Sample selection and event study estimation

17 Pages·2009·0.28 MB·English

by  Kenneth R. Ahern

JournalofEmpiricalFinance16(2009)466–482
ContentslistsavailableatScienceDirect
Journal of Empirical Finance
journal homepage: www.elsevier.com/locate/jempfin
☆
Sample selection and event study estimation
⁎
Kenneth R. Ahern
RossSchoolofBusiness,UniversityofMichigan,701TappanStreet,AnnArbor,MI48109,UnitedStates
a r t i c l e i n f o a b s t r a c t
Articlehistory: Theanomaliesliteraturesuggeststhatpricingisbiasedsystematicallyforsecuritiesgroupedby
Received23April2008 certaincharacteristics.Ifthesecharacteristicsarerelatedtoselectioninaneventstudysample,
Receivedinrevisedform15January2009 imprecisepredictionsofaneventstudymethodmayproduceerroneousresults.Thispaper
Accepted23January2009
performssimulationstocompareabatteryofshort-runeventstudypredictionandtesting
Availableonline31January2009
methods wheresamples aregrouped bymarketequity, prior returns,book-to-market, and
earnings-to-priceratios.Significantstatisticalerrorsarereportedforbothstandardandnewer
JELclassification: methods,includingthree-andfour-factormodels.Acharacteristic-basedbenchmarkmodel
G30
producestheleastbiasedreturnswiththeleastrejectionerrorsinallsamples.
C14
©2009ElsevierB.V.Allrightsreserved.
C15
Keywords:
Eventstudies
Nonparametricteststatistics
Multifactormodels
Characteristic-basedbenchmarkmodel
1.Introduction
Thewidevarietyofapplicationsandtherichnessofdataavailablehavemadeeventstudiescommonplaceineconomic,finance,
andaccountingresearch.Thestrengthoftheeventstudymethodologyisthatabnormalreturnsduetoafirm-specific,buttime-
independenteventmaybepreciselyestimatedbyaggregatingresultsovermanyfirmsexperiencingasimilareventatdifferent
times. Brown and Warner (1985) (BW) conduct simulated event studies of random samples and find that simple estimation
techniquesarewellspecified.Inparticular,estimatesfromordinaryleastsquares(OLS)withamarketindextestedwithparametric
statistical tests arewell-specified using non-normally distributed daily data and in the presence of non-synchronous trading.
Moreover,BWshowsthatabnormalreturnsmeasuredwithsimplerestimationproceduressuchasmarket-adjustedandmean-
adjustedreturnsdisplaynosignificantmeanbias.
ThispapersuggeststhattheresultsofBWmaynotholdinactualeventstudies.BWshowresultsfordatathatarerandomly
selectedfromallsecurities,whereaseventstudiestypicallyhavedatathatarecharacteristicallynon-representativeoftheoverall
marketandoftengroupedbyunderlyingtraitssuchassize,momentum,andvaluation.Forinstance,firmsthatinitiatedividends,
split their stock, or make acquisitions are likely to be largewith high prior returns. Under these conditions, it should not be
assumedthatthemarketaverageresultsofBWshouldhold.
ThispapersimulateseventstudiessimilartoBW,butdrawsamplesnon-randomly.Inparticular,samplesaredrawnfromthe
highest and lowest deciles of market equity, prior returns, book-to-market, and earnings-to-price ratios, where deciles are
computedfromallNYSEfirms.Irunahorseracebetweeneightpredictionmethods,includingacharacteristic-basedbenchmark
model,amarketmodel,FamaFrenchThree-FactorandCarhartFour-Factormodels,andfourteststatistics,bothparametricand
☆ IthankRichardRollforencouragingmetodevelopthistopic.IespeciallyappreciatethecommentsofStephenBrownandJeroldWarner.IalsothankJean-
LaurentRosenthal,AntonioBernardo,J.FredWeston,DukeBristow,andRaffaellaGiacominiforhelpfulcomments.Thecommentsoftheassociateeditorandtwo
anonymousrefereessubstantiallyimprovedthispaper.ThispaperisanextensionofChapter1ofmydissertationcompletedattheEconomicsDepartmentofUCLA.
⁎ Tel.:[phone].
E-mailaddress:[email protected].
0927-5398/$–seefrontmatter©2009ElsevierB.V.Allrightsreserved.
doi:10.1016/j.jempfin.2009.01.003
K.R.Ahern/JournalofEmpiricalFinance16(2009)466–482 467
non-parametric,todeterminewhichmethodhastheleastmeanbiasandthebestpowerandspecificationofthetestsindifferent
samples.Ialsoinvestigatetheeffectofusingpost-eventversuspre-eventdatatoestimatemodelparameters,theeffectofevent-
inducedvariance,andtherelationbetweenbiasandsamplesize.
Theresultssuggestthatstandardeventstudymethodsproducestatisticalbiasesingroupedsamples.Themostsignificanterrors
arefoundtobefalsepositiveabnormalreturnsinsamplescharacterizedbysmallfirmsandfirmswithlowpriorreturns.Falsenegative
abnormalreturnsarefoundinsamplescharacterizedbylargefirmsandbyfirmswithhighpriorreturns.Thecharacteristic-based
benchmarkmodel,wherestockreturnsareadjustedbyamatchedsize-returnportfolioofcontrolstocks,displaystheleastbiasofall
themodels.Multifactormodelsproduceonlymarginalbenefitsoverastandardmarketmodelinpredictingeventdaynormalreturns,
buttheygeneratelessskewedabnormalreturnsthatarebettersuitedforstatisticaltests.Usingpost-eventestimationwindowsalso
reducesforecasterrorbias.Eventwindowvarianceincreasespresentabiggerproblemthanpricingbias,butmaybecorrectedusing
thesigntest.
Thoughthemeanbiasreportedinthepredictionmodelsissmall,itsignificantlyaffectstheTypeIandTypeIIerrorsofthemodels.
Withtheexceptionofthecharacteristic-basedbenchmarkmodel,allofthemodelsover-rejectthenullhypothesisofnoabnormal
returnsinlowertailtestsinsamplesoflargefirms.Likewise,theyunder-rejectinuppertailtestsforlargefirms.Insamplesgroupedby
priorreturns,themodelsover-rejectinuppertailtestsinsamplesoffirmswithlowpriorreturnsandinlowertailtestsofsamplesof
firmswithhighpriorreturns.Thecharacteristic-basedbenchmarkmodelalsosuffersfromthisproblem,buttoalesserdegreethanthe
standardmethods.Thebiasinrejectionratesacrosstailsaffectstheasymmetryofthepowerofthetestsinsimilarways.
Samplescomposedofmultivariatedistributionsacrossthemis-pricingcharacteristicsmaybepronetocompoundbiasesin
some cases, or may have reduced errors if the pricing biases cancel out. Running simulations where samples are matched to
distributionsofsizeandpriorreturnsfrom564stocksthatcompletedreverse-stocksplitsduring1972to2002,Ifindaslightmean
biasinallmodels,buttheeconomicsignificanceissmall.Inaddition,othersimulationsareperformedwithsamplesintendedto
matchlargeacquirers,distressedfirms,firmswithnewexchangelistings,andfirmsmakingseasonedequityofferings.Biasesarenot
largeingeneral,butwhenactualeventreturnsaresmallorzero(suchasinacquisitions),oriftheeventwindowislongtoaccount
fornoisymeasuresofeventdates,thebiaseswilldominatetheactualreturns.Inaddition,thereturnbiasesleadtolargeabnormal
dollarreturnbiasesinsamplesoflargefirms($45millionoverthreedaysforafirmatthe85thNYSEpercentileofmarketequity).
Otherstudieshaveaddressedsampleselectionbias.Brownetal.(1995)presentatheoreticalmodelofsurvivorshipwhere
volatilityispositivelyrelatedtoanexpostbiasproducedbyalowerboundonpricesforsurvivingfirms.Thus,thisbiaswouldbe
severeforfirmsinfinancialdistress,forexample,whichmaybeanexplicitinclusionconditioninaneventstudyofcorporate
restructuring.Brownetal.alsoshowthatasampleofstocksplitswillbeconditionedontheoccurrenceofpositivereturnsinthepre-
eventperiod.BiasintroducedbyasizeeffectisanalyzedinDimsonandMarsh(1986)forthecaseofpressrecommendationson
stockreturns.Theyalsonotethatstockpricerun-upsmayattracttheattentionofthepressandleadtomorerecommendationsfor
firmswithhighpriorreturns.Fama(1998)statesthatevenriskadjustmentwiththetrueassetpricingmodelcanproducesample
specificanomaliesifsamplespecificpatternsarepresent.Thesestudiessuggestthattheunderlyingcharacteristicsoffirmsselected
foraneventstudysamplemayleadtobiasedpredictionsifanon-robustpredictiontechniquesuitableformarketaveragefirmsis
used.
