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Abstract

The aim of this paper is to examine the empirical usefulness of two new MSF – Scalar BEKK(1,1) models of n-variate volatility. These models formally belong to the MSV class, but in fact are some hybrids of the simplest MGARCH and MSV specifications. Such hybrid structures have been proposed as feasible (yet non-trivial) tools for analyzing highly dimensional financial data (large n). This research shows Bayesian model comparison for two data sets with n = 2, since in bivariate cases we can obtain Bayes factors against many (even unparsimonious) MGARCH and MSV specifications. Also, for bivariate data, approximate posterior results (based on preliminary estimates of nuisance matrix parameters) are compared to the exact ones in both MSF-SBEKK models. Finally, approximate results are obtained for a large set of returns on equities (n = 34).
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Abstract

Often daily prices on different markets are not all observable. The question is whether we should exclude from modelling the days with prices not available on all markets (thus loosing some information and implicitly modifying the time axis) or somehow complete the missing (non-existing) prices. In order to compare the effects of each of two ways of dealing with partly available data, one should consider formal procedures of replacing the unavailable prices by their appropriate predictions. We propose a fully Bayesian approach, which amounts to obtaining the marginal posterior (or predictive) distribution for any particular day in question. This procedure takes into account uncertainty on missing prices and can be used to check validity of informal ways of ”completing” the data (e.g. linear interpolation). We use the MSF-SBEKK structure, the simplest among hybrid MSV-MGARCH models, which can parsimoniously describe volatility of a large number of prices or indices. In order to conduct Bayesian inference, the conditional posterior distributions for all unknown quantities are derived and the Gibbs sampler (with Metropolis-Hastings steps) is designed. Our approach is applied to daily prices from six different financial and commodity markets; the data cover the period from December 21, 2005 till September 30, 2011, so the time of the global financial crisis is included. We compare inferences (on individual parameters, conditional correlation coefficients and volatilities), obtained in the cases where incomplete observations are either deleted or forecasted.
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Abstract

We develop a fully Bayesian framework for analysis and comparison of two competing approaches to modelling daily prices on different markets. The first approach, prevailing in financial econometrics, amounts to assuming that logarithms of prices behave like a multivariate random walk; this approach describes logarithmic returns most often by the VAR(1) model with MGARCH (or sometimes MSV) disturbances. In the second approach, considered here, it is assumed that daily price levels are linked together and, thus, the error correction term is added to the usual VAR(1)–MGARCH or VAR(1)–MSV model for logarithmic returns, leading to a reduced rank VAR(2) specification for logarithms of prices. The model proposed in the paper uses a hybrid MSV-MGARCH structure for VAR(2) disturbances. In order to keep cointegration modelling as simple as possible, we restrict to the case of two prices representing two different markets. The aim of the paper is to show how to check if a long-run relationship between daily prices exists and whether taking it into account influences our inference on volatility and short-run relations between returns on different markets. In the empirical example the daily values of the S&P500 index and the WTI oil price in the period 19.12.2005 – 30.09.2011 are jointly modelled. It is shown that, although the logarithms of the values of S&P500 and WTI oil price seem to be cointegrated, neglecting the error correction term leads to practically the same conclusions on volatility and conditional correlation as keeping it in the model.
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Abstract

The first so-called hybrid MSV-MGARCH models were characterized by the conditional covariance matrix that was a product of a univariate latent process and a matrix with a simple MGARCH structure (Engle’s DCC or scalar BEKK). The aim was to parsimoniously describe volatility of a large group of assets. The proposed hybrid models, similarly as pure MSV specifications (and other models based on latent processes), required the Bayesian approach equipped with efficient MCMC simulation tools. The numerical effort has payed – the hybrid models seem particularly useful due to their good fit and ability to jointly cope with large portfolios. In particular, the simplest hybrid, now called the MSF-SBEKK model, has been successfully used in many applications. However, one latent process may be insufficient in the case of a highly heterogeneous portfolio. Thus, in this study we discuss a general hybrid MSV-MGARCH model structure, showing its basic characteristics that explain greater flexibility of such hybrid structure with respect to the corresponding MGARCH class. From the empirical perspective, we advocate the GMSF-SBEKK specification, which uses as many latent processes as there are relatively homogeneous groups of assets. We present full Bayesian inference for such models, with the use of an efficient MCMC simulation strategy. The approach is used to jointly model volatility on very different markets. Joint modelling is formally compared to individual modelling of volatility on each market.
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Abstract

The s-period ahead Value-at-Risk (VaR) for a portfolio of dimension n is considered and its Bayesian analysis is discussed. The VaR assessment can be based either on the n-variate predictive distribution of future returns on individual assets, or on the univariate Bayesian model for the portfolio value (or the return on portfolio). In both cases Bayesian VaR takes into account parameter uncertainty and non-linear relationship between ordinary and logarithmic returns. In the case of a large portfolio, the applicability of the n-variate approach to Bayesian VaR depends on the form of the statistical model for asset prices. We use the n-variate type I MSF-SBEKK(1,1) volatility model proposed specially to cope with large n. We compare empirical results obtained using this multivariate approach and the much simpler univariate approach based on modelling volatility of the value of a given portfolio.
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Abstract

Hybrid MSV-MGARCH models, in particular the MSF-SBEKK specification, proved useful in multivariate modelling of returns on financial and commodity markets. The initial MSF-MGARCH structure, called LN-MSF-MGARCH here, is obtained by multiplying the MGARCH conditional covariance matrix Ht by a scalar random variable gt such that{ln gt, t ∈ Z} is a Gaussian AR(1) latent process with auto-regression parameter φ. Here we alsoconsider an IG-MSF-MGARCH specification, which is a hybrid generalisation of conditionally Student t MGARCH models, since the latent process {gt} is no longer marginally log-normal (LN), but for φ = 0 it leads to an inverted gamma (IG) distribution for gt and to the t-MGARCH case. If φ =/ 0, the latent variables gt are dependent, so (in comparison to the t-MGARCH specification) we get an additional source of dependence and one more parameter. Due to the existence of latent processes, the Bayesian approach, equipped with MCMC simulation techniques, is a natural and feasible statistical tool to deal with MSF-MGARCH models. In this paper we show how the distributional assumptions for the latent process together with the specification of the prior density for its parameters affect posterior results, in particular the ones related to adequacy of thet-MGARCH model. Our empirical findings demonstrate sensitivity of inference on the latent process and its parameters, but, fortunately, neither on volatility of the returns nor on their conditional correlation. The new IG-MSF-MGARCH specification is based on a more volatile latent process than the older LN-MSF-MGARCH structure, so the new one may lead to lower values of φ – even so low that they can justify the popular t-MGARCH model.
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