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. ...
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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.
... Taking logs is used widely in applied (time series) econometrics for linearizing relations or stabilizing variances. It has become a standard transformation for time series in numerous economic and financial applications; see among others Andersen, Bollerslev, and Huang (2011), Bauer andVorkink (2011), Brechmann, Heiden, andOkhrin (2018), Golosnoy, Okhrin, and Schmid (2012), Hautsch (2012), Lütkepohl and Xu (2012), Mayr and Ulbricht (2015), or Gribisch (2018). Indeed, models in logs often turn out to be better suited for both estimation and forecasting. ...
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... 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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This paper introduces the model confidence set (MCS) and applies it to the selection of models. An MCS is a set of models that is constructed so that it will contain the best model with a given level of confidence. The MCS is in this sense analogous to a confidence interval for a parameter. The MCS acknowledges the limitations of the data; uninformative data yield an MCS with many models whereas informative data yield an MCS with only a few models. The MCS procedure does not assume that a particular model is the true model; in fact, the MCS procedure can be used to compare more general objects, beyond the comparison of models. We apply the MCS procedure to two empirical problems. First, we revisit the inflation forecasting problem posed by Stock and Watson (1999) and compute the MCS for their set of inflation forecasts. Second, we compare a number of Taylor rule regressions and determine the MCS of the best in terms of in-sample likelihood criteria.
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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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Parametric families of continuous bivariate distributions with given margins that include independence and perfect positive dependence are compared on the basis on some important properties. Since many such families exist, the comparisons are helpful for deciding on suitable models for multivariate data. The study of the properties has motivation from applications in extreme value inference. One property considered for bivariate families is whether they extend to multivariate families, and extensions are given when possible. Several new bivariate and multivariate families are included and some open research problems in the area of multivariate families are mentioned.
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We present a new matrix-logarithm model of the realized covariance matrix of stock returns. The model uses latent factors which are functions of lagged volatility, lagged returns and other forecasting variables. The model has several advantages: it is parsimonious; it does not require imposing parameter restrictions; and, it results in a positive-definite estimated covariance matrix. We apply the model to the covariance matrix of size-sorted stock returns and find that two factors are sufficient to capture most of the dynamics.
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Building on the work of Bedford, Cooke and Joe, we show how multivariate data, which exhibit complex patterns of dependence in the tails, can be modelled using a cascade of pair-copulae, acting on two variables at a time. We use the pair-copula decomposition of a general multivariate distribution and propose a method for performing inference. The model construction is hierarchical in nature, the various levels corresponding to the incorporation of more variables in the conditioning sets, using pair-copulae as simple building blocks. Pair-copula decomposed models also represent a very flexible way to construct higher-dimensional copulae. We apply the methodology to a financial data set. Our approach represents the first step towards the development of an unsupervised algorithm that explores the space of possible pair-copula models, that also can be applied to huge data sets automatically.
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This paper is about how to estimate the integrated covariance 〈X,Y〉T of two assets over a fixed time horizon [0,T], when the observations of X and Y are “contaminated” and when such noisy observations are at discrete, but not synchronized, times. We show that the usual previous-tick covariance estimator is biased, and the size of the bias is more pronounced for less liquid assets. This is an analytic characterization of the Epps effect. We also provide the optimal sampling frequency which balances the tradeoff between the bias and various sources of stochastic error terms, including nonsynchronous trading, microstructure noise, and time discretization. Finally, a two scales covariance estimator is provided which simultaneously cancels (to first order) the Epps effect and the effect of microstructure noise. The gain is demonstrated in data.
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Both volatility clustering and conditional non-normality can induce the leptokurtosis typically observed in financial data. In this paper, the exact representation of kurtosis is derived for both GARCH and stochastic volatility models when innovations may be conditionally non-normal. We find that, for both models, the volatility clustering and non-normality contribute interactively and symmetrically to the overall kurtosis of the series.
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Applied researchers often test for the difference of the Sharpe ratios of two investment strategies. A very popular tool to this end is the test of Jobson and Korkie (1981), which has been corrected by Memmel (2003). Unfortunately, this test is not valid when returns have tails heavier than the normal distribution or are of time series nature. Instead, we propose the use of robust inference methods. In particular, we suggest to construct a studentized time series bootstrap confidence interval for the difference of the Sharpe ratios and to declare the two ratios different if zero is not contained in the obtained interval. This approach has the advantage that one can simply resample from the observed data as opposed to some null-restricted data. A simulation study demonstrates the improved finite sample performance compared to existing methods. In addition, two applications to real data are provided.
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We show how pre-averaging can be applied to the problem of measuring the ex-post covariance of financial asset returns under microstructure noise and non-synchronous trading. A pre-averaged realised covariance is proposed, and we present an asymptotic theory for this new estimator, which can be configured to possess an optimal convergence rate or to ensure positive semi-definite covariance matrix estimates. We also derive a noise-robust Hayashi–Yoshida estimator that can be implemented on the original data without prior alignment of prices. We uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high-frequency equity data.
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To estimate the parameters of a stationary univariate fractionally integrated time series, the unconditional exact likelihood function is derived. This allows the simultaneous estimation of all the parameters of the model by exact maximum likelihood. Issues involved in obtaining maximum likelihood estimates are discussed. Particular attention is given to efficient procedures to evaluate the likelihood function, obtaining starting values, and the small sample properties of the estimators. Limitations of previous estimation procedures are also discussed.
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Volatility has been one of the most active and successful areas of research in time series econometrics and economic forecasting in recent decades. This chapter provides a selective survey of the most important theoretical developments and empirical insights to emerge from this burgeoning literature, with a distinct focus on forecasting applications. Volatility is inherently latent, and Section 1 begins with a brief intuitive account of various key volatility concepts. Section 2 then discusses a series of different economic situations in which volatility plays a crucial role, ranging from the use of volatility forecasts in portfolio allocation to density forecasting in risk management. Sections 3–5 present a variety of alternative procedures for univariate volatility modeling and forecasting based on the GARCH, stochastic volatility and realized volatility paradigms, respectively. Section 6 extends the discussion to the multivariate problem of forecasting conditional covariances and correlations, and Section 7 discusses volatility forecast evaluation methods in both univariate and multivariate cases. Section 8 concludes briefly.
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Misperceptions about extreme dependencies between different financial assets have been an im- portant element of the recent financial crisis. This paper studies inhomogeneity in dependence structures using Markov switching regular vine copulas. These account for asymmetric depen- dencies and tail dependencies in high dimensional data. We develop methods for fast maximum likelihood as well as Bayesian inference. Our algorithms are validated in simulations and applied to financial data. We find that regime switches are present in the dependence structure of various data sets and show that regime switching models could provide tools for the accurate description of inhomogeneity during times of crisis.
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Regular vine distributions which constitute a flexible class of multivariate dependence models are discussed. Since multivariate copulae constructed through pair-copula decompositions were introduced to the statistical community, interest in these models has been growing steadily and they are finding successful applications in various fields. Research so far has however been concentrating on so-called canonical and D-vine copulae, which are more restrictive cases of regular vine copulae. It is shown how to evaluate the density of arbitrary regular vine specifications. This opens the vine copula methodology to the flexible modeling of complex dependencies even in larger dimensions. In this regard, a new automated model selection and estimation technique based on graph theoretical considerations is presented. This comprehensive search strategy is evaluated in a large simulation study and applied to a 16-dimensional financial data set of international equity, fixed income and commodity indices which were observed over the last decade, in particular during the recent financial crisis. The analysis provides economically well interpretable results and interesting insights into the dependence structure among these indices.