Some cautions on the use of panel methods for integrated series of macro-economic data

Testo completo

(1),N+, NWV,+5WAv W5AWANA, #,+A,A 6 ,W5. ,NW `Lh!?} @Tih , L 2fff%2f. 5L4i @|L?t L? |i Nti Lu @?i* i|L_t uLh W?|i}h@|i_ 5ihit Lu @UhL,UL?L4U #@|@ AAa+B BAjijje Btt6,BAN BiWj,,AN @?_ BiB tOB|. #W 6W,5wc 5 #,W E6W.

(2) ** h}|t hitihi_ L T@h| Lu |t T@Tih 4@) Mi hiThL_Ui_ ? @?) uLh4 |L| Tih4ttL? Lu |i @|Lh. U 2fff @?ihiic @hUi**?L @?_ tM@| h?|i_ ? W|@*) ? Li4Mih 2fff ,hLTi@? N?iht|) W?t|||i @_@ 6itL*@?@ WDffS 5@? #L4i?UL E6W W|@*).

(3) Some Cautionsonthe U se ofP anelM ethod sfor Integrated SeriesofM ac ro-E c onom ic Data¤ Anind ya B anerjeey. M assim iliano M arc ellinoz. Chiara O sb atx. T hisversion: J uly 2 0 0 0. A bstract W e show how the use ofpaneldata methods such as those proposed in single equations by Kao(1 999)and P edroni (1 999)orin systems by L arsson and L yhagen (1 999)to investigate economic hypotheses such as purchasing powerparity orthe term structure ofinterestrates may be a®ected by the existenceofcross-unitcointegratingrelations.T heexistingliteratureassumes thatsuch relations,thattie the units ofthe paneltogether,are notpresent. U singempiricalexamples from apanelofO ECD countries weshowthatthis assumptionisverylikelytobeviolated.Simulationsofthepropertiesofpanel cointegration tests in the presence ofcross-unitrelations are then presented todemonstratetheserious costofassumingaway such relations. Some¯xes areproposed as away ofdealingwith thesemoregeneralscenarios.. ¤. W e are very gratefultothe European U niversity Institute forpartially fundingthis research underresearch project\ M odellingand ForecastingEconomicT imeSeries in theP resenceofH igh O rders ofIntegration and StructuralB reaks in D ata". y Corresponding author. D epartment of Economics, European U niversity Institute, 50 0 1 6 San D omenico di Fiesole. P hone: + 39-0 55-4685-356. Fax: + 39-0 55-4685-20 2. E-mail: banerjee@ datacomm. iue. it z Istitutodi EconomiaP oliticaand IG IER ,U niversita'B occoni,V iaSalasco3,20 1 36 M ilano x D epartmentofEconomics,European U niversity Institute,50 0 1 6 San D omenicodi Fiesole.

(4) 1. In trod uc tion. Inrec ent years,the use ofpaneld ata techniquesto test m ac roec onom ic hypotheses hasb ec om e increasingly c om m on.T w o exam plesofthe m ost popular usesto w hic h these tec hniqueshave b eenput are testingf or c onvergence ingrow th ratesofG DP 1 and tests f or purc hasing pow er parity2 . E ac h ofthese hypotheses is f ormulated essentially astestingf or a unit root inthe estim ated resid ualsofa single-equation regressionand the argum ent presented inf avour ofpaneld ata m ethod sisthat they have greater pow er thanstand ard unit root tests, b y virtue ofthe red uc tionin noise c aused b y the averaging or pooling ac rossthe unitsofthe panel. Anad d ed ad vantage of tenproposed isthat the asym ptotic d istrib ution(und er the null) of som e ofthese tests, eveninthe presence ofintegrated d ata, isnorm al. Inf erence is thereb y insom e sense m ad e \stand ard ". T he em phasis inthis literature has theref ore prim arily takenthe f orm ofc onsid eringthe asym ptotic propertiesofpanel d ata estim atorsand test statistic sasT and N go to in¯nity, possib ly at ratesof d ivergence that are c ontrolled appropriately. T ec hniquesd eveloped b y M c K oskey and K ao (19 9 8), K ao (19 99 ) and P ed roni (199 9 ) inter alia have extend ed E ngle (19 87) type static single equationregressionsto pooled panelstatic regressionsand prod uc ed test statistic sand c ritic alvaluesto test f or c ointegrationinpaneld ata. Im plic it inthese analysesisthe assum ptionofunique c ointegrating vec tors, alb eit heterogeneous,ac rossthe units. Severalissues are w orth noting w ithinthe c ontext ofthis stud y f or the use ofpanelsinm ac ro-ec onom etric researc h. It isim portant ¯rstly to m ake a c lear d istinctionb etw eentheir use here and their use inm ic ro-ec onom etric research.T he m ac roec onom ic d ata sets(relatingto grow th,d istrib ution,term struc ture ofinterest ratesor P P P ) typic ally c onsist ofsam plesofup to 50 c ountriesw ith ob servations 1 2. See e. g.Evans and Karras (1 996)and L ee,P esaran,and Smith (1 997) O n panelstudies ofP P P,see Frankeland R ose (1 996)and P apell(1 997).. 1.

(5) f or thirty or f orty yearsquarterly or annually.T husthe d im ensionofthe numb er ofc ross-sec tionunitsN , although som etim esquite large, isvery of tend om inated b y the tim e d im ensionT . T he usesofpaneld ata m ethod sinm ic ro-ec onom etric s onthe other hand hasa muc h longer history and involve lookingat settingsw here the tim e d im ensionT issigni¯c antly sm aller thanN . T hus, w hile anim pressive am ount ofresearc h has f oc used onm od elling the heterogeneity ac ross the units b y m eansofc om m ontim e e®ec ts, or ¯xed - and rand om -e®ec ts, and onm ethod s ofoverc om ing w hat isc om m only know nasthe \incid entalparam etersprob lem " (usually b y m eansoftransf orm ationsofthe m od elor b y instrum entalvariab lesor G M M. m ethod s), the issue isd ealt w ith b y c onsid ering large-T, relatively-large-N. sam plesinthe c ase ofm ac ro-ec onom etric d ata sets.T hisisnot to saythat incid ental param eter prob lem sd o not have anyrelevance f or our analysis,since the assum ption that w e have su± c ient d ata to estim ate allthe param etersofour m od elisf ar f rom innoc uousand w e shallvisit thisissue b elow . Sec ond ly,inthe c ontext ofanalysisofmultivariate (> 2 ) d ata setsit isnatural to f oc usonm ethod sthat relax the assum ptionofa unique c ointegrating vec tor. G roenand K leib ergen(19 9 9),Larssonand Lyhagen(199 9 ) and (2 0 0 0 ),and Larsson, Lyhagen, and LÄothgren(19 9 8) have theref ore d eveloped tec hniquesanalogousto (J ohansen19 9 5) m aximum likelihood m ethod sto allow f or multiple c ointegrating vec tors inthe c ross-sec tionunits. Since the assum ptionofunique c ointegrating vec torsm ayb e thought ofasb eingunnec essarilyrestric tive and unrealistic ,system sm ethod sf or panelshave represented a notab le ad vance. How ever,third ly,and m ost im portantly,none ofthe m ethod sc ited ab ove takes any ac c ount ofthe possib ility oflong-rund epend ence am ong the variab lesac ross the unitsc om prisingthe panel(exc ept through short-rune®ec tsor through c orrelationsam ongthe errorsac rossthe units).Asw e show b elow ,partic ularly inm ac ro-. 2.

(6) ec onom ic stud iesw ith integrated variab les,thisisa seriousrestric tion.W hile G LS m ethod ssuc h asthose used b y O Connell(19 98) go som e w ay tow ard sallow ingthe unitsto b e related b y m eansofa non-d iagonalvariance-c ovariance m atrix ofthe system ,these d o not ac c ount f or the existence oflong-runor c ointegratingrelationshipsac rossvariab lesinthe unitsofthe panel.Nor isthe incorporationofshort-run d epend ence su± c ient f or the analysisofm ac roec onom ic paneld ata sets. T hislast ob servationlead sonto im portant c om plic ationsf or the researc h agend a involvingthe use ofm ac roec onom ic paneld ata.W ithinthe c om passofthisc ritique lie allthe pre-existingtestsf or unit rootsand c ointegrationinpanels,to the extent ofvitiating entirely their use.T w o signi¯c ant f eaturesinhib it their applic ab ility the assum ptionsofthe sam e m aximum c ointegrating rank ac rossthe unitsand no long-rund epend ence ac rossthe units. O ur paper takesLarssonand Lyhagen(19 9 9) (hencef orth LL) asitsstarting point, since it provid esone ofthe m ost generaltreatm entsofpanelc ointegration (sub jec t to the restric tionshighlighted ab ove).T he m otivationf or and b ackground to our stud y isthe c onsid erationofsm all- to m ed ium -siz ed m ac ro-ec onom ic panels, w ith the tim e d im ensiontypic ally ranging f rom 10 0 to 2 0 0 . O ur m ethod ology is b ased onsimulations. InSec tion2 , w e b rie°y present the m aximum likelihood f ram ew ork f or analyz ing paneld ata asexem pli¯ed b y LL w ho also d erive the null d istrib utionsofthe test statistic s.InSec tion3, w e use anem piric alexam ple w ith tw o unitsinthe panelto illustrate the c onsid erab le d i± c ultiesf or the d irec t use of thism ethod .B y extension,any ofthe single-equationm ethod sc ited ab ove m ay b e show nto b e insim ilar troub le.B y m eansofsim ple stud iesoflong- and short-run interest ratesina sm allpanelw here the c ross-c ountry c ointegrationpossib ilities are self -evid ent, w e m otivate strongly the need to m od if y pre-existing m ethod sof lookingat c ointegrated panels.. 3.

(7) Inord er to b ench-m ark our simulationstud y, inSec tion4 w e simulate d ata und er the LL spec i¯c ationsofthe d ata generationproc essand show that use ofthe asym ptotic d istrib utionsand c ritic alvaluesd erived b yLL lead sto testsw ith c orrec t siz e and good pow er.O ur point ofd eparture f rom thisf ram ew orkisthento simulate d ata w ith c ross-unit c ointegratingrelationsor unitsw ith d i®erent c ointegratingrank and to show the propertiesofthe LL testsintheir presence.Notab ly, inSec tions 5 and 6 w e present resultsd em onstrating ¯nd ingsofincorrec t rank und er sim ple c hangesofthe proc essgeneratingthe d ata. W e also suggest and evaluate m od i¯c ationsofthe testing f ram ew ork to allow f or c ross-unit c ointegrationand to im prove inf erence.Anob viousroute w e explore is to pre-test the units ofthe panelf or c ointegrating relations w ithineach unit usingthe J ohansenM L m ethod .T he LL m ethod isapplic ab le ifand only ifone is entitled to allow f or the sam e m aximum c ointegrating rank ac rossthe unitsofthe sam ple and ,m oreover,c ointegrationoc c ursonly w ithinunitsand not ac rossunits. E stim atingthe system unit b yunit usingJ ohansenM L isa w ayofverif yingthe ¯rst assum ption.Ifthisisveri¯ed ,one m ay thenextrac t the c om m ontrend sim plied b y the c ointegratingvec torsand test f or c ointegrationam ongthe c om m ontrend sµa la G onzalo and G ranger (19 9 5) to rule out the existence ofc ross-unit relations.O nly thenisthe LL approac h (and the spec ialc asesofthisapproach,such asK ao (199 9 ) and P ed roni(199 9 )) justi¯ed ,and lead sto gainsine± c iencyover estim atingthe f ull system (c onsistingofallthe variab lesinallthe unitsofthe panel).T hissequential testing proc ed ure isim plem ented inSec tion7inord er to analyz e inm ore d etail the interest rate d ata. T he analysisc learly ind ic atesthat the panelc ointegration tec hniquesare not suited to thisc ontext,and c anlead to incorrec t c onclusions. T he lim itations ofour c urrent stud y must ofc ourse b e b orne inm ind . O ur investigationislim ited to lookingat only the sim plest possib le sc enarios.W e d o not. 4.

