ar, ma, arma, & all that

73 Pages·2011·4.21 MB·English

6
TIME SERIES MODELS:
AR, MA, ARMA, & ALL THAT
Time series are ubiquitous in everyday manipulations of financial data. They are
especially well suited to the nature of financial markets, and models and methods
havebeendevelopedtocapturetimedependenciesandproduceforecasts.Thisisthe
mainreasonfortheirpopularity.Thischapterisdevotedtoageneralintroductionto
thelineartheoryoftimeseries,restrictedtotheunivariatecase.Laterinthebook,we
willconsiderthemultivariatecase,andwewillrecasttheanalysisoftimeseriesdata
intheframeworkofstatespacemodelsinordertoconsiderandanalyzenonlinear
models.
6.1 NOTATION AND FIRST DEFINITIONS
The goal of time series analysis is to analyze data containing finite sequences of
measurements,eachcomingwithatimestamp,thesetimestampsbeingorderedin
anaturalfashion.Thepurposeoftheanalysisistoquantifythedependenciesacross
time,andtotakeadvantageofthesecorrelationstoexplaintheobservationsathand,
andtoinferpropertiesoftheunobservedvaluesoftheseries.
Wehavealreadyencounteredmanyinstancesoftimeseries(recall,forexample,
the coffee futures data as plotted in Figure 3.4). In most cases, we transformed the
datatoreducetheserialcorrelationtoaminimum,andweusedstatisticaltechniques
completelyindifferenttotheorderofthedata:inthisway,wedidnotuseanypossi-
bleserialdependenceinthedata.Itisnowtimetoinvestigatethevariouswaysone
can model this dependence, and take advantage of the properties of these models.
However,therewasoneinstancewherewehadtodealwiththeeffectofthedepen-
denciesovertimeofthedataentries.Thiswasthecaseoftheutilityindexes,whose
datawereproducenow.
ENRON.index DUKE.index UTILITY.index
01/04/1993 135.0000 104.2857 99.94170
01/05/1993 135.3714 103.5714 99.49463
01/06/1993 132.8571 104.2857 99.86034
304 6 TIMESERIESMODELS:AR,MA,ARMA,&ALLTHAT
01/07/1993 130.7143 103.5714 98.70023
...... ...... ...... ......
12/28/1993 166.4571 125.0000 107.15202
12/29/1993 170.7429 123.9429 106.75023
12/30/1993 169.3143 124.2857 106.12351
12/31/1993 165.7143 121.0857 104.95227
Thefirstcolumngivesasetofdates,someformofdailytimestamps,whilethenextthree
columnscontainthenumericalvaluesofthethreeindexesonthesedates.Thisisthetypical
structureoftimeseriesdatawhichweconsiderinthischapterandthenext.
6.1.1 Notation
Moststatisticalproblemsdealwithdataintheform
x ,x ,...,x . (6.1)
0 1 n
In the regression applications considered so far, the order in which the observations were
collecteddidnotplayanyrole.Wearenowinterestedinapplicationsforwhichtheorderof
thex ’splaysacrucialroleintheinterpretationofthedata,aswellasinthedefinitionofthe
i
inferentialproblemsweconsider.
Intheapplicationsweareconsideringnow,thelabelnoftheobservationx corresponds
n
toatimestamp,sayt ,givingthetimeatwhichthemeasurementwastaken.Asalways,itis
n
convenienttoviewtheobservations(6.1)asrealizationsofrandomvariablesX ,X ,...,X
0 1 n
whichweshallsometimesdenoteX ,X ,...,X whenwewanttoemphasizetheroleof
t0 t1 tn
thetimestamps.Thesen+1randomvariableswillmostoftenberegardedasasubsetofa
(possiblyinfinite)sequence{X }ofrandomvariables.Thex ’s(andhencetheX ’s)canbe
t i i
scalarsasinthischapter,inwhichcasewetalkaboutunivariatetimeseries,orvectorsasin
thenextchapter,inwhichcasewetalkaboutmultivariatetimeseries.Asbefore,wetrytouse
regularfontsforscalarsandbold-facefontsforvectors.
Mostofthischapterisdevotedtotheanalysisoftimeseriesmodels.Amodel isaset
ofprescriptionsforthejointdistributionsoftherandomvariables(orrandomvectorsinthe
multivariatecase)
X ,X ,...,X
i1 i2 ik
forallpossiblechoicesofthefiniteorderedsequencei < i < ··· < i oftimestamps.
