Markov Chain Monte Carlo Estimation of Regime Switching Vector Autoregressions

Financial time series data are typically found to possess leptokurtic frequency distributions, time varying volatilities, others and correlation structures inconsistent with linear generating processes, nonlinear dependence, and dependencies between series that are not stable over time. Regime Switching Vector Autoregressions are of interest because they are capable of explaining the observed features of the data, can capture a variety of interactions between series, appear intuitively reasonable, are vector processes, and are now tractable. This paper considers a vector autoregression subject to periodic structural changes. The parameters of a vector autoregression are modelled as the outcome of an unobserved discrete Markov process with unknown transition probabilities. The unobserved regimes, one for each time point, together with the regime transition probabilities, are determined in addition to the vector autoregression are modelled as the outcome of an unobserved discrete Markov process with unknown transition probabilities. The unobserved regimes, one for each time point, together with the regime transition probabilities, are determined in addition to the vector autoregression parameters within each regime. A Bayesian Markov Chain Monte Carlo estimation procedure is developed which efficiently generates the posterior join density of the parameters and the regimes. The complete likelihood surface is generated at the same time, enabling estimation of posterior model probabilities for use in non-nested model selection. The procedure can readily be extended to produce joint prediction densities for the variables, incorporating both parameter and model uncertainty. Results using simulated and real data are provided. A clear separation of the variance between a stable and an unstable regime was observed. Ignoring regime shifts is very likely to produce misleading volatility estimates and is unlikely to be robust to outliers. A comparison with commonly used models suggests that Regime Switching Vector Autoregressions provide a particularly good description of the observed data. KEYWORDS Vector regime switching, joint parameter density, outliers, Gibbs sampler, Metropolis-Hasting algorithm, Markov Chain Monte Carlo, Posterior model probabilities, model selection