Thepresentpaperprovidesanumberofcontributionstotheeventstudymethodsliterature.Thoughothershaverecognized
that firm characteristics associated with pricing anomalies may be correlated with corporate events, there has not been a
comprehensivesimulationstudyofgroupedsamplestodetermineifthepotentialbiasesaresmallenoughtobeignored.Second,
therandomsampleresultsofBWwillbesignificantlyupdated.ThedatainBrownandWarnercovertheseventeenyearsfrom1963
to1979,ofwhichonlysevenincludeNASDAQfirms(1973–1979).ThetimeperiodinmystudyextendsthedataofBWtoalmost
40years(1965–2003),with31yearsincludingtheNASDAQ.Thiswillprovideamuchbroaderuniverseofsecuritiesforwhichto
test the specification of event study methods. Finally, this is the first paper to study the benefits of using multifactor and
characteristic-based benchmark models in short-run event studies. Combining a much larger dataset with a comprehensive
collectionofpredictionmodelsandteststatisticsinaneventstudysimulationusingdailyreturnsdatawillbringevidencetobear
onthepotentialbiasesineventstudymethodologies.Itshouldbenotedthattheappropriatechoiceofmethodologywilldepend
uponthesettingofastudyandthattheresultsofthispaperdonotprovidea‘best’methodsuitedtoallstudies.
2.Methodologicalissuesofeventstudies
2.1.Selectionbysecuritycharacteristics
Ifeventstudysamplesecuritiesarecharacterizedbyfactorsrelatedtopricingbiases,thentheabnormalreturnsestimatedby
theeventstudyarepotentiallybiased.Thesepredictionbiasesariseduetoskewnessinreturns,evenafterincludinganintercept
term.Takeforexamplethewellknownsmall-firmbiasintheCAPM(Banz,1981).Supposethetruepopulationmodelis:
R =α+β Rm+β SMB +e; ð1Þ
i 1 i 2 i i
whereR isanindividualfirmreturn,Rmisthemarketreturn,andSMBisthesizefactorinFamaandFrench(1993).However,
i
insteadofthetruemodelweestimate:
R =α+β Rm+e: ð2Þ
i 1 i
468 K.R.Ahern/JournalofEmpiricalFinance16(2009)466–482
FollowingWooldridge(2000),theestimatedcoefficientsofβ andαare
1
βˆ =β SST +β Xn (cid:2)Rm− R¯m(cid:3)SMB + Xn (cid:2)Rm− R¯m(cid:3)e ð3Þ
1 1 1 2 i i i i
i=1 i=1
αˆ = R¯ −βˆ R¯m ð4Þ
i 1
whereSST isthesumofsquareddeviationsinRm.Theomittedvariablebiaswillleadtothefollowingincorrectestimatesofαand
1 i
β :
1
(cid:2) (cid:3)
E βˆ =β +βδˆ ð5Þ
1 1 2
(cid:2) (cid:3)
E αˆ = R¯ − βˆ R ð6Þ
i 1 m
where^δîstheestimatedcoefficientfromaregressionofSMBonR andaconstant.However,evenwiththesecoefficientbiases,the
m
estimateoftheaveragefirmreturnisunbiased,
¯
EðRÞ= R ð7Þ
i i
Sincetheslopecoefficientisbiased,theintercepttermadjustssothatthefittedlinepredictsthemeanfirmreturngiventhe
meanmarketreturn.Ifonanygiveneventdayweexpecttoobservetheaveragemarketreturn,thenthisfittedlinewouldcorrectly
predictthefirmreturnandnopredictionerrorwouldoccur.Moreover,predictionerrorsthatoccurwhenthemarketreturnis
higherorlowerthanitsaveragewouldcanceleachotherout.
However,itisincorrecttoexpecttoobservetheaveragemarketreturnonanygivendaybecausereturnsarepositivelyskewed.
Thisimpliesthatthemeanislargerthanthemodeinthedistributionandthatformorethanhalfofasampleofrandomlychosen
days,theobservedreturnwillbelessthanthemeanreturn.Becauseoftheskewness,omittedvariablebiaswillgenerateincorrect
predictionsonaverageevenwhenthemodelallowsforanendogenouslydeterminedinterceptterm.Thisbiaswillpersistevenin
largesamples.Thus,becausebiasesinstandardassetpricingmodelsaregeneratedbyomittedvariables,itmakessensetolookat
samplesgroupedbythesecharacteristics.
Asmentioned,Banz(1981)findsthattheCAPMpredictsreturnsthataretoolowforsmallfirms.Basu(1983)showsthatprice
toearningsisnegativelyrelatedtoreturns,controllingformarketbeta,suggestingthattheCAPMwillpredictreturnstoohighfor
firmswithhighP/Eratios.Itishypothesizedthatasimplemarketmodeleventstudyprocedurewillmakethesamemistakes,
leadingtoafalsefindingofpositiveabnormalreturnsforsmallfirmsandnegativeabnormalreturnsforfirmswithhighP/Eratios.
Pricinganomaliesduetomomentumalsohavebeendocumented.JegadeeshandTitman(1993)findthatsecuritiesexhibiting
recent(pastyear)levelsofhigh(low)returnshavepredictablyhigh(low)returnsinthefollowingthreemonthsafteraccounting
for systematic risk and delayed reactions to common factors. Thus short-runpricing models under-price securities with high
returnsintherecentpast,andover-pricesecuritieswithlowreturns.Thismayleadtotheappearanceofpositiveabnormalreturns
topositivemomentumfirmsandnegativeabnormalreturnstonegativeorlowmomentumfirms.Ifreturnsaremean-reverting,
however,thebiaswouldhavetheoppositesign.
Book-to-marketratiosalsohavebeenfoundtopredictreturnssystematically.FamaandFrench(1992)findapositiverelation
betweenaveragereturnandbook-to-marketequityratiosafteraccountingforbeta.Thissuggestseventstudiesusingaprediction
modelwithonlyamarketindexasanexplanatoryvariablewilltendtofindfalsenegativeabnormalreturnsforfirmswithlow
book-to-marketratiosandfalsepositiveabnormalreturnsforfirmswithhighbook-to-marketratios.
Thesepricinganomaliesmayconfoundeventstudyresultsifsamplesaredominatedbysecuritiescharacterizedbytheabove
factors.Priorstudieshaveshownthatfirmsthatundergoparticularcorporateeventsoftenhavecommoncharacteristicsofsize,
momentum,andvaluationratiosdifferentfrommarketaverages.Table1presentsasummaryofthesamplecharacteristicsofprior
eventstudies.1Samplesoflargefirmswithhighpriorreturnsandlowbook-to-marketratiosaretypicalinstudiesofacquisitions
andstocksplits.Bothsamplesofnewexchangelistingsandacquisitiontargetstendtohavesmallfirmswithhighpriorreturnsand
lowbook-to-marketratios.Seasoned-equityofferings,dividendinitiationsandomissions,bankruptcy,andothercorporateevents
alsohavesamplesthatdifferfrommarketaveragesacrossthesepricinganomalyfactors.Giventhenon-randomsamplesofprior
eventstudies,itisrelevanttodetermineifabnormalreturnsgeneratedbystandardeventstudymethodsaresystematicallybiased
whensamplesaregroupedbytheabovecharacteristics.
2.2.Predictionandtestingmethods
ThispaperperformsBrownandWarnereventstudysimulationswhensamplesaregroupedbycharacteristicsassociatedwith
the possible inclusion into an actual event study, namely market equity (ME), prior returns (PR), book-to-market (BM), and
1 Earnings-to-priceratiosarenotreportedinTable1becausetheyaretypicallynotreportedineventstudies.However,itisreasonabletoassumethatevent
firmshavenon-randomE/Pratiosgiventhepreponderanceofnon-randomsamplesacrosssize,momentum,andbook-to-marketratios.ThusE/Pisincludedin
thisstudyasapotentialsourceofbias.
K.R.Ahern/JournalofEmpiricalFinance16(2009)466–482 469
Table1
Samplecharacteristicsacrosspricinganomalies.
