Measures of conditional dependence for nonlinearity, asymmetry and beyond

Abstract

Detecting the correlation between two random variables is widely used in many empirical problems in economics. Among them, Pearson’s correlation can be used to quantify the degree of dependence between variables. However, it cannot handle asymmetric correlations. To deal with this situation, we proposed a pair of widely applicable measures of conditional dependence ($MCDs$), which can not only account for the asymmetry but also the linear or nonlinear conditional dependencies in the presence of multiple variables. We give instances: when the paired measures are the same, resulting in symmetric correlation measures that are equivalent to the square of the Pearson coefficient; when no condition variables are given, $MCDs$ are used to assess the relationship between two variables. Consequently, Pearson’s correlation is a particular instance of $MCDs$. Theoretical attributes of $MCDs$ show that they have wide applicability. In statistical inference, we develop the joint asymptotics of kernel-based estimators for $MCDs$, which can be applied to determine whether two randomly generated variables exhibit symmetric conditional dependence in the presence of confounding variables. In the simulation, we verify the efficacy of the proposed $MCDs$. Then we use real data to analyze the asymmetric impact of $MCDs$ on stock market movements.