ERIC ED610086: Testing Autocorrelation and Partial Autocorrelation: Asymptotic Methods versus Resampling Techniques

2018·0.5 MB·English

by  ERIC

Runninghead: TESTINGAUTOCORRELATIONANDPARTIALAUTOCORRELATION 1
TestingAutocorrelationandPartialAutocorrelation: AsymptoticMethodsversusResampling
Techniques
ZijunKe
SunYat-SenUniversity
Zhiyong(Johnny)Zhang
UniversityofNotreDame
Ke, Z., & Zhang, Z. (2018). Testing Autocorrelation and Partial Autocorrelation:
Asymptotic Methods versus Resampling Techniques. British Journal of Mathematical
and Statistical Psychology, 71(1), 96–116.
This research was supported by the grant program on Statistical and Research
Methodology in Education from the Institute of Education Sciences of U.S.
Department of Education ( ).
Grant no: R305D140037
AuthorNote
CorrespondenceshouldbeaddressedtoZijunKe,DepartmentofPsychology,SunYat-Sen
University,HigherEducationMegaCenter,Guangzhou,Guangdong510006,China. Email:
[email protected].
TESTINGAUTOCORRELATIONANDPARTIALAUTOCORRELATION 2
Abstract
Autocorrelationandpartialautocorrelation,whichprovideamathematicaltooltounderstand
repeatingpatternsintimeseriesdata,areoftenusedtofacilitatetheidentificationofmodelorders
oftimeseriesmodels,e.g.,movingaverage(MA)andautoregressive(AR)models. Asymptotic
methodsfortestingautocorrelationandpartialautocorrelationsuchasthe1/T approximation
methodandtheBartlett’sformulamethodmayfailinfinitesamplesandarevulnerableto
nonnormality. Resamplingtechniquessuchasthemovingblockbootstrapandthesurrogatedata
methodarecompetitivealternatives. Inthisstudy,weusedaMonteCarlosimulationstudyanda
realdataexampletocompareasymptoticmethodswiththeaforementionedresampling
techniques. Foreachresamplingtechnique,weconsideredboththepercentilemethodandthe
bias-correctedandacceleratedmethodforintervalconstruction. Simulationresultsshowedthat
thesurrogatedatamethodwithpercentileintervalsyieldedbetterperformancethantheother
methods. AnRpackagepautocorrisdevelopedanddemonstratedtocarryouttestsevaluated
inthisstudy.
TESTINGAUTOCORRELATIONANDPARTIALAUTOCORRELATION 3
TestingAutocorrelationandPartialAutocorrelation: AsymptoticMethodsversusResampling
Techniques
Growinginterestinstudyingbehavioralchangeviatimeseriesanalysishasbeenobserved
insocialandbehavioralsciences(Browne&Nesselroade,2005;Borckardtetal.,2008;Ferrer&
Zhang,2009). Bycollectingrepeatedobservationsofthesameindividual,researchersareableto
studypatternsofchangeuniquetotheindividualandmodellaggedeffectsofthesamevariableor
amongvariables. Amongtheclassictimeseriesmodelsfordelineatinglaggedlinearrelationsare
autoregressivemodels(AR),movingaverage(MA)models,andthecombinationofthesetwo
models,i.e.,autoregressivemovingaverage(ARMA)models(e.g.,Shumway&Stoffer,2006).
InanARmodeloforderp,abbreviatedasAR(p),thecurrentvalueoftheseriescanbepredicted
asafunctionoftheppastvalues. Likewise,inanMAmodeloforderq,abbreviatedasMA(q),
thecurrentvalueoftheseriescanbepredictedasafunctionoftheq pastshockvariables
(unpredictedfactors). AstheintegrationofARandMAmodels,thecurrentvalueofanARMA
modelcanbeexplainedasafunctionoftheppastvaluesandq pastshockvariables.
TobuildARMAmodels,onekeystepistoidentifymodelorders,i.e.,togetpreliminary
valuesofpandq. Traditionally,researchersdeterminethevaluesofpandq bytestingthe
significanceoftheautocorrelation/partialautocorrelationfunction(ACF/PACF).Similarto
correlationcoefficientsforindependentdata,theACFisthecorrelationbetweenthecurrentvalue
oftheseriesandvaluesatdifferenttimepoints. Inotherwords,itquantifiesthesimilarity
betweenobservationsasafunctionofthetimelagbetweenthem. ThePACF,asimpliedbyits
name,isa“partialcorrelation”versionoftheACF.Specifically,thePACFisthecorrelation
betweenthecurrentvalueoftheseriesandvaluesatdifferenttimepointsafterpartiallingoutthe
effectsofthevaluesin-between. StatisticianshavefoundthatthebehavioroftheACFandPACF
ofstationaryandinvertibleARMAmodels1 islinkedtothevaluesofpandq. Morespecifically,
1Therearestrictandweakformsofstationarity. Strictstationarityensuresthatthecumulativedistributionfunction
ofeverycollectionofvaluesisinvariantacrosstimepoints. Weakstationaritymeansthatstatisticalpropertiessuchas
mean,variance,andcovarianceagreewiththeircounterpartsintheshiftedsets. Inthispaper,stationarityreferstothe
latter. Invertibilityensuresmodeluniqueness. TwodifferentMA(ARMA)modelscanproducethesametimeseries
(e.g., Shumway & Stoffer, 2006, pp. 91-92; du Toit & Browne, 2007, Tables 6-7). In particular, one of these MA
(ARMA)modelscanbe“inverted”tobeanARmodelofaninfiniteorderandthusisdefinedasbeinginvertible. In
thislight,invertibilityhelpssolvingtheproblemofnonuniqueness.Moreformally,anARMA(p,q)modelisstationary
TESTINGAUTOCORRELATIONANDPARTIALAUTOCORRELATION 4
forAR(p)models,theACFtailsoffandthePACFcutsoffafterlagp;forMA(q)models,the
PACFtailsoffandtheACFcutsoffafterlagq;andforARMA(p,q)models,boththeACFand
PACFtailoff. Makinguseoftheassociationbetweenmodelordersandthe(P)ACF,researchers
candeterminethevaluesofpandq bystudyingthebehaviorofthe(P)ACF.However,inrealdata
analysis,theACFandPACFaresampleestimates. Consequently,researchershavetorelyon
hypothesistestingtoexaminethesignificanceoftheACFandPACF.
Twowell-establishedasymptoticmethodsfortestingautocorrelationsarethe1/T
approximation(e.g.,Shumway&Stoffer,2006,p.519-520)andtheBartlett’sformulamethods
(Bartlett,1955,p.289;Shumway&Stoffer,2006,p.519-520;Zhang&Browne,2010). The1/T
approximationmethodhasbeenadaptedforpartialautocorrelationswhereastheBartlett’smethod
hasnot. Giventhelimitationsofasymptoticmethodsinsmallsamples(e.g.,unwarrantedfinite
sampleperformanceandthevulnerabilitytononnormalityinsmallsamples),resampling
techniqueshavebeendevelopedforassessing(partial)autocorrelations. Usually,thebootstrap
comesintoplaywhenananalyticalsolutionisdifficulttobeadaptedtosmallsamples. Toaddress
theproblemofthelackofindependenceintimeseriesdata,themovingblockbootstraphasbeen
proposedforinferenceon(partial)autocorrelations(Künsch,1989;Efron&Tibshirani,1994,
p.99-102). Whilethebootstrapresamplesfromadistributionassumedtogeneratetheobserved
data,thesurrogatedatamethodresamplesunderthenullhypothesis(Theileretal.,1992). In
otherwords,thebootstrapmethodtriestopreservethetimedependenceofthedatawhereasthe
surrogatedatamethoddestroysthetimedependencetoresemblethenull. Thejackknifeis
anothercommonlyusedresamplingtechniqueforestimatingthebias,thestandarderrorandthe
influencefunction(Efron,1994). Tostudythebiasandthevariabilityoftheestimator,the
jackknifecreatesreplicationsbysequentiallydeletingoneormoreobservations. Becauseonly
oneorafewobservationsarealtered,estimatesbasedontheresultingreplicationsarecorrelated
witheachotherandthusthevariability(difference)acrossjackknifereplicationsismuchsmaller
andinvertibleonlywhentherootsofA(z) = 1 p a zj andB(z) = 1+ q b zj witha sandb sbeing
≠ j=1 j j=1 j j j
ARandMAweightsrespectivelylieoutsidetheuniqtcircle. Practically,withpopuqlationajsandbjs,onemaycheck
thestationarityandinvertibilityofanARMAmodelbyconstructingtheAmatrixinduToit&Browne(2007,Equ.
