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

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1 Tesig Lieariy i Coiegraig Relaios Wih a Applicaio o Purchasig Power Pariy Seug Hyu HONG Korea Isiue of Public Fiace KIPF), Sogpa-ku, Seoul, Souh Korea Peer C. B. PHILLIPS Yale Uiversiy, New Have, CT 06520, Uiversiy of Aucklad, Aucklad, New Zealad, Uiversiy of Souhampo, Souhampo, U.K., ad Sigapore Maageme Uiversiy, Sigapore, Sigapore This aricle shows ha whe applied o osaioary ime series, he coveioal Regressio Error Specificaio Tes RESET) leads o severe size disorio ad is asympoic disribuio ivolves a mixure of oceral χ 2 disribuios. Nosaioariy iroduces bias erms i he limi disribuio, ad appropriae correcios for he bias are preseed leadig o a modified RESET es ha has a ceral χ 2 limi disribuio. I simulaios, his modified es is show o have power o oly agais oliear coiegraio bu also agais he absece of coiegraio. I a empirical illusraio, he liear purchasig power pariy PPP) specificaio is esed usig five Orgaizaio for Ecoomic Cooperaio ad Developme OECD) couries. KEY WORDS: Noceral χ 2 disribuio; Noliear coiegraio; RESET es; Specificaio es.. INTRODUCTION 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 was developed 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. Empirical applicaios, however, ofe simulae a ieres i oliear specificaios ad, as a 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 ad Dijk 999), Lo ad Zivo 200), ad Teräsvira ad Eliasso 200) amog ohers. However, 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., Haug ad Basher 2003). Such exesios also awai a correspodig developme i ess of specificaio. Neglecig he possible 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. Cosideraio of he possibiliies suggess 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 e.g., Grager ad Hallma 989). So, radiioal liear coiegraio aalysis requires a addiioal es of specificaio o address his paricular issue of fucioal form. However, i he absece of more appropriae specificaio ess, applied ecoomiss have reaed exisig coiegraio ess as ess for liear coiegraio ad all subseque aalysis ress o his assumpio. Furhermore, exisig lieariy ess also fail o provide ay reliable guidace cocerig he ype of relaioship ha may be prese bewee osaioary ime series Grager 995; Lee, Kim, ad Newbold 2005). I is o surprisig o fid ha exisig lieariy ess developed for saioary processes work poorly wih osaioary ime series ad his was well recogized earlier i he case of he Regressio Error Specificaio Tes RESET). The RESET es by 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 ad Kashyap 984) or coais a deermiisic ime red Leug ad Yu 200). Usig simulaio, Porer ad Kashyap showed ha he presece of serially correlaed disurbaces combied wih a AR) regressor leads o size disorios, ad he more auocorrelaed he regressor is, 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 200 America Saisical Associaio Joural of Busiess & Ecoomic Saisics Jauary 200, Vol. 28, No. DOI: 0.98/jbes

2 Hog ad Phillips: Tesig Lieariy i Coiegraig Relaios 97 regressor has a auoregressive ui roo ad he errors are ypically serially depede. This aricle aalyzes he source of his es failure ad shows how he origial es ca be modified for empirical use wih osaioary ime series usig asympoic ools from Park ad Phillips 999, 200). The res of he aricle is orgaized as follows. The ex secio iroduces he model ad he maiaied assumpios ad shows how osaioariy of he daa chages he limi heory of exisig ess. Secio 3 discusses he modificaios ha are eeded whe he RESET es is applied o evaluae coiegraig relaios. Secio 4 discusses he behavior of he modified es uder aleraives. Secio 5 summarizes simulaio resuls ad Secio 6 preses a empirical applicaio of he modified es o purchasig power pariy PPP). Secio 7 cocludes ad he proofs are colleced i he Appedix. 2. MODELS AND BACKGROUND IDEA Suppose ha we wa o es he liear codiioal mea specificaio H 0 : P[EY X ) = θ X ]=, wih a specific oliear aleraive model i mid, such as f X ). The oe ca use some coveioal ess such as a Wald or Lagrage muliplier LM) es of H 0 : θ 2 = 0i Y = θ X + θ 2 f X ) + u. ) I may pracical cases, however, heory fails o provide a specific fucioal form, ad he focus of aeio is some coveie liear model such as ha implied by purchasig power pariy cosideraios) wih o specific oliear aleraive. Numerous specificaio ess have bee developed so far, ad oe of he mos frequely used approaches is he so-called residual based procedure. The residuals from regressig Y o X, if he ull is rue, should o coai ay sysemaic par of f X) ad he may residual based ess arise from differe mehods of deecig such lefover sigals i residuals. For example, he KPSS Kwiakowski e al. 992) es ad he CUSUM es Ploberger ad Kramer 992 for srucural chage, Xiao ad Phillips 2002 for coiegraio) are based o he excess variaios ad he Park 990) es ad ANN Arificial Neural Nework; Whie 992; Lee, Whie, ad Grager 993) esare based o he approximaio o a uspecified f ). I he case of he approximaio-based ess, firs he uspecified oliear fucio f X ) is replaced wih is parial sum approximaio ˆf k X ) = k j= β j F j X ) for some give basis fucios {F j x)} ha form a complee se i L 2. Now we proceed o es he validiy of he liear specificaio by esig wheher a liear combiaio of {F j )} k ca deec ay olieariy i he regressio residuals û i Y = θx + u ad û = k β j F j X ) + e. 2) I geeral, here are wo ways o es he liear specificaio i his seig. For a approximaio based es wih kj= β j F j X ), we ca eiher direcly es is saisical sigificace wih H 0 : β j = 0, j Ramsey 969; Whie 992; Kapeaios 2003; DeBeedicis ad Giles 998), or fid wheher he opimal k = 0, usig a versio of order selecio crierio Eubak ad Har 992). I his aricle, we ake he firs approach wih j= he polyomial basis fucios F j X ) = X j+ for he RESET es. Noe ha esimaio of 2) ivolves workig wih he sample momes of oliearly rasformed iegraed ime series 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 codiios: u = c j ε j = CL)ε, v = d j η j = DL)η, j=0 where ζ = η +,ε ) is a saioary ad ergodic marigale differece sequece wih aural filraio F = σ{ ζ s } ) saisfyig. sup E ζ r F )< a.s. for some r > 4 2. E ζ i, 2 ζ j, l ) = 0 for all i, j ad for all l 3. ε is iid wih E ε r < for some r > 8, is disribuio is absoluely coiuous wih respec o Lebesgue measure ad is characerisic fucio ϕ saisfies ϕλ) = o λ δ ) as λ for some δ>0. I addiio, {c j, d j } saisfy he summabiliy codiios: j=0 j d j <, j=0 j /2 c j <, ad D) 0. These assumpios o he iovaio processes are fairly sadard ad are saisfied by a wide class of processes; for example, a iverible Gaussia auoregressive movig average ARMA) model. Similar codiios are employed i derivig he resuls of Park ad Phillips 999, 2000, 200) ad Chag, Park, ad Phillips 200). However, de Jog s 2002) more relaxed codiios are sufficie for he modificaio of he RE- SET es preseed i his aricle. Uder Assumpio A, he ivariace priciple holds for ζ so ha /2 [r] ˆζ d BM ). Usig he Beveridge Nelso decomposiio Phillips ad Solo 992), we ca show ha a similar resul holds for he ime series ζ =[v +, u ], [r] ) ζ d Bx r) Br) BM ), B u r) ) vv wih = vu. j=0 uv uu Here he covariace marix = h= Ɣ ζ h), where Ɣ ζ h) = Eζ 0 ζ h ). Also, defie he oe-sided log-ru covariace marices wih similar pariios as = h= Ɣ ζ h) ad = Ɣ ζ 0) +. Amog he wide variey of possible oliear fucios, Park ad Phillips 999, 200) provide ools for asympoic aalysis wih some classes of fucios of iegraed processes) saisfyig cerai regulariy codiios. The simple basis fucios {X} j of a Taylor series expasio fall wihi he so-called H-regular class or Class H). Fucios i his class have homogeeous, asympoically domiaig compoes, ha is, f λx) = κλ)hx), ha are locally iegrable. Hx) is referred as he asympoic homogeeous fucio of Fx) ad κλ) as

