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Structuralchanges,commonstochastictrends,andunitrootsinpaneldataJushanBaiDepartmentofEconomicsNewYorkUniversityJosepLluísCarrion-i-SilvestreDepartmentofEconometrics,StatisticsandSpanishEconomyUniversityofBarcelonaFirstdraft:July21th,2002Revised:December10th,2003AbstractInthispaperweproposeanewteststatisticthatconsidersmultiplestructuralbreakstoanalysethenon-stationarityofapaneldataset.
Themethodologyisbasedonthecommonfactoranalysisinanattempttoallowforsomesortofdependenceacrosstheindividuals.
Thusallowingformultiplestructuralbreaksinthe"PanelAnalysisofNon-stationarityinIdiosyncraticandCommoncomponents"(PANIC)methodologyincreasesthedegreeofheterogeneitywhenassessingthestochasticpropertiesofthepaneldataset.
Keywords:multiplestructuralbreaks,commonfactors,paneldataunitroottest,principalcomponentsJELcodes:C12,C22,C3,C51IntroductionNowadays,theincreasingapplicationofthepaneldatatechniquestothedeter-minationoftimeseries'stochasticpropertieshasledtothedevelopmentofawiderangeofnewproposalsintheeconometricliterature.
Theshorttimepe-riod'scoveragethato¤ermostoftheavailablemacroeconomictimeseriesmaybethoughtasthemainreasonbehindthisexplodingphenomenon.
Thislackofinformation,intermsoftimeobservations,impliesalossinthepowerofunitroot,stationarityandcointegrationtests.
Thecombinationoftheinformationinthetimeandcross-sectiondimensionstocomposeapaneldatasetofindivid-uals,i.
e.
countriesorregions,ontowhichperformtheanalysisofthestochasticpropertieshasrevealedasapromisingwaytoincreasethepowerofthesetests.
1Thus,againinpowerisexpectedwhenperformingastatisticalinference–unitroot,stationarityorcointegrationtest–usingapaneldatasetmadeupofin-dividualsthatshare,at…rst,somesimilarities.
BreitungandMeyer(1994),Im,PesaranandShin(1997),MaddalaandWu(1999)andLevin,LinandChu(2002),ontheunitroottests,andPedroni(1995)andPhillipsandMoon(1999),onthecointegrationanalysis,aresomeofthemostrelevantpapers.
Compre-hensivesurveysofthe…eldcanbefoundinBanerjee(1999),Baltagi(2001)andBaltagiandKao(2001).
Althoughthedeterministiccomponentshouldnotbeofinterestwhenanalysingtheorderofintegrationofthetimeseries,itsmisspeci…cationcandrivetomisleadingconclusions.
Thus,astationarytimeseriesthatevolvesaroundabreaking-trendmodelmightbecharacterizedasanon-stationaryprocessiftheorderofintegrationanalysisfailstoconsiderthestructuralbreaks–seePerron(1989)fortheunivariatetimeseriesframeworkandCarrion-i-Silvestre,delBar-rioandLópez-Bazo(2001)forthepaneldataframework.
Ourproposalfocusonthepresenceofmultiplestructuralbreaksa¤ectingthepaneldataset,sothattakingintoaccountthepresenceofthesestructuralbreaksovercomestheinter-ferencesthatcancausethemisspeci…cationerrorinthestochasticpropertiesofthepanel.
Inthispaperweanalysethepresenceofmultiplestructuralbreakswhentestingfortheunitroothypothesisinapaneldataframework.
Someoftherecentproposalsinthepaneldatabasedunitrootandstationaritytestshaveaddressedthisquestionbydevelopingsuitabletests–seeIm,LeeandTieslau(2002)fortheLMtestandCarrion-i-Silvestreetal.
(2001)fortheDFwithonestructuralbreak,andCarrion-i-Silvestreetal.
(2002)fortheKPSStestswithmultiplestructuralbreaks.
However,ourapproachovercomesthecriticismthathasraisedtheassumptionofcross-sectionindependenceinwhichmostofthepaneldatabasedtestsrely,andmodelsthecross-sectiondependenceintermsofthecommonfactorsasinBaiandNg(2001,2004).
Brieyspeaking,theideaistoestablishadistinctionbetweencomovementsandidiosyncraticshocksthatmaybea¤ectingtheindividualtimeseries.
Filteringoutthecomovementswillreducethenoiseinthesystem,sothat,theanalysiswillfocusonthoseshocksthatarespeci…cforeachindividual.
Moreover,notethatthecross-sectionin-dependenceismorelikelytobeful…lledwhenusingtheseidiosyncraticshocksthanwhenusingtherawdata.
Therestofthepaperisorganizedasfollows.
Section2describesthemodelandthetwodeterministicspeci…cationsthatareconsideredalongthepaper.
Thesemodelsarisebecauseofthedi¤erente¤ectsthatthestructuralbreaksmaycauseonthedeterministicpartofthemodel.
Section4presentsdi¤erentpooledtests,whileinSectionweanalysethe…nitesampleperformance.
Finally,Section6concludes.
AllproofsarepresentedintheAppendix.
22PanelunitroottestwithmultiplestructuralbreaksLetusde…nethepaneldatamodelgivenby:Xi;t=Di;t+Fti+ei;t;(1)(IL)F0t=C(L)ut;(2)(1iL)ei;t=Hi(L)"i;t;(3)t=1;T,i=1;N,whereC(L)=P1j=0CjLjandHi(L)=P1j=0Hi;jLj.
Di;tdenotesthedeterministicpartofthemodel,Ftisa(l1)-vectorthatac-countsforthecommonfactorsthatarepresentinthepanelandei;tistheidiosyncraticdisturbanceterm.
Ouranalysisisbasedonthesamesetofas-sumptionsinBaiandNg(2004).
LetM0,(iii)P1j=0jkCjk0;(ii)E("i;t"j;t)=i;jwithPNi=1ji;jj·Mforallj;(iii)E1pNPNi=1["i;s"i;tE("i;s"i;t)]4·M,forevery(t;s).
AssumptionD:Theerrors"i;t,ut,andtheloadingsiarethreemutuallyindependentgroups.
AssumptionE:EkF0k·M,andforeveryi=1;N,Ejei;0j·M.
AssumptionAensuresthatthefactorloadingsareidenti…able.
AssumptionBestablishestheconditionsontheshortandlong-runvarianceofFt–i.
e.
positivede…niteshort-runvarianceandlong-runvariancethatcanbeofreducedrankinordertoaccomodatelinearcombinationsofI(1)factorstobestationay.
AssumptionC(i)allowsforsomeweakserialcorrelationin(1iL)ei;t,whereasC(ii)andC(iii)allowforweakcross-sectioncorrelation.
Finally,AssumptionEde…nestheinitialconditiononei;t.
ThismodelexpressesthestochasticprocessXi;tasthesumofuptothreedi¤erentcomponents,sothatwecanfocusoneachofthesecomponentstochar-acterizeXi;tintermsofitsstochasticproperties.
Notethatthenon-stationarityofXi;tcanbeduetothenon-stationarityofeitherFtorei;t,sothatwehavetwopotentialsourcesofnon-stationaritywithdi¤erenteconomicinterpreta-tions.
