联立方程计量经济学模型案例

第六章联立方程计量经济学模型案例1、下面建立一个包含3个方程的中国宏观经济模型，已经判断消费方程式恰好识别的，投资方程是过度识别的。对模型进行估计。样本观测值见表6.1'C=a+aY+aC+u<tIO=p；pY:/1t、 tY=I+C+Gt表6.1中国宏观经济数据单位：亿元年份YICG年份YICG19783606137817594691991212807517103163447197940741474200559519922586496361246037681980455115902317644199334501149981568238211981490115812604716199446691192612081066201982548917602868861199558511238772694576891983607620053183888199668330268673215293111984716424693675102019977489428458348551158119858792338645898171998790032954636921125361986101333846517511121999826733070239334126371987117844322596115012000893413250042896139451988147045495763315762001985933746145898152341989164666095852418472001107514423554853516624199018320644491132763（1）用狭义的工具变量法估计消费方程选取方程中未包含的先决变量G作为内生解释变量Y的工具变量，过程如下:

结果如下：DependentVariable:C01Method:Two-St^geLeastSquaresDate:07AJ4/08Time:20:39Sample(adjusted):19792002Includedobsen/atians:24afteradjustmentsInstrumentlist:CGC01(■1)VariableCoefficientStd.Errort-StatisticProb.C682.2761192.86963.0190150.0065¥0.27485G0.07330S37493240.00120.4321240.1702202.53S6150.0191R-squared0.998744Meandependentvar17685.46AdjustedR-squared0.998624S.D.dependentvar16106.62S.E.ofregression597.4201Sumsquaredresid7495127.Durbin-Watsonstat0.851369Second-stageSSR3B660181所以，得到结构参数的工具变量法估计量为：&=582.2761,&=0.274856,&=0.4321240 1 2（2）用间接最小二乘法估计消费方程消费方程中包含的内生变量的简化式方程为：fC=兀+兀C+兀G+£

\Y=兀+兀C+兀G+£Vt20 21t-1 22t 21参数关系体系为:‘兀一a兀-a=0〈兀-a-a兀=0兀一a兀=0I12 122用普通最小二乘法估计，结果如下：DependentVariable:C01Method:LeastSquaresDate:07^04^18Time:20:48Sample(adjusted):19792002Includedobservations:24afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.C1135.937440.3420 2.5796700.0175C01(-1)0.6197S20.273112 2.2693310.0339G1.2398980.751061 1.6508610.1136R-squared0.993521Meandependentv/ar17685.46AdjustedR-squared0.992904S.D.dependentvar16106.62S.E.ofregression135S.820Akaikeinfocriterion17,38014Sumsquaredresid386601S1Schwarzcriterion17,52740Loglikelihood-205.5617F-statistic1610.049Durbin-Watsonstat0.661227Prob(F-statistic)0.000000DependentVariable:YMethod:LeastSquaresDate:07/T]4^iSTime:20:49Sample(adjusted):19792002Includedobsen/ations:24afteradjusimentsVariableCoefficientStd.Errort-StatisticProb.C2014.3681036.7351.9429920.0655C01(-1)0.6827500.6430121.0618000.3004G4.5110841.76S2B92.5511010.01S6R-squared0.992382Meandependentvar37485.33AdjustedR-squared0.991656S.D.dependentvar3497177S.E.ofregression3194.479Akaikeinfacriterion19,09270Sumsquaredresid2.14E-HJ6Schwarzcriterion19,23996Laglikelihood-226.1125F-statistic1367765Durbin-Watsonstat0729172Prob(F-statistic)0.000000所以参数估计量为：元=1135.937,元=0.619782,元=1.23989810 11 12元20:2014.368,元烈=0.682750,元22:4.511084所以，得到间接最小二乘估计值为：八a=，=0.2748561 7122&=元-0C元=0.4321242 11 121&=元-0C元=582,2758

0 10 120（3）用两阶段最小二乘法估计消费方程第一阶段使用普通最小二乘法估计内生解释变量的简化方程，得到丫=2014.368+0.68275c+4.511084Gt t—1 t用Y的预测值替换消费方程中的Y，直接用OLS估计消费方程，过程如下:

