Download presentation
1 -

IntroductionThispaperconsiderstheextensionofthesingleequationdynamicor


truehomogeneouscointegrationvectorInthesesituationscombiningcross-sectionalandtime-seriesinformationintheformofapanelcanprovidemuchmoreprecisepointestimatesofthecointegrationvectorwithreasonablyaccura

quinn's Recent Documents

Plant Based Foods Association
Plant Based Foods Association

Certified Plant BasedClaim CertificationProgramVersion 1 Issue 1Published November2018Plant Based FoodsClaim CertificationProgram21 BackgroundThe Plant Based Foods Associations mission is to promote t

published 0K
US Department of Labor
US Department of Labor

Wage and Hour Division February 2013Fact Sheet month periodunder the Fa

published 0K
The 13 British Colonies
The 13 British Colonies

ENS400200400kmBRITISH TERRITORYATLANTICQUEBECTERRITORYSavannahCharlestonNew BernTrentonPortsmouthNew HavenNew York CityProvidenceWilliamsburgCAROLINANORTHCAROLINADELAWAREMARYLANDPENNSYLVANIAYORKMASSAC

published 0K
IntroductionThispaperconsiderstheextensionofthesingleequationdynamicor
IntroductionThispaperconsiderstheextensionofthesingleequationdynamicor

truehomogeneouscointegrationvectorInthesesituationscombiningcross-sectionalandtime-seriesinformationintheformofapanelcanprovidemuchmoreprecisepointestimatesofthecointegrationvectorwithreasonablyaccura

published 0K
BER OF PARKI
BER OF PARKI

MansfieldUniversityParkingMapLegendVisitor parkingisavailable inthefollowinglotsvisitor permitsmustbedisplayedPermitsareavailableattheAdmissionsOfficeinSouthHall or UniversityPoliceDepartmentinDoaneCe

published 0K
nrrrrr   r rr   rrr rrn   r
nrrrrr r rr rrr rrn r

nrn////0 1111002 r 344440r5r6r57236rrrrr08rr96/rr0/rr0/rr0/rr00r 000066r06r06r06r08x-1678-800000AAnrnr8r8r8r86BCnDx0000E76BCnDx0000E76BCnDx0000E76BCnDx0000E7r F95Dx0000r F95Dx0000r F95Dx0000r F95Dx000

published 0K
Protestatt haveevery retoitograti a
Protestatt haveevery retoitograti a

themselves Thus Orangeism and false Na-tionalism are settled by the same blow Citi-sensof Lr remember the majority of 74for F WW Rueland freedom over the fiercestoan tmy As to the gallant MAjor heSir

published 0K
KCARev BasicANNEXBETWEENTHE NATIONAL AERONAUTICS AND SPACE ADMINISTRAT
KCARev BasicANNEXBETWEENTHE NATIONAL AERONAUTICS AND SPACE ADMINISTRAT

This Annex shall be for the purpose of NASA KSC providing SpaceX withNASA Operational Lab Support SpaceX willReimburse NASA in accordance with the NASA KSC cost estimate set forth in Article 3 for the

published 0K
Download Section

Download - The PPT/PDF document "" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.






Document on Subject : "IntroductionThispaperconsiderstheextensionofthesingleequationdynamicor"— Transcript:

1 IntroductionThispaperconsiderstheextensi
IntroductionThispaperconsiderstheextensionofthesingleequationdynamicordinaryleastsquares(DOLS)methodofSaikkonen()andStockandWatson(993)forestimatingandtestinghypothesesaboutacointegratingvectortopaneldata.WecalltheestimatorpanelDOLS.Wediscussitslimitdistributionandapplyittoestimatethelong-runmoneydemandfunctionusingapaneldatasetof9countrieswithannualobservationsspanningfrom957to996.PanelDOLSisfullyparametricandoersacomputationallyconvenientalternativetothepanel‘fullymodied’OLSestimatorproposedbyPedroni(997)andPhillipsandMoon(999a).Proper-tiesofpanelDOLSwhentherearexed-eectsinthecointegratingregressionhavebeendiscussedbyKaoandChiang(2000).Wetakethistobethestartingpointforouranalysis.Inourenvironment,thecointegratingvectorishomogeneousacrossindividualsbutweallowforindividualheterogeneitythroughdisparateshort-rundynamics,individual-specixedeectsandindividual-specictimetrends.Moreover,wepermitalimiteddegreeofcross-sectionaldependence(CSD)throughthepresenceoftime-speciects.WepresenttwolimitdistributionsforpanelDOLS.Therstoneisobtainedforaxednumberofcross-sectionalunitsandletting.Inthiscase,panelDOLSconvergesindistributiontoafunctionofBrownianmotionsandtheWaldstatisticfortestingasetoflinearconstraintshasalimiting)distribution.Thislimittheoryseemswellsuitedformanyappliedmacroeconomicorinternationalproblems.Here,researchersoftenhaveavailablepaneldatasetsofmoderatebutmuchlarger.Withthepassageoftime,thesedatasetswillgaintime-seriesobservationsbuttheyareunlikelytoacquiremanymorecross-sectionalunits.Wealsoobtainthesequentiallimitdistributionbyrstlettingfor,thenlettingasproposedbyPhillipsandMoon(999a).Here,panelDOLShasalimitingGaussiandistributionandasinthecasetheWaldstatistichasalimitingchi-squaredistri

2 bution.Intheabsenceoflineartrendsintheco
bution.Intheabsenceoflineartrendsinthecointegratingregression,thesequentiallimitingnormalityoftheestimatoristheoreticallyinterestingbuthaslesspracticalimportbecausethelimitdistributionoftheteststatisticsisidenticaltothedistributionwithxed.However,whenlineartrendsarepresent,thesequentiallimittheoryproducesconsiderablesimplications.Here,theestimatorofthecointegrationvectorandthetime-trendslopecoecientsremaincorrelatedforbutareasymptoticallyuncorrelatedwhenthenSincesingleequationcointegrationvectorestimatorsaresuperconsistent,itisnaturaltoaskwhatistobegainedbyusingthepanelestimator.Theansweristhatsuperconsistencymeansonlythatconvergencetowardstheasymptoticdistributionoccursatratebutitsaysnothingaboutthesamplingvariabilityoftheestimatorforaxedvalueof.Infact,thestatisticalpropertiesofsingle-equationcointegration-vectorestimatorscanbequitepoorwhenappliedtosamplesizesassociatedwithmacroeconomictimeseriestypicallyavailabletoresearchers[e.g.,Inder(993),StockandWatson(993)].Moreover,evenlimitedamountsofheterogeneityintheshort-rundynamicsacrossindividualscangenerateconsiderabledisparitiesinsingle-equationDOLSestimatesoftheForexample,iftheobservationalunitisanationaleconomy,thetotalnumberofcountriesmayuctuateovertime,butisunlikelytogotoinnity.Whilethebreak-upoftheSovietUnioncreatedseveralneweconomiesbuttheoppositetrendisatworkinEuropewheretheEMUmayeventuallycombinetoformasingleeconomicunit.Butbeyondthis,researcherstypicallychoosetogroupcountriesintoclassesthatsharecommoncharacteristicssuchasincomelevels,stagesofdevelopmentorgeographywhichoftenresultinpanelswith5to20individuals. truehomogeneouscointegrationvector.Inthesesituations,combiningcross-sectionalandtime-seriesinformationintheformofapanelcanprovi