Event Samplefirmcharacteristics
Marketequity Priorreturns Book-to-market Source
Acquisition High High Low Rhodes-Kropfetal.(2005)
High N/A Low MitchellandStafford(2000)
N/A High N/A Asquith(1983)
Bankruptcy Low Low — Campbelletal.(2005)
Dividendinitiation High High — Lipsonetal.(1998)
High High N/A Michaelyetal.(1995)
Dividendomission High Low N/A Michaelyetal.(1995)
Newexchangelisting Low High Low DharanandIkenberry(1995)
Seasonedequityoffering Low High Low Bravetal.(2000)
Low N/A Low MitchellandStafford(2000)
Sharerepurchase Low N/A Low MitchellandStafford(2000)
Stocksplit High High Low Ikenberryetal.(1996)
Targetofacquisition Low N/A Low Rhodes-Kropfetal.(2005)
Low High Low Schwert(2000)
N/A Low N/A Asquith(1983)
Thistablepresentstherelationshipbetweensamplefirmsinprioreventstudiestomarketaveragesforthecharacteristicsofmarketequity,priorreturns,andbook-
to-marketratios.High(Low)indicatessamplefirmsareabove(below)marketaveragesforaparticularcharacteristic.Nodifferencebetweenthemarketaverage
andthesampleaverageisindicatedby—.N/Aindicatesthattheinformationwasnotreportedinthesourcearticle.
earnings-to-priceratios(EP).Foreachsamplecriterion,thepropertiesofabatteryofpredictionmodelsandteststatisticswillbe
examined.
Thepredictionmodelstestedinthisstudyincludetraditionalmethodsaswellaslesscommonlyusedmethodsforcomparison.
Thesimplestmethodusedtopredictanormalreturnistosimplysubtractasecurity'stimeseriesaveragefromaneventdatereturn
(mean-adjustedreturn),denotedhereasMEAN.Themostcommonlyusedpredictionmethodisthemarketmodel,wherefirm
returnsareregressedonaconstanttermandamarketindex,eitherequal-orvalue-weighted(MMEWandMMVW).Asimilar
procedure, the market-adjusted return method, subtracts the market index from an event date security return (MAEW and
MAVW).Inboththemarketmodelandthemarket-adjustedreturnproceduresresearchersneedtochooseamarketindex.Because
thecriteriaforthischoicearenotwelldefined,thispaperanalyzesbothequal-andvalue-weightedindexesforcomparison.
InresponsetothepricinganomaliesoftheCAPMdiscussedabove,alternativepricingmodelshavebeendeveloped,though
theiruseinshort-runeventstudieshasbeenlimited.Inparticular,FamaandFrench(1996)useathree-factormodelincludinga
marketindex,sizeindex,andbook-to-marketindextoexplainstockreturns(FF3F).Carhart(1997)usesafour-factormodelwhich
appendstheFama–French three-factormodelwithashort-runmomentumindex (FF4F).Bothof thesemodelswillbetested
alongsidethemorecommonpredictionmodels.
Though multifactor regression models may alleviate the omitted variable bias of a simple market model, they may also
introduce additional estimation error (Fama and French,1997). An alternative to regression models is a characteristic-based
benchmark(CBBM)estimateasusedinthemutualfundperformanceliterature(Danieletal.,1997).Inthismodel,theevent
returnsofasamplestockareadjustedbytheequally-weightedreturnsofaportfoliooftencontrolstocksmatchedbysizeand
prior-returndeciles.Thismodelhastheadvantagethatnoregressorsneedtobeestimated,whichreducesestimationerror.In
addition,thismodeldoesnotrequirearesearchertochoosea‘normal’estimationperiod,eitherpre-orpost-event.Thedetailsof
allmodelsareprovidedinAppendixA.
The four leading test statistics used in event studies, t-statistic, standardized t-statistic, rank, and the sign statistics, are
comparedinthisstudy.Thet-statisticiscomputedasinBrownandWarner(1985).Thethreeremainingstatisticsarecalculatedas
describedinCorradoandZivney(1992).Thestandardizedt-statisticnormalizeseachabnormalreturnbythefirm'stimeseries
standarddeviation.Theranktestorderstheabnormalreturnsovertheentireperiodthatincludesboththeestimationandevent
windows and assigns a corresponding value between zero and one for each day's observation. The sign test assigns either a
negativeone,apositiveone,orzerotoeachday'sobservationforabnormalreturnsthatareabove,below,orequaltothemedian
abnormalreturn,respectively.Thus,ifaneventdatehasanaveragerankedabnormalreturnacrossfirmsclosetoone,oramajority
ofabnormalreturnsabovethemedianabnormalreturn,thentherankandsigntestswillrejectthenullhypothesisofnoabnormal
returns.Becausethenon-parametrictestsarebasedonmediansanddonotrequiredistributionalassumptions,theymaybemore
appropriatefortheskewedandhighlykurtoticnatureofdailystockreturns.DetailsofallteststatisticsareprovidedinAppendixA.
Inresponsetotheproblemsofpricinganomaliesineventstudies,researchershaveusedestimationperiodsotherthanthe
periodimmediatelypriortotheeventperiod,thoughthishastypicallybeendoneinlimitedcircumstancesandgenerallyforlong-
runstudiesusingmonthlydata.2Mandelker(1974)addressesthisissuebyseparatelyestimatingparametercoefficientsusingboth
2 IamgratefultoJeroldWarnerforsuggestingthisapproach.
470 K.R.Ahern/JournalofEmpiricalFinance16(2009)466–482
pre-andpost-eventestimationperioddataonmergers.CopelandandMayers(1982)usepost-eventdatainordertominimizebias
associatedwithabnormalpriorreturnsinthepre-eventperiodforfirmsrankedbyValueLine.Agrawaletal.(1992)andGregory
(1997)usepost-eventestimationdatainlong-runstudiesofmergers.Thepresentstudyestimatesallmodelswithseparatepre-
andpost-eventestimationwindows.Unlessotherwisenoted,allresultspresentedinthispaperwillbegeneratedusingpre-event
data,sincethisisthemorecommonprocedure.
FollowingBWandothersimulationstudies,thispaperartificiallyintroducesabnormalreturnsaswellasvarianceincreasesto
theeventdatereturnsforeachcharacteristicsample.This facilitatescomparisons betweenthepredictionmodel-teststatistic
combinationstodeterminewhichmethodsarethebestspecifiedandhavethemostpowertodetectabnormalreturns.
This study will concentrate only on short-run event study methods, restricting analysis to a one-day event window. This
providesthebestcomparisonofthevariousmethodsbecausetheshortertheeventwindow,themoreprecisearethetests.Ifatest
doesnotperformwellforaone-dayeventwindow,itwillonlyperformworseforlonger-runstudies.Thusifsmallerrorsare
presented in this study, they will be compounded in long-run studies (Fama, 1998; Kothari and Warner, 2005). Moreover,
recognizingtheproblemofpredictingnormalreturnsoveralonghorizon,long-runeventstudiesusedifferentmethodologiesthan
thosepresentedhere.3
3.Experimentaldesign
Thisstudysimulates1000samplesof250securitieseachbyrandomselectionwithreplacementfromasubsetofsecuritiesin
theCRSPDailyStockdatasetbetweenJanuary1965andDecember2003,wheresubsetsarebasedonsize,momentum,andtwo
measuresofvaluation.Abnormalreturnsaregeneratedandtestedbytheintroductionofartificialperformanceandvarianceon
eventdatereturns.Eachofthesetopicsisdiscussedindetailbelow.
3.1.Datarequirements
To be included in thisstudya security must meetthe following requirements. It must be an ordinarycommonshare of a
domesticorforeigncompany(CRSPSHRCD=10,11,or12).ThisexcludesADRs,SBIs,closed-endfunds,andREITs.Furthermore,it
mustnotbesuspendedorhalted(CRSPEXCHCD=0,1,2,or3).Foreachsecurity-eventdate,(day0),thedailyreturnsarecollected
overamaximumperiodof489days(−244,+244)wherethepre-eventestimationperiodisdefinedas(−244,−6),theevent
periodis(−5,+5),andthepost-eventestimationperiodis(+6,+244).However,ifafirmhasatleast50non-missingreturnsin
thepre-eventestimationperiod,atleast50non-missingreturnsinthepost-eventestimationperiod,andnomissingobservations
intheperiod(−15,+15),thenitisincludedinthesample.IfanobservationisCRSPcoded-99,thecurrentpriceisvalid,butthe
previouspriceismissing,whichmeansthatthenextreturnisoveraperiodlongerthanoneday,soobservationsfollowinga-99
codearecountedasmissingobservations.
