Testing Linearity in Cointegrating Relations with an Application to Purchasing Power Parity

Transcription

1 Tesig Lieariy i Coiegraig Relaios wih a Applicaio o Purchasig Power Pariy Seug Hyu Hog Deparme of Ecoomics Cocordia Uiversiy ad Peer C. B. Phillips Cowles Foudaio, Yale Uiversiy Uiversiy of Aucklad ad Uiversiy of York Jauary 3, 2006 Absrac This paper develops a lieariy es ha ca be applied o coiegraig relaios. We cosider he widely used RESET specificaio es ad show ha whe his es is applied o osaioary ime series is asympoic disribuio ivolves a mixure of oceral χ 2 disribuios, which leads o severe size disorios i coveioal esig based o he ceral χ 2. Nosaioariy is show o iroduce wo bias erms i he limi disribuio, which are he source of he size disorio i esig. Appropriae correcios for his asympoic bias leads o a modified versio of he RESET es which has a ceral χ 2 limi disribuio uder lieariy. The modified es has power o oly agais oliear coiegraio bu also agais he absece of coiegraio. Simulaio resuls reveal ha he modified es has good size i fiie samples ad reasoable power agais may oliear models as well as models wih o coiegraio, cofirmig he aalyic resuls. I a empirical illusraio, he liear purchasig power pariy PPP) specificaio is esed usig US, Japa, ad Caada mohly daa afer Breo Woods. While commoly used ADF ad PP coiegraio ess give mixed resuls o he presece of liear coiegraio i he series, he modified es rejecs he ull of liear PPP coiegraio. JEL Classificaio: C2, C22 Key words ad phrases: oliear coiegraio, specificaio es, RESET es, oceral χ 2 disribuio Deparme of Ecoomics, Cocordia Uiversiy, 455 de Maisoeuve Blvd. Wes, Moreal, Quebec H3G M8, Caada; shhog@alcor.cocordia.ca Cowles Foudaio for Research i Ecoomics, Yale Uiversiy, Box 20828, New Have, CT , USA; peer.phillips@yale.edu. Parial suppor from NSF gra is ackowledged.

2 Iroducio Ecoomic ime series are ofe believed o exhibi oliear behavior ad ecoomiss usually formulae his olieariy i oe of he followig wo ways: i) by buildig oliear dyamics io he model for a idividual ime series; or ii) by allowig for olieariy i he relaioship bewee ime series. This paper ivesigaes issues associaed wih he secod approach ad does so from a model specificaio perspecive. Sice he iroducio of he coiegraio cocep, liear models have domiaed pracical work i coiegraio aalysis. This emphasis has arise, o so much because he uderlyig ecoomic heory suggess lieariy, bu raher because he coiegraio cocep ad associaed ecoomeric mehodology has bee developed very largely for liear models of iegraed processes. Correspodigly, he ools of ecoomeric aalysis are available i his case ad here is grea coveiece i compuaio for applied work. I coras, uil recely, here has bee a lack of appropriae aalyic ools for cosiderig oliearly rasformed iegraed ime series ad a absece of a asympoic heory of iferece. Empirical applicaios ofe simulae a ieres i oliear specificaios ad, i cosequece, may oliear models ad almos as may specificaio ess) have bee developed for saioary ime series modelig. May rece oliear model applicaios of osaioary ime series have focused o oliear shor-ru dyamics aroud liear log-ru equilibria i error correcio models ECM), as i Berbe & Dijk 999), Lo & Zivo 200), ad Teräsvira & Eliasso 200) amog ohers. Few aemps have bee made o sudy oliear coiegraig relaios direcly ad he mehods ha have bee ried i pracical work ofe require resricive codiios o he DGP e.g. Basher & Haug, 2003). Such exesios also awai a correspodig developme i ess of specificaio. Neglecig he possibiliy of olieariy i a log-ru relaioship ca be paricularly derimeal i osaioary cases. For saioary ime series, liear models ca ofe provide workable approximaios a leas locally o oliear models. Ulike mea-reverig saioary processes, osaioary ime series have a edecy o wader wih o fixed mea or localiy i he sample space ad, like radom walks, revisi pois disa from he origi a ifiie umber of imes. I such cases, local liear approximaios ca oly poorly represe he global characerisics of he process, producig a high risk of fauly iferece abou a misspecified log-ru equilibrium. Cosideraios of he possibiliies sugges hree cases liear coiegraio, some form of oliear coiegraio, or complee absece of coiegraio. Exisig coiegraio ess esseially presume a form of liear coiegraio ad do o effecively disiguish bewee liear ad oliear coiegraio Grager & Hallma, 989). So, liear coiegraio aalysis requires a addiioal es of specificaio o address his paricular issue. Exisig lieariy ess also fail o provide ay guidace cocerig he ype of relaioship ha may be prese bewee osaioary ime series if i were oliear Grager, 995). Accordigly, i is o surprisig o fid ha lieariy ess which have bee developed for saioary processes work poorly wih osaioary ime series. This was well recogized earlier i he case of he RESET es. The RESET es Ramsey, 969) is a coveie device for esig geeral misspecificaio e.g. Vialiao, 987; Baghesai, 99; Peers, 2000, amog ohers), bu is kow o o be robus o auocorrelaed disurbaces, especially whe he regressor is iself highly auocorrelaed Porer & Kashyap, 984) or coais a deermiisic ime red Leug & Yu, 200). Usig simulaio experimes, Porer & Kashyap 984) show ha he presece of serially correlaed 2

3 disurbaces combied wih a AR) regressor leads o size disorios, ad he more auocorrelaed is he regressor he less robus he RESET es is o error auocorrelaio. Naurally, we migh expec his size disorio problem o become worse i he coiegraig case where he regressor has a auoregressive ui roo ad he errors are ypically serially depede. I he absece of more appropriae specificaio ess, applied ecoomiss have reaed exisig coiegraio ess as ess for liear coiegraio. I oher words, if evidece for coiegraio is foud, i is coveioally assumed wihou furher esig ha such coiegraio is liear i aure ad all subseque aalysis ress o his assumpio. The prese paper develops a direc esig mehod ha ca aswer he simple bu impora quesio: do he daa suppor a liear coiegraig specificaio? To do so, we modify he widely used regressio error specificaio es RESET), which is a lieariy es based o geeral approximaio heory. The RESET es implicily uses a Taylor series approximaio o capure uspecified oliear forms by seekig o deec olieariies ha remai i he liear regressio residuals usig a liear combiaio of polyomial fucios. Our approach here is o use recely developed asympoic ools for oliearly rasformed iegraed ime series from Park & Phillips 999, 200) o modify he RESET es i a appropriae maer so ha i ca be applied o osaioary ime series o es direcly for lieariy i coiegraig relaios. The res of his paper is orgaized as follows. The ex secio iroduces he model ad he maiaied assumpios ad shows how he esig mehod is relaed o a uderlyig heory of oliear approximaio. I addiio, we show how he osaioariy of he daa chages he limiig heory of he exisig es usig sample covariace asympoics of oliearly rasformed iegraed processes. Secio 3 discusses he modificaios ha are eeded whe he RESET es is applied o coiegraig relaios. The asympoic disribuio of he modified es saisic is discussed boh for he ull ad various aleraive hypoheses. Simulaio resuls o he fiie sample size ad he power of he modified es are repored i Secio 4. Secio 5 preses a empirical applicaio of he modified es o purchasig power pariy PPP). Secio 6 cocludes ad addiioal assumpios, lemmas ad proofs are colleced ogeher i he Appedix. 2 Model ad Backgroud Ideas The RESET es uilizes a approximae represeaio based o a power series expasio o deermie wheher he liear specificaio leaves ayhig uexplaied i he regressio residuals ha ca be deeced by a liear combiaio of polyomial basis fucios. The idea ca be aurally exeded o oher families of basis fucios ad, as we will show, uilized i he coex of osaioary daa applicaios. For a arbirary fucio fx) lyig i a suiably defied L 2 fucio space over a cerai domai, i is possible o cosruc a orhogoal series represeaio of he followig form fx) β j F j x) or i wo pars fx) j= k β j F j x) + j= = ˆf k x) + error j=k+ β j F j x) i erms of some basis fucios {F j x)} ha form a complee se ad where sigifies L 2 covergece. I he special case of a coverge power series Taylor) represeaio, we may 3

