"Frontmatter". In: Analysis of Financial Time Series

380 MULTIVARIATE VOLATILITY MODELSwhere standard errors of the parameters in the first equation are 0.016 and 0.017,those of the parameters in the second equation are 0.052, 0.059, and 0.021, and thoseof the parameters in the third equation are 0.050, 0.057, and 0.022, respectively. Allestimates are statistically significant at about the 1% level. As expected, the meanequations for r 1t and r 2t are essentially the same as those in the bivariate case.The three-dimensional time-varying volatility model becomes a bit more complicated,but it remains manageable asg 11,t = 0.006 + 0.050b 2 1,t−1 + 0.943g 11,t−1q 21,t = 0.277 + 0.824q 21,t−1 − 0.035a 2,t−1 (9.33)g 22,t = 0.178 + 0.082b2,t−1 2 + 0.889g 22,t−1q 31,t = 0.039 + 0.973q 31,t−1 + 0.010a 3,t−1q 32,t = 0.006 + 0.981q 32,t−1 + 0.004a 2,t−1g 33,t = 1.188 + 0.053b 2 3,t−1 + 0.687g 33,t−1 − 0.019g 22,t−1 ,where b 1t = a 1t , b 2t = a 2t − q 21,t b 1t , b 3t = a 3t − q 31,t b 1t − q 32,t b 2t ,andstandard errors of the parameters are given in Table 9.2. Except for the constantterm of the q 32,t equation, all estimates are significant at the 5% level.Let ã t = (a 1t / ˆσ 1t , a 2t / ˆσ 2t , a 3t / ˆσ 3t ) ′ be the standardized residual series, whereˆσ it = √ ˆσ ii,t is the fitted conditional standard error of the ith return. The Ljung–Boxstatistics of ã t give Q(4) = 34.48(0.31) and Q(8) = 60.42(0.70), where the degreesof freedom of the chi-squared distributions are 31 and 67, respectively, after adjustingfor the number of parameters used in the mean equations. For the squared standardizedresidual series ã 2 t ,wehaveQ(4) = 28.71(0.58) and Q(8) = 52.00(0.91).Therefore, the fitted model appears to be adequate in modeling the conditionalmeans and volatilities.The three-dimensional volatility model in Eq. (9.33) shows some interesting features.First, it is essentially a time-varying correlation GARCH(1, 1) model becauseonly lag-1 variables are used in the equations. Second, the volatility of the daily logreturn of S&P 500 index does not depend on the past volatilities of Cisco or Intelstock return. Third, by taking the inverse transformation of the Cholesky decomposition,the volatilities of daily log returns of Cisco and Intel stocks depend on the pastTable 9.2. Standard Errors of Parameter Estimates of a Three-Dimensional VolatilityModel for the Daily Log Returns in Percentages of S&P 500 Index and Stocks of CiscoSystems and Intel Corporation from January 2, 1991 to December 31, 1999. The Orderingof the Parameter Is the Same As That Appears in Eq. (9.33).Equation Standard error Equation Standard errorg 11,t 0.001 0.005 0.006 q 21,t 0.135 0.086 0.010g 22,t 0.029 0.009 0.011 q 31,t 0.017 0.012 0.004g 33,t 0.407 0.015 0.100 0.008 q 32,t 0.004 0.013 0.001