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1、CHAPTER65serialautocorrelationinreturnsInthisChapter...•measuringserialautocorrelation•modelingserialautocorrelation•thetelegraphequation65.1INTRODUCTIONOneofthemanyassumptionsunderlyingtherandomwalksusedinfinance,atleastforequities,isthatofindependentreturns.Theretur
2、nononedayisindependentofwhateverhappenedinthepast.Thefactthatastockpricehasrisentendaysinarowisirrelevant,thenextreturnisacompletelyfreshstart.Nowthismaybetrueforacointossbutshoulditbetrueforfinancialseries?Andifit’snot,howcanweincorporatethisintoamodel?65.2EVIDENCECa
3、lculatingserialautocorrelationissimple.It’slikecalculatingcorrelationbutyouonlyuseonetimeseries.Supposeyouhaveaseriesofreturnsfromstockpricedata,andyou’vesubtractedoffthemeananddividedbythestandarddeviationtomakeaseries,Ri,withzeromeanandunitstandarddeviation.Theseri
4、alautocorrelationwithalagofonedayisthencalculatedasN−1s1=RiRi+1.i=11046PartFiveadvancedtopics0.4252-daySACinSPX0.30.20.1026-Oct-22-Jul-8517-Apr-12-Jan-08-Oct-04-Jul-9631-Mar-25-Dec-20-Sep-17-Jun-8288919399010407−0.1−0.2−0.3Figure65.1Moving252-day,one-daylag,serialau
5、tocorrelationinSPX.Youcansimilarlycalculatetheautocorrelationwithatwo-daylagetc.:N−2s2=RiRi+2.i=1InExcelthisisparticularlysimpletodo.JustusetheworksheetfunctionCORREL.Thistakesintwoarraysasitsarguments.Thefirstarraywillbethereturnsseries,notincludingthefirstone.Thesec
6、ondarraywillbethereturnsseriesfromthefirstuntilthepenultimate.Inotherwords,calculatethecorrelationbetweenaseriesandtheserieslaggedbyoneday.Tocalculatethetwo-daylaggedautocorrelationissimilar,justshifttheseriesbytwodaysinsteadofoneintheCORRELfunction.Figure65.1showsthe
7、moving252-day,one-daylagged,serialautocorrelationinSPX.Theaveragelevelofthiscorrelationisabout0.02.However,priortotheyear2000theaveragelevelhadbeenabove4%.Isthissignificant,istheresomethingstructuralgoingoncausingserialautocorrelationtoventureintonegativeterritoryofla
8、te?Oristhisalljustsamplingerror,andreallythereisn’tanycorrelationatall?Youcaninterpretpositiveserialautocorrelationasbeingtrendlike
