X-Differencing and Dynamic Panel Model Estimation - PDF document

X-DifferencingandDynamicPanelModelEstimationChirokHanKoreaUniversityPeterC.B.PhillipsYaleUniversity,UniversityofAucklandUniversityofSouthampton&SingaporeManagementUniversityDonggyuSul 1IntroductionThereisnowavastempiricalliteratureondynamicpanelregressionscoveringawidearenaofdatasetsandapplicationsthatextendbeyondeconomicsacrossthesocialsciences.Muchoftheappealofpaneldatastemsfromitspotentialtoaddressgeneralsocio-economicissuesinvolvingdecisionmakingovertime,sothatdynamicsplayanimportantroleinmodelformulationandestimation.Totheextentthatthereiscommonalityindynamicbehavioracrossindividuals,itisnaturaltoexpectthatpoolingcrosssectiondatawillbeadvantageousinregression.However,sinceNickell(1981)pointedtotheincidental-parameter-inducedbiaseffectsinpooledleastsquaresregression,therehasbeenanongoingsearchforimprovedstatisticalprocedures.ProminentamongthesealternativemethodsisGMMestimation,whichisnowthemostcom-monapproachinpracticalempiricalworkwithdynamicpanelregression.ThepopularityofGMMismanifestintheextensivecitationofarticlessuchasArellanoandBond(1991)whichdevelopedageneralGMMapproachtodynamicpanelestimation.GMMisconvenienttoimplementinem-piricalresearchanditswidespreadavailabilityinpackagedsoftwareenhancestheusabilityofthismethodology.Ontheotherhand,itisnowwellunderstoodthattheoriginalrstdifferenceIV(AndersonandHsiao,1982)andmoregeneralGMMapproachestotheestimationofautoregres-siveparametersindynamicpanelsoftensufferfromproblemsofinefciencyandsubstantialbias,especiallywhenthereisweakinstrumentationasinthecommonlyoccurringcaseofpersistentornearunitrootdynamics.Solutionstotheweakinstrumentproblemhavefollowedseveraldirec-tions.Oneapproachfocusesonthelevelsequation,wherethereisnolossofsignalintheunitrootcase,combinedwiththeuseofdifferencedlaggedvariablesasinstrumentsundertheassump-tionthatthexedeffectsareuncorrelatedwiththeidiosyncraticerrors,asdevelopedbyArellanoandBover(1995)andBlundellandBond(1998).Anotherapproachcorrectsforthebiasofleastsquaresestimatorsbasedonparametricassumptions,leadingtoimprovedestimationprocedures.Forexample,Kiviet(1995)proposedabiascorrectionthatisbasedonNickell’s(1981)biascal-culationsforthepanelAR(1);andHahnandKuersteiner(2002)modiedthepooledleastsquares(LSDV)methodtoremovebiasuptoorder,whereisthetimedimension.Otherrecentworksuggestsalternativemethodsofbias-freeparametricestimation.Forinstance,Hsiao,PesaranandTahmiscioglu(2002)andKruiniger(2008)proposetheuseofquasi-maximumlikelihoodondifferenceddataundersomeparametricassumptionsonthedistributionoftheidiosyncraticer-rors,whichappearstoreducebiaswithoutmakinganexplicitbiascorrection.HanandPhillips(2010a)suggestasimpleleastsquaresprocedureappliedtoadifference-transformedpanelmodel thateffectivelyreducesbiasinthepanelAR(1)caseandleadstoanasymptotictheorythatiscontinuousastheautoregressivecoefcientpassesthroughunity.Whiletherstapproachmakesmomentassumptionsontheunobservableindividualeffects,theotherapproacheseffectivelymakeparametricassumptionsontheidiosyncraticerrorprocess.Themethodsdevelopedinthepresentpaperbelongtothesecondcategoryabovebuttheyintroduceanoveltechniqueofsystematicdifferencing,whichwecall“X-differencing”,thatelim-inatesxedeffectswhileretaininginformationandsignalstrengthincasesofpracticalimportancewherethereisanautoregressiverootatornearunity.Theresulting“panelfullyaggregated”esti-mator(PFAE)isobtainedbyapplyingleastsquaresregressiontothefullsystemofX-differencedequations.Themethodissimpletoimplement,isasymptoticallyfreefrombiasforallparametervalues,andintheunitrootcasehashigherasymptoticefciencythanbias-correctedLSDVes-timation,therebyretainingsignalstrengthandresolvingmanyofthedifcultiesassociatedwithweakinstrumentationanddynamicpanelregressionbias.Inthestationarycase,bothPFAEandthebias-correctedLSDVestimatorarelarge-cientThegeneralmodelconsideredhereisalineardynamicpanelmodelwithAR()idiosyncraticerrorsandexogenousvariables,sotheframeworkiswellsuitedtoawiderangeofmodelsusedinappliedwork.UnliketheHahnandKuersteiner(2002)biascorrectedLSDVestimator,thePFAEmethoddoesnotrequirelargeforconsistency.ThePFAEprocedurealsosupersedestheHanandPhillips(2010a)leastsquaresmethodbygeneralizingittoAR()modelsandbyconsiderablyimprovingitsefciencybothinstationaryandunitrootcases.SincethePFAEisaleastsquaresestimator,thereisnodependenceondistributionalassumptionsbesidestimeseriesstationarity,andnoneofthecomputationalburdenandpotentialsingularitiesthatexistinnumericalproceduressuchasdifferenceMLE(Hsiaoetal,2002;Kruiniger,2008).Moreover,sinceX-differencingeliminatesxedeffects,theasymptoticdistributionofthePFAEestimatordoesnotdependonthedistributionoftheindividualeffects,whereasGMMinlevels(ArellanoandBover,1995)andsystemGMM(BlundellandBond,1998)arebothknowntosufferfromthisproblem(Hayakawa,2008).Fur-thermore,becausetheautoregressivecoefcientsareconsistentlyestimated,itisstraightforwardtoimplementparametricpanelGLSestimationinasecondstageregression(e.g.,generalizingBhar-gavaetal,1982,topanelAR()models).Finally,notethatX-differencingremovesxedeffectsandattainsatthesametimestrongidenticationbymakinguseofmomentconditionsimpliedbytheAR()errorstructureandthestationarityofthedifferenceddata.Thus,theprocedurerequiresxedeffectsbeadditiveinthemodelandthattheprocessesbetemporallystationary.Thecurrentpaperrelatestoacompanionworkbytheauthors(Han,PhillipsandSul,2011;HPS hereafter),whichintroducedthe‘time-reversal’technologyusedheretodesigntheX-differencingtransformationsthateliminatexedeffectsandcorrectforautoregressiveestimationbias.Us-ingthismethodology,thecompanionpaperdevelopedanew“fullyaggregated”estimator(FAE)specicallyforthetimeseriesAR(1)model.ThatpaperfocusedontheprocessofinformationaggregationinX-differencedequationsystemstoenhanceefciencyintimeseriesregressionandtoretainasymptoticnormalityforinferencepurposes,whilethecurrentpaperemphasizesbiasre-movalandefciencyimprovementinthepanelcontext.ThepresentpaperalsoextendstheHPStechnologytoAR(p)panelregressionsandtomodelswithexogenousvariables.Theremainderofthepaperisorganizedasfollows.Section2providesthekeymotivatingideasandsomeheuristicsthatexplaintheX-differencingprocessandhowthenewestimationmethodworksinthesimplepanelAR(1)model.Section3extendsthemethodologytothepanelAR(p)model,developstheX-differencedequationsystem,veriesorthogonality,anddiscussesimple-mentationofthePFAEprocedure.Section4presentsthelimittheoryofthePFAEandprovidescomparisonswithothermethodssuchasbiascorrectedLSDVandrstdifferenceMLE(FDMLE).Thissectionalsodiscussesissuesoflaglengthselectioninthecontextofdynamicpanelswithun-knownlaglength.Section5reportssomesimulationresultswhichcomparethenitesampleperformanceofthenewprocedurewithexistingestimators.Section6concludes.Somemoregenerallimittheory,proofs,andsupportingtechnicalmaterialaregivenintheAppendices.2KeyIdeasandX-DifferencingWestartbydevelopingsomekeyideasandprovideintuitionforthenewprocedureusingthesimplepanelAR(1)modelwithxedeffectswith>⎩⎩⎩>>⎩⎩⎩>>wheretheinnovationsoverThemodelcanbewritteninalternativeformwhichcorrespondstotheconventionaldynamicpanelAR(1)modelWhen,theindividualeffectsareeliminatedbydifferencingandboth(1)and(2)reducetoTheAR(1)specicationisusedonlyforexpositorypurposesandisreplacedbyAR()dynamicsintherestofthepaper,wherewealsorelaxtheconditionsontheinnovations .Initialconditionsaresetintheinnitepastinthestablecaseandatsomeinitializationwhen,althoughvariousothersettings,whilenotourconcernhere,arepossibleandcanbetreatedasinPhillipsandMagdalinos(2009).Observethatthereisnorestrictiononin(1),whereasin(2)iseffectivelyrestrictedtotheregioninwhichcasethesystemhasadeterministicexplosivecomponentincontrastto(1).Thisimplicitrestrictionin(2)isnotcommonlyrecognizedintheliteraturebut,asmentionedlaterinthepaper,itisimportantincomparingdifferentestimationprocedureswheresomemayberestrictedintermsoftheirsupportbutnotothers.NodistributionalassumptionsareplacedontheindividualeffectsSothemodelcorre-spondstoaxedeffectsenvironmentwheretheincidentalparametersneedtobeestimatedoreliminated.Variousapproacheshavebeendevelopedintheliterature,includingthewithin-group(regression)transformation,rstdifferencing,recursivemeanadjustment,forwardltering,andlong-differencing.However,allofthesemethodsleadtonalestimatingequationsforinwhichthetransformed(dynamic)regressoriscorrelatedwiththetransformederror.Inthesimpletimeseriescase,wheretheinterceptisttedinleastsquaresregressionleadingtoademeaningtrans-formation,theeffectsofbiasintheestimationofhavelongbeenknowntobeexacerbatedbydemeaning(e.g.,OrcuttandWinokur,1969)andinthepanelcasethesebiaseffectspersistasymp-toticallyasforxed(Nickell,1981).Accordingly,variousestimationmethodshavebeenproposedtoaddressthedifcultysuchasinstrumentalvariableandGMMmethods,directbiascorrectionmethods,andthevarioustransformationandquasi-likelihoodmethodsdiscussedintheIntroduction.Theessenceofthetechniqueintroducedinthepresentpaperisanoveldifferencingprocedurethatsuccessfullyeliminatestheindividualeffects(likeconventionaldifferencing)whileatthesametimemakingtheregressorandtheerroruncorrelatedafterthetransformation(whichothermethodsfailtodo).AkeyadvantageisthatthenewapproachdoesnotsufferfromtheweakidenticationandinstrumentationproblemsthatbedevilIV/GMMmethodsbasedonrstdifferenced(orforwardltered)equationswhenthedynamicsarepersistent.ThisfailureofGMMinunitrootandnearunitrootcasesproducessomeundesirableperformancecharacteristicsintheGMMestimatorandpoorapproximationbytheusualasymptotictheory.Atthesametime,becausetheareeliminated, StationaryinitializationintheinnitepastforisalsoassumedinlevelsGMM(ArellanoandBover,1995;BlundellandBond,1998),andismorerestrictivethanerrorserialuncorrelatednesswhichisassumedbyAndersonandHsiao(1981)andArellanoandBond(1991).Hahn(1999)discusseshowassumptionsaboutinitialconditionsmayaffectefciency.Forinstance,thenitesamplevarianceoftherstdifferenceGMMestimatorinthestationarycaseincreases