4 use simple polyomials as a basis. Give fx), he accuracy of he fiie sum approximaio ˆf k x) = k j= β jf j x) or he size of he error erm depeds o he umber of erms k) icluded i he sum ad he properies of he fucio f, o which here is a huge lieraure i Fourier series aalysis e.g. Tolsov, 976). Suppose ha we wa o es he liear codiioal mea specificaio H 0 : P[EY X ) = θ X ] =, usig a regressio specificaio Y = θ X + θ 2 fx ) + u. ) If oe has a specific oliear aleraive model i mid, such as ) for some give f X ), ad was o es ha specific model agais he liear model, he oe ca use ess such as a Wald or LM es of H 0 : θ 2 = 0 o decide which model fis he daa beer. I may pracical cases, however, heory fails o provide a specific fucioal form, ad while i is possible o come up wih alerae oliear models, hese ofe seem raher arbirary. Also, if he focus of aeio is some coveie liear model such as ha implied by purchasig power pariy cosideraios) wih o specific oliear aleraive, he i is of paricular ieres o es wheher he liear specificaio is accepable. A lieariy es based o approximaio heory seems appropriae i hese siuaios. Replacig he uspecified oliear fucio fx ) wih a parial sum approximaio ˆf k X ), we may proceed o es he validiy of he liear specificaio by esig wheher a liear combiaio of a fiie umber of suiable basis fucios {F j )} k ca deec ay olieariy i he regressio residuals. This procedure ivolves esig H 0 : β j = 0, j i he regressio Y = θx + u ad û = k β j F j X ) + e, 2) where he û are he regressio residuals ad he basis fucios are chose o be F j X ) = X j+ for he RESET es. As is appare, usig his approach here ca be as may ess of lieariy as here are approximaio mehods 2. Here we will focus o he RESET es i view of is populariy i applied work. Noe ha esimaio of 2) ivolves workig wih he sample momes of oliearly rasformed iegraed ime series of he form F jx )û ad F jx )F i X ), whose asympoic behavior mus be characerized. Before examiig hese quaiies, we firs specify he daa geeraig processes ad some assumpios ha will faciliae he developme of a limi heory. Assumpio A: Le X = v ad u be geeral liear processes saisfyig he followig While he origial es by Ramsey 969) ad he similar ess by Keea 985) ad Tsay 986) use he fied value Ŷ, Thursby & Schmid 977) propose usig he polyomials of X isead for a higher power. 2 For example, DeBeedicis & Giles 998) use a Fourier series approximaio idea ha has beer global fiig capabiliy ha a Taylor series approximaio, while Keea 985), Tsay 986) ad Barahoa & Poo 996) use variaios of Volerra series expasios. Whie 989) preses a eural ework NN) es based o he cdf of he logisic disribuio ad Blake & Kapeaios 2000) develop aoher NN es usig radial basis fucios for arificial eural eworks. j= 4

5 codiios. u = c j ε j = CL)ε v = ζ = j=0 η+ d j η j = DL)η j=0 ) is a saioary ad ergodic marigale differece sequece ε ) ) wih aural filraio F = σ { ζ s } σ σ ad variace marix Σ = 2 σ 2 σ 22 ad where {c j, d j } saisfy he codiios D) 0, j d j <, ad j=0 j /2 c j < These assumpios o he iovaio processes are fairly sadard, alhough i some cases below he liear process X is assumed o be predeermied i he sese ha EF j X ) F ) = F j X ). Codiios similar o hese assumpios ad Assumpio A i he Appedix a addiioal echical mome codiio) are employed i derivig he resuls of Park & Phillips 999, 2000, 200) ad Chag, Park & Phillips 200). However, De Jog s 2002) more relaxed codiios are sufficie for he modificaio of he RESET es preseed i his paper. Uder Assumpios A ad A, he followig ivariace priciple holds [r] ζ d Wr) j=0 ) W r) BM Σ), W 2 r) ad usig he Beveridge-Nelso decomposiio Phillips & Solo 992)), we ca show ha a similar resul holds for he ime series ζ = [v, u ]. [r] ζ d Br) ) Bx r) BM Ω). B u r) Here he covariace marix Ω = h= Γ ζh), where Γ ζ h) = Eζ 0 ζ h ). I is coveie o pariio Ω coformably wih ζ as ) Ωvv Ω Ω = vu, 3) Ω uv ad o defie ad pariio he oe-sided log-ru covariace marix ) Λvv Λ Λ = Γ ζ h) = vu, 4) h= Ω uu Λ uv Λ uu vv = Γ ζ 0) + Λ = vu uv uu ). 5) Amog he wide variey of possible oliear fucios, Park & Phillips 999, 200) provide asympoic ools for cerai classes of fucios of iegraed processes) saisfyig some regulariy codiios. The simple basis fucios {X j } of a Taylor series expasio fall wihi he H-regular or Class H) class, which is defied as follows. 5

