Linear Cointegration of Nonlinear Time Series with an Application to Interest Rate Dynamics

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Linear Cointegration of Nonlinear Time Series with an Application to Interest Rate Dynamics Barry E. Jones and Travis D. Nesmith 2007-03 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Linear Cointegration of Nonlinear Time Series with an Application to Interest Rate Dynamics BarryE.Jones BinghamtonUniversity TravisD.Nesmith (cid:3) BoardofGovernorsoftheFederalReserveSystem November29,2006 Abstract Wederiveade(cid:2)nitionoflinearcointegrationfornonlinearstochasticprocesses usingamartingalerepresentationtheorem. Theresultshowsthatstationarylinear cointegrationscanexhibitnonlineardynamics,incontrastwiththenormalassumptionoflinearity. Weproposeasequentialnonparametricmethodtotest(cid:2)rstfor cointegrationandsecondfornonlineardynamicsinthecointegratedsystem. We applythismethodtoweeklyUSinterestratesconstructedusingamultirate(cid:2)lter rather than averaging. The Treasury Bill, Commercial Paper and Federal Funds ratesarecointegrated,withtwocointegratingvectors. Bothcointegrationsbehave nonlinearly. Consequently,linearmodelswillnotfullyreplicatethedynamicsof monetarypolicytransmission. JELClassi(cid:2)cation:C14;C32;C51;C82;E4 Keywords:cointegration;nonlinearity;interestrates;nonparametricestimation Correspondingauthor:20thandCSts.,NW,MailStop188,Washington,DC20551 (cid:3) E-mail:travis.d.nesmith@frb.gov MelvinHinichprovidedtechnicaladviceonhisbispectrumcomputerprogram. HermannBierensprovided technicaladviceonimplementinghisnonparametriccointegrationtests.WethankWilliamBarnett,Florenz Plassmann,EricVerhoogenandseminarparticipantsatGeorgeWashingtonUniversity,theIMFInstitute, the2006MidwestMacroeconomicsConference,andthe2006UdineWorkshopforhelpfulcommentsand suggestions. WealsothankSamiaHusainforresearchassistance. Theviewspresentedaresolelythoseof theauthorsanddonotnecessarilyrepresentthoseoftheFederalReserveBoardoritsstaff. Anyremaining errorsaretheresponsibilityoftheauthors. 1

1 Introduction Cointegrationistheprimaryeconometricmodelofsystemdynamicsfornonstationary time series. Cointegration is normally de(cid:2)ned as the existence of a stationary linear combinationofnonstationarytimeseries. Thefactthatthecombinationislineardoes not necessarily imply linear dynamics for the resulting stationary stochastic process. Cointegrationisneverthelessstronglyassociatedwithlineardynamics,becausecointegrationwasinitiallydevelopedwithinthelinearBox-Jenkinsframework. Inparticular, the standard model of cointegration(cid:151)the vector error correction model (VECM)(cid:151) doesassumelineardynamics. Linearitymakeseconometricmodelstractable,butlinearmodelscanonlyreproducearestrictedclassofdynamicbehavior. Mosteconomic modelsarenonlinear: producingricherdynamics. Abroaderde(cid:2)nitionofcointegrationisnecessaryinordertoincorporatenonlinear dynamics. Ourmotivationforbroadeningtheclassofdynamicsisbasedonthesimple observation that nonlinearity is a dominant property in the sense that a linear combinationofnonlinearprocessesisitselfgenerallynonlinear. Nonstationarityissimilarly dominant.Cointegrationisaspecialcasewhereaddingtwoormorenonstationaryprocessestogetherresultsinastationaryprocess. Butifanyofthecointegratedseriesare nonlinear, the linear combination generally produces a nonlinear stationary process. Forexample,letx x " sothatx isarandomwalkandlet y x z where t t 1 t t t t t D (cid:0) C D C z isastationaryandnonlinearstochasticprocess. Then[1 1]T isalinearcointet (cid:0) gratingvectorfor[y x ]T as y x z andz isstationary. Sincez ,isnonlinear, t t t t t t t (cid:0) D thecointegratingrelation y x isanonlinearstochasticprocess. t t (cid:0) We developed this motivation in Barnett, Jones, and Nesmith (2000), mainly as a critique of Johansen’s maximum likelihood cointegration estimator which assumes linearity.1 Althoughwestartfromthesameobservation(cid:151)thatlinearcombinationsof nonlinearseriesaregenerallynonlinear(cid:151)thispaperismoreconstructive. Wederivea de(cid:2)nitionoflinearcointegrationfromHallandHeyde’s(1980)martingalerepresentation theorem for stationary stochastic processes. This extended de(cid:2)nition, suggested by Bierens (1997), does not a priori restrict the dynamic behavior to be linear as did previousde(cid:2)nitionsofcointegration. Asaresult,thestationarydynamicsofthecointegrated system may exhibit nonlinear dynamics. We also develop an asymptotically validproceduretotestlinearcointegrationsforthenonlinearstationarydynamics. Ourde(cid:2)nitionofcointegration,andtheassociatedconceptofnonlineardynamics, differs from nonlinear cointegration introduced by Granger (1991). Intuitively, non- 1Johansen’sestimatorassumeslinearitybecauseitisbasedonaVECMmodeldrivenbyGaussianinnovations.CalzaandSousa(2006)citeourcritiqueintheirrejectionofJohansen’sestimator. 2

linear cointegration occurs when a nontrivial nonlinear combination of nonstationary time series is stationary. In contrast, our extended de(cid:2)nition still uses linear combinationstoproducestationarity. Thede(cid:2)nitionofnonlinearcointegrationactuallysays nothingaboutthedynamicsoftheresultingstationaryprocess,whichisourfocus. In practice, nonlinear cointegration has been de(cid:2)ned by a VECM model with a nonlinear error correction term. Consequently, nonlinear cointegration has been predicated on the assumption that the stationary dynamics are linear. The martingale based definitioncouldbefurtherextendedtoallowfornonlinearcointegrationwithpotentially nonlinear stationary dynamics, but testing for such complicated dynamics would be challenging. To test for the presence of nonlinear dynamics in a linearly cointegrated system, we implement a sequential procedure. In the (cid:2)rst stage, cointegrating vectors are estimated using Bierens’s (1997, 2005) nonparametric test for cointegration. Although Bierens(1997)assumedlinearityforclearerexposition,thetestisstillvalidwhenapplied to nonlinear processes. If a cointegration is found, it de(cid:2)nes a new stationary processrepresentingthelong-runeconomicequilibrium. Inthesecondstage,thestationarycointegrationistestedfornonlinearity. Atthisstagewearetestingasystemof economicvariables,oranequilibriumeconomicrelation,fornonlineardynamicseven thoughexistingtestsfornonlinearityareunivariate. Thenonlinearitytestusedinthesecondstageshouldbeconservative. Aconservativetestreducesthechancesof(cid:2)ndingnonlinearityduetoinappropriatelyacceptingthe hypothesisofstationarityinthe(cid:2)rststage. Thepossibilitythatanonstationarylinear timeseriescouldbeidenti(cid:2)edasnonlinearisnotanewproblem. Testsfornonlinearity requirestationarityasamaintainedassumption. Giventhestrongevidencethatmany economicseriesarenonstationary, thisrequirementimpliesthattestsfornonlinearity are almost always conditional on the correct removal of nonstationarity, for example throughcorrectdetrendingordifferencing. Failuretoremoveanynonstationaritycan leadtospuriousacceptanceofnonlinearity(Lee,Kim,andNewbold,2005/6). BasedonMonteCarlocomparisonsofvarioustestsfornonlinearity,weuseHinich’s (1982)nonparametricbispectraltest,whichwasfoundtobeconservative.2 Wedonot, however, implement the surrogate data and bootstrap methods introduced by Hinich, Mendes, and Stone (2005) to improve the power of Hinich’s test, as the theoretical validityofthesequentialtestingrestsonanasymptoticargument. Furthermore,bootstrappingonlythesecond-stageestimatorwouldbeinappropriate.3 2SeeBarnett,Gallant,Hinich,Jungeilges,Kaplan,andJensen(1995),Barnett,Gallant,Hinich,Jensen, andJungeilges(1996)andBarnett,Gallant,Hinich,Jungeilges,Kaplan,andJensen(1997), 3There are also potential problems with applying surrogate methods to testing for nonlinearity. The 3