EventdatesarerandomlychosenoveralltradingdaysbetweenJanuary1,1965andDecember31,2003,sothateachmonth
anddayoftheyearisequallyrepresentedinthesample.Forarandomlyselectedeventdate,asecurityisrandomlychosenfrom
thegrouped samples.If thissecurity does notmeetthe requirements listed above,a newsecurity is selected usingthe same
event date.If no security in the samplemeets the requirements for inclusionon a particularevent date, a neweventdate is
chosen. This is done to ensure that event dates are evenly distributed over the 39 years, even though there are many more
possiblesecurity-eventdatesinlateryearsduetoagreaternumberoffirmsinthedataset.Sincethefocusofthispaperismean
bias,event-dateclusteringisnotinvestigated.Becausecross-sectionaldependenceofsamplesgroupedbysize,priorreturns,or
valueislargerthanrandomsamples,biasesinstandarderrorsusingOLSinclusteredeventdateswillbelargerinthegrouped
samples. However, using a portfolio approach or seemingly-unrelated-regressions will correct the bias, just as in random
samples(Campbelletal.,1997).
3.2.Samplecharacteristics
For each security that meets the share type and active trading requirements listed above, samples are formed on the
characteristicsofmarketcapitalization(ME),priorreturns(PR),book-to-market(BM),andearnings-to-priceratios(EP).These
measuresarecalculatedasinFamaandFrench(1992)andKennethFrench'sWebsite.4Eachsecurityisthenassignedaquadruple
decile according to the New York Stock Exchange (NYSE) breakpoints provided on Kenneth French's Web site for the four
characteristicsofME,PR,BM,andEP,wheretheBMandEPdecilesareassignedyearlyandtheMEandPRdecilesareassigned
monthly.Foreachyearlydecileassignment,thecorrespondingtwelvemonthsareassignedthesamedecile.Thuseachsecurityis
assignedadecileforeachofthefourcharacteristicsforeachmonthwheredataareavailable.Iftheaccountingorreturnsdatado
notallowoneormoreofthecharacteristicstobecomputedandassignedtoaNYSEdecile,thesecurityisstilleligibleforinclusion
inasample,thoughitwillnotbeincludedinasamplegroupedbyamissingdecileassignment.
3 SeeMitchellandStafford(2000)foradiscussionofthevariousissuesthatariseinling-runeventstudies.AlsoseeBarberandLyon(1996),Kothariand
Warner(1997),Fama(1998),Lyon,Barber,andTsai(1999),Brav(2000),LoughranandRitter(2000),AldersonandBetker(2006),andGur-Gershgoren,Hughson,
andZender(2008)forfurtherresearchonlong-runeventstudymethodologies.
4 http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.
K.R.Ahern/JournalofEmpiricalFinance16(2009)466–482 471
Table2
Samplereturnpropertiesoverestimationperiod.
Sample Mean Median Standarddeviation t-statformean Skewness Kurtosis Studentizedrange
AllCRSP 0.0008 −0.0002 0.0338 0.384 0.629 9.858 8.239
ME High 0.0008 0.0001 0.0179 0.664 0.158 5.791 7.305
Low 0.0008 −0.0002 0.0413 0.314 0.832 11.769 8.604
PR High 0.0024 −0.0001 0.0380 0.995 0.897 10.391 8.027
Low −0.0010 −0.0007 0.0475 −0.319 0.451 11.020 8.714
BM High 0.0014 −0.0001 0.0411 0.538 0.917 12.514 8.677
Low 0.0009 −0.0004 0.0392 0.344 0.564 8.956 8.175
EP High 0.0010 −0.0001 0.0313 0.515 0.740 10.369 8.405
Low 0.0010 −0.0004 0.0446 0.365 0.770 10.218 8.341
Percentagepoints
250observationsfromanormalpopulation
Variable 0.01 0.05 0.95 0.99
t-statistic −2.596 −1.970 1.970 2.596
Skewness −0.360 −0.251 0.251 0.360
Kurtosis 2.42 1.55 3.52 3.87
Studentizedrange 4.544 4.812 6.624 7.140
Sources:PearsonandHartley(1966,Tables34BandC),LundandLund(1983).
PropertiesofdailyeventstudyperformancemeasureswheresamplesarerandomlydrawnfromeithertheentireCRSPdatabase,orareselectedbasedonone
ofthecharacteristicsofmarketequity(ME),one-yearpriorreturns(PR),book-to-marketratio(BM),orearnings-to-priceratio(EP).Therandomlyselectedevent
datescovertheperiod1965–2003.Theperformancemeasuresarecomputedovertheestimationperiod(−244,−6).Measuresaretakenfrom1000samplesof
250securities.Foreachparameter,thetablereportsthemeanof250,000estimatesindecimalformat.HighindicatessamplesdrawnfromthetopNYSEdecile.
LowindicatessamplesdrawnfromthebottomNYSEdecile.Thepercentagepointslistedatthebottomdescribethecriticalvaluesofeachstatisticalmeasurefor
asampleof250standardnormalrandomvariables,correspondingtothedaysoftheestimationperiod.Forexample,akurtosisof3.52occursrandomly5%of
thetimeinsuchasample.
Characteristic samples are chosen by selecting securities that are assigned to a particular decile or group of deciles for a
particularcharacteristic.Thusforagivenrandomlyselectedeventdate,asamplefirmisselectedrandomlyfromallfirmsthatmeet
thedecilerequirementforaparticularcharacteristicforthepreviousmonth-end.
4.Results
4.1.Estimationperiodreturns
Table 2 displays the distribution properties of sample returns over the pre-event estimation period (−244, −6) by
characteristicsamples.Foreachcharacteristic,twosamplegroupingsareformed.Highindicatesasampleformedbyonlyincluding
firmsrankedinthetopNYSEdecileforaparticularcharacteristic.LowsamplesareformedfromthebottomNYSEdecile.Each
numberreportedisthemeanperformancemeasureovertheestimationperiodofthe250,000firm-datesampleobservations.
As is well documented, random sample returns are non-normally distributed with positive skewness, leptokurtosis, and a
studentizedrangelargerthannormal.Averagedailyreturnsare0.08%.Therawreturnsdatapresentedherefrom1965to2003
haveahighermean,standarddeviation,kurtosis,andstudentizedrange,andlessskewnessthantheearlierperiodreturnsover
1963–1979,aspresentedinBW.Thisisnotsurprising,sincethepresentdataincludemanymorelistingsofsmallfirms.
Meanreturnsareidenticalbetweenlargeandsmallfirmsandequaltherandomsamplemeanreturnsof0.08%.However,high
MEfirmshavereturnswithsmallerstandarddeviations,skewness,kurtosis,andstudentizedrangesthantherandomsample
firms.Highpriorreturnfirmshaveatime-seriesaveragereturnof0.24%,threetimesaslargeasrandomsamples.Lowpriorreturn
firms have negative daily returns of −0.1% on average. High PR firms also have smaller standard deviations, kurtosis, and
studentizedrangesthanlowPRfirms,butgreaterpositiveskewness.
High BM firms are characterized by performance measures above the random sample benchmarks, including high mean
returns,standarddeviations,skewness,kurtosis,andstudentizedrange.ThelowBMfirmsexhibitreturnsverysimilartorandom
samplereturns.BoththehighestandlowestEPdecileshavereturnswithmeansabovetherandomsamplemean,thoughthereisa
negativerelationshipbetweenEPandstandarddeviation.Theotherstatisticalmeasuresofskewness,kurtosis,andstudentized
rangearequitesimilarbetweenthetwodecilesandareabovethevaluesoftherandomsamplemeasures.Thissuggeststhatthe
performancemeasuresofEPgroupedfirmreturnsdisplaynon-linearpatternsacrossNYSEdeciles.
4.2.Predictionmodelperformanceonday0
Table3 presentscross-sectional results forreturnsand prediction modelabnormalreturnson day ‘0.’ Thesevalues reflect
theabilityofthepredictionmodeltoaccuratelypredicta‘normal’returnintheeventperiod.PanelAofTable3presentsthe
472 K.R.Ahern/JournalofEmpiricalFinance16(2009)466–482
Table3
Meanperformancemeasuresatday0.
