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5.1²²²²5.2Y=BX+N(5-1)BY+¡X=N(5-2)Y=AertK®L¯5.1(1)(2)-84-(3)80805%01198010-85-5.2t;F;Â2-86-RMSRMSi=vuutnXt=1e2it=neit=yit¡^yityit(5-3)RERE=yit¡^yityitTeii(1)1TTXi=1ei(2)vuut1TTXi=1e2i(3)TT¡1PTi=1(ei¡ei¡1)2PTi=1e2iN^YNRMSXtequilibriumerrorcorrectionmechanismECMFriedman-87-5.25.1(ErrorCorrectionModel)Davidson!Hendry!SrbaYeo1978DHSYXt®®TXt=0Xt®T±XtequilibriumerrortermXtlong¡runequilibrium¯¯TXtXt:®T±Xt=¡°¯TXt¡1+t¯TXt¡1errorcorrectiontermt(1;1)yt=¯0+¯1zt+¯2yt¡1+¯3zt¡1+t¢yt=¯0+¯1¢zt+(¯2¡1)yt¡1+¯3zt¡1+¯1zt¡1+t=¯0+¯1¢zt+(¯2¡1)(y¡¯1+¯31¡¯2z)t¡1+ty¡¯1+¯31¡¯2z¯1+¯31¡¯2zytztyt±ytytztyt=®ztytztyt=¯1+¯31¡¯2z±yt±yt=¯0+¯1±Zt+°ecm+t-88-5.1ECM1datacosuming;2inputcosuming@@;3datalines;4184236437803896107013311746223626412834297231383397361153791;datagdp;6inputgdp@@;7datalines;83624.14038.24517.84862.45294.75934.57171.08964.410202.2911962.514928.316909.218547.921617.826638.134634.446759.41058478.167884.674462.678345.282067.589468.197314.8;datatemp;11mergegdpcosuming;12title'ErrorCorrectionMechanism';title2'Estimationcointegration13vector:Residuals';14procregdata=temp;15cgdp:modelcosuming=gdp;16outputout=abcresidual=r;17run;datatemp(keep=d1cd1glr);18settempabc;19d1c=dif(cosuming);20d1g=dif(gdp);21lr=lag(r);22output;23run;title2'ECMsRegression';procregdata=temp;24ECM:modeld1c=d1glr/dw;25run;5.3AndrewsAndrews;1971GodfreyWickensGodfrey&Wickens;1981Y=f(X;®)+²(5-4)(5-4)Taylor-89-5.3ST(®)=kY¡f(X;®)k2=(Y¡f(X;®)0(Y¡f(X;®))(5-5)Amemiya;1985;pp:125¡135f(5-5)²»N(0;P)Lik(®;XjY)=1(2¼)T=2jPj1=2exp(¡12(Y¡f(X;®))0(X)¡1(Y¡f(X;®)))(5-6)®PBunke;1980;pp:609¡6113®OLS3(1)(2)YY¡!Y(1)Pareto(2)²²(outliers)²H0:²tt=1;2;¢¢¢WShapiro,S.S.&Wilk,M.B.,1965W=(PT=2i=1®T¡i+1(^eT¡i+1¡^ei))2PTi=1(^ei¡¹e)2(5-7)1Q=®+¯1P+¹(5-8)-90-CESY=A(±1K¡½+±2L¡½)¡1½(5-9)Yi=f(Xi;B)+ui(5-10)|S(^¯)=nXi=1fyi¡f(xi;¯)g2(5-11)f(xi;¯)|S(^¯)5.1()IQPL-91-5.45.21datacountrymen;2inputyearIQPL@@;3cards;4197862.45100.0100.031.52197979.30107.5122.1531.90198096.50109.0130.835.021981107.65115.36138.536.921982120.80128.4141.538.051983142.407138.4147.843.401984185.85155.4153.758.8819858238.70160.7166.967.131986285.52166.1177.675.2291987343.80175.8198.981.301988442.60182.6244.61086.111989495.30188.3281.384.981990524.66202.611274.086.741991559.30210.1268.489.061992613.6612223.5277.597.651993743.49241.0314.7109.98199413979.39261.7440.3119.6419951271.16290.2527.9127.071419961567.33317.5550.1130.2819971721.71333.7525.315135.27;run;I=AQ®1P®2L®3(5-12):DW=0:8955.45.2()yt=®t+¯txt+ut(5-13)OLS®t=®0+®1pt¯t=¯0+¯1pt5.31procregdata=countrymen;2modellog(I)=log(Q)log(P)log(L)/dwrp;3outputout=tempr=residualp=yhat;4run;56procautoregdata=countrymen;7modelI=QPL/nlag=2;8run;-92-½1·t·n0pt=0tn0pt=1n05.1(Chow)Chow1960(5-13)yt=®0+¯0xt+¹1tt=1;¢¢¢;n0yt=(®0+®1)+(¯0+¯1)xt+¹2tt=n0+1;¢¢¢(5-14)5.2(Gujarati)Gujarati1970yt=(®0+®1D)+(¯0+¯1D)xt+¹t5.2()1964¡1981Gujarati1964¡¡1981:yt=®0+®1+(®2¡®1)D+¯1Xt+(¯2¡¯1)DXt+ut(5-15)Chow1964¡19721973¡1981n0Var(uit=Var(u2t)n0(5-14)n0Var(uit6=Var(u2t)n0(5-14)u1t»N(0;¾21)u2t»N(0;¾22)GoldfeldQuandt1973n0lnL(¯;¾2jn0)=¡n2ln(2¼)¡n0ln¾1¡(n¡n0)ln¾2¡12¾21n0Pt=1(yt¡®0¡¯0xt)2¡12¾22nPt=n0+1(yt¡(®0+®1)¡(¯0+¯1)xt)2n0=1;2;¢¢¢;nn0®t=®0+t¯t=¯0+´t(5-16)t;´ty=®+¯x+!-93-5.45.41datatwincome;2inputincomesaving@@;3datalines;48.80.3659.40.21610.00.08710.60.20811.00.10911.90.121012.70.411113.50.501214.30.431315.50.591416.70.901517.70.951618.60.821719.71.041821.11.531922.81.942023.91.752125.21.99;22datatwincome;23retainyear1963;24settwincome;25year+1;26output;27run;5.5Chowmethod1datachow1;2settwincome;3ifyearle1972;4run;procprint;run;datachow2;5settwincome;6whereyear=1973;7run;procregdata=chow1;8modesaving=income;9run;procregdata=chow2;10modesaving=income;11run;-94-5.6Gujaratimethod1dataGujarati;2settwincome;3ifyearle1972thend=1;4elsed=0;5dincome=d¤income;6output;7run;procregdata=gujarati;8modelsaving=incomeddincome/covb;9run;!t=t+¹t+´txtE!t=0E(xt!t)=E(txt+¹txt+´tx2t)=0var(!t)=E(t+¹t+´t!t)2=E(¹2t)+E(2t)+E(´txt)2=(2+x2t)¾25.3(Hildreth-Houck)HildrethHouck1968yt=¯0t+¯1tx1t+¯2tx2t+¢¢¢+¯ktxkt+¹tt=0;1;¢¢¢;n¯jt=¯j+jt;Var(jt)=¾2jj=1;¢¢¢;k®t=®+µpt+t¯t=¯+±pt+´tyt=®+µpt+¯xt+±ptxt+¹t+t+´t+´txt5.4()yt=®t+¯txt+¹t®t®t=®t¡1+t¡1E(t)=0Var(t)=¾2-95-5.4¯t=¯®t®t=®0+±ptpt=½pt¡1+t®t=®0+±½pt¡1+±tpt½¼1®t=®t¡1+t®t0yt=®t+¯xt+¹t=®t¡1+¯xt+¹t+t¡1=®+¯xt+!t®n+1=®®n=®¡n¢¢¢¢¢¢¢¢¢¢¢¢®1=®¡PVar(¹t)=¾2¹Var(t)=¾2Cov(!)=26666641+n¸(n¡1)¸¢¢¢3¸2¸¸(n¡1)¸1+(n¡1)¸¢¢¢3¸2¸¸..................2¸2¸¢¢¢2¸1+2¸¸¸¸¢¢¢¸¸1+¸3777775¾2¹¸=¾2±¾2¹¸GLS5.5()(1)yt=®0+®1t+®2t2+¢¢¢+®ktk-96-(2)yt=abt(3)yt=k+abt(4)(Logistic)(5)(Gompertz)B:Gompertz1825yt=KabtLogisticSVerhulst1845yt=K1+eÁ(t)Á(t)=a0+a1t+a2t2+¢¢¢+aktkyt=K1+ae¡bt(5-17)(5-17)²Y(0;K)K²Logistic(5-17)1y=1K+aKe¡bt(5-18)KK100%60%KKKa!b5.3(Logistic)bK;at=1;2;¢¢¢;rt=r+1;r+2;¢¢¢;2rt=2r+1;2r+2;¢¢¢;n(5-18)S1=rPt=11yt=rK+aKe¡b(1¡e¡rb)1¡e¡bS2=2rPt=r+11yt=rK+aKe¡(r+1)b(1¡e¡rb)1¡e¡bS3=3rPt=2r+11yt=rK+aKe¡(2r+1)b(1¡e¡rb)1¡e¡b-97-5.5D1=S1¡S2;D2=S2¡S3D1=aKe¡b(1¡e¡rb)21¡e¡bD2=aKe¡(r+1)b(1¡e¡rb)21¡e¡bD1/D2=erbb=1r(lnD1¡lnD2)ybD1¡D2=aKe¡b1¡e¡b(1¡e¡rb)3D21=a2K2e¡2b(1¡e¡b)2(1¡e¡rb)4D21±(D1¡D2)=aK¢e¡b1¡e¡b(1¡e¡rb)=S1¡rKK=rS1¡D21/(D1¡D2)a=KD21D1¡D2¢eb¡11