Thus,thematrixFtcollectsthecommone¤ectsthatarepresentacrossthecross-sectiondimensionand,therefore,thenon-stationarityofFtwillmeanthatallindividualsinthepanelarecommonnon-stationary.
Thesee¤ectsa¤ecttheindividualswithdi¤erentmagnitude(i).
However,evenifXi;tisdrivenbyacommonnon-stationarycomponent(Ft),theidiosyncratice¤ectmaybe3ei;tI(0).
Thiswillmeanthatthestochasticshocksthatonlya¤ecteachin-dividualarestationary.
Hence,thenon-stationarityanalysiscanbeperformedthroughtheapplicationofunitroottestsonFtandei;t.
Regardingthedeterministiccomponent,thespeci…cationthatisadoptedinthemodelisquitegeneraltoallowforthepresenceofmultiplestructuralbreaks.
Speci…cally,weformulate:Di;t=i+it+miXk=1i;kDUi;k;t+miXk=1°i;kDT¤i;k;t;(4)thatis,weallowformistructuralbreaksa¤ectingthemeanofthetimeseries.
Thedummyvariablesarede…nedasDUi;k;t=1andDT¤i;k;t=tTib;kfort>Tib;kand0elsewhere,whereTib;kdenotesthek-thdateofthebreakforthei-thindividual,k=1;mi,mi1.
Infact,equation(4)neststwodi¤erentspeci…cationsdependingonthee¤ectofthestructuralbreaksonthedeterministiccomponents.
Ontheonehand,wecanintroducetheconstrainti=°i;k=0,8i;k,in(4)toanalysethestochasticpropertiesofpaneldatasetsformedbynon-trendedvariables–forinstance,thePPPhypothesisoughttobetestedusingthisspeci…cation.
Hereafter,theconstrainedmodelisdenotedasModel1.
Formallyspeaking,Model1impliesthefollowingdeterministicspeci…cation:Di;t=i+miXk=1i;kDUi;k;t;whichincludesindividuale¤ectsandindividualshiftinge¤ects.
Ontheotherhand,wewilldenotetheunconstrainedmodelgivenby(4)asModel2,aspeci-…cationthatissuitablefortrendedvariablesthatmaybea¤ectedbystructuralbreaksthatshiftboththeindividualandthespeci…ctimetrend–forinstance,theanalysisoftheunitroothypothesisinGDPshouldbebasedonthisspeci-…cation.
Noticethatbothmodelsassumethatthestructuralbreaksareidiosyncraticfortheindividuals,since(i)theycanbepositionedatdi¤erentdatesforeachindividual,(ii)theymayhavedi¤erentmagnitudeand(iii)eachindividualmayhavedi¤erentnumberofstructuralbreaks.
Therefore,ourspeci…cationtakesintoaccountahighdegreeofindividual'sheterogeneity.
Oncethemodelhavebeende…nedinageneralway,nowwearegoingtoaddresstheunitrootnullhypothesistestingthroughtheconsiderationoftwosituations:…rst,weassumethattherearenocommonfactors,i=08iin(1)and,second,weallowforthepresenceofsuchcommonfactors,i6=0in(1),i=1;N.
Foreaseofexposition,at…rstwetakethedateofthebreaksasknown.
Oncethelimitdistributionsarederived,weintroducethediscussionabouttheproceduresthatcanbeappliedinordertoestimatethem.
42.
1IndividualsareassumedtobeindependentacrossiFromatheoreticalpointofview,itisofinteresttoconsiderthesimpli…edsituationinwhichi=08iin(1)andfei;tgisastochasticprocessindependentacrossi=1;N.
InordertotestthenullhypothesisthatXi;tI(1),8i,i=1;N,wesuggesttocomputethesquareofthemodi…edSargan-Bhargava(MSB)teststatisticde…nedinStock(1999):MSBi(i)=T2PTt=1~X2i;t1~2i;(5)where~Xi;t=Xi;t~Di;tand~2iisthelong-runvarianceof~Xi;t.
Wehavemadeexplicitthedependencyofthetestonthestructuralbreaksthroughtheconsiderationofiinthenotation,wherei=(i;1;i;mi)0,i;k=Tib;k=T,k=1;mi,istheso-calledvectorofbreakfractionparameters.
Thelimitdistributionof(5)forthetwodi¤erentmodelsconsideredinthepaperisgiveninthefollowingTheorem.
Theorem1LetXi;t,i=1;N,t=1;T,bethestochasticprocessgeneratedby(1)withi=08iandi=1in(3).
AsT;Tib;k!
1inawaythati;k=Tib;k=Tremainsconstant,8i;k;i=1;N,k=1;mi,thenthetestin(5)convergesto:(1)Model1:MSBi(i))Pmi+1k=1(i;ki;k1)2R10Vi;k(b)2db(2)Model2:MSBi(i))Pmi+1k=1(i;ki;k1)2R10Vi;k(b)2dbwhere)denotesweakconvergenceoftheassociatedmeasureofprobability,Vi;k(b)=Wi;k(b)R10Wi;k(s)ds;andVi;k(b)=Wi;k(b)(46b)R10Wi;k(s)ds(6+12b)R10sWi;k(s)ds,withWi;k(b)thestandardBrownianmotion,andi;0=0andi;mi+1=1.
Theorem1showsthatthelimitdistributionoftheMSBi(i)testisfunctionofBrownianmotionsandtwonuisanceparameters–i.
e.
thebreakfractionparameters(i)andthenumberofstructuralbreaks(mi).
MoreoverandasshownintheAppendix,whenthereisonlyonestructuralbreak,mi=1,thelimitdistributionofthetestissymmetricaroundi=0:5.
Finally,notethatformi=0thelimitdistributionsinTheorem1coincidewiththeonesgiveninStock(1999).
Besides,althoughthesituationinwhichN=1canbeunderstoodasaspecialcase,thisisofgreatinterestprovidedthatitgeneralisestheproposalinPerron(1997)andLumsdaineandPapell(1997)throughtheconsiderationofmultiplestructuralbreaksinthenon-stationarityanalysis.
Thus,ourcanbeappliedtotestthenullhypothesisofunitrootonasingletimeseriesallowingforthepresenceofmultiplestructuralbreaksbothunderthenullandalternativehypotheses.
Asmentionedabove,Theorem1indicatesthatthelimitdistributionoftheMSBtestdependsbothonthenumberofstructuralbreaks(mi)andtheirloca-tion(i).
Thisgivesrisetotwopossiblesituations.
First,practitionersshouldbe5willingtoassumethatthenumberanddatesofthestructuralbreaksareknown.
Forinstance,theGermanreuni…cationandtheEurocurrency'sbirtharetwoeventsforwhichtheexogenousnatureofthestructuralbreakscanbeassumed.
However,thissituationisrarelyfoundinpractice,sothatthecomputationoftheMSBwillrequiretheapplicationofaconsistentestimationproceduretodeterminethenumberofstructuralbreaksandtherespectivevectorofbreakfractionparameters.
Thisde…nesthesecondsituationofinterest.
Letusnowfocusonthe…rstsituationinwhichboththenumberandthepositionofthestructuralbreaksareknown.
TheMSBtestcanbecomputedandcomparedtothecriticalvaluesdrawnfromthelimitdistributionsinTheorem1.
However,webelievethattheavailabilityoftheassossiatedp-valuecouldbemoreinformativewhenperformingthestatisticalinference.
