Modeling Dependencies with Copulae

The modeling and estimation of multivariate distributions is one of the most critical issues in financial and economic applications. The distributions are usually restricted to the class of multivariate elliptical distributions. This limits the analysis to a very narrow class of candidate distribution and requires the estimation of a large number of parameters. Two further problems are illustrated in Figure 1.1. The scatter plot in the first figure shows realizations of two Gaussian random variables, the points are symmetric and no extreme outliers can be observed. In contrast, the second picture exhibits numerous outliers. The outliers in the first and third quadrants show that extreme values often occur simultaneously for both variables. Such behavior is observed in crisis periods, when strong negative movements on financial markets occur simultaneously. In the third figure we observe that the dependency between negative values is different compared to positive values. This type of non-symmetric dependency cannot be modeled by elliptical distributions, because they impose a very specific radially symmetric dependency structure. Both types of dependencies are often observed in financial applications. The assumption of Gaussian distribution is therefore rarely consistent with the empirical evidence and possibly leads to incorrect inferences from financial models. Moreover, the correlation coefficient is equal for all three samples, despite clear differences in the dependencies. This questions the suitability of the correlation coefficient as the key measure of dependence for financial data.