relation between price changes and trading volume a survey

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JOURNALOFFINANCIALANDQUANTITATIVEANALYSISVOL.22,NO.1,MARCH1987TheRelationbetweenPriceChangesandTradingVolume:ASurveyJonathanM.Karpoff*AbstractThispaperreviewspreviousandcurrentresearchontherelationbetweenpricechangesandtradingvolumeinfinancialmarkets,andmakesfourcontributions.First,twoempiri-calrelationsareestablished:volumeispositivelyrelatedtothemagnitudeofthepricechangeand,inequitymarkets,tothepricechangeperse.Second,previoustheoreticalresearchontheprice-volumerelationissummarizedandcritiqued,andmajorinsightsareemphasized.Third,asimplemodeloftheprice-volumerelationisproposedthatisconsis-tentwithseveralseeminglyunrelatedorcontradictoryobservations.Andfourth,severaldirectionsforfutureresearchareidentified.I.IntroductionThispaperisareviewofempiricalandtheoreticalresearchintotheprice-1volumerelationinfinancialmarkets.Thereareatleastfourreasonswhytheprice-volumerelationisimportant.First,itprovidesinsightintothestructureoffinancialmarkets.Themodelsdiscussedbelowpredictvariousprice-volumere-lationsthatdependontherateofinformationflowtothemarket,howtheinfor-mationisdisseminated,theextenttowhichmarketpricesconveytheinforma-tion,thesizeofthemarket,andtheexistenceofshortsalesconstraints.Empiricalrelationsbetweenpricesandvolumecanhelpdiscriminatebetweendifferinghypothesesaboutmarketstructure.Second,theprice-volumerelationisimportantforeventstudiesthatusea2combinationofpriceandvolumedatafromwhichtodrawinferences.Ifprice*TheauthorwouldliketothankLindaBamber,T.WakeEpps,LarryHarris,AlanHess,Rob-ertJennings,AviKamara,GeorgeTauchen,RalphWalkling,andtheJFQAManagingEditorandanonymousrefereesformanyhelpfulcommentsandsuggestions.FinancialsupportwasprovidedbytheUniversityofWashington'sCenterfortheStudyofBankingandFinancialMarkets.1Tokeepthistaskmanageable,theprice-volumerelationisnarrowlydefined.Asubstantialamountofempiricalandtheoreticalworkalsorelatestothecostsoftransactinginsecuritiesmarketsandtheprice-settingbehavioroffloorspecialistsanddealers(seeCohen,Maier,Schwartz,andWhitcomb1979forasurvey,andalso[27],[14],and[7]).Epps([21])examinestheeffectoftrans-actioncostsonvolume.Volumehasalsobeenusedtoexaminetheexistenceofdividendclienteles([1],[58])arbitragearoundex-dividenddays([47],[32]),theeffectsofinformationuncertainty[4],andarbitrageactivity[50].2Volumehasbeenusedtoinferwhetheraneventhad"informationalcontent"andwhetherinvestors'interpretationsoftheinformationweresimilarordifferent([5],[47],[25],[53],[59],[57],and[2],[3];foracritique,see[63]).HarrisandGurel[37]usepriceandvolumeeffectstoexaminethepricepressurehypothesis.Thedividendstudiescitedinnote1areotherexamples.109 110JournalofFinancialandQuantitativeAnalysischangesandvolumearejointlydetermined,incorporatingtheprice-volumerela-tionwillincreasethepowerofthesetests.Forexample,Richardson,Sefcik,andThompson[58]examinetradingvolumeandpricechangestotestfortheexis-tenceofdividendclienteles.Inothertests,pricechangesareinterpretedasthemarketevaluationofnewinformation,whilethecorrespondingvolumeiscon-sideredanindicationoftheextenttowhichinvestorsdisagreeaboutthemeaning3oftheinformation.Theconstructionoftestsandvalidityoftheinferences4drawndependonthejointdistributionofpricechangesandvolume.Third,theprice-volumerelationiscriticaltothedebateovertheempiricaldistributionofspeculativeprices.Whensampledoverfixedcalendarintervals(e.g.,days),ratesofreturnappearkurtoticcomparedtothenormaldistribution.Twocompetinghypothesestoexplainthisare(1)ratesofreturnarebestcharac-terizedbyamemberofaclassofdistributionswithinfinitevariance(thestableParetianhypothesis),and(2)thedistributionofratesofreturnappearskurtoticbecausethedataaresampledfromamixtureofdistributionsthathavedifferentconditionalvariances(themixtureofdistributionshypothesis).Price-volumetestsgenerallysupportthemixtureofdistributionshypothesis.This,inturn,hasseveralimplications.Asanexample,itappearspricedataaregeneratedbyaconditionalstochasticprocesswithachangingvarianceparameterthatcanbeproxiedbyvolume.Knowledgeoftheprice-volumerelationcanthenbeusedineventstudiestomeasurechangesinthevarianceofthepriceprocessfromnon-5eventtoeventtime.Andfourth,price-volumerelationshavesignificantimplicationsforre-searchintofuturesmarkets.Pricevariabilityaffectsthevolumeoftradeinfu-turescontracts([17],[49]).Thishasbearingontheissueofwhetherspeculationisastabilizingordestabilizingfactoronfuturesprices[61].Thetimetodeliveryofafuturescontractaffectsthevolumeoftrading,andthroughthiseffect,possi-blyalsothevariabilityofprice[29].Theprice-volumerelationcanalsoindicatetheimportanceofprivateversuspublicinformationindetermininginvestors'de-mands[56].Theplanofthispaperisasfollows.PartIIprovidesabriefreviewofearlyresearchintotheprice-volumerelation.Insubsequentresearch,twoempiricalrelationsemergeas"stylizedfacts:"(1)Thecorrelationbetweenvolume(V)andtheabsolutevalueofthepricechange(|Ap|)ispositiveinbothequityandfuturesmarkets;and(2)thecorrelationbetweenvolumeandthepricechange6perse(Ap)ispositiveinequitymarkets.PartIIIexaminesresearchthatsup-portstheformer,whiletheworkcoveredinPartIVsupportsthelatterrelation.3AstatementofthishypothesisisfoundinBeaver([5],p-69):"Animportantdistinctionbe-tweenthepriceandvolumetestsisthattheformerreflectschangesintheexpectationsofthemarketasawholewhilethelatterreflectschangesintheexpectationsofindividualinvestors.''4Inferencesfromvolumedataalsodependonthetheoreticalconnectionbetweeninformationandvolume(see[68],[44]).Hypothesesoftheprice