3 98 Joural of Busiess & Ecoomic Saisics, Jauary 200 he asympoic order of Fx). Park ad Phillips 999) provided various examples ha belog o his class, such as fiie order polyomials, ad disribuio-like fucios, icludig heir liear combiaios ad producs. The polyomial basis fucios {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. All pdf-like fucios belog o his class. See Park ad Phillips 999) for furher deails o hese classificaios. 2. Noliear Sample Covariaces & RESET Tesig a liear specificaio wih H 0 : β j = 0, j i 2) ivolves he sample covariace bewee u ad he polyomials of X. The followig lemma is a special case of de Jog 2002) ad shows he asympoic behavior of his sample quaiy. Lemma. Suppose Assumpio A holds. For m, he sample covariace bewee X m ad u saisfies m+)/2 X m u d B m x db u.x + uv vv + m vu B m x db x B m x, 3) where vu = h= Ev 0 u h ) ad he Browia moio B u.x = B u uv vv B x is idepede of B x ad has variace uu.v = uu uv vv vu. The limi of he sample covariace 3) has wo compoes B m x db u.x, a mea zero Gaussia mixure, ad he remaiig wo erms ha correspod o he so-called edogeeiy bias ad serial correlaio bias of liear coiegraio heory Phillips ad Hase 990). These secod-order bias erms sem from he osaioariy of X ad shif he mea of he limi disribuio away from zero. I he simples case of sricly exogeous X, Ev u s ) = 0 for all, s so ha he wo bias erms are zero wih uv = uv = 0. I relaed work, de Jog 2002) examied oliear sample covariace asympoics uder codiios ha are less sric o he iovaio processes, bu more resricive i erms of fucioal forms; ad a geeral semimarigale approach o esablishig limi resuls of his ype was developed i Ibragimov ad Phillips 2004). The effecs of he wo bias erms i 3) ca be subsaial o he disribuio of he RESET es saisic. As is well kow e.g., Muirhead 982, heorem.4.5), a quadraic form x Ax i he Gaussia radom vecor x Nξ, V) follows a oceral χ 2 disribuio, χ 2 k,ν), where k = rakav) is he degrees of freedom ad ν = ξ Aξ is he oceraliy parameer. Leig x be he limi of û F afer appropriae ormalizaio, we ca show ha he es saisic R i 4) follows a mixure of oceral χ 2 disribuios uder suiable codiioig. Due o he presece of he fied residual û isead of u, some addiioal bias erms appear i he limi, i addiio o he wo erms show i 3). The followig heorem summarizes his resul. Theorem 2. Uder Assumpio A, he RESET es saisic R asympoically has a mixure of oceral χ 2 k,ν) disribuios wih k degrees of freedom ad he radom oceraliy parameer ν = ξ Aξ. Tha is, he RESET es saisic ) ) ) R = û F ˆ uu.v F F û F = û F ˆ uu.v F F) F û a χ 2 k,ν) 4) for he auxiliary regressors F =[X 2 X k+ ], F =[F,...,F ], ad he regressio residuals F =[ F,..., F ] wih F = F X X X ) X F. The radom vecor ξ is k wih m )h eleme defied as, for m 2, ξm ) = uv vv B m x db x + m vu B m x vu B 2 x ) B m+ x, 5) wih B m x = Bm x B x B 2 x ) B m+ x ad A is he iverse of a k k limi variace marix, vis-a-vis, 2 B2 x B2 Bk+ x x A =. uu.v ) Bk+ x B 2 x Bk+ 2 x Whe X is sricly exogeous, all bias erms disappear wih Eu v s ) = 0,, s, ad he es saisic R asympoically has a mixure of ceral χ 2 disribuios codiioal o F x = σb x r), 0 r ). Sice he limi disribuio is idepede of F x, we deduce ha R coverges o χ 2 k) ucodiioally. I geeral, R χ 2 k,ν) asympoically ad his oceral disribuio ca be approximaed by a muliple of he ceral χ 2 disribuio, aχ 2 b), where he wo cosas are give by Johso ad Koz 970) a = + ν ν2 ad b = k + k + ν k + 2ν k. Therefore, codiioal o F x, he probabiliy of rejecig he liear ull hypohesis ca be show o be a leas as grea as he omial size α asympoically, vis-a-vis, [ P[R >χα 2 ] P + ν k + ν P [χ k 2 + ν2 k + 2ν ) χ 2 b)>χ 2 α ] ) ] >χα 2 α, ad his explais he large size disorios i Porer ad Kashyap 984). 3. BIAS CORRECTION & MODIFIED TEST The previous secio shows ha osaioariy of X iroduces bias erms i he limi disribuio of he sample covariace bewee X m ad û, leadig o he oceral limi disribuio of he RESET saisic R. These bias erms are he mai source of he large size disorios of he es ad he followig heorem preses a mehod o remove hem similar o he direc oparameric correcio mehod i fully modified FM) regressio Phillips ad Hase 990; Phillips 995).

