异方差与序列相关性练习

1、一、异方差检验与修正（一）建立初始回归模型相关命令：data x yscat x yls y c x模型一：Dependent Variable: YMethod: Least SquaresDate: 10/23/14 Time: 10:46Sample: 1 20Included observations: 20VariableCoefficientStd. Errort-StatisticProb.  C272.3635159.67731.7057130.1053X0.7551250.02331632.386900.0000R-squared0.983129 

2、;   Mean dependent var5199.515Adjusted R-squared0.982192    S.D. dependent var1625.275S.E. of regression216.8900    Akaike info criterion13.69130Sum squared resid846743.0    Schwarz criterion13.79087Log likelihood-134.9130

3、60;   F-statistic1048.912Durbin-Watson stat1.301684    Prob(F-statistic)0.000000（二）异方差的四种检验方法及其分析右击resid选择Object Copy，输入e得到初始回归模型的残差序列；1. 图示法：scat x e22. 模型检验法：ls e2 c xDependent Variable: E2Method: Least SquaresDate: 10/23/14 Time: 10:52Sample: 1 20Included observ

4、ations: 20VariableCoefficientStd. Errort-StatisticProb.  C-65281.6621544.58-3.0300730.0072X16.493443.1458955.2428430.0001R-squared0.604286    Mean dependent var42337.15Adjusted R-squared0.582302    S.D. dependent var45279.67S.E. of regression29264.05

5、    Akaike info criterion23.50075Sum squared resid1.54E+10    Schwarz criterion23.60032Log likelihood-233.0075    F-statistic27.48740Durbin-Watson stat1.029463    Prob(F-statistic)0.0000553. GQ假设检验法首先，点击工具按钮proc选择sort cu

6、rrent page，输入X，按升序排序；去掉中间约n/4个样本点，然后对前后两个子样本分别进行回归；子样本模型一：Dependent Variable: YMethod: Least SquaresDate: 10/23/14 Time: 10:57Sample: 1 8Included observations: 8VariableCoefficientStd. Errort-StatisticProb.  C1277.1611540.6040.8290000.4388X0.5541260.3114321.7792870.1255R-squared0.345397

7、60;   Mean dependent var4016.814Adjusted R-squared0.236296    S.D. dependent var166.1712S.E. of regression145.2172    Akaike info criterion13.00666Sum squared resid126528.3    Schwarz criterion13.02652Log likelihood-50.02663&

8、#160;   F-statistic3.165861Durbin-Watson stat3.004532    Prob(F-statistic)0.125501子样本模型二：Dependent Variable: YMethod: Least SquaresDate: 10/23/14 Time: 10:57Sample: 13 20Included observations: 8VariableCoefficientStd. Errort-StatisticProb.  C212.2118530.8

9、8920.3997290.7032X0.7618930.06034812.625050.0000R-squared0.963723    Mean dependent var6760.477Adjusted R-squared0.957676    S.D. dependent var1556.814S.E. of regression320.2790    Akaike info criterion14.58858Sum squared resid615472.0 

10、;   Schwarz criterion14.60844Log likelihood-56.35432    F-statistic159.3919Durbin-Watson stat1.722960    Prob(F-statistic)0.000015根据得到的RSS1与RSS2，求得F检验统计量值。F= RSS2/RSS1=615472.0/126528.3=4.86;查F分布表，确定临界值F0.05（6,6）；若F> F0.05（6,6）则拒绝H0，认为原初始模型的随

11、机误差项存在显著的异方差；反之则认为不存在显著的异方差问题。4. 怀特检验法：打开初始模型一，点击View工具按钮，选择residual tests右拉列表选择White Heteroskedasticity Test(cross terms)White Heteroskedasticity Test:F-statistic14.63595    Probability0.000201Obs*R-squared12.65213    Probability0.001789Test Equation:Dependen

12、t Variable: RESID2Method: Least SquaresDate: 10/23/14 Time: 11:24Sample: 1 20Included observations: 20VariableCoefficientStd. Errort-StatisticProb.  C-180998.9103318.2-1.7518580.0978X49.4284628.939291.7080060.1058X2-0.0021150.001847-1.1447420.2682R-squared0.632606    Me

