Copula diagnostics for asymmetries and conditional dependence

Vine copulas are constructed from a sequence of trees to represent dependence and conditional dependence, and a set of bivariate copulas that are applied to univariate distributions in tree 1 and to conditional univariate distributions in subsequent trees. Diagnostic methods based on measures of dependence and tail asymmetry are proposed to guide the choice of parametric bivariate copula families assigned to the edges of the trees in the vine and to assess whether a copula is constant over the conditioning value(s) for trees 2 and higher. The measures are conditional measures applied to bivariate conditional distributions in trees 2 and higher. If the diagnostic methods suggest the existence of reflection asymmetry, permutation asymmetry and possible asymmetric tail dependence, then three- or four-parameter bivariate copula families might be needed. Moreover, if the conditional dependence measures or asymmetry measures in trees 2 and up are not constant over the conditioning value(s), then non-constant copulas should be considered. We illustrate the use of the diagnostic methods for a gamma factor model and two real datasets. The examples show that better models are attained by using asymmetric and non-constant copulas under the guidance of the diagnostic tools.