协整与误差修正模型
数据来源于wind。
上证50指数和上证综合指数的协整与误差修正模型（日收盘价，2005.6. 6-2012.11.14，共1814组数据）
第一步，检验上证50指数与上证综合指数的平稳性
单位根检验：
结果表明，上证综合指数是非平稳时间序列
Null Hypothesis: SZZS has a unit root Exogenous: Constant
Lag Length: 0 (Automatic based on SIC, MAXLAG=24)
t-Statistic -1.638128 -3.433755 -2.862931 -2.567557
Prob.* 0.4628
Augmented Dickey-Fuller test statistic Test critical values:
1% level 5% level 10% level
*MacKinnon (1996) one-sided p-values.
但一阶差分之后变成平稳的，如下
Null Hypothesis: D(SZZS) has a unit root Exogenous: None
Lag Length: 0 (Automatic based on SIC, MAXLAG=24)
t-Statistic
Prob.*
Augmented Dickey-Fuller test statistic Test critical values:
1% level 5% level 10% level
-42.20904 -2.566239 -1.940998 -1.616582
0.0001
*MacKinnon (1996) one-sided p-values.
下面检验结果表明，上证50指数也是非平稳时间序列
Null Hypothesis: SZ50 has a unit root Exogenous: Constant
Lag Length: 0 (Automatic based on SIC, MAXLAG=24)
t-Statistic -1.610216 -3.433755 -2.862931 -2.567557
Prob.* 0.4771
Augmented Dickey-Fuller test statistic Test critical values:
1% level 5% level 10% level
*MacKinnon (1996) one-sided p-values.
但一阶差分后，变成平稳的，如下
Null Hypothesis: D(SZ50) has a unit root Exogenous: None
Lag Length: 0 (Automatic based on SIC, MAXLAG=24)
t-Statistic -42.01825 -2.566239 -1.940998 -1.616582
Prob.* 0.0000
Augmented Dickey-Fuller test statistic Test critical values:
1% level 5% level 10% level
*MacKinnon (1996) one-sided p-values.
第二步，建立回归方程 进行协整检验
建立方程，SZZS = C(1) + C(2)*SZ50，估计结果如下
Dependent Variable: SZZS Method: Least Squares Date: 11/14/12 Time: 16:57 Sample: 1 1814
Included observations: 1814
Variable C SZ50
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
Coefficient 251.0439 1.207469
Std. Error 7.132941 0.003308
t-Statistic 35.19500 364.9603
Prob. 0.0000 0.0000 2650.256 1017.397 12.37863 12.38470 133196.0 0.000000
0.986579 Mean dependent var 0.986571 S.D. dependent var 117.8988 Akaike info criterion 25187049 Schwarz criterion -11225.42 F-statistic 0.013360 Prob(F-statistic)
生成残差序列，对残差进行单位根检验： 残差图
单位根检验结果表明，残差序列是平稳的。因此，上证50指数与上证综合指数存
在协整关系。
Null Hypothesis: RESID01 has a unit root Exogenous: None
Lag Length: 1 (Automatic based on SIC, MAXLAG=24)
t-Statistic -2.931306 -2.566239 -1.940998 -1.616582
Prob.* 0.0033
Augmented Dickey-Fuller test statistic Test critical values:
1% level 5% level 10% level
*MacKinnon (1996) one-sided p-values.