1 2 k
Thesejointdistributionsarecompletelydeterminedbythemodelinsomecases,whileinother
cases,onlypartialinformationisprovidedbytheprescriptionsofthemodel.
6.1.2 RegularTimeSeriesandSignals
Regulartimeseries aresetsofmeasurementstakenatregulartimeintervals.Inotherwords,
the time stamps {t } are of the form t = t +j∆t for j = 0,1,...,n. Such
j j=0,1,...,n j 0
asequenceoftimesisdeterminedbyitsstartt ,itslengthn+1,andthetimeinterval∆t
0
betweentwosuccessivetimes.Notethat,insteadofgivingthesamplinginterval∆t,onecan
equivalentlygivethe samplingfrequency,orthetimeofthefinalmeasurement.Oncethetime
sequencehasbeendefined,onecanthengivethesequenceofcorrespondingmeasurements
separately.Thischaracterizationofthesequenceoftimestampsofaregulartimeseriesby
threecharacteristicsisfundamentalinthetimeseriespackagesimplementedinmostofthe
statisticalsoftwarecomputerpackages.
6.1 NotationandFirstDefinitions 305
Figure6.1givesanexampleofsucharegulartimeseries.Itisaspeechsignalwhichwe
createdbyrecordingtheshortsentence”how are you”,digitizingthesoundfile,andcol-
lectingtheresultingnumericalvaluesinanRnumericalvectorwhichwecalledHOWAREYOU.
Figure6.1wasproducedwiththecommand:
Speech Signal HOWAREYOU
0.5
OU
WAREY 0.0
O
H
-0.5
0 1000 2000 3000 4000 5000
Index
Fig.6.1.Plotofthesound”HowAreYou”digitizedat8000Hz
plot(HOWAREYOU,type="l",main="Speech Signal HOWAREYOU")
Insuchaplot,thetimestampsusedtolabeltheelementsofthesignalaresimplysuccessive
integersstartingfromone.TheyarereferredtoasINDEXintheplot.Thisshouldbecontrasted
withwhatcomesnext.
In part because of their frequent occurrence in applications to signal analysis (as tradi-
tionallyperformedbyelectricalengineers),regulartimeseries areoftencalledsignals.The
librarystatsofRprovidesobjectsofclasststomanipulatesignals,butweshallnotuse
them.
6.1.3 CalendarandIrregularTimeSeries
Mostofthefinancialtimeseriesdonothavethegoodtastetoberegularinthesensegiven
above.Theydifferfromtheregulartimeseriesdiscussedaboveinseveralways,andmostly
bythefactthatthetimestampsaregivenbydatesandtimes,thustheirnamecalendartime
series.Eventhoughcalendartimeseriesareparticular casesofalargerclassofirregulartime
series,theywillbetheonlyonesconsideredhere. Oftentimes,thesedataaredaily,andgaps
duetoweekendsandholidayscreateirregularities.Figure6.2givesthedailyclosingpricesof
theS&P500indexontheNewYorkStockExchange(NYSEforshort)fortheperiodranging
fromJanuary3,1950toAugust20,2010.
306 6 TIMESERIESMODELS:AR,MA,ARMA,&ALLTHAT
Time Series Plot of DSP.ts
0
0
5
1
0
0
0
1
0
0
5
0
1960 1970 1980 1990 2000 2010
Fig.6.2.timeSeriesplotofthedailyclosingvaluesoftheS&P500indexproducedby
thecommandplot(DSP.ts).
Hereisa(very)smallsubsetofthedatausedtoproducetheplot.
Date Open High Low Close
..... ..... ..... ..... .....
17-Sep-01 1092.54 1092.54 1037.46 1038.77
10-Sep-01 1085.78 1096.94 1073.15 1092.54
7-Sep-01 1106.40 1106.40 1082.12 1085.78
6-Sep-01 1131.74 1131.74 1105.83 1106.40
5-Sep-01 1132.94 1135.52 1114.86 1131.74
4-Sep-01 1133.58 1155.40 1129.06 1132.94
31-Aug-01 1129.03 1141.83 1126.38 1133.58
30-Aug-01 1148.60 1151.75 1124.87 1129.03
29-Aug-01 1161.51 1166.97 1147.38 1148.56
28-Aug-01 1179.21 1179.66 1161.17 1161.51
..... ..... ..... ..... .....
