4.6
1.进行最小二乘估计,结果为： Dependent Variable: Y Method: Least Squares Sample: 1 20
Included observations: 20
Variable C X R-squared
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
Coefficient 29.03227 0.806178 Std. Error 47.26019 0.016951 t-Statistic 0.614307 47.55956 Prob. 0.5467 0.0000 642.2246 11.07494 11.17451 2261.912 0.000000
0.992105 Mean dependent var 2188.500 0.991666 S.D. dependent var 58.62795 Akaike info criterion 61870.26 Schwarz criterion -108.7494 F-statistic 2.066630 Prob(F-statistic)
Y = 29.03227493 + 0.8061776361*X 2.异方差检验 散点图如下
X与Y散点分布的相关图显示随机项存在递增的异方差 3.用Goldfeld-Quandt检验分析异方差性
假设：H0：随机项是同方差的 H1：随机项是递增异方差的 第一个子样本数据为
进行最小二乘估计(1~10)，
Dependent Variable: Y Method: Least Squares Sample: 1 10
Included observations: 10
Variable C X
R-squared
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
Coefficient -90.22196 0.869812
Std. Error 123.2315 0.058442
t-Statistic -0.732134 14.88346
Prob. 0.4850 0.0000 186.8839 10.23695 10.29747 221.5174 0.000000
0.965144 Mean dependent var 1735.600 0.960787 S.D. dependent var 37.00713 Akaike info criterion 10956.22 Schwarz criterion -49.18477 F-statistic 2.939370 Prob(F-statistic)
Y = -90.22196172 + 0.8698118059*X 表中的残差平方和为10956.22
第二个子样本数据(11~20),进行最小二乘估计为
Dependent Variable: Y Method: Least Squares Sample: 11 20
Included observations: 10
Variable C X
R-squared
Adjusted R-squared
Coefficient -79.77156 0.835176
Std. Error 108.5232 0.032573
t-Statistic -0.735065 25.64025
Prob. 0.4833 0.0000 616.4322
0.987978 Mean dependent var 2641.400 0.986475 S.D. dependent var
S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 71.68987 Akaike info criterion 41115.50 Schwarz criterion -55.79716 F-statistic 1.971395 Prob(F-statistic) 11.55943 11.61995 657.4227 0.000000 Y = -79.77156004 + 0.8351763428*X
表中的残差平方和为41115.50 从而G-Q检验的F统计量为F=ESS2/ESS1=3.752708507， F0.05（10-1-1，10-1-1）=F0.05（8，8）=3.44 F〉F0.05（8，8） 随机项存在递增的异方差 4.异方差的修正
变量代换后最小二乘估计的Eviews输出表
Dependent Variable: Y Method: Least Squares Sample: 1 20
Included observations: 20 Weighting series: 1/X Variable C X
Weighted Statistics R-squared
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Unweighted Statistics R-squared
Adjusted R-squared S.E. of regression Durbin-Watson stat Coefficient 45.46337 0.799834
Std. Error 51.69134 0.021266 t-Statistic 0.879516 37.61075 Prob. 0.3907 0.0000 55.15445 10.96478 11.06435 0.773548 0.390713 642.2246 62357.86
0.041204 Mean dependent var 2044.832 -0.012062 S.D. dependent var 55.48609 Akaike info criterion 55416.72 Schwarz criterion -107.6478 F-statistic 2.090260 Prob(F-statistic)
0.992043 Mean dependent var 2188.500 0.991601 S.D. dependent var 58.85852 Sum squared resid 2.072106 得到原模型对应参数的加权对小二乘估计。 回归方程为，Y = 45.4633662 + 0.7998344672*X 4.7证明：Yi=bXi+εi Var(εi)=σ2Xi2
线性：因为参数的最小二乘估计量的表达式只依赖于残差平方和最小这一原则，与随机项εi 的古典假设无关，因此线性特征任成立。
?xy?xYY?x??b??xx???xiiii2iiii2i2iiiii
?x??X??X?0iiiii 令
Ki?xi??KY ，则b?iii2xi?ii根据经典假设8，因为xi不全为零，所以Ki不全为零，故线性关系成立。
无偏性：参数的最小二乘估计量的无偏性依赖于解释变量的非随机变量和随机项的零均值假定，与同方差假定无关，因此随机项存在异方差时，并不影响到无偏性的成立。
?)?E[[XTX]?1XTY]?[[XTX]?1XT]E(XB??)?B E(B最小方差性： 在模型的所有线性无偏估计两种，最小二乘估计量的方差最小的前提条件是随机项εi 是同方差的，如果随机项是异方差的，就不能保证最小二乘估计量的方差最小。
?)?Var(b?K?)?Var(b?ii11i?i2?x2iX2ii?xi2其值随xi而变化，会夸大或缩小真实的方差和协方差，因此无法确定最小
i方差性。 5.7.