ProvidedthattheMSBtesthasanon-standardlimitdistribution,thep-valueshavetobeapproximatedbysimulations.
MacKinnon(1994),AddaandGonzalo(1996),Hansen(1997),andBaiandNg(2003)computedasymptoticp-valuesforteststatisticwithnon-standarddistribution.
HerewefollowMacKinnon(1994)andestimateasetofresponsesurfacestoapproximatethep-valuesoftheMSBtest.
However,wegeneralisethepreviousproposalsandestimateresponsesurfacesforthep-valuesthattakeintoaccountthesamplesize.
Theestimationismadeassumingaprobitmodelforthep-value(pi)asafunctionofpowersofthequantile(qi),thesamplesizeandthebreakfractionparameters,logpi1pi=g(qi;T;i).
Wehaveessayeddi¤erentfunctionalformsusingtheNewey-Westrobustcovarianceestimatortoanalysetheindividualsigni…canceoftheparameters.
Inconcrete,forthesituationinwhichmi=0theresponsesurfaceisgivenby:logpi1pi=1Xj=00j+1jqi+2jq1=2i+3jq1=3i+4jq1=4i1Tj+ui;(6)where,foreachsamplesize(T),1,000quantiles,i=1;1000,hasbeencom-putedfromtheempiricaldistributiontoestimatethemodel.
WehaveconductedaMonteCarloexperimenttoobtaintheempiricaldistributionoftheMSBtestforT={30,35,40,45,50,55,60,65,70,75,80,85,90,100,125,150,175,200,225,250,300,350,400,450,500,2000}using50,000replications.
Thep-valuesresponsesurfacesarecollectedinPanelAofTable1.
SimilarresponsesurfacesarepresentedinPanelAofTable2formi=1.
Notethat(6)doesnotproduceadirectestimateofpi.
Theestimateofpiisobtainedfrom^pi=expf^g(qi;T;i)g1+expf^g(qi;T;i)g:Letusnowfocusontheproceduresthatarebasedontheendogenousde-terminationofthebreakingpoints.
TheproposaldescribedinBaiandPerron(1998)isveryconvenientforthespeci…cationinModel2,providedthatboththenumberanddatesofthebreakscanbeconsistentlyestimatedunderthenullhy-pothesistakingthe…rstdi¤erenceofyt.
Therefore,theproblemreducestothe6Table1:Responsesurfacesforthep-valuesestimationform=0PanelAPanelBModel1Model2Model1Model2^0019.
646551.
438010.
707120.
6574^103.
863412.
94791.
05863.
7891^20-13.
0719-24.
3411-8.
9767-13.
5487^3076.
9588168.
684346.
763480.
4365^40-81.
4440-189.
8098-47.
6969-85.
3760^01178.
3351968.
0575-41.
0104^1129.
6877123.
17549.
507741.
7794^21-191.
4098-586.
9201-31.
2927-130.
8045^311098.
99503963.
188063.
8206587.
8564^41-1093.
0750-4307.
8610-484.
0637PanelAcorrespondstoTheorem1andPanelBcorrespondstoTheorem2.
Thefunctionalformisgivenbylogpi1pi=P1j=00j+1jqi+2jq1=2i+3jq1=3i+4jq1=4i1Tj+ui.
TheR2ofalltheseestimationswere0.
99.
Theincludedparametersweresigni…cantatthe5%level-weusedtheNewey-Westrobustestimatortocomputethes.
e.
identifycationoflevelshiftsonyt,astationaryvariable,onwhichthedynamicoptimizationalgorithminBaiandPerron(1998)canbeapplied.
Notwithstand-ing,fortheModel1wehavetofollowadi¤erentapproachgiventhattakingthe…rstdi¤erenceofytwillimplydatingimpulseoutliers–additiveoutliers(AO)–andthissituationisnotcoveredinBaiandPerron(1998).
ThestandardwaytodealwithAOoutliersrequirestheestimationofafullyparametrisedARMAmodelonwhichtheoutlierdetectionanalysisisperformedusingatstatisticinaniterativefashion–seeTsay(1986)andChenandLiu(1993),amongothers.
ThisiterativeapproachwasfollowedinFransesandHaldrup(1994)toallowforAOoutliersintheADFtest.
However,twomaindrawbackscanbehighlighted.
First,itrequirestocontrolthedynamicstructure–i.
e.
estimationofafullyparametrisedARMAmodel–and,second,thetstatisticthatisusedtodetectthepresenceofoutliersreliesonthedistributionalassumptionsabouttheerrorterm.
Instead,wecouldestimatetheshiftdatesusingtheproposalsinPerronandVogelsang(1992)andVogelsang(1998).
Brieyspeaking,PerronandVogelsang(1992)datethebreakingpointsintheadditivespeci…cationthroughthemin-imisationofthesigni…cancetestofthedummyparameters.
Ontheotherhand,Vogelsang(1998)usesthesupPSTtestwhichdoesnotrelyonthedynamicofthesystemand,hence,serial-correlationparametersdoesnothavetobeesti-7Table2:Responsesurfacesforthep-valuesestimationform=1PanelAPanelBModel1Model2Model2^0035.
490111.
352629.
997^1013.
7152.
5935277.
791^20-19.
677-38.
27770-17.
269^30129.
744299.
7168110.
703^40-142.
760-356.
3048-120.
846^011029.
6526.
959164592.
798^113.
955-16.
29266-12.
452^21-571.
7449.
835470-411.
448^313928.
595235.
56532625.
848^41-4326.
511-591.
7060-2787.
674^'1010.
181374.
39209.
407^'20-33.
43418898.
49-33.
580^'3046.
504-44424.
2748.
346^'40-23.
25228018.
83-24.
173^'1173.
704-5029.
99039.
579^'21-213.
64812255.
89-132.
099^'31279.
888-8638.
537185.
039^'41-139.
94428817.
84-92.
519^#1-96.
699-69590.
05-76.
999^#2555.
555444.
876^#3-917.
712-735.
753^#4458.
856367.
876PanelAcorrespondstoTheorem1andPanelBcorrespondstoTheorem2.
Thefunctionalformisgivenbylogpi1pi=P1j=00j+1jqi+2jq1=2i+3jq1=3i+4jq1=4i1Tj+P1j=0'1ji+'2j2i+'3j3i1Tj+P3j=1#jqiji+ui.
TheR2ofalltheseestimationswere0.
99.
Theincludedparametersweresigni…cantatthe5%level-weusedtheNewey-Westrobustestimatortocomputethes.
e.
8mated.
However,theseproposalsdonotprovideagoodapproximation.
Ontheonehand,PerronandVogelsang(1992)showthatthedateofthebreakisnotidenti…edunderthenullalternativeofunitroot.
Ontheotherhand,thetestinVogelsang(1998)isnotconsistentwhenytI(1)sinceithasthesamelimitingdistributionunderthenullandthealternativehypothesis.
Therefore,thistestshouldnotbeusedtoestimatethelocationofthelevelshift.
Toovercometheselimitationsweproposetheuseoftheprocedurede…nedinCarrion-i-Silvestre(2003),whichconsistsontheidenti…cationofAO'sinthe…rstdi¤erencedtimeserieswithouthavingtospecifyafullyparametrisedmodelasrequiredintheexistingproposals.
