ADF Test Statistic -2.090711 1% Critical Value* -4.3260
5% Critical Value -3.2195
10% Critical Value -2.7557
*MacKinnon critical values for rejection of hypothesis of a unit root.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(X1)
Method: Least Squares
Date: 07/01/05 Time: 12:54
Sample(adjusted): 1992 2001
Included observations: 10 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
X1(-1) -0.380698 0.182090 -2.090711 0.0749
D(X1(-1)) 0.919158 0.293195 3.134968 0.0165
C 0.032264 0.016274 1.982530 0.0879
R-squared 0.586641 Mean dependent var -0.003640
Adjusted R-squared 0.468538 S.D. dependent var 0.016227
S.E. of regression 0.011829 Akaike info criterion -5.793128
Sum squared resid 0.000980 Schwarz criterion -5.702353
Log likelihood 31.96564 F-statistic 4.967214
Durbin-Watson stat 1.727237 Prob(F-statistic) 0.045409
ADF Test Statistic -1.774646 1% Critical Value* -4.4613
5% Critical Value -3.2695
10% Critical Value -2.7822
*MacKinnon critical values for rejection of hypothesis of a unit root.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(X2)
Method: Least Squares
Date: 07/01/05 Time: 12:44
Sample(adjusted): 1993 2001
Included observations: 9 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
X2(-1) -0.052738 0.029717 -1.774646 0.1361
D(X2(-1)) 0.833179 0.273405 3.047422 0.0285
D(X2(-2)) -0.505207 0.238098 -2.121839 0.0873
C 8260.113 2639.968 3.128868 0.0260
R-squared 0.811418 Mean dependent var 7699.467
Adjusted R-squared 0.698269 S.D. dependent var 2998.985
S.E. of regression 1647.343 Akaike info criterion 17.95282
Sum squared resid 13568700 Schwarz criterion 18.04047
Log likelihood -76.78768 F-statistic 7.171234
Durbin-Watson stat 2.594088 Prob(F-statistic) 0.029258
ADF Test Statistic -7.584454 1% Critical Value* -4.3260
5% Critical Value -3.2195
10% Critical Value -2.7557
*MacKinnon critical values for rejection of hypothesis of a unit root.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(Y)
Method: Least Squares
Date: 07/01/05 Time: 12:58
Sample(adjusted): 1992 2001
Included observations: 10 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
从表中看出,只y是平稳的,x1、x2都有可能是非平稳序列。
回归如下:
Dependent Variable: Y
Method: Least Squares
Date: 07/01/05 Time: 12:27
Sample: 1990 2001
Included observations: 12
Variable Coefficient Std. Error t-Statistic Prob.
C 4884.729 8538.390 0.572090 0.5813
X1 -195491.6 71636.27 -2.728948 0.0233
X2 1.381533 0.071401 19.34894 0.0000
R-squared 0.983602 Mean dependent var 69465.50
Adjusted R-squared 0.979958 S.D. dependent var 41050.42
S.E. of regression 5811.429 Akaike info criterion 20.38536
Sum squared resid 3.04E+08 Schwarz criterion 20.50659
Log likelihood -119.3122 F-statistic 269.9303
Durbin-Watson stat 1.124471 Prob(F-statistic) 0.000000
ARCH Test:
F-statistic 16.63761 Probability 0.002189
Obs*R-squared 8.261959 Probability 0.016067
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 07/01/05 Time: 12:34
Sample(adjusted): 1992 2001
Included observations: 10 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
C 5963322. 10085707 0.591265 0.5729
RESID^2(-1) -0.626371 0.441135 -1.419908 0.1986
RESID^2(-2) 2.508035 0.436696 5.743201 0.0007
R-squared 0.826196 Mean dependent var 29821600
Adjusted R-squared 0.776538 S.D. dependent var 50360420
S.E. of regression 23806285 Akaike info criterion 37.05212
Sum squared resid 3.97E+15 Schwarz criterion 37.14290
Log likelihood -182.2606 F-statistic 16.63761
Durbin-Watson stat 2.200372 Prob(F-statistic) 0.002189
White Heteroskedasticity Test:
F-statistic 2.999141 Probability 0.097505
Obs*R-squared 7.578148 Probability 0.108312
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 07/01/05 Time: 12:35
Sample: 1990 2001
Included observations: 12
Variable Coefficient Std. Error t-Statistic Prob.