也可以用工具变量法估计消费方程，过程如下:结果如下:DependentVariable:C01Method:Two-StageLeastSquaresDate:07/04/08Time:21:03Sample(adjusted):19792002Includedobseivations:24afteradjustmentsInstrumentlist:CYFC01(-1)VariableCoefficientStd.Errort-StatisticProb.C582.27G1192.06963.0190150.0065Y0.2748560.0733003.7493240.00120.4321240.1702202.5306150.0191R-squared0.998744Meandependentvar17665.46AdjustedR-squared0.998624S.D.dependentvar16106.62S.E.ofregression597.4201Sumsquaredresid7495127.Durbin-Watsonstat0.851389Second-stageSSR38660181综上所述，可知道，对于恰好识别方程，三种方法得到的结论是一样的。（4）用两阶段最小二乘法估计投资方程，过程同上。（5）投资方程是过度识别的方程，也可以用GMM估计，选择的工具变量为先决变量C01、G。估计结果如下:■Equation:UNTITLED¥orkflie:弟一题二二Uirt [yiew][procllobject][PrintIfName][Freeze!fE5timate][Ferecsst[stats[ResidsDependentVariable:IMethod:GeneralizedMethodofMomentsDate:05/04Z09Time:22:14Sample(adjusted):19792002Includedobservations:24afleradjustmentsKernel:Bartlett,Bandwidth:Fixed(2),NaprewhiteningSimultaneousweightingmatrix&coefficientiterationConvergenceachievedafter:8weightmatrices,9totalcoefiterationsInstrumentlist:CC01(-1)GVariableCoeificientStd.Errort-StatisticProb.C-139.5888157.1465-0.8882720.3840V0.3828140.005292 72.343360.0000R-squared0.996353Meandependentwar14318.54AdjustedR-squared0.996187S.D.dependentvar13464.79S.E.ofregression831.4584Sumsquaredresid15209108Durbin-Watsonstat0744531J-statistic0.039126与2SLS结果比较，结构参数估计量变化不大。残差平方和由变为，显著减少。为什么？利用了更多的信息。2.以表6.2所示的中国的实际数据为资料，估计下面的联立模型。y=p+^m+yc+yI+uM=a+aY+yP+u表6.2年份货币于准货币M2/亿元国内生产总值GDP/亿元居民消费价格指数P(1978为100居民消费CONS/亿元固定投资I/亿元199015293.418319.5165.29113.24517199119349.921280.4170.810315.95594.5199225402.225863.7181.712459.88080.1199334879.834500.7208.415682.413072.3199446923.546690.7258.620809.817042.1199560750.558510.5302.826944.520019.3199676094.968330.4327.932152.322913.5199790995.374894.2337.134854.624941.11998104498.579003.3334.436921.128406.21999119897.982673.1329.739334.429854.72000134610.389112.533142911.932917.7建立工作文件后，进行如下步骤:®EVie»61

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E-ObjectViewProcQuickO^ticmsWindowHelpNewObject...FetchfrumDB...UfidatefrumDB...SturetuDB...CopyUbject...N：diiie...Delete妈Rasa画区SSSMSTorkfile:UITTITLED::Untitled\jFreeze]Defaultv(Sort)Tr•日n5P归dit+/J5mpi+/-][ln£CONS11996302.50001996327.90001997337.10001998334.40001999329.70002口口口331.0000FreezeOutputPrintViewUptiore15682.4020809.8026944.5032152.3034854.6036921.1039334.4042911.9013072.3017042.1020019.3022913.5024941.1028406.2029854.7032917.70建立联立模型，并命名为MYNevObjectTj^ieci£objectN：djrieforubjectEj-sierTiMY|EquationGraphGroupLc-gLMatrik-Vector-Coe:ModelFoolE：uiipleSeriesSeriesLinkSeriesAlphaSSpace【郎一1C；iiicelSystemTableTentValM^pW在SYSTEM窗口里面定义联立方程组和使用的工具变量。DSyste>:IT Torkfile:UKTITLED:ilTntitledY 口叵]回YiE网][Prcii:][0bjEd:] [riErqrT已笈1:]归5匕并己1:4]}口「匚][51：0|：弓][区)引11|5]gdp=c(1)+cp)*m2+c(3)*ccns+c(4)*im2=c[5}+c(E)*gclp+c(7昨instconsipc|一选择两阶段最小二乘法进行估计。•口匚[Objed]print:flethoi：IlcrrLelins+rirriHintetiirLinRarQuadrAndrews^3]2|I'lentityweightingrnatriKinestimation.C2SLScoe£s&GUNgdn二c1〕+cQ『m2+eg]*cnns+c：C4『iEstimatJE==d-+l^ttBandwidth1*■Fikhd得到如下输出结果:System:MYEstimationMethod:Two-StageLeastSquaresDate:07^05/08Time:12:38Sample:19902000Includedobservations:11Totalsysiem(balanced)observations22CoeificientStd.Errori-StatisticProb.C⑴-1306.303726.9939-1.7968560.0925-0.1506320.028877-5.2127950.0001C⑶2.063E400.15396713,403160.0000酒口一期心花0.1664904.1224240.0009C⑸43243.344524.6759.5572250.0000C⑹2.9381200.09B57929.B04780.0000C[7)-511.412735,63530-14.351260.0000Determinantresidualcovariance5.S4E+11Equation:GDP=C(1>C(2)*M2-bCp)*CQNS+C(4riInstruments:CONSIPCObservations:11R-squared0.999590Meandependentvar54470.82AdjustedR-squared0.999425S.D.dependentvar26264.90S.E.ofregression630.2510Sumsquaredresid2700522.Durbin-Watsonstat2.201170Equation:M2=C(5)+C(B)*GDP-bC(7)*PInstruments:CONSIPCObservations:11R-squared0.990410Meandependentvar66245.11AdjustedR-squared0.990023S.D.dependentvar41923.75S.E.ofregression1064.293Sumsquaredresid27004720Durbin-Watsonstat1.290522所以得到联立方程计量经济学模型的估计表达式为：Y=-1306.3-0.151M+2.064C+0.6861M=43243.34+2.938Y—511.413P3、以Klein（克莱因）联立方程模型为例介绍两阶段最小二乘估计。首先建立工作文件，数据如表7。表6.3 Klein联立方程模型数据年份CCPPWPIIKKXXWGGGTTAA192039.812.728.82.7180.144.92.22.43.4-11192141.912.425.5-0.2182.845.62.73.97.7-1019224516.929.31.9182.650.12.93.23.9-9192349.218.434.15.2184.557.22.92.84.7-8192450.619.433.93189.757.13.13.53.8-7192552.620.135.45.1192.7613.23.35.5-6192655.119.637.45.6197.8643.33.37-5