Model Random ME PR BM EP
sample
High Low High Low High Low High Low
PanelA:Modelsindependentofestimationperiod
Returns 0.0010⁎⁎⁎ 0.0004⁎⁎⁎ 0.0013⁎⁎⁎ 0.0008⁎⁎⁎ 0.0012⁎⁎⁎ 0.0016⁎⁎⁎ 0.0007⁎⁎⁎ 0.0012⁎⁎⁎ 0.0011⁎⁎⁎
CBBM 0.0001 −0.0000 0.0001 −0.0001 −0.0001 −0.0000 0.0001 0.0000 0.0001
MAEW 0.0002⁎⁎ −0.0004⁎⁎⁎ 0.0004⁎⁎⁎ −0.0001 0.0003⁎⁎ 0.0007⁎⁎⁎ −0.0002⁎ 0.0003⁎⁎⁎ 0.0003⁎⁎⁎
MAVW 0.0006⁎⁎⁎ −0.0000 0.0009⁎⁎⁎ 0.0004⁎⁎⁎ 0.0007⁎⁎⁎ 0.0011⁎⁎⁎ 0.0003⁎⁎⁎ 0.0007⁎⁎⁎ 0.0007⁎⁎⁎
PanelB:Abnormalreturnsgeneratedusingthepre-eventestimationperiod
MEAN 0.0002⁎⁎ −0.0003⁎⁎⁎ 0.0005⁎⁎⁎ −0.0016⁎⁎⁎ 0.0021⁎⁎⁎ 0.0002 −0.0001 0.0002⁎ 0.0001
MMEW 0.0002⁎⁎ −0.0004⁎⁎⁎ 0.0005⁎⁎⁎ 0.0016⁎⁎⁎ 0.0020⁎⁎⁎ 0.0002 −0.0002⁎⁎ 0.0001 0.0001
MMVW 0.0002⁎⁎ −0.0003⁎⁎⁎ 0.0004⁎⁎⁎ 0.0016⁎⁎⁎ 0.0020⁎⁎⁎ 0.0001 −0.0002⁎ 0.0002⁎ 0.0001
FF3F 0.0002⁎ −0.0003⁎⁎⁎ 0.0004⁎⁎⁎ 0.0016⁎⁎⁎ 0.0019⁎⁎⁎ 0.0001 −0.0002⁎⁎ 0.0001 0.0001
FF4F 0.0002⁎ −0.0003⁎⁎⁎ 0.0004⁎⁎⁎ 0.0016⁎⁎⁎ 0.0019⁎⁎⁎ 0.0001 −0.0002⁎⁎ 0.0001 0.0002
PanelC:Abnormalreturnsgeneratedusingthepost-eventestimationperiod
MEAN 0.0001 −0.0001 0.0000 0.0001 −0.0004⁎⁎⁎ 0.0003 0.0000 0.0001 0.0000
MMEW 0.0001 0.0000 0.0000 −0.0000 −0.0004⁎⁎⁎ −0.0000 0.0000 0.0001 0.0000
MMVW 0.0001 −0.0000 0.0000 0.0000 −0.0004⁎⁎⁎ −0.0001 0.0000 0.0001 −0.0000
FF3F 0.0001 −0.0000 0.0001 0.0000 −0.0003⁎⁎ −0.0000 0.0001 0.0001 0.0000
FF4F 0.0001 −0.0000 0.0001 0.0000 −0.0003⁎⁎ −0.0000 0.0000 0.0001 0.0000
Cross-sectionalpropertiesonday0oversamplesrandomlydrawnfromeithertheentireCRSPdatabaseorthehighestorlowestNYSEdecileofeachcharacteristic
marketequity(ME),one-yearpriorreturns(PR),book-to-marketratio(BM),orearnings-to-priceratio(EP).Therandomlyselectedeventdatescovertheperiod
1965–2003.Modelsaredescribedintheappendix.Eachnumberreportedisbasedon1000valuesofsamplemeanmeasures,wherethesamplemeanmeasureis
theaverage(adjusted)returnoverthe250securitiesineachsample.Significantdeviationsfromzeroarecomputedwithat-testatthe10%,5%,and1%levelsand
areindicatedby⁎,⁎⁎,and⁎⁎⁎,respectively.
unadjustedandmarket-adjustedreturnsforeachsamplegroupingaswellasthecharacteristic-basedbenchmarkmodel.5PanelB
presentsadjustedreturnsofthemarketmodelsandthethree-andfour-factormodelswheremodelcoefficientsareestimated
usingpre-eventestimationobservations.PanelCpresentsthesesamemodelsestimatedwithdatainthepost-eventestimation
period.
As in BW,all theprediction modelscorrectly predict analmost zero abnormal returnwhensamples arerandomlydrawn.
However,onlyCBBMisstatisticallyzero,whereastheothermodelsaresignificantlydifferentthanzero,exceptwhenpost-event
dataisusedtoestimatethemodelparameters.Thestatisticalsignificancedoesnotimplyeconomicsignificancehowever.Though
the methods are statistically biased in general, the economic bias in random samples is only 0.02%. The distinction between
statisticalandeconomicbiasisrelevantforalloftheresultsofthispapersincefindingstatisticalsignificanceisnotunlikelywith
1000simulations.
Acrossthenon-randomsamplegroupings,theMEandPRsamplesproducesignificantbiasesformostmodelsusingpre-event
data. In particular low ME samples lead to significant positive deviations from zero and high ME samples lead to significant
negativedeviationsfromzeroforallpredictionmodelsexceptCBBM.Themultifactormodelsdonotprovideanyimprovement
oversimplermodels,withallthemodelsinPanelBfindingpositiveabnormalreturnsofabout0.04%forlowMEfirms.Post-event
datareducesthebiasinPanelCtozero.Thus,thereisastatisticallysignificantsizeeffectwhensamplesareformedusingonlyfirms
inthesmallestorlargestNYSEdecilesusingpre-eventestimationdatatoestimateabnormalreturns.
TheresultsofTable3showthatsamplesgroupedbyhighpriorreturnspredict‘normal’returnsthataretoohigh,leadingto
findings of significantly negative abnormal returns (−0.16%) for models based onpre-event data. Post-event data erases this
problembutcreatessignificantnegativereturnsforlowPRsamplesofabout−0.04%.LowPRsamplesestimatedwithpre-event
dataexhibitsignificantpositivebias.AsinthesmallMEsamples,themultifactormodelsdonotprovidesubstantialimprovements
overthe market models in the PR samples. The valuation-based samples display much less bias. Only low BM firms produce
significantbiasamongthecommonestimationprocedures.
ThebiasesinthePRsamplesaredrivenbyareversiontothemeanbetweentheestimationperiodandtheeventperiod.Prior
studiesofmomentumhaveformedportfoliosofpastwinnersandloserswheresecuritiesshareacommoncalendar(Jegadeesh
andTitman,1993;Carhart,1997).Incontrast,thisstudygroupsfirmsintosamplesbasedonpriorperformancethoughatrandom
dates.ThusasecurityinthetoppriorreturnsNYSEdecilein1973mayhaveamuchloweraveragereturnthandoesasecurityinthe
same decile in 1998. The aggregationover time used in thisstudy is appropriate toeventstudies, but will generate different
momentumeffectsthanwillacalendar-timeportfolio.Therefore,thefindingsinthisstudydonotnecessarilycontradictorsupport
the notion of persistence. Likewise, though the three- and four-factor models are designed to capture omitted explanatory
variables, their performance in an event-study setting is not directly comparable to settings where returns share a common
calendar.Moreover,thedailyfactorreturnsarefromportfolioswhichareconstructedonayearlybasis,whereasmysamplesofME
5 Unadjustedreturnsonday0donotalwayshavethesamemeanasintheestimationperiod,sincethereturnsarehighlykurtoticandwidelydispersed.Thusit
isnotunreasonablethattheaverageonanyoneparticulardayshouldbedifferentthanthetime-seriesaverageover239days.
K.R.Ahern/JournalofEmpiricalFinance16(2009)466–482 473
Table4
RejectionfrequencyofMEsamples.
