In many practical applications, the dynamics of different quantities is reasonably well described by linear equations. In economics, such linear dynamical models are known as vector autoregressive (VAR) models. These linear models are, however, only approximate. The deviations of the actual value of each quantity from the predictions of the linear model are usually well described by normal or Student-t distributions. To complete the description of the joint distribution of all these deviations, we need to supplement these marginal distributions with the information about the corresponding copula. To describe this dependence, in the past, researchers followed the usual idea of trying copulas from several standard families: Gaussian, Student, Clayton, Frank, Gumbel, and Joe families. To get a better description, we propose to also use convex combinations of copulas from different families; such convex combinations are known as mixed copulas. On the example of the dynamics of US macroeconomic data, including GDP, unemployment, consumer price index, and the real effective exchange rate, we show that mixed copulas indeed lead to a better description of the actual data. Specifically, it turns out that the best description is obtained if we use a convex combination of Student and Frank copulas.

中文翻译：

---

用于计量经济分析的基于混合 Copula 的向量自回归模型

在许多实际应用中，线性方程可以很好地描述不同量的动力学。在经济学中，这种线性动态模型被称为向量自回归 (VAR) 模型。然而，这些线性模型只是近似的。每个量的实际值与线性模型预测的偏差通常可以用正态分布或 Student-t 分布很好地描述。为了完成对所有这些偏差的联合分布的描述，我们需要用对应的 copula 的信息来补充这些边际分布。为了描述这种依赖性，过去，研究人员遵循通常的想法，即尝试来自几个标准家族的 copula：Gaussian、Student、Clayton、Frank、Gumbel 和 Joe 家族。为了得到更好的描述，我们建议也使用来自不同家族的 copula 的凸组合；这种凸组合称为混合 copula。以美国宏观经济数据的动态为例，包括 GDP、失业率、消费者价格指数和实际有效汇率，我们表明混合 copula 确实可以更好地描述实际数据。具体来说，如果我们使用 Student 和 Frank copulas 的凸组合，则可以得到最好的描述。