192756.219.837.94.2203.464.43.646.7-4192857.321.139.23207.664.53.74.24.2-3192957.821.741.35.1210.66744.14-219305515.637.91215.761.24.25.27.7-1193150.911.434.5-3.4216.753.44.85.97.50193245.6729-6.2213.344.35.34.98.31193346.511.228.5-5.1207.145.15.63.75.42193448.712.330.6-320249.7646.83193551.31433.2-1.319954.46.14.47.24193657.717.636.82.1197.762.77.42.98.35193758.717.3412199.8656.74.36.76193857.515.338.2-1.9201.860.97.75.37.47193961.61941.61.3199.969.57.86.68.9819406521.1453.3201.275.787.49.69194169.723.553.34.9204.588.48.513.811.610建立Klein联立方程组，模型如下：CC=a°+a1PP+a2PP(—1)+aJWP+WG) (消费方程)II=P°+P1PP+P2PP(-1)+P3KK (投资方程)WP=y0+y1XX+y2XX(-1)+y3AA (私人工资方程)XX=CC+II+GG (均衡需求恒等式)PP=XX-TT-WP (私人利润恒等式)KK—KK(-1)+II (私人存量恒等式)使用的工具变量是：WGGGTTAAPP(-1)KKXX(-1)C过程如下:^EVieTS工iL曰Edit01ject白曰廿 史口ickU£ticnsjVindowHelj口Torkf|NewObject...队加eLwfDEJ回国c?erLsrateS匕r1・[viewllProcfEre:diLinks...Et匚h[sture]口号1号1:&[Gerir白|Range:1Sample:1FetchtrumDIL..DisplayFilter:*KyHaaUpiiiateselectedfrorTiHE...■StoreselectedtoDB...SccCopye&1ected...SggEerL：arieselected...SkkBeltiteeelected^PP必residPrintSelwutK翦旧Vu'pSxx选择System,并起名为KleinModelKevObject旦犯之forotjsetSyztem一£闻口品1|Equatilir<Tf：iplLGrcmiiLogLNatriSctor-C口白:ModelPoolS：ajripleSeriesSeriesLinkSeriesAlphaSSpace1要1C：iXLcelSystemTableTestValMapYAK在窗口空白处输入方程指令，只要求写行为方程（前3个方程），不需定义方程（后3个方程），最后一行命令列出的是所用工具变量。DSyst»:ELEinODEL Torkfile:CASET::Case7\□回国[Vi日冏]0bjenlj[print •日日2可加日「q日丁exl][Estmate][弓口曰匚][5tatw)R日二ick]cc=c(1)-fc(2)*p口十二⑶*pp(-1)十c(4)*(wp+wg)ii=c(5)+c值「口口十 )十c(Grkkwp=c⑶十c(10)*xx+c(11)*x虻1)十c(12)*aainstwgggttaapp(-1)kkxx(-1)c|对联立方程进行估计：点击system窗口上的estimate键选择2TSLS即两阶段最小二乘估计Eitimationsettingslisthod：Eitimationsettingslisthod：Urdin:aryLe：ie+.Equare三Ordin=±t-y-Least5qn：=Lree“elghtnilL.S.(equationwelghtejSeeminp;lyUrn-elatedRepi-eseiunTwo-StatreLeast5如打电三W^igh-tei Lem三tSquar匕二Ttii-titi-Stagt：LtastEiiu:iTtiEF】iL1InfciTiiticTL LikelihcGNM-CraeeSecti-：inlytLitecov.JGTM1-Ti。54inQ£A门TimeseriesHACspeciticatiuIFr^'i'i'tiiteruingbyTAJi(KernelB：=lttleitQuadraticBarulvid七k■„. ,IINuiriberorITA1-*-Fixed|nwI£*A-riilrews『ariwbl且一B电*刊「中比SystgmEstiMationEstimationMethodJteratiOILOptions确定 取消得到如下Klein联立方程的估计结果：CoefficientStd.Errort-StatisticProb.