计量经济学第五章异方差性参考答案讲解(20210310104313)

1、第五章异方差性课后题参考答案5.1因为，所以取弘=十，用幽乘给定模型两端得丄 xli2X2i上述模型的随机误差项的方差为一固定常数，即Ui12Var(*)2Var(uJ *X2iX2i(2)根据加权最小二乘法，可得修正异方差后的参数估计式为* * *必=丫 -?2X 2 -?3X 3? (ZWiyx2WiXL(迟W2iyj(xZWi*XXi二77二*2；W2i X2i JW2i X3i-乂2 X2i X3i?3(ZW2iy*iW2iX$厂(瓦W2iy(XM*Xi*XiW2i X*2i li* 2* 2iW2i X2iW2i X3i -其中*X2、W2iX 2*WiX i 3X3 -*、Wi/_

2、、W,iX2i- X2X3 二X3 - X3y= iY y5.2(1)Yy_x氏 Y_X宾丁 (u)= ( J1u 二 ln(u) 1又；Eln(u) =0.E(J) =Eln(u) 1 =Eln( u) 1 =1(2)Eln( J)HX R In 7 =ln 1 丨 =0= i 丨 叮=1E()八二叮=1不能推导出|1 / =1所以E)=1时，不一定有E( ln J =0(3)对方程进行差分得：lnYi-lnYi-1 =nX % )+(ln-ln 叫 J则有：E（ln 7 _ln 叫）=05.3(1) 该模型样本回归估计式的书写形式为：Y = 11.44213599 + 0.62678299

3、62*X(3.629253)(0.019872)t= 3.15275231.540972 2R 0. 94 49 1 1 R =0. 94 39 6 1 S.E.=9.158900DW=1.597946F=994.8326(2) 首先，用Goldfeld-Quandt法进行检验。a.将样本X按递增顺序排序，去掉中间1/4的样本，再分为两个部分的样本，即厲2 =22。b.分别对两个部分的样本求最小二乘估计，得到两个部分的残差平方 和，即瓦 e2 = 624.3004 瓦 ef = 2495.840J求F统计量为Z2电2495.840F=3 9978e2 624.3004 3.9978给定:=0.

4、05，查F分布表，得临界值为F0.05(20,20) =212。C.比较临界值与F统计量值，有f =4.1390Fo.o5(20,20)=212，说明该模型 的随机误差项存在异方差。其次，用White法进行检验。具体结果见下表White Heteroskedasticity Test:F-statistic6.105557Probability0.003958Obs*R-squared10.58597Probability0.005027给定:=0.05，在自由度为2下查卡方分布表，得2=5.9915。比较临界值与卡方统计量值，即nR2 =10.8640 2 =5.9915，同样说明模型中的

5、随机误差项存在异方差。(2)用权数W =1/|e|，作加权最小二乘估计，得如下结果Depe ndent Variable: YMethod: Least SquaresDate: 05/28/07 Time: 00:20Sample: 1 60In cluded observati ons: 60Weight ing series: 1/RVariableCoefficientStd. Errort-StatisticProb.C27.500006.09E-084.52E+080.0000X0.5000007.16E-106.98E+080.0000Weighted StatisticsR-s

6、quared1.000000varMean depe ndent70.01964Adjusted1.000000S.D. depe ndent379.890R-squaredvar9S.E. of regressi on8.44E-10Akaike info-38.916criteri on22Sum squared4.13E-17Schwarz criterio n-38.846residLog likelihood1169.487F-statistic414.88E+170.00000Durbi n-Wats on0.786091Prob(F-statistic)stat0Un weigh

7、tedStatisticsR-squared0.883132Mean depe ndent119.666var7Adjusted0.881117S.D. depe ndent38.6898R-squaredvar4S.E. of regressi on13.34005Sum squared10321.5resid0Durbi n-Wats on0.377804statWhite检验：White Heteroskedasticity Test:F-statistic2.357523Probability0.1038220.101063Obs*R-squared4.584017Probabilit

