Analysis of Panel Vector Error Correction Models Using Maximum Likelihood, the Bootstrap, and Canonical-Correlation Estimators.

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1 Anderson, Qan and Rasche Februar 5, 006 DRAFT PLEASE DO NOT CITE OR QUOTE WITHOUT PERMISSION Analss of Panel Vecor Error Correcon Models Usng Maxmum Lkelhood, he Boosra, and Canoncal-Correlaon Esmaors Rchard Anderson* Halong Qan** Rober Rasche*** Ths Verson: Februar 5, 006 Abrac In hs sud, we examne he use of he Box-Tao canoncal correlaon sasc as an alernave o boh lkelhood rao nference and error correcon-model resdual-based conegraon nference n mulvarae models. I s well-known ha he Johansen MLE, whle havng erhas he bes sascal roeres among lkelhood-based ess, has small-samle dsrbuons ha dffer sharl from her asmoc counerars. Furher, he dsrbuons of economc and fnancal me seres end o dsla fa als, heeroskedasc and skewness nconssen wh dsrbuonal assumons of lkelhood-based ess. The nonaramerc Box-Tao es shows romse as an addonal es for he resence of conegraon and ess of s rank n mulvarae ssems. * Economs and vce resden, Research Dearmen, Federal Reserve Bank of S. Lous. Conac: Anderson@sls.frb.org **Assocae rofessor, Dearmen of Economcs, San Lous Unvers, and Research Fellow, Research Dearmen, Federal Reserve Bank of S. Lous. Conac: qan@slu.edu *** Senor vce resden and drecor of research, Research Dearmen, Federal Reserve Bank of S. Lous. Conac: rasche@sls.frb.org Vews exressed heren are solel he auhors, and are no necessarl hose of he Federal Reserve Bank of S. Lous, he Board of Governors of he Federal Reserve Ssem, nor her saffs. We hank man colleagues for commens bu hold none resonsble for an foolsh asserons, mehods or conclusons n hs analss.

2 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006. Inroducon Inference regardng he number of long-run equlbrum relaonshs (ha s, he conegraon rank) among a se of economc, fnancal or socal varables s mos-ofen based on maxmum lkelhood (ML) esmaon and relaed asmoc dsrbuons, erhas wh a small-samle Barle correcon. Among, lkelhood rao ess, hs has been shown o have he bes sascal roeres; see for examle, Johansen (995, 000, 00), Phls (995), Sock and Wason (988). However, s now well-known va smulaon sudes ha he asmoc dsrbuons of ML-based esng sascs for conegraon ranks are no good aroxmaons o he rue dsrbuons of he esng sascs when he samle sze s small o moderae; see for examle, Toda (995), Jacobson (995), and Haug (996, 00). Because Johansen s maxmum lkelhood aroach s based on he assumon ha he rue daa generang rocess (DGP) s ndeenden and dencall dsrbued (..d.) mulvarae normal, s of neres o exlore alernave rocedures for esng conegraon ranks and esmaon of vecor error-correcon (VEC) models ha are robus o dearures from hese assumons. I s also well known, for examle, ha he dsrbuons of economc and fnancal daa ofen fa als, heeroscedasc, and skewness. A sgnfcan leraure has arsen focused on resdual-based ess of conegraon, boh n unvarae and anel models. In hese sudes, nference regardng conegraon s conduced va resdual-based ess va a mang beween he f of a resdual-based regresson and he resence of conegraon. The nuon n hs sud s smlar, as we exlan below. In he resence of nonsaonar, lagged values of a me seres should have redcve ower for fuure values; n he resence of saonar, he wll no. Accurae nference regardng he conegrang rank n mulvarae models s moran. If he CI rank s ncorrecl nferred due o large sze dsorons and/or low ower of ML-based CI rank es sascs, he long-run coeffcen marx of a vecor errorcorrecon model (VECM) s mssecfed. Ths n urn resuls n an ncorrec esmaon of he number of common sochasc rends of he ssem and subsequenl causes Johansen (000, 00) furnsh addonal references. McCoske and Kao (00), Weserlund (005), and references heren.

3 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 erroneous esmaes of shor-run coeffcens. Ths mssecfcaon of CI rank wll have serous consequences for emrcal alcaons, esecall for alcaons o macroeconomc models ha rescrbe olc recommendaons. To ncrease ower and reduce small-samle sze dsorons, nonsaonar anel daa models have recenl become ver oular; see, for examle, Pedron (996, 004), McCoske and Kao ( 00), Kao (999), Banerjee (999) Banerjee e al. (004), and Kao and Chang (000). However, he curren leraure on anel conegraon ess and esmaon usuall assumes ha he number of cross-seconal uns s large and does no allow for () cross-seconal deendence n he error erms, () he neracon of shor-run dnamcs beween cross-secons, () he dfference n conegraon ranks across crosssecons, or (v) he ossbl ha long-run equlbrum relaonshs exs beween dfferen cross-secons (hereafer referred o as beween-conegraon). If an of hese four ossbles holds, he conclusons drawn from he exsng anel conegraon leraure wll be lkel msleadng and erroneous. 3 To be more secfc abou he weakness of he exsng anel conegraon leraure, le us consder he oular examle of esng for urchasng ower ar (PPP) among G-7 economes. For hs secal examle, () f G-7 economes are affeced b he same nernaonal economc, fnancal and olcal shocks, hen we exec ha he error erms of dfferen regresson equaons are conemoraneousl correlaed. () If emorar changes n radng arners domesc rces and exchange raes also affec each oher n he shor-run, hen a anel vecor error-correcon model should also allow for he neracon of shor-run dnamcs across cross-secons. () Snce dfferen counres ado dfferen monear and fscal olces, s also ver naural o allow for dfferen number of conegraon (or long-run equlbrum) relaonshs among he varables for a gven counr. Fnall, (v) f we use U.S. as he reference counr, hen n he regressons of he logarhm of a domesc rce on exchange rae (measured as he domesc rce er US dollar) and he logarhm of US rce, US rce aears n ever regresson and s obvousl (rvall) conegraed wh self across dfferen cross-seconal regressons. In 3 A racce recenl nroduced n he resdual-based esng leraure s he removal of common me effecs as a subsue for allowng conegraon across anel uns. Essenall, hs enals subsracng from each seres a mean value calculaed across all anels members a each dae. See Weserlund (005) for examle.