-volumerelationcanalsoyieldinferencesfromempiricalteststhatareotherwiseunobtainable(see,forexamples,[70],[52]).5TheproblemofchangingreturnvarianceineventtimeisdiscussedbyChristie[9J.UsingvolumetoproxyforthevarianceofthepriceprocessisimpliedbydiscussionsinRogalski[60]andHarris[34].6Throughoutthispaper,theterm"pricechange"isused,althoughmostoftheempiricalworkusespricechangerelatives,computedasthefirstdifferenceinthelogpriceorthepercentagepricechange. Karpoff1117Mosttheoreticalmodelsareunabletoexplainbothprice-volumerelations,butinPartVwearguethatthesetwoempiricalfindingscanbemutuallyconsistent.Finally,inPartVI,someunsolvedquestionsanddirectionsforfurtherresearcharediscussed.II.EarlyResearchAcademictreatmentofaprice-volumerelationcanbetracedtoOsborne[54],whoattemptedtomodelthestockpricechangeasadiffusionprocesswithvariancedependentonthenumberoftransactions.ThiscouldimplyapositivecorrelationbetweenVandAp,aslaterdevelopedbyClark[10],TauchenandPitts[65],andHarris[34].However,byassumingtransactionsareuniformlydistributedintime,Osbornewasabletoreexpressthepriceprocessintermsoftimeintervals,anddidnotdirectlyaddressthevolume-priceissue.Anearlyempiricalexaminationofthevolume-pricerelationwasconductedbyGrangerandMorgenstern[30].Usingspectralanalysisofweeklydatafrom1939-1961,theycoulddiscernnorelationbetweenmovementsinaSecuritiesandExchangeCommissioncompositepriceindexandtheaggregatelevelofvol-umeontheNewYorkStockExchange.Datafromtwoindividualstocksalsodisplayednoprice-volumerelation.In1964,Godfrey,Granger,andMorgen-sternpresentednewevidencefromseveraldataseries,includingdailyandtrans-actiondataforindividualstocks.Butonceagaintheycouldfindnocorrelationbetweenpricesortheabsolutevaluesofpricedifferencesandvolume.AnotherfindingbyGodfrey,Granger,andMorgensternisthatdailyvolumecorrelatespositivelywiththedifferencebetweenthedailyhighanddailylow.Thisissupportedbyalaterfinding[31]thatdailyvolumecorrelateswiththesquareddifferencebetweenthedailyopenandclose.Theauthorsattributethiscorrelationtoinstitutionalfactorssuchasstop-lossandbuy-above-marketordersthatincreasevolume"asthepricedivergesfromitscurrentmean"([28],p.20).However,EppsandEpps[24]havesuggestedthatvolumemoveswithmeasuresofwithin-daypricevariabilitybecausethedistributionofthetransactionpricechangeisafunctionofvolume.ThefailureofGodfreyetal.touncoveraprice-volumerelationmotivatedtheempiricaltestsofYing[72]andCrouch[19].Yingappliedaseriesofchi-squaredtests,analysesofvariance,andcross-spectralmethodstosix-year,dailyseriesofpriceandvolume.PricesweremeasuredbytheStandardandPoor's500compositeindexadjustedfordividendpayouts,andvolumebytheproportionofoutstandingNYSEsharestraded.Thefollowinglistisasubsetofhisfindings:"(1)Asmallvolumeisusuallyaccompaniedbyafallinprice.(2)Alargevolumeisusuallyaccompaniedbyariseinprice.(3)Alargeincreaseinvolumeisusuallyaccompaniedbyeitheralargeriseinpriceoralargefallinprice."([72],p.676).7ExceptionsincludeJennings,Starks,andFellingham[41],Karpoff[43],Harris[36].Morethanoneoftheresearcherswhoseworkiscitedbelowseemedunawareofeithertheabsolutevalueorthepersecorrelations(forexamples,seeMorgan[51],pp.505-536,orKelles[45]). 112JournalofFinancialandQuantitativeAnalysis8Ying'sempiricalmethodsareeasilycriticized,butnotethatitems(1)and(2)suggestVandA/?arepositivelycorrelated,anditem(3)isconsistentwithacorrelationbetweenVandAp.Asdiscussedbelow,eachoftheseinterpretationshasbeensupportedinsubsequenttests.Thus,Yingwasthefirsttodocumentbothprice-volumecorrelationsinthesamedataset.III.VolumeandtheAbsoluteValueofthePriceChangeA.EmpiricalEvidenceItisanoldWallStreetadagethat"Ittakesvolumetomakepricesmove."Althoughonecanquestiontheassertedcausality,numerousempiricalfindingssupportwhatwillbecalledherea''positivevolume-absolutepricechangecorre-lation."Crouch([18],[19])foundpositivecorrelationsbetweentheabsolutevaluesofdailypricechangesanddailyvolumesforbothmarketindicesandindi-vidualstocks.Clark[10]foundapositiverelationbetweenthesquareofamea-sureofthepricechangeandaggregatedvolumeusingdailydatafromthecottonfuturesmarkets.Usingfour-dayintervalandmonthlydatafromatotalof51stocks,Morgan[51]foundthatinallcasesthevarianceofpricechangewaspositivelyrelatedtotradingvolume.Westerfield[69]foundthesamerelationinasampleofdailypricechangesandvolumesfor315commonstocks,asdidTauchenandPitts[65]usingdailydatafromtheTreasurybillfuturesmarket.EppsandEpps[24]foundapositiverelationbetweenthesamplevariancesofpricechangesatgivenvolumelevelsandthevolumelevelsusingtransactionsdatafrom20stocks,andWood,Mclnish,andOrd[71]alsoreportapositivecorrelationbetweenvolumeandthemagnitudeofthepricechangeatthetransac-tionslevel.JainandJoh[38]documentasimilarcorrelationoverone-hourinter-vals,usingdatafromamarketindex.Cornell[17]foundpositiverelationsbe-tweenchangesinvolumeandchangesinthevariabilityofprices,eachmeasuredovertwo-monthintervals,foreachof17futurescontracts.Therelationwasal-mostentirelycontemporaneous,asmostleadingandlaggedrelationswerestatisticallyinsignificant.GrammatikosandSaunders[29]alsofoundvolumetobepositivelycorrelatedwithpricevariability,butforforeigncurrencyfutures.Rutledge"{61]foundsignificantcorrelationsbetweendailyvolumeandtheabso-lutevalueofthedailypricechangefor113outof136futurescontractsanalyzed.Comiskey,Walkling,andWeeks[12]foundasimilarcorrelationusingyearlydataonindividualcommonstocks.Richardson,Sefcik,andThompson[58]foundthattradingvolumeincreaseswiththesquareofameasureofabnormalreturnaroundannouncementsofdividendchanges.AndHarris[34]foundaposi-tivecorrelationbetweenvolumeandthesquareofthepricechangeusingdailydatafrom479commonstocks.Thestrengthofthecorrelationvariedacrosssecu-rities[36]andthecorrelationwasalsofoundtobestrongerfordailythanfortransactionsdata[35].8OneproblemarisesbecauseYing'spriceseries(S&P's500index)andvolumeseries(NYSEpercentagevolume)arenotnecessarilycomparable.AsecondproblemarisesfromhisadjustmentstothedatafordividendsandtotalNYSEsharesoutstanding.Ying'sdailypriceserieswasadjustedbyquarterlydividenddata,andthedailyvolumeserieswasadjustedbymonthlydataonthenumberofoutstandingshares,eachusinglinearinterpolations.Also,severalofYing'sfindings(notlistedhere)areinconsistentwithweakformmarketefficiency. Karpoff113TheseempiricalresultsaresummarizedinTable1.Theypromptthreeob-servationsthatwillbediscussedbelow.First,theV,Apcorrelationappearsinboththeequityandfuturesmarkets.Second,despitethealmostuniversalfindingofapositivecorrelation,someofthesetestsindicatethatthecorrelationisweak.Forexample,theaveragesquaredcorrelationcoefficientobtainedbyCrouchwas0.20amongthestockindicesand0.23amongtheindividualfirms.ItisarguedinPartVthatthisstemsfromheteroskedasticerrortermsthataregeneratedwhenastraightlineisfittodatafrommarketsinwhichshortsalesarerelativelycostly.Third,thiscorrelationappearswithpriceandvolumedatameasuredoverallcal-endarintervals,butitappearstobeweakerintransactionsdata.TABLE1SummaryofEmpiricalStudiesfromwhichInferencesCanbeMadeabouttheCorrelationoftheAbsoluteValueofthePriceaChange(|Ap|)withTradingVolume(V)YearSampleSampleDifferencingSupportPositiveAuthor(s)ofStudyDataPeriodInterval(|Ap|,^Correlation?Godfrey,Granger,1964Stockmarketaggregates,1959-62,weekly,daily,NoandMorgenstern3commonstocks1951-53,63transactionsYing1966Stockmarketaggregates1957-62dailyYesCrouch19701'SI5commonstocks1963-67dailyYesCrouch1970H3]Stockmarketaggregates,1966-68hourlyandYes3commonstocksdailyClark1973Cottonfuturescontracts1945-58dailyYesEppsandEpps197620commonstocksJan.,1971transactionsYesMorgan197617commonstocks,and1962-65,4-days,Yes44commonstocks1926-68monthlyWesterfield1977315commonstocks1968-69dailyYesCornell1981Futurescontractsfor1968-79daily"Yes17commoditiesHarris198316commonstocks1968-69dailyYesTauchenandPitts1983T-billfuturescontracts1976-79dailyYesComiskey,Walkling,1984211commonstocks1976-79yearlyYesandWeeksHarris198450commonstocks1981-83transactions,YesdailyRutledge1984Futurescontractsfor1973-76dailyYes13commoditiesWood,Mclnish,1985946commonstocks,1971-72,minutesYesandOrd1138commonstocks1982Grammatikos1986Futurescontractsfor1978-83dailyYesandSaunders5foreigncurrenciesHarris1986479commonstocks1976-77dailyYesJainandJon1986Stocksmarketaggregates1979-83hourlyYesRichardson,Sefcik,1987106commonstocks1973-82weeklyYesandThompson»Thistablesummarizesthegeneralconclusionsofthesestudiesaboutthecorrelationof|Ap|andV.Resultsthatindicatenosignificantcorrelationarelistedasnotsupportingapositivecorrelation.Thesestudiesemployvariousmeasuresofthepricechangeandtradingvolume.Formoreprecisedescriptionsofthedataandvariabletransformations,thereaderisreferredtotheoriginalpapers.bThedailydataaretransformedintoaseriesofestimatedaveragedailyvolumesanddailyreturnvariancesforsuccessivetwo-monthintervals.9B.TheoreticalExplanationsCopeland([15],[16])hasconstructeda"sequentialarrivalofinformation"9Anearlybutflawedattempttoexplainthepositivevolume-pricecorrelationisinCrouch[18].Alltradingoccursthroughadealer.Inoneversionofthetheory,thedealerirrationallysatisfiesalldemandstotradeeventhoughheexpectstoloseoneachtrade(i.e.,thedealerhasaninfinitesupplyofsecuritiesandcash).Amendingthisversion,Crouchassumesthatinvestors'demandschangeat 114JournalofFinancialandQuantitativeAnalysismodelinwhichinformationisdisseminatedtoonlyonetraderatatimeandthatimpliesapositivecorrelationbetweenVandAp.Theinformationcausesaone-timeupwardshiftineach"optimist's"demandcurvebyafixedamount8andadownwardshiftof8ineach"pessimist's"demandcurve.Tradingoccursaftereachtraderreceivestheinformation,butuninformedtradersdonotinferthecon-tentoftheinformationfrominformedtraders'actions.Also,shortsalesarepro-hibited.WithNtraders,therewillingeneralbekoptimists,rpessimists,andN-k-runinformedinvestorsatanypointintimebeforeallinvestorsbecomeinformed.Thevaluesofkandrdependontheorderinwhichinvestorsbecomeinformed.Becauseoftheshortsalesprohibition,volumegeneratedbyapessimistisgener-allylessthanthatgeneratedbyanoptimist(i.e.,thepessimistcannotsellshortuponreceivingtheinformation).Sothepricechangeandtradingvolumewhenthenexttraderbecomesinformeddependuponboth(i)thepreviouspatternofwhohasbeeninformedand(ii)whetherthenexttraderisanoptimistorpessim-ist.Likewise,thetotalvolumeafteralltradersbecomeinformeddependsonthepathbywhichthefinalequilibriumisreached.Itisarandomvariablewithanexpectedvalueequaltoaweightedaverageofthetotalvolumesundereachpos-siblepathofinformationdispersion.SimulationtestsindicatethatVishighestwheninvestorsarealloptimistsorallpessimists.AlsoApislowestatthesamepercentageofoptimistsatwhichVislowest,andriseswithV.ThissupportsapositivecorrelationofVandAp.Thismodelisopentoatleasttwocriticisms.Firstistheassumptionthatprohibitstradersfromlearningfromthemarketpriceasothertradersbecomeinformed.Secondistheimplicationthatvolumeisgreatestwhenallinvestorsagreeonthemeaningoftheinformation.Thisiscontrarytotheinferencedrawnfromhighmeasuresofvolume([10],[5],[46],[26],[2],[3]),andisinconsistentwiththeempiricalfindingsofComiskey,Walkling,andWeeks[12].Copelandattributesthistotheshortsalesconstraint,butthatisonlypartofthestory.Alsoimportantistheratherpeculiarinterpretationofdisagreementamongtraders