第三步，误差修正模型
建立误差修正模型 D(SZZS) = C(1) + C(2)*D(SZ50) + C(3)*ECM(-1)
Dependent Variable: D(SZZS) Method: Least Squares Date: 11/14/12 Time: 19:42 Sample (adjusted): 2 1814
Included observations: 1813 after adjustments
Variable C D(SZ50) ECM(-1)
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
Coefficient -0.002223 1.186166 -0.006198
Std. Error 0.318894 0.006962 0.002709
t-Statistic -0.006970 170.3710 -2.287863
Prob. 0.9944 0.0000 0.0223 0.563177 56.04263 8.056365 8.065471 14530.57 0.000000
0.941369 Mean dependent var 0.941304 S.D. dependent var 13.57755 Akaike info criterion 333673.0 Schwarz criterion -7300.095 F-statistic 1.643504 Prob(F-statistic)
常数项C不显著，剔除后建立误差修正模型 D(SZZS) = C(1)*D(SZ50) + C(2)*ECM(-1)
Dependent Variable: D(SZZS) Method: Least Squares Date: 11/14/12 Time: 19:43 Sample (adjusted): 2 1814
Included observations: 1813 after adjustments
Variable D(SZ50) ECM(-1)
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood
Coefficient 1.186166 -0.006198
Std. Error 0.006960 0.002708
t-Statistic 170.4272 -2.288494
Prob. 0.0000 0.0222 0.563177 56.04263 8.055262 8.061332 1.643504
0.941369 Mean dependent var 0.941337 S.D. dependent var 13.57380 Akaike info criterion 333673.0 Schwarz criterion -7300.095 Durbin-Watson stat
残差的相关图如下
显然不是白噪声，因此
继续建立误差修正模型 DSZZS = C(1)*DSZ50 + C(2)*DSZZS(-1) + C(3)*DSZ50(-1) + C(4)*ECM(-1)
Dependent Variable: DSZZS Method: Least Squares Date: 11/14/12 Time: 21:21 Sample (adjusted): 3 1814
Included observations: 1812 after adjustments
Variable DSZ50 DSZZS(-1) DSZ50(-1) ECM(-1)
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood
Coefficient 1.191449 0.181502 -0.203319 -0.007768
Std. Error 0.006881 0.023242 0.028379 0.002672
t-Statistic 173.1479 7.809306 -7.164428 -2.906926
Prob. 0.0000 0.0000 0.0000 0.0037 0.565386 56.05803 8.023269 8.035415 2.002233
0.943372 Mean dependent var 0.943278 S.D. dependent var 13.35103 Akaike info criterion 322275.9 Schwarz criterion -7265.082 Durbin-Watson stat
2
显然，此模型调整的最大，同时AIC、SC最小。
且其残差检验显示为白噪声（见最下表），所以此模型最优。
1短从上述估计的模型中，我们可以看到上证指数的短期变动分为四部分：○2上一期上证指数的影响，3上一期上证50指数的影期上证50指数变动的影响，○○4由于前一期上证指数偏离长期均衡关系（响，○
）的影响。
误差修正模型残差检验：
协整与误差修正模型
数据来源于wind。
上证50指数和上证综合指数的协整与误差修正模型（日收盘价，2005.6. 6-2012.11.14，共1814组数据）
第一步，检验上证50指数与上证综合指数的平稳性
单位根检验：
结果表明，上证综合指数是非平稳时间序列
Null Hypothesis: SZZS has a unit root Exogenous: Constant
Lag Length: 0 (Automatic based on SIC, MAXLAG=24)
t-Statistic -1.638128 -3.433755 -2.862931 -2.567557
Prob.* 0.4628
Augmented Dickey-Fuller test statistic Test critical values:
1% level 5% level 10% level
*MacKinnon (1996) one-sided p-values.
但一阶差分之后变成平稳的，如下
Null Hypothesis: D(SZZS) has a unit root Exogenous: None
Lag Length: 0 (Automatic based on SIC, MAXLAG=24)
t-Statistic
Prob.*
Augmented Dickey-Fuller test statistic Test critical values:
1% level 5% level 10% level
-42.20904 -2.566239 -1.940998 -1.616582
0.0001
*MacKinnon (1996) one-sided p-values.
下面检验结果表明，上证50指数也是非平稳时间序列
Null Hypothesis: SZ50 has a unit root Exogenous: Constant
Lag Length: 0 (Automatic based on SIC, MAXLAG=24)
t-Statistic -1.610216 -3.433755 -2.862931 -2.567557
Prob.* 0.4771
Augmented Dickey-Fuller test statistic Test critical values:
1% level 5% level 10% level
*MacKinnon (1996) one-sided p-values.
但一阶差分后，变成平稳的，如下
Null Hypothesis: D(SZ50) has a unit root Exogenous: None
Lag Length: 0 (Automatic based on SIC, MAXLAG=24)
t-Statistic -42.01825 -2.566239 -1.940998 -1.616582
Prob.* 0.0000
Augmented Dickey-Fuller test statistic Test critical values:
1% level 5% level 10% level
*MacKinnon (1996) one-sided p-values.