4 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 fac, when he nomnal U.S. rce s negraed of order one (.e. I()), acs lke a common sochasc rend n he anel regressons. O Connel (999) showed hrough smulaons ha gnorng he cross-seconal deendence n he error erms can cause large sze dsorons and sgnfcan loss of ower for exsng anel un roo ess. However, no aer ublshed so far has examned he effecs on he sze and ower of anel conegraon ess when here s cross-seconal neracon of shor-run dnamcs, crossseconal dfferences n conegraon rank, or cross-seconal conegraon. The curren aer has hree man objecves. Frs, we seek o relax hose resrcve assumons ha are rounel made n he exsng anel conegraon leraure. Secfcall, we wll exlcl allow for cross-seconal deendence among shocks (.e. model dsurbances), cross-seconal neracons n shor-run dnamcs, dfferences n conegraon rank across cross-seconal uns, as well as he exsence of long-run equlbrum relaonshs beween dfferen cross-secons. Second, n order o relax he dsrbuonal assumons of Johansen s (995) ML-based aroach, we roose usng Box and Tao s (977) canoncal correlaon (CC) analss o es for conegraon rank and esmae conegraon vecors. Box and Tao s CC-based nferences and esmaors of long-run arameers do no requre an dsrbuonal assumons of he daa generang rocess and are found o have beer dsrbuonal roeres; see Bewle, Orden, Yang and Fsher (994) and Bewle and Yang (995). Thrd, snce we do no make an dsrbuonal assumons of he rue DGP (exce he usual regular condons abou he exsence of relevan momens) nor do we assume ha he samle avalable s large, we use a boosra mehod o fnd he daa-deenden and emrcall correc fne-samle dsrbuons for our conegraon rank ess and arameer esmaors. 4 The aer s organzed as follows. In he nex secon, we descrbe he resrced anel VEC models commonl used b he curren leraure on anel conegraon and hen resen he unresrced anel VEC model consdered b he curren aer (essenall, ha of Larsson and Lhagen, 999). In Secon 3, we movae he value of an unresrced anel VEC model secfcaon va Mone Carlo smulaons of he sze and ower roeres of a resdual-based anel conegraon es sasc. Secon 4 nroduces 4 E.g. Goncalves and Whe (004). 3

5 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 he Box and Tao s (977) canoncal correlaon sasc and s exenson o esng for conegraon rank. Secon 5 rovdes an emrcal alcaon usng a anel VEC model for he deermnaon of M veloces n U.S. and Canada. We conclude n he las secon wh some commens.. Panel Vecor Error-Correcon Models Le = (,,, )' be a vecor of neres for cross-secon n erod. Suose ha follows a nonsaonar VAR(k) rocess: (A) = δ d k + Φ j= j, j + ε, =,,..., T; =,,..., N, where d s a vecor of deermnsc comonens; ha s, d = or (, ), δ s a or marx of arameers. Thus δ d s a vecor wh he j-h elemen equal o δ j or δ + reresenng he deermnsc comonen of he model. In hs aer, we j δj assume ha he number of cross-secons (N) s fxed and he number of me erods (T) s relavel large. Gven (A), we can also equvalenl reresen as a VECM: (B) Δ = δ d + Π k, + ΓjΔ, j + ε, =,,..., T; =,,..., N, j= where k Γj = Φ s for j=,,..., (k-) and Π = ( I m Φ j). s= j+ k j= Now, defne: (A) Γ = Γ, Γ,, Γ ) (,k (B) = ( Δ ', Δ ',, Δ ')'. X,,, (k ) Then, (B) can be rewren as: 4

6 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, (3), X d ε + + Γ + Π = δ Δ. For a gven, model (3) can be sacked over cross-secons o oban: (4A) Π Π Π + δ δ δ = Δ Δ Δ N,,, N N N d ε ε ε + Γ Γ Γ + N N N X X X, or more comacl, (4B) X d ε + + Γ + Π = δ Δ, for =,,..., T, where (5A) = N, Δ Δ Δ = Δ N, = N X X X X, ε ε ε = ε N (5B) )' ', ', ', ( N δ δ δ = δ (5C) Π Π Π Π = N (5D) Γ Γ Γ Γ = N. (4B) s he usual form of VEC models, wh coeffcen marces resrced b (5C) and (5D).

7 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 We now make he followng assumon: (A.) ε s d, wh mean equal o a zero vecor and covarance marx equal o (6) Ω Ω = Ω N Ω Ω N NN s a N N osve defne marx, wh Ω j var( ε ). Here we arcularl noce ha he covarance marx n (6) allows for arbrar crossseconal deendence across cross-secons, whch s a sgnfcan relaxaon of he crossseconal ndeendence assumon made b almos all of he curren nonsaonar anel daa leraure. Now, suose ha he long-run coeffcen marx, rank decomoson: Π, has he followng reduced (7) Π = α β ', where α and β are of dmenson, wh r = rank( Π ). Here we noe ha we r < allow he conegraon rank o be dfferen among cross-secons, whch s also an exenson of he exsng anel conegraon leraure, snce he curren leraure on anel conegraon alwas assumes ha dfferen cross-secons have he same conegraon rank: as: r = r for all. Gven (7), we can facor he long-run coeffcen marx Π of (4B) (8) Π = αβ', where 6

8 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 (9) α α = α α N, β β = β β N. Then, model (4B) above can be exressed as a famlar anel VEC model: (0) Δ = δd + αβ' + ΓX + ε, for =,,..., T. Ths s he model call secfed b almos all of he aers n he relavel new leraure on anel VEC models; see for examle, Groen and Klebergen (000) and Larsson and Lhagen (999). In hs secfcaon: () shor-run dnamcs are assumed o be unrelaed beween cross-secons; ha s, he marx Γ s assumed o be block dagonal, as gven n (5D). () There are no long-run equlbrum relaonshs amongcross-secons; n oher words, cross-seconal conegraon s no ermed snce β s resrced o be block dagonal, as gven n (9) above. () The conegraon ranks are assumed o be he same for all cross-secons. And (v) Temorar devaon from long-run equlbrum n one cross-secon s no allowed o nfluence he oher members of he anel; ha s, he adjusmen marx α s assumed o be block dagonal, as gven n (9) above. We beleve ha hese four assumons are unrealsc and ver resrcve. Thus, n hs aer, we seek o relax hese resrcve assumons. More recsel, we wll allow he shor-run dnamc marx ( Γ ), he adjusmen marx ( α ) and he conegraon marx (β ) unresrced, as follows. (A) Γ Γ Γ = Γ N Γ Γ Γ N Γ Γ Γ N N NN 7

