协整检验的临界值

2023-02-27 来源：画鸵萌宠网

QED

Queen’sEconomicsDepartmentWorkingPaperNo.1227

CriticalValuesforCointegrationTests

JamesG.MacKinnonQueen’sUniversity

DepartmentofEconomicsQueen’sUniversity94UniversityAvenueKingston,Ontario,Canada

K7L3N6

1-2010

CriticalValuesforCointegrationTests

JamesG.MacKinnon

DepartmentofEconomics

Queen’sUniversityKingston,Ontario,Canada

K7L3N6

jgm@econ.queensu.ca

http://www.econ.queensu.ca/faculty/mackinnon/

Abstract

Thispaperprovidestablesofcriticalvaluesforsomepopulartestsofcointegrationandunitroots.Althoughthesetablesarenecessarilybasedoncomputersimulations,theyaremuchmoreaccuratethanthosepreviouslyavailable.Theresultsofthesimulationexperimentsaresummarizedbymeansofresponsesurfaceregressionsinwhichcriticalvaluesdependonthesamplesize.Fromtheseregressions,asymptoticcriticalvaluescanbereadoﬀdirectly,andcriticalvaluesforanyﬁnitesamplesizecaneasilybecomputedwithahandcalculator.Addedin2010version:Anewappendixcontainsadditionalresultsthataremoreaccurateandcovermorecasesthantheonesintheoriginalpaper.

January1990

ReissuedJanuary2010withadditionalresults

ThispaperoriginallyappearedasUniversityofCaliforniaSanDiegoDiscussionPaper90-4,whichisapparentlynolongeravailableontheweb.ItwaswrittenwhileIwasvisitingUCSDinthefallof1989andearly1990andsupported,inpart,bygrantsfromtheSocialSciencesandHumanitiesResearchCouncilofCanada.IamgratefultoalloftheeconometriciansatUCSDforprovidingahospitableresearchenvironmentandtoRobEngleandthelateCliveGrangerforcommentsonanearlierversion.

Forward

Ispenttheacademicyearof1989-1990onsabbaticalattheUniversityofCaliforniaSanDiego.SinceRobertEngleandCliveGrangerweretherepermanently,andDavidHendry,SørenJohansen,andTimoTer¨asvirtawerealsovisiting,therewasinevitablymuchdiscussionaboutcointegration.Iwassurprisedtoﬁndthat,formostofthetests,accuratecriticalvaluesdidnotthenexist.IthereforesetouttocalculatethemfortheDickey-FullerandEngle-Grangertests,andtheresultwasthispaper.ThepaperoriginallyappearedasUniversityofCaliforniaSanDiegoDiscussionPaperNo.90-4.Formanyyears,abitmappedPDFofitthatwasmissingthecoverpagecouldbefoundontheUCSDEconomicswebsite,butitseemstohavevanishedsometimeduring2009.ThepaperwaslaterpublishedinabookeditedbyRobEngleandCliveGranger;seethereferences.

IhavemadethisversionavailableasaQueen’sEconomicsDepartmentWorkingPapersothatresearcherswhosearchfortheUCSDworkingpaperonthewebwillbeabletoﬁndit.IhavealsoaddedanappendixinwhichIprovidenewresultsthataremuchmoreaccurateandcovermorecasesthantheonesintheoriginalpaper.Theenormousincreasesincomputingpoweroverthepasttwentyyearsmadethisquiteeasytodo.Ihavealsoaddedsomeadditionalreferencestoworksthatdidnotexistwhenthepaperwasoriginallywritten.

AfterIwrotethispaper,Idevelopedmoreadvancedmethodsforcalculatingapproxi-matedistributionfunctionsofteststatisticssuchastheonesdealtwithinthispaper;see,inparticular,MacKinnon(1994,1996,2000),MacKinnon,Haug,andMichelis(1999),andEricssonandMacKinnon(2002).MacKinnon(1996)providesreasonablyaccurateresultsforDickey-FullerandEngle-Grangertestswhichcoverthesamecasesasthoseinthenewappendix,althoughtheyarenotquiteasaccurate.The“numericaldistributionfunctions”obtainedinthatpapercanbeusedtocomputePvaluesaswellascriticalvalues.Nevertheless,theresultsofthispapercontinuetobeusedfarmoreoftenthantheonesfromthe1996paper.Perhapsthatisbecausetheydonotrequiretheuseofaspecializedcomputerprogramtocalculatecriticalvalues.Ihopethat,infuture,researcherswhoprefertheapproachofthispaperwillusethemoreaccurateandmorewidelyapplicableresultsthatarenowinTables2,3,and4.ThisversionofthepaperisdedicatedtothelateSirCliveGranger,1934–2009,withoutwhosefundamentalcontributionsitcouldneverhavebeenconceived.

January2010

–1–

1.Introduction

EngleandGranger(1987)suggestedseveraltechniquesfortestingthenullhypothesisthattwoormoreseries,eachofwhichisI(1),arenotcointegrated.Thispaperisconcernedwiththemostpopularofthesetechniques,whichIshallrefertoasEngle-Granger(orEG)testseventhoughtheywerenottheonlytestsproposedbythoseauthors.EGtestsarecloselyrelatedtosomeofthetestssuggestedbyFuller(1976)andDickeyandFuller(1979)totesttheunitroothypothesis;IshallrefertotheseasDickey-FullerorDFtests.EGandDFtestsareveryeasytocalculate,buttheysuﬀerfromoneseriousdisadvantage:Theteststatisticsdonotfollowanystandardtabulateddistribution,eitherinﬁnitesamplesorasymptotically.

