Coinegraion in Frequenc Domain* Daniel Lev Deparmen o Economics Bar-Ilan Universi Rama-Gan 59 ISRAEL Tel: 97-3-53-833 Fax: 97-3-535-38 LEVDA@MAIL.BIU.AC.IL and Deparmen o Economics Emor Universi Alana, GA 33 USA Tel: 44 77-94 Fax: 44 77-4639 ECONDL@EMOR.EDU Augus 8, Revised: April, Kewords: common sochasic rend; coinegraion: requenc domain anlsis; cross-specrum; ero-requenc. * The manuscrip was compleed when I was a Visiing Associae Proessor a Bar-Ilan Universi. I am graeul o he anonmous reeree or imporan suggesions, which led o he generaliaion o he resuls repored in his paper, and o And oung or commens. All errors are mine.
Coinegraion in Frequenc Domain* Asrac Exisence o a coinegraion relaionship eween wo ime series in he ime domain imposes resricions on he series ero-requenc ehaviour in erms o heir squared coherence, phase, and gain, in he requenc domain. I derive hese resricions suding cross-specral properies o a coinegraed ivariae ssem. Speciicall, I demonsrae ha i wo dierence saionar series, and, are coinegraed wih a coinegraing vecor [ ] and hus share a common sochasic rend, hen a he ero requenc, he squared coherence o L and L will equal one, heir phase will equal ero, and heir gain will equal.
. Inroducion Since he inroducion o coinegraion and common rend analsis in economerics and saisics Engle and Granger 987 and Sock and Wason 988, inegraion and coinegraion ess have now ecome an essenial par o he applied economericians and macroeconomiss sandard ool ki. These ess are rouinel applied o economic ime series ecause he noion o coinegraion has a naural economic inerpreaion: exisence o a coinegraion relaionship eween wo variales indicaes ha he series move ogeher in he long run, and so he share a common sochasic rend, alhough in he shor run he series ma diverge rom each oher. Since man economic heories make hese kinds o long-run and shor-run dierenial predicions aou economic ime series co-movemens, man economic models and paricularl macroeconomic models lend hemselves naurall o coinegraion esing Engle and Grange, 987. The coinegaion proper is a long-run proper, and hereore in requenc domain i reers o he ero-requenc relaionship o he ime series. Thereore, here is a requenc-domain equivalen o he ime domain coinegraion proper. Speciicall, exisence o a coinegraion relaionship eween wo ime series in he ime domain imposes resricions on he series ero-requenc ehavior in erms o heir cross specral measures in he requenc domain. The purpose o his paper is o use a ivariae seing o derive hese requenc-domain resricions in erms o he ime series squared coherence, phase and gain, which are he measures praciioners picall consider when suding cross specral properies o ime series. Squared coherence is analogous o he square o he correlaion coeicien and measures he degree o which one series can e represened as a linear uncion o he oher. Phase measures he phase dierence or he iming i.e., lead or lag eween he
3 requenc componens o he wo series. Gain indicaes how much he specrum o one series has een ampliied o approximae he corresponding requenc componen o he oher. I is esseniall he regression coeicien o one series on anoher a requenc. Thus, he squared coherence, phase and gain are requenc-domain equivalens o he correlaion coeicien, ime-dela lag, and regression coeicien, respecivel, and, hereore, he have a naural inerpreaion in erms o he sandard ime domain regression analsis. The paper proceeds as ollows: I derive cross specral properies o a coinegraed ivariae ssem eginning wih wo non-saionar ime series ha are coinegraed wih a coinegraion vecor [ ], and using sandard Fourier Transorm mehods and marix algera, I derive requenc domain properies o he series comovemen in erms o heir squared coherence, phase and gain. Speciicall, I show ha he squared coherence eween such series, aer dierencing, will equal one, heir phase will equal ero, while heir gain will equal. The paper ends wih a rie conclusion in Secion 3.. Cross-Specral Properies o a Coinegraed Bivariae Ssem Le he ime series o and e dierence saionar. Thus, le ~ I and ~ I, so ha he can e wrien as u and v,
4 respecivel, where u ~ I, and v ~ I. Moreover, le us assume ha and are coinegraed wih he coinegraion vecor [ ], so ha he sais µ, where µ ~I. Then, and processes share a common sochasic rend and, hereore, can e wrien in a marix noaion T x T where T is he common sochasic rend wih he proper L T, ~ iid, σ is a whie noise process, x ~ I, and ~ I. process Appling dierence operaor L o ields a ivariae saionar L L x L L wih he special marix 3 The diagonal elemens o he marix are he specral densi uncions o L and L, deined : π i e d 4a
5 π i e d 4 where and deined are he auocovariance uncions o L and L, E[ µ µ ] 5a E[ µ µ ], 5 respecivel, where µ and µ denoe he means o L and L, respecivel. The o-diagonal elemens o he marix are he cross-specral densi uncions o L and L, deined π i e d 6a π i e d 6 where and L, and L and L L and are he crosscovariance uncions o, deined E[ µ µ ] 7a E[ µ µ ], 7 respecivel.