9 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 (B) α α α = α N α α α N α α α N N NN, β β β = β N β β β N β β β N N NN, where boh marces α and β are of dmenson N r, wh r r + r + + r N. N < Under hs secfcaon of unresrced marces Γ, α and β, he anel VECM (0) allows: () neracons of shor-run dnamcs beween cross-secons, () nfluence of one cross-secon s emorar long-run equlbrum error on oher members of he anel, () he dfference n conegraon ranks across cross-secons, and (v) cross-seconal conegraon. Usng secfcaon (0)-(), we can also es wheher he convenonal block-dagonal resrcons on he shor-run coeffcen marx ( Γ ), he conegrang marx (β ) and he adjusmen marx ( α ) are vald once we have esmaed he unresrced marces Γ, β and α. 3. Mone Carlo Smulaons To movae our unresrced anel VECM secfcaon, we conduced Mone Carlo smulaons o examne he effec of cross-seconal correlaon and/or cross-seconal conegraon on he sze and ower of a anel conegraon es. For smlc, we use he resdual-based anel KPSS es for conegraon, whch s a drec exenson of he resdual-based unvarae KPSS es for conegraon; see, for examle, Shn (994). 5 More secfcall, he anel KPSS esng sasc for he null of conegraon s calculaed b: N T S LM =, N T = = σˆ ε 5 Erksson (004) emhaszes ha Mone Carlo exermens ha comare ML esmaors o a KPSS-sle ess are more reasonable han he more-common exermens based on Dcke-Fuller or Phlls-Perron ess. Boh ML (lkelhood rao) and KPSS ess have null hoheses of conegraon, whle Dcke-Fullere ess have a null hohess of no conegraon, makng suble an conclusons drawn from he exermens usng DF or PP ess. 8

10 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 N T where S = εˆ j and σˆ ε = εˆ, wh ˆε beng he esmaed resdual of crosssecon a me. j= NT = = The smulaon desgn for he daa generang rocesses (DGPs) s as follows: = α + x β + z γ + ε α ~ U[0, 0], β ~ U[0, ], γ ~ U[0, ] x x + v =,, v ~ IN(0, σ ), σ ~ U[0.5,.5] z = z + e, e ~ IN(0, λ ), λ ~ U[0.5,.5] ε = θ u + u k= k ( u,..., u N ) = ( η,..., ηn ) L η ~ IN(0, δ ), δ ~ U[0.5,.5], where α s he fxed effec for cross-secon, x s he I() regressor of cross-secon ha vares over cross-secons, z s he common sochasc rend across cross-secons ha caures he cross-seconal conegraon among he regressors of dfferen crosssecons. The arameer θ conrols he degree of nonsaonar n he regresson error erms, and he lower rangular marx L ( N N ) conrols he cross-seconal correlaon. The arameer values used n he smulaons are: Samle Sze: T={50, 00} No. of cross-secons: N={, 5, 0} θ {0.00, 0.05, 0.0, 0.5, 0.0}. McCoske and Kao (00) use a smlar smulaon desgn, hough he do no consder he effecs on KPSS ess of cross-seconal correlaon or cross-seconal conegraon. When θ = 0, hen ε = u, whch s saonar. Thus, θ = 0 mles a conegraon relaonsh beween, x and z. Also, f γ 0 for all, here s no cross-seconal conegraon beween he regressors of dfferen cross-secons; so β 0 and γ = 0 (for all ) corresonds o he case of whn conegraon onl. 6 = 6 Banerjee, Marcellno and Osba (004) consder smlar arameerzaons. 9

11 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 We consder boh he dnamc OLS (DOLS) and he dnamc seemngl unrelaed regressons (DSUR) esmaon mehods n our smulaons. Noe: DSUR s asmocall equvalen o MLE f all he members of he anel are conegraed; see Mark, Ogak and Sul (005). For DOLS, he cases consdered are: Case : wh no cross-seconal correlaon or cross-seconal conegraon; ha s: L =, γ 0 for all. I N = Case : wh cross-seconal correlaon bu wh no cross-seconal conegraon; ha s: L s a lower rangular marx, and γ = 0 for all. Case 3: wh no cross-seconal correlaon bu wh cross-seconal conegraon; ha s: L =, γ 0 for some of he. I N Case 4: wh boh cross-seconal correlaon and cross-seconal conegraon; ha s: L s a lower rangular marx, and γ 0 for some of he. For DSUR, he cases consdered are: Case : wh cross-seconal correlaon bu wh no cross-seconal conegraon, Case : wh no cross-seconal correlaon bu wh cross-seconal conegraon, Case 3: wh boh cross-seconal correlaon and cross-seconal conegraon. Our smulaons are conduced n GAUSS 3.6 and he number of relcaons used (.e. R) s 5,000. The smulaon resuls are reored n Tables A-4B. A bref summar of our man fndngs from he smulaons s as follows. A. Tables A-B ndcae ha: () When here s cross-seconal correlaon, he anel KPSS esng sasc (.e. LM-bar) s severel over-szed; ha s, over-rejecs anel conegraon. On he oher hand, he KPSS sasc aled searael o each cross-secon has he roer sze. () When here s cross-seconal conegraon, he LM-bar and he ndvdual LM sascs are all severel under-szed; ha s, he over-acce anel conegraon. 0

12 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 () When here are boh cross-seconal correlaon and conegraon, he LM-bar and he ndvdual LM sascs connue o be severel under-szed bu he degree of sze dsoron s less han n case 3. B. Tables A-B ndcae ha: Boh cross-seconal correlaon (case ) and cross-seconal conegraon (case 3) cause severe loss of ower. However, case 3 s much more severe han case, esecall when he error erm s nearl saonar (ha s, when he value of hea s low). C. Tables 3A-4B ndcae ha: The huge sze dsoron and severe ower loss of he LM sascs based on DOLS ha we found n Tables A-B are exacerbaed for dnamc GLS esmaor (or MLE) because of he cross-equaon conamnaon. In summar, cross-seconal conegraon causes more severe sze dsoron and ower loss for he ooled (.e. anel) and ndvdual LM-sascs han does cross-seconal correlaon. Our fndng warns raconers of he lmaons of he exsng anel conegraon esng rocedures, snce almos all of hese rocedures neglec he ossbl of long-run equlbrum relaonshs among cross-secons. 7 For examle, on he ongong debae abou wheher PPP holds or no, raconers usuall do no a an aenon o he obvous cross-seconal conegraon: he same rce level of he reference counr aears n all equaons of he anel. Fnall, a word on our smulaon desgn, our smulaon s conduced for sngle equaons no for a VAR or a VECM. Ths s manl for smlc n desgnng he DGPs and rogrammng he relevan calculaons. However, we beleve ha our fndngs based on hs relavel smle DGP desgn wll carr over o VAR or VEC models, snce exsng esmaon and ess for anel conegraon almos alwas neglec cross-seconal deendence and/or long-run cross-seconal equlbrum relaonshs. Our smulaon resuls ndcae ha when here s cross-seconal correlaon or cross-seconal conegraon, exsng anel conegraon ess have large sze dsoron and low ower, whle he esmaes for long-run arameers ma be nconssen f crossseconal conegraon s negleced. Now he challengng queson s how o fnd a vald esmaon and esng rocedure for anel conegraon models ha have cross-seconal 7 Smlar resuls are obaned and warnngs made b Banerjee, Marcellno and Osba (004).

