Conducting event studies on a small stock exchange

Transcription

1 WORKING PAPER F Jan Barholdy, Denns Olson & Paula Peare Conducng even sudes on a small sock exchange Fnance Research Group

2 Conducng even sudes on a small sock exchange Jan Barholdy * jby@asb.dk Aarhus School of Busness Deparmen of Busness Sudes Fuglesangs Allé 4 DK-80 Aarhus V Denns Olson dolson@ausharjah.edu Amercan Unversy of Sharjah Paula Peare ppe@asb.dk * Correspondng auhor.

3 Conducng even sudes on a small sock exchange Absrac hs paper analyses wheher s possble o perform an even sudy on a small sock exchange wh hnly rade socks. he man concluson s ha even sudes can be performed provded ha ceran adjusmens are made. Frs, a mnmum of 5 evens appears necessary o oban accepable sze and power n sascal ess. Second, rade o rade reurns should be used. hrd, one should no expec o conssenly deec abnormal performance of less han abou % (or perhaps even %), unless he sample conans prmarly hckly raded socks. Fourh, nonparamerc ess are generally preferable o paramerc ess of abnormal performance. Ffh, researchers should presen separae resuls for hckly and hnly raded sock groups. Fnally, when nonnormaly, even nduced varance, unknown even day, and problems of very hn radng are all consdered smulaneously, no one es sasc or ype of es sasc domnaes he ohers. Keywords: Even sudes, hn radng. INRODUCION Followng he semnal arcles by Ball and Brown (968) and Fama, Fsher, Jensen and Roll (969), even sudes have become one of he mos wdely used emprcal echnques n fnance and accounng. hey are normally desgned o deec abnormal prce changes n fnancal asses n he me perod around varous evens. Crucal o he process s he ably for he researcher o accuraely deermne wha consues abnormal performance, regardless of he nsuonal seng. Alhough even sudy mehods are well developed and ofen used o es fnancal heores for he US and oher well-esablshed sock exchanges, here s some concern regardng effcency when appled o small sock exchanges domnaed by hnly raded socks (.e. socks ha do no rade every day). Henkel and Kraus (988), Campbell and Wasley (993) and Cowan and Sergean (996) have dscussed varous modfcaons n even sudy echnques o adjus for hn radng. Alhough even sudes dae back o he 930s, he papers by Ball and Brown (968) and Fama e. al. (969) nroduced he mehods used oday. MacKnlay (997) conans an excellen descrpon of he hsory and mplemenaon of even sudes.

4 However, hese sudes are based upon US sock reurn daa derved from he CRSP apes he prmary source of daa for mos even sudes. he nsuonal seng n oher counres dffers, and on some sock exchanges he problem s no jus hn radng, bu one of very hnly raded socks. In such suaons, quesons arse as o wheher even sudes can be relably conduced wh daly daa, or wheher researchers need o use weekly or monhly reurns daa. Maynes and Rumsey s (993) sudy of hn, moderae, and hckly raded Canadan socks provdes a good framework for conducng even sudes on a small sock exchange. Usng rade o rade reurns o adjus for hn radng and nonparamerc es sascs o deal wh nonnormaly, hey were able o successfully deec daly abnormal reurns n all socks for all hree radng frequences. her fndngs are encouragng, bu s no known f her resuls generalse o even sudes ousde of Norh Amerca. A parcular concern s ha Maynes and Rumsey s (993) sample of hnly raded socks averaged 3.67 days beween rades. Even he average sock on some small exchanges may rade less frequenly han hs. For example, he average number of days beween rades for a sock on he Copenhagen Sock Exchange (CSE) was 6.86 days n 990 and 4.7 days n 00. Durng varous years from 990 o 00, he radng frequency for hnly raded CSE socks ranged from o 3 days. he purpose of hs paper s o provde furher evdence abou he effcency of even sudy echnques when appled o hnly raded socks by examnng daa from he Copenhagen Sock Exchange (CSE). I s a small sock exchange where hn (or very hn) radng s exensve. he sudy apples Brown and Warner s (980, 985) well-even sudy mehods o hree samples of Dansh socks dfferenaed by her relave radng frequences (hck, medum, and hn). Followng Maynes and Rumsey (993), we use rade o rade reurns o deal wh hn radng and apply a baery of paramerc and nonparamerc ess, as dscussed by Brown and Warner (985) o deec abnormal performance. In addon, we apply he procedures of Boehmer, Musumec, and Poulsen (99) o examne even nduced varance and consder he case where he even day s unceran. Specfcally, we smulae an even sudy by randomly selecng even days for varous socks. On he even day, daly reurns are ncreased by ncremens rangng from zero o wo percen o smulae he mpac of new nformaon. hs procedure s repeaed 000 mes for porfolos of 0, 5, and 50 socks o deermne he power and sze of varous paramerc and nonparamerc es sascs. Abnormal performance n boh frequenly and nfrequenly raded Dansh socks was successfully deeced--provdng evdence ha even sudes probably can be successfully conduced on oher small sock exchanges.

5 . CALCULAION OF REURN UNDER HIN RADING he mpac of new nformaon on he value of a gven sock s measured by he dfference beween he acual reurn a me, r, and he expeced reurn E(r ). hs dfference s called he abnormal reurn, A, and s gven by: A = r E[ r] Expeced reurn can be obaned from esmaon of he marke model so Er [ ] ˆ ˆ = α + βrm ] where r m s he marke reurn a me and ˆα and ˆ β are obaned from OLS esmaon of he regresson r = + rm + α β ε durng he esmaon perod. he sandard esmaon perod s beween 00 and 50 observaons;.e. abou a year of radng pror o a hree day even wndow. Esmaon of abnormal reurns for frequenly raded socks s herefore relavely sraghforward. 3 Nonradng, and he subsequen problem of mssng reurn observaons, s no encounered n mos even sudes. Mos emprcal evdence for even sudes n fnance comes from a sngle daa source he CRSP daa fles for US socks. Daly prce and reurns daa are readly avalable for large socks and for mos small socks. If a sock does no rade on a gven day, prce s ypcally recorded on he CRSP apes as he average of he bd-ask quoes. Provded hese quoes are realsc, a daly reurn seres wh no mssng observaons can be calculaed. In conras, here may be no meanngful bd or ask prces n he order book on any gven day for hnly raded socks on oher sock exchanges. here s no sandard mehod o calculae daly Brown and Warner (980, 985) used wo addonal mehods for measurng expeced reurn: mean adjused reurn and marke adjused reurn. For mean adjused reurn, expeced reurn s se equal o he mean of he reurn over he esmaon perod (equvalen o seng β = 0 n he marke model). For marke adjused reurn, he mean of he marke reurn over he esmaon perod s used for expeced reurn (equvalen o seng α = 0 and β = n he marke model). hese mehods are smpler han he marke model and sgnfcanly reduce compung requremens- -a major concern n he 980s. Snce he marke model ncludes boh of hese mehods as specal cases, and compung requremens are no longer a concern, only he marke model s consdered here. 3 As poned ou by Scholes and Wllams (977), OLS bea esmaes are based downward for secures more hnly raded han he marke ndex. However, OLS probably remans he bes echnque for esmang abnormal reurns n he presence of hn radng. Cowan and Sergean (996) repor ha he Scholes-Wllams correcon for nonsynchronous radng provdes no sgnfcan benef n copng wh he problems caused by hn radng. In fac, he Scholes-Wllams (977) and Dmson (979) correcons for nonsynchroncy can even make maers worse. For example, n a sudy of he bases nduced by hn radng n New Zealand, Barholdy and Rdng (994) found ha OLS bea esmaes were less based, more effcen, and a leas as conssen as esmaes made usng he Dmson or Scholes-Wllams correcons. 3