(8) asyet have su± c ient evid ence f or larger d im ensionalsystem s. Nor d o w e present here the resultsf or d ata generationproc essesw ith d eterm inistic c om ponentssuc h asc onstantsor trend sor b reakd um m ies,c om plic ationsthat are very likely to oc c ur inprac tic e.Inthislatter c ase,provid ed these d eterm inistic variab lesenter into the system unrestric ted ly, w e m ay c onsid er our analysisasproxying f or asifw e w ere w orkingw ith d etrend ed or d em eaned d ata.Nevertheless,b ased onthe c onsid eration ofeventhese sim ple c ases,w e are d raw ninexorab lyto the c onclusionthat ingeneral onlypanelm ethod sthat allow f or f ullsystem m aximum likelihood analysisare likely to lead to the \right"answ er.How ever,d ata lim itations,w ith the im plied d egreesof f reed om restric tion,w illnot allow suc h a f ullsystem to b e estim ated unrestric ted ly inallc irc um stances.W e theref ore c onclud e that,partic ularly inintegrated panels, great c are must b e takeninusing and interpreting the resultsofm ac roec onom ic panels.Short-c utsor restric tionsofthe kind im plic it inLL, or ¯xessuc h asthose w hic h w ould enab le the w ithin-unit c ointegratingrelationsto b e estim ated ¯rst and thenim posed inord er to lookat the c ross-unit c ointegratingrelations,c ould insom e c irc um stancesb e usef ulb ut prove to b e prof ound ly m islead inginothers.. 2. T he P anelCoin tegrationFram ew ork. T he LL panelc ointegrationm od elc orrespond sto a restric ted c ointegrated VAR . T he m od elund er c onsid erationis. 0. ¢ Y t= ® ¯ Y t¡1 + w here ® and ¯ have d im ensionN p £. mX¡1 k= 1. PN. ¡ k¢ Y t¡k + "t. i= 1 ri, N. isthe numb er ofunits, p isthe. d im ensionofeac h sub -system Y it = (yi1t;yi2 t;:::;yipt)0;i= 1;2 ;:::N , t= 1;2 ;:::;T , 5.

(9) and ri isthe rank ofeac h sub -system . T he vec tor Y t isgivenb y stac king the N vec torsY it.T he m atric es® and ¯ have the f ollow ingstruc ture: 0. B ® 11 ® 2 2 B B B ®21 ®22 B ® = B B . B . . B @ ®N 1 ®N 2. 1. ¢ ¢ ¢ ® 1N C C C C C C ; C ... . . . C C A ¢ ¢ ¢®N N. 0. B ¯ 11 0 B B B 0 ¯22 B ¯ = B B . B . . B @ 0 0. 1. ¢ ¢ ¢ 0 .... C C C C C C : . C . . C C A. ¢ ¢ ¢¯N. N. T he variance-c ovariance m atrixof"t is 0. B § 11 § 12 B B B §21 §22 B § = B B . B . . B @ §N 1 §N 2. 1. ¢ ¢ ¢ § 1N C C C :::: § 2 N C C C C ... . . . C C A ¢ ¢ ¢§ N N. T he m od eltheref ore allow sf or interac tionam ong the unitsthrough the long-run ad justm ent c oe± c ients® ,the short-runm atric es¡ k,and the o®-d iagonalelem ents of§ ; b ut the restric tion¯ ij = 0 8i6 = jrulesout c ointegratingrelationshipsac ross the units.A f urther restric tionisthat ofri b eingassum ed to b e the sam e f or eac h unit.T hese tw o restric tionsare the m ost im portant issuesto b e investigated ,since there are m any exam plesw here they m ay b e inconsistent w ith b oth theory and d ata.3 T he estim ationalgorithm f or the ind ivid ualc ointegratingrelationsisa seriesof red uc ed rank regressionsinw hic h eac h ¯ ii isestim ated b y c oncentrating out the n ¡1 rem aining ones.Hence, ¯ 11 to ¯ N. N. 3. are estim ated at eac h iteration, and the. N otethatinpractice,asN grows,theamountofcorrelationamongtheunitsmustbecontrolled apriori byimposingrestrictions on the® ;¡and§ matrices in ordertosatisfydegrees offreedom restrictions implied by thedata.. 6.

(10) proc ed ure isrepeated untilc onvergence.4 T he trac e test f or c ointegrating rank ofri = q versusri = p, f or eac h iand f or q = 0 ;:::;p ¡1,isd erived inLL.T he rank istested und er the b loc k-d iagonality restric tionon¯ .T he asym ptotic d istrib utionofthe test statistic isthe c onvolutionof the stand ard J ohansenLR ranktest and anind epend ent Â 2 w ith N (N ¡1)(p ¡q)q d egreesoff reed om .T hism eansthat the test f or ri = 0 isthe sam e asthe J ohansen test,w hile f or rank larger thanz ero there isanad d itionalc om ponent inthe d istrib utionw hic h ac c ountsf or the ad d itionalz ero restric tionsim posed on¯ .Note that testingf or ri = q versusri = p inthisf ram ew orkc orrespond sto testingf or rankN q versusN p ina f ullsystem c ontext,w hereastestingf or rankinthe J ohansenf ram ew ork w ould allow f or allthe interm ed iate possib ilities.For exam ple, ifN = p = 2 , LL testsf or 0 versus4 f ollow ed b y 2 versus4 . J ohansenw ould test f or 0 versus4 , 1 versus4 and so on. F inally, note that w henp = 1 and ¡ k isb loc k d iagonalf or eac h k, the null hypothesisofthe LL test isthat allthe variab lesare I(1), against the alternative that they are stationary w ith heterogenousstationary roots.T hisresult istheref ore the panelversionofthe ADF test inthe single unit c ase, and inthissense the LL f ram ew ork nestspanelunit root tests(see e.g. Levinand Lin(19 93), Im and M .H.and Shin(19 9 7)).. 3 E m piric alE xam pl es T o illustrate the kind ofprob lem one c ould encounter inusing the LL panelc ointegrationm od elinem piric alm od elling, w e present tw o sim ple exam plesinvolving short- and long-term interest rates.5 W e estim ate the tw o system sc onsistingofG er4. Forstartingvalues,L L proposeusingthe¯ ii estimatedfrom astandardcointegrationanalysis on each unitseparately.W einstead usetheinitialvalues suggested in (Johansen 1 995),p.1 1 0 . 5 T hevariables,takenfrom theO ECD database,arethree-monthinterestrates(code6225D )and longterm rates (codes 6253D forG ermany,6261 D forD enmarkand 6269D forthe N etherlands).. 7.

(11) m any and the Netherland s, and G erm any and Denm ark b y f ullsystem m aximum likelihood , using the J ohansenapproach.It turnsout that b oth pairsofc ountries exhib it a c ointegrationpatternthat isnot c onsistent w ith the struc ture ofthe LL m od el.6. 3. 1. G erm an y an d T he Netherl and s. T he sam ple period f or the G erm an-Dutch m od elgoesf rom M ay 19 9 0 to J anuary 19 9 9 . T he c hosenspec i¯c ationf or the unrestric ted VAR involves f our lagsand anunrestric ted c onstant.Diagnostic testsf or thissim ple m od elpoint tow ard sits c ongruence.T here isonly a m arginalrejec tionofthe nullhypothesisofnorm ality inthe system , w hic h isnot likely to a®ec t the outc om e ofthe c ointegrationtests, see e.g.G onzalo (19 94 ). T he J ohansentrac e and m aximum eigenvalue c ointegrationtest statistic sinthis m od elind ic ate that the c ointegrating rank isequalto one, see the ¯rst panelof T ab le 1.T hisisanind ic ationthat usingthe LL m od elw ould b e inappropriate. T he restric ted c ointegrationvec tor c orrespond ingto a rankofone involvesshortterm ratesf rom b oth c ountries, w ith hom ogeneousc oe± c ientsequalto one.O nly the G erm anshort-term interest rate ad juststo d isequilib rium .T he LR test c orrespond ingto thisrestric ted spec i¯c ationisd istrib uted asa Â 2(6) and takesa value of 4 :84 ; c orrespond ingto a tailprob ab ility value of0 :56:T he estim ated c oe± c ientsare reported inthe sec ond panelofT ab le 1.It isevid ent that w ith thisc ointegration struc ture it w ould b e inappropriate to estim ate the LL panelc ointegrationm od el. T his partofthe analysis was conducted usingP cFiml9. 21 D oornikand H endry (1 997) 6 W e are very gratefultoJohan L yhagen forprovidingus with the L arsson and L yhagen panel dataprogramswritteninG A U SS 3. 0 .T hiscodewassubsequentlytranslatedbyustoO X (D oornik (1 999))andmodi¯edandextendedfortheuses madein this paperandis freelyavailablefrom us.. 8.

(12) 3. 2. G erm an y an d Den m ark. For the G erm an-Danish m od el, the sam ple period c onsid ered goesf rom M ay 199 0 to J anuary 19 9 9.W e ad opt the sam e spec i¯c ationasab ove, w ith f our lagsand an unrestric ted c onstant.Inthissam ple, there isevid ence ofparam eter instab ility in 19 9 3 inthe Danish short-term rate and inthe G erm anlong-term rate. How ever, the only evid ence ofm isspec i¯c ationis the rejec tionofnorm ality inthe Danish short- and long-term ratesand inthe f ullsystem .Ifw e analyz e thism od elonthe sam ple starting f rom O c tob er 19 9 3, w e ¯nd no evid ence ofparam eter instab ility. T he c ointegrationanalysissuggestsa rank ofone inb oth sam ples, and w e report the resultsf or the longer sam ple only. T he J ohansenc ointegrationtests,reported inthe ¯rst panelofT ab le 2 ,ind ic ate that the rank isone inthisb ivariate system asw ell. T he restric ted c ointegrating vec tor involvesthe short-term ratesonly, and the G erm anlong-term rate isw eakly exogenous. T he LR test f or thisspec i¯c ationis d istrib uted asa Â 2(3) and takesthe value of4 :92 ,c orrespond ingto a tailprob ab ility of0 :18:P aram eter estim atesf or the restric ted m od elare reported inthe sec ond panelofT ab le 2 . Insum m ary, inb oth c ases the b loc k d iagonality of¯ that charac teriz es the panelc ointegrationf ram ew orkisviolated .W e w illevaluate the c onsequencesofthis violationonthe perf orm ance ofthe LL test inSec tion5, w hile the next sec tion exam inesthe siz e and pow er ofthe test, assum ingthat itsund erlyingassum ptions are satis¯ed .. 9.

