  We propose a new measure related with tail dependence in terms of
correlation: quantile correlation coefficient of random variables X, Y. The
quantile correlation is defined by the geometric mean of two quantile
regression slopes of X on Y and Y on X in the same way that the Pearson
correlation is related with the regression coefficients of Y on X and X on Y.
The degree of tail dependent association in X, Y, if any, is well reflected in
the quantile correlation. The quantile correlation makes it possible to measure
sensitivity of a conditional quantile of a random variable with respect to
change of the other variable. The properties of the quantile correlation are
similar to those of the correlation. This enables us to interpret it from the
perspective of correlation, on which tail dependence is reflected. We construct
measures for tail dependent correlation and tail asymmetry and develop
statistical tests for them. We prove asymptotic normality of the estimated
quantile correlation and limiting null distributions of the proposed tests,
which is well supported in finite samples by a Monte-Carlo study. The proposed
quantile correlation methods are well illustrated by analyzing birth weight
data set and stock return data set.
