第五章练习题及参考解答12

第五章练习题及参考解答 5.1 设消费函数为

i i i i u X X Y +++=33221βββ

式中，i Y 为消费支出；i X 2为个人可支配收入；i X 3为个人的流动资产；i u 为随机误差项，

并且2

22)(,0)(i i i X u Var u E σ==（其中2σ为常数）。试回答以下问题：

(1)选用适当的变换修正异方差，要求写出变换过程；

(2)写出修正异方差后的参数估计量的表达式。

【练习题5.1参考解答】

（1）因为2

2()i i f X X =，所以取221

i i

W X =

，用2i W 乘给定模型两端，得 312322221i i i

i

i i i Y X u X X X X βββ=+++ 上述模型的随机误差项的方差为一固定常数，即

22221

(

)()i i i i

u Var Var u X X σ==

（2）根据加权最小二乘法，可得修正异方差后的参数估计式为

***12233ˆˆˆY X X βββ=-- ()()()()()()()

***2****

2223232232

2

*2*2**2223223ˆi i i i i i i i i i i i i i i i i i W y x W x W y x W x x W x W x W x x β-=

-∑∑∑∑∑∑∑

()()()()

()()()

***2****2322222233

2

*2*2**

2223223ˆi i i i i i i i i i i i i i i i i i W y x W x W y x W x x W

x W x W x x β-=

-∑∑∑∑∑∑∑

其中

22232***23222,

,

i i

i i

i i

i

i

i

W X W X

W Y X X Y W

W

W

=

=

=

∑∑∑∑∑∑

****

**222333

i i i i i x X X x X X y Y Y =-=-=-

5.2 对于第三章练习题3.3家庭书刊消费与家庭收入及户主受教育年数关系的分析，进一步作以下分析:

1）判断模型123i i i i Y X T u βββ=+++是否存在异方差性。 2。如果模型存在异方差性，应怎样去估计其参数？

3）对比分析的结果，你对第三章练习题3.3的结论有什么评价? 【练习题5.2参考解答】

建议学生自己独立完成 答案：（1）根据WHITE 检验,利用P 值决策，P=0.0340<0.05,拒绝原假设，存在异方差。

Heteroskedasticity Test: White

F-statistic

4.868935 Prob. F(5,12)

0.0115 Obs*R-squared 12.05690 Prob. Chi-Square(5) 0.0340 Scaled explained SS

14.17359 Prob. Chi-Square(5) 0.0145

（2）利用加权最小二乘法去修正，并且估计参数。

Heteroskedasticity Test: White

F-statistic

4.868935 Prob. F(5,12)

0.0115 Obs*R-squared 12.05690 Prob. Chi-Square(5) 0.0340 Scaled explained SS

14.17359 Prob. Chi-Square(5) 0.0145

Test Equation:

Dependent Variable: RESID^2 Method: Least Squares Date: 06/06/15 Time: 21:01

Sample: 1 18

Included observations: 18

Variable Coefficient Std. Error t-Statistic Prob. C 13145.36 12596.22 1.043596 0.3172 T -4134.531 2338.281 -1.768192 0.1024 T^2 381.1302 121.1794 3.145173 0.0085 T*X -2.208704 1.122873 -1.967012 0.0727 X 11.05181 8.449834 1.307932 0.2154 X^2

0.003059

0.004393

0.696318

0.4995

R-squared 0.669828 Mean dependent var 3082.837 Adjusted R-squared 0.532256 S.D. dependent var

5836.878

S.E. of regression 3991.946 Akaike info criterion 19.68315 Sum squared resid 1.91E+08 Schwarz criterion 19.97994 Log likelihood -171.1483 Hannan-Quinn criter. 19.72407 F-statistic 4.868935 Durbin-Watson stat 2.315725 Prob(F-statistic) 0.011547

分析过程如下试一试不同的权重

Dependent Variable: Y

Method: Least Squares

Date: 06/06/15 Time: 22:01

Sample: 1 18

Included observations: 18

Weighting series: X

Weight type: Inverse standard deviation (EViews default scaling)

Variable Coefficient Std. Error t-Statistic Prob.

X 0.062345 0.028715 2.171170 0.0464

T 54.31872 5.143047 10.56158 0.0000

C -20.92859 49.21098 -0.425283 0.6767

Weighted Statistics

R-squared 0.962642 Mean dependent var 824.4223 Adjusted R-squared 0.957661 S.D. dependent var 567.0282 S.E. of regression 60.70470 Akaike info criterion 11.20093 Sum squared resid 55275.92 Schwarz criterion 11.34933 Dependent Variable: Y

Method: Least Squares

Date: 06/06/15 Time: 22:01

Sample: 1 18

Included observations: 18

Weighting series: X^2

Weight type: Inverse standard deviation (EViews default scaling)

Variable Coefficient Std. Error t-Statistic Prob.

X 0.050647 0.029772 1.701181 0.1095

T 53.42191 5.325817 10.03075 0.0000

C 25.26657 50.52396 0.500091 0.6243

Weighted Statistics

R-squared 0.966601 Mean dependent var 896.1108 Adjusted R-squared 0.962148 S.D. dependent var 973.2092 S.E. of regression 60.92597 Akaike info criterion 11.20821 Sum squared resid 55679.60 Schwarz criterion 11.35660