Finally,forfurtherpurposesitwouldbeusefultoderivethemeanandvarianceofthelimitdistributionofMSBforModels1and2.
Speci…cally,thesetwomomentsareusedtode…neoneofthepooledtestsinSection4.
TheyarepresentedinthefollowingProposition.
Proposition1LetMSBi(i)=~2iT2PTt=1~X2i;t1betheteststatisticwithlimitdistributiongiveninTheorem1.
Moreover,leti=E(MSBi(i))and&2i=V(MSBi(i))bethemeanandvarianceofMSBi(i)respectively.
Then,asT;Tib;k!
1inawaythati;k=Tib;k=Tremainsconstant,8i;k;i=1;N,k=1;mi,(1)Model1:i=16Pmi+1k=1(i;ki;k1)2and&2i=145Pmi+1k=1(i;ki;k1)4;(2)Model2:i=115Pmi+1k=1(i;ki;k1)2and&2i=116300Pmi+1k=1(i;ki;k1)4;wherei;0=0andi;mi+1=1.
Notethatthesemomentsarefunctionofthebreakfractionparameters.
Besides,whentherearenostructuralbreakstheycoincidewiththemeanandthevarianceofthelimitdistributioninStock(1999).
TheseresultsagreewiththelimitdistributionsinTheorem1.
2.
2AllowingforcommonfactorsLetusnowweakentheframeworkthathasbeenconsideredintheprevioussectiontakingintoaccountthepresenceofcommonfactorsinthepaneldata.
Obviously,themaindicultycomesfromthefactthatthefactorsandtheidiosyncraticcomponentsareunobservedsothat,the…rststepoftheanalysisliesingettingaconsistentestimateofbothcomponents.
FollowingBaiandNg(2001,2004),inordertoestimatetheseunobservedcommonfactorsweapplytheprincipalcomponentstechniquetothedi¤erenced-detrendedmodelwhich,expressedinmatrixnotation,isgivenby:MiXi=MiFi+Mieixi=fi+zi;(7)9whereXi=(Xi;2;Xi;3;Xi;T)0andei=(ei;2;ei;3;ei;T)0aretwo((T1)1)-vectorsforthei-thindividual,F=[F1F2:::Fl]isa((T1)l)-matrixbeingFj=(Fj;2;Fj;3;Fj;T)0,j=1;l,a((T1)1)-vector,andi=(i;1;i;l)0isthe(l1)-vectorofloadingparametersforthei-thindividual,i=1;N.
Ontheotherhand,wede…neMi=IT1ai(a0iai)1a0i,withai;t=hDTib;1tDTib;mitiforModel1,beingDTib;kt=1fort=Tib;k+1and0elsewhere,k=1;mi,andai;t=hDTib;1tDTib;mit;DUi;1;t;DUi;mi;tiforModel2.
Miistheusualidempotentmatrixofprojectionintothespacespannedbyai;t.
Theestimatedfactors^f1;t;fl;taretheleigenvectorsthatcorrespondstothellargesteigenvaluesofthe(T1T1)matrixxx0,beingx=[x1;xN].
Thematrixofestimatedweights,^=(^1;N)0,isgivenby^=x0^ft.
Asaresult,wecanobtainanestimateofzifrom^zi=xi^f^i,that,aftercomputingitscumulatedsum,producesaconsistentestimationoftheidiosyncraticdisturbanceterm,~ei;t=Ptj=1^zi;j=Ptj=1(Mi^ei)j.
Now,thenullhypothesisofunitrootintheidiosyncraticstochasticelement,i.
e.
ei;tI(1),canbetestedthroughthecomputationoftheMSBtestusing~ei;t:MSBi(i)=T2PTt=1~e2i;t1~2i;(8)where~2iisanestimationofthelong-runvarianceoff~ei;tg.
ThefollowingTheoremgivestheasymptoticdistributionof(8).
Theorem2LetfXi;tgN;Ti=1;t=1thestochasticprocessgeneratedby(1)withi6=08i.
Ifi=1in(3),andT;Tib;k!
1inawaythati;k=Tib;k=Tremainsconstant,8i;k;i=1;N,k=1;mi,thenthetestin(8)convergesto:(1)Model1:MSBi(i))R10W2i(r)dr(2)Model2:MSBi(i))Pmi+1k=1(i;ki;k1)2R10V2i;k(b)db;whereWi(r)isthestandardBrownianmotion,Vi;k(b)=Wi;k(b)bWi;k(1)isaBrownianbridge,andi;0=0andi;mi+1=1.
Theorem2showsthatthelimitingdistributionoftheMSBtestforModel1doesnotdependonthepresenceofthestructuralbreaks,sincethee¤ectoftheimpulsedummyisasymptoticallynegligible.
ThisresultisalsofoundinImetal.
(2002)fortheLMpaneldatabasedunitroottest.
However,thisisnottrueforthemodelthatallowforstructuralbreaksa¤ectingthetimetrend.
Thus,theasymptoticdistributionofthetestforModel2dependsonthesetofnuisanceparametersde…nedbythebreakfractionparameters.
Moreover,theasymptoticdistributionoftheMSBtestformi=1issymmetricaroundi=0:5forModel2,afeaturethathasalsobeenhighlightedintheprevioussection.
Theresponsesurfacesforthep-valuesestimationarecollectedinPanel10BofTables1and2formi=0andmi=1respectively.
ThemeanandvarianceofthelimitdistributionofMSBforModels1and2arepresentedinthefollowingProposition.
Proposition2LetMSBi(i)=~2iT2PTt=1~e2i;t1betheteststatisticwithlimitdistributiongiveninTheorem2.
Moreover,leti=E(MSBi(i))and&2i=V(MSBi(i))bethemeanandvarianceofMSBi(i)respectively.
Then,asT;Tib;k!
1inawaythati;k=Tib;k=Tremainsconstant,8i;k;i=1;N,k=1;mi,(1)Model1:i=12and&2i=13;(2)Model2:i=16Pmi+1k=1(i;ki;k1)2and&2i=145Pmi+1k=1(i;ki;k1)4;wherei;0=0andi;mi+1=1.
SeeLevinandLin(1992)fortheproofofstatement1andtheAppendixfortheproofofthestatement2ofProposition2.
3Asimpli…edteststatisticInthisSectionweproposeasimpli…edtestthatexploitsthefactthatthelimitingdistributionsinTheorems1and2areweigthedsumsofindependentfunctionalsofBrownianmotions.
WefollowBusettiandHarvey(2001)andcomputetheMSBtestasaweightedsumofpartialsumprocessessothatwegetridofthebreakfractionparametersinthelimitdistributions.
Thissimpli…cationreducestheamountofcomputatione¤ortthathastobemadetoprovidepractitionerswithsuitablesetsofp-valuesforlargemi.
However,thisapproachisprimarilyaddressedforpanelswithlargeTprovidedthattheapproximationisforthelimitdistribution.
Firstofall,letusfocusonthesituationwheretherearenotcommonfactors,thatis,i=08i,i=1;N.
TheweightedMSBtest,MSB¤i(i),isgivenby:MSB¤i(i)=Pmi+1k=1Tib;kTib;k12PTib;kt=Tib;k1~X2i;t1~2i;(9)i=1;N,withTib;0=0andTib;mi+1=T.
