复旦大学计量经济学讲义

1.1EconometricR:FreshEconometricsR:Frish1926Biometrics193012»29Econometrica1933(20-40)17Newton¡Leibniz191809Gauss182119Galton20FisherR:1890¡1962NeymanJ:D:Frischbiometrics19261930FrischTinbergenFisher19331Econometrica304050-701950KoopmanKoopman¡Hood-1-1.1Klein1921»194119501928»19501955207070LinkProject1987782L:Klein198020306080-9070Granger¡Newbold1974Box¡Jenkins1967Cooper1972(1)(2)(3)Dickey¡Fuller1979DFADFPhillips¡Perron1988ZSargan1964Hendry¡Anderson(1977)Davidson(1978)Hendry1980SimsVAR1987Engle¡GrangerGranger1988¡1992JohansenVEC1960-2-197919841980Klein199881.1()CCA:Cowles(CC)CCCCCC:(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)1I0CCCCCCCC19601950CCCCLucas(1976)Lucas:T:Sarget(1976)Lucas2Lucas-3-1.2Sims(1980)VARVARCCCC1I(0)1CCCC1960(1)CCT:HaavelmoOLS(2)Haavelmo1943(3)TKoopmans1949(4)KoopmansRubinLeipnik(5)Lucas1976CLM(6)Sims1980VAR1.2123121.3-4-1.1()(1)(2)(3)(4)(5)1.2()1.3()1.4()(1)(2)(3)(4)(5)(6)(7)(8)1.2()-5-1.4-1.4(1)(2)30(1)(2)(1)-6-(2)MAARARMA(3)(4)D.F.HendryHendry506070(5)C-DGMM;GeneralizedMethodofMomentsMM;MethodofMoments-7-1.4PanelDataModelwithDiscretedependentVariableProbitLogitLimitedDependentVariableDurationModelClassicalEconometrics2070R:FrishT:HaavelmoL:R:Klein20702000penalJ:HeckmanD:McFaddan-8-2.1µ°=g(µ)g(_)°^°=g(^µ)g(µ)Var(^°)°=WTµ^°=WT^µVar(WT^µ)=E(WT(^µ¡µ)(^µ¡µ)TW)=WTE((^µ¡µ)(^µ¡µ)T)W=¾2WT(XTX)¡1Wg(µ):f0(x)=f(b)¡f(a)b¡af(a+h)=f(a)+hf0(a+¸h)(2-1)hpf(a+h)=f(a)+p¡1Xi=1hii!f(i)(a)+hpp!f0(a+¸h)f(a+h)»=f(a)+hf0(a)+h22f0(a)f(x)[a;a+h]p^µpnn12(^µ¡µ0)!N(0;V(^µ))(2-2)g(^µ)µ0^°»=g(µ0)+g`(µ0)(^µ¡µ0)(2-3)n!1nn1=2:n1=2(^°¡°0)ag`(µ0)n1=2(^µ¡µ0)(2-4)^°S°´¯¯¯g0(^µ)¯¯¯Sµ(2-5)2.2COLSRichmand1974-9-2.3Y=AK®L¯e¡ulnY=lnA+®lnK+¯lnL¡uln^Y=(^a¡^u)+^®lnK+^¯lnL^u=max(lnYi¡ln^Y)lnY¤=^u+ln^Y2.3Q=nXi=1(Yi¡^Yi)2(2-6)^¯0^¯1#=·TPxtPxtPx2t¸¡1·PytPxtyt¸=1TPx2t¡(Pxt)2·Px2t¡Pxt¡PxtT¸·PytPxtyt¸(2-7)2.4FeasibleGeneralizedLeastSquaresY=XB+N(2-8)E(N)=0Cov(NN0)=E(NN0)=¾2­­=26664w1w12¢¢¢w1nw21w2¢¢¢w2n...wn1wn2¢¢¢wn37775=DD0(2-9)D¡1D¡1Y=D¡1XB+D¡1N(2-10)Y¤=X¤B+N¤(2-11)­OLS­:^B=(X0^­¡1X)¡1X0^­¡1Y(2-12)FGLS-10-­µ­=266641½¢¢¢½n¡1½1¢¢¢½n¡2............½n¡1½n¡2¢¢¢137775(2-13)µ­FGLS­OLS­^­=26664~e21~e1~e2¢¢¢~e1~en~e2~e1~e22¢¢¢~e2~en............