4 Hog ad Phillips: Tesig Lieariy i Coiegraig Relaios 99 Theorem 3. Suppose Assumpio A holds. If {X, Y } are liearly coiegraed, he followig modified RESET saisic has a limiig ceral χ 2 k) disribuio MR ={û FD E S } ˆ uu.v D F FD ) {D F û E S } a χ 2 k), where û is a vecor of residuals from he liear coiegraio regressio 2) wih F ad F as i Theorem 2. 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), E m ) = ˆ uv ˆ vv [{ ) m X v m ˆ vv S m ) = ˆ vu m ) X v ˆ vv X X ˆ vu 2 X 2 X ) ) m } ) m+ )], 7) ) m 2 X 2 ) X ) m+. 8) Sice his ype of es is based o a fiie approximaio, is power aurally depeds o he adequacy of he approximaio uder he aleraive specificaio. The goodess of approximaio depeds o he give oliear fucioal form ha is approximaed ad wo compoes ha ca be corolled he ype ad he umber of basis fucios icluded i he augmeed regressors. A good approximaio will defiiely help i deecig olieariy whe i is prese, bu eve poor approximaios ca be effecive for esig purposes. This is because he ull hypohesis requires ha all coefficies be zero, β j = 0forj =,...,k, ad he es will rejec he ull hypohesis if a leas oe coefficie deviaes eough from zero, ha is, if a leas oe basis fucio is able o cach some par of he olieariy. 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 possible 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. Whe he liear model is rejeced ad he aleraive polyomial model is esimaed, he correcio mehods i Theorem 3 ca be applied agai o correc he biases i he leas-squares LS) coefficie esimaors. For example, suppose we esimae he followig oliear coiegraio model Y = θx m + u, =,...,. The he correspodig FM esimaor of θ is ) { } θ m = X 2m X m Y m+)/2 [E m + S m ] wih he correcio erms E m ˆ vu m S m ˆ uv ˆ vv X { ) m, X ) m v ˆ vv m X ) m } ad we ca show ha he modified esimaor has a mixed Gaussia asympoic disribuio abou he rue value, ha is, m+)/2 θ m θ) d B 2m x ) B m x db u.x. Whe m =, θ m is simply he FM esimaor i a ypical liear coiegraio model. Wih saioary ime series, he auxiliary regressor se {X j+ } k j= ofe suffers from mulicollieariy, i which case pricipal compoes ca be used isead. If his is he case, he bias correcio erms also eed o be adjused accordigly, ad he modified es saisic usig pricipal compoes ca be cosruced as follows: {û F E G S G} ˆ uu.v F F ) {F û G E G S } a χ 2 k), where G is he k k marix whose colums are he eigevecors of F F divided by he correspodig eigevalues, ad F = FD G is k ormalized marix wih he jh pricipal compoe i he jh colum. The k eigevecors are chose accordig o he k larges eigevalues. This mulicollieariy problem is maily due o he meareversio propery of saioary ime series for which he variaio of X aroud zero is dampeed by polyomial rasformaios. However, his is o he case for iegraed X, which speds lile ime aroud he origi ad whose variaios are ypically magified by polyomial rasformaios as icreases. 4. MODIFIED TEST UNDER ALTERNATIVES 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. This secio examies es power i hese aleraive scearios.

5 00 Joural of Busiess & Ecoomic Saisics, Jauary No Coiegraio Case Lee, Kim, ad Newbold 2005) examied six widely used lieariy ess ad fid ha evidece of spurious olieariy icreases wih he sample size. The followig heorem shows ha our modified es saisic also diverges whe i is applied o wo idepede I) processes. However, divergece of he es saisic should o be ierpreed as evidece of spurious olieariy bu raher simply as a rejecio of he liear coiegraio specificaio wih wo possible aleraive cases. Therefore, he divergig es saisics i he o-coiegraio case correcly poi ou he absece of liear coiegraio. To deermie if he rejecio is due o olieariy, a furher specificaio es is required. Theorem 4. Suppose X ad Y are o coiegraed so ha Y = θx + u, =,..., wih he I) process u saisfyig /2 u =[ ] d B u ).Ihis 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 coiegraio as well. Thus, he modified RESET es ca serve as a omibus es for he liear coiegraio specificaio ha has power agais boh o coiegraio ad oliear coiegraio. A similar idea i he coex of deecig ui roos is prese i Park s 990) ui roo es by variable addiio. This es uses deermiisic polyomials 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. 4.2 Noliear Coiegraio Case Amog he may ypes of possible olieariies i coiegraed sysems, we cosider here models i he followig oliear form Y = f X,θ)+ u, =,..., 9) wih f ) beig eiher H-regular or I-regular. Theorem 5. Suppose he rue model has he oliear form 9) ad {X, u } saisfy he codiios of Theorem 3. For give M, 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. Obviously, he power of he modified RESET es depeds o he rue 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 rasformaio of X ha remai i he liear coiegraio residuals {û } become egligible relaive o he variaios of X as icreases. 5. SIMULATIONS Moe Carlo resuls are preseed i his secio o show he size disorio of he origial RESET es ad o ivesigae how saisfacory he suggesed modificaios are i achievig he omial asympoic size i fiie samples. We also repor some simulaios o he power of he modified RESET es agais some specific oliear models, choosig he followig seve oliear models i addiio o he liear coiegraio model as he referece case: ): Y =.X + u, 2): Y = log X +) + u, 3): Y = X 2 + u, 4): Y =.2exp X 2 ) + u, 5): Y =.X I { /2 X 0.6} 0.8X I { /2 X <0.6} + u, 6): Y = / X +) + u, 7): Y =. X +) /2 + u, 8): Y =. X +) 3/2 + u. 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, ) N 0, I 2 ). Noe ha he iovaio processes are cosruced i such a way ha X is predeermied, as specified i Assumpio A. Samples of five 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 5. Size of he Tes Figure 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) a varyig umber of observaios for a give level of serial correlaio ρ = 0.6) ad b) a 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 i

6 Hog ad Phillips: Tesig Lieariy i Coiegraig Relaios 0 a) Varyig umber of observaios wih ρ = 0.6adk = 3. b) Varyig serial correlaio wih = 000 ad k = 3. Figure. The RESET es saisics before ad afer modificaio uder H 0. Empirical disribuios of he es saisic show above are from 0,000 simulaed samples wih k = 3. The badwidh for he kerel esimaor of log-ru co)variace is chose auomaically followig Adrews 99). χ 2 -disribuio i a hick solid lie represes he limi disribuio of es saisics from a ceral χ 2 k) disribuio. he regressio error, he RESET es wihou correcio erms shows severe size disorios ha become eve worse as he sample size icreases. For a omial asympoic 5% size, he probabiliy of a Type I error rises up o wih = 000. This resul may be regarded as a exreme versio of he earlier fidigs i Porer ad Kashyap 984). Corary o he severe size disorios i he origial es, he modified es i he righ pael of Figure a) exhibis oly a mior size disorio, which vaishes as icreases ad, a he same ime, shows a relaively fas covergece o he limi disribuio. Figure b) shows how he bias correcio erms work for differe ρ values. The lef pael cofirms he severe size dis-