To demonstrate the two-stage nonparametric testing method, we test a system of short-termU.S.interestrates: speci(cid:2)callytheratesforshort-termCommercialPaper, short-term Treasury Bills, and Federal Funds. Short-term interest rates on Federal Funds and on unsecured corporate and government debts are frequently included in studies of the business cycle, money demand, and the monetary transmission mechanism. Since short-term interest rates are likely to respond more quickly to monetary policythanothereconomicvariables,thedynamicinteractionbetweenFederalFunds andothershort-termratesiscriticaltounderstandinghowchangesinmonetarypolicy aretransmittedthroughtheeconomy. Besidestheireconomicrelevance,interestratesareavailableonadailybasisfora longperiodoftimeandthenonparametrictestsperformbetterwithmoredata. Using thebusinessdaydatadoes,howevercausedif(cid:2)cultieswithmissingvaluesduetoholidays. Toavoidthisproblem,wesamplethedailydataataweeklyfrequency,butonly afterappropriately(cid:2)lteringthedatatopreventaliasing. Thismultirate(cid:2)lter,produced by applying the anti-aliasing (cid:2)lter and resampling, appears to be a new approach in econometricsandimprovestheperformanceofbothBierens’andHinich’stests. Correctingforaliasingmayalsobethereasonweunequivocally(cid:2)ndthatU.S.interest rates contain a unit root. Whether or not interest rates contain unit roots has been heavily debated. Although many authors have found that U.S. interest rates are integrated (Nelson and Plosser, 1982, Psaradakis, Sola, and Spagnolo, 2006, Rapach andWeber,2004,Rose,1988),otherresearchhassuggestedthatinterestratesarebetterdescribedaslong-memoryorfractionallyintegratedseries(BackusandZin,1993, Gil-Alana, 2004, Tsay, 2000). The empirical case for long-memory is usually based oncon(cid:3)ictingresultsfromvarioustestsoftheunitrootandstationarityhypotheses. In contrast,we(cid:2)nduniformagreementamongavarietyofunivariateteststhatthelevels oftheinterestratesarenonstationaryandthedifferencesarestationary. Otherauthors, suchasPfann,Schotman,andTschernig(1996)andMaki(2003),havesuggestedthat interest rates exhibit nonlinear dynamics which affect the power of stationarity tests. Ourresultsdonotseemtosufferfromlowpower,despite(cid:2)ndingevidenceofnonlinearity. Since the interest rates are both integrated and nonlinear, we apply our two-stage method. Bierens’ nonparametric test shows that the weekly interest rates are linearly cointegrated. Forcomparison,wealsoperformJohansen(1988)’sstandardparametric tests. Thenonparametricandparametricresultsareverysimilar: identifyingthesame twocointegratingvectors. Inaddition,thecointegrationestimatesarerobusttoremovmethoddevelopedbyHinichetal.(2005)solvestheseproblemsforalargesubsetofunivariatelinearprocesses.Buttheirmethodneedstobeextendedbeforeitcanbeappliedtosystemsofcointegratedvariables. 4

ingtheperiodstartingnearthethirdquarterof1979throughthe(cid:2)rstquarterof1984 when the Federal Reserve shifted its monetary policy instrument away from interest rates (Rudebusch, 1995). Estimates for the prior and subsequent sub-sample (cid:2)nd the samecointegratingvectorsastheestimatesforthefullsample. After identifying two cointegrations in the (cid:2)rst stage, we subsequently test each cointegrationfornonlinearity.Linearityisrejectedforbothusingthefulldataset.This result is not completely robust as linearity can be accepted for the (cid:2)rst sub-sample. ThisresultmaystemfromthereductioninthepowerofHinich’stestthatstemsfrom the relatively short span of data. However linearity can be rejected for the second sub-sample, which suggests that the nonlinearity is not produced only by switching regimes.4 We conclude that stationary interest rate dynamics are nonlinear. A simple explanationisthattheadjustmentmechanismthatcorrectsdeviationsfromthelong-run interest rate equilibrium is nonlinear. Since we (cid:2)nd two cointegrations, it is possible thatnonlinearityalsodescribesmovementswithinthecointegrationspace. Regardless ofwhethernonlinearitycanbeisolatedasadisequilibriumphenomenaornot,theequilibriumdynamicsarenotsimplycharacterizedbytheindividualdynamicsaswouldbe expected from a linear system. This complexity points to the need for further work modelingtheinterestratedynamics. Thepaperisorganizedasfollows.Section2clari(cid:2)esthedifferencebetweenlinearityandnonlinearityforstationaryprocesses.Wealsodiscussthebispectrumtoprovide intuitionforHinich’stest.Section3containsthetheoreticalcontribution.Usingamartingalerepresentationforintegratedprocesses,wederiveade(cid:2)nitionofcointegration thatisapplicabletononlinearstochasticprocesses. Thisextendedde(cid:2)nitionoflinear cointegration is compared to the standard VECM model of both linear and nonlinear cointegration. Section 4 contains the empirical results and Section 5 concludes. The appendixreviewsthealiasingproblemandourmultirateanti-aliasing(cid:2)lterdesign. 2 Nonlinear Processes Whetheraprocessislinearornonlinearisdeterminedbyitsserialdependencestructure. For stationary processes, the difference between linear and nonlinear dynamics can be clari(cid:2)ed by looking at the restrictions implied by linearity for both the Wold decompositionandtheVolterrarepresentation. Thediscussioninthissectionassumes 4Severalauthorshavesuggestedregimeshiftsasasourceonnonlinearityininterestrates. SeePfann etal.(1996)foradiscussion. 5

stationarity. Wedeferformallyde(cid:2)ningstationarityuntilthediscussionofintegration andcointegrationinthenextsection. Undermildregularityconditions,astationarystochasticprocessX hasarepresent tationoftheform: 1 X g (cid:15) ; (2.1) t u t u D (cid:0) u DX(cid:0)1 where g is a sequence of coef(cid:2)cients, and (cid:15) is a serially uncorrelated white noise u t inputsequence. ThisisaconsequenceoftheWolddecompositiontheorem. TheWold decompositionthereforeshowsthatastationaryprocess,suchasthatproducedbycointegration,canberepresentedastheoutputofamovingaverage(cid:2)lterappliedtouncorrelatedwhitenoiseinput. At (cid:2)rst glance, this representation seems to suggest that every stationary process can be represented as an in(cid:2)nite-order moving average process. This impression is misleading. The process may be nonlinear because the input process is uncorrelated butisnotnecessarilystochasticallyindependent. X canberepresentedastheoutputof t atime-invariantlinear(cid:2)lterappliedtowhitenoiseinput,butX isalinearprocessonly t if" isstochasticallyindependent.5 Ingeneral,whitenessisnotsuf(cid:2)cientforstochastic t independenceunlessthewhitenoisesequenceisGaussian. Forlinearmodels,thecoef(cid:2)cientsofthemovingaveragerepresentationcompletely characterize the effect of a shock. The response of a linear sequence to a shock is completelycharacterizedbythetransferfunctionofthe(cid:2)lter: G.f/ 1 g e i.2(cid:25)f/u: (2.2) u (cid:0) D u DX(cid:0)1 If the input to a linear sequence is a sine wave of frequency f, the output will also be a sine wave with frequency f. The amplitude will be scaled by G.f/, and the j j phasewillbeshiftedbytan 1.ImG.f/=ReG.f//wheretheoperator denotesthe (cid:0) j j complexmodulus. Ageneralmodelforastationarystochasticprocessis X h.:::;" ;" ;" ;" ;" ;:::/ (2.3) t t 2 t 1 t t 1 t 2 D (cid:0) (cid:0) C C where,unliketheWoldrepresentation" isstochasticallyindependent. If X iscausal, t t itdoesnotdependonthefuturevaluesof" (makingthiscommonassumptionwould t notsubstantivelyaffectourdiscussion). Ifh isawell-behavedfunctionitcanberep- 5SeeHinich(1982),HinichandPatterson(1989)andPriestley(1988,pp.13-16). 6

