复旦大学计量经济学讲义06联立方程计量经济学模型理论与方法

6.1²²²²²simultaneous¡equationsmodelE(X`u)6=0OLSOLS6.1()endogenousvariable6.2()exogenousvariable6.3()predeterminedvariable(LaggedEndogenousVariables)6.4()structuralmodel6.5()reduced¡formequations6.6()recursivesystem-100-6.2SimultaneousEquationsModelsI:Walras²²²²²=complete)²²(structuralform)(reducedform)(recursivemodel)(seeminglyunrelatedmodel)Yt=BYt+¡Xt+tBB(1)(2)-101-6.2SimultaneousEquationsModels(3)(4)(5)OLSI¡BYtYt=(I¡B)¡1¡Xt+(I¡B)¡1t=¦Xt+VtPhillips_Wt=®0+®1UNt+®2_Pt+u1t_Pt=¯0+¯1_Wt+¯2_Rt+¯3_Mt+u2tUN=_R=_M=_W_PUN_R_M-102-µ1¡®2¡¯11¶µ_Wt_Pt¶+µ¡®0¡®100¡¯00¡¯2¡¯3¶0BBB@1UNt_Rt_Mt1CCCA=µu1tu2t¶6.7[simultaneousbias]OLSOLSOLSy1=®1y2+¯1z1+u1y2=®2y1+¯2z2+u2y2u1OLS(3.3)AIDPSEXPAIDAIDINC;POP;PS-103-6.3(identi¯cation)6.1SAS:1PROCIMPORTOUT=LI.HAUSMAN2DATAFILE=D:nMYDOCUMENTnDATASETnPINDYCKnEX73.XLSREPLACE;3GETNAMESYES;4RUN;PROCREGDATA=LI.HAUSMAN;5MODELAID=PS;6OUTPUTOUT=WR=WHAT;7RUN;DATATEMP(KEEP=WHATPOPINCAIDEXPPS);8MERGELI.HAUSMANW;9RUN;PROCREGDATA=TEMP;10MODELEXP=AIDINCPOPWHAT;11RUN;6.2SAS:HAUSMAN12PROCMODELDATA=TEMPOUT=B;3ENDOGENOUSEXPAID;4EXP=A1+A2¤AID+A3¤INC+A4¤POP;5AID=B1+B2¤EXP+B3¤PS;6FITEXPAID/OLSSUR2SLS3SLSFIMLHAUSMAN;7INSTRUMENTSINCPOPPS;8RUN;PROCPRINTDATA=B;RUN;6.3(identi¯cation)OLSSEMy1=®1y2+¯1z1+u1y2=®2y1+¯2z2+u2z1z2(exclusiontestrictions)²²²-104-JustIdentificationOveridentificationOLSOLSYt=BYt+¡Xt+²t(6-1)Yt=¦Xt¡Vt(6-2)¦=(I¡B)¡1¡(I¡B)¦=¡(6-3)ordercondition¸¡1rankcondition=¡16-1(identifiable)(6-3)(exactlyidentification)(over¡identificaiton)-105-6.3(identi¯cation)(1)(2)(3)(4)XX=X1[X2X1X2X1X2X2X2SEM²SEM²²card(A)Acard(X2)¸g¡1;g=(6-4)card(X1)=Mcard(X2)=Ng¡1RZ1t+SZ2t=!t-106-²Ng¡1²N¸g¡1rank(S)=g¡1N¸g¡1;rank(S)=g¡1(6-5)1111²²²²fh(Yi;ZXi;®)=²ih;h=1;2;¢¢¢;g;i=1;2;¢¢¢;T(6-6)gk®SAS=ETSSYSLIN6.4Yt=BYt+¡Xt+t(6-7)-107-6.5EstimationMethods(1)B(2)tOLS(1)(2)OLSZeller(6-7)B=0OLSZellner(1)SUR(2)(3)126.5EstimationMethodsILSOLSOLS¦=A¡1BILSOLSILSOLS-108-(1)(2)(3)(1)(2)(3)OLS(4)(1)(2)ILS1(1)(2)(3)(4)IV(1)(2)n(3)IV(1)(2)(3)(4)-109-6.5EstimationMethodsIVOLS2sLSILSIVH:TheilL:Basman(1)(2)(3)lim1TX0X0(4)(5)(1)OLS(2)OLS2SLS6.18:Ct=®0+®1Yt+®2Ct¡1+¹t1It=¯0+¯1Yt+¹2tYt=It+Ct+Gt(6-8)1978¡19966.3SAS:2SLS1libnameli'd:nMydocumentndatasetnsas';datatemp;2setli.macroeco;3CL=lag(C);4ifN1;5output;6run;procmodeldata=temp;7C=a0+a1¤Y+a2¤CL;8¯tC/2SLSFSRSQ;9instrumentsYGCL;10run;-110-3sLSH:TheilA:Zeller19622sLS2sLS3sLS2sLS2sLSGLS3sLS(1)(2)(3)(4)(5)3sLS2sLS3SLS2SLS(1)2SLS(2)2SLS(3)yiT£1=YibiT£(gi¡1)£1+Xi°iT£ki£1+iT£1=¡YiXi¢µbi°i¶+i=Zi±i+iX=¡X1X2¢¢¢Xg¢y=X0­0BBB@y1y2...yg1CCCAZ=X0­0BBB@Z10¢¢¢00Z2¢¢¢0............00¢¢¢Zg1CCCA=0BBB@12...g1CCCA3sLS-111-6.5EstimationMethods(1)2sLS±i^i^±i=(Z0iX(X0X)¡1X0Zi)¡1Z0iX(X0X)¡1X0yi^i=yi¡Zi^±i(2)^§^R^§=1T0BBB@^1^2...