thenewmethodisunaffectedbytherelativevarianceratiobetweentheindividualeffectsandtheidiosyncraticerrors,which,iflarge,makesthesystemGMMestimator(BlundellandBond,1998)performpoorly(seeHayakawa,2008).Hence,weexpectthatthenewprocedureshouldoffersubstantialgainsoverbothGMMandsystemGMMmethods,whilestillhavingtheadvantageofeasycomputation.Thenewprocedurebeginsbycombining(2)withtheimpliedforwardlookingregressionequa-withandwherethe‘future’variableisontherighthandside,asopposedtotheoriginal‘backwardlooking’equation(2).Importantlyinboththebackwardandtheforwardlookingequations,theregressorsareuncorrelatedwiththecorrespondingregressionerrors.Thatis,in(2)(2)|Ω+1(|Ω+1−|Ω⎛1)]=in(3),underthefollowingconditions:(i)forall(aconditionthatisnotactuallyrequiredinoursubsequentdevelopmentbecausetheareeliminated-seeequation(6)below);(ii)iswhitenoiseoverand(iii).Theproofof(4)isgiveninAppendixA.Ifthenthelastequalityof(4)isnottrue,butthisrestrictionisremovedinthenaltransformation(see(7)below).Theorthogonality(4)isacriticalelementinthedevelopmentofthenewestimationprocedureinvolvingsystematicdifferencing.Importantly,theorthogonality(4)stillholdsifwereplacewithany,i.e.,forany.Theimplicationisthattheoriginalbackwardlookingregressorisuncorrelatedwiththeforwardlookingregressionerrorsaslongas.Thatis,undertheconditionsiswhite-noiseover,and,wehavee|Ω⎛1(|Ω+1−|Ω⎛1)]+EforanyAgaintheconditionthatisnotrequiredinthenaltransformationstepshownbelowin(7).Results(4)and(5)canbeusedtoeliminatethexedeffects.Bysimplysubtracting(3)from(2),wegetthenewregressionequation ratherthandecreasesasincreases(see,AlvarezandArellano,2003;Hayakawa,2008)incontrasttothepredictionofasymptotictheory. wheretheregressorisuncorrelatedwiththeerroraslongas⎨forall.NotethatwenowallowfortheunitrootcaseandthisrelaxationisjustiinLemma1below.Thus,formodel(2),ifiswhite-noiseover,thenthekeyorthogonalityconditionforallholdsformodel(6),therebyvalidatingtheuseofpooledleastsquaresregressiontechniques.Wecallthedatatransformationinvolvedinsettinguptheregressionequation(6)“X-differencing”.Observethatthedependentvariabledifferencedwhereastheregressordifferenced.So,theregressionequationisstructuredwithvari-abledifferencing:thedifferencingvariesinasystematicandcriticalwaybetweenthedependentvariableandtheregressor.Further,wewanttoallowforthedifferencingrateitselftochange,isavariable.Hence,theterminologyX-differencing.Fig.1showshowX-differencingcombinesobservations(usingcrossratherthanparallelcombinations)toeliminatexedeffectsincomparisonwithothermethods.First differencingLong differencingX-Differencing X t X t 1 X t 2 X t X t1 X t 2 X t X t 1 X t 2 X t 5 X t4 X t 3 X 2 X 3 1 X t 2 X t3 X 3 X 2 X 1 differencingcombinesshortandlongdifferencingorthogonalityThisisnotpossiblewithotherdifferencingmethodsThesimpleX-differencingtransformationthatleadsto(6)eliminatesthenuisanceparametersjustlikeordinarydifferencing,butithastheadditionaladvantagethattheregressionequationesafundamentalorthogonalitycondition:thereisnocorrelationbetweentheregressorandtheerrorin(6).Asaresult,X-differencingisverydifferentfromexistingdifferencingmethods thathavebeenusedintheliterature.Inonewayitisfundamentallysimpler–becauseoftheappealingorthogonalitypropertysatisedby(6).Inanotherwayitismorecomplete–becausethedifferencingrateisvariable,sothatitispossibletothinkof(6)asasystemofequationsovereachequationofwhichcarriesusefulinformationabouttheautoregressivecoefcientItisinterestingtocompare(6)withotherdifferencingtransformationsthathavebeenusedintheliterature.First,itisdifferentfromlongdifferencing(Hahn,HausmanandKuersteiner,2007),whichtransformsequation(2)to)+(,whereasourmethod)yields)+(sothepositionsofandareswitched,theequationerrorisdifferentandourapproachallowstovary.Second,X-differencing)isalsodistinguishedfromsimplerstdifferencing,whichgivestheequation.Inourmodel,wereplaceonthelefthandsidewiththeequationerrorisdifferent,andagainweallowforhigherorderdifferences.AlsoX-differencingisquitedifferentfromforwardorthogonaldeviations(ArellanoandBover,1995).Whileforwardorthogonaldeviationspreserveserialorthogonalityinthetransformederrors,X-differencingmaintainsorthogonalitybetweenthetransformedregressorsandthecorrespondingerrors.Third,when,thetransformedequation(6)inourmodelcanbewrittenas.ThisequationcanusefullybecomparedwiththeAR(1)bias-correctiontransformationmodel+errorthatwasusedinPhillipsandHan(2008)andHanandPhillips(2010a).InthenewX-differencingapproach,thepresentmethodreplacestheterminmodel(9)with.This“tem-poralbalancing”aroundthelaggeddifferenceisasubtlebutimportantbreakthroughthatleadstothevariableX-differencinggeneralizationof(9)and,asweshallsee,leadstoconsiderableciencygainsandfurtherallowsforconvenientgeneralizationfromAR(1)toAR()models.Importantly,anyvaluessuchthat⎨satisfy(7)underthestatedregularity,sothatthenewregressionequation(6)isvalidacrossallthesevalues.Tomakefulluseofallthisinformation,weproposetostacktheregressionequations(6)forallpossiblevalues.Butweexcludebecauseinthiscasethecorrespondingregressorin(6)iszeroedout.Thus,weproposetouse equation(6)for.Theresultingstackedandpooledleastsquaresestimatorhasthefollowingsimpleform andisthepanelfullyaggregatedestimator(PFAE)ofinthepanelAR(1)model(2).InthetimeseriescasewherereducestotheFAEestimatorintroducedinHPS(2011).Inviewof(7),thereis,infact,exactuncorrelatednessbetweentheregressoranderrorin(6),whichturnsouttobeimportantinproducinggoodlocationpropertiesofthePFAEestimator.Asshowninthesimulationsreportedlater(seeTable1),theestimatorhasvirtuallynobiasfor.Inthelimit,consistencyholdsprovidedthetotalnumberofobservationstendstonity—irrespectiveoftheratio–indicatingthattheestimatorwillbeusefulinshortandlongpanels,aswellasnarrowandwidepanels,makingitappealinginbothmicroeconometricandmacroeconometricdatasets.Thisresult,togetherwiththeasymptoticdistributiontheoryandassociatedtoolsforinference,willbedevelopedinthefollowingsectionsinthecontextofthegeneralAR()panelmodel.3ThePanelAR()ModelwithFixedEffectsThissectionextendstheaboveideasonX-differencingandfullyaggregatedestimationtothegeneralcaseofadynamicpanelAR()model.Ourprimaryconcernistheestimationofthecommonautoregressiveparametersinthefollowingpanelmodelwithxedeffectsandautoregressiveerrors>⎩⎩⎩>>is,foreach,amartingaledifferencesequence(mds)underthenaturalltrationwith,and.AsintheAR(1)casewehavetheequivalentspecication(atleastinthestationaryandunitrootcases,c.f.thediscussionfollowing(2)above)Wemaintaintheassumptionthathasatmostoneunitroot.When,thelongrunARcoefcientisandwewrite)=(1wheretherootsof areoutsidetheunitcircle.Inthisevent,in(12)andthereisnodriftintheprocess.Initialconditionsformaybesetintheinnitepastinthestationarycase.Intheunitrootcase,wecanwrite andsettheinitialconditionsforthestationaryAR()processintheinnitepast.SinceourestimationprocedurereliesonlyonX-differenceddata,itisnotnecessarytobeexplicitaboutinitialconditionsforInfact,ourresultswillholdfordistantandinnitelydistantinitializations(wherecanbe forsomewhichmaytendtoinnitywithaswellasinitializations(seePhillipsandMagdalinos,2009,fordiscussionoftheseinitialconditions).FollowingthesamemotivationasintheAR(1)case,toconstructtheX-differencedequationsystemwerewrite(12)inforwardlookingformatas.Then,bysubtractingthisequationfromtheoriginalbackwardlookingequation(12),weconstructtheX-differencedequationsystemjustasintheAR(1)case.Thesystemmayalsobewrittenasandisfreeofxedeffects.Observethatthevariablesappearingin(13)involvedifferencesfor>⎩⎩>⎩Theregressorsin(13)arealluncorrelatedwiththeregressionerrorintheequation,asshowninLemma1below.Importantly,thisorthogonalityconditionholdsforthefullsystemofequationsgivenin(13)—thatisforallLemma1forall,forallInstackingthesystem(13)forestimationpurposes,weuseallpossiblevaluesupto.Toputtheestimatorinaconciseform,letΩ>andThen,(13)canbeexpressedasThePFAEforissimplytheleastsquaresestimatorbasedonthestacked(overandpooled(oversystem(14),viz., Thedegreesoffreedomconditionisrequiredfortheexistenceof,sothatonemoretimeseriesobservationisneededthanforotherestimatorssuchasLSDVandGMM.Notethatasingle>suchthatleadstoregressorsingularityin14),makingitappearasifisrequired.Buttheregressormatrixstackedoverallpossiblehasfullcolumnrankaslongas.Forexample,for+3=5,wehavethethree)+()+()+(forwhichtheregressorsandtheerrorsareuncorrelated.Thetworegressorsofeachoftheseequa-tionsarelinearlydependent,buttheyjointlyidentifywhenstacked.