6 Defiiio A rasformaio F ) is said o be H-regular iff F λx) = κ λ)h x) + R x, λ) where H ) is locally iegrable, ad R, ) is such ha R x, λ) aλ) P x), where lim sup λ aλ)/κ λ) = 0 ad P ) is locally iegrable, or R x, λ) b λ)qλx), where lim sup λ b λ)/κλ) < ad Q ) is locally iegrable ad vaishes a ifiiy, i.e. Qx) 0 as x Fucios i his class have homogeous, asympoically domiaig compoes κλ)hx) ha are locally iegrable. Hx) is referred as he asympoic homogeous fucio of Fx) ad κλ) as he asympoic order of Fx). Park & Phillips 999) provide various examples ha belog o his class, such as fiie order polyomials, logarihmic fucios, ad disribuio-like fucios, icludig heir liear combiaios ad producs. The basis fucios {F m = X m+ } from a Taylor series expasio belog o his class wih Hx) = x m+ ad κλ) = λ m+. Aoher impora class of oliear rasformaio is he I-regular or Class I) rasformaio. Roughly speakig, fucios i his class are bouded, iegrable ad piecewise) smooh see Park & Phillips 999) for furher deails). All pdf-like fucios belog o his class. 2. A Lieariy Tes ad Sample Covariaces of Noliearly Trasformed X As a firs sep i he developme, cosider he simples case where X is sricly exogeous so ha Assumpio A holds wih Ev u s ) = 0 for all, s ad he log-ru covariaces Ω uv = Λ uv = 0. Boh OLS ad FM-OLS Phillips ad Hase, 990) esimaors of θ i 2) he yield cosise ad asympoically mixed ormally disribued ˆθ, ad he RESET es saisic for H 0 : β j = 0, j follows a limiig ceral χ 2 k) disribuio, as we ow show. Tha is wih R = F = F X û F ) ˆΩ uu.v F F ) û F ) ) [ X X X F for F = A χ 2 k), 6) ] X 2 X k+, 7) ad where ˆΩ uu.v is a cosise esimaor of Ω uu uder exogeeiy. The es saisic R is a quadraic form of sample covariaces bewee a oliearly rasformed iegraed process ad he fied residuals, viz., X )m û = X ) m u d B m x db u, 8) where he ilde over a variable implies ha i is he residual from a liear regressio of ha variable o X for a fiie sample, ad equivalely for he limi processes where we wrie he projecio residuals as B m x = B m x B x B 2 x) B m+ x, where deoes iegraio over [0, ] 6

7 wih respec o Lebesgue measure. The laer oaio is used hroughou he paper. The limi 8), codiioal o F x = σb x r), 0 r ), is a mea-zero Gaussia mixure B x m db u N 0, ) Ω uu B x Fx m 2, so ha, combied wih he limi of he sample variace X ) m ) X m B x m 2, ad a cosise esimaor for Ω uu, he RESET es saisic 6) has he followig limi R d ) ) B x m db u Ω uu B x m 2 B m x db u ) Fx χ 2 k) codiioal o F x. Sice he limi disribuio is idepede of F x, we deduce ha R χ 2 k) ucodiioally. Nex, we discard he srog exogeeiy codiio o X so ha Assumpio A holds wih Ω uv 0. Abadoig he srog exogeeiy codiio chages he previous resul i several ways. Firs, he leas squares esimaor of θ ow has wo secod order bias erms i he limi, viz., ) ˆθ θ = 2 d ) X 2 X u ) { B x db u.x + Ω uv Ω vv B 2 x B x db x + uv }, 9) where he Browia moio B u.x = B u Ω uv Ω vv B x is idepede of B x ad has variace Ω uu.v = Ω uu Ω uv Ω vv Ω vu. I 9) he erm B x db u.x is a mea zero Gaussia mixure, bu he oher wo erms i braces shif he mea of he limi disribuio away from zero. These erms correspod o he so-called edogeeiy bias ad serial correlaio bias of liear coiegraio heory Phillips & Hase, 990), ad sem from he osaioariy of X. Similar effecs also geerally arise wih he sample covariace of u wih oliear rasformaios of X. The followig lemma summarizes he effecs whe he sample covariaces ivolve polyomial fucios. Lemma 2 Uder Assumpios A ad A, he sample covariace bewee X m m+)/2 X m u d Bx m db u.x + Ω uv Ω vv B m x db x + m vu ad u saisfies B m x 0) wih vu = h=0 E v 0u h ). 7

8 Lemma 2 shows ha, as i he liear coiegraio case 9), sample covariaces of oliear fucios ad saioary processes have limis ha also ivolve wo bias erms ha produce ozero locaio effecs i he limi disribuio of {ˆβ j } i 2). We refer o hese effecs as bias erms because boh erms shif he locaio of he limi disribuio of ˆβ j away from he rue value β j. The hree compoes i he limi disribuio i 0) are a mea-zero Gaussia mixure ad wo bias erms semmig from edogeeiy ad serial correlaio effecs, he laer beig radom whe m >. De Jog 2002) examies his oliear sample covariace asympoics uder codiios ha are less sric o he iovaio processes, bu more resricive i erms of fucioal forms. He shows ha for a H-regular fucio F ) wih coiuously differeiable asympoic homogeeous fucio H ), he sample covariace of FX ) wih u saisfies /2 κ FX )u d HB x )db u + uv H B x ) ) where he asympoic order of H ) is κ = κ ). Wih Fz) = z m, we have Hz) = z m ad κ = m/2, so ha ) he reduces o 0) i Lemma 2. A rece ad much more geeral semimarigale approach o esablishig limi resuls such as ) is developed i Ibragimov ad Phillips 2004). The effecs from he bias erms i Lemma 2 ca be subsaial i he RESET es saisic 6). As is well kow e.g., Muirhead, 982, heorem.4.5), a ecessary ad sufficie codiio for he quadraic form x A x i he Gaussia radom vecor x N ξ, V ), where V is osigular, o follow a oceral χ 2 disribuio is ha AV be idempoe. I his eve x A x is oceral χ 2 k, ν), where k = rak AV ) is he degrees of freedom ad ν = ξ A ξ is he oceraliy parameer. Here, we ca replace x wih he limi of ûf afer a appropriae ormalizaio ad suiable codiioig, ad hereby show ha he es saisic R i 6) follows a mixure of oceral χ 2 disribuios, givig he followig asympoic resul for he es whe he exogeeiy codiio o X does o hold. Theorem 3 Uder Assumpios A ad A, he RESET es saisic R has asympoically a mixure oceral χ 2 disribuio wih k degree of freedom ad radom oceraliy parameer ν = ξ A ξ. Tha is, R = û F ) ˆΩ uu.v F F ) ) = û F ˆΩuu.v F F F û A χ 2 k, ν) û F ) for F defied i 7), F = [ F,, F ], F = [ F,, F ], ad where ˆΩuu.v is a cosise esimaor of Ω uu.v. The radom oceraliy vecor ξ is k wih m ) h eleme defied as ) ξm ) = Ω uv Ω vv Bx m db x + m vu Bx m Λ vu Bx m+, B 2 x 8

