Outer power transformations of hierarchical Archimedean copulas: Construction, sampling and estimation

Outer power (OP) transformations of Archimedean generators are suggested to increase the modeling flexibility and statistical fitting capabilities of classical Archimedean copulas restricted to a single parameter. For OP-transformed Archimedean copulas, a formula for computing tail dependence coefficients is obtained, as well as two feasible OP Archimedean copula estimators are proposed and their properties studied by simulation. For hierarchical extensions of OP-transformed Archimedean copulas under the sufficient nesting condition, a new construction principle, efficient sampling and parameter estimation for models based on a single one-parameter Archimedean family are addressed. Special attention is paid to the case where the sufficient nesting condition simplifies to two types of restrictions on the corresponding parameters. By simulation, the convergence rate and standard errors of the proposed estimator are studied. Excellent tail fitting capabilities of OP-transformed hierarchical Archimedean copula models are demonstrated in a risk management application. The results show that the OP transformation is able to improve the statistical fit of exchangeable Archimedean copulas, particularly of those that cannot capture upper tail dependence or strong concordance, as well as the statistical fit of hierarchical Archimedean copulas, especially in terms of tail dependence and higher dimensions. Given how comparably simple it is to include OP transformations into existing exchangeable and hierarchical Archimedean copula models, OP transformations provide an attractive trade-off between computational effort and statistical improvement.