  In the instrumental variable quantile regression (IVQR) model of Chernozhukov
and Hansen (2005), a one-dimensional unobserved rank variable monotonically
determines a single potential outcome. Even when multiple outcomes are
simultaneously of interest, it is common to apply the IVQR model to each of
them separately. This practice implicitly assumes that the rank variable of
each regression model affects only the corresponding outcome and does not
affect the other outcomes. In reality, however, it is often the case that all
rank variables together determine the outcomes, which leads to a systematic
correlation between the outcomes. To deal with this, we propose a nonlinear IV
model that allows for multivariate unobserved heterogeneity, each of which is
considered as a rank variable for an observed outcome. We show that the
structural function of our model is locally identified under the assumption
that the IV and the treatment variable are sufficiently positively correlated.
