Quantile correlation coefficient: a new tail dependence measure

A quantile correlation coefficient is newly defined as the geometric mean of two quantile regression slopes—that of X on Y and that of Y on X—in the same way that the Pearson correlation coefficient is related to regression coefficients. The quantile correlation is a measure of overall sensitivity of a conditional quantile of a random variable to changes in the other variable. The proposed quantile correlation can be compared across different tails within a given distribution to provide meaningful interpretations, for example, that there is stronger dependence in the left tail than overall. It can also be compared with the Pearson correlation. Neither of these two comparability within a given distribution is enabled by the existing tail-dependence correlation measures. Moreover a test for differences in the quantile correlations at different tails is proposed. The asymptotic normality of the estimated quantile correlation and the null distribution of the proposed test are established and are well supported by a Monte-Carlo study. The proposed quantile correlation methods are illustrated well by an analysis of stock return price data sets, yielding a clear indication of stronger left-tail dependence than overall dependence and stronger overall dependence than right-tail dependence.