New-KeynesianModelsandMonetaryPolicy:AReexaminationoftheStylizedFacts∗UlfSo¨derstro¨mPaulSo¨derlindAndersVredinAugust2003AbstractUsinganempiricalNew-Keynesianmodelwithoptimaldiscretionarymon-etarypolicy,wecalibratekeyparameters—thecentralbank’spreferencepa-rameters;thedegreeofforward-lookingbehaviorinthedeterminationofinfla-tionandoutput;andthevariancesofinflationandoutputshocks—-estratesmoothing,:Interestratesmoothing,centralbankobjectives,:E52,E58.∗So¨derstro¨m:IGIER,Universita`Bocconi,ViaSalasco5,20136Milano,Italy,-strom@;So¨derlind:SBF,,Rosenbergstrasse52,,Switzerland,@;Vredin:SverigesRiksbank,SE-10337Stock-holm,Sweden,@“CanaCalibratedNew-KeynesianModelofMonetaryPolicyFittheFacts?”.WearegratefultoSusantoBasu,FabioCanova,RichardDennis,CarloFavero,BenjaminFriedman,PaoloGiordani,KevinLansing,EricLeeper,JesperLinde´,MarianneNesse´n,AthanasiosOrphanides,GlennRude-busch,andseminarparticipantsatSverigesRiksbank,UppsalaUniversity,HarvardUniversity,.
1IntroductionManyanalysesofmonetarypolicypresentedinrecentyearshavebeenbasedonthehypothesisthattheprivatesector’,-Keynesianframeworkinboththeoreticalandappliedworkiseasytograsp:whilebeingsimilarinstructuretotraditionalmodelsusedforpolicyanalysis(suchastheIS/LMmodel),itcan(undercertainassumptions),andtheempiricalliteraturehasnotyetreachedaconsensusaboutkeyparametersinthemodel,,however,whetherthesecalibratedmodelsaresuitableforpolicyanalysis,thatis,,therefore,(basedonRudebusch,2002a)wheretheprivatesector’’spreferencesortheprivatesector’sbehavioraffectthetime-seriespropertiesofinflation,thenominalinterestrateandtheoutputgap,,,empiricalestimatesofTaylorrulestypicallysuggestthatcentralbankshaveverystrongpreferencesforsmoothingnominalinterestrates1EarlyreferencesonhowtoderivesuchmodelsfromoptimizingbehaviorincludeMcCallumandNelson(1999)andRotembergandWoodford(1997).Claridaetal.(1999)
(.,Claridaetal.,2000).Althoughargumentsforwhysuchapolicymaybeopti-malhavebeenpresented(see,.,Cukierman,1991;Goodfriend,1991;Woodford,1999),ithasalsobeenarguedthatthereislittledeliberateinterestratesmoothinginpracticeandthattheempiricalTaylorrulesaremisspecifieddescriptionsofmon-etarypolicy(seeRudebusch,2002b).Inourmodelwithoptimalmonetarypolicy,,-versusbackward-lookingbehaviorinprivateagents’-looking(.,RudebuschandSvensson,1999;Dennis,2001b;FaveroandRovelli,2001),,purelyforward-lookingmodelshavebeenshowntohavedifficultiesexplainingthepersistenceinthedata(.,FuhrerandMoore,1995;EstrellaandFuhrer,2002).Ouranalysisstressesthefactthatthetime-seriespropertiesofinflation(andothervariablesintheeconomy)areaffectedbybothprivatebehaviorandmonetarypolicy;,weshowthatthedegreeofforward-lookingbehaviorinprice-settinghasimportantconsequencesnotonlyforthepersistenceofinflation,,theabilitytoexplainthepersistenceofinflationdependsnotonlyonthedegreetowhichpricesettersareforward-looking,,,intermsofstandarddeviationsandautocorrelationsforinflation,theoutputgap,
Figure1:Dataseries,1987Q4–1999Q4Sources:BureauofEconomicAnalysis(inflation);(outputandinterestrate).,(2002b)(sea-sonallyadjusted),(measuredinchained1996dollars,seasonallyadjusted)frompotentialGDP,ascalculatedbytheCongressionalBudgetOffice(seeCongressionalBudgetOffice,1995).Theinterestrateisthe(annualized),(intermsofstandarddeviations)(theinstrumentofmonetarypolicy)
