"Frontmatter". In: Analysis of Financial Time Series

GARCH MODELS FOR BIVARIATE RETURNS 367ing trend in volatility. The unconditional innovational variance of the Hong Kongmarket is about 1.76 and that of the Japanese market is 1.31.Turning to bivariate GARCH models, we obtain two models that fit the data well.The mean equations of the first bivariate model arer 1t =−0.118r 1,t−6 + a 1tr 2t = a 2t ,where the standard error of the AR(6) coefficient is 0.044. The volatility equationsof the firstmodelare[ ]σ11,t=σ 22,t⎡ ⎤ ⎡⎤0.275 0.112 ·[ ⎢(0.079)⎥⎣ 0.051 ⎦ + ⎢(0.032)⎥ a2]1,t−1⎣ · 0.091 ⎦ a2,t−12(0.014)(0.026)⎡⎤0.711 ·[ ]+ ⎢(0.068)⎥ σ11,t−1⎣ · 0.869 ⎦ , (9.20)σ 22,t−1(0.028)where the numbers in parentheses are standard errors. The estimated correlationcoefficients between a 1t and a 2t is 0.226 with standard error 0.047.Let ã t = (ã 1t , ã 2t ) ′ be the standardized residuals, where ã it = a it / √ σ ii,t .TheLjung–Box statistics of ã t give Q(4) = 22.29(0.10) and Q(8) = 34.83(0.29),where the number in parentheses denotes p value. Here the p values are based onchi-squared distributions with 15 and 31 degrees of freedom, respectively, becausean AR(6) coefficient is used in the mean equation. The Ljung–Box statistics for theã 2 t process give Q(4) = 9.54(0.85) and Q(8) = 18.58(0.96). Consequently, thereare no serial correlations or conditional heteroscedasticities in the bivariate standardizedresiduals of model (9.20). The unconditional innovational variances of the tworesiduals are 1.55 and 1.28, respectively, for the Hong Kong and Japanese markets.The model in Eq. (9.20) shows two uncoupled volatility equations, indicating thatthe volatilities of the two markets are not dynamically related, but they are contemporaneouslycorrelated. We refer to the model as a bivariate diagonal constantcorrelationmodel.The mean equations of the second bivariate GARCH model arer 1t =−0.143r 1,t−6 + a 1tr 2t = a 2t ,where the standard error of the AR(6) coefficient is 0.042, and the volatility equationsof the second model are