(Wecanverifythisforstationaryandintegratedseparately.)Ingeneral,thedenominatorof(15)isnonsingularaslongasandThedoublesummationin(15)foreachindicatesthatthecomputationalburdenincreasesatrate,asisthecasefortheconventionalGMMestimators.Butwecanusetheidentity(32)intheAppendixtoreducecomputationtoanrateofincrease.ThePFAEmaybeconvenientlyconstructedinanalternativemannerasfollows.Foragiven,letbetheX-differencingestimatorbasedontheequationHere,istheminimumlagallowedinPFAestimation(toavoidperfectcollinearity),andisthemaximumlag.Letbetheregressormatrixforthislag(forallandforallpossible)andletbethecorrespondingregressandvector.Whenweregress,wegetlag-estimator.ThenthePFAEiswhichisaweightedaverageofalllag-estimators,wheretheweightsareassignedaccordingtothemagnitudeofthelag-signalmatrix.Notethatallsinglelag-estimatorsarethemselvesindividuallyconsistentasthesamplesizeincreases.TheeffectofaggregatingisfullydiscussedinHPS(2011)inthetimeseriescontext,whereitisshownthatthereisatradeoffbetweenuniform asymptoticGaussianityandefciency/rateofconvergence.Whenpaneldataareavailable,ontheotherhand,asymptoticnormalityisdrivenbythepowerofcrosssectionvariationandsoitisunnecessarytopartiallyaggregatethelagsunlessissmallandallpossiblelagsareemployed.TheorthogonalityconditioninLemma1holdsifiswhitenoiseforeach.However,thedevelopmentofanasymptotictheoryforrequiresstrongerregularityconditionsthatvalidatelawsoflargenumbers(LLNs),centrallimittheorems(CLTs)andfunctionalCLTsasandtoinnity.Ourtheoryincludesbothxedxedcases.Forthesedevelopments,weassumethefollowing.ConditionA(i)with,whereacrossss(ΠΩ)4+]∙forallandsome;(ii)isastationaryandergodicmartin-galedifferencesequence(mds)overforallsuchthatandwithunitconditionalvariances)=E((iii)convergetonitelimitsas1.Weallowcross-sectionheterogeneityin(i)byconsideringascaledversionofanrandomsequence()foreach.Thisassumptionisnotcrucialbutitsimpliestheanalysisconsiderably.Generalizationtonon-identicallydistributed(across)innovationsispossiblebutinvolvesfurthertechnicalities,includingsomeexplicitconditionsforthirdandfourthmomentsandtheLindebergcondition.2.Condition(ii)isabidirectionalconditionandcorrespondstoaconventionalwhitenoiseassumption.Thisconditionisweakerthanrequiringindependenceinover,butisstrongerthanaunidirectionalcondition.3.Higherorderserialdependence(over)maybeallowedaslongasConditionA(ii)issatis-ed.Ifxedandislarge,noconditionsontheserialdependenceofarerequiredotherthanforall4.ConditionA(iii)seemsquiteweak,althoughitisnotimpliedbyConditionA(i).WhenA(iii)holds,theaveragemomentsconvergetonitepositivelimitsinviewofConditionA(i). Whenxedand,werequirethefollowingregularityforthestandardizederrorse-quencesowemayestablishstandardasymptoticsforthePFAE.ConditionBForanygiven,(i)isnonsingular,wherearedenedin(14);(ii)IndevelopingaCLTforthenumeratorofacentredformof(15),onlyConditionAisrequired.ConditionB(i)isrelevantforestablishingthestandardnormallimitgiveninTheorem2below.ConditionB(ii)isusefulfortheestimationofthevariance-covariancematrixofthelimitdistribution.Whenisindependentandpossiblyheterogeneousacross,asufcientconditionforB(ii)isgiveninPhillipsandSolo(1992,Theorem2.3). When,thetemporaldependencestructuremattersandaffectsthelimittheoryandratesofconvergence.InthegeneralAR()modelwithaunitroot,thereisanasymptoticsingularityinthesamplemomentmatrixbecauseofthestrongersignalinthedataintheunitrootdirection,justasinthetimeseriescase(ParkandPhillips,1988).SingularitiesaretreatedbyrotatingtheregressorspaceandreparameterizationasdetailedinAppendixA.4AsymptoticTheoryThissectiondevelopsanasymptotictheoryforthePFAETechnicalderivationsandageneraltheoryaregiveninAppendixA.Tomaketheresultsofthepapermoreaccessible,onlythemainndingsthatareusefulforempiricalresearcharereportedhere.Westartwiththefollowingnota- P21P01 sothatforallbyLemma1,wecanexpectthepanelestimatortobeconsistentandasymptoticallynormalunderregularityconditionsthatensuresuitablebehaviorforthesample.Inparticular,ifisstationary,thenconsistencyandasymptoticnormalitywillhold,providedthetotalnumberofobservationsintheregressionislarge,i.e.ifSo,noconditiononthebehavioroftheratio isrequiredinthelimittheory.Ifispersistent(sothatthelongrunARcoefcientisunity)andnite,thenlarge-asymptoticsareagainstandardbecauseanyspecialbehaviorinthecomponents(e.g.nonstandardconvergenceratesandlimitbehaviorassociatedwithnonstationarity)occursonlywhenNext,ifispersistentand,theestimatorisconsistentandstillasymptoticallynormalwhen,againirrespectiveoftheratio.Inthiscase,thecorrespondingestimateofthelongrunARcoefcient(which,becauseofpersistence,is)hasafasterconvergenceratestemmingfromthestrongersignalinthenonstationarycomponentofthedata,therebyproducingasingularityinthejointasymptoticnormaldistributionofwithonecomponent(inthedirection)convergingfastertoitsnormaldistributionthantheothercomponents.Whenxedandinthepersistentcase,thenthelimitdistributionofisagainsingularnormal(whenbutthereisafasterrateofconvergenceinthedirectionandthelimitdistributionisnonstandardinthatdirection.ThelatterresultisrelatedtothelimittheoryofthetimeseriesFAEestimatorgiveninHPS(2011)forthespecialcasewhereTheorem5inAppendixAprovidesacompletestatementforinterestedreadersofthislimittheory,coveringthegeneralpanelAR(caseinauniformwayforlargeandaswellasbothxedandxedcases.TheremainderofthissectionfocusesonpracticalaspectsofthislimittheoryandtheusabilityofthePFAEinappliedworks.Forinferenceandpracticalimplementation,Theorem2belowpresentsafeasibleversionofthemainpartofTheorem5inAppendixAthatholdsuniformlyforallvaluesincludingbothstationaryandunitrootcases.Forconvenience,weusethemodel(1)formulationinwhich,whereisanAR()processasdenedin(10).Theorem2isAR()asdenedin(10).UnderConditionA,foranysuchthat,wherearedenedin(17).Theconvergence(18)holdsas,andasinallcases(thatis,for,eitherniteorincreasingtoinnity,nomatterhowfast).Thelimitdistributionofispartlynormalandpartlynonstandard.ItisgiveninTheorem5(d)inAppendixA. 1.NotethatcrosssectionheterogeneityispermittedinTheorem2underConditionA.Thema-inthetheoremaredesignedtobeheteroskedasticityrobustsothat(18)providesacentrallimittheoremsuitableforimplementationuponestimationofasdiscussedbelow.Theasymptoticformofthestandardizationmatrix(18)isgivenin(49)inAppendixAandshowsexplicitlytheconvergenceratesintermsofaswellasthetransformationmatrixinvolvedinarrangingdirectionsoffasterandslowerconvergencewhenthereisaunitrootinthesystem.2.Forstatisticaltesting,itisnecessarytoreplacebyafeasiblestatistic.Inviewof(17)andtheconsistencyof,wecanusetheresidualsΩ>inplaceof.TheasymptoticcovariancematrixestimateestimatePΩΩ]⎛1PΩbΩb0Ω[PΩΩ]⎛1maythenbeusedininference.Simulationsshowthatthischoiceworkswellwhenlarge.Ifisnotsolarge,inferencesbasedonthismethodstillshowreasonableperformanceandmaybeimprovedbymodicationofthelimitdistributionoftheassociated(scalar)teststatisticstoaStudentdistributionwithdegreesoffreedomasproposedinHansen(2007)iftherandomvariablesare3.Forpracticalwork,itmaybeusefultoprovideestimatesoftheremaining(nondynamic)parametersinthemodel(10).Consistentestimationoftheautoregressivecoefcientsin(10)enablesestimationofthexedeffects,thevarianceofthexedeffectsandthatoftherandominnovationsinastandardway.Forexample,thetransformedxedeffectscanbeestimatedbytheindividualsamplemean,oftheresidualsandtherandomidiosyncraticinnovationscanbeestimatedbythequantity.Theaveragevariancesofandcanthenbeestimatedbythesamplevariancesofafterthedegreesoffreedomcorrection),respectively.AsymptoticsfortheseadditionalestimatesfollowinastandardwayfromtheusuallimittheoryforsamplemomentsandtheconsistencyofthettedautoregressivecoefWenowprovidesomefurtherdiscussionofefciency.Atpresentthereisnogeneraltheoryofasymptoticefciencyforpaneldatamodelsthatappliesformulti-indexasymptoticsandpossiblenonstationarity.TheusualHájek-LeCamrepresentationtheory(Hájek,1972;LeCam,1972)holdsforlocallyasymptoticallynormal(LAN)familiesandregularestimatorsinthecontextofsingle indexand