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5.6 (1)

a)用Eviews模型分析得： Dependent Variable: Y Method: Least Squares Date: 12/24/15 Time: 19:16 Sample: 1978 2011 Included observations: 34

Variable Coefficient Std. Error t-Statistic X 0.746241 0.019120 39.03027 C 92.55422 42.80529 2.162215 R-squared 0.979426 Mean dependent var

Adjusted R-squared 0.978783 S.D. dependent var S.E. of regression 173.1597 Akaike info criterion Sum squared resid 959497.2 Schwarz criterion Log likelihood -222.4566 Hannan-Quinn criter. F-statistic 1523.362 Durbin-Watson stat Prob(F-statistic) 0.000000

得回归模型为：

Y=0.746241 X+92.55422

b)检验是否存在异方差：

①用Goldfeld-Quanadt检验如下：

1）当定义区间为1-13时，由软件分析得： Dependent Variable: Y Method: Least Squares Date: 12/24/15 Time: 19:27 Sample: 1 13 Included observations: 13

Variable Coefficient Std. Error t-Statistic X 0.967839 0.026879 36.00771 C -18.86861 8.963780 -2.104984 R-squared 0.991587 Mean dependent var

Adjusted R-squared 0.990823 S.D. dependent var S.E. of regression 12.17039 Akaike info criterion Sum squared resid 1629.301 Schwarz criterion Log likelihood -49.84742 Hannan-Quinn criter. F-statistic 1296.555 Durbin-Watson stat Prob(F-statistic) 0.000000

2 得∑e1i=1629.301

Prob. 0.0000 0.0382 1295.802 1188.791 13.20333 13.29311 13.23395 1.534491

Prob. 0.0000 0.0591 280.1377 127.0409 7.976527 8.063442 7.958662 1.071505

2）当定义区间为1-13时，由软件分析得：

Dependent Variable: Y Method: Least Squares Date: 12/24/15 Time: 19:34 Sample: 22 34 Included observations: 13

Variable Coefficient Std. Error t-Statistic Prob. X 0.719567 0.058312 12.33998 0.0000 C 179.3950 202.8764 0.884258 0.3955 R-squared 0.932629 Mean dependent var 2496.127

Adjusted R-squared 0.926504 S.D. dependent var 1022.591 S.E. of regression 277.2250 Akaike info criterion 14.22817 Sum squared resid 845390.4 Schwarz criterion 14.31509 Log likelihood -90.48313 Hannan-Quinn criter. 14.21031 F-statistic 152.2752 Durbin-Watson stat 1.658418 Prob(F-statistic) 0.000000

2 得∑e2i=845390.4

3）根据Goldfeld-Quanadt检验，F统计量为： F=∑e2i2 /∑e1i2 =845390.4/ 1629.301=518.8669

在α=0.05水平下，分子分母的自由度均为11，查分布表得临界值F0.05（11,11）=4.47，因为F=518.8669> F0.05（11,11）=4.47，所以拒绝原假设，此检验表明模型存在异方差。

②White检验

用EViews软件分析得：

Heteroskedasticity Test: White

F-statistic 10.36759 Prob. F(2,31)

Obs*R-squared 13.62701 Prob. Chi-Square(2) Scaled explained SS 76.13635 Prob. Chi-Square(2)

Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 12/24/15 Time: 19:56 Sample: 1 34 Included observations: 34

Variable Coefficient Std. Error t-Statistic C 11581.11 26117.11 0.443430 X -27.69901 27.86540 -0.994029 X^2 0.012230 0.005156 2.371861 R-squared 0.400795 Mean dependent var

Adjusted R-squared 0.362136 S.D. dependent var S.E. of regression 81255.15 Akaike info criterion Sum squared resid 2.05E+11 Schwarz criterion Log likelihood -431.0554 Hannan-Quinn criter. F-statistic 10.36759 Durbin-Watson stat Prob(F-statistic) 0.000357

从上图中可以看出，nR2=13.62701，比较计算的nR2=13.62701>异方差。

用以上两种方法，可以检验模型是存在异方差的。

c)修正模型

0.0004 0.0011 0.0000 Prob. 0.6605 0.3279 0.0241 28220.51 101738.9 25.53267 25.66735 25.57860 3.021651

统计量的临界值，因为

0.05（2）=5.9915，所以拒绝原假设，不拒绝备择假设，表明模型存在

1）用加权二乘法修正异方差现象步骤如下：

①当权数w1=1/x时，用软件分析得： Dependent Variable: Y Method: Least Squares Date: 12/24/15 Time: 20:22 Sample: 1 34 Included observations: 34 Weighting series: W1

Variable Coefficient Std. Error t-Statistic X 0.821013 0.016866 48.67993 C 17.69318 6.283256 2.815926 Weighted Statistics R-squared 0.986676 Mean dependent var

Adjusted R-squared 0.986260 S.D. dependent var S.E. of regression 37.91285 Akaike info criterion Sum squared resid 45996.29 Schwarz criterion Log likelihood -170.8132 Hannan-Quinn criter. F-statistic 2369.735 Durbin-Watson stat Prob(F-statistic) 0.000000

Unweighted Statistics R-squared 0.968070 Mean dependent var

Adjusted R-squared 0.967072 S.D. dependent var S.E. of regression 215.7175 Sum squared resid Durbin-Watson stat 1.079107

得方程模型为：

Y=0.821013X-17.69318

t=（48.67993）（2.815926）

R2=0.986676 F=2369.735 DW=0.605852

Prob. 0.0000 0.0083 457.8505 41.70384 10.16548 10.25527 10.19610 0.605852

1295.802 1188.791 1489089.

对此模型进行White检验如下： Heteroskedasticity Test: White

F-statistic 1.348072 Prob. F(2,31) 0.2745

Obs*R-squared 2.720457 Prob. Chi-Square(2) 0.2566 Scaled explained SS 1.221901 Prob. Chi-Square(2) 0.5428

Test Equation: Dependent Variable: WGT_RESID^2 Method: Least Squares Date: 12/24/15 Time: 20:29 Sample: 1 34 Included observations: 34 Collinear test regressors dropped from specification

Variable Coefficient Std. Error t-Statistic Prob. C 1678.870 416.5417 4.030498 0.0003 WGT^2 -32.13071 187.6175 -0.171257 0.8651 X*WGT^2 -0.484040 1.279449 -0.378319 0.7078

R-squared 0.080013 Mean dependent var 1352.832

Adjusted R-squared 0.020659 S.D. dependent var 1382.825 S.E. of regression 1368.467 Akaike info criterion 17.36487 Sum squared resid 58053732 Schwarz criterion 17.49955 Log likelihood -292.2027 Hannan-Quinn criter. 17.41080 F-statistic 1.348072 Durbin-Watson stat 1.199640 Prob(F-statistic) 0.274545

2

从上图中可以看出，nR=2.720457，比较计算的统计量的临界值， 因为nR2=2.720457<响。

0.05（2）=5.9915，所以接受原假设，即该模型消除了异方差的影