8、yTest Equati on:Depe ndent Variable: STD_RESID2Method: Least SquaresDate: 05/28/07 Time: 00:27Sample: 1 60In cluded observati ons: 60VariableCoefficientStd. Errort-StatisticProb.C3.86E-191.73E-192.2337560.0294X3.21E-212.16E-211.4895320.1419XA2-7.59E-246.18E-24-1.2296410.2239R-squared0.076400Mean dep

9、e ndent6.88E-1var9Adjusted0.043993S.D. depe ndent1.56E-1R-squaredvar9S.E. of regressi on1.52E-19Sum squared1.32E-3resid6F-statistic2.357523Durbi n-Wats on1.19153stat1Prob(F-statistic)0.1038225.4令Y表示农业总产值，X1-X5分别表示农业劳动力、灌溉面积、化肥用量、户 均固定资产和农机动力。建立模型：丫八0X 泊23X34X45X5回归结果如下：丫？= 4.717198 0.039615-0.036895

10、X2 0.263256X3 0.013463X4 0.025469X5t =(0.516910)(1.452697) ( -0.474813)(0.479104)(2.712997)(1.625993)R2 =0.974539 R2=0.953321 DW=1.969898 F=45.93047从回归结果可以看出，模型的R2和R2值都较高，F统计量也显著。但是除X4的系数显著之外，其他系数均不显著，模型可能存在多重共线性。计算各解释变量的相关系数。相关系数矩阵X1X2X3X4X5X11.0000000.8518670.9631730.4569130.892506X20.8518671.0000

11、000.8435410.5493900.856933X30.9631730.8435411.0000000.5830480.924806X40.4569130.5493900.5830481.0000000.543765X5 0.8925060.8569330.9248060.5437651.000000由相关系数矩阵可以看出，解释变量之间的相关系数较高，存在多重共线性。 采用逐步回归的办法，来解决多重共线性问题。分别做丫对XI、X2、X3、X4、X5的一元回归，结果如下表所示：一元回归结果变量X1X2X3X4X5参数估计值0.0840780.4567671.5264100.0352770.0

12、78269t统计量8.0976515.09937111.621322.9913268.197929R20.8676760.7222500.9310610.4722410.870476R20.8544430.6944750.9241670.4194650.857524其中加入X3的方程R2最大，以X3为基础，顺次加入其他变量逐步回归，结果如下：加入新变量的回归结果（一）变量X1X2X3X4X5R2X3，X10.0026361.4819090.915816(0.089770)(2.879293X3，X2)0.0669091.3602910.9212040.7899585.456584X3，X41.

13、3522910.0096910.9444929.7767642.159071X3，X51.1156800.0235520.929684(3.355936)(1.335921)经比较,新加入X4的方程R2 =0.944492，改进最大且从经济意义来看，户均固定资产对农业总产值有影响，因此保留 X4，再加入其他变量逐步回归，结果 如下：加入新变量的回归结果（二）变量X1X2X3X4X5R2X3，X4 X10.0354380.6966510.0128870.949360(1.365712)(1.399128)(2.638461X3，X4 X2丿0.0474861.241502)0.0092960.9

14、40595(1.487193(5.528062(1.984375)X3， X4，0.9519240.0095940.0230590.952585X5(3.375236(2.312344(1.592574)加入X1后方程的R2增大，但是t值不显著；加入X2后R2降低，且系数不显著;假如X5后方程的R2增大，但是t值不显著。修正多重共线性影响的回归结果为： =14.74802 +1.352291X3 +0.009691X4t =1.8354419.7767642.1590712 2R2=0.954584 R 2 =0.944492 DW=2.482223 F=94.58409White检验：nR2

15、 =4.132927 瞪05(5) =11.0705接受原假设，模型不存在异方差。5.5(1) 建立样本回归模型。7=1 92.99 4 40 X03 1 9( 0.1 9 48)( 3.83 )R2 =0.4 7 8s,= . .27159. 1 5,1 4.66 92(2) 利用White检验判断模型是否存在异方差。White Heteroskedasticity Test:F-statistic3.057161Probability0.076976Obs*R-squared5.212471Probability0.073812给定:=0.05和自由度为2下，查卡方分布表，得临界值2=5.

