dsge high school students‘ dsge modelling

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1、HighSchoolStudents'DSGEModellingWithDynareWeijieChen1DepartmentofPoliticalandEconomicStudiesUniversityofHelsinkiApril1,20121Anyquestions,pleasesendemailtoweijie.chen@helsinki..AbstractThisthesimplestDSGEmodelever,evenahighschoolstudentcanunder-standwhatthisis.Thepurposeofthisnoteis

2、toillustrateatwo-equationlinearrationalexpectationmodelandhowtoemployDynaretoplaywithit.IfyoudonothavefaithonclimbingthemountainofDSGEyet,thisshallbeyourrststep.WewillworkthroughsolutionandBayesianestimationinthisnote.1SimplestDSGEModelWeonlyhavetwoendogenousvariablesinthislinearDS

3、GE(LRE)model,yandx.Andwewillstartwithasingleexogenousvariablerst,u.Wespecifyanagentbehaviourrule,yt=Etyt+1+xt(1)andanexogenousMarkovprocesstodescribethebehaviourofnature,x=x+""iid(0;2);jj<1(2)tt1ttytisforward-lookingvariableandxtbackward-lookingvariable.Thebasicideaofsolving(

4、1)istondadierenceequationtodescribethedynamicofytbyitsownhistoryanditsshocks".1xtisalreadyfullydescribedbyitshistoryandnature,namelyxt1and",thuswedon'tneedtoworryaboutxt.Wesolveytforwards,moveoneperiodforwards,yt+1=Et+1yt+2+xt+1Takeconditionexpectation,Etyt+1=Et[Et+1yt+2]+Etxt+

5、1=Etyt+2+Etxt+1(3)wherethesecondequationmakesuseofiteratedlawofexpectation.Sub-stitutebackto(1),y=(Ey+Ex)+x=2Ey+Ex+x(4)ttt+2tt+1ttt+2tt+1tThenmove(1)forwardstwoperiods,yt+2=Et+2yt+3+xt+21ThisisthecoreideaofsolvingDSGEmodel.1Takeconditionalexpectation,Etyt+2=Et[Et+2yt+3]+Etxt+2=

6、Etyt+3+Etxt+2Substituteinto(3),Ey=2(Ey+Ex)+Ex+xtt+1tt+3tt+2tt+1t=3Ey+2Ex+Ex+xtt+3tt+2tt+1tPerformrecursivesubstitutioninnitetimes,wegetX1y=Eix(5)ttt+ii=0wherewehaveusedthetransversalityconditionlimEtyt+i=0i!1Wendthetimepathofytdescribedbyx,weneedtotakeadeeperlookintoxt,move(2)

7、oneperiodforwardsandtakeconditionalexpectation,Etxt+1=Et[xt+"t]=xtMovetwoperiodsandtakeconditionalexpectation,Ex=E[x+"]=Ex=2xtt+2tt+1t+2tt+1tThenwecandeductthatEx=ixtt+itSubstituteto(5),X1X1X1y=iEx=iix=xiittt+itti=0i=0i=0wherethethirdequationholdsbecausexthasnoianymore.Ifwem

8、akeakeyassumptionhe