Table1:()()()()()()()()()()()()Note:,1987Q4–,usingaNeweyandWest(1987):EstimatedTaylorruleCoefficientStatistics2γγγρRσ0piyiζ()()()()Note:,1987Q4–,usingaNeweyandWest(1987),whichshowstheresultsfromestimatingaTaylor-typeruleoftheformi=(1−ρ)[γ+γp¯i+γy]+ρi+ζ,(1)ti0pitytit−1t∑3whereiisthe3-monthinterestrate,p¯i=1/4piisthefour-quarterinflationttt−jj=,respectively,buttheshort-runeffects2aredominatedbythelaggedinterestrate,-Keynesianmodel(surveyedbyClaridaetal.,1999)isattractiveinthatitisderivedfrom2Whiletheseresultsarefairlystandardintheliterature(.,2000),,,
,itiswell-knownthatthissimplemodelhasproblemswhenconfrontedwiththedata,andadditionalinertiaisoftenintroducedinpracticalapplications(see,.,Claridaetal.,1999;EstrellaandFuhrer,2002).(2002),whichisanempiricalmodelintheNew-Keynesiantradition,-terminterestratetominimize(underdiscretion),themodelisgivenbythe4followingthreeequations:4∑pi=µEp¯i+(1−µ)αpi+αy+ε,(2)tpit−1t+3pipijt−jyt−1tj=12∑y=µEy+(1−µ)βy−β[i−Ep¯i]+η,(3)tyt−1t+1yyjt−jrt−1t−1t+3tj=1minVar[p¯i]+λVar[y]+νVar[∆i].(4)ttt{i}tEquation(2)isanempiricalversionofaNew-KeynesianPhillipscurve(orag-gregatesupplyequation),whereinflationdependsonexpectedandlaggedinflation,theoutputgapofthepreviousperiod,andthe“cost-pushshock”ε.Equation(3)tisanaggregatedemandequation(orconsumptionEulerequation)thatdeterminestheoutputgapasafunctionoftheexpectedandlaggedoutputgap,therealshort-terminterestrateofthepreviousperiod,andthedemandshockη.Equation(4)tistheobjectivefunctionformonetarypolicy;thecentralbankactstominimizetheweightedunconditionalvariancesofinflation,theoutputgap,,whilethatofoutputisgivenbythepotentiallevel,(2002a)allowsforforward-lookingbehaviorinthedeter-minationofinflation,(1997),Ireland(2001),Christianoetal.(2001),andSmetsandWouters(2002),amongothers,,theydonotmodelcentralbankbehaviorasoptimizing,,RotembergandWoodford(1997)andIreland(2001)(2002b)usesasimilarmodel,withforward-lookingbehaviorinbothinflation,output,,sincehismodelincludesexpectedfutureinterestrates,andthereforecannotbeeasilymappedintothestandardlinearREframework(see,.,So¨derlind,1999).Dennis(2001a)
Table3:ParametervaluesInflationOutputgapαβα−β−αβαα:ParametersestimatedbyRudebusch(2002a),1968Q3–,althoughweassumethatthecentralbankactsunderdiscretion,thereis5noinflationbias,-Keynesianmodelsandcer-tainfacts(.,thevolatilityandpersistenceofinflation,outputandtheinterestrate),,wechoosetoestimatethoseparametersthatarecru-cialforthemomentsofinterest:theimportanceofforward-lookingbehaviorinthedeterminationofinflationandoutput(µandµ),therelativeweightsinthecen-piytralbank’sobjectivefunction(λandν),andthestandarddeviationofsupplyanddemandshocks(σ,σ).Fortheotherparameterswesimplyusetheestimatesofpiy6Rudebusch(2002a),(see,.,Ireland,2001;SmetsandWouters,2002;orSo¨derlind,1999).Thisreflectsthefactthattheoreticalmodelscontainmanyparametersthatareredundantwhenitcomesto5Thereremainsadifferencebetweenthediscretionaryoutcomeandthatundercommitment,however:becausediscretionarypolicycannotexploitprivateagents’expectations,“stabilizationbias”(seeSvensson,1997,orWoodford,1999),andisanalyzedindetailbyDennisandSo¨derstro¨m(2002).6TheseparametervalueswereestimatedbyRudebusch(2002a)–1996Q4(withsurveydataforinflationexpectations).Stabilityteststypicallycannotrejectthehypothesisofnostructuralbreaksinsuchestimatedequations(Rude-buschandSvensson,1999;Rudebusch,2002a;Dennis,2001b);µ,butrestrictsµαandβparametersareobtainedpiyrestrictingalsoµtozero(RudebuschandSvensson,1999;Rudebusch,2001),orusingFIMLorpiSURtechniques(Dennis,2001b).6