asymptotics.PanelLANasymptoticsweredevelopedforthestationaryGaussianAR(1)casebyHahnandKuersteiner(2002)allowingforxedeffectsundercertainrateconditionspassingtoinnity.Buttheirresultdoesnotapplywhenthereisaunitrootinthesystem.Anysuchfurtherextensionofexistingoptimalitytheorywouldrequirethatbecauseforxed(andinparticular=1)thelikelihooddoesnotbelongtotheLANfamilybutisofthelocallyasymptoticallyBrownianfunctionalfamily(Phillips,1989;Jeganathan,1995),forwhichthereisnopresenttheoryofoptimalestimationorasymptoticefciency.Moreover,itisnowknownfromtheresultsofHPS(2011)thatimprovementsinbothbiasandvarianceovertheMLEandbiascorrectedMLEarepossibleinlocalneighborhoodsofunityinthetimeseriescase(Forthepurposeofthepresentstudy,weundertakeamorelimitedinvestigationofefciencyandconsiderthesimplepanelAR(1)model(1)withGaussianerrors.Normalityisnotneededforthelimittheorybutonlyforthediscussionofoptimalityinthestationarycase(c.f.HahnandKuersteiner,2002).Forthismodel,thefollowingresultholdsandshedslightontherelativeciencypropertiesofthePFAEprocedure,includingboththestationaryandunitrootcases,inrelationtotheMLE.Theorem3SupposethatΩΩforsome.Then>1.Asymptoticsforthestationarycase(20)holdasregardlessofthecrosssectionaldimension.Wefurthernotethatasymptoticnormalitydoesnotrequirelarge.However,theformoftheasymptoticvariancegivenin(20)doesrequire.Inthiscase,LANasymptoticsapplyasandthevarianceattainstheCramérRaobound,whichisthesameasinthestationarytimeseries(=1)case.So,when,thePFAEisasymptot-icallyefcientas.ThisresultcorrespondstothendinginHahnandKuersteiner(2002,theorem3)thatwhenthebiascorrectedMLEattainsthe(semiparametric)ciencyboundfortheestimationofthecommonautoregressivecoefcientinthepresencexedeffectsundertheratecondition> 2.HahnandKuersteiner(2002,theorem4)showthatwhen>the(biascorrected)LSDVestimatoroisasymptoticallydistributedaso µ0 ¶ Thus,thePFAEestimatorhassmallerasymptoticvariancethanthebias-correctedLSDVestimatorandthePFAErequiresnobiascorrection.ObservethattheLSDVestimatoristheGaussianMLEcorrectedforitsasymptoticbias.So,theimprovementofthePFAEoverthebiascorrectedLSDVestimatoratisanalogoustotheimprovementoftheFAEestimatorovertheMLEinthetimeseriesunitrootcaseshowninHPS(2011).Inthatcase,correctingforthebiasbyre-centeringtheMLEestimatoraboutitsmeandoesnotreducevariation,whereasHPS(2011)showthattheFAEestimatorreducesboththeasymptoticbiasandthevarianceoftheMLEnotonlyatbutalsointhevicinityofunity,whilehavingthesamelimittheoryinthestationarycase.Thelimitresult(21)revealsthattheimprovementoftheFAEoverthe(levels)MLEatunityinthetimeseriescasecarriesovertothepanelcasewhere3.TheimprovementofthePFAEoverthebiascorrectedLSDVestimatormightbeconsideredcounter-intuitivebecausedifferencingisusuallyregardedasinferiorintermsofefciencytolevelsestimationandtheuseofawithin-grouptransformationtoeliminateindividualef-fects(unlessGLSormaximumlikelihoodisappliedtothedifferenceddata).However,theconsiderableadvantageofthePFAEtechniqueisthatitremovesindividualeffectsbysys-tematicX-differencingand,inaddition,becauselongdifferencesareincludedinthestackedsystemestimation,anystrongsignalinformationinthedataisretainedbyvirtueofthefullaggregationthatisbuiltintotheestimator.Theresultisimprovedestimationintermsofbothbiasandefciencyoverregression-baseddemeaningofthelevelsdataandbias-correctioninMLestimation.4.Similarly,fortheAR()panelmodel,whenisstationary,thePFAEisapproximatelyequivalenttothebias-correctedLSDVestimator.Inthiscasebiasrapidlydisappearsasthetotalsamplesizeincreases.Whenhasaunitroot,thePFAEhassubstantiallysmallerbiasandanefciencygaincomparedwiththeLSDVestimator.5.WhenthereisasimplerelationshipbetweenthePFAEandthebiascorrectedMLEorLSDVestimator.Inparticular,asshowninAppendixD,when wehave 1)= o ¶ 3P1³P1´22PP21 PP21+ Accordingto(23),maybeinterpretedasaedversionofthebiascorrectedformofoThemodicationisimportantbe-causethesecondtermof(23)contributestothelimitdistributionandleadstoareduc-tioninthelimitingvarianceoftheLSDVestimator.Inparticular,itisthe(negative)cor-relationofthesecondtermwiththersttermof(23)thatreducestheasymptoticvari-anceofLSDV,Avar ˆ =51,totheasymptoticvarianceofPFAE,Avar .Infact,thisnegativecorrelationmakesitpossibletolowertheasymptoticvariancefurther,asshowninAppendixDatleastfor6.ForthepanelAR(1)modelwhen,usingsequentiallimitsasfollowedby,Kruiniger(2008)showedthattherstdifferenceGaussianquasi-MLE(calledFDMLE;seealsoHsiaoetal.,2002)hastheasymptoticdistributionoThelimitdistributionoftheFDMLEforocomparableto(20).ButwhenthevarianceofthelimitdistributionoftheFDMLEissmallerthanthatofthePFAEThisreductioninvarianceisexplainedbythefactthattheFDMLEisarestrictedmaximumlikelihoodestimator.TheFDMLEiscomputedusingaquasi-likelihoodthatisdenedonlyfor (seeKruiniger,2008).Sorestrictedbytheupperboundofthisregionatwhichpointthequasi-likelihoodbecomesned.Weusetheterm“quasi-likelihood”indescribingtheFDMLEbecauseitisthetruelikelihood.Infact,nodatageneratingmechanismisgiveninKruiniger(2008)forthecaseandthequasilikelihoodisconstructedoverthatregionsimplybytakingananalyticextensiontotheregion oftheGaussianlikelihoodbasedonthedensityofthedifferenceddataoverthestationaryregionTheconsequentialrestrictionindomain,andhenceinestimation,playsakeyroleinthevariancereductionoftheFDMLE.Thisreductionisborneoutinsimulations.Forexample,simulationswith=200=50showthevarianceofFDMLEtobeapproximately87%ofthevarianceofPFAE,whichcorrespondswellwiththelimittheoryvarianceratioof.Also,inviewofthesingularityinthequasilikelihoodattheupperlimitofthedomainofdenition,numericalmaximizationofthelog-likelihoodfrequentlyencountersconvergencedifcultiesinthecomputationoftheFDMLE.Numericaloptimizationcanfailisnotlarge.Forexample,insimulationswith=10=50,wefoundthatatotal32outof1000iterationsfailedtoconvergetoalocaloptimizer.TheserestricteddomainandconvergenceissuesassociatedwiththeFDMLEprocedurearediscussedmorefullyinseparatework(HanandPhillips,2010b). 7.AsymptoticsfortheFDMLEprocedurearedevelopedinKruiniger(2008)onlyforthepanelAR(1)modelandcomputationismuchmoredifcultinthecaseofthepanelAR(p)model.TheselimitationsmakeitdesirabletohaveasimpleunrestrictedestimatorlikePFAEwithnitesampleandasymptoticpropertiesthatcanbeeasilyimplementedingeneralpanelAR(p)models.8.Intheunitrootcasewith,thelimitdistribution(21)holdsforboth>,butnoconditionisrequiredontheratio.ForweknowfromtheresultsinHPS(2011)thatthe(timeseries)MLEbasedonlevelsisnotefcientandthatremainstrueevenwhenwebiascorrecttheMLE.Infact,asshowninHPS(2011),theFAEissuperiortotheMLEinthewholevicinityofunitywhenSo,wecanatleastconcludethatthePFAEissuperiortotheMLEforWeexpectbutdonotprovethatthisconclusionholdsforallxedThelimittheoryforthe(restricteddomain)FDMLEestimatoratindicatesthattheremaybescopeforimprovingestimationefciencyatandpossiblyintheimmediateneigh-borhoodofunity.Thisissueiscomplexand,asindicatedearlier,thereiscurrentlynogeneraloptimalestimationtheorythatcanbeappliedtostudythisproblem.InAppendixDweprovethatasmallmodicationtothePFAEprocedurecanindeedreducevarianceforthecase.Thecationisofsomeindependentinterestbecauseitmakesuseoftherelationship(23)betweenPFAEandthebias-correctedLSDVestimatorofHahnandKuersteiner(2002).Inparticular,inthesimplepanelAR(1)model(1),themodiedestimatorisobtainedbytakingthefollowinglinearcombinationforsomescalarweight+(1o o sothatthecentredandscaledestimatorhastheform1)=o o ThePFAEcorrespondsto.Inthiscase,the(negative)correlationofthesecondtermwithrsttermof(25)reducestheasymptoticvarianceofo,whichisdowntotheasymptoticvarianceof,whichis9,asdiscussedinRemark5above.Thevariancecanbeloweredfurtherbychoosinganoptimal.AccordingtothecalculationsshowninAppendixD,gives,whichistheminimalvarianceattainablebyadjustingintherelationship(25). ThemodiedestimatorcanalsobeunderstoodasaGMMestimatorbasedonthetwomomentconditionsEand,whereo ,i.e., 32P3P1h((1i2( 22P1£ù¡3 with,and.Notethattherstobservationsareignoredinforalgebraicsimplicityandtheireffectisasymptoticallynegligiblewhen.Inviewoftheidentity(seeHPS,2011) anyweightedGMMestimatorcanbeexpressedintheform+(1)(ˆ forsometherebyleadingbacktotheoriginalformulation(24).ThemodiedPFAEwithattainsanefciencylevelof325=096%efciency)relativetotherestrictedFDMLE.However,thisargumentcannotbeusedforvaluesbecauseo doesnotcorrectthebiasif.Thisisevidentfromthefactthat o (2+ isthebiascorrectedestimatorproposedbyHahnandKuersteiner(2002,p.1645)forthestationarycase,i.e., o suchthat.Ofcourse,whenwealsohave o o(1)sointhiseventthebiasissmallbecausesofast.Toclosethissection,wenowdiscusssomeremainingpracticalissuesinvolvedinmodelingandtheuseofX-differencing.First,practicalapplicationsoftencallfordatadeterminationofthelaglengthintheautoregression.Consistentpanelinformationcriteriamaybeconstructedorageneral-cmodelingalgorithmcanbeusedforthispurpose.OnesuchpossibilityisconsideredinHPS(2010).SeealsoLee(2010b).Next,themomentconditionsusedin(7)requirethatisstationaryoverSowhentheinnovationsaretemporallyheteroskedasticortherearenonstationaryinitialconditions,thePFAEmaybeinconsistent.Forexample,forthepanelAR(1)with,from(32)intheappendixwe ndthatplim asbefore.Thus,ifismoderate( )andtemporalheteroskedasticityissoseverethatishuge,thenthePFAEmaybemorebiasedthantheLSDVestimator.Finally,extensiontomodelswithexplanatoryvariablesisstraightforwardiftheyappearasinwith.ForthismodelthepersistenceparameterscanbeidentiedbyX-differencingforgiven,whiletheslopeparametersatisestheusualorthogonalityconditionsiffΨΩΩ]=0forrelevant.Atwostepprocedurecanthenbeused,effectivelygeneralizingBhargavaetal.’s(1982)feasibleGLSproceduretothepanelAR(isstrictlyexogenous.Ontheotherhand,itisunclearifandhowwecanderivemomentconditionsforthedynamicmodel.Thisimportanttopicisleftforfutureresearch.5SimulationsThissectionreportssimulationswhichshedlightonthenitesamplepropertiesofourproceduresinrelationtoexistingmethodsofdynamicpanelestimation.Inparticular,wecomparethePFAEprocedurewithexistingestimatorssuchasArellanoandBond’s(1991)differenceGMMestimatorandBlundellandBond’s(1998)systemGMMestimatorforapanelAR(2)model.