Nowthecomputationofthetestdistinguishesamongmi+1subperiodswhicharerescaledbythesquareofthecorrespondingnumberofobservations.
ThelimitdistributionoftheMSB¤i(i)testforthemodelswithoutcommonfactorsispresentedinthefollowingCorollary.
Corollary1LetXi;t,i=1;N,t=1;T,bethestochasticprocessgeneratedby(1)withi=08iandi=18iin(3).
AsT;Tib;k!
1inaway11thati;k=Tib;k=Tremainsconstant,8i;k;i=1;N,k=1;mi,thenthetestin(9)convergesto:(1)Model1:MSB¤i(i))Pmi+1k=1R10Vi;k(b)2db(2)Model2:MSB¤i(i))Pmi+1k=1R10Vi;k(b)2dbwhere)denotesweakconvergenceoftheassociatedmeasureofprobability,Vi;k(b)=Wi;k(b)R10Wi;k(s)ds;andVi;k(b)=Wi;k(b)(46b)R10Wi;k(s)ds(6+12b)R10sWi;k(s)ds,withWi;k(b)thestandardBrownianmotion.
TheprooffollowsfromTheorem1and,hence,isomitted.
Similardevelop-mentscanbemadeforthespeci…cationinModel2withcommonfactors.
Forthismodel,theMSB¤i(i)testshouldbecomputedas:MSB¤i(i)=Pmi+1k=1Tib;kTib;k12PTib;kt=Tib;k1~e2i;t1~2i(10)withthelimitingdistributiongiveninthefollowingCorollary.
Corollary2LetfXi;tgN;Ti=1;t=1thestochasticprocessgeneratedby(1)withi6=08i.
Ifi=18iin(3),andT;Tib;k!
1inawaythati;k=Tib;k=Tremainsconstant,8i;k;i=1;N,k=1;mi,thenthetestin(10)convergesto:MSB¤i(i))mi+1Xk=1Z10V2i;k(b)db;whereVi;k(b)=Wi;k(b)bWi;k(1)isaBrownianbridge.
TheprooffollowsfromTheorem2and,hence,isomitted.
Notethatthede…nitionoftheweightedMSBtestmakesfreethelimitdistributionofthebreakfractionparameters,althoughitstilldependsonthenumberofstructuralbreaks–infact,theybelongtothefamilyofCramér-vonMisesdistributionswith(mi+1)-degreesoffreedom.
Theasymptoticp-valuesofthelimitdistributionsinCorollaries1and2canbecomputedfromtheresponsesurfacesinTable3–PanelAforthelimitdistributionsinCorollary1andPanelBfortheoneinCorollary2.
Theyarecomputedusingthemethodologydescribedaboveusinguptomi=15structuralbreakswithT=2;000toapproachthestepsand50,000replications.
ItcanbeshownthattheresponsesurfacesinTable3providesagoodapprox-imationofthecriticalvaluesfortheCramér-vonMissesdistributioncomputedinCanovaandHansen(1995)andNyblomandHarvey(2000).
Forinstance,fortheCramér-vonMissesdistributionwithtwodegreesoffreedomde…nedbydemeanedBrownianmotions,theseauthorssetthe95%quantileas0.
749-seethesecondrowofTable1inCanovaandHansen(1995).
Usingthisquantile(^qi=0:749)withmi=1intheresponsesurfacefortheModel2–PanelBofTable3–weobtain^pi=0:94965.
12Table3:Responsesurfacesforthep-valuesestimationforthesimpli…edteststatisticsPanelAPanelBModel1Model2Model2^0031.
087^017.
28834.
7623.
789^02-1.
826-15.
523-27.
683^032.
17473.
602154.
984^04-62.
667-156.
524^109.
934-22.
413-9.
316^112.
4890.
301^12-18.
193-6.
255^1388.
838-16.
629^14-81.
85722.
62724.
046^202.
5974.
8622.
872^21-0.
135-0.
037^22-6.
817-8.
695-6.
145^2328.
81340.
73426.
669^24-24.
517-36.
618-23.
212^30-0.
041-0.
0980.
001^310.
0010.
007^32-0.
114-0.
184-0.
219^330.
1510.
2720.
559^34-0.
349PanelAcorrespondstoCorollary1andPanelBcorrespondstoCorollary2.
Thefunctionalformisgivenbylogpi1pi=P3j=00j+1jqi+2jq1=2i+3jq1=3i+4jq1=4imji+ui.
TheR2ofalltheseestimationswere0.
99.
Theincludedparametersweresigni…cantatthe5%level-weusedtheNewey-Westrobustestimatortocomputethes.
e.
13Theperformanceofthesimpli…edtestin…nitesamplesmightnotshowgoodproperties.
ThestatementsinCorollaries1and2arevalidasT!
1,whichpreventtheuseoftheP-valuefunctionsthathavebeenestimatedabovein…nitesamples.
ThevalueofTforwhichtheasymptoticresultsareofapplianceissomethingtobeaddressedintheMonteCarloanalysis,butweshouldmentioninadvancethatthesimpli…edtestshowsanempiricalsizedistortionevenforT=300.
Thus,wewouldliketomakeavailableateststatisticthatcanbeappliedin…nitesamples,allowingformultiplestructuralbreaks,andforwhichthecomputationofsuitablep-values(orcriticalvalues)wouldnotrepresentahighcost.
Thepointhereisthecomputationofthese…nitesamplep-values.
NotethatthelimitingdistributionsinCorollaries1and2donotdependonthebreakfractionparameters,butjustonthenumberofbreaks.
ThisisbecauseasT;Tib;k!
1inawaythati;k=Tib;k=Tremainsconstant,8i;k;i=1;N,k=1;mi,thenthelimitingdistributionscanbeexpressedasthesumofmiindependentfunctionalsofBrownianmotions.
Whenapplyingthisstrategytothe…nitesampleframeworkwe…ndthatitisimpossibletogetridofthenumberofobservationsthatareinvolvedineachregime.
Thus,weshouldcomputethe…nitemomentsusing…nitevaluesforT.
Onepossiblesolutionconsistontheuseofanapproximate…nitesampledistribution.
Thus,wecande…nebyTiapprox=T=(mi+1)the…nitesamplesizeforthei-thindividualandapproximatethe…nitesampledistributionusingTiapprox.
Thissimpli…cationisspeciallyappealingprovidedthatthis…nitesampledistributionwillconvergetothelimitingdistributionasinCorollaries1and2T!
1.
TablepresentstheestimatesfortheP-valuefunctionsthatcanbeusedtoobtainthecorresponding…nitesamplep-valuesforuptomi=15structuralbreaks.
4PoolingtheindividualtestsTheresultscontainedinPropositions1and2de…nethe…rstwayofpoolingtheindividualinformation,whichgivesrisetothefollowingteststatistic:Z=pNMSB()&N(0;1);whereMSB()=N1PNi=1MSBi(i),with=N1PNi=1iand&2=N1PNi=1&2icomputedusingthestatementsinPropositions1and2.
ThestandardnormaldistributionisobtainedfromtheapplicationoftheLindberg-LévyCentralLimitTheorem(CLT).
AsmentionedinBaiandNg(2001),thiswayofpoolingcandrivetounsatisfactoryresults,speci…callywhentheasymp-toticdistributionoftheindividualtestsisskewed,asthisisthecase.