fittingthemodeltodata,“best”estimatefortheparametersofinterest,,,weregarditasausefulalternativetomethodsthatfocusononesingle“best”(λ,ν,µ,µ,σ,σ)λ,ν∈{0,,,,1,2,5};µ,µ∈piy{,,,,,,1};andσ,σ∈{,,,,,1,},re-piysultingin117,(standarddevi-8ationsandautocorrelations)ofinflation,,,wechoosetoidentifythoseconfigurationswherethestandardde-viationsandautocorrelationsofinflation,output,andtheinterestrateliewithin±,,theta-9blealsoincludesaparameterizationthatismorecommonintheliterature,where(λ,ν,µ,µ)=(,,,),buttheshockvariancesareleftattheircali-piybratedvalues(σ,σ)=(,).,ourmodelneedstobecharacterizedby•afairlysmallpreferenceforoutputstabilization:λ≤;•alargepreferenceforinterestratesmoothing:≤ν≤2;7′ˆˆIntheGMMestimationweminimizethefunction(ξ−ξ)(ξ−ξ)overourgridofparameterjjˆconfigurations,whereξisthevectorofmomentsforconfigurationjandξ,,thereducedformofthemodel,λandνinthe“typical”parameterizationareusedasabenchmarkbyRudebuschandSvensson(1999)andRudebusch(2001),whilethevaluesforµandµaresimilartothosepiyusedbyRudebusch(2002a).7
Table4:.λνµµσσ±.•asmalldegreeofforward-lookingbehaviorinprice-setting:µ≤;andpi•alargedegreeofforward-lookingbehaviorinconsumption/aggregatedemand:µ≥,authorsoftenassumealargerpreferenceforoutputstabilitythanforinterestratesmoothing,,ourcalibrationindicatesthatcentralbankbehavior(atleastthatoftheFederalReserve),(2001b)estimatesthepreferenceparametersoftheFederalReserveusingfullin-formationmaximumlikelihood(FIML)fortheperiod1979–2000,andobtainsesti-10matesof(λ,ν)=(,).FaveroandRovelli(2001)useGMMfortheperiod1980–98andobtain(λ,ν)=(,).ThedifferencesbetweentheseresultsseemtobemainlyduetoFaveroandRovelli(2001)usingafinitepolicyhorizon(offourquarters),whileDennis(2001b)usesaninfinitehorizon(asinourmodel).Nevertheless,bothstudiesfindamoreimportantroleforinterestratesmoothingthanforoutputstabilization,,output,andthechangeintheinterestrate,Dennisobtains(λ,ν)=(,),butthisparameterizationimpliesavarianceoftheinterestratewhichisalmosttwiceaslargeasinthedata(Dennis,2001b,Appendix2).8
-lookingspecificationoftheNew-Keynesianmodel(withµ=µ=1)isatoddspiywiththedata(EstrellaandFuhrer,2002),andthereisalargeliteratureestimat-ingversionsoftheNew-KeynesianPhillipscurveinequation(2).Gal´ıandGertler(1999)arguethatthebackward-lookingtermisnotquantitativelyimportant,butmanyotheranalysestendtofavorprimarilybackward-lookingspecifications,andestimateµ,´(2002)estimatesversionsofequations(2)and(3)–97,andobtainsµ=µ=,LansingandTrehan(2001)findthatalargeµ,Table5showsthestandarddevia-tionsandautocorrelations,,,,inflationistoovolatileandtoopersistentcomparedwiththedata,,,Figure2showshowtheecon-omyrespondstounitshockstothethreevariablesatt=0,usinga“baseline”-correlationsinthedata,(inthefirstrow),theinterestrateisslowlymovedbacktoneutral,-tion(seeequations(2)–(3)),thereisnoimmediateresponseofinflationoroutputtothepolicyshock,butfromt=1onwards,(approximately−%)comesafter11See,.,Fuhrer(1997),Roberts(2001),RuddandWhelan(2001),Linde´(2002),orJondeauandLeBihan(2001).12Theinterestratedisturbanceisnotpartofthemodel,,anartificialinterestrateshockcanbeconstructedbyassumingthattheinterestrateisunexpectedlyraisedbyonepercentagepointforoneperiod,andthatthesystemfollowsthereduced-formafterwards(asinSvensson,2000).