(13) 4. Simul ation sw henthe LL f ram ew ork isc orrec t. For d ata generationproc essesthat satisf y the LL spec i¯c ationrestric tions,the d istrib utionofthe LL trac e test f or c ointegratingrankisasgiveninLL (and d esc rib ed b rie°y ab ove). W e w ere ab le to simulate the c ritic alvaluesim plied b y the d istrib ution.7 Inpartic ular, inthese experim entsw e assum e N = 2 ; a b loc k-d iagonal¯ m atrixand the sam e c ointegratingrank inallunitsofthe panel.T he hom ogeneity ofthe c ointegrating vec tor isalso assum ed , f or sim plic ity. R esultsf or N = 4 and N = 8 are sum m ariz ed inSec tion6 b elow . T he d ata generationproc essesare asgiveninT ab le 3. DG P 1 isthe sim plest null, w ith rank z ero inb oth units. DG P 2 A hasrank 1 inb oth units, w ith the load ingm atrix® c onstruc ted suc h that the equilib rium c orrec tionterm sentersinto onlyone equationofeac h unit.InDG P 2 B ,w ith rank1 inb oth units,the ¯rst equilib rium c orrec tionterm entersone equationeac h ofboth unitsw hereasthe sec ond equilib rium c orrec tionterm entersonly one equationofthe sec ond unit.DG P 2 A and DG P 2 B are c onstruc ted inord er to have the sam e am ount ofintegration(as re°ec ted inthe ab solute valuesofthe non-z ero rootsofthe c om panionm atrix) w hile havingd i®erent equilib rium c orrec tionproperties.DG P 2 C isa variant ofDG P 2 A w ith lessintegration.8 InT ab le 4 w e report the siz e ofnot only the LL testsb ut also ofthe J ohansen proc ed ure unit b y unit, jointly, and inf ullsystem s. T husthe c olum nhead ed LL givesthe rejec tionf requenciesofthe nullhypothesisofri = rLL = 0 , i = 1;2 ; at the 5% signi¯c ance levelw henthe LL statistic isc alculated f or DG P 1 and LL c ritic alvaluesare used . For DG P 2 A to DG P 2 C the rejec tionf requenciesofthe nullhypothesisofri = rLL = 1 at the 5% signi¯c ance levelisgiven, againw ith 7. T hevalues coincidewith thosein L L and,forthespecialcasewhere thecointegratingrankis 0 ,with thosepresented in Johansen (1 995). 8 O therexperimentswithdi®erentcon¯gurationsfor® yieldedqualitativelysimilarresults,while with smallerstationary roots the statistics have bettersize and higherpower.. 10.

(14) Larssonand Lyhagen(19 9 9 ) c ritic alvalues. T he c olum nshead ed \unit 1"and \unit 2 "provid e the c orrespond ingrejec tion f requencies, f or eac h unit ind ivid ually, at the 5% signi¯c ance levelusing J ohansen (19 9 5) c ritic alvalues. T he rejec tionf requenciesinthe c olum nhead ed \joint" are c alculated b y settingthe signi¯c ance levelofthe unit b y unit testsat 1 ¡(0 :9 5)1=2 , so that the signi¯c ance ofthe joint test (i.e.the prob ab ly ofrejec tingrank0 inb oth unitsinDG P 1, and rank 1 inb oth unitsinDG P 2 A to 2 C) is5% . T he c olum n head ed \J system "givesthe rejec tionf requencies(ofrJ = 0 f or DG P 1 and rJ = 2 f or DG P s2 A to 2 C) w henthe f ullf our-d im ensionalsystem , isestim ated w ithout restric tion.Note that rLL ind ic atesthe c ointegratingrank inthe LL sense, and rJ that inthe f ullsystem . T he ¯nalc olum nofT ab le 4 givesthe rejec tionf requenciesofc ointegrationam ong the c om m ontrend sd erived f rom the unit b y unit c ointegrationanalysis, w henever the nullhypothesisisac c epted ineac h unit.Since f or allthe d ata generationproc essesc onsid ered inT ab le 3 there isno c ross-unit c ointegration, the ¯nalc olum n theref ore reportsthe rejec tionf requenciesofr = 0 w henever f our c om m ontrend s (tw o f rom eac h unit) are extrac ted inthe sam plesgenerated b y DG P 1, b ased on testingf or c ointegratingrank unit b y unit,and w henever tw o c om m ontrend s(one f rom eac h unit) are extrac ted inthe sam plesgenerated b y DG P 2 A to 2 C.T hisproc ed ure isequivalent to that proposed b y G onzalo and G ranger (19 95) f or checking f or c ointegrationam ongsub -system sofc ointegrated system s.G onzalo and G ranger suggested that the asym ptotic d istrib utionofthe c ointegrationtest isnot a®ec ted b y havingestim ated the c om m ontrend sinthe ¯rst stage.Hence,the c ritic alvalues used are takenf rom J ohansen(19 9 5). T he resultsonsiz e,asreported inT ab le 4 ,are encouraging.E xc ept f or the joint test, no sub stantialsiz e d istortionsare evid ent. T he d istortionsofthe joint test. 11.

(15) d isappear asthe sam ple siz e increasesto 4 0 0 b ut,partic ularly f or DG P 2 A and 2 B , they are present and im portant at ec onom ic ally typic alsam ple siz esof10 0 or 2 0 0 . T he d istortionsare greater w ith m ore f eed b ac k (c om pare DG P 2 B w ith 2 A) and low er w ith low er integration(c om pare DG P 2 A w ith 2 C). T he G onzalo and G ranger proc ed ure isslightly und er-siz ed w henever c ointegrating vec torsare estim ated , asinDG P 2 A to 2 C, evenat large sam ple siz esof4 0 0 . T hisislikely a c onsequence ofusing c ritic alvaluesthat are stric tly not applic ab le w ith c onstruc ted asopposed to raw series.It isinprinciple possib le to ad just the c ritic alvaluesf or siz e b ut since suc h c orrec tionsw ould b e ad hoc , the siz e d istortionsare not great,and the generalpointsc anb e m ad e inthe ab sence ofsuch siz e c orrec tions,w e d o not pursue thisf urther. T ab le 5 (und er id entic alhead ings) provid esthe pow er ofthe test proc ed uresf or DG P 2 A to 2 C.T husf or DG P 2 A to 2 C the rejec tionf requenciesofrLL = 0 are given inthe c olum nhead ed LL,w hile the rem ainingc olum nsprovid e the sam e inf orm ation f or the other tests.Allthe testshave pow er approac hing1 assam ple siz e increases. B road ly speakinghow ever,the LL test isseento have the b est pow er propertiesf or a m ajority ofc ases, b oth at low , m ed ium and large sam ple siz es. T he test b ased onestim ating the J ohansenf ullsystem isthe least pow erf ul(although increasing rapid ly to 1) f or allb ut one ofthe c ases.T hisisto b e expec ted ,since w henthe LL nullissatis¯ed , estim ationofthe J ohansenf ullsystem involvesestim ating a large numb er ofunnec essary param eters.T hislead sto a lossine± c iency inestim ation ofim portant param etersand lossofpow er.9 E stim atingthe system unit b y unit is partly b ene¯c ial(b y c utting d ow nonthe numb er ofparam etersto b e estim ated ) although c ross-unit c onnec tionsvia the ® m atrix are not takeninto ac c ount.T he joint test theref ore oc c upiesthe m id d le ground interm sofpow er perf orm ance. T ab le 6 provid esd etailsofthe b iasand stand ard d eviationofthe estim atesof 9. See discussion ofresults ofT able 6 below.. 12.

(16) the c ointegratingvec torsf or eac h ofthe three d i®erent m ethod s. Interm sofb ias, evenf or T = 10 0 , the LL test and the J ohansentrac e statistic unit b y unit perf orm quite w ell.T he latter isslightly b etter thanthe f orm er, also interm sofe± c iency, i.e., low er stand ard errorsf or the estim ated c oe± c ients.T he f ullsystem J ohansen ranksthird f or b oth b iasand e± c iency.. 5 Simul ation sw henthe LL f ram ew orkisnot c orrec t From the previoussec tion,the LL test appearsto have the proper siz e,good pow er c om pared to other c ointegrationproc ed ures,and to yield som e e± c iency gainsw hen estim atingthe c ointegratingc oe± c ients.Y et,asw e stressed inthe introd uc tion,the LLf ram ew ork(and the evenm ore restric ted m od elsinthe panelunit root literature), m ay b e inappropriate inapplic ationsw ith m ac roec onom ic d ata. Inpartic ular, in thisc ontext there c anexist c ointegrating relationships ac rossthe units, and the unitsc anb e d rivenb y a d i®erent numb er ofc om m ontrend s.W e now evaluate the c onsequencesofthese tw o f eaturesonthe perf orm ance ofthe LL test,and c onsid er som e \d iagnostic tests"to spot their presence. W e use simulationsto und ertake thispart ofthe analysis,instead oflookingf or analytic alexpressionsofthe d istrib utionsund er the alternative.T he d i± c ultiesof d eriving d ensitiesinthe presence ofc ross-unit c ointegrationare noted b y P hillips (19 9 9 ) w ho c om m ent that w henthere are strongc orrelationsam ongthe unitsofa panel(asw ould apply inour c asesofinterest), stand ard c entrallim it theory and law soflarge numb ersw illno longer apply. M oreover our interest isnot solelyinthe lim it d istrib utions(w ith N and T approachingin¯nity) b ut w e w ish also to lookat the b ehaviour ofthese statistic sf or sm allN and m ed ium -siz ed T.T he pred ic tions. 13.

(17) ofasym ptotic theory are thusoflim ited use here.. 5. 1. Coin tegrationac rossthe un its. W henthe unitsare related b y c ointegratingrelationships,the hypothesisofa b lock d iagonal¯ is violated . Severalstruc tures f or ¯ are now possib le, and w e f oc us onthree ofthese inour simulationexperim ents. T he d ata generationproc esses are d esc rib ed inT ab le 7.InDG P 3 there existsonly one c ointegratingrelationship ac rossthe units,so that rLL = 0 and rJ = 1.T he DG P s3A,3B and 3C d i®er f or the load ingm atrix,® ,w hile in3D there ism ore stationarity.InDG P 4 w e c onsid er the c ase rLL = 0 and rJ = 2 , nam ely, tw o c ointegrating relationshipsac rossthe units, and none w ithinthe units.InDG P 5w e also allow f or w ithin-unit c ointegration,and keep one c ross-unit relationship, so that rLL = 1 and rJ = 3.Severalsub -c asesof DG P 4 and 5 are analyz ed ,w hic h d i®er f or the struc ture ofthe ® m atrix(ind ic ated b y A,B ,C) and f or the m agnitud e ofthe roots(D). T he rejec tionf requenciesofthe LL test are reported inthe LL c olum nsofT ab le 8.For DG P 3,w here rLL = 0 ; the prob ab ility ofrejec tingrLL = 0 quic kly increases tow ard sone,w hile that ofrejec tingrLL = 1 isinthe range 0 :2 6¡0 :2 8 w henT = 4 0 0 . For DG P 4 also the prob ab ility ofrejec tingrLL = 1 increasesto one,and the sam e istrue f or DG P 5. T hese resultsind ic ate a m assive rejec tionofthe proper null hypothesisofthe LL test inthe presence ofc ross-unit c ointegration.Cointegration ac rossunitsisw rongly attrib uted to c ointegrationw ithinunits. It isinteresting to evaluate w hether thisisalso a prob lem inthe unit b y unit analysisusingthe J ohansentrac e test.Ind eed ,f rom the c olum ns\unit 1"and \unit 2 "ofT ab le 8,there are c asesw henthe siz e ofthe test isseverelyb iased upw ard s,even ifthe highest rejec tionprob ab ilitiesofrLL = 0 are muc h sm aller thanthose f rom LL, ab out :38 f or T = 4 0 0 .T he d istortionisrelated to the c ointegrating relationships. 14.

