Quantifying directed dependence via dimension reduction

Studying the multivariate extension of copula correlation yields a dimension reduction principle, which turns out to be strongly related with the ‘simple measure of conditional dependence’ $T$ recently introduced by Azadkia and Chatterjee (2021). In the present paper, we identify and investigate the dependence structure underlying this dimension reduction principle, provide a strongly consistent estimator for it, and demonstrate its broad applicability. For that purpose, we define a bivariate copula capturing the scale-invariant extent of dependence of an endogenous random variable $Y$ on a set of $d\ge 1$ exogenous random variables $X=(X_1,\dots ,X_d)$, and containing the information whether $Y$ is completely dependent on $X$, and whether $Y$ and $X$ are independent. The dimension reduction principle becomes apparent insofar as the introduced bivariate copula can be viewed as the distribution function of two random variables $Y$ and $Y^{\prime }$ sharing the same conditional distribution and being conditionally independent given $X$. Evaluating this copula uniformly along the diagonal, i.e., calculating Spearman’s footrule, leads to an unconditional version of Azadkia and Chatterjee’s ‘simple measure of conditional dependence’ $T$. On the other hand, evaluating this copula uniformly over the unit square, i.e., calculating Spearman’s rho, leads to a distribution-free coefficient of determination (also known as ‘copula correlation’). Several real data examples illustrate the importance of the introduced methodology.