6 sock reurns on he days when a sock does no rade, bu four possble echnques could be employed o adjus for nonradng. Probably he leas sasfacory means of dealng wh nonradng s o calculae smple reurns for each sock only for days for whch consecuve prces are avalable. hen, subrac he marke reurn on hese days o oban daly abnormal reurn and gnore ndvdual sock and marke reurns on oher days. hs echnque gves unbased esmaes of abnormal reurns on he days calculaed, bu fals o use nformaon abou reurns on oher days and esmaors based on hese reurns are herefore no effcen. he Copenhagen Sock Exchange and oher small exchanges generally ls he las observed ransacon prce as a sock s prce on nonradng days. Calculang daly reurns from he recorded prce seres herefore gves zero reurns for nonradng days and relavely large posve or negave reurns on days when a sock rades. Maynes and Rumsey (993) refer o reurns generaed n hs manner as lumped reurns because all of he mul-perod reurn s allocaed o he radng day, or he las day of a mul-perod nerval. he numerous zeroes n he reurn seres lead o underesmaes of he varance of reurns and bas he es sascs used o judge abnormal performance. In spe of hese problems, lumped reurns are easy o calculae and s perhaps he mos frequenly used mehod of adjusng for hn radng. he unform mehod s a hrd echnque for handlng hn radng. Insead of fllng n unknown reurn days wh zeroes, calculaes oal reurn beween rades and hen allocaes he average daly reurn o each day over he mul-perod nerval beween rades. he same reurn s recorded for all of he nonradng days, as well as on he day a rade occurs. hs echnque s more effcen han usng smple reurns, bu he es sascs are poenally based jus as when calculang lumped reurns. 4 In addon, unform reurns gve no parcular emphass o he acual rade day, meanng ha some nformaon s gnored when usng hs mehod. Maynes and Rumsey (993) fnd ha he unform reurn mehod performs abou he same as lumped reurns. Calculaon of rade o rade reurns represens a fourh echnque for dealng wh nonradng. he frs sep s o calculae an ndvdual sock s reurn beween he days when ransacons acually ake place. hen, rade-o-rade reurns for he marke ndex are calculaed over he same perod as for he sock. hese wo ses of rade-o-rade reurns are used o 4 If he dfference beween he flled n value and he underlyng unobservable rue value s whe nose hen boh he lumped and unform mehods provde an unbased esmae of reurns. Also, he bas n he lumped reurn mehod may no be oo large f volume and reurns are posvely correlaed. If a lack of volume mples small prce changes, a zero reurn on a nonradng day mgh be a reasonable esmae of he rue unobserved reurn for ha day. See Karpoff (987) for a survey of he relaonshp beween changes n prce and volume. 4

7 esmae he marke model o oban abnormal reurns for he sock over hs perod. Snce emprcal esmaon of abnormal reurn by hs mehod s more dffcul han for lumped reurns, he deals are presened n Appendx I. he rade o rade mehod uses all avalable nformaon abou oal sock and marke reurns over me and no bas s nroduced by aempng o esmae unobserved daly sock reurns. Alhough gnores nformaon abou daly marke reurns over nonradng perods, he small reducon n effcency relave o he lumped sum or unform mehods s more han offse by he desrable propery of unbasedness. For our Dansh daa, we fnd ha rade o rade reurns are he bes way o adjus for he problem of hn radng. hs s conssen wh Maynes and Rumsey (993), who also fnd ha he rade o rade mehod o ou performs he lumped or unform reurn echnques. For brevy, n he remander of he paper, we only repor resuls usng rade o rade mehod of adjusng for hn radng. 3. DAA AND ES SAISICS Prce and volume daa are from he Copenhagen Sock Exchange (CSE). Daly dvdend and spl adjused daa were exraced from Børsdaabasen, whch was mananed by he Århus School of Busness. Børsdaabasen sars n 985, bu daa for he man ndex on he exchange, he KFX (Københavns Fondsbørs Indeksndex), s no avalable unl 990. hroughou hs sudy he KFX reurns seres s reaed as he marke ndex. 5 herefore, he sample sars on he frs radng day n 990 and ends n md-00 he las avalable dae for he daabase. Please nser able abou here. 3. Descrpve sascs Dansh socks are sored no one of hree groups each year (hck, medum, or hn) based on radng frequency. A sock s defned o rade f volume s posve. A hck-raded sock s defned as one radng on more han 80% of radng days, or an average of more han four days per week. Socks n he medum radng group show ransacons on 40% o 80% of all radng 5 Real me values for he KFX are provded by several daa vendors. For example, s lsed as ^KFX on Yahoo- Fnance. he KFX s a value-weghed ndex of he 0 larges Dansh blue chp socks seleced from a ls of he 5 mos acvely raded (lqud) socks durng a monh perod endng n November of he prevous year. Alhough he ndex s no nended o reflec all major ndusry groups, s a good gauge of he Dansh economy. hroughou he sample perod he ndex has generally ncluded frms n food producon, bankng, nsurance, echnology, wnd energy, ransporaon, pharmaceucal, and he real secors. Lke mos ndexes, does no ncluded dvdends. Snce s composed of he mos lqud socks on CSE, s unlkely o suffer from hn radng problems. herefore, s a beer marke ndex han an equal or value-weghed ndex of all socks on he CSE. 5

8 days, or abou wo o four days per week. hnly raded socks rade on less han 40% of all radng days or on average less han wo days per week. able llusraes he growh of radng acvy on he Copenhagen Sock Exchange over he decade of he 990s. he oal number of socks lsed on he exchange ncreases from 04 n 990 o 4 n 00. Parcularly noeworhy s he large ncrease n he percenage of socks radng each day. In 990, abou 60% of all Dansh socks were classfed as hnly raded (70% n 99), whle by md 00 only abou 6% of socks were hnly raded. Smlarly, he percenage of all socks n he hckly raded group rose from abou 5% n 990 o 49% n 00. Even wh he ncrease n radng acvy on he CSE, hn radng remans an mporan ssue n all years. Resrcng a sample o medum and hck raded socks, as ofen done n US even sudes, may no be feasble for smaller sock exchanges. o oban enough even daes and large enough samples for sascal nference, a Dansh even sudy would lkely need o nclude he hnly raded socks whch hsorcally comprsed 4% o 70% of all CSE socks. he average number of days beween rades for CSE hnly raded socks has ranged from abou o 3 days, whle he average across all hree radng groups on he CSE has ranged from 4.7 days n 00 up o.7 days n 99. Noe ha he average Dansh sock rades less frequenly han Maynes and Rumsey s (993) average hnly raded Canadan sock (radng frequency of 3.67 days). herefore, our ess of he Maynes and Rumsey (993) approach under condons of more exreme hn radng condons may be more represenave of he suaon n mos emergng markes and on oher small sock exchanges. Please nser able abou here. able shows he descrpve sascs of sock reurns for each of he hree radng frequency groups usng he rade o rade adjusmen for hn radng. Some unusual feaures of he Dansh daa are ha he average coeffcen of skewness s hgher for he hckly raded group han for he medum raded group and ha hnly raded socks dsplay less excess kuross han her hckly raded counerpars. he daa smply show ha Dansh sock reurns, even among hckly raded socks, devae consderably from normaly. he average R-squared obaned from esmang he marke model (equaon b n Appendx I) over a 47 day perod s que low--rangng from % for he medum raded group o 7% for he hck-raded group. However, here s a relavely large varaon over me n hese numbers (no repored n he ables). For example, he average R-squared n 990 for he hck group was around 5%, bu has fallen o 3% o 5% n he laer years of he sample. he 6