,whoareforcedintoabinaryresponsetonewinformation(demandsshiftupby810ordownby8).Copeland'smodelhasbeenextendedbyJennings,Starks,andFellingham([42];seesectionIV.B.below)andbyMorse[52],whoderivesfromthesequen-tialarrivalofinformationprocessthehypothesisthattradingvolumewillbeab-normallyhighduringthesameperiodsinwhichtheabsolutevaluesofreturnsaredifferenttimes,andthenecessarysupplyofsecurities(whendemandsincrease)orcash(whende-mandsdecrease)comesfromothersellersorbuyers.WhilethischangeanticipatesCopeland'snotionofsequentialinformationarrival,italsogutsCrouch'stheory,whichboilsdowntoanassertionthatwhensomeinvestors'demandschange,theresultingrealignmentofsecuritiescausesasimultaneousincreaseinvolumeandapricerevision.10Theimplicationthatvolumeincreaseswiththedegreeofhomogeneityacrossinvestorsfol-lowsbecause,onceanoptimisthastraded(bybuyingunitsfromotherinvestors),alaterpessimistwillhavefewerunitstosellwhenitishisturntotrade,whereasalateroptimisthasnofewerunitsavailableforpurchase.Likewise,ifanoptimistfollowsapessimist,hehasfewerunitsavailableforpurchase(sincethepessimistpreviouslysoldsomeofhisunits).Ontheotherhand,anoptimistwhofollowsotheroptimistshasanincreasednumberofsuppliersfromwhomtobuy,whileapessimistwhofollowsotherpessimistshasanincreasedsupplyofassetstotradeaway.Asaresult,amixofoptimistsandpessimistsleadstolowerVandsmaller|Apthanwhenonetypeoftraderpredominates. Karpoff115seriallycorrelated.JenningsandBarry[40]consideranextensionofthesequen-tialinformationarrivalmodelbypermittinginformedtraderstotakespeculativepositions.Speculationcausespricestoadjustmorequicklytonewinformation,buttheeffectontradingvolumeisambiguous.TheirmodelimpliesapositivecorrelationbetweenVandApforagiveninvestor'strade,butthiscorrelationcangetobscuredwithdatasampledovertimeintervalsaslongasaday.AnotherexplanationofthepositivecorrelationbetweenVand|Ap|comesfromresearchintothedistributionofspeculativeprices.Dailypricechangesofspeculativeassetsappeartobeuncorrelatedwitheachotherandsymmetricallydistributed,butthedistributioniskurtoticrelativetothenormaldistribution([25],[10]).Oneexplanationisthatdailypricechangesaresampledfromasetofdistributionsthatarecharacterizedbydifferentvariances.Thisisthe"mixtureofdistributionshypothesis"(MDH).InoneformoftheMDH,EppsandEpps[24]deriveamodelinwhichthevarianceofthepricechangeonasingletransactionisconditionaluponthevol-umeofthattransaction.Transactionpricechangesarethenmixturesofdistribu-tionswithvolumeasthemixingvariable.InasecondformoftheMDH([10],[65],[34]),thedailypricechangeA/?isthesumofavariablenumbermofinde-pendentwithin-daypricechanges.Foragivenm,theCentralLimitTheoremimpliesthatApisapproximatelynormalwithvarianceproportionaltom.Foravariablem,however,theCentralLimitTheoremisinapplicableandthedistribu-ntionofA/?issubordinatetothedistributionofm.Itisintuitivelyattractivetointerpretmasthenumberofwithin-dayinformationarrivals,sotheconditionalvarianceofApisconsideredtobeanincreasingfunctionoftherateatwhichnewinformationentersthemarket.TheV,Apcorrelationresultsbecausevolumeisalsoanincreasingfunctionofthenumberofwithin-daypricechanges.TheEppsandEppsmodelissimilartothesequentialinformationarrivalmodelinthatitplacesaparticularstructureonthewayinvestorsreceiveandrespondtoinformation.EppsandEppsprovideempiricalsupportfortheircon-tentionthataV,|Ap|correlationoccursatthetransactionlevel,afindingthatisconfirmedbyWood,Mclnish,andOrd[71].However,Harris[37]findsthatthiscorrelation,aswellasotherobservedpropertiesofdailydata,arenotcharacteris-ticsoftransactionsdata.ThecentralpropositionofthemodelsbyClark,TauchenandPitts,andHar-risisthattransactiontimeintervalsarevariable.Thereisalsosomeempiricalsupportforthiscontention.Clark'stestsusevolumeasaproxyvariableforthenumberoftransactionsvariablem,andshowthattheleptokurtosisintheempiri-caldistributionofdailypricechangeslargelydisappearswhenthechangesaregroupedbyvolumeclasses.ThisfindingissupportedbyMorgan[51]andWes-terfield[69].Byassumingthevariationofmvariesacrosssecurities,Harris[36]derivesinferencesthatsamplemeasuresofpricechangekurtosis,volumeskew-ness,andcorrelationofsquaredpricechangewithvolumeshouldallbeposi-tivelycorrelatedacrosssecurities.Thesehypothesesaresupportedinhistests.AdditionalsupportisprovidedbyUptonandShannon[66],butconflictingevi-dencecomesfromWood,Mclnish,andOrd[71],whofindthattheabsolutesize11See[10]or[69]fordiscussionsofsubordinatedprocesses.Loosely,thedistributionofthedailypricechangeis"subordinate"tothatofmbecauseitsparametersarefunctionsofm. 116JournalofFinancialandQuantitativeAnalysisofthepricechangeatthetransactionslevelispositivelyrelatedtothelengthoftimebetweentransactions.Thisisinconsistentwithacentralassumptionofthesemodelsthatthedistributionofthetransaction-levelpricechangeisindependentofthenumberoftransactionswithinafixedcalendarinterval.Thehypothesisthattransactionstimediffersfromcalendartimeprovidesinsightsintoseveralrelatedmarketphenomena.TheTauchenandPittsmodelimpliesthattheV,|Ap|correlationincreaseswiththevarianceofthedailyrateofinformationflow,andthat,asthenumberoftradersincreases,thevolumeoftradeincreasesandpricevariabilitydecreases.ThislatterpredictionisconsistentwithevidencefromtherelativelynewTreasurybillsfuturesmarket.AnotherextensionhasbeenmadebyGrammatikosandSaunders[29],whoarguethatthevarianceofthepricechangeprocessinfuturescontractsdecreaseswiththetimetomaturityofthecontract.LehvariandLevy[48]havedemonstratedthatempir-icalestimatesofsystematicriskaresensitivetothecalendartimeintervalsoverwhichreturnsarecalculated.CarpenterandUpton[8