Aswecansee,thetimestampscanbeveryirregularlyspacedattimes.Asillustratedinthis
snapshot,theregularityofthemeasurementscanbeaffectedbyunexpectedevents.However,
thescaleofatypicalplotofamulti-yeardataserieswouldnotallowustoseethegapsdueto
weekendsandholidaysandextraordinarymarketclosureevents.
6.1.4 CreatingandPlottingtimeSeriesObjectsinR
The manipulation of calendar time series in R in done via objects of class timeSeries.
Theseobjectscontainaslotpositionsforthetimestamps,andaslotdatafortheac-
tualvaluesofthenumericalmeasurements. Typically,positionsisavectorofdatesor
datesandtimes,whiledataisanumericalmatrixwithonerowforeachentryofthevec-
torpositions.OnecreatesanobjectofclasstimeSerieswiththeconstructorfunction
timeSeries whose use is illustrated below. The numeric vector of the weekly values of
6.1 NotationandFirstDefinitions 307
theS&P500indexwasusedearlierinChapter2whenwecomputedtheweeklyreturnson
theindex.WemakeitintoatimeSeriesobjectbyattachingthetimestampsgivingthe
weeksofthemeasurements.WefirstcreatethevectorSPWEEKSofdateswiththefunction
timeSequence,andwebundlethisvectorofdateswiththevectorWSPofclosingvalues
intoatimeSeriesobjectWSP.tswhichwethenplotwiththecommandplot.Allthis
isdonewiththefollowingcommands:
SPWEEKS <- timeSequence(from = "1950-01-03",
by = "week",length.out = 3163)
WSP.ts <- timeSeries(positions=SPWEEKS,data=WSP)
plot(WSP.ts)
The generic method plot can be used with timeSeries objects. The resulting plot is
giveninFigure6.3.Atthisscale,itisdifficulttodifferentiatethisplotfromtheplotofthe
dailyclosingvaluesoftheindex.
Time Series Plot of WSP.ts
0
0
5
1
0
0
0
1
0
0
5
0
1960 1970 1980 1990 2000 2010
Fig. 6.3. timeSeries plot of the weekly closes of the S&P 500 index produced by the
commandplot(WSP.ts).
6.1.5 HighFrequencyData
The widespread availability of high frequency data changed the landscape of financial data
analysis, and spurred for better or worse the development of high frequency trading. Our
discussionofthemorningandafternoonindicatorsinChapter5onnonparametricregression
wasafirstexampleinvolvinghighfrequencydata.Here,weconsideranotherexamplemore
inlinewiththecurrentdiscussionoftimeseries.
Highfrequencydataaretheresultofadifferentdatacollectionprocess:arecordisadded
tothedatafileeachtimeanewtransactiontakesplace.Thesedataarealsocalledtransaction
data, or tick data. They offer a unique insight into actual trading processes and market mi-
crostructure.Inthissubsection,westudythemostimportantfeaturesofhigh-frequencytime
seriesdata,identifyingstrikingdifferenceswithlower-frequencydata,andintroducingnew
toolsandnewmethodstailoredtothenewchallengespresentedbythisnewtypeofdata.
308 6 TIMESERIESMODELS:AR,MA,ARMA,&ALLTHAT
Tick-by-tickdataareavailableforliquidfuturescontracts,andinthissection,weconsider
theexampleofafuturescontractsontheS&P500indexforthesakeofillustration.Hereis
thewaythedataoftheSeptember1998contractlooklike.
”date” ”time” ”close”
1997100814:53:381013.20
1997101710:59:16 986.00
1997102710:02:13 960.00
1997110310:28:51 968.00
1997110509:08:44 975.00
1997110610:59:21 969.00
1997112412:52:52 986.90
1997120910:58:181015.00
1997121009:22:051005.70
1997122409:27:21 968.00
... ... ...