3 demuchmoreprecisepointestimatesofthecoin
demuchmoreprecisepointestimatesofthecointegrationvectorwithreasonablyaccurateasymptoticapproximationstotheexactsamplingdistribution.InaseriesofMonteCarloexperiments,westudythesmallsampleperformanceofpanelDOLSandcompareittosingle-equationDOLS.PanelDOLSgenerallyperformswellundertheshort-rundynamicdesignsthatweconsiderandcanattainastrikingimprovementinestimationprecisionoverthatofsingle-equationDOLSwithevenamodestnumberofcross-sectionalunits.WethenapplypanelDOLStoestimatethelong-rundemandforMmoney.ThecountriesinourstudyareAustria,Australia,Belgium,Canada,Denmark,France,Finland,Germany,Iceland,Ireland,Japan,Norway,NewZealand,theNetherlands,Portugal,Spain,Switzerland,theUnitedKingdom,andtheUnitedStates.Here,webuildonthetime-seriescontributionsbyBaba,HendryandStarr(992),Ball(998),Homan,Rasche,andTieslau(995),Lucas(988)andStockandWatson(993),andthecross-sectionalstudiesbyMulliganandSala-i-Martin(992),andMulli-gan(997),mostofwhichhasfocusedonU.S.data.Thestudiescitedabovereportconictingresultsalongthreedimensions.First,pointestimatesfromtime-seriesstudiesexhibitsubstantialdependenceonthesampleperiodofthedata.IncomeelasticityestimatesfrompostWWIIU.S.datatypicallyliewellbelow–whichimpliesexistenceofeconomiesofscaleinmoneymanagement–whereasestimatesobtainedfrompre-WWIIobservationsorthosethatcombinepre-andpost-warobservationsaregenerallycloseto.UsingannualU.S.dataspanningfrom903to987,StockandWatson’s(993)DOLSestimateoftheincomeelasticityis0.97.Whenthesamplespansfrom903to945,theirestimateis0.89butdropsto0.27whenthedataspanfrom946to987.Ball(998)extendsthesedataandobtainsanestimateof0.42whentheobservationsspanfrom946to996.Usingannualobservationsfrom900to958,Lucas’s(988)estimateoftheM(permanent)incomeelastic

4 ityis.06andhisestimateofthe(short-term)i
ityis.06andhisestimateofthe(short-term)interestratesemi-elasticityis-0.07.Usingdataspanningfrom958to985,hisincomeelasticityestimatedropsto0.2andhisinterestsemi—elasticityestimateis-0.0.Second,thereistensiongeneratedbythelargedierencebetweentheestimatesfromtime-seriesstudiesandthosefrompost-warcross-sectionstudies.MulliganandSala-i-Martin’sestimatesfroma989cross-sectionaldatasetfromtheSurveyofConsumerFinancesrangebetween0.82to.37.Mulligan(997)runsOLSonforapanelof2000rmsobservedfrom992andobtainsanincome-elasticityestimateof0.83.Third,thereissubstantialcross-countryvariationevenamongsteconomiesofsimilarincomelevelsandnancialmarketdevelopment.Inouranalysis,single-equationDOLSwithtrendgivessuchdisparateincomeelasticityestimatesas-.23forNewZealandand2.42forCanada.Thecorrespondinginterestratesemi-elasticityestimatesrangefrom0.02forIreland(whichhasthewrongsign)to-0.09fortheUK.Whentrendsareomitted,theincomeelasticityestimatesrangefrom0.3forBelgiumto2.64forNorwayandtheinterestsemi-elasticityestimatesrangefromrangefrom0.02forIrelandto-0.6forNorway.Withonly40annualobservations,theheterogeneitythatweobserveinthepointestimatesmayplausiblyhavebeengeneratedfromanunderlyingdatageneratingprocesswithahomogeneouscointegrationvectorandheterogeneousshortrundynamics.WhenweincludeheterogeneouslineartrendsandestimatethecointegratingvectorbypanelDOLS,weobtainapointestimateoftheincomeelasticityof.08(asymptotics.e.=0.26)andapointestimateoftheinterestsemi-elasticityLessrecentcross-sectionalstudiesincludeMeltzer(1963)andGandolandLothian(1976). of-0.02(asymptotics.e.=0.0).Moreover,theseestimates,whicharemoreinlinewiththosefromcross-sectionalstudiesonU.S.data,arestableasthespanofthetime-seriesdimensionisvariedandarereasona

5 blyrobusttotheinclusionoromissionofheter
blyrobusttotheinclusionoromissionofheterogeneouslineartrends.Theremainderofthepaperisorganizedasfollows.Thenextsectiondescribestherepresentationofthenonstationarypaneldataandregularityconditionsassumedinthepaper.SectionIIIdescribesthepanelDOLSestimatoranddiscussesitsasymptoticproperties.SectionIVreportstheresultsofaMonteCarloexperimenttoexaminethesmallsampleperformanceofthepanelDOLSestimatorandtheaccuracyoftheasymptoticapproximations.InSectionVwepresentourlong-runmoneydemandstudyandSectionVIconcludesthepaper.Proofsofpropositionsandsupplementaryresultsfromthemoneydemandstudyarecontainedinanappendixwhichisavailableuponrequestfromtheauthors.RepresentationofCointegratedObservationsinPanelDataConsiderabalancedpanelofindividualsindexedbyobservedovertimeperiods.Vectorsareunderlinedandmatricesappearinboldface.)isavectorstandardBrownianmotionfor0,and[]denotesthelargestintegervalueoffor0.Asisstandardintheliterature,wewilldropthenotationaldependenceonandwriteRRR)asWdW.ScaledvectorBrownianmotionsaredenotedbywhereisascalingmatrix.ForanymatrixdenotestheEuclidiannorm,[Tr()].Wewillbeworkingwithdoubleindexedsums.Insomeinstances–todealwithindividual-specixedeectsorcommontimeeects–thesesumswillinvolvetransformationsoftheoriginalobservations.Tohandlesuchsituations,wegenericallydenotethesamplecross-momentmatrixaveragedoverandlettheprecisedenitiondependupontheparticularmodelunderconsideration.Also,wegenericallydenotethelimitofthemomentmatrixasforgiven.AsneednotconvergetoaconstantandwedenotethislimitasSimilarly,ourgenericnotationforthesamplecross-productvectorbetweentheregressorsandtheequilibriumerroris,andthelimitforNTNSincethemodelwestudyallowsforindividualspeciects,perhapsitwouldbemoreaccuratetocalltheestimator

6 dynamicLSDV.However,intheinterestsofsimp
dynamicLSDV.However,intheinterestsofsimplicity,wewillrefertotheestimatoraspanelDOLS.TriangularRepresentationy,xbea()dimensionalvectorofobservationswhereisascalarandisadimensionalvector.Observationsoneachindividualobeythetriangularrepresentationitiitititwhere()isacointegratingvectorbetweenandthatisidenticalacrossindividuals.Thecompositeequilibriumerrorispotentiallycomprisedofanindividual-speciect ,anindividual-speciclineartrend,andacommontime-specicfactor.Theremainingiitidiosyncraticerrorisindependentacrossbutpossiblydependentacross.Analternativerepresentationfor(2)allowstohaveanindividual-specicvectorofdrifttermsandforthetrendin()tobeinducedbythisdrift.Withsomeminormodications,theresultsofthispapercontinuetoholdinthisalternativeenvironment.Inadditiontoindividual-specixed-eectsandlineartrends,potentiallydisparateshort-rundynamicsofthecovariancestationaryerrorprocessu,vintroducesanadditionalititsourceofheterogeneousbehavioracrossindividuals.TheunderlyingerrordynamicsaregiveninAssumptionErrorDynamics.)isindependentacross,...,N,andhasthemovingaveragerepresentation†††itiit†††††††wherei.i.d.withE)=0isaitititititiijdimensionalmatrixlagpolynomialinthelagoperatorL,whereandisthem,nthelementofthematrixijij†††Ourassumptionthatisindependentacrossindividuals(E[]=0$"#itit)followstherecenteconometricsliteratureonnonstationarypaneldata[e.g.,PhillipsandMoon(999a,b),KaoandChiang(2000),andPedroni(997)].Unliketheseauthors,weassumethatthecoecientsinthe)polynomialarexedforagivenalthoughtheycandieracrossindividuals.bea-dimensionalstandardBrownianmotion.ByAssumption,itfollowsthatforeachobeysthefunctionalcentrallimittheoremmTr]X#D†††itii,whereB,B)isascaledmixedBrownianmotionandiui††††††††uu,ivu,i=E[)]==iii,ij,ij,ivv,