]havesuggestedthatthenumberofinformationarrivalsmisameasureof"effectivetime"intheevolu-tionofthepriceprocess,soestimatesofbetawillalsobesensitivetom.Thissuggestsarolefortheuseofvolumeintheadjustmentofreturnsforrisk,sincevolumeisaproxyvariablefortherateofinformationflowm.AswasmentionedinPartI,anotherapplicationoftheMDHmaybetoindicatemethodstoadjustteststatisticsineventstudiestotakeintoaccountchangesintheconditionalvari-anceofthepricechangeprocess.Onedrawbackofthesemodelsisthattheyimplynorelationbetweenvol-umeandthepricechangeperse.Thisisinconsistentwithmostoftheevidencediscussedinthefollowingsection,inwhichitissuggestedthatapositivecov(Ap,V)arisesbecauseofshortsalesrestrictions.TheTauchenandPittsandHarrismodelsexplicitlyassumeawayshortsaleconstraints,anditisunclearwhethertheirinclusionwouldimplyacorrelationbetweenApandV.AnotherexplanationforapostivecorrelationbetweenVandtheApisim-pliedinamodelbyPfleiderer[56],whichextendspreviousworkoninformationaggregationinmarkets.Arationalexpectationsequilibriumisestablishedinwhichspeculators'privateinformationisonlypartiallyaggregatedbythemarketpricebecauseofnoiseintroducedbylife-cycletrading.Speculativetradingin-creaseswiththeprecisionofprivateinformationandisuncorrelatedwithAp,butlife-cycletradingrandomlyaffectsthesupplyavailabletospeculators.Thevolumeoflife-cycletradingthushasaneffectonthemagnitudeofpricechanges.ThismodelalsoimpliesthatthestrengthofthecorrelationbetweenVandApincreaseswiththerelativeimportanceoflife-cycletradinginthemarket.Itshouldbenotedthat,whiletheEppsandEppsmodelrequiresallinvestorstoreceiveinformationsimultaneously,theClark,TauchenandPitts,Harris,andPfleiderermodelscanbemutuallyconsistentwithsequentialinformationalar-rival.Whilethesemodelsimplysimultaneousdispersionofaninformationbit,theydonotrequireit.Thesuccessiveequilibriapresumedbythesemodelscanresultfromagradualdisseminationofasinglebitofinformation,asinthese-quentialinformationarrivalmodel,orfromaprocessinwhichinvestorsreceiveinformationsimultaneously.Thesemodelsarealsomoregeneralthanthese-quentialarrivalofinformationmodel,fortworeasons.First,theyareconsistent Karpoff117witheithersimultaneousorgradualinformationdissemination,whileCopeland'smodelimpliesanegativeV,|Ap|correlationwhensimultaneousinformationar-rivalisimposed.Andsecond,theyexplainagreaternumberofphenomena.TheMDHisconsistentwiththeempiricaldistributionofpricechangesandthediffer-enceintheV,|Ap|correlationoverdifferentfrequencies,whilePfleiderer'smo-ndelconsiderstheinformationalcontentofthemarketprice.IV.VolumeandthePriceChangePerSeA.EmpiricalEvidenceAnotherfamiliarWallStreetadageisthatvolumeisrelativelyheavyinbullmarketsandlightinbearmarkets.Assupport,Eppsdevelopedtests,firstfromthebondmarket[20],thenfromthestockmarket[22],whichindicatethattheratioofVtoApisgreaterfortransactionsinwhichthepriceticksupthanfortransactionsondownticks.ThiswasfoundtoholdevenwhenVandApweremeasuredoverdailyintervals[22],andwithoutregardforthegeneralmovementinprices[33].ConflictingevidencewasfoundbyWood,Mclnish,andOrd[71],whofoundthattheratioofVApishigherfordownticks.AndSmirlockandStarks[64]foundtherelationtoholdonlyduringperiodsinwhichtheycoulddistinguishthearrivalofinformationexante.Inotherperiods,theyfoundslightevidencethattheratioofVto|Ap|islowerforupticksthanfordownticks,whichtheyattributetopositivetransactioncostsandthelackofinformationarrival.However,usinghourlydatafromabroadmarketindex,JainandJoh[38]findthatvolumeispositivelyrelatedtothemagnitudeofthepricechange,butthatvolumeismoresensitivetopositivethannegativepricechanges.ThefindingsofEpps,Hanna,JainandJoh,andpartsofSmirlockandStarkscouldimplyapositivecorrelationbetweenvolumeandthepricechangeperse(Ap).SuchacorrelationisimpliedbyYing'sitems(1)and(2),andseveralresearchershavedirectlytestedandfoundapositivecorrelation.Usingmonthlydatafrom10stocksand10warrants,Rogalski[60]foundacontemporaneouscorrelationbetweenpricechangeandvolume,butnolaggedcorrelations.Mor-gan[51]andHarris([35],[37])eachfoundapositivecorrelationbetweenpricechangesandvolumeeventhoughitappearstheywerenotlookingforone,asdidRichardson,Sefcik,andThompson[58].Comiskey,Walkling,andWeeks[12]foundpositivecross-sectionalcorrelationsbetweenannualmeasuresofturnoverandpricechange.However,JamesandEdmister[39]foundnosuchcross-sec-13tionalcorrelation.ThisempiricalevidenceissummarizedinTable2.Twofeaturesofthesetestswillbefurtherdiscussedbelow.First,andunliketheempiricalcorrelationsbetweenVand|Ap|reportedinPartIII,thesefindingsareallreportedfromstockorbondmarketdata.Thiscorrelationhasnotbeenreportedinfuturesmarkets.Second,andlikesomeofthefindingsreportedinPartIII,manyofthestatistical12Eachmodelalsoyieldsadditionalinsights.Forexamples,thesequentialinformationarrivalandMDHmodelsarebothconsistentwithincreasingvolumeasthenumberofmarketagentsin-creases(see[65],[15])andwithpositiveskewnessinthedistributionofvolume([34],[15]),whilethePfleiderermodelisconsistentwithanincreaseinfuturestradingasthedeliverydateapproaches.13Incurrentresearch,thisauthorhasfoundpositivecorrelationsbetweenreturnsandtradingvolumeusingdailydatafrombroadmarketindices. 