resentedasaVolterraseries:6 1 1 1 X g " g " " t u t u u;v t u t v D (cid:0) C (cid:0) (cid:0) u u v DX(cid:0)1 DX(cid:0)1 DX(cid:0)1 1 1 1 g " " " ::: (2.4a) u;v;w t u t v t w C (cid:0) (cid:0) (cid:0) C u v w DX(cid:0)1 DX(cid:0)1 XD(cid:0)1 IfX islinearthenonlythe(cid:2)rsttermintheVolterrarepresentationexists;forlinear t processestheWoldandVolterrarepresentationsareidenticalimplyingthattheimpulse processintheWolddecompositionmustbeindependentinthiscase. Theexistenceofhigher-ordertermsintheVolterraexpansionimpliesthat X isa t nonlinearprocess. Unlikealinearprocess,theresponseofthenonlinearsequencetoa shockwilldependongeneralizedtransferfunctionsoftheform: G.f/ 1 g e i.2(cid:25)f/u; G.f;g/ 1 1 g e i2(cid:25).fu gv/; ::: (2.5) u (cid:0) u;v (cid:0) C D D u u v DX(cid:0)1 DX(cid:0)1 DX(cid:0)1 If the input to a nonlinear sequence contains components with frequencies f and g, thentheoutputwillcontaincomponentswithfrequencies f,g,.f g/,2f,2g,2.f C C g/, 3f, 3g, 3.f g/;:::, and the amplitudes and phases of these components will C dependonthegeneralizedtransferfunctions. TestsforlinearityandGaussianitycanbebasedonthepropertiesofthesegeneralizedtransferfunctionsasre(cid:3)ectedinaprocess’higher-orderpolyspectra. Ingeneral, thekth-orderpolyspectrumistheFouriertransformofthekth-ordercumulantfunction. The(cid:2)rstthreecumulantsarede(cid:2)nedbyc .t/ E[X ],c .t ;t / E[X X ],and X D t XX 1 2 D t1 t2 c .t ;t ;t / E[X X X ].7 Strict stationarity (or even third-order stationar- XXX 1 2 3 D t1 t2 t3 ity) implies c .t/ 0 for all t, c .t ;t / is a function only of (cid:28) .t t /, and X XX 1 2 1 2 D D (cid:0) c .t ;t ;t /isafunctiononlyof(cid:28) .t t /and(cid:28) .t t /. Thesecondand XXX 1 2 3 1 1 2 2 2 3 D (cid:0) D (cid:0) third-ordercumulantfunctionsforstationaryprocessescanbedenotedbyc .(cid:28)/and XX c .(cid:28) ;(cid:28) /respectively. Thesefunctionsareassumedtobeabsolutelysummable. XXX 1 2 Thepowerspectrumisthende(cid:2)nedastheFouriertransformofc .(cid:28)/: XX P .f/ 1 c .(cid:28)/e i2(cid:25)f(cid:28); f < 1 : (2.6) X XX (cid:0) D j j 2 (cid:28) DX(cid:0)1 where f denotes the frequency measured in units of inverse time.8 The bispectrum is 6FordetailsonVolterrarepresentations,seeSchetzen(1980,1981)andRugh(1981). 7Cumulantsandmomentsareequivalentuptothethird-order.Thisisnottrueforhigherorders. 8Multiplyingthesefrequenciesby2(cid:25)convertsthemtoradians. 7

de(cid:2)nedasthesecond-orderFouriertransformofc .(cid:28) ;(cid:28) /: XXX 1 2 B .f;g/ 1 1 c .(cid:28) ;(cid:28) /e i2(cid:25).f(cid:28)1 g(cid:28)2/; (2.7) X XXX 1 2 (cid:0) C D (cid:28)1XD(cid:0)1 (cid:28)2XD(cid:0)1 .f;g/ D .f;g/ 0 < f < .1=2/;g < f;2f g < 1 which is called the 2 D f j C g principal domain (Hinich and Messer, 1995). If the second and third-order cumulant functionsareabsolutelysummable,thenthepowerspectrumandthebispectrumexist andarewellde(cid:2)ned. Theintegralofthepowerspectrumisequaltothevarianceofthe sequence, c .0/, and the power spectrum can be interpreted as a decomposition of XX thevariancebyfrequency. Similarly,thebispectrumdecomposestheskewnessofthe sequence,c .0;0/,bypairsoffrequencies. XXX De(cid:2)ne the skewness function, 0 .f;g/, as the normalized square modulus of the X bispectrum: B .f;g/2 X 0 X .f;g/ j j : (2.8) D P .f/P .g/P .f g/ X X X C Let" beastochasticallyindependentsequence,then P .f/ c .0/and B .f;g/ t " "" " D D c .0;0/forall.f;g/ D. Thisimpliesthatalinearprocesshasaconstantskewness """ 2 function equal to 0 .f;g/ c .0;0/2c .0/ 3, because P .f/ G.f/2P .f/ X """ "" (cid:0) X " D D j j andB .f;g/ G.f/G.g/G .f g/B .f;g/.Ifthestochasticallyindependentinput X (cid:3) " D C sequenceisalsoGaussianthenc .0;0/ 0and0 .f;g/willbeidenticallyzero. """ X D ThesepropertiesformthebasisofHinich’s(1982)testsofGaussianityandlinearity. The intuition is that the skewness function will be (cid:3)at for linear processes and identicallyzeroforGaussianprocesses. Iftheskewnessfunctionissigni(cid:2)cantlyrough thenlinearityisrejected. Hinich’s test is conservative, not only in practice but also in theory. The test is conservative in theory because the null hypothesis that 0 .f;g/ is constant for all X frequency pairs is a necessary, but not suf(cid:2)cient condition for linearity. Nonlinearitycouldbedetectedinhigherorderpolyspectra,evenifthenormalizedbispectrumis (cid:3)at.9 Nevertheless,Ashley,Patterson,andHinich(1986)foundthatHinich’sbispectral testhadsubstantialpoweragainstmanycommonnonlineartimeseriesmodelsincluding bilinear models, nonlinear moving-average and autoregressive models, and linear andnonlinearthresholdautoregressivemodels. AkeyaspectofHinich’stestsisthat(atleastthird-order)stationarityisassumed. However,economictimeseriesoftenappeartobesubjecttopermanentshocks,andit 9Testsofhigher-orderpolyspectraaregenerallynotapplicableineconometrics,becausemosteconomic timeseriesarenotlongenoughforconsistentestimationofeventhefourth-orderpolyspectrum. 8

hasbecomeastandardpracticetomodelthesetimeseriesasnon-stationaryintegrated processes. As is the norm in testing for nonlinearity, if the process is nonstationary Hinich’stestcanfalselyrejectlinearity. Consequently,individualeconomicseriesare usuallydifferencedordetrendedbeforebeingtestedfornonlinearity.Cointegrationcan providearichermodelofnonstationarityandanalternatemethodtorecoverstationary dynamicsforasystemofeconomicvariables. 3 Integration and Cointegration Cointegrationasitisnormallyde(cid:2)nedisincompatiblewithnonlineardynamics. Cointegrationwasdevelopedwithintheframeworkofvectorerror-correctionmodels. Linearityofthestationarydynamicswasexplicitlyassumed,becausetheVECMmodelis linearandtheinnovationprocesswasassumedtobeindependentorGaussian. However,thereisnocompellingreasonforthisrestriction. Using Hall and Heyde’s martingale representation, we show that the innovation process of a integrated series is not in general a linear stochastic process. It is then straightforward to de(cid:2)ne cointegration for a vector of integrated processes using the martingalerepresentation. Forclarity,ourmartingale-basedde(cid:2)nitioniscontrastedwiththestandardVECM de(cid:2)nitionofcointegrationincludingtheextensiontononlinearcointegration. Therepresentationtheoremshowsthatnonlinearityismoregeneralthanjustnonlinearcointegration,asnonlineardynamicscanbepresentevenwhenthecointegratingrelationship islinear. Initially,weestablishsomede(cid:2)nitionsandnotationalconventions. Thede(cid:2)nitions are standard and can be found in a number of references. For all time periods, let S t denoteaq-dimensionalvectorrandomsequence, S .S ;:::;S /T onaprobability t 1t qt D space.(cid:127); ;(cid:22)/. F MartingaleDe(cid:2)nition. Avectormartingaleisanadaptedsequence.S ; /where t t t F F isanincreasingsequenceof(cid:27)-algebrascontainedin suchthat S isintegrableand t F satis(cid:2)es E.S / S a:s: t t 1 t 1 jF(cid:0) D (cid:0) foreveryt:The(cid:2)rstdifferenceofamartingale,1S S S Y isreferredtoas t t t 1 t D (cid:0) (cid:0) D amartingaledifferencesequence;itisintegrableandsatis(cid:2)es E.Y / 0a:s: t t 1 jF(cid:0) D LetT : (cid:127) (cid:127)denoteaonetooneergodicmeasure-preservingshifttransforma- ! tion. If X .!/isarandomvariableontheprobabilityspace,then X .!/ X .Tt!/ 0 t 0 D 9