EngleandGranger(1987),EngleandYoo(1987),Yoo(1987),andPhillipsandOuliaris(1990)allprovidetablesforoneormoreversionsoftheEGtest.Butthesetablesarebasedonatmost10,000replications,whichmeansthattheyarequiteinaccurate.Moreover,theycontaincriticalvaluesforonlyafewﬁnitesamplesizes;asymptoticcriticalvalues,whichareinmanycasesthemostinterestingones,arenotprovided.ThispaperprovidestablesofcriticalvaluesfortwoversionsoftheEGtestandthreeversionsoftheDFtest.Althoughtheyarebasedonsimulation,theyshouldbeaccurateenoughforallpracticalpurposes.Theresultsofthesimulationexperimentsaresummarizedbymeansofresponsesurfaceregressions,inwhichcriticalvaluesarerelatedtosamplesize.Thecoeﬃcientsoftheresponsesurfaceregressionsaretabulatedinsuchawaythatasymptoticcriticalvaluescanbereadoﬀdirectly,andcriticalvaluesforanyﬁnitesamplesizecaneasilybecomputedwithahandcalculator.

2.Engle-GrangerandDickey-FullerTests

Engle-Grangertestsareconceptuallyandcomputationallyquitesimple.Letthevectoryt≡[yt1,...,ytN]󰀂denotethetthobservationonNtimeseries,eachofwhichisknowntobeI(1).Ifthesetimeseriesarecointegrated,thereexistsavectorαsuchthatthestochasticprocesswithtypicalobservationzt≡[1yt]󰀂αisI(0).Iftheyarenotcointegrated,therewillexistnovectorαwiththisproperty,andanylinearcombinationofy1throughyNandaconstantwillstillbeI(1).

ToimplementtheoriginalformoftheEGtest,oneﬁrsthastorunthecointegratingregression

N󰀂

yt1=α1+αjytj+ut,(1)

j=2

ˆ≡[1−αforasampleofsizeT+1,thusobtainingavectorofcoeﬃcientsαˆ1...−αˆN]󰀂.

Onethencalculates

ˆ=y1t−αzˆt=[1yt]󰀂αˆ1−αˆ2yt2...−αˆNytN

andteststoseeifzˆtisI(1)usingaprocedureessentiallythesame(exceptforthedis-tributionoftheteststatistic)astheDFtest.Thenullhypothesisofnon-cointegration

correspondstothenullhypothesisthatzˆtisI(1).Ifonerejectsthenull,oneconcludesthaty1throughyNarecointegrated.

–2–

TotestwhetherzˆtisI(1),onemayeitherruntheregression

zˆt=ρzˆt−1+εt

andcalculatetheordinarytstatisticforρ=1,orruntheregression

∆ˆzt=γzˆt−1+εt,

(3)(2)

where∆ˆzt≡zˆt−zˆt−1,andcalculatetheordinarytstatisticforγ=0.Ineithercase,onedropstheﬁrstobservation,reducingthesamplesizetoT.Thesetwoproceduresevidentlyyieldidenticalteststatistics.Becausethereisaconstanttermin(1),thereisnoneedtoincludeonein(2)or(3).Theregressandzˆtandregressorzˆt−1wouldeachhavemeanzeroifbothwereobservedoverobservations0throughT.However,becausetheregressiondoesnotmakeuseoftheﬁrstobservationonzˆtorthelastobservationonzˆt−1,thatwillnotbequitetrue.ButtheyshouldbothhavemeanveryclosetozeroexceptwhenTissmallandeitherzˆ0orzˆTisunusuallylargeinabsolutevalue.Henceaddingaconstantto(2)or(3)wouldgenerallyhaveanegligibleeﬀectontheteststatistic.1

ThewaytheEGtestiscomputedissomewhatarbitrary,sinceanyoneoftheyjcouldbegiventheindex1andmadetheregressandofthecointegratingregression(1).Asaresult,thevalue(butnotthedistribution)oftheteststatisticwilldiﬀerdependingonwhichseriesisusedastheregressand.Onemaythereforewishtorepeattheprocedurewithdiﬀerentchoicesofyjservingasregressand,thuscomputinguptoNdiﬀerentteststatistics,especiallyiftheﬁrstoneisnearthechosencriticalvalue.

IfN=1,thisprocedureisequivalenttoonevariantoftheordinaryDFtest(seebelow),inwhichonerunstheregression

∆zt=α1+γzt−1+εt

andtestsforγ=0.Asseveralauthorshaveshown(seeWest(1988)andHyllebergandMizon(1989)),thelatterhastheDickey-Fullerdistributiononlywhenthereisnodriftterminthedata-generatingprocessforzt,sothatα1=0.Whenα1=0,theteststatisticisasymptoticallydistributedasN(0,1),andinﬁnitesamplesitsdistributionmayormaynotbewellapproximatedbytheDickey-Fullerdistribution.TheoriginalversionoftheEGtestlikewisehasadistributionthatdependsonthevalueofα1;sincealltabulatedcriticalvaluesassumethatα1=0,theymaybequitemisleadingwhenthatisnotthecase.

Thereisasimplewaytoavoidthedependenceonα1ofthedistributionoftheteststatistic.Itistoreplacethecointegratingregression(1)by

yt1=α0t+α1+

N󰀂j=2

αjytj+ut,

(4)

Somechangesweremadeinthisparagraphinthe2010versiontocorrectminorerrors

intheoriginalpaper.Theconclusionisunchanged.

–3–

thatis,toaddalineartimetrendtothecointegratingregression.Theresultingteststatisticwillnowbeinvarianttothevalueofα1,althoughitwillhaveadiﬀerentdistributionthantheonebasedonregression(1).2Addingatrendtothecointegratingregressionoftenmakessenseforanumberofotherreasons,asEngleandYoo(1990)discuss.TherearethustwovariantsoftheEngle-Grangertest.The“no-trend”variantuses(1)asthecointegratingregression,andthe“with-trend”variantuses(4).Insomecases,thevectorα(oratleastα2throughαN)maybeknown.Wecanthenjustcalculatezt=yt1−α2yt2...−αNytNanduseanordinaryDFtest.Inthiscase,itiseasiesttodispensewiththecointegratingregressions(1)or(4)entirelyandsimplyrunoneofthefollowingtestregressions:

∆zt=γzt−1+εt∆zt=α1+γzt−1+εt

∆zt=α0t+α1+γzt−1+εt.

