A simple extension of Azadkia & Chatterjee's rank correlation to a vector of endogenous variables
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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 ≥ 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 Y = (Y_1,…,Y_q) on a number of exogenous variables X = (X_1, …,X_p), p≥1, characterizes independence between X and Y as well as perfect dependence of Y on X and hence fulfills the desired properties of a measure of predictability for (p+q)-dimensional random vectors ( X, 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].
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