COWLES FOUNDATION FOR RESEARCH IN ECONOMICS New Haven, Connecticut 065 - PDF document

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS New Haven, Connecticut 06520-8281 http://cowles.econ.yale.edu/ INTERNATIONALECONOMICREVIEWVol.52,No.1,February2011EXPLOSIVEBEHAVIORINTHE1990sNASDAQ:WHENDIDEXUBERANCEESCALATEASSETVALUES?C.B.P ANDYU 3/77 5/81 7/85 9/89 11/93 1/98 2/02 0 200 400 600 800 1000 1200 Period Real Nasdaq Dividends TIMESERIESPLOTSOFREALNASDAQPRICEANDREALNASDAQDIVIDENDFROMFEBRUARYTOJUNEBOTHSERIESARENORMALIZEDTOATTHEBEGINNINGOFTHESAMPLEManyresearchersattributetheepisodetoÞnancialbubbles.ExamplesincludeGreenspan(1996),Thaler(1999),Shiller(2000),TheEconomist(2000),Cooperetal.(2001),RitterandWelch(2002),OfekandRichardson(2002),LamontandThaler(2003),andCunadoetal.Amongthemanyreferences,theremarkbyGreenspan(1996)onDecember5,1996,isthemostcelebrated,involvingasitdidthecoiningofthephraseÒirrationalexuberanceÓtocharacterizeherdstockmarketbehavior,aphrasethatremainsthemostoft-quotedremarkoftheformerchairmanoftheFederalReserveBoard.TheremarkhasbeeninßuentialinthinkingaboutÞnancialmarketsandherdbehavior,anditalsohadsomeshort-termmarketeffects.Indeed,immediatelyafterGreenspancoinedthephraseinadinnerpartyspeech,stockmarketsfellsharplyworldwidethenextday.However,inspiteofthiscorrection,theGreenspanremarkdidnothaltthegeneralupwardmarchoftheUnitedStatesmarket.Onthecontrary,overthefulldecadeofthe1990s,theNasdaqindexrosetothehistoricalhighof5,048.62pointsonMarch10,2000from329.80onOctober31,1990(seeFigure1).OnepurposeofthepresentarticleistoexamineempiricallytheNasdaqmarketperformanceinrelationtothemarketperceptionsofexuberancebyGreenspanandothercommentators.Inparticular,itisofinteresttodeterminewhethertheGreenspanperceptionofexuberancewassupportedbyempiricalevidenceinthedataorifGreenspanactuallyforesawtheoutbreakofexuberanceanditsdangerswhenhemadetheremark.Toachievethisgoal,weÞrstdeÞneÞnancialexuberanceinthetimeseriescontextintermsofexplosiveautoregressivebehaviorandthenintroducesomeneweconometricmethodologybasedonforwardrecursiveregressiontestsandmildlyexplosiveregressionasymptoticstoassesstheempiricalevidenceofexuberantbehaviorintheNasdaqstockmarketindex.Inthiscontext,theapproachiscompatiblewithseveraldifferentexplanationsofthisperiodofmarketactivity,includingtherationalbubbleliterature,herdbehavior,andexuberantandrationalresponsestoeconomicfundamentals.Allthesepropagatingmechanismscanleadtoexplosivecharacteristicsinthedata.Hence,theempiricalissuebecomesoneofidentifyingtheorigination,termination,andextentofSomeeconomistshavealsosoughttorationalizetheequityboomusingavarietyofeconomicvariables,includinguncertaintyaboutÞrmproÞtability(PastorandVeronesi,2006),decliningmacroeconomicrisk(Lettauetal.,2008),highandvolatilerevenuegrowth(SchwartzandMoon,2000),learning(PastorandVeronesi,2009),andotherfundamentals.Forexample,thestockmarketsinFrankfurt,HongKong,London,Toyko,andtheUnitedStatesfellby4%,3%,4%,3%,and2%,respectively. EXPLOSIVEBEHAVIORINTHESNASDAQtheexplosivebehavior.AlthoughwithtraditionaltestproceduresÒthereislittleevidenceofexplosivebehaviorÓ(Campbelletal.,1997,p.260),withtherecursiveprocedure,wesuccessfullydocumentexplosiveperiodsofpriceexuberanceintheNasdaq.Amongthepotentialexplanationsofexplosivebehaviorineconomicvariables,themostprominentareperhapsmodelswithrationalbubbles.Accordingly,werelateouranalysisofexplosivebehaviortotherationalbubbleliterature,whereitiswellknownthatstandardecono-metrictestsencounterdifÞcultiesinidentifyingrationalassetbubbles(FloodandGarber,1980;FloodandHodrick,1986;Evans,1991).Theuseofrecursivetestsenablesustolocateexplod-ingsubsamplesofdataanddetectperiodsofexuberance.TheeconometricapproachutilizessomenewmachinerythatpermitstheconstructionofvalidasymptoticconÞdenceintervalsforexplosiveautoregressiveprocessesandtestsofexplosivecharacteristicsintimeseriesdata.Thisapproachcandetectthepresenceofexuberanceinthedataanddatestamptheoriginationandcollapseofperiodsofexuberance.WeapplyoureconometricapproachtotheNasdaqindexoverthefullsampleperiodfrom1973to2005andsomesubperiods.Usingtheforwardrecursiveregressiontechnique,wedatestamptheoriginandconclusionoftheexplosivebehavior.ToanswerthequestionraisedbyGreenspanintheÞrstepigraphinthisarticle,wematchtheempiricaltimestampoftheoriginationagainstthedatingofGreenspanÕsremark.Thestatisticalevidencefromthesemeth-odsindicatesthatexplosivenessstartedin1995,therebypredatingandprovidingempiricalcontenttotheGreenspanremarkinDecember1996.TheempiricalevidenceindicatesthattheexplosiveenvironmentcontinueduntilsometimebetweenSeptember2000andMarchIfthediscountrateistimeinvariant,theidentiÞcationofexplosivecharacteristicsinthedataisequivalenttothedetectionofastockbubble,asarguedinDibaandGrossman(1987,1988).UsingstandardunitroottestsappliedtorealU.S.StandardandPoorÕsCompositeStockPriceIndexdataovertheperiod1871Ð1986,DibaandGrossman(1988)testedlevelsanddif-ferencesofstockpricesfornonstationarity,Þndingsupportinthedatafornonstationarityinlevelsbutstationarityindifferences.Sincedifferencesofanexplosiveprocessstillmanifestexplosivecharacteristics,theseÞndingsappeartorejectthepresenceofamarketbubbleinthedata.AlthoughtheresultswerelessdeÞnitive,furthertestsbyDibaandGrossman(1988)providedconÞrmationofcointegrationbetweenstockpricesanddividendsoverthesamepe-riod,supportingtheconclusionthatpricesdidnotdivergefromlong-runfundamentalsandtherebygivingadditionalevidenceagainstbubblebehavior.Evans(1991)criticizedthisap-proach,showingthattimeseriessimulatedfromanonlinearmodelthatproducesperiodicallycollapsingbubblesmanifestsmorecomplexbubblecharacteristicsthataretypicallynotuncov-erablebystandardunitrootandcointegrationtests.Heconcludedthatstandardunitrootandcointegrationtestsareinappropriatetoolsfordetectingbubblebehaviorbecausetheycannoteffectivelydistinguishbetweenastationaryprocessandaperiodicallycollapsingbubblemodel.Patternsofperiodicallycollapsingbubblesinthedatalookmorelikedatageneratedfromaunitrootorstationaryautoregressionthanapotentiallyexplosiveprocess.Recursivetestsofthetypeundertakeninourarticlearenotsubjecttothesamecriticismand,asdemonstratedinouranalysisandsimulationsreportedbelow,arecapableofdistinguishingperiodicallycollapsingbubblesfrompureunitrootprocesses.Theremainderofthearticleisorganizedasfollows.Section2deÞnesmarketexuberance,discussesmodelspeciÞcationissues,andrelatesexuberancetotheearlierliteratureonrationalbubbles.Section3discussessomeeconometricissues,suchasÞnitesampleestimationbiasandtheconstructionofvalidasymptoticconÞdenceintervalsformildlyexplosiveprocesses.Section4describesthedatausedinthisstudy.TheempiricalresultsarereportedinSection5.Section6documentstheÞnitesamplepropertiesofourtestsanddevelopssomeasymptoticpropertiesoftheEvans(1991)modelofperiodicallycollapsingbubbles.SimulationswiththesemodelsareconductedandtheÞnitesamplepropertiesofthetestsareanalyzed.Section7concludes.TheAppendixprovidesamathematicalanalysisoftheperiodicallycollapsingbubblemodelof