9 ad A is a k k covariace marix A = Ω uu.v B2 x 2. Bk+ x B x 2 B2 x Bk+ x Bk+ x wih B m x = B m x B x B 2 x) B m+ x. Remarks. I geeral, he mea ad variace of a quadraic form x A x wih a oceral χ 2 k, ν) are e.g., Johso & Koz, 978) Ex A x) = k + 2 ξ A ξ ad varx A x) = 2k + ξ A ξ). So, codiioal o F x = σb x r), 0 r ), he firs wo momes of R ca be wrie as k + 2 ν ad 2k + ν) respecively, ad hey are greaer ha he ceral χ2 k) couerpar. This implies a higher probabiliy of Type I errors, which explais he large size disorios observed i he simulaio work by Porer & Kashyap 984). We ca check his by approximaig he oceral χ 2 k, ν) disribuio by a muliple of a ceral χ 2 disribuio, aχ 2 b), where he wo cosas are give by see Johso & Koz, 978) a = + ν k + ν ad b = k + ν2 k + 2ν k. Therefore, codiioal o F x, he probabiliy of rejecig he lieariy ull hypohesis ca be wrie approximaely as [ P[R > χ 2 α] P + ν ) ] ) ] χ 2 b) > χ 2 α P [χ k 2 + ν2 > χ 2 α α, k + ν k + 2ν which is always a leas as grea as he omial size α. 2. Origially, Ramsey 969) iroduced he RESET es as a F-es, where he es saisic ca be wrie as F = û û ê ê)/k ê ê/ K k) = û I M F )û/k û M F û/ K k), 2) where û ad ê are he vecors of regressio residuals from 2), K is he colum umber of X, ad he k k projecio marix M F = I M X FF M X F) F M X wih M X = I XX X) X ad X = [X,, X ]. The umeraor i 2) is equal o he χ 2 versio of he RESET es R i Theorem 3 muliplied by ˆΩ uu.v /k. Tha is, kû I M F ) û = ˆΩ uu.v k û F ) ˆΩuu.v F M X F F û A Ω uu.v χ 2 k, ν). k 9

10 The deomiaor of 2) also ca be show o coverge o Ω uu.v / K k) imes a mixure of oceral χ 2 radom variable χ 2 K k, ν ) wih oceraliy parameer ν = Eû M F û X). Therefore, codiioal o F x, he raio of wo ormalized oceral χ 2 radom variables follows a doubly oceral F-disribuio, deoed as Fk, K k; ν, ν ), ad F Fk, K k; ν, ν ) wih ν = ξ A ξ ad ν = ξ I A)ξ, 3) for ξ ad A defied i Theorem 3. Johso & Koz 978) show ha his doubly oceral F-disribuio ca be approximaed by a ceral F-disribuio ) Fk, K k; ν, ν + ν/k ) + ν Fdf, df 2 ), / K k) wih wo degrees of freedom defied by df = k + ν)2 k + 2ν ad df 2 = K k + ν ) 2 K k + 2ν. Usig his approximaio, he probabiliy of rejecig he lieariy ull hypohesis ca be wrie as ) + ν/k P[F > F α ] P[Fdf, df 2 ) > F α /C] wih C = + ν, / K k) wih radom oceraliy parameers ν ad ν. Noe ha as log as ξ 0, C + ν/k as ad he oceraliy ca herefore produce a subsaial size disorio i he es. 3 Bias Correcio ad a Modified RESET Tes The previous secio shows ha osaioariy of X iroduces wo bias erms i he limi disribuio of he sample covariace bewee X m ad û, so ha he RESET saisic R is a limiig mixure of oceral χ 2 disribuios. These bias erms are he mai source of he large size disorios i he es ad we ow prese a mehod o remove hem, leadig o he modified RESET es whose limi disribuio is ceral χ 2. The correcio mehod is similar o ha used i FM regressio Phillips ad Hase, 990). Afer ideifyig sample quaiies ha coverge o he bias erms, oparameric correcios are implemeed i he es saisic o elimiae hem. For he firs sep, he followig lemma iroduces sample quaiies ha have he same limis as he bias erms. Lemma 4 Le Assumpios A ad A hold. For m, m ) m X ˆ vu d m vu ad X ) m v ˆ vv m X ) m d B m x, 4) B m x db x, 5) where ˆ vu ad ˆ vv are cosise esimaes of he log ru covariace quaiies vu ad vv. 0

11 Remarks. Lemma 4 provides sample quaiies ha coverge o he asympoic bias erms show i Lemma 2. I liear coiegraio, he correspodig erms for he wo biases i 9) are give by ˆΩ uv ˆΩ vv ) X v ˆ vv d Ω uv Ω vv B x db x ad ˆ uv p uv Deoig hese wo compoes as E ad S, respecively, he sample covariace ow becomes a mea zero Gaussia mixure i a liear case X u E S d B x db u.x, ad FM esimaio simply applies hese correcios, givig ) ˆθFM θ = 2 d ) { } X 2 X u E S ) B x db u.x N 0, Ω uu.v Fx B 2 x B 2 x ) ) 6) We will use he same idea here o correc he bias erms i he es saisic R. 2. De Jog 2002) also recogizes he presece of wo biases i oliear coiegraig regressios ad gives he same expressio as ours for he serial correlaio bias correcio i ), bu akes a differe approach o correc he edogeeiy bias. Noig ha FM regressio correcs he edogeeiy bias by replacig B x db u wih B x db u.x, De Jog suggess a direc correcio o he regressio errors by usig u ˆΩ uv ˆΩ vv v isead of u. The sample quaiies ad heir limis show i Lemma 4 are closely relaed o he oceraliy vecor ξ defied i Theorem 3. Sice he es saisic R is a quadraic form ivolvig sample covariaces of oliear fucios, ad he oceraliy parameer of is limi disribuio correspodigly ivolves a quadraic form of ξ, we may elimiae he oceraliy by subracig he sample quaiies ha coverge o ξ from he oliear sample covariaces. The followig heorem explais how o accomplish his modificaio of he RESET es ad remove he oceraliy. Theorem 5 Suppose Assumpios A ad A hold. If {X, Y } are liearly coiegraed, he followig modified RESET saisic has a limiig ceral χ 2 disribuio wih degrees of freedom k MR = { û F D E S } ˆΩuu.v D F F D ) {D F û E S } A χ 2 k),

12 where û is vecor of residuals from he liear coiegraio regressio 2), F is a k marix wih he m, ) eleme X m+, ad F is he regressio residual from regressig F o X as i Theorem 3. The k k ormalizaio marix D ad he m ) h elemes of he wo k correcio vecors E = [ E ),, E k)] ad S = [ S ),, S k)] are defied as D = diag 3/2, 4/2,, k+2)/2 ) [{ ) m E m ) = ˆΩ X ˆΩ v uv vv ˆ m vv ) X v ˆ vv 2 S m ) = Remarks ˆ vu m X ) m ˆΛ vu 2 X 2 X ) m } X 2 ) ) X ) m+. X ) m+ ),. Alhough he RESET es is usually hough of as a geeral lieariy es wihou specific aleraives, i also ca be ierpreed as a LM es, where he basis fucios are reaed as possible aleraive oliear specificaios. By cosrucio, he es has highes power agais such aleraives. Furhermore, if he es rejecs lieariy, he esimaed oliear coiegraio relaioship provides a aleraive oliear model, or more specifically a parial approximaio o a aleraive oliear model for he daa, a leas whe he relaioship is o spurious. 2. Sice his ype of es is based o a fiie approximaio mehod, power aurally depeds o he adequacy of he approximaio uder he aleraive. Noe ha he goodess of approximaio depeds o he give oliear fucioal form ha is approximaed, ad he wo compoes ha ca be corolled he ype ad umber of basis fucios icluded i he augmeed regressors. A good approximaio will help i deecig olieariy whe i is prese, bu eve poor approximaios ca be effecive. This is because he ull hypohesis requires ha all k coefficies be zero, β j = 0 for j =,, k, ad he es will rejec if a leas oe coefficie deviaes eough from zero, i.e. if oe basis fucio is able o cach some par of he olieariy. Therefore, RESET es resuls should be ierpreed coservaively: failure o rejec he lieariy hypohesis H 0 does o ecessarily cofirm a liear specificaio bu raher ha he relaioship does o coai ay olieariy ha ca be deeced hrough he basis fucios {F j : j =,...k}. The relaioship bewee he power of he es ad he choice of k is examied i he ex secio usig Moe Carlo simulaio. 3. The F-es versio of he RESET es asympoically follows a mixure of doubly oceral F-disribuios as i 3), where boh radom oceraliy parameers are quadraic forms i ξ. Sice our modified es saisic i Theorem 5 implies ha E + S ξ, he bias correcio mehod give above agai ca be used o cosruc a modified versio of he F-es ha has a limiig ceral F-disribuio i a similar way. 2