(5)(6)(7)

Thetstatisticsforγ=0inthesethreeregressionsyieldtheteststatisticsthatFuller(1976)referstoasτˆ,τˆµ,andτˆt,respectively;heprovidessomeestimatedcriticalvaluesonpage373.Wewillrefertotheseteststatisticsasthe“no-constant”,“no-trend”,and“with-trend”statistics,respectively.Notethatthetabulateddistributionoftheno-constantstatisticdependsontheassumptionthatz0=0,whilethoseoftheothertwoareinvarianttoz0.Thetabulateddistributionoftheno-trendstatisticdependsontheassumptionthatα1=0(seeWest(1988)andHyllebergandMizon(1989)),whilethatofthewith-trendstatisticdependsontheassumptionthatα0=0.Uptothispoint,ithasbeenassumedthattheinnovationsεtareseriallyindependentandhomoskedastic.Theseratherstrongassumptionscanberelaxedwithoutaﬀectingtheasymptoticdistributionsoftheteststatistics.Theteststatisticsdonotevenhavetobemodiﬁedtoallowforheteroskedasticity,since,asPhillips(1987)hasshown,heteroskedasticitydoesnotaﬀecttheasymptoticdistributionofawideclassofunitrootteststatistics.Theydohavetobemodiﬁedtoallowforserialcorrelation,however.TheeasiestwaytodothisistouseAugmentedDickey-Fuller,orADF,andAugmentedEngle-Granger,orAEG,tests.Inpractice,thismeansthatonemustaddasmanylagsof∆ˆzttoregressions(2)or(3),orof∆zttoregressions(5),(6),or(7),asarenecessarytoensurethattheresidualsforthoseregressionsappeartobewhitenoise.Adiﬀerentapproachtoobtainingunitrootteststhatareasymptoticallyvalidinthepresenceofserialcorrelationand/orheteroskedasticityofunknownformwassuggestedbyPhillips(1987)andextendedtothecointegrationcasebyPhillipsandOuliaris

ˆtstatistic(1990).TheasymptoticdistributionsofwhatPhillipsandOuliariscalltheZ

areidenticaltothoseofthecorrespondingDF,ADF,EG,andAEGtests.PhillipsandOuliaristabulatecriticalvaluesfortwoformsofthisstatistic(correspondingtotheno-trendandwith-trendversionsoftheDFandEGstatistics)forseveralvaluesofN.Unfortunately,thesecriticalvaluesarebasedononly10,000replications,sothat

Itwillnotbeinvarianttothevalueofα0,however.Toachievethat,onewouldhavetoaddt2totheregression;seetheappendix.

–4–

theysuﬀerfromconsiderableexperimentalerror.Moreover,theyarefor500ratherthananinﬁnitenumberofobservations,sothattheyarebiasedawayfromzeroasestimatesofasymptoticcriticalvalues.AscanbeseenfromTable1below,thisbiasisbynomeansnegligibleinsomecases.

3.TheSimulationExperiments

Insteadofsimplyprovidingtablesofestimatedcriticalvaluesforafewspeciﬁcsamplesizes,aspreviouspapershavedone,thispaperestimatesresponsesurfaceregressions.Theserelatethe1%,5%and10%lower-tailcriticalvaluesfortheteststatisticsdis-cussedabove,forvariousvaluesofN,tothesamplesizeT.3RecallthatTreferstothenumberofobservationsintheunitroottestregression,andthatthisisonelessthanthetotalnumberofobservationsavailableandusedinthecointegratingregres-sion.Responsesurfaceswereestimatedforthirteendiﬀerenttests:theno-constant,no-trendandwith-trendversionsoftheDFtest,whichareequivalenttothecorres-pondingEGtestsforN=1,andtheno-trendandwith-trendversionsoftheEGtestforN=2,3,4,5and6.Thusatotalofthirty-nineresponsesurfaceregressionswereestimated.

TheDFtestswerecomputedusingtheone-stepproceduresofregressions(5),(6),and(7),whiletheEGtestswerecomputedusingtwo-stepproceduresconsistingofregressions(1)or(4)followedby(3).Thesearetheeasiestwaystocalculatethesetests.Notethatthereisaslightdiﬀerencebetweenthedegrees-of-freedomcorrectionsusedtocalculatetheregressionstandarderrors,andhencetstatistics,fortheDFtests(N=1)andfortheEGtests(N≥2).Iftheno-trendandwith-trendDFtestswerecomputedinthesamewayasthecorrespondingEGtests,theywouldbelargerby

󰀃󰀁1/2󰀃󰀁1/2

factorsof(T−1)/(T−2)and(T−1)/(T−3),respectively.Conceptually,eachsimulationexperimentconsistedof25,000replicationsforasinglevalueofTandasinglevalueofN.4The1%,5%and10%empiricalquantilesforthesedatawerethencalculated,andeachofthesebecameasingleobservationintheresponsesurfaceregression.Thenumber25,000waschosentomakethebiasinestimatingquantilesnegligible,whilekeepingthememoryrequirementsoftheprogrammanageable.

ForallvaluesofNexceptN=6,fortyexperimentswererunforeachofthefollowingsamplesizes:18,20,22,25,28,30,32,40,50,75,100,150,200,250,and500.ForN=6,thesamplesizeswere20,22,25,28,30,32,36,40,50,100,250,and275.Mostofthesamplesizeswererelativelysmallbecausethecostoftheexperimentswasslightlylessthanproportionaltothesamplesize,andbecausesmallsamplesizes

Theuppertailisnotofanyinterestinthiscase,andthevastmajorityofhypothesistestsareatthe1%,5%,or10%levels.