ANDYU 20 40 60 80 100 0 5 Simulated AR(1) with 20 40 60 80 100 0 10 Simulated AR(1) with 20 40 60 80 100 0 50 100 Simulated AR(1) with Time TYPICALSTATIONARYRANDOMWALKANDEXPLOSIVEAUTOREGRESSIVETRAJECTORIESSPECIFICATIONISSUESWhenGreenspancoinedÒirrationalexuberance,ÓthephrasewasnotdeÞnedÑseetheÞrstepigraphinthisarticle.Instead,theappellationcanbeinterpretedasatypicallycrypticwarningthatthemarketmightbeovervaluedandinriskofaÞnancialbubble.Intheevent,asthesecondepigraphinthisarticleindicates,thesubsequentriseandfallofinternetstockstotheextentof$8trillionofshareholderwealthrenewedalong-standinginterestamongeconomistsinthepossibilityofÞnancialbubbles.TheoreticalstudiesonrationalbubblesinthestockmarketincludeBlanchard(1979),BlanchardandWatson(1982),Shiller(1984),Tirole(1982,1985),Evans(1989),EvansandHonkapohja(1992),andOlivier(2000),amongmanyothers,andempiricalstudiesincludeShiller(1981),West(1987,1988),CampbellandShiller(1987,1989),DibaandGrossman(1988),FrootandObstfeld(1991),andWu(1997).FloodandHodrick(1990)andGurkaynak(2005)surveyexistingeconometricmethodologiesandtestresultsforÞnancialbubbles.Itiswellknownintherationalbubbleliteraturethatbubbles,iftheyarepresent,shouldmanifestexplosivecharacteristicsinprices.ThisstatisticalpropertymotivatesadeÞnitionofexuberanceintermsofexplosiveautoregressivebehaviorpropagatedbyaprocessoftheformwhereforcertainsubperiodsofthedata1.Figure2givestypicaltimeseriesplotsforstationary(9),randomwalk,andexplosiveprocesses(withintercept0andinputs.ThedifferencesinthetrajectoriesarequiteTheconceptofrationalbubblescanbeillustratedusingthepresentvaluetheoryofÞnancewherebyfundamentalassetpricesaredeterminedbythesumofthepresentdiscountedvaluesofexpectedfuturedividendsequence.Mosttestsbeginwiththestandardnoarbitragecondition istherealstockprice(ex-dividend)attimeistherealdividendreceivedfromtheassetforownershipbetween1and,andisthediscountrate(0).Thissectiontobetimeinvariant.However,makingthediscountratestationaryandtime-varyingdoesnotchangetheimplicationofsubmartingale(explosive)behaviorgivenin(4)below,butcomplicatestheanalysisoftherationalbubblesolution. EXPLOSIVEBEHAVIORINTHESNASDAQWefollowCampbellandShiller(1989)bytakingaapproximationof(1),whichyieldsthefollowingsolutionthroughrecursivesubstitution: 1Š+)i=0iEtdt+1+i,(3)bt=iEtpt+i,Et(bt+1)=1 bt= Špt,(4)t=t),dt=t)R)=1/ )),with beingtheaveragelogdividendÐpriceratio,and)log Obviously,01.Followingconvention,wecall,whichisexclusivelydeterminedbyexpecteddividends,thefundamentalcomponentofthestockprice,and,whichsatisÞesthedifferenceequation(5)below,therationalbubblecomponent.Bothcomponentsareexpressedinnaturallogarithms.Asexp( 0,therationalbubbleisasubmartingaleandisexplosiveinexpectation.Equation(4)impliesthefollowingprocess: btŠ1+b,tg)btŠ1+b,t,EtŠ1(b,t)=0,(5)=1 Š1= 0isthegrowthrateofthenaturallogarithmofthebubbleisamartingaledifference.Asevidentfrom(2),thestochasticpropertiesofaredeterminedbythoseof.Intheabsenceofbubbles,i.e.,,wewillhave,andisdeterminedsolelybyandhenceby.Inthiscase,from(3),weobtain arebothintegratedprocessesoforderone,denotedbyI(1),then(6)impliesthatarecointegratedwiththecointegratingvector[1Ifbubblesarepresent,i.e.,0,since(5)impliesexplosivebehaviorinwillalsobeexplosivebyEquation(2),irrespectiveofwhetherisanintegratedprocess,I(1),orastationaryprocess,denotedbyI(0).Inthiscase,isalsoexplosiveandthereforecannotbeAlthoughlog-linearapproximationsofthistypeaboutthesamplemeanarecommonlyemployedinboththeoreticalandempiricalwork,weremarkthattheymaybelesssatisfactoryinnonstationarycontextswherethesamplemeansdonotconvergetopopulationconstants.Wethereforeusedtheseriesbothinloglevelsandinlevelsinourempiricalworkandfoundverysimilarresultsforbothcases. ANDYUstationary.ThisimplicationmotivatedDibaandGrossman(1988)tolookforthepresenceofbubblebehaviorbyapplyingunitrootteststo.Findinganempiricalrejectionofthenullofaunitrootin,DibaandGrossman(1988)concludedthatwasnotexplosiveandthereforetherewasnobubbleinthestockmarket.InthecasewhereisI(1)andhenceisI(0),Equation(6)motivatedDibaandGross-man(1988)tolookforevidenceoftheabsenceofbubblesbytestingforacointegratingrelationbetween.Inthepresenceofbubbles,isalwaysexplosiveandhencecan-notco-moveorbecointegratedwithisitselfnotexplosive.Therefore,anempiricalÞndingofcointegrationbetweenmaybetakenasevidenceagainstthepresenceofEvans(1991)questionedthevalidityoftheempiricaltestsemployedbyDibaandGrossman(1988)byarguingthatnoneofthesetestshavemuchpowertodetectperiodicallycollapsingbubbles.Hedemonstratedbysimulationthatthelowpowerofstandardunitrootandcoin-tegrationtestsinthiscontextisduetothefactthataperiodicallycollapsingbubbleprocesscanbehavemuchlikeanI(1)processorevenlikeastationarylinearautoregressiveprocessprovidedthattheprobabilityofcollapseofthebubbleisnotnegligible.Asaresult,Evans(1991,p.927)claimedthatÒperiodicallycollapsingbubblesarenotdetectablebyusingstandardtests.ÓEquations(5)and(2)suggestthatadirectwaytotestforbubblesistoexamineevidenceforexplosivebehaviorinwhenthediscountrateistimeinvariant.Ofcourse,explosivecharacteristicsincouldinprinciplearisefromandthetwoprocesseswouldthenbeexplosivelycointegrated.However,ifisdemonstratedtobenonexplosive,thentheexplosivebehaviorinwillprovidesufÞcientevidenceforthepresenceofbubblesbecausetheobservedbehaviormayonlyarisethroughthepresenceof.Ofcourse,itseemslikelythatinpracticeexplosivebehaviorinmayonlybetemporaryorshort-lived,asinthecaseofstockmarketbubblesthatcollapseafteracertainperiodoftime.Someofthesepossibilitiescanbetakenintoaccountempiricallybylookingatsubsamplesofthedata.Lookingdirectlyforexplosivebehaviorinandnonexplosivebehaviorinviaright-tailedunitroottestsisoneaspectoftheempiricalmethodologyofthisarticle.Althoughthisapproachisstraightforward,ithasreceivedlittleattentionintheliterature.OnepossibleexplanationistheconsensusviewthatÒempiricallythereislittleevidenceofexplosivebehaviorÓinstockprices,asnotedinCampbelletal.