Sample: 1 957
Included observations: 957
Autocorrelation | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Partial Correlation | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
AC PAC Q-Stat Prob -0.023 -0.023 1
2 -0.001 -0.002 3 -0.002 -0.002 4 0.002 5 0.001 6 0.025 7 0.070 8 0.001 9 0.055 10 0.069 11 0.007 12 0.025 13 0.030 14 0.007
0.002 0.001 0.025 0.071 0.004 0.056 0.073 0.011 0.026 0.029 0.004
0.5267 0.5279 0.5304 0.5333 0.5336 1.1177 5.8808 5.8815 8.8505 13.489 13.533 14.122 14.992 15.039 15.062
0.468 0.768 0.912 0.970 0.991 0.981 0.554 0.660 0.451 0.198 0.260 0.293 0.308 0.376 0.447
15 -0.005 -0.007
图4.8 ARCH(2)模型残差平方的自相关图
Date: 12/16/14 Time: 08:54 Sample: 1 957
Included observations: 957
Autocorrelation | | |* | | | | | | | | | |* | | | | | | | | | | | | | | | | |
Partial Correlation | | |* | | | | | | | | | |* | | | | | | | | | | | | | | | | |
AC PAC Q-Stat Prob 1 -0.000 -0.000 2 0.109 3 0.001 4 0.027 5 0.005 6 0.028 7 0.087 8 0.010 9 0.043 10 0.063 12 0.040 13 0.047
0.109 0.001 0.015 0.005 0.023 0.087 0.005 0.025 0.062 0.026 0.043
0.0002 11.411 11.413 12.101 12.126 12.862 20.108 20.212 21.998 25.905 25.929 27.454 29.603 29.617 29.645
0.990 0.003 0.010 0.017 0.033 0.045 0.005 0.010 0.009 0.004 0.007 0.007 0.005 0.009 0.013
11 0.005 -0.005
14 0.004 -0.013 15 -0.005 -0.017
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图4.9 ARCH(1)模型残差平方的自相关图
Date: 12/16/14 Time: 08:55 Sample: 1 957
Included observations: 957
Autocorrelation | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Partial Correlation | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
AC PAC Q-Stat Prob 1 -0.031 -0.031 2 0.045
0.044
3 -0.029 -0.027 4 -0.024 -0.027 5 -0.017 -0.016 6 -0.002 -0.001 7 0.065 9 0.050 10 0.051
0.065 0.043 0.059
8 -0.013 -0.011
0.9430 2.8653 3.6894 4.2329 4.5182 4.5219 8.5997 8.7598 11.144 13.696 14.097 14.103 14.358 14.393 15.226
0.332 0.239 0.297 0.375 0.477 0.606 0.283 0.363 0.266 0.187 0.228 0.294 0.349 0.421 0.435
11 -0.020 -0.019 12 -0.002 -0.004 13 0.016
0.023
14 -0.006 -0.006 15 -0.029 -0.029
图4.10 GARCH(1,1)模型残差平方的自相关图
ARCH(2)模型和GARCH(1，1)模型残差平方序列不存在自相关性，而ARCH(1)模型残差平方序列存在自相关性，故ARCH（1）模型不适合。下面进行正态性检验。
140120100806040200-4-3-2-101234Series: Standardized ResidualsSample 1 957Observations 957Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-BeraProbability 0.000873-0.019032 4.141549-3.934093 1.000522 0.183238 4.317777 74.59976 0.000000 图4.11 ARCH(2)模型的柱形统计图
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140120100806040200-4-3-2-101234Series: Standardized ResidualsSample 1 957Observations 957Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-BeraProbability 0.002827-0.021716 4.097276-4.558566 1.002241 0.137876 4.454455 87.38517 0.000000 图4.12 GARCH(1,1)模型柱形统计图
由以上结果可知，均不满足正态分布。再进行ARCH效应的检验。
表4.7 ARCH（1）模型残差ARCH效应检验
Heteroskedasticity Test: ARCH F-statistic
0.173845 Prob. F(3,950)
0.9141 0.9137
Obs*R-squared
0.523445 Prob. Chi-Square(3)
表4.8 GARCH（1,1）模型残差ARCH效应检验
Heteroskedasticity Test: ARCH F-statistic
1.154565 Prob. F(3,950)
0.3261 0.3252
Obs*R-squared
3.465643 Prob. Chi-Square(3)
LM检验的P值均大于5%，故不存在ARCH效应。下面对三个模型进行比较。
表4.9 不同模型结果对比
AIC SC 残差检验 ARCH(2) 3.336256 3.351503 ARCH(1) 3.350173 3.360337 GARCH(1,1) 3.326751 3.341998 无自相关性，无ARCH效应，不满足正态性 存在自相关性，无ARCH效应，不满足正态性 无自相关性，无ARCH效应，不满足正态性 由上表对比结果可知，GARCH(1，1)效果最好，故在此选择GARCH(1，1)模型。
4.5 不同GARCH模型的对比分析
