Local Identification in Instrumental Variable Multivariate Quantile Regression Models

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. In practice, when researchers are interested in multiple outcomes, it is common to estimate separate IVQR models for each of them. This approach implicitly assumes that the rank variable in each regression affects only its associated outcome, without influencing others. In reality, however, outcomes are often jointly determined by multiple latent factors, inducing structural correlations across equations. To address this limitation, we propose a nonlinear instrumental variable model that accommodates multivariate unobserved heterogeneity, where each component of the latent vector acts as a rank variable corresponding to an observed outcome. When both the treatment and the instrument are discrete, we show that the structural function in our model is locally identified under a sufficiently strong positive correlation between the treatment and the instrument.