(1997,p.260)forinstance.However,asEvans(1991)noted,explosivebehaviorisonlytemporarywheneconomicbubblesperiodicallycollapse,andinsuchcasestheobservedtrajectoriesmayappearmorelikeanI(1)orevenstationaryseriesthananexplosiveseries,therebyconfoundingempiricalevidence.HedemonstratedbysimulationthatstandardunitroottestshaddifÞcultiesindetectingsuchperiodicallycollapsingbubbles.Inorderforunitroottestprocedurestobepowerfulindetectingexplosiveness,weproposetheuseofrecursiveregressiontechniquesandshowbelowbyanalyticmethodsandsimulationsthatthisapproachiseffectiveindetectingperiodicallycollapsingbubbles.Usingthesemethods,thepresentarticleÞndsthatwhenrecursivetestsareconductedanddatafromthe1990sareincludedinthesample,somestrongevidenceofexplosivecharacteristicsinOurtestsareimplementedasfollows.Foreachtimeseries(logstockpriceorlogdividend),weapplytheaugmentedDickeyÐFuller()testforaunitrootagainstthealternativeofanexplosiveroot(theright-tailed).Thatis,weestimatethefollowingautoregressivespeciÞcationbyleastsquaresNID(0WealsoimplementedthePhillips(1987)andPhillipsandPerron(1988)testsandobtainedresultsverysimilartotheADFtest. EXPLOSIVEBEHAVIORINTHESNASDAQforsomegivenvalueofthelagparameter,whereNIDdenotesindependentandnormalInourempiricalapplicationweusesigniÞcanceteststodeterminethelagorder,assuggestedinCampbellandPerron(1991).Theunitrootnullhypothesisis1andtheright-tailedalternativehypothesisisInforwardrecursiveregressions,model(7)isestimatedrepeatedly,usingsubsetsofthesampledataincrementedbyoneobservationateachpass.IftheÞrstregressioninvolvesinvolvesnr0]observations,forsomefractionofthetotalsamplewherewheresigniÞestheintegerpartofitsargument,subsequentregressionsemploythisoriginatingdatasetsupplementedbysuccessiveobservationsgivingasampleofsizeizenr]forr0 r 1.DenotethecorrespondingADFandhencecorrespondstothefullsample.UnderthenullwehaveADFWdW r0,1]ADFFr0,1]r0WdW isthestandardBrownianmotionand isdemeanedBrownianComparisonofsupADFwiththeright-tailedcriticalvaluesfromfromr0,1]r0WdWmakesitpossibletotestforaunitrootagainstexplosive-ness.However,thistestingprocedurecannotdatestamptheemergenceorcollapseofexuberance.Tolocatetheoriginandtheconclusionofexuberance,onecanmatchthetimeseriesoftherecursiveteststatisticADF,withhr0,1],againsttheright-tailedcriticalvaluesoftheasymptoticdistributionofthestandardDickeyÐFuller-statistic.Inparticular,iftheoriginationdateandisthecollapsedateofexplosivebehaviorinthedata,weconstructestimatesofthesedatesasfollows:ADFADFistheright-sidecriticalvalueofADFcorrespondingtoasigniÞcancelevel.Inpractice,itisconventionaltosetthesigniÞcancelevelinthe1Ð5%range.Buttoachieveconsistentestimationofthedatestamps,thesigniÞcancelevelneedstoapproachzeroasymptotically,andcorrespondinglymustdivergetoinÞnityinordertoeliminatethetypeIerroras.Wethereforeletdependonintheaboveformulas.Inourpracticalworkreportedbelowitisconvenienttouseadirectsettingandexpansionrateforthecriticalvalueofinsteadofanexplicitsettingfor.Thesettingemployedis100.Forthesamplesizesconsideredinourempiricalapplication,thissettingleadstocriticalvaluesaroundthe4%signiÞcancelevel.ThisdatestampingprocedureTheasymptotictheorydevelopedbelowdoesnotrequirethenormalityassumption,whereasthebiascorrectionexplainedlaterdoesusethedistributionalassumption.Observethat,giventhelimitingBrownianmotionprocessprocess0,1],thelimitingvariateWdWcorrespondingtoADFisastochasticprocessthatevolveswith.However,theÞnitedimensionaldistributionofisthesameforall0andistheusualunitrootlimitdistributionWdW ANDYUhassomegoodpropertiesand,inparticular,enablestheconsistentestimationoforiginationandcollapsedates,asdiscussedbelow.Ingeneral,ofcourse,thelowertheactual-valueoftheADF,thestrongertheempiricalevidenceforexplosivebehavior.Ifthesetestsleadtoarejectionofinfavorof,thenwemayconstructavalidasymptoticconÞdenceintervalforusingsomeneweconometrictheoryfortheexplosivecase,asexplainedinSection3.ECONOMETRICISSUESEconometricAnalysisofExplosiveProcesses.RecentworkbyPhillipsandMagdalinos(2007a,b)hasprovidedanasymptoticdistributiontheoryformildlyexplosiveprocessesthatcanbeusedforconÞdenceintervalconstructioninthepresentcontext.Thesepapersdealwithanexplosivemodeloftheform whichisinitializedatsome independentof,andwhereisasequenceincreasingtosuchthat.Theerrorprocessmaycompriseeitherindependentandidenticallydistributedrandomvariablesoraweaklydependenttimeserieswith0anduniformÞnitesecondmomentssothatsup.Model(9)doesnotincludeaninterceptinordertoavoidthepresenceofadeterministicallyexplosivecomponentinThesequence 1islocaltotheorigininthesensethat1as,butforanyÞniteitinvolvesmoderatedeviationsfromaunitroot,i.e.,deviationsthataregreaterthantheconventionaldeviationsforwhichunitroottestshavenontriviallocalpowerproperties(seePhillips,1987)andunitroottypedistributionsapply.Thecorrespondingtimeseries(whichisstrictlyspeakinganarrayprocess)in(9)ismildlyexplosive.Importantly,maybewithinaslowlyvaryingfactorof,forinstancelog,sothatwemayhaveModelsoftheform(9)seemwellsuitedtocapturingtheessentialfeaturesofeconomicandÞnancialtimeseriesthatundergomildlyexplosivebehavior.Theyalsoseemappropriateforcapturingperiodicallycollapsingbubblebehaviorwherethebubblemayappearoverasubpe-riodoflength.Thesemildlyexplosivemodelshavetheveryinterestingandsomewhatunexpectedproperty,establishedinPhillipsandMagdalinos(2007a,b),thattheyareamenabletocentrallimittheory.Moreover,thelimittheoryturnsouttobeinvarianttotheshortmem-orypropertiesoftheinnovations,sothatinferentialproceduresbasedonthislimittheoryarerobusttomanydifferentdeparturesfromsimplei.i.d.errors.Thismeansthatthemodelsandthelimittheorymaybeusedasabasisforstatisticalinferencewithprocessesthatmani-festmildlyexplosivetrajectories.ForeconomicandÞnancialdata,thistypicallymeansvaluesthatareintheregion[105].Inparticular,if200,wehave 1.002,1.053]forc[0.1,2].Undersomegeneralregularityconditions,PhillipsandMagdalinosshowthattheleastsquaresregressionestimatorhasthefollowinglimittheoryformildlyexplo-siveprocessesoftheform(9): 2c(önŠn),n)n isastandardCauchyrandomvariable.Itfollowsthata100(1)%conÞdenceintervalisgivenbytheregion EXPLOSIVEBEHAVIORINTHESNASDAQ isthetwo-tailedpercentilecriticalvalueofthestandardCauchydistribution.For90%,95%,and99%conÞdenceintervals,thesecriticalvaluesareasfollows:ThesevaluescanbecomparedwiththecorrespondingGaussiancriticalvaluesof1.645,1.96,TheconÞdenceintervalsandlimittheoryareinvarianttotheinitialconditionbeinganyÞxedconstantvalueorrandomprocessofsmallerasymptoticorderthan.TheconÞdenceintervalsandlimittheoryalsoremainunchangedifthedata-generatingprocessisaunitrootmodelfollowedbyamildlyexplosiveautoregressionsuchas(9).Thesepropertiesprovidefurtherrobustnesstotheprocedure.FiniteSampleBiasCorrectionviaIndirectInferenceEstimation.Leastsquares(LS)regressioniswellknowntoproducedownwardbiasedcoefÞcientestimatesintheÞrst-orderautoregression(AR).ThisbiasdoesnotgotozeroastheARcoefÞcient0andthebiasincreasesasgetslarger.ItisthereforehelpfultotakeaccountofthisbiasinconductinginferenceonautoregressivecoefÞcientssuchasin(9).Severalstatisticalproceduresareavailablefordoingso,includingtheuseofasymptoticexpansionformulas(Kendall,1954),jackkniÞng(Quenouille,1956;Efron,1982),medianunbiasedestimation(Andrews,1993),andindirectinference(MacKinnonandSmith,1998;Gourierouxetal.,2000).IndirectinferencewasoriginallysuggestedandhasbeenfoundtobehighlyusefulwhenthemomentsandthelikelihoodfunctionofthetruemodelaredifÞculttodealwith,butthetruemodelisamenabletodatasimulation(Smith,1993;Gourierouxetal.,1993).Infact,theprocedurealsoproducesimprovedsmallsamplepropertiesandhasthecapacitytoreduceautoregressivebias,asshownbyMacKinnonandSmith(1998)andGourierouxetal.(2000)inthetimeseriescontextandGourierouxetal.