Article

A Multivariate Volatility Vine Copula Model

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Abstract

This paper proposes a dynamic framework for modeling and forecasting of realized covariance matrices using vine copulas to allow for more flexible dependencies between assets. Our model automatically guarantees positive definiteness of the forecast through the use of a Cholesky decomposition of the realized covariance matrix. We explicitly account for long-memory behavior by using ARFIMA and HAR models for the individual elements of the decomposition. Furthermore, our model incorporates non-Gaussian innovations and GARCH effects, accounting for volatility clustering and unconditional kurtosis. The dependence structure between assets is studied using vine copula constructions, which allow for nonlinearity and asymmetry without suffering from an inflexible tail behavior or symmetry restrictions as in conventional multivariate models. Further, the copulas have a direct impact on the point forecasts of the realized covariances matrices, due to being computed as a nonlinear transformation of the forecasts for the Cholesky matrix. Beside studying in-sample properties, we assess the usefulness of our method in a one-day ahead forecasting framework, comparing recent types of models for the realized covariance matrix based on a model confidence set approach. Additionally, we find that in Value-at-Risk (VaR) forecasting, vine models require less capital requirements due to smoother and more accurate forecasts.

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... For more details, see Cooke (2001, 2002), Aas et al. (2009), and Kurowicka and Joe (2011). Vine copulas find many applications in financial econometrics such as the estimation of Valueat-Risk, see Brechmann and Czado (2013), the modeling and forecasting of realized covariance matrices, see Brechmann et al. (2018) as well as the modeling of serial dependence in stationary time series, for example, see Loaiza-Maya, Smith, and Maneesoonthorn et al. (2018). ...
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... See also Simard et al. (2015). For the multivariate case, Brechmann et al. (2016) proposed a dynamic framework for modeling and forecasting realized covariance matrices using vine copulas. This vine-copula approach is still based on bivariate copulas to connect residuals of the univariate models for each element of the Cholesky factors. ...
Preprint
Multivariate volatility modeling and forecasting are crucial in financial economics. This paper develops a copula-based approach to model and forecast realized volatility matrices. The proposed copula-based time series models can capture the hidden dependence structure of realized volatility matrices. Also, this approach can automatically guarantee the positive definiteness of the forecasts through either Cholesky decomposition or matrix logarithm transformation. In this paper we consider both multivariate and bivariate copulas; the types of copulas include Student's t, Clayton and Gumbel copulas. In an empirical application, we find that for one-day ahead volatility matrix forecasting, these copula-based models can achieve significant performance both in terms of statistical precision as well as creating economically mean-variance efficient portfolio. Among the copulas we considered, the multivariate-t copula performs better in statistical precision, while bivariate-t copula has better economical performance.
... A simulation and forecasting exercise was conducted to highlight the importance of modelling both long memory and tail dependence to capture extreme events. More recently, Brechmann et al. (2018) studied a dynamic framework for forecasting realised covariances using vine copulas. The results obtained showed that for HAR models, using a vine structure significantly improves statistical loss, mean-variance efficient portfolios and VaR; while for ARFIMA-based vine models, the gains are less apparent, except for forecasting daily VaRs. ...
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... This method has been widely applied in finance and economics. For example, Riccetti (2013) applies vine copula method to the macroasset allocation of portfolios containing a commodity component, and Arreola Hernandez (2014) fits vine copula models and portfolio optimization methods with respect to five risk measures to investigate the dependence risk and resource allocation characteristics of two 20-stock coal-uranium and oil-gas sector portfolios from the Australian market in the context Brechmann et al. (2015) and Huang et al. (2016), who analyze the real interest rate-stock market link using vine copula models. However, multivariate dependence among commodity prices, the real value of the US dollar and the US real interest rate has not been addressed yet in the literature, which is explored in the current paper using vine copula methods. ...
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... However, it is possible to first apply a DCC decomposition approach and a CD on the correlation matrix thereafter. In general, the nonlinear dependence of the elements in the decomposition can also be an advantage, as the dependency structure between the Cholesky elements can be studied and used for forecasting, see e.g. Brechmann et al. (2015). ...
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C.Dolans-Dade1 and P.A.Meyer1 (1) Sminaire de Probabilits, Univerit de Strasbourg, Strasbourg, France Intgrales Stochastiques par Rapport aux Martingales Locales
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