时间序列分析第四次作业房青B新编

时间序列分析第四次作业 ——房青 B0712094 1071209153

1. ARMA-GARCH modeling of SSE Composite Index. Use the recent 1000 obervations on the log return of the SSECI.

(1) Use PACF to identify an ARCH model of the series. In terms of correlations, is this model adequate for the modeling of the conditional heteroskedasicity? And what about the conditional mean?

Lag

P a r t i a l A C F 0

510

15202530

-0.05

0.0

0.05

0.1

0 Series : difflogsh

通过PACF 图，可以确定ARCH 模型的阶数为24。

Ljung-Box test for standardized residuals:

Statistic P-value Chi^2-d.f. 28.83 0.004172 12

Ljung-Box test for squared standardized residuals:

Statistic P-value Chi^2-d.f. 3.974 0.9839 12

根据上述检验结果可以看出，在5%显著性水平下模型残差具有显著自相关性，说明ARCH （24）对条件异方差的拟合能力并不好。

-4-2

2

4

-2

2

Quantiles of gaussian distribution

S t a n d a r d i z e d R e s i d u a l s

QQ-Plot of Standardized Residuals

Jarque-Bera P-value 104.9 0

从QQ 图和Jarque-Bera 检验中可以得出，模型残差不符合正态分布，说明模型还需要改进。

以上的结论可以看出，需要加入ARMA 部分来优化模型。

(2) Estimate an ARMA(1,6)-ARCH(p) model of the series, where p is determined above. Is this model adequate?

Ljung-Box test for standardized residuals:

Statistic P-value Chi^2-d.f. 13.59 0.3278 12 Ljung-Box test for squared standardized residuals:

Statistic P-value Chi^2-d.f. 12.68 0.3925 12

从上述检验结果中可以看出，在5%显著性水平下模型残差以及残差的平方都已经是白噪声过程，不具有自相关性。说明该模型的拟合效果有很大的提高。

-5

5

-2

2

Quantiles of gaussian distribution

S t a n d a r d i z e d R e s i d u a l s

QQ-Plot of Standardized Residuals

Jarque-Bera P-value 2205 0

虽然仍没有通过Jarque-Bera 检验，但是从QQ 图上来看，残差对正态分布的趋近程度

比上个模型大大提高了。

说明加入了ARMA 部分后，模型的拟合能力提高很大。

(3) Estimate a GARCH(1,1) model of the series. Is this model adequate for the conditional heteroskedasicity? What about the conditional mean?

Ljung-Box test for standardized residuals: Statistic P-value Chi^2-d.f. 30.93 0.002018 12 Ljung-Box test for squared standardized residuals:

Statistic P-value Chi^2-d.f. 10.4 0.5807 12

根据上述检验结果可以看出，在5%显著性水平下模型残差具有显著的自相关性，说明该

模型对条件异方差的拟合能力并不好。

Jarque-Bera P-value

146.2 0

-4-2

2

4

-2

2

Quantiles of gaussian distribution

S t a n d a r d i z e d R e s i d u a l s

QQ-Plot of Standardized Residuals

从QQ 图和Jarque-Bera 检验中可以得出，模型残差不符合正态分布，说明模型还需要改进。

以上的结论可以看出，需要加入ARMA 部分来优化模型。

(4) Estimate an ARMA(1,6)-GARCH(1,1) model of the series. Plot(i) Conditional Standard Deviations, sigma_t (ii) ACF of Standardized Residuals, \hat\varepsilon_t (iii) QQ-Plot of Standardized Residuals.

Ljung-Box test for standardized residuals:

Statistic P-value Chi^2-d.f.

13.73 0.3181 12 Ljung-Box test for squared standardized residuals:

Statistic P-value Chi^2-d.f. 9.298 0.6773 12

从上述检验结果中可以看出，在5%显著性水平下模型残差以及残差的平方都已经是白噪声