(2010)inthedynamicpanelcontext.Weshalluseindirectinferenceinthepresentapplicationbecauseofitsknowngoodperformancecharacteristicsandconvenienceinautoregressivemodelestimation.Toillustrate,supposeweneedtoestimatetheparameterinthesimpleAR(1)model(i.e.,0inmodel(7)fromobservations,wherethetruevalueof.Someautoregressivebiasreductionmethods,suchasKendallÕs(1954)procedure,requireexplicitknowledgeoftheÞrsttermoftheasymptoticexpansionofthebiasinpowersof.Suchexplicitknowledgeofthebiasisnotneededinindirectinference.Instead,indirectinferencecalibratesthebiasfunctionbysimulation.Theideaisasfollows.WhenapplyingLStoestimatetheAR(1)modelwiththeobserveddata,weobtaintheestimateandcanthinkofthisestimateanditsproperties(includingbias)asbeingdependentonthroughthedata.Givenaparameterchoice,letbedatasimulatedfromthetruemodel,forbeingthetotalnumberofsimulatedpaths.Thesesimulationsrelyonthedistributionalassumptionmadein(10).LettheLSestimatorbasedonthethsimulatedpath,given,bedenotedby0,weneedtoaugmentmodel(10)accordingly. ANDYUTheindirectinferenceestimatorisdeÞnedastheextremumestimator issomeÞnitedimensionaldistancemetricandistheparameterspace,whichiscompact.InthecasewheretendstoinÞnity,theindirectinferenceestimatorbecomes))istheso-calledbindingfunction.Inthiscase,assumingthefunctiontobeinvertible,theindirectinferenceestimatorisgivenbyTheprocedureessentiallybuildsinasmall-samplebiascorrectiontoparameterestimation,withthebiasbeingcomputeddirectlybysimulation.Itcanbeshownthattheasymptoticdistributionofisthesameasthatof.SotheasymptoticconÞdenceintervalderivedintheprevioussectionappliesequallywelltotheindirectinferenceestimatorandwillbeimplementedinwhatfollows.EstimatingOriginationandCollapseDates.Asexplainedearlier,datestampingthebeginningandconclusionofexplosivebehaviorinthedataisbasedonthecriteria(8),leadingtothepointestimates.Itisclearlydesirableforthesepointestimatestobeconsistentforthetruevaluesasthesamplesize.Asymptoticanalysisofdependsontheformofthetruemodelunderboththenullandthealternativehypothesis.Sincethenullisthatofaunitrootmodelwithnoperiodofexplosivebehavior,itisthealternativehypothesisthatisofprimaryinterest.Notethatunderthenullhypothesisofnoexplosivebehavior,if0as,then.ItfollowsthatunderthatnullADFWdW Hence,inthelimitasunderthenull,therewillbenooriginationpointforanexplosiveperiodinthedata.Inordertoconsistentlyestimatetheoriginationandcollapsedatesofexplosivenessunderthealternative,wemustspecifyamodelthatallowsforregimesthatswitchbetweentheunitrootandmildlyexplosiveepisodes.Forthepurposeofthediscussionthatfollows,weuseadata-generatingmechanismthatallowsforthepossibilityofasingleexplosiveepisode,viz., n ,c�0(0,1), EXPLOSIVEBEHAVIORINTHESNASDAQisiinre]istheoriginationdate,andandnrf]isthecollapsedateoftheexplosiveepisode.Ifthereisnomildlyexplosiveepisode,then0and1.Model(14)startswithaunitrootmodelbutallowsforswitchesinregimeat(totheexplosiveepisode)(backtounitrootbehavior).Whentheexplosiveperiodcomestotheend,theinitialvalueofthenewunitrootperioddiffersfromtheendvalueoftheexplosiveperiod.SothespeciÞcationcapturesbothexuberanceandcollapseandinvolvesreinitializationoftheprocessunderthecollapse.Withthereinitializationat,theprocessjumpstoadifferentlevelThenewinitialvaluemayberelatedtotheearlierperiodofmartingalebehaviorintheprocess,perhapswithsomerandomdeviation,inwhichcasewewouldhaverandomquantity.AdetailedanalysisofthismodelandtheasymptoticbehaviorofatestprocedurefordatestampingexplosivebehaviorisgiveninPhillipsandYu(2009).WesummarizethoseÞndingsinwhatfollowshere.AlthoughthemechanismofcollapseisverysimpleinModel(14),thespeciÞcationmaybefurtheradaptedtoallowforashortperiodtransitionaldynamic,whichcouldbemeanrevertingtothelevelRecursiveregressionsarerunwith(14)usingthedata,..., nr]withr r0,sothattheminimumamountofdatausedfortheregressionsisisnr0].9Accordingto(8),wedatetheoriginationoftheexplosiveepisodeasöönöre]whereöre=infs r0 s:ADF)istheright-side100%criticalvalueofthelimitdistributionWdWoftheADFstatisticbasedononns]observationsandisthesizeoftheone-sidedtest.itiseasytoshowthat0,as,justasunderthenull.Denote.Whenhennr]andr�re,weÞndbyexaminingthedominantcomponentsinthenumeratoranddenominatoroftherecursivecoefÞcientestimator 2cön( )Šn =Š( Š e)n n+1/2 j= exjŠ1j 1+op(1) 2cŠ2( Š e)n wherethelimitisaCauchyvariate(cf.,theorem4.3ofPhillipsandMagdalinos,2007a).Then,0wehave Forconvenienceofpresentation,itisassumedinthissectionthatthelaglength ANDYUandtheDF-statisticis ö 2 1/2(ön( )Š Š2 j=1x2jŠ1 ö 2 1/2 (ön( )Šn 2( Š e)n 22cx2 e 2( Š e)nre 2c2rx2 e1/2n1Š 1+op(1)}=n1Šc3/2r3/2 whereöistheusualleastsquaresresidualvarianceestimator.Wededucefrom(18)thatforallllnr]andr�rePADF 0.Accordingto(15)wehaveöADF.ItfollowsthatforanyADF1forall0.Since0isarbitraryandsince0asshownearlier,wededucethat0as,provided cvn(r)+cvn(r) forallllr0,1].Hence,asADFFr0,1]pre.Condition(20)seemsamildconditiononthecriticalvalue.Inparticular,)isrequiredtogotoinÞnity(toensurethetypeIerrorisnegligibleasymptotically)andataslowerrate.Accordingly,anyslowlyvaryingexpansionrateforsuchas,willsufÞce,forallConditionalonÞndingsomeoriginatingdateöforexplosivebehavior,wedatethecollapseoftheexplosiveepisodebyöönörf]whereADFUsingaanalysissimilartothatforö,PhillipsandYu(2009)showthatö,providedforallllr0,1].Hence,undersomemildregularityandrateconditions,therecursiveADFprocedureconsistentlyestimatestheoriginationandcollapsedatesofexplosiveItcanalsobeusefultoimposeaminimumdurationrequirementsuchas incondition(21),sothatonlybubblesofreasonableduration(i.e.,greaterthanaverysmallinÞnityas)aredetectedinthetest. EXPLOSIVEBEHAVIORINTHESNASDAQTESTINGFOREXPLOSIVEBEHAVIORINTHENASDAQINDEXFROMFEBRUARYTOJUNE r0,1]ADF Logprice8262LogdividendCriticalValuesfortheExplosiveAlternative 1%06024%0011081441 :Thistablereportsandsupupr0,1]ADFtestsofthenullhypothesisofaunitrootagainstthealternativeofanexplosiveroot,where10.TheoptimallaglengthfortheADFtestisselectedaccordingtotop-downsequentialsigniÞcancetesting,assuggestedbyCampbellandPerron(1991),withthemaximumlagsetto12andthesigniÞcantlevelsetto5%.TheseriesarethelogrealNasdaqpriceindexandlogrealNasdaqdividend.ThesampleperiodisFebruary1973toJune2005with389monthlyobservations.ThecriticalvaluesfortheADFstatisticandsupsupr0,1]ADFareobtainedbyMonteCarlosimulationwith10,000replications.behavior.Thisresultisnotsurprising.Thereasonisthatwhencollapsestoalevelwithinanneighborhoodofthesignalinthedatafromtheexplosiveperiodisstrongenoughtodeterminetheasymptotics,asshowninPhillipsandMagdalinos(2007a)inasimilarcontext.Infact,inthiscase1whenbutthereisadownwardbiasinthelimitingdistribution.Theformulation(14)isrelatedtotheMarkovswitchingmodelofHalletal.