Prb0.05 PrN0.95 Prb0.05–PrN0.95
High Low Difference High Low Difference High Low
(1) (2) (1)−(2) (3) (4) (3)−(4) (1)−(3) (2)−(4)
t-statistics
CBBM 0.049 0.052 −0.003 0.049 0.049 0.000 0.000 0.003
MEAN 0.072⁎⁎⁎ 0.032⁎⁎⁎ 0.040⁎⁎⁎ 0.025⁎⁎⁎ 0.064⁎⁎ −0.039⁎⁎⁎ 0.047⁎⁎⁎ −0.032⁎⁎⁎
MAEW 0.098⁎⁎⁎ 0.037⁎ 0.061⁎⁎⁎ 0.025⁎⁎⁎ 0.063⁎ −0.038⁎⁎⁎ 0.073⁎⁎⁎ −0.026⁎⁎⁎
MAVW 0.052 0.031⁎⁎⁎ 0.021⁎⁎ 0.049 0.082⁎⁎⁎ −0.033⁎⁎⁎ 0.003 −0.051⁎⁎⁎
MMEW 0.087⁎⁎⁎ 0.036⁎⁎ 0.051⁎⁎⁎ 0.027⁎⁎⁎ 0.068⁎⁎ −0.041⁎⁎⁎ 0.060⁎⁎⁎ −0.032⁎⁎⁎
MMVW 0.095⁎⁎⁎ 0.034⁎⁎ 0.061⁎⁎⁎ 0.023⁎⁎⁎ 0.067⁎⁎ −0.044⁎⁎⁎ 0.072⁎⁎⁎ −0.033⁎⁎⁎
FF3F 0.100⁎⁎⁎ 0.039 0.061⁎⁎⁎ 0.020⁎⁎⁎ 0.063⁎ −0.043⁎⁎⁎ 0.080⁎⁎⁎ −0.024⁎⁎
FF4F 0.101⁎⁎⁎ 0.039 0.062⁎⁎⁎ 0.024⁎⁎⁎ 0.065⁎⁎ −0.041⁎⁎⁎ 0.077⁎⁎⁎ −0.026⁎⁎
Standardizedt-statistics
CBBM 0.057 0.078⁎⁎⁎ −0.021⁎ 0.054 0.055 −0.001 0.003 0.023
MAEW 0.127⁎⁎⁎ 0.070⁎⁎⁎ 0.057⁎⁎⁎ 0.022⁎⁎⁎ 0.077⁎⁎⁎ −0.055⁎⁎⁎ 0.105⁎⁎⁎ −0.007
MAVW 0.061 0.044 0.017⁎ 0.057 0.098⁎⁎⁎ −0.041⁎⁎⁎ 0.004 −0.054⁎⁎⁎
MMEW 0.095⁎⁎⁎ 0.043 0.052⁎⁎⁎ 0.032⁎⁎⁎ 0.080⁎⁎⁎ −0.048⁎⁎⁎ 0.063⁎⁎⁎ −0.037⁎⁎⁎
MMVW 0.096⁎⁎⁎ 0.043 0.053⁎⁎⁎ 0.032⁎⁎⁎ 0.080⁎⁎⁎ −0.048⁎⁎⁎ 0.064⁎⁎⁎ −0.037⁎⁎⁎
FF3F 0.100⁎⁎⁎ 0.047 0.053⁎⁎⁎ 0.036⁎⁎⁎ 0.077⁎ −0.041⁎⁎⁎ 0.064⁎⁎⁎ −0.030⁎⁎
FF4F 0.104⁎⁎⁎ 0.051 0.053⁎⁎⁎ 0.038⁎⁎⁎ 0.084⁎⁎ −0.046⁎⁎⁎ 0.066⁎⁎⁎ −0.033⁎⁎
Rankstatistics
CBBM 0.045 0.061 −0.016 0.061 0.042 0.019⁎ −0.016 0.019⁎⁎⁎
MAEW 0.080⁎⁎⁎ 0.048 0.032⁎⁎⁎ 0.027⁎⁎⁎ 0.063⁎ −0.036⁎⁎⁎ 0.053⁎⁎⁎ −0.015
MAVW 0.089⁎⁎⁎ 0.050 0.039⁎⁎⁎ 0.028⁎⁎⁎ 0.065⁎⁎ −0.037⁎⁎⁎ 0.061⁎⁎⁎ −0.015
MMEW 0.088⁎⁎⁎ 0.042 0.046⁎⁎⁎ 0.021⁎⁎⁎ 0.060 −0.039⁎⁎⁎ 0.067⁎⁎⁎ −0.018⁎
MMVW 0.082⁎⁎⁎ 0.048 0.034⁎⁎⁎ 0.021⁎⁎⁎ 0.063⁎ −0.042⁎⁎⁎ 0.061⁎⁎⁎ −0.015
FF3F 0.084⁎⁎⁎ 0.042 0.042⁎⁎⁎ 0.023⁎⁎⁎ 0.058 −0.035⁎⁎⁎ 0.061⁎⁎⁎ −0.016
FF4F 0.088⁎⁎⁎ 0.046 0.042⁎⁎⁎ 0.025⁎⁎⁎ 0.056 −0.031⁎⁎⁎ 0.063⁎⁎⁎ −0.010
Signstatistics
CBBM 0.045 0.059 −0.014 0.057 0.048 0.009 −0.012 0.011
MAEW 0.068⁎⁎ 0.044 0.024⁎⁎ 0.031⁎⁎⁎ 0.058 −0.027⁎⁎⁎ 0.037⁎⁎⁎ −0.014
MAVW 0.081⁎⁎⁎ 0.046 0.035⁎⁎⁎ 0.032⁎⁎⁎ 0.051 −0.019⁎⁎ 0.049⁎⁎⁎ −0.005
MMEW 0.085⁎⁎⁎ 0.040 0.045⁎⁎⁎ 0.032⁎⁎⁎ 0.060 −0.028⁎⁎⁎ 0.053⁎⁎⁎ −0.020⁎⁎
MMVW 0.076⁎⁎⁎ 0.034⁎⁎ 0.042⁎⁎⁎ 0.033⁎⁎ 0.064⁎⁎ −0.031⁎⁎⁎ 0.043⁎⁎⁎ −0.030⁎⁎⁎
FF3F 0.078⁎⁎⁎ 0.038⁎ 0.040⁎⁎⁎ 0.028⁎⁎⁎ 0.054 −0.026⁎⁎⁎ 0.050⁎⁎⁎ −0.016⁎
FF4F 0.080⁎⁎⁎ 0.049 0.031⁎⁎⁎ 0.030⁎⁎⁎ 0.054 −0.024⁎⁎⁎ 0.050⁎⁎⁎ −0.005
Probabilityofrejectionofthenullhypothesisthatabnormalreturnsequalzero,whennoabnormalperformanceisintroduced.Samplesaredrawnfromthehighest
andlowestmarketequitydeciles.Modelsaredescribedintheappendix.Criticalvaluesof0.05and0.95aregeneratedfromastandardnormaldistribution.
Numbersinparenthesesundercolumns(1),(2),(3),and(4)indicatethep-valuefromatwo-tailedexactbinomialtest.Numbersinparenthesesundercolumnsof
differencesindicatethep-valuefromatwo-tailedFisherexacttest.Significanceatthe10%,5%,and1%levelsandareindicatedby⁎,⁎⁎,and⁎⁎⁎,respectively.
andPRfirmsaregroupedmonthly.SincetheCBBMmodelconstructsabenchmarkportfoliooffirmsfrommatchingmonthly
decilesitperformsbetterthanthemulti-factormodels.
Thoughstatisticallysignificant,thedegreeofbiasinthemarketandmultifactormodelsissmallineconomicmagnitudefor
mostsamples.However,comparedtoanormaldailyaveragereturnof0.1%,the−0.16%and+0.20%biasesofthePRsamplesare
quitelarge.Inunreportedtests,overathreedaywindow,thehighPRfirmsgenerateanegativebiasofabout−0.45%usingeither
marketormultifactormodels.Thelongertheeventwindowthegreaterwillbethebias.Overafive-daywindowthebiasisabout
−0.75%.Onlythecharacteristic-basedmodelproducesnosignificantbiasinanysample.
4.3.Statisticalpowerofthetests
Inlargesamples,underthenullhypothesiseachoftheteststatisticsshouldapproximateastandardnormalrandomvariable.
However,thetruetestofastatisticisitsempiricalrejectionfrequencies.TheminimizationofTypeIandTypeIIerrorsor,inother
words,theabilitytoacceptthenullhypothesiswhenitistrueandtorejectitwhenitisfalse,arethetwocriteriabywhichatest
statisticisjudged.ThenextsectionsreportTypeIandTypeIIerrorresultsforeachofthesamplegroupings,startingwithsamples
formedbymarketequity.6
6 Unreportedresultsfromrandomlyselectedsamplesconfirmfindingsintheeventstudymethodologyliterature(BW,Corrado,1989;CorradoandZivney,
1992).Inparticular,allthepredictionmodelsgenerateabnormalreturnsinsignificantlydifferentthanzero,andthenon-parametricrankandsignstatistics
provideconsiderablymorepowerthandotheparametrictandstandardizedt-statistics.Performancemeasuresoftheteststatisticsarenotreported,butare
availableuponrequest.