Infact,resultsforN=2,3,4,and5werecomputedtogethertosavecomputertime.ResultsforN=1werecomputedseparatelybecausethecalculationswereslightlydiﬀerent.ResultsforN=6werecomputedseparatelybecauseitwasnotdecidedtoextendtheanalysistothiscaseuntilaftermostoftheothercalculationshadbeencompleted.

–5–

providedmoreinformationabouttheshapeoftheresponsesurfaces.5However,afewlargevaluesofTwerealsoincludedsothattheresponsesurfaceestimatesofasymptoticcriticalvalueswouldbesuﬃcientlyaccurate.Thetotalnumberofreplicationswas12millionin480experimentsforN=1and15millionin600experimentsfortheothervaluesofN.6

Usingacorrectfunctionalformfortheresponsesurfaceregressionsiscrucialtoob-tainingusefulestimates.Afterconsiderableexperimentation,thefollowingformwasfoundtoworkverywell:

−2−1

+β2Tk+ek.Ck(p)=β∞+β1Tk

(8)

HereCk(p)denotestheestimatedp%quantilefromthekthexperiment,Tkdenotes

thesamplesizeforthatexperiment,andtherearethreeparameterstobeestimated.Theparameterβ∞isanestimateoftheasymptoticcriticalvalueforatestatlevelp,since,asTtendstoinﬁnity,T−1andT−2bothtendtozero.TheothertwoparametersdeterminetheshapeoftheresponsesurfaceforﬁnitevaluesofT.

Theabilityof(8)toﬁtthedatafromthesimulationexperimentswasremarkablygood.Totestitsadequacy,itwascomparedtothemostgeneralspeciﬁcationpossible,inwhichCk(p)wasregressedon15dummyvariables(12whenN=6),correspondingtothediﬀerentvaluesofT.ItwasrejectedbytheusualFtestinonlyaveryfewcaseswhereitseemedtohavetroubleﬁttingtheestimatedcriticalvaluesfortheverysmallestvalue(s)ofT.Theadequacyof(8)wasthereforefurthertestedbyaddingdummyvariablescorrespondingtothesmallestvaluesofT,andthistestprovedslightlymorepowerfulthantheﬁrstone.Wheneithertestprovidedevidenceofmodelinadequacy,theoﬀendingobservations(T=18andinonecaseT=20aswell)weredroppedfromtheresponsesurfaceregressions.

Severalalternativefunctionalformswerealsotried.Addingadditionalpowersof1/Tneverseemedtobenecessary.Infact,inseveralcases,fewerpowerswerenecessary,sincetherestrictionthatβ2=0appearedtobeconsistentwiththedata.Inmostcases,onecouldreplaceT−2byT−3/2withouthavinganynoticeableeﬀectoneithertheﬁtoftheregressionortheestimateofβ∞;thedecisiontoretainT−2ratherthanT−3/2in(8)wasbasedonveryslimevidenceinfavoroftheformer.Ontheotherhand,replacingTbyeitherT−NorT−N−1,thenumbersofdegreesoffreedomfor

Theexperimentswouldhaverequiredroughlyninehundredhoursona20Mh.386personalcomputer.AllprogramswerewritteninFORTRAN77.About70%ofthecomputationsweredoneonthePC,usingprogramscompiledwiththeLaheyF77L-EM/32compiler.Someexperiments,representingroughly30%ofthetotalcomputa-tionalburden,wereperformedonothercomputers,namely,anIBM3081G,whichwasabout7.5timesasfastasthePC,andanHP9000Model840,whichwasabout15%faster.

TheexperimentsforN=6weredonelaterthantheothersandweredesignedinthelightofexperiencewiththem.ItwasdecidedthattheextraaccuracyavailablebydoingmoreexperimentsforlargevaluesofTwasnotworththeextracost.

–6–

thecointegratingregressionintheno-trendandwith-trendcases,respectively,often(butnotalways)resultedinadramaticdeteriorationintheﬁtoftheresponsesurface.Theresidualsekinregression(8)wereheteroskedastic,beinglargerforthesmallersamplesizes.Thiswasparticularlynoticeablewhen(8)wasrunforlargervaluesofN.TheresponsesurfaceregressionswerethereforeestimatedbyfeasibleGLS.Asaﬁrststep,Ck(p)wasregressedon15(or12)dummyvariables,yieldingresidualse´k.Thefollowingregressionwasthenrun:

−1

e´2+δ2(Tk−d)−2+error,k=δ∞+δ1(Tk−d)

(9)

wheredisthenumberofdegreesoffreedomusedupinthecointegratingregression,

andtherearethreecoeﬃcientstobeestimated.7Theinversesofthesquarerootsoftheﬁttedvaluesfrom(9)werethenusedasweightsforfeasibleGLSestimationof(8).ThefeasibleGLSestimatesweregenerallymuchbetterthantheOLSonesintermsofloglikelihoodvalues,butthetwosetsofestimateswerenumericallyveryclose.TheﬁnalresultsofthispaperarethefeasibleGLSestimatesofregression(8)for39setsofexperimentaldata.TheseestimatesarepresentedinTable1.Theestimatesofβ∞provideasymptoticcriticalvaluesdirectly,whilevaluesforanyﬁniteTcaneasilybecalculatedusingtheestimatesofallthreeparameters.Therestrictionthatβ2=0

ˆ2waslessthanoneinabsolutevalue.hasbeenimposedwheneverthetstatisticonβ

ˆ∞butnotforβˆ1orβˆ2,sincethelatterEstimatedstandarderrorsarereportedforβ

areofnointerest.Whatisofinterestisthestandarderrorof

ˆ∞+βˆ1T−1+βˆ2T−2,ˆ(p,T)=βC

theestimatedcriticalvalueforatestatthep%levelwhenthesamplesizeisT.

