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Applicability of Pearson's correlation as a measure of explained variance is by now well understood. One of its limitations is that it does not account for asymmetry in explained variance. Aiming to develop broad applicable correlation measures, we study a pair of generalized measures of correlation (GMC) that deals with asymmetries in explained variances, and linear or nonlinear relations between random variables. We present examples under which the paired measures are identical, and they become a symmetric correlation measure that is the same as the squared Pearson's correlation coefficient. As a result, Pearson's correlation is a special case of GMC. Theoretical properties of GMC show that GMC can be applicable in numerous applications and can lead to more meaningful conclusions and improved decision making. In statistical inference, the joint asymptotics of the kernel-based estimators for GMC are derived and are used to test whether or not two random variables are symmetric in explaining variances. The testing results give important guidance in practical model selection problems. The efficiency of the test statistics is illustrated in simulation examples. In real-data analysis, we present an important application of GMC in explained variances and market movements among three important economic and financial monetary indicators. This article has online supplementary materials.
Zheng et al. (Mon,) studied this question.