第二步，建立回归方程 进行协整检验
建立方程，SZZS = C(1) + C(2)*SZ50，估计结果如下
Dependent Variable: SZZS Method: Least Squares Date: 11/14/12 Time: 16:57 Sample: 1 1814
Included observations: 1814
Variable C SZ50
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
Coefficient 251.0439 1.207469
Std. Error 7.132941 0.003308
t-Statistic 35.19500 364.9603
Prob. 0.0000 0.0000 2650.256 1017.397 12.37863 12.38470 133196.0 0.000000
0.986579 Mean dependent var 0.986571 S.D. dependent var 117.8988 Akaike info criterion 25187049 Schwarz criterion -11225.42 F-statistic 0.013360 Prob(F-statistic)
生成残差序列，对残差进行单位根检验： 残差图
单位根检验结果表明，残差序列是平稳的。因此，上证50指数与上证综合指数存
在协整关系。
Null Hypothesis: RESID01 has a unit root Exogenous: None
Lag Length: 1 (Automatic based on SIC, MAXLAG=24)
t-Statistic -2.931306 -2.566239 -1.940998 -1.616582
Prob.* 0.0033
Augmented Dickey-Fuller test statistic Test critical values:
1% level 5% level 10% level
*MacKinnon (1996) one-sided p-values.
第三步，误差修正模型
建立误差修正模型 D(SZZS) = C(1) + C(2)*D(SZ50) + C(3)*ECM(-1)
Dependent Variable: D(SZZS) Method: Least Squares Date: 11/14/12 Time: 19:42 Sample (adjusted): 2 1814
Included observations: 1813 after adjustments
Variable C D(SZ50) ECM(-1)
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
Coefficient -0.002223 1.186166 -0.006198
Std. Error 0.318894 0.006962 0.002709
t-Statistic -0.006970 170.3710 -2.287863
Prob. 0.9944 0.0000 0.0223 0.563177 56.04263 8.056365 8.065471 14530.57 0.000000
0.941369 Mean dependent var 0.941304 S.D. dependent var 13.57755 Akaike info criterion 333673.0 Schwarz criterion -7300.095 F-statistic 1.643504 Prob(F-statistic)
常数项C不显著，剔除后建立误差修正模型 D(SZZS) = C(1)*D(SZ50) + C(2)*ECM(-1)
Dependent Variable: D(SZZS) Method: Least Squares Date: 11/14/12 Time: 19:43 Sample (adjusted): 2 1814
Included observations: 1813 after adjustments
Variable D(SZ50) ECM(-1)
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood
Coefficient 1.186166 -0.006198
Std. Error 0.006960 0.002708
t-Statistic 170.4272 -2.288494
Prob. 0.0000 0.0222 0.563177 56.04263 8.055262 8.061332 1.643504
0.941369 Mean dependent var 0.941337 S.D. dependent var 13.57380 Akaike info criterion 333673.0 Schwarz criterion -7300.095 Durbin-Watson stat
残差的相关图如下
显然不是白噪声，因此
继续建立误差修正模型 DSZZS = C(1)*DSZ50 + C(2)*DSZZS(-1) + C(3)*DSZ50(-1) + C(4)*ECM(-1)
Dependent Variable: DSZZS Method: Least Squares Date: 11/14/12 Time: 21:21 Sample (adjusted): 3 1814
Included observations: 1812 after adjustments
Variable DSZ50 DSZZS(-1) DSZ50(-1) ECM(-1)
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood
Coefficient 1.191449 0.181502 -0.203319 -0.007768
Std. Error 0.006881 0.023242 0.028379 0.002672
t-Statistic 173.1479 7.809306 -7.164428 -2.906926
Prob. 0.0000 0.0000 0.0000 0.0037 0.565386 56.05803 8.023269 8.035415 2.002233
0.943372 Mean dependent var 0.943278 S.D. dependent var 13.35103 Akaike info criterion 322275.9 Schwarz criterion -7265.082 Durbin-Watson stat
2
显然，此模型调整的最大，同时AIC、SC最小。
且其残差检验显示为白噪声（见最下表），所以此模型最优。
1短从上述估计的模型中，我们可以看到上证指数的短期变动分为四部分：○2上一期上证指数的影响，3上一期上证50指数的影期上证50指数变动的影响，○○4由于前一期上证指数偏离长期均衡关系（响，○
）的影响。
误差修正模型残差检验：