7 02 Joural of Busiess & Ecoomic Saisics, Jauary 200 orios due o he serially correlaed errors. For a omial asympoic 5% size, he probabiliy of a Type I error reaches up o 70% for ρ = 0.8, while icludig wo correcio erms brigs i back o 4.99%. 5.2 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 may oliear possibiliies for a wide rage of ρ values. The probabiliies of rejecig he lieariy ull are over 90% i mos cases excep for 4) ad 7). As expeced, he modified RESET es is mos powerful agais polyomial ype olieariy 3) bu also shows good powers agais logarihmic 2), hreshold 5), ad reciprocal 6) olieariies ad a small deviaio from liear model 8) as well. Noe also ha he origial RESET es i he secod par also shows a similar paer. The low power agais 4) ad 7) is due o fac ha i hese cases, he oliear rasforms ed o suppress he variaios of X, while he polyomials of he RESET es ed o magify he variaios. Therefore, he asympoic forms of he fucio e X2 ad X +) /2 whe X = O p /2 ) for large are o well capured by he asympoic form of he polyomial erms X m = O p m/2 ) for m 2. Table 2 shows aoher direcio for he aleraive case of 2 idepede I) variables. As discussed i Theorem 4, 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, /4, /3, /2 ad he usual daa-depede auomaic badwidh Adrews 99) for a Parze kerel. Two aspecs of he resuls i Table 2 cofirm Theorem 4. Firs, he rejecio probabiliy eds o be 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, especially 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. 5.3 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 coiegraio fucioal forms. Oce he 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 ukow oliear coiegraig fucio f X ). If here exiss a se of coefficies {β j } k j= such ha k j= β j F j X ) is close o f X ) over a wide eough domai, he i is clear ha we ca expec he es o rejec liear coiegraio i favor of some form of oliear coiegraio, correspodig o he ozero {β j } esimaes. Oce he ype of basis fucios {F j } is seleced, he umber of hem, k, eeds o be chose. Alhough 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, 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 Type I error uder he ull as well as a decrease i degrees of freedom i he regressio. 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 j= ˆβ j F j X ). Simulaios o repored here) sugges 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. Aoher impora facor ha is o show explicily i he regressio Equaio 2) is he choice of badwidh parameer M for kerel esimaio. As discussed i Theorem 4 ad show i Table 2, he power agais he o-coiegraio aleraive depeds o /M. The es saisic uder he same 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 a more cocree resul. A popular choice for badwidh selecio is he daa depede mehod of Adrews 99). The Parze kerel is used i he simulaios show bu, while o repored here, oher kerels wih heir auomaic badwidhs gave similar power ad size properies. Fially, a impora bu ucorollable facor ha affecs he power of he es is he acual oliear fucioal form uder he aleraive. Alhough geeral approximaio mehods, icludig he power series approximaios ha uderlie he RE- SET es, ca provide reasoable approximaios for a wide class of oliear fucios, here are oliear rasformaios ha are o well approximaed by hese mehods. Low power of he modified RESET es agais I-regular-ype olieariy ca be udersood i his coex. This problem ca be alleviaed by ui roo esig, which ca someimes provide iformaio abou he ype of olieariy. For example, if he depede variable is a rigoomeic fucio of a I) process, his fucio will behave like a saioary AR) process e.g., Ermii ad Grager 993) ad ui roo ess prior o coiegraio aalysis will help o ideify he depede variable as saioary while he idepede variables are osaioary. Cerai exesios o polyomial or raioal) approximas are eeded i order o produce global approximaios for such iegrable fucios over he whole real lie. Phillips 983) suggesed a class of exeded raioal approximas ha have good global approxima performace over he whole real lie o iegrable fucios, which may herefore be useful i his coex. Oe useful feaure of he approximaio-based lieariy es is ha he esimaed liear combiaios of he basis fucios ca sugges possible oliear aleraives whe he liear specificaio is rejeced due o olieariy. I his case, he modified RESET es ca be ierpreed as a LM es ha compares a liear coiegraio model agais a esimaed approximaio o some ukow oliear coiegraio model. So, if he ull is rejeced, we ca wrie dow a aleraive oliear model wih addiioal basis fucios uil he approximaio errors become I0) ad reesimae his model usig he

8 Modified RESET es Table. Probabiliy of rejecig H 0 of liear coiegraio Origial RESET es Fucio Type ) 2) 3) 4) 5) 6) 7) 8) ) 2) 3) 4) 5) 6) 7) 8) ρ = 0.2: = = = = = ρ = 0.4: = = = = = ρ = 0.6: = = = = = ρ = 0.8: = = = = = ρ = 0.9: = = = = = NOTE: ) 8) deoe he fucioal forms defied i he begiig of simulaio. The probabiliies are calculaed from 0,000 simulaed samples wih k = 3 ad he badwidh is chose auomaically followig Adrews 99) for Parze widow. Hog ad Phillips: Tesig Lieariy i Coiegraig Relaios 03

9 04 Joural of Busiess & Ecoomic Saisics, Jauary 200 Table 2. Probabiliy of rejecig lieariy/coiegraio whe X ad Y are o coiegraed Number of basis fucios k) Badwidh M = /5 = = = = M = /4 = = = = M = /3 = = = = M = /2 = = = = Auomaic = = = = NOTE: The rejecio probabiliies are calculaed from 0,000 replicaios for he omial 5% es. The auomaic daa-deermied badwidh choice i he boom pael is based o Adrews 99). FM regressio mehod preseed i he previous secio. This approach has clear advaages i empirical research over oher residual-based geeral specificaio ess like he CUSUM ype es, which idicae oly wheher he ull hypohesis is rejeced or o. Of course, he aleraive oliear model is valid oly if he rejecio of he ull hypohesis is due o olieariy. If he rejecio is due o complee lack of coiegraio, he he approximaio error will o become I0). Also, fidig a saisfacory oliear aleraive specificaio by way of approximaioivolveschoosig a suiablevalue of k for he regressio so ha he approximaio error is reduced while o aempig o overfi he daa. Such complex issues ecessarily ivolve model selecio ad are beyod he scope of he prese aricle. 6. EMPIRICAL APPLICATION The iroducio of ui roo limi heory ad coiegraio mehods has 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 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. 6. PPP Models PPP is a simple, iuiively appealig empirical proposiio daed a leas o he 6h 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 is sric sese, he idea is someimes udersood as a exesio of he law of oe price LOP), P i, = S P i,, wih 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, P i,. Aggregaig his relaioship over raded goods, PPP saes ha P i, = S P i,. 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, 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, radiioal 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 empirical 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 log-horizo daa, fidig sroger suppor for PPP e.g., Lohia ad Taylor 996) by his mehod. Usig cross-coury daa o improve he power of ui roo ess has also eded o produce sroger suppor for PPP, bu wih some criicism for eglecig cross-coury depedece e.g., O Coell 998). While hese 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/rasporaio coss or rade barriers e.g., Sercu, Uppal, ad va Hulle 995 ad Michael, Nobay, ad Peel 997). These marke fricios are usually formulaed i erms of oliear adjusmes o pariy, ad some varias of hreshold models are beig suggesed o fid sroger empirical evidece i suppor of hese models Saikkoe ad Choi 2004). I addiio o he oliear shor-ru adjusme erms associaed wih he log-ru liear equilibrium, Haug ad Basher 2003) posied a oliear PPP relaioship ad apply a simple oliear coiegraio es developed by Breiug 200), bu failed o fid ay liear ad oliear coiegraio relaioship amog he G0 couries. We use heir model, P ) S = α + f + u, 0) P i