(1999).AnimportantdifferencebetweenthetwoapproachesisthatwedonotspecifythemechanismforregimeswitchingwhereasinHalletal.,natureselectstheregime(orstate,asrepresentedby0or1)atdatewithaprobabilitythatdependsonwhatregimetheprocesswasinatdate1.OurapproachallowsustoestimatetheoriginationandconclusiondateswhereastheMarkovswitchingmodelcanestimatetheÞlteredorsmoothedprobabilityofthestatevariable.Itisreasonabletobelievethatthe-valueoftheone-sidedADFtestisnegativelyrelatedto)),theÞlteredprobabilityofbeingintheexplosivestateintheMarkovregimeswitchingmodel.OurdataaretakenfromDatastreamInternational.WecollectmonthlyobservationsontheNasdaqcompositepriceindex(withoutdividends)andtheNasdaqcompositedividendyields,andcomputetheNasdaqcompositedividendseriesfromthesetwoseries.WeusetheConsumerPriceIndex(CPI),whichisobtainedfromtheFederalReserveBankofSt.Louis,toconvertnominalseriestorealseries.OursamplecoverstheperiodfromFebruary1973toJune2005andcomprises389monthlyobservations.Figure1plotsthetimeseriestrajectoriesoftheNasdaqrealpriceandrealdividendindices.Bothseriesarenormalizedto100atthebeginningofthesample.Ascanbeseen,bothpriceanddividendgrewsteadilyfromthebeginningofthesampleuntiltheearly1990s.Thepriceseriesthenbegantosurge,andthesteepupwardmovementintheseriescontinueduntilthelate1990sasinvestmentinDotComstocksgrewinpopularity.Earlyintheyear2000thepriceabruptlydroppedandcontinuedtofalltothemid-1990slevel.Thedividendseries,ontheotherhand,remainedsteadythroughoutthesampleperiod.TESTINGANDDATINGEXUBERANCETable1reportstheandsupsupr0,1]ADFteststatisticsforboththelogNasdaqrealpriceandlogNasdaqrealdividendindicesforthefullsamplefromFebruary1973toJune2005,where10(i.e.,theinitialstart-upsamplehas39observations).Alsoreportedarethevarious ANDYU 05/80 07/84 09/88 11/92 01/97 03/01 05/05 0 1 2 3 Period ADF Stat for Dividend Critical Values July 1995 March 2001 TIMESERIESOFADFSTATISTICFORTHELOGARITHMICREALNASDAQPRICEANDTHELOGARITHMICREALNASDAQDIVIDENDFROMAPRILTOJUNEADFSTATISTICISOBTAINEDFROMTHEFORWARDRECURSIVEREGRESSIONWITHTHEFIRSTOBSERVATIONINFEBRUARYcriticalvaluesforeachofthetwotests.Forthetest,theasymptoticcriticalvaluesareobtainedfromMonteCarlosimulationandareconsistentwiththosereportedbyFuller(1996,table10.A.2).Forsupsupr0,1]ADF,thecriticalvaluesareobtainedusingMonteCarlosimulationbasedon10,000replications.Severalconclusionsaredrawnfromthetable.First,ifweweretofollowtheconventionandapplytheADFtesttothefullsample(February1973toJune2005),thetestscouldnotrejectthenullhypothesis1infavoroftheright-tailedalternativehypothesis1atthe5%signiÞcancelevelforthepriceseries,andthereforeonewouldconcludethattherewerenosigniÞcantevidenceofexuberanceinthepricedata.Ifonebelievesinaconstantdiscountrate,theresultisconsistentwithDibaandGrossman(1988)andissubjecttothecriticismleveledbyEvans(1991)becausestandardunitroottestsforthefullsamplenaturallyhavedifÞcultyindetectingperiodicallycollapsingbubbles.Second,thesupupr0,1]ADFtest,ontheotherhand,providessigniÞcantevidenceofexplosivenessinthepricedataatthe1%level,suggestingthepresenceofpriceexuberance,butnoevidenceinthedividenddata.However,supupr0,1]ADFcannotrevealthelocationoftheexuberance.Tolocatetheoriginandtheconclusionofexuberance,Figure3plotstherecursiveADFticsforthelogrealpriceandthelogrealdividend.Alsoplottedisthecurvelog(log(log(log(.1,1]and389,thatisusedforthecriticalvalues.Sincerangesbetween39and389,log(log(100rangesbetween0.013and0.018.Thesevaluesofturnouttobeclosetothe4%signiÞcancelevelcriticalpoint.Obviously,thesecriticalvaluesgotoinÞnityasaslowerratethan.TheoptimallaglengthisdeterminedusingtheproceduresuggestedbyCampbellandPerron(1991).Startingwith12lagsinthemodel,coefÞcientsaresequentiallytestedforsigniÞcanceatthe5%level,leadingtotheselectionofWehavealsoconductedthetestsusingpriceanddividendseriesininsteadofinnaturallogarithms.Theresultsaresimilarandtheconclusionsremainqualitativelyunchanged.Theyarenotreportedtoconservespaceandareavailableuponrequest.ThesameremarkappliestoFigures4and5discussedbelowaswell.TheprocedureofCampbellandPerroninvolvestwosteps.Followingasuggestionofareferee,weestimatedthelaglengthandtheautoregressiveparametersinonestepviatheLasso-typemethod(KnightandFu,2000;CanerandKnight,2008)andfoundthepointestimatesofin(7)arenearlyidenticalbythetwomethods.Tothebestofourknowledge,theasymptotictheoryisnotyetknownfortheLassoestimatorofinamixedsetofunitrootandexplosivevariables,therebyinhibitinginferencewiththisprocedure. EXPLOSIVEBEHAVIORINTHESNASDAQ 8/83 10/87 12/91 2/96 4/00 6/04 0 2 4 Period ADF Stat for Dividend 5% CV July 1995 Sep 2000 TIMESERIESOFADFSTATISTICFORTHELOGARITHMICREALNASDAQPRICEANDTHELOGARITHMICREALNASDAQDIVIDENDFROMJUNETOJUNEADFSTATISTICISOBTAINEDFROMTHEROLLINGREGRESSIONWITHTHEFIRSTSAMPLERANGINGFROMFEBRUARYTOJUNE1979(,77OBSERVATIONSAREUSEDINEACHREGRESSIONthemodelforwhichthecoefÞcientofthelastincludedlagissigniÞcantatthe5%level.initialstart-upsamplefortherecursiveregressioncoverstheperiodfromFebruary1973toApril1976(10%ofthefullsample).TheforwardrecursiveregressionsgivesomeinterestingnewÞndings(seeFigure3).Thedividendseriesisalwaysnonexplosive.Thestockpriceseriesisalsotestedtobenonexplosivefortheinitialsample,whichsuggestsnoevidenceofexuberanceintheinitialdata.ThisfeatureismaintaineduntilJune1995.InJuly1995,thetestdetectsthepresenceofexuberanceinthedata,andtheevidenceinsupportofpriceexuberancebecomesstrongerfromthispointonandpeaksinFebruary2000.TheexuberanceisdetectedascontinuinguntilFebruary2001,andbyMarch2001,thereislittleevidenceofexuberanceinthedata.InApril2001,theevidenceofexuberanceshowsupagaininthedataandpersistsuntilJuly2001.InAugust2001,nofurtherevidenceofexuberanceispresentinthedata.Interestingly,theÞrstoccurrencedateforpriceexuberanceinthedataisJuly1995,whichismorethanoneyearbeforeGreenspanÕshistoricremarkofÒirrationalexuberanceÓmadeinDecember1996.Followingasuggestionofthereferees,wecheckedtherobustnessoftheempiricalresultsbyrunningrollingregressions,inwhicheachregressionisbasedonasubsampleofsize(ofsmallerorderthanandwiththeinitializationrollingforward.Forthisparticulardata,we77,whichis20%ofthefullsample,andhencetheÞrstsampleperiodisfromFeburary1973toJune1979.Figure4plotstherollingrecursiveADFstatisticforthelogrealpriceandthelogrealdividend.Alsoplottedisthe5%asymptoticcriticalvalue.doesthetestbasedonforwardrecursiveregression,thetestbasedtherollingregressionsdetectsexplosivenessinpriceinthe1990s.Inparticular,thetestindicatesthatexuberanceinthe1990sstartsinJuly1995andendsinSeptember2000.TheestimatedoriginationdateisthesameasinFigure3.SotheempiricalidentiÞcationofexuberanceinthe1990sandtheNgandPerron(1995)demonstratethattooparsimoniousamodelcanhavelargesizedistortions,whereasanoverparameterizedmodelmayresultinreductionoftestpower.Theyshowthatmethodsbasedonsequentialtestshaveanadvantageoverinformation-basedrulesbecausetheformerhavelesssizedistortionsandhavecomparablepower.Twoalternativemovingwindowsizes,60and120(5and10years,respectively)weretriedandverysimilarresultswereobtained.AlthoughthesamplesizeisÞxedat77inalltherollingregressions,theasymptoticcriticalvaluesareveryclosetothecriticalvalueswhenthesamplesizeis77. ANDYUTESTINGFOREXPLOSIVEBEHAVIORINTHENASDAQINDEXINTHE r0,1]ADF95%ConÞdenceInterval PanelA:SamplePeriod:January1990toDecember1999 Logprice309289410251033[1.016,1.050]Logdividend6260 PanelB:SamplePeriod:January1990toJune2000 Logprice975297510361040[1.033,1.047]Logdividend6260 CriticalValuesfortheExplosiveAlternative 1%06024%0011081441 :Thistablereportsandsupupr0,1]ADFtestsofthenullhypothesisofaunitrootagainstthealternativeofanexplosiveroot,where10.TheoptimallaglengthfortheADFtestisselectedaccordingtotop-downsequentialsigniÞcancetesting,assuggestedbyCampbellandPerron(1991),withthemaximumlagsetto12andthesigniÞcantlevelsetto5%.TheseriesarethelogrealNasdaqpriceindexandlogrealNasdaqdividend.PanelAreportstheresultsfortheperiodJanuary1990toDecember1999;PanelBreportstheresultsfortheperiodJanuary1990toJune2000whenexplosivebehaviorisdetectedtobethestrongest.ThecriticalvaluesfortheADFstatisticandsupsupr0,1]ADFareobtainedbyMonteCarlosimulationwith10,000replications.empiricallydetermineddateoforiginationofexuberanceappearrobusttothechoiceoftheregressionschemes.However,theestimatedcollapsedateisafewmonthsearlierinthedata.Interestingly,thenewtestalsodetectssomeexplosivebehaviorbeforethe1987crash,althoughthisexuberanceisveryshort-lived.TohighlighttheexplosivebehaviorintheNasdaqduringthe1990s,wecarryouttheanalysisusingtwosubsamples.TheÞrstsubsampleisfromJanuary1990toDecember1999,the10-yearperiodthatrecentresearchershavefocusedon(e.g.,PastorandVeronesi,2006;OfekandRichardson,2003;BrunnermeierandNagel,2004).PanelAofTable2reportsthetestresults.Asabove,weapplytheandsupADFtestsforaunitrootagainstthealternativeofanexplosiveroottoboththelogrealpriceandlogrealdividendseries.Wealsoobtaintheleastsquaresestimate,theindirectinferenceestimate,the95%asymptoticconÞdenceintervalofbasedon,andcriticalvaluesfortheunitroottests.Alltheresultsgivestrongevidenceofexplosivenessin.Forexample,forthelogrealNasdaqpriceindex,thestatisticforthefullsampleis2.309,farexceedingthe1%criticalvalueof0.60.SimilarresultsoccurwiththesupADFtest.Wethereforerejectthenullhypothesisofaunitrootatthe1%signiÞcancelevelinfavorofexplosivebehaviorfortheNasdaqstockindex.Incontrast,thereisnoevidencethatthelogrealdividendseriesexhibitsexplosivebehavior.Figure5graphsthetrajectoryoftheADFstatisticstogetherwithlog(log(100,withwith.1,1]and186,asthecriticalvaluesforsampleobservationsfromJanuary1990totheendofthesample.Asforthefullsample,wechoose10.SimilartoFigures4and5,weagaindatethestartofpriceexuberanceinJuly1995,sotheempiricallydetermineddateoforiginationoftheexuberanceappearsrobusttothechoiceoftheinitialsample.TherecursiveAlso,thenonmonotonicityinADFforthe1990episodeinFigure4ismoreapparentthaninFigure3.Forexample,weÞndweakevidenceofaunitrootinJanuary1997andinAugust-October1998,wheretheADFstatisticisslightlysmallerthanthecorrespondingcriticalvaluesatthesedates.Thelaglengthchosenbysequentialtestingfor0forboththelogpriceandlogdividendseries.The5%and1%criticalvaluesforaunitrootagainstthestationaryalternativeare2.86and3.42,respectively.Basedonthesecriticalvalues,thetestwillindeedrejectthenullhypothesisofaunitrootinfavorofthealternativeofstationarityforthedividendseries. EXPLOSIVEBEHAVIORINTHESNASDAQ 09/9210/9411/9612/9801/0102/03 -10-6-22 ADF Stat for Dividend Critical Values Oct 2000 July 1995Period TIMESERIESOFADFSTATISTICFORTHELOGARITHMICREALNASDAQPRICEANDTHELOGARITHMICREALNASDAQDIVIDENDFROMJUNETOJUNEADFSTATISTICISOBTAINEDFROMTHEFORWARDRECURSIVEREGRESSIONWITHTHEFIRSTOBSERVATIONINJANUARYregressionsdetecttheconclusionofexuberanceinOctober2000,somewhatearlierthanthatreportedinFigure3butverysimilartothatreportedinFigure4.TheautoregressiongivestheARcoefÞcientestimate025instockprice.Assumingthattheerrortermintheregressionfollowsani.i.d.normaldistributionand0,weobtaintheindirectinferenceestimate033viasimulationwith10,000replications.Theassociated95%asymptoticconÞdenceintervalforis[1050].Thisimpliesthatthelogstockpricewillgrowattheexplosiverateof3.3%permonth.Sincethedividendseriesisnotexplosive,withaconstantdiscountratethefundamentalpriceisalsonotexplosive,beingdeterminedexclusivelybydividendsaccordingto(3).Therefore,from(2),(thelogbubble)mustalsobeexplosivewithagrowthrateatleastashighasthegrowthrateofstockprice,3%permonth.With95%conÞdence,thetruegrowthrateliesintherangebetween1.6%and5%permonth.Undertheassumptionofconstantdiscountrate,thisprovidessufÞcientconditionsforthepresenceofbubble.Tounderstandtheimplicationoftheestimatedexplosiverateforstockprice,supposethattheNasdaqindexwereovervaluedbyaround10%whenGreenspanmadehisÒirrationalexuberanceÓcommentinDecember1996.Thentheinitialsizeofthelogbubblewouldbe0953inDecember1996.Usingtheindirectinferenceestimateofthegrowthrate033,wemaycalculatethat,byMarch2000whentheNasdaqindexreacheditshistorichigh(39monthslater),theexpectedloglevelofthepricebubblewouldhaverisento338,andtheratiooftheexpectedNasdaqpricetoitsfundamentalvaluewouldhavebeen40.Inotherwords,after39months,theexpectedNasdaqindexwouldhavebecomearound40%overvaluedrelativetoitsfundamental.Noticethat033reportedinPanelAofTable2givesanunbiasedestimateoftheexplosiverootforthestockpriceprocess,whichcanbeconsideredalowerboundoftheexplosiverootoftheunobservablebubbleprocess.Thereasonisasfollows.From(2),weknowthattheactualstockpriceconsistsofthefundamentalcomponentandthebubblecomponent.UndertheassumptionthatthefundamentalcomponentiseitherI(1)orI(0)and ANDYUthebubblecomponentisexplosive,ifabubblelastsforasufÞcientlylongperiodoftime,thebubblecomponentwilldominatethefundamentalcomponentandtheactualstockpricewillgrowataroundthesamespeedasthebubblecomponentdoes.However,withinalimitedtimeperiodwhenabubbleisÞrstdeveloping,themagnitudeofthebubblecomponentmaybesmallrelativetothefundamentalcomponenteventhoughtheprocessisexplosive,andthereforeemployingthestockpriceseriesforestimationwillunderestimatethetruegrowthrateoftheToprovideamorerealisticestimateofthegrowthrateofthebubble,sincetheNasdaqindexkeptrisingafterDecember1999,weimplementthetestbyextendingtheÞrstsubsampletoJune2000whenthetestdetectsexplosivepricebehaviorwiththemostsigniÞcantADFteststatistic.PanelBofTable2reportstheleastsquaresestimateforthissample,whichyieldstheindirectinferenceestimateThisimpliesagrowthrate4%permonth.Althoughthisisstillalowerboundestimateofthegrowthrateofthebubbleprocess,itisplausibletothinkofitastheclosesttothetruegrowthrate.SupposethattheNasdaqindexwereovervaluedby10%whenourtestÞrstdetectedthebubbletostartinJune1995;thentheinitialsizeofthebubbleis0953.Usingtheaboveunbiasedestimateofthespeedofbubble,byJune2000(60monthslater)whenourtestdetectedthebubbletobethestrongest,theexpectedsizeofthebubblewouldhavebecome0025.ThisimpliesthattheratiooftheexpectedNasdaqpricetoitsfundamentalvaluewouldhavebeenInotherwords,theexpectedvalueoftheNasdaqindexwouldhavebeen173%overpricedrelativetoitsfundamentalvalueafter60months.TheactualNasdaqindexpeakedat5,048.62pointsonMarch10,2000,thendroppedto1,950.4byDecember31,2001andto1,335.31byDecember31,2002.Iftheyear2001endvalueisconsideredclosetotheÒfundamentalÓvalue,thentheNasdaqindexwouldbe159%overpricedatthepeak(5049/19502.59).Ontheotherhand,iftheyear2002endvalueisconsideredtheÒfundamentalÓvalue,thepeakvaluewouldbe278%overpriced(5049/13353.78).Therefore,theaboveestimateofthegrowthrateofthebubblematchestheactualNasdaqpricedynamicsreasonablywell