"Frontmatter". In: Analysis of Financial Time Series

VECTOR VOLATILITY MODELS 379large-cap companies tend to be affected by the past behavior of the market. However,the market return is not significantly affected by the past returns of individualcompanies.Turning to volatility modeling and following the suggested procedure, we startwith the log return of S&P 500 index and obtain the modelr 1t = 0.078 + 0.042r 1,t−1 − 0.062r 1,t−3 − 0.048r 1,t−4 − 0.052r 1,t−5 + a 1tσ 11,t = 0.013 + 0.092a 2 1,t−1 + 0.894σ 11,t−1, (9.29)where standard errors of the parameters in the mean equation are 0.016, 0.023,0.020, 0.022, and 0.020, respectively, and those of the parameters in the volatilityequation are 0.002, 0.006, and 0.007, respectively. Univariate Ljung–Box statisticsof the standardized residuals and their squared series fail to detect any remainingserial correlation or conditional heteroscedasticity in the data. Indeed, we haveQ(10) = 7.38(0.69) for the standardized residuals and Q(10) = 3.14(0.98) for thesquared series.Augmenting the daily log returns of Cisco stock to the system, we build a bivariatemodel with mean equations given byr 1t = 0.065 − 0.046r 1,t−3 + a 1tr 2t = 0.325 + 0.195r 1,t−2 − 0.091r 2,t−2 + a 2t , (9.30)where all of the estimates are statistically significant at the 1% level. Using the notationof Cholesky decomposition, we obtain the volatility equations asg 11,t = 0.006 + 0.051b 2 1,t−1 + 0.943g 11,t−1q 21,t = 0.331 + 0.790q 21,t−1 − 0.041a 2,t−1 (9.31)g 22,t = 0.177 + 0.082b2,t−1 2 + 0.890g 22,t−1,where b 1t = a 1t , b 2t = a 2t − q 21,t b 1t , standard errors of the parameters in the equationof g 11,t are 0.001, 0.005, and 0.006, those of the parameters in the equation ofq 21,t are 0.156, 0.099, and 0.011, and those of the parameters in the equation of g 22,tare 0.029, 0.008, and 0.011, respectively. The bivariate Ljung–Box statistics of thestandardized residuals fail to detect any remaining serial dependence or conditionalheteroscedasticity. The bivariate model is adequate. Comparing with Eq. (9.29), wesee that the difference between the marginal and univariate models of r 1t is small.The next and final step is to augment the daily log returns of Intel stock to thesystem. The mean equations becomer 1t = 0.065 − 0.043r 1,t−3 + a 1tr 2t = 0.326 + 0.201r 1,t−2 − 0.089r 2,t−2 + a 2t (9.32)r 3t = 0.192 − 0.264r 1,t−1 + 0.059r 3,t−1 + a 3t ,