Instead,theysuggesttofollowtheproposalinMaddalaandWu(1999)andChoi(2001)thatpoolthep-valuesassociatedtotheindividualtests-henceforth,wedenotethesep-valuesaspi,i=1;N.
Undertheassumptionofcross-sectioninde-pendence,2lnpi22,aresultsthatwasusedinMaddalaandWu(1999)to14de…netheFisher-typeteststatistic:P=2NXi=1lnpi22N:NoticethatthisstatementdoesnotrequireN!
1tobesatis…ed,sothisteststatisticisofapplianceforpanelswithsmallcross-sectiondimension.
Besides,Choi(2001)proposesthefollowingtestwhenN!
1:Pm=2PNi=1lnpi2Np4NN(0;1);wherethestandardnormallimitdistributionisobtainedfromtheapplicationoftheLindberg-LévyCLT.
Asaresult,thePmtestissuitableforthosepanelswithlargeN.
Thisspeci…cationwaschoseninBaiandNg(2004)totestthenullhypothesisofnon-stationarypanelusingtheDFtest.
Whilethemainadvantageofthep-valuespoolingstrategycomesfromthefactthatthede…nitionofthetestcanbeadaptedtothecross-sectiondimension,itsmaindrawbackreliesontheavailabilityofthep-values.
TheyareprovidedbytheresponsesurfacesestimatedinSection2.
5FinitesampleperformanceWeanalysetheperformanceofthepaneldataunitroottestintwodi¤erentsituations.
Firstweconsiderthecaseinwhichtherearenocommonfactors,thatis,westudythepropertiesofthetestassumingthattheindividualsarecross-sectionindependent.
Afterthat,wewillfocusonthosepanelswherethecross-sectiondependenceisdrivenbythepresenceofuptothreecommonfactors.
Inallthesesimulationsweassumethatthedateofthebreaksareknown.
ThreevaluesforthenumberofindividualsN=f20;40;100ghavebeenconsidered,withasamplesizeequaltoT=100.
Thenumberofreplicationsisr=5;000.
TheDGPisgivenbyequations(1)to(3)withiU[0;1],iU[0:2;0:5],i;kU[10;3]and°i;kU[0:3;0:9],whereU[]denotestheUniformdistri-bution.
WehaveallowedonestructuralbreakrandomlypositionedaccordingtoiU[0:15;0:85].
Underthenullhypothesisei;tI(1)havebeengeneratedasarandomwalkwithoutdriftde…nedbythecumulatedsumofiidN(0;1)processes.
Thecommonfactorsarede…nedfollowingtheAR(1)model:Fj=Fj1+Fut;where=f0:5;0:8;0:9;0:95gand2F=f0:5;1;10g,j=1;l,withthefactorloadingsgivenbyjN(1;1).
Thesimulationshavespeci…edl=1andl=3commonfactors.
Thenumberofcommonfactorsare…xedusingthepanelBICinformationcriterioninBaiandNg(2002)withlmax=6asthemaximumnumberoffactors.
Table4reportsthesamplesizeofthethreedi¤erentstatisticswhentherearenocommonfactors.
Thetestbasedonthestandarisationpresentasize15Table4:Empiricalsize.
KnownbreaksandnocommonfactorsModel1Simpli…edtestNZPmPZPmP200.
5400.
0510.
0420.
4750.
0580.
048400.
8600.
0510.
0450.
8080.
0590.
0531000.
9970.
0440.
0400.
9930.
0580.
053Model2Simpli…edtestNZPmPZPmP200.
1320.
0660.
0580.
1500.
0610.
050400.
2710.
0730.
0620.
2650.
0590.
0511000.
5680.
0750.
0660.
5480.
0530.
051distortionthatincreseswiththenumberofindividuals.
ThisisinaccordancewithBaiandNg(2004),whereitismentionedthatpoolinginthiswaycanleadtounsatisfactoryresultsspeciallywhentheasymptoticdistributionoftheindividualtestsisskewed,asthisisthecase.
Onthecontrary,thetestsbasedonthecombinationoftheindividualp-valuesshowhaveanempiricalsizeclosetothenominalone.
Notethatthisisalsotrueforthesimpli…edtest,whichindicatestheusefulnessofourproposalinappliedresearch.
Thepicturechangeswhenweanalysethepaneldatasetthatallowsforcommonfactors.
ForModel1allthreeteststatisticsshowgoodperformanceintermsofempiricalsize.
TheexceptionisthePtest,whichinsomesituationspresentsempiricalsizedistortionsthatleadtounderrejectthenullhypothesis–seeTable5.
ForModel2thePmtestistheonewiththemoststableempiricalsize,providedthattheZandPtestsunderrejectthenullhypothesis.
Thisisalsotrueforalltheversionofthesimpli…edtests–seeTable6.
6ConclusionsInthispaper,wehaveproposednewproceduresfortestingnon-stationarityofpaneldatainthepresenceofmultiplestructuralbreaksanddynamiccommonfactors.
Intheabsenceofcommonfactors,thelimitingdistributionsareshowntobeweightedsumofindependentandidenticallydistributedBrownianmotions(demeanedordetrended).
Theseresultsareofspecialinterestforthesingletimeseriesanalysis–i.
e.
panelswithN=1individual–providedthattheyextendtheproposalsinPerron(1997)andLumsdaineandPapell(1997),amongothers,andallowtotesttheunitroothypothesiswithmultiplestructuralchanges.
Whendynamicfactorsarepresent,thePANICapproachofBaiandNg(2004)isusedtoestimatethemodel.
Thelimitingdistributionsoftheteststatisticsareinvarianttomeanbreaks.
Forbreaksinthelineartrend,thelimitingdistributinosare16Table5:Empiricalsizeformodel1andN=40.
Knownbreakswithcommonfactorsr=1r=32FZPmPZPmP0.
500.
040.
040.
040.
040.
030.
030.
50.
50.
050.
040.
030.
040.
030.
030.
50.
80.
050.
050.
040.
040.
040.
040.
50.
90.
050.
050.
040.
050.
040.
040.
50.
950.
040.
050.
040.
050.
050.
04100.
040.
050.
040.
040.
040.
0310.
50.
040.
040.
040.
030.
040.
0310.
80.
050.
040.
040.
040.
050.
0410.
90.
050.
050.
040.
050.
040.
0310.
950.
040.
040.
040.
040.
040.
031000.
060.
050.
040.
030.
030.
03100.
50.
050.
040.
040.
040.
040.
03100.
80.
060.
050.
040.
040.
040.
04100.
90.
050.
050.
040.
040.
040.
04100.
950.
050.
050.
040.
050.
050.
04showntobeweightedsumofiidBrownianbridges.
Wefurtherintroducedasimpli…edteststatistic,andshowedthatthelimitingdistributionisinvarianttobothmeanandtrendbreaks.
Pooledteststatisticisalsostudied.
Responsesurfacesforp-valuesofallteststatisticsarecomputed.
17Table6:Empiricalsizeformodel2andN=40.
Knownbreakswithcommonfactorsr=1Simpli…edtests2FZPmPZPmP0.
500.
020.
050.
040.
010.
030.
030.
50.
50.
020.
060.
060.
020.
040.
030.
50.
80.
030.
070.
060.
020.
040.
030.
50.
90.
030.