(18) a®ec tingseveralvariab lesofthe system (c asesB and C).Asa c onsequence,the joint proc ed ure b ased onthe c omb inationofthe unit b y unit resultsalso presentssiz e d istortionsinthese c ases(c olum n\joint").10 Inthe light ofthese results, it seem spartic ularly im portant either to use a f ull system approac h, or at least to evaluate w hether there isc ross-unit c ointegration. T he f orm er approac h though isnot f easib le w ith a larger numb er ofunits, so that the latter b ec om esevenm ore im portant.T he last three c olum nsofT ab le 8 report the siz e and pow er ofthe G onzalo and G ranger proc ed ure.T he pow er ofthe test (c olum nsrC = 0 f or allDG P sand also rC = 1 f or DG P 4 ) isingeneralrather low f or T = 10 0 ,b ut quic kly increasesw ith T and isof tenc lose to one f or T = 4 0 0 .T he siz e (c olum nsrC = 1 f or DG P s3 and 5, and c olum nrC = 2 f or DG P 4 ) isslightly low er thanthe nom inalvalue also f or T = 4 0 0 ,asinthe previoussec tion. O verall, these resultssuggest that the LL test, and to a c ertainextent unit b y unit analysis, c anlead to over-ac c eptance ofw ithin-unit c ointegrationw henthere existsc ross-unit c ointegration.O nthe other hand ,the latter isd etec ted quite w ell b y the G onzalo and G ranger test,at least f or large enough sam ple siz es.. 5. 2. Di®eren t c oin tegratingranksineac h un it. T he c asesofd i®erent w ithin-unit c ointegratingranksw e c onsid er are listed inT ab le 9. InDG P 6 it isr1 = 1, r2 = 0 ; f or DG P 7r1 = 2 , r2 = 0 ; f or DG P 8, r1 = 2 , r2 = 1.For eac h DG P w e also c onsid er d i®erent load ingm atric es® (sub c asesA,B , C),and low er stationary roots(D). T he rejec tionf requenciesofthe LL test are reported inthe LL c olum nsofT ab le 10 .W henT = 4 0 0 , rLL = 0 isrejec ted w ith prob ab ility one inallc ases, and also 10. N ote that,in ordertoreduce uncertainty, the same random numbers are used forallexperiments.H ence some values forthe unitby unitanalysis arethe same (e. g.those forunit2,D G P s 3A ,3B ,3D ,4A ,4B ,4D ).. 15.

(19) rLL = 1 isrejec ted w ith prob ab ility c lose to one f or DG P s7and 8. W henT is sm aller, the rejec tionf requenciesare low er insom e c ases, b ut overallthese results also ind ic ate a tend ency ofthe LL test to sub stantially und er rejec t w ithin-unit c ointegration. T he unit b y unit c ointegrationanalysesprovid e anind ic ationab out d i®erences inthe c ointegratingranks(c olum ns\unit 1"and \unit 2 "ofT ab le 10 ).Inf ac t w hen T = 4 0 0 ,the unit b y unit testsingeneralhave pow er c lose to one,and siz e c lose to the nom inallevel.T he exc eptionsare DG P s6C and 7C, w henthe siz e ofthe test islarger,ab out 0 :14 .T hisisthe c ase w here the c ointegratingvec torsf rom the ¯rst unit also a®ec t the sec ond unit. W henT issm aller, the pow er ofthe testsinsom e DG P sislow , inpartic ular those oftype A.Inthese c asesthe perf orm ance ofthe joint test (c olum n\joint") is also unsatisf ac tory,w ith sub stantialover rejec tionofthe c orrec t null. Insum m ary, w henthe c ointegrating ranks are d i®erent ac ross units, the LL test tend sto over-ac c ept the presence ofw ithin-unit c ointegration.Inother w ord s, c ointegrationinone unit b iasesthe test tow ard snonrejec tionofc ointegrationin the other unit.U nit b y unit analysisisac c urate w henT islarge enough, and c an provid e a usef uld iagnostic test f or d i®erent c ointegratingranksac rossunits. T he suggestionf or em piric alanalysisthat em ergesf rom thissec tionisto use f ull-system estim ationw henever possib le.Ifthisisnot f easib le,the ¯rst step should b e a unit b y unit c ointegrationanalysis.T he sec ond step isto test f or the ab sence ofc ross-unit c ointegrationb y m eans ofthe G onzalo and G ranger proc ed ure. If thisisac c epted , and the unit b y unit analysisd oesnot ind ic ate the presence of d i®erent ranksac rossunits,the third step isto applythe LL statistic ,that c anyield e± c iency gainsinterm sofhigher pow er and low er stand ard errorsf or the estim ated c ointegrating c oe± c ients. T hissequentialem piric alproc ed ure isim plem ented in. 16.

(20) Sec tion7, to evaluate inm ore d etailthe presence ofc ointegrationinthe interest rate d ata.. 6 Sum m ary ofresul tsf or m ore units Since the c ase ofN = 2 m ight b e thought to b e und uly restric tive, inthissec tion w e sum m ariz e the resultsofsimulationsw henthe numb er ofunitsinthe panelis increased to 4 or 8.Conventionalpanelstud iesof tenc onsid er c asesw here the num b er ofunitsisevenlarger, and a lim itationofthe highly param eteriz ed m aximum likelihood m ethod sc onsid ered inthispaper isthat thisc annot b e d one, unlessT isvery large w ith respec t to N or the d epend ence ac rossthe unitsisseverely restric ted .Im posingthese restric tionslead sto the f ram ew ork c onsid ered b y Larsson et al.(19 9 8),or b y K ao (19 9 9 ) and P ed roni (19 99 ) inter alia,w here the latter papersm ake the f urther assum ptionofone c ointegratingvec tor.T he trad eo® b etw een higher d im ensionality and a priori restric tionsisanissue that m eritsf urther investigation.Nevertheless,a panelof4 or 8 unitsisa reasonab le siz e w henc onsid ering f or exam ple d ata f rom the G 7c ountries,or f rom the largest ec onom iesofthe E uropeanU nion, or ec onom ic groupingssuc h asNAF T A.T he key ec onom ic ind ic ators ofthe m ainsec torsofanec onomy,ofgeographic ally d i®erentiated regionsw ithina c ountry,or ofthe d i®erent segm entsofthe lab our m arket c ould also b e investigated w ithinthisc ontext. Inord er to evaluate the siz e and pow er ofthe c om peting approac hes, w e m ad e use ofDG P sw hic h replic ated the struc turesused f or N = 2 . T hus, f or exam ple,. 17.

(21) DG P 2 A f or N = 4 ,took the f orm 0. 1. 0 0 C B ¡:1 0 B C B C B 0 0 0 0 C B C B C B C ¡:1 0 0 C B 0 B C B C B 0 0 0 0 C B C C ; ® = B B C B 0 0 ¡:1 0 C B C B C B C B 0 0 0 0 C B C B C B C B 0 0 0 ¡:1 C B C @ A 0 0 0 0. 0. B B B B B B B B B B B B ¯= B B B B B B B B B B B B @. 1. 0. 0. 1. 0 C C C ¡1 0 0 0 C C C C 0 1 0 0 C C C 0 ¡1 0 0 C C C : C 0 0 1 0 C C C C 0 0 ¡1 0 C C C C 0 0 0 1 C C A 0 0 0 ¡1. T he extensionsf or the other DG P sstud ies,nam ely,3to 8,are sim ilarly straightf orw ard .M oreover,f or DG P s3-5 w e also c onsid ered c asesw ith m ore thantw o c ointegratingvec torsac rossthe units,and f or DG P s6-8 d i®erent d egreesofc ointegration w hithineac h unit.A d etailed ac c ount ofthese results(availab le uponrequest) isnot presented here f or spac e c onstraints.T he f ollow ingim portant f eaturesare how ever w orth noting: 1) Asthe numb er ofunitsinincreased , the siz e ofthe LL testsb ec om esd istorted ifasym ptotic c ritic alvaluesare used .For exam ple,w hend ata are generated w ith ri = 1 f or eac h i, N = 4 , T = 2 0 0 , and the largest stationary rootsare 0 .8, the LL test rejec tsthe nullw ith f requency of0 .19 at a c on¯d ence levelof5% .T his d istortionb ec om esvery severe f or N = 8, w ith the rejec tionf requency increasing to 0 .90 .T he d istortionalso increasesw ith the m agnitud e ofthe largest stationary roots, ifthe other f eaturesofthe DG P are lef t unchanged . T hus, w ith ri = 1 f or eac h i, N = 4 , T = 2 0 0 , and largest stationary rootsof0 .9, the c orrespond ing rejec tionf requency is0 .32 .. 18.

(22) T he issue ofsiz e d istortionsisevid ent inthe d isc ussioninLL oftheir simulations, lead ingthem to suggest the use of\B artlett c orrec tions"f or the test statistic .T he f orm ofthe c orrec tionisgivenb y E(C T ) CeT = C1 ; E(C 1 ) w here C 1 isthe asym ptotic c ritic alvalue and E(C T ) and E(C 1 ) are,respec tively, the expec tationsofthe ¯nite sam ple and asym ptotic d istrib utionsofthe test statistic . B oth E(C T ) and E(C 1 ) c anb e approxim ated b y simulations. O ur results suggest that the use ofCeT ise®ec tive inc orrec ting the siz e ofthe tests. Inother w ord s, CeT isc lose inm agnitud e to the c orrespond ingem piric alquantile ofthe d is-. trib utionofthe test statistic .T he pow er ofthe test w henusing CeT issatisf ac tory, c lose to one evenf or T = 2 0 0 .. 2 ) W e also investigated the perf orm ance ofthe LL proc ed ure inthe presence of c ross-unit c ointegrationand d i®erent c ointegrating ranks. Asb ef ore, w e ¯nd that eventhe presence off ew c ross-unit c ointegrating relationshipssub stantially b iases the LL test tow ard srejec tionofno w ithin-unit c ointegration, e.g., w ith T = 2 0 0 , ri = 0 f or eac h i,and one c ointegratingrelationship am ongallunits,the prob ab ility ofrejec tingthe nullri = 0 f or eac h iis0 .56w henN = 8.T hisc anb e c om pared w ith the ¯gure of0 .62 f or DG P 3A inT ab le 8.W e c onclud e f rom thisthat the im portance ofc ross-unit c ointegratingrelationshipsd oesnot d ec rease evenw henthe numb er of unitsincreases.For the c ase ofd i®erent ranksac rossthe units,the over-ac c eptance ofw ithin-unit c ointegrationreported inT ab le 10 and d isc ussed inSec tion5.2 is c on¯rm ed .T he d egree ofover-ac c eptance increasesw ith the numb er ofunitsw hic h have c ointegration. For exam ple, ifN = 4 , T = 2 0 0 , the rejec tionf requenciesof ri = 0 f or eac h iare 0 .18,0 .44 ,and 0 .69 f or,respec tively,(r1 = 1;r2 = r3 = r4 = 0 ), (r1 = r2 = 1;r3 = r4 = 0 ),and (r1 = r2 = r3 = 1;r4 = 0 ). 19.

