problem subject to an additional
mean-independence condition. This paper provides a new set of results on VQR
beyond the case with correct specification which had been the focus of previous
work. First, we show that even under misspecification, the VQR problem still
has a solution which provides a general representation of the conditional
dependence between random vectors. Second, we provide a detailed comparison
with the classical approach of Koenker and Bassett in the case when the
dependent variable is univariate and we show that in that case, VQR is
equivalent to classical quantile regression with an additional monotonicity
constraint.