.FINITESAMPLEPROPERTIESUnitRootTestsforanExplosiveBubble.AlthoughstandardunitroottestshavebeenappliedtotestforunitrootsagainstexplosivenessinthepriceseriesinDibaandGrossman(1988)andEvans(1991),bothpapersonlyexaminedtheÞnitesampleperformanceofthestandardunitroottestsforthebubble(seeSectionVIinDibaandGrossmanandSectionIIIinEvans).Naturally,however,itismoreinformativetoverifytheÞnitesampleperformanceofthestandardunitroottestsinthepriceseriesitselfbecauseinpracticethepriceseriesisobservedbutthebubbleseriesisnot.Considerthefollowingdata-generatingprocess,wherethefundamentalpricefollowsaran-domwalkwithdriftandthebubbleprocessisalinearexplosiveprocesswithoutcollapsing:),and).WeuseNasdaqpriceindexdatafromFebruary1973toDecember1989(i.e.,beforethe1990sexplosivepriceperiodstarted)toestimatetheThelaglengthchosenbysequentialtestingfor5forthelogpriceseriesand0forthelogdividendseries.TheOLSestimate,1.036,impliesonly121%overpricedindexlevel.Hence,thecompoundingeffectarisenfromtheestimationbiasiseconomicallysigniÞcant. EXPLOSIVEBEHAVIORINTHESNASDAQPOWEROFTHE PanelA InitialValue0.00(size) 0.000.0490.1070.4580.8060.9340.020.0490.1110.4640.8100.9370.040.0490.1150.4760.8180.9350.060.0490.1190.4950.8280.9510.080.0490.1250.5220.8480.9540.100.0490.1340.5500.8660.961 PanelB InitialValue 0.000.6520.8170.9010.9300.9420.020.8220.8510.9050.9330.9440.040.9720.9110.9240.9340.9480.060.9990.9620.9360.9450.9500.081.0000.9880.9530.9520.9550.101.0000.9980.9680.9610.961 :ThistablereportstheempiricalpoweroftheADFtestforanexplosivestockmarketbubbleatthe5%nominalsizelevelwith120observationsand10,000MonteCarloreplications.Themodelusedfortheexperimentis,withparametervalues0324estimatedbasedontheNasdaqpriceindexdataasdescribedinthetext.Theseparametervaluesareusedtoconductsimulationsunderdifferentassumptionsaboutthespeedparameterandtheinitiallevelofthebubble.ResultsarereportedinPanelA.PanelBdisplaysresultswithdifferentvaluesassignedtoandthebubbleinnovationstandarddeviationwhenthespeedparameterissettoitsempiricallyÞttedvalueof0.04.fundamentalprocess,assumingthattherewasnobubbleduringthisperiodsothat.Thisestimationyieldsthevalues00227and05403.Wethenusethesetwoparametervaluesalongwith04(basedontheindirectinferenceestimateofinPanelBofTable2)toobtaintheestimateofthebubbleinnovation0324byemployingdatafortheexplosiveperiodJanuary1990toJune2000viatheKalmanÞlter,asinWu(1997).Theseparameters,andareusedtoconductsimulationsunderdifferentassumptionsaboutthespeedandtheinitiallevelofthebubblewith120observationsand10,000replications.ThesimulationresultsarereportedinPanelAofTable3.PanelBdisplaystheresultsfordifferentvaluesforthebubbleinnovationstandarddeviation,whereasthespeedparameterissetto0.04,whichistheindirectinferenceestimateof1reportedinPanelBofTable2.ItisknownfromDibaandGrossman(1988)thatstandardunittestscandetectexplosivecharacteristicsin.Oursimulationresultssuggestthatthestandardunitroottestscanalsode-tecttheexplosivecharacteristicsinwhenbubblesappearintheempiricallyrealisticsettingsaslongasthebubblesarenotperiodicallycollapsing.PanelAofTable3clearlydemonstratesthatthehigherthetestpoweris,thelargeristhegrowthrateand/orthelargeristheinitial-.Whenthegrowthrateislargerthan0.01,thetesthassubstantialpoweragainsttheexplosivealternative,andwhen04(theindirectinferenceestimateusingtheNasdaqstockindexduringthebubblyperiod),thetesthasnearlyperfectpoweragainsttheexplosive-nessalternativeregardlessoftheinitiallevelofthebubble.PanelBofTable3showsthatsmallervaluesofthestandarddeviationleadtogreatertestpowerprovidedtheinitialvalueisnottoosmall(here03).Overall,thepowerisnotverysensitivetotheinnovationstandarddeviationortotheinitialvalueofthebubbleandisquitehighwiththegrowth ANDYURecursiveUnitRootTestsandPeriodicallyCollapsingBubbles.Theabovesimulationdesigndoesnotallowforthepossibilityofperiodicallycollapsingbubbles,animportantclassofbubblesthatseemmorerelevantinpracticaleconomicandÞnancialapplications.Evans(1991)proposedamodeltosimulatesuchcollapsingbubblesandshowedthatstandardunitroottestshadlittlepowertodetectthistypeofbubbles.Inthissection,weÞrstdesignasimulationexperimenttoassessthecapacityofourrecursiveregressionteststodetectthistypeofperiodicallycollapsingbubbles.Weshowthatalthoughthetestsareinconsistentinthecontext,inÞnitesamplesthetestshavegoodpower.Evans(1991)suggestedthefollowingmodelforabubbleprocessthatcollapsesperiodi-periodi-+Š1(1+g)t+1BtŠ(1+g)Š1 ]b,t+1,ifBt� ,2)withisanexogenousBernoulliprocessthattakesthevalue1withprobabilityand0withprobability1.Evans(1991)speciÞeshismodelinlevelsandsoprice,dividend,andbubbleareinlevelsandareexpressedinupper-caseletters.Thismodelhasthepropertythat,analogousto(4).Themodelgeneratesbubblesthatsurviveaslongastheinitialboundingconditionapplies(say)andthereafteronlyaslongasthesuccessionofidenticalrealizations,hold.ThebubbleburstswhenTofacilitatecomparisonsbetweenoursimulationresultswiththoseofEvans(1991),weusethesamesimulationdesignandparametersettingsashis.Inparticular,abubbleprocessof100observationsissimulatedfromthemodel(24)and(25)withtheparametersettings5,and05,andisaBernoulliprocessthattakesthevalue1withprobabilityand0withprobability1.When0,thebubblebursts.Wechoosethevalue999,0.99,0.95,0.85,0.75,0.5,0.25.Inaddition,adividendseries(inlevels)of100observationsissimulatedfromthefollowingrandomwalkmodelwithdrift:3.Consequently,thefundamentalpriceisgeneratedandthesimulatedpriceseriesfollowsas.Inthesimulationsreported,isscaledupwardsbyafactorof20,assuggestedinEvans(1991).Table4reportstheempiricalpoweroftheandsupADFstatisticsfortestinganexplosivebubblebasedonthe5%criticalvaluereportedinTable1and10,000replications.Weshouldemphasizethat,unlikeEvans(1991),whoassumedthatisobservedandtestedtheexplosivenessin,weapplythetesttothepriceseriesitself.Severalinterestingresultsemergefromthetable.First,thepowerofthetestdependscriticallyon.When999or0.99,thetesthasconsiderablygoodpower(0.914and0.460respectively).95,thetesthasessentiallynopower.TheseresultsareconsistentwiththosereportedinEvans(1991,table1).Second,thepowerofthesupADFstatisticalsodepends,butinamuchlessdrasticway.Forexample,when25,itstillhasconsiderableBlanchard(1979),FloodandGarber(1980),andBlanchardandWatson(1982)ÞrstproposedstochasticbubblesthatcanburstwithaÞxedprobability.Burmeisteretal.