C 1.23E+08 1.87E+08 0.656433 0.5325
X1 -1.11E+09 4.64E+09 -0.240222 0.8170
X1^2 8.20E+09 2.89E+10 0.284316 0.7844
X2 -4697.178 3699.568 -1.269656 0.2448
X2^2 0.050797 0.036053 1.408943 0.2017
R-squared 0.631512 Mean dependent var 25329530
Adjusted R-squared 0.420948 S.D. dependent var 46753534
S.E. of regression 35577322 Akaike info criterion 37.90665
Sum squared resid 8.86E+15 Schwarz criterion 38.10870
Log likelihood -222.4399 F-statistic 2.999141
Durbin-Watson stat 2.585745 Prob(F-statistic) 0.097505
回归结果如下:
Dependent Variable: LOG(Y)
Method: Least Squares
Date: 07/01/05 Time: 13:06
Sample: 1990 2001
Included observations: 12
Variable Coefficient Std. Error t-Statistic Prob.
C -2.912563 0.363818 -8.005561 0.0000
LOG(X1) -0.104478 0.052415 -1.993287 0.0774
LOG(X2) 1.252898 0.036680 34.15767 0.0000
R-squared 0.993932 Mean dependent var 10.93112
Adjusted R-squared 0.992584 S.D. dependent var 0.748034
S.E. of regression 0.064419 Akaike info criterion -2.434484
Sum squared resid 0.037349 Schwarz criterion -2.313257
Log likelihood 17.60690 F-statistic 737.1016
Durbin-Watson stat 1.000566 Prob(F-statistic) 0.000000
ARCH Test:
F-statistic 2.619451 Probability 0.141496
Obs*R-squared 4.280533 Probability 0.117623
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 07/01/05 Time: 13:07
Sample(adjusted): 1992 2001
Included observations: 10 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
C 0.002133 0.002072 1.029565 0.3375
RESID^2(-1) -0.657373 0.634981 -1.035265 0.3350
RESID^2(-2) 1.433592 0.635484 2.255904 0.0587
R-squared 0.428053 Mean dependent var 0.003698
Adjusted R-squared 0.264640 S.D. dependent var 0.005199
S.E. of regression 0.004458 Akaike info criterion -7.744744
Sum squared resid 0.000139 Schwarz criterion -7.653968
Log likelihood 41.72372 F-statistic 2.619451
Durbin-Watson stat 2.362453 Prob(F-statistic) 0.141496
从以上结果看Probability的值,拒绝H0犯错误概率较大,同时残差序列的系数的t值并不显著,应该接受残差序列系数为零的原假设,即为模型不存在异方差. 另一方面,从White检验看,也不存在异方差.
White Heteroskedasticity Test:
F-statistic 1.573377 Probability 0.281511
Obs*R-squared 5.681125 Probability 0.224261
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 07/01/05 Time: 13:07
Sample: 1990 2001
Included observations: 12
Variable Coefficient Std. Error t-Statistic Prob.
C 0.438517 0.928484 0.472294 0.6511
LOG(X1) -0.004873 0.083923 -0.058066 0.9553
(LOG(X1))^2 -0.000283 0.015544 -0.018211 0.9860
LOG(X2) -0.087720 0.174478 -0.502757 0.6306
(LOG(X2))^2 0.004285 0.008235 0.520299 0.6189
R-squared 0.473427 Mean dependent var 0.003112
Adjusted R-squared 0.172528 S.D. dependent var 0.004898
S.E. of regression 0.004455 Akaike info criterion -7.695083
Sum squared resid 0.000139 Schwarz criterion -7.493039
Log likelihood 51.17050 F-statistic 1.573377
Durbin-Watson stat 2.429343 Prob(F-statistic) 0.281511
C LOG(X1) LOG(X2)
C 0.132363 -0.001369 -0.012518
LOG(X1) -0.001369 0.002747 0.000788
LOG(X2) -0.012518 0.000788 0.001345