A Nonlinear Panel Unit Root Test under Cross Section Dependence

Transcription

1 A onlnear Panel Un Roo Tes under Cross Secon Dependence Maro Cerrao a,chrsan de Pere b, cholas Sarans c ovember 007 Absrac We propose a nonlnear heerogeneous panel un roo es for esng he null hypohess of un-roo processes agans he alernave ha allows a proporon of uns o be generaed by globally saonary ESTAR processes and a remanng non-zero proporon o be generaed by un roo processes. The proposed es s smple o apply and accommodaes cross secon dependence. Mone Carlo smulaons shows ha our es holds correc sze and under he hypohess ha daa are generaed by globally saonary ESTAR processes has a beer power han he recen lnear es proposed n Pesaran (005). An applcaon o a panel of blaeral real exchange raes wh he US Dollar from he 0 major OECD counres s provded. JEL Classfcaon: C, C5, C, C3; F3 Key Words: on-lnear panel un roo ess; Cross-secon dependence. ac Cenre for Inernaonal Capal Markes, London Meropolan Busness School, London Meropolan Unversy, UK. b Deparmen of Economcs, Unversy of Evry-Val-d'Essonne, France. Correspondng auhor: Maro Cerrao, Cenre for Inernaonal Capal Markes, London Meropolan Busness School, London Meropolan Unversy, 84 Moorgae, London ECM 6SQ, UK. Emal: m.cerrao@londonme.ac.uk. Phone:

2 . Inroducon There s now a large leraure on esng for un roos n economc and fnancal varables employng a varey of me seres and panel ess. The growh n ha area s manly due o emprcal applcaons on, for example, Purchasng Power Pary (PPP) and Growh (see Cerrao and Sarans, 007a, 007b; and Emerson and Kao, 006, amongs ohers). A weakness of he exsng unvarae and panel un roo ess s ha hey are based on he assumpon ha he underlyng varable follows a lnear process. However economc heory suggess ha many varables exhb nonlnear behavour. For example, a number of heorecal models n nernaonal macroeconomcs formalse he noon of nonlnear exchange rae behavour due o ransacon coss (e.g. Dumas, 99; Sercu e al, 995; O Connell, 998; Goswam e al, 00), whle ohers descrbe currency and fnancal crses as non-lnear processes (e.g. Jeanne and Masson, 000; Chang and Velasco, 00). In growh economcs, a number of heorecal models sugges ha economc growh s a nonlnear process wh he economy bouncng back and forh beween dfferen regmes (e.g. Zlbo, 995; Pereo, 999; Masuyama, 999; Galor e al, 000) 3. Theorecal models n fnance hghlgh heerogeneous expecaons (e.g. Brock and Hommes, 998; De Grauwe and Grmad, 005), heerogeney n nvesors objecves (e.g. Peers, 994), and herd behavour (e.g. Lux, 995) as some of he sources of nonlneary n asse prces. If economc and fnancal varables exhb nonlnear behavour, he sandard un roo ess ha are based on a lnear AR process wll have low power. Two recen papers, Solls e al (00) and Kapeanos e al (003), address hs ssue by developng formal un roo ess agans he alernave of nonlnear mean reverson. Boh papers examne he un roo hypohess agans he nonlnear STAR (smooh ranson auoregressve) alernave and show ha, under he null hypohess, he dsrbuon of he respecve ess s no normal. As a resul he wo papers employ Mone Carlo smulaons o oban crcal values. The man dfference beween he wo ess s ha Solls e al use a logsc ranson funcon (LSTAR) whle Kapeanos e al use an exponenal ranson funcon (ESTAR). For a revew of he varous un roo ess see, for example, Breung and Pesaran (007) and Cerrao and Sarans (007b). For emprcal sudes on nonlnear exchange rae models, see Mchael e al, (997), Sarans (999), Taylor e al (00), and Rapach and Wohar (003), among ohers. 3 A number of auhors have also underaken emprcal nvesgaons of nonlnear growh models; see, for example, Fasch and Lavezz (007), Lu and Sengos (999) and Durlauf and Johnson (995).

