异方差性的检验和补救一、研究目的和要求表1列出了1998年我国主要制造工业销售收入与销售利润的统计资料，请利用统计软件Eviews建立我国制造业利润函数模型，检验其是否存在异方差，并加以补救。

表1 我国制造工业1998年销售利润与销售收入情况二、参数估计EVIEWS 软件估计参数结果如下Dependent Variable: Y Method: Least Squares Date: 06/01/16 Time: 20:16 Sample: 1 28Included observations: 28Variable Coefficient Std. Error t-Statistic Prob. C 12.03349 19.51809 0.616530 0.5429 X0.1043940.008442 12.366580.0000R-squared 0.854694 Mean dependent var 213.4639 Adjusted R-squared 0.849105 S.D. dependent var 146.4905 S.E. of regression 56.90455 Akaike info criterion 10.98938 Sum squared resid 84191.34 Schwarz criterion 11.08453 Log likelihood -151.8513 Hannan-Quinn criter. 11.01847 F-statistic 152.9322 Durbin-Watson stat 1.212781 Prob(F-statistic)0.000000用规范的形式将参数估计和检验结果写下2ˆ12.033490.104394(19.51809)(0.008442) =(0.616530) (12.36658)0.854694152.9322iY X t R F =+ = =三、 检验模型的异方差（一） 图形法 1. 相关关系图X YX Y 相关关系图2. 残差图形生成残差平方序列22e resid ，做2e 与解释变量 X 的散点图如下。

05,00010,00015,00020,00025,000X E 22e 与 X 散点图3. 判断由图可以看出，残差平方 2e 对解释变量 X 的散点图主要分布在图形中的下三角部分，大致看出残差平方 2e 随 X 的变动呈增大的趋势，因此，模型很可能存在异方差。

但是否确实存在异方差还应通过更进一步的检验。

（二） Goldfeld-Quanadt 检验 1. 排序使用 Sort X 命令对解释变量 X 进行排序。

2. 构造子样本区间，建立回归模型样本容量 n=28，去掉中间 c=8 个样本值，得到两个样本区间 1~10、19~28的两组样本值。

1~10区间回归估计Dependent Variable: Y Method: Least Squares Date: 06/01/16 Time: 20:35 Sample: 1 10Included observations: 10Variable Coefficient Std. Error t-Statistic Prob. C 15.76466 14.82022 1.063727 0.3185 X0.0858940.019182 4.4779370.0021R-squared 0.714814 Mean dependent var 77.06400 Adjusted R-squared 0.679166 S.D. dependent var 31.70225 S.E. of regression 17.95685 Akaike info criterion 8.790677 Sum squared resid 2579.587 Schwarz criterion 8.851194 Log likelihood -41.95338 Hannan-Quinn criter. 8.724289 F-statistic 20.05192 Durbin-Watson stat 2.280129 Prob(F-statistic)0.00206119~28区间回归估计Dependent Variable: Y Method: Least Squares Date: 06/01/16 Time: 20:36 Sample: 19 28Included observations: 10Variable Coefficient Std. Error t-Statistic Prob. C -11.99687 138.6642 -0.086517 0.9332 X0.1105520.039367 2.8082090.0229R-squared 0.496413 Mean dependent var 369.2440 Adjusted R-squared 0.433465 S.D. dependent var 118.6175 S.E. of regression 89.28163 Akaike info criterion 11.99833 Sum squared resid 63769.67 Schwarz criterion 12.05884 Log likelihood -57.99163 Hannan-Quinn criter. 11.93194 F-statistic 7.886037 Durbin-Watson stat 2.489267Prob(F-statistic)0.0229063. F 统计量值 对样本 1~10回归分析102112579.587ii e==∑对样本 19~28 回归分析1022163769.67ii e==∑222163769.6724.722579.587i ie F e===∑∑4. 判断取显著性水平 0.05α=，子样本个数为 10，变量个数为 2，因此子样本的残差平方和的自由度为 8，查 F 分布表得 0.05(8,8) 3.44F =0.0524.72 3.44F F =>=所以拒绝原假设，表明模型确实存在异方差性。