Table5:(a)Inflation:(b)Outputgap:(c)Interestrate:(λ,ν,µ,µ,σ,σ)=(,,,,,).piypiy10
Figure2::(λ,ν,µ,µ,σ,σ)=(,,,,,).piypiytwoquarters,whilethatoninflation(approximately−%)comeslater,(see,.,Christianoetal.,1999).Aftershockstoinflationandoutput,monetarypolicyrespondsonlygradually,13sincethecentralbankdislikeslargeswingsintheinterestrate(νislarge).Afteraninflationdisturbance,themonetarypolicyresponseopensupanegativeoutputgap,whichisthenclosedveryslowly(sincetheweightonoutputstabilizationλissmall).Afteranoutputdisturbance,thecentralbankmustchangethepositiveoutputgapintoanegativegapinordertofighttheinflationaryimpulse,andthisisdonefairlyquickly(againbecauseofthesmallλ).Inbothcasesthequarterlyinflationratedisplaysarathervolatilepattern,,aparameterizationthatismorecommonintheliteratureisclearlyatoddswiththedatawhendescribingthebehaviorof13Notethatthecentralbankaimstostabilizeannualinflationratherthanquarterlyinflation(whichisshowninthefigure),(´n,2002).11
:ThekeyparametersTosomeextent,,,sinceourapproachaimsatmatch-ingmomentsfromdifferentparameterconfigurationswiththoseinactualdata,,(–5)inoneparameterdimensionatatimeandcalculatetheresultingstandarddevia-14tionandfirst-orderautocorrelationofinflation,,-sultsarereportedinFigures3–,verticallinesrepresentthebaselineparametervalues,’spreferenceparameters,wewouldaprioriexpectthatincreasingtheweightofonevariableintheobjectivefunctionwouldmakethatvari-ablemorestableandlesspersistent,,–bshowsthatthisintuitionholdswhenvaryingtheweightonoutputstabilization,λ.Asλincreases(keepingtheotherpa-rametersfixed),outputbecomesmorestableandlesspersistent,whileinflationand(tosomeextent)-flationandoutput,increasingλmakesthestandarddeviationandautocorrelationsmoveawayfromthevaluesinthedata,-ingλtowardszerohasnoimportanteffectonthebehaviorofanyvariable;sinceνislarge,azeroweightonoutputstabilizationdoesnotleadtomuchvolatilityintheinterestrateandoutput,aswouldhavebeenthecasewithν=,Figure3indicatesthatlargervaluesofλ
Figure3:VaryingpreferenceparametersfromthebaselineconfigurationUnconditionalstandarddeviationsandfirst-orderautocorrelationsastheweightonoutputsta-bilizationλ(upperpanels)andinterestratesmoothingν(lowerpanels)varyfromthebaselineconfiguration().Verticallinesrepresentbaselinevalues,,ν,Figure3c–ν,inflationbecomesmorestableandtheinterestratemorevolatile,,thevolatilityinoutputincreasesslightly:thedirecteffectoftheincreasedinterestratevolatilityonoutputseemstodominatethelargerweightonoutputstability(relativetointerestratesmoothing)νfalls,ν,aslongasν>νapproacheszero,theinterestratebecomesextremelyvolatile(forν=0,%),,smallervaluesofν-lookingbehaviorinthedeterminationofinfla-tionandoutput,wewouldaprioriexpectthatmoreforward-lookinginthedeter-minationofonevariablewouldmakethatvariablelesspersistent,whileitisdifficult13
