We perform Bayesian joint estimation of a multivariate GARCH model where the dependence structure of the innovations across the univariate time series is given by a D-vine copula. Vine copulas are a flexible concept to extend bivariate copulas to the multivariate case. It is based on the idea that a multivariate copula can be constructed from (conditional) bivariate copulas.
In particular it is possible to allow for symmetric dependence between some pairs of margins by
using e.g. bivariate Student t or Gaussian copulas and asymmetric dependence between other
pairs using e.g. bivariate Clayton or Gumbel copulas. A further advantage of D-vine copulas
is that the resulting correlation matrix is always positive definite without imposing restrictions on the parameters. In contrast to likelihood based estimation methods a Bayesian approach always allows to construct valid interval estimates for any quantity which is a function of the model parameters. This provides the possibility to assess the uncertainty about Value at Risk (VaR) predictions. In a simulation study and two real data examples with up to 5 dimensions we compare the proposed model to a benchmark multivariate GARCH model with dependence structure of the innovations governed by a multivariate Student t copula. The proposed model shows a clearly better fit according to the DIC. The choice between the two models also affects the VaR predictions. We further study the error introduced by the widely used two step estimation approach in the VaR prediction. This shows that the two step estimation approach leads to an underestimation of the uncertainty of VaR predictions for simulated and real data.

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