16)usinga sorb sandexaminingwhethertheabsoluteeigenvaluesofthematrixarelessthenone. Infinitesamples,
j j
the augmented Dickey-Fuller test is often used to test whether differencing is needed to make the observed series
stationary(e.g.,Brockwell&Davis,2002,pp. 193-196).
TESTINGAUTOCORRELATIONANDPARTIALAUTOCORRELATION 5
thanthatofthesamplingdistribution. Totakeintoaccountthenon-independence,thejackknife
estimatorsofthebiasandthestandarderroruseacorrectionfactortoenlargethedifferenceorthe
variability. Becausefordifferentstatistics,thecorrectionfactormayormaynotbevalid,the
bootstrapisusuallymoregeneralthanthejackknife.
AlthoughhowtodeterminemodelordersofARMAmodelshasbeenextensivelydiscussed
intheliterature,twoissueshavecomplicatedtheimplementationoftheseexistingmethodsin
reality. First,smallsamplesizeandnonnormalityposeathreattothevalidityofasymptotic
methods. Samplesizeinpsychologicalstudiesisusuallysmalltomoderate. Thusasymptotic
methodsthatarebasedonlargesamplebehaviormayfailinsmallsamples. Inaddition,
asymptoticmethodsusuallyassumenormalitytoachievemathematicalsimplicity. However,
psychologicaldatararelyfollownormaldistributions(Micceri,1989). Therefore,whether
asymptoticmethodsfortesting(partial)autocorrelationsarerobusttosmallsamplesizeand
nonnormalityawaitsfurtherinvestigation.
Second,althoughbootstrapping(partial)autocorrelationstomakeinferencehasbeen
largelyreliedonthemovingblockbootstrap,itisnotnecessarythatthemovingblockbootstrapis
superiortothesurrogatedatamethodwhentheinterestistotest(partial)autocorrelations. A
relatedstudyontestsofcorrelationsforindependentdatashowedthatunivariatebootstrapwhich
resamplesunderthenulloutperformedbivariatebootstrapwhichresamplesunderthealternative
whentestingcorrelations(Lee&Rodgers,1998). Inaddition,amongtheseveralinterval
constructionmethodsforresamplingtechniques,bias-correctedandaccelerated(BCa)intervals
areoftenrecommendedbecausetheyaresecond-orderaccurate,asopposedtothefirstorder
accuratepercentileintervals(Efron&Tibshirani,1994,p.187)2. Aswewillshowinthisstudy,
BCaintervalsmaynotalwaysbesuperiortopercentileintervals. Thisexceptionislikelyrelated
tothecomputationofthebias-correctionfactorduringtheconstructionofBCaintervalsforthe
surrogatedatamethod.
Insum,becausetherelativeperformanceoftheaforementionedtestsof(partial)
autocorrelationsislargelyunknown,theobjectiveofthisstudyis1)tocomparetherelativefinite
sampleperformanceofvarioustests(i.e.,the1/T approximationmethod,theBartlett’sformula
2Second-orderaccuratemeansthattheerrorinmatchingtheintendedmiss-coverageratesgoestozeroatarateof
1/T intermsofsamplesizeortimeserieslengthT. Forfirst-orderaccurate,therateis1/ÔT.
