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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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