Wenoticethatthetimestampscanbeverysparse.Thisisduetothefactthatthesetrades
occurredondaysveryfarfromthematurityofthecontract:speculatorsareactivelytrading
contractsclosertodelivery!However,thesituationchangesdramaticallywhenwelookatthe
datalaterinthelifeofthecontract.Indeed,oneseesthatnotonlythetransactionsaremore
frequent,butalsothatalargenumberoftransactionsappearstohappensimultaneouslysince
theyhavethesametimestamp.Giventhefactthateachrowcontainsonlyonenumberbesides
thedateandtime,weshallassumethatthisnumberisthepriceatwhichthetransactionwas
settled, not a bid or ask price. This information is not always given by the data provider,
andthedataanalystmaybeforcedtomakethissortofassumption.Oneoftheunexpected
surpriseswithhigh-frequencyfinancialdataisthefactthatthenotionofpriceisnotclearly
defined.Strangelyenough,therearemanyreasonsforthat.Thefirstoneisclearfromthedata
reproducedbelow:differentvaluescanbequotedwiththesametimestamp,sowhatisthe
priceatthattime?
”date” ”time” ”close”
... ... ...
1998080411:05:001103.50
1998080411:05:001103.00
1998080411:06:001102.80
1998080411:06:001102.60
1998080411:06:001102.50
1998080411:06:001102.40
1998080411:06:001102.20
1998080411:06:001102.00
... ... ...
1998080411:06:001102.50
1998080411:06:001102.40
1998080411:06:001102.20
1998080411:06:001102.00
1998080411:07:001101.70
... ... ...
Whetheronelooksatthisparticularportionofthedatasetornot,ithappensveryoftenthat,
manysecondsdonotappearbecausethereisnotransactionatthesetimes.Anotheridiosyn-
6.1 NotationandFirstDefinitions 309
crasyofhigh-frequencydataisthefactthatthebidandaskpricesdonotmakesenseallthe
time.Indeed,whenthefrequencyishighenough,thetimeintervalbetweentwotransactionsis
sosmallthatthepricecannotmoveoutofthebid-askspread,muddyingboththedefinitionof
thenotionofpriceandofbid-askspreadatthesametime.Weavoidthisissuebyconsidering
thatthetickvaluegivenbythedataprovideristhesettlementprice.
ThedatadiscussedaboveareincludedinthelibraryRsafd.Theyarecontainedinthe
dataframeSPsep98whosetopwereproducebelow:
head(SPsep98)
date time close
1 19971008 14:53:38 1013.2
2 19971017 10:59:16 986.0
3 19971027 10:02:13 960.0
4 19971103 10:28:51 968.0
5 19971105 09:08:44 975.0
6 19971106 10:59:21 969.0
Thisdataframehasthreecolumns.Thefirstone,nameddate,givesthedateofthequote.
Noticetheformat!Thesecondcolumn,namedtime,givesthetimeofthedayatwhichthe
quotewasprovided,andfinally,thethirdcolumn,namedclosegivestheactualthequote
(soweassume).
InordertocreateanobjectofclasstimeSerieswiththesedata,wefirstcreateavector
ofpositionswiththedaysandtimesgiveninthefirsttwocolumnsofthedataframe.Wefirst
usethefunctionmakeDatewiththecolumndate,specifyingtheformatfromwhichthe
dateshouldbereadintheparameterin.format.Inthiscase,theuppercase”Y”indicates
thatthefirstfourcharactersshouldbeunderstoodastheyear,thelowercase”m”indicatesthat
thenexttwocharactersshouldbeunderstoodasanintegergivingthemonthoftheyear,andthe
lowercase”d”indicatesthatthelasttwocharactersshouldbeunderstoodasanintegergiving
thedayoftheweek.ThecommandusingthefunctiontimeDategivesanotherexampleof
formattedreading,moreintheunixstylethistime.Inanycase,itsoutputisavectorwhich
canbeusedasavectorofpositionsforanobjectofclasstimeSeries.Theextraparameter
unitsintheconstructortimeSeriesisusedtospecifyanameforthenumericalvalues
ofthedatacomponentofthetimeseries.
SPsep98day <- makeDate(SPsep98[,1],in.format="Ymd")
SPsep98POS <- timeDate(paste(SPsep98day,SPsep98[,2]),
format = "%Y-%m-%d %H:%M:%S")
SPsep98.ts <- timeSeries(positions=SPsep98POS,
data=SPsep98[,3], units="Sep98")
head(SPsep98.ts)
Sep98
1997-10-08 14:53:38 1013.2
1997-10-17 10:59:16 986.0
1997-10-27 10:02:13 960.0
1997-11-03 10:28:51 968.0
1997-11-05 09:08:44 975.0
1997-11-06 10:59:21 969.0
plot(SPsep98.ts)
310 6 TIMESERIESMODELS:AR,MA,ARMA,&ALLTHAT
ThecorrespondingtimeSeriesplotisreproducedinFigure6.4.