de(cid:2)nesastrictlystationaryergodicsequence. Astochasticsequenceissaidtobeintegratedoforderone, I.1/,ifthe(cid:2)rstdifferenceofthesequenceisstrictlystationary.10 Amartingaledifferencesequenceisstrictlystationarybyde(cid:2)nition,somartingalesare I.1/:Theconceptofintegrationcanbeextendedtohigherorders. HallandHeyde(1980,pp.136)provethefollowingrepresentationtheorem: HallandHeydeRepresentationTheorem. Anystationaryergodicsequence X can t berepresentedintheform: X Y Z Z ; (3.1) t t t t 1 D C (cid:0) C whereY .!/ Y .Tt!/isastationarymartingaledifferencesequence,and Z .!/ t 0 t D D Z .Tt!/suchthat Z .!/isinL1.11 Explicitformulasfortherepresentationaregiven 0 0 by: 1 Y .E[X F ] E[X F ]/ (3.2) 0 k 0 k 1 D j (cid:0) j (cid:0) I k DX(cid:0)1 and, 1 1 (cid:0) Z .E[X F ]/ .X E[X F ]/; (3.3) 0 k 1 k k 1 D j (cid:0) (cid:0) (cid:0) j (cid:0) k 0 k XD DX(cid:0)1 where F isthe(cid:2)ltrationgeneratedbytheshifttransform. s f g From the Hall and Heyde representation, we derive a representation for an I.1/ sequence: I.1/RepresentationCorollary. Ifthestationary(cid:2)rst-differenceofan I.1/sequence isergodic,thenthenonstationaryleveloftheintegratedsequenceisrepresentedby t S Y Z Z S : (3.4) t s t 1 1 0 D (cid:0) C C C s 0 XD Proof. Ifthestationary(cid:2)rst-differenceofan I.1/sequenceisergodic,fromtherepresentationtheoremithasthefollowingrepresentation: S Y Z Z ; t t t t 1 D C (cid:0) C where Y is a stationary vector martingale difference sequence and Z is a stationary t t 10EngleandGranger(1987)addtheconditionthatthestationarymovingaveragerepresentationofthe (cid:2)rstdifferencebeinvertible. 11HallandHeydealsorequire E X0 < . Alternatively,thetheoremcanbeprovedundermixingale j j 1 assumptions,seeDavidson(1994,pp.247-252). 10

vector sequence. Equation (3.4) is derived by solving this representation of the (cid:2)rst difference for S , advancing the index one period and recursively substituting for t 1 (cid:0) S . t 1 C Remark1. TheleveloftheI.1/sequenceisdominatedbytheaccumulatedmartingale differencesequencewhichgivesrisetothepermanentshocks. Remark2. Thecomponentsof1S ,the(cid:2)rstdifferenceofS ,havetheform: t t 1S Y Z Z : (3.5) jt jt jt j;t 1 D C (cid:0) C From(3.2)and(3.3),bothY andZ exhibitdependence,althoughY isamartingale jt jt jt differenceandisnon-forecastableinthemeansquaremetric,seeHinichandPatterson (1987). A system of integrated time series is cointegrated if some linear combinations of thetimeseriesarestationary. Cointegrationcanbede(cid:2)nedasareducedrankcondition involvingthecovariancematrixofthevectormartingaledifference. Weneedthefollowinglemmafortheformofthecovariancematrixforavectormartingaledifference sequence. Lemma1. Thecovariancematrixofamartingaledifferencesequencehastheform: CCT ifs t E[Y YT] D (3.6) t s D8 0 ifs t < 6D Proof. Vectormartingaledifferencesare:seriallyuncorrelatedandhavepositivesemide(cid:2)nitecovariancematrices. Cointegratingvectorsforan I.1/sequencethatarebasedonthemartingalerepresentationarede(cid:2)nedby: Theorem1(MartingaleCointegration). IfCin(3.6)hasreducedrank,.q r/,then (cid:0) therewillexistr non-trivialvectors(cid:12) ;:::;(cid:12) ,calledcointegrationvectorswhere,the 1 r linear combinations (cid:12)TS , called cointegration relations, are stationary for all j j t D 1;:::;r. Proof. Choose(cid:12);aqbyr matrix (cid:12) (cid:12) :::(cid:12) thatspansthenullspaceofC.Thenby 1 2 r de(cid:2)nition(cid:12)TC 0T, forall j 1;:::;r:Thesevectorsde(cid:2)nestationaryprocesses j D D (cid:2) (cid:3) 11

because t t t t T T E (cid:12)TY (cid:12)TY (cid:12)TE Y Y (cid:12) 0 j s j s D j 0 s s j (3.7) (cid:20)(cid:16)X s D 0 (cid:17)(cid:16)X s D 0 (cid:17) (cid:21) (cid:20)(cid:16)X s D 0 (cid:17)(cid:16)X s D 0 (cid:17) (cid:21) (cid:12)TCCT(cid:12) 0T0 0: D j j D D Theproofalsosupportsthefollowingcorollary: Corollary2. Denotetheq by.q r/orthogonalcomplimentmatrixof(cid:12) by(cid:12) , so (cid:0) ? that(cid:12) hastheproperty(cid:12)T(cid:12) 0. Thecommonstochastictrend(cid:12)TS , whichhas t ? ? D ? dimension .q r/, is integrated but not cointegrated. The q-dimensional sequences (cid:0) 1S t and (cid:12)T (cid:12)T1 S t are both stationary. In the absence of cointegration, the ? twotransfhormationsareiequivalent. Ifr 0,then(cid:12) isfullrankandcanbetakenas D ? theidentitymatrix. In contrast to extant de(cid:2)nitions, the martingale based de(cid:2)nition of cointegration doesnotrequireindependence,Gaussianity,orlinearityofthestationarycomponents oftheprocess.Previousde(cid:2)nitionsoflinearcointegrationareaspecialcase,muchlike independenceisaspecialcaseofthemartingaleproperty. Thedifferencecanbemade clearerbylookingattheexpectationofthecointegrationrelations, E (cid:12)TS E (cid:12)T.Z Z / (cid:12)TS : (3.8) 0 j t D 0 j 1 (cid:0) t C 1 C j 0 h i h i These expectations have been purged of the effects of the permanent shocks generatedbythemartingaledifferenceandarestationary. Whenviewedasanewstochastic process,therearenorestrictionsonthedependencestructureof(cid:12)TS ,asidefromstaj t tionarityandergodicity. Ourmethodcontrastswiththestandardapproachtocointegration.Stationarylinear combinations of integrated variables are usually speci(cid:2)ed to follow a linear ARMA processorareincludedinlinearstructuralmodels. ThestandardlinearVECMhasthe form: p 1 1S (cid:11)(cid:12)TS (cid:0) 0 1S " : (3.9) t t 1 j t j t D (cid:0) C (cid:0) C j 1 XD Ifthemodeliscointegratedthentheq byr parametermatrices,(cid:11) and(cid:12),haverankr. The cointegration relations enter the model linearly, through (cid:11). The error-correction model is estimated under the assumption that " is stochastically independent, which t impliesthatthecointegrationrelationsarelinearstochasticprocesses. Ourdiscussion 12

showsthatcointegrationdoesnotgenerallyimplylinearity,therefore,thereisnoreason toexpect" tobeeitherGaussianorindependent. t Granger(1991)proposesthreenonlineargeneralizationsofcointegration. The(cid:2)rst generalizationisthatnonlinearfunctionsofthetimeseriesmaybecointegratedinthe sensethatg .x /andg .x /haveadominantpropertythatthelinearcombinationof 1 1t 2 2t nonlinearlytransformedvariablesz g .x / Ag .x /doesnotexhibit. Asecond t 1 1t 2 2t D (cid:0) generalizationistoallowtime-varyingcointegrationvectors. Athirdgeneralizationis nonlinear error correction, in which the cointegration relations would enter the errorcorrectionmodelthroughanonlinearfunction f,i.e. p 1 1S f.(cid:12)TS / (cid:0) 0 1S " : (3.10) t t 1 j t j t D (cid:0) C (cid:0) C j 1 XD Granger(1991)givesconditionsunderwhich f.z/isstationary. A natural nonlinear error-correction speci(cid:2)cation is to allow mean reversion only forlargedeviations,sothat f hastheform: z if z >k f.z/ (cid:0) j j : (3.11) D8 0 if z k < j j(cid:20) In this case, z (cid:12)TS behaves like : a unit root in a neighborhood of its mean, but t t D exhibitsmeanreversionwhenitisoutsidetheneighborhood. Thismodelisastraightforward generalization of the standard error-correction model that exhibits nonlinear dynamics,butthelinearcombination(cid:12)TS isnotstationary.12 t Although, the extended de(cid:2)nition of cointegration could be further extended to allowfornonlinearcointegration,welimitourselvestothecasewherethelinearcombination is stationary. Such stationary linear combinations can exhibit nonlinear dynamics. Differentiating between nonlinear error correction and stationary nonlinear dynamicsislikelydif(cid:2)cultinpractice. Ourproposedmethodfortestingforwhetheracointegrationisnonlinearissequential. Thissequentialmethodallowsustotestthestationarycomponentsofthesystem fornonlineardynamics. We(cid:2)rstestimatecointegratingvectorsusingBierens’(1997) non-parametrictest. Bierens’testisasymptoticallyvalidforanonlineardatagenerating processes due to Hall and Heyde’s representation theorem. We then test the estimatedcointegratingrelationsforGaussianityandlinearityusingHinich’s(1982)tests. 12Forexample,theprocessde(cid:2)nedby(3.11)isnonstationaryandbehaveslikeaunitrootwhennearits mean. 13