10 Hog ad Phillips: Tesig Lieariy i Coiegraig Relaios 05 ad es for lieariy i his coiegraig relaioship direcly. 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). Secod, seig a regressio equaio i he geeral form of 0) allows for a more flexible ierpreaio. Apar from providig a esable form of PPP, 0) ca be hough of as a geeral model of omial exchage rae deermiaio i erms of ecoomic fudameals. Alhough Meese ad Rogoff 983) foud 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 ad oupu of he wo couries, ad he risk premium. Usig he price raio o reflec he ecoomic fudameals, 0) 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 P = S S ) P P 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. Also, oe ha our modified es becomes equivale o he radiioal RESET es as boh he bias ad he correcio erms vaish asympoically for saioary ime series. 6.2 Daa We cosider five couries U.S., Japa, Caada, Mexico, ad U.K.) formig he four pairs: U.S. Japa, U.S. Caada, U.S. Mexico ad U.S. U.K. Boh borderig he U.S., Caada ad Mexico had srog ecoomic ies o he U.S. eve before he Norh America Free Trade Ac NAFTA) came io effec i Jauary 994. Accordig o WTO saisics, i he year 2005, 36.7% of expors ad 26.8% of impors of he oal merchadise rade of he U.S. are wih hese wo couries. Those proporios are as high as 83.9% of expors ad 56.5% of impors i Caada wih he U.S., ad 85.8% of expors ad 53.6% of impors i Mexico wih he U.S. While boh couries deped heavily o radig wih he U.S., heir experieces wih he U.S. are quie differe i our sample period, which will be discussed laer. Due o he geographic proximiy as well as previous rade agreemes icludig NAFTA, we expec ha he marke fricios rasporaio cos, rade barriers, ad so o hamperig he ieraioal arbirage are a he lowes level amog hese couries. The U.K. is oe of he bigges ecoomies i he EU as well as i he world ad has a log hisory of close coecio wih he U.S. Alhough abou half of is merchadise impors ad expors are wih oher EU couries, he ex larges radig parer is he U.S., accouig for. 4.7% of expors ad 8% of impors i U.K. merchadise rade i Aoher ieresig coury is Japa, which used o be he secod larges ecoomy i he world excludig he EU. Like he U.K., i s sill oe of he bigges players i world rade ad is bigges rade parer is he U.S. 22.9% of expors ad 2.7% of impors). Japa aloe akes 6.% of expors ad 8.2% of impors i U.S. merchadise rade. Boh he U.K. ad Japa are geographically far from he U.S. compared wih Caada ad Mexico, bu Japa is i geeral very differe from he oher four couries socioecoomically, so ha we expec such differeces will cause movemes i relaive price levels as well as differeces i exchage raes. Our 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)/wholesale price idex WPI) a a mohly frequecy. The daa spa he period from 97: o 2004:2, yieldig 34 years or 408 mohly observaios excep Mexico s PPI series which sars from 98:. A mohly average marke rae is used for he omial exchage rae ad boh he CPI ad PPI/WPI are used o calculae price raios. The daa are ploed agais ime i Figure 2. The lef colum shows he absolue PPP i levels) omial exchage rae solid), CPI raio dashed), ad PPI raio dash-doed) ad he righ colum shows he relaive PPP i he same maer i chages calculaed by year-oyear raios, ha is, for he omial exchage rae S ), S /S 2 ad for he CPI or PPI P ), P /P 2 )/P /P 2 ). 6.3 Variaios ad Two Sample Periods The fac ha exchage raes are much more volaile ha he price measures has bee posied as oe of he reasos why i is hard o fid empirical suppors for PPP, ofe leadig o he models of facioally iegraed real exchage rae series, or oher oliear models for PPP. We firs calculae he sadardized variaios. For a arbirary mohly series {X }, defie a hree-year rollig sadardized variaio of X a by where V = X = 36 8 j= j= 7 X +j X ) 2 X, X +j for = 7,..., 8, which is he raio of a sadard deviaio i earby hree-year period 36 mohs) o is local mea for ha period. Therefore, V is a ui-free measure of he size of variaios i X durig he hree years i he eighborhood of, proporioal o is level. Figure 3 shows hese proporioal sadardized variaios of exchage raes, CPIs ad PPIs boh i levels i lef colum) ad i raios i righ colum). The righ y-axis scale is used for Mexico ad he res of he couries follow he lef y-axis scale. Oe hig ha is clear from hese plos is a decliig volailiy i he case of price measures, especially from early 980s, bu he exchage raes do o show ay clear paer. Oly Mexico is a excepioal case, where price levels become much more volaile durig 980s ad he, oly afer early 990s, hey become sabilized a a lower level bu sill cosiderably

11 06 Joural of Busiess & Ecoomic Saisics, Jauary 200 Figure 2. Nomial exchage raes ad price raios: U.S. Caada, U.S. Japa, U.S. Mexico, ad U.S. U.K. The sample spas from 97: o 2004:2. Lef figures plo daase i levels ad righ figures plo i he chages, S /S 2 ad P /P 2 )/P /P 2 ) where he omial exchage rae is ploed i he solid lie, CPI raio i he dashed lie, ad PPI raio i he dash-doed lie. higher ha hose of he oher couries. This period of early 980s roughly coicides wih he so-called Volcker period durig which he Fed srogly fough for he worldwide high iflaio raes, ad we cosider his a subsample period where he volailiies of price measures are sigificaly lower ha hose from he oher period. I our aalysis, we firs cosider he whole sample period 97M 2004M2: Period hereafer) wih 408 mohly

