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 & 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 $d \geq 1$ exogenous random variables ${\bf X} = (X_1, \dots, X_d)$, and containing the information whether $Y$ is completely dependent on ${\bf X}$, and whether $Y$ and ${\bf X}$ 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 ${\bf X}$. Evaluating this copula uniformly along the diagonal, i.e. calculating Spearman's footrule, leads to 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 (a.k.a. copula correlation). Several real data examples illustrate the importance of the introduced methodology.