Recognizing and visualizing copulas: an approach using local Gaussian approximation

Copulas are much used to model nonlinear and non-Gaussian dependence between stochastic variables. Their functional form is determined by a few parameters, but unlike a dependence measure like the correlation, these parameters do not have a clear interpretation in terms of the dependence structure they create. In this paper we examine the relationship between a newly developed local dependence measure, the local Gaussian Correlation, and standard copula theory. We are able to describe characteristics of the dependence structure in different copula models in terms of the local Gaussian correlation. In turn, these characteristics can be effectively visualized. More formally, the characteristic dependence structure can be used to construct a goodness-of-fit test for bivariate copula models by comparing the theoretical local Gaussian correlation for a specific copula and the estimated local Gaussian correlation. A Monte Carlo study reveals that the test performs very well compared to a commonly used alternative test. We also propose two types of diagnostic plots which can be used to investigate the cause of a rejected null. Finally, our methods are used on a ”classic”  insurance data set.