13 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 correlaon and/or cross-seconal conegraon. Ths s he man movaon for our unresrced anel VECM esmaon and esng aroach o be defned n he nex secon. 4. Canoncal Correlaon Analss Box and Tao (977) consder he redcabl of lnear combnaons of a mulvarae me seres from he hsor of he lnear combnaons concerned. More secfcall, suose ha we have a random samle of T observaons on a dmensonal me-seres = (,,..., )', whch has mean zero and varance equal o Σ 00. Box and Tao consder he redcabl of z c' based on he as values of z, where c s a vecor of consans. Then, f he lnear combnaon z s saonar and ndeendenl dsrbued (for examle, a whe nose rocess), he as values of z are no nformave for forecasng he curren value of z. On he oher hand, f z s ver erssen over me (for examle, z s nonsaonar or has long memor) hen he as values of z can forecas he curren value of z ver well. Precsel he same movaon/ nuon underles he recen error-correcon, resdual-based conegraon ess of Weserlund (005, 006) and Weserlund and Edgeron (005). More recsel, defne he lnear rojecon of on s own hsor as () ŷ = E(,, ) = Γ, k = where he rojecon coeffcen marces Γ s are. Then, we have: (3) = ŷ + e, where e s he rojecon error ha s uncorrelaed wh Defne: ŷ (or he lagged values of ). (4) Ω ( ) = var( ), Ω ( ŷ ) = var(ŷ ), Σ = var( e ).

14 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 Assume ha Ω ) s osve defne and Ω ŷ ) and Σ are osve sem-defne. ( ( Now consder forecasng he lnear combnaon c '. Box and Tao (977) use (5) μ var(c' ŷ var(c' ) ) o measure he forecasabl of he lnear combnaon c '. Noce ha (6) c' Ω(ŷ )c μ =, c' Ω( )c hen, under he normalzaon c' Ω ( )c =, s eas o show ha he c ha mnmzes (maxmzes) μ (measurng he degree of redcabl of he followng frs order condon: c ' based on c' ŷ ) sasfes (7) Ω ŷ )c = λω( ) c, ( where λ s an egenvalue of Ω ŷ ) n he merc of Ω ). 8 Thus, he c ha mnmzes ( (maxmzes) μ s jus he egenvecor corresondng o he smalles (larges) egenvalue of Ω ŷ ) n he merc of Ω ). Noce ha (7) also mles: ( ( ( (8) c' Ω (ŷ )c = λc' Ω( ) c or equvalenl: (9) c' Ω(ŷ )c λ =. c' Ω( )c 3

15 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 Thus, he mnmum (maxmum) value of μ acheved also equals he smalles (larges) egenvalue. We now wan o show ha he maxmum value of μ defned n (5) above also equals he squared maxmum canoncal correlaon (CC) coeffcen beween ŷ. To hs end, le us suose ha max and λ s he maxmum egenvalue of Ω ŷ ) n he merc of Ω ( ) and v max s he corresondng egenvecor. Le a = v max ', â = v max ' ŷ and u a â =. Then, usng (6) and (9), we have: ( (0) var(â ) λ max = μ =. var(a ) On he oher hand, he maxmum canoncal correlaon coeffcen beween gven b: and ŷ s cov( a,â ) cov(â + u,â ) () ρ max = / / [var(a ) var(â )] [var(a ) var(â )] var(â ) = [var(a ) var(â )] / [var(â )] = [var(a )] / / where we use a = â + u and cov( â,u ) = 0. Comarng (0) and (), we oban λ = ; ha s, he maxmum (mnmum) egenvalue of Ω ŷ ) n he merc of max ρ max Ω ( ) equals he squared maxmum (mnmum) canoncal correlaon beween and he VECM for We now wsh o exend he above dea o he hohess esng for he CI rank of Ω ) are ordered as: (. To hs end, we assume ha he egenvalues of Ω ŷ ) n he merc of λ λ λ and he corresondng egenvecors are: ( ( ŷ. 8 See, for examle, Dhrmes (978),.7. Hamlon (994) exlans ha hs language reflecs no more han he mos-common normalzaon used when calculang egenvalues. Here, for recson, we rean he classc language. 4

16 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 wh normalzaon v,, v,, v () v ' Ω ( )v, for =,,...,. = Now, we esablsh Lemma, as follows. Lemma : () The egenvalues of Ω ŷ ) n he merc of Ω ) sasf 0 λ for all. () The egenvecors v, v,, v are lnearl ndeenden. Proof: See Aendx. ( ( We now defne: (3a) Λ = dag( λ, λ,, ), wh 0 λ for =,,...,, λ (3b) (3c) M = (v, v,..., v )', z = M. Then, (4) z z v' = =. z v ' Thus, usng = ŷ + e we have: (5) z = zˆ + q, where ẑ = Mŷ = E(z z, z,...) and q = Me. (5) mles: 5

17 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 (6) Ω ( z ) =Ω ( zˆ ) + var( q ), snce ẑ and q are uncorrelaed. Now, we are read o esablsh Lemma, as follows. Lemma : We can ransform he orgnal vecor z = such ha z = zˆ + q (ha s, (5)), where M no a canoncal vecor () var( z ) = I ; () cov( z,ẑ ) = var(ẑ ) = Λ dag( λ, λ,..., λ ), wh λ λ λ ; and 0 () var( r ) I Λ. = Proof: () Under he normalzaon (), we can easl verf ha Ω z ) = MΩ( )M' ( = I. () Snce he columns M are he egenvecors of Ω ŷ ) n he merc of Ω ), we have: ( ( (7) Ω ŷ )M' = Ω( )M' Λ (. Then, remullng b M and usng M Ω ( )M' = I, we oban: (8) MΩ (ŷ )M' = Λ. Tha s, var( ẑ ) = MΩ(ŷ )M' = Λ. () Usng Ω ( z ) =Ω ( zˆ ) + var( q ), we oban: (9) var( q ) =Ω( z ) Ω ( zˆ ) = I Λ, b resuls () and () above. Ths comlees he requred roof. 6