060.
050.
020.
030.
030.
50.
950.
020.
060.
060.
010.
040.
03100.
030.
080.
070.
020.
050.
0510.
50.
020.
060.
050.
020.
030.
0310.
80.
040.
050.
040.
020.
030.
0310.
90.
020.
060.
050.
020.
030.
0310.
950.
020.
060.
050.
020.
030.
031000.
030.
060.
060.
020.
040.
03100.
50.
030.
060.
060.
020.
040.
03100.
80.
020.
060.
050.
020.
030.
03100.
90.
030.
060.
050.
020.
030.
03100.
950.
030.
070.
060.
010.
040.
03r=3Simpli…edtests2FZPmPZPmP0.
500.
020.
050.
040.
020.
020.
020.
50.
50.
020.
040.
040.
010.
020.
020.
50.
80.
030.
050.
050.
020.
030.
020.
50.
90.
030.
050.
050.
010.
030.
030.
50.
950.
030.
060.
050.
020.
040.
03100.
010.
040.
030.
010.
020.
0210.
50.
020.
040.
040.
010.
020.
0210.
80.
020.
050.
040.
020.
030.
0310.
90.
020.
060.
060.
010.
030.
0310.
950.
020.
050.
050.
010.
040.
031000.
020.
050.
040.
010.
020.
02100.
50.
020.
050.
040.
010.
030.
02100.
80.
030.
060.
050.
020.
020.
02100.
90.
020.
050.
050.
010.
030.
03100.
950.
030.
080.
070.
020.
050.
05187Appendix:ProofofTheorem17.
1Proofofstatement(1)Themodelthatisconsideredinthisstatementistheonefornon-trendedvari-ableswherei=°i;k=0,8i;kin(4).
Inaddition,theconstraintini=08iisimposedinordertoavoidthepresenceofcommonfactorsthatdrivethebehaviouroftheindividualtimeseries.
Fromthisspeci…cation,theestimatedOLSresidualsofthemodelareob-tainedfrom~ei=Miei,withMi=ITai(a0iai)1a0i.
Notethatformistructuralchangesthedeterministicpartofthemodelgivenin(4)canbeexpressedintermsoforthogonalregressorsde…ningablockdiagonalmatrix.
Theelementsinthediagonalaregivenbyvectorsk=(1;1)0ofdimensionTib;kTib;k11,k=1;mi+1,withTib;0=0andTib;mi+1=T.
Thus,thecross-productmatrixofregressorsa0iaiisgivenbya0iai=2666664Tib;10Tib;2Tib;1.
.
.
Tib;miTib;mi10TTib;mi3777775;UsingthefactthatTib;k=i;kTandde…ningthe(mimi)-diagonalrescalingmatrixPi=diagT1=2T1=2,P0ia0iaiPicanbeexpressedasP0ia0iaiPi=diag(i;1;(i;2i;1)i;mii;mi1);(1i;mi)).
Ontheotherhand,underthenullhypothesisthateiI(1),T1P0ia0iei)iRi;10Wi(s)ds;iRi;2i;1Wi(s)ds;iRi;mii;mi1Wi(s)ds;iR1i;miWi(s)ds0.
Thismeansthatfort·Tib;1T1=2~ei;t)iWi(r)i1i;1Zi;10Wi(s)ds;0kand0elsewhere,withk=Tib;k=T,k=1;mi.
ThelimitexpressionofT1=2~ei;tgivenby(15)involvestwodi¤erentkindofelements:(i)theBrownianmotion,Wi(r),and(ii)thedi¤erenceofBrownianmotions,dWi(k),k=1;mi.
FollowingPerron(1997),thee¤ectofthesedi¤erencescanbeunderstoodasneg-ligiblecomparedtoWi(r),sothat,wecanconsiderthatT1=2~ei;t)iWi(r).
Therefore,theteststatisticconvergesto:MSBi(i))Z10W2i(r)dr;providedthat~2i!
2i.
Noticethatafterconsideringthenegligiblee¤ectofthedWi(k)terms,k=1;mi,theasymptoticdistributionofthetestdoesnotdependonthebreakfractionparametersk,thatis,thetestisinvarianttothepresenceofstructuralbreaksa¤ectingthemeanofthetimeseries.
8.
2Proofofstatement(2)Letusnowfocusonthespeci…cationgivenbyModel2,thatis,themodelfortrendedregressorswherei6=°i;k6=0,8i;kin(4).
Asinthepreviousproof,thecomputationofthepartialsumprocesscanbedonefrom(13).
However,wehavetoassessthatT1=2°°°Pts=2^fsdi°°°=op(1).
NotethatT1=2°°°Pts=2^fsdi°°°·T1=2°°°Pts=2^fs°°°kdik.
FromBaiandNg(2003),kdik=op(1),andT1=2tXs=2^fs=T1=2tXs=2^fsHfs+Hfs=T1=2tXs=2vs+HT1=2tXs=2fs=op(1)+HT1=2tXs=2fs:TodeterminetheorderinprobabilityofT1=2Pts=2fswerewritethematrixofdeterministicelementsaiina(T(2mi+1))quasiblockdiagonalmatrix:ai=DU1iDTib;1DU2iDTib;2DU3i:::DUmiiDTib;miDUmi+1i¤;whereDUki=1forTib;k1Tib:Thecomputationofthepartialsumprocessinvolves:T1=2tXs=21TDU1DU10sFs=8Tib;whichisOp(1).
Thesameresultisfoundfortheproductinvolvingthe…fthelementof(16).
Thesecondelementof(16)gives:(1+)T1T+TDTibDTib0F=((1+)T1T+TFTib+1t=Tib+10t6=Tib+1;sothatthepartialsumprocessisT1=2tXs=2(1+)T1T+TDTibDTib0sFs=(0t·Tib(1+)T1=21T+TFTib+1t>Tib;withOp(1)asorderinprobability.
Forthethirdelementwehave11T+TDTibDU20F=FTFTib+1t=Tib+10t6=Tib+1;sothatT1=2tXs=211T+TDTibDU20sFs=8Tib;25whichisalsoOp(1).
Thefourthelementis11T+TDU2DTib0F=0t·Tib11T+TFTib+1t>Tib:Thus,T1=2tXs=211T+TDU2DTib0sFs=(0t·TibFTib+1(tTib)(1T+T)T1=2t>Tib;whichisOp(1).
Finally,the…fthelement11T+TDU2DU20F=(0t·Tib11T+TFTFTib+1t>Tib;withcumulatedsumT1=2tXs=211T+TDU2DU20sFs=8Tib;whichisalsoOp(1).
Therefore,allthepartialsumprocessesinvolvingPiFareOp(1),aresultthatcanbestraightforwardlyextendedtothosesituationsthatallowformultiplebreaks.
Consequently,T1=2°°°°°tXs=2^fsdi°°°°°·T1=2°°°°°tXs=2^fs°°°°°kdik·Op(1)op(1);whichmeansthatT1=2°°°Pts=2^fsdi°°°=op(1).
Asinthepreviousproof,thepartialsumprocessoftheestimatedresid-ualsisgivenby(14).
Now,thecumulativeprocessaregivenbytheprevi-ousexpressionsbutreplacingFbyei.
