introduction to financial time series analysis-filtering and smoothing techniques

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1、Chapter3IntroductiontofinancialtimeseriesanalysisFordetailsseetextbooksbyMakridakisetal.[1989],Brockwelletal.[1991]andTsay[2002].3.1PrologueAtimeseriesisasetofmeasurementsrecordedonasingleunitovermultipletimeperiods.Moregenerally,atimeseiesisasetofstatistics,usuallycollectedatregularint

2、ervals,andoccurringnaturallyinmanyapplicationareassuchaseconomics,finance,environmental,medecine,etc.Inordertoanalyseandmodelpriceseriestodevelopefficientquantitativetrading,wedefinereturnsasthedifferencesofthelogarithmsoftheclosingpriceseries,andwefitmodelstothesereturns.Further,toconstru

3、ctefficientsecurityportfoliosmatchingtheriskprofileandneedsofindividualinvestorsweneedtoestimatethevariouspropertiesofthesecuritiesconsititutingsuchaportfolio.Hence,modellingandforecastingpricereturnandvolatilityisthemaintaskoffinancialresearch.Focusingonclosingpricesrecordedattheendofeac

4、htradingday,wearguethatitisthetradingdayratherthanthechronologicaldaywhichisrelevantsothatconstructingtheseriesfromavailabledata,weobtainaprocessequallyspacedintherelevanttimeunit.WesawinSection(2.1.5)thatafirststeptowardsforecastingfinancialtimeserieswastoconsidersometypeoftechnicalindi

5、catorsormathematicalstatisticswithpriceforecastingcapability,hopingthathistorytrendswouldrepeatitself.However,followingthisapproachwecannotassesstheuncertaintyinherentintheforecast,andassuch,wecannotmeasuretheerrorofforecast.Analternativeistoconsiderfinancialtimeseriesanalysiswhichiscon

6、cernedwiththeoryandpracticeofassetvaluationovertime.Whilethemethodsoftimeseriesanalysispre-datethoseforgeneralstochasticprocessesandMarkovChains,theiraimsaretodescribeandsummarisetimeseriesdata,fitlow-dimensionalmodels,andmakeforecasts.Eventhoughitisahighlyempiricaldiscipline,theoryform

7、sthefoundationformakinginference.However,bothfinancialtheoryanditsempiricaltimeseriescontainsomeelementsofuncertainty.Forinstance,therearevariousdefinitionsofassetvolatility,andinaddition,volatilityisnotdirectlyobservable.Consequently,statisticaltheoryandmethodsplayanimportantroleinfina