7 ivu,i"#"#†††uuuv†††ititjituu,j,ivu,j,i=E
ivu,i"#"#†††uuuv†††ititjituu,j,ivu,j,i=E()=E=j,iititvuvvvv,j,iititvu,j,iThetrendin(1)canbeinducedbyadriftin.Ifinstead,thereisadriftinbutnoneinitititwherewithgiven,thenrepeatedsubstitutiongiveswhereitiitiitiiititit000isadriftlessvectorI(1)process.ThecointegratingregressionbecomesInthis012case,alloftheensuinganalysisistobedoneusingthestatisticalpropertiesof.Incomputations,onecanfollowtherecommendationsofPhillipsandMoon(1999b)toobtainanestimateofrstbyestimatingthedrift)andthenuseitiiTitiTi Theissuesinvolvedinpanelcointegrationvectorestimationandtestingparallelsthatinthesin-gleequationenvironment.Forasingleequation,OLSisaconsistentestimatorofthecointegratingvectorbutitsasymptoticdistributiondependsonthelong-runcovariancebetweenand.ThisnuisanceparameterdependencyinvalidatesstandardhypothesistestingintheOLSframeworkwith-outmodications.DOLS,dynamicGLS,andfullymodiedOLSareexamplesofsuchmodications.Similarly,inpaneldata,PhillipsandMoon(999a)andPedroni(997)showthatfor,thepooledOLSestimatorisaconsistentestimatorofthecointegratingvectorasandcanbeusedinarstpassingettingpointestimates.Inpaneldata,however,theproblemsofsecond-orderasymptoticbiasandnuisanceparameterdependencearecompoundedandarepotentiallymoreseri-OLSousinthesensethatthebiasaccumulateswiththesizeofthecross-section.Inparticular,iftheOLSestimatorforthepooledcross-sectiontime-seriesdata,onecannotruleoutthepossibilityOLSthat)divergesasthen.ItfollowsthatthedistributionforaWaldstatisticfortestinglinearrestrictionsbecomesevenlessusefulasthecross-sectionaldimensionofthepanelgrowssinceittoocandiverge.IIIPanelDOLSWeconsiderthepanelDOLSestimatorofthevectorofslopecoecientsanditslimitdistri-butionforvarioussubcasesofthemodelin()and(2).Wetakethemodelwithin

8 dividual-speciectsasourstartingpoint.Sec
dividual-speciectsasourstartingpoint.Section(i)discussesthebaselinexed-eectsmodel.Onceweobtainthelimitdistributionforthisbaselinecase,thelimittheoryformoregeneralversionsofthemodelwithheterogeneouslineartrendsandcommontimeeectsfollowinananalogousmanner.Insection(ii)weaddheterogeneoustrendstothexed-eectsmodel,andsection(iii)considersthemodelwithxed-eects,trends,andcommontimeeects.FixedEffectsInappliedwork,theresearcherwillalmostalwaysneedtoincludeindividual-specicconstantsintheregression.Tohandlethissituation,webeginbysetting=0forallin(whichwewriteasitiAssumethatiscorrelatedwithatmostleadsandlagsof.Tocontrolforthisititendogeneity,projectontotheseleadsandlags000itititi,sitsi,sitsiit000whereisavectorofprojectioncoecients,)isai,sii,pi,i,p0000dimensionalvectorandx,...,x,...)isa(2pititsionalvectorofleadsandlagsoftherstdierencesofthevariables.TheprojectionerrorKaoandChiang(2000)derivethesequentiallimitdistribution()forpanelDOLSinamodelwithindividual-speciects.Theydonotconsiderthexedthelimittheorywithxed,nordotheyallowfortimetrendsortime-speciects. byconstruction,orthogonaltoallleadsandlagsof.Itfollowsfromassumptionthatbecauseisindependentacrossweprojectonlyontoleadsandlagsofforindividualandnotititontoleadsandlagsoftheotherindividuals(Substitutingtheprojectionrepresentationfor(6)into(5)yieldsitiititiitTheprojectiondenesthenewcovariancestationaryvectorprocess,u,v)whereforeachuu,iititvv,iobeysthefunctionalcentrallimittheoremmTr]X#D&w!B=#(#)W,iitiiwhereB,Bareindependent,anduiuiivivi·¸·¸uu,iuu,i=E[)]==vv,ivv,ivv,iTakingthetime-seriesaverageof(7)givesTTTTXXXX####itiitititTTTTSubtracting(8)from(7)eliminatesandgivesitititiitwherea‘tilde’denotesthedeviationofanobservationfromitstime-seriesaverage,PPPPTTTT11

9 11,and˜ititititititititititititTTTTToset
11,and˜ititititititititititititTTTTTosetuptheestimationproblem,let˜bethe2)dimensionalvectorwhoserstelementsare˜,elements+(2))++(2))are˜and0selsewhere.Thatis,=(˜...=(˜z...=(˜...NtNt0000Letthegrandcoecientvectorbe)andwritethecompactformoftheregression as˜.ThepanelDOLSestimatorfortheectsmodelis,whereitit"#"#NTNTXXXX)=˜NTitititWeexploitthefactthatthelimitingbehavioroftheregressionerror˜,isidenticaltothatofititSomealgebrarevealsthat˜BdWitituu,iuiititviviviPPPTTT111###.Wealsohave˜ititituiuu,iuiTTTsoweareabletouseestimatedvaluesof˜toobtainaconsistentestimateof.Theituu,ilimittheorywithforpanelDOLSwithindividual-specixedeectsisgiveninProposition(Fixedwithxedeects.)LetForthepanelDOLSviviviestimator(0),forxedandareindependentforeach,where,andNviviBdWuu,iuiNviiNTR(%$%)][RDRRNTR(%$%)]!!(s),whereisanrestrictionNTNT˜˜˜˜matrix,MVMandNNNNuuiviviviviNNih³´ibbb,whereMVMNTNNTNTNTititNTNTbbbandisaconsistentestimatorofNTuuiuuiuu,iititTheasymptoticindependenceof)and)asfollowsforthesamereasonsasinthesingle-equationenvironmentandbecause)convergesindistributiontoamixednormalrandomvector,thelimitingchi-squaredistributionofthequadraticforminpart()ofthepropositionalsofollowsbythestandardargument.Theasymptoticcovariancematrixcanbeconsistentlyestimatedbyanditfollowsthatunderthenullhypotheses,theWaldstatisticcNT(R%$r)][RDRRNT(R%$r)]!!(s)(##)NTNTNTforanygivenThesequentiallimitdistributionofisobtainedbyshowingthatthesequenceviviobeysalawoflargenumbersforindependentbutheterogeneouslydistributedobservationsandthatthesequenceBdWobeysacentrallimittheoremforindependentbutheteroge-uuiuineouslydistributedobservations.ThesequentiallimittheoryforpanelDOLSisgivenin Proposition(Sequentiallimitdistribution,xedeects.)ForthepanelDOLSestim