13 { I pracice, he regressor se X j+ } k j= may suffer from mulicollieariy, so ha i is a commo pracice o use heir pricipal compoes as he regressors. 3 I his case, he bias correcio erms eed o be adjused accordigly, ad he modified es saisic usig pricipal compoes ca be cosruced as i he followig corollary. Corollary 6 Suppose he codiios i Theorem 5 hold. Le G be he k k marix wih k eigevecors of F F i is colums, afer dividig by he correspodig eigevalues. The modified es saisic MR based o hese pricipal compoes follows a limiig ceral χ 2 k) disribuio as follows {û F E G S G } ˆΩuu.v F F ) {F û G E G } A S χ 2 k), where F = F D G is k ormalized marix wih he j h pricipal compoe i he j h colum. The k eigevecors are chose such ha he correspodig eigevalues are he k bigges oes. 3. The Modified RESET Tes uder Aleraives As discussed earlier, cosiderig olieariy ogeher wih osaioariy gives rise o hree possible scearios. Our modified es ess he ull hypohesis of liear coiegraio agais boh oliear coiegraio ad he absece of coiegraio, he laer icorporaig boh he coveioal spurious regressio case ad omied variable cases. As show above, he es has a limiig ceral χ 2 disribuio uder lieariy, ad his subsecio examies es power i hese aleraive scearios. 3.. The Case of No Coiegraio Kim, Lee, & Newbold 2003) show ha may exisig lieariy ess ed o fid spurious olieariy whe hey are applied o wo idepede I) processes. They examie six widely used lieariy ess Ramsey s 969) RESET es, Whie s 989) NN es, he Keea 985) es, he McLeod ad Li 983) es, he Whie 992) dyamic iformaio marix es, ad Hamilo s 2000) flexible oliear es ad fid ha evidece of spurious olieariy icreases wih he sample size. The followig Theorem shows ha our modified es saisic also diverges whe i is applied o wo idepede I) processes. However, divergece of he es should o be ierpreed as evidece of spurious olieariy bu raher simply a rejecio of he liear coiegraio specificaio wih wo possible aleraive cases. For osaioary ime series, a lieariy es ess he liear coiegraio) specificaio agais o oly oliear coiegraio models bu also absece of coiegraio. Therefore, a divergig es saisic i he o-coiegraio case correcly pois ou he absece of liear coiegraio. A furher specificaio es is eeded o deermie if he rejecio is due o olieariy. 3 While his procedure is ofe ecessary whe he es is applied o saioary X, mulicollieariy seldom arises whe X is osaioary. I coras o mea-reverig saioary ime series for which he variaio of X aroud zero are dampeed by polyomial rasformaios, iegraed X sped lile ime aroud he origi ad heir variaios are ypically magified by polyomial rasformaios as icreases. 3

14 Theorem 7 Suppose X ad Y are o coiegraed so ha Y = θ X + u =,, wih he I) process u saisfyig /2 u =[ ] d B u ). I his case he modified RESET saisic diverges a he rae of /M, where M is he badwidh parameer used i kerel esimaio of he log-ru co)variaces. This resul is of some pracical ieres. The RESET es was origially developed for esig lieariy of he model bu, whe applied o coiegraig relaios, he es has power agais lack of coiegraig as well. Thus, he modified RESET es ca serve as a omibus es for he ull of liear coiegraio agais he aleraives of boh o coiegraio ad oliear coiegraio. A similar idea i he coex of deecig ui roos is prese i Park990) s ui roo es by variable addiio. This es uses polyomials of a deermiisic process as added variables o deec he presece of lefover sochasic reds), he RESET es uses polyomials of he sochasic regressors isead, which have a aural advaage whe here is oliear coiegraio ivolvig hese variables. Sice he rae of divergece depeds o he relaive size of he badwidh parameer ad he umber of observaios, he choice of M ca grealy affec he power of he es agais he lack of coiegraio. Similar issues arise i oher ess ha rely o oparameric esimaes, such as he KPSS es for saioariy. We will discuss his issue i he ex secio ogeher wih oher pracical issues relaed o applyig he modified RESET es The Noliear Coiegraio Case Amog he may ypes of possible olieariies i coiegraed sysems, we cosider here models ha ivolve rasformaios belogig o he H-regular ad I-regular classes iroduced earlier. I paricular, we suppose he rue coiegraed sysem has he followig oliear form Y = fx ) + u, =,, 7) where X ad u saisfy Assumpios A ad A wih f ) belogig eiher o he H-regular or I-regular oliear rasformaio class. Theorem 8 If he rue model has he oliear form 7) ad {X, u } saisfy he codiios of Theorem 5, he he modified es saisic MR diverges a he rae M i he H-regular oliear case, bu does o diverge i he I-regular oliear case. Thus, he power of he modified RESET es depeds o he oliear fucioal form. For H-regular olieariies, he es saisic diverges a he rae O p M ), jus as i he case of o coiegraio. Noe ha his resul icludes he case of a hreshold model aleraive, where he H-regular rasformaio is based o idicaor fucios. The asympoic order i his case is κ =, as i he case of liear coiegraio, bu he es saisic sill diverges i his case a he rae /M. Corary o he H-regular case, he modified es has paricularly low power agais I- regular ype olieariy. This is because he variaios from he I-regular ype oliear 4

