"Frontmatter". In: Analysis of Financial Time Series

424 MCMC METHODSht0 20 40 601970 1980 1990 2000yeargarch0 20 40 601970 1980 1990 2000Figure 10.5. Time plots of fitted volatilities for monthly log returns of S&P 500 index from1962 to 1999. The upper panel shows the posterior means of a Gibbs sampler with 5000iterations. The lower panel shows the results of a GARCH(1, 1) model.yearThe posterior mean of α 1 is 0.685, which is smaller than that obtained by Jacquier,Polson, and Rossi (1994) who used daily returns of the S&P 500 index. But it confirmsthe strong serial dependence in the volatility series. Finally, we have used differentinitial values and 3100 iterations for other Gibbs sampler, the posterior meansof the parameters change slightly, but the series of posterior means of h t are stable.10.7.2 Multivariate Stochastic Volatility ModelsIn this subsection, we study multivariate stochastic volatility models using theCholesky decomposition of Chapter 9. We focus on the bivariate case, but the methodsdiscussed also apply to the higher dimensional case. Based on the Choleskydecomposition, the innovation a t of a return series r t is transformed into b t such thatb 1t = a 1t , b 2t = a 2t − q 21,t b 1t ,where b 2t and q 21,t can be interpreted as the residual and least squares estimate ofthe linear regressiona 2t = q 21,t a 1t + b 2t .