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ParsimoniousModelingofYieldCurvesCharlesR.Nelson;AndrewF.SiegelTheJournalofBusiness,Vol.60,No.4.(Oct.,1987),pp.473-489.StableURL:=0021-9398%28198710%2960%3A4%3C473%3APMOYC%3E2.0.CO%3B2-6TheJournalofBusinessiscurrentlypublishedbyTheUniversityofChicagoPress.YouruseoftheJSTORarchiveindicatesyouracceptanceofJSTOR'sTermsandConditionsofUse,availableat://@jstor.org.:522007CharlesR.NelsonAndrewF.SiegelUniversityofWashingtonParsimoniousModelingofYieldCurves*I.IntroductionTheneedforaparsimoniousmodeloftheyieldcurvewasrecognizedbyMiltonFriedman(1977,p.22)whenhestated,Studentsofstatisticaldemandfunctionsmightfinditmoreproductivetoexaminehowthewholetermstructureofyieldscanbedescribedmorecompactlybyafewparameters.Thepurposeofthispaperistoin-troduceasimple,parsimoniousmodelthatisflexibleenoughtorepresenttherangeofshapesgenerallyassociatedwithyieldcurves:mono-tonic,humped,andSshaped.TheabilityofthemodeltofitU.S.Treasurybillyieldsandtopre-dictthepriceofalong-termTreasurybondsug-geststousthatthemodelsucceedsintheobjec-tivesetbyFriedman.Potentialapplicationsofparsimoniousyieldcurvemodelsincludede-TheauthorswishtothanktheCenterfortheStudyofBankingandFinancialMarketsattheUniversityofWashing-tonforsupportingthisresearch.Nelsonalsoreceivedsup-portfromtheNationalScienceFoundationunderagranttotheNationalBureauofEconomicResearch,whichisac-knowledgedwiththanks.ResearchassistancewasprovidedbyFrederickJoutzandAnnKremer.WearegratefultoVanceRoleyforobtainingthedatasetusedinthisstudy.ThanksareduetoEdwardBomhoff,J.HustonMcCulloch,PatrickMinford,VanceRoley,GaryShea,RobertShiller,RichardStehle,MarkWatson,andananonymousrefereeforhelpfulcriticismandsuggestionswithoutimplicationofre-sponsibilityfortheresult.(JournalofBusiness,1987,vol.60,no.4)01987byTheUniversityofChicago.Allrightsreserved.0021-9398187/6004-0002$01.SOThispaperintroducesaparametricallypar-simoniousmodelforyieldcurvesthathastheabilitytorepresenttheshapesgenerallyassociatedwithyieldcurves:monotonic,humped,andSshaped.Wefindthatthemodelexplains96%ofthevariationinbillyieldsacrossmaturitiesdur-ingtheperiod1981-83.ThemovementoftheparametersthroughtimereflectsandconfirmsachangeinFederalReservemone-tarypolicyinlate1982.Theabilityofthefittedcurvestopredictthepriceofthelong-termTreasurybondwithacorrelationof.96sug-geststhatthemodelcapturesimportantat-tributesoftheyield1maturityrelation.474JournalofBusinessmandfunctions(Friedmanhadinmindmoneydemand),testingoftheoriesofthetermstructureofinterestrates,andgraphicdisplayforinformativepurposes.ThefittingofyieldcurvestoyieldlmaturitydatagoesbackatleasttothepioneeringeffortsofDavidDurand(1942),whosemethodoffittingwastodrawamonotonicenvelopeunderthescatterofpointsinawaythatseemedtohimsubjectivelyreasonable.Yieldmaybetransformedtopresentvalue,andJ.HustonMcCulloch(1971,1975)hasproposedapproximatingthepresentvaluefunctionbyapiecewisepolynomialsplinefittedtopricedata.GaryShea(1982,1985)hasshownthattheresultingyieldfunctiontendstobendsharplytowardtheendofthematurityrangeobservedinthesample.Thiswouldseemtobeamostunlikelypropertyofatrueyieldcurverelationandalsosuggeststhatthesemodelswouldnotbeusefulforpredictionoutsidethesamplematurityrange.Otherresearchershavefittedavarietyofparametricmodelstoyieldcurves,includingCohen,Kramer,andWaugh(1966),Fisher(1966),EcholsandElliott(1976),Dobson(1978),HellerandKhan(1979),andChambers,Carleton,andWaldman(1984).Someofthesearebasedonpolynomialregression,andallincludeatleastalineartermthatwouldforceextrapolatedverylongtermratestobeunboundedlylarge(eitherpositiveornegative)despitetheirabilitiestofitcloselywithintherangeofthedata.VasicekandFong(1982)haverecommendedexponentialsplinesasanalternativetopolynomialsplines.Inacomparisonofthetwosplinemethodologies,Shea(1984)findsthatexponentialsplinesaresubjecttothesameshortcomingsthatpolynomialsplinesare,essentiallybecausepolynomialsplinesareusedafterachangeofvariables.Studentsofthetermtomaturitystructureofinterestrateshaveinvariablydescribedyieldcurvesthatareessentiallymonotonic,humped,or,occasionally,Sshaped.Thisconsistencyisstrikinglyevi-dentinthelonghistoricalrecordofsubjectivelydrawncurvespre-sentedbyWood(1983).AsimilarconsistencyisshownbytheyieldcurvesplottedinMalkiel(1966,pp.13,14)andintheTreasuryBulle-tin.Thisistrueevenofyieldcurvesbas