计量经济学第二版第四章答案.docx
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计量经济学第二版第四章答案.docx
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计量经济学第二版第四章答案
4.1
(1)存在
。
因为
当
之间的相关系数为零时,离差形式的
有
同理有:
(2)
因为
,且
,
由于
,则
则
(3)存在
。
因为
当
时,
同理,有
4.3
(1)建立中国商品进口额为Y与国内生产总值x1、居民消费价格指数x2得回归模型
估计模型参数,结果为
DependentVariable:
LNY
Method:
LeastSquares
Date:
05/16/12Time:
19:
15
Sample:
19852007
Includedobservations:
23
Variable
Coefficient
Std.Error
t-Statistic
Prob.
C
-3.
0.
-9.
0.0000
LNX1
1.
0.
17.96703
0.0000
LNX2
-1.
0.
-4.
0.0001
R-squared
0.
Meandependentvar
9.
AdjustedR-squared
0.
S.D.dependentvar
1.
S.E.ofregression
0.
Akaikeinfocriterion
-1.
Sumsquaredresid
0.
Schwarzcriterion
-1.
Loglikelihood
18.10482
F-statistic
1275.093
Durbin-Watsonstat
0.
Prob(F-statistic)
0.
参数估计结果如下:
(2))数据中有多重共线性,居民消费价格指数的回归系数的符号不能进行合理的经济意义解释,且其简单相关系数呈现正向变动。
(3)
DependentVariable:
LNY
Method:
LeastSquares
Date:
05/16/12Time:
19:
17
Sample:
19852007
Includedobservations:
23
Variable
Coefficient
Std.Error
t-Statistic
Prob.
C
-4.
0.
-10.64579
0.0000
LNX1
1.
0.
34.62222
0.0000
R-squared
0.
Meandependentvar
9.
AdjustedR-squared
0.
S.D.dependentvar
1.
S.E.ofregression
0.
Akaikeinfocriterion
-0.
Sumsquaredresid
0.
Schwarzcriterion
-0.
Loglikelihood
8.
F-statistic
1198.698
Durbin-Watsonstat
0.
Prob(F-statistic)
0.
DependentVariable:
LNY
Method:
LeastSquares
Date:
05/16/12Time:
19:
18
Sample:
19852007
Includedobservations:
23
Variable
Coefficient
Std.Error
t-Statistic
Prob.
C
-5.
1.
-4.
0.0003
LNX2
2.
0.
11.68091
0.0000
R-squared
0.
Meandependentvar
9.
AdjustedR-squared
0.
S.D.dependentvar
1.
S.E.ofregression
0.
Akaikeinfocriterion
1.
Sumsquaredresid
4.
Schwarzcriterion
1.
Loglikelihood
-14.57192
F-statistic
136.4437
Durbin-Watsonstat
0.
Prob(F-statistic)
0.
DependentVariable:
LNX1
Method:
LeastSquares
Date:
05/16/12Time:
19:
19
Sample:
19852007
Includedobservations:
23
Variable
Coefficient
Std.Error
t-Statistic
Prob.
C
-1.
0.
-1.
0.0636
LNX2
2.
0.
16.81400
0.0000
R-squared
0.
Meandependentvar
10.87007
AdjustedR-squared
0.
S.D.dependentvar
1.
S.E.ofregression
0.
Akaikeinfocriterion
0.
Sumsquaredresid
1.
Schwarzcriterion
0.
Loglikelihood
-2.
F-statistic
282.7107
Durbin-Watsonstat
0.
Prob(F-statistic)
0.
单方程拟合效果都很好,回归系数显著,可决系数较高,GDP和CPI对进口分别有显著的单一影响,在这两个变量同时引入模型时影响方向发生了改变;GDP对CPI进行回归分析,回归系数显著,判定系数较高,说明GDP和CPI有很强的线性关系,这正是原模型多重共线性的原因。
(4)如果仅仅是作预测,可以不在意这种多重共线性,但如果是进行结构分析,还是应该引起注意。
4.6
(1)建立对数线性多元回归模型,引入全部变量建立对数线性多元回归模型如下:
变量对数线性多元回归,结果为:
DependentVariable:
LNY
Method:
LeastSquares
Date:
05/16/12Time:
19:
29
Sample:
19852007
Includedobservations:
23
Variable
Coefficient
Std.Error
t-Statistic
Prob.
C
3.
2.
1.
0.2228
LNX1
11.83820
2.
5.
0.0001
LNX2
-11.33780
1.
-5.
0.0000
LNX3
-0.
0.
-0.
0.6132
LNX4
0.
0.
1.
0.1688
LNX5
-0.
0.
-1.
0.1042
LNX6
0.
0.
0.
0.4681
LNX7
1.
0.
2.
0.0193
R-squared
0.
Meandependentvar
11.78641
AdjustedR-squared
0.
