  Recently, Chatterjee (2023) recognized the lack of a direct generalization of
his rank correlation $\xi$ in Azadkia and Chatterjee (2021) to a
multi-dimensional response vector. As a natural solution to this problem, we
here propose an extension of $\xi$ that is applicable to a set of $q \geq 1$
response variables, where our approach builds upon converting the original
vector-valued problem into a univariate problem and then applying the rank
correlation $\xi$ to it. Our novel measure $T$ quantifies the scale-invariant
extent of functional dependence of a response vector $\mathbf{Y} =
(Y_1,\dots,Y_q)$ on predictor variables $\mathbf{X} = (X_1, \dots,X_p)$,
characterizes independence of $\mathbf{X}$ and $\mathbf{Y}$ as well as perfect
dependence of $\mathbf{Y}$ on $\mathbf{X}$ and hence fulfills all the
characteristics of a measure of predictability. Aiming at maximum
interpretability, we provide various invariance results for $T$ as well as a
closed-form expression in multivariate normal models. Building upon the
graph-based estimator for $\xi$ in Azadkia and Chatterjee (2021), we obtain a
non-parametric, strongly consistent estimator for $T$ and show its asymptotic
normality. Based on this estimator, we develop a model-free and
dependence-based feature ranking and forward feature selection for
multiple-outcome data. Simulation results and real case studies illustrate
$T$'s broad applicability.
