"Frontmatter". In: Analysis of Financial Time Series

368 MULTIVARIATE VOLATILITY MODELS[ ]σ11,t=σ 22,t⎡ ⎤ ⎡⎤0.378 0.108 ·[ ⎢(0.103)⎥⎣ · ⎦ + ⎢(0.030)⎥ a2]1,t−1⎣ · 0.172 ⎦ a2,t−12(0.035)⎡⎤· 0.865[ ]+ ⎢ (0.109)⎥ σ11,t−1⎣ 0.321 0.869 ⎦ , (9.21)σ 22,t−1(0.135) (0.028)where the numbers in parentheses are standard errors. The estimated correlationcoefficient between a 1t and a 2t is 0.236 with standard error 0.045. Definingthe standardized residuals as before, we obtain Q(4) = 24.22(0.06) andQ(8) = 35.52(0.26) for the standardized residuals of the prior model and Q(4) =17.45(0.29) and Q(8) = 24.55(0.79) for the squared standardized residuals. TheseLjung–Box statistics are insignificant at the 5% level, and hence the model inEq. (9.21) is also adequate. The unconditional innovational variances of the priormodel are 1.71 and 1.32, respectively, for the Hong Kong and Japanese markets.In contrast with model (9.20), this second bivariate GARCH(1, 1) model showsa feedback relationship between the two markets. It is then interesting to comparethe two volatility models. First, the unconditional innovational variances of model(9.21) are closer to those of the univariate models in Eqs. (9.18) and (9.19). Second,Figure 9.3 shows the fitted volatility processes of model (9.20), whereas Figure 9.4shows those of model (9.21). Because model (9.20) implies no dynamic volatilitydependence between the two markets, Figure 9.3 is similar to that of Figure 9.2.In contrast, Figure 9.4 shows evidence of mutual impacts between the two markets.Third, the maximized log likelihood function for model (9.20) is −535.13 fort = 8,...,469, whereas that of model (9.21) is −540.32; see the log probabilitydensity function in Eq. (9.6). Therefore, model (9.20) is preferred if one usesthe likelihood principal. Finally, because practical implications of the two bivariatevolatility models differ dramatically, further investigation is needed to separate them.Such an investigation may use a longer sample period or include more variables (e.g.,using some U.S. market returns).Example 9.2. As a second illustration, consider the monthly log returns, inpercentages, of IBM stock and the S&P 500 index from January 1926 to December1999 used in Chapter 8. Let r 1t and r 2t be the monthly log returns for IBM stockand the S&P 500 index, respectively. If a constant-correlation GARCH(1, 1) modelis entertained, we obtain the mean equationsr 1t = 1.351 + 0.072r 1,t−1 + 0.055r 1,t−2 − 0.119r 2,t−2 + a 1tr 2t = 0.703 + a 2t ,where standard errors of the parameters in the first equation are 0.225, 0.029, 0.034,and 0.044, respectively, and that of the parameter in the second equation is 0.155.