9 explanaon probably les n he change n he composon of he hck-raded group over me. In he early years of he sample he hck raded group conssed of a few large frms (e.g., 6 frms n 990) ha were generally members of he KFX--he marke ndex. By he end of he sample perod, he hck raded group conssed of 8 frms, ncludng many smaller frms no par of he KFX. Snce he hck raded group and he marke have become less smlar over me, he drop n average R-squared values over me seems undersandable. Fnally, he average Durbn-Wason sascs ndcae ha auocorrelaon s unlkely o pose a sgnfcan problem n even sudes. 3. es sascs. Paramerc es sascs for abnormal performance on even days are based on a sandard es of he dfference beween wo means. he numeraor of he es sasc measures he absolue mpac of some even relave o he reurn expeced usng some knd of marke model. he denomnaor scales hs number by some measure of esmaed varance. As dscussed n Brown and Warner (985), paramerc es sascs dffer from one anoher prmarly n he way hey adjus for problems encounered n he daa. hree paramerc es sascs frequenly used n even sudes are calculaed for Dansh socks n he smulaons n he nex secon. Deals abou all es sascs used n hs paper, ncludng formulas, are presened n Appendx II. he sascs examned are: 3 -es wh adjused cross seconal ndependence (Brown and Warner, 985 and Paell, 976) -es wh sandardsed abnormal reurn (Brown and Warner, 985) -es wh adjused sandardsed abnormal reurn (Brown and Warner, 985 and Paell, 976) For he es sasc, he abnormal reurn on he even day s assumed o be ndependen across socks. (An alernave es sasc can be derved f one beleves here s cross seconal dependence n he daa.) In s unadjused form (shown n Appendx II), he varance of he es sasc s he sum of he varances of abnormal reurns of he ndvdual socks. However, o mprove upon he performance of he es sasc, Paell (976) recommends esmang abnormal reurns based on forecass from he marke model, and hen 7

10 calculang he varance of he es sasc as he sum of he varance of he forecas errors for he ndvdual socks. 6 For, abnormal reurns for each sock are scaled by her ndvdual sandard devaons and added ogeher o produce he es sasc. I s a -sasc based on sandardsed abnormal reurns. An adjused verson of hs es sasc, 3, s calculaed usng he sandard devaon of he forecas errors (raher han acual sandard devaons) o scale he abnormal reurns. he second se of ess s nonparamerc and no based upon he assumpon of normaly. Gven ha he Dansh reurns daa n able show consderable devaons from normaly, hese sascs should be more relable han he paramerc measures of abnormal performance. he nonparamerc sascs are: 4 Rank es (Corrado, 989) 5 Sgn es (Corrado and Zvney, 99) 6 Generalsed sgn es (Cowan, 99 and Cowan and Sergean, 996) he rank es, 4, s from Corrado (989) and Corrado and Zvney (99). he sgn es, 5, s from Corrado and Zvney (99), whle he generalsed sgn es, 6, s based upon Cowan (99) and Cowan and Sergean (996). he sgn es, 5, assumes ha he probably of observng eher a negave or posve abnormal reurn s 0.5, whereas for 6 hs probably s esmaed from acual reurns over he esmaon perod. A fnal se of ess s used o deec abnormal performance when here s a change n varance around he even day. Boehmer, Musumec and Poulsen (99) developed a paramerc es and a smple adjusmen o he rank es, 4 provdes a non-paramerc alernave. 7 8 Paramerc es wh varance adjusmen (Boehmer, Musumec and Poulsen, 99) Rank es wh varance adjusmen (Corrado and Zvney, 99 and Maynes and Rumsey, 993) For he prevous sascs ( 6 ) he varance or he varance of he forecas error was esmaed over he esmaon perod-- ha s, pror o he even wndow. In conras 7 and 8 are adjused for changes n varances over he even wndow. 6 es sascs for cross seconal dependence and cross seconal ndependence, as presened n Brown and Warner {980, 985) were also calculaed. hese sascs are relaed o, bu are known have lower power. Our resuls confrmed ha hey dd no perform as well as. Furher descrpon of es sascs and resuls from he lumped reurn adjusmen are avalable from he auhors upon reques. 8

11 Snce he exac even dae ofen s no known, a wndow s assgned o he even and Cumulave Average Reurns (CAR) are calculaed and esed over he wndow. he mos common wndow s plus/mnus one day around he expeced even day. For he smulaons n he nex secon, an even wndow of hree days s used. he even day s randomly assgned (usng a unform dsrbuon) o one of hese days and he hree day CAR measure = + +, ' ' CAR A A0 A ' + s appled o all es sascs o accoun for even day uncerany. 3.3 Smulaon he performance of he varous es sascs s examned usng he smulaon approach of Brown and Warner (980, 985). A oal of 000 porfolos of 0, 5, and 50 socks are generaed by randomly selecng an even day, and hen randomly selecng socks for ncluson n he porfolo. o be ncluded n he porfolo, he sock mus rade every day n he even wndow and have a mnmum of 0 observed prces n he esmaon perod. he lengh of he esmaon perod s 47 days, or approxmaely a year of radng pror o he hree day even wndow. Specfcally, he esmaon perod s from = - 49 o -, he even wndow s from = o =+, and he acual even s a = 0. o smulae he mpac of an even 0, 0.5% and % are added o he abnormal reurn on he even day. he condon ha he sock has o rade every day n he even wndow replcaes he selecon creron ypcally used n even sudes where one wans o ensure ha acual prces are used o measure he effec of an even. However, mposng hs condon can cause problems n real-world even sudes. Socks affeced by an even have a hgher han usual probably of radng durng he even wndow, bu socks no nfluenced by an even are no more lkely o rade han on any oher day. Socks for whch an even has no effec wll be dropped from he sample, hereby basng he sudy n favour of fndng a sgnfcan effec. In erms of a smulaon sudy, a posve correlaon of volume and reurn means ha socks sasfyng he radng condon are more lkely o have abnormal reurns n he even wndow hereby basng he es sascs n favour of fndng sgnfcan an effecs. 4. RESULS 4. Properes of es sascs under he null hypohess Please nser able 3 abou here. 9