10 ator(0),thenwhereMVMkNNandNvv,iNuu,ivv,i
ator(0),thenwhereMVMkNNandNvv,iNuu,ivv,i,whereisdenedinproposition.d.NTNNTControllingforxedeectsresultsinashrinkageofthesequentiallimitasymptoticvariance,comparedtowhentherearenoxedeects.Inthecasewithoutxedeectswhere=0forallandNvv,iNuu,ivv,i(ii)FixedEffectsandHeterogeneousTrendsWenowadmitbothindividual-specixedeectsandheterogeneoustimetrendsintothespeci-cation.Uponsubstitutionoftheprojectionrepresentationfortheequilibriumerror(6)into((with=0forall)wehave,itiiititiitTakingthetime-seriesaverageof(2)yieldsTTTTXXXX####itiiititiitTTTwhereweusethefactthat2.Tocontrolforthexed-eectssubtract(3)from2)togetitiititiitwhereagainweusea‘tilde’todenotethedeviationofanobservationfromitstime-seriesaverage,PPPPTTTT1111,andititititititititititititTTTTTosetuppanelDOLS,let)bethevectoroftrendslopecoecients,000)bethegrandcoecientvector,andde000=(˜000=(˜00˜000=(˜NtNtThenthepanelDOLSestimatorof"#"#NTNTXXXXititNTitit For,as)isindependentof)and)fortheiiNNstandardreasonsbut)and)remaincorrelated.ThelimittheorywithforthexedeectsmodelwithtrendisgiveninProposition(Fixedxedeectsandtrends.)LetForthepanelviviviDOLSestimator(6),forxedisindependentofandforeachiiNN,whereBB,,NNvivih³´³´iRRRR1111˜˜˜˜,rBandvNvNBdWuu,iuihhihiipRpRrdW···rdWuu,uu,NuNWhenthen,thepanelDOLSestimatorofthetrendslopecoecientsandthecointegrationvectorareindependentwhichresultsinconsiderablesimplication.ThesequentiallimittheoryforpanelDOLSinthiscaseisgiveninProposition(Sequentiallimits,xedeectsandtrends.)ForthepanelDOLSestimator(6),thenandareindependent.,whereMVMand,Nvv,i,nuu,ivv,ibbb,whereMVMNTNNTNTNTititNTNTibbbandisaconsistentestimatorofuuiuu,iuu,iititNoticethatthesequentiallimitdistributionfor)isidenticaltothatobtainedinproposition2intheabsenceoftrends.Con

11 structionofaWaldtestunderthesequentialli
structionofaWaldtestunderthesequentiallimittheorycanproceedasinsection(i). (iii)FixedEffects,HeterogeneousTrends,andCommonTimeEf-fectsTheasymptoticdistributiontheorythatweemployrequiresthatobservationsareindependentacrossindividualsbutinapplications,onetypicallyencounterssomedegreeofCSD.Inthissectionwetakeupthecompletemodel()whichallowsustomodelalimitedformofCSDinwhichtheequilibriumerrorforeachindividualisdriveninpartbyBeginbysubstitutingtheprojectionrepresentationforinto()togetitiitititiitControllingforthecommontimeeectrequiresananalysisofthecross-sectionalaverageoftheobservations.Becauseweadmitheterogeneityintheprojectioncoecientsacross,theresultingcross-sectionalaverageswillinvolvesumssuchaswhichcomplicatesestimationofthejjtcoecients.Theestimationproblemcanbesimpliedbyproceedingsequentiallyandaddressingtheendogeneitycorrectionseparatelyfromcointegrationvectorestimation.Todothis,letbetheerrorfromprojectingeachelementofonto,t,z)andbethevectorofprojectionerrorsfromprojectingeachelementofontoititititwhereisa(+2)matrixofprojectioncoecients.Substitutingtheprojectionrepresentationsforandinto(7)givestitititWenowworkwith(8)sinceforthepurposesofestimatinganddrawinginferenceaboutitisequivalentto(7).Nowtakethecross-sectionalaverageof(8)togetNNNXXX###tjtjtjtNNNSubtracting(9)from(8)eliminatesthecommontimeeectgiving(20)ititwherea‘star’denotesthedeviationofanobservationfromitscross-sectionalaverage.Thatis,ititjtititjtitjtThepanelDOLSestimatorof"#"#NTNTXXXXxxxy.ititititAsinthecaseofthexed-eectsmodelwithlineartrends,thepanelDOLSestimatorofthegrandcoecientvectorconvergestoamixednormalrandomvectorbut)and areasymptoticallycorrelatedforxed.Weomitastatementofthelimittheoryforthiscase.Asthen,however,)and)areinde

12 pendentandthelimittheoryforthiscaseisgiv
pendentandthelimittheoryforthiscaseisgiveninProposition(Sequentiallimitdistribution.)ForthepanelDOLSestimator(2),asthenandareindependent.,whereMVMand,Nvv,i,Nuu,ivv,iwhereMVMxx,NTNNT,NT,NTitit,NT,NTxx,andisaconsistentestimatorof,NTuu,iuu,iuu,iititNoticethatthelimitdistributionofproposition5isidenticaltothesequentiallimitdistributionofproposition4.Controllingforxedeectsagainproducesashrinkageoftheasymptoticvariance,whilecontrollingforthecommontimeeectrequirestakingthedeviationfromthecross-sectionalaverage.Thesecross-sectionaltransformationshavenoeectonthesequentialasymptoticvarianceoftheestimator.IfthemodicationstoOLSaresuccessfulinremovingthecorrelationbetweentheequilibriumerrorandleadsandlagsofforbutthetime-speciectsdonotfullyaccountforCSD,thentheresidualcross-sectionalcorrelationintheprojectionerrorchangesonlytheformulafortheasymptoticstandarderrors.Thisisafeasibleestimationstrategyforsmalltomoderate.Butifthereremainscorrelationbetweentheequilibriumerrorandleadsandlagsotherequationx,ithenpanelDOLSexhibitsthesamesortofsecond-orderasymptoticbiasaspooledOLSasdiscussedinsectionII.Forsmalltomoderate,afeasiblesolutiontothisproblemistoincludeleadsandlagsofx,jintheprojection(6).Weclosethissectionbynotingthatforlarge,modelingCSDinpaneldataisitselfanactiveareaofresearchandonethathasshownitselftobeathornyproblem.Whatoneseeksinthiscaseisasimpleparametricstructurethatdoesanadequatejobofcapturingthelongruncovariancestructure.BaiandNg(200),MoonandPerron(2002),andPhillipsandSul(2002)studymodelsinwhichtheerrortermsindynamicpaneldataregressionshaveafactorstructure,buttheimplicationsforsuchfactormodelshavenotbeenstudiedinthepanelcointegrationcontext.MonteCarloExperimentsInthissection,wepresentsomeMonteCarloex

13 perimentstoinvestigatesomesmallsamplepro
perimentstoinvestigatesomesmallsamplepropertiesofpanelDOLSandtocomparethemtosingle-equationDOLSinthepresenceofindividualInthiscase,theasymptoticvarianceofpanelDOLSisconsistentlyestimatedby¡¢¡¢¡¢PPPTTT000XXXXXXwherex,...,x)andisaconsistentttttuu,TtNtestimatorofthelong-runcovariancematrixof,...,N ectsandCSD.Ourdatageneratingprocess(DGP)includestworegressorsinthecointegratingrelationandisgivenbyiti,it,itit,iti,it,it,it000Letting,v,v²,²,²),ande,e,e),theshort-run,it,it,it,it,it,ititititdynamicsaregivenbyititittitiidiidwherefor,...,Nj,ittjtWedesignedtheDGPtoprovideaconnectiontotheempiricalworkonmoneydemandofthenextsection,wheretheregressorsarerealincome(whichhasadrift),andthenominalinterestrate(whichdoesnot).Accordingly,weinduceatrendintothecointegratingrelationthroughthedrifttermfortherstregressorandspecifythesecondregressortobeadriftlessI()process.,it,itInaddition,wemodeltheequilibriumerror,toadmitamoregeneralformofCSDthaniitthecommontimeeectmodelconsideredintheprevioussection.Thissingle-factormodeloftheshort-runinnovationsisofthetypeconsideredbyPhillipsandSul(2002).CSDintheequilibriumerrorsisinducedby,whileandinducecross-sectionalendogeneitybetweenandtj,kti,kThesefeatureswerenotexplicitlyaccountedforinthetheoreticalanalysis,butmaybeencounteredinempiricalwork.InthepresenceofheterogeneousCSD,subtractingothecross-sectionalaveragedoesnotcompletelyeliminateCSD.Ourinteresthereisinevaluatingtheseriousnessoftheresultingdistortions.ThedegreeofCSDismodulatedbythesizeofThetruevalueofthecointegrationvectoris()=().Foreachindividual,thevaluesofandarerstobtainedbyadrawfromtheuniformdistributionthenheldiijixedthroughouttheexperiment.Thepersistenceintheshort-rundynamicsarecontrolledbyvaryingthesupportoftheunifo

