Testing for Pairwise Serial Independence via the Empirical Distribution Function
Author(s): Yongmiao Hong
Source: Journal of the Royal Statistical Society. Series B (Statistical Methodology), Vol. 60, No.
2 (1998), pp. 429-453
Published by: Wiley for the Royal Statistical Society
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J.R. Statist.
Soc. B (1998)
60, Part2,pp.429-453
Testing forpairwise serial independence via the
empiricaldistributionfunction
Yongmiao Hongt
CornellUniversity,
Ithaca,USA
[ReceivedJuly1996. RevisedMay1997]
Summary.Builton Skaug and Tj0stheim's
approach,thispaperproposesa newtestforserial
independenceby comparing
the pairwiseempirical
distribution
functions
of a timeseries with
the productsof its marginals
forvariouslags, wherethe numberof lags increaseswiththe
samplesize anddifferent
lagsare assigneddifferent
weights.
themorerecentinformation
Typically,
receivesa largerweight.
The testhas some appealingattributes.
Itis consistent
againstall pairwise dependencesand is powerful
againstalternatives
whose dependencedecays to zero as
thelag increases.Although
theteststatistic
is a weighted
sumofdegenerateCram6r-von
Mises
ithas a nullasymptotic
statistics,
N(O,1) distribution.
The teststatistic
and itslimit
distribution
are
invariant
to anyorderpreserving
transformation.
The testappliesto timeserieswhosedistributionscan be discreteor continuous,
withpossiblyinfinite
moments.
theteststatistic
Finally,
only
involvesranking
theobservations
and is computationally
simple.Ithas theadvantageofavoiding
smoothednonparametric
A simulation
estimation.
is conductedto studythe finite
experiment
sampleperformance
oftheproposedtestin comparison
withsome relatedtests.
Keywords:
Asymptotic
normality;
Cram6r-von
Misesstatistic;
Empirical
diitribution
function;
Hypothesis
testing;
Serialindependence;
Weighting
1. Introduction
The detectionof serialdependenceis importantfornon-Gaussianand non-lineartimeseries.
Tests forserialindependenceare usefuldiagnostictools in fitting
non-lineartimeseriesand
identifying
appropriatelags, especiallywhen the time serieshas zero autocorrelation(see
Grangerand Terasvirta(1993)). For instance,to detectnon-linearity
we can testwhetherthe
residualsfroma linearfitare independent.
We can also testtherandomwalk hypothesis
fora
timeseriesbytesting
whether
itsfirst
differences
are seriallyindependent.
Testsforindependence
are usefulin othercontextsas well (see Robinson (1991a)). Independenceis stilla testable
in populationswithinfinite
hypothesis
second-order
moments.
In practice,correlationtests(e.g. Box and Pierce (1970) and Ljung and Box (1978)) are
widelyused to testserialindependence,but theyare not consistentagainstalternatives
with
zero autocorrelation.
The lack of consistencyis unsatisfying
fromboth theoreticaland practical viewpoints.Also, thesetestsrequiresecond-or higherordermoments,excludingtheir
applicationsto timeserieswithinfinite
second-ordermoments.
There have been manynonparametric
testsfor serialindependence,e.g. Chan and Tran
(1992), Delgado (1996), Hjellvikand Tj0stheim(1996), Pinkse(1997), Robinson(1991a) and
tAddressfor correspondence:
Departmentof Economics and Departmentof StatisticalSciences,College of Arts
and Sciences,Uris Hall, CornellUniversity,
Ithaca, NY 14853-7601,USA.