TESTINGAUTOCORRELATIONANDPARTIALAUTOCORRELATION 6
method,themovingblockbootstrapandthesurrogatedatamethod)of(partial)autocorrelations
withnormalandnonnormaldata,2)todevelopaneasy-to-useRpackagepautocorrto
implementtheaforementionedtests,and3)toprovideapracticalguidelineonapplicationsof
thosetestsundervariouscircumstances.
Therestofthearticleisorganizedasfollows: thetwoasymptotictestsaredescribedfirst,
followedbyasectiononresamplingtechniques. Asimulationstudyispresentedtoevaluatethe
performanceofthosetests. AnRpackageandarealdataexampleareincludedtoillustratetheuse
ofdifferenttestsof(partial)autocorrelations. Finally,thisarticleendswithconcludingcomments.
AsymptoticMethods
The1/T approximationmethodfor(partial)autocorrelationtestingismotivatedbythefact
thatunderthenullhypothesiswhichassumesthatallautocorrelationsarezero,timeseriesdata
reducetouncorrelateddata. Therefore,thewell-establishedstandarderrorestimatorfor
correlation,1/ÔT whereT isthesamplesize,canbeuseddirectlyfor(partial)autocorrelations.
Moreformally,lety ,y ,...,y beastationarytimeseriesoflengthT whosestatistical
1 2 T
propertiessuchasmean,varianceandcovarianceareinvariantacrossdifferenttimepoints. The
concurrentvariance“ = Var(y )andthelaggedcovariances
0 t
“ = Cov(y ,y ) = E[(y µ)(y µ)]areestimatedbytheirsamplecounterpartsusing
l t+l t t+l t
≠ ≠
1 t=T l
≠
“ˆl = (yt+l y¯)(yt y¯)Õ
T ≠ ≠
t=1
ÿ
wherey¯isthesamplemeany¯= 1 T y . The1/T approximationmethodassumesthatunder
T t=1 t
thenull,autocorrelationestimatesfqollowanormaldistributionflˆ N 0, 1 wherefl = “l is
l ≥ T l “0
thepopulationautocorrelationatlagl andflˆ = “ˆl isitssamplecounter1part. 2
l “ˆ0
Thepartialautocorrelation„ forstationaryseriesisdefinedas„ = fl and
l 1 1
„ = Corr y yt l+1,y yt l+1 forl 2. Hereyt l+1 denotesthebestlinearpredictionof
l t ≠ t≠ t≠l ≠ t≠≠l Ø t≠
1 2
y basedony ,y ,...,y 3. Inplainlanguage,„ isthecorrelationbetweeny andy
t t 1 t 2 t (l 1) l t t l
≠ ≠ ≠ ≠ ≠
afterremovingtheassociationattributabletoy ,y ,...,y . Underthenull,„ reduceto
t 1 t 2 t (l 1) l
≠ ≠ ≠ ≠
fl becausebothy andy areunrelatedwith y ,y ,...,y :
l t t l t 1 t 2 t (l 1)
≠ ≠ ≠ ≠ ≠
Ó Ô
3Theoretically,ytt≠l+1canbeobtainedbyregressingytonyt 1,yt 2,...,yt (l 1)inthepopulationandcalculat-
≠ ≠ ≠ ≠
ingthepredictedvaluefory usingtheobtainedregressioncoefficients.
t
TESTINGAUTOCORRELATIONANDPARTIALAUTOCORRELATION 7
„ = Corr y yt l+1,y yt l+1 = Corr(y ,y ) = fl . Inotherwords,underthenullthat
l t ≠ t≠ t≠l ≠ t≠≠l t t≠l l
datapoints1areuncorrelatedwitheach2other,thesamplingdistributionof„ˆ,theestimateof„ ,
l l
shouldbethesameasthatofflˆ. Hence,the1/T approximationmethodassumesthesame
l
distributionfor„ˆ,„ˆ N 0, 1 . Putittogether,the1/T approximationmethodcomputesthe
l l ≥ T
teststatistic flˆl or „ˆl a1ndco2mparesittoacriticalvalueofZ orZ where
1/ÔT 1/ÔT –/2 1≠–/2
Z N (0,1)and– isthechosensignificancelevel.
