计量经济学模型分析方法.doc

计量经济学上机模型分析方法总结
一、随机误差项的异方差问题的检验与修正
模型一：
Dependent Variable: LOG(Y)

Method: Least Squares

Date: 07/29/12 Time: 09:03

Sample: 1 31

Included observations: 31

Variable
Coefficient
Std. Error
t-Statistic
Prob.  

C
1.602528
0.860978
1.861288
0.0732
LOG(X1)
0.325416
0.103769
3.135955
0.0040
LOG(X2)
0.507078
0.048599
10.43385
0.0000

R-squared
0.796506
    Mean dependent var
7.448704
Adjusted R-squared
0.781971
    S.D. dependent var
0.364648
S.E. of regression
0.170267
    Akaike info criterion
-0.611128
Sum squared resid
0.811747
    Schwarz criterion
-0.472355
Log likelihood
12.47249
    F-statistic
54.79806
Durbin-Watson stat
1.964720
    Prob(F-statistic)
0.000000

（一）异方差的检验
1、GQ检验法
模型二：
Dependent Variable: LOG(Y)

Method: Least Squares

Date: 07/29/12 Time: 09:19

Sample: 1 12

Included observations: 12

Variable
Coefficient
Std. Error
t-Statistic
Prob.  

C
3.744626
1.191113
3.143804
0.0119
LOG(X1)
0.344369
0.082999
4.149077
0.0025
LOG(X2)
0.168904
0.118844
1.421228
0.1890

R-squared
0.669065
    Mean dependent var
7.239161
Adjusted R-squared
0.595524
    S.D. dependent var
0.133581
S.E. of regression
0.084955
    Akaike info criterion
-1.881064
Sum squared resid
0.064957
    Schwarz criterion
-1.759837
Log likelihood
14.28638
    F-statistic
9.097834
Durbin-Watson stat
1.810822
    Prob(F-statistic)
0.006900

模型三：
Dependent Variable: LOG(Y)

Method: Least Squares

Date: 07/29/12 Time: 09:20

Sample: 20 31

Included observations: 12

Variable
Coefficient
Std. Error
t-Statistic
Prob.  

C
-0.353381
1.607461
-0.219838
0.8309
LOG(X1)
0.210898
0.158220
1.332942
0.2153
LOG(X2)
0.856522
0.108601
7.886856
0.0000

R-squared
0.878402
    Mean dependent var
7.769851
Adjusted R-squared
0.851381
    S.D. dependent var
0.390363
S.E. of regression
0.150490
    Akaike info criterion
-0.737527
Sum squared resid
0.203824
    Schwarz criterion
-0.616301
Log likelihood
7.425163
    F-statistic
32.50732
Durbin-Watson stat
2.123203
    Prob(F-statistic)
0.000076

进行模型二和模型三两次回归，目的仅是得到出去中间7个样本点以后前后各12个样本点的残差平方和RSS1和RSS2，然后用较大的RSS除以较小的RSS即可求出F统计量值进行显著性检验。
2、怀特检验法（White）
模型一的怀特残差检验结果：
White Heteroskedasticity Test:

F-statistic
4.920995
    Probability
0.004339
Obs*R-squared
13.35705
    Probability
0.009657

Test Equation:

Dependent Variable: RESID^2

Method: Least Squares

Date: 05/29/13 Time: 09:04

Sample: 1 31

Included observations: 31

Variable
Coefficient
Std. Error
t-Statistic
Prob.  

C
3.982137
2.882851
1.381319
0.1789
LOG(X1)
-0.579289
0.916069
-0.632364
0.5327
(LOG(X1))^2
0.041839
0.066866
0.625710
0.5370
LOG(X2)
-0.563656
0.203228
-2.773514
0.0101
(LOG(X2))^2
0.040280
0.013879
2.902173
0.0075

R-squared
0.430873
    Mean dependent var
0.026185
Adjusted R-squared
0.343315
    S.D. dependent var
0.038823
S.E. of regression
0.031460
    Akaike info criterion
-3.933482
Sum squared resid
0.025734
    Schwarz criterion
-3.702194
Log likelihood
65.96898
    F-statistic
4.920995
Durbin-Watson stat
1.526222
    Prob(F-statistic)
0.004339

一方面，根据上面的Obs*R2=31*0.430873=13.35705＞χ2(4)，说明存在显著的异方差问题；另一方面，根据下面的辅助回归模型可以看出LOG(X2) 与(LOG(X2))^2均通过了t检验，说明异方差的形式可以用LOG(X2) 与(LOG(X2))^2的线性组合表示，权变量可以简单确定为1/LOG(X2)。
（二）加权最小二乘法（WLS）修正
1、方法原理：具体参见教材。
2、回归结果分析
模型四：
Dependent Variable: LOG(Y)

Method: Least Squares

Date: 07/29/12 Time: 09:06

Sample: 1 31

Included observations: 31

Weighting series: 1/LOG(X2)

Variable
Coefficient
Std. Error
t-Statistic
Prob.  