12 Hog ad Phillips: Tesig Lieariy i Coiegraig Relaios 07 Figure 3. Chages i sadarized variaios. Graphs show he chages i he sadardized variaios i exchage raes, CPI raios, ad PPI raios for a hree-year rollig widow. Lef figures plo he variaios of variables i he levels, ha is, variables for he absolue PPP, ad righ figures plo he same for he relaive PPP. observaios ad he he pos-volcker period 983M 2004M2: Period 2 hereafer) wih 264 mohly observaios. These wo sample sizes are roughly equal o he wo sample sizes 250 ad 500) we cosidered i he simulaio, so ha we expec our es saisic will show a similar performace as show i he simulaio sudy. As we have see already, wih a moderae level of serial correlaio i he error, our modified es is sigificaly beer ha he origial es i he es size ad relaively good powers as well. I Figures 2 ad 3, his Volcker period is show i shade, ad our Period 2 covers he sample afer his period. 6.4 Tradiioal Coiegraio Aalysis We sar wih he radiioal coiegraio aalysis ad will compare he resuls wih our modified RESET es resuls. Firs, we apply augmeed Dickey Fuller ADF) ess o deermie wheher our daase coais iegraed processes. 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). The Phillips Perro es gives similar resuls. Secod, we apply ADF ad KPSS ess o he regressio residuals from regressig he exchage rae S ) o a cosa ad he price raio P /P ), wih varyig sample periods, o check wheher hese residual-based coiegraio ess fid ay meaigful liear) coiegraio relaioship see Table 3). For he ADF es, various specificaios of Dickey Fuller regressio are used wih differe lagged erms ad boh ) cosa or 2) cosa ad liear reds. Noe ha he ull hypoheses of he wo ess are differe: o liear coiegraio for he ADF es ad liear coiegraio for

13 08 Joural of Busiess & Ecoomic Saisics, Jauary 200 Table 3. Residual-based coiegraio ess of absolue PPP: ADF ad KPSS ADF es Number of lags KPSS es A) PPP wih cosumer price idex Period : U.S. Caada ) ) U.S. Japa ) ) U.S. Mexico ) ) U.S. U.K. ) ) Period 2: U.S. Caada ) ) U.S. Japa ) ) U.S. Mexico ) ) U.S. U.K. ) ) B) PPP wih wholesale/producer price idex Period : U.S. Caada ) ) U.S. Japa ) ) U.S. Mexico a U.S. UK ) ) Period 2: U.S. Caada ) ) U.S. Japa ) ) U.S. Mexico ) ) U.S. U.K. ) ) NOTE: The coiegraio regressio is esimaed for Period 97M 2004M2) ad Period 2 983M 2004M2) wih a cosa ad a liear red. The umber of lags i he colum shows he umber of lagged erms i he Dickey Fuller regressio for he regressio residuals. The ADF es saisics wih a cosa erm are repored i ) ad saisics wih boh a cosa ad a liear ime red are abulaed i 2). * s show he ull hypohesis rejeced. Oe aserisk meas rejecio a a 0% sigificace level, 2 ad 3 aserisks imply 5% ad %, respecively. a Sice PPI series for Mexico is available oly afer 98, coiegraio is esed oly for he secod period. he KPSS es. As much previous research has repored, coveioal liear coiegraio ess show somewha mixed resuls.. The ADF es o U.S. Caada ad U.S. 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. However, ess fid sigifica liear coiegraio for U.S. Mexico ad U.S. U.K. wih CPI for he whole sample period ad hese coiegraio relaioships become less sigifica if we look a Period 2. While he sylized fac from he exisig empirical sudies shows more favorable evidece wih PPI, he ADF es wih our sample does o show such paer. 2. Ulike he ADF es, which fids oly a few cases of liear coiegraio, he KPSS es fids may liear coiegraio relaios for he whole sample period, bu some of hese are o suppored by ess for Period 2, especially wih PPI, ad his is exacly opposie o he empirical sylized fac. Alhough U.S. Mexico is he mos srogly suppored liear coiegraio by he ADF es, he KPSS es

14 Hog ad Phillips: Tesig Lieariy i Coiegraig Relaios 09 shows sroger suppor for U.S. U.K., fidig liear coiegraio i all four cases. For U.S. Japa, however, excep he whole period wih PPI, he KPSS does o fid ay liear coiegraio a all. 6.5 The Modified RESET Tes Whe hese wo popular residual-based coiegraio ess produce ambiguous fidigs, we ow apply our modified RE- SET es ad compare he resuls wih he origial RESET es. Table 4 summarizes he resuls from he modified es as well as he origial RESET for boh absolue PPP lef) ad relaive PPP righ) wih varyig badwidhs ad umbers of polyomials k. The upper par A) is he case wih CPI ad he lower par B) is he case wih PPI, ad each par is divided io wo sample periods. While he origial RESET es eds o fid ha mos relaioships are liear, he modified RESET es shows lile suppor for a liear coiegraio specificaio excep oe special case of U.S. Mexico where he liear coiegraio is srogly suppored for all cases. Oe ieresig poi is ha his eds o be opposie o he origial RESET es, which fids lile or o evidece for a liear relaioship bewee he U.S. ad Mexico compared wih oher coury pairs where he modified es cao fid a liear coiegraio relaioship. There are a few oher cases where our modified RESET es foud some evidece for a liear relaioship such as U.S. U.K. wih CPI, U.S. Japa wih PPI, or U.S. Caada wih PPI. These fidigs look cosise wih Figure 2, bu i seems ha heir coiegraio relaioships are o as sable as he U.S. Mexico case. Table 4 shows a few addiioal ieresig resuls. Alhough may empirical sudies fid ha PPP works beer wih PPI ha CPI Froo ad Rogoff 995), here seems o be o sigifica differece bewee CPI ad PPI i our modified es resul, ad wo radiioal ess eve fid CPI is more supporive of liear specificaio ha PPI. Regardig he wo sample periods, wo radiioal ess show cosiderable differeces bewee hese wo periods while our modified es does o show such differeces. I he case of he relaive PPP, where our es becomes a lieariy es isead of a liear coiegraio es, our modified es does o fid ay sigifica liear relaioship while he origial RESET es foud ha all he relaioships are liear. Our resuls are also more cosise wih Figure 2 ha he origial es resuls. 7. CONCLUSION Usig some recely developed asympoic ools i Park ad Phillips 999, 200), his aricle preses how osaioariy combied wih olieariy ierferes wih he RESET es ad we aalyze he resulig severe size disorio ha makes he es usuiable for empirical applicaio. The appropriae modificaios o he RESET es are proposed o elimiae he biases ha cause hese size disorios ad he proposed modificaios are show o lead o a correced es saisic ha has a limiig ceral χ 2 disribuio. The proposed modified es saisic has good power agais boh oliear coiegraio ad o coiegraio aleraives so ha i ca be used o assess he adequacy of a liear coiegraig relaio agais cerai forms of oliear coiegraio ad he aleraive of o coiegraio. Some relaed work is i progress. Sice he power of he es depeds o he choice of basis fucios, we are developig a se of lieariy ess usig differe basis fucios. This seems paricularly appropriae whe we wa o allow for fucios whose behavior is poorly approximaed by polyomials, such as iegrable fucios ha aeuae he ifluece of iegraed regressors. A he same ime, here is scope for developig a lieariy es ha is o based direcly o a approximaig family, so ha he power ad he size of he es do o deped o so may choices, such as he basis fucios, he umber of basis fucios, ad a badwidh parameer. APPENDIX The followig proofs skech he mai seps i he argumes ad deails are provided i a earlier versio of he aricle Hog ad Phillips 2007). The proofs frequely use sadard limi heorems for oliearly rasformed iegraed processes. These are based o lemma 5 of Chag, Park, ad Phillips 200), uless specified oherwise. Proof of Lemma See de Jog 2002) or Ibragimov ad Phillips 2004). Proof of Theorem 2 The proof is he same as Theorem 3 excep ha he secodorder bias erms are o correced bu are colleced ogeher o form he oceraliy parameer. Proof of Theorem 3 The es saisic is a quadraic form i D F û ad wo bias correcio erms, wih he weigh marix ˆ uu.v D F FD ) as meric i he form. We prove his heorem i wo seps. Firs, codiioal o F x = σb x r), 0 r ), we show ha D F û becomes a zero mea Gaussia vecor afer bias correcios i he limi; ad secod, ha is variace marix is he limi of he weigh marix. The m )h eleme i D F û is m+)/2 = X m û m+)/2 X m u m+)/2 X u B m x db εc) + m vu = B x db u + vu ) B m x db u + m vu X 2 ) B m x B 2 x ) B m x vu X m+ B m+ x B 2 x ) B m+ x