Time Series Plot of SPsep98.ts
1200
1150
1100
1050
1000
950
Nov Jan Mar May Jul Sep
Fig. 6.4. timeSeries plot of the high frequency quotes of the S&P 500 futures contract
maturingonSeptember1998asproducedbythecommandplot(SPsep98.ts).
Thisplotshowsclearlythespecificfeaturesofthedatawhichweidentifiedearlier.Theleft
partoftheplotcontainsonlyafewtransactions,andbecausetheplottingprograminterpolates
linearlybetweenpoints,weseeanartificialpiecewiselinearpatternforthepriceofthefutures
contract.Therightpartoftheplotismoretypicalofvolatilefinancialtimeseries.
Remark on Quantized Ticks. Price changes from one transaction to the next are quoted
inmultiplesofticksize.Thisticksizevariesfromoneexchangetoanother.Typicalvalues
are (or used to be) one eighth and one sixteenth of a dollar. This practice is obsolete on a
certainnumberofexchanges.Forexample,allNewYorkStockExchange(NYSE)andNew
YorkMercantileExchange(NYMEX)stocksaretradedindecimalssinceJanuary29,2001.
Nevertheless,practitionersshouldbeawareofthefactthathigh-frequencydata,andespecially
historical high-frequency raw data which has not been pre-processed, quite often take only
discretevalues:thiscanintroducenumericalartifacts,andinparticularspuriouscorrelation.
6.1 NotationandFirstDefinitions 311
Histogram of FRACT
0
0
8
0
0
6
y
c
n
e 0
u 0
q 4
e
Fr
0
0
2
0
0.0 0.2 0.4 0.6 0.8
FRACT
Fig.6.5.HistogramofthefractionalpartoftheJune15,1998quotesoftheS&P500Septem-
ber1998futurescontract.Thediscretenatureofthedatashowsclearly.
WeillustratethisfactwiththeS&P500dataconsideredinthissubsection.Wecomputed
the fractional parts of the transaction prices (obtained by removing the integer parts to the
actualprices),andweplottedtheirhistograminFigure6.5.Thequantificationeffectappears
clearly.
6.1.6 TimeDateManipulations
It is very easy to develop specific functions satisfying the needs of most time series data
analysis.Inparticular,thelibraryRsafdcontainsacertainnumberof”homegrown”func-
tionswhichwewrotetomaketimeDatemanipulationseasy.Amongthem,thefunctions
begdayandnoonextractthebeginningofagivenday,andnoonofthesameday.Inother
words,givenatimeDate,thefirstfunctionextractstheday,andreturnsatimeDatein-
cludinghours,minutesandsecondsofthebeginningofthatsameday.
ThefollowingcommandsillustratethealgebraicmanipulationsontimeDateobjects,and
showhowonecanextractsubsetsofatimeseries.vskip4pt
DAY <- timeDate("08/12/1998")
SP081298 <- SPsep98.ts[seriesPositions(SPsep98.ts) >= DAY
& seriesPositions(SPsep98.ts)<DAY +24*3600]
plot(SP081298)
andtheresultingplotisreproducedinFigure6.6.
312 6 TIMESERIESMODELS:AR,MA,ARMA,&ALLTHAT
Time Series Plot of SP081298
1090
1088
1086
1084
1082
1080
1078
10:00 12:00 14:00
Fig.6.6.PlotofonedayextractedfromthetimeSeriesobjectSPsep98.ts.
Onecouldaswellextractthreeconsecutivedaysinsteadofone.
SPDAYS <- SPsep98.ts[seriesPositions(SPsep98.ts)>=DAY
& seriesPositions(SPsep98.ts)<DAY+3*24*3600]
plot(SPDAYS)
andtheresultingplotisgiveninFigure6.7.Asexplainedearlier,theplottingfunctionper-
Time Series Plot of SPDAYS
1090
1080
1070
1060
Thu Fri
Fig. 6.7. Plot of the short timeSeries object obtained by extracting three consecutive
daysfromthetimeSeriesobjectSPsep98.ts.
formsalinearextrapolationtojointhelastquoteofonedaytothefirstquoteofthefollowing
day.However,thescaleofthehorizontalaxisisuniformthroughoutthetimedomainandno