C
1.478085
0.817610
1.807811
0.0814
LOG(X1)
0.377915
0.096925
3.899044
0.0006
LOG(X2)
0.473471
0.048398
9.782864
0.0000

Weighted Statistics

R-squared
0.872646
    Mean dependent var
7.423264
Adjusted R-squared
0.863550
    S.D. dependent var
0.436598
S.E. of regression
0.161276
    Akaike info criterion
-0.719639
Sum squared resid
0.728274
    Schwarz criterion
-0.580866
Log likelihood
14.15440
    F-statistic
49.27256
Durbin-Watson stat
2.036239
    Prob(F-statistic)
0.000000

Unweighted Statistics

R-squared
0.789709
    Mean dependent var
7.448704
Adjusted R-squared
0.774688
    S.D. dependent var
0.364648
S.E. of regression
0.173088
    Sum squared resid
0.838862
Durbin-Watson stat
2.028211

加权修正以后的模型四怀特检验结果如下：
White Heteroskedasticity Test:

F-statistic
6.555091
    Probability
0.000870
Obs*R-squared
15.56541
    Probability
0.003661

可以看出并没有消除异方差性，加权修正无效。
下面采用1/abs(e)权变量进行WLS回归，结果如下：
模型五：
Dependent Variable: LOG(Y)

Method: Least Squares

Date: 07/29/12 Time: 09:10

Sample: 1 31

Included observations: 31

Weighting series: 1/ABS(E)

Variable
Coefficient
Std. Error
t-Statistic
Prob.  

C
1.227929
0.297268
4.130708
0.0003
LOG(X1)
0.375748
0.056830
6.611734
0.0000
LOG(X2)
0.510120
0.017781
28.68847
0.0000

Weighted Statistics

R-squared
0.999990
    Mean dependent var
7.558578
Adjusted R-squared
0.999989
    S.D. dependent var
12.31758
S.E. of regression
0.041062
    Akaike info criterion
-3.455703
Sum squared resid
0.047210
    Schwarz criterion
-3.316930
Log likelihood
56.56339
    F-statistic
1960.131
Durbin-Watson stat
2.487309
    Prob(F-statistic)
0.000000

Unweighted Statistics

R-squared
0.794514
    Mean dependent var
7.448704
Adjusted R-squared
0.779836
    S.D. dependent var
0.364648
S.E. of regression
0.171099
    Sum squared resid
0.819694
Durbin-Watson stat
2.007122

对加权以后的模型五进行怀特检验如下：
White Heteroskedasticity Test:

F-statistic
0.199645
    Probability
0.936266
Obs*R-squared
0.923778
    Probability
0.921125

可以看出，模型已经不再存在异方差问题，模型五可以作为修正以后的最终模型。
二、随机误差项序列相关性问题的检验与修正
模型一：
Dependent Variable: Y

Method: Least Squares

Date: 07/29/12 Time: 09:48

Sample: 1991 2011

Included observations: 21

Variable
Coefficient
Std. Error
t-Statistic
Prob.  

C
178.9755
55.06421
3.250305
0.0042
X
0.020002
0.001134
17.64157
0.0000

R-squared
0.942463
    Mean dependent var
922.9095
Adjusted R-squared
0.939435
    S.D. dependent var
659.3491
S.E. of regression
162.2653
    Akaike info criterion
13.10673
Sum squared resid
500270.3
    Schwarz criterion
13.20621
Log likelihood
-135.6207
    F-statistic
311.2248
Durbin-Watson stat
0.658849
    Prob(F-statistic)
0.000000

初始回归模型一经济意义合理，统计指标较为理想，但DW值偏低，模型可能存在序列相关性。
（一）序列相关性的检验方法
1、自回归模型检验法

Dependent Variable: E

Method: Least Squares

Date: 07/29/12 Time: 09:49

Sample (adjusted): 1992 2011

Included observations: 20 after adjustments

Variable
Coefficient
Std. Error
t-Statistic
Prob.  

