GARCH copulas, v-transforms and D-vines for stochastic volatility
The bivariate copulas that describe the dependencies and partial dependencies of lagged variables in strictly stationary, first-order GARCH-type processes are investigated. It is shown that the copulas of symmetric GARCH processes are jointly symmetric but non-exchangeable, while the copulas of processes with symmetric innovation distributions and asymmetric leverage effects have weaker h-symmetry; copulas with asymmetric innovation distributions have neither form of symmetry. Since the true bivariate copulas are typically inaccessible, due to the unknown functional forms of the marginal distributions of GARCH processes, a new class of approximating copulas is proposed. These rely on copula density constructions that combine standard bivariate copula densities for positive dependence with two uniformity-preserving transformations known as v-transforms. The construction is shown to be particularly effective when applied to the density of the copula of the absolute values of a spherical t distribution. Tractable simplified D-vines incorporating the new pair copulas are developed for applications to time series showing stochastic volatility. The resulting models are shown to provide better fits to simulated data from GARCH processes, and to a dataset of financial exchange-rate returns, than have previously been obtained using vine copulas.