118JournalofFinancialandQuantitativeAnalysisresultsareweak.Forexample,Rogalski'scorrelationswerelow,theaveragebeing0.395amongthestockdataand0.318amongthewarrantdata.Inaddition,therewereseveralfindingsinconsistentwithapositivecorrelation.TABLE2SummaryofEmpiricalStudiesfromwhichInferencesCanbeMadeabouttheCorrelationofthePriceChange(Ap)withaTradingVolume(V)YearSampleSampleDifferencingSupportPositiveAuthor(s)ofStudyDataPeriodInterval(Ap,V)Correlation?Granger1963Stockmarketaggregates,1939-61weeklyNoandMorgenstern2commonstocksGodfrey,Granger,1964Stockmarketaggregates,1959-62,weekly,daily,NoandMorgenstern3commonstocks1951-53,63transactionsYing1966Stockmarketaggregates1957-62dailyYesEpps197520NYSEbondsJan.,1971transactionsYesMorgan197617commonstocks,and1962-65,4-days,Yes44commonstocks1926-68monthlyEpps197720commonstocksJan.,1971transactions,Yes"dailyHanna197820NYSEbondsMay,1971transactionsYesRogalski197810commonstocksand1968-73monthlyYes10associatedwarrants0JamesandEdmister1983500commonstocks1975,77-79dailyNoComiskey,Walkling,1984211commonstocks1976-79yearlyYesandWeeksHarris198450commonstocks1981-83transactions,YesdailySmirlockandStarks1985131commonstocks1981transactionsYes"Wood,Mclnish,1985946commonstocks1971-72,minutesNoandOrd1138commonstocks1982Harris1986479commonstocks1976-77dailyYesJainandJoh1986Stocksmarketaggregates1979-83hourlyYesRichardson,Sefcik,1987106commonstocks1973-82weeklyYesandThompsonaThistablesummarizesthegeneralconclusionsofthesestudiesaboutthecorrelationofApandV.Resultsthatindicatenosignificantcorrelationarelistedasnotsupportingapositivecorrelation.Thesestudiesemployvariousmeasuresofthepricechangeandtradingvolume.Formoreprecisedescriptionsofthedataandvariabletransformations,thereaderisreferredtotheoriginalpapers.bSupportforapositivecorrelationbetweenApandVatthetransactionsleveldependsonthetreatmentofvolumeovertransactionswithnopricechanges.0Stocksaregroupedintodecilesrankedbyaveragedailyvolume.Decilerankingiscomparedwithmeandailyreturn.dThedataareconsistentwithapositivecorrelationbetweenApandVondaysinwhichthereisknowninformationarrival.Onotherdays,thecorrelationappearsinsignificantornegative.B.TheoreticalExplanationsSeveralauthorshaveattemptedtoexplainthesefindings.Morgan[51]sug-geststhatvolumeisassociatedwithsystematicrisk,andthroughthis,tostockreturns.Harris([35],[36])pointsoutthatthemixtureofdistributionshypothesisimpliesapositiveV,Apcorrelationiftheconditionalmeanofthestockpriceprocessisproportionaltothenumberofinformationarrivals.Thenthemeanofthepricechangeprocessissubordinatetothesameparameterasthemeanofthevolumeprocess.Butitisunclearhoweitheroftheseconnectionswouldwork.Infact,theMDHwithmeansubordinationisinconsistentwithmarketequilibrium,sinceitimpliestheexpectedpricechangefromaninformationarrivalispositive.Epps[20]hasconstructedamodelthatimpliesthatvolumeontransactionsinwhichthepricechangeispositiveisgreaterthanfornegativepricechanges.Heassumestwogroupsofinvestors—"bulls"and"bears."Thekeydistinc- Karpoff119tionsarethat"bulls"aremoreoptimisticaboutthevalueoftheassetattheendofthetradingperiod,andtheyreactonlytopositiveinformationabouttheasset'svalue.Thepessimistic"bears"reactonlytonegativeinformation.Thetransac-tiondemandcurveinthismarketconsistsonlyofthedemandpricesof"bulls,"while"bears"comprisethetransactionsupplycurve.Eppsdemonstratesthattherelativeoptimismofthe"bulls,"combinedwithappropriateassumptionsaboutinvestors'utilityfunctions,impliesthemarketdemandcurveissteeperthanthesupplycurve.Becauseofthis,theratioofvolumetoapositivepricechange(whenbulls'demandsincrease)isgreaterthantheabsolutevalueoftheratioofvolumetoanegativepricechange(whenbears'demandsdecrease).Whilethismodelhasbeenusedbyotherresearchers,e.g.,Kelles[45]andSmirlockandStarks[64],itmustbeseriouslyquestioned,sinceitrequiresallinvestorstosystematicallyandselectivelyignorepertinentinformation.Thisim-pliesinvestorirrationality.Withoutadditionalrestrictions,themodelalsoim-pliesasituationinwhich"bulls"acquireincreasinglylargenumbersofshares14from"bears,"whoholdincreasinglynegativequantitiesofshares.InanextensionofCopeland'ssequentialinformationarrivalmodeltoincor-poraterealworldmarginconstraintsandshortselling,Jennings,Starks,andFell-ingham(JSF)[42]provideanalternatetheoryconsistentwiththecorrelationbe-tweenVandAp.Thekeyinnovationisthatshortpositionsarepossiblebutaremorecostlythanlongpositions,whichimpliesthatthequantitydemandedofaninvestorwithashortpositionislessresponsivetopricechangesthanthequantitydemandedofaninvestorwithalongholding.JSFareabletoshowthat,formany(butnotall)cases,thevolumethatresultswhenapreviouslyuninformedtraderinterpretsthenewspessimisticallyislessthanwhenthetraderisanoptimist.Sinceprice(marginally)decreaseswithapessimist(whosells)andincreaseswithanoptimist(whobuys),itisarguedthatvolumeisrelativelyhighwhenthepriceincreasesandlowwhenthepricedecreases.WhileitisconsistentwiththeempiricalcorrelationbetweenVandAp,thismodelissubjecttothesamecriticismsasCopeland's:itreliesonapeculiarinter-pretationofheterogeneityacrossinvestors,itprohibitsuninformedinvestorsfromlearningfromthetradesofinvestorswhoareearlyintheinformationqueue,anditreliesonabehavioraldistinctionbetweengroupsofinvestors.Nev-ertheless,theabsenceofdocumentationofapositivecorrelationbetweenVandA/?infuturesmarkets,wherethecostsoftakinglongandshortpositionsaresymmetric,indicatesthatthedifferentialcostofshortsalesisverylikelyonekeytoatheoryofthevolume-pricechangecorrelation.Thisinsightisadoptedinapaperbythisauthor[43],inwhichamodelisconstructedthatdependsonasymmetriesinthecostsofgoinglongandshort.Costlyshortsalesrestrictsomeinvestorsfromactingontheirinformationwhentheeffectistodecreasetheirdemands.Thisdecreasesthevarianceofinterperiodshiftsintransactionsupplyrelativetothatfortransactiondemand,whichinturncreatesapositivecovariancebetweenvolumeandpricechangeovertheperiod.Consistentwiththiscostlyshortsaleshypothesis,empiricaltestsarepresentedthatrevealthattheempiricalrelationbetweenApandVfoundinstockandbondmarketdataisabsentinfuturesmarketdata.14Inaddition,seethecritiqueofthismodelbySchneller[62]andtheresponsebyEpps[23]. 