≥
Becausepartialautocorrelationsremovethedependencebetweeny andy dueto
t t l
≠
y ,y ,...,y ,estimating„ isnotasstraightforwardasestimatingfl . The
t 1 t 2 t (l 1) l l
≠ ≠ ≠ ≠
ÓDurbin-LevinsonalgoritÔhmisawidelyusediterativealgorithmforcomputing„ˆ forstationary
l
series. Let„ denotestheregressioncoefficientofy intheone-step-aheadprediction
lk t k
≠
y „ y +„ y +...+„ y . Bythepropertyofbestlinearprediction,stationarity,and
t l1 t 1 l2 t 2 ll t l
≥ ≠ ≠ ≠
matrixalgebra,itcanbeshownthat„ = „ 4. TheDurbin-Levinsonalgorithmisusedtocompute
ll l
theregressioncoefficientsintheone-step-aheadprediction. Inparticular,with„ˆ = 0,forl 1,
0
Ø
thealgorithmcomputes„ˆ aswellasotherregressioncoefficientsasfollows(e.g.,seeShumway
l
&Stoffer,2006,pp.113-114),
flˆ l 1 „ˆ flˆ
„ˆll = „ˆl = l1≠≠qk≠lk=≠=111„ˆl≠l≠11,k,kflˆl≠kk
whereforl 2andk = 1,2,...l 1, q
Ø ≠
„ˆ = „ˆ „ˆ„ˆ .
lk l 1,k l l 1,l k
≠ ≠ ≠ ≠
Clearly,theDurbin-Levinsonalgorithmcomputes„ˆ usingsampleautocorrelationsflˆ ratherthan
l l
rawdata. Inthisstudy,whenrawseriesarenotavailable,werelyontheDurbin-Levinson
algorithmtoobtainsamplepartialautocorrelations5.
TheBartlett’sformulawasoriginallydevelopedforobtainingtheasymptoticdistribution
forautocorrelationsundergeneralconditions,notonlyunderthenullhypothesis. Accordingto
theBartlett’sformula(Bartlett,1955,p. 289),theasymptoticdistributionofÔT (flˆ fl )is
l l
≠
N (0,W )where
l
W = Π2fl2fl2 2fl fl (fl +fl )+fl2 +fl fl . (1)
l l u ≠ l u u+l u≠l u u≠l u+l
u=ÿ≠ŒÓ Ô
4DetaileddiscussioncanbefoundinExercise3.12inShumway&Stoffer(2006).
5Thefirstauthorhastestedthattheresultingsamplepartialautocorrelationswereexactlythesameasthoseobtained
fromthebuilt-infunctionpacfinthewidelyusedstatisticalsoftwareR.
TESTINGAUTOCORRELATIONANDPARTIALAUTOCORRELATION 8
Inshort,testsbasedontheBartlett’sformulafirstconstructtheconfidenceintervalforfl using
l
flˆ Z Wˆl whereWˆ isobtainedbyreplacingpopulationautocorrelationsinW withtheir
l ± –/2 T l l
Ò
samplecounterparts,andthenexaminewhetherthetestvalue,0,fallsintotheobtained
confidenceinterval. OnepracticalconcernofthismethodisthatinEquation(1)theasymptotic
varianceofflˆ isafunctionofconcurrentandlaggedcorrelationsuptoinfinity. Inpractice,we
l
needtosetanupperlimitforthenumberofsummedterms. Studiesindicatedthat30works
satisfactorily(Zhang&Browne,2010). Thusinthisstudy,weuse30. Accordingtoour
knowledge,theBartlett’sformulahasnotbeenadaptedforpartialautocorrelations. Thus,when
evaluatingtestsofpartialautocorrelations,wewouldnotconsidertheBartlett’sformulamethod.