1. 进行最小二乘估计
Dependent Variable: Y Method: Least Squares Sample: 1 20
Included observations: 20
Variable C X
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)
Coefficient -1.454750 0.176283
Std. Error 0.214146 0.001445
t-Statistic -6.793261 122.0170
Prob. 0.0000 0.0000 24.56900 2.410396 -1.972991 -1.873418 -1.953553 0.734726
0.998792 Mean dependent var 0.998725 S.D. dependent var 0.086056 Akaike info criterion 0.133302 Schwarz criterion 21.72991 Hannan-Quinn criter. 14888.14 Durbin-Watson stat 0.000000
回归方程为，Y = -1.454750 + 0.17628*X
2. D-W检验分析随机项的一阶自相关性
作散点图如下
相邻2期残差散点分布的自相关图显示随机项存在一阶的正自相关 利用D-W方法检验随机项的自相关。在表中，D-W检验的d统计量值为0.734726，
dL=1.20 dU=1.41 0 （1）估计一阶自相关系数 根据Eviews软件计算残差关于其滞后项回归的Eviews输出表如下 Dependent Variable: R Method: Least Squares Sample (adjusted): 2 20 Included observations: 19 after adjustments Variable R(-1) R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient 0.631164 Std. Error 0.183294 t-Statistic 3.443450 Prob. 0.0029 0.001371 0.085825 -2.527598 -2.477891 -2.519186 0.396971 Mean dependent var 0.396971 S.D. dependent var 0.066648 Akaike info criterion 0.079954 Schwarz criterion 25.01218 Hannan-Quinn criter. 1.739540 从表中得到一阶自相关系数的估计值为0.631164 （2）广义差分变换及广义最小二乘估计 Dependent Variable: YY Method: Least Squares Sample (adjusted): 2 20 Included observations: 19 after adjustments Variable C XX R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) Coefficient -0.394117 0.173758 Std. Error 0.167230 0.002957 t-Statistic -2.356732 58.76793 Prob. 0.0307 0.0000 9.391862 0.932492 -2.464342 -2.364928 -2.447518 1.650243 0.995102 Mean dependent var 0.994814 S.D. dependent var 0.067154 Akaike info criterion 0.076665 Schwarz criterion 25.41125 Hannan-Quinn criter. 3453.669 Durbin-Watson stat 0.000000 得到原模型对应系数的广义最小二乘估计值 YY = -0.3941168467 + 0.17375831191*XX （4）直接用差分法估计回归模型参数 Dependent Variable: YY Method: Least Squares Date: 11/22/14 Time: 22:57 Sample (adjusted): 2 20 Included observations: 19 after adjustments Variable C XX R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) Coefficient 0.040528 0.158783 Std. Error 0.022642 0.007248 t-Statistic 1.789959 21.90756 Prob. 0.0913 0.0000 0.411579 0.344146 -2.514302 -2.414888 -2.497477 1.748834 0.965791 Mean dependent var 0.963778 S.D. dependent var 0.065498 Akaike info criterion 0.072929 Schwarz criterion 25.88587 Hannan-Quinn criter. 479.9410 Durbin-Watson stat 0.000000 YY = 0.0405279643212 + 0.158783078331*XX 5.6 Yt?b0?b1X1t?b2X2t??t （1） ?iYt?1??1b0??1b1X1t?1??1b2X2t?1??1?t?1 （2） ?2Yt?2??2b0??2b1X1t?2??2b1X1t?2??2?t?2 （3） （1）-（2）-（3）得， Yt??1Yt?1??2Yt?2?b0(1??1??2)?b1(X1t??1X1t?1??2X1t?2)?b2(X2t??1X2t?1??2X2t?2) Y*t?a0?a1X1t?a2X2t?ut 满足经典假设，估计参数，并推导出原方程参数估计值。 **