15 rasformaio of X ha remai i he liear coiegraio residuals {û } become egligible relaive o he variaios of X as icreases, while he variaios i he basis fucios F j X ) remai sigifica regardless of. Sice H-regular ad I-regular classificaios do o exhaus all ypes of oliear rasformaios, here will be oher ypes of oliear rasformaios ha he modified es fails o deec. Wheher a cerai ype of oliear coiegraio is well deeced by he modified es or o is relaed o he effeciveess of he parial sum approximaio refleced i he augmeed regressors {F j } k j=. Agai, however, his es does o require a good fi o deec olieariy. If ay polyomial erm caches eough of he olieariy o make a leas oe of he fied β j coefficies sigifica, he modified es will have power i ha direcio o rejec he ull of a liear coiegraio relaioship. 3.2 Implicaios for Noliear Regressio wih Iegraed Processes The wo bias erms i he liear coiegraio regressio 9) are called secod-order i he sese ha hey cause bias oly i he limi disribuio, wihou affecig he cosisecy of LS esimaor. The same argume applies o oliear coiegraio case. As FM regressio 6) correcs he wo biases usig sample momes ad sample esimaes of he log-ru co)variaces, Theorem 5 ca be applied o correc he wo biases i he LS coefficie esimaor i he oliear coiegraio regressio. Suppose we esimae a oliear regressio of he followig form Y = θfx ) + u =,, where fx ) = X m. From Lemma 2 ad Lemma 4 we ca correc he secod-order biases m+)/2 θm ) θ ) { = X 2m } m+ X m m+)/2 u E m) S m) ) d Bx m db u x B 2m x so ha he modified esimaor has a Gaussia asympoic disribuio aroud he rue value. The wo correcio erms are defied as follows E m) ˆ m ) { m ) m X vu, S m) ˆΩ X ˆΩ v uv vv ˆ m ) } m X vv. Whe m =, θ m is simply he FM esimaor i a ypical liear coiegraio model ad he wo correcio erms E m) ad S m) reduce o he usual form ha appear i 6). 4 Simulaios Moe Carlo resuls are preseed i his secio o show he size disorio of he RESET es caused by osaioariy ad o ivesigae how saisfacory he suggesed modificaios are i achievig he omial asympoic size i fiie samples. We also repor some simulaios 5

16 o he power of he modified RESET es agais some specific oliear models, choosig he followig four models i addiio o he liear coiegraio model: ) : Y =.X + u 2) : Y = log X + ) + u 3) : Y = X 2 + u 4) : Y =.2 exp X 2 ) + u 5) : Y =.X I { /2 X 0.6} 0.8X I { /2 X <0.6} + u Here, he liear model ) is used as he referece case, 2) is a moooically icreasig, cocave rasformaio i R + ha is symmeric abou he origi, 3) is a H-regular ype oliear rasformaio ha is ofe used o check he power of a cerai es agais oliear models, 4) is bell-shaped I-regular ype oliear rasformaio, ad 5) is a hreshold model of a ype ha is commoly used i pracical models of ecoomic ime series. The regressio error {u } ad he iegraed regressor X are geeraed from he desig X = v = e 2, + 0.4e 2, 2, u = ρu + 2 e, + e 2, ), where ρ [0.2, 0.4, 0.6, 0.8] corols he level of serial correlaio i he error erm, ad {e,, e 2, )} are idepedely ad ideically disribued as ) e, N 0, I 2 ). 8) e 2, Noe ha he iovaio processes are cosruced i such a way ha X is predeermied, as specified i Assumpio A. Samples of 5 differe sizes = 50, 00, 250, 500, 000) are draw wih 0,000 replicaios o examie boh small sample properies ad rae of covergece o he limi 4. Size of he Tes Fig. compares wo RESET ess before ad afer bias correcios whe X ad Y are liearly coiegraed. The four graphs summarize he es performace uder H 0 from Table wih a) varyig umber of observaios for a give level of serial correlaio ad b) varyig level of serial correlaio for a give umber of observaios. As show i he upper paels a), wih a moderae level of serial correlaio AR coefficie is 0.6) i he regressio error, he RESET es wihou correcio erms shows severe size disorios ha become eve worse as he sample size icreases. 4 For a omial asympoic 5% size, he acual probabiliies of a ype I error are =50), =00), =250), =500) ad =000). This weakess of he RESET es is already well kow i he saioary ad highly auocorrelaed X case from work of Porer & Kashyap 984), ad he resuls here for he case of a coiegraig 4 The probabiliy of a ype I error icreases whe he regressio errors are more serially correlaed as show i b). I he exreme case of idepede) I) errors, he es saisic diverges, as repored i Kim, Lee & Newbold 2003). 6

17 relaio may be regarded as a exreme versio of hese earlier fidigs. While he es wihou correcio erms suffers from icreasig ype I errors, he modified RESET es i he righ pael of Fig. a) exhibis oly a small size disorio, which vaishes as icreases, ad a he same ime, shows a relaively fas covergece o he limi disribuio. The probabiliies of he ype I errors for he omial 5% es are =50), =00), =250), =500) ad =000). Fig. b) shows how he bias correcio erms work for differe ρ values. The lef pael cofirms he severe size disorios due o he serially correlaed errors. For a omial asympoic 5% size, he probabiliy of a ype I error reaches up o 70% for ρ = 0.8, while icludig wo correcio erms brig i back o 4.99%. These figures are based o Table which compares wo ess for differe ρ s ad s wih k = Power of he Tes Table also repors he power of he modified RESET es agais some specific oliear models. Wih liear coiegraio as he referece case i ), simulaio resuls show ha he modified RESET es is quie sesiive o all he olieariies excep 4) for a wide rage of ρ values. The probabiliies of rejecig he lieariy ull are over 90% i mos cases excep for 4). As expeced, he modified RESET es is mos powerful agais polyomial ype olieariy always higher ha 99% i case 3)) bu also shows good powers agais logarihmic 2) ad hreshold 5) olieariies. Noe also ha he origial RESET es i he secod par also shows he similar paer. The low power agais 4) is due o fac ha he regressio fucio is a iegrable rasform of X, which is poorly capured by he polyomial basis erms i he RESET es. I paricular, he asympoic form of he fucio e X2 whe X = O p ) for large is o capured by he asympoic form of he polyomial erms X j = O p j/2 ) i he RESET basis. Table 2 shows he probabiliy of rejecig he liear coiegraio ull hypohesis whe he modified es is applied o wo I) variables ha are o coiegraed, i.e. Y =.X + u wih u I) As discussed i Theorem 7, he modified es saisic diverges a he rae /M so ha he rejecio rae is sesiive o he choice of he badwidh parameer M. We repor five cases, correspodig o M = /5, M = /4, M = /3, M = /2 ad he usual daa-depede auomaic badwidh Adrews, 99) for a Parze kerel i Table 2. Two aspecs of he resuls i Table 2 cofirm Theorem 7. Firs, he rejecio probabiliy eds o higher for he smaller badwidh choices for give k ad. Secod, he rejecio probabiliy icreases wih as well as wih he umber of augmeed regressors k i geeral, espeically for smaller badwidhs. For M = /3, he effec of icreasig k o he rejecio probabiliy is o as large as i he case of M = /5, ad eve decreases for M = /2. Whe a auomaic badwidh rule is employed, icreasig k has a more sigifica effec o power for a give ha icreasig for a give k. 4.3 Limiaios ad Pracical Issues The limiaios of he modified RESET es are relaed o he approximaio mehod ha he es is based o ad he aure of he oliear coiegraio fucioal forms. As meioed 7