（三） White 检验对前文参数检验的结果进行 White 检验，结果如下图Heteroskedasticity Test: WhiteF-statistic3.607090 Prob. F(2,25)0.0420 Obs*R-squared 6.270439 Prob. Chi-Square(2) 0.0435 Scaled explained SS7.630696 Prob. Chi-Square(2) 0.0220Test Equation:Dependent Variable: RESID^2 Method: Least Squares Date: 06/01/16 Time: 20:38Sample: 1 28Included observations: 28Variable Coefficient Std. Error t-Statistic Prob. C -3279.669 2857.119 -1.147894 0.2619 X^2 -0.000871 0.000653 -1.334033 0.1942 X5.6706873.1093661.8237440.0802R-squared 0.223944 Mean dependent var 3006.833 Adjusted R-squared 0.161860 S.D. dependent var 5144.454 S.E. of regression 4709.748 Akaike info criterion 19.85361 Sum squared resid 5.55E+08 Schwarz criterion 19.99635 Log likelihood -274.9506 Hannan-Quinn criter. 19.89725 F-statistic3.607090 Durbin-Watson stat2.576402Prob(F-statistic)0.042040故 2 6.2704nR =，取0.05α=，则220.056.2704>(2) 5.99nR χ==，所以拒绝原假设，表明模型确实存在异方差性。

四、异方差的修正（加权最小二乘法）1. 权数将权数分别设置为123211,,tt t t t w w w x x ===2. 最小二乘估计 在 Eviews 命令窗口输入211/21/3(1) (2) (3) genrW X genrW X genrW LS W W Y C X LS W W Y C X LS W W Y C X======得到如下结果Dependent Variable: Y Method: Least Squares Date: 06/01/16 Time: 21:39 Sample: 1 28Included observations: 28 Weighting series: W1Weight type: Inverse standard deviation (EViews default scaling)Variable Coefficient Std. Error t-Statistic Prob. C 5.988351 6.403691 0.935141 0.3583 X 0.1086050.008156 13.31659 0.0000Weighted StatisticsR-squared0.872130 Mean dependent var 123.4049 Adjusted R-squared 0.867212 S.D. dependent var 31.99804 S.E. of regression 32.07267 Akaike info criterion 9.842635 Sum squared resid26745.07 Schwarz criterion9.937792Log likelihood -135.7969 Hannan-Quinn criter. 9.871726 F-statistic 177.3317 Durbin-Watson stat 2.386165 Prob(F-statistic) 0.000000 Weighted mean dep. 67.92073Unweighted StatisticsR-squared 0.853094 Mean dependent var 213.4639 Adjusted R-squared 0.847443 S.D. dependent var 146.4905 S.E. of regression 57.21696 Sum squared resid 85118.31 Durbin-Watson stat 2.472027权数为W1 时的最小二乘估计结果Dependent Variable: YMethod: Least SquaresDate: 06/01/16 Time: 21:46Sample: 1 28Included observations: 28Weighting series: W2Weight type: Inverse standard deviation (EViews default scaling)Variable Coefficient Std. Error t-Statistic Prob.C 6.497148 3.486625 1.863449 0.0737X 0.106890 0.010991 9.724824 0.0000Weighted StatisticsR-squared 0.784362 Mean dependent var 67.92073 Adjusted R-squared 0.776068 S.D. dependent var 75.51949 S.E. of regression 21.39500 Akaike info criterion 9.032941 Sum squared resid 11901.39 Schwarz criterion 9.128098 Log likelihood -124.4612 Hannan-Quinn criter. 9.062031 F-statistic 94.57221 Durbin-Watson stat 2.826376 Prob(F-statistic) 0.000000 Weighted mean dep. 36.45271Unweighted StatisticsR-squared 0.854180 Mean dependent var 213.4639 Adjusted R-squared 0.848571 S.D. dependent var 146.4905 S.E. of regression 57.00507 Sum squared resid 84489.02 Durbin-Watson stat 2.489641权数为W2 时的最小二乘估计结果Dependent Variable: YMethod: Least SquaresDate: 06/01/16 Time: 21:48 Sample: 1 28Included observations: 28 Weighting series: W3Weight type: Inverse standard deviation (EViews default scaling)Variable Coefficient Std. Error t-Statistic Prob. C 8.639271 11.18768 0.772213 0.4470 X 0.1061530.007746 13.70430 0.0000Weighted StatisticsR-squared0.878396 Mean dependent var 165.8409 Adjusted R-squared 0.873718 S.D. dependent var 67.13183 S.E. of regression 42.63779 Akaike info criterion 10.41211 Sum squared resid 47267.52 Schwarz criterion 10.50727 Log likelihood -143.7695 Hannan-Quinn criter. 10.44120 F-statistic 187.8079 Durbin-Watson stat 2.423771 Prob(F-statistic)0.000000 Weighted mean dep. 123.4049Unweighted StatisticsR-squared 0.854451 Mean dependent var 213.4639 Adjusted R-squared 0.848853 S.D. dependent var 146.4905 S.E. of regression 56.95205 Sum squared resid 84331.95Durbin-Watson stat2.493962权数为W3 时的最小二乘估计结果3. 判断由上述三个结果可以看出，W1 的 t 检验均显著，F 检验也显著，即对异方差的修正效果最好。