12 Under he null hypohess of zero abnormal reurn, all es sascs presened n Secon 3 are assumed o have a sandard normal dsrbuon. Properes of he es sascs, under he null hypohess of zero abnormal performance, are presened n able 3. he emprcal dsrbuons are based on 000 values of he es sascs for porfolos of 5 socks. Under he null hypohess, he mean of each es sasc should be zero. Insead, all es sascs a all radng frequences have a posve mean. hs bas wll lead o over rejecon of he null hypohess meanng ha we may deec even sgnfcance n some nsances when here s none. he lkely cause of hs problem was dscussed n he prevous secon. All es sascs consdered ( o 8 ) have lower means for he hckly raded group han for he medum and hnly raded sock groups. As a consequence, researchers are more lkely o ncorrecly denfy sgnfcan effecs n samples ha conan a large proporon of medum and hnly raded socks. I s herefore mporan o repor resuls for socks across dfferen radng frequences and confrm ha he fndngs are conssen among radng groups. For example, he fndng of a sgnfcan effec for hnly raded socks and no effec for hckly raded socks may sugges problems arsng from samplng echnque. Alhough no sngle es sasc s superor across all radng frequences, he non-paramerc es sascs generally ou perform he paramerc es sascs. he sandard devaon of each of he es sascs should be one under he null hypohess. A sandard devaon of less han one means he null hypohess wll be rejeced oo ofen, whereas a sandard devaon greaer han one leads o less frequen rejecons (falng o fnd an effec when here s one). For he hckly raded group, excep for and 3, he sandard devaon for all he es sascs dffers from uny by less han 0%. For he medum and hnly raded groups, he sandard devaons of he es sascs are generally closer o one for he nonparamerc group of sascs. hs s conssen wh Campbell and Wasley (993) who fnd ha he rank sasc s beer specfed han paramerc sascs under he null hypohess. he sasc, 8, ha accouns for even nduced volaly has a sandard devaon near one and performs well n our smulaons. Skewness and excess kuross of he es sascs should boh be zero under he null hypohess. Wh nonzero skewness, he rejecon frequences for he null hypohess dffer for posve and negave evens. If excess kuross s posve, hen he als are oo hck leadng o ncorrec (generally oo hgh) rejecon frequences under he null hypohess. For he hnly raded group, boh skewness and kuross are a problem for paramerc es sascs; so he researcher should generally rely upon nonparamerc es sascs o denfy he sgnfcance of evens on hese socks. hs s also he case for he hckly and medum raded socks, bu o a lesser exen. 0

13 In general all es sascs are based snce he mean s greaer han one and he bas ncreases wh lower radng frequences. hus, one s more lkely o rejec he null hypohess when s rue for hnly raded socks han for frequenly raded socks. he non-paramerc ess generally appear o be beer specfed han he paramerc ess, bu no sngle ess sasc s superor across all radng frequences and specfcaon crera. 4. Performance of he es sascs: sze and power Please nser able 4 abou here. o deermne he performance of he dfferen es sascs n erms of sze and power, porfolos of socks wh arfcal abnormal reurns of 0%, 0.5%, and % n he even wndow were randomly generaed. he resuls obaned from hese smulaons for porfolos of 50 socks are presened n able 4. For he hckly raded group, he rejecon rae for a 5% sgnfcance level s generally around 5% for all es sascs when no abnormal reurn s added. All of he es sascs perform reasonably well n erms of sze. In erms of power,.e., deecng an effec when exra reurn has been added on he even day, he four non-paramerc ess have he bes performance. For example, hey rejec he null hypohess beween 78% and 88% of he me when 0.5% has been added on he even day. hs s sgnfcanly beer han he paramerc ess for whch rejecon of he null hypohess ranges from a low of 4% for o a hgh of abou 7% for and 7. hese resuls are smlar o hose obaned by Corrado (989) for US daa. However, unlke Maynes and Rumsey s (993) resuls wh Canadan daa, he rank ess do beer han he oher paramerc or nonparamerc es sascs. For he medum radng group, he non-paramerc es sascs perform beer, n erms of boh sze and power han he group of paramerc es sascs. For example, when 0.5% s added o he reurn, he non-paramerc es sascs rejec he null a leas 90% of he me whle he paramerc sascs rejec he null, a mos, 7% of he me. 7 he performance of he paramerc ess s no as good as n Cowan and Sergean (996) for US daa; bu he nonparamerc ess perform smlarly for boh US and Dansh daa. 7 Alhough resuls for he lumped reurn adjusmen for hn radng are no repored n he paper for sake of brevy, lumped reurns acually provde slghly beer resuls n erms of power and sze for he medum raded group. rade o rade reurns do beer for boh hnk and hnly raded socks. Neverheless, snce he lumped reurn adjusmen s relavely easy o mplemen, researchers facng me consrans mgh consder.

14 For he hn radng group, he paramerc es sascs perform very badly n erms of sze when no abnormal reurn s added. Rejecon raes range from.9% for up o 3% for when he null hypohess s rue. Power vares consderably across es sascs when a 0.5% abnormal reurn s added wh rejecon raes rangng from 0% up o 73%. he suaon s slghly beer for nonparamerc es sascs. he rejecon rae when a zero abnormal reurn s added s beween 6.6% and 8.4%. As was he case for he medum radng group, he nonparamerc ess are noceably more powerful han he paramerc ess. Rejecon raes range from abou 77% o 8%. Fnally, we noe ha for an abnormal reurn of % on he even day, all non-paramerc ess have a rejecon rae of 00%. hs s reassurng for deecng abnormal performance n hnly raded socks. o summarze, he resuls n able 4 provde srong suppor for relyng prmarly on nonparamerc sascs when aempng o deec abnormal performance n he presence of hn radng. Please nser able 5 abou here. Please nser able 6 abou here. ables 5 and 6 presen he same nformaon as able 4, bu for porfolos of 5 and 0 socks, respecvely. Wh abnormal reurns of 0% or.0% nduced on even day, he es sascs look smlar o hose n able 4 n erms of boh sze and power. However, he power of all of he ess drops dramacally for he case of 0.5% nduced abnormal reurns. Power of he es was above 80% for all nonparamerc es sascs for 50 sock porfolos, bu power drops o beween 50% and 80% for 5 sock porfolos. Also, noce ha he reduced power s observed for all hree radng groups hnk, medum, and hnly raded socks. A smlar resul s observed for 0 sock porfolos, excep ha he power drops o beween 0% and 50%. Agan, hs s observed for all hree groups of socks. Smlar resuls regardng he dramac reducon n power of es sascs have been observed n even sudy smulaons wh US daa. In general, he power and sze of es sascs n ables 4-6 are no sgnfcanly lower han found n US sudes--suggesng ha researchers may be able o perform even sudes on small sock exchanges usng daly daa. Even n he presence of very hn radng, seems possble o deec abnormal performance. Some recommendaons for oher researchers nclude: ry o denfy a leas 5 occurrences of an even, bu preferably fnd a leas 50 evens. hs s based on he dea ha porfolos of 50 socks provde good sze and power for es