13、an dependent var42337.15Adjusted R-squared0.589384    S.D. dependent var45279.67S.E. of regression29014.92    Akaike info criterion23.52649Sum squared resid1.43E+10    Schwarz criterion23.67585Log likelihood-232.2649    

14、F-statistic14.63595Durbin-Watson stat2.081758    Prob(F-statistic)0.000201首先根据上方假设检验统计量及其伴随概率可知，Obs*R-squared=12.65，判断与2个自由度的卡方统计量临界值的大小关系，得出具体假设检验结果，原理类似于F检验。（二）异方差的修正方法及其分析加权最小二乘法WLS 首先点击主菜单QuickEstimate Equation，在空白区域输入模型形式Y C X，点击右上方Option按钮，选中左侧中间的WLS法，在W空白区域输入权变量1/abs(e)，回车

15、即可得到加权以后的回归模型。Dependent Variable: YMethod: Least SquaresDate: 10/23/14 Time: 11:12Sample: 1 20Included observations: 20Weighting series: 1/ABS(E)VariableCoefficientStd. Errort-StatisticProb.  C415.6603116.97913.5532880.0023X0.7290260.02242932.503490.0000Weighted StatisticsR-squared0.999895

16、    Mean dependent var4471.606Adjusted R-squared0.999889    S.D. dependent var7313.160S.E. of regression77.04831    Akaike info criterion11.62138Sum squared resid106856.0    Schwarz criterion11.72096Log likelihood-114.21

17、38    F-statistic1056.477Durbin-Watson stat2.367808    Prob(F-statistic)0.000000Unweighted StatisticsR-squared0.981664    Mean dependent var5199.515Adjusted R-squared0.980645    S.D. dependent var1625.275S.E. of regressi

18、on226.1101    Sum squared resid920263.9Durbin-Watson stat1.886959对加权修正以后的模型进行怀特异方差检验，以确定异方差问题是否消除，步骤同前。White Heteroskedasticity Test:F-statistic0.032603    Probability0.967983Obs*R-squared0.076420    Probability0.962511Test Equation:Depende

19、nt Variable: STD_RESID2Method: Least SquaresDate: 10/23/14 Time: 11:25Sample: 1 20Included observations: 20VariableCoefficientStd. Errort-StatisticProb.  C6196.48111798.680.5251840.6062X-0.1653233.304793-0.0500250.9607X24.80E-060.0002110.0227450.9821R-squared0.003821    

20、;Mean dependent var5342.798Adjusted R-squared-0.113377    S.D. dependent var3140.196S.E. of regression3313.430    Akaike info criterion19.18684Sum squared resid1.87E+08    Schwarz criterion19.33620Log likelihood-188.8684   &#

21、160;F-statistic0.032603Durbin-Watson stat2.153876    Prob(F-statistic)0.967983非常明显地判断出异方差性问题已经消除，上面加权修正后的模型即可作为最终模型。二、随机误差项序列相关性问题的检验与修正（一）建立初始回归模型相关命令：data x yscat x yls y c x 模型一：Dependent Variable: YMethod: Least SquaresDate: 07/29/12 Time: 09:48Sample: 1991 2011Included obser

22、vations: 21VariableCoefficientStd. Errort-StatisticProb.  C178.975555.064213.2503050.0042X0.0200020.00113417.641570.0000R-squared0.942463    Mean dependent var922.9095Adjusted R-squared0.939435    S.D. dependent var659.3491S.E. of regression162.2653&

23、#160;   Akaike info criterion13.10673Sum squared resid500270.3    Schwarz criterion13.20621Log likelihood-135.6207    F-statistic311.2248Durbin-Watson stat0.658849    Prob(F-statistic)0.000000 初始回归模型一经济意义合理，统计指标较为理想，但DW值偏低，模型

24、可能存在序列相关性。（二）序列相关性的四种检验方法及其分析右击resid选择Object Copy，输入e得到初始回归模型的残差序列；1. 图示法：scat e(-1) e散点图形略2. 自回归模型检验法一阶自回归为：ls e e(-1)Dependent Variable: EMethod: Least SquaresDate: 07/29/12 Time: 09:49Sample (adjusted): 1992 2011Included observations: 20 after adjustmentsVariableCoefficientStd. Errort-StatisticPr