18 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 Remark : The egenvalues ( λ, λ,, λ ) are jus he squared canoncal correlaon coeffcens beween and ŷ and are ordered from he smalles o he larges; for examle, λ s he square of he smalles canoncal correlaon coeffcen beween he covaraes z = v' and v' ŷ ẑ =, and λ s he square of he larges canoncal correlaon coeffcen beween he covaraes z = v ' and v ' ŷ ẑ =. A large canoncal correlaon beween z = v ' and v ' ŷ ẑ = mles ha he canoncal covarae z = v ' s hghl redcable accordng o Box and Tao (977); n oher words, f λ s ver close o, hen he canoncal covarae z = v ' s hghl redcable. Then, usng he fac ha an I() rocess s hghl redcable, we can now use he hohess of λ = o es wheher he canoncal covarae z = v ' s I() or no. On he oher hand, f he canoncal covarae z = s hghl v' unredcable, hen λ mus be ver close o zero or a leas sgnfcanl less han. Thus, we can also use he hohess of λ o es wheher he frs canoncal covarae z = s I(0) or no. v' Remark : The number of egenvalues (ha s, he λ s) ha are close o s he same < as he number of lnear combnaons ha can be almos erfecl forecased; see, Box and Tao (977). Thus, he number of egenvalues ha are close o s he same as he number of common sochasc rends n he VAR ssem for Bewle and Yang (995). ; see for examle, Remark 3: The dfference beween Johansen s MLE-based mehod and he canoncal correlaon mehod s ha Johansen s mehod calculaes he canoncal correlaon beween Δ and, whle he canoncal correlaon analss uses he canoncal correlaon beween and. The man reason for hs dfference s ha Johansen s MLE aroach uses he ror nformaon ha he elemens of whle he canoncal correlaon analss does no. Remark 4: Based on Remark above, f he CI rank equals r, hen we have: λ λ λ <, and λ r + = λ r+ = = λ =. 0 r are I(), 7

19 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 Thus, we can drecl es he hoheses abou he number of common rends of a VAR ssem based on hoheses abou he number of egenvalues of Ω ŷ ) (n he merc of Ω ) ) ha are equal o one. Gven he fac ha he egenvalues are alread ranked from ( he smalles o he larges, we can es he hohess ha here are (k-r) common rends based on he followng null and alernave: H : λ r + = v.s. H : λ r + <. 0 Bewle and Yang (995) roosed several CI ess based on hs dea. In he nex secon, we wll roose wo new sascs for esng CI rank. Remark 5: Noce ha λ = v ' Ω(ŷ )v / v ' Ω( ) v z = v ' s almos unredcable, z = v ' s almos erfecl redcable, ( = var( v ' ŷ ) / var(v ' ) var(ẑ ) / var(z ). Then f = λ mus be less han. On he oher hand, f λ mus be equal o (or a leas ver close o). Thus, we can es he hohess ha λ < based on esng wheher he corresondng canoncal covarae z v ' s saonar or no. Therefore, we can al KPSS esng sasc o he canoncal covarae z = v '. Ths aroach s analogous o he resdual-based es for CI; see, for examle, Shn (994). Smlarl, we can es wheher λ = or no based on esng wheher he corresondng canoncal covarae z v ' has a un roo or no, whch can be execued b alng Dcke- Fuller (DF) or augmened DF (ADF) un-roo es o z. As an llusraon, le us now examne he VAR() model n deal. Suose ha he vecor follows VAR(): (30) = φ + u, where φ s he coeffcen marx. Then, usng he noaon of hs secon, we have: (3) = ŷ + u, 8

20 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 where ŷ = E(,,...) = φ. Thus, Ω ( ) = Ω(ŷ ) + Σ, where Ω ( Γ and Ω ŷ ) = var( φ ) = φγ φ'. ) = E( ') 0 ( 0 Le λ λ λ be he ranked egenvalues of Ω ŷ ) = φγ φ' n he 0 ( 0 merc of Ω ( ) = Γ0 and, v,, v v be he corresondng egenvecors ha are lnearl ndeenden. As n he revous secon, le M = (v,, v )'. Then, remullng (30) b M, we have: (3) z = φ z + r zˆ + q, where z = M, ~ φ = MφM and q = Mu, and where, b Lemma above, var( z ) =, cov( z I,ẑ ) = var(ẑ ) = Λ = dag( λ, λ,, λ ), var( q ) = I Λ. The VAR() model n (3) s usuall referred o as he canoncal model. Now, we urn o examnng he roeres of φ ~ when some of he egenvalues aroach he un crcle. Secfcall, suose ha (-r) egenvalues of φ aroach ons on he un crcle. Paron z, q and ~ φ as: z z =, z q q = q ~ ~ ~ φ φ, φ = ~ ~, φ φ where ~ z and q are r vecors and φ s r r. Box and Tao (977) showed he followng moran resuls: () If, and onl f, (-r) egenvalues of φ (or equvalenl, φ ~ ) aroach values on he un crcle, hen (-r) egenvalues of Ω ( ŷ ) = φγ0 φ' n he merc of Ω ( ) = Γ0 aroach. () The canoncal model for z n (3) becomes, n he lm, 9

21 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 (33) z φ 0 z, q z = z + φ φ, q where z follows a saonar VAR() rocess and z s non-saonar. Thus, he canoncal ransformaon ( z = M ) decomoses he orgnal -dmensonal vecor no wo subvecors: one saonar subvecor ( z : r ) and he oher non-saonar subvecor ( z : ( r) ) ha also deends on z,. Remark 6: Exresson (33) can be hough of as he rangular reresenaon for he vecor of canoncal covaraes reresenaon s for he vecor z = M' self.. In conras, Phlls (99) rangular Remark 7: Gven (33) above, we can also es hoheses abou he CI rank (for examle, H 0 : CI rank = r v.s. H : CI rank < r) based on esng: ~ H 0 : φ = 0 v.s. ~ H : φ 0, ~ because φ 0 holds f, and onl f, (-r) of he egenvalues of Ω ŷ ) = φγ φ' n = he merc of 0 ( 0 ~ Ω ( ) = Γ are equal o ; ha s, φ 0 holds f, and onl f (-r) of he canoncal correlaon coeffcens beween and ŷ are equal o. Moreover, ~ he esng sascs for φ 0 are jus he Wald-e sascs and are usuall = dsrbued as a Ch-square. = 0

