Sector identification in a set of stock return tim

arXiv:cond-mat/0508122v1[cond-mat.dis-nn]4Aug2005SectoridentificationinasetofstockreturntimeseriestradedattheLondonStockExchange∗C.Coronnello×,M.Tumminello+,F.Lillo×,+,†,S.Miccich`e×,+andR.N.Mantegna×,+×INFM-CNR,Unit`adiPalermo,Palermo,Italy+DipartimentodiFisicaeTecnologieRelative,Universit`adegliStudidiPalermo,VialedelleScienze,Edificio18,I-90128,Palermo,Italy†SantaFeInstitute,1399HydeParkRoad,SantaFe,NM87501,USAWecomparesomemethodsrecentlyusedintheliteraturetodetecttheexistenceofacertaindegreeofcommonbehaviorofstockreturnsbelongingtothesameeconomicsector.Specifically,wediscussmethodsbasedonran-dommatrixtheoryandhierarchicalclusteringtechniques.WeapplythesemethodstoaportfolioofstockstradedattheLondonStockExchange.Theinvestigatedtimeseriesarerecordedbothatadailytimehorizonandata5-minutetimehorizon.Thecorrelationcoefficientmatrixisverydiffer-entatdifferenttimehorizonsconfirmingthatmorestructuredcorrelationcoefficientmatricesareobservedforlongtimehorizons.Alltheconsid-eredmethodsareabletodetecteconomicinformationandthepresenceofclusterscharacterizedbytheeconomicsectorofstocks.Howeverdifferentmethodspresentadifferentdegreeofsensitivitywithrespecttodifferentsectors.Ourcomparativeanalysissuggeststhattheapplicationofjustasinglemethodcouldnotbeabletoextractalltheeconomicinformationpresentinthecorrelationcoefficientmatrixofastockportfolio.PACSnumbers:89.75.FbStructuresandorganizationincomplexsystems,89.75.HcNetworksandgenealogicaltrees,89.65.GhEconomics;econophysics,fi-nancialmarkets,businessandmanagement1.IntroductionMultivariatetimeseriesaredetectedandrecordedbothinexperimentsandinthemonitoringofawidenumberofphysical,biologicalandeconomic∗Presentedat...(1)2MantegnaKrakowprintedonFebruary2,2008systems.Afirstinstrumentintheinvestigationofamultivariatetimeseriesisthecorrelationmatrix.Thestudyofthepropertiesofthecorrelationmatrixhasadirectrelevanceintheinvestigationofmesoscopicphysicalsystems(1),highenergyphysics(2),informationtheoryandcommunication(3;4;5),investigationofmicroarraydatainbiologicalsystems(6;7;8)andeconophysics(9;10;11;12;13;14;15).Multivariatestockreturntimeseriesarecharacterizedbyacorrelationmatrixwhichiscarryinginformationabouttheeconomicsectorsoftheconsideredstocks(11;16;17;18;19;20;21;22;23;24;25).Recentempiricalandtheoreticalanalysishaveshownthatthisinforma-tioncanbedetectedbyusingavarietyofmethods.Inthispaperwere-viewsomeofthesemethodsbasedonRandomMatrixTheory(RMT)(18),correlationbasedclustering(11),andtopologicalpropertiesofcorrelationbasedgraphs(25).Thecommonanddifferentaspectsofthesemethodsarediscussedbyconsideringtheresultsofananalysisinvestigatingthesetofn=92stocksbelongingto“SET1”oftheLondonStockExchange(LSE).Thetimeperiodofthetimeseriesistheentire2002yearandtheanalysisisperformedattwodifferenttimehorizons.Specifically,weinvestigatethe5-minutetimehorizonandthedailytimehorizontoshowthedifferencesdetectedinthestructureofthecorrelationmatrixofhighfrequencyanddailyreturns.Thepaperisorganizedasfollows:inSection2wediscussthemethodsusedtoextracteconomicinformationfromacorrelationmatrixofastockportfoliobyusingconceptsandtoolsofRMTandhierarchicalclustering.Theinvestigatedcorrelationbasedclusteringproceduresarethesinglelink-ageandaveragelinkage.Wealsoconsideragraphobtainedbyimposingthetopologicalconstraintofplanarityduringitsconstructionalongawelldefinedalgorithmicprocedure.ThisgraphhasbeennamedbyauthorsasthePlanarMaximallyFilteredGraph(PMFG).InSection3wepresenttheempiricalresultsobtainedfordailyreturnsofthe92stocksbelongingto“SET1”oftheLSErecordedin2002.Section4presentsthetheempiricalresultsobtainedfor5-minutereturnsofthesamesetofdata.InSection5wedrawourconclusions.2.MethodsInthissectionwereviewseveralmethodsusedtoselectpartofthecontentofthecorrelationcoefficientmatrixwhichisrobustwithrespecttostatisticaluncertaintyandcarryingeconomicinformation.ThecorrelationcoefficientbetweenthetimeevolutionoftwostockreturnMantegnaKrakowprintedonFebruary2,20083timeseriesisdefinedasρij(Δt)=hrirji−hriihrjiq(hr2ii−hrii2)(hr2ji−hrji2)i,j=1,...,n(1)wherenisthenumberofstocks,iandjlabelthestocks,riisthelogarithmicreturndefinedbyri=lnPi(t)−lnPi(t−Δt),Pi(t)isthevalueofthestockpriceiatthetradingtimetandΔtisthetimehorizonatwhichonecomputesthereturns.Inthisworkthecorrelationcoefficientiscomputedbetweensynchronousreturntimeseries.Thecorrelationcoefficientmatrixisann×nmatrixwhoseelementsarethecorrelationcoefficientsρij(Δt).WestartourreviewofmethodsbydiscussingtheapplicationofconceptsofRMTwhichhavebeenusedtoselecttheeigenvaluesandeigenvectorsofthecorrelationmatrixlessaffectedbystatisticaluncertainty.Thenweconsidertwodifferentcorrelationbasedclusteringprocedures.Correlationbasedclusteringproceduresareusedtoobtainareducednumberofsimi-laritymeasuresrepresentativeofthewholeoriginalcorrelationmatrix.Thefilteringprocedureassociatedwithareductionoftheconsideredsimilaritymeasuresistypicallygoingfromn(n−1)/2distinctelementstoanumberofsimilaritymeasuresoftheorderofn.Thefirstclusteringprocedureweconsiderhereisthesinglelinkageclusteringmethodthathasbeenrepeat-edlyusedtodetecta