3 However boh hese nonlnear un roo ess are unvarae and, consequenly, wll sll suffer from low power n he case of small samples. In hs paper we exend he Kapeanos e al nonlnear un roo es o a panel conex n order o address he low power problem of unvarae ess. Snce heerogeneous cross-secon dependence ends o be mporan n mos emprcal applcaons, we employ he Pesaran (005) panel un roo framework ha enables us o accoun for heerogeneous cross-secon dependence n a novel way. Pesaran (005) shows ha he ndvdual CADF (Cross Augmened Dckey Fuller) and he panel sasc (CIPS) have non-normal dsrbuons, so her crcal values (for dfferen and T) are obaned by Mone Carlo smulaons. The panel un roo es proposed by Pesaran (005) dffers from oher ess such as Cho (00) and Hadr (000) n ha whle he laer all assume ha ndvdual me seres are ndependen, Pesaran (005) shows ha cross seconal dependence can be accouned for by augmenng he sandard DF regresson wh he cross secon averages of lagged levels and frs dfferences of he ndvdual seres. In hs paper we propose a novel nonlnear panel un roo es ha exends boh he unvarae nonlnear ess and he lnear panel un roo ess, hus fllng an mporan gap n he exsng leraure. Snce he panel nonlnear sasc has a non-normal dsrbuon, we use Mone Carlo smulaons o analyse he sze and power of he es under dfferen scenaros, and we calculae crcal values whch can be used n fuure applcaons of he es. We also llusrae he applcably of our es by applyng o a panel of blaeral real exchange raes. The res of he paper s organsed as follows. Secon specfes he nonlnear dynamc panel model wh cross-secon dependence. Secon 3 derves he ndvdual and panel nonlnear un roo ess, and hen uses sochasc smulaons o oban he dsrbuons of hese sascs and crcal values. Secon 4 analyses he sze and power of he panel nonlnear un roo es under alernave scenaros and compares he resuls o he performance of he lnear Pesaran (005) es. Secon 5 repors he resuls from an applcaon o real exchange raes, whle Secon 6 concludes.. A onlnear Dynamc Panel wh Cross-Secon Dependence Suppose he observaon y on he h cross-secon un a me s generaed accordng o he dynamc nonlnear heerogeneous panel ESTAR model below: y y u, =,, T, =,,, (), y, Z( ; y, d ) 3

4 where nal value, y 0, s gven, and he error erm, u, has he one-facor srucure: u f.. d.(0, ) () n whch f s he unobserved common effec, and s he ndvdual-specfc (dosyncrac) error. Followng he leraure on STAR models, he ranson funcon adoped here s of he exponenal form,.e., Z( ; y ) exp( y ) (3), d, d where we assume ha 0, and d s he delay parameer. To begn wh we assume ha y s a mean zero sochasc process. We dscuss processes wh nonzero mean and laer. To smplfy he model and followng he exsng leraure, he delay parameer d s se o be equal o one and ()-(3) are re-wren n frs dfference form as: y y [ exp( y )] f,,,, (4) where ( ). If y s assumed o follow a un roo process n he mddle regme, hen 0, 4 and equaon (4) can be re-wren as: y [ exp( y )] f,, (5) Usng (5), we are neresed n esng he hypoheses: Ho : 0 for all (5a) agans he possbly heerogeneous alernaves, Ha : 0 for,..., ; 0 for,,..., (5b) 4 I follows he pracce n he leraure (e.g. Balke and Fomby, 997, n he conex of TAR models and Mchael e al., 997 n he conex of ESTAR models). 4

5 Remark : The alernave hypohess above mples ha some uns are generaed by a saonary ESTAR model bu also allows a proporon of uns beng a un roo process. The followng assumpons are nroduced: Assumpon : / q as, wh 0 q under he alernave hypohess. 5 Assumpon : are ndependenly dsrbued for all mean, consan varance, and fne fourh order momen.,..., and,..., T, wh zero Assumpon 3: f s serally uncorrelaed wh zero mean, consan varance f, and fne fourh momen. (Whou loss of generaly f wll be se equal o uny.) Assumpon 4:, f, and are ndependenly dsrbued for all. Assumpon 5: Followng Pesaran (004a), we defne he weghs { } havng he followng properes: ( ) ; ; K for K ; ( ). Assumpon 6: Le. We suppose 0 for a fxed and for. j j 3. onlnear Un Roo Tess wh Serally Uncorrelaed Errors Assumpons and ogeher mply ha he compose error, u, s serally uncorrelaed. Ths resrcon wll be relaxed n Secon Indvdual CADF Tes Tesng he null hypohess (5a) drecly s no feasble, snce s no denfed under he null. 6 To overcome hs problem, we follow Luukkonen e al. (988), and derve below a - ype es sasc. Usng Taylor expanson on (5), under he null hypohess, he followng auxlary regresson s obaned: a b y f e 3,. (6) 5 As noed n Im, Pesaran and Shn (003) hs condon s necessary for he conssency of he panel un roo ess. 6 See for example Daves (987). 5