15 sascs, whle porfolos of 5 socks only provde accepable sze and power n some nsances. When he number of evens s 5 or less, abnormal reurns of % or greaer may be needed for he es sascs o relably denfy abnormal performance. If he number of evens s small, researchers can only denfy evens havng major mpacs on sock prces. Snce es sascs for hnly raded socks are based under he null hypohess, he researcher should separae he sample no groups or caegores based on radng frequency, and repor any dfferences n resuls across he frequency groups. he rade o rade mehod of adjusng for hn radng, as done n hs sudy, s preferable o oher hn radng adjusmen mehods, bu lumped reurns perform nearly as well and could be used f an even sudy mus be done quckly. Emphasze nonparamerc, raher han paramerc es sascs for judgng sgnfcance. However, snce no ndvdual es s superor o he ohers for all porfolo szes and radng frequences, researchers may wsh o calculae a baery of es sascs, as done n hs sudy. 5. SOME EXENSIONS 5. Increase n varance around he even day Boh he reurn and varance of a sock can change as a resul of he release of new nformaon on he fnancal markes. heores n behavoural fnance argue ha nvesors need me o process and prce new nformaon, leadng o ncreased reurns volaly durng hs perod. Alernavely, new nformaon may lead o an ncrease n sysemac rsk whch also ncreases volaly around he even day. For example, dfferen socks do no respond n he same way o an even such as a posve or negave earnngs announcemen because he response depends upon he dfference beween expeced and acual earnngs. he effec of hs dfference s wofold: () snce an addonal random amoun s added o he abnormal, he varance of he abnormal reurn for an ndvdual sock ncreases on he even day; and () because he effec s dfferen across companes, he cross-seconal varance s ncreased around he even day compared o he esmaon perod. Whou adjusng for even nduced volaly, he varance esmaed over he esmaon perod s lkely o undersae he varance n he even wndow causng he null hypohess (of no effec) o be rejeced oo ofen. Alhough examnaon of he causes of ncreased varance around an even day s self an neresng opc, he focus of hs paper s on he effec of ncreased varance, regardless of orgn, on he varous es sascs ypcally used n even sudes. 3

16 he procedures from he prevous secon can be modfed o accoun for even nduced volaly. Followng Boehmer e. al. (99), he varance on he even day s assumed o be 50% larger han over he esmaon perod. In able 7, abnormal reurns of 0%,.5%, and % have been added o even day reurns for porfolos of 5 socks. Please nser able 7 abou here. Rejecon raes can be compared wh hose n able 5 for porfolos of 5 socks whou even nduced varance. Wh even nduced volaly, boh paramerc ess and nonparamerc rank ess lead o hgh false rejecon raes of he null when here s no abnormal performance (0% abnormal reurn added). For he hck radng group, he lowes rejecon raes usng he rank ess are abou %, versus 5% n able 5. he wo sgn ess ( 5 and 6 ) have god performance n erms of sze. As expeced, he 7 es desgned specfcally o handle he problem of even nduced varance, also performs que well. For he medum and hnly raded sock groups, he performance of he paramerc ess (and o a lesser exen he rank ess) are worse han for hckly raded socks. For example, he sandardsed abnormal reurn es, has a rejecon rae of 37% for hnly raded socks when here s no abnormal reurn. Boehmer e. al. (99) dd no consder hnly raded secures, bu our resuls for he hckly raded group are n lne wh her fndngs. From he evdence above, large ncreases n even nduced varance appear o lead o a msspecfcaon of paramerc es sascs and nonparamerc rank sascs o such an exen ha hey are no relable n even sudes. Examnaon of rejecon raes for 0% addonal abnormal reurn for he sgn ess ( 5 and 6 ) and he paramerc es desgned o deal wh ncreased varance around he even day ( 7 ) suggess ha all hree ess are reasonably well specfed n erms of sze. For an nduced abnormal performance of 0.5%, he power of he sgn ess are 6% and 0% for he hckly raded group and even smaller for he Medum and hnly raded groups. For 7 he power s only margnally beer han for he sgn ess. For 0.5% abnormal reurns, he power of all of he ess s such ha one s unlkely o fnd any effec, even f one exss. For an nduced abnormal performance of % n he hckly raded group, power s above 90% for he for he sgn ess ( 5 and 6 ) and above 97% for 7 and 8. he power for all four es sascs drops for he Medum and hnly raded groups. he performance of 7, n parcular, deeroraes for he less frequenly raded socks. However, he power of 8 does no drop below 89% for any of he radng groups. Overall, 7, he varance adjused sandardsed abnormal reurns es proposed by Boehmer e al (99), appears o be he bes es when here s 4

17 an ncrease n varance around he even day. he sgn ess, 5 and 6, also have accepable sze and power and can be used o verfy he resuls obaned usng Unknown even day. o deermne performance of he es sascs for he suaon where he even day s no known wh cerany, he smulaons were repeaed for an even wndow of hree days where he even day was randomly assgned (usng a unform dsrbuon) o one of he hree days. he mos obvous effec of an unknown even day s ha 33% of he socks are expeced o experence an even on a gven day, so he average abnormal reurn on he even days s dlued 8. he queson s wheher he CAR es can compensae for hs dluon by summng he reurns over he hree days. he resuls from he smulaons made for porfolos of 5 socks are repored n able 8. he op panel of able 8 shows he average daly resuls, whle he boom panel presens rejecon raes based upon cumulave average abnormal reurns. Please nser able 8 abou here. Relave o a known even day, here s a marked drop n rejecon raes across all radng frequency groups for.5% and % nduced addonal abnormal reurns. For example, for he medum raded group, rejecon raes for he bes es sascs drop from around 66% n able 5 o 7% n he op panel of able 8 for an nduced addonal abnormal reurn of 0.5%. For a % arfcal abnormal reurn, he drop n rejecon raes across all radng frequences s from around 99% n able 5 down o 66% (a bes). he wo sandardsed abnormal reurn es sascs, and 3, perform he bes overall n erms of power of he es. hs s n conras o he beer performance of he rank sascs ( 4 and 8 ) n able 5 and he superor performance of he sgn sascs ( 5 and 6 ) and he 7 sasc n able 7. hus, he nroducon of a random even day no only reduces power across es sascs, may also nfluence he researcher s choce of opmal es sascs. Examnaon of he CARs n he boom panel of able 8 ndcaes ha he power of ess ncreases subsanally relave o he same sascs n he op panel of he able. Rejecon raes for deecon of a % nduced abnormal reurn ncrease o abou 95% for he and 3 sascs 8 A normal dsrbuon could be used o assgn even days. I assgns more evens o he orgnal even day and leads o less dluon of abnormal reurns. We chose he unform dsrbuon mehod o ensure a more srngen es of he mpac of an unknown even day. 5