^g1CCCA^R=^§­(X0X)(3)y=Z±+µ12¶GLS^±=(Z0^R¡1Z)¡1Z0^Ry²²²SURZELLER6.8()(seeminglyunrelatedregressionSUR)²²SUR3SLS-112-SUR26664Y1Y2...YG37775=26664X10¢¢¢00X2¢¢¢0............00¢¢¢XG3777526664¯1¯2...¯G37775+26664u1u2...uG37775(6-9)E(uiu0i)=26664¾ij0¢¢¢00¾ij:::0............00¢¢¢¾ij37775=¾ijI(6-10)G­=E(uu0)=26664E(u1u01)E(u1u02)¢¢¢E(u1u0G)E(u2u01)E(u2u02):::E(u2u0G)............E(uGu01)E(uGu02)¢¢¢E(uGu0G)37775=26664¾11I¾12I¢¢¢¾1GI¾21I¾22I:::¾2GI............¾G1I¾G2I¢¢¢¾GGI37775(6-11)­^¯=(X0­¡1X)¡1(X0­¡1Y)(6-12)E[(^¯¡¯)(^¯¡¯)0]=(X0­¡1X)¡1(6-13)­GOLS^¾ij=^ui^u0jp(N¡Ki)(N¡Kj)(6-14)i;ji6=j;¾ij=0SUROLSSURP(samplingvariability)6.4SAS:SUR1libnameli'd:nMydocumentndatasetnsas';datatemp;2setli.macroeco;3CL=lag(C);4ifN1;5output;6run;procmodeldata=temp;7C=a0+a1¤Y+a2¤CL;8I=b0+b1¤Y;-113-6.5EstimationMethods9Y=C+I+G;10¯tCI/surcovs;11run;y1=Y2b+X1°+1(y1Y2)µ1¡b¶=X1°+1(6-15)(1)OLSOLSM1:M1=(y1Y2)0(I¡X1(X01X1)¡1X01)(y1Y2)(2)OLSOLSMM=(y1Y2)0(I¡X(X0X)¡1X0)(y1Y2)(3)(M1¡¸M)¯=0¸1^¯1^¯1=Ã1¡^b!=^¯^bbLIML(4)°LIMLµ^°0¶=(X0X)¡1X0Y^¯LIML²ILS²²E(^bLIML)=16.2½Y1t=b12Y2t+°11X1t+1tY2t=b21Y1t+°22X2t+°23X3t+2t-114-t=(1t2t)0Y0Y=µ146610¶Y0X=µ230210¶X0X=0@1000100011Ab12;°11OLSOLSM1M1=¡y1Y2¢0(I¡X1(X01X1)¡1X01)¡y1Y2¢=Y0Y¡Y0x1(x01x1)¡1x01Y=µ146610¶¡µ22¶(I)¡1¡22¢=µ10226¶OLSOLSMM=¡y1Y2¢0(I¡X(X0X)¡1X0)¡y1Y2¢=Y0(I¡X(X0X)¡1X0)Y=Y0Y¡Y0X(X0X)¡1X0Y=µ146610¶¡µ230210¶I0@2231001A=µ1¡1¡15¶(M1¡¸M)¯=0µ10¡¸2+¸2+¸6¡5¸¶¯=µ00¶¸min=1¯=(1¡^b12)0^b12=3µ^°110¶=(X0X)¡1X0Y^¯^°11=¡42sLSY2OLS-115-6.5EstimationMethodsX1;X2;X3^y2=X(X0X)¡1X0y2=X0@2101Ay2OLS2sLSÃ^b12^°11!=µ^y02^y02^y02x1x01^y2x01x1¶¡1µ^y02y1x01y1¶=µ5221¶¡1µ72¶=µ3¡4¶OLSÃ^b12^°11!=µy02y2y02x1x01y2x01x1¶¡1µy02y1x01y1¶=µ10221¶µ62¶=µ1343¶k{H:Theil(1958)k¡¡OLS;2sLS;LIMLy1=Y1b+X1°+1(6-16)k1;g1k2Ã^b(k)^°(k)!=µY01Y1¡k^01^1Y01X1X01Y1X01X1¶¡1µY01¡k^01X01¶y1^1,(I¡X(X0X)¡1X0)Y1(6-17)Y1²k=0;1OLS2sLS²kLIML²kplimk=1kKinal(1980)²2sLSL,k2¡(g1¡1)L²0k1kT¡(k1+g1¡1)²k1kFIML²²-116-²AYt=¡Xt+tt=1;2¢¢¢;T(6-18)tN(0;§)f(1t;2t;¢¢¢;gt)=((2¼)g=2j§j1=2)¡1exp(¡120t§¡1t)(6-19)Lik=jAj(2¼)Tg2j§jT2exp(¡12TXi=1(AYt¡¡Xt)0§¡1(AYt¡¡Xt))6.6²OLS²OLS²²6.9excludingarelevantvariableunderspecifyingthemodely=¯0+¯1x1+¯2x2+u~y=~¯0+~¯1x1~¯1=nPi=1(xi1¡¹x1)yinPi=1(xi1¡¹x1)2yi=¯0+¯1xi1+¯2xi2+ui~¯1E(~¯1)=¯1+¯2nPi=1(xi1¡¹x1)xi2nPi=1(xi1¡¹x1)2¯2x2x1-117-6.76.10()omittedvariablebias:E(~¯1)¡¯1=¯2nPi=1(xi1¡¹x1)xi2nPi=1(xi1¡¹x1)2=¯2±²²²²OLS^¯1=¯1+n¡1nPi=1(xi1¡¹x1)uinPi=1(xi1¡¹x1