The…rstelementofthepartialsumprocess,whichinvolvesthe…rstsetofstepdummyvariables,convergestoT1=2Pts=21TDU1DU10sei;s)(r=)Wi(i)fort·TibandT1=2Pts=21TDU1DU10sei;s)Wi(i)fort>Tib.
Thesecondele-mentproduces(1+)T1=21T+Tei;Tib+1)dWi(i),anelementthatvanishasymp-totically.
Thethirdelementisop(1),whereasthefourthelement,whichin-volvesthesecondsetofstepdummyvariables,11T+Tei;Tib+1tTib)ri1idWi(i),anotherelementthatvanishasymptotically.
Finally,the…fthelementisT1=2Pts=211T+TDU2DU20sei;s)ri1i(Wi(1)Wi(i))fort>Tiband0elsewhere.
26Therefore,T1=2~ei;t)iWi(r)i(r=i)Wi(i)forr·iandT1=2~ei;t)iWi(r)iWi(i)iri1i(Wi(1)Wi(i))forr>i,andtheMSBi(i)teststatisticconvergesto:T2PTt=1~e2i;t1~2i=T2PTibt=1~e2i;t1~2i+T2PTt=Tib+1~e2i;t1~2i)Zi0[Wi(r)(r=i)Wi(i)]2dr+Z1i·Wi(r)Wi(i)ri1i(Wi(1)Wi(i))2dr;providedthat~2i!
2i.
However,thelimitdistributionofMSBi(i)canbeexpressedasthesumoftwoindependentintegrals.
Letusde…neb=r=i;1sothat0UsingthepropertiesoftheBrownianmotionsthelimitdistri-butioncanbewrittenintermsofbasWi(r)(r=i)Wi(i)=piWi;1(b)bpiWi;1(1)=pi(Wi;1(b)bWi;1(1)),sothatZi0[Wi(r)(r=i)Wi(i)]2dr=2iZ10[Wi;1(b)bWi;1(1)]2db=2iZ10V2i;1(b)db;whereVi;1(b)denotesthedemeanedBrownianmotion.
Forthesecondinte-gral,letusnowde…neb=(ri)=(1i),sothat0Now,thelimitdistributioncanbereexpressedintermsofbasWi(r)Wi(i)ri1i(Wi(1)Wi(i))=Wi;2(b)bWi;2(1),whichimpliesthatZ1i·Wi(r)Wi(i)ri1i(Wi(1)Wi(i))2dr=(1i)2Z10V2i;2(b)db;whereVi;2(b)denotesthedemeanedBrownianmotion.
Therefore,theasymp-toticdistributionofthetestwhenmi=1isgivenbyMSBi(i))2iZ10V2i;1(b)db+(1i)2Z10V2i;2(b)db;(17)whereVi;1(b)andVi;2(b)aretwoindependentBrownianbridges.
Notealsothesymmetryoftheasymptoticdistributionaroundi=0:5.
Asshownabove,wecaninterchangeiand(1i)in(17)andobtainthesameasymptoticdistribution.
Ingeneral,fork=1;mi+1wehaveTbi;k1Letusnowde-…neb=(ri;k1)=(i;ki;k1)sothat0Asbefore,the27limitdistributionofthepartialsumprocessesisgivenby~1iT1=2~ei;t)pi;ki;k1(Wi;k(b)bWi;k(1)),andtheteststatisticMSBi(i)=~2iT2PTt=1~e2i;t1=~2iT2·PTib;1t=1~e2i;t1PTib;kt=Tib;k1+1~e2i;t1++PTt=Tib;mi+1~e2i;t1iwithlimitdistributiongivenby:MSBi)2i;1Z10V2i;1(b)dbi;ki;k1)2Z10V2i;k(b)db1i;mi)2Z10V2i;mi+1(b)db;whereVi;k(),k=1;mi+1,denotesthedemeanedBrownianmotionandprovidedthat~2i!
2i–seebelowtheproofoftheconsistencyofthenon-parametriclong-runvarianceestimation.
ThelimitdistributionofMSBi(i)istheweightedsumof(mi+1)independentCramér-vonMisesdistributions.
TheexpectationsoftheseCramér-vonMisesdistributionsareEhR10V2i;k(b)dbi=1=6wherethevarianceareVhR10V2i;k(b)dbi=1=45,8k=1;mi+1-seeLevinandLin(1992).
Therefore,E[MSBi(i)]=(1=6)Pmi+1k=1(i;ki;k1)2andV[MSBi(i)]=(1=45)Pmi+1k=1(i;ki;k1)4.
8.
3Proofoftheconsistencyofthelong-runvariancees-timationLetusde…netheAR(1)regressionontheestimatedidiosyncraticresiduals:~ei;t=bi~ei;t1+i;t;(18)whichunderthenullhypothesisofunitrootimpliesthat~bi1=Op(1=T).
From(18)wecanexpresstheerrortermas:~i;t=~ei;t+1~bi~ei;t1;wherefrom(13)itfollowsthat~i;t=zi;t+vtH1i^ftdi+1~bi~ei;t1=zi;t+wi;t;withzi;t=(Miei)tandwi;t=vtH1i^ftdi+1~bi~ei;t1.
Forarbitrarytimeseriesatandbtde…ne:dNWab=1TTXt=1atbt+JXj=1K(j)"1TTjXt=1(atbt+j+at+jbt)#;28withK(j)=1j=(J+1).
ThendNWziziistheNewey-Westestimatorofthelong-runvarianceofzi=Miei.
InordertoprooftheconsistencyofthisestimatorweneedtoshowthatdNW~i~idNWzizi=op(1):From~i;t=zi;t+wi;twehavethatdNW~i~i=dNWzizi+2dNWziwi+dNWwiwi:WenextshowthatifJ!
1andJ=±NT!
0,±NT=min[N;T],thendNWziwi=op(1)anddNWwiwi=op(1):First,noticethatdNWziwi·1TTXt=1z2i;t!
1=21TTXt=1w2i;t!
1=2+JXj=1K(j)241TTjXt=1z2i;t!
1=21TTjXt=1w2i;t+j!
1=2+1TTjXt=1z2i;t+j!
1=21TTjXt=1w2i;t!
1=235:Noticethat1TPTt=1z2i;t=Op(1).
Ontheotherhand,jwi;tj2·4kvtk2°°H1i°°24°°°^ft°°°2kdik2+41~bi2~e2i;t1;sothat1TTXt=1jwi;tj2·41TTXt=1kvtk2°°H1i°°241TTXt=1°°°^ft°°°2kdik2+41~bi21TTXt=1^e¤2i;t1:FromLemmas1(a)and1(c)inBaiandNg(2003),1TPTt=1kvtk2=Op±2NTandkdik2=Op±2NTrespectively,and1~bi21TPTt=1~e2i;t1=T1~bi21T2PTt=1~e2i;t1=Op(1=T).
Therefore,1TPTt=1jwi;tj2·Op±2NT.
Thesein-termediateresultsenableustoestablishthatdNWziwi·(J+1)Op±1NT!
0:Moreover,since1TPTt=1jwi;tj2·Op±2NTthendNWwiwi·(J+1)Op±2NT!
0:Thus,wehaveshownthatthelong-runvariancecanbeconsistentlyestimatedthroughtheapplicationofthenon-parametricNewey-Westestimationproce-dure,thatis,weproposetouse~2i=dNW~i~i.
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