We propose new copula-based models for multivariate time series having continuous or discrete distributions, or a mixture of both. These models include stochastic volatility models and regime-switching models. We also propose statistics for testing independence between the generalized errors of these models, extending previous results of Duchesne, Ghoudi and Remillard (2012) obtained for stochastic volatility models. We define families of empirical processes constructed from lagged generalized errors, and we show that their joint asymptotic distributions are Gaussian and independent of the estimated parameters of the individual time series. Moebius transformations of the empirical processes are used to obtain tractable covariances. Several tests statistics are then proposed, based on Cramer-von Mises statistics and dependence measures, as well as graphical methods to visualize the dependence. In addition, numerical experiments are performed to assess the power of the proposed tests. Finally, to show the usefulness of our methodologies, examples of applications for financial data and crime data are given to cover both discrete and continuous cases. ll developed methodologies are implemented in the CRAN package IndGenErrors.

翻译：本文提出了新的基于Copula的多元时间序列模型，适用于连续分布、离散分布或两者混合的情形。该模型体系包含随机波动率模型和状态转换模型。我们进一步提出了用于检验这些模型广义误差间独立性的统计量，扩展了Duchesne、Ghoudi和Remillard（2012）针对随机波动率模型获得的研究结果。我们构建了基于滞后广义误差的经验过程族，并证明其联合渐近分布服从高斯分布且与各时间序列的估计参数无关。通过引入经验过程的莫比乌斯变换，我们得到了可处理的协方差结构。基于Cramer-von Mises统计量与依赖度量，我们提出了多种检验统计量，并提供了可视化依赖关系的图形方法。此外，通过数值实验评估了所提出检验方法的功效。最后，为展示本方法体系的实际应用价值，我们分别以金融数据和犯罪数据为例，涵盖了离散与连续两种情形。所有开发的方法均已在CRAN软件包IndGenErrors中实现。