6 Lemma : If Assumpons ()-(6) are sasfed, hen he common facor f approxmaed by: can be f w 3 y (7) Proof: see Appendx. Therefore, follows ha Equaon (6) can be wren as he followng nonlnear crossseconally augmened DF (CADF) regresson: a b y 3 3, c d y e (8) The dea s, gven he framework above, o develop a un roo es n heerogeneous panel model based on Equaon (8). Exendng he dea n Kapeanos e al. (003), we sugges usng model (8) and -sasc on b, ha s denoed by bˆ L (, T), (9) s. e.( bˆ) where bˆ s he OLS esmae of b, and s.e.( bˆ) s assocaed sandard error. Denoe he suden sasc on he rao of b n Equaon (8) as: ' y, M L (, T) (0) / ( M ) ( y M y ' ', /, ) where y,,..., )', y ( y, y,..., y )', X,, y )', M he (, T,,0,, T (, projecon marx ono (X ), he orhogonal complemen of he span of X, ' (,,...,) _ j _ j, y, y j, j and. The crcal values of he CADF es can be 6

7 compued by sochasc smulaon for any fxed T > 3, and for gven dsrbuonal assumpons for he random varables (ε,f ). To accommodae sochasc processes wh nonzero means, we need he followng modfcaons. In he case where he daa has nonzero mean,.e., where x = μ+ y, we use he de-meaned daa y = x x, where x s he sample mean. In hs case he asympoc dsrbuon of he L sasc s bascally he same as (0), excep ha daa are replaced by he de-meaned daa. 7 Fgure dsplays he smulaed cumulave dsrbuon funcon of he ndvdual CADF sasc under he null hypohess usng 50,000 replcaons for = 00 and T = 500. For comparson he smulaed cumulave dsrbuon funcon of Pesaran CADF sasc s also provded. The seres y,=y,-+f+u,, for =,,,00, and = -50,-49,,,,, 500 were frs generaed from y,-50 = 0, wh f and u, as..d. (0,). Then 50,000 CADF regressons of Δy, on y 3,-, Δy, and y 3 -. Δy and y 3 - were compued over he sample =,,,500. Fgure plos he ordered values of he OLS -raos of y 3,- n hese regressons. o surprsngly he nonlnear CADF dsrbuon, as he Pesaran s CADF dsrbuon, s more skewed o he lef as compared o he sandard DF dsrbuon. Ths s clearly refleced n he crcal values of he dsrbuons summarzed n Table.Crcal values of he ndvdual nonlnear CADF dsrbuon for values of T and n he range of 0 o 00 are gven n Appendx. The nonlnear CADF dsrbuon, lke he Pesaran s CADF dsrbuon and he sandard DF dsrbuon, depars from normaly n wo mporan respecs: has a subsanally negave mean and s sandard devaon s less han uny, alhough no by a large amoun. The smulaed densy funcons of he sandardzed CADF, compued wh = 00, T = 500, and 50,000 replcaons are dsplayed n Fgure. The mean, sandard devaon, skewness and Kuross -3 coeffcens of he CADF and he Pesaran s CADF dsrbuons are repored n Table. They are que small, alhough sascally hghly sgnfcan. Snce cross-seconal dependence n panel daa s wdely known now o be a serous problem, n he nex secons we shall be usng model (6) o develop a un roo es o es for he null hypohess of un roo agans an ESTAR saonary alernave. 7 Smlarly, for he case wh nonzero mean and nonzero lnear rend,.e., where x = μ+ δ +y, we use he demeaned and de-rended daa y = x ˆ ˆ, where ˆand ˆare he OLS esmaors of μand δ. ow he assocaed asympoc dsrbuons are such ha W(r) s replaced by he de-meaned and de-rended sandard Brownan moon W (r). 7

