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AppliedTimeSeriesAnalysis AppliedTimeSeries Analysis APracticalGuidetoModelingand Forecasting TerenceC.Mills
LoughboroughUniversity,Loughborough,UnitedKingdom
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Introduction 0.1 Datatakingtheformoftimeseries,whereobservationsappearsequentially,usuallywithafixedtimeintervalbetweentheirappearance(every day,week,month,etc.,),areubiquitous.Manysuchseriesarefollowed avidly:forexample,theDowJonesIndustrialstockmarketindexopened 2017withavalueof20,504,closingtheyearon24,719;ariseof20.6%, oneofthelargestannualpercentageincreasesonrecord.ByJanuary26, 2018,theindexhadreachedanintradayhighof26,617beforedeclining quicklytocloseonFebruary8at23,860;avalueapproximatelyequaltothat oftheindexattheendofNovember2017andrepresentingafallof10.4% fromitspeak.Fivedayslater,itclosedonFebruary13at24,640;3.3% abovethis“localminimum.”BytheendofMay2018,theindexstoodat 24,415;littlechangedovertheensuing3months.
This“volatility,”whichwasthesubjectofgreatmediaand,ofcourse, stockmarketattention,wassurpassedbythebehaviorofthepriceofthe crypto-currencybitcoinduringasimilarperiod.Bitcoinwaspricedat$995 atthestartof2017and$13,850attheendoftheyear;anastonishingalmost 1300%increase.Yet,duringDecember17,justafortnightbefore,ithad reachedanevenhigherpriceof$19,871;analmost1900%increasefromthe startoftheyear.Thedeclinefromthishighpointcontinuedintothenew year,thepricefallingto$5968onFebruary6(a70%declinefromthepeak pricelessthan2monthsprior),beforereboundingagaintocloseat$8545on February13.Sincethenthepricehasincreasedto$11,504onMarch4 beforefallingbackto$6635onApril6.AttheendofMay2018,theprice stoodat$7393.
0.2 Whilefinancialtimeseriesobservedathighfrequencyoftendisplay suchwildlyfluctuatingbehavior,therearemanyothertimeseries,often fromthephysicalworld,whichdisplayinterestingmovementsoverlonger periods. Fig.I.1 showsthedecadalaveragesofglobaltemperaturesfromthe 1850sonward.Thebehavioroftemperaturetimeseries,whetherglobalor regional,havebecomethesubjectofgreatinterestand,insomequarters, greatconcern,overthepastfewdecades. Fig.I.1 showswhy.Globaltemperatureswererelativelyconstantfromthe1850stothe1910sbefore increasingoverthenextthreedecades.Therewasthenasecond“hiatus” betweenthe1940sand1970sbeforetemperaturesbegantoincreaserapidly
from 1961 to 1990
Averagedecadalglobaltemperatures,1850 2010.
fromthe1980sonward.Suchbehavior,inwhichtrendsareinterspersedwith periodsofrelativestability,isafeatureofmanytimeseries.
0.3 Howtomodeltimeseriesstatisticallyistheconcernofthisbook,in which,asitstitlesuggests,Iemphasizethepractical,applied,aspectsofstatisticaltimeseriesmodeling.Whileaformaltheoreticalframeworkis,aswe shallsee,anunavoidablenecessity,Idonotoverburdenthereaderwith “technicalniceties”thathavelittlepracticalimpact—myaimisalwaysto providemethodsthatmaybeusedtoanalyzeandunderstandtimeseriesthat occurinthe“realworld”thatresearchersface.1 Examplesare,therefore, drawnfromavarietyoffieldsthatIamfamiliarwithandhaveindeed researchedin.