22 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, Alcaon: A Panel Conegraon Model for M Veloces 5.. Tesng for Conegraon Rank usng Johansen s ML-based Mehod In hs subsecon, we al Johansen s ML-based mehod o he anel VEC model for M veloc deermnaon n he Uned Saes and Canada. More secfcall, we seek o esmae he followng anel daa model for M veloc: (34) log( GDP / M ) = β0 + β(/ R ) + ε, where equals (for U.S.) and (for Canada), ndexes annual daa from 99 o ear 00, R s a long erm nomnal neres rae, and ε s he error erm of he model. Usng he noaon of Secons 3 and 4, we have: (35A) (35B) (35C) = (GDPUS, / MUS,,/ R US, )', = (GDPCan, / MCan,,/ R Can, )', = ( ', ')' = (GDP US, / M US,,/ R US,,GDP Can, / M Can,,/ R Can, )'. We now noce ha, b he defnons of = ( Δ ', Δ ',, Δ ' )' n X,,, ( ) (B) and X = (X ', X ',, X ' )' n (5A), we can roae he coeffcen marx Γ and N he lagged-dfference vecor can be rewren n he famlar form: X n VECM (0) of Secon 3 so ha he anel VECM (0), j= * (36) Δ = δd + αβ' + Γj Δ j + ε where all he coeffcen marces are unresrced exce he reduced rank decomoson of he long-run coeffcen marx. Then, under he normal and d assumon for ε, usual maxmum lkelhood based esmaon and nference mehods are drecl alcable o he unresrced VEC models defned for resecvel; see for examle, Johansen (995). of (35A), of (35B) and of (35C),

23 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 Usng ADF ess, we found ha all he observable varables (ncludng he M veloces, GDP/M) aearng n (35A)-(35C) are I() a 5% sgnfcance level. Thus, regresson model (34) oss a conegraon relaonsh beween M veloc and nverse long rae. However, when we use Johansen s (995) ML-based race and maxmumegenvalue sascs o es for he conegraon ranks n VEC models defned for (35A), of of (35B) and of (35C), resecvel, we found ver lle emrcal evdence for eher conegraon whn US and Canada or conegraon n he ooled anel VEC model. More secfcall, f we assume ha he varables do no conan lnear rends and allow a consan erm n conegraon relaonshs, or ha he varables conan lnear rends and hus allow a consan erm n boh he conegraon relaonshs and he vecor error-correcon regressons, we were no able o rejec: () he hohess ha he CI rank for US s zero, when he number of lagged dfferences ncluded n he VECM s less han or equal o 0; () he hohess ha he CI rank for Canada s zero, when he number of lagged dfferences ncluded n he VECM s less or equal o. If we ool US and Canada no a anel daa se and al Johansen s CI es o he ooled VECM of dmenson four, we found ha: () he CI rank s zero f he number of lagged dfference ncluded s less han or equal o 7, regardless of he secfcaon of he deermnsc comonens; () when he number of lagged dfferences ncluded s equal o 8, () he CI rank s based on he race sasc, regardless of he secfcaon of he deermnsc comonens; () he CI rank s equal o based on he maxmum-egenvalue sasc, f here s no lnear rend n he daa bu allowng a consan n CI relaons; () he CI rank s margnall equal o 3 based on he maxmum-egenvalue sasc, f here are lnear rends n he daa and allowng consan erms n CI relaons; and (3) he CI rank s equal o 3 based on he race or maxmum-egenvalue sascs f he number of lagged dfference ncluded n he VECM s equal o 9, regardless of he secfcaon of he deermnsc comonens. Because he daa used n our emrcal analss above are annual daa, we srongl beleve ha he long lag lengh requred (a leas 7 lags) for Johansen s ML-based aroach o deec an conegraon relaonshs n he unresrced anel VEC models for he vecors defned n (35A)-(35C) s unreasonable. Thus, s far o sa ha Johansen s MLbased CI ess rovde ver weak evdence for eher whn or cross-counr conegraon. Ths ma be caused b he naccurac of Johansen s asmoc aroxmaon for he

24 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 dsrbuons of hs esng sascs when he small samle sze s small; n our alcaon he samle sze used s 8. See, for examle, Toda (995), Jacobson (995) and Haug (996). One wa o overcome he naccurac of asmoc dsrbuonal aroxmaon s o boosra Johansen s esng sascs. Table 5 rovdes he boosra resuls for Johansen s race and maxmum-egenvalue esng sascs for esng he null hohess ha he conegraon rank s 3. Based on he boosraed dsrbuon n anel A of Table 5, we acce he null hohess ha he conegraon rank s equal o 3. The oher aroach o overcomng he dffcules wh Johansen s asmocbased ess for CI rank s o use Box and Tao (977) and Bewle and Yang (995) canoncal correlaon mehod for esng for CI rank. 5.. Boosraed Canoncal Correlaon Aroach Suose ha we have a VECM of dmenson wh he squared canoncal correlaons ordered as λ λ λ (see Secon 4 above), and ha we wsh 0 o es wheher he conegraon rank s r or no; ha s, we wsh o es he followng hohess, H 0 : CI rank = r and H : CI rank > r. As exlaned n Secon 4, hs s equvalen o esng H 0 : λ r + = λ r+ = = λ = agans H : λ r + λ r+ λ r+ j < for some j<(n-r). We now roose he followng four CC-based esng sascs: r ) = r+ The frs race sasc: λ = T ( λ r ) = r+ The second race sasc: λ = T ln( λ The frs mnmum-egenvalue sasc: λ = ( λ ) mn T r+ The second mnmum-egenvalue sasc: λ = ln( λ ). mn T r+ The frs race sasc and he frs mnmum-egenvalue sasc are consdered b Bewle and Yang (995), whle he second race sasc and he second mnmumegenvalue sasc are newl roosed and are also analogous o Johansen s (995) race and maxmum-egenvalue sascs, resecvel. Snce we do no wan o assume ha he rue DGP s mulvarae normal wh he same covarance marx over me, nor do we wan o assume ha he samle sze used s large, we choose o boosra he four esng sascs above o fnd emrcall correc 3

