CCOPULA Procedure

There are three basic types of dependence measures: linear correlation, rank correlation, and tail dependence. Linear correlation is given by

The linear correlation coefficient contains very limited information about the joint properties of the variables. A well-known property is that zero correlation does not imply independence, whereas independence implies zero correlation. In addition, there are distinct bivariate distributions that have the same marginal distribution and the same correlation coefficient. These results suggest that caution must be used in interpreting the linear correlation.

Another statistical measure of dependence is rank correlation, which is nonparametric. For example, Kendall’s tau is the covariance between the sign statistics and , where is an independent copy of :

The sign function (sometimes written as sgn) is defined as

Spearman’s rho is the correlation between the transformed random variables:

The variables are transformed by their distribution functions so that the transformed variables are uniformly distributed on . The rank correlations depend only on the copula of the random variables and are indifferent to the marginal distributions. Like linear correlation, rank correlation has its limitations. In particular, different copulas result in the same rank correlation.

A third measure, tail dependence, focuses on only part of the joint properties between the variables. Tail dependence measures the dependence when both variables have extreme values. Formally, they can be defined as the conditional probabilities of quantile exceedances. There are two types of tail dependence:

Tail dependence is hard to detect by looking at a scatter plot of realizations of two random variables. One graphical way to detect tail dependence between two variables is by creating the chi plot of those two variables. The chi plot, as defined in Fisher and Switzer (2001), has characteristic patterns that depend on the dependence structure between the variables. The chi plot of the random variables X and Y is a scatter plot of the pairs for each data point . is a measure of the distance of the data point from the center of the data as measured by the median values of , and is a correlation coefficient between dichotomized values of X and Y. A positive means that and are either both large with respect to their median values or both small. A negative means that or is large with respect to its median and that the other value is small. Signs of tail dependence manifest as clusters of points that are significantly far from the axis around values of 1. If X and Y are uncorrelated, the values cluster around the axis.

A Kendall plot enables you to compare the multivariate cumulative distribution function (CDF) for observed data to the CDF for a reference distribution. The Kendall plot is a multidimensional analog to the Q-Q plot that is used to compare distributions in one dimension. In both Q-Q plots and Kendall plots, you visually assess the similarity between the distributions of two random samples, A and B, by creating a scatter plot of the ordered values of A versus B. In the unpacked version of Kendall plots output by the CCOPULA procedure, three reference curves are displayed in addition to the observed, or simulated data. The horizontal reference curve is a Kendall plot for variables with a correlation of -1. The diagonal curve is Kendall plot for an uncorrelated bivariate uniform distribution. The arced curve is a Kendall plot for variables with a correlation of +1.