18 across all hree radng groups. hese paramerc es sascs appear o ou perform he nonparamerc ess n hs suaon, alhough he nonparamerc es sascs are sll preferred on he bass of sze. ha s, he sze of he es sascs for and 3 s above 7%, whle he sze of he rank sascs s below 6%. he CAR mehod has good power and s able o deec abnormal reurns above % when he even day s unceran. However, all es sascs are raher small for nduced abnormal reurns of 0.5%. he rejecon raes are less han 50% across all radng groups. Neverheless, by usng he rank and sandardsed ess ogeher, CAR analyss seems o have reasonable power and sze o denfy abnormal reurns above % for random even days. 5. CONCLUSION hs paper has analysed he effcency of even sudy mehods n he presence of hn radng usng daa from he Copenhagen Sock Exchange. he man concluson s ha even sudes usng daly daa can be meanngfully performed on a small exchange provded ha ceran adjusmens are made o accoun for he problems caused by very hn radng. Frs, researchers should adop less resrcve selecon crera han used n US even sudes n order o have enough observaons or evens o relably deec abnormal performance. For he Dansh marke, a mnmum of 5 evens appears necessary o oban accepable sze and power n sascal ess. If possble, 50 or more evens provde even beer sze and power n he es sascs. Second, an adjusmen for hn radng s necessary o oban daly reurns. he rade o rade adjusmen mehod s preferred, bu a lumped reurn adjusmen would perform almos as well f even sudy resuls were needed quckly. hrd, researchers should no expec o be able o conssenly deec abnormal performance of less han % (or perhaps even less han %), unless he sample conans prmarly hckly raded socks. Fourh, due o he nonnormaly of Dansh sock reurns, nonparamerc ess are generally preferable o paramerc ess of abnormal performance excep n he cases of even nduced volaly and unknown even day. Ffh, researchers should presen separae resuls for hckly and hnly raded sock groups. If here are noceable dfferences beween groups, furher research can be underaken o deermne he cause of he nconssences. Fnally, when nonnormaly, even nduced varance, unknown even day, and problems of very hn radng are all consdered smulaneously, no one es sasc or ype of es sasc domnaes he ohers. A researcher can use a varey of paramerc and nonparamerc ess o deec abnormal performance. If all ess agree, he researcher can be farly confden of resuls. When here are dfferences beween ess, researchers can ry o deermne he source of problems and perhaps refer o resuls n hs paper regardng whch ess are lkely o be more relable n he presence of varous problems wh he daa. 6

19 REFERENCES Ball, R. and Brown, P. (968) An emprcal evaluaon of accounng numbers, Journal of Accounng Research, 6(), Barholdy, J. and Rdng, A. (994) hn radng and he esmaon of beas: he effcacy of alernave echnques, Journal of Fnancal Research, 7(), Boehmer, E., Musumec, J. and Poulsen, A. (99) Even-sudy mehodology under condons of even-nduced varance, Journal of Fnancal Economcs, 30(), Brown, S. J. and Warner, J.B. (980) Measurng secury prce performance, Journal of Fnancal Economcs, 8(3), Brown, S. J. and Warner, J.B. (985) Usng daly sock reurns: he case of even sudes of even-nduced varance, Journal of Fnancal Economcs, 4(), 3-3. Campbell, C. J. and Wasley, C. E. (993) Measurng secury prce performance usng daly NASDAQ reurns, Journal of Fnancal Economcs, 33(), Corrado, C. J. (989) A nonparamerc es for abnormal secury-prce performance n even sudes, Journal of Fnancal Economcs, 3(), Corrado, C.J. and Zvney,.L. (99) he specfcaon and power of he sgn es n even sudy hypohess ess usng daly sock reurns, Journal of Fnancal and Quanave Analyss, 7(3), Cowan, A. (99) Nonparamerc even sudy ess, Revew of Quanave Fnance and Accounng, (4), Cowan, A. and Sergean, A. (996) radng frequency and even sudy es specfcaon, Journal of Bankng & Fnance, 0(0), Dmson, E. (979) Rsk measuremen when shares are subjec o nfrequen radng, Journal of Fnancal Economcs, 6(), Fama, E., Fsher, L., Jensen, M. and Roll, R. (969) he adjusmen of sock prces o new nformaon, Inernaonal Economc Revew, 0(), -. Henkel, R. and Kraus, A. (988) Measurng even mpacs n hnly raded socks, Journal of Fnancal and Quanave Analyss, 3(), MacKnlay, G. (997) Even sudes n economcs and fnance, Journal of Economc Leraure, 35(), Karpoff, J. M. (987) he relaon beween prce changes and radng volume: A survey, Journal of Fnancal and Quanave Analyss, (), Lyon, J.D, Barber, B.M. and C. sa, C. (999) Improved mehods for ess of long-run abnormal sock reurns, Journal of Fnance, 54(),

20 Maynes, E. and Rumsey, J. (993) Conducng even sudes wh hnly raded socks, Journal of Bankng and Fnance, 7(), Paell, J. M. (976) Corporae forecass of earnngs per share and sock prce behavor: Emprcal ess, Journal of Accounng Research, 4(), Salnger, M. (99) Sandard Errors n Even Sudes, Journal of Fnancal and Quanave Analyss, 7(), Scholes, M. and Wllams, J. (977) Esmang beas from nonsynchronous daa, Journal of Fnancal Economcs, 5(3),

21 APPENDIX I: ESIMAION OF RADE O RADE ABNORMAL REURNS Followng Maynes and Rumsey (993), for a gven sock, he observable rade-o-rade reurn beween me -n and me (where here are no rades on he n- days n beween) s gven by: P R = ln. P n However, jus because a sock does no rade does no mean does no have a value. Leng P ˆs denoe he unobserved (closng) prce for he sock on day s where no rade akes place he radeo-rade reurn can be wren as: P Pˆ R L Pˆ ( n ) = ln = r + r r ( n ) Pˆ ˆ P P n where r s s he unobserved reurn for day s. hus, he rade-o-rade reurn s he sum of he n unobserved daly reurns whch occur because here are n- days wh no rades 9. If he unobserved daly reurns are generaed by he marke model (where he parameers are consan over he n days wh no observed daly reurns) hen we have: r = α + βr + ε s = ( n ), K,. s ms s where r ms s he observed reurn on he marke for day s. hus: ε R = r = ( α + βr + ε ) = nα + β r + s. s ms s ms s= ( n ) s= ( n ) s= ( n ) s= ( n ) he rade o rade reurn for he marke beween day -n and day s equal o he sum of he n (n hs case observed) daly reurns. ha s, R m = r ms s = ( n ) so we have m R = nα + βr + ε (a) s= ( n ) s By assumpon, he marke model error erms ε s are ndependenly and dencally dsrbued. he error erm n (a), s = ( n ) ε, herefore depends on he number of erms n he sum, namely s 9 If here are no rades on he n- days beween day -n and day, hen he n daly reurns for day s -(n-) o day are unobserved. 9

22 n. Aggregaon of error erms nroduces heeroscedascy no he model, so for esmaon purposes (a) s dvded by n o remove he nduced heeroscedascy gvng:. R = nα + β Rm + ε s (b) n n n From (a) expeced reurn s gven by: [ ] ˆ α ˆ s= ( n ) E R = n + βr () m where ˆα and ˆ β are obaned from esmaon of (b) usng OLS. herefore abnormal reurn s gven by: [ ] ˆ α ˆ A = R E R = R n + βr. (3a) m Dvdng by n agan removes he nroduced heeroscedascy and solves he problem for esmaon purposes. ha s, he followng equaon s used for esmaon: ' A ˆ ˆ = A = R nα + βr m. n n n (3b) 0