E-mail:[email protected]
? 1998 RoyalStatistical
Society
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1369-7412/98/60429
430
Y. Hong
Skaug and Tj0stheim(1993a, b, 1996). These testsare consistentagainstserialdependences
densityestimation.Both
up to a finiteorder.Some of themrequiresmoothednonparametric
theoryand simulationstudies(e.g. Skaug and Tj0stheim(1993b, 1996)) suggestthatthefinite
testsmay dependheavilyon the choice of
of smootheddensity-based
sampleperformances
methods
thesmoothingparameters.Resamplingmethodssuchas bootstrapand permutation
have been used to obtainaccuratesizes (e.g. Chan and Tran (1992), Delgado (1996), Hjellvik
and Tj0stheim(1996) and Skaug and Tj0stheim(1993b, 1996)).
is Skaugand Tj0stheim's
estimation
testthatavoidssmoothednonparametric
One important
whichextendsthetestof Blumet al.
function,
(1993a) testbased on theempiricaldistribution
way. The teststatisticis invariantto any order
(1961) to a timeseriessettingin a significant
The distributiongeneratingthe data can be continuousor dispreservingtransformation.
crete;when it is continuous,the test is distributionfree.No momentis required;this is
as oftenarisesin economicsand high
attractivefortimeserieswhose variancesare infinite,
frequencyfinancialtimeseries(e.g. Fama and Roll (1968) and Mandelbrot(1967)). Finally,
thestatisticdependsonlyon rankingtheobservationsand is simpleto compute.We notethat
functionvia an alternativeapproach.
Delgado (1996) also used the empiricaldistribution
Builton Skaug and Tj0stheim's(1993a) approach,thispaper proposesa new testforserial
functionsof a timeserieswith
independence.It comparesthepairwiseempiricaldistribution
the productsof its marginalsforvarious lags. The testhas some new attributes.First,it is
consistentagainstall pairwisedependencesforcontinuousrandomvariables.Thus, the test
is available. It may also be expectedto have good
maybe usefulwhenno priorinformation
power against long memoryprocessesbecause a long lag is used (see Robinson (1991b)).
largerweightsare givento lower
lags; typically
weightsare givento different
Second,different
orderlags. Non-uniform
weightingis expectedto givebetterpowerthanuniformweighting
whosedependencedecaysto zero as thelag increases,as is oftenobserved
againstalternatives
forseasonallyadjustedstationarytimeseries.Third,althoughour statisticis a weightedsum
of Cramer-vonMises statistics,it has a null asymptoticone-sidedN(O, 1) distributionno
matterwhetherthe data are generatedfroma discreteor a continuousdistribution.We
consider some 'leave-one-out'test statistics,which improve size performancesin small
samples.We also presenta slightlymodifiedversionof Skaug and Tj0stheim's(1993a) test
thatimprovesthe size in small sampleswhen relativelymanylags are used. We emphasize
thatthenew testshouldbe viewedas not competingwithbut as a complementto Skaug and
regimesand have theirown merits.
Tj0stheim's(1993a) test,because theyworkin different
fewlags are tested,whereasthe
Skaug and Tj0stheim's(1993a) testis relevantwhenrelatively
testproposedappliesforrelativelymanylags. In addition,thepresenttestis builton Skaug
and Tj0stheim's(1993a) approach,and one mayperformbetterin some situationswhilethe
betterin othersituations.Simulationshowsthatthenewtesthas good power
otherperforms
againstlinear- bothshortand longmemory- processesand somenon-linearprocesses,but
like othertestsbased on the empiricaldistributionfunctionsit has relativelylow power
against Engle's (1982) autoregressiveconditionalheteroscedasticprocess. For this class of
tests(e.g. Skaug and Tj0stheim(1996)) maybe expected
smootheddensity-based
alternatives,
to performbetter.
Section 2 introducesthe test statisticand derivesits asymptoticnormality.Section 3
establishesconsistencyagainst all pairwisedependencesand derivesan optimalweighting
schemeforvarious lags. In Section 4, we use simulationmethodsto compare the new test
withSkaug and Tj0stheim's(1993a) test,and Skaug and Tj0stheim's(1996) density-based
sketchedin AppendixA; detailsare available fromtheauthoron
test.The proofsare briefly
request.