(TheFDMLEmethodisnotincludedbecauseofcomputationaldifcultieswiththisprocedureandthefactthatitisarestrictedestimator,asdiscussedearlier.)Wethenconsiderpanelswithnonstationaryinitialconditionstoexaminetheeffectofdeparturesfromstationarity.I.Comparisonofbiasandefciency:AR(1).rstcomparethepropertiesofthePFAEwiththeLSDVestimator(whichisinconsistent),HahnandKuersteiner’sbias-correctedLSDVestimator(HK),theone-steprstdifferenceGMM(GMM1/DIF),andthetwo-stepsystemGMM(GMM2/SYS),forthepanelAR(1)model.Themodelisstandardnormalvariablesandisalsonormalwitharbitrarilysetto2.Whengeneratingthedata,theprocessesareinitializedatsuchthat100:=0,andthenobservationsforarediscarded.ThenormalvariatesaregeneratedusingthefunctionofStata.ThedifferenceGMMandthesystemGMMareestimatedbythe‘xtabond’and the‘xtdpdsys’commandsofStatarespectively,andthePFAEisobtainedbydirectcalculationusingformula(16).Weconsider=100and,whereisthesmallesttimedimensionthatallowsfortheX-differencingestimation,whiletheotherestimators(LSDV,HK,GMMestimators)arealsocalculableforTable1reportsthesimulatedmeansoftheestimatorsfrom1,000replications.TheLSDVestimatorisobviouslybiaseddownward,asperNickell(1981).The(smallsample)biasesoftherstdifferenceandsystemGMMestimatorsdependonthedistributionof.Ontheotherhand,PFAEshowsverylittlebiasforallparametervaluesandisconsiderablysuperiortoHK.Table1alsopresentssimulatedvariancesoftheestimators.Whenissmall(and=10),PFAEislessefcientthanthebias-correctedLSDVestimator(HK),butwhenislarger=20)andislargeandinoursimulation),PFAEisasefcientormorecientthanHK.TheinefciencyofPFAErelativetoHKforisduetothesmallerdegreesoffreedomofPFAE,butitisalsonotablethattheMSEisconsiderablysmallerforPFAEforallandforallincluding.Withlargervalues,PFAEattainstheasymptoticvariance,asdoestheHKestimator.For=20,wenoticethatPFAEappearslessefcientthanLSDVat,whichlookscontrarytotheasymptoticndingsthatoandwithorespectivelydenotingtheLSDVandPFAEestimators.Thisoutcomeoccursbecause=20isnotlargeenoughfortheasymptoticstobeaccurateenoughforthedistinctiontomanifest.For,theasymptoticvarianceof,whichisapproximatelywith=100=20.Thistheoreticalvalueisclosetothesimulatedvariance.Asincreasesfurther,sothatiscloseto1andtheasymptoticsforthePFAEissufcientlyaccurate,weexpectthehigherasymptoticefciencyofPFAErelativetoLSDVtobecomeevidentinsimulations.Table2revealsthatthisexpectedimprovementoccursforforallvaluesofTheperformanceoftheGMMestimatorsdiffersaschanges.ComparingPFAEandGMM,PFAEperformsuniformlybetterthantheGMMestimatorsinoursimulationsexceptforwith=10.ItishoweverworthnotingthattheGMMestimatorsarebasedonmomentconditionsdifferentfromthoseusedbyPFAEandLSDV,andthattheperformanceoftheGMMestimatorsalsodependsontheinitialcrosssectionalvarianceoftheidiosyncraticerrors.ItwouldalsobeworthcomparingtheperformanceofthePFAEandtheindirectinference(II)procedure(Gouriérouxetal.,2010).SomecomparisonsinthetimeseriescasewereundertakeninHan,PhillipsandSul(2011)-bothestimatorshavenegligiblebiasbutIIhassmallervarianceintheunitrootcase.Forthepanelmodel,interestedreadersarereferredtoGouriérouxetal.’sTable2, thoughcautionisneededinthiscomparisonbecausethesizesofdonotexactlymatchthoseusedhereandthegeneratedsamplesaredifferent.Lookingattheseresultsinthepanelcase,itappearsthatbothIIandPFAEhavenegligiblebiasandIIhassmallervariance.AfullcomparisonbetweenthetwoproceduresisnotyetpossiblebecausethelimittheoryforpanelIIisnotyetavailableinunitrootandnearunitrootcases.Thislimittheoryhasonlyrecentlybeenobtainedforthetimeseriesmodel(Phillips,2011)andthepanelextensionisleftforasubsequentcontribution.II.Comparisonofbiasandefciency:AR(2).WenextconsideranAR(2)dynamicpanelmodel(i.e.,.ExceptforbeingAR(2),allotherset-tingsarethesameasintheprevioussimulation.Weset,and.Thepanelsarestationarywhen,andareintegratedwhenTable3reportsthesimulatedmeansandvariancesoftheestimatesof.HahnandKuer-steiner’s(2002)estimatorisevaluatedbyapplyingtheirTheorem2totheAR(1)representationofAR(2)ratherthantheirAR(1)correctionformulaasbiascorrectionbasedonthemisspecimodelcanexacerbatethebias(Lee,2010a).TheLSDVestimatorisagainbiaseddownward,andthePFAEexhibitsverylownitesamplebias.TheGMMestimatorperformancedependsonthevarianceof.Again,LSDV,HKandPFAEarefreefromtheeffectsofthe,whilethetwoGMMestimatorsarenot.ThePFAEperformswellinallconsideredcases.AsremarkedinthediscussionoftheAR(1)simulations,itisnoteworthythattheaccuracyoftheGMMestimatorsdependsonthevarianceoftheinitialidiosyncraticerrorsaswell.III.Inference.WenextinvestigatethepropertiesoftheestimatedvarianceofthePFAE,wherewithnedrightafterLemma1andfoundin(19).Becauseallthestatisticsarefreefromindividualeffects,wecaneliminatefromthedatagenerationprocess.WefocusonthepanelAR(1)model,wherewith.Wetest(i)and(ii).Wepresenttestsizesforthenullhypothesisthattheparameteristhesameasthetrueparameterusedinthedatageneration.Gausswasusedforthesimulations.Weusethecriticalvaluesintesting,asrecommendedbyHansen(2007).Table4reportstheempiricalsizesfromasimulationof5,000replications.Exceptforaslightover-rejectioninsmallsampleswithhigh,sizeperformanceisreasonablygood.Thesimulated powersforthenullhypotheses(left)and(right)arepresentedinTable5.ThispartofthesimulationisintendedtobeillustrativeasitsmainpurposeistoexhibitgeneralperformancecharacteristicsofinferencewiththePFAEprocedure.Thoroughcomparisonswithotherestimatorswouldrequireamoresystematicsimulationstudy.IV.Departuresfromstationarity.Finally,weexaminetheperformanceofthePFAEwhenthestationarityassumptionisviolated.AstheexampleofanAR(1)withshowsattheendoftheprevioussection,thebiasofthePFAEcanbemadearbitrarilylargebycorrespondinglylargeheterogeneityintheerrorvariances.Variousotherdeparturesfromstationarityarepossibleandinthissectionweconsiderthecaseofnonstationaryinitialconditions,leavingotherdeparturestoseparateresearch.Specically,thedataaregeneratedbyΩΩasinpartIabove,butthistimeweset(insteadof100).Wedeliberatelyusethismodeltomaketheindividualmeansinvariantovertime.Table6reportssimulationresultsforLSDV,HK,differenceGMM,systemGMM,andPFAE.TheresultsaresimilartopartI,andforthisspecicDGP,nonstationarityofdoesnotintroduceseriousbiastoPFAE,butwestillobserveslightlymorebiasformoderatevaluescomparedwithpartI.Ifthemeanofchangesoverorifheteroskedasticityiswilder,thentheX-differenceestimatorsmaybemorebiasedthanGMMestimatorswhichdonotdependonthestationarityof(or6ConclusionTheestimationmethodintroducedinthispaperforlineardynamicpanelmodelsusesanewdiffer-encingprocedurecalledX-differencingtoeliminatexedeffectsandasimpletechniqueofstackedandpooledleastsquaresonthefullsystemofX-differencedequations.Themethodisthereforestraightforwardtoimplementinpracticalwork.Itisalsofreefrombiasforallparametervaluesandavoidsweakinstrumentationproblemsinunitrootandnearunitrootcases.Theasymptotictheoryshowsgainsinefciencyintheunitrootcaseoverbias-correctedmaximumlikelihoodandequivalentefciencyinthestationarycasebutthenewmethodhasnoneedforbiascorrection.Theasymptoticsalsoapplyirrespectiveoftheratioas>.Theseadvantagesmakethenewestimationprocedureattractiveforempiricalresearch,especiallyincasesofdatapersistenceanddispersedindividualeffectswhereothermethodscanperformpoorly.Thendingsofthepresentpaperpointthewaytofurtherresearch.First,thereisaneedfor atheoryofoptimalestimationinpanelmodelswhichallowsforrootsinthevicinityofunityanddualindexasymptotics.Whilethereis,asyet,nooptimalestimationtheoryintimeseriesautoregressionthatincludestheunitrootcase,theprocessofcrosssectionaveraginginpanelestimationleadstoimportantsimplicationsinthelimittheorythatmakesuchanoptimalitytheoryfeasible.Inparticular,thelimittheorybelongstoanasymptoticallynormal(asdistinctfromanonstandarddistribution)familywhenButthelimitdistributioncanalsobedegeneratewithasingularityinthecovariancestructureandachangeintheconvergenceratewhenthereisanautoregressiveunitroot.Thesefeaturesofthelimittheoryandtheirimpactonoptimalityinestimationdeservedetailedstudy.Asindicatedearlier,thereisalsoscopeforfurtherworkonmodelselectionindynamicpanels,includinganextensivenumericalstudyofsequentialtestingrulesandafurtheranalysisoftheasymptoticbehaviorofvariousinformationcriteria.Second,consistentestimationofpanelautoregressionsusingX-differencingandPFAEmeth-odsisusefulintheestimationofmoregeneralpanelmodelswithadditionalregressors.Forexam-ple,inparametricmodelswithexogenousregressorsandAR()errorssuchaswith,wecanconsistentlyestimateusingPFAEandresidualsbasedonapreliminaryconsistentestimateofThen,aparametricfeasibleGLSesti-matecanbeconductedasanaturalextensionofBhargava,FranziniandNarendranathan’s(1982)treatmentoftheAR(1).Suchstepwiseestimationofmaybeiterateduntilconvergence,combiningmomentconditionsforbasedonassumedexogeneityofandthemomentcondi-tionsimpliedbyLemma1usingforgivenFinally,asnotedabove,theconsistencyofX-differenceestimatorsreliesonthestationarityof(orisintegrated)over.Asaresult,whenthevarianceoftheinnovationsvariesovertimeortherearenonstationaryinitialconditions,theX-differenceestimatorsmaynotbeconsistent.Whileimportant,theseissuesintroducenewcomplicationsthathavenotbeenaddressedproperlyunderthexedeffectsenvironment.Fullexplorationofthemisleftforfutureresearch.AppendixA:TechnicalResultsandProofsProofof(4).,wehaveReplacingtheontherighthandsidewith,weget