HP-filter

KnowledgeBriefforBankStaffPRMED•TheWorldBank•1818HStNW•WashingtonDC20433TheHodrick-PrescottFilterTrend-cycledecompositionsareroutineinmodernmacroeconomics.Thebasicideaistodecomposetheeconomicseriesofinterest(e.g.,thelogofGDP)intothesumofaslowly-evolvingseculartrendandatransitorydeviationfromitwhichisclassifiedas‘cycle’:xt=τt+ζtObservedSeries=PermanentTrend+Cycle(1)However,astheseconstituentparts—trendandcycle—arenotreadilyobserved,anydecompositionmustnecessarilybebuiltonaconceptualartifact.Thus,anydetrendingmethod,muststartoutbysomehowarbitrarilydefiningwhatshallbecountedastrendandascycle,beforetheseelementscanbeestimatedfromthedata.ThemostcommonmethodusedtoextractthetrendfromatimeseriesistheHodrick-Prescott(HP)filter(HodrickandPrescott,1997).TheHPfilterextractsthetrend,τt,bysolvingthefollowingstandard-penaltyprogram:min{τt}TXt=1(xt−τt)2|{z}GoodnessofFit+λT−1Xt=2[(τt+1−τt)−(τt−τt−1)]2|{z}PenaltyforRoughness(2)wherethesmoothingparameterλcontrolsthesmoothnessoftheadjustedtrendseries,ˆτt—i.e.,asλ→0,thetrendapproximatestheactualseries,xt,whileasλ→∞thetrendbecomeslinear.WhileHodrickandPrescott(1997)suggestvaluesforλ,MarcetandRavn(2003)recasttheprogram(2)asaconstrainedminimizationprogramtodeterminethevalueofλendogenously.Forannualdata,λshouldbebetween6and7;seeRavnandUhlig(2002),andMaravall(2004).NotethattheHPprogram(2)canbewrittenmoresuccintlyas:1min{τt}TXt=1ζ2t+λTXt=3(∇2τt)2(3)WhichindicatesthattheHPfilterattemptstomaximizethefitofthetrendtotheseries—i.e.,minimizethecyclecomponentin(1)—whileminimizingthechangesinthetrend’sslope.1Where∇=(1−B)isthestandarddifferencingoperatorandBisthestandardbackshift(lag)operator,suchthatBjxt=xt−j,and∇xt=xt−xt−1.Alsodefinetheforwardshiftingoperators:F=B−1andΔ=(1−F).Version:December17,2006(1989)showthatthesolutionto(2)isgivenby:ˆτ=[I+λK0K]−1x(4)wherex=[x1,...,xT]0,τ=[τ1,...,τT]0,IisaT×Tidentitymatrix,andK={kij}isa(T−2)×Tmatrixwithelementsgivenbykij=(1ifi=jori=j+2,−2ifi=j+1,0otherwise.thatis:K=1−2100···00001−210···000...........................00000···1−21Considerx=Iτ+ζ,thenageneralized-ridgeregressionrulewouldestimatethetrendasˆτ=[I0I+λA]−1I0x=[I+λA]−1xwhereAisasymmetric,positive-definitematrix(HoerlandKennard,1970).MakingA=K0K,itbecomesapparentthattheformulation(4)isaparticularcaseofthisapproach.ThisimpliesthattheHPtrendcanbeinterpretedasthefittedvaluesofaridgeregression.Relatedly,inaBayesianframework,whereζ∼N(0,σ2),ifwespecifiythenaturalconjugateprior(τ|σ)∼N(0,(σ2/λ)(K0K)−1),thentheposteriormeanofτisgivenby(4).1.1.ExampleAsanillustration,letusconsider,e.g.,T=5,inwhichcase:I+λK0K=1+λ−2λ−λ00−2λ1+5λ0−λ0−λ01+6λ0−λ0−λ01+5λ2λ00−λ2λ1+λMakingλ=7,wecompute[I+7K0K]−1,andweobtainˆτ1ˆτ2ˆτ3ˆτ4ˆτ5=0.6440.3750.156−0.014−0.1610.3750.3220.2160.100−0.0140.1560.2160.2540.2160.156−0.0140.1000.2160.3220.375−0.161−0.0140.1560.3750.644x1x2x3x4x52Afewthingsareworthnoting:(i)Foreacht,alltheweightsaddtounity,andtheydonotdependonthedata.(ii)Negativeweightsoccurforsomeobservations(thosemorethan3periodsapartfromtinthisexample).(iii)Thethirdobservation—sinceithasequalnumberofobservationsbeforeandafterit—istheonlyonethatwillbefilteredbyasymmetricfilter.(iv)Theendpointswillplaceaverylargeweight(0.649)ontheirobservedvaluesfordeterminingthecorrespondingtrendvalue,andthefilterisone-sidedattheendpoints.(v)Thetrendfortheobservationsnexttotheendpoints,however,willputalargerweightonthefirstandlastobservations(0.375)thanonthemselves(0.322).Thelasttwopoints,(iv)–(v),illustratetheoriginoftheendpointsampleproblem.WhenTgetsrevised,orwhenT+kbecomesavailable,thereisasubstantialeffectontheestimatesforτT−1andτT.-10-8-6-4-202400.10.20.30.40.50.6hpfilter.nb1Fig.1.HPWeightsforT=11,andλ=7.Horizontalaxis:k=0,±1,±2,...;Verticalaxis:weightonxt+kforestimatingτt2.Model-BasedInterpretationsIncontrasttotheoriginaladhocformulationoftheHPfiltergivenby(2),theHPfilterhasdifferentinterpretations;itcanbeseenasaparticularcaseoftheButterworthfamilyoffilters(G´omez,1999),itcanalsobeobtainedinthecontextofanunobserved-componentsformulation(HarveyandJaeger,1993andKingandRebelo,1993),orasaWiener-Kolmogorovfilter(KaiserandMaravall,2001).Theunobserved-components(UC)representationisfairlygeneral,asmanypopulardecompo-sitions,includingtheHPfilter,canbeformulatedwithinitsframework.TheUCrepresentationisgenerallygivenbyObservedSeries:xt=τt+ζt(5)Trend:τt=μ+τt−1+εt,εt∼iidN(0,σ2ε)(6)Cycle:ζt∼stationaryandergodic(7)3Often,thecycle,ζt,isassumedtohaveanARMA(p,q)representation,ϕp(B)ζt=ϑq(B)at;thestandardUC-ARMAdecompositionassumesthattheshockstothetrendandthecycleareuncorrelated—i.e.,itsetsσaε=0.WhenthisrestrictionisrelaxedthisdecompositionleadstotheBeverige-Nelsondecomposition(Morley,NelsonandZivot,2003).AsnotedbyHarveyandJaeger(1993)andKingandRebelo(1993),theHPfiltercanbeinterpretedastheoptimalestimatorintheUCmodelgivenby(5),withζt∼iidN(0,σ2c),and∇2τt=εt,εt∼iidN(0,σ2ε)(8)ThisformulationimpliesanIMA(2,2)modelforxt(KaiserandMaravall,1999),andtherefore:∇2xt=εt+∇2