14 rmdistributionfromwhichtheelementsofared
rmdistributionfromwhichtheelementsofaredrawn.WeconsiderthreelevelsofpersistenceandthreealternativedegreesofCSD.Persistencelevelscanbelow(),medium(),orhigh(),anddegreesofCSDarere.3,0.5][07][0eithernone(=0),low(3)orhigh(7).AssignmentoftheremainingparametervaluesKaoandChiang(2000)comparedthesmall-sampleperformanceofpanelDOLSandpanelfullymodiedOLSwithxedeectsinthecaseofasingleregressor.TheyfoundthatpaneldynamicOLSperformedmuchbetterthanpanelfullymodiedOLSinremovingnitesamplebiassowedonotincludepanelfullymodiedOLSinthecomparison.Inthecommontimeeectspecication,thecross-sectionalcorrelationbetweenindividualsandisidenticalforalli,j.ThishomogeneousCSDisobtainedherebysettingtobeidenticalacross.ThatallowingforheterogeneityinresultsinheterogeneousCSDcanbeseeninthecaseofanAR(1)whereandall,iiotherelementsofaresettozero.ThenitcanbeshownthatCorr(,whereitjtijE²²,jt©¡¢¡¢ª¡¢¡¢2222E²E²+(1$#$+(1,jt aredeterminedby,U,AU,AA"0.05,0.05][05][05][0,iii.0,0.04][2353][133][034][2.Thelong-runvarianceisestimatedbytheprewhitenedquadraticuu,ispectral(QSPW)methodsuggestedbySul,Phillips,andChoi(2002).Eachexperimentconsistsof5,000randomsamplesof=40=200observationson0or=20individuals.Thenumberofleadsandlagsofincludedare2(=40),3(00),and4=200).Weorganizetheexperimentsaccordingtothefollowingthreecases.Case(NoCSD,variablepersistence).Setting=0yieldsnoCSD.Persistencelevelsarelow,medium,andhigh.Case2:(HomogeneousCSDandPersistence)SettingyieldsthehomogeneousCSDasinthecommontimeeectspecication.WeconsiderhighandlowCSDandlow,mediumandhighlevelsofpersistence.Case3:(HeterogeneousCSDandPersistence)AllowingyieldsheterogeneousCSD.WeconsiderhighandlowCSDandlow,mediumandhighlevelsofpersistence.Webeginwiththeeectivesizeofnominal5%and0%sizedtests

15 ofthehypothesis.Toprovideapointofcompari
ofthehypothesis.Toprovideapointofcomparison,Tabledisplaystheeectivesizeof(single-equation)DOLStests.Table2showsthepanelDOLSsizeresultsforCase.Underlowandmediumlevelsofpersistence,thetestsarereasonablysized.Sizeaccuracyisseentoimprovewithincreasingsamplesizebothinthetimeseriesaswellasinthecross-sectionaldimensions.Underhighlevelsofpersistence,thetestforremainsreasonablysizedbutthetestforbecomesslightlymis-sized.Thismis-sizingworsenssomewhatasthecross-sectionincreases(e.g.,for00,the5%testhassizeof6%for0and25%for=20).IncomparisontoDOLS,thetestforisbettersizedwhereasthetestforisroughlyequivalent.Table3reportstheeectivesizeofpanelDOLStestsunderCase2.ForalowdegreeofCSD,thesizeofthetestforimproveswithpersistenceandisaccuratewhenthelevelofpersistenceishigh.Thesizeofthetime-seriesisrelativelyunimportant.SimilarresultsareobtainedforthetestonunderlowCSD.UnderhighCSD,thereissomemis-sizingoftheteston,whichiscomparabletothesizeoftheDOLStest.Fortheteston,sizeaccuracyimproveswiththesizeofthecross-sectionandoverallsizedistortionismodest.ectivesizeperformanceofpanelDOLStestsundercase3,showninTable4,isverysimilartothatundercase2.Subtractingothecross-sectionalaverageworksreasonablywellasacontrolfortheheterogeneousCSDconsideredhere.Table5reportsquantilesofˆfromDOLSandpanelDOLSunderCase3.Here,itisseenthatdramaticprecisiongainsoversingle-equationDOLScanbeattainedinsmallsamples.For=400underhighpersistenceandhighCSD,theinter-95percentilerangeforDOLSis(-0.304;2.495)whileforpanelDOLSis(0.883;52).Precisiongainscontinuetoaccruewhen=200.For0,underhighpersistenceandhighCSD,thepanelDOLSinter-95percentilerangeof(0.979;.028)whereasforDOLSitis(0.89;22).PrecisionadvantagesarealsoseentoThisislargelyafeatureofSul,PhillipsandChoi

16 ’sQSPWestimatorofthelong-runvariancewhic
’sQSPWestimatorofthelong-runvariancewhichworkswellunderhighpersistence. accruefromenlargingthecross-sectionaldimension.UnderhighpersistenceandhighCSD,=40,theinter-95percentilerangeshrinksfrom(0.883;52)for0to(0.924;02)for=20.Table6displaysanalogousquantileinformationforˆ.Here,thebenetsfromthecross-sectiondimensionarelargelyobtainedwith0.For=40,underhighpersistenceandCSD,theinter-95percentilerangeofpanelDOLSis(0.096;0.0),whichisanimprovementoverthe(0.054;0.70)rangeforDOLS.WesummarizetheMonteCarloresultswithfourgeneralobservations.First,forgiven,theempiricalsizeofthepanelDOLSt-testsworsensslightlywhenisincreasedfrom0to20.Second,sizedistortion,whilenotparticularlysevereat=40andisreasonablysmallat=200.Third,subtractingthecross-sectionalaveragetocontrolforCSDworksreasonablywelleveninthepresenceofheterogenousCSD.Fourth,panelDOLSismuchmoreprecisethansingle-equationDOLS.Long-RunMoneyDemandWenowemploypanelDOLStoestimatecoecientsofthelong-runMdemandfunction.Economistshavelongbeeninterestedinobtainingpreciseestimatesofmoneydemandforatleasttworeasons.First,knowingtheincomeelasticityofmoneydemandhelpsindeterminingtherateofmonetaryexpansionthatisconsistentwithlong-runpricelevelstability.Second,knowingtheinterestelasticityofmoneydemandaidsincalculatingtheareaunderthedemandcurveandtoassessthewelfarecostsoflong-runination[Baily(956)].Additionally,becauseastablemoneydemandfunctionisabuildingblockoftheIS-LMmodel,economistshavehistoricallybeeninterestedinknowinghowwellthisparticularaspectofthemodelperformed.Whilethismotivehasbecomelessimportantintheeraofdynamicgeneralequilibriummodels,Lucas(988)showsthatsuchaneoclassicalmodelwithacashinadvanceconstraintgeneratesastandardmoneydemandfunction.WefollowStockandWatson