120JournalofFinancialandQuantitativeAnalysisV.ASynthesisofPreviousResearchIthasbeenalternativelyconcludedthat(1)novolume-pricecorrelationex-ists;(2)acorrelationexistsbetweenVand|Ap|;(3)thecorrelationisbetweenVandAp;and(4)Vishigherwhenpricesincreasethanwhenpricesdecrease.Certainly,items(3)and(4)canbemutuallyconsistent.Butitisarguedherethat(2),(3),and(4)areprobablyalltrue,atleastinmarketsinwhichshortpositionsaremorecostlythanlongpositions.Thereasonfortheseseeminglyinconsistentfindingsisthatmosttestsarebasedonimplicitassumptionsthattheprice-volumerelationsarefunctionaland/ormonotonic,whenitislikelythattheV,AprelationisnotmonotonicandtheV,Aprelationisnotaone-onefunction.Thus,aresearchermayfindweaksupportforanyoneoftheabovehypothesesandstoplookingfortheothers.DefinethesetsoftransformationsW={w=w(Ap)|w'(Ap)>0}X={x=x(Ap)|x'(Ap)>0}Y={y=y(V)|y'(V)>0}.2Examplesincludew(Ap)=Apory(V)=lnV.Empiricaltests,have,ingeneral,specifiedmonotonic,linearrelationsbetweeneitherweWandyeYorbetweenxeXandysY.(ExceptionsincludethenonparametrictestsofEpps[20],SmirlockandStarks[64],andHarris[35],[36]).Butitispossiblethatvolume(elementsofthesetY)correlatespositivelywiththeelementsofbothsetsWandX.Toillustrate,considerFigure1,inwhichitisassumedforsimplicitythatw(Ap)=|Ap|,x(Ap)=Ap,y(V)=V,andtheexpectedvolumeisrelatedline-arlytothepricechange(thesolidline),butwithadiscontinuityatAp=0suchthattherelationisnotmonotonic.Forpositivepricechanges,theconditional+expectedvolume-pricerelationisV=f(ApAp5*0);fornegativepricechanges,itisV~=g(ApAp=£0).If/'>g'isassumed,thenforanygivenexpectedlevelofvolumeE(V)=E(V+)=E(V"),E(V+/Ap)>E{V~/Ap).ThiswouldbeconsistentwiththefindingsofEpps[20],[22],Hanna[33],Smir-lockandStarks[64],andJainandJoh[38].Similarly,atestforlineardepen-dencebetweenVandApperse,althoughmisspecified,woulddiscoverapositivecorrelation.Thisisrepresentedbythedashedline,andisconsistentwiththefindingsofYing(items1and2),Morgan[51],Rogalski[60],Harris[35],[36],Richardson,Sefcik,andThompson[58],andComiskey,Walkling,andWeeks[12].Theassumptionthat/'>|g'|alsoimpliestherelationbetweenVandApisnotaone-onefunction,asdemonstratedwhentheconditionalexpectedvolumeonnegativepricechangesisplottedinthefirstquadrant,V~—h(Ap\Ap=s0).AtestforlineardependencebetweenVand|Ap|,alsomisspecified,wouldyieldpositiveresults,asdatawouldbegeneratedalongthesolidanddottedlinesinthefirstquadrant.ThisisconsistentwithmostofthefindingslistedinTable1.How-ever,noticethatbothofthespecifiedlinearrelationsarewrong,soonewouldexpectempiricalteststhatspecifylinearrelationswouldyieldstatisticallyweakresults.Aspreviouslynoted,thisisalsoconsistentwithseveralofthestudiesreported,andcanaccountforthefailureofaminorityofresearcherstouncoverstatisticallysignificantvolume-pricecorrelations. Karpoff121V=f(Ap|Ap>0)V=g(Ap|Ap<0)V=h(|Ap||Ap<0)ApFIGURE1IllustrationofanAsymmetricVolume-PriceChangeRelationCallFigure1arepresentationofan"asymmetricvolume-pricechangehy-pothesis,"indicatingtherelationisfundamentallydifferentforpositiveandneg-ativepricechanges.Itimpliesthefollowingempiricalpropositions,eachofwhichisapparentfrominspectionofFigure1.(1)Thecorrelationbetweenvolumeandpositivepricechangesispositive.(2)Thecorrelationbetweenvolumeandnegativepricechangesisnegative.(3)Testsusingdataonvolumeandtheabsolutevalueofpricechangeswillyieldpositivecorrelationsandheteroskedasticerrorterms.(4)Testsusingdataonvolumeandpricechangespersewillyieldpositivecorrelations.Whenrankedbythepricechange,theresidualsfromalinearre-gressionofvolumeonpricechangeswillbeautocorrelated.VI.Conclusions:IssuesforFurtherResearchItislikelythatobservationsofsimultaneouslargevolumesandlargepricechanges—eitherpositiveornegative—canbetracedtotheircommontiestoin-formationflows(asinthesequentialinformationarrivalmodel),ortheircom-montiestoadirectingprocessthatcanbeinterpretedastheflowofinformation(asinthemixtureofdistributionshypothesis).Andtherelativelylargecostoftakingashortpositionprovidesanexplanationfortheobservationthat,inequitymarkets,thevolumeassociatedwithapriceincreasegenerallyexceedsthatwithanequalpricedecrease,sincecostlyshortsalesrestrictsomeinvestors'abilitiestotradeonnewinformation.Thissummarizesmuchofwhatisknownabouttheprice-volumerelation.Weconcludebyidentifyingseveralissuesthatmeritfur-therinquiry.(1)Istheprice-volumerelationasymmetric,asproposedinPartV?Anasymmetricrelationexplainsthetwomajorempiricalfindingsreportedinthissurvey,butitisnottheonlyplausibleexplanation.Ying[72],Morgan[51],Harris[35],[36],andRichardson,Sefcik,andThompson[58]eachfoundbothcorrelationsinthesamedataset.Thisprovidessomesupportforthehypothesisofanasymmetricrelation,butthesomewhatanomalousfindingsofWood,Mclnish,andOrd[71]indicatethattheasymmetryoccursintheoppositedirec-tion.(2)Whatwouldaccountfortheasymmetricprice-volumerelation?Ifthe 122JournalofFinancialandQuantitativeAnalysiskeyisshortsaleconstraints,thenfuturesmarketdatawouldrevealnocorrelationbetweenVandA/?,asreportedinKarpoff[43].Totheextentorganizedoptiontradingreducesthecostoftakingnetshortpositions,theasymmetryshouldbeattenuatedinpriceandvolumedatafromoptionablesecurities.Thisproposition15hasnotbeentested.(3)Doesthesizeofthemarketaffecttheprice-volumerelation?Thisanaly-sissuggeststwopossibleeffectsofmarketsize.First,inequitymarkets,heavilytradedissuesaremorelikelytobeoptionableinorganizedexchanges(seetheprecedingparagraph).ThesecondpossibleeffectissuggestedbyTauchenandPitts'results,whichpredictthatthecovarianceofvolumeandthesquaredpricechangeincreaseswiththenumberofinvestors,butatadecreasingrate.