S.D.dependentvar
0.
S.E.ofregression
0.
Akaikeinfocriterion
-3.
Sumsquaredresid
0.
Schwarzcriterion
-3.
Loglikelihood
51.17824
F-statistic
350.8771
Durbin-Watsonstat
1.
Prob(F-statistic)
0.
从修正的可决系数和F统计量可以看出,全部变量对数线性多元回归整体对样本拟合很好,,各变量联合起来对能源消费影响显著。
可是其中的lnX4、lnX6对lnY影响不显著,而且lnX2、lnX3、lnX5的参数为负值,在经济意义上不合理。
所以这样的回归结果并不理想。
(2)解释变量国民总收入(亿元)X1(代表收入水平)、国内生产总值(亿元)X2(代表经济发展水平)、工业增加值(亿元)X3、建筑业增加值(亿元)X4、交通运输邮电业增加值(亿元)X5(代表产业发展水平及产业结构)、人均生活电力消费(千瓦小时)X6(代表人民生活水平提高)、能源加工转换效率(%)X7(代表能源转换技术)等很可能线性相关,计算相关系数如下
变量
LNX1
LNX2
LNX3
LNX4
LNX5
LNX6
LNX7
LNX1
1
0.
0.
0.
0.
0.99717
0.
LNX2
0.
1
0.
0.
0.
0.
0.
LNX3
0.
0.
1
0.
0.
0.
0.71606
LNX4
0.
0.
0.
1
0.
0.
0.
LNX5
0.
0.
0.
0.
1
0.
0.
LNX6
0.99717
0.
0.
0.
0.
1
0.
LNX7
0.
0.
0.71606
0.
0.
0.
1
可以看出lnx1与lnx2、lnx3、lnx4、lnx5、lnx6之间高度相关,许多相关系数高于0.900以上。
如果决定用表中全部变量作为解释变量,很可能会出现严重多重共线性问题。
(3)因为存在多重共线性,解决方法如下:
DependentVariable:
Y
Method:
LeastSquares
Date:
05/16/12Time:
19:
49
Sample:
19852007
Includedobservations:
23
Variable
Coefficient
Std.Error
t-Statistic
Prob.
C
-76917.33
.4
-0.
0.4671
X1
15.23223
4.
3.
0.0052
X2
-15.90504
4.
-3.
0.0029
X3
-2.
3.
-0.
0.4817
X4
26.26439
11.12634
2.
0.0322
X5
0.
3.
0.
0.9840
X6
890.4204
364.5072
2.
0.0274
X7
2155.185
1498.804
1.
0.1710
R-squared
0.
Meandependentvar
.9
AdjustedR-squared
0.
S.D.dependentvar
51806.33
S.E.ofregression
6477.323
Akaikeinfocriterion
20.65821
Sumsquaredresid
6.29E+08
Schwarzcriterion
21.05316
Loglikelihood
-229.5694
F-statistic
198.9049
Durbin-Watsonstat
1.
Prob(F-statistic)
0.
由图可以看出还是有严重多重共线性。
我会采用逐步回归的办法,去检验和解决多重共线性问题:
DependentVariable:
Y
Method:
LeastSquares
Date:
05/16/12Time:
19:
59
Sample:
19852007
Includedobservations:
23
Variable
Coefficient
Std.Error
t-Statistic
Prob.
C
79949.57
2951.120
27.09126
0.0000
X1
0.
0.
26.31049
0.0000
R-squared
0.
Meandependentvar
.9
AdjustedR-squared
0.
S.D.dependentvar
51806.33
S.E.ofregression
9098.624
Akaikeinfocriterion
21.15258
Sumsquaredresid
1.74E+09
Schwarzcriterion
21.25131
Loglikelihood
-241.2546
F-statistic
692.2419
Durbin-Watsonstat
0.
Prob(F-statistic)
0.
DependentVariable:
Y
Method:
LeastSquares
Date:
05/16/12Time:
19:
57
Sample:
19852007
Includedobservations:
23
Variable
Coefficient
Std.Error
t-Statistic
Prob.
C
79577.18
3085.516
25.79056
0.0000
X2
0.
0.
25.24391
0.0000
R-squared
0.
Meandependentvar
.9
AdjustedR-squared
0.
S.D.dependentvar
51806.33
S.E.ofregression
9471.027
Akaikeinfocriterion
21.23280
Sumsquaredresid
1.88E+09
Schwarzcriterion
21.33154
Loglikelihood
-242.1772
F-statistic
637.2550
Durbin-Watsonstat
0.
Prob(F-statistic)
0.
DependentVariable:
Y
Method:
LeastSquares
Date:
05/16/12Time:
19:
59
Sample:
19852007
Includedobservations:
23
Variable
Coefficient
Std.Error
t-Statistic
Prob.
C
81615.09
2696.634
30.26555
0.0000
X3
1.