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PairwiseSerialIndependence
431
2. The approach and test statistics
Consider a real-valuedstrictlystationarytime series {X,}I-,? with marginaldistribution
Fj(x,y) = P(X, < x, X,- <, y), (x,y) E R',
function
G(x) = P(X, < x) and pairwisedistribution
Suppose thata randomsample{X,},= of size n is observed.We are interested
j = 0, 1,.
in testingthe null hypothesisthat {X,} is seriallyindependent.
Existingtests for serial independenceare all based on some measure of dependence.
Because X, and X,< are independentifand onlyifg(X,) and g(X,-) are independentforany
continuousmonotonicfunctiong, it is desirableto use a measurethatis invariantto g (see
discussion).Hoeffding(1948) proposedthemeasure
Skaug and Tj0stheim(1996) forfurther
D2fj)=J {F)(u, v) - G(u) G(v)}2 dFj(u, v).
(1)
By Hoeffding's(1948) theorem3.1,
transformation.
This is invariantto any orderpreserving
D2(j) = 0 ifand onlyif
functions,
density
whenFj(u,v) has continuouspairwiseand marginal
consistentagainstall
are
measure
(1)
In
on
based
this
case,
tests
X, and X,< are independent.
is
thatD2(j) = 0 but
it
If
possible
however,
is
discontinuous,
pairwisedependences. Fj(u, v)
In thiscase,
for
an
example.
548,
p.
Hoeffding
(1948),
See
X, and X,< are not independent.
all
dependences.
pairwise
against
not
consistent
on
measure
are
(1)
testsbased
Hoeffding
(1948) used a UMeasure(1) or itsanaloguehas been used to testindependence.
betweentwoidentically
statistic
estimatorforan analogueof measure(1) to testindependence
randomvariables.Blum et al. (1961) used an empiricaldistridistributed
and independently
estimatorforan analogue of measure(1) to testindependenceamong
butionfunction-based
distributedrandom vector.Also see
the componentsof an identicallyand independently
Deheuvels(1981) and Carlstein(1988).
For j = 0, 1, . . . define the pairwise empirical distributionfunction of (X,, X,<)
Fj(x, y) = (n _])'
, 1(X,< x) 1(X,< < y),
I=j+l
where1(.) is the indicatorfunction.A consistentestimatorformeasure(1) can be givenby
(j)=
(
_j)_ ,
1=1+1
{F(X,, X,<)-F
(X,, oo) Ft(oo,X,..) }2.
(2)
Skaugand Tj0stheim(1993a) werethefirstto use measure(2) to testforserialindependence.
Theyconsideredtheteststatistic
STI a= (n -
(3)
1) Fn(j).
j=1
and identicallydistributed,
When {X,}InAare independently
ST I a
ZE XZJ
Aij
00
00
i=1j=1
(P)
x2 randomvariableswithp degrees
in distribution
as n -* oo, wheretheX (p) are independent
random
For
continuous
variables,Aij= (ij2)-2 , and the
the
are
of freedomand
weights.
A,
the
discrete
random
free.For
variables, Ai dependon thedata-generating
testis distribution
free.
distribution
test
is
not
processand mustbe estimated;the
extensionof Box and Pierce's
natural
Statistic(3) can be viewedas an appropriateand
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432
Y. Hong
(1970) correlationtest.It can detectpairwisedependencesup to lag p, includingthosewith
Some evidence(Skaug and Tj0stheim(1993a), p. 600) suggeststhatthis
zero autocorrelation.
approachthattestsjointindependence
pairwiseapproachhas betterpowerthanan alternative
of (X,-1,. . ., X,_p).It also gives a much simplerlimitdistributionthan the joint testing
assetofstatistic
approach.Skaug and Tj0stheim(1993a) documentedthatthemostsignificant
(3) is that it is very close to correlationtests in power against linear processes and is
considerablybetterthancorrelationtestsin poweragainsta varietyof non-linearprocesses,
although it has relativelylow power against Engle's (1982) autoregressiveconditional
process. In addition,the size of statistic(3) is quite accurateformoderate
heteroscedastic
sample sizes comparedwithexistingsmootheddensity-basedtestsfor serial independence,
especiallywhenp is small. For largep, Skaug and Tj0stheim(1993a) pointedout that the
lead to a testwitha levelthatis too high
criticalvaluesofstatistic(3) progressively
asymptotic
methods,advocatedby Skaug and
as p increasesforsmalln. The bootstrapand permutation
tests,can be
Tj0stheim(1993b, 1996)to producethecorrectlevelsforsmootheddensity-based
applied to statistic(3) as well.