Figure4:Varyingthedegreeofforward-lookingbehaviorfromthebaselineconfig-urationUnconditionalstandarddeviationsandfirst-orderautocorrelationsasthedegreeofforward-lookingbehaviorininflationµ(upperpanels)andinoutputµ(lowerpanels)varyfromthebaselinepiyconfiguration().Verticallinesrepresentbaselinevalues,–bshowsthatincreasingµquicklyre-piducesthepersistenceininflation,,µapproachesunity(acommoncaseinthetheoreticalliterature),theinterestratepibecomesverystable,-lookingelementsininflationtomatchthepersistenceofinflation,,largervaluesofµ–dshowthatalargerdegreeofforward-lookinginoutputmakestheoutputgaplessvolatileandpersistent,µfallsyalsoinflationandtheinterestratebecomemorevolatile,
Figure5:VaryingthestandarddeviationofshocksfromthebaselineconfigurationUnconditionalstandarddeviationsandfirst-orderautocorrelationsasthestandarddeviationofinflationshocksσ(upperpanels)andoutputshocksσ(lowerpanels)varyfromthebaselinepiyconfiguration().Verticallinesrepresentbaselinevalues,,whichincreasesconsiderablyasµ,lettingµapproachzerohasnoimportanteffectoninflationyyoroutput,whileitleadstoaveryvolatileinterestrate(%whenµ=0).Asoutput(andconsumption)becomesmoreinertialandylessforward-looking,µ,thebehaviorofoutputisnotmuchaffectedbychangesinµ.yFinally,σleadstomorevolatilityinallvariables,piincreasingσ-offtopolicymakers,,ontheotherhand,σservestomatchthepi15ThisisconsistentwithLansingandTrehan(2001),whoarguethatasmalldegreeofforward-lookinginoutputleadstoveryaggressivepolicybehavior,.,
volatilityinallvariableswhereasσ,-neousequations,(andthusmonetarypolicy)weneednotonlyalargeweightoninterestratesmoothing,-seriespropertiesofinflationcanbeexplainedonlyifourmodelincludesasmalldegreeofforward-lookingbehaviorinprice-setting(asnotedelsewhere),,tomatchthebehavioroftheoutputgapweneedasmallweightonoutputstabilizationandasmalldegreeofforward-lookinginprice-setting,-equationanalysesmaynotbesufficienttopindownthevalueofanyoneparameter;-gardingdata,parametervalues,,ourcalibratedparameterconfigurationsarechosensothatthemomentsofthemodelliewithin±±1to±:whenweincreasetheacceptedrange,±,theresultsareverymuchthesameasinourpreferredcalibrations,whileabove±2standarderrors,νandµcoveralmostytheentirepermittedinterval(exceptν=0).However,ν=0isonlypickedoutfor±;looselyspeakingthehypothesisofnointerestratesmoothingis“rejected”,whilecalibratingthemodeltomatchdatafrom1987to1999,weuse16
Figure6:Parameterrangesforvaryingcriteria17
Table6:Parametervalues,1987–2001InflationOutputgapαβα−β−αβαα:ParametersestimatedbyCastelnuovo(2002),1987Q3–µandµ–,soiftheparametersestimatedbyRudebusch(2002a)arenottrulystructural,,Table6showsparameterestimatesobtainedbyCastelnuovo(2002)whenestimatingthepurelybackward-lookingversionofthemodel(µ=µ=0)forthesample1987Q3–:,weobtainsimilarresultsforλ,σandσ,evenlargerpiyvaluesforν(above2),andsmallervaluesforµ().Mostinterestingly,ythiscalibrationresultsininflationbeingeitherpurelybackward-looking(µ=0)pior,moreoften,purelyforward-looking(µ=1).Thissuggeststhattheparameterspiintheinflationandoutputequationsareimportantwhencalibratingthedegreeofforward-lookingbehavior,,,theCBO’smethodologytocalculatepotentialoutputleadstoarathersmoothseries,λandµandasmallerσ.Still,sincethestandarddeviationoftheoutputgapisnotpipitheonlymomentthattiesdownthecalibration(thesmallvalueofλisalsorelatedtothevolatilityofinflationandthepersistenceintheoutputgap),,estimatedTaylorrulesareoftenusedtodiscusstheissueofinterest18