0.4 Chapter1,TimeSeriesandTheirFeatures,initiatesouranalysisby consideringsomeofthefeaturesthatmaybeexhibitedbyanindividualtime seriesoragroupoftimeseries,andthisleadsnaturallytoChapter2, TransformingTimeSeries,whereavarietyoftransformationsareintroduced thatenableobservedtimeseriestobecomemoreamenabletostatisticalanalysis.Chapter3,ARMAModelsforStationaryTimeSeries,introducesthe basicformalconceptsofstochasticprocessesandstationaritythatunderpin allstatisticaltimeseriesmodels.Itthendevelopsthebasicclassofunivariate timeseriesmodels,theautoregressive-movingaverage(ARMA)process, whichisthecoremodelusedthroughoutthebook.Noteverytimeseriesis
FIGUREI.1
stationary,however:indeed,manyrequiretransformingtostationarity. Integratedprocesses,whichmaybemadestationarybydifferencing,arean importantclassofnonstationaryprocessesandChapter4,ARIMAModels forNonstationaryTimeSeries,thusextendsARMAmodelstotheARintegrated-MA(orARIMA)classofprocesses.2
Animportantaspectofappliedmodelingisthatofdeterminingwhethera timeseriesisstationaryornonstationaryand,iffoundtobethelatter,of determiningwhatformofnonstationarityittakes.Chapter5,UnitRoots, DifferenceandTrendStationarity,andFractionalDifferencing,considers thisproblemwithinthecontextoftestingforunitroots,discriminating betweendifferenceandtrendstationarity,andinvestigatingwhetherfractionaldifferencingisrequired.TheanalysisisextendedinChapter6, BreakingandNonlinearTrends,toconsidermodelshavingbreakingand nonlineartrends.
Animportantuseoftimeseriesmodelsisinforecastingfuture,andhence unobserved,valuesofaseries.Chapter7,AnIntroductiontoForecasting WithUnivariateModels,thusintroducesthetheoryofunivariatetimeseries forecastingbasedontherangeofmodelsintroducedsofar.Chapter8, UnobservedComponentModels,SignalExtraction,andFiltersand Chapter9,SeasonalityandExponentialSmoothing,bothfocusonothertypes ofunivariatemodels,theformeronunobservedcomponentmodels,thelatter onexponentialsmoothingtechniques,whichareparticularlyusefulwhen seasonality,whichisalsodiscussedinthischapter,isafeatureofthetime series.
Chapter10,VolatilityandGeneralizedAutoregressiveConditional HeteroskedasticProcesses,considersvolatility,characterizedbyatimevaryingvariance,andconsequentlyintroducesthegeneralizedautoregressive conditionalheteroskedastic(GARCH)processtomodelsuchvolatility.A time-varyingvariancemayberegardedasaformofnonlinearity,butthere aremanyothertypesofnonlinearprocessesandthesearereviewedin Chapter11,NonlinearStochasticProcesses.
Theremainderofthebookdealswithmultivariatemodels,beginningin Chapter12,TransferFunctionsandAutoregressiveDistributedLag Modeling,withanintroductiontotransferfunctionsandautoregressivedistributedlag(ARDL)models,inwhichan“endogenous”timeseriesis influencedbyoneormore“exogenous”series.Theendogenous/exogenous distinctionmayberelaxedtoallowagroupoftimeseriestoallbeconsideredendogenous.Thisleadstothevectorautoregressive(VAR)modeland therelatedconceptof(Granger)causalitywhichisthetopicofChapter13, VectorAutoregressionsandGrangerCausality.
Thesetwochaptersarerestrictedtoanalyzinggroupsofstationarytime series,butwhenintegratedseriesareallowed,someimportantnewconcepts needtobeintroduced.TheinclusionofintegratedseriesinanARDLmodel
requiresconsiderationoftherelatedconceptsoferrorcorrectionandcointegration,asoutlinedinChapter14,ErrorCorrection,SpuriousRegressions, andCointegration.Chapter15,VARsWithIntegratedVariables,Vector ErrorCorrectionModels,andCommonTrends,extendstheseconceptstoa VARsetting,whichleadstothevectorerrorcorrectionmodel(VECM)and theconceptofcommontrends.
Chapter16,CompositionalandCountTimeSeries,focusesonboth themodelingoftimeseriesthatcomeas“counts,”thatis,smallinteger valuesforwhichtheusualassumptionofacontinuousmeasurementscale isuntenable,andtheimplicationsofmodelingasetoftimeseriesthatform a“composition.”Theseareseriesthataresharesofawholeandmust, therefore,satisfythetwinconstraintsoftakingonlyvaluesbetweenzero andunityandofsummingtounityforeveryobservation,constraintswhich must,forexample,besatisfiedbyfor ecastsoffutureshares.Chapter17, StateSpaceModels,introducesageneralsetupknownasthestatespace form,whichenablesmanyofthemodelsintroducedinthebooktobecast inasingleframework,whichmaybeanalyzedusingapowerfultechnique knownastheKalmanfilter.Chapter18,SomeConcludingRemarks, providessomeconcludingremarksaboutthenatureofappliedtimeseries modeling.