ThisvarieswithTandtendstobesmallestforsamplesizesintherangeof80to150.ExceptforverysmallvaluesofT(lessthanabout25),thestandarderrorof

ˆ∞,sothat,ifˆ(p,T)wasalwayslessthanthestandarderrorofthecorrespondingβC

ˆ∞wereaccurate,theycouldberegardedasupperboundsthestandarderrorsoftheβ

ˆ(p,T)formostvaluesofT.forthestandarderrorsofC

ˆ∞reportedinTable1areundoubtedlytoosmall.However,thestandarderrorsforβ

Theproblemisthattheyareconditionalonthespeciﬁcationoftheresponsesurfaceregressions.Althoughthespeciﬁcation(8)performedverywellinallcases,otherspeciﬁcationsalsoperformedwellinmanycases,sometimesoutperforming(8)in-signiﬁcantly.Estimatesofβ∞sometimeschangedbyasmuchastwicethereportedstandarderrorasaresultofminorchangesinthespeciﬁcationoftheresponsesurfacethatdidnotsigniﬁcantlyaﬀectitsﬁt.Thusitisprobablyreasonabletothinkofthe

Considerableexperimentationprecededthechoiceofthefunctionalformforregression(9).Itwasfoundthatomittingdhadlittleeﬀectontheﬁtoftheregression,althoughonbalanceitseemedpreferabletoretainit.Inthisrespect,regression(9)isquitediﬀerentfromregression(8),whereusingTk−dratherthanTksometimesworsenedtheﬁtsubstantially.

–7–

ˆ∞asbeingabouttwiceaslargeasthereportedones.actualstandarderrorsontheβ

Evenso,itseemslikelythatfewifanyoftheestimated1%criticalvaluesinTable1diﬀerfromthetruevaluebyasmuchas.01,andextremelyunlikelythatanyoftheestimated5%and10%criticalvaluesdiﬀerfromtheirtruevaluesbythatmuch.

4.Conclusion

ItishopedthattheresultsinTable1willproveusefultoinvestigatorstestingforunitrootsandcointegration.Althoughthemethodsusedtoobtaintheseresultsarequitecomputationallyintensive,theyareentirelyfeasiblewithcurrentpersonalcomputertechnology.Theuseofresponsesurfaceregressionstosummarizeresultsisvaluablefortworeasons.First,thisapproachallowsonetoestimateasymptoticcriticalvalueswithoutactuallyusinginﬁnitelylargesamples.Second,itmakesitpossibletotabulateresultsforallsamplesizesbasedonexperimentalresultsforonlyafew.Similarmethodscouldbeemployedinmanyothercaseswhereteststatisticsdonotfollowstandardtabulateddistributions.

–8–

Table1.ResponseSurfaceEstimatesofCriticalValues

Variantnoconstant

Level1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%

Obs.600600560600600600600600600600600600600560600560560600600600600560560600600560560520560600600600600480480480480480480

β∞−2.5658−1.9393−1.6156−3.4336−2.8621−2.5671−3.9638−3.4126−3.1279−3.9001−3.3377−3.0462−4.3266−3.7809−3.4959−4.2981−3.7429−3.4518−4.6676−4.1193−3.8344−4.6493−4.1000−3.8110−4.9695−4.4294−4.1474−4.9587−4.4185−4.1327−5.2497−4.7154−4.4345−5.2400−4.7048−4.4242−5.5127−4.9767−4.6999

(s.e.)(0.0023)(0.0008)(0.0007)(0.0024)(0.0011)(0.0009)(0.0019)(0.0012)(0.0009)(0.0022)(0.0012)(0.0009)(0.0022)(0.0013)(0.0009)(0.0023)(0.0012)(0.0010)(0.0022)(0.0011)(0.0009)(0.0023)(0.0012)(0.0009)(0.0021)(0.0012)(0.0010)(0.0026)(0.0013)(0.0009)(0.0024)(0.0013)(0.0010)(0.0029)(0.0018)(0.0010)(0.0033)(0.0017)(0.0011)

β1−1.960−0.398−0.181−5.999−2.738−1.438−8.353−4.039−2.418−10.534−5.967−4.069−15.531−9.421−7.203−13.790−8.352−6.241−18.492−12.024−9.188−17.188−10.745−8.317−22.504−14.501−11.165−22.140−13.641−10.638−26.606−17.432−13.654−26.278−17.120−13.347−30.735−20.883−16.445

β2−10.04

1notrend

1withtrend

2notrend

2withtrend

3notrend

3withtrend

4notrend

4withtrend

5notrend

5withtrend

6notrend

−29.25−8.36−4.48−47.44−17.83−7.58−30.03−8.98−5.73−34.03−15.06−4.01−46.37−13.41−2.79−49.35−13.13−4.85−59.20−21.57−5.19−50.22−19.54−9.88−37.29−21.16−5.48−49.56−16.50−5.77−41.65−11.17−52.50−9.05

6withtrend

–9–

ExplanationofTable1

N:NumberofI(1)seriesforwhichnullofnon-cointegrationisbeingtested.Level:Levelofone-tailtestoftheunitrootnullagainstthealternativeofstationarity.Obs.:Numberofobservationsusedinresponsesurfaceregression.Possiblevaluesare

600,560,520,and480.IfObs.=600,theregressionused40observationsfromeachofT=18,20,22,25,28,30,32,40,50,75,100,150,200,250,and500.IfObs.=560,observationsforT=18werenotused.IfObs.=520,observationsforT=18andT=20werenotused.IfObs.=480,theregressionused40observationsfromeachofT=20,22,25,28,30,32,36,40,50,100,250,and275.β∞:Estimatedasymptoticcriticalvalues(withestimatedstandarderrorsinparen-theses).β1:CoeﬃcientonT−1inresponsesurfaceregression.