,Section3focusesonmatchingonlythevolatilityofkeyeconomicvariables,λ,whileforν,µandµ,westillgetalargevalueforν:tomatchthedegreeofpersistenceintheTaylorrule,wemustallowforaverylargeweightoninterestratesmoothing(evenlargerthaninSection3).Furthermore,,inadditiontointroducingmorelagstothetheoreticalversionoftheNew-Keynesianmodel,theRudebusch(2002a)(see,.,RotembergandWoodford,1997,orChristianoetal.,2001),ithasimportantconsequencesforthedynamicsofthemodel(DennisandSo¨derstro¨m,2002).Wethereforeexaminealsothemorestandarddatingofexpectationsusingthespecification4∑pi=µEp¯i+(1−µ)αpi+αy+ε,(5)tpitt+3pipijt−jyt−1tj=12∑y=µEy+(1−µ)βy−β[i−Ep¯i]+η.(6)tytt+1yyjt−jrt−1t−1t+3tj=1CalibratingthismodelgivesvirtuallyidenticalresultstothoseinSection3,themaindifferencebeingthatthiscalibrationalwaysyieldsλ=0andν>-lookingininflationandoutputmaynotbeentirelyrobust:,ontheotherhand,-ibrationsyieldalargerpreferenceforoutputstabilizationthaninSection3,thepreferenceforinterestratesmoothingisalwaysatleastaslargeas(andinmostparameterizationsmanytimeslargerthan),sincecali-19
-Keynesianfeatures,,frequentlyusedcalibratedmodelsoftenassumethatthecentralbank’—likesomeearlierpapers—,(virtuallyzero)preferenceforoutputstabilization,ourNew-Keynesianmodelcanhardlymatchthelowvolatilityandpersistenceininflation,,,-ically,thisisneededtomatchthelowvolatilityintheinterestrate:withamorepersistentoutputgap,,(ahighλinrelationtoν),itseemstobebasedonthesimpleobservationthatcentralbanksdonotpursue“strict”inflationtargeting(soλandνarenotbothzero):inflationistoopersistentandinterestratesaretoostabletobeconsistentwith“strict”“flexible”-20
,,therehavebeenmanyactionstakenwiththeexplicitintenttopromotefinancialstability(seeEstrella,2001,foranoverview).Itshouldbestressedthatmuchworkremainsbeforecentralbanks’’-more,,theyseemtochangeinterestratesinastep-wisefashion,-ered,,withinthelinear-quadraticframeworkcommonlyapplied,policyshouldbedescribedintermsofahighνinrelationtoλ,-seriespropertiesofnominalinterestrates(inrelationto,.,inflation).-gregatedemandequationstemfromtherepresentativeindividual’sdesiretosmoothconsumptionovertime(anEulerequation).Withthederegulationsandinnovationsinfinancialmarketsthathavetakenplacesincethe1980s,-lookingintheaggregatesupplyequationsuggestedbyour16See,.,Cukierman(1991)andGoodfriend(1991).Lorenzoni(2001)presentsatheoreticalanalysisofthedualobjectivesofpricestabilityandpaymentsystemstability,
-Keynesianmodelconsistentwiththedata,butthereisnoconvincingtheoreticalargumentforaverylowµ.,,intheformofeconometricallyestimatedNew-Keynesianmodelsofaggregatesupply,aggregatedemandandmonetarypolicy,haverecentlybeenpresentedbyDennis(2001b),FaveroandRovelli(2001),andLinde´(2002).Theirresultsarenotentirelyconsistent,however,