18 previously, oce he oliear coiegraig fucio is give, he size of he approximaio error is deermied by he ype ad umber of he basis fucios {F j } k j=. These choices deermie how well a liear combiaio of he basis fucios ca approximae some oliear coiegraig fucio of X. If here exiss a se of coefficie { } k β j j= such ha k j= β jf j X ) is close o f X ) over a wide eough domai sice a I ) process like X visis all pois of he space a ifiie umber of imes), he we ca expec he es o rejec liear coiegraio i favor of some form of oliear coiegraio, correspodig o he o-zero { } β j esimaes. Oce he basis fucios {F j } k j= are seleced, k eeds o be chose. While larger k may produce a improved approximaio o f ), i a fiie sample esig framework here exis some rade-offs. O he oe had, larger values of k will, a leas o a cerai poi 5, geerally icrease he power of he es by virue of heir improved approximaio capabiliy. O he oher had, larger k icreases he risk of spurious olieariy resulig i a higher probabiliy of a ype I error uder he ull. Moreover, o rejec he ull hypohesis H 0 : β = = β k = 0, a leas oe sigifica coefficie will suffice, a codiio ha is less resricive ha requirig a good fi o f X ) by k ˆβ j= j F j X ). Simulaios o repored here) suggess ha he use of k = 2 or 3 geerally produces good size ad reasoable power, while icreasig k o k = 3 or 4 adds power wihou oo much compromise i size. Alhough o show explicily i he regressio equaio 2), he choice of badwidh parameer M for kerel esimaio of log-ru co)variace ca be aoher impora eleme ha affecs he size ad he power of he es, especially i small sample. As discussed i Theorem 7 ad show i Table 2, he power agais he o-coiegraio aleraive depeds o /M. The es saisic uder he some aleraives diverges faser as M/ becomes smaller, bu his makes he es saisic uder he ull coverge o he asympoic disribuio a a slower rae. Therefore, i addiio o he choice of k, i is recommeded o apply he es wih differe combiaios of k ad M o ge more a cocree resul. Oe popular choice for he badwidh selecio is he daa depede mehod i Adrews 99). He proposes he auomaic badwidh choices for various kerels, ad for he Parze kerel we use, he auomaic badwidh is [ 4ˆρ2ˆσ 2 ) ˆσ 4 )] /5 M = ˆρ) 8 ˆρ) 4 where ˆρ is he AR) coefficie esimae i û = ρû + e ad ˆσ 2 is he variace esimae of e. Aoher impora facor ha affecs he power of he es is he acual oliear fucioal form. Alhough geeral approximaio mehods, icludig he power series approximaios ha uderlie he RESET es, ca provide reasoable approximaios o a wide class of oliear fucios, here are oliear rasformaios ha cao be well approximaed by hese mehods. I paricular, cerai exesios o polyomial or raioal) approximas are geerally eeded i order o produce global approximaios o fucios over he whole real lie. Phillips 983) suggesed a class of exeded raioal approximas ha have good global approxima 5 For k very large, he regressor marix F ca maifes mulicollieariy ad pricipal compoes may be used. I may cases, he firs few pricipal compoes ed o explai mos of variaio i F ad icreasig k he leads o lile improveme i he power of he es. Noe ha icreasig k also leads o a decrease i degrees of freedom i he regressio. 8

19 performace over he whole real lie o iegrable fucios. Oe has o keep i mid ha accepig he ull of liear coiegraio leaves ope he possibiliy of some udeeced oliear effecs especially if hese are of he small ype ha would be delivered by iegrable rasformaios). Rejecig he ull suggess ha here may be oliear models ha ouperform he liear model or ha here may be o meaigful coiegraig relaio. Esimaed liear combiaios of he basis fucios ca sugges a possible oliear aleraive if he rue relaioship is oliear. I his case, as discussed i he previous secio, he modified RESET es ca be ierpreed as a LM es which compares a liear coiegraio model agais a esimaed approximaio o some ukow oliear coiegraio model. Whe he es rejecs he ull, we ca wrie dow a aleraive oliear model wih addiioal basis fucios ad re-esimae his model usig FM regressio. This leaves he remaiig issues of choosig a suiable value of k for he regressio so ha he approximaio error is reduced while o aempig o overfi he daa. These issues are complex ad are beyod he scope of he prese paper. 6 5 Empirical Applicaio o PPP The iroducio of ui roo limi heory ad coiegraio mehods have led o a vas umber of empirical sudies wih osaioary ime series, may of hem coduced wihou furher aeio o specificaio esig beyod wha is implied by ui roo ad coiegraio ess. This secio cosiders he purchasig power pariy PPP) relaioship bewee omial exchage raes ad he foreig-domesic price raio ad applies he modified RESET lieariy es o check wheher he radiioal liear coiegraio specificaio is appropriae i his coex. 5. PPP Models PPP is a simple, iuiively appealig empirical proposiio daed a leas o he sixeeh ceury i Spai Dorbusch, 987). The heory posulaes ha oce covered o a commo currecy, he price level of raded goods should be equalized across couries due o arbirage. I his sric sese, he idea is someimes udersood as a exesio of he law of oe price LOP). For a omial exchage rae, S, a domesic price of a raded good i a ime, P i,, ad he foreig price for he same good, Pi,, he LOP saes ha he same good should be sold a he same price i differe couries if prices are covered io a commo currecy Rogoff, 996) P i, = S Pi,. Aggregaig his relaioship over raded goods, PPP saes ha P i, = S i For a variey of reasos, his exac form of PPP, he so-called absolue PPP, does o hold ad a weaker versio of PPP is commoly used o provide a defiiio of he real exchage rae as q = s + p p, 6 Of course, rejecig he ull of liear coiegraio may be due eiher o olieariy or o lack of coiegraio. Developig a approximae oliear coiegraed sysem will be valid oly whe he rejecio is due o olieariy. i P i,. 9

20 where q ad s are log rasforms of real ad omial exchage raes, ad p ad p are log rasforms of foreig ad domesic price levels. Iuiively acceped as providig a log-ru equilibrium relaioship amog price levels ad exchage raes, PPP has bee esed i various frameworks, leadig o some mixed empirical fidigs. 7 There have bee may aemps o explai, usig boh ecoomic ad saisical argumes, he failure o fid cocree empirical evidece for PPP. 8 For example, i he weaker versio of PPP, he log of he real exchage rae q is usually divided io wo pars: a raded goods compoe ad a bilaeral differece bewee he relaive price of raded o o-raded goods, viz., q = s + p p = { s + p T p T } + { α p N p T ) αp N p T ) } where he superscrips T ad N sad for raded ad o-raded respecively. The price idices are geerally assumed o be geomeric averages of raded ad o-raded goods, p = α)p T + αp N ad p = α )p T + α p N, ad, defiig P = s + p T p T ad P2 = α p N p T ) αp N p T ), he real exchage rae is saioary eiher if P ad P2 are saioary, or if P ad P2 are osaioary bu coiegraed. Accepig PPP as a log-ru equilibrium relaioship, P is saioary ad i is o a all surprisig ha may fid he real exchage rae o be osaioary cosiderig he presece of he possibly osaioary compoe P2. Tradiioal ui roo/coiegraio approaches have bee he mos widely used mehod i PPP empirical sudies, bu hese mehods have ofe failed o fid ay srog suppor for PPP. These failures have led o he use of may ew mehods i searchig for evidece of PPP, icludig loger daases, pael ui roo evaluaios, ad he use of oliear models. Noicig he low power of ui roo ess i small samples, researchers have esed PPP usig loghorizo daa, fidig sroger suppor for PPP e.g. Lohia & Taylor, 995) by his mehod. May empirical researchers have foud ha he floaig exchage rae sysem iroduced wih he Breo Woods sysem has led o larger deviaios from PPP e.g. Taylor, 2002). Usig cross-coury daa o improve he power of ui roo ess has also eded o produce sroger suppor for PPP, bu hese mehods have also bee criicized by O Coell 998) ad ohers for eglecig cross coury depedece. While hese firs wo mehods have ivolved he use of differe daases o improve ess of PPP, he las approach akes io accou he possibiliy of differe model specificaios. Noliear specificaios are ofe obaied from marke fricios like rasacio coss, e.g. Dumas 992), Sercu, Uppal & va Hulle 995), ad Michael, Nobay & Peel 997). Because of marke fricios, here exiss a iacive rage aroud pariy i which ieraioal arbirage does o work ad adjusmes o pariy sar o occur oly whe he exchage rae moves ou of his rage. This oliear adjusme o pariy ca be formulaed usig variaios of he hreshold model e.g. STAR, ESTAR) ad some sigifica empirical 7 Froo & Rogoff 995) provide a discussio of he evoluio of PPP ess ad Rogoff 996) surveys empirical sudies i he area. 8 See Grilli & Kamisky 99), Pedroi 200) ad Ng & Perro 2002) for some saisical argumes ad Sercu, Uppal & va Hulle 995) ad Rogoff 996) for popular ecoomic explaaios. 20