isthesameforalland,wehave Proofof(7).Itissimplertoworkwith,where.Weshallshow:=E(.For,wehave=E(wherethelastequalityisderivedbyexpandingWhenisstationary,soisobviouslyzero.If,thensowhen,wehaveasclaimed. WeproveLemma1using.Notethatwhereiswhite.Wealsohave.Werstestablishthefollowinggenerallemma.Lemma4beapanelAR()processsuchthatisstationaryAR()forsomenon-negativeinteger,where:=1ThenforallsuchthatProof.FirstconsiderthecasewhereiscovariancestationaryAR(),i.e.,.Let.Let.WehavebytheYule-Walkerequationswhenasclaimed.Nowforgeneral,wehave,where)=1andtherootsofareoutsidetheunitcircle.Firstnotethat,sousing:=1,wehave=:˜ Thatis,,where.Furthermore,isstationaryAR(byassumption,andbytheresultforthestationarycase,wehaveforall⎨Theresultfollowsbywriting Lemma1isnowstraightforward.ProofofLemma1.foralland,weshallprovethat)=0forall⎨.Becauseforalland,itsufcestoshowthatforsuch.Ifisstationary),thenthisholdsbecauseofLemma4with.Ifandisstationary),wehaveforallbyLemma4.But(because)andand).Inbothcasesandtheresultfollows. NextweproveTheorem2.rstintroducesomeusefulnotationandtransformationsthatfacilitateanalysisoftheunitrootcase.Let Ω> ,wherearedenedin(14).Denethetransformationmatrixanditsinversefollows001000Notethatforany,andforany.Thesetransformationmatricesareneededfortheunitrootcase.Alsoletdiag(>and Forauniformdevelopmentoftheasymptotictheory,wederivethelimitdistributionofthestandardizedandcenteredquantityinwhatfollows.Notethat X1011 Theorem5isstationaryAR()orifisstationaryAR(),thenunderConditionsAandB(i),thefollowingresultshold:(a)Ifwhere:=lim=plim:=lim=plim(b)If>where=lim=lim,and=lim=lim(c)Ifxed,andifisstationaryAR(where,and(d)Ifxed,andif(1)isstationaryAR( (0)P2 −ZΩ()]>whereareindependentstandardBrownianmotions,isthevariance-covariancematrixof,andareindependent. Theproofof(a)isstraightforwardandisgivenrst.Let):=limProofofTheorem5(a).Weconsiderthenumeratoranddenominatorof(28)separately.(i)Denominator:Notethat:=Eisidenticalforall.Alsoniteduetothenitefourthmomentassumptionfor.So XÃ1 ):=limTheuniformboundednessofimpliesthathatΩ(m>nisboundeduniformlyoverallforall,wherem>nm>nelementof,sovar m>n Ω(m>nThusthedenominatorconvergestotherighthandsideof(30)inmeanandthereforeinprobability.Theequivalenceofandisalsoimpliedstraightforwardly.(ii)Numerator:WehavebyLemma1ConditionAimpliestheconvergenceof.TheLindebergconditionholdssinceisboundedun-dertheuniformnitefourthmomentcondition.Thus,where.Theresultforfollowsimmediately.=plimisimpliedbyConditionB(ii). TheremainingpartsofTheorem5involve,andweproceedbyapproximatingthecomponentsofbysimplerterms.Let.Letwhereisa-vectorwithunitelements.Letandbedenedby(26)and(27),respectively.Let=diag(1.Alsolet >m>n>⎩⎩⎩>>,where> (Thuswehave.)Letfornotationalbrevity.rstapproximateasshowninthefollowingresult: P=+1 >Ω P1 m>n>Ω 2P=+)21 2"P=+)#21 n⎨ 2P=+)21 2"P=+)#2+1 +1PP=+(++)(2+ +P1 +P+P(31 P+P++P1 Proofof(32)..Let.Wederivetherstlineof(32)forgivenand.Letomittingthesubscript.Wehavem>>Thesecondandthirdidentitiesaboveareobtainedbylettingrespectively,andthenremovingthedashes.Therstidentityofthesecondlineisobtainedbyrearrangingterms,andthelastidentityisobtainedbyswappingandandthennoting>.Therighthandsideontherstlineandtherighthandsidetermonthesecondlinetogether Wehave+1))=(+1) (+−2m−+1)Hence,for⎨,wehave+1+andforNotethatand.RecoverandletTransformingbythenremovingthedashesfromand,weget>⎨,wehaveThuswhenResult(32)isobtainedbysubtractingandaddingandthendivid-ingthroughbyinthiscase.Theidentityholdsformnaswellbecausen>m.The casewithissimilarlyhandled.Finally,thesecondlineof(32)isderivedbymeansoftheidentity Allthetermsin(32)turnouttobenegligiblecomparedwiththeothertermswhenconsideringeithertimeseriesorpanelasymptoticswithlarge.Moreprecisely,thedenominatorandnumeratorin(29)abovemaybeapproximatedasshowninthefollowinglemma,wheretheapproximationholdsbothforstationaryandintegratedLemma6UnderConditionA,wehave X10011+1 where,andaredenedin(29),forsomesuchthatasgivenin(41)and(42)below.Proof.Letk>Ωandk>Ωk>Ωk>Ωaredenedin(32).(i)Denominator:For(36),thesecondlineof(32)implies=(1andstandsfortheHadamard(element-wise)product.Becauseandwith=diag(,wehave P10(0+0)1+3P  Theexpectationoftheabsolutevalueofthersttermis,whichcanbeobtainedbywritingandnotingthathasauniformlyboundedrstmoment.Wecanalsoshowthatk>ΩforallbyLemma9inAppendixB.Thus(36)andtherstpartof(38)follow.(ii)Numerator:For(37)andthesecondpartof(38),weuse(31)andthesecondlineof(32)again,givingk>Ω.Thislastexpressionand(40)implythatt(10+10)¯Î¨0ΩΨΩ]ρ+3Xk=1(k>Ω−k>Ω,wehave.Using,itfollowsthat 12X100+3X k>Ω.(Notethatsubtractingmeansisvalidbecause.)Lemma10showsthatthevariance-covariancematrixofthelasttermontherighthandsideis,andrsttermis,wherethesecondmomentofisbounded.Theresultfollows. Withtheseresultsinhand,theproofofTheorem5(c)forthestationarycasewithlargeandsmallisnowstraightforward.ProofofTheorem5(c).Inthiscase,notethatxed,isstationary(over),and.UnderConditionA,wehaveforeachisindependentofinviewofConditionA(i).Fromthisresultand(36),wehaveplim P2¶0 (seePhillipsandSolo,1992,Theorem3.16).Also,whichtogetherwith(37)impliesthat Theresultfollowsimmediately. Intheunitrootcasewithlarge,weusethestandardizationmatrix=diag(>andcoordinatetransformationThedenominatorcanbehandledusing(36).Forthenumerator,wehave k∼k2]=(1)>andΩ=(1>Ω>2>Ω>Ωwith P11 2P1P1 2P211 >⎩⎩⎩>>duetoLemma11.Thelargeasymptotics(forsmallorlarge)areobtainedbyevaluatingProofofTheorem5(d).Notethat:=0withoutlossofgeneralitybecausetheestimatorisexpressedintermsofdifferences.Otherwisewecouldsimplyreplacewith(i)Denominator:Therstdiagonalelementof 2X11 P1¸22 >):=>wheretheareindependentstandardBrownianmotions.(SeePhillips,1987,Theorem3.1,orusetheBNdecompositionin(51)below.)Theotherelementsoftherstrow(andthecolumn)are,whichareandthusconvergetozeroas.Theremainingelementsofthematrixcorrespondtothestationaryseries>⎩⎩⎩>andthismatrixconvergesinprobabilityto,wherethevariance-covariancematrixof.Wethereforehavediag> foreach,wherethecoefcientappearsinthelimitbecauseofLemma7below.(ii)Numerator:Dueto(45)andLemma7,wehave Z10()()10()Z10()22 whichisalsotheweaklimitoftherstelementof.From(44)and(46),thevectorofthesecondtolastelementsof,denotedby(anotationusedonlyinthisproof),denotesthecolumnsof,andelementsof.Thus,hus,2Ω(0ρ)⎛1\Ω>2Ω02Ω]0.Finally,toseetherelationshipbetweenthelimitsofand,wenotethatthesam-plerandomfunctioncorrespondingtoo]=1Ωandthethelementof.ThejointGaussianityofisstraightforward,andthecovari-ancebetween>Ω>Ωiszerounderthebi-directionalmartingaledifferenceassumption.Soareindependent.Combiningtheseresultswith(47)and(37),andnotingthat,wegetthestatedresult. Nextweprovethepanellimittheorywhere.HeretheLLNandCLTareestablishedusingvariationacrossProofofTheorem5(b).):=limasbeforeand):=lim(i)Stationarycase:Wehave.Forthedenominator,wehave,wherethus X01=1 Forthenumerator,bythemartingaleCLTwehave XE Theresultfollowsstraightforwardlyas>(ii)Integratedcase:Weworkwiththerotatedvariables.Forthedenominator,letm>nbethe m>nelementof,whichistheleadingtermof(36).Then1)= PP211 PP1¸22) ⎠(1>1)]=anditsvarianceisbyLemma8below..ThisisalsotheprobabilitylimitoftheelementofLemma6.Theremainingelementsintherstrow(andtherstcolumn)ofthedenominatormatrixare >⎩⎩⎩>>rstmomentisbyLemma8(iii)andsecondmomentisbyLemma8(vii).Soforall,whichiss⎠(1>m)].Finally,form>n whichis>m>n,byevaluatingthemeanandthevarianceagain.So,wherethelimitistakenas>Forthenumerator,weuse(44),(45)and(46).Lemma12showsthatthevarianceoftherstel-ementofconverges,anditslimitisthesameasthevarianceofthecorrespondingweaklimitobtainedinTheorem5(d).ThevarianceoftheremainingtermsofandthecovariancesarealsostraightforwardlyshowntoconvergetothelimitvarianceandcovarianceofthecorrespondingweaklimitsinTheorem5(d).ConvergenceofthevarianceandtheboundednessofimplytheLindebergcondition forallvectors,whichensurestheCLTforTheseargumentsjustifyjointlimitsas>asdiscussedinPhillipsandMoon(1999)generaltreatmentofpanelasymptotics. ProofofTheorem2.Theorem5(a)–(c)implythatplim where1 P1011 andtheprobabilitylimitsaretakenas,orasforanysequence)when(1).Thus,plim.Foranysuchthat,i.e.,suchthatwehave(HereweusedtheLyapunovconditionA(i)andthehighlevelconditionB(ii).SeePhillipsandSolo,1992,fortheconvergenceof.)Theresultfollowsbyletting ProofofTheorem3.rstresultisimmediatefromCorollary4(i)ofHPS(2011).ThesecondresultfollowsfromthedirectevaluationofthemeanofthedenominatorandthevarianceoftheexpressioninthenumeratorofCorollary4(ii)ofHPS(2011). AppendixB:SupplementaryLemmasThissectiongatherstogethersometechnicallemmas.Since,theareuniformlybounded,andthequantitiesandareconvergent,theheteroskedasticitymaybeignoredinthecalculationsgivenhere.Hence,insteadofintroducingnewnotationforthestandardizedquantities,wesimplylet:=1sothatthecomponentrandomvariablesare.WealsomaintainConditionsAandBthroughout,andassumethat:=0withoutlossofgeneralityif;otherwisewecould simplyreplacealltheintheproofswith.ThistranslationisjustiedbythatfactthatthePFAEisexpressedintermsofdifferences.WefrequentlyusethefollowingBNdecomposition(PhillipsandSolo,1992,Lemma2.1;PhillipsandMoon,1999,Lemma2):Let.Then.IntheAR()case,,where):=,soforany,thusforanyand(PhillipsandSolo,1992).Therefore,,andNotethatforthestationarycasehasadifferentmeaningthanthesamenotationforthecase.Thisduplicatedusageofonenotationwillnotcauseanyconfusionbecausethesetermsdonotappeartogether.For,thefollowingistrue.Lemma7(1)=,where=(1Proof.When,wehave)=(1.So)+(1implyingthat(1)=(1)= Someresultsfortheunitrootcaseareprovidednext.TheseareusefulinanalyzingtermswhenLemma8Under(50),if,then(i)(ii))(P=1Ω⎛1)2](13)ρ(1)⎛2;(iii)E(P=1Ω⎛1∆Ω⎛m)=()forall (vi)E[((vii)E[(forallProof.