Asymptotically,Hinich’stestisalsovalidasthecointegratingvectorsarestationary.In practice,asisthenorm,theresultsofHinich’stestsareconditionalonwhetherthe(cid:2)rst stageestimatesdoeliminateanynonstationarity. 4 Empirical Results Weapplyoursequentialproceduretoasystemofshort-termU.S.interestrates. Shortterminterestratesareavailableatahighfrequencyoveranextendedtimeperiod: constituting a larger sample size than many other business cycle variables, such as real outputandin(cid:3)ation. Inaddition,interestratesdirectlycapturethedynamicscausedby monetarypolicychanges. Weusebusinessdailydatafortheinterestratesonone-monthCommercialPaper (CP), the secondary market rate on one-month Treasury Bills (TB), and the Federal Funds(FF)from4=08=1971to8=29=1997. ThecommercialpaperandFederalFunds rates are available from the Federal Reserve Board’s website. The commercial paper rateserieswasdiscontinuedinAugust1997. TheFederalReserveBankofSt. Louis provideduswiththesecondarymarketrateonone-monthTreasuryBills. Theseinterestratesareconvertedtoone-monthholdingperiodyieldsonabondinterestbasis,and arepassedthroughananti-aliasing(cid:2)lter. Theanti-aliasing(cid:2)lterisdesignedtoremove thehigh-frequencypowerinthedailyrateseriestominimizethebiascausedbyconvertingthedailytimeseriestoweeklytimeserieseitherbydirectsamplingorweekly averaging. Thedailyratesareconvertedtoweeklyratesbysamplingthe(cid:2)ltereddaily ratesonceperweek. Figure1onthefollowingpagedisplaysthenaturallogarithmsof the(cid:2)lteredinterestrates. Correcting for aliasing does not impact the asymptotics of the cointegration estimator,becausecointegrationisrelatedtothelong-rundynamicswhilealiasingdistorts higher frequency dynamics. Nevertheless, correcting for aliasing might improve the powerofthecointegrationestimatorsina(cid:2)nitesample. Inaddition,HinichandPatterson(1985,1989)showedthataliasingdoesbiastestsfornonlinearitytowardsaccepting linearity. Aliasingisdiscussedintheappendix. Afterapplyingthemultirate(cid:2)lter,wetestthisdatawithourtwo-stagemethod:(cid:2)rst testing for cointegration and then testing for nonlinearity. Two cointegrating vectors arefoundforthesystemofthreeinterestratesovertheperiod1971 1997. Wethen (cid:0) run several tests on each cointegrating relation. We (cid:2)rst test the cointegrations for an alternative form of nonstationarity considered by Hinich and Wild (2001). This alternativetypeofnonstationarityisrejected, sowetestforGaussianity. Gaussianity 14

Figure1: LogarithmofInterestRates 1971-1997 3.6 2.4 1.2 0.0 71 73 75 77 79 81 83 85 87 89 91 93 95 97 One-Month Treasury Bill Rate 3.6 2.4 1.2 0.0 71 73 75 77 79 81 83 85 87 89 91 93 95 97 One-Month Commercial Paper Rate 3.6 2.4 1.2 0.0 71 73 75 77 79 81 83 85 87 89 91 93 95 97 Federal Funds Rate ofboththerealandimaginarypartsofthebispectrumisstronglyrejected. Finally,we testfornonlinearity. We(cid:2)ndstrongevidencethatthecointegrationsexhibitnonlinear dynamics. 4.1 UnivariateTests Beforeestimatingcointegrationrelations,werunabatteryofunivariatetests. We(cid:2)rst test the unit root and stationarity hypotheses on ln.CP/, ln.TB/, ln.FF/, and their (cid:2)rst differences .1/ using several tests with different nulls. These tests include: an augmented Dickey-Fuller (ADF) test and a Phillips-Perron (PP) test of the unit root hypothesesagainstthealternativeofstationarity;theKPSStestofthenullofstationarityagainstthealternativeofnonstationarity;andtheBierens(1997)non-parametrictest fortheexistenceofcointegrationrunasaunivariatetestoftheunitrootwithdrifthy- 15

Table1: UnivariateStationarityTests Variable ADF PP KPSS1 Bierens ln.CP/ 1:9132 8:6587 0:9272 1:1441 (cid:0) (cid:0) ln.TB/ 1:8289 9:4498 1:0186 0:7741 (cid:0) (cid:0) ln.FF/ 1:8174 8:8747 1:0027 1:0203 (cid:0) (cid:0) 1ln.CP/ 7:4386 740:9803 0:1120 0:0000 (cid:0) (cid:0) 1ln.TB/ 8:0123 877:6077 0:1210 0:0000 (cid:0) (cid:0) 1ln.FF/ 7:3217 2118:753 0:1249 0:0000 (cid:0) (cid:0) H0: UR UR S URD H1: S S NS TS 5%c.v. < 3:86 < 14:0 >:436 <:025 (cid:0) (cid:0) 10%c.v. < 2:57 < 11:2 >:347 <:006 (cid:0) (cid:0) pothesisagainsttrendstationarityoneachvariable.13 FortheADFtest,thelaglength, p,ischosenbytheformula p 5.n/:25. ForthePPandKPSStests,thetruncationlag D fortheNewey-Westestimatorisalsosetwiththisformula. Thetestresultsareshown in Table 1 along with a mnemonic for the tests’ hypotheses and the 5 and 10 percent criticalvalues. TheloggedinterestratesareclearlyI.1/processes:everytestrejectsstationarityof thelevelsatwellabovethe95%con(cid:2)dencelevelandfailstorejectstationarityofthe (cid:2)rst differences even at the 80% con(cid:2)dence level. The consistency of the test results isimportant,becausedifferencesinthesetestscanbeinterpretedasevidenceoflongmemory rather than integration. For example, Karanasos, Sekioua, and Zeng (2006) interprettheirsimultaneousrejectionofboththeunitroothypothesisandstationarity asevidenceforfractionalintegrationandlong-memoryinrealU.S.interestrates. Our resultsarenotopentosuchinterpretation. Given the results of the stationarity tests, we test the stationary (cid:2)rst difference of each interest rate for nonlinearity. We pre-whiten each of the components using an AR.6/(cid:2)ltertoeliminatebiasinthespectralestimationpriortotestingandtodecrease thelikelihoodoffalselyrejectingthenulloflinearity. Thetests(availableonrequest) provideoverwhelmingevidenceofnonlineardynamicsforthe(cid:2)rstdifferencesofthese short-terminterestratesoverthefullsample. Wealsotestedthe(cid:2)rstdifferencesfornonlinearityovertwosub-periods: Sept. 13, 1974 through Sept. 19, 1979 and March 1, 1984 through Dec. 31, 1996. These are periodsoverwhichatargetforthefederalfundsratecanbeconstructed,seeRudebusch 13Stationaritycanbeviewedasaspecialcaseoftrendstationaritywiththetrendrestrictedtobezero. Consequently, running versions of the ADF, PP, and KPSS tests that test for trend stationarity produces resultsconsistentwithBierens’test. 16