17 (993),Ball(999),andHomanet.al.995)andapp
(993),Ball(999),andHomanet.al.995)andapproachlong-runmoneydemandasacointegratingrelationship.Ouranalysissuggeststhatinstabilityexhibitedbytime-seriesestimatesfromtheliteraturedonotreectunderlyingshiftsinbehavioralrelationshipsbutinsteadindicateinherentdicultiesassociatedwithestimationusingrelativelyshortsamplespansinenvironmentswithpersistentshortrundynamics.CombiningobservationsacrosscountriesallowsustoobtainrelativelysharpandstableestimatesofmoneydemandelasticitiesandthepanelcointegrationapproachseemswellsuitedtotakeupKing’s(988)suggestiontoextendthemoneydemandanalysisbeyondtheUnitedStates.Inhiswords,“theresultsofsuchinvestigationswouldprovideuswithsharperestimatesofthelongrunvaluesofFriedman’s(956)‘numericalconstantsofmonetarybehavior’whenweapproachthedicultproblemoftheshortrundemandformoney.”Theequationthatweestimateis,ln=(22)iityitritfor9,whereisanMmeasureofmoney,isthepricelevel,isrealGDP,andititititisanominalshortterminterestrate.Datadenitionsandsourcesareavailableintheunpublishedappendix.Inadditiontocountryspeciects,,weallowforpossiblyheterogeneouslineartrendsandcommontimeeects.Thesetrendsareincludedtocapturechangesinthenancialtechnologythataectsmoneydemandindependentlyofincomeandtheopportunitycostofholding money.Pre-testing:CointegrationandHomogeneityRestrictionsPanelDOLSestimationof(22)requiresthatthetheequilibriumerrorsarestationaryandthatthecointegratingvectorsforeachcountrymustbeidentical.Toinvestigatethestationarityoftheequilibriumerrors,weemployPedroni’s(999)panel-test.Thisresultsintherejectionofthenullhypothesisofno-cointegrationatthe0.%levelwhetherornotheterogeneouslineartrendsandcommontimeeectsareincluded.Next,weconductaWaldtestofthehomogeneityrestrictionsonthecointegratingv

18 ector.Whentrendsareomittedfromtheregress
ector.Whentrendsareomittedfromtheregression,theevidenceagainsthomogeneityismixed.Theasymptotictestrejectstherestrictionsinthiscase,butinsomeunreportedMonteCarloexperiments,wefoundmoderatesizedistortionintheWaldtestforsamplesizesof9and=40.Usingasizeadjustmentfromtheseexperiments,thehomogeneityrestrictionsonincomeisrejectedatthe5%levelbutnotfortheinterestrate(p-value=0.30).However,whenweimposehomogeneityontheinterestrateslope,thetestforslopehomogeneityonincomeisnotrejected(p-value=0.70).Whenheterogeneouslineartrendsareincluded,theevidencesupportinghomogeneitystrengthens.Here,weobtainap-valueof0.63forthetestofhomogeneityontheincomecoecientandap-valueof0.22forthetestofhomogeneityontheinterestratecoecient.(ii)Comparisonbetweensingle-equationandpanelDOLSestimatesOurpanelDOLSestimatesuse2leadsand2lagsofintheregressions.Pointititestimatesandasymptoticstandarderrorsarereportedintable7.SingleequationDOLSestimatesareseentodisplaysuchcross-sectionalvariabilitythattheyareculttointerpret.InDOLSregressionswithouttrend,theincomeelasticitiesareallpositive,rangingfrom0.34(Belgium)toawhopping2.64(Norway),buttheinterestsemi-elasticityhasthewrongsignforBelgium,France,Ireland,andJapan.Whenatrendisincludedintheregression,incomeelasticityestimatesarenegativeforFinland,Iceland,Norway,andNewZealand,andinterestsemi-elasticityestimatesarepositiveforFinland,France,andIceland.Ifwemaintainanunderlyingbeliefthatthenancialsystemsandtransactionstechnologiesacrossmoderneconomiesareessen-tiallysimilar,thecross-sectionalvariabilityintheseestimatesmustreecttheinherentdicultyofobtaininggoodestimatesratherthanevidenceofdisparateeconomicbehavior.PanelDOLSestimatesareshownatthebottomoftable7.Whenthepanelregressionomitstrends,weestim

19 ate0.86(asymptotics.e.=0.09)andtheintere
ate0.86(asymptotics.e.=0.09)andtheinterestsemi-elasticitytobe-0.02(asymp-totics.e.=0.0).Whenweincludeheterogeneoustrends,weestimatetheincomeelasticitytobe(asymptotics.e.=0.26)andtheinterestsemi-elasticitytobe-0.02(asymptotics.e.=0.0).ResultsobtainfromcontrollingforCSDareverysimilar.Tofurtherillustratetheproblemofestimationinstabilityinthetimedimension,weconstructedrecursivesingle-equationDOLScoecientestimatesfortheUS,UK,France,andJapanandpanelWealsoconrmedthesecointegrationtestresultsbyusingIm,PesaranandShin(1997)andMaddalaandWu(1999)panelunitroottestsundertheassumptionthatthecointegratingvectorisknowntobe(105).Thejusticationforusingthesevaluesisthat10isatypicalvalueoftheincomeelasticityestimatedintheliteraturewhileacommonestimateoftheinterestratesemielasticity05. DOLSforall9countries.RecursiveDOLSestimatesoffrom979to995forboththeincomeelasticityandinterestsemielasticityexhibitsubstantiallymorevariabilitythantherecursivepanelDOLSestimatesandinseveralinstancesevenchangesign.Theseresultsarealsocontainedintheunpublishedappendix.ConclusionsHeterogeneityandpersistenceinshortrundynamicscancreatesubstantialvariabilityinsingle-equationcointegrationvectorpointestimates.Theresultisthattheseestimatorscanbequitesensitivetotheparticulartimespanoftheobservationsaswellastotheparticularindividualbeingstudied.Thissmallsamplefragilitycanbeencounteredinspiteofthesuperconsistencyoftheseestimators.Intheseenvironments,panelDOLScanprovidemuchmorepreciseestimates.PanelDOLSisstraightforwardtocomputeandrelevantteststatisticshavestandardasymptoticdistributions.Theasymptoticdistributionswerefoundtoprovidereasonablycloseapproximationstotheexactsamplingdistributionsinsmallsamples.WeappliedthepanelDOLSmethodtoestimatethel