(23) 3) Asf ar asthe variousversionsofthe J ohansentest are c oncerned (unit b y unit,joint,and system ),these retaintheir good siz e and pow er propertiesw henthe unitsd o not c ointegrate w ith eac h other.Inthe presence ofc ross-unit c ointegration, how ever, asf or N = 2 , the unit b y unit analysis(and theref ore the joint test) c an rejec t the nullofno c ointegrationtoo of ten.T he siz e ofthe G G test rem ainsslightly low er thanthe nom inalvalue,at ab out 0 .04 .Itspow er isrelatively low w henthere isone c ross-unit relationship, b ut quic kly increasesw ith the am ount ofc ross-unit c ointegrationand the numb er ofob servations. Insum m ary,it issatisf yingto c onclud e that once the B artlett c orrec ted c ritic al valuesare used f or the LLtest,the overallperf orm ance ofthe varioustestsc onsid ered rem ainsqualitatively unaltered f rom that reported ind etailf or N = 2 .. 7 E m piric alE xam pl esR evisited T he ¯rst step inthe sequentialproc ed ure w e suggested ab ove isunit b y unit analysis.Hence,w e spec if y sim ilar m od elsf or eac h c ountry to estim ate the unit-spec i¯c c ointegratingvec tors.W e thenc om plem ent the analysisb y extrac tingthe c om m on trend sf rom eac h unit and looking f or c ointegrationam ong these. F inally, w e run the LL test to evaluate w hether the outc om e isinline w ith the simulationresults. Detailed resultsare reported f or N = 2 , and a sum m ary ofthe m ain¯nd ingsis presented f or N = 8.. 7. 1. G erm an y an d T he Netherl and s. T he c hosenspec i¯c ationf or G erm any involves4 lagsand anunrestric ted c onstant. T he availab le sam ple period goesf rom M ay19 9 0 to J anuary19 99 ,f or a totalsam ple siz e of10 5 ob servations. Diagnostic stestsf or thisspec i¯c ationpoint tow ard sits c ongruence since only heterosked astic ity and norm ality ofthe resid ualsofthe short20.

(24) term rate are m arginally rejec ted .Cointegrationtestsf or eac h rankare reported in T ab le 11,¯rst panel,alongw ith the c orrespond ingc ritic alvalues. T he statistic sind ic ate that the c ointegratingrank inthe m od elf or G erm any is z ero. O ne c anthenuse b oth the short-and the long-term rate f or G erm any ina suc c essive step ofthe analysisb ased onthe G onzalo and G ranger m ethod . T he c hosenspec i¯c ationf or the Netherland sinvolves4 lagsand anunrestric ted c onstant.T he availab le sam ple period goesf rom M ay 199 0 to J anuary 19 9 9, f or a totalsam ple siz e of10 5ob servations.Diagnostic stestsf or thisspec i¯c ationind ic ate no evid ence ofm isspec i¯c ation. Cointegrationtestsf or each rank are reported in T ab le 11,sec ond panel,alongw ith the c orrespond ingc ritic alvalues. T he statistic sind ic ate that the c ointegratingrankinthe m od elf or T he Netherland sisalso z ero, w hic h isc onsistent w ith the f ull-system c ointegrationanalysis. Asa c onsequence,the analysisf or c ointegrationam ongG erm anand Dutc h interest ratesw ould have to b e c ond uc ted onthe f ullsystem .From the resultsinSec tion3, w e know that there existsone c rossc ountry c ointegrating relationship.Hence, w e expec t the LL test to b e b iased tow ard rejec tionofthe nullhypothesisrLL = 0 . For the LL m od elthe sam ple goesf rom M ay 19 9 0 to J anuary 19 9 9 ,w e leave the c onstant unrestric ted ,and use 4 lags.T he panelLR testsare reported inT ab le 12 , ¯rst panel. Inline w ith our expec tations,the tab le ind ic atesrejec tionofrLL = 0 at the 10 % , though there ism arginalnon-rejec tionofthe hypothesisat the 5% .Notic e also that w hatever c hoic e the investigator m akesonthe b asisofthe LL test, the resulting m od elw illb e m isspec i¯ed . IfrLL = 0 isc hosen, the system w illb e analyz ed in ¯rst d i®erences,thereb y overlookingthe existence ofc ointegrationb etw eenthe tw o c ountries.Ifinstead rLL = 1 isselec ted , the w rong c ointegrating relationshipsw ill b e includ ed inthe E CM m od els.. 21.

(25) 7. 2. G erm an y an d Den m ark. T he ¯rst tentative spec i¯c ationf or Denm ark involves4 lagsand anunrestric ted c onstant. T he availab le sam ple period goesf rom M ay 199 0 to J anuary 199 9 , f or a totalsam ple siz e of10 5 ob servations. W e are led to perf orm the ¯nalanalysis onthe red uc ed sam ple startinginO c tob er 19 9 3 b ec ause large outliersin19 9 3 provoke heterosked astic ity and non-norm ality inthe Danish short-term rate.T here is no evid ence ofm isspec i¯c ationinthe red uc ed sam ple going f rom O c tob er 19 9 3 to J anuary 199 9 ,exc ept f or non-norm ality inthe short-term rate. T he c ointegrationpropertiesd o not c hange inthe tw o sam ples, since the rank inb oth isf ound to b e z ero.B oth setsofresultsare reported inT ab le 11,third and f ourth panels.T he ¯nd ing ofz ero c ointegrating rank inDenm ark im pliesthat the G onzalo and G ranger analysisf or c ointegrationam ongG erm anand Danish interest ratesw ould also have to b e c ond uc ted onthe f ullsystem .From Sec tion3,w e know that there existsonly one c rossc ountry c ointegrating relationship, so that the LL test should againover-rejec t the nullofno c ointegration. T o avoid the troub lesom e period , w e only estim ate the LL m od elonthe sec ond sub sam ple, w ith 4 lagsand anunrestric ted c onstant.T he panelLR testsare reported inT ab le 12 , sec ond panel.T histim e the LL test rejec tsa c om m onunitb y unit rank ofz ero at 5% , asw ellasa c om m onrank ofone. T hisw ould lead the investigator to estim ate the m od elim posinga c om m onrankof2 ,w ith no c rossc ountryc ointegration.Since the onlyc ointegrationvec tor inthe f ullsystem involves variab lesf rom b oth c ountries,the resultsw ould prob ab ly b e very m islead ing.. 7. 3 M ore un its W e now increase the numb er ofunitsinthe panelto 8, b y includ ing France, Italy, U K , Spainand Austria inthe analysis. Asab ove, w e start b y analyz ing the f ull 22.

(26) system , w hic h includ es the short- and long-term interest rates f or eac h c ountry, giving a totalof16 variab les.O nly one lagw asinclud ed inthe unrestric ted VAR , b oth f or the sake ofparsim ony and b ec ause the resulting m od eld id not present any m ajor evid ence ofm isspec i¯c ation.11 T he J ohansentrac e test suggests the presence of9 c ointegratingrelationshipsac rossthe variab les.W e d o not attem pt to id entif ythem ,b ut w e note that thisoutc om e alread ysuggeststhat either som e units have c ointegratingrank larger thanone or that there exist c ross-unit c ointegrating relationships. Inord er to evaluate the ¯rst possib ility,w e perf orm ed a unit-b y-unit c ointegrationanalysis.Inline w ith our resultsf or G erm any,T he Netherland sand Denm ark, there appearsto b e no w ithin-unit c ointegrationinthe other c ountries, w ith the possib le exc eptionofSpain. T husthe ¯rst requirem ent f or the applic ationofthe LL test issatis¯ed . How ever, w e stillneed to verif y the presence or ab sence of c ross-unit c ointegration. G iventhat the nullhypothesisofz ero rank ineac h unit isac c epted , the G G test c anb e runw ith the originalset ofvariab lesand isequivalent to the trac e test inthe f ull-system analysis.T he outc om e ofsuc h a test c annow b e interpreted as ind ic ating the presence ofm any c ross-unit c ointegrating relationships. From the simulationexperim entsinSec tion5.1 and 6, w e know that the presence ofc rossunit c ointegrationc ansub stantially b iasthe LL test tow ard srejec tionofthe no c ointegrationhypothesis(ri = 0 f or each i, w hic h issupported b y the unit-b y-unit analysis) evenw henthe numb er ofunitsislarge.T he b iasisinf ac t m ore serious thanthat c aused b y the violationofnon-unif orm rank ineac h ind ivid ualunit. U sing c orrec ted c ritic alvaluesc alib rated to our m od el(N = 8, T = 10 0 , and the largest stationary rootsare 0 .6), w e ¯nd that the hypothesisri = 0 f or each i isrejec ted at the 10 % level, and ri = 1 f or eac h iisalso strongly rejec ted . T hus, 11. Fulldetails oftheestimations reported in this section are available from us upon request.. 23.

(27) w e have another exam ple w here the applic ationofthe panelc ointegrationtestsis prob lem atic b ec ause the m ainund erlying hypothesis ofno long rund epend ence ac rossthe unitsisviolated , lead ing usto \spuriously" c onclud e inf avour ofthe presence ofw ithin-unit c ointegration.. 8. Concl usions. U singd ata f or short- and long-term interest ratesina sm allpanelofO E CD c ountries, the restric tionsim plied b y the b loc k d iagonality of¯ that c harac teriz e the panelc ointegrationf ram ew ork (not only insystem sm ethod ssuc h asLL b ut also inthe panelanalogue ofE ngle-G ranger m ethod sasd eveloped b y K ao (199 9 ) and P ed roni(19 99 ) inter alia) c anb e very easily show nto b e violated .Since w e suspec t that thisisvery likely to b e the c ase w henanalyz ing m ac roec onom ic tim e series ac rossc ountries,w e should b e c aref ulinusingthese m ethod s. O ur simulationresultsf or N = 2 ;4 and 8 ind ic ate that w henthe hypotheses und erlying the LL f ram ew ork are satis¯ed , the LL test hasgood siz e and pow er properties, and of tenyield sgainsine± c iency relative to f ullsystem analysisf or estim ationofthe c ointegratingparam eters. T he c onsequencesofviolationsofthe LL assum ptionsc anhow ever b e serious. O ur results suggest that the LL test c anlead to sub stantialover-ac c eptance of w ithin-unit c ointegrationw henthere exists c ross-unit c ointegrationor w henthe ranksare d i®erent ac rossunits. Since testsf or unit rootsinpanelsand other existing testsf or c ointegrationc anb e re-interpreted w ithinthe generaliz ed system f ram ew ork, it f ollow sthat suc h testsalso rejec t non-stationarity too of ten. T hus, the c om m onem piric al¯nd ing inthe P P P and c onvergence literature, w hentested inpanelsofc ountries,nam ely that P P P hold sand c ountry G DP sc onverge,m ay b e a spuriousc onsequence ofsuc h violations.W e are takinga c loser look at thisissue 24.

(28) inon-goingresearc h. U nit b yunit analysishow ever isac c urate w henT islarge enough,and c anprovid e a usef uld iagnostic test f or d i®erent c ointegratingranksac rossunits,w hile c ross-unit c ointegrationisd etec ted b y the G onzalo and G ranger test,at least f or large enough sam ple siz es. T he suggestionf or em piric alanalysisthat theref ore em ergesisto use f ullsystem estim ationw henever possib le.Ifthisisnot f easib le,the ¯rst step should b e a unit b y unit c ointegrationanalysis.T he sec ond step isto test f or the presence ofc rossunit c ointegrationb y m eansofthe G onzalo and G ranger proc ed ure.Ifthisisrejec ted , and the unit b y unit analysisd oesnot ind ic ate the presence ofd i®erent ranksac ross units,the third step isto apply the LL statistic since thisc anyield e± c iency gains interm sofhigher pow er and low er stand ard errorsf or the estim ated c ointegrating c oe± c ients.. 25.