0.5 Eachchaptercontainsappliedexamples,someofwhichare“developed”overseveralchapters.Allcomputationsusethe EconometricViews, Version10 (EViews10)softwarepackageandfulldetailsonhowthecomputationswereperformedareprovidedinthecomputingexercisesthataccompanyeachchapter. EViews10 wasusedasitisanexcellentandpopular packagedevelopedspecificallyforthetimeseriestechniquesdiscussedin thebook.However,Iamawarethatmanyresearchersoutsidetheeconomics andfinanceareausethestatisticalprogramminglanguageR,and EViews10 containslinksbetween EViews routinesandRroutines,whichthose researchersmaywishtoconsult.
0.6 Itisassumedthatthereaderhasabackgroundinintroductorystatistics (atthelevelofMills,2014,say)andsomebasicknowledgeofmatrixalgebra(Mills,2013a,Chapter2,providesaconvenientpresentationofthematerialrequired).
0.7 Abriefwordonnotation.Ascanbeseen,chaptersectionsaredenoted x.y,where x isthechapterand y isthesection(thischapterisnumbered 0). Thisenablesthelattertobecross-referencedas §x.y.Matricesandvectors arealwayswritteninboldfont,uppercaseformatrices,lowercasefor vectorswhereverpossible.Thelatterareregardedascolumnvectorsunless otherwisestated:thus, A denotesamatrixwhile a isavector.
ENDNOTES 1.Thereareseveralhighlytechnicaltreatmentsoftimeseriesanalysis,mostnotablyBrockwell andDavis(1991)andHamilton(1994),whichtheinterestedreadermaywishtoconsult.
2.Acronymsaboundintimeseriesanalysisandhaveevenpromptedajournalarticleonthem (Granger,1982),althoughinthealmostfourdecadessinceitspublicationmany,manymore havebeensuggested!
1.1 AsstatedintheIntroduction,timeseriesareindeedubiquitous,appearinginalmosteveryresearchfieldwheredataareanalyzed.However,their formalstudyrequiresspecialstatisticalconceptsandtechniqueswithout whicherroneousinferencesandconclusionsmayalltooreadilybedrawn,a problemthatstatisticianshavefoundnecessarytoconfrontsinceatleast UdnyYule’sPresidentialAddresstotheRoyalStatisticalSocietyin1925, provocativelytitled“Whydowesometimesgetnonsense-correlations betweentimeseries?Astudyinsamplingandthenatureoftimeseries.”1
1.2 Ingeneral,atimeseriesonsomevariable x willbedenotedas xt , wherethesubscript t representstime,with t 5 1beingthefirstobservation availableon x and t 5 T beingthelast.Thecompletesetoftimes t 5 1; 2; ; T willoftenbereferredtoasthe observationperiod. Theobservationsaretypicallymeasuredatequallyspacedintervals,sayeveryminute, hour,orday,etc.,sotheorderinwhichobservationsarriveisparamount. Thisisunlike,say,dataonacrosssectionofapopulationtakenata given pointintime,wheretheorderingofthedataisusuallyirrelevantunless someformofspatialdependenceexistsbetweenobservations.2
1.3 Timeseriesdisplayawidevarietyoffeaturesandanappreciationof theseisessentialforunderstandingboththeirpropertiesandtheirevolution, includingcalculatingfutureforecastsand,therefore,unknownvaluesof xt at, say,times T 1 1; T 1 2; ...; T 1 h,where h isreferredtoasthe forecast horizon
Fig.1.1 showsmonthlyobservationsofanindexoftheNorthAtlantic Oscillation(NAO)between1950and2017.TheNAOisaweatherphenomenonintheNorthAtlanticOceanandmeasuresfluctuationsinthedifference 1
FIGURE1.1 NAOindex:monthly,January1950 December2017. NAO,NorthAtlanticOscillation. DatafromClimatePredictionCenter,NOAACenterforWeatherandClimatePrediction.
ofatmosphericpressureatsealevelbetweentwostableairpressureareas, theSubpolarlowandtheSubtropical(Azores)high.Strongpositivephases oftheNAOtendtobeassociatedwithabove-normaltemperaturesineastern UnitedStatesandacrossnorthernEuropeandwithbelow-normaltemperaturesinGreenlandandacrosssouthernEuropeandtheMiddleEast.These positivephasesarealsoassociatedwithabove-normalprecipitationover northernEuropeandScandinaviaandwithbelow-normalprecipitationover southernandcentralEurope.Oppositepatternsoftemperatureandprecipitationanomaliesaretypicallyobservedduringstrongnegativephasesofthe NAO(seeHurrelletal.,2003).
