"Frontmatter". In: Analysis of Financial Time Series

428 MCMC METHODSTable 10.2. Estimation of Bivariate Volatility Models for Monthly LogReturns of IBM Stock and the S&P 500 Index from January 1962 to December1999. The Stochastic Volatility Models Are Based on the Last 1000 Iterationsof a Gibbs Sampling with 1300 Total Iterations.(a) Bivariate GARCH(1, 1) model with time-varying correlationsParameter β 01 β 02 α 10 α 11 α 12 α 20 α 21 γ 0Estimate 1.04 0.79 3.16 0.83 0.10 10.59 0.04 0.35Std. Error 0.31 0.20 1.67 0.08 0.03 0.93 0.02 0.02(b) Stochastic volatility modelParameter β 01 β 02 α 10 α 11 σ 2 1vα 20 σ 2 2vγ 0 σ 2 uPost.Mean 0.86 0.84 0.52 0.86 0.08 1.81 0.39 0.39 0.08Std. Error 0.30 0.18 0.18 0.05 0.03 0.11 0.06 0.03 0.02These prior distributions are relatively noninformative. We ran the Gibbs samplingfor 1300 iterations, but discarded results of the first 300 iterations. The randomsamples of g ii,t were drawn by Griddy Gibbs with 400 grid points in the intervals[0, 1.5s 2 i ],wheres2 iis the sample variance of the log return r it . Posterior means andstandard errors of the “traditional” parameters of the bivariate stochastic volatilitymodel are given in Table 10.2(b).To check for convergence of the Gibbs sampling, we ran the procedure severaltimes with different starting values and numbers of iterations. The results are stable.For illustration, Figure 10.7 shows the scatterplots of various quantities for two differentGibbs samples. The first Gibbs sample is based on 300 + 1000 iterations, andthe second Gibbs sample is based on 500 + 3000 iterations, where M + N denotesthat the total number of Gibbs iterations is M + N, but results of the first M iterationsare discarded. The scatterplots shown are posterior means of g 11,t , g 22,t , q 21,t ,σ 22,t , σ 21,t , and the correlation ρ 21,t . The line y = x is added to each plot to showthe closeness of the posterior means. The stability of the Gibbs sampling results isclearly seen.It is informative to compare the GARCH model with time-varying correlationsin Eqs. (10.33)–(10.36) with the stochastic volatility model. First, as expected, themean equations of the two models are essentially identical. Second, Figure 10.8shows the time plots of fitted volatilities for IBM stock return. The upper panelis for the GARCH model, and the lower panel shows the posterior mean of thestochastic volatility model. The two models show similar volatility characteristics;they exhibit volatility clusterings and indicate an increasing trend in volatility. However,the GARCH model produces higher peak volatility values. Third, Figure 10.9shows the time plots of fitted volatilities for the S&P 500 index return. The GARCHmodel produces an extra volatility peak around 1993. This additional peak does notappear in the univariate analysis shown in Figure 10.5. It seems that for this particularinstance the bivariate GARCH model produces a spurious volatility peak. This spuriouspeak is induced by its dependence on IBM returns and does not appear in thestochastic volatility model. Indeed, the fitted volatilities of S&P 500 index return by