theorythatgivesa simpleand reasonableapproxIn thispaper,we developa distribution
imationforstatistic(3) whenp is large.We show thatforlargep it is possibleto obtain an
forstatistic(3), afterproperstandardization.However,forpower,
N(0, 1) limitdistribution
we maywantto checkmorelags as n increases.This ensuresconsistencyagainstall pairwise
weightscan be givento
dependencesforcontinuousrandomvariables.In addition,different
i.e. to
different
lags. In particular,more weightscan be givento more recentinformation,
lowerorderlags. This may improvepoweragainststationarytimeserieswhose dependence
decays to zero as the lag increases.These considerationssuggestthe statistic
=
n-I
k
-_j) 23(j),
k2(j/p)(n
j=l
(4)
the followingassumption.
wherek is a kernelfunctionsatisfying
continuousat 0 and all excepta finitenumber
1. k: R -+ [-1, 1] is symmetric,
Assumption
<
=
dz
oo
and
<
of points,withk(O) 1, Jfok2(z)
Ik(z)l CIzI-b as z -* oo forsome b > ^ and
0 < C<
oo.
This assumptionhelpsto ensurethatstatistic(4), afterdivisionby n,convergesin probability
a consistenttestagainstall pairwisedependencesforcontinuous
to Zj??1D2(j), thusdelivering
of k withk(O) = 1 ensuresthatthebias of statistic(4)
randomvariables.Here,thecontinuity
vanishes.The conditionJork2(z)dz < oo impliesthatk(z) - 0 as z -* oo. This ensuresthat
Daniell,Parzen,quadBartlett,
varianceofstatistic(4) vanishes.The truncated,
theasymptotic
raticspectraland Tukeykernels(e.g. Priestley(1981), pages 441-442) all satisfyassumption
weights.Note that all n - 1 lags
1. Except for the truncatedkernel,all have non-uniform
are used in statistic(4) if k has unboundedsupport.
Statistic(4) is a weightedsum of von Mises statistics.When the truncatedkernel(i.e.
k(z) = 1(IzJ< 1)) is used, we obtain
ST2a = L (n-])
n( j)
(5)
j=l
whichis asymptotically
equivalentto statistic(3). Thus, we can see thatboth statistic(3) and
statistic(5) are based on the truncatedkernelor uniformweighting.The use of n -j in
statistic(5) is similarin spiritto Ljung and Box's (1978) modifiedversionof Box and Pierce's
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PairwiseSerialIndependence
433
(1970) test.Our simulationstudylatershowsthatstatistic(5) has bettersize thanstatistic(3)
forlargep in small samples.
We also considera leave-one-outempiricaldistribution
function
(XS<
E
,s#t
s=j+l
Fj(,)(X,,X,-j)=(n-j-l)-'
< X,<j),
X1)l(XS_
wherethe sum excludesthe tthobservation.This yieldsthealternativestatistic
n-2
V* = S:k
j=I
(n -j -1)n
2(jlp)
(6)
2(j),
where
b*2(j)
=
(n _j)_
{t](,)(X,,K,.)- F()(X,, co)FP(,)(oo,
X_j)}2-
t=j+l
but asymptoticanalysis
Both statistic(4) and statistic(6) have the same limitdistribution,
shows that statistic(6) has a smallerapproximationerrorfor the limitdistribution.
Thus,
in our simulationstudy.
statistic(6) may give bettersizes in small samples,as is confirmed
Similarly,we can consider
STlb =(n
-
ST2b =
(n -j
E
j=l
l)
j=1
-
(7)
n)2()
(8)
1) Jn*2(j)
as statistics(3) and (5).
