异方差性作业

一、 问题的提出和模型的设定

为了给国内生产总值的研究寻找依据，分析比较国内生产总值和居民消费水平的关系，建立国内生产总值和居民消费水平的回归模型。假定国内生产总值和居民消费水平之间满足线性约束，则理论模型设定为

Yi=β1+β2Xi+ui

其中，Yi表示国内生产总值，Xi表示居民消费水平。由《中国统计年鉴》得到1988-2008年的相关数据。 1988-2008年中国国内生产总值与居民消费水平 居民消国内生产费水平总值Y/亿（X1)/年份 元 元 1988 15042.8 714 1989 16992.3 788 1990 18667.8 833 1991 21781.3 932 1992 26923.5 1116 1993 35333.9 1393 1994 48197.9 1833 1995 60793.7 2355 1996 71176.6 2789 1997 78973 3002 1998 84402.3 3159 1999 89677.1 3346 2000 98000.5 3632 2001 108068.2 3869 2002 119095.7 4106 2003 135174 4411 2004 159878.3 4925 2005 183217.4 5463 2006 211923.5 6138 2007 257305.6 7103 2008 300670 8183 二、 参数估计 进入EViews软件包，确定样本范围，编辑输入数据，选择估计方程菜单，估计一下样本回归函数。 估计样本回归函数 Dependent Variable: Y Method: Least Squares Date: 05/27/10 Time: 15:08 Sample: 1988 2008 Included observations: 21 Variable ∧ C X R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient -22997.51 37.44105 Std. Error 4413.145 1.119365 t-Statistic -5.211139 33.44846 Prob. 0.0000 0.0000 101966.4 81193.29 21.49633 21.59580 1118.800 0.000000 0.983301 Mean dependent var 0.982422 S.D. dependent var 10764.71 Akaike info criterion 2.20E+09 Schwarz criterion -223.7114 F-statistic 0.130034 Prob(F-statistic) 估计结果为

Yi*=-22997.51+37.44105Xi

(-5.211139) (33.44846) 括号内为T统计量值 R2=0.983301，F=1118.8 三、检验模型的异方差

生成残差平方（e^2）序列

Last updated: 05/27/10 - 15:09 Modified: 1988 2008 // e2=(resid)^2 127857387.187195 109961782.879649 109765834.828736 97688616.985904 66207547.16643 38143361.3624126 6584198.13043462 19205945.2640684 105041477.803645 108733133.860987 118297446.13547 158839119.595988 224636533.393974 190266390.82371 135483482.244349 48733766.6299123 2314527.37376778 2803809.81355403 26090125.2535279 206190536.496472 298854278.666403 e^2与x的散点图

3.20E+082.80E+082.40E+082.00E+08E21.60E+081.20E+088.00E+074.00E+070.00E+00020004000X6000800010000 由以上数据和图表可知,模型很可能存在异方差性.下面用GOLDFILED-QUANADTA检验 以递增型排序

obs 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 Y 15042.8 16992.3 18667.8 21781.3 26923.5 35333.9 48197.9 60793.7 71176.6 78973 84402.3 89677.1 98000.5 108068.2 119095.7 135174 159878.3 183217.4 211923.5 257305.6 300670 X 714 788 833 932 1116 1393 1833 2355 2789 3002 3159 3346 3632 3869 4106 4411 4925 5463 6138 7103 8183 构建子样本区间建立回归模型。在本例中，样本容量是21删除中间的1/4，余下的部分平分为两个样本区间1-8he 14-21,它们的个数均是8个其回归结果如表A和表B

表A: Dependent Variable: Y Method: Least Squares Date: 05/27/10 Time: 15:16 Sample: 1988 1995 Included observations: 8 Variable C X R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient -4772.583 28.29324 Std. Error 658.1054 0.484093 t-Statistic -7.252004 58.44590 Prob. 0.0003 0.0000 30466.65 16494.47 16.27971 16.29957 3415.923 0.000000 0.998247 Mean dependent var 0.997954 S.D. dependent var 746.0231 Akaike info criterion 3339303. Schwarz criterion -63.11883 F-statistic 1.649502 Prob(F-statistic)