8 3. Panel onlnear CADF Tes Followng Pesaran (005), we sugges usng he -sasc n Equaon (0) o consruc a panel un roo es by averagng he ndvdual es sascs: (, T) (, T) () L L Ths s a nonlnear cross-seconally augmened verson of he IPS es based (CIPS). The es sasc defned n Equaon () can also be exended o he case where seral correlaon s presen n he daa. In hs parcular case, one may nclude, n he model, lags of he lef hand sde varable afer usng an nformaon crera o selec he lag order. We smulaed he dsrbuon of CIPS seng = 00, T = 500, and usng 50,000 replcaons. The smulaed densy funcons of he CIPS and he Pesaran s CIPS Sascs are dsplayed n Fgure 3. Boh he denses show marked deparures from normaly. The densy shows a grea degree of deparure from normaly. The skewness and Kuross -3 coeffcens of he CIPS and he Pesaran s CIPS dsrbuons are repored n Table 4. The crcal values of he nonlnear CIPS es are gven n Appendx. 3.3 The Serally Correlaed Errors Case Seral correlaon can be ncorporaed n he model n a varey of dfferen ways. In wha follows, we use he model n Equaon (4) and specfy he seral correlaon srucure as: u u, () We frs model seral correlaon as above and hereafer cross secon dependence as (3) f Usng Equaon (6) jonly wh () above we oban: a ( ) b ( ) y (4) 3,, 8

9 And subsung (3) no (4) a ( ) b ( ) y f (5) 3,, Fnally by mposng he un roo null on Equaon (5): a ( ) f (6), Usng Equaon (6) and he same approach as n Appendx, one can oban proxes for f. We sugges n hs case usng he followng non-lnear CADF regresson: a b y 3 p p 3, c y, d j j j, j e (4) (7) j 0 j Informaon crera can be used o choose he lengh of p. 4. Small Sample Analyss In hs secon we assess he sze of he nonlnear panel es defned n Equaon () under dfferen scenaros. Frsly, we look a power of he es n he case of weak and srong cross seconal dependence bu no movng average srucure for he error erm. In he nex secon, we generalse hs scenaro by allowng a movng average specfcaon for he error erm and weak-srong cross seconal dependence. For comparson, n all he above expermen we also repor he sze of he Pesaran (005) es when a nonlnear DGP s consdered. The daa generang process (DGP) consdered s he followng Panel ESTAR: y [ exp( y )] f,, (8) wh =,,, ; =-5, -50,,,,, T; f ~.. d. (0,) ; ~.. d. (0, ) ; ~.. d. U[0.5,.5]. We consder wo scenaros for cross seconal dependence, namely low cross seconal dependence ~.. d. U[0,0.0], and hgh cross seconal dependence ~.. d. U[,3]. 9

10 4. Sze Dsoron Analyss In our sze analyss below, we generae daa by seng 0 for all. Sze s compued a he 5% nomnal sgnfcance level. The number of replcaons s se o 5,000. The sandard error of he compued sze s Resuls for he sze are repored n Table 4 below. The es seems o have an accepable sze for large cross secon dmenson and somehow slghly underszed wh respec o he Pesaran (005) es. 4. Power Analyss In hs secon we assess he power of he es defned n Equaon () under he same DGP as above bu we consder he cases of weak and srong alernaves, namely we assume for he weak alernave: 0 for,,..., / 0. 0 for /,...,, (9a) whle for he srong alernave: 0 for,,..., / for /,...,. (9b) The power s compued a he 5% nomnal sgnfcance level, and resuls are repored n Tables 5 and 6. The es we propose seems o have sronger power han he Pesaran (005) es when he rue DGP s nonlnear. 4.3 Seral Correlaed Errors Case In hs secon we analyze sze and power of he proposed es when seral correlaon s ncorporaed no he DGP. We consder posve seral correlaon. The errors ε were generaed as: ε = ρ ε,- + ζ, ζ ~..d.(0,σ ), σ ~..d.u[0.5;.5], ρ ~..d.u[0.; 0.4] n he case of posve correlaon, ρ ~..d.u[-0.4; -0.] n he case of negave correlaon. (0a) (0b) (0c) (0d) (0e) 0

11 We only consder here for he power analyss he case where 0 for,,..., /, for /,...,, (a) and hgh cross-seconal dependence: ~.. d. U[,3]. (b) The sze and power are compued a 5% nomnal sgnfcance level and are based on he followng non-lnear CADF regresson: a b y 3 3, c y d,0 d,,, e (),,..., ;,,..., T, y y. The es s compued as: (, T) (, T ) (3) L L L where (, T ) s he OLS -rao of b n he above non-lnear ADF regresson. The number of smulaon s se equal o 5,000. Table 7 below shows he resuls. Boh ess have a good sze wh he Pesaran (005) beng conssenly overszed. In Table 80 we show resuls on he power of he es n he case when posve as well as negave seral correlaon s presen n he DGP. For panels of a moderae sze, he gan n power from usng he non-lnear panel un roo es wh respec o he Pesaran (005) es s evden.