25 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 crcal values and -values. To hs end, we follow he boosra rocedure of van Gersbergen (996). More secfcall, akng he frs race sasc as an examle, we follow he followng sx ses o boosra s fne samle dsrbuon. r ) = r+ Se : For a gven samle, calculae he emrcal value of λ = T ( λ. Se : Esmae arameer values under he jon null hohess H ' 0 : CI rank = r and μ = 0 b runnng he resrced VECM regresson: k Δ = αβ' + Γ Δ + ε, j= j j where α and β are of dmenson r. We esmaed he CI vecor, β, b Box and Tan s canoncal mehod. Le α ~, β ~ ~ ~, Γ,..., Γ k be he resrced coeffcen esmaes and ~ ε be he corresondng resduals. Le ~ ε = T T be he samle mean of he resduals. / Le ˆε be he scaled and cenered resduals, ε ˆ [T /(T (k ))] ( ~ ~ = ε μ), whch are μ ~ = saonar under he null hohess H 0. Thus, we can now use saonar boosra mehod o resamle hese adjused resduals. Se 3: Le { ε * : =,..., T} be T resamled resduals from he adjused resduals, { ε ˆ : =,..., T}. Gven a resamle of { ε * : =,..., T}, generae a boosra samle { * : =,..., T} from he resrced model: k * * ~ * * Δ = αβ ~~ ' + Γ Δ + ε, for =,,..., T, j= j j where he nal values are gven b = for s = (-k), (-k),...,, 0. * s s Se 4: Use he boosraed samle { * : =,..., T} from se 3 o calculae a boosra realzaon of he frs race sasc, denoed b * λ rb. Se 5: Reea ses 3 and 4 a large number of mes, sa B mes, for b=,,..., B. Se 6: For a gven sgnfcance level α, we rejec he null hohess of conegraon rank equal o r, f he emrcal value (calculaed n se ) s larger han he [( α)b] -h larges boosra realzaon; ha s, we rejec he null f λ r > λ * r[( α)b], where λ < λ < < λ. * r[] * r[] * r[b] 4

26 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 Our boosra rocedure s mlemened n GAUSS 6.0 and he number of boosra (ha s, B n se 5 above) used s 0,000. Snce our daa s annual, we choose o nclude one lagged dfference; ha s, we esmae a VEC() model; n fac, we dd no deec an seral correlaon beond one lag. Table 6 rovdes he boosraed dsrbuons of he race and mnmum-egenvalue sascs for esng he null hohess ha he conegraon rank s 3. Based on he boosra dsrbuons gven n Panel A of Table 6, we see ha he null hohess ha he CI rank s equal o 3 s acceed for an sgnfcance level α 0%. The hree normalzed conegraon vecor esmaors are: βˆ = = 0 0 βˆ, βˆ 0 0 = β ˆ = 0.060, βˆ 0 = 0 β ˆ 34 0 = , where we normalzed he conegraon vecors n such a wa ha ˆβ s he whn-u.s. conegraon vecor, ˆβ s he whn-canada conegraon vecor, and ˆβ 3 s he beweenconegraon vecor (ha s, he conegraon vecor beween U.S. and canada). Also, b Panel B of Table 6, he 95% confdence nerval for he second elemen of he frs conegraon vecor (ha s, β ) s [0.007, 0.043], whch does no nclude 0; hus β s sgnfcan. Smlarl, we see ha he fourh elemen of he second conegraon vecor (.e. β 4 ) and he fourh elemen of he hrd conegraon vecor (.e. β 34 ) are boh sgnfcan. Therefore, s no dffcul o see he ossble grave consequences on arameer esmaon and nferences f he beween-conegraon relaonsh (.e. β 3 ) s negleced, whch unforunael s he common racce of he curren anel VECM leraure, esecall for alcaons wh he number of cross-secons moderae o large. More moranl, usng he nferences based on he boosraed dsrbuons of canoncal correlaon esmaors, we successfull avoded he ambgu encounered above when we al Johansen s asmoc-based mehod o es for he conegraon rank n VECM (36). Thus, based on hs small emrcal alcaon, aears ha he boosraed canoncal correlaon analss aroach s sueror o he ML-based aroach for esng hoheses of conegraon rank and for esmang conegraon vecors, snce 5

27 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 he boosraed canoncal correlaon mehod does no deend on dsrbuonal assumons of he rue DGP, does no requre he covarance marx of he VECM errors o be homoscedasc (n fac, our boosra rocedure accommodaes for ossble heeroscedasc n he VECM errors), nor does requre ha he samle sze used s large. 6. Conclusons Gven he oor small samle erformance of Johansen s ML-based asmoc aroach o esng for conegraon ranks of vecor error correcon models and s crcal deendence on he dsrbuonal assumons, we beleve ha here s a genune need o fnd alernave mehods for esng conegraon ranks ha do no deend on he resrcve dsrbuonal assumons or naccurae asmocs. In hs aer, we consder four sascs (wo of whch are roosed b us for he frs me) for esng conegraon ranks based on Box and Tao s (977) canoncal correlaon aroach. To ensure ha our canoncal correlaon based ess have correc emrcal sze, we use boosra mehod o fnd he fne-samle dsrbuons of he four esng sascs. The curren leraure on anel conegraon ess almos alwas assumes ha () here s no conemoraneous cross-seconal correlaon n he error erms; () here s no neracon of shor-run dnamcs beween cross-secons; () dfferen cross-secons have he same conegraon rank; and (v) here are no long-run equlbrum relaonshs beween dfferen cross-secons. In hs aer, we argue ha cross-seconal deendence n shor-run s ervasve snce dfferen cross-seconal uns are nfluenced b he same es of domesc and nernaonal macro shocks, and ha long-run equlbrum relaonshs beween cross-secons are also ver common snce dfferen economes (or cross-secons) end o move ogeher n he long-run, esecall n hs age of globalzaon. Our Mone Carlo smulaons demonsrae ha he resence of shor-run and/or long-run cross-seconal deendence causes ver severe sze dsorons and ower loss for he anel KPSS conegraon es. To overcome he weakness of he curren anel conegraon ess, we roose n hs aer an unresrced anel VECM ha allows for arbrar conemoraneous correlaon, cross-seconal neracon of shor-run dnamcs, 6

