台湾清华大学 DSGE模型讲义 lecture4

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1、KalmanFilterandMaximumLikelihoodEstimationofLinearizedDSGEModelsDr.Tai-kuangHoAssociateProfessor.DepartmentofQuantitativeFinance,NationalTsingHuaUniversity,No.1StatespaceformandtheKalman…lter1.1Statespaceformxt:thestatevariableszt:theobservablevariablesTransitionequation101,Se

2、ction2,Kuang-FuRoad,Hsinchu,Taiwan30013,Tel:+886-3-571-5131,ext.62136,Fax:+886-3-562-1823,E-mail:tkho@mx.nthu.edu.tw.xt+1=Fxt+G!t+1!t+1N(0;Q)Measurementequationz0t=Htxt+ttN(0;R)Wewanttowritethelikelihoodfunctionofzt.1.2SomeusefulpropertiesofnormaldistributionAssumethat"#"#"#

3、!XjwxxxxyZjw=N;YjwyyxyyTheniffollowsXjY;wNx+11xyyy(Yy);xxxyyyyx1.3Kalman…lterzt1fz1;z0;:::;zt1gxt1,thehistoryoftheobserv-tjt1:therandomvariablesxtconditionalonzablevariablesxtjt1Extjzt10tjt1Extxtjt1xtxtjt1jzt1ztjt1Eztjzt1

4、0tjt1Eztztjt1ztztjt1jzt1zzt10t1tjt1Etjz=EHtxt+tjz=H0txtjt10t1tjt1Eztztjt1ztztjt1jz0=EH0xH0t1ttxtjt1+ttxtxtjt1+tjz0=EH0xt10t1ttxtjt1xtxtjt1Htjz+Ettjz=H0ttjt1Ht+R0t1Eztztjt1xtxtjt1jz00t1=EHtxtxt

5、jt1+txtxtjt1jz=H0ttjt11.4Kalman…lter:…rstiterationx0j10j1(symmetricmatrix)Assumethat"#"#"#!x01x0j10j10j1H0jzN0;00z0HH0x0j100j1H00j1H0+Rz1fz1gTheniffollowshix10jz;z0Nx0j0;0j01xH0z00j0=x0j1+0j1H000j1H0+R0H0x0j11H0H00j0=0j10j1H00

6、0j1H0+R00j1z0fz1;z0gItfollowsthatxx001j0=E1jz=EFx0+G!1jz=Fx0j0001j0=Ex1x1j0x1x1j0jz00=EFx0+G!1x1j0Fx0+G!1x1j0jz00=EFx0+G!1Fx0j0Fx0+G!1Fx0j0jz0=EFx000000x0j0x0x0j0Fjz+EG(!1)(!1)Gjz=F0+GQG00j0Fzz0001j0=E1jz=EH1x1+1jz=H01x1j000

7、1j0=Ez1z1j0z1z1j0jz000000=EH1x1+1H1x1j0H1x1+1H1x1j0jz0=EH0x00011x1j0x1x1j0H1jz+E(1)(1)jz=H011j0H1+RWehavemovedasfollows:x0j1;0j1!z0!x0j0;0j0!x1j0;1j0;z1j0;1j01.5Kalman…lter:thealgorithmxtjt1tjt1zt1xH0z0tjt=xtjt1+tjt1Htttjt1Ht+Rt

8、Htxtjt11H0H0tjt=tjt