~en~e1~en~e2¢¢¢~e2n37775(2-14)2.5PartitionedRegressionY=X1B1+X2B2+N·X01YX02Y¸=·x01x1x01x2x02x1x02x2¸^B1^B2#(2-15)^B1=(X01X1)¡1X01Y¡(X01X1)¡1X01X2^B2=(X01X1)¡1X01(Y¡X2^B2)(2-16)X01X2=0^B1=(X01X1)¡1X01Y^B2=(X02X2)¡1X02Y-11-2.6PartialRegression2.6PartialRegressionX02Y=X02X1^B1+X02X2^B2X02Y=X02X1((X01X1)¡1X01Y¡(X01X1)¡1X01X2^B2)+X02X2^B2=X02X1(X01X1)¡1X01Y¡X02X1(X01X1)¡1X01X2^B2+X02X2^B2X02(I¡X1(X01X1)¡1X01)Y=X02(I¡X1(X01X1)¡1X01)X2^B2(X02(I¡X1(X01X1)¡1X01)X2)¡1X02(I¡X1(X01X1)¡1X01)Y=^B2(I¡X1(X01X1)¡1X01)=M^B2=(X02MX2)¡1(X02MY)M^B2=(X02M0MX2)¡1(X02M0MY)=(X¤2X¤2)¡1(X¤2Y¤)Y¤=MY=(I¡X1(X01X1)¡1X01)Y=Y¡X1(X01X1)¡1X01YYX1X2X1^X2=X1((X01X1)¡1X01X2)X2¡^X2=X2¡X1((X01X1)¡1X01X2)=(I¡X1(X01X1)¡1X01)X2=MX2=X¤2(2-17)2.72.1()y1=¯0+¯1y2+¯2z1+u1y2z1z2z3exclusionrestrictionIVIVy2y2reducedformequationy2=Á0+Á1z1+Á2z¡2+Á3z3+v2OLS^y2y2IVIVtwostageleastsquaresestimatorz2z3IV-12-2.8AcrossRegression0lnq=®0+®1lnI+®2lnp+¹lnqj=a+®1lnIj+¹jj=1;2;¢¢¢;m^®1lnqt=®0+®1lnIt+®2lnpt+¹tyt=lnqt¡^®1lnItyt=®0+®2lnpt+¹t^®1-13-2.92.9GMM:MLMLnnn2.2()PDFPDFPDFf(y;µ)=nYt=1f(yt;µ)(2-18)MLMMMLMLEY=X¯+UU»N(0;¾2I)X(2-19)XYN(X¯;¾2)YtPDF:ft(yt;¯;¾2)=1¾p2¼EXP(¡(yt¡Xt¯)22¾2(2-20)¶(y;¯;¾)=¡n2log2¼¡n2log¾2¡12¾2(y¡X¯)T(y¡X¯)(2-21)¯¾ML¾¶(y;¯;¾)¾¾concentratedloglikelihoodfunction¯¯MLOLS¯MLOLSMLMLML£(µ)-14-MLg(y;^µ)=0g(y;µ)gi(y;µ)´@¶(y;µ)@µi=nXt=1@¶(yt;µ)@µiMLµ(j+1)=µj¡H¡1(j)g(j)(2-22)µ(j+1)=µj+®D¡1(j)g(j)(2-23)MLML2.1(Jensen)Xh(_)E(h(X)·h(E(X))h(_)XXL(µ¤)=L(µ0)µ0µ¤E0logL(µ¤)L(µ0)logE0L(µ¤)L(µ0)(2-24)E0µ0DGPyE0L(µ¤)L(µ0)=ZL(µ¤)L(µ0)L(µ0)dy=1E0logL(µ¤)L(µ0)=E0¶(µ¤)¡E0¶(µ0)0(2-25)()plimn!11n¶(µ¤)·plimn!11n¶(µ0)(2-26)µ¤6=µ0MLE¶(µ)plimn!11n¶(^µ)¸plimn!11n¶(µ0)(2-27)plimn!11n¶(^µ)=plimn!11n¶(µ0):µ¤6=µ0plimn¡1¶(µ¤)6=plimn¡1¶(µ0)kµ:f(yn;µ)=nYt=1f(ytjyt¡1;µ)(2-28)-15-2.9¶(yn;µ)=nXt=1¶t(yt;µ)(2-29)yn¶tµiytytn£kG(y;µ)Gti(yt;µ)´@¶t(yt;µ)@µi(2-30):gi(y;µ)=nXt=1Gti(yt;µ)(2-31)G(y;µ)G(y;µ)yµ(LIML;LimitedInformationMaximumLikelihood)AndersonRubin1949BY+¡X=NY1=¯12Y2+¯13Y3+¢¢¢+¯1g1Yg1+°11X1+°12X2+¢¢¢+°1k1Xk1+N1Y1=(Y0;X0)µB0¡0¶+N1Y0=£Y2Y3¢¢¢Yg1¤=26664y21y31¢¢¢yg11y22y32yg12.........y2ny3nyg1n37775X0=hX1X2¢¢¢Xk1i=26664x11x21¢¢¢xk11x12x22xk12.........x1nx2nxk1n37775B0=26664¯12¯13...¯1g137775¡0=26664°11°12...°1k137775-16-Y1=26664y11y12...y1n37775N1=26664¹11¹12...¹1n37775(Y10;X0)µB10¡0¶+N1=0Y10=(Y1;Y0)B10=µ¡1B0¶(2-32)Y10=X¦10+E10(2-33)LnL(Y10)=c+n2ln¯¯­¡10¯¯¡12tr(­¡10)(Y10¡X¦10)0(Y10¡X¦10)(2-34)(Y10¡X¦10)0(Y10¡X¦10)(2-35)^¦10(FullInformationMaximumLikelihood;FIML)RothenbergLeenders1964FIMLFIMLML2.10(Bayes)T:R:Bayes192050H:Robbins²:²:²:-17-2.11²:g(µjY)=f(Yjµ)g(µ)f(Y)(2-36)2.11[16]2.3()T1T2¢¢¢TNTiXTiXT=fXT1;¢¢¢;XTNgTN!12.1limE[XT¡E(XT)]2=0TVar(XT)TVar(^X)!¾2OoOp;op:O;ofangfbng2.4an=O(bn)anObnjan=bnjnKn(K)nn(K)janjKjbnjfangfbng2.5an=o(bn)anobnjan=bnj809n()nn()janjjbnjfangfbngfbngfangfbngbn=n¡1;bn=n¡1=2;bn=n;bn=nlognN(1)fangn(2)kankankank=sXia2ni(2-37)(2.4)(2.5)fang(3)can=O(bn)an=O(cbn)(4)an=o(1)an!