经典线性回归模型的诊断与修正

经典线性回归模型的诊断与修正1、普通最小二乘法回归结果如下：[=]Equation:EQ01Wbrkfile:04:;Untitled\-DX[viewjProc□bject||Print]Name[Freeze[Estimate[ForecaitstatiDependentVariably.GDPMethod'LeastSquaresDate:12/Oft'17Time:14:50Sample.19962015Includedobservations-20VariableCoefficrentStd.Errort-StatistirPrabc兀906.別39^150638.405953O.ODOOPI1.1753630.03633132.351130.0000R-squared09030&2Meandependentvar2S37&B3AdjustedR-squared0.982163S.D.dependentvar20B3&3.1SEofregression270270^1Akaikeinfocnterion2340010Sumsquaredresid139E+10Schwarzcriterion23.49967Loglikdihaod-2320010Hannan-Quinnenter.23.41954Fstatistic1046.599Durtiin-Watson5tat0.365276ProbfF-statistic^0.000000方程初步估计为：GDP=+1数据来源于国家统计局网站年度数据

7F=DW=2、异方差的检验与修正从上图可以看出，残差平方和与解释变量的散点图主要分布在图形的下半部分，有随PI的变动增大的趋势，因此，模型可能存在异方差。但是否确定存在异方差，还需作进一步的验证。G-Q检验如下：去除序列中间约1/4的部分后，1996-2003年的OLS估计结果如下所示：[=]Equation:UNTITLEDWorkfile:O4：:Unti-tted\■口淇[vfe^lProcObjectPrint]Namefreeze|Estimateforecast[statsResideD即endentVariable:GDPMethod:LeastSquaresDate:01；04/18lime:20:59Sample:19B€2OD3Ineludedobservations'8VariableCoefficientStd.Errort-StalisticProb.C29637.515357.5665一53IT990.0015PI2.0357530.14935913.63&050.0000Rsquared0.968714Meandependentvar99696.^9AdjustedRsquared0.963500S.D.dependentvar22372.04St.ofregression4274.201AkaikeinfDcriterion19.7rOSOSumsquaredresid1.1OE+O0Schwarzcriterion1979076Loglikelihood-7708360Hannan-Quinnertter19.&3695F-statistic1857783Durbin-Waisonstat1.097903Prob(F-5tati5tiD}0.000010残差平方和RSS]=.

2008-2015年的OLS估计结果如下：[=]Equalion:UNTITLEDWor^ile:&4:Untilied\-nxVfewkrocOtjtctPrin:NameFreezeEstimateForecastSlatsResideDependentVariableGDPf4ethod:LeastSquaresCate:01/04/1BTiine:21:C,Sample:2&D82015Includedobservations:8VanableCoefficientStd.Lrrort-StatisticPrab.C16237Z120372.637.09463200002PI0.9583220.05354117.89873.O.OOQOR-squared0.991616Meandependentvar&0494S.7AdjustedR-squared0.97B662S.D.dependentvar136279.0S.E.cfregression19958.44Afcaikeinfocriterion22.8&301Sumsquaredresid2.39E+09Schwarzcriterion22.87287Loglikelihood-8941204Hmnnan-Quinnenter.22.719061-statistic320.3646DurbirrWatsonstat1.1276&2Probil-statistic}0.000002残差平方和RSS2=+09.F=':-■-根据G-Q检验，F统计量为F=':-■->F0.I”|----因此，在5%的显著性水平下拒绝两组子样本方差相同的假设，即存在异方差。Gleiser检验结果如下[=1Equation:UNTITlED\Atarkfile:G4::Lntided\-nx|View][Proc]Object|[3rinljNameFreeze|Estimate[Forecast[s-tats{Relids]DependentVariable:LOG(E2)P/lethod:LeastSquaresDateD1/O4?18Time204SSample(adjusted)'200-12011Includedobservations:1afteradjustmentsVariableCoefficienLStd.Errort-SlatieticProb.C-1&014677.50763325060050.0541LOG(PI)2.3020050.63156037988820.0128R-squared0.741IB7TMeandependentvar9.606314AdjustedR-squHrsd06&00133Ddependent,var1.504728S.L.ofregression0.037700Akaikeinfoenterion2.71893+Sumsquaredresid3.509375Schfwarzcrrterian2.703J79Loglikelihood-7.515917Hannan-Quinnerrter.2.527822F-statistic11435563Durbin-Watsnnstat2093248FTDb(F-statistjc)0.012773参数的估计值显著地不为0，则可以认定模型存在着异方差。