(1995). Effectively, we are dropping the period when the Federal Reserve shifted its intermediate target away from interest rates. This period is also when many interest rateswerederegulated. For these sub-samples, we can accept the null of linearity in the (cid:2)rst sub-sample, butrejectlinearityinthesecond. Thenumberofdatapointsforthe(cid:2)rstsub-sampleis 258versus669forthesecondsub-sampleand1;363forthefullsample. Theevidence reportedinAshleyetal.(1986)wouldindicatethatthepowerofthesetestsissubstantiallyhigheroverboththesecondsub-sampleandoverthefullsample. Thisprovides oneexplanationfortheinabilitytorejectlinearityinthe(cid:2)rstsub-sample.Anotherpossibleexplanationfor(cid:2)ndingnonlinearityonlyinthesecondsub-samplecouldbethat deregulationofinterestratestransformedthedynamicsgoingforward. Regardless for the reasons for accepting linearity in the (cid:2)rst sub-sample, (cid:2)nding evidenceofnonlinearityinthesecondsub-sampleiscrucial. Iflinearitywasrejected forbothsub-samples,itwouldappearthatthenonlinearityfoundoverthefullsample wasdrivensolelybyaregimeshift. Rejectinglinearityinthesecondsub-sampledoes notruleoutabreakinthedynamicsduetothepolicy,butitdoesruleouttheshiftbeing theonlysourceofnonlinearity. Consequently,wecontinueanalyzingthefullsample, althoughwealsochecktheresultsforthetwosub-samples. 4.2 Cointegration Thecointegrationanalysisusedthesystem S [ln.CP1M/;ln.TB1M/;ln.FF/]T : t D The cointegration analysis is conducted in two steps: rank identi(cid:2)cation and estimation. Therankidenti(cid:2)cation,whichdeterminesthenumberofcointegrationrelations, isbasedonthenon-parametrictestproceduredevelopedbyBierens(1997,2005). The number of cointegration relations is determined by a set of hypothesis tests, called (cid:21)-min tests, that are essentially non-parametric versions of the well-known Johansen (1988)parametric(cid:21)-maxtests. The(cid:21)-mintestsarenon-parametricbecausethematrices involved are constructed from the data independently of the data-generating process. Thenumberofcointegrationrelationscanalsobeestimatedusingafunctionof theeigenvaluesg .r/. Thevalueofr thatminimizesg .r/isaconsistentestimateof m m O O thetruenumberofcointegrationrelations. The number of cointegrations determined by both the (cid:21)-min test and estimating g .r/ is 2. The (cid:21)-min tests are reported in Table 2 on the following page. M is the m O 17

Table2: NonparametricCointegrationTests Hypothesis TestStat CriticalRegion M Conclusion H0:r 0 0:00000 20%.0;:006/ 3 Reject D H1:r 1 10%.0;:017/ 4 Reject D 5%.0;:008/ 4 Reject H0:r 1 0:00054 20%.0;:077/ 3 Reject D H1:r 2 10%.0;:034/ 3 Reject D 5%.0;:017/ 3 Reject H0:r 2 0:76618 20%.0;:341/ 3 Accept D H1:r 3 10%.0;:187/ 3 Accept D 5%.0;:111/ 3 Accept Misthesmoothingparameterforthenonparametricestimator smoothingparameter;thevalueissetoptimallyforthedifferentcon(cid:2)dencelevelsfollowingBierens(1997). Thetestsareruninsequence,startingwiththenullhypothesis thatthenumberofcointegratingvectorsiszero,followedbyatestofthenullhypothesisthatthereisonecointegratingvector, andsoonuntilthenullcannotberejected. We (cid:2)nd thatr 0 (no cointegration) is decisively rejected, as is the hypothesis that D r 1(onecointegratingvector),butwecannotrejectthehypothesisthatr 2(two D D cointegratingvectors). Forcomparison, wealsoestimatetheparametricmaximumlikelihood(cid:21)-maxand trace tests of Johansen (1988) using the CATS package (Hansen and Juselius, 2006). The I.1/ maximum likelihood method estimates a (cid:2)nite-order VECM, as in (3.9), wherethecoef(cid:2)cientmatrices5;0 ;:::;0 are3 3. Ifthesystemiscointegrated 1 p 1 (cid:0) (cid:2) then the matrix 5 has reduced rankr < 3, and can be decomposed into 5 (cid:11)(cid:12)T. D The matrices (cid:11) and (cid:12) are full rank 3 by r matrices, and the columns of (cid:12) are the cointegrationvectors. Pantula (1989) and Johansen (1992) suggested a procedure to jointly identify the deterministiccomponentsandtherankof5.Theideaistotestthemodelssequentially, beginning with the most restrictive model considered. Each hypothesis can be tested usingeitherthetraceor(cid:21)-maxteststatistics. Weconductedthesetestsforasetoflag lengths p 4;5;:::;20. These tests uniformly (cid:2)nd that there are two cointegration D vectors and that the correct deterministic component is a constant that is restricted to the cointegration space. This speci(cid:2)cation is therefore extremely robust to the lag length and agrees with the rank determination of the non-parametric test. Table 3 on thenextpagereportsthesetestsforalaglengthof p 6.14 D 14WecomputedvariousinformationcriteriafortheVECM.TheSchwartzcriteriaindicatedalaglength of4andtheAkaikecriterionindicatedalengthof20.Weestimatedthemodeloverthisrangeoflags.The 18

Table3: ParametricCointegrationTests Hypothesis (cid:21) max 90%c.v. Trace 90%c.v. 95%c.v. (cid:0) H0:r 0 rest. const 103:48 14:09 158:63 31:88 34:78 D H0:r 0 const. 103:47 13:39 158:60 26:70 29:38 D H0:r 0 const.,trend 114:26 16:13 172:02 39:08 42:20 D H0:r 1 rest. const 50:59 10:29 55:15 17:79 19:99 D H0:r 1 const. 50:59 10:60 55:13 13:31 15:34 D H0:r 1 const.,trend 51:41 12:39 57:77 22:95 25:47 D H0:r 2 rest. const 4:55 7:50 4:55 7:50 9:13 D The results from the nonparametric and parametric estimators are very similar. Thenon-parametricestimateofthecointegrationvectorsis(cid:12) [(cid:12) (cid:12) ], NP 1;NP 2;NP D where(cid:12) .1; 1:075;0/T and(cid:12) .0;1; 0:863/T. Theparametricesti- 1;NP 2;NP D (cid:0) D (cid:0) mateofthecointegrationvectorsis(cid:12) [(cid:12) (cid:12) ],where(cid:12) .1; 1:031; 0/T P 1P 2P 1P D D (cid:0) and (cid:12) .0; 1; 0:913/T.15 The parametric estimate is statistically equivalent to 2P D (cid:0) the nonparametric estimate. For both estimators, the (cid:2)rst basis vector (cid:12) re(cid:3)ects the 1 near stationarity of the spread between the logarithms of the Commercial Paper and TreasuryBillrates. Similarly,thesecondbasisvector(cid:12) re(cid:3)ectsthenearstationarity 2 of the spread between the Treasury Bill rate and Federal Funds rates.16 The nonparametric estimates of the two cointegration relations are shown in Figure 2 on the followingpage. Thedifferencesbetweenthenonparametricandparametricestimates, alsoincludedinthe(cid:2)gure,areanorderofmagnitudesmaller. Thisconsistencyofthenonparametricandparametriccontrastswiththeresultsof CoakleyandFuertes(2001)andCalzaandSousa(2006)wheretheparametricandnonparametricresultsdiffer. Inthesepapers,theauthorsargueforacceptingthenonparametricresultsbecauseBierensestimatorisvalidforabroaderrangeofdatagenerating processes. In particular, Coakley and Fuertes (2001) argue that the maximum likelihoodestimatesaredistortedduetononlinearmeanreversioninexchangerateswhich wouldimplynonlinearcointegration. Theconsistencybetweenournonparametricand parametric estimates reveals no evidence of nonlinear cointegration between interest rates. Bierens (1997) argued that hypothesis tests in the parametric model have higher resultswerenotgreatlyaffectedbythechoiceoflaglengthwithinthisrange.Themodelwithp 6isfairly D parsimoniousandpassedtestsforabsenceof(cid:2)rstandfourthorderauto-correlation. 15Thebasisforthecointegrationspacehasbeentransformedintoabasiswithonezeroineachvectorand theestimatedrestrictedconstantissubtractedfromthecointegrationrelations.Thistransformationdoesnot changeanyresults. 16Chi-squaredtestsintheVECM.6/acceptthehypothesisthat(cid:12)1 .1; 1;0/T butrejectthehypothesis that(cid:12)2 .0;1; 1/T.Thevaluesoftheteststatisticsare1:31and10 D :87res (cid:0) pectively.Thesetestsare(cid:31)2.1/. D (cid:0) 19