E(-1)
0.717080
0.201852
3.552497
0.0021

R-squared
0.398929
    Mean dependent var
2.801737
Adjusted R-squared
0.398929
    S.D. dependent var
161.7297
S.E. of regression
125.3870
    Akaike info criterion
12.54939
Sum squared resid
298716.2
    Schwarz criterion
12.59918
Log likelihood
-124.4939
    Durbin-Watson stat
1.080741

说明模型一的随机误差项至少存在一阶正序列相关性，结合该自回归模型的DW值为1.08，怀疑存在更高阶的序列相关，继续引入e(-2)如下：
Dependent Variable: E

Method: Least Squares

Date: 07/29/12 Time: 09:49

Sample (adjusted): 1993 2011

Included observations: 19 after adjustments

Variable
Coefficient
Std. Error
t-Statistic
Prob.  

E(-1)
1.094974
0.178768
6.125108
0.0000
E(-2)
-0.815010
0.199977
-4.075513
0.0008

R-squared
0.692885
    Mean dependent var
7.790341
Adjusted R-squared
0.674819
    S.D. dependent var
164.5730
S.E. of regression
93.84710
    Akaike info criterion
12.02051
Sum squared resid
149723.7
    Schwarz criterion
12.11993
Log likelihood
-112.1949
    Durbin-Watson stat
1.945979

由于e(-2)的t检验显著，说明模型一的随机误差项确实存在二阶正序列相关性，结合该二阶自回归模型的DW值为1.95，基本确定不存在更高阶的序列相关。
Breusch-Godfrey Serial Correlation LM Test:

F-statistic
0.888958
    Probability
0.431668
Obs*R-squared
1.998924
    Probability
0.368077

可以看出二阶自回归模型的随机误差项不存在序列相关性，论证了原模型仅存在二阶序列相关。
2、DW检验法
0<DW<dL 存在正自相关(趋近于0)
DL<DW<dU 不能确定
DU<DW<4－dU 无自相关（趋近于2）
3、LM检验法
原理：一方面，根据上面的假设检验结果判断是否存在序列相关性，即根据(n-p)*R2统计量值与卡方检验临界值χ2(P)进行比较，其中n为原模型样本容量，P为选择的滞后阶数，R2为下面辅助回归模型的可决系数。若(n-p)*R2﹥χ2(P)，则拒绝不序列相关的原假设，说明模型存在显著的序列相关性；另一方面，结合下面的辅助回归模型中残差滞后变量是否通过t检验及DW值判断序列相关的具体阶数，方法与上面的自回归模型检验法相同。
选择滞后一阶检验：
Breusch-Godfrey Serial Correlation LM Test:

F-statistic
13.15036
    Probability
0.001931
Obs*R-squared
8.865308
    Probability
0.002906

Test Equation:

Dependent Variable: RESID

Method: Least Squares

Date: 07/29/12 Time: 09:51

Presample missing value lagged residuals set to zero.

Variable
Coefficient
Std. Error
t-Statistic
Prob.  

C
-14.24472
43.18361
-0.329864
0.7453
X
0.000714
0.000907
0.786617
0.4417
RESID(-1)
0.763263
0.210477
3.626342
0.0019

R-squared
0.422158
    Mean dependent var
1.30E-13
Adjusted R-squared
0.357953
    S.D. dependent var
158.1566
S.E. of regression
126.7275
    Akaike info criterion
12.65352
Sum squared resid
289077.4
    Schwarz criterion
12.80274
Log likelihood
-129.8619
    F-statistic
6.575179
Durbin-Watson stat
1.159275
    Prob(F-statistic)
0.007183

说明原模型确实存在一阶序列相关性，结合该辅助回归模型的DW值为1.16，怀疑存在更高阶的序列相关，引入滞后二阶检验如下：
Breusch-Godfrey Serial Correlation LM Test:

F-statistic
20.49152
    Probability
0.000030
Obs*R-squared
14.84303
    Probability
0.000598

Test Equation:

Dependent Variable: RESID

Method: Least Squares

Date: 07/29/12 Time: 09:51

Presample missing value lagged residuals set to zero.

Variable
Coefficient
Std. Error
t-Statistic
Prob.  