ResamplingTechniques
Requiringfewerassumptionsandspurredbytheadvanceofcomputertechnology,
resamplingtechniqueshavebecomeincreasinglypopular. Tworesamplingtechniquesproposed
fortestingautocorrelationandpartialautocorrelationarethebootstrapmethodandthesurrogate
datamethod. Becausethesetwomethodsoftenmakestatisticalinferencethroughconfidence
intervals,wefirstbrieflyreviewtwowidelyusedintervalconstructionmethods. Wethenmoveon
tothebootstrapmethod. Thesurrogatedatamethodisnext.
IntervalConstructionMethods
Twowidelyusedintervalconstructionmethodsforresamplingtechniquesarethepercentile
methodandthebias-correctedandaccelerated(BCa)method. Interestedreaderscanreferto
Efron&Tibshirani(1994)andEfron(1987)fordetaileddiscussion. Here,wesummarizethe
mainideasfromthebookandthepaper. Let◊ˆı(–/2) bethe100–thpercentileof(partial)
2
autocorrelationestimatesfromB resamplingreplications. Thepercentileintervalofintended
coverage1 – isobtainedbyfindingthecorrespondingpercentiles,i.e.,[◊ˆı(–/2),◊ˆı(1 –/2)]. For
≠
≠
example,supposethatwedecidetouseacertainresamplingprocedure,e.g.,thebootstrap.
Followingthisprocedure,weformareplicationsampleandwecomputetheestimateforthis
replication. Thisprocessisrepeatedfor2000timesand2000estimatesareobtained. The
percentileintervalendpointsarethe50thandthe1950thorderedvaluesofthe2000estimates.
AccordingtoEfron(1987),percentileintervalsareidealforstatisticsinwhichthereexistsa
TESTINGAUTOCORRELATIONANDPARTIALAUTOCORRELATION 9
g(◊ˆ) g(◊)
monotonictransformationg suchthat ≠ N (0,1)where‡ istheconstantstandard
‡g(◊ˆ) ≥ g(◊ˆ)
errorofg ◊ˆ . Arelatedexampleisthesamplecorrelationcoefficientforbivariatenormally
1 2
distributeddata. Thesamplingdistributionofasamplecorrelationisskewedifthetrue
correlationisfarawayfromzero. Inaddition,thestandarderrorofthesamplecorrelationgets
largerasthetruecorrelationgetsclosertozero. Fisher’sztransformationisoneoftheg
transformationmentionedabovebecauseitnormalizesthesamplingdistribution(thetransformed
statisticfollowsanormaldistribution)andstabilizesthestandarderror(thestandarderror
becomesindependentofthetruecorrelation). Analytically,researchersmayfindouttheinterval
endpointsforg ◊ˆ first,andthentransformthembacktothemetricof◊ˆtoobtaininterval
estimatesfor◊ˆ.1As2pointedoutbyEfron(1987),thebootstrapautomaticallygoesthroughthis
processandthereisnoneedtofindouttheexacttransformationg.
BCaintervals,however,arepreferabletopercentileintervalswhenthetransformationg can
onlynormalizethestatisticbutfailtocorrectbiasandstabilizethestandarderror. More
specifically,thetransformedstatisticg ◊ˆ isallowedtohavebiasandanon-constantstandard
g(◊ˆ) g(◊) 1 2
error, ≠ N ( z ‡,‡2)where‡ = 1+cg(◊)(Efron,1987). Thetwoquantitiesz andc
‡g(◊ˆ) ≥ ≠ 0 0
arecalledbiasandaccelerationconstants. Obviously,z isproportionaltothedifferencebetween
0
theexpectation(ormedian)ofg ◊ˆ andg(◊). Thisishowz getsitsname. Toestimatez ,Efron
0 0
1 2
&Tibshirani(1994)suggestedassessingtheamountofmedianbiasbycountingtheproportionof
replicationslessthantheoriginalestimate. Thecorrespondingpercentileinthestandardnormal
distributionreflectsthesizeofmedianbiasonthescaleofthestandardnormaldistributionand
thuscanbeusedaszˆ ,theestimateofz . Similarly,accelerationcgetsitsnamebecauseitisthe
0 0
rateofchangeofthestandarderrorwithrespecttothetrueparametervalue. Efron(1987)
proposedtoapproximatecbyone-sixthofthe“skewness”oftheinfluencefunction. Theinfluence
functiongenerallymeasureshowtheestimatorchangesifthedistributionofdatachangesslightly.