16、9915， 而White统计量nR2 =5.2125，有nR 爲(2)，则不拒绝原假设，说 明模型中不存在异方差。(3) 有Glejser检验判断模型是否存在异方差。经过试算，取如 下函数形式e = P/X得样本估计式 =6.4435 .X(4.5658) R2 =0.2482由此，可以看出模型中随机误差项有可能存在异方差。(4) 对异方差的修正。取权数为 wN/X，得如下估计结果Y?二-243.4910 0.0367X(-1.7997)(5.5255)2R =0.1684, s.e. =694.2181, F = 30.53090.6561425.6回归结果如下：Y? =0.89 0.237

17、200Xit =(4.356086) (15.89724)se=(0.204312)( 0.014921 )2 2R2 =0.933511 R2=0.929817 DW=1.363966 F=252.7223 e： -0.3 、 e22i =2.024= 6.7467F0.05(8,)=3.44拒绝原价设，模型存在异方差。取权数为Y? =0.752923 +0.249487Xise=(0.098255)( 0.011723)Wx，加权后回归结果:t = (7.662934)(21.28124)2 2R2 =0.765382 R2=0.752348 DW=1.240480 F=452.89145

18、.7(1) 求回归估计式。Y?=4.61030. X5 7 4(4.2495)(5.0516)R2 =0 . 5 8 6ste严.3.F3910,25.5 183作残差的平方对解释变量的散点图3025-2015-10-51030RESIDE由图形可以看出，模型有可能存在异方差。(2) 去掉智利的数据后，回归得到如下模型W = 6.73810.X2 15(2.8254)(0.3987)2R =0 . 0 0 9s,= . . 3 ,F3z90 6,作残差平方对解释变量的散点图0.1589RESIDE从图形看出，异方差的程度降低了。（3）如果去掉智利数据后得出不存在异方差的结论，则说明异方差性还会

19、 因为异常值的出现而产生。5.8（1）回归结果如下:Y = 12.12542711 + 0.1043661755*X(19.51012(0.008439)t= (0.621494)(12.36777)2R =0.854718R2 =0.849130S.E.=56.89947DW=1.212859F=152.9617销售收入每增长一元，销售利润平均增长0.104366元给定。=0.05，t =12.36777t.025（26） =2.056，拒绝原假设，说明销售收入对销售 利润有显著性影响。F =152.9617 a F0.o5（1,26） = 4.23 R2 = 0. 8 5 4 6 9表明方

20、程显著，且拟和程度较 好（2）图形法：500040003000X200010000 -200 -100 0 100RESID从图中可以看出，e2i有随着X增大而增大的趋势，所以模型可能存在异方差 用Glejser检验模型是否存在异方差。经过试算，取如下函数形式e = ： 2、X得样本估计式| ?|=1.049787 Xt =8.075394R2=0.306629系数显著不为0,由此，可以看出模型中随机误差项有可能存在异方差。White检验：White Heteroskedasticity Test:F-statistic3.609579Probability0.041959Obs*R-squa

21、red6.273796Probability0.043417nR2 =6.273796 720.05(2) =5.91147拒绝原假设，模型存在异方差。(3) 对异方差的修正。取权数为 W二/x2，得如下估计结果Depe ndent Variable: YMethod: Least SquaresDate: 05/28/07 Time: 00:16Sample: 1 28In cluded observati ons: 28Weight ing series: 1/XA2VariableCoefficie ntStd. Errort-StatisticProb.C6.4548963.48563

22、41.8518570.0754X0.1070750.0109849.7481670.0000Weighted StatisticsR-squared0.922863varMean depe ndent67.93474Adjusted0.919896S.D. depe ndent var75.46572R-squaredS.E. of regressi on21.35880Akaike info criteri on9.029554Sum squared11861.15Schwarz criteri on9.124711residLog likelihood-124.4137F-statisti

23、c95.02675Durbi n-Wats on1.909174Prob(F-statistic)0.000000statUn weightedStatisticsR-squared0.854132Mean depe ndent213.4650varAdjusted0.848522S.D. depe ndent var146.4895R-squaredS.E. of regressi on57.01397Sum squared resid84515.42Durbi n-Wats on1.244888statWhite检验：White Heteroskedasticity Test:F-stat