C
14.06463
32.40987
0.433961
0.6698
X
-0.000628
0.000742
-0.846303
0.4091
RESID(-1)
1.108488
0.176127
6.293696
0.0000
RESID(-2)
-0.918175
0.226004
-4.062643
0.0008

R-squared
0.706811
    Mean dependent var
1.30E-13
Adjusted R-squared
0.655072
    S.D. dependent var
158.1566
S.E. of regression
92.88633
    Akaike info criterion
12.07027
Sum squared resid
146673.8
    Schwarz criterion
12.26923
Log likelihood
-122.7379
    F-statistic
13.66102
Durbin-Watson stat
1.950263
    Prob(F-statistic)
0.000087

由于e(-2)的t检验显著，说明模型一的随机误差项确实存在二阶正序列相关性，结合该二阶自回归模型的DW值为1.95，基本确定不存在更高阶的序列相关。
当然可以继续引入滞后三阶检验如下：
Breusch-Godfrey Serial Correlation LM Test:

F-statistic
12.85743
    Probability
0.000157
Obs*R-squared
14.84303
    Probability
0.001956

Test Equation:

Dependent Variable: RESID

Method: Least Squares

Date: 07/29/12 Time: 09:52

Presample missing value lagged residuals set to zero.

Variable
Coefficient
Std. Error
t-Statistic
Prob.  

C
14.06467
33.40734
0.421005
0.6794
X
-0.000628
0.000765
-0.820934
0.4237
RESID(-1)
1.108206
0.271327
4.084401
0.0009
RESID(-2)
-0.917559
0.499523
-1.836870
0.0849
RESID(-3)
-0.000601
0.431119
-0.001395
0.9989

R-squared
0.706811
    Mean dependent var
1.30E-13
Adjusted R-squared
0.633514
    S.D. dependent var
158.1566
S.E. of regression
95.74504
    Akaike info criterion
12.16551
Sum squared resid
146673.8
    Schwarz criterion
12.41421
Log likelihood
-122.7379
    F-statistic
9.643071
Durbin-Watson stat
1.950030
    Prob(F-statistic)
0.000363

可以看出并不存在三阶序列相关。
（二）广义差分法修正
1、方法原理
参考教材自己推导二元线性回归模型存在二阶序列相关时的广义差分模型。
2、上机实现结果分析
模型二：
Dependent Variable: Y

Method: Least Squares

Date: 07/29/12 Time: 09:55

Sample (adjusted): 1992 2011

Included observations: 20 after adjustments

Convergence achieved after 8 iterations

Variable
Coefficient
Std. Error
t-Statistic
Prob.  

C
160.0892
182.8917
0.875323
0.3936
X
0.021469
0.003072
6.988975
0.0000
AR(1)
0.730078
0.203352
3.590223
0.0023

R-squared
0.964570
    Mean dependent var
958.0450
Adjusted R-squared
0.960402
    S.D. dependent var
655.9980
S.E. of regression
130.5388
    Akaike info criterion
12.71870
Sum squared resid
289686.3
    Schwarz criterion
12.86806
Log likelihood
-124.1870
    F-statistic
231.4107
Durbin-Watson stat
1.116066
    Prob(F-statistic)
0.000000

Inverted AR Roots
      .73

由于AR(1)通过t检验，说明模型一确实至少存在一阶序列相关，结合DW值为1.12，怀疑存在更高阶序列相关性， LM检验结果如下：
Breusch-Godfrey Serial Correlation LM Test:

F-statistic
6.380262
    Probability
0.009885
Obs*R-squared
9.193288
    Probability
0.010086

Test Equation:

Dependent Variable: RESID

Method: Least Squares

Date: 07/29/12 Time: 09:57

Presample missing value lagged residuals set to zero.

Variable
Coefficient
Std. Error
t-Statistic
Prob.  

C
80.86347
145.2643
0.556665
0.5860
X
-0.003554
0.002602
-1.365556
0.1922
AR(1)
-0.572841
0.437314
-1.309909
0.2099
RESID(-1)
1.029157
0.339541
3.031022
0.0084
RESID(-2)
-0.187923
0.598223
-0.314136
0.7577

R-squared
0.459664
    Mean dependent var
-7.24E-11
Adjusted R-squared
0.315575
    S.D. dependent var
123.4773
S.E. of regression
102.1528
    Akaike info criterion
12.30313
Sum squared resid
156527.8
    Schwarz criterion
12.55207
Log likelihood
-118.0313
    F-statistic
3.190131
Durbin-Watson stat
2.021