25、ob.  E(-1)0.7170800.2018523.5524970.0021R-squared0.398929    Mean dependent var2.801737Adjusted R-squared0.398929    S.D. dependent var161.7297S.E. of regression125.3870    Akaike info criterion12.54939Sum squared resid298716.2

26、60;   Schwarz criterion12.59918Log likelihood-124.4939    Durbin-Watson stat1.080741说明模型一的随机误差项至少存在一阶正序列相关性，结合该自回归模型的DW值为1.08，怀疑存在更高阶的序列相关，继续引入e(-2)如下：ls e e(-1) e(-2)Dependent Variable: EMethod: Least SquaresDate: 07/29/12 Time: 09:49Sample (adjusted): 1993 2011In

27、cluded observations: 19 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.  E(-1)1.0949740.1787686.1251080.0000E(-2)-0.8150100.199977-4.0755130.0008R-squared0.692885    Mean dependent var7.790341Adjusted R-squared0.674819    S.D. dependen

28、t var164.5730S.E. of regression93.84710    Akaike info criterion12.02051Sum squared resid149723.7    Schwarz criterion12.11993Log likelihood-112.1949    Durbin-Watson stat1.945979由于e(-2)的t检验显著，说明模型一的随机误差项确实存在二阶正序列相关性，结合该二阶自回归模型的DW值为1.95，基本确

29、定不存在更高阶的序列相关。Breusch-Godfrey Serial Correlation LM Test:F-statistic0.888958    Probability0.431668Obs*R-squared1.998924    Probability0.368077可以看出二阶自回归模型的随机误差项不存在序列相关性，论证了原模型仅存在二阶序列相关。3. DW检验法0<DW<dL 存在正自相关(趋近于0) DL<DW<dU 不能确定 DU<DW<4dU 无自相关（

30、趋近于2）4. LM检验法原理：一方面，根据上面的假设检验结果判断是否存在序列相关性，即根据(n-p)*R2统计量值与卡方检验临界值2(P)进行比较，其中n为原模型样本容量，P为选择的滞后阶数，R2为下面辅助回归模型的可决系数。若(n-p)*R22(P)，则拒绝不序列相关的原假设，说明模型存在显著的序列相关性；另一方面，结合下面的辅助回归模型中残差滞后变量是否通过t检验及DW值判断序列相关的具体阶数，方法与上面的自回归模型检验法相同。打开初始模型一，点击View工具按钮，选择residual tests右拉列表选择Serial Correlation LM Test，在出现的对话框中选择滞后的

31、阶数，即检验模型的resid取到滞后多少期。选择滞后一阶检验：Breusch-Godfrey Serial Correlation LM Test:F-statistic13.15036    Probability0.001931Obs*R-squared8.865308    Probability0.002906Test Equation:Dependent Variable: RESIDMethod: Least SquaresDate: 07/29/12 Time: 09:51Presample miss

32、ing value lagged residuals set to zero.VariableCoefficientStd. Errort-StatisticProb.  C-14.2447243.18361-0.3298640.7453X0.0007140.0009070.7866170.4417RESID(-1)0.7632630.2104773.6263420.0019R-squared0.422158    Mean dependent var1.30E-13Adjusted R-squared0.357953 &#

33、160;  S.D. dependent var158.1566S.E. of regression126.7275    Akaike info criterion12.65352Sum squared resid289077.4    Schwarz criterion12.80274Log likelihood-129.8619    F-statistic6.575179Durbin-Watson stat1.159275  &

34、#160; Prob(F-statistic)0.007183说明原模型确实存在一阶序列相关性，结合该辅助回归模型的DW值为1.16，怀疑存在更高阶的序列相关。重复上述操作，引入滞后二阶检验如下：Breusch-Godfrey Serial Correlation LM Test:F-statistic20.49152    Probability0.000030Obs*R-squared14.84303    Probability0.000598Test Equation:Dependent Vari

35、able: RESIDMethod: Least SquaresDate: 07/29/12 Time: 09:51Presample missing value lagged residuals set to zero.VariableCoefficientStd. Errort-StatisticProb.  C14.0646332.409870.4339610.6698X-0.0006280.000742-0.8463030.4091RESID(-1)1.1084880.1761276.2936960.0000RESID(-2)-0.9181750.226004-4.