Figure 2: Nonparametric cointegrations and the difference from the parametric estimates CP-TBill Cointegration 0.20 0.15 0.10 0.05 -0.00 -0.05 -0.10 -0.15 -0.20 71 73 75 77 79 81 83 85 87 89 91 93 95 97 TBill-Fed Funds Cointegration 0.20 0.15 0.10 0.05 -0.00 -0.05 -0.10 -0.15 -0.20 71 73 75 77 79 81 83 85 87 89 91 93 95 97 Deviations of Parametric from Nonparametric Estimate (Different Scale) 0.020 CP_TB_DEV 0.015 TB_FF_DEV 0.010 0.005 0.000 -0.005 -0.010 -0.015 -0.020 71 73 75 77 79 81 83 85 87 89 91 93 95 97 powerthancomparabletestsinthenon-parametricmodel.Thisargumentdoesnotnecessarilyholdbecausetheargumentandthehypothesistestsarepredicatedonlinearity. Despite the parametric estimator’s consistency with the nonparametric estimates, the parametricestimatorislikelymisspeci(cid:2)edsincetheindividualinterestratesarenonlinear. Since the nonparametric and parametric cointegrations are indistinguishable, we cansafelysidesteptheissueofmisspeci(cid:2)cationbyfocusingsolelyonthenonparametricresults. 4.2.1 Robustness As already discussed, our results are robust to the type of estimator and lag length. Beforemovingtothesecondstageofourapproachandtestingfornonlinearity,wealso tested the results for robustness to the Federal Reserve’s choice of policy instrument, 20

Table4: NonparametricCointegrationTestsforSub-Samples Hypothesis TestStat: TestStat: CriticalRegion M Conclusion 1974 1979 1984 1996 (cid:0) (cid:0) H0:r 0 0:00000 0:00000 20%.0;:006/ 3 Reject D H1:r 1 10%.0;:017/ 4 Reject D 5%.0;:008/ 4 Reject H0:r 1 0:00523 0:00008 20%.0;:077/ 3 Reject D H1:r 2 10%.0;:034/ 3 Reject D 5%.0;:017/ 3 Reject H0:r 2 1:33438 2:33982 20%.0;:341/ 3 Accept D H1:r 3 10%.0;:187/ 3 Accept D 5%.0;:111/ 3 Accept Misthesmoothingparameterforthenonparametricestimator by examining the integration and cointegration properties of the data over two subperiods: Sept. 13,1974throughSept. 19,1979andMarch1,1984throughDec. 31, 1996. The results of the non-parametric cointegration tests for the two sub-samples are reportedinTable4. Theresultsshowthattherankidenti(cid:2)cationsareconsistentwith thosefromthefullsample. Theparametricestimatorsalsoidenti(cid:2)edtwocointegrating vectorsforeachsub-period. Further,theestimatedcointegrationvectorsareconsistent withtheestimatedvectorsfromthefullsample;wecannotrejectthejointhypothesis, H : (cid:12) .1; 1:031;0/T and (cid:12) .0;1; 0:913/T, for either sub-sample. 0 1P 2P D (cid:0) D (cid:0) Thesetestsare(cid:31)2.1/. Forthe1974-1979sub-sample,theteststatisticis3:26(p-value of:2),andforthe1984-1996sub-sample,theteststatisticis:38(p-valueof:83). 4.3 TestsforNonlinearityoftheCointegrationRelations Thestationarycomponentsofthesystemconsistofthetwocointegrationrelationsand the(cid:2)rstdifferenceofthecommonstochastictrend. Wetesttheestimatedcointegration relationsfornonlinearserialdependenceusingthebispectrumtests. Thecointegration vectors, (cid:12) and (cid:12) are basis vectors for the cointegration space, so that any linear 1 2 combination of (cid:12) and (cid:12) are also stationary. Thus, evidence of nonlinearity in one 1 2 of the cointegration relations is actually evidence that the stationary components of the system are nonlinear. Prior to testing for nonlinearity, each of the cointegration relationsispre-whitenedbyan AR.6/(cid:2)ltertoeliminatebiasinthespectralestimation priorandtodecreasethelikelihoodoffalselyrejectingthenulloflinearity. Forrobustness,wetesttheserelationsforstationarityusingthefrequencydomain 21

Table5: Stationarity,GaussianityandTimeReversabilityTests.1971 1997/ (cid:0) Cointegration HW Gauss1 Gauss2 Nonparametric (cid:12)T S 25:2276.0:8620/ 638:6265.0:0000/ 707:1962.0:0000/ 1;NP t (cid:12)T S 34:2187.0:4572/ 387:4622.0:0000/ 465:5005.0:0000/ 2;NP t HWteststatisticis(cid:31).34/under H0:stationarity Gauss1teststatisticis(cid:31).34/under H0:Re.B.f;g// 0 .f;g/ D D 8 2 Gauss2teststatisticis(cid:31).34/under H0:Im.B.f;g// 0 .f;g/ D D 8 2 test derived by Hinich and Wild (2001). The Hinich and Wild (HW) test checks for residual non-stationarity due to the existence of a waveform with random phase and amplitude. Thistesthasaverydifferentalternativehypothesisthanthecointegration test,andshoulddetectnonstationarityatseasonalfrequencies.Thetestis(cid:31)2.34/under thenullofstationarity. TheHW-stationaritytests,reportedinTable5,con(cid:2)rmthatthe cointegrationrelationsarestationary. If the time series are Gaussian, then the real and imaginary components of the bispectrum are zero. The test statistics for these two hypotheses, called Gauss1 and Gauss2respectively,arealsoreportedinTable5. Ifeithertherealorimaginarycomponentsofthebispectrumarenon-zerothenGaussianityisrejected. Iftheimaginary component is non-zero then the sequence is not time-reversible. The results indicate thatthestationarycomponentsofthesystemarehighlynon-Gaussianandarenottimereversible. Rejecting Gaussianity is necessary but not suf(cid:2)cient to reject linearity. Table 6 gives the results of Hinich’s test for nonlinearity over the full sample. The Z test (cid:26) statisticsareindependentandnormallydistributedunderthenulloflinearity, andwe treatthesetestsastwo-tailed,asAshleyetal.(1986)foundthatone-tailedtestsmayfail to detect certain types of nonlinearity The tests are computed for the non-parametric estimatesofthecointegrations(cid:12)T S and(cid:12)T S : 1;NP t 2;NP t Table6: LinearityTests.1971 1997/ (cid:0) Cointegration Z Z Z Z Z Z :1 :2 :4 :6 :8 :9 Nonparametric (cid:12)T S -2.81 -4.09 -5.39 -2.65 2.78 2.19 1;NP t (cid:12)T S -2.26 -2.66 -0.93 -0.38 1.29 1.94 2;NP t Z is N.0;1/under H0: B.f;g/isconstant .f;g/ D: (cid:26) 8 2 Linearityisrejectedif Z exceedsthecriticalvalue. (cid:26) c.v. 1:65.90%/1:96.95%/2:58.99%/ (cid:12) (cid:12) (cid:12) (cid:12) 22

Figure 1 Skewness Function Figure3:TSBk-eCwPn Ceosisntfeugnrcattiioonn woifthC SPR- TDBynilalmciocisntegration The strongest evidence of nonlinearity is found in the (cid:2)rst cointegration relation. Linearityisrejectedfor(cid:12)T S by Z , Z , Z , Z ,and Z usingthe99%critical 1;NP t :1 :2 :4 :6 :8 valuesandby Z usingthe95%criticalvalues. Flatnessoftheskewnessfunctionis :9 anecessaryconditionforlinearity. Figure3showsthattheskewnessfunctionforthe (cid:2)rst cointegration is clearly far from (cid:3)at, which is what should be expected from the statisticaltests. Evidence of nonlinearity is also found in the second cointegration, although this evidence is somewhat weaker. Linearity is rejected for (cid:12)T S by Z and by Z 2;NP t :1 :9 using the 95% critical values, and by Z using the 99% critical values. Figure 4 on :2 thefollowingpageshowstheskewnessfunctionforthesecondcointegration. Again, theskewnessfunctionisnot(cid:3)at,butitis(cid:3)atterthantheskewnessfunctionofthe(cid:2)rst cointegration, re(cid:3)ecting weaker evidence of nonlinearity in the second cointegration. However,nonlinearityineithercointegrationimpliesthatthecointegratedsystemexhibitsnonlineardynamics. 4.3.1 Robustness Structural shifts over the long period being analyzed could be mistaken for nonlinear dynamics. As previously discussed, to address this issue we delete the period 1979 1983 and consider the two sub-samples. Table 7 on the next page presents (cid:0) 23