(i)From(51),wehave 2P21=2 2P+(1)1 (ii)From(52),wehave 3E"µP1¶2#=2 31P)2+(1)1 (iii)Wehave,so.Soforall(iv)and(v):By(52),(1).So(1) (1)=(1)(1)+8(¨implyingthat(vi)Wehaveforsomeuniformlyniteconstant.Thustheexpectationoftheabovedisplayedequationis.(Forthesecondterm,usetheCauchy-Schwarzinequality.) (vii)Wehave⎨e2Ω⎛1(∆Ω⎛m)2]∙forsomeniteandtheresultfollows.(Forthesecondterm,usetheCauchy-Schwarzinequality.) Nowweshowthattheremaindertermsk>Ωinthedenominatorarenegligibleunderlargeasymptotics(whetherislargeorsmall).Lemma9Under(50),,wherearenedin(26)and(27)andaredenedin(39).Proof.Wewillshowthatatmostforallm>n,wherearedenedin(33)–(35).(i):Letthethreecomponentsofbedenotedbyand,soaswrittenin(33).For,wehave 2"1X)#2 Takingexpectationsandaveragingacross (1)atmostbyLemma8(iv).For>Ω,considerΩ> m>n:=(Ω>notationisusedonlyinthispartoftheproof.)Becauseoftheinequality P||1 P2E2)121 P2·1 P2¸12 wehaveΩ> Ph()2i·1 Becausethisboundholdsforany,wehavem>n(1)>Ω(ii):Thiscaseisclearbecauserunsfrom(iii):Werstshowthatforgivenand,whichistrue P¯¯¯1 P2)122121 whereweusedthefactthat(0)andLemma8(i).Theresultfollowsbecausese,where Wederivesimilarresultsforthenumerator.Here,theremaindertermsdisappearinLemma1010k⎛1√0(k>Ω]=0Proof.For,wewillgett((m>n)k>Ω−E(m>n)k>Ω)2]∙E(m>n)k>Ω2=(1)becausethenhen⎛1√0(.For,wewillestablishasharperboundaryfortherotatedandrescaledremainderremainder:Againnotethat>Ω>ΩasintheproofofLemma9.For,wehavem>n byLemma8(v).For,wehavem>n 42P£()2¤+1 whichis(1)forgivenand(small)becauseisanitesumofstationarytermsforgivenirrespectiveoftheexistenceoftheunitroot,soitsfourthmomentsareuniformly(over)bounded.issimilarlyhandled.(ii):Thiscaseisstraightforwardbecausexedandsmall. (iii):Wehaveisthematrixwhosem>nelementisisthevectorwhoseelementisisthevectorwhosethelementis,andistheHadamardproduct.Because)+¨m>n,whereisthethelementofisthethelementofm>nisthem>nelementof,wehave+1¨.Sowhereweusetherelationforcolumnvectors.Because,whereistherstcolumnof,wehave,thenallthetermsin(55)areeasytohandle:thevariancesdisappearasbecausethevarianceofanddisappearatanrate.Nowlet(1).Thersttermof(55)isnullbecause.Forthesecondtermof(55),wehave.Sotheelementofandsatis 2P11E11)¸21 wherethelastordercanbeobtainedusing(52).Them>nelementsofforareeasilyhandledbecausetheyinvolveonlydifferences(whicharestationary)andinitialvalues.Thevarianceofthethirdtermontherighthandsideof(55)is.Thelasttermof(55)containsonlyonenonzeroelement,whichistherstelementequalto.Itsvarianceasshownin(56). Nextweapproximate.Therstelementof.Oftheseterms,thetermsinthelasttermcanbereplacedbyinthesensethat 2P1=1 wherethelast(1)termisnegligibleinthesense,andallthetermscanbereplacedwiththeleadingtermof(51),i.e.,with.Also,thevectorofthesecondtolastelementsisapproximatedbyby∆Ω⎛1>⎩⎩⎩>∆Ω⎛+1]0becausetheremainingtermsarenegligibleinthesenseasshownlater.Thus,wehavethefollowingresult:Lemma11Let.Thenwith>Ω>Ω,where 1 P11 2µP1P+1 2P21¸21 ,andProof.Let P11 2µP1P+P =(˜.Werstshowthat,where.Let>Ωbetherstelementofandtheremainingelements,sothat.Then ,wehave notationisusedonlyinthisproof.)But 4Ph21()2i+2 UsingtheBNdecomposition(51),wecanapproximate(1).Thenthersttermontherighthandsideofthelastexpressionisandthesecond termisalso.Becauseisanitesumof,wehaveshownthatNext,wehave 32P¨1¸P1 istheelementsof.Becauseisstationary,thevarianceofthersttermofandthesecondtermalsohasanvariance-covariancematrix,whichcanbeshownusing(51).ThecovariancealsodisappearsduetoHölder’sinequality.Sofar,wehaveapproximatedwith(inthesense).Nowweshowthat.Thispartcanbedoneusing(51)andLemma7.Moreprecisely,because(1)byLemma7,wehave:=˜>Ω P2¨1)1 2P0¨1)·P1 Thesecondmomentsoftherstandsecondtermsare,andforthelastterm,weagainapply(51)andshowthatitssecondmomentis Lemma12,under(50),Proof.Let,and.Then 1 P1¯¯+1 (anotationusedonlyinthisproof).Using,wehave (1) 2+1 Forthesecondmoment,wehave +E¯ 4E"µP21¶2#2 E¯¯P1+2 3EP1P21¸2 2E¯¯P21¸=1+2+3+4+5+6 .For,wehave,so+(1 Fortherest,notethatwhere(57)isobvious,(58)uses,and(59)isobtainedbyrearrangingthetermsafterexpanding.Now,for,from(59),wehave 4+1 3=7 Using(57)and(58),wehave From(57)and(59),wehave 3P1P 3P)4Z10)2 Finally,from(58)and(59),wehave 321 2+5 6+7 34 3=1 ,whichimpliestheresult. AppendixC:UnitRootAsymptoticsforaModiedPFAEProofof(23).Theorem3ofHPS(2011)givesarepresentationoftheFAEestimatorintermsofthepooledOLSestimator.ThisrelationshipinthepanelcontextgivesthefollowingrelationshipbetweenthePFAEestimatorandtheLSDVestimatoroo PP21+Pn1212(1+2)P1o ,andwhereo withItfollowsthatwhen 0 o1)+ 2PP21 PP21+ n1212(1+2)P1o PP21= o PP21 PP21!+µ ¶= o 3PP21+³PP213PP21´ PP21+ o ¶+ PP213PP21 (1) o ¶+ 3P12³P1´22PP21 (1)givingthestatedrelationshipbetweenthetwoestimatorso WenowproceedtoderiveasymptoticsforthemodiedPFAEgivenby(24)as>.Notethatwecanset:=0withoutlossofgeneralitywhen.Letrstidentityof(25)implies1)=o o +b where=b1·1 12XX1ù3 ,and 12X1 2X21+121+2 2X13 (Fortheexpressionfor,seeHPS,2011,Theorem3.)Itisstraightforwardtoshowthat.Next,HahnandKuersteiner(2002)showthatandtheasymptoticvarianceofSotheasymptoticvarianceofForthevarianceof,wenotethat 12X1 2X213 X21#+(12)=2 standardWienerprocesses.Notethatandweneedtocalculatethevarianceof=4E+9EForthersttermof(61),wehave>ZeZΩ()−ZΩ()]>=2Z10Z0³EZΩ()4+E[ bydirectcalculation,wherethesecondidentityholdsbecausebecauseZΩ()−ZΩ()]ZΩ()3=0.Forthesecondtermof(61),afterlongandtediousalgebra,wehave 30 Forthethirdtermof(61)wenotethat,sothat 9Ehi=1 9× Thus,theasymptoticvarianceof 301 3= implyingthattheasymptoticvarianceofis24/5.Torecapitulate,whatwehaveobtainedsofarisAvar)=51,andAvar)=24.WealsohaveAvarbyTheorem3,and1)=Avar=Avar()+Avar()+2Acov(9=515+245+2Acov(,implyingthatAcov(Itthereforefollowsfrom(60)thatAvar=Avar(Acov(Avar( 6+ Thisasymptoticvarianceisminimizedat,wheretheminimumvarianceattainedis8+(24=33340=8 ReferencesAhn,S.C.andP.Schmidt(1995)Efcientestimationofmodelsfordynamicpaneldata.JournalofEconometrics,68,5–27.Alvarez,J.andM.Arellano(2003)Thetimeseriesandcross-sectionasymptoticsofdynamicpaneldataestimators.Econometrica,71(4),1121–1159.Anderson,T.W.andC.Hsiao(1981)Estimationofdynamicmodelswitherrorcomponents.JournalofAmericanStatisticalAssociation,76,598–606.Arellano,M.(1987)Computingrobuststandarderrorsforwithin-groupsestimators.OxfordBul-letinofEconomicsandStatistics,19,431–434.Arellano,M.andS.Bond(1991)Sometestsofspecicationforpaneldata:MonteCarloevidenceandanapplicationtoemploymentequations.ReviewofEconomicStudies,58,277–297.Arellano,M.andO.Bover(1995)Anotherlookattheinstrumentalvariableestimationoferror-componentsmodels.JournalofEconometrics,68,29–51.Bauer,P.,Pötscher,B.M.,andP.Hackl(1988)Modelselectionbymultipletestprocedures.,19,39–44.Bertrand,M.,E.Duo,andS.Mullainathan(2004)Howmuchshouldwetrustdifferences-in-differencesestimates?QuarterlyJournalofEconomics,249–275.Bhargava,A.,L.FranziniandW.Narendranathan(1982)Serialcorrelationandthexedeffectsmodel.ReviewofEconomicStudies,49(4),533–549.Blundell,R.andS.Bond(1998)Initialconditionsandmomentrestrictionsindynamicpaneldatamodels.JournalofEconometrics,87,115–143.Davidson,J.(1994)StochasticLimitTheory.OxfordUniversityPress,NewYork.Gouriéroux,C.,P.C.B.PhillipsandJ.Yu(2010)Indirectinferencefordynamicpanelmodels,JournalofEconometrics,157,68–77.Hahn,J.(1999)Howinformativeistheinitialconditioninthedynamicpanelmodelwithxedeffects?,JournalofEconometrics,93,309–326. Hahn,J.,J.Hausman,andG.Kuersteiner(2007)Longdifferenceinstrumentalvariablesestima-tionfordynamicpanelmodelswithxedeffects.JournalofEconometrics,140(2),574–617.Hahn,J.andG.Kuersteiner(2002)Asymptoticallyunbiasedinferenceforadynamicpanelmodelxedeffectswhenbothandarelarge.Econometrica,70(4),1639–1657.Hájek,J.(1972)Localasymptoticminimaxandadmissibilityinestimation,ProceedingsofSixthBerkeleySymposiuminMathematicalStatisticsandProbability,1,175–194.Han,C.andP.C.B.Phillips(2006)GMMwithmanymomentconditions.,74,Han,C.andP.C.B.Phillips(2010a)GMMestimationfordynamicpanelswithxedeffectsandstronginstrumentsatunity.EconometricTheory,26,119–151.Han,C.andP.C.B.Phillips(2010b)FirstDifferenceMLEandDynamicPanelEstimation.Unpublishedmanuscript.Han,C.,P.C.B.Phillips,andD.Sul(2010)X-DifferencingandDynamicPanelModelEstima-tion.CowlesFoundationDiscussionPapers1747,YaleUniversity.Han,C.,P.C.B.Phillips,andD.Sul(2011)Uniformasymptoticnormalityinstationaryandunitrootautoregression.EconometricTheory,forthcoming.Hansen,C.B.