Missing observations in daily returns - Bayesian inference within the MSF-SBEKK model

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 unavailable observations are either deleted or forecasted.