β2:CoeﬃcientonT−2inresponsesurfaceregression.Itwasomittedifthetstatistic

waslessthanoneinabsolutevalue.ForanysamplesizeT,theestimatedcriticalvalueis

β∞+β1/T+β2/T2.

Forexample,whenT=100,the5%criticalvalueforthewith-trendEGtestwhenN=5is

−4.7154−17.432/100−16.50/1002=−4.8914.

–10–

Appendix(addedin2010version)

Inthetwentyyearssincethispaperwaswritten,computertechnologyhasadvancedenormously.Ontheoccasionofreissuingthepaper,itthereforemakessensetoreportnewresultsbasedonamuchlargernumberofreplicationsthatcoverabroaderrangeofcases.However,forcomparabilitywiththeoriginalpaper,themethodologyislargelyunchanged.ThesimulationresultscouldalsohavebeenusedtoprovidesomewhatmoreaccuratenumericaldistributionfunctionsthanthoseofMacKinnon(1996),whichallowonetocomputePvaluesandcriticalvaluesfortestsatanylevel,butnoattemptismadetodosohere.

Onemajordiﬀerencebetweentheoriginalexperimentsandthenewonesisthatthelatterinvolvefarmorecomputation.Insteadof25,000replications,eachsimulationnowinvolves200,000.Insteadof40simulationsforeachsamplesize,therearenow500.Andinsteadofthe12or15samplesizesusedoriginally,thereare30.Thesamplesizesare20,25,30,35,40,45,50,60,70,80,90,100,120,140,160,180,200,250,300,350,400,450,500,600,700,800,900,1000,1200,and1400.SomeofthesevaluesofTaremuchlargerthanthelargestonesusedoriginally.Thisincreasestheprecisionoftheestimatesbutmadetheexperimentsmuchmoreexpensive.

Thetotalnumberofobservationsfortheresponsesurfaceregressionsisusually15,000(thatis,30times500),although,inafewcases,thesimulationsforT=20weredroppedbecausetheresponsesurfacedidnotﬁtwellenough.Inaveryfewcases,thesimulationsforT=25weredroppedaswell.Thusafewoftheestimates,alwaysforlargervaluesofN,arebasedoneither14,500or14,000observations.

Becausethenewresponsesurfaceregressionsarebasedon200timesasmanysimula-tionsastheoriginalones,anymisspeciﬁcationbecomesmuchmoreapparent.Itwasthereforenecessaryinmostcasestoaddanadditional(cubic)termtoequation(8),whichthusbecomes

−1−2−3

Ck(p)=β∞+β1Tk+β2Tk+β3Tk+ek.

(A.1)

Toavoidaddingthecubicterm,itwouldusuallyhavebeennecessarytodropone

ormoreofthesmallersamplesizes(oftenquiteafewofthem).Thiswastriedinseveralcases,andtheestimateofβ∞didnotchangemuchwhenenoughsmallvaluesofTkweredroppedsothattheestimateofβ3becameinsigniﬁcantatthe10%level.

ˆ∞waslowerwhenequation(A.1)wasestimatedIngeneral,thestandarderrorofβ

usingallthedatathanwhenequation(8)wasestimatedwithouttheobservationscorrespondingtosomeofthesmallervaluesofTk.

Thesecondmajordiﬀerencebetweentheoriginalexperimentsandthenewonesisthatthelatterdealwithabroaderrangeofcases.ThevaluesofNnowgofrom1to12insteadoffrom1to6.Moreover,anadditionalvariantoftheDFandEGtestsisincluded,inwhicht2isaddedtoeitherthecointegratingregression(4)ortheDFtestregression(7).ThisvariantofthetestswasadvocatedbyOuliaris,Park,andPhillips(1989).InthenotationusedbyMacKinnon(1996),thetestsforwhichcriticalvaluesarecomputedaretheτc,τct,andτctttests.Theseinvolve,respectively,aconstant

–11–

term,aconstantandatrend,andaconstant,trend,andtrendsquaredineitherthecointegratingregressionortheDFtestregression.ForN=1only,asinTable1,resultsarealsopresentedfortheτnctest,inwhichtheDFtestregressiondoesnotcontainaconstantterm.Useofthistestisnotrecommended,however,becauseitrequireshighlyunrealisticassumptions.

Table2containsresultsfortheτctestforN=1to12andalsofortheτnctestforN=1.Table3containsresultsfortheτcttestforN=1to12,andTable4containsresultsfortheτctttestforN=1to12.ThesetablesaretobereadinexactlythesamewayasTable1,exceptthatthereis(usually)onemorecoeﬃcienttotakeintoaccount.Notethattheβ3coeﬃcient√wassetto0(andomittedfromthetable)when

ˆ3waslessthan2inabsolutevalue.thetstatisticonβ

ThefeasibleGLSmethodofestimatingequation(A.1)discussedinthebodyofthis

paperyieldsidenticalresultstotheGMMmethoddiscussedinMacKinnon(1996)whenthevariancesoftheekareestimatedinthesameway.TheestimatesinTables2,3,and4wereactuallyobtainedusingthelattermethod.OneadvantageofthisapproachisthatitautomaticallyyieldsaGMMoveridentiﬁcationteststatisticwhichwouldrevealmisspeciﬁcationifitwereatallserious.