0.
28.34793
0.0000
R-squared
0.
Meandependentvar
.9
AdjustedR-squared
0.
S.D.dependentvar
51806.33
S.E.ofregression
8461.964
Akaikeinfocriterion
21.00749
Sumsquaredresid
1.50E+09
Schwarzcriterion
21.10623
Loglikelihood
-239.5862
F-statistic
803.6049
Durbin-Watsonstat
0.
Prob(F-statistic)
0.
DependentVariable:
Y
Method:
LeastSquares
Date:
05/16/12Time:
19:
59
Sample:
19852007
Includedobservations:
23
Variable
Coefficient
Std.Error
t-Statistic
Prob.
C
79251.87
3030.263
26.15346
0.0000
X4
13.21408
0.
25.79385
0.0000
R-squared
0.
Meandependentvar
.9
AdjustedR-squared
0.
S.D.dependentvar
51806.33
S.E.ofregression
9275.342
Akaikeinfocriterion
21.19105
Sumsquaredresid
1.81E+09
Schwarzcriterion
21.28979
Loglikelihood
-241.6971
F-statistic
665.3228
Durbin-Watsonstat
0.
Prob(F-statistic)
0.
DependentVariable:
Y
Method:
LeastSquares
Date:
05/16/12Time:
20:
00
Sample:
19852007
Includedobservations:
23
Variable
Coefficient
Std.Error
t-Statistic
Prob.
C
82253.98
5537.916
14.85288
0.0000
X5
10.92177
0.
13.47603
0.0000
R-squared
0.
Meandependentvar
.9
AdjustedR-squared
0.
S.D.dependentvar
51806.33
S.E.ofregression
17071.46
Akaikeinfocriterion
22.41115
Sumsquaredresid
6.12E+09
Schwarzcriterion
22.50988
Loglikelihood
-255.7282
F-statistic
181.6035
Durbin-Watsonstat
0.
Prob(F-statistic)
0.
DependentVariable:
Y
Method:
LeastSquares
Date:
05/16/12Time:
20:
00
Sample:
19852007
Includedobservations:
23
Variable
Coefficient
Std.Error
t-Statistic
Prob.
C
66876.70
3935.724
16.99222
0.0000
X6
679.2253
30.41199
22.33413
0.0000
R-squared
0.
Meandependentvar
.9
AdjustedR-squared
0.
S.D.dependentvar
51806.33
S.E.ofregression
10657.87
Akaikeinfocriterion
21.46893
Sumsquaredresid
2.39E+09
Schwarzcriterion
21.56766
Loglikelihood
-244.8926
F-statistic
498.8135
Durbin-Watsonstat
0.
Prob(F-statistic)
0.
DependentVariable:
Y
Method:
LeastSquares
Date:
05/16/12Time:
20:
00
Sample:
19852007
Includedobservations:
23
Variable
Coefficient
Std.Error
t-Statistic
Prob.
C
-.
.8
-4.
0.0004
X7
19372.59
4118.642
4.
0.0001
R-squared
0.
Meandependentvar
.9
AdjustedR-squared
0.
S.D.dependentvar
51806.33
S.E.ofregression
37002.72
Akaikeinfocriterion
23.95831
Sumsquaredresid
2.88E+10
Schwarzcriterion
24.05705
Loglikelihood
-273.5206
F-statistic
22.12419
Durbin-Watsonstat
0.
Prob(F-statistic)
0.
分别作一元回归得到:
变量
X1
X2
X3
X4
X5
X6
X7
参数估计值
0.7345
0.7365
1.7332
13.2141
10.9218
679.2253
19372.59
t统计量
26.3105
25.2439
28.3479
25.7939
13.4760
22.3341
4.7036
0.9706
0.9681
0.9745
0.9694
0.8963
0.9596
0.5130
0.9692
0.9666
0.9733
0.9679
0.8914
0.9577
0.4898
以X1为基础加入其他变量,结果为:
DependentVariable:
Y
Method:
LeastSquares
Date:
05/16/12Time:
20:
01
Sample:
19852007
Includedobservations:
23
Variable
Coefficient
Std.Error
t-Statistic
Prob.
C
83843.32
2740.212
30.59738
0.0000
X1
7.
1.
3.
0.0019
X2
-6.
1.
-3.
0.0044
R-squared
0.
Meandependentvar
.9
AdjustedR-squared
0.
S.D.dependentvar
51806.33
S.E.ofregression
7574.958
Akaikeinfocriterion
20.82419
Sumsquaredresid
1.15E+09
Schwarzcriterion
20.97230
Loglikelihood
-236.4782
F-statistic
504.5147
Durbin-Watsonstat
0.
Prob(F-statistic)
0.
DependentVariable:
Y
Method:
LeastSquares
Date:
05/16/12Time:
20:
02
Sample:
19852007
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