23 able. Descrpve sascs Our analyss uses an esmaon plus even perod of 47 days and descrpve sascs for each sock are calculaed over welve 47 day nervals. Whn each nerval or year, a sock s assgned o a radng frequency group. Snce socks move from one radng frequency o anoher over me, one canno smply calculae he sascs for each sock over he whole perod. he repored sascs are averages of he ndvdual sascs for each sock whn each group over he nervals. hn radng s accouned for usng he rade o rade adjusmen procedure. Bea, R-squared and he Durbn Wason sascs are from equaon (b): R n R. = α+β n n m+ ε n s. s= n radng frequency hck Medum hn Daly reurn (%) Sandard devaon (%) Coeffcen of Skewness Coeffcen of Kuross Bea Sandard error of he resduals (%) R-squared Durbn Wason sasc.0.3.7

24 able. radng sascs for dfferen radng groups Socks n he sample are dvded no radng groups based on her radng frequency. Socks n he hck radng group rade more han 80% of he me.e. on average, more han 4 days a week. Socks n he medum radng group rade beween 40% and 80% of he me.e., on average, beween and 4 days a week, and socks n he hn radng group rade han 40% of he me,.e. on average, on less han days a week. Year radng Frequency oal number of hck Medum hn socks Number of socks % of socks Average number of days beween rades Average radng Frequency (%) Number of socks % of socks Average number of days beween rades Average radng frequency (%) Number of socks % of socks Average number of days beween rades Average radng frequency (%) 6 5, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Noe: he number of lsed companes n able s he average number of lsed socks over he year. he number of socks here s he oal number of socks over he year n each group. hus he sum of he number of socks here correspond s o he maxmum number of socks lsed n a gven year and s herefore larger han he average number provded n able.

25 able 3. Properes of es sascs under he null hypohess. Under he null hypohess of zero abnormal reurn, he es sascs are assumed o have a sandard normal dsrbuon. Emprcal dsrbuons are based on 000 values of he es sascs for porfolos of 5 socks. he wo bes performng sascs have been shaded. es sasc Mean hck radng Medum radng hn radng Sd. Skew- Kuro- Mean Sd. Skew- Kuro- Mean Sd. Skew- Dev ness ss Dev ness ss Dev ness Paramerc ess Adj. cross sec. nd Sand. Ab. Re Adjused s.a.r Non-paramerc ess 4 - Rank es Sgn es Generalsed sgn es Even nduced varance adjused ess 7 - Paramerc Rank es Kuross

26 able 4. Average rejecon raes for porfolos of 50 socks Porfolos of 50 socks were randomly generaed and arfcal reurns of 0, 0.5% and % were added durng n he even wndow. he repored numbers are he rejecon raes for a 5% sgnfcance level. he wo bes performng es sascs have been shaded. es sascs hck radng Medum radng hn radng Level of arfcal nduced reurn Paramerc ess - Adjused cross sec. nd Sand. abnormal reurn Adjused s.a.r Non-paramerc ess 4 - Rank es Sgn es Generalsed sgn es Even nduced varance 7 - Paramerc es adjused ess 8 - Rank es

27 able 5. Average rejecon raes for porfolos of 5 socks Porfolos of 50 socks were randomly generaed and arfcal reurns of 0, 0.5% and % were added durng he even wndow. he repored numbers are he rejecon raes for a 5% sgnfcance level. he wo bes performng es sascs have been shaded. es sascs hck radng Medum radng hn radng Level of arfcal nduced reurn Paramerc ess - Adjused cross sec. nd Sand. abnormal reurn Adjused s.a.r Non-paramerc ess 4 - Rank es Sgn es Generalsed sgn es Even nduced varance 7 - Paramerc es adjused ess 8 - Rank es

28 able 6. Average rejecon raes for porfolos of 0 socks Porfolos of 0 socks were randomly generaed. Arfcal reurns of 0, 0.5% and % were added on he even day. he repored numbers are he rejecon raes for a 5% sgnfcance level. es sascs hck radng Medum radng hn radng Level of arfcal nduced reurn Paramerc ess - Adjused cross sec. nd Sand. abnormal reurn Adjused s.a.r Non-paramerc ess 4 - Rank es Sgn es Generalsed sgn es Even nduced varance 7 - Paramerc es adjused ess 8 - Rank es

29 able 7. Average rejecon raes for porfolos of 5 socks wh ncreased varance on he even day Porfolos of 5 socks were randomly generaed. Abnormal performance added o he even day s a random varable drawn from N(x,.5σ) where x s 0, 0.5% and %. σ s he varance of he abnormal reurn for he secury over he even perod. hck radng Medum radng hn radng Level of arfcal nduced reurn Paramerc ess - Adjused cross sec. nd Sand. abnormal reurn Adjused s.a.r Non-paramerc ess 4 - Rank es Sgn es Generalsed sgn es Even nduced varance 7 - Paramerc adjused ess 8 - Rank es

30 able 8. Average rejecon raes for porfolos of 5 socks wh random even day CAR analyss. Porfolos of 5 socks were randomly generaed wh a random even day drawn over he nerval [-, ] assumng a unform dsrbuon. Arfcal reurns of 0, 0.5% and % were added on he even day. he repored numbers are he rejecon raes for a 5% sgnfcance level. es sascs hck radng Medum radng hn radng Level of arfcal nduced reurn Even sudy assumng ha he even day s = 0 Paramerc ess - Adjused cross sec. nd Sand. abnormal reurn Adjused s.a.r Non-paramerc ess 4 - Rank es Sgn es Generalsed sgn es Even nduced varance 8 - Paramerc adjused ess 6 - Rank es Cumulave or CAR Analyss Paramerc ess - Adjused cross sec. nd sand. abnormal reurn Adjused s.a.r Non-paramerc ess 4 - Rank es Sgn es Generalsed sgn es Even nduced varance 8 - Paramerc adjused ess 6 - Rank es

31 APPENDIX II DESCRIPION OF HE ES SAISICS -- -es wh adjused cross seconal ndependence (Brown and Warner, 980, 985) he cross-seconal average of abnormal reurns s gven by: A N ' ' = Aj N j = () he varance of he average s he average of he ndvdual varances, so he es sasc (as ye unadjused for forecas error) s: ' A0 (unadj.) = ~ N(0,). ' S(A ) he degrees of freedom are large, so ha he es sasc can be assumed o be un normal, and he sandard devaon s gven by: ' ( ) S A N ' A. N = = he varance of each secury s calculaed separaely. Snce reurns are assumed o be ndependenly dsrbued, he sandard devaon of he cross-seconal average reurn on he even day s he average of he ndvdual sandard devaons. Noce ha S(A ) s an esmae of he sandard devaon of abnormal reurns over he esmaon perod, or even wndow. Snce hese abnormal reurns are he forecass from he marke model, s necessary o adjus S(A ) o accoun for he varance of he forecas error. he sandard devaon of abnormal reurns for secury on he even day (ncludng he adjusmen for forecas error) s hen gven by 0 : S A = = ' [ A ] + + ( R R ) j m,0. ( ) R R Wh adjusmens for forecas error, he sandard devaon used n he es sasc s: S A N ' ( A ) = [ A ] + ' N ( Rm,0 Rm ) j ( R R ) = j m m +. m m m 0 hs s ofen referred o as Paell s (976) adjusmen.