Clearly,beingabletoidentifyrecurringpatternsintheNAOwouldbe veryusefulformedium-tolong-rangeweatherforecasting,but,as Fig.1.1 illustrates,noreadilydiscerniblepatternsseemtoexist.
AUTOCORRELATIONANDPERIODICMOVEMENTS 1.4 Suchaconclusionmay,however,beprematurefortheremightwellbe internalcorrelationswithintheindexthatcouldbeusefulforidentifying interestingperiodicmovementsandforforecastingfuturevaluesofthe index.Thesearetypicallyreferredtoasthe autocorrelations betweena currentvalue, xt ,andprevious,orlagged,values, xt k ,for k 5 1; 2; ....The lag-k(sample)autocorrelation isdefinedas
arethesamplemeanandvarianceof xt ,respectively.Thesetofsampleautocorrelationsforvariousvaluesof k isknownasthe sampleautocorrelation function (SACF)andplaysakeyroleintimeseriesanalysis.Anexamination oftheSACFoftheNAOindexisprovidedinExample3.1.
1.5 Asecondphysicaltimeseriesthathasamuchmorepronounced periodicmovementistheannualsunspotnumberfrom1700to2017as shownin Fig.1.2.Ashasbeenwell-documented,sunspotsdisplaya periodiccycle(theelapsedtimefromoneminimum(maximum)tothe next)ofapproximately11years;see,forexample,Hathaway(2010). TheSACFcanbeusedtocalculatean estimateofthelengthofthiscycle, asisdoneinExample3.3.
1.6 Fig.1.3 showsthetemperatureofahospitalwardtakeneveryhourfor severalmonthsduring2011and2012(seeIddonetal.,2015,formore detailsanddescriptionofthedata).Herethereisalongcyclicalmovement—anannualswingthroughtheseasons—superimposeduponwhichare short-termdiurnalmovementsaswellasaconsiderableamountofrandom
FIGURE1.2 Annualsunspotnumbers:1700 2017. DatafromWDC-SILSO,RoyalObservatory ofBelgium,Brussels.
Observation t
FIGURE1.3 Hourlytemperaturesin CofawardinBradfordRoyalInfirmaryduring2011 and2012. DatafromIddon,C.R.,Mills,T.C.,Giridharan,R.,Lomas,K.J.,2015.Theinfluence ofwarddesignonresiliencetoheatwaves:anexplorationusingdistributedlagmodels.Energy Build.,86,573 588.
fluctuation(knownas noise),typicallytheconsequenceofwindowsbeing leftopenonthewardforshortperiodsoftimeandmorepersistentmovementswhicharerelatedtoexternaltemperaturesandsolarirradiation (sunshine).
SEASONALITY 1.7 Whenatimeseriesisobservedatmonthlyorquarterlyintervalsan annual seasonalpattern isoftenanimportantfeature. Fig.1.4 showsquarterlyUnitedKingdombeersalesbetween1997and2017thathaveapronouncedseasonalpatternthathasevolvedovertheyears:firstquartersales arealwaysthelowest,fourthquartersalesareusually,butnotalways,the highest,whiletherelativesizeofsecondandthirdquartersalesseemsto fluctuateovertheobservationperiod.
STATIONARITYANDNONSTATIONARITY 1.8 Animportantfeatureof Figs.1.1 1.3 istheabsenceofanysustained increaseordeclineinthelevelofeachseriesovertheobservationperiod;in otherwords,theyfluctuatearoundaconstantmeanlevelwhichisclearlyto beexpectedfromthephysicalnatureofeachseries.Aconstantmeanlevelis one,butnottheonly,conditionforaseriestobe stationary.Incontrast,beer
FIGURE1.4 UnitedKingdombeersales(thousandsofbarrels):quarterly,1997 2017. Data fromtheBritishBeer&PubAssociation.
FIGURE1.5 $ d Exchangerate:dailyobservations,January1975 December2017. Data fromBankofEngland.
salesin Fig.1.4 showadeclinethroughouttheobservationperiod,most noticeablyfrom2004onwards.Clearlybeersaleshavenotfluctuatedaround aconstantmeanlevel;rather,themeanlevelhasfalleninrecentyearswhich hasbeenawell-documentedconcernofUKbrewers.
1.9 Ifthemeanlevelcannotberegardedasconstantthenaseriesissaidto be nonstationary.Nonstationarity,however,canappearinmanyguises. Fig.1.5 plotsdailyobservationsontheUSdollar UKsterling($ d)
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