(29) R ef erences D oorni k,J. A. (19 9 9 ): O bjec t-O riented M atrixP rogramm ingU singO x, 3rd E d . T imb erlake ConsultantsP ress,Lond on.w w w .nu®.ox.ac .uk/U sers/Doornik. D oorni k,J. A.,and D . F. H e nd ry (19 97): M od ellingDynam ic SystemsU sing P c F im l9 for W ind ow s.T imb erlake Consultants,Lond on. Engle ,R .F.and G range r,C .W .(19 87): \Cointegrationand E rror-Correc tion: R epresentation,E stim ation,and T esting,"E conom etrica,55,2 51{2 76. Ev ans,P.,and G . K arras (19 96): \Convergence R evisited ," J ournalofM onetary E conomic s,37,2 4 9 {2 65. Franke l,J.,and A. R ose (19 9 6): \A P anelP rojec t onP urchasing P ow er P arity: M eanR eversionB etw eenand W ithinCountries," J ournalofInternational E conomic s,4 0 ,2 0 9{2 2 4 . G onzalo,J.(19 9 4 ): \F ive Alternative M ethod sofE stim atingLong-R unE quilib rium R elationships,"J ournalofE conom etric s,60 ,2 0 3{2 33. G onzalo,J.,and C . W . J. G range r (19 9 5): \E stim ationofCom m onLongM em ory Com ponentsinCointegrated System s," J ournalofB usinessand E conomic Statistic s,13,2 7{36. G roe n,J. J.,and F. R . K le i be rge n (19 9 9): \Likelihood -B ased Cointegration AnalysisinP anelsofVec tor E rror Correc tionM od els,"Disc ussionP aper Disc ussionP aper 55,T inb ergenInstitute. Im,K. S.,P.,and Y. M . H .and Sh i n(19 9 7): \T estingf or U nit R ootsinHeterogeneousP anels,"Disc ussionpaper,Departm ent ofApplied E c onom ic s,U niversity ofCamb rid ge. 26.

(30) Joh anse n,S. (199 5): Likelihood -B ased Inference inCointegrated Vec tor Autoregressive M od els.O xf ord U niversity P ress,O xf ord and New Y ork. K ao,C .(199 9 ): \SpuriousR egressionand R esid ual-B ased T estsf or Cointegration inP anelData,"J ournalofE conom etric s,9 0 ,1{4 4 . L arsson,R ., and J. L yh age n (19 99 ): \Likelihood -B ased Inf erence inM ultivariate P anelCointegrationM od els," Disc ussionP aper W orking P aper Seriesin E c onom ic sand F inance 331,Stoc kholm Sc hoolofE c onom ic sand F inance. (2 0 0 0 ): \T esting f or Com m onCointegrating R ank inDynam ic P anels," Disc ussionP aper W orkingP aper SeriesinE c onom ic sand F inance 378,Stoc kholm Sc hoolofE c onom ic s. Ä L arsson,R .,J. L yh age n,and M . L o th gre n (19 9 8): \Likelihood -B ased CointegrationT estsinHeterogeneousP anels,"Disc ussionP aper W orkingP aper Series inE c onom ic sand F inance 2 50 ,Stoc kholm Sc hoolofE c onom ic s. L e e ,K .E.,M .H .Pe saran,and R .Smi th (199 7): \G row th and Convergence in a M ulti-Country E m piric alStoc hastic Solow M od el," J ournalofApplied E conometric s,12 ,357{39 2 . L e v in,A.,and C . F. L i n(19 9 3): \U nit R oot T estsinP anelData: New R esults," Disc ussionP aper Disc ussionP aper 56,U niversity ofCalif ornia at SanDiego. M cKoske y,S., and C . K ao (19 9 8): \A R esid ual-B ased T est f or the Nullof CointegrationinP anelData,"E conom etric R eview s,17,57{84 . O C onne ll,P. (19 98): \T he overvaluationofpurchasing pow er parity," J ournal ofInternationalE conomic s,4 4 ,1{19. Pap e ll,D . (19 9 7): \Searc hing f or Stationarity: P urc hasing P ow er P arity und er the Current F loat,"J ournalofInternationalE conom ic s,4 3,313{32 3. 27.

(31) Pe d roni ,P. (19 9 9): \Critic alValues f or CointegrationT ests inHeterogeneous P anelsw ith M ultiple R egressors," O xford B ulletinofE conom ic sand Statistic s, 61 Spec ialIssue,653{70 . Ph i lli p s,P. C . B.and M oon,H . R . (19 9 9 ): \Linear R egressionLim it T heory f or Nonstationary P anelData,"E conom etrica,67,10 57{1111.. 28.

(32) Append ix. T ab le 1: Cointegrationanalysisf or G erm any and the Netherland s Ho: rank = r m axeigenvalue 95%. trac e. 9 5%. r= 0. 33:16¤¤. 2 7:1. 4 6:74. 4 7:2. r ·1. 10 :4 5. 2 1:0. 13:58. 2 9 :7. r ·2. 3:117. 14 :1. 3:12 7. 15:4. r ·3. 0 :0 10 7. 3:8. 0 :0 10 7 3:8. Variab l es. ® c oe± c ien ts. G erm any short. ¡0 :2 8589. 1. G erm any long. 0. 0. Netherland sshort. 0. ¡1. Netherland slong. 0. 0. (stan d ard errors). (0 :0 52 4 18). ¯ c oe± c ien ts. LR test ofrestric tions: Â 2(6) = 4 :84 36 [0 :564 0 ]. 29.

(33) T ab le 2 : Cointegrationanalysisf or G erm any and Denm ark Ho: rank= r M axeigenvalue 9 5%. T rac e. 9 5%. r= 0. 4 0 :0 4 ¤¤. 2 7:1. 56:77¤¤. 4 7:2. r ·1. 19:65. 2 1:0. 2 2 :83. 2 9:7. r ·2. 7:2 0 9. 14 :1. 7:2 88. 15:4. r ·3. 0 :0 7885. 3:8. 0 :0 7885 3:8. Variab l es. ® c oe± c ien ts. G erm any short. ¡0 :0 4 34 39. G erm any long. 0. Denm ark short. ¡0 :4 8171. 1. Denm ark long. ¡0 :0 9 12 65. 0. (stan d ard errors). ¯ c oe± c ien ts ¡1:19. (0 :0 18386). 0 (0 :11189 ). (0 :0 2 74 68). LR test ofrestric tions: Â 2(3) = 4 :9 176 [0 :1779 ]. 30.

(34) T ab le 3: Data G eneratingP roc essesf or Siz e and P ow er: B lock-d iagonal¯ DG P 1. 2A. 2B. 2C. rLL. rJ. 0. 0. 1. 1. 1. 2. 2. 2. ® 0. 0. 1. 0 C B ¡0 :1 B C B C B 0 0 C B C B C B C B 0 ¡0 :1 C B C @ A 0 0 0 1 0 C B ¡0 :1 B C B C B 0 0 C B C B C B C B 0 :1 ¡0 :1 C B C @ A 0 0 0 1 0 C B ¡0 :2 B C B C B 0 0 C B C B C B C B 0 ¡0 :2 C B C @ A 0 0. ¯0. R oots. 0. (1,1,1,1). 0. 1. (1,1,0 .9,0 .9). 0. 1. (1,1,0 .9,0 .9). 0. 1. (1,1,0 .8,0 .8). B 1 ¡1 0 0 C A @ 0 0 1 ¡1. B 1 ¡1 0 0 C @ A 0 0 1 ¡1. B 1 ¡1 0 0 C @ A 0 0 1 ¡1. rLL : Cointegratingrank f or eac h unit insense ofLarssonand Lyhagen rJ: Cointegratingrank f or f ullsystem insense ofJ ohansen. 31.

(35) T ab le 4 : Siz e oftestsw ith b loc k d iagonal¯ m atrix DG P. T. LL(a). J unit b y unit (b) unit 1 unit 2. 1. 2A. 2B. 2C. J system. (d ). G G. (e). joint(c). 10 0. 0 .064. 0 .04 8. 0 .058. 0 .056. 0 .065. 0 .051. 2 00. 0 .059. 0 .051. 0 .051. 0 .056. 0 .060. 0 .04 5. 400. 0 .057 0 .052. 0 .051. 0 .052. 0 .057. 0 .04 5. 10 0. 0 .09 8. 0 .04 8. 0 .04 9. 0 .815. 0 .02 2. 0 .02 1. 2 00. 0 .087 0 .055. 0 .052. 0 .14 2. 0 .051. 0 .02 4. 400. 0 .069. 0 .051. 0 .052. 0 .051. 0 .056. 0 .02 8. 10 0. 0 .118. 0 .04 9. 0 .035. 0 .887. 0 .02 4. 0 .038. 2 00. 0 .09 3 0 .055. 0 .04 6. 0 .384. 0 .04 5. 0 .033. 400. 0 .069. 0 .051. 0 .04 8. 0 .055. 0 .055. 0 .031. 10 0. 0 .09 8. 0 .054. 0 .054. 0 .136. 0 .057. 0 .02 7. 2 00. 0 .072. 0 .054. 0 .050. 0 .054. 0 .058. 0 .02 9. 400. 0 .060. 0 .050. 0 .053. 0 .050. 0 .053. 0 .031. (a): R ejection frequencies ofnullhypothesis at5% signi¯cance levelusingL L (1 999)asymptoticcriticalvalues,when L L statisticis calculated.. (b): R ejection frequencies, unit by unit, ofnullhypothesis at5% signi¯cance level, using Johansen (1 995)asymptoticcriticalvalues.. (c ): Calculated by setting the signi¯cance levelofthe unitby unitJohansen tests at 1 ¡ (0 :9 5)1=2 ,sothatthesigni¯canceofthe jointtestis 5%. (d ): R ejection frequencies at5% signi¯cance levelusingJohansen (1 995)asymptoticcritical values whensystem is estimatedwithoutrestriction(apartfrom thosenecessaryforidenti¯cation).. (e): R ejection frequencies ofcointegration amongthe common trends derived from the unit by unitcointegration analysis when the nullhypothesis is accepted in each unit.. 32.

(36) T ab le 5: P ow er oftestsw ith b lock d iagonal¯ m atrix DG P. T. LL(a). J unit b y unit (b) unit 1 unit 2. 2A. 2B. 2C. J system. joint(c). 10 0. 0 .513 0 .576. 0 .575. 0 .22 9. 0 .136. 200. 0 .978. 0 .982. 0 .985. 0 .914. 0 .635. 400. 1.00 0. 1.00 0. 1.00 0. 1.00 0. 0 .99 9. 10 0. 0 .69 9. 0 .576. 0 .367. 0 .151. 0 .165. 200. 0 .99 9. 0 .982. 0 .792. 0 .667. 0 .52 2. 400. 1.00 0. 1.00 0. 0 .998. 0 .99 5. 0 .974. 10 0. 0 .983 0 .985. 0 .983. 0 .917. 0 .658. 200. 1.00 0. 1.00 0. 1.00 0. 1.00 0. 0 .99 9. 400. 1.00 0. 1.00 0. 1.00 0. 1.00 0. 1.00 0. (a) ¡(d ): See notesto T ab le 4. 33. (d ).

