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.

翻译：本文研究了严格平稳一阶GARCH类过程中滞后变量的相依性与偏相依性所对应的二元Copula。研究表明：对称GARCH过程的Copula具有联合对称性但不可交换；具有对称创新分布与非对称杠杆效应的过程其Copula仅具有较弱的h对称性；而具有非对称创新分布的Copula则不具备任何对称性。由于GARCH过程的边缘分布函数形式未知，真实的二元Copula通常难以直接获取，为此本文提出了一类新的近似Copula。该类Copula基于密度构造方法，将描述正相依的标准二元Copula密度与两种保持均匀性的变换——v变换相结合。研究证明，当该构造方法应用于球面t分布绝对值所对应Copula的密度时效果尤为显著。针对呈现随机波动性的时间序列应用，本文开发了包含新型配对Copula的简化D-vine模型。实验表明，相较于以往采用vine copula的模型，新模型在GARCH过程模拟数据及金融汇率收益率数据集上均表现出更优的拟合效果。