对此模型进行White检验得： Heteroskedasticity Test: White
F-statistic 1.003964 Prob. F(2,31) 0.3780
Obs*R-squared 2.068278 Prob. Chi-Square(2) 0.3555 Scaled explained SS 1.469638 Prob. Chi-Square(2) 0.4796
Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 12/24/15 Time: 21:45 Sample: 1 34 Included observations: 34
Variable Coefficient Std. Error t-Statistic Prob. C 0.039547 0.046759 0.845753 0.4042 LNX -0.011601 0.014012 -0.827969 0.4140 LNX^2 0.000932 0.001028 0.906774 0.3715 R-squared 0.060832 Mean dependent var 0.004950
Adjusted R-squared 0.000240 S.D. dependent var 0.006365 S.E. of regression 0.006364 Akaike info criterion -7.192271 Sum squared resid 0.001255 Schwarz criterion -7.057592 Log likelihood 125.2686 Hannan-Quinn criter. -7.146342 F-statistic 1.003964 Durbin-Watson stat 2.022904 Prob(F-statistic) 0.378027
从上图中可以看出，nR2
=2.068278，比较计算的统计量的临界值，nR2=2.068278<0.05（2）=5.9915，所以接受原假设，此模型消除了异方差。
综合两种方法，改进后的模型最好为：
LnY=0.946887 LNX+0.201861
为因 (2)
1）考虑价格因素，首先用软件三者关系进行分析如下： Dependent Variable: Y Method: Least Squares Date: 12/24/15 Time: 21:51 Sample: 1 34 Included observations: 34
Variable Coefficient Std. Error t-Statistic X 0.741684 0.019905 37.26095 P 0.235025 0.271701 0.865012 C 43.41715 71.22946 0.609539 R-squared 0.979911 Mean dependent var
Adjusted R-squared 0.978615 S.D. dependent var S.E. of regression 173.8449 Akaike info criterion Sum squared resid 936883.7 Schwarz criterion Log likelihood -222.0511 Hannan-Quinn criter. F-statistic 756.0627 Durbin-Watson stat Prob(F-statistic) 0.000000
1）用Goldfeld-Quanadt检验如下： ①当样本为1-13时，进行回归分析： Dependent Variable: P Method: Least Squares Date: 12/24/15 Time: 21:59 Sample: 1 13 Included observations: 13
Variable Coefficient Std. Error t-Statistic X -0.170484 0.203868 -0.836247 Y 0.458660 0.209755 2.186646 C 59.50496 7.385841 8.056627 R-squared 0.956255 Mean dependent var
Adjusted R-squared 0.947506 S.D. dependent var S.E. of regression 8.466678 Akaike info criterion Sum squared resid 716.8464 Schwarz criterion Log likelihood -44.51063 Hannan-Quinn criter. F-statistic 109.2993 Durbin-Watson stat Prob(F-statistic) 0.000000
2 得∑e1i=716.8464
Prob. 0.0000 0.3937 0.5466 1295.802 1188.791 13.23830 13.37298 13.28423 1.681521
Prob. 0.4225 0.0536 0.0000 135.3231 36.95380 7.309328 7.439701 7.282530 0.637181
②当样本为22-34时，做回归分析得： Dependent Variable: Y Method: Least Squares Date: 12/24/15 Time:22:07 Sample: 22 34 Included observations: 13
Variable Coefficient Std. Error t-Statistic Prob. X 0.641197 0.092678 6.918569 0.0000 P -1.206222 1.114278 -1.082514 0.3044 C 795.6887 603.8605 1.317670 0.2170 R-squared 0.939696 Mean dependent var 2496.127
Adjusted R-squared 0.927635 S.D. dependent var 1022.591 S.E. of regression 275.0847 Akaike info criterion 14.27121 Sum squared resid 756715.7 Schwarz criterion 14.40158 Log likelihood -89.76286 Hannan-Quinn criter. 14.24441 F-statistic 77.91291 Durbin-Watson stat 1.128778 Prob(F-statistic) 0.000001
2 得∑e2i=756715.7
③根据Goldfeld-Quanadt检验，F统计量为： F=∑e2i2 /∑e1i2 =756715.7/ 716.8464=1055.6176
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（10,10）=2.98，因为F=1055.6176> F0.05（10,10）=2.98，所以拒绝原假设，此检验表明模型存在异方差。
2）用White检验，软件分析结果为： Heteroskedasticity Test: White
F-statistic 7.312529 Prob. F(5,28) Obs*R-squared 19.25463 Prob. Chi-Square(5) Scaled explained SS 119.3072 Prob. Chi-Square(5)
Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 12/24/15 Time: 22:18 Sample: 1 34 Included observations: 34
Variable Coefficient Std. Error t-Statistic C 79541.08 112647.3 0.706107 X 209.4964 63.90400 3.278298 X^2 -0.024133 0.010712 -2.252841 X*P -0.235137 0.106647 -2.204822 P -1175.326 1156.253 -1.016495 P^2 1.637366 2.600020 0.629751 R-squared 0.566313 Mean dependent var
Adjusted R-squared 0.488869 S.D. dependent var S.E. of regression 77206.44 Akaike info criterion Sum squared resid 1.67E+11 Schwarz criterion Log likelihood -427.5874 Hannan-Quinn criter. F-statistic 7.312529 Durbin-Watson stat Prob(F-statistic) 0.000171
从上图中可以看出，nR2=19.25463，比较计算的nR2=19.25463>异方差。
0.0002 0.0017 0.0000 Prob. 0.4860 0.0028 0.0323 0.0358 0.3181 0.5340 27555.40 107990.9 25.50514 25.77450 25.59700 2.787044
统计量的临界值，因为
0.05（5）=11.0705，所以拒绝原假设，不拒绝备择假设，表明模型存在
2）修正
①建立对数模型，用软件分析如下： Dependent Variable: LNY Method: Least Squares Date: 12/24/15 Time: 22:24 Sample: 1 34 Included observations: 34
Variable Coefficient Std. Error t-Statistic Prob. LNX 0.939605 0.013645 68.86088 0.0000 LNP 0.026821 0.028454 0.942609 0.3532 C 0.108230 0.126322 0.856784 0.3981 R-squared 0.995646 Mean dependent var 6.687779
Adjusted R-squared 0.995365 S.D. dependent var 1.067124 S.E. of regression 0.072652 Akaike info criterion -2.322188 Sum squared resid 0.163625 Schwarz criterion -2.187509 Log likelihood 42.47720 Hannan-Quinn criter. -2.276259 F-statistic 3544.292 Durbin-Watson stat 0.930109 Prob(F-statistic) 0.000000