474 K.R.Ahern/JournalofEmpiricalFinance16(2009)466–482
Table5
PowerofteststatisticsofMEsamples.
Abnormalperformance(%)
Lowertail Uppertail
High ME Low ME High ME Low ME
Statistic/model −1 −1 −1 −1 +1 +1 + 1 +1
2 2 2 2
t-statistics
CBBM 99.8 100.0 42.8 90.0 99.4 100.0 44.4 90.6
MEAN 99.3 100.0 42.4 90.2 98.2 100.0 52.4 94.2
MAEW 99.9 100.0 43.9 90.6 99.1 100.0 52.6 94.5
MAVW 99.9 100.0 38.5 87.8 99.8 100.0 56.4 95.6
MMEW 100.0 100.0 43.6 90.6 99.3 100.0 52.8 94.8
MMVW 100.0 100.0 43.3 90.8 99.6 100.0 52.7 94.6
FF3F 100.0 100.0 44.3 91.2 99.9 100.0 52.9 94.2
FF4F 100.0 100.0 44.1 91.2 99.7 100.0 52.5 94.3
Standardizedt-statistics
CBBM 100.0 100.0 74.0 99.8 100.0 100.0 68.4 99.7
MEAN 100.0 100.0 82.0 100.0 99.7 100.0 88.4 100.0
MAEW 100.0 100.0 83.4 100.0 100.0 100.0 83.7 100.0
MAVW 100.0 100.0 74.9 100.0 100.0 100.0 86.6 100.0
MMEW 100.0 100.0 84.5 100.0 100.0 100.0 89.8 100.0
MMVW 100.0 100.0 84.2 100.0 100.0 100.0 89.4 100.0
FF3F 100.0 100.0 84.1 100.0 100.0 100.0 89.5 100.0
FF4F 100.0 100.0 84.0 100.0 100.0 100.0 89.3 100.0
Rankstatistics
CBBM 100.0 100.0 87.7 100.0 100.0 100.0 86.8 100.0
MAEW 100.0 100.0 99.9 100.0 100.0 100.0 100.0 100.0
MAVW 100.0 100.0 98.8 100.0 100.0 100.0 98.2 100.0
MMEW 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
MMVW 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
FF3F 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
FF4F 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Signstatistics
CBBM 100.0 100.0 86.7 100.0 100.0 100.0 84.0 100.0
MAEW 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
MAVW 100.0 100.0 99.4 100.0 100.0 100.0 99.1 100.0
MMEW 100.0 100.0 100.0 100.0 99.9 100.0 100.0 100.0
MMVW 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
FF3F 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
FF4F 100.0 100.0 99.9 100.0 100.0 100.0 100.0 100.0
ProbabilityofrejectionofthenullhypothesisthatabnormalreturnsequalzeroforsamplestakenrandomlyfromthehighestandlowestMEdeciles.Valuesinthe
tablereportthepercentageof1000samplesinwhichthenullhypothesisofnoabnormalreturnsisrejected.Criticalvaluesof5%and95%aregeneratedfroma
standardnormaldistribution.Modelsaredescribedintheappendix.Readingacrosseachrowpresentsthepoweratthe5%significancelevelsofeachofthe
predictionmodel–statisticaltestcombinationforloweranduppertailtests.
4.3.1.Marketequitysamples
Table4presentsrejectionfrequenciesoftheteststatistic-predictionmodelcombinationswhennoabnormalperformanceis
artificiallyintroducedandsamplesaretakenfromeitherthehighestorlowestNYSEdecileofmarketequity.7Unlessotherwise
stated,alltherejectionfrequenciesandpowermeasuresreportedarecomputedusingapre-eventestimationperiod,sincethisis
mostcommon.Correctlyspecifiedstatisticswillrejectthenullhypothesiswithafrequencyequaltothenominalsizeofthetest.
Columns(1)and(3)presentlower-andupper-tailedtestsatthe5%level,respectively.8
HighMEsamplesareincorrectlyrejectedbyalmostallofthepredictionmodel-teststatisticcombinationsinlowertailtests.In
uppertailtests,themodelsacceptthenullofnoabnormalreturnstoooften.Thisisconsistentwiththenegativemeanbiasinhigh
MEsamplespresentedinTable3.Therearenoeconomicmagnitudeshere,butinsteadtheseresultsshowthatevenwithverylittle
meanbias,thetestswillover-orunder-rejectinsamplesofhighMEfirms.Theover-rejectioninthelowertailissubstantial.The
commonlyemployedmarketmodel-tstatisticcombinationrejectsthenullalmosttwiceasoftenasitshould(0.095vs0.050).This
findingisconsistentacrosstheteststatistics,thoughusingthevalue-weightedindexwiththet-statisticsreducestheseerrors.
Skewed returns data or prediction models with biased means may lead to skewed test statistics such that the rejection
frequenciesareunbalancedbetweenthetails.ThelastcolumnofTable4,reportsthathighMEsamplesyieldatendencytoover-
7 Themean-adjustedmodeltestedwiththesignandranktestsarebiasedbyconstructionandproducegreatlymisspecifiedresultsinallsamplegroupings.For
thisreason,theseresultsarenotpresentedinthefollowingtables.
8 Testsatthe1%levelalsowereconducted.Theseresultsareavailableuponrequest.
K.R.Ahern/JournalofEmpiricalFinance16(2009)466–482 475
Table6
RejectionfrequencyofPRsamples.
Prb0.05 PrN0.95 Prb.05–PrN.95
High Low Difference High Low Difference High Low
(1) (2) (1)−(2) (3) (4) (3)−(4) (1)−(3) (2)−(4)
t-statistics
CBBM 0.030⁎⁎⁎ 0.067⁎⁎ −0.037⁎⁎⁎ 0.028⁎⁎⁎ 0.076⁎⁎⁎ −0.048⁎⁎⁎ 0.002 −0.009
MEAN 0.101⁎⁎⁎ 0.013⁎⁎⁎ 0.088⁎⁎⁎ 0.009⁎⁎⁎ 0.169⁎⁎⁎ −0.160⁎⁎⁎ 0.092⁎⁎⁎ −0.156⁎⁎⁎
MAEW 0.028⁎⁎⁎ 0.039 −0.011 0.027⁎⁎⁎ 0.083⁎⁎⁎ −0.056⁎⁎⁎ 0.001 −0.044⁎⁎⁎
MAVW 0.020⁎⁎⁎ 0.028⁎⁎⁎ −0.008 0.047 0.098⁎⁎⁎ −0.051⁎⁎⁎ −0.027⁎⁎⁎ −0.070⁎⁎⁎
MMEW 0.098⁎⁎⁎ 0.013⁎⁎⁎ 0.085⁎⁎⁎ 0.010⁎⁎⁎ 0.158⁎⁎⁎ −0.148⁎⁎⁎ 0.088⁎⁎⁎ −0.145⁎⁎⁎