(2)and(3)oneperiod:µpipi=E[pi+pi+pi+pi](A1)t+1tt+1t+2t+3t+44+(1−µ)[αpi+αpi+αpi+αpi]+αy+ε,pipi1tpi2t−1pi3t−2pi4t−3ytt+1y=µEy+(1−µ)[βy+βy]t+1ytt+2yy1ty2t−1[]1−βi−E(pi+pi+pi+pi)+η.(A2)rttt+1t+2t+3t+4t+14Thensolvefortheforward-lookingvariablesEpiandEyandtakeexpectationstt+4tt+2asofperiodt:()µµµµpipipipiEpi=1−Epi−Epi−Epitt+4tt+1tt+2tt+34444−(1−µ)[αpi+αpi+αpi+αpi]−αy(A3)pipi1tpi2t−1pi3t−2pi4t−3ytβrµEy+Epi=Ey−(1−µ)[βy+βy]ytt+2tt+4tt+1yy1ty2t−14[]1+βi−E(pi+pi+pi),(A4)rttt+1t+2t+34andreintroducethedisturbancesviathe(predetermined)variablespi=Epi+ε,(A5)t+1tt+1t+1y=Ey+η.(A6)t+1tt+1t+117Definean(n×1)vector(n=11)ofpredeterminedstatevariablesas11′x={pi,pi,pi,pi,y,y,y,y,i,i,i},(A7)1ttt−1t−2t−3tt−1t−2t−3t−1t−2t−3an(n×1)vector(n=4)offorward-lookingjumpvariablesas22′x={Epi,Epi,Epi,Ey},(A8)2ttt+1tt+2tt+3tt+1andan(n×1)vectorofshockstothepredeterminedvariablesas1{}′′′v=ε,0,η,0.(A9)1ttt3×16×1Wecanthenwritethemodelincompactformasxx1t+11tA=A+Bi+v,(A10)011tt+1Exxt2t+12t17Theadditionallagsoftheoutputgapandtheinterestratearenotstatevariables,
wherev1t+1v=,(A11)t+10n2×1andwherethematricesA,Σ,whichisadiagonalmatrixwithdiagonal1tv1{}2′2′σ,0,σ,0andzeroselsewhere.ε3×1η6×1−118Toobtaintheusualstate-spaceform,premultiply(A10)byAtoget0xx1t+11t=A+Bi+v,(A12)tt+1Exxt2t+12t−1−119whereA=AAandB=’sobjectivefunction(4),itisconvenienttodefineavectoroftargetvariablesas′z={p¯i,y,∆i},(A13)ttttwhichcanbecalculatedbyz=Cx+Ci.(A14)txtitThecentralbank’speriodlossfunctionin(4)canthenbewrittenas222L=¯+λy+ν(∆i)tttt′=zKz,(A15)(A14),thelossfunctioncanbeexpressedas′L=zKzttt′[][]Cxtx′′=KxiCCxitt′Citi′′′′′′′′=xCKCx+xCKCi+iCKCx+iCKCixtitxtittxtxtiti′′′′′=xQx+xUi+iUx+iRi,(A16)tttttttt18ThismeansthatAmustbenon-singular,.,µ,µ =−119NotethatAv=vsinceAisblockdiagonalwithanidentitymatrixasitsupperleftt+1t++124
wherex1tx=,(A17)tx2tandwhere′Q=CKC,(A18)xx′U=CKC,(A19)ix′R=CKC.(A20)iiThusthecentralbank’scontrolproblemisgivenbytheconventionalBellmanequa-tion′′′′′J(x)=min{xQx+xUi+iUx+iRi+δEJ(x)},(A21)ttttttt+1ttttitsubjecttothetransitionequation(A12),andtheoptimalpolicyrulecanbecalcu-latedusingstandardmethods(seeSo¨derlind,1999,foranoverview).Theoptimalpolicyunderdiscretionisarulefortheinterestrateasalinearfunctionofthepredeterminedvariables:i=Fx,(A22)t1tresultinginthereducedformx=Mx+v,(A23)1t+11t1t+1x=Nx.(A24)2t1tSeeSo¨derlind(1999)=Cx,(A25)t1twhereC=C+CN+CF.(A26)x1x2iThereducedform(A23)impliesthattheunconditionalvariance-covariancema-trixofxsatisfies1t′Σ=MΣM+Σ,(A27)x1x1v1andusingthevecoperatorandsolvingforvec(Σ),wegetx1−1vec(Σ)=(I−M⊗M)vec(Σ).(A28)x1v125
Thecovariancematrixofxisthengivenby2t′Σ=NΣN,(A29)x2x1andthatofzist′Σ=CΣC.(A30)zx1BResponsestoaninterestrateshockInordertomodelamonetarypolicyshock,.,aone-timeshocktotheinterestrate,supposethecentralbankchangestheinterestrateattimet=0bydi,andtfromthenonfollowsitsoptimalpolicyrulei=Fxforallt>
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