12 5. An Emprcal Applcaon: Real Exchange Raes In hs secon we apply our es o real exchange raes agans he US dollar for weny OECD counres over he perod 973Q-998Q. The daa se s he same used by Murray and Papell (00, 004). Snce he long-run Purchasng Power Pary (PPP) relaonshp s one of he man componens of heorecal nernaonal macroeconomc models, a large number of sudes have esed hs relaonshp by applyng un roo ess o real exchange raes. Mos of hese sudes show evdence of un roo behavour n real exchange raes, whch has become a puzzle n nernaonal fnance. The growng leraure on nonlnear exchange raes argues ha ransacon coss and frcons n fnancal markes may lead o nonlnear convergence n real exchange raes. Consequenly, he non-mean reverson repored by lnear un roo ess may be due o he fac hese ess are based on a ms-specfed sochasc process. The ndvdual sascs for our un roo es are shown n Table 9. For comparson purposes, we also repor he sascs for he Pesaran (005) es whch accouns for cross secon dependence bu no for nonlneary. The Pesaran (005) es rejecs he un roo null hypohess n only ou of 0 cases a all levels of sgnfcance. By conras, he nonlnear es rejecs he null n cases a he % sgnfcance level, and n 5 cases a he 5% and 0% level. Hence our es rejecs he un roo null more frequenly and herefore yelds sronger suppor for he long-run PPP. As we argued above, unvarae ess have low power and hs problem s overcome by employng panel un roo ess. The resuls for our panel un roo es and he Pesaran panel un roo es are shown n Table 0. The conras beween he wo panel sascs s raher srong. The Pesaran (005) es fals o rejec he un roo null a all levels of sgnfcance, hus mplyng non-mean reverson n he whole panel of real exchange raes. On he oher hand, our nonlnear panel es rejecs he un roo null for he panel of real exchange raes a all levels of sgnfcance, gvng suppor o he long-run PPP for he whole panel of OECD counres. Ths evdence of nonlnear mean reverson n he OECD real exchange raes may sugges ha prevous evdence of non-mean reverson n real exchange raes s due o usng lnear un roo ess.

13 6. Concluson A number of panel un roo ess allowng for cross-secon dependence have been proposed n he leraure. In hs paper we propose a nonlnear heerogeneous panel un roo es for esng he null hypohess of un-roo processes agans he alernave ha allows a proporon of uns o be generaed by globally saonary ESTAR processes and a remanng non-zero proporon o be generaed by un roo processes. The proposed es s smple o apply and accommodaes boh nonlneary and cross seconal dependence. Our es s compared o Pesaran s (005) lnear es va Mone Carlo smulaon exercses, and s found ha our es holds correc sze and under he hypohess ha daa are generaed by globally saonary ESTAR processes has a beer power han he Pesaran es. We also calculae crcal values for varyng cross secon and me dmensons whch can be used n fuure applcaons of our es. We provde an applcaon o a panel of blaeral real exchange rae seres wh he US dollar from he 0 major OECD counres. In conras o he evdence obaned by lnear ess, we fnd evdence of nonlnear mean-reverson n he real exchange raes for he whole OECD panel ha gves suppor o he long-run PPP hypohess. Gven he mporance of he PPP n nernaonal macroeconomc models, our evdence suggess ha he employmen of nonlnear panel un roo ess may provde a soluon o he PPP puzzle. Gven he growng leraure of nonlnear models, we beleve ha he developmen of panel nonlnear un roo ess has large poenal n macroeconomc and fnancal applcaons. Evdence ndcaes ha dfferen me seres may follow dfferen nonlnear specfcaons. Consequenly, one could consder un roo ess wh dfferen ypes of ranson funcons ha allow for asymmerc dynamc adjusmen. Anoher exenson would be o allow for dfferen ranson varables. Furher applcaons of our ess and heorecal exensons are lef for fuure work. 3