15 Table 4. p-values of he modified ad origial RESET ess Absolue PPP Relaive PPP Modified RESET Origial RESET Modified RESET Origial RESET Badwidh Choice of k: ) Period : 97M 2004M2 A) PPP wih cosumer price idex U.S. Caada M = / M = 2/ Auo U.S. Japa M = / M = 2/ Auo U.S. Mexico M = / M = 2/ Auo U.S. U.K. M = / M = 2/ Auo ) Period 2: 983M 2004M2 U.S. Caada M = / M = 2/ Auo U.S. Japa M = / M = 2/ Auo U.S. Mexico M = / M = 2/ Auo U.S. U.K. M = / M = 2/ Auo Joural of Busiess & Ecoomic Saisics, Jauary 200

16 Absolue PPP Table 4. Coiued) Relaive PPP Modified RESET Origial RESET Modified RESET Origial RESET Badwidh Choice of k: B) PPP wih wholesale/producer price idex ) Period : 97M 2004M2 U.S. Caada M = / M = 2/ Auo U.S. Japa M = / M = 2/ Auo U.S. Mexico a U.S. U.K. M = / M = 2/ Auo ) Period 2: 983M 2004M2 U.S. Caada M = / M = 2/ Auo U.S. Japa M = / M = 2/ Auo U.S. Mexico M = / M = 2/ Auo U.S. U.K. M = / M = 2/ Auo NOTE: The modified RESET es resuls wih badwidhs M = /3 ad M = 2/3 ad auomaic badwidh are repored. The p-values from he origial RESET es wihou bias correcios are repored i he righ pael for compariso. Sice PPI series for Mexico is available oly afer 98, coiegraio is esed oly for he secod period. Hog ad Phillips: Tesig Lieariy i Coiegraig Relaios

17 2 Joural of Busiess & Ecoomic Saisics, Jauary 200 = B m x db u.x + uv vv B m x db Proof of Theorem 5 x + vu m B m x B 2 x ) ) B m+ x wih B m x = Bm x B x B 2 x ) B m+ x. Noe ha his limi is he sum of hree elemes a zero mea Gaussia mixure, he edogeeiy bias, ad he serial correlaio bias. Wih cosise esimaors of vv, vu, ad vu, 2 bias correcio erms 7) ad 8) coverge o he correspodig bias erms i he above limi, so ha he m )h eleme of he sample covariace {D F û E S } becomes m+)/2 X m u E m ) S m ) d B m x db u.x, ) Uder he aleraive specificaio of oliear coiegraio, he modified es saisic chages oly hrough û, ha is, hrough D F û, ˆ vu, ˆ uv ad ˆ uu.v. We will examie he chages i he saisic by checkig he orders of each of hese erms. Firs, if we esimae he followig misspecified liear regressio by LS Y = θx + u, where he rue relaioship is oliear 9), he coefficie esimae ˆθ ca be show o be eiher coverge o zero i he I-regular case assumig x k f x) is iegrable) or of he order of /2 κ for he H-regular case usig Chag, Park, ad Phillips 200). Wih affixes I ad H o desigae hese cases, we have ˆθ I) B 2 x ) o p ) + B x db u + vu ) θ I), which follows N 0, uu.x B m x B m x ), codiioal o Fx. For he weigh marix, he i, j) eleme of D F FD ) has he followig limi i+j+2)/2 X i+ X j+ d. 2) Bi+ x B j+ x From ) ad 2), i follows ha he modified RESET saisic is a quadraic form wih a limiig χ 2 disribuio as {û FD E S } ˆ uu.v D F FD ) {D F û E S } ) ) d a B x,k db u.x A B x,k db u.x χ 2 k), where B x,k db u.x =[ B2 x db u.x,..., Bk+ x db u.x ] ad a iverse covariace marix A as defied i 6) of Theorem 2.Noe ha he es saisic follows a ceral χ 2 k) ucodiioally. Proof of Theorem 4 Noe ha if u is I), he D F u = O p ) sice [ ) ] m X u d B m x B u. Also, sice ˆ vu ad ˆ vu are O p M) see Xiao ad Phillips 2002, lemma ), he wo bias correcio erms i 7) ad 8) diverge a he rae of O p M) as well. Therefore, he sample covariace, augmeed by he correcio erms, diverges a he rae of, ha is, D F û E S ) = O p ), ad he variace marix erm diverges a he rae of M, ha is, ˆ uu.v D F FD ) = O p M) from Xiao ad Phillips 2002, lemma ). Combiig hese wo, he modified RESET es saisic diverges a he rae of /M. ˆθ H) κ θ H), B 2 x ) HB x )B x + κ B x db u + ) vu κ Noe ha whe he rue model is a liear coiegraio we have HB x ) = θb x ad he ˆθ H) θ = O p ), as usual. We firs cosider he H-regular case. The m )h eleme of he ormalized oliear sample covariace D F û is O p κ ) sice κ { )m } X û { = ) m X [ f X ) + u ˆθ H) ] } X κ B m x hb x) + O p /2 κ ) θ H) B m+ x. 3) The orders of he wo bias correcio erms 7) ad 8) deped o he asympoic order of he kerel esimaors ˆ vu ad ˆ uv. Leig Kj/M) be he lag kerel, we may decompose each of hese esimaes as follows: ˆ uv = = M j= M M j= M + ){ j K M } û +j v ){ j K u+j v M f X+j )v ˆθ H) X+j v }. 4) The firs erm i braces i 4) iso p ) ad he oher wo are O p /2 κ ), so ha he overall maximum asympoic orders of