32 Noce ha s he number of observed reurns n he esmaon wndow, so approaches zero for large. Also, snce he value of he denomnaor ( R ( Rm,0 Rm) ( Rm, Rm) m Rm ) ncreases wh, approaches zero for large. Gven ha s ypcally over 00 for hckly raded socks, hs adjusmen s ofen gnored. For hnly raded socks calculaed usng rade o rade reurns, even hough he esmaon perod exceeds 00 days, here may be only 0 o 0 observed reurns over he perod. Hence, s no approprae o gnore he above adjusmen. Afer adjusng for forecas error, he es sasc becomes: A = ~ N(0,). S 0 A ' ( A ) he same es sasc appled o he unadjused cumulave abnormal reurns (CARs) s ' A CAR = (unadj.) = ~ N(0,). ' ( ) 3 S A When Paell s adjusmen s used, he es sasc s CAR = = - 3S A A ' ' ( A ). -- -es wh sandardsed abnormal reurn (Brown and Warner, 985) Sandardsed (un varance) abnormal reurns for secury are gven by: ' s A A =, S A where he es sasc for he even day s: S ( ) ' ( A ) = ( A ) = N s = ( Aj ) ~ N( 0,). N =.

33 Assumng un varance, he es sasc based on a hree day even wndow s: N N N s s s Aj + Aj0 + Aj+ j= j= j= ( ) CAR = ~ N 0,. 3N es wh adjused sandardsed abnormal reurn (Brown and Warner, 985 and Paell, 976) As was he case for, s possble o adjus for he forecas errors, so ha where A As ' A =, A S ( A ) ( Rm,0 Rm) A ' S ( A ) = A + + = ( Rm, Rm) he adjused es sasc for he even day s gven by: 3 = N j= N As A j j 4. he Cumulave Average Resduals for a hree day wndow are CAR j = Aj =. he CARs are smply he unadjused abnormal reurns. Usng he sandard errors gven by Salnger (99, p. 4) for a 3-day even wndow, we consruc a es sasc for he adjused CARs as follows: A CAR 3 = N j σ + + j N Car j r m m = = 3 σ r m r.

34 4 -- Rank es (Corrado, 989 and Corrado and Zvney, 99) hs es corresponds o 3 n Corrado and Zvney [99]. Le K denoe he rank of excess reurn A n secury s esmaon and even perod of 50 days: ( A ), = 48,.. + K = rank. he frs 47 observaons are used as an esmaon perod and observaon on each sde of day 0 are used as he even wndow. o accoun for mssng observaons from hn radng Corrado and Zvney (99) sandardsed each rank by he number of non-mssng abnormal reurns as follows: U K =, ( + ) where s he number of non-mssng abnormal reurns for secury over he enre perod. Snce he sum of values of K s ( + ), he average value of U s he es sasc s gven by: 4 N = U S N j= j ( K ) ( + ) + U = =. where S( K) N = U 50 = 48 N = and N represens he number of non-mssng reurns n a cross-secon of N-frms a me. Under he null hypohess he rank of he abnormal reurn s drawn from he unform dsrbuon, so Corrado and Zvney (99) sugges ha he es sasc converges o a sandard normal. he es sasc for CARs usng ranks s gven by: CAR 4 = = N U j N j= ( ) 3 S K 5 -- Sgn es (Corrado and Zvney, 99)

35 Le he medan abnormal reurn n secury s me seres of abnormal reurns be denoed by medan (A ). For each day n he sample perod, he sgn of each excess reurn s calculaed as: ( A medan( A )), = 48,.., + G = sgn, where sgn(x) s +, - or zero dependng on wheher x s posve, negave or zero. he expeced value of G under he null hypohess s zero and he es sasc s gven by: where 5 = N 0 S N0 j= ( G) G j0 ( ) s G 5 N = G 50 = 44 N = and N s he number of non-mssng reurns on day.. he es sasc for CARs usng sgns s gven by: CAR 5 = N0 G j N0 j=. = 3S G ( ) 6 -- Generalsed sgn es (Cowan, 99 and 996) For he radonal sgn es, 5, he expeced number of posve abnormal reurns under he null hypohess s 0.5, whereas for hs es he expeced number of abnormal reurns s esmaed from he esmaon perod across me and socks. he fracon of posve abnormal reurns under he null hypohess s gven by: where ϕ = f N N j = j ˆp = ϕ, j j = A > 0, and ϕ = 0 oherwse. he generalsed sgn es sasc s: 6 = w Nxpˆ Nxp pˆ. Agan, he CAR verson of he es sasc s he sum of he hree es sascs durng he even wndow dvded by he square roo of hree.

36 7 - Varance adjused sandardsed abnormal reurns (Boehmer, Musumec and Poulsen, 99) Ofen an even leads o ncreases n he varance of he reurns around he even due o a emporary ncrease n sysemac rsk, uncerany regardng he effecs of he even, ec. In hs suaon, he sandard devaon calculaed over he usual esmaon perod under saes he sandard devaon of reurns expeced durng he hree-day even wndow. o adjus for hs problem, he resduals are frs sandardsed and hen varance s esmaed durng he even wndow. he resulng es s gven by: where 9 = ( ) N = N As As A A N = N N N A As and A As A = A S ' ( A ) ( Rm,0 Rm) A ' S ( A ) = A + + = ( Rm, Rm) he CAR verson of hs es sasc s calculaed usng Salnger [99] as for 3 above Rank es of adjused sandardsed abnormal reurns Corrado and Zvney [99] and Maynes and Rumsey, (993.) hs es s equvalen o 4 excep ha makes a cross-seconal varance adjusmen and follows Corrado and Zvney [99]. he sandardsed reurns are gven by where S(A j ) s gven by A s = A S( A)

37 ' ( ) = ( A) S A = for he esmaon perod ( Rm,0 Rm) ' ( ) = A + + n he even wndow = S A ( Rm, Rm). he cross-seconal varance adjusmen s hen appled: where X s A, 0 = A, = 0 S( A ) s s 0 N s 0 0 N = s s ( ) S( A ) = A A 0 he es sasc s hen derved n he same manner as 4.

38 Workng Papers from Fnance Research Group F F F F F F F F F Jan Barholdy, Denns Olson & Paula Peare: Conducng even sudes on a small sock exchange. Jan Barholdy & Cesáro Maeus: Deb and axes: Evdence from bankfnanced unlsed frms. Esben P. Høg & Per H. Frederksen: he Fraconal Ornsen-Uhlenbeck Process: erm Srucure heory and Applcaon. Charloe Chrsansen & Angelo Ranaldo: Realzed bond-sock correlaon: macroeconomc announcemen effecs. Søren Wllemann: GSE fundng advanages and morgagor benefs: Answers from asse prcng. Charloe Chrsansen: Level-ARCH shor rae models wh regme swchng: Bvarae modelng of US and European shor raes. Charloe Chrsansen, Juanna Schröer Joensen and Jesper Rangvd: Do more economss hold socks? Mchael Chrsensen: Dansh muual fund performance - selecvy, marke mng and perssence. Charloe Chrsansen: Decomposng European bond and equy volaly.

39 ISBN Deparmen of Busness Sudes Aarhus School of Busness Fuglesangs Allé 4 DK-80 Aarhus V - Denmark el Fax