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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.

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In this study, we introduce a mixed copula-based vector autoregressive (VAR) model for investigating the relationship between random variables. The one-step maximum likelihood estimation is used to obtain point estimates of the autoregressive parameters and mixed copula parameters. More specifically, we combine the likelihoods of the marginal and mixed Copula to construct the full likelihood function. The simulation study is used to confirm the accuracy of the estimation as well as the reliability of the proposed model. Various mixed copula forms from a combination of Gaussian, Student-t, Clayton, Frank, Gumbel, and Joe copulas are introduced. The proposed model is compared to the traditional VAR model and single copula-based VAR models to assess its performance. Furthermore, the real data study is also conducted to validate our proposed method. As a result, it is found that the one-step maximum likelihood provides accurate and reliable results. Also, we show that if we ignore the complex and nonlinear correlation between the errors, it causes significant efficiency loss in the parameter estimation, in terms of Bias and MSE. In the application study, the mixed copula-based VAR is the best fitting Copula for our application study.