匚⑴16.5M761.46797911,277250.0000C⑵0.0173020.1312050.1318720.895EC⑶0.21E2340.1192221.0137140.075EC⑷0.8101B30.04473510,110690.0000源20,278218.3832492.4188960.0192C⑹0.1502220.19253407802370.43890.6169440.1809263.4043900.0013-0.1577880.040152-3.9297510.0003涧1.5002971.2756061.1760700.2450C(10)0.4380590.03960311,001550.0000C(11)0.1466740.0431643.3900630.0013C(12)0.1303960.0323084.0260010.0002Determinantresidualcovariance0.2877UEquation:CC=C(1)+C(2rPP+CprPP(-1)+C(4)*(WP+WG)Instruments:WGGGHMPP(-1)KKXX(-1)CObservations:21R-squared0.976711Meandependentvar53,99524AdjustedR-squared0.972601S.D.dependentvar6.860866S.E.ofregression1.135659Sumsquaredresid21,92525Durbin-Watsonstat1.485072Equation:H=C[5>C(BfPP+Cpj*PP(-1)+C(8)*KKInstruments:WGGGnAAPP(-1)KKXX(-1)CObseivatiors:21R-squared0.884BB4Meandeperdentvar1.266667AdjustedR-squared0.864569S.D.dependentvar3.551948S.E.ofregression1.307149Sumsquaredresid29,0468EDurbin-Watsonstat2.086334Equaiion:WP=C(9)+CC166<+311)%(+1〕+式12月以Instruments:WGGGnMPP(-1)KKXX(-1)CObservations:21R-squared0.907414Meandependentvar36,36190AdjustedR-squared0.985193S.D.dependentvar6.304401S.E.ofregression0.767155Sumsquaredresid10,00496Durbin-Watsonstat1.963416上述输出结果与线性单方程分析相同。1.对联立方程组进行预测联立方程的预测是以上述估计结果为基础进行的。主要分为3步:第1步：建立模型需EVievs-[Syst»:ELEIN1ODELforkfile:CASE7::CeFileElitOlijectViewProc取iLckOptionsView乒而31口昧仕][PririH[Qime[「r比比][MerqeT已工t]EstimateWiniowHelpJEp&zJEta信][r■已引dvS'FileElitOlijectViewProc取iLckOptionsView乒而31口昧仕][PririH[Qime[「r比比][MerqeT已工t]EstimateWiniowHelpJEp&zJEta信][r■已引dvS'黄EstDatSarInclfiliiieReziiiu：il5MiikeErLiiogyrLunsGroupM：=LkeModeltSquaresE3CoeificientStd.Errort-StatisticProb.C⑴16,554761.46797911,277250.0000C(2)0.0173020.1312050.1318720.8956c⑶0.2162340.1192221.8137140.0756Update匚口已frtmSystem出现如下对话框:TotalsysiemtDaianceajODserv-anons□Model:UNTITLEDTorkfile:CASET::CaseT\□回匠科随口匚]01：|缸1:||Prinl:[NaiTe|『rBEEE|[5口I中日|归quaticinEjN日ri日bl「二Equations:3国KLEINMODELTestBaselineEqT.cc,ii,wp=F(aa,kk,pp,wg,wp,mm)输入需要加入的定义方程:第2步：输入定义方程由于在设定联立方程时没有输入定义方程，因此在求解模型时应该加入，否则，模型只识别System中设定的内生变量。加入定义方程的方法如下：IodelSourceEditEntei-oneqifrioxelirL^s；sx=cc+ii+ggpp=KH-tt-wpkfe=kkFl〕+ii|□回国N厄国J(PrciuJ[口bj日武](PrintNanreJFreeze)[Solve F Equatici*VariablesT日即Equations:6Baseline这时模型窗口如下:TKTTKTTKT=F(co,gg,ii)=F(tt,wp,KK)=F(ii,kk)ii,wp=F(aa,kk,pp,wg,wp,kx)□lodel:UNTITLEDWorkfile:CASE?::Case7\"xx=cc+ii+gg"Eq1:"pp=kk-tt-wp"Eq2PP"kk=kk(-1)+ii"Eq3:kkKLEINMODELEq4..CC,第3步：求解模型口回因[view][procObject][PrintNameJFreeze)[Solve)|EquationsVari