14 Appendx Proof of Lemma We assume ha he error erm auo-covarance gven by u u n (8) follows a saonary process, for all, wh summable a l, l l0, wh, beng a zero mean random varable wh varance marx defned by Il and fne fourh order momen. The varance of u s fne and gven by: l0 Var( u ) a l. (4) Frs noe ha, afer usng cross seconal averages, Equaon (8) can be wren as: 3 f u (5) Wh ; y 3 3 y, ; ; and u a. Assumng ha 0, hen f can be approxmaed as follows: f 3 u y, (6) 4

15 5 And snce 0 ) ( ) ( a u Var and ) ( ) ( ) ( u Var, I follows ha as,. 0 ) ( u E consequenly he facor f can now be approxmaed by: 3, y y f (7)

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19 Fgure : Cumulave Dsrbuon Funcon of Pesaran s Cross-Seconally Augmened DF, and nonlnear Cross-Seconally Augmened DF Sascs Fgure : Smulaed Densy Funcon of he Sandardzed CADF and he Sandardzed Pesaran s CADF Dsrbuons as Compared o he ormal Densy 9

20 Fgure 3: Smulaed Densy Funcon of he CIPS Sasc and he Pesaran s CIPS Dsrbuons 0

21 Table : Crcal Values of he DF, Pesaran s CADF, and nonlnear CADF Dsrbuons (=00,T=500, 50,000 replcaons) Level DF CADF CADF % % % % Table : Momens of he CADF Dsrbuons Pesaran s CADF CADF Mean Sandard devaon Skewness Kuross Table 3: Momens of he CIPS dsrbuons Pesaran s CIPS CIPS Mean Sandard devaon Skewness Kuross

22 Table 4: Sze of onlnear Cross-Seconally Augmened Panel Un Roo Tess o Seral Correlaon, Low and Hgh Cross Secon Dependence Case Low Cross Secon Dependence /T Tes CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS Hgh Cross Secon Dependence /T Tes CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS

23 Table 5: Power of Cross-Seconally Augmened onlnear Panel Un Roo Tess o Seral Correlaon, Low and Hgh Cross Secon Dependence Case Weak Alernave Low Cross Secon Dependence /T Tes CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS Hgh Cross Secon Dependence /T Tes CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS

24 Table 6: Power of Cross-Seconally Augmened onlnear Panel Un Roo Tess o Seral Correlaon, Low and Hgh Cross Secon Dependence Case Srong Alernave Low Cross Secon Dependence /T Tes CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS Hgh Cross Secon Dependence /T Tes CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS

25 Table 7: Sze of Cross-Seconally Augmened onlnear Panel Un Roo Tess Srong alernave, Hgh Cross Secon Dependence Case Posve Seral Correlaon /T CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS egave Seral Correlaon /T CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS

26 Table 8: Power of Cross-Seconally Augmened onlnear Panel Un Roo Tess Srong alernave, Hgh Cross Secon Dependence Case Posve Seral Correlaon /T CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS egave Seral Correlaon /T CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS CIPS

27 Table 9: Indvdual Un Roo Tess for Real Dollar Exchange Raes Counry Lag Cerrao e al (CADF) Pesaran (CADF) Ausrala Ausra Belgum Canada Denmark Fnland France Germany Greece Ireland Ialy Japan eherlands Zealand orway Porugal Span Sweden Swzerland UK Crcal Values (=0, T=00): % 5% 0% Rejecon Raes of he Panel Un Roo Tess Cerrao e al(007) Pesaran (005) H0 H H0 H % 90% 0% 95% 5% 5% 75% 5% 95% 5% 0% 75% 5% 95% 5% 7

28 Table 0: Panel Un Roo Tess Cerrao e al (CIPS) Pesaran (CIPS) Crcal Values (=0, T=00): % 5% 0%

29 Appendx Crcal values A: Crcal Values of Indvdual CADF Dsrbuon T %.5 % 5 % 0 % T %.5 % 5 % 0 %

30 B: Crcal Values of Average of Indvdual onlnear Cross-Seconally Augmened Dckey-Fuller Dsrbuon T %.5% 5% 0% T %.5% 5% 0%