"Frontmatter". In: Analysis of Financial Time Series

374 MULTIVARIATE VOLATILITY MODELŜΣ 888 (1) =[ ]71.09 21.83,21.83 17.79where the covariance is obtained by using the constant correlation coefficient 0.614.For the time-varying correlation model in Eqs. (9.24) and (9.25), we have a 1,888 =3.287, a 2,888 = 4.950, σ 11,888 = 83.35, σ 22,888 = 28.56, and ρ 888 = 0.546. The1-step ahead forecast for the covariance matrix iŝΣ 888 (1) =[ ]75.15 23.48,23.48 24.70where the forecast of the correlation coefficient is 0.545.In the second method, we use the Cholesky decomposition of Σ t to modeltime-varying correlations. For the bivariate case, the parameter vector is Ξ t =(g 11,t , g 22,t , q 21,t ) ′ ; see Eq. (9.13). A simple GARCH(1, 1)-type model for a t isg 11,t = α 10 + α 11 b 2 1,t−1 + β 11g 11,t−1q 21,t = γ 0 + γ 1 q 21,t−1 + γ 2 a 2,t−1 (9.26)g 22,t = α 20 + α 21 b1,t−1 2 + α 22b2,t−1 2 + β 21g 11,t−1 + β 22 g 22,t−1 ,where b 1t = a 1t and b 2t = a 2t −q 21,t a 1t . Thus, b 1t assumes a univariate GARCH(1,1) model, b 2t uses a bivariate GARCH(1, 1) model, and q 21,t is auto-correlated anduses a 2,t−1 as an additional explanatory variable. The probability density functionrelevant to maximum likelihood estimation is given in Eq. (9.15) with k = 2.Example 9.2. (continued). Again we use the monthly log returns of IBMstock and the S&P 500 index to demonstrate the volatility model in Eq. (9.26).Using the same specification as before, we obtain the fitted mean equations asr 1t = 1.364 + 0.075r 1,t−1 − 0.058r 2,t−2 + a 1tr 2t = 0.643 + a 2t ,where standard errors of the parameters in the first equation are 0.219, 0.027, and0.032, respectively, and that of the parameter in the second equation is 0.154. Thesetwo mean equations are close to what we obtained before. The fitted volatility modelisg 11,t = 3.714 + 0.113b 2 1,t−1 + 0.804g 11,t−1q 21,t = 0.0029 + 0.9915q 21,t−1 − 0.0041a 2,t−1 (9.27)g 22,t = 1.023 + 0.021b1,t−1 2 + 0.052b2 2,t−1 − 0.040g 11,t−1 + 0.937g 22,t−1 ,