28 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 heerogeneous conegraon ranks across cross-secons, as well as conegraon beween dfferen cross-secons. In our emrcal alcaon of an unresrced anel VECM for he long-run deermnaon of M veloces n U.S. and Canada, usng boosra mehod, we unequvocall fnd hree conegraon relaonshs; wo for whn-counr conegraons (one for each counr) and he oher for beween-counr conegraon. Ths s n shar conras o he confusng resuls from he Johansen s ML-based asmoc aroach, hough we also clearl confrm ha he conegraon rank s 3 when we boosra Johansen s esng sascs. The unresrced anel VECM aroach advocaed n hs aer can be easl aled o man oher neresng economc and fnancal roblems; for examles, he esng of economc convergence of OECD counres and he esmaon of consumon and nvesmen funcons across regons and saes. Foonoe O Connel (999) and Maddala and Wu (999) consdered he effec of cross-seconal correlaon on he sze and ower of anel-un roo ess. Aendx Proof of Lemma : () See Proosons 6 and 64 of Dhrmes (978,. 7-74). () Because Ω ( s a real smmerc marx, here exs orhogonal / / ) Ω(ŷ ) Ω( ) egenvecors, sa b, b,..., b. Le B = (b,, b ). Then, (A.) / / Ω( ) Ω(ŷ ) Ω( ) B = BΛ. Premullng b / Ω ( ), we have: (A.) / / Ω(ŷ )[ Ω( ) B] = Ω( )[ Ω( ) B] Λ, 7

29 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 where Λ = dag( λ, λ,, ). Ths means ha he columns of Ω ( ) / B are he λ egenvecors of Ω ŷ ) n he merc of Ω ) and he dagonal elemens of Λ are he ( ( corresondng egenvalues. Now, le / M' = Ω( ) B. Then, (A.3) / / M Ω( )M' = B' Ω( ) Ω( ) Ω( ) B = B' B = I, snce he columns of B are orhogonal. Thus, rank(m)=, whch mles ha he egenvecors of Ω ŷ ) n he merc of Ω ) mus be lnearl ndeenden. Ths ( comlees he requred roof. ( 8

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32 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 Karlsson, S., and M. Lohgren (000), On The Power and Inerreaon of Panel Un Roo Tess, Economcs Leers 66, Larsson R. and J. Lhagen (999), Lkelhood-Based Inference n Mulvarae Panel Conegraon Models, workng aer No. 33, Sockholm School of Economcs. Larsson R., J. Lhagen and M. Lohgren (00), Lkelhood-based Conegraon n Heerogeneous Panels, Economercs Journal 4, Levn, A. and C.F. Ln (993), Un roo es n Panel Daa: Asmoc and Fne- Samle Proeres, Dscusson Paer 93-56, Dearmen of Economcs, UCSD. Lukeohl, H. and D. Posk (998), Conssen esmaon of he number of conegraon relaons n a vecor auoregressve model. In Economercs n Theor and Pracce, Galan, R. and H. Kuchenhoff Eds.; Phsca-Verlag: Hedelberg, Lhagen, J. (000), Wh no use sandard anel un roo es for esng PPP, workng aer no. 43, Sockholm School of Economcs. Maddala, G.S., and S. Wu (999), A Comarave Sud of Un Roo Tess Wh PanelDaa and a New Smle Tes, Oxford Bullen of Economcs and Sascs, Mark, N., M. Ogak and D. Sul (005), Dnamc Seemngl Unrelaed Conegrang Regresson, Revew of Economc Sudes, 7(3), McCoske, S., and C. Kao (00), "A Mone Carlo Comarson of Tess for Conegraon n Panel Daa," Journal of Proagaons n Probabl and Sascs,, Moon, R. and B. Perron (004), Tesng for A Un Roo n Panels wh Dnamc Facors, Journal of Economercs, Seember, (), Ng, S. and P. Perron (997), Esmaon and Inference n Nearl Unbalanced Nearl Conegraed Ssems, Journal of Economercs, O Connell, P. (998), The overvaluaon of urchasng ower ar, Journal of Inernaonal Economcs, -9. Pedron, P. (999), Crcal Values for Conegraon n Heerogeneous Panels wh Mulle Regressors, Oxford Bullen of Economcs and Sascs, 6, Pedron, P. (004), Panel Conegraon: Asmoc and Fne Samle Proeres of Pooled Tme Tess wh an Alcaon o he PPP Hohess, Economerc Theor, 0,

33 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 Pesaran, M.H., Y. Shn and R. Smh (000), Srucural Analss of VECMs wh exogenous I() Varables, Journal of Economercs, Augus, 97(), Pesaran, M.H., Y. Shn and R. Smh (999), Pooled Mean Grou Esmaon of Dnamc Heerogeneous Panels, Journal of Amercan Sascal Assocaon 94, Phlls, P.C.B. (99), Omal Inference n Conegraed Ssems, Economerca 59, Phlls, P.C.B. (995), Full Modfed Leas Squares and Vecor Auoregresson, Economerca 63, Phlls, P. and H. Moon (000), Nonsaonar Panel Daa Analss: An Overvew of Some Recen Develomens, Economerc Revews, 9(3), Shn, Y. (994), A resdual-based es of he null of conegraon agans he alernave of no conegraon, Economerc Theor 0, 9-5. Sock, J. and M. Wason (988), Tesng for Common Trends, Journal of he Amercan Sascal Assocaon 83, Talor, M. and L. Sarno (998), The Behavor of Real Exchange Raes Durng he Pos- Breon Woods Perod, Journal of Inernal Economcs 46, 8-3. Toda, H.Y. (995), Fne Samle Performance of Lkelhood Rao Tess for Conegrang Ranks n Vecor Auoregressons, Economerc Theor, van Gersbergen, N.P.A. (996), Boosrang he Trace Sasc n VAR Models: Mone Carlo Resuls and Alcaons, Oxford Bullen of Economcs and Sascs 58(), Weserlund, Joakm (005). New Smle Tess for Panel Conegraon, Economerc Revews, 4(3), (006). Tesng for Panel Conegraon wh Mulle Srucural Breaks, Oxford Bullen of Economcs and Sascs, forhcomng. Weserlund, Joakm and Davd Edgeron (005). Panel Conegraon Tess wh Deermnsc Trends and Srucural Breaks, Workng Paer 005:4, Dearmen of Economcs, Lund Unvers, Ocober. 3

34 Anderson, Qan and Rasche Panel Conegraon Draf Please Do No Quoe Whou Permsson Februar 5, 006 Esmaon Mehod: Dnamc OLS (DOLS) Samle Sze (T)=50 No. of Relcaons (R)=5000 Table A: Acual Sze of LM Tes Nomnal Sze: 0% 5% Case Case case 3 case 4 Case Case case 3 case 4 N= ooled un un N=5 ooled un un un un un N=0 ooled un un un un un un un un un un