ˆ∞reportedinthetablesareundoubtedlytoosmall,becauseThestandarderrorsofβ

theyignoreuncertaintyaboutthespeciﬁcationoftheresponsesurface.Nevertheless,itisinterestingtocomparethestandarderrorsinTable1withtheonesinthethreenewtables.Forexample,considertheτcttestswithN=2andN=3.InTable1,thestandarderrorsforthe.05asymptoticcriticalvaluesofthosetwotestsare0.0013and0.0011,respectively.InTable3,theyare0.000054and0.000066.ThestandarderrorislargerforN=3thanforN=2becauseβ3hastobeestimatedintheformercasebutnotinthelatter.Inmostcases,thestandarderrorsseemtobesmallerbyfactorsofbetween15and20.

Usingthetables,itiseasytocalculateacriticalvalue(strictlyvalidonlyundertheassumptionthattheerrorsareIIDnormal)foranyﬁnitesamplesizeT.Theestimatedcriticalvalueissimply

β∞+β1/T+β2/T2+β3/T3.Forexample,whenT=100,the5%criticalvaluefortheτcttest(thatis,theEGtestwithtrend)whenN=5is

−4.71537−17.3569/100−22.660/1002+91.359/1003=−4.89111.

Thisisveryclosetothevalueof−4.8914thatwascalculatedusingtheresultsinTable1;seetheexplanationfollowingthattable.

–12–

Table2.CriticalValuesforNoTrendCase(τncandτc)

N111111222333444555666777888999101010111111121212

Variantτncτncτncτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτcτc

Level1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%

Obs.15,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00014,50015,00015,00015,00014,50015,00015,00015,00015,00015,000

β∞−2.56574−1.94100−1.61682−3.43035−2.86154−2.56677−3.89644−3.33613−3.04445−4.29374−3.74066−3.45218−4.64332−4.09600−3.81020−4.95756−4.41519−4.13157−5.24568−4.70693−4.42501−5.51233−4.97684−4.69648−5.76202−5.22924−4.95007−5.99742−5.46697−5.18897−6.22103−5.69244−5.41533−6.43377−5.90714−5.63086−6.63790−6.11279−5.83724

(s.e.)(0.000110)(0.000074)(0.000059)(0.000127)(0.000068)(0.000043)(0.000102)(0.000056)(0.000044)(0.000123)(0.000067)(0.000043)(0.000101)(0.000055)(0.000043)(0.000101)(0.000055)(0.000043)(0.000124)(0.000068)(0.000054)(0.000126)(0.000068)(0.000054)(0.000126)(0.000068)(0.000053)(0.000126)(0.000069)(0.000062)(0.000128)(0.000068)(0.000054)(0.000145)(0.000068)(0.000055)(0.000127)(0.000069)(0.000054)

β1

−2.2358−0.26860.2656−6.5393−2.8903−1.5384−10.9519−6.1101−4.2412−14.4354−8.5631−6.2143−18.1031−11.2349−8.3931−21.8883−14.0406−10.7417−25.6688−16.9178−13.1875−29.5760−19.9021−15.7315−33.5258−23.0023−18.3959−37.6572−26.2057−21.1377−41.7154−29.4521−24.0006−46.0084−32.8336−26.9693−50.2095−36.2681−29.9864

β2

−3.627−3.365−2.714−16.786−4.234−2.809−22.527−6.823−2.720−33.195−10.852−3.718−37.972−11.175−4.137−45.142−12.575−3.784−57.737−17.492−5.104−69.398−22.045−6.922−82.189−24.646−7.344−87.365−26.627−9.484−102.680−30.994−7.514−106.809−30.275−4.083−124.156−32.505−2.686

β331.22325.364−79.433−40.040

47.43327.982

88.63960.00727.877164.295110.76167.721256.289144.47994.872248.316176.382172.704389.330251.016163.049352.752249.994151.427579.622314.802184.116

–13–

Table3.CriticalValuesforLinearTrendCase(τct)

N111222333444555666777888999101010111111121212

Level1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%

β∞−3.95877−3.41049−3.12705−4.32762−3.78057−3.49631−4.66305−4.11890−3.83511−4.96940−4.42871−4.14633−5.25276−4.71537−4.43422−5.51727−4.98228−4.70233−5.76537−5.23299−4.95405−6.00003−5.46971−5.19183−6.22288−5.69447−5.41738−6.43551−5.90887−5.63255−6.63894−6.11404−5.83850−6.83488−6.31127−6.03650

(s.e.)(0.000122)(0.000066)(0.000051)(0.000099)(0.000054)(0.000053)(0.000126)(0.000066)(0.000053)(0.000125)(0.000067)(0.000054)(0.000123)(0.000068)(0.000054)(0.000125)(0.000066)(0.000053)(0.000125)(0.000067)(0.000054)(0.000126)(0.000068)(0.000054)(0.000125)(0.000069)(0.000054)(0.000127)(0.000069)(0.000063)(0.000125)(0.000069)(0.000055)(0.000126)(0.000068)(0.000074)

β1

−9.0531−4.3904−2.5856−15.4387−9.5106−7.0815−18.7688−11.8922−9.0723−22.4694−14.5876−11.2500−26.2183−17.3569−13.6078−29.9760−20.3050−16.1253−33.9165−23.3328−18.7352−37.8892−26.4771−21.4328−41.9496−29.7152−24.2882−46.1151−33.0251−27.2042−50.4287−36.4610−30.1995−54.7119−39.9676−33.2381

β2

−28.428−9.036−3.925−35.679−12.074−7.538−49.793−19.031−8.504−52.599−18.228−9.873−59.631−22.660−10.238−75.222−25.224−9.836−84.312−28.955−10.168−96.428−31.034−10.726−109.881−33.784−8.584−120.814−37.208−6.792−128.997−36.246−5.163−139.800−37.021−6.606

β3−134.155−45.374−22.380

21.892104.24477.33235.40351.31439.64754.10950.64691.35976.781202.253132.03094.272245.394182.342120.575335.920220.165157.955466.068273.002169.891566.823346.189177.666642.781348.554210.338736.376406.051317.776