24、istic3.143257Probability0.060574Obs*R-squared= 5.626143 =Probability0.0600202 、. 2nR =5.626143 V 尤 0.05( 2)=5.91147不存在异方差.5.9(1)建立样本回归函数。7=4 3.89670.X 104(2.1891)(37.7771)2R =0 .9 8 5s4s, = . .60F49 20,1 42 7.1 1 2从估计的结果看，各项检验指标均显著，但从残差平方对解释变量散点图可以看 出，模型很可能存在异方差。1500X 1MO500-500010000150002Q00DRESID

25、E(2)用White检验判断是否存在异方差。White Heteroskedasticity Test:F-statistic9.509463Probability0.001252Obs*R-squared11.21085Probability0.0036782由上表可知，nR =11.2109，给定:=0.05，在自由度为2下，查卡方分布表， 得临界值为2 =5.9915，显然，n R2=11.2109 5.15，则拒绝原假设，说 明模型存在异方差。进一步，用ARCH检验判断模型是否存在异方差。经试算选滞后阶数为1，则ARCH检验结果见下表ARCH Test:F-statistic9.394

26、796Probability0.006109Obs*R-squared7.031364Probability0.0080092由上表可知，(n -P)R =7.0314,在=0.05和自由度为1下，查卡方分布表，2 2 2得临界值为 金=3.8415，显然，(n - p)R =7.0314人点1)3.8415，则说明 模型中随机误差项存在异方差。(3)修正异方差。取权数为 W=1/e2，得如下估计结果Depe ndent Variable: YMethod: Least SquaresDate: 05/28/07 Time: 02:15Sample: 1978 2000In eluded ob

27、servati ons: 23Weighti ng series: W2VariableCoefficie ntStd. Errort-StatisticProb.C6.6590270.25376126.241330.0000X0.8686910.000985881.79380.0000Weighted StatisticsR-squared1.000000Mean depe ndent var224.0761Adjusted R-squared1.000000S.D.dependent var988.1865S.E. of regressi on0.206384Akaike info cri

28、teri on-0.235219Sum squared resid0.894478Schwarz criteri on-0.136481Log likelihood4.705022F-statistic777560.3Durb in -Watson stat1.281139Prob(F-statistic)0.000000Un weighted StatisticsR-squared0.980282Mean depe ndent var633.0004Adjusted R-squared0.979343S.D.dependent var490.5345S.E. of regressi on70

29、.50182Sum squared resid104380.6Durb in -Watson stat0.279924Y = 6.65902728 + 0.8686910728*XWhite检验White Heteroskedasticity Test:F-statistic1.021337Probability0.378144Obs*R-squared2.131389Probability0.344489经检验异方差的表现有明显的降低。5.10剔除物价上涨因素后的回归结果如下：Y? =36.66453 0.753946Xt =(3.516150) (22.43266)R2 =0.959941

30、 R2 =0.958033 F=503.2243其中丫代表实际消费支出，X代表实际可支配收入 用White方法来检验模型是否存在异方差：White Heteroskedasticity Test:F-statistic1.647288Probability0.2176470.196625Obs*R-squared3.252914ProbabilityTest Equati on:Depe nde nt Variable: RESIDEMethod: Least SquaresDate: 05/07/07 Time: 17:14Sample: 1978 2000In cluded observa

31、ti ons: 23VariableCoefficientStd. Errort-StatisticProb.C105.9636475.51820.2228380.8259X10.2642903.0373300.0870140.9315X1A20.0009670.0043920.2200610.8281R-squared0.141431Mean depe ndent275.881var8Adjusted0.055574S.D. depe ndent272.672R-squaredvar6S.E. of regressi on264.9875Akaike info14.1183criteri on5Sum squared1404368.Schwarz criterio n14.2664resid6Log likelihood-159.361F-statistic1.6472808Durbi n-Wats on1.456581Prob(F-statistic)0.21764stat7n R2 =3.252914 沪0.05(2) =5.9915，表明模型不存在异方差。G-Q检验：Depe ndent Variable: Y1Method: Least SquaresDate: 05/07/07 Time: 17:18Sample: 1978 1986I