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We propose a direct and natural extension of Azadkia & Chatterjee's rankcorrelation $T$ introduced in [4] to a set of $q \geq 1$ endogenous variables.The approach builds upon converting the original vector-valued problem into aunivariate problem and then applying the rank correlation $T$ to it. The novelmeasure $T^q$ then quantifies the scale-invariant extent of functionaldependence of an endogenous vector ${\bf Y} = (Y_1,\dots,Y_q)$ on a number ofexogenous variables ${\bf X} = (X_1,\dots,X_p)$, $p\geq1$, characterizesindependence of ${\bf X}$ and ${\bf Y}$ as well as perfect dependence of ${\bfY}$ on ${\bf X}$ and hence fulfills all the desired characteristics of ameasure of predictability. Aiming at maximum interpretability, we providevarious general invariance and continuity conditions for $T^q$ as well as novelordering results for conditional distributions, revealing new insights into thenature of $T$. Building upon the graph-based estimator for $T$ in [4], wepresent a non-parametric estimator for $T^q$ that is strongly consistent infull generality, i.e., without any distributional assumptions. Based on thisestimator we develop a model-free and dependence-based feature ranking andforward feature selection of multiple-outcome data, and establish tools foridentifying networks between random variables. Real case studies illustrate themain aspects of the developed methodology.

A simple extension of Azadkia $\&$ Chatterjee's rank correlation to a vector of endogenous variables