We assume that we have multiple ordinal time series and we would like to specify their joint distribution. In general it is difficult to create multivariate distribution that can be easily used to jointly model ordinal variables and the problem becomes even more complex in the case of time series, since we have to take into consideration not only the autocorrelation of each time series and the dependence between time series, but also cross-correlation. Starting from the simplest case of two ordinal time series, we propose using copulas to specify their joint distribution. We extend our approach in higher dimensions, by approximating full likelihood with composite likelihood and especially conditional pairwise likelihood, where each bivariate model is specified by copulas. We suggest maximizing each bivariate model independently to avoid computational issues and synthesize individual estimates using weighted mean. Weights are related to the Hessian matrix of each bivariate model. Simulation studies showed that model fits well under different sample sizes. Forecasting approach is also discussed. A small real data application about unemployment state of different countries of European Union is presented to illustrate our approach.

翻译：我们假设存在多个序数时间序列，并希望确定其联合分布。一般而言，构建能够方便地联合建模序数变量的多元分布较为困难，而在时间序列情形下问题更为复杂，因为不仅需要考虑每个时间序列的自相关性及序列间的依赖性，还需考虑交叉相关性。从两个序数时间序列的最简情形出发，我们提出使用连接函数（copula）来设定其联合分布。我们将该方法推广至高维情形，通过复合似然——特别是条件成对似然——来近似完全似然，其中每个二元模型均由连接函数定义。为避免计算复杂性问题，我们建议独立最大化每个二元模型，并通过加权平均综合各独立估计值。权重与各二元模型的海森矩阵相关。模拟研究表明，该模型在不同样本量下均能良好拟合。本文亦讨论了预测方法，并以欧盟各国失业状态的小型实际数据应用为例，展示了所提方法的有效性。