20 ong-runmoneydemandfunctionusingapanelof9
ong-runmoneydemandfunctionusingapanelof9countrieswithannualdatafrom957to996.Theestimatesinwhichwehavethemostcondenceareanincomeelasticitynearandaninterestratesemi-elasticityof-0.02. ReferencesBaba,Yoshihisa,DavidF.Hendry,andRossM.Starr992).‘TheDemandforMintheU.S.A.,960—988,’ReviewofEconomicStudies,59,pp.25—6Bai,JushanandSerenaNg(200).“DeterminingtheNumberofFactorsinApproximateFactorModels,”forthcominginEconometricaBailey,MartinJ956).‘TheWelfareCostofInationaryFinance,’JournalofPoliticalEcon-,64,(April)PP93—Ball,Laurence998).‘AnotherLookatLong-RunMoneyDemand,’NBERWorkingPaper.No.W6597.Friedman,Milton956).‘TheQuantityTheoryofMoney—ARestatement,’inMiltonFriedman(ed.),StudiesintheQuantityTheoryofMoney.Chicago:UniversityofChicagoPress.Gandolfi,ArthurE.,andJamesR.Lothian976).‘TheDemandforMoneyfromtheGreatDepressiontothePresent,’AmericanEconomicReview,66(2),pp.46—5Hoffman,DennisL.,RobertH.Rasche,andMargieA.Tieslau995).‘TheStabilityofLong-RunMoneyDemandinFiveIndustrialCountries,’JournalofMonetaryEconomics,35,pp.7—339.Im,KyungSo,M.HashemPesaranandYongcheolShin997),‘TestingforUnitRootsinHeterogeneousPanels,’DiscussionPaper,UniversityofCambridge.Inder,Brett993).“EstimatingLong-RunRelationshipsinEconomics:AComparisonofDif-ferentApproaches,JournalofEconometrics,57(-3),May—June,pp.53—68.Kao,ChihwaandMin-HsienChiang(2000).‘OntheEstimationandInferenceofaCointe-gratedRegressioninPanelData,’inAdvancesinEconometrics:NonstationaryPanels,PanelCointegrationandDynamicPanels,5,pp79—222.King,RobertG988).‘MoneyDemandintheUnitedStates:AQuantitativeReview:AComment,’Carnegie-RochesterConferenceSeriesonPublicPolicy,29,pp.69—72.Lucas,RobertE.Jr.988).‘MoneyDemandintheUnitedStates:AQuantitativeReview,’Carnegie-RochesterConf

21 erenceSeriesonPublicPolicy,29,pp.37—68.M
erenceSeriesonPublicPolicy,29,pp.37—68.Maddala,G.S.andShaowenWu999)‘AComparativeStudyofUnitRootTestswithPanelDataandaNewSimpleTest,’OxfordBulletinofEconomicsandStatistics,63-652.Meltzer,AllanH.963).‘TheDemandforMoney:ACross-SectionStudyofBusinessFirms,’QuarterlyJournalofEconomics77,pp.405-422.Moon,HyungsikR.,andPierrePerron(2002.)“TestingforaUnitRootinPanelswithDynamicFactors,”mimeoUniversityofSouthernCalifornia.Mulligan,CaseyB.997).‘ScaleEconomies,theValueofTime,andtheDemandforMoney:LongitudinalEvidencefromFirms,’JournalofPoliticalEconomy05,pp.079.Mulligan,CaseyB.,andXavierSala-i-Martin992).‘U.S.MoneyDemand:SurprisingCross-SectionalEstimates,’BrookingsPapersonEconomicActivitypp.285—329.Pedroni,Peter997).‘FullyModiedOLSforHeterogeneousCointegratedPanelsandtheCaseofPurchasingPowerParity,’mimeo,DepartmentofEconomics,IndianaUniversity. Pedroni,Peter999).‘CriticalValuesforCointegrationTestsinHeterogeneousPanelswithMultipleRegressors,’OxfordBulletinofEconomicsandStatistics,(November),pp.653—670.Phillips,PeterC.B.andHyungsikR.Moon999a).‘LinearRegressionLimitTheoryforNonstationaryPanelData,’Econometrica67(5),pp.057—####Phillips,PeterC.B.andHyungsikR.Moon999b).‘NonstationaryPanelDataAnalysis:AnOverviewofSomeRecentDevelopment,’mimeo,DepartmentofEconomics,UniversityofCaliforniaSantaBarbara,999.Phillips,PeterC.B.andDonggyuSul(2002).“DynamicPanelEstimationandHomogeneityTestingUnderCrossSectionDependence,”mimeoYaleUniversity.Saikkonen,Pentti).‘AsymptoticallyEcientEstimationofCointegrationRegressions,’EconometricTheoryStock,JamesH.andMarkW.Watson993).‘ASimpleEstimatorofCointegratingVectorsinHigherOrderIntegratedSystems,’Econometrica:783—820.Sul,Donggyu,PeterC.B.Phillips,andChi-YoungChoi.(2002).“Prewhiteni

22 ngBiasinHACEstimation,”mimeo,YaleUnivers
ngBiasinHACEstimation,”mimeo,YaleUniversity.White,Halbert984.AsymptoticTheoryforEconometricians,AcademicPress,NewYork. TableectiveSizeofDOLStests.0102Persistence5%0%5%Low0.0970.520.0930.40Medium0.80.790.40.High0.790.24750.239Low0.000.550.0930.00Medium0.00.700.040.High0.840.2500.820.248Low0.0670.280.0670.200Medium0.07320.0740.High0.70.40.Table2:EectivesizeofpanelDOLStests.Case:NoCSD,variablepersistence0102Persis-N=0N=20N=0N=20Ttence5%0%5%0%5%0%5%Low0.0920.540.0870.420.0960.560.0820.40Medium0.0760.330.0650.260.030.690.040.High0.0560.0980.0390.0740.00.800.550.237Low0.0720.220.0590.090.07260.0640.00Medium0.0620.###0.0540.060.0800.360.0720.High0.050.0930.0420.0930.840.590.250Low0.0600.050.0600.00.0580.30.0640.200Medium0.0590.020.0580.080.0620.0.0670.High0.0450.0970.0440.0850.0900.670.470.228 Table3:EectivesizeofpanelDOLStests.Case2:HomogeneousCSD.0102Persis-N=0N=20N=0N=20TCSDtence5%0%5%0%5%0%5%Low0.030.600.###820.060.650.0940LowMedium0.0870.470.0880.500.270.200.0830.High0.0690.30.050.0920.650.2450.0690.Low0.0670.280.0790.360.0730.290.0650.00LowMedium0.0630.80.0730.240.0840.430.0590.High0.0500.0940.0530.0980.400.2270.0670.Low0.0680.0.0630.50.0740.280.0500.098200LowMedium0.0680.220.0570.00.0750.270.0450.090High0.0470.0980.0460.0970.200.2040.0530.Low0.050.670.0.2340.680.2400.990.26740HighMedium0.0960.520.00.670.2460.High0.0650.20.0820.390.550.2300.0880.Low0.0940.550.330.2060.000.240.00HighMedium0.0870.470.230.860.30.790.30.High0.0640.50.070.730.500.240.0980.Low0.020.640.300.980.0890.450.09200HighMedium0.050.630.280.2000.0900.520.0830.High0.0870.470.240.340.260.0800. Table4:EectivesizeofpanelDOLStests.Case3:HeterogeneousCSD.0102Persis-N=0N=20N=0N=20TCSDtence5%0%5%0%5%0%5%Low0.040.660.0980.590.0940Low

23 Medium0.0830.340.0820.400.090.720.0960.H
Medium0.0830.340.0820.400.090.720.0960.High0.0590.040.0500.0850.980.240.Low0.0750.340.0820.400.07280.0680.00LowMedium0.0690.0.0760.0.0850.440.0770.High0.050.0990.050.0990.20.870.380.229Low0.0420.0920.0630.0400.0880.0550.200LowMedium0.0400.0820.0570.20.0460.0880.0560.High0.0400.0760.0420.090.0660.00.Low0.40.720.470.270.520.220.2060.28340HighMedium0.0940.420.80.900.470.250.670.234High0.0580.0980.0530.0940.420.230.0890.Low0.0980.650.380.2020.0980.650.370.20700HighMedium0.0880.990.080.730.High0.0600.0820.380.460.2250.0950.Low0.0900.500.280.990.0680.90.020.200HighMedium0.0880.470.450.2070.0740.020.High0.0700.240.90.900.080.880.0870. Table5:Quantilesforrstregressorslope(Case3):PanelDOLSandDOLS0N=0N=20Persis-DOLSPanelDOLSPanelDOLSCSDtence2.5%50%97.5%2.5%50%97.5%2.5%50%97.5%T=40Low0.4600.99920.952.00.0480.976.000.025NoneMed0.2250.999.7690.934.00.0700.967.002.038High-0.02.4200.890.007370.937.008.078Low0.4560.999.5290.95.000.0520.973.000.029LowMed0.2060.996.7820.93.00.0750.963.003.044High-0.259.0062.4350.882500.936###.090Low0.4320.999.5360.956.000.0470.97.000HighMed0.0.996.7830.936.00.0730.960.003.049High-0.304.0042.4950.883###520.924Low0.906.000.0940.988.00020.994.000.006NoneMed0.852.000460.982.00080.99.000High0.692.002.3470.966.003.0390.98.002.022Low0.904.000.0950.985.00060.99.000.009LowMed0.850.000500.978.000.0230.988.00High0.682.3690.96.005.0520.978.004.030Low0.90.000.0990.986.00040.99.000.009HighMed0.8470.999540.978.000.0220.987.00High0.666.000.3780.957.005.0560.974.004.036T=200Low0.970.000.0290.996.000.0040.998.000.002NoneMed0.955.000.0450.994.000.0060.997.000.004High0.898.000070.987.0040.993.000.008Low0.970.000.0300.993.000.0070.996.000.004LowMed0.955.000.0470.990.00000.995.000.006High0.895.000