Figure 3 Skewness Function Figure4: SkewTnBe-FsFs fCuoninctteiogrnatfioorn TwBithil Sl-RF eDdynFaumnidcsscointegration thetestfornonlinearityoverthesetwosub-samples. Similartotheunivariateresults, linearitycannotberejectedforthe(cid:2)rstsubsamplebutcanforthesecond.Aspreviously mentioned, Hinich’s test has relatively low power for the (cid:2)rst sub-sample. Rejecting linearity for the second sub-sample implies that the shift in policy regime does not causethenonlinearityperse. Theresultscouldalternativelybeinterpretedastheresultofinterestratederegulationratherthanlowpower. Post-deregulation,theinterest rate dynamics seem to become more complex, even though the long-run equilibrium Table7: LinearityTestsforSub-Samples Cointegration Z Z Z Z Z Z :1 :2 :4 :6 :8 :9 Sub-Sample#1: 1974 1979 (cid:0) (cid:12)T S -1.07 -1.39 -1.99 -0.36 0.98 1.12 1;NP t (cid:12)T S -1.15 -1.72 0.50 0.79 0.15 0.67 2;NP t Sub-Sample#2: 1984 1996 (cid:0) (cid:12)T S -2.00 -2.95 -4.43 -3.88 1.95 1.96 1;NP t (cid:12)T S -1.99 -2.82 -2.14 0.05 2.40 1.91 2;NP t Z is N.0;1/under H0: B.f;g/isconstant .f;g/ D: (cid:26) 8 2 Linearityisrejectedif Z exceedsthecriticalvalue. (cid:26) c.v. 1:65.90%/1:96.95%/2:58.99%/ (cid:12) (cid:12) (cid:12) (cid:12) 24

relationswereunchanged. There is strong evidence of nonlinearity in the stationary components of the system. Althoughtheevidenceisnotcompletelyrobusttothesampleofdatatested,the nonlinearitydoesnotstemsolelyfromastructuralbreakcausedbythechangeintargetingapproachintheearly1980sorthederegulationofinterestrates, asthesecond sub-sampleshowsstrongevidenceofnonlinearity. 5 Conclusion Wehaveshownthatcointegrationrelationsin I.1/systemsgenerallyproducenonlineardynamics. Ourapproachfollowsadvancementsinprobabilitytheorywheremany resultsthatrequiredindependence,andthereforeimpliedlinearity,havebeenextended usingmartingalestoallowformoregeneraldependenceornonlinearity. Becausethe cointegrationrelationsderivedfrom I.1/systemsarestationary,theycanbetestedfor nonlinear serial dependence using standard polyspectral techniques. A feature of our two-stagemethodisthatittestsasystemofeconomicvariables,oranequilibriumeconomic relation, for nonlinearity even though existing tests for nonlinearity, including thebispectrumtest,areunivariate. Tests for the existence of nonlinear dynamics require large sample sizes and may beadverselyaffectedbyaliasingandotherproblemsassociatedwithtimeaggregation. Interestratesaremeasuredwithhighfrequencyandaliasingcanbecontrolledbyadequateattention to(cid:2)lterdesign. Forthesereasons, theconditionsare morefavorable to testing interest rate data for nonlinear dynamics than for most other variables that are important to the business cycle, money demand, and the monetary transmission mechanism. We found that short-term US interest rates are cointegrated and that the stationary components of the system are nonlinear. The Hinich nonlinearity test is conservative,whichstrengthensour(cid:2)ndingofnonlinearinterestratedynamics. These results suggest that the untested assumption of linearity may be incorrect. Thefailureto(cid:2)ndrobustevidenceofnonlinearityinlowerfrequencymacroeconomic time series may be due to the small sample sizes that can be obtained for those time series, in addition to problems associated with sampling and time aggregation. Our particular example shows that the spreads between the Commercial Paper, Treasury Bill, and Federal Funds rates exhibit nonlinear dynamics. Our results are consistent with work that suggests there are asymmetric effects of monetary policy on interest rates, such as Choi (1999). Our results suggest that better forecasts of these spreads mightbeobtainedwithnonlinearmodels,suchasbilinearmodels. 25

Appendix: Aliasing and Constructing Anti-Aliasing Filters Let X be a stationary continuous-time series that is sampled at regular intervals of t time,0;(cid:14)T;2(cid:14)T;:::;.N 1/(cid:14)T. (cid:14)T iscalledthesamplinginterval,and1=(cid:14)T isthe (cid:0) samplingrate. Thesampledsequenceisdenoted X ,k 0;:::;N 1. k(cid:14)T D (cid:0) Thepowerspectrumofthecontinuous-timeseriesis g.f/ 1 c .(cid:28)/e i.2(cid:25)f/(cid:28): XX (cid:0) D Z(cid:0)1 The power spectrum of the discrete-time sampled sequence, g .f/, is given by the (cid:14)T following: 1 j g .f/ g.f / (5.1) (cid:14)T D C (cid:14)T j DX(cid:0)1 for f .1=2(cid:14)T/(seeKoopmans,1975,pp.66-73). Thefrequency f .1=2(cid:14)T/is N j j(cid:20) D calledtheNyquistfoldingfrequency.Ifg.f/ 0forall f f thenthepowerspec- N D (cid:21)j j trumofthecontinuous-timeseriesandthediscrete-timesampledsequenceareequal. If the continuous-time series does not have this property then the power spectrum at frequency, f, ofthesampledsequenceisequaltothesumofthevaluesofthepower spectrumofthecontinuous-timeseriesatallfrequenciesoftheform f .j=(cid:14)T/for C j 0; 1; 2;:::. Thus, the low frequency harmonics are made indistinguishable D (cid:6) (cid:6) from the combined power of higher frequency harmonics because of sampling. This phenomenoniscalledaliasing. ItisveryimportanttoeliminateanypowerinatimeseriesatfrequenciesthatexceedtheNyquistfoldingfrequencypriortosampling,becausefailuretodosowilllead to biased estimation due to aliasing. Aliasing has traditionally been described in the frequency domain, but Hinich and Rothman (1998) showed that aliasing corrupts the impulseresponsefunctionsinthetimedomainandthereforeleadstoseriousidenti(cid:2)cationproblems. The same problem results if a discrete-time sequence is sampled at a lower frequency, such as sampling a daily interest rate at weekly intervals. In this case, the samplingintervalis(cid:14)T 7andtheNyquistfoldingfrequencyis.1=2(cid:14)T/ .1=14/. D D Ifthedailyinterestrateshavepoweratfrequenciesexceeding.1=14/thenaliasingwill occur. The solution to this problem is to (cid:2)lter the daily interest rates in such a way thatthepowerspectrumofthe(cid:2)lteredrateswillbezeroatfrequenciesexceedingthe Nyquist. If g arethe(cid:2)lterweightsthenthepowerspectrumofthe(cid:2)lteredsequence j (cid:8) (cid:9) 26

equals the power spectrum of the underlying sequence multiplied by the gain of the (cid:2)lter G.f/2 whereG.f/ 1 g e i.2(cid:25)f/j. Thesolutiontothealiasingproblem j (cid:0) j j D j wouldbetodesigna(cid:2)lterwithDgP (cid:0)ai1n: 1 if f f G.f/2 j j(cid:20) N : (5.2) j j D8 0 if f > f < j j N Thisgainfunctioncorrespondstotheid:ealsymmetriclow-pass(cid:2)lterwithweights sin.2(cid:25) f =(cid:25)k k 1; 2;::: N g D(cid:6) (cid:6) (5.3) j D8 2f 1=(cid:14)T k 0 < N D D whichcannotberealizedwit:ha(cid:2)nitedatasample. Infact,therateofdecreaseofthe (cid:2)lterweightsistooslowtosimplytruncatethe(cid:2)lteratsome(cid:2)nitenumberofleadsand lags. The usual solution is to taper the weights of the ideal (cid:2)lter. We taper the ideal weightsusingaHanningcosinetaper. This(cid:2)lterisreferredtoasananti-aliasing(cid:2)lter inthetext. Applyingtheanti-aliasing(cid:2)lterproducesabusinessdayseriesthatshouldnotcontain power at frequencies higher than every two weeks. This series still suffers from problems created by missing values caused by holidays. We subsequently resample ourseriesoneveryWednesdaytoavoidthemissingvalueproblem. SincetheNyquist frequencyistheneverytwoweeks, theresultingweeklyseriesshouldavoidaliasing. Thecombinationofapplyingtheanti-aliasing(cid:2)lterandthendecimatingtotheweekly sampleproducesamulti-rate(cid:2)lter. The common approach in economics is to report unweighted weekly averages of dailyinterestrates.Weeklyaveragesarealsoeffectivelyproducedbyamulti-rate(cid:2)lter: combininga(cid:2)lterwithdecimation. The(cid:2)lterisanunweightedaveraging(cid:2)lterthathas awidermainlobandmuchlargersidelobesthantheanti-aliasing(cid:2)lterweuse.Weekly averagesthereforepotentiallyarestronglyaliased.Monthlyandquarterlyaveragesthat areoftenusedinstudiesoftherealinterestratearesimilarlyaliased. 27

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