(2007)Asymptoticpropertiesofarobustvariancematrixestimatorforpaneldataislarge.JournalofEconometrics,141,597–620.Hayakawa,K.(2007)ASimpleEfcientInstrumentalVariableEstimatorinPanelAR()Mod-els.Hi-StatDiscussionPaperSeriesd07-213,InstituteofEconomicResearch,HitotsubashiUniversity.Hsiao,C.,M.H.Pesaran,andA.K.Tahmiscioglu(2002)Maximumlikelihoodestimationofxedeffectsdynamicpaneldatamodelscoveringshorttimeperiods.JournalofEconometrics109,107–150.Imbs,J.,H.Mumtaz,M.O.Ravn,andH.Rey(2005)PPPstrikesback:Aggregationandtherealexchangerate,QuarterlyJournalofEconomics,120,1–43.Jeganathan,P.(1995)Someaspectsofasymptotictheorywithapplicationstotimeseriesmodels,EconometricTheory11,818-867. Keane,M.,andD.Runkle(1992)Ontheestimationofpanel-datamodelswithserial-correlationwheninstrumentsarenotstrictlyexogenous.JournalofBusiness&EconomicStatistics10(1),1–9.Kezdi,G.(2002)RobustStandardErrorEstimationinFixed-EffectsPanelModels.WorkingPaper,UniversityofMichigan.Kiefer,N.M.(1980)Estimationofxedeffectmodelsfortimeseriesofcrosssectionwitharbi-traryintertemporalcovariance.JournalofEconometrics,14,195–202.Kiviet,J.F.(1995)Onbias,inconsistency,andefciencyofvariousestimatorsindynamicpaneldatamodels.JournalofEconometrics,68,53–78.Kruiniger,H.(2008)MaximumlikelihoodestimationandinferencemethodsforthecovariancestationarypanelAR(1)/unitrootmodel.JournalofEconometrics,144,447–464.LeCam,L.(1972)Limitsofexperiments.ProceedingsoftheSixthBerkeleySymposiuminMath-ematicalStatisticsandProbability,1,245-261.Lee,Y.(2010a)BiasinDynamicPanelModelsunderTimeSeriesMisspecication.Unpublishedmanuscript,UniversityofMichigan.Lee,Y.(2010b)ModelSelectioninthePresenceofIncidentalParameters.Unpublishedmanu-script,UniversityofMichigan.Nickell,S.(1981)Baisesindynamicmodelswithxedeffects.Econometrica,49,1417–1426.Orcutt,G.H.andH.S.Winokur(1969)Firstorderautoregression:inference,estimationandEconometrica,37,1-14.Park,J.Y.andP.C.B.Phillips(1988)Statisticalinferenceinregressionswithintegratedprocesses:Part2,EconometricTheory5,95-131.Phillips,P.C.B.(1987)Timeseriesregressionwithaunitroot.,55,277–301.Phillips,P.C.B.(1989)Partiallyidentiedeconometricmodels,EconometricTheory5,181–240.Phillips,P.C.B.(2011)Folkloretheorems,implicitmaps,andnewunitrootlimittheory,Econo- Phillips,P.C.B.andC.Han(2008)GaussianinferenceinAR(1)timeserieswithorwithoutunitroot.EconometricTheory,24,631–650.Phillips,P.C.B.andT.Magdalinos(2007)Limittheoryformoderatedeviationsfromaunitroot.JournalofEconometrics,136,115–130.Phillips,P.C.B.andT.Magdalinos(2009)Unitrootandcointegratinglimittheorywheninitial-izationisintheinnitepast.EconometricTheory,25,1682-1715.Phillips,P.C.B.andH.R.Moon(1999)Linearregressionlimittheoryfornonstationarypaneldata.,67(5),1057–1111.Phillips,P.C.B.andW.Ploberger(1994)Posterioroddsfortestingforaunitrootwithdata-basedmodelselection.EconometricTheory,10,774–808.Phillips,P.C.B.andV.Solo(1992)Asymptoticsforlinearprocesses,AnnalsofStatistics,20(2),Pötscher,B.M.(1983)OrderestimationinARMA-modelsbyLagrangianMultipliertests.nalsofStatistics,11,872–885.Qian,H.,andP.Schmidt(2003)PartialGLSregression.EconomicsLetters,79,385–392.White,H.(1980)Aheteroskedasticity-consistentcovariancematrixandadirecttestforhet-eroskedasticity.Econometrica,48,817–838.Wooldridge,J.M.(2002)EconometricAnalysisofCrossSectionandPanelData.MITPress, Table1:SimulationforpanelAR(1),1000replications,=100ΩΩ LSDV GMM2 GMM2 PFAE DIF DIF SYS -0.1111 0.010 -0.0116 -0.0035 -0.1719 0.310 -0.0226 -0.0006 -0.0062 -0.0064 -0.2161 -0.0059 0.510 -0.0353 -0.0037 -0.0095 -0.0122 -0.2631 -0.0169 0.710 -0.0562 -0.0099 -0.0161 -0.0202 -0.3126 -0.0325 -0.0295 0.910 -0.0905 -0.0215 -0.0170 -0.0338 -0.0323 -0.0321 -0.3386 -0.0151 -0.0154 1.010 -0.1141 -0.0123 -0.0122 -0.0533 -0.0327 -0.0318 *HKLSDV1)+1 (Table1continued)Variance LSDV GMM2 GMM1 GMM2 PFAE DIF 5.812 16.016 10.258 33.820 22.485 10.5360.010 1.431 2.103 2.500 3.430 1.503 0.550 0.827 0.768 1.494 0.557 7.083 26.429 12.681 74.392 23.779 13.7100.310 1.498 2.364 3.401 3.267 1.593 0.545 0.867 0.857 1.242 0.551 7.912 37.652 13.884 22.584 15.9190.510 1.450 2.485 4.300 3.250 1.545 0.510 0.885 0.900 1.123 0.512 8.666 56.160 15.235 21.632 18.1560.710 1.338 2.414 5.934 3.270 1.442 0.445 0.763 0.977 0.984 0.439 9.314 19.408 22.419 20.4250.910 1.201 2.261 10.11 2.712 1.367 0.350 0.691 1.345 0.882 0.345 9.560 14.933 18.269 21.5421.010 1.138 30.37 0.760 1.369 0.279 0.681 4.214 0.682 0.273 Table2:VarianceofLSDVandPFAEforAR(1)with,10,000replicationsΩΩ=50=100=200 LSDVPFAE LSDVPFAE LSDVPFAE 0.49100.5526 0.24480.2769 0.12310.1392 0.12460.1252 0.06380.0643 0.03100.0319 0.03280.0305 0.01590.0153 0.00780.0073 0.00800.0074 0.00400.0036 0.00200.0018 Note:TheLSDVestimatorisunbiasedfor.TheslightdisparitybetweenTables1and2=100and=20isascribedtothefactthatStataisusedforonetableandGaussfortheother,thegeneratedsamplesaredifferent,andthereplicationnumbersaredifferent. Table3:SimulationforfromAR(2),1000replications,=100ΩΩ 1 LSDV GMM1GMM2 GMM1GMM2 PFAE DIFSYS DIFSYS -0.1994 -0.03580.0082 -0.06760.1559 0.210 -0.0255 -0.01990.0033 -0.02410.0976 -0.0056 -0.0126-0.0002 -0.01330.0825 -0.2452 -0.0407-0.0067 -0.09690.0865 0.510 -0.0369 -0.0238-0.0020 -0.03160.0601 -0.0080 -0.0147-0.0081 -0.01610.0428 -0.2843 -0.0446-0.0149 -0.12510.0411 0.710 -0.0515 -0.0275-0.0057 -0.04130.0380 -0.0110 -0.0171-0.0125 -0.01990.0193 -0.3340 -0.0531-0.0201 -0.16370.0025 0.910 -0.0800 -0.0348-0.0105 -0.06260.0136 -0.0178 -0.0211-0.0172 -0.0280-0.0020 -0.3997 -0.1172-0.0252 -0.2525-0.0237 1.110 -0.1407 -0.0676-0.0188 -0.1166-0.0162 -0.0425 -0.0355-0.0250 -0.0533-0.0271 -0.4414 -0.8647-0.0139 -0.8791-0.0137 1.210 -0.1978 -0.4500-0.0075 -0.4494-0.0075 -0.0896 -0.2215-0.0233 -0.2217-0.0226 (Table3continued)Variance 1 LSDV GMM1GMM2 GMM1GMM2 PFAE DIFSYS DIFSYS 4.435 22.87410.249 50.30127.440 8.7260.210 1.481 2.5372.095 2.8974.858 1.444 0.546 0.7070.751 0.7381.504 0.517 4.856 28.3599.722 73.09919.188 8.6450.510 1.576 2.6752.149 3.3143.319 1.401 0.562 0.7120.758 0.7621.024 0.516 5.059 34.3629.411 98.81615.715 8.1580.710 1.648 2.8542.195 3.9802.909 1.362 0.576 0.7160.712 0.7930.863 0.517 5.161 46.8829.156 134.72713.798 7.7530.910 1.729 3.2232.208 5.5512.751 1.341 0.600 0.7350.711 0.8790.818 0.521 5.149 112.01910.432 189.90213.019 7.8731.110 1.777 5.1892.260 9.1722.583 1.396 0.650 0.9220.779 1.2560.880 0.542 5.135 418.2098.470 406.4918.496 8.0531.210 1.713 32.591.818 32.431.827 1.479 0.672 4.4880.914 4.5300.906 0.569 Table4:SimulatedsizesforAR(1),5000replications truthvs 0.00.30.50.70.91.0 2510 0.06580.06520.06560.06720.07540.07702520 0.05920.06280.06400.06500.06660.07262540 0.05340.05340.05520.05720.06060.0710 5010 0.05820.05900.06420.06380.06520.06305020 0.04540.04680.04960.05300.05660.06285040 0.05300.05040.05220.05400.05760.0618 10010 0.05380.05200.05340.05120.05400.052210020 0.05060.05320.05460.05340.05140.061410040 0.04860.05100.05020.05580.05620.0610 20010 0.04800.04980.05500.05580.05300.055620020 0.04820.05020.04640.05040.05180.052220040 0.04700.04980.05080.04660.05120.0514 Table5:SimulatedpowerforforAR(1)model,5000replications ,0:1:6 ,0:1:6 0.0000.0250.0500.075 0.9250.9500.9751.000 2510 0.06580.07420.11260.1768 0.22340.14400.09680.07702520 0.05920.08740.18140.3380 0.60180.33400.14560.07262540 0.05340.12140.33080.6156 0.98980.84660.37260.0710 5010 0.05820.07900.15600.2748 0.32740.17940.08920.06305020 0.04540.11340.30460.5822 0.88660.57600.20760.06285040 0.05300.17960.55620.8888 1.00000.99160.59720.0618 10010 0.05380.10060.24900.4826 0.55940.29640.12040.052210020 0.05060.18380.54000.8734 0.99480.85980.34780.061410040 0.04860.33200.86420.9932 1.00001.00000.88860.0610 20010 0.04800.14780.45100.7910 0.83840.49520.16880.055620020 0.04820.31080.83060.9916 1.00000.99360.58660.052220040 0.04700.57240.98661.0000 1.00001.00000.99640.0514 Table6:Simulationfornonstationaryinitialconditions,1000replications,=100ΩΩ LSDV GMM2 GMM2 PFAE DIF DIF SYS -0.1124 0.010 -0.0132 -0.0027 -0.1891 0.310 -0.0262 -0.0006 -0.0057 -0.0059 -0.2640 -0.0086 0.510 -0.0445 -0.0045 -0.0102 -0.0117 -0.3411 -0.0083 -0.0122 0.710 -0.0798 -0.0115 -0.0208 -0.0191 -0.3661 -0.0013 -0.0102 0.910 -0.1286 -0.0170 -0.0123 -0.0523 -0.0359 -0.0293 -0.3345 -0.0031 1.010 -0.1140 -0.0131 -0.0133 -0.0535 -0.0452 -0.0441 *HKLSDV1)+1 (Table6continued)Variance LSDV GMM2 GMM1 GMM2 PFAE DIF 5.718 14.163 29.605 20.622 0.010 1.403 1.925 2.295 3.317 1.480 0.581 0.832 0.772 1.642 0.586 7.141 24.831 12.427 63.064 22.796 12.3810.310 1.513 2.252 3.181 3.193 1.596 0.584 0.939 0.895 1.400 0.590 8.001 42.910 12.484 20.574 13.7480.510 1.502 2.424 4.018 3.132 1.543 0.541 0.932 0.941 1.245 0.537 8.659 16.084 15.3770.710 1.436 2.253 5.398 2.889 1.423 0.462 0.854 0.963 1.005 0.442 9.159 18.1290.910 1.281 11.234 13.137 1.335 0.354 0.938 1.357 0.900 0.317 9.316 20.1251.010 1.143 28.754 28.130 1.336 0.265 0.739 3.989 0.771 0.268