(4)Dopropertiesoftherateofinformationflowaffecttheprice-volumerelation?Allofthemodelsdiscussedinthisreviewusethenotionthatinforma-tion,orsomethingthatisinterpretedasinformation,drivesthemarkettoitssuc-cessiveequilibria.Price-volumerelationsthenarisebecauseofassumptionsaboutthenatureofinvestors'reactions.InthemixtureofdistributionsmodelsofTauchenandPitts[65]andHarris[34],thecorrelationofthesquareddailypricechangeandthedailyvolumeincreaseswiththevarianceofthedailyrateofinfor-mationflow.Thisimpliesthatthisprice-volumerelationisstrongestinmarketsorattimesinwhichtheflowofinformationismostvolatile.Ontheotherhand,Pfleiderer[56]arguesthatthestrengthoftherelationincreasesasmoretradingoccursfornoninformationalreasons.(5)Aretheprice-volumerelationsreportedhereidenticaloverdifferentfre-quencies?TauchenandPitts'modelimpliesthatVandApareindependentatthetransactionlevel.Epps'[20]modelimpliesapositivecorrelationatthetransac-tionlevel,whiletheEppsandEpps'andJenningsandBarrymodelsimplyacorrelationbetweenVandApatthetransactionlevel.Copeland'smodelimpliesthissamecorrelation,butoverthelengthoftimenecessaryforeachpieceofinformationtoreachallinvestors.TheempiricalworktodateindicatesthattheempiricalcorrelationofVand|Ap|isstrongeroverfixedtimeintervalsthanoverafixednumberoftransactions([35],[6]).(6)Cantheprice-volumerelationbeexploitedtoimproveeventstudystatis-tics?Themodelsthathypothesizedailypricechangesaremixturesofdistribu-tionsthatimplytheconditionalvarianceofthepricechangeisproportionaltovolume.Volume,inturn,proxiesforanentitythatcanbeinterpretedasinforma-tionflow.Iftherateofinformationflowtendstobehigheraround''event''peri-ods,thenthevarianceofthe"true"priceprocessishigheraroundtheeventdate,indicatingthatstatisticaltestsofabnormalreturnsaroundtheeventshouldbedonewithasamplevarianceadjustedfortherateofinformationflow.Witha15Exactlyhowoptionstradingshouldaffecttheprice-volumeasymmetryisnotobvious.Inves-torswithunfavorableinformationcannowbuyputsorwritecallsratherthansellthestockshort.Thiswouldfurtherdecreasestocktradingvolumeoverintervalswithunfavorableinformation.However,'arbitrageurswouldsellthestocktomaintainput-callparity,increasingthevolumeoftrade.Theneteffectshouldbeanincreaseintradingvolumeassociatedwithunfavorableinformation,sincethecostoftradingonsuchinformationislower.Arefereehaspointedoutthatwhetheroptionsaretradeddependsoncharacteristicsofthesecuritythatmayalsoaffectthecorrelationofpriceandvolume,e.g.,thesizeofthemarketorinformationflows(seepoints3and4below).Thus,atestforwhetherthecorrelationbetweenVandApisattenuatedinoptionablesecuritiesrequiresfurtherconstraintstodistinguishbetweenfutureresearchpoints2,3,and4. Karpoff123higherrateofinformationflowaroundeventdates,testsforstatisticalsignifi-16canceshouldbemorestringentthanwhatisfrequentlyused.(7)Isinformationarrival,ingeneral,sequentialorsimultaneous?SmirlockandStarks[63]findsupportforsequentialoversimultaneousinformationarrival,buttheirhypothesisistestedjointlywithatestofCopeland'smodel.Theissueof"sequentialorsimultaneous"isinpartasemanticissue.Empiricalresearchin-dicatesthatpriceadjustmenttonewinformationis"veryquick,"(e.g.,[56]),but"veryquick"canbeinterpretedasnearlyinstantaneousorassupportinggradualinformationdissemination(theinterpretationgivenbyJenningsandBarry[40],[41]).Morepuzzlingisevidencethatabnormallyhighvolumeper-sistsforsometimeafterinformationalevents,afterthetimeperiodoverwhich17priceeffectsaremeasured.Doesthisindicatetradingbyinvestorswhoarelateinthequeueofagradualinformationreleaseprocess?Ifso,doesitmeanthat"uninformed"investorsdonotknowthattheyarelateinthequeue?Ifabnor-mallyhighvolumearoundeventsrepresentschurningby"uninformed"traders,thiscastsdoubtontheinterpretationofvolumestatisticsasmeasuresof"infor-18mationcontent"ineventstudies.(8)Isthetheorythatguidesempiricalworkinthisareaadequate?Themod-elsreviewedinthispapersacrificegeneralityfortractability,andthishasledtosomeartificialbehavioralclassificationssuchas"optimists/pessimists"and"bulls/bears."Animportantbutdifficultareaforfurtherworkinthisareaistodevelopatheoreticalunderstandingofmarketsthatincorporatesmanydiverseagents,eachofwhommaximizesamultiperiodobjectivefunctionsubjecttoastochasticenvironment.Thejointdistributionofpriceandvolumecouldemergeinsuchamodelasaresultofidiosyncraticshocksthatimpingeonindividual19traders,andaggregateshocksthataffectallmarketagents.16Pincus[57]usesmeasuresofabnormalvolumeandpricechangetoidentifytheperiodofadjustmenttoearningsannouncements,andexaminesthevarianceofthepricechangeduringthisperiod.17Forexample,see[5]and[53].Karpoff[44]arguesthatthismaybeduetomarketfrictionsthatkeepalldemandsfrominstantaneouslyclearing.18Forexamplesofusesofvolumeineventstudies,seefootnote3.19AnimportantstepinthisdirectionistheworkbyPfleiderer[56].AnotherrecentexampleisamodelbyVarian[67]inwhichtradesaremotivatedbydifferencesinopinions(priors).However,theoneunambiguousconclusionreachedabouttheprice-volumerelationisthat,asbeliefsaremoredispersed(andthederivativeofinvestors'risktoleranceislessthan1),equilibriumpricesdecreasewhilevolumeincreases.Thiscannotexplainmostoftheempiricalevidencecitedinthissurvey,whichsupportsapositivecorrelationbetweenVandA/J.Finally,anotherpossiblesourceforfutureworkontheprice-volumerelationaremarketmicrostructuremodelsalongthelinesofGarman[27],whichfocusonthemarketdealer'sinventoryandbid-askquotedecisions. 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