21 evidece has bee foud i suppor of hese models, a rece coribuio beig Saikkoe ad Choi 2004) who use smooh oliear rasiios. I addiio o he oliear shor-ru adjusme erms icluded i log-ru liear equilibrium, Basher & Haug 2003) posi a oliear PPP relaioship ad apply a oliear coiegraio es developed by Breiug 200), bu fail o fid ay liear ad oliear coiegraio relaioship amog he G0 couries. We use heir model ) P S = α + f + u 9) ad es for lieariy i his coiegraig relaioship bewee he omial exchage rae ad he raio of foreig ad domesic prices. No havig a specific fucioal form for f ) offers some advaages. Firs, eve if he hreshold model had srog heoreical jusificaio for oe radable good, aggregaig over all goods ad usig a geeral price level ieviably obscures he form of he implied olieariy for he aggregae relaioship for isace, because of he maifold hreshold pois ha appear i he aggregaio). Seig a regressio equaio i he geeral form of 9) allows for a more flexible ierpreaio. Apar from providig a esable form of PPP, 9) ca be hough of as a geeral model of omial exchage rae deermiaio i erms of ecoomic fudameals. Alhough Meese & Rogoff 983) fid ha o exisig srucural model ouperforms a simple radom walk model i predicio, he moeary model has bee he sadard model for exchage rae deermiaio. This model s mai implicaio is ha he omial exchage rae is deermied by some ecoomic fudameals like moey m) ad oupu y) of he wo couries, ad he risk premium ρ). Frakel & Rose 994) show he followig expressio, firs give by Mussa 976), ca be derived usig moey marke equilibrium, PPP, ad ucovered ieres pariy: s = [m m ) βy y ) + ρ ] + α E s ) + ε d Usig he price raio o reflec he ecoomic fudameals, 9) ca be regarded as expressig omial exchage raes as some ukow fucio of uderlyig fudameals. I addiio o he PPP i levels or absolue PPP), we also es relaive PPP which ca be wrie as Rogoff, 996) ) P S P = P S Sice he price idex is he relaive value o a base year ad we do o kow how big he deviaio from absolue PPP was a he base year, his relaive versio of PPP requires he relaioship o hold oly i erms of chages. I his case, sice he logarihms of he price ad exchage rae raios are saioary, we eed o ierpre empirical resuls appropriaely Daa ad Empirical Resuls We cosider hree couries US, Japa ad Caada) formig he wo pairs: US-Japa ad US- Caada. We focus o hese wo pairs, which represe respecively a relaioship bewee wo 9 Noe ha if he variables are saioary, our modified es becomes equivale o he radiioal RESET es as he bias ad correcio erms vaish asympoically. P P 2

22 big ecoomies ad a relaioship bewee couries wih fewer rade barriers ad rasporaio coss. Our US, Japa ad Caadia daase is ake from he IMF s Ieraioal Fiacial Saisics IFS) CD-ROM ad coais omial exchage raes, he cosumer price idex CPI) ad producer price idex PPI) a a mohly frequecy. The daa spas he period from 97: o 2004:2, yieldig 34 years or 408 mohly observaios. A mohly average marke rae is used for he omial exchage rae ad boh he CPI ad PPI are used o calculae price raios. The daa are ploed i Figure 2. The lef colum shows he US-Caada: omial exchage rae solid), CPI raio dashed) ad PPI raio dash-doed), ad he righ colum shows he US-Japa i he same maer. The upper paels plo he omial exchage raes, CPI raios ad PPI raios i levels o i logs). The lower paels plo he same series bu i chages calculaed by year-o-year raios, i.e. for he omial exchage rae S ), S /S 2 ad for he CPI or PPI P ), P /P 2 )/P /P 2 ). Firs, we apply augmeed Dickey-Fuller ADF) ess o deermie wheher he ime series ploed i Figure 2 are iegraed processes he Phillips-Perro es gave similar resuls). Tes resuls o repored here) idicae ha he omial exchage rae, CPI, ad PPI are all ui roo osaioary i levels for absolue PPP) ad saioary i chages for relaive PPP). Secod, we apply ADF ad KPSS ess o he regressio residuals wih varyig sample periods, o check wheher hese ess fid ay meaigful liear coiegraio relaioship. As much previous research has repored, coveioal liear) coiegraio ess show somewha mixed resuls Table 3).. We firs cosider he whole sample period 97M 2004M2: Period ) ad he he pos-volcker period 983M 2004M2: Period 2). 2. The ADF es applied o US-Caada ad US-Japa does o fid evidece of ay liear) coiegraio relaioship bewee omial exchage rae ad he raio of price levels absolue PPP) wih eiher CPI or PPI. 3. The KPSS es fids liear coiegraio relaios for he whole sample period bu some of hese are o suppored by ess for he differe sample period. Alhough o repored here, i is o hard o fid a subsample period where he ADF es fids evidece of a coiegraio relaio. 4. Depedig o he ype of coiegraio es ad he sample period, you may or may o fid he coiegraio relaioship for he same sample. Sice hese wo popular residual based coiegraio ess produce ambiguous fidigs, we apply our modified RESET es o check wheher he relaioship is liear. The modified es is used for boh absolue PPP ad relaive PPP wih varyig badwidhs M ad umbers of polyomial erms k. Table 4 summarizes he resuls from he modified es as well as he origial RESET es before modificaio. While he origial RESET es eds o sugges suppor for a liear relaioship much of he ime excep for absolue PPP usig he CPI), he modified RESET es shows lile suppor for a liear coiegraio specificaio excep i he case of absolue PPP wih PPI ad k = 2 i Japa-US). These fidigs corroborae some exisig empirical work o real exchage raes repored i Froo & Rogoff 995). Firs, accordig o ha work, coiegraio is rejeced more ofe i 22