–14–

Table4.CriticalValuesforQuadraticTrendCase(τctt)

N111222333444555666777888999101010111111121212

Level1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%1%5%10%

Obs.15,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00015,00014,50014,50015,00015,00015,00015,00015,00015,00015,00015,00014,50015,00014,50015,00015,00014,500

β∞−4.37113−3.83239−3.55326−4.69276−4.15387−3.87346−4.99071−4.45311−4.17280−5.26780−4.73244−4.45268−5.52826−4.99491−4.71587−5.77379−5.24217−4.96397−6.00609−5.47664−5.19921−6.22758−5.69983−5.42320−6.43933−5.91298−5.63704−6.64235−6.11753−5.84215−6.83743−6.31396−6.03921−7.02582−6.50353−6.22941

(s.e.)(0.000123)(0.000065)(0.000051)(0.000124)(0.000067)(0.000052)(0.000125)(0.000068)(0.000053)(0.000125)(0.000069)(0.000053)(0.000125)(0.000068)(0.000054)(0.000126)(0.000067)(0.000054)(0.000125)(0.000067)(0.000062)(0.000143)(0.000067)(0.000054)(0.000125)(0.000069)(0.000054)(0.000125)(0.000070)(0.000054)(0.000145)(0.000069)(0.000062)(0.000124)(0.000070)(0.000063)

β1

−11.5882−5.9057−3.6596−20.2284−13.3114−10.4637−23.5873−15.7732−12.4909−27.2836−18.4833−14.7199−30.9051−21.2360−17.0820−34.7010−24.2177−19.6064−38.7383−27.3005−22.2617−42.7154−30.4365−24.9686−46.7581−33.7584−27.8965−50.9783−37.056−30.8119−55.2861−40.5507−33.8950−59.6037−44.0797−36.9673

β2

−35.819−12.490−5.293−64.919−28.402−17.408−76.924−32.316−17.912−78.971−31.875−17.969−92.490−37.685−18.631−105.937−39.153−18.858−108.605−39.498−17.910−119.622−44.300−19.688−136.691−42.686−13.880−145.462−48.719−14.938−152.651−46.771−9.122−166.368−47.242−10.868

β3−334.047−118.284−63.55988.88472.74166.313184.782122.70583.285137.871111.817101.920248.096194.208136.672393.991232.528174.919365.208246.918208.494421.395345.480274.462651.380346.629236.975752.228473.905316.006792.577487.185285.164989.879543.889418.414

–15–

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Note:Referencesthatwerenotintheoriginalpaperaremarkedwithanasterisk.Dickey,D.A.andW.A.Fuller(1979),“Distributionoftheestimatorsforautoregres-sivetimeserieswithaunitroot,”JournaloftheAmericanStatisticalAssociation,84,427–431.Engle,R.F.andC.W.J.Granger(1987),“Co-integrationanderrorcorrection:Rep-resentation,estimationandtesting,”Econometrica,55,251–276.Engle,R.F.andB.S.Yoo(1987),“Forecastingandtestinginco-integratedsystems,”JournalofEconometrics,35,143–159.Engle,R.F.andB.S.Yoo(1991),“Cointegratedeconomictimeseries:Anoverviewwithnewresults,”Chapter12inLong-RunEconomicRelationships:ReadingsinCointegration,ed.R.F.EngleandC.W.J.Granger.Oxford,OxfordUniversityPress.Ericsson,N.R.andJ.G.MacKinnon(2002),“Distributionsoferrorcorrectiontestsforcointegration,”EconometricsJournal,5,285–318.[∗]Fuller,W.A.(1976),IntroductiontoStatisticalTimeSeries.NewYork,Wiley.Hylleberg,S.andG.E.Mizon(1989),“Anoteonthedistributionoftheleastsquaresestimatorofarandomwalkwithdrift,EconomicsLetters,29,225–230.MacKinnon,J.G.(1991),“Criticalvaluesforcointegrationtests,”Chapter13inLong-RunEconomicRelationships:ReadingsinCointegration,ed.R.F.EngleandC.W.J.Granger.Oxford,OxfordUniversityPress.[∗]MacKinnon,J.G.(1994),“Approximateasymptoticdistributionfunctionsforunit-rootandcointegrationtests,”JournalofBusinessandEconomicStatistics,12,167–176.[∗]MacKinnon,J.G.(1996),“Numericaldistributionfunctionsforunitrootandcointe-grationtests,”JournalofAppliedEconometrics,11,601–618.[∗]MacKinnon,J.G.(2000),“Computingnumericaldistributionfunctionsineconomet-rics,”inHighPerformanceComputingSystemsandApplications,ed.A.Pollard,D.Mewhort,andD.Weaver.Amsterdam,Kluwer,455–470.[∗]MacKinnon,J.G.,A.A.Haug,andL.Michelis(1999),“Numericaldistributionfunc-tionsoflikelihoodratiotestsforcointegration,”JournalofAppliedEconometrics,14,563–577.[∗]Ouliaris,S.,J.Y.Park,andP.C.B.Phillips(1989),“Testingforaunitrootinthepresenceofamaintainedtrend,”inAdvancesinEconometrics,ed.B.Raj.Boston,KlumerAcademicPublishers,7–28.[∗]Phillips,P.C.B.(1987),“Timeseriesregressionwithaunitroot,”Econometrica,55,277–301.

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Phillips,P.C.B.andS.Ouliaris(1990),“Asymptoticpropertiesofresidualbasedtestsforcointegration,”Econometrica,58,165–193.Yoo,B.S.(1987),“Co-integratedtimeseries:Structure,forecastingandtesting,”unpublishedPh.D.Dissertation,UniversityofCalifornia,SanDiego.West,K.D.(1988),“Asymptoticnormality,whenregressorshaveaunitroot,”Econo-metrica,56,1397–1417.

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