Canova(XXXX)DSGE,VAR_SS

MatchingDSGEmodels,VARs,andstatespacemodelsFabioCanovaEUIandCEPRSeptember2012Outline•AlternativerepresentationsofthesolutionofaDSGEmodel.•FundamentalnessandfiniteVARrepresentationofDSGEsolutions.•LinkingDSGEandVARswithsignrestrictions.•Someempiricalconcerns.•Dealingwithnon-fundamentalnessReferencesCanova,F.andPaustian,M.(2011).Businesscyclemeasurementwithsometheory,JournalofMonetaryEconomics,48,345-361.Christiano,L.,Eichenbaum,M.andViggfusson,R.2006,AssessingstructuralVARs,NBERMacroeconomicAnnual,1-125.Chari,V.V.,Kehoe,P.andMcGrattan,E.(2008)ArestructuralVARswithlongrunrestrictionsusefulindevelopingbusinesscycletheory?,JournalofMonetaryEconomics,55,1337-1352.Dupor,W.andHan,J.,(2011)Handlingnon-invertibility:theoryandapplications,OhioStatemanuscript.Kascha,C.andK.Martens(2009)BusinesscycleanalysisandVARMAmodels,JournalofEconomicDynamicsandControl,33,267-282Ravenna,F.(2007)VectorautoregressionsandreducedFormRepresentationsofDSGEmodels,JournalofMonetaryEconomics,54,2048-2064.Ramey,V.(2011),IdentifyingGovermentSpendingshocks:Itisallinthetiming!,QuarterlyJournalofEconomics,.Gambetti,L.andForni,M.(2011)Fiscalforesigntandtheeffectofgovermentspending,UABmqanuscript..Merten,K.andRavn,M.(2010).Measuringtheimpactoffiscalpolicyinthefaceofanticipation,EconomicJournal,120,395-413.FernandezVillaverde,J.,RubioRamirez,J,Sargent,T.andWatson,M.(2007)TheA,B,CandDforunderstandingVARs,AmericanEconomicReview,97,1021-1026.Sims,E.(2011)News,Non-invertibilitiesandstructuralVARs,universityofNotreDame,manuscript1SolutionofDSGEmodels•Typical(Log-)linearizedsolutionofaDSGEmodelisoftheform:2=A22()2−1+A21()3(1)1=A11()2−1+A12()3(2)2=states(endogenousandexogenous),1=controls,3shocks.-A()=12aretimeinvariant(reducedform)matricesanddependon,thestructuralparametersoftechnologies,policies,etc.-Therearecrossequationrestrictionssince=1appearsinmorethanoneentryofthesematrices.-(1)-(2)isastatespacemodel,with(1)beingthetransitionequationand(2)themeasurementequation.-Ifboth2and1areobservables(1)-(2)isalsoarestrictedVAR(1).-Restrictionsareonthelaglengthandontheentriesofthecoefficientmatrix.Letting=[21]0=3A0=210012#−1A()=210012#−1220110#Thenthesystemis=A()−1+(3)-Singularsystem:dim(3)(1+2).2Alternativerepresentationsfor11)IfA12isinvertible3=A−112(1−A112−1)2=A222−1+A21A−112(1−A112−1)(1−(A22−A21A−112A11))2=A21A−1121IfA22−A21A−112A11hasalleigenvalueslessthan12=(1−(A22−A21A−112A11))−1A21A−1121and1=A11(1−(A22−A21A−112A11))−1A21A−1121−1+A123(4)-Ifonly1isobservable,thesolutionsolutionisaVAR(∞).-DoesitmakesensetoassumethatA12isinvertible?Weneedtohaveasmanycontrolsasshocks.Typically,thisconditionisnotsatisfied.Addmeasurementerrortomakethisconditionwork.-NeedinvertibilityconditionsontheeigenvaluesofA22−A21A−112A11.2)IfA11isinvertible2−1=A−111(1−A123)A−111(1+1−A123+1)=A22A−111(1−A123)+A2131+1=A11A22A−1111+(A11A21−A11A22A−111A12)3+A123+1)1+1=A11A22A−1111+(+(A11A21A−112−A11A22A−111))4+1where4≡A123.-Ifonly1isobservablesolutionisaVARMA(1,1)-DoesitmakesensetoassumethatA11isinvertible?Needasmanycontrolsasstates.-NoneedtoimposerestrictionsontheeigenvaluesofA22−A21A−112A11.3)FinalformcomputationsFrom2)1=11−1++1−1.Thenforany(1121)∙1−11−12−211−22¸∙1121¸=∙1−1112211+22¸∙12¸Notethat(())=(1+11)(1+22)−12212.Then:∙1+22−21−121+11¸∙1−11−12−211−22¸∙1222¸=(())∙12¸Underthisalternativerepresentation,thesolutionfor1isaVARMA(2,2).•If()isofreducedrank,write()=1+1111211+21#some.Then(())=1+(11+21).Inthiscasethesolutionfor1isaVARMA(2,1).2.1AnimportantdigressionTheaboverepresentationsassumethatthestate2isobservable.Whatifitisnot?-Typically,oneneedstocomputeKalmanfilterforecastofit,i.e.ˆ2=(22−11)ˆ2−1+ˆ1whereistheKalmangainwhichdependsontheobservables1.-Addingandsubtracting11ˆ2to(2)wehave1=11ˆ2−1+(5)=11(2−1−ˆ2−1)+123(6)•Ifthestatesarenotobservablesandtheobservable1failtoperfectlyrevealthem,i.e.2−16=ˆ2−1,theinnovationsofthemodelincludenotonlythestructuralshocksbutalsotheforecasterrorsmakeinpredictingthenon-observablestates.Whyisthisrelevant?If,forexample,newsshocksarepresent,theywillbestatevariables,buttheyarenotobservable.Thusaneconometricianneedstoforecastthemusingtheobservables.-Theconditionthattheeigenvaluesof(22−21−11211)arealllessthanoneinmodulusimpliesthatΣ=12Σ3012andthattheforecasterrorsdisappear-Thisconditionisequivalenttosayingthatwiththeobservablesvariableswecanperfectlyreconstructtheunobservablestates.Example1Supposethatthecapitalstockisastateanditisunobservable.ThentheeffectofcapitalasastatecanberecoveredfromaVARifhevariablesincludedallowtoperfectlypredictit.Thus,forexample,VARswithoutinvestmentwillnotsatisfytheconditionontheeigenvaluesof(22−21−11211)•ThinkaboutthepotentialstructuralmodelwhichhasgeneratedthedatabeforechoosingthevariablesofaVAR!Whentheeigenvalueconditionisnotsatisfi