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

In the present paper, we propose a direct and simple extension of Azadkia $\&$ Chatterjee's rank correlation $T$ introduced in [4] to a set of $q \geq 1$ endogenous variables where me make use of the fact that $T$ characterizes conditional independence. The approach is exceptional in that we convert the original vector-valued problem into a univariate problem and then apply the rank correlation measure $T$ to it. The new measure $T^q$ then quantifies the scale-invariant extent of functional dependence of an endogenous vector ${\bf Y} = (Y_1,\dots,Y_q)$ on a number of exogenous variables ${\bf X} = (X_1, \dots,X_p)$, $p\geq1$, characterizes independence between ${\bf X}$ and ${\bf Y}$ as well as perfect dependence of ${\bf Y}$ on ${\bf X}$ and hence fulfills the desired properties of a measure of predictability for $(p+q)$-dimensional random vectors $({\bf X},{\bf Y})$. For the new measure $T^q$ we study invariance properties and ordering properties and provide a strongly consistent estimator that is based on the estimator for $T$ introduced in [4].