FF3F 0.093⁎⁎⁎ 0.017⁎⁎⁎ 0.076⁎⁎⁎ 0.009⁎⁎⁎ 0.164⁎⁎⁎ −0.155⁎⁎⁎ 0.084⁎⁎⁎ −0.147⁎⁎⁎
FF4F 0.096⁎⁎⁎ 0.017⁎⁎⁎ 0.079⁎⁎⁎ 0.010⁎⁎⁎ 0.164⁎⁎⁎ −0.154⁎⁎⁎ 0.086⁎⁎⁎ −0.147⁎⁎⁎
Standardizedt-statistics
CBBM 0.058 0.088⁎⁎⁎ −0.030⁎⁎⁎ 0.041 0.067⁎⁎ −0.026⁎⁎⁎ 0.017⁎ 0.021⁎
MEAN 0.160⁎⁎⁎ 0.014⁎⁎⁎ 0.146⁎⁎⁎ 0.018⁎⁎⁎ 0.218⁎⁎⁎ −0.200⁎⁎⁎ 0.142⁎⁎⁎ −0.204⁎⁎⁎
MAEW 0.057 0.076⁎⁎⁎ −0.019⁎ 0.057 0.080⁎⁎⁎ −0.023⁎⁎ 0.000 −0.004
MAVW 0.044 0.053 −0.009 0.077⁎⁎⁎ 0.091⁎⁎⁎ −0.014 −0.033⁎⁎⁎ −0.038⁎⁎⁎
MMEW 0.166⁎⁎⁎ 0.012⁎⁎⁎ 0.154⁎⁎⁎ 0.019⁎⁎⁎ 0.222⁎⁎⁎ −0.203⁎⁎⁎ 0.147⁎⁎⁎ −0.210⁎⁎⁎
MMVW 0.165⁎⁎⁎ 0.013⁎⁎⁎ 0.152⁎⁎⁎ 0.018⁎⁎⁎ 0.220⁎⁎⁎ −0.202⁎⁎⁎ 0.147⁎⁎⁎ −0.207⁎⁎⁎
FF3F 0.163⁎⁎⁎ 0.013⁎⁎⁎ 0.150⁎⁎⁎ 0.018⁎⁎⁎ 0.219⁎⁎⁎ −0.201⁎⁎⁎ 0.145⁎⁎⁎ −0.206⁎⁎⁎
FF4F 0.168⁎⁎⁎ 0.014⁎⁎⁎ 0.154⁎⁎⁎ 0.018⁎⁎⁎ 0.211⁎⁎⁎ −0.193⁎⁎⁎ 0.150⁎⁎⁎ −0.197⁎⁎⁎
Rankstatistics
CBBM 0.024⁎⁎⁎ 0.109⁎⁎⁎ −0.085⁎⁎⁎ 0.077⁎⁎⁎ 0.029⁎⁎⁎ 0.048⁎⁎⁎ −0.053⁎⁎⁎ 0.080⁎⁎⁎
MAEW 0.135⁎⁎⁎ 0.009⁎⁎⁎ 0.126⁎⁎⁎ 0.012⁎⁎⁎ 0.130⁎⁎⁎ −0.118⁎⁎⁎ 0.123⁎⁎⁎ −0.121⁎⁎⁎
MAVW 0.119⁎⁎⁎ 0.009⁎⁎⁎ 0.110⁎⁎⁎ 0.016⁎⁎⁎ 0.123⁎⁎⁎ −0.107⁎⁎⁎ 0.103⁎⁎⁎ −0.114⁎⁎⁎
MMEW 0.133⁎⁎⁎ 0.009⁎⁎⁎ 0.124⁎⁎⁎ 0.014⁎⁎⁎ 0.107⁎⁎⁎ −0.093⁎⁎⁎ 0.119⁎⁎⁎ −0.098⁎⁎⁎
MMVW 0.116⁎⁎⁎ 0.010⁎⁎⁎ 0.106⁎⁎⁎ 0.015⁎⁎⁎ 0.112⁎⁎⁎ −0.097⁎⁎⁎ 0.101⁎⁎⁎ −0.102⁎⁎⁎
FF3F 0.132⁎⁎⁎ 0.014⁎⁎⁎ 0.118⁎⁎⁎ 0.015⁎⁎⁎ 0.101⁎⁎⁎ −0.086⁎⁎⁎ 0.117⁎⁎⁎ −0.087⁎⁎⁎
FF4F 0.132⁎⁎⁎ 0.018⁎⁎⁎ 0.114⁎⁎⁎ 0.018⁎⁎⁎ 0.100⁎⁎⁎ −0.082⁎⁎⁎ 0.114⁎⁎⁎ −0.082⁎⁎⁎
Signstatistics
CBBM 0.028⁎⁎⁎ 0.094⁎⁎⁎ −0.066⁎⁎⁎ 0.079⁎⁎⁎ 0.019⁎⁎⁎ 0.060⁎⁎⁎ −0.051⁎⁎⁎ 0.075⁎⁎⁎
MAEW 0.109⁎⁎⁎ 0.012⁎⁎⁎ 0.097⁎⁎⁎ 0.026⁎⁎⁎ 0.124⁎⁎⁎ −0.098⁎⁎⁎ 0.083⁎⁎⁎ −0.112⁎⁎⁎
MAVW 0.101⁎⁎⁎ 0.011⁎⁎⁎ 0.090⁎⁎⁎ 0.027⁎⁎⁎ 0.118⁎⁎⁎ −0.091⁎⁎⁎ 0.074⁎⁎⁎ −0.107⁎⁎⁎
MMEW 0.095⁎⁎⁎ 0.016⁎⁎⁎ 0.079⁎⁎⁎ 0.019⁎⁎⁎ 0.095⁎⁎⁎ −0.076⁎⁎⁎ 0.076⁎⁎⁎ −0.079⁎⁎⁎
MMVW 0.084⁎⁎⁎ 0.016⁎⁎⁎ 0.068⁎⁎⁎ 0.025⁎⁎⁎ 0.097⁎⁎⁎ −0.072⁎⁎⁎ 0.059⁎⁎⁎ −0.081⁎⁎⁎
FF3F 0.089⁎⁎⁎ 0.019⁎⁎⁎ 0.070⁎⁎⁎ 0.027⁎⁎⁎ 0.077⁎⁎⁎ −0.050⁎⁎⁎ 0.062⁎⁎⁎ −0.058⁎⁎⁎
FF4F 0.101⁎⁎⁎ 0.022⁎⁎⁎ 0.079⁎⁎⁎ 0.021⁎⁎⁎ 0.082⁎⁎⁎ −0.061⁎⁎⁎ 0.080⁎⁎⁎ −0.060⁎⁎⁎
Probabilityofrejectionofthenullhypothesisthatabnormalreturnsequalzero,whennoabnormalperformanceisintroduced.Samplesaredrawnfromthehighest
andlowestpriorreturndeciles.Modelsaredescribedintheappendix.Criticalvaluesof0.05and0.95aregeneratedfromastandardnormaldistribution.Numbers
inparenthesesundercolumns(1),(2),(3),and(4)indicatethep-valuefromatwo-tailedexactbinomialtest.Numbersinparenthesesundercolumnsof
differencesindicatethep-valuefromatwo-tailedFisherexacttest.Significanceatthe10%,5%,and1%levelsandareindicatedby⁎,⁎⁎,and⁎⁎⁎,respectively.
rejectinthelowertailcomparedtotheuppertail.ThedifferencescolumnsinTable4showthattheasymmetrybetweenuppertail
andlowertailtestsaswellastheasymmetricrejectionsbetweenhighandlowMEfirmsisstatisticallysignificant.
Ofallthemodels,onlytheCBBMmodelmakesveryfewTypeIerrors.Thismodelincombinationwiththet-statisticorthesign
statisticiscorrectlyspecifiedinthesize-basedsamples.Itneitherover-orunder-rejectsthenullandisnotasymmetricinthetails
either.Whenusedwithastandardizedt-statisticitslightlyover-rejectsinthelowertailforsmallfirms.Whenusedwiththerank
statisticitexhibitsaslightasymmetrybetweenhighandlowMEfirmsintheuppertailtest,rejectinghighMEfirmsmoreoften
thanlowMEfirms.Insummary,usingstandardeventstudymethodswithsamplesoflargefirmswillleadtofalsefindingsof
negativereturns,thoughacharacteristic-basedbenchmarkapproachdoesnotsufferfromthiserror.
Teststatisticsalsomustbeabletorejectthenullwhenitisfalse.Followingprevioussimulationstudies,abnormalperformance
isartificiallyintroducedintothereturnsdatabyaddingafixedreturntotheobservedreturnintheamountsof−0.005,−0.010,
+0.005,and+0.010.Thetestsarerunidenticallyasbefore.Atestwithhighpowershouldmoreoftenrejectthenullhypothesisfor
everysamplesimulated.TheactualrejectionfrequenciesunderabnormalperformancearereportedinTable5.
ThemorewidelydisperseddistributionsofthelowMEfirms,comparedtothehighMEfirmsappearstoresultinlowerpower
forlowMEfirms.Theequal-weightedmarketmodelwithat-statisticrejectsapositive0.005abnormalperformance99.3%ofthe
timeforhighMEfirms,butonlyrejectsatarateof52.8%forlowMEfirms.Thisproblemismostacuteinthetandstandardizedt
testsandisalleviatedthemostbyusingsignstatistics.Itisalsothecasethatthetandstandardizedttestsaremorelikelytodetect
positiveabnormalperformancethannegativeabnormalperformanceinlowMEfirms.ThissupportstheTypeIresultsthatsmall
firmsaremorelikelytofalselyexhibitpositiveabnormalreturns.
Powerisincreasedtremendouslybyusingtherankandsigntestscomparedtothetandstandardizedt.UsingtheMMEW
model,theranktestcorrectlydetectsabnormalperformanceof−0.005overtwiceasofteninlowMEfirmsasdoesthet-statistic
(100%vs.43.6%).Powerisalsoimprovedslightlyinalltheteststatisticswiththeuseofmultifactormodels.Thisisevidencedby