24 70.983.002.0240.99.00Low0.969.000.030.99
70.983.002.0240.99.00Low0.969.000.030.994.000.0060.996.000.004HighMed0.952.000.0480.990.00000.994.000.007High0.890220.979.002.0280.987.002 Table6:Quantilesforsecondregressorslope(Case3):PanelDOLSandDOLS0N=0N=20Persis-DOLSPanelDOLSPanelDOLSCSDtence2.5%50%97.5%2.5%50%97.5%2.5%50%97.5%T=40Low0.0720.000.290.0980.000.020.0990.000.NoneMed0.0650.420.0980.000.030.0980.000.High0.0500.050.720.0970.020.070.0940.0980.Low0.0750.000.270.0980.000.020.0990.000.LowMed0.0680.400.0980.030.0980.000.High0.0520.050.720.0970.020.080.0940.0980.Low0.0760.000.260.0980.000.020.0980.000.HighMed0.0690.390.0970.040.0970.000.High0.0540.050.700.0960.030.00.0920.0980.Low0.0930.000.070.0990.000.000.000.NoneMed0.09000.###0.0990.000.0.0990.000.High0.0850.020.280.0990.020.0980.0990.Low0.0940.000.060.0990.000.000.000.LowMed0.0920.000.00.0990.000.0.0990.000.High0.0860.020.250.0990.030.0980.0990.Low0.0940.000.060.0990.000.0.0990.000.HighMed0.0920.000.00.0990.000.0.0990.000.High0.0870.020.240.0990.040.0970.0990.T=200Low0.0970.000.030.000.000.000.000.000.NoneMed0.0960.000.050.000.000.000.000.000.High0.0940.30.000.000.0.0990.000.Low0.020.000.0980.000.000.000.000.000.LowMed0.030.000.0960.000.000.000.000.000.High0.050.0990.0890.000.000.0990.000.Low0.020.000.0980.000.000.000.000.000.HighMed0.030.000.0960.000.000.0990.000.000.High0.050.0990.0900.0.0990.0980.000.099 Table7:Single-equationandPaneldynamicOLSestimatesoflongrunmoneydemandNoTrendWithTrendCountryˆ(s.e.)ˆ(s.e.)ˆ(s.e.)ˆ(s.e.)TrendyRyRAustria0.90(0.39)-0.009(0.029).552(0.349)-0.037(0.026)-0.00Belgium0.34(0.28)0.009(0.039)83(0.444)-0.033(0.026)0.002Denmark.460(0.70)-0.043(0.009)0.684(0.32)-0.036(0.006)0.009Finland9(0.634)-0.006(0.4)-0.740(0.88)0.009(0.0)-0.005France0.677(0.23)0.00(

25 0.020)0.842(0.699)0.004(0.03)0.003German
0.020)0.842(0.699)0.004(0.03)0.003Germany.548(0.033)-0.09(0.008)(0.97)-0.023(0.009)0.003Iceland0.594(0.)-0.00(0.005)-0.45.093)-0.004(0.008)0.004Ireland0.507(0.69)0.022(0.022).670(2.805)0.05(0.029)0.005Netherlands2(0.###)-0.045(0.020)0.309(0.45)-0.0(0.022)0.003Norway2.64(0.450)-0.60(0.046)-0.676(2.54)-0.092(0.060)0.0Portugal0.57(0.36)-0.037(0.0.624(0.379)-0.043(0.0)0.0Spain.203(0.09)-0.030(0.008).203(0.90)-0.030(0.009)0.003Switzerland.020(0.208)-0.062(0.02.447(0.482)-0.053(0.02)0.0.738(0.097)-0.089(0.008)2.28(0.726)-0.089(0.008)0.0Japan0.889(0.599)0.009(0.200).798(0.45)-0.076(0.06)0.0Australia0.926(0.36)-0.043(0.02)0.068(0.329)-0.048(0.007)0.002NewZealand.349(0.539)-0.076(0.026)-.233(49)-0.084(0.08)-0.00Canada.245(0.29)-0.057(0.024)2.420(0.903)-0.078(0.024)-0.0US0.428(0.074)-0.035(0.008).022(0.47)-0.039(0.007)-0.00Panel0.860(0.092)-0.020(0.007).079(0.264)-0.022(0.006)–Panel0.820(0.05)-0.07(0.005)0.986(0.336)-0.06(0.005)–Note:controlsforcommontimeeect. Cointegration Vector Estimation by Panel DOLS and Long-Run Money DemandNBER Technical Working Paper No. 287December 2002 We study the panel DOLS estimator of a homogeneous cointegration vector for a balanced panel of Nindividuals observed over T time periods. Allowable heterogeneity across individuals includeindividual-specific time trends, individual-specific fixed effects and time-specific effects. The estimatoris fully parametric, computationally convenient, and more precise than the single equation estimator.For fixed N as T approaches infinity, the estimator converges to a function of Brownian motions andthe Wald statistic for testing a set of linear constraints has a limiting chi-square distribution. Theestimator also has a Gaussian sequen

26 tial limit distribution that is obtained
tial limit distribution that is obtained first by letting T go to infinitythen letting N go to infinity. In a series of Monte Carlo experiments, we find that the asymptoticdistribution theory provides a reasonably close approximation to the exact finite sample distribution.We use panel dynamic OLS to estimate coefficients of the long-run money demand function from apanel of 19 countries with annual observations that span from 1957 to 1996. The estimated incomeelasticity is 1.08 (asymptotic s.e.=0.26) and the estimated interest rate semi-elasticity is -0.02(asymptotic s.e.=0.01).Nelson C. MarkDonggyu SulDepartment of EconomicsDepartment of EconomicsOhio State UniversityUniversity of Auckland410 Arps HallPrivate Bag 92019Columbus, OH 43210Auckland, New Zealandand NBERD.sul@auckland.ac.nzmark.1@osu.edu TECHNICAL WORKING PAPER SERIESTechnical Working PaperCambridge, MA 02138December 2002This paper was previously circulated under the title “A Computationally Simple Cointegration VectorEstimator for Panel Data.” For valuable comments on earlier drafts, we thank Ronald Bewley, Roger Moon,Peter Phillips, seminar participants at Georgetown University, Ohio State University, the 2001 New ZealandEconometric study group meeting, the University of California at Santa Barbara, the University of SouthernCalifornia, and an anonymous referee. The usual disclaimer applies. The views expressed in this paper arethose of the authors and not necessarily those of the National Bureau of Economic Research.© 2002 by Nelson C. Mark and Donggyu Sul. All rights reserved. Short sections of text, not to exceed twoparagraphs, may be quoted without explicit permission provided that full credit, including © notice, is g