18 Hog ad Phillips: Tesig Lieariy i Coiegraig Relaios 3 ˆ uv ad ˆ vu are all /2 Mκ. Thus, combiig 4) ad 3) we fid ha D F û E S = O p /2 κ ) Op /2 Mκ ) Op /2 Mκ ) has order O p /2 κ ) sice M/ 0. For he variace erm ˆ uu.v D F FD, he order ow depeds o he order of ˆ uu.v, sice he remaiig facor is of order O p ) uder boh he ull ad he aleraive hypoheses. The kerel esimaor ˆ uu ca be show o be of he maximum order of Mκ 2 bu O p) as usual uder he ull hypohesis). I paricular, wih ˆ uu = = M j= M M j= M )[ j K M )[ j K M ] û û +j f X )f X +j ) + f X )u +j ˆθ H) f X )X +j + f X +j )u + u u +j ˆθ H) X +j u ˆθ H) f X +j )X ˆθ H) X u +j + ˆθ ] H)2 X X +j he maximum order of each erm i he square bracke ca be deermied as follows.. By virue of he Cauchy iequaliy, he maximum orders of followig erms are O p κ 2 ): f X )f X +j ), ˆθ H) f X )X +j, ˆθ H) f X+j )X. 2. The followig erms are all of he same order O p /2 κ ): 3. f X )u +j, f X+j )u, ˆθ H) X+j u, ˆθ H) X u +j. u u +j = O p ) ad ˆθ H)2 X X +j = O p κ 2) O p ) = O p κ 2). Combiig hese resuls, he modified RESET es saisic is a quadraic form i a vecor of O p /2 κ ) elemes wih a weigh marix of order O p M κ 2 ), so ha he overall order of he es saisic is a mos O p /M). For he I-regular case, we ca show ha he sample covariace does o diverge ) m X [ f X ) + u ˆθ I) ] X O p ) + B m x db u θ I) = O p ). j= M B m+ x The kerel esimaor of he log-ru co)variace is M )[ j ˆ uv = K u+j v M + f X+j )v ˆθ I) uv + O p M/ 3/4 ) O p M/) = O p max {, M/ 3/4 }), X+j v ] so ha he wo correcio erms E ad S do o diverge eiher, as log as M/ 3/4 0. Therefore, D F û E S = O p max {, M/ 3/4 }). The variace erm, as i he H-regular case, ca be show o have he followig order: ˆ uu uu +O p M/ 3/4 ) +O p M/) = O p max {, M/ 3/4 }), so ha wih ˆ uu.v = ˆ uu ˆ uv ˆ vv ˆ vu, he es saisic is O p ). ACKNOWLEDGMENTS We hak hree aoymous referees ad he edior for helpful commes ad suggesios. Mos of his work was doe while Hog was affiliaed wih Cocordia Uiversiy, Moreal, Caada. Hog ackowledges parial fiacial suppor from a Cocordia Uiversiy GRF gra ad Phillips ackowledges parial fiacial suppor from NSF Gras SES ad [Received Augus Revised February 2008.] REFERENCES Adrews, D. A. K. 99), Heeroskedasiciy ad Auocorrelaio Cosise Covariace Marix Esimaio, Ecoomerica, 59 3), [0-04] Baghesai, H. 99), Applicaio of he RESET Tes o he Origial Aderse Jorda Equaio, Joural of Macroecoomics, 3 ), [96] Berbe, R., ad Dijk, D. 999), Ui Roo Tess ad Asymmeric Adjusme: A Reassessme, Research Repor ET-9902/A, Ecoomeric Isiue. [96] Breiug, J. 200), Rak Tess for Noliear Coiegraio, Joural of Busiess ad Ecoomic Saisics, 9 3), [04] Chag, Y., Park, J., ad Phillip, P. C. B. 200), Noliear Ecoomeric Models Wih Coiegraed ad Deermiisically Tredig Regressors, Ecoomerics Joural, 4, 36. [97,09,2] DeBeedicis, L. F., ad Giles, D. E. A. 998), Diagosic Tesig i Ecoomerics: Variable Addiio, RESET ad Fourier Approximaios, i Hadbook of Applied Ecoomic Saisics, eds. A. Ullah ad D. E. A. Giles, New York: Marcel Dekker, pp [97] De Jog, R. M. 2002), Noliear Esimaors Wih Iegraed Regressors bu Wihou Exogeeiy, mimeo, Uiversiy of Michiga, Ecoomics Dep. [97,98,09]

19 4 Joural of Busiess & Ecoomic Saisics, Jauary 200 Dorbusch, R. 987), Purchasig Power Pariy, i The New Palgrave: A Dicioary of Ecoomics, eds. J. Easwell, M. Milgae, ad P. Newma, Lodo: MacMilla/New York: Socko Press, pp [04] Ermii, L., ad Grager, C. W. J. 993), Some Geeralizaios o he Algebra of I) Processes, Joural of Ecoomerics, 58, [02] Eubak, R. L., ad Har, J. D. 992), Tesig Goodess-of-Fi i Regressio via Order Selecio Crieria, The Aals of Saisics, 20 3), [97] Froo, K., ad Rogoff, K. 995), Perspecives o PPP ad Log-Ru Real Exchage Raes, i Hadbook of Ieraioal Ecoomics, eds. G. Grossma ad K. Rogoff, The Neherlads: Norh Hollad, pp [09] Grager, C. W. J. 995), Modellig Noliear Relaioships Bewee Exeded-Memory Variables, Ecoomerica, 63 2), [96] Grager, C. W. J., ad Hallma, J. 989), Noliear Trasformaios of Iegraed Time Series, Joural of Time Series Aalysis, 2, [96] Haug, A. 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