Becausethejackknifeslightly“changes”theempiricaldistributionofdatabydeletingoneora
fewobservationssequentially,thereisacloseconnectionbetweenthejackknifeandtheinfluence
function. Thus,thejackknifeisoftenusedtoestimatetheinfluencefunctionandhenceis
recommendedforobtainingcˆ,theestimateofc(Frangos&Schucany,1990). Lety bethe
t
≠
originalsamplewithy omittedand◊ˆ betheestimateof(partial)autocorrelationsfromsample
t (t)
TESTINGAUTOCORRELATIONANDPARTIALAUTOCORRELATION 10
y . Define◊ˆ = T ◊ˆ /T . Theaccelerationcˆcanbeobtainedusingthefollowingequation
t () t=1 (t)
≠ ·
q T ◊ˆ ◊ˆ 3
cˆ= t=1 (·) ≠ (t) .
6 qT 1◊ˆ ◊ˆ 22 3/2
t=1 (·) ≠ (t)
5 1 2 6
q
Withzˆ andcˆ,theBCaintervalofintendedcoverage1 – isobtainedbyfindingthefollowing
0
≠
percentiles[◊ˆı(–L),◊ˆı(–U)]where
zˆ +z(–/2)
– =� zˆ + 0
L A 0 1 cˆ(zˆ0 +z(–/2))B
≠
zˆ +z(1 –/2)
– =� zˆ + 0 ≠ .
U A 0 1 cˆ(zˆ0 +z(1≠–/2))B
≠
Herez(–/2) isthe100–/2thpercentileofthestandardnormaldistributionand�( )isthe
·
cumulativedistributionfunctionofthestandardnormaldistribution.
Whenz = c = 0,BCaintervalsreducetopercentileintervals. Onemayexpectcomparable
0
performanceofthesetwomethods. Whenz andcarenonzero,Efron(1987)hasshownthatBCa
0
intervalsaresecond-orderaccuratewhereaspercentileintervalsareonlyfirst-orderaccurate. In
otherwords,empiricalmiss-coverageratesofBCaintervalsconvergetotheintendedcoverage
ratefasterthandopercentileintervalsasthetimeserieslengthincreases. Intheexampleof
correlation,ifthetruecorrelationiszero,whetherornotthebivariatenormalityassumptionis
violated,Fisherztransformationofsamplecorrelationapproximatelyshowszerobiasandits
standarderrorisindependentofthetruecorrelationvalue(e.g.,Hawkins,1989). Hence,inthis
situation,percentileintervalsarecomparabletoBCaintervals,providedthatourestimatesofz
0
andcareaccurate. Whenthetruecorrelationisnonzeroandthebivariatenormalityassumptionis
violated,thestandarderrorofthetransformedcorrelationdependsonthetruecorrelationvalue
(e.g.,Hawkins,1989). Inthissituation,BCaintervalsarelikelybetterthanpercentileintervals.
Similarresultscanbeexpectedforautocorrelationsandpartialautocorrelations.
MovingBlockBootstrap
Thebootstrap,originallyproposedforindependentdata,isacomputerbasedmethodof
statisticalinferencethatcanavoidtheformidablewallofmathematics(Efron&Tibshirani,
1994): Itsampleswithreplacementfromtheoriginaldatapointstoformbootstrapsamplesand
makesinferenceoftheoriginalsamplebasedonbootstrapsamplesinordertomodeltheinference