(37) T ab le 6: E stim atesof¯ w ith b loc k d iagonal¯ m atrix DG P. 2A. T. 100. 20 0. 40 0. 2B. 100. 20 0. 40 0. 2C. 100. 20 0. 40 0. LL. ¯ ¯ ¯d¯ ¯ ¯ 12¯ ¯¯ 1 1 ¯. mean 1 . 1 69. (a). ¯ ¯ ¯d ¯ ¯¯ 22 ¯ ¯¯ 21 ¯. J unitby unit(b) ¯ ¯ ¯d¯ ¯ ¯ 12¯ ¯¯ 1 1 ¯. ¯ ¯ ¯d ¯ ¯¯ 22 ¯ ¯¯ 21 ¯. J system (c). ¯ ¯ ¯d ¯ ¯¯ 1 2 ¯ ¯¯ 1 1 ¯. ¯ ¯ ¯d ¯ ¯¯ 22 ¯ ¯¯ 21 ¯. 0. 961. 1. 0 58. 1. 001. 1. 1 41. 1. 005. s. d. 8. 837 1 0 . 730. 2. 71 6. 2. 783. 1 7. 1 50. 20 . 564. mean 1 . 014. 1. 0 29. 1. 002. 1. 004. 1. 011. 0. 871. s. d. 1 . 0 26. 3. 0 20. 0. 1 59. 0. 1 61. 4. 779. 5. 60 5. mean 0 . 999. 1. 000. 0. 999. 1. 000. 0. 781. 1. 0 52. s. d. 0 . 0 68. 0. 0 70. 0. 0 65. 0. 0 64. 20 . 829. 1 8. 631. mean 1 . 0 52. 1. 0 58. 1. 0 58. 1. 371. 1. 0 92. 0. 937. s. d. 2. 884. 7. 0 68. 2. 71 6. 33. 670. 9. 922. 1 8. 448. mean 1 . 002. 0. 823. 1. 002. 1. 0 36. 1. 1 50. 0. 540. s. d. 0 . 1 26. 1 8. 0 62. 0. 1 60. 4. 81 0. 10. 389. 1 5. 1 84. mean 1 . 000. 1. 000. 0. 999. 1. 0 45. 1. 007. 0. 731. s. d. 0 . 0 52. 0. 0 75. 0. 0 65. 0. 1 26. 1. 117. 1. 437. mean 1 . 004. 1. 015. 1. 002. 1. 005. 0. 772. 0. 81 8. s. d. 0 . 334. 0. 839. 0. 1 53. 0. 1 73. 23. 71 3. 6. 589. mean 1 . 000. 0. 999. 1. 000. 1. 000. 0. 942. 0. 91 2. s. d. 0 . 0 69. 0. 112. 0. 0 66. 0. 0 63. 3. 276. 3. 499. mean 1 . 000. 1. 000. 1. 000. 1. 000. 1. 363. 0. 496. s. d. 0 . 0 32. 0. 0 30. 0. 0 31. 0. 0 30. 38. 559. 37. 520. (a): M onteCarlomeanandstandarddeviationofestimatednormalisedcointegratingparameterin each unitusingL L method.. (b): M onteCarlomeanandstandarddeviation ofestimatednormalisedcointegratingparameterin each unitusingJohansen unitby unit.. (c ):M onteCarlomean and standard deviation ofestimated normalised cointegratingparameterin each unitusingJohansen fullsystem.. 34.

(38) T ab le 7: Desc riptionofData G eneratingP roc essesf or R ejec tionFrequencies: Nonb loc k-d iagonal¯ DG P 3A 3B 3C 3D. 4A. 4B. 4C. 4D. 5A. 5B. 5C. 5D. rLL. rJ. 0. 1. 0 0 0. 0. 0. 0. 0. 1. 1. 1. 1. 1 1. µ µ. 1. 2. 2. 2. 2. 3. 3. 3. 3. 0. µ. ®0 ¡0 :1 0 0 0. ¶. ¡0 :1 ¡0 :1 0 0 ¡0 :1 0 0 ¡0 :1 µ ¶ ¡0 :2 0 0 0. ¶ ¶ 1. ¡0 :1 0 0 C A ¡0 :1 0 0 0 0 1 ¡0 :1 0 0 C B 0 @ A ¡0 :1 ¡0 :1 0 0 0 1 ¡0 :1 0 0 C B 0 @ A ¡0 :1 0 ¡0 :1 0 0 1 ¡0 :2 0 0 C B 0 @ A ¡0 :2 0 0 0 0 1 0 0 C B ¡0 :1 0 B C B C B 0 C 0 ¡ 0 : 1 0 B C @ A 0 0 0 0 :1 1 0 0 ¡0 :1 C B ¡0 :1 0 B C B C C B 0 0 ¡ 0 : 1 0 C B A @ 0 0 0 0 :1 0 1 0 ¡0 :1 C B ¡0 :1 0 C B B C C B 0 0 ¡ 0 : 1 0 B C A @ 0 0 ¡0 :1 0 :1 0 1 0 0 C B ¡0 :2 0 35C B B C B 0 C 0 ¡ 0 : 2 0 B C @ A B @. 0. µ µ µ µ. 0. ¯0 1 0 ¡1 0 1 0 ¡1 0 1 0 ¡1 0 1 0 ¡1 0. B 0 @ 1 0 B 0 @ 1 0 B 0 @ 1 0 B 0 @ 1 0 B 1 B B B 0 B @ 0 0 B 1 B B B 0 B @ 0 0 B 1 B B B 0 B @ 0 0 B 1 B B B 0 B @. 1. 0. 0 ¡1 1. 0. 0 ¡1 1. 0. 0 ¡1 1. 0. 0 ¡1 ¡1 0 0. 1. 1. 0. ¡1 0 0. 1. 1. 0. ¡1 0 0. 1. 1. 0. ¡1 0 0. 1. ¶ ¶ ¶ ¶. 1. ¡1 C A 0 1 ¡1 C A 0 1 ¡1 C A 0 1 ¡1 C A 0 1 0 C C C ¡1 C C A ¡1 1 0 C C C ¡1 C C A ¡1 1 0 C C C ¡1 C C A ¡1 1 0 C C C ¡1 C C A. R oots (1 ,1 ,1 ,0 . 9) (1 ,1 ,1 ,0 . 9) (1 ,1 ,1 ,0 . 9) (1 ,1 ,1 ,0 . 8). (1 ,1 ,0 . 9,0 . 9). (1 ,1 ,0 . 9,0 . 9). (1 ,1 ,0 . 9,0 . 9). (1 ,1 ,0 . 8,0 . 8). (1 ,0 . 9,0 . 9,0 . 9). (1 ,0 . 9,0 . 9,0 . 9). (1 ,0 . 9,0 . 9,0 . 9). (1 ,0 . 8,0 . 8,0 . 8).

(39) T ab le 8: R ejec tionf requenciesoftestsw ith non-b loc k d iagonal¯ m atrix DG P. LL(a). T. J unitby unit(b). unit 1. 3A. 3B. 3C. 3D. 4A. 4B. 4C. 4D. unit 2. G G. (d ). joint. rLL = 0. rLL = 1. rJ= 0. rJ= 1. rJ= 0. rJ= 1. r= 0. r= 1. r= 2. 10 0. 0 .231. 0 .10 5. 0 .054. 0 .00 4. 0 .054. 0 .00 6 0 .055 0 .210. 0 .02 0. -. 200. 0 .62 1. 0 .235. 0 .051 0 .00 4. 400. 0 .993. 0 .281. 0 .053 0 .00 3 0 .051 0 .00 4. 10 0. 0 .583. 0 .22 1. 0 .263 0 .02 6 0 .054. 200. 0 .970. 0 .288. 0 .32 4. 400. 1.00 0. 0 .275. 10 0. 0 .580. 200. 0 .053 0 .00 5 0 .056 0 .60 5 0 .036 0 .050. -. 0 .99 3 0 .04 1. -. 0 .50 2. 0 .037. -. 0 .02 6 0 .053 0 .00 5 0 .276 0 .960. 0 .039. -. 0 .369. 0 .02 8 0 .051 0 .00 5 0 .32 0. 1.00 0. 0 .038. -. 0 .218. 0 .054. 0 .00 4. 0 .265 0 .02 3 0 .22 1. 0 .50 0. 0 .035. -. 0 .969. 0 .285. 0 .051 0 .00 4. 0 .34 1 0 .030. 0 .960. 0 .04 2. -. 400. 1.00 0. 0 .264. 0 .053 0 .00 3 0 .382. 0 .02 9. 0 .333 1.00 0. 0 .031. -. 10 0. 0 .62 8. 0 .236. 0 .056 0 .00 5 0 .054. 0 .00 6 0 .067 0 .613 0 .038. -. 200. 0 .995. 0 .285. 0 .060. 0 .00 5 0 .053 0 .00 5 0 .059. 0 .99 4. 0 .04 0. -. 400. 1.00 0. 0 .271. 0 .059. 0 .00 3 0 .051 0 .00 5 0 .052. 1.00 0. 0 .038. -. 10 0. 0 .512. 0 .350. 0 .063 0 .00 8 0 .054. 200. 0 .977. 0 .937 0 .063 0 .00 6 0 .053 0 .00 5 0 .058. 400. 1.00 0. 1.00 0. 0 .058 0 .00 6 0 .051 0 .00 5 0 .04 5 1.00 0. 10 0. 0 .696. 0 .416. 0 .164. 0 .02 5 0 .054. 200. 0 .998. 0 .861. 0 .22 2. 0 .034. 0 .053 0 .00 5 0 .171. 400. 1.00 0. 1.00 0. 0 .264. 0 .034. 0 .051 0 .00 5 0 .19 7 1.00 0. 10 0. 0 .84 3. 0 .638. 0 .14 2. 0 .02 3 0 .265 0 .02 3 0 .24 1. 200. 0 .997. 0 .99 3. 0 .180. 0 .02 6 0 .34 1 0 .30 2. 400. 1.00 0. 1.00 0. 0 .20 3 0 .02 6 0 .382. 0 .02 9. 10 0. 0 .979. 0 .939. 0 .066 0 .00 6 0 .054. 200. 1.00 0. 1.00 0. 0 .070. 400. 1.00 0. 1.00 0. 0 .070. 0 .00 6 0 .219. 0 .288. 0 .00 6 0 .058. 0 .49 5 0 .115 0 .016 0 .976 0 .632 1.00 0. 0 .038 0 .033. 0 .00 6 0 .12 6 0 .663 0 .12 1 0 .014 0 .99 7 0 .470. 0 .02 6. 0 .966 0 .02 0. 0 .79 8 0 .24 0. 0 .02 0. 0 .30 6 0 .99 9. 0 .837 0 .02 6. 0 .358. 1.00 0. 0 .02 1. 0 .00 6 0 .057 0 .978 0 .630. 0 .036. 1.00 0. 0 .00 5 0 .053 0 .00 5 0 .054 1.00 0 36 0 .00 7 0 .051 0 .00 5 0 .04 7 1.00 0. 0 .99 9. 0 .032. 1.00 0. 0 .02 6.