异方差的修正：运用加权最小二乘法对异方差进行修正[=]Equation:JMT1TLEDWorkfi1e:04::Untiiled\-BXViewProc|ObjEctPrintNameFreezeEstimateFare(:日wtStatsResidsDepentlentVariableGDPMethod.LeastSquare^Date-D1W10Time'21'11Sample.2008201｛；ndudedobservations:9Weightingseries.WWeighttype'Inversestandarddeviationdetfaulrscaling)Whiteheteroskedastitity-cjnsistentstandarderrors&covarian匚^VariableCoefficientStd.Errorl-StatisticPnab.C144704.013141.2811.011410.0000Fl10128790.<M163724.326550.0000WeightedStatisticsR-squared0.976332Meandependentvar455593.2AdjustedR-squared0972445SCdependentvar6482590S.E.ofregression30746.47Akaikeinfocriterion2£.93046Sumsquaredresid258E+09Schwarzcriterion2295032Loglikelihood-30.72193Hannan-QuinnLriter.22.79651F-statistic240D4O2Durbin-Watsonstat1.540263Prob(F-slatiStic)0.000004Weightedmeandep.412667.7WaldF-statistic5917000Prob(WaldF-statistic)0.000000UnweightedStatisticsR-squaredAdjustedR-squaredSEofregressionIDurbin-Watsonstat09783290.97471721669.(M1.040448MeandependentvarS.D.dependsntvarSumsquaredresi-5049+5.713627D.02«2E*09对加权后的模型进行异方差检验，结果如下：Proc|ObjectPnntNameFreez-sEstimateForecastStatsResidsHetero5keda5ticrtyTest.White[=1Equation：UNTITLEDVtorkfile:04^Jntitled\Hetero5keda5ticrtyTest.WhiteF-statistjcOb5*R-squaredScaledexplained£S2.1J12453.&B15O71.232342Prob.F(2?5)Prob.Chi-Square^ProbChi-&quare(2)0.2.1410.158705400TestEquationDependentVariable:WGT_RJESIDX2Miethod.LeastSquaresDateD1/M/19Time:21:13Sample:200B2015Ineludedobservations.ElWhiteheteroskedasticity-consist-entstandarderrors&cowarianceColhn&artestregressorsdroppedfromspecificationVariableCoefficientStd「「rart-StatisticProbCPPWGP2WGTX2-t58E+091206315-1,55E+098.07E^O8-11.75677559254Q52.0350167.43E+0S-2.0934910.13-93009740,0905R-squaredAd|ustedR-squaredS.Eofregr-essionSumsquaredresidLoglikelihoodF-stgtisticPiiobfF-statistie)O46D1090.24426+J.27I-+065.3EE+17-W<j.J2212131245G214G95MeandependentvarS.D.dependentvarAkaikeinfocnienonSchwarzcriterionHannan-Quinncriter.Durbin-Watson5t3t3.23E4083.76E+0B42.33053423603242.129602994540View已知White统计量nF'=3-6H2,由于：.-:'：'>,因此，可以判断在给定显著性水平的情况下，加权后的模型不再存在异方差，说明异方差性已经消除。3、序列相关性的检验与修正序列相关性的检验如下：做残差与残差滞后一期的散点图：[ill]Graph:6RAPH02Workfile:04-::Untitled\-OXView|pracObjectPrintNameFre^zsOptionsUpdateAddTeKtLine/shadeRemove50DOOD40,000-30.000-20,000-•二W.OOO-0--10,000--2D.DOO--30.000--11111-60,000-4&,00G-20,000020,00040:00060.0W可以看出，E与E（-1）逐渐合拢，因此残差与其滞后一期的残差存在序列相关性。D-W检验：从0LS估计结果中可直接得到DW值为，给定a=，已知n=20,k=2，查DW检验临界值表可得，出丄L.]_41,由〈可知模型存在正自相关。序列相关性的修正：”.-■.3653利用广义差分法，由于，有估计结果如下:旦Equation:EQ04Wortcfile:34::Unlitled\[jiewProclobject"intNameFreeze三stimateForecastStatsResideDependentVariableDYMethod'LeastSquaresDate:12r07/17Time:09:57Sample(adjusted}:199?2015Includedobservations:19afterad|ustjnentsVariableCaefficierrtStdFrrart-StatisticProb22631.611.009500&002.S70^1.6237050.Q665S21&.161670.0003Q.OOODR-squaredAdjusledR-squaredFirnfregressionSumsquarednesicLoglikelihoodFstatisticProb(F-st3tistic)0.93114009270S914355143.50£^0&-2077636229.3763-O.OUDOOOMeandependentvarSDdependentvarAkaikeinfocriterionSchwarzcritenonHannan-Quinncrite匚Durtoin-Watsonslat79730.6953163172200090221603222.097731.576941在新序列估计结果下，丸-•〈DW二〈4-九-：-•，：•--■-^,故不再存在序列相关性，自相关已经得到消除。运用Cochrane-Orcutt迭代法进行自相关的修正：r[=]Equation:EQO2Workfile:D4::Untitled\_nxVieir\:PrueObjectPrintNameFreez