表B： Dependent Variable: Y Method: Least Squares Date: 05/27/10 Time: 15:21 Sample: 2001 2008 Included observations: 8 Variable C X R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient -63023.51 44.78756 Std. Error 3104.628 0.544091 t-Statistic -20.29986 82.31624 Prob. 0.0000 0.0000 184416.6 68356.82 18.43908 18.45894 6775.963 0.000000 0.999115 Mean dependent var 0.998968 S.D. dependent var 2196.105 Akaike info criterion 28937275 Schwarz criterion -71.75631 F-statistic 1.433715 Prob(F-statistic) 求F统计量值：由表A和表B的残差平方和的值，有F= Sum squared resid(1988-1995)/ Sum squared resid(2001-2008)=8.66566显然子样本子、之间存在异方差性.。 四、异方差性的修正

在运用加权最小二乘法（WLS）估计过程中，分别选用了权数W1t=1/X, W2t=1/X^2,W3t=1/sqr(X)。 用权数w1的结果： Dependent Variable: Y Method: Least Squares Date: 05/27/10 Time: 19:10 Sample: 1988 2008 Included observations: 21 Weighting series: W1 Variable C X R-squared Adjusted R-squared S.E. of regression Coefficient 9152.175 32.42343 Std. Error 1509.768 0.976166 t-Statistic 6.061974 33.21507 Prob. 0.0000 0.0000 54779.62 9134.275 20.14275 Weighted Statistics 0.659178 Mean dependent var 0.641240 S.D. dependent var 5471.120 Akaike info criterion

Sum squared resid Log likelihood Durbin-Watson stat R-squared Adjusted R-squared S.E. of regression Durbin-Watson stat

5.69E+08 Schwarz criterion -209.4988 F-statistic 1.327333 Prob(F-statistic) Unweighted Statistics 20.24223 1103.241 0.000000 101966.4 81193.29 4.71E+09 0.964300 Mean dependent var 0.962421 S.D. dependent var 15739.48 Sum squared resid 0.107792 用权数w2的结果： Dependent Variable: Y Method: Least Squares Date: 05/27/10 Time: 19:12 Sample: 1988 2008 Included observations: 21 Weighting series: W2 Variable C X R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat R-squared Adjusted R-squared S.E. of regression Durbin-Watson stat

Coefficient 6391.108 29.93823 Std. Error 554.5822 0.619152 t-Statistic 11.52418 48.35358 Prob. 0.0000 0.0000 30182.36 20115.05 17.36831 17.46778 2338.069 0.000000 101966.4 81193.29 8.90E+09 Weighted Statistics 0.995616 Mean dependent var 0.995385 S.D. dependent var 1366.514 Akaike info criterion 35479839 Schwarz criterion -180.3672 F-statistic 2.119407 Prob(F-statistic) Unweighted Statistics 0.932483 Mean dependent var 0.928929 S.D. dependent var 21645.41 Sum squared resid 0.078262 用权数w3的结果： Dependent Variable: Y Method: Least Squares Date: 05/27/10 Time: 19:14 Sample: 1988 2008 Included observations: 21 Weighting series: W3 Variable C X R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat R-squared Adjusted R-squared S.E. of regression Durbin-Watson stat

Coefficient 13340.99 34.54781 Std. Error 2762.403 1.076811 t-Statistic 4.829488 32.08344 Prob. 0.0001 0.0000 76321.45 38571.85 21.07040 21.16988 1029.347 0.000000 Weighted Statistics 0.951671 Mean dependent var 0.949127 S.D. dependent var 8699.896 Akaike info criterion 1.44E+09 Schwarz criterion -219.2392 F-statistic 1.118407 Prob(F-statistic) Unweighted Statistics 101966.4 81193.29 2.98E+09 0.977429 Mean dependent var 0.976242 S.D. dependent var 12514.95 Sum squared resid 0.130199 从以上三种估计结果中知道，运用用权数w2的结果最好，运用用权数w2的

结果如下：

Y = 6391.107567 + 29.93823178*X

（11.52418） （48.35358）

R2=0.995616，DW=2.119407,F=2338.069 括号中数据为t统计量值。

可以看出运用加权最小二乘法消除了异方差性后，参数t检验均显著，可决系数大幅度提高，F检验也显著，并说明人均消费水平每增长1元，国内生产总值将增长29.93823178亿元。

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