(40) (c ontinued ). 5A 10 0. 0 .771 0 .364. 0 .584. 0 .04 9. 2 00. 1.00 0. 0 .84 2. 0 .984. 0 .056 0 .986 0 .062. 400. 1.00 0. 1.00 0. 1.00 0. 0 .051 1.00 0. 10 0. 0 .954. 0 .42 4. 0 .584. 0 .04 9. 2 00. 1.00 0. 0 .89 6 0 .984. 0 .056 0 .99 3 0 .083 0 .12 1. 0 .90 5 0 .013 -. 400. 1.00 0. 1.00 0. 0 .051 1.00 0. 1.00 0. 10 0. 0 .99 3 0 .687 0 .584. 0 .04 9. 2 00. 1.00 0. 0 .99 7 0 .984. 400. 1.00 0. 1.00 0. 1.00 0. 5D 10 0. 1.00 0. 0 .84 9. 0 .985 0 .053 0 .99 1 0 .071 0 .12 0. 2 00. 1.00 0. 1.00 0. 1.00 0. 0 .054. 1.00 0. 0 .063 0 .04 6 1.00 0. 0 .02 0. 400. 1.00 0. 1.00 0. 1.00 0. 0 .04 9. 1.00 0. 0 .061 0 .039. 0 .017 -. 5B. 5C. 1.00 0. 0 .59 5 0 .061 0 .80 3 0 .52 0 0 .131. 0 .039. -. 0 .956 0 .02 3 -. 0 .058 0 .04 1. 1.00 0. 0 .018 -. 0 .715 0 .068 0 .751. 0 .714. 0 .02 4. 0 .10 3 0 .068. 0 .911 0 .172. 0 .00 9. -. -. 0 .681. 0 .698 0 .016 -. 0 .056 0 .99 9. 0 .22 1 0 .20 4. 0 .92 8 0 .00 6 -. 0 .051 1.00 0. 0 .270. 1.00 0. 0 .199. 0 .00 3 -. 0 .956 0 .02 3 -. 1.00 0. -. (a):R ejectionfrequencies ofrLL = 0 or1 at5% signi¯cancelevelusingL L (1 999)asymptotic criticalvalues,when L L statisticis calculated. (b):R ejectionfrequencies,unitbyunit,ofrank0 or1 at5% signi¯cancelevel,usingJohansen (1 995)asymptoticcriticalvalues.. (c ): R ejection frequencies ofnullhypothesis using jointtest, where the signi¯cance levelof the unitby unitJohansen tests is setat1 ¡(0 :9 5)1=2 .. (d ) : R ejection frequencies ofcointegration amongcommon trends derived from unitby unit analysis when the nullhypothesis is accepted in each unit.. 37.

(41) T ab le 9: Desc riptionofData G eneratingP roc essesf or R ejec tionFrequencies: B lock d iagonal¯ m atrixb ut d i®erent rank ineac h unit DG P. rJ. 6A. 1. 6B 6C 6D. 1 1. µ µ. 1. 7A. 2. 7B. 2. 7C. 2. 7D. 2. 8A. 3. 8B. 3. 8C. 3. 8D. 3. 0. µ. ®0 ¡0 :1 0 0 0. ¶. ¡0 :2 ¡0 :1 0 0 ¡0 :1 0 ¡0 :1 0 µ ¶ ¡0 :2 0 0 0. ¶ ¶ 1. 0 0 0 C B ¡0 :1 A @ 0 ¡0 :1 0 0 0 1 B ¡0 :1 0 :1 0 0 C @ A 0 ¡0 :1 0 0 0 1 0 0 C B ¡0 :1 0 :1 @ A 0 ¡0 :1 0 :1 0 0 1 0 0 0 C B ¡0 :2 @ A 0 ¡0 :2 0 0 1 0 0 0 0 C B ¡0 :1 C B C B C B 0 ¡ 0 : 1 0 0 B C @ A 0 0 ¡0 :1 0 0 1 0 0 C B ¡0 :1 0 :1 B C B C B 0 C ¡ 0 : 1 0 0 B C @ A 0 0 ¡0 :1 0 0 1 0 0 C B ¡0 :1 0 :1 B C C B B 0 C ¡ 0 : 1 0 : 1 0 B C @ A 0 0 ¡0 :1 0 :1 0 1 0 0 0 C B ¡0 :2 B 3C 8 B C B 0 C ¡ 0 : 2 0 0 B C @ A. µ µ µ µ. 0. ¯0 1 ¡1 0 0 1 ¡1 0 0 1 ¡1 0 0 1 ¡1 0 0. B 1 @ 0 0 B 1 @ 0 0 B 1 @ 0 0 B 1 @ 0 0 B 1 B B B 0 B @ 0 0 B 1 B B B 0 B @ 0 0 B 1 B B B 0 B @ 0 0 B 1 B B B 0 B @. ¶ ¶ ¶ ¶. 1. 0 0 0 C A 1 0 0 1 0 0 0 C A 1 0 0 1 0 0 0 C A 1 0 0 1 0 0 0 C A 1 0 0 1 0 0 0 C C C 1 0 0 C C A 0 1 0 1 0 0 0 C C C 1 0 0 C C A 0 1 0 1 0 0 0 C C C 1 0 0 C C A 0 1 0 1 0 0 0 C C C 1 0 0 C C A. R oots (1 ,1 ,1 ,0 . 9) (1 ,1 ,1 ,0 . 9) (1 ,1 ,1 ,0 . 9) (1 ,1 ,1 ,0 . 8). (1 ,1 ,0 . 9,0 . 9). (1 ,1 ,0 . 9,0 . 9). (1 ,1 ,0 . 9,0 . 9). (1 ,1 ,0 . 8,0 . 8). (1 ,0 . 9,0 . 9,0 . 9). (1 ,0 . 9,0 . 9,0 . 9). (1 ,0 . 9,0 . 9,0 . 9). (1 ,0 . 8,0 . 8,0 . 8).

(42) T ab le 10 : R ejec tionf requenciesoftestsw ith b loc k d iagonal¯ m atrix b ut d i®erent rank ineac h unit DG P. T. LL. (a). J unitby unit(b) unit1. 6A. 6B. 6C. 6D. 7A. 7B. unit2. joint(c). rLL = 0. rLL = 1. rJ= 0. rJ= 1 rJ= 0. rJ= 1. 100. 0. 224. 0. 0 62. 0. 584. 0. 0 49. 0. 443. 0. 0 22. 0. 592. 20 0. 0. 61 5. 0. 0 93. 0. 984. 0. 0 56. 0. 0 53. 0. 005. 0. 101. 40 0. 0. 994. 0. 0 96. 1. 000. 0. 0 51. 0. 0 51. 0. 004. 0. 0 51. 100. 0. 962. 0. 1 26. 1. 000. 0. 0 55. 0. 0 58. 0. 006. 0. 0 60. 20 0. 1. 000. 0. 108. 1. 000. 0. 0 54. 0. 0 53. 0. 005. 0. 0 59. 40 0. 1. 000. 0. 0 99. 1. 000. 0. 0 55. 0. 0 51. 0. 005. 0. 0 54. 100. 0. 587. 0. 115. 0. 584. 0. 0 49. 0. 1 31. 0. 017. 0. 61 8. 20 0. 0. 969. 0. 116. 0. 984. 0. 0 56. 0. 1 38. 0. 015. 0. 1 57. 40 0. 1. 000. 0. 102. 1. 000. 0. 0 51. 0. 1 45. 0. 014. 0. 1 27. 100. 0. 635. 0. 100. 0. 985. 0. 0 53. 0. 0 54. 0. 006. 0. 0 96. 20 0. 0. 995. 0. 0 94. 1. 000. 0. 0 54. 0. 0 53. 0. 005. 0. 0 58. 40 0. 1. 000. 0. 0 90. 1. 000. 0. 0 49. 0. 0 51. 0. 005. 0. 0 50. 100. 0. 1 31. 0. 0 56. 0. 60 3. 0. 40 7 0 . 0 54. 0. 006. 0. 788. 20 0. 0. 499. 0. 30 6. 0. 998. 0. 987 0 . 0 53. 0. 005. 0. 0 94. 40 0. 0. 997. 0. 981. 1. 000. 1. 000. 0. 0 51. 0. 005. 0. 0 27. 100. 0. 21 8. 0. 0 79. 0. 796. 0. 41 7 0 . 0 54. 0. 006. 0. 767. 20 0. 0. 744. 0. 328. 1. 000. 0. 922. 0. 0 53. 0. 005. 0. 228. 40 0. 1. 000. 0. 866. 1. 000. 1. 000. 0. 0 51. 0. 005. 0. 0 26. 39.

(43) (c ontinued ). 7C. 7D. 8A. 8B. 8C. 8D. 100. 0. 493. 0. 232. 0. 796. 0. 41 7 0 . 119. 0. 014. 0. 769. 20 0. 0. 972. 0. 81 4. 1. 000. 0. 922. 0. 1 37 0 . 015. 0. 259. 40 0. 1. 000. 1. 000. 1. 000. 1. 000. 0. 1 43. 0. 014. 0. 102. 100. 0. 51 8. 0. 322. 0. 997 0 . 985. 0. 0 54. 0. 006. 0. 0 89. 20 0. 0. 997 0 . 984. 1. 000. 1. 000. 0. 0 53. 0. 005. 0. 0 31. 40 0. 1. 000. 1. 000. 1. 000. 1. 000. 0. 0 51. 0. 005. 0. 0 27. 100. 0. 225. 0. 0 86. 0. 60 3. 0. 40 7 0 . 20 9. 0. 0 31. 0. 975. 20 0. 0. 849. 0. 41 0. 0. 998. 0. 987 0 . 657 0 . 0 48. 0. 559. 40 0. 1. 000. 0. 982. 1. 000. 1. 000. 0. 999. 0. 0 53. 0. 0 38. 100. 0. 334. 0. 114. 0. 796. 0. 41 7 0 . 20 9. 0. 0 31. 0. 970. 20 0. 0. 949. 0. 365. 1. 000. 0. 922. 0. 657 0 . 0 48. 0. 625. 40 0. 1. 000. 0. 865. 1. 000. 1. 000. 0. 999. 0. 0 37. 100. 0. 475. 0. 227 0 . 796. 0. 41 7 0 . 0 94. 0. 017 0. 985. 20 0. 0. 989. 0. 81 0. 1. 000. 0. 922. 0. 235. 0. 0 33. 0. 892. 40 0. 1. 000. 1. 000. 1. 000. 1. 000. 0. 737 0 . 0 51. 0. 452. 100. 0. 864. 0. 422. 0. 997 0 . 985. 0. 670. 0. 0 52. 0. 545. 20 0. 1. 000. 0. 984. 1. 000. 1. 000. 0. 999. 0. 0 51. 0. 0 33. 40 0. 1. 000. 1. 000. 1. 000. 1. 000. 1. 000. 0. 0 52. 0. 0 27. (a) ¡(c ): Seenotes toT able8. 40. 0. 0 53.

(44) T ab le 11: U nit b y unit c ointegrationanalysis Ho: rank= r M axeigenvalue 9 5% G erm an y,19 9 0 -19 9 9 r= 0 4 :86 14 :1 r ·1 0 :0 2 539 3:8 Ho:rank= r M axeigenvalue 9 5% T he Netherl an d s,19 9 0 -19 9 9 r= 0 2 :9 84 14 :1 r ·1 0 :10 33 3:8 Ho: rank= r M ax.eigenvalue 9 5% Denm ark,19 9 0 -19 9 9 r == 0 3:30 6 14 :1 r ·1 0 :2 79 8 3:8 Ho: rank= r M ax.eigenvalue 9 5% Denm ark,19 9 3-19 9 9 r= 0 7:59 6 14 :1 r ·1 0 :579 1 3:8. 41. T rac e 95% 4 :886 15:4 0 :0 2 5 3:8 T rac e 9 5% 3:0 87 15:4 0 :0 9 5 3:8 T rac e 9 5% 3:967 15:4 0 :30 9 3:8 T rac e 9 5% 8:175 15:4 0 :579 3:8.

(45) T ab le 12 : P anelLL tests ran k. LR test. 9 5% c ritic alval ues. G erm any - T he Netherland s,199 0 -19 9 9 rLL = 0. 4 6:737. 4 7:4 9. rLL = 1 11:658. 18:2 1. ran k. 9 5% c ritic alval ues. LR test. G erm any - Denm ark,19 9 3-199 9 rLL = 0. 62 :74 8. 4 7:4 9. rLL = 1 2 2 :888. 18:2 1. 42.

(46)