6 To compue he elemens o he marix, irs compue he auocovariancecrosscovariance marix o, which is given : [ ] 8, x x x x E x x E where suscrips and denoe x and, respecivel, or noaional simplici, he diagonal elemens o he las marix in 8 are he auocovariance uncions, and, and he o-diagonal elemens are he crosscovariance uncions and, respecivel, as deined in 5a-5 and 7a-7. Appling he Fourier Transorm o oh sides o equaion 8, mulipling hrough π, and using he specrum and cross-specrum deiniions provided 4a-4 and 6a-6, we ge he special marix 9 which can e rewrien as
7 The cross specrum in can e wrien in Caresian orm ecause he specral marix is in general a complex valued uncion. Thus, or example, we can wrie, mn mn mn q i c where mn c denoes he cospecral densi uncion o m and n, and mn q denoes he quadraure specral densi uncion o m and n. Thereore, using Priesle s 98, p. 668, Equaion 9..53 resul ha nm mn, can e rewrien as where ar denoes a complex conjugae. Comining wih Caresian represenaion o and, 3a q i c and 3 q i c ields 4 c c Now, consider he value o he specral marix a requenc, which using 3 and 4 can e wrien as,
8 5. c c Recall ha is a whie noise process, and hereore, is heoreical specrum is la and equals π σ / or all requencies π π. In addiion, x and are I, and hereore heir ero-requenc specral densi, cross specral densi and cospecral densi uncions equal ero. Thus, ever elemen o he second marix o he righ hand side o 5 vanishes, and hereore he specral marix, evaluaed a requenc, ecomes 6. π σ π σ π σ π σ To see he implicaions o his resul or he ehavior o he heoreical squared coherence, phase and gain, recall rom polar represenaion o ha [ ] 7 K [ ] 8, ] Re[ Im arcan φ and
9 Γ [ ], 9 where Im [ ] and Re [ ] [ ] Α, and K, denoe he imaginar and real pars o φ, and Γ denoe he squared coherence, phase, and he gain o L and L, respecivel Jenkins and Was, 968. Then, using he marix 6 along wih he deiniions o squared coherence, phase, and gain provided in 7, 8 and 9, we ge ha a he ero requenc he ollowing equaliies hold. For he squared coherence o L and L, 6 and 7 impl ha a requenc, K σ σ σ π π π To deermine he phase o L and L, noe ha rom he Caresian represenaion o, we can wrie c i q. However, rom 6 we know ha a ero requenc σ π. Rewrie or, c i q 3
and compare he resuling equaion 3 o equaion. The equali o he wo equaions requires ha heir righ hand sides e equal. However, we know ha or a complex numer o equal a real numer, i is necessar ha he imaginar par o he complex numer e ero. In oher words, i is necessar ha he imaginar par o he complex numer e ero. In oher words, or equali o and 3, i is necessar ha a he requenc, he cospecrum o L and L sais c Re[ ] σ π 4 and he quadraure specrum o L and L sais q Im[ ]. 5 Susiuing 4 and 5 ino he deiniions o phase 8 or he requenc, we ind ha φ Im arcan Re arcan, [ ] [ ] 6 Finall, o deermine he gain o L and L, we need o comine 6 and 9 and evaluae he resul or he requenc. This ields
Γ σ π, σ π 7 where is he coeicien ha measures he exen o he long run relaionship eween and. Tha is, is he coeicien in he coinegraion relaionship, µ, where µ ~ I. Equaions, 6 and 7 esalish he main resuls o his paper: i wo dierence saionar series, and, are coinegraed wih he coinegraing vecor [ ], hen he ero requenc squared coherence, phase, and gain o L and L will equal one, ero, and, respecivel. This is a generaliaion o Lev, which onl ocuses on he ehaviour o squared coherence and gain, and onl or he case. 3. Conclusion The conigraion proper is a long-run proper, and hereore in he requenc domain, i reers o he ero-requenc relaionship o he ime series. Thereore here is a requenc-domain equivalen o he ime-domain coinegraion proper: exisence o a coinegraion relaionship eween wo ime series in he ime domain, imposes resricions on he series ero-requenc ehavior in erms o heir squared coherence, phase, and gain in he requenc domain. In his paper, I derive hese requencdomain resricions in a ivariae seing. Speciicall, I demonsrae ha i wo dierence saionar series, and, are coinegraed wih he coinegraing vecor [ ], hen he ero requenc squared coherence, phase, and gain o L L will equal one, ero, and, respecivel. and
I is well known ha he sandard ime series coinegraion ess have a low power. The resuls derived in his paper sugges ha i ma e useul o es or coinegraion in he requenc domain. Fuure work should examine limiing null disriuions and inie sample properies o such ess, in order o assess heir pracical useulness.
3 Reerences Engle, R.E. and Granger, C.J.W. 987 Coinegraion and error correcion: represenaion, esimaion, and esing, Economerica 55, 5-76. Jenkins, G.M. and Was, D.G. 968 Specral Analsis and is Applicaions. San Francisco: Holden Da. Lev, D. Invesmen-saving comovemen and capial moili: evidence rom cenur-long US ime series, Review o Economic Dnamics 3, -36. Priesle, M.B. 98 Specral Analsis and Time Series. New ork: Academic Press. Sock, J.M. and Wason, M.W. 988 Tesing or common rends, Journal o American Saisical Associaion 83, 97-7.