(1983)showtheequivalenceofaclassofdifferent-lookingstochasticbubbleprocesses. EXPLOSIVEBEHAVIORINTHESNASDAQPOWEROFTHEHEr0,1]ADFTESTSUNDERTHEEVANS 0.9990.990.950.850.750.500.25 0.9140.4600.0690.0220.0160.0260.0440.044r0,1]ADF0.9920.9270.7140.4320.3510.3420.340 :ThistablereportstheempiricalpoweroftheADFtestforanexplosivebubbleatthe5%nominalsizelevelwith100observationsand10,000Monte-Carloreplications.Themodelusedfortheexperimentis)andcollapsesperiodicallyaccordingtoo+Š1(1+g)t+1(BtŠ(1+g)Š1)]b,t+1,ifBt� ,beingaBernoulliprocessthattakesthevalue1withprobabilityand0withprobability1.WesetWechoosedifferentvaluesforpower(0.340).Forempiricallymorerelevantcases,saywhen95,thepowerofsupADFbecomesmuchhigher(0.714).Clearlytheperformanceofthetestsisdeterminedbythetimespanofabubble.IntheAppendix,weformallyshowthatthemaximumtimespanofacollapsingbubbleinEvansÕ(1991)modelis,whichisveryshortrelativetothefullsamplesize,sothatstandardunitroottestscannotbeexpectedtoperformverywell.ThisAppendixfurthershowsthatinaregressionofobservationsfromanexplosiveperiod,thesignalintheregressionhasthemaximumorderof2log( )Š 2 ).Whenlog( )1+ 2 ,thissignalissmallerthanthatofanintegratedprocesswhosesignalis)andsigniÞcantlylessthanthatofanexplosiveprocess.TheseÞndingsexplainthefailureofconventionalunitrootteststodetectbubblesofthistype,conÞrmingthesimulationsinEvans(1991)andinourTable4.Inrecursiveregressions,thesignalwillbecomparativelystrongerbecausethedatasetisshorteranditwillbeemphasizedwhentheendpointintherecursionoccurstowardtheendofabubble.Thisargumentsuggeststhattherewillbesomestatisticaladvantagetotheuseofrecursiveregressiontechniquesandtheuseofasuptestinassessingtheevidenceforperiodicallycollapsingbubbles,asconÞrmedinTable4.However,inarecursiveregressionusingsamplesofsizesizenr]for0,themaximumlengthofthebubbleisstill)andthisisstillnotlongenoughrelativetoforarecursivetesttobeconsistentessentiallybecausethesignalisnotstrongenough.Thislimitationshowsupinthesimulationsasthetestperformsworsewhengetssmaller,althoughthepowerforthesuptestisclearlynontrivialandsubstantiallybetterthanthatofconventionaltests.Wemightexpectsomeadditionalgainfromtheuseofarollingregressioninconductingthetest,wherethesamplesize()usedfortheregressionhassmallerorderthan,forinstance,instance,n]forsome1,oreven).WhenWhenn],forinstance,thesignalfromtheexplosivepartofthedata,whichstillhasthetimespanofwilldominateprovidedthat2log( )Š 2 .However,inthecaseofrollingregressionsofthistype,testsgenerallyhavedifferentlimitdistributionsfromthosestudiedalreadyintheunitrootandstructuralbreakliterature,forexamplebyBanerjeeetal.(1993),whererollingregressionsoflengthproportionaltothesamplesizeareused.Thisarticlehasproposedanewapproachtotestingforexplosivebehaviorinstockpricesthatmakesuseofrecursiveregression,right-sidedunitroottests,andanewmethodofconÞdenceintervalconstructionforthegrowthparameterinstockmarketexuberance.Simulationsreveal ANDYUthattheapproachworkswellinÞnitesamplesandhasdiscriminatorypowertodetectexplosiveprocessesandperiodicallycollapsingbubbleswhenthediscountrateistimeinvariant.TheempiricalapplicationofthesemethodstotheNasdaqexperienceinthe1990sconÞrmstheexistenceofexuberanceanddatestampsitsoriginationandcollapse.Asthesecondepi-graphindicates,theexistenceofexuberanceorÒbubbleÓactivitymaybeself-evidenttosomeeconomistsinviewofthesheersizeofthewealthcreatedandsubsequentlydestroyedintheNasdaqmarket.Ofprimaryinterestthereforeareitsparticularcharacteristicssuchastheorig-inationdate,whichweÞndtobemid-1995,thepeakinFebruary2000,andtheconclusionsometimebetweenSeptember2000andMarch2001.ComparisonofthisstatisticaloriginationtothetimingofthefamousremarkbyGreenspaninDecember1996afÞrmsthatGreenspanÕsperceptionswereactuallysupportedbyempiricalevidenceofexuberanceinthedataatthatGreenspanÕsremarksareoftentakentoindicateforesightconcerningthesubsequentpathofNasdaqstocks.ThepresentÞndingsindicatethathisremarkswerealsosupportedinsomemeasurebythetrackrecordofempiricalexperienceuptothattime.Thus,GreenspanÕsper-spectiveconcerningirrationalexuberanceinstockpricesandfutureproÞtabilityinDecember1996showedhindsightaswellasforesightconcerningtheimpendingescalationintechnologyassetvalues.Thisarticlehasnotattemptedtoidentifyexplicitsourcesofthe1990sexuberanceininternetstocks.Severalpossibilitiesexist,includingthepresenceofarationalbubble,herdbehavior,orexplosiveeffectsoneconomicfundamentalsarisingfromtimevariationindiscountrates.IdentiÞcationoftheexpliciteconomicsourceorsourceswillinvolvemoreexplicitformulationofthealternativemodelsandsuitablemodeldeterminationtechniquestoempiricallydistinguishbetweensuchmodels.Thepresenteconometricmethodologyshowshowthedatamaybestudiedasamildlyexplosivepropagatingmechanism.TheresultsconÞrmstrongempiricalsupportforsuchactivityintheNasdaqdataoverthe1990s.Themethodologycanalsobeappliedtostudyrecentphenomenainrealestate,commodity,foreignexchange,andequitymarkets,whichhaveattractedattention.Theresultswillbereportedinfuturework.PROPERTIESOFEVANSWemaywritetheinitialstoppingtimeforwhichtheboundaryvalueisattainedasSubsequentstoppingtimesaredeterminedinthesamewayaftertheinitialbubblecollapses.Thedurationofeachofthebubblesdependsonthesestoppingtimesplusthenumberofrepeatedsubsequentdrawsof1.Itisknown(e.g.,Schilling,1990)thatthemaximumruntime,,forasequenceofidenticalBernoullidrawsinasampleofsizehasmean )andvarianceVar( 6log .Itfollowsthat).Hence,themaximumtimespanofacollapsingbubbleoverthefullsamplewill).Todeterminethelengthofthestoppingtime,observethattheconditionin(24)requires whichholdsif 2 2 , EXPLOSIVEBEHAVIORINTHESNASDAQ isastandardBrownianmotion,thisconditioncanberewrittenas 1+g)Š 2,A=1 ThetimespanoftheÞrstcomponentinthebubble(24)isthereforethepassagetimeuntilastandardBrownianmotionwithdrifthitstheboundaryvalue.ThatisItiswellknown(e.g.,BorodinandSalminen,1996,p.223)thatthispassagetimesatisÞesand,sinceforsmallvaluesofandwith1wehave0,itfollowsthathasmomentgeneratingfunction(BorodinandSalminen,1996,p.223)sothattheexpectedhittingtime 1+g)Š isÞnite,asisthevariance.Itfollowsthatthemaximumtimespanofacollapsingbubblegeneratedby(24)and(25)overthefullsampleisand,ingeneral,thetimespanwillbeshorterthanbecausethemaximumruntimewillnotusuallybeThisÞndingexplainsthefailureofconventionalunitrootteststodetectbubblesofthistype,conÞrmingthesimulationsinEvans(1991).Ineffect,eventhemaximumtimespanofforthesecollapsingbubblesissoshortrelativetothefullsamplesizethatfullsampletestsforexplosivebehaviorareinconsistent.Heuristically,thisisbecausethesignalfromtheexplosivepartofthetrajectoryisgenerallynotstrongenoughtodominatetheregressionbeforethe ANDYUbubblecollapses.Inparticular,ifdatawereavailable,thesignalfromanexplosiveperiodinitializedatandofdurationintheregressionofhasorder 2t 2T0+T +Rns=T0+T +1us=OpRn+T0+T t=T0+T +11+g 2t 2T0+T +Rns=T0+T +1us=OpRn+T0+T t=T0+T +11+g 2t 2!T0+T +Rns=T0+T +1ysŠRn 2 2"#=OpRn+T0+T t=T0+T +11+g 2t 2!T0+T +Rns=T0+T +1ysŠRn 2 2"#=OpRn+T0+T t=T0+T +1$1+g 2t 2 W(T0+T +Rn)ŠW(T0+T )ŠRn 2 2&'=OpRn+T0+T t=T0+T +1$1+g 2t 2Op( Rn ŠRn 2 2&'=OpRn+T0+T t=T0+T +1$1+g 2t 2Š 2 2&'=OpRn+T0+T t=T0+T +11+g 2t×OpeŠ 2 2 =Op1+g 2Rn×OpeŠ 2 2log )Š 2 ).Thesignalfromastationaryautoregressionis)andfromaunitrootautoregressionis)sothatthesignalfromtheexplosivecomponentabovewillbeofmaximalorder2log( )Š 2 ),whichisstillapowerlawinandnogreaterthanthatofanintegratedprocess,whosesignalis),when 1 2+ 2 andnogreaterthanthatofapolynomialinanintegratedprocessingeneral,therebyexcludingexplosivebehavior.,D.,ÒExactlyMedian-UnbiasedEstimationofFirstOrderAutoregressive/UnitRootModels,Ó61(1993),139Ð65. 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