Testing Conditional Independence via Quantile Regression Based Partial Copulas

The partial copula provides a method for describing the dependence between two random variables $X$ and $Y$ conditional on a third random vector $Z$ in terms of nonparametric residuals $U_1$ and $U_2$. This paper develops a nonparametric test for conditional independence by combining the partial copula with a quantile regression based method for estimating the nonparametric residuals. We consider a test statistic based on generalized correlation between $U_1$ and $U_2$ and derive its large sample properties under consistency assumptions on the quantile regression procedure. We demonstrate through a simulation study that the resulting test is sound under complicated data generating distributions. Moreover, it is competitive with other state-of-the-art conditional independence tests in terms of power, and it has superior level properties compared to conditional independence tests based on conventional residuals obtained through conditional mean regression.