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尝试建立不同的GARCH模型形式，TARCH模型、EGARCH模型、ARCH-M模型。
表4.10 TARCH模型的估计结果
Dependent Variable: R
Method: ML - ARCH (Marquardt) - Normal distribution Date: 12/16/14 Time: 22:15 Sample: 1 957
Included observations: 957
Convergence achieved after 11 iterations Presample variance: backcast (parameter = 0.7)
GARCH = C(1) + C(2)*RESID(-1)^2 + C(3)*RESID(-1)^2*(RESID(-1)<0) + C(4)*GARCH(-1)
Variable
C RESID(-1)^2
RESID(-1)^2*(RESID(-1)<0)
GARCH(-1)
R-squared
Coefficient
Std. Error
z-Statistic 2.124299 3.899567 1.278813 44.86304
Prob. 0.0336 0.0001 0.2010 0.0000 0.010480 1.292140 3.328008 3.348338 3.335751
Variance Equation 0.051813 0.035207 0.014738 0.927946
0.024391 0.009028 0.011525 0.020684
-0.000066 Mean dependent var 0.000979 S.D. dependent var 1.291507 Akaike info criterion 1596.268 Schwarz criterion -1588.452 Hannan-Quinn criter. 2.020182
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
由γ系数不显著 ，因此不能利用非对称模型对样本数据进行估计。
表4.11 EGARCH模型的估计结果
Dependent Variable: R
Method: ML - ARCH (Marquardt) - Normal distribution Date: 12/16/14 Time: 22:18 Sample: 1 957
Included observations: 957
Convergence achieved after 12 iterations Presample variance: backcast (parameter = 0.7)
LOG(GARCH) = C(1) + C(2)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(3) *RESID(-1)/@SQRT(GARCH(-1)) + C(4)*LOG(GARCH(-1))
Variable
Coefficient
Std. Error
z-Statistic - 13 -
Prob.
Variance Equation C(1) C(2) C(3) C(4)
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
-0.063973 0.109035 -0.014034 0.967261
0.015847 0.022411 0.010169 0.016573
-4.036994 4.865277 -1.379975 58.36433
0.0001 0.0000 0.1676 0.0000 0.010480 1.292140 3.328490 3.348819 3.336233
-0.000066 Mean dependent var 0.000979 S.D. dependent var 1.291507 Akaike info criterion 1596.268 Schwarz criterion -1588.682 Hannan-Quinn criter. 2.020182
由γ系数C（3）不显著，因此不能利用非对称模型对样本数据进行估计。
表4.12 ARCH-M模型的估计结果
Dependent Variable: R
Method: ML - ARCH (Marquardt) - Normal distribution Date: 12/16/14 Time: 22:31 Sample: 1 957
Included observations: 957
Convergence achieved after 10 iterations Presample variance: backcast (parameter = 0.7) GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*GARCH(-1)
Variable @SQRT(GARCH)
C RESID(-1)^2 GARCH(-1)
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
Coefficient 0.009831
Std. Error 0.032322
z-Statistic 0.304144 2.065507 4.182745 48.14421
Prob. 0.7610 0.0389 0.0000 0.0000 0.010480 1.292140 3.328742 3.349072 3.336485
Variance Equation 0.046265 0.038536 0.934829
0.022399 0.009213 0.019417
0.000115 Mean dependent var 0.000115 S.D. dependent var 1.292066 Akaike info criterion 1595.978 Schwarz criterion -1588.803 Hannan-Quinn criter. 2.020552
均指项不显著，因此不考虑ARCH-M模型，也不考虑该模型与推广其他GARCH模型的结合形式。综上可知，GARCH(1，1)效果最好，故在此选择GARCH(1，1)模型。这也说明沪深300指数的收益率影
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响中不存在非对称性。 4.6 模型的预测
选择GARCH(1,1)模型进行预测，预测结果如下。
3210-1-2-3930Forecast: RF1Actual: RForecast sample: 930 960Included observations: 28Root Mean Squared Error Mean Absolute Error Mean Abs. Percent Error Theil Inequality Coefficient Bias Proportion Variance Proportion Covariance Proportion 935940RF1945950?2 S.E.9559602.0610611.503282100.00001.0000000.180765NANA1.551.501.451.401.351.301.251.20930935940945950955960Forecast of Variance
图4.13 GARCH(1,1)模型
预测值为0，而且预测协方差误与方差误没有显示。
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