Estimation of Agent-Based Models the case of an As

EstimationofAgent-BasedModels:thecaseofanAsymmetricHerdingModel∗SimoneAlfaranoa,FriedrichWagnerbandThomasLuxaJuly5,2004aDept.ofEconomics,UniversityofKiel,GermanybDept.ofPhysics,UniversityofKiel,GermanyAbstractThebehavioraloriginsofthestylizedfactsoffinancialreturnshavebeenaddressedinagrowingbodyofagent-basedmodelsoffinancialmarkets.Whilethetraditionalefficientmarketviewpointexplainsallstatisticalpropertiesofreturnsbysimilarfeaturesofthenewsarrivalprocess,themorerecentbehav-ioralfinancemodelsexplainthemasimprintsofuniversalpatternsofinteractioninthesemarkets.Inthispaperwecontributetothisliteraturebyintroducingaverysimpleagent-basedmodelinwhichtheubiquitousstylizedfacts(fattails,volatilityclustering)areemergentpropertiesoftheinteractionamongtraders.Thesimplicityofthemodelallowsustoestimatetheunderlyingparameters,sinceitispossibletoderiveaclosedformsolutionforthedistributionofreturns.Weshowthatthetailshapecharacterizingthefatnessoftheunconditionaldistributionofreturnscanbedirectlyderivedfromsomestructuralvari-ablesthatgovernthetraders’interactions,namelytheherdingpropensityandtheautonomousswitchingtendency.Keywords:HerdBehavior;SpeculativeDynamics;FatTails;VolatilityClustering.JELClassification:G12;C61.Correspondingauthor:Dept.ofEconomics,UniversityofKiel;Olshausenstr.40,D-24118Kiel,Germany.E-mail:[e-mail]∗Earlierversionsofthispaperhavebeenpresentedatthe11thSymposiumoftheSocietyofNonlinearDynamicsandEconometrics,FlorenceMarch2003,the8thSpringMeetingofYoungEconomists,LeuvenApril2003,the8thWorkshoponEconomicswithHeterogeneousInteractingAgents,KielMay2003,the27thcongressofAssociazioneperlaMatematicaApplicataalleScienzeEconomicheeSociali,CagliariSeptember2003;researchseminarsatthedepartmentofEconometrics,UniversityofGeneva,March2003,andatthedepartmentofPhysics,UniversityofCagliari,May2003,andhavegainedconsiderablyfromcommentsbymanyparticipantsintheseevents.WeareparticularygratefultoManfredGilliandTonyHeforhelpfulsuggestionsandintensediscussionsduringthepreparationofthispaper.1IntroductionResearchinempiricalfinancehascomeupwithanumberofstylizedfactsofempiricaldata,involvingconditionalandunconditionalpropertiesofthetimeseries.Startingwiththeformer,thenon-Gaussiancharacteroftheunconditionaldistributionofreturnsisawell-knownubiqui-tousfindingintheliterature[26,23,31].Inparticular,ithasbeenobservedthatpracticallyallfinancialtimeseriesareleptokurtic[26],i.e.theyarecharacterizedbyahigherconcentrationofprobabilitymassinthecenterandinthetailofthedistributionthantheGaussian.However,kurtosisisnotafullyadequatemeasureofdeviationsfromnormality,sinceexistenceoftheforthmomentitselfdependsonthetailbehaviorofthedistribution.IthasbeenrecentlyrecognizedthattheextremepartofthedistributioniswellapproximatedbyaParetolaw[31,27](incon-trasttotheexponentialdecayoftheGaussian).Thisindicatesthatamoreappropriatewaytomeasurethedispersionofreturnsisthetailindex,definedastheorderofthehighestfiniteabsolutemoment.Ananalysisofestimatedindicesacrossdifferentmarketsandtimeresolutions[22,31]showsarelativelysmallintervalofvariabilityoftheseestimates,whichtypicallyhoverbetween2.5and4.Gopikrishnanetal.[13]goevenfurtherclaimingthatwiththetypicalestimatesaround3aninversecubiclawholdsforfinancialfluctuations.Anotheruniversalfeaturethatisstronglyrelatedtotheleptokurticshapeofthedistributionofreturnsistheintermittentbehaviorofthevolatility,orvolatilityclustering,phenomenologi-callydescribedas“periodsofquiescenceandturbulencetendingtoclustertogether”[31,35].Whileserialcorrelationisabsentinrawreturns(whichisinharmonywithinformationalefficiencyoffinancialmarkets),volatilityclusteringleadstopositiveautocorrelationforabsoluteandsquaredreturnsoveranextendedtimehorizon[31].Theslowdecayoftheauto-correlationofallmeasuresofvolatility1isagainwellrepresentedbyapowerlaw,whichisthedefiningpropertyoflongmemorystochasticprocesses.Ithasbeenshownthatthehyperbolicdeclineoftheautocorrelationofthevolatilityisanotherextremelyrobustempiricalphenomenon[21].Anadditionalimportantitemtothelistofempiricalregularitiesisthesocalledunitrootproperty[31,35]:oneisnotabletorejectthehypothesisthatthepricesfollowarandomwalkoramartingaleprocess.Theresultinglackofpredictabilityofthefuturepriceisinaccordancewithinformationalefficiencyofthemarket.Despitethehugeamountofempiricalworkonthestatisticalpropertiesoffinancialdata,theoriginoftheuniversalityoffattailsandvolatilityclusteringinfinancialmarketsisstillobscure.Onecoulddistinguishtwocompetinghypothesesfortheirorigin:onederivedfromthetraditionalEfficientMarketHypothesis(EMH)andarecentalternativewhichwemightcalltheInteractingAgentHypothesis(IAH).TheEMHstatesthatthepricefullyandinstantaneouslyreflectsanynewinformation:themarketis,therefore,efficientinaggregatingavailableinfor-mationwithhisinvisiblehand.Theagentsareassumedtoberationalandhomogeneouswithrespecttotheaccessandtheirassessmentofinformation;and,asaconsequence,interactionsamongthemcanbeneglected.Thereturnsare,therefore,amerereflectionofforthcominginformationsothat,consequently,theirempiricalregularities