36、0626430.0008R-squared0.706811    Mean dependent var1.30E-13Adjusted R-squared0.655072    S.D. dependent var158.1566S.E. of regression92.88633    Akaike info criterion12.07027Sum squared resid146673.8    Schwarz criterion

37、12.26923Log likelihood-122.7379    F-statistic13.66102Durbin-Watson stat1.950263    Prob(F-statistic)0.000087由于e(-2)的t检验显著，说明模型一的随机误差项确实存在二阶正序列相关性，结合该二阶自回归模型的DW值为1.95，基本确定不存在更高阶的序列相关。当然可以继续引入滞后三阶检验如下：Breusch-Godfrey Serial Correlation LM Test:F-statistic12.857

38、43    Probability0.000157Obs*R-squared14.84303    Probability0.001956Test Equation:Dependent Variable: RESIDMethod: Least SquaresDate: 07/29/12 Time: 09:52Presample missing value lagged residuals set to zero.VariableCoefficientStd. Errort-StatisticProb. &

39、#160;C14.0646733.407340.4210050.6794X-0.0006280.000765-0.8209340.4237RESID(-1)1.1082060.2713274.0844010.0009RESID(-2)-0.9175590.499523-1.8368700.0849RESID(-3)-0.0006010.431119-0.0013950.9989R-squared0.706811    Mean dependent var1.30E-13Adjusted R-squared0.633514   

40、; S.D. dependent var158.1566S.E. of regression95.74504    Akaike info criterion12.16551Sum squared resid146673.8    Schwarz criterion12.41421Log likelihood-122.7379    F-statistic9.643071Durbin-Watson stat1.950030   

41、0;Prob(F-statistic)0.000363 可以看出并不存在三阶序列相关。（二）广义差分法修正1、方法原理参考教材自己推导二元线性回归模型存在二阶序列相关时的广义差分模型。2、上机实现结果分析主窗口命令区域输入ls y c x ar(1) 模型二：Dependent Variable: YMethod: Least SquaresDate: 07/29/12 Time: 09:55Sample (adjusted): 1992 2011Included observations: 20 after adjustmentsConvergence achieved after 8 it

42、erationsVariableCoefficientStd. Errort-StatisticProb.  C160.0892182.89170.8753230.3936X0.0214690.0030726.9889750.0000AR(1)0.7300780.2033523.5902230.0023R-squared0.964570    Mean dependent var958.0450Adjusted R-squared0.960402    S.D. dependent var655

43、.9980S.E. of regression130.5388    Akaike info criterion12.71870Sum squared resid289686.3    Schwarz criterion12.86806Log likelihood-124.1870    F-statistic231.4107Durbin-Watson stat1.116066    Prob(F-statistic)0.000000I

44、nverted AR Roots      .73由于AR(1)通过t检验，说明模型一确实至少存在一阶序列相关，结合DW值为1.12，怀疑存在更高阶序列相关性。点击模型二的View工具按钮，选择residual tests右拉列表选择Serial Correlation LM Test，LM检验结果如下： Breusch-Godfrey Serial Correlation LM Test:F-statistic6.380262    Probability0.009885Obs*R-squar

45、ed9.193288    Probability0.010086Test Equation:Dependent Variable: RESIDMethod: Least SquaresDate: 07/29/12 Time: 09:57Presample missing value lagged residuals set to zero.VariableCoefficientStd. Errort-StatisticProb.  C80.86347145.26430.5566650.5860X-0.0035540.002602-1

46、.3655560.1922AR(1)-0.5728410.437314-1.3099090.2099RESID(-1)1.0291570.3395413.0310220.0084RESID(-2)-0.1879230.598223-0.3141360.7577R-squared0.459664    Mean dependent var-7.24E-11Adjusted R-squared0.315575    S.D. dependent var123.4773S.E. of regression102.1528