搜索
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!0an=O(1)Knjanj·Kjanj(5)an=O(an)-18-(6)O(an)O(bn)=O(anbn)O(an)o(bn)=o(anbn)o(an)o(bn)=o(anbn)(2-38)(7)n2.1eenen=(1+1n)n(2-39)log(en)=nlog(1+n¡1)f(t)=log(1+t)f(t)=f(0)+f0(0)t+o(t)=t+o(t)(2-40)log(en)=nlog(1+n¡1)=n(n¡1+o(n¡1)=1+o(1)(2-41)log(en)!1(2-42)2.2fengeenf(t)xnxn=log(1+n¡1)(2-43)f(n+c)xng(2-44)cxnxn=f(n¡1)=n¡1¡12n¡2+o(n¡2)(2-45)(n+c)xn=(n+c)(n¡1¡12n¡2+o(n¡2)=1+cn¡121n¡12cn¡2+no(n¡2)+co(n¡2)=1+(c¡12)n¡1+o(n¡1)(2-46)c=12(n+c)xn1o(1)o(n¡1)e¤ne¤n=(1+n¡1)n+12(2-47)en-19-2.11f(n+c+dn¡1gf(t)=log(1+t)f(t)=t¡12t2+13t3+o(t3)(2-48)(n+c+dn¡1)xn=(n+c+dn¡1)(n¡1¡12n¡2+23n¡3+o(n¡3)(2-49)c=12d=¡112fXngFn(x)=PfXn·xg(2-50)XnFn(x)Xn2.6()XnFnXFlimn!1Fn(x)=F(x)(2-51)FxXnXCCcFc(x)=½0xc1xc(2-52)Fnx=c2.7()0limn!1PfjXn¡cj·g=1(2-53)fXngcXn¡!pc2.2CPfC=cg=1L[Xn]!L[C]Xn¡!pcFn!FFFn(x)!F(x)2.2XnPfXn=kg=(1¡pn)k¡1pn(2-54)Gn(x)Gn(x)=PfXn6xg=½1¡(1¡pn)[x]x00x60(2-55)p¡1p=pn=¸n¡1;¸0n!1E(Xn)=¸¡1n2.3n!1Xn!1-20-Yn=n¡1Xn(2-56)YnFn(x)=PfYn·xg=PfXn·nxg=Gn(nx)(2-57)Fn(x)x·0Fn(x)!0x0Fn(x)=1¡(1¡¸n¡1)nx(1¡¸n¡1)[nx]¡nx(2-58)(1¡¸n¡1)¡1¸(1¡¸n¡1)[nx]¡nx¸1(2-59)x0(1¡¸n¡1)[nx]¡nx!1(2-60)n!1(1¡¸n¡1)[nx]¡nx=1+o(1)(2-61)(1¡¸n¡1)nx!e¡¸x(2-62)Fn(x)!1¡e¡¸x(2-63)MGFX=(X1;¢¢¢;XTXXX(t1;¢¢¢;tT)=E(et1X1+¢¢¢+tTXT)(2-64)2.3(2-64)0=(0;¢¢¢;0)(t1;¢