These two testshave the same limitdistribution
We firstgiveour genericteststatistics.
Theorem1. Suppose that assumption(1) holds, and p
0 < c < oo. Define
n-In2
=
cnV for some 0 < v < 1 and
2o E k 4(
M =Zk2(J
)
{(2Inj)b2(j)AO}/A
k4(i;)}
Mb = Zk {(n--)2b*2(j)
(ip)
where
Ao= [n
A0=
n
(n-2
Om
(X,){1-
n
Z
VE[{min(X,,
t=l s=1
)
Xs)}-G(X,)
(Xs)]
\2
with
(u) = n-1E 1(X,< u).
t=1
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,
434
Y. Hong
If {X,}'I
are identicallyand independently
then Ma
distributed,
-
and Ma -* N(O, 1) and Mb -* N(O, 1) in distribution.
Mb -* 0 in probability,
Theorem 1 applies to discrete,continuousor mixeddistributions.
If G(x) is knownto be
continuous,we can use the followingsimplified
statistics.
Theorem2. Suppose thattheconditionsof theorem1 hold. Define
Ma = 90
E k2Q)
Mb= 90Z k2 (J){(n
{(nD-j)J 3(j)
_j
l}/{2
- 1) f)*2( j)
2 k4(J
)
, k
j
/2
_
}1/
If {X,}', are identicallyand independently
distributedcontinuousrandomvariables,then
Ma
-
Mb -* 0 in probability, and Ma -* N(0, 1) and Mb -* N(0, 1) in distribution.
We note that theorem2 does not apply to discretedistributions,
because the mean and
asymptoticvarianceof (n -j) DnL ) cannotbe computedin discretecases.
Althoughthevon Mises statistic
(n -j) Dn(j) does notfollowan asymptotic
x2-distribution,
theorems1 and 2 showthata weightedsum of (n _j) i52 (j) has an asymptotic
N(0, 1) distributionforlargep aftercentring
and scaling.To obtainan intuitive
idea, considerforexample
theuse of thetruncatedkernel.In thiscase, Ma of theorem2 becomes
Ma =
E
{(n-_j)
2l(j)-3
}/
2
)
As shownin Skaug and Tj0stheim(1993a),
(n-j) 12(j)
E
k=1 1=1
(kir)2(lr)2 X21(1)
in distribution
as n -* oo, thushavingmean 1/36 and asymptoticvariance2/902.In addition,cov{(n i)12(i), (n -_j) J2(j)} -_ 0 for i 7'-j as n -* oo, suggestingthat(n - i)13)(i)
and (n -j) I5 (j) are asymptotically
uncorrelatedor independent.Therefore,{ (n -j) 32(]j),
j = 1, . . ., p} can be viewedas an asymptotically
identicallyand independently
distributed
sequencewithmean 1/36and variance2/902.The sum of thissequence,afterdifferencing
the
mean and dividingby the standarddeviation,will convergeto N(0, 1) in distribution
as p
becomeslarge.Our proof,ofcourse,does not dependon thissimplisticintuition.Instead,we
developa dependentdegenerateV-statistic
projectiontheoryto approximatestatistic(4) as a
weightedsum of degenerateV-statistics
overlags,which,afterstandardization,
is thenshown
to be asymptotically
N(0, 1) by usingan appropriatemartingalelimittheorem(e.g. Brown
(1971)). See AppendixA formoredetails.For degenerateV-statistics
(or relatedU-statistics)
of dependentprocesses,see (forexample)Carlstein(1988) and Sen (1963).
The N(0, 1) approximation
is convenient
forinference.
For smalln or smallp, however,itmay
notbe accurate.As a practicalalternative,
thebootstrapor permutation,
as advocatedin Skaug
and Tj0stheim(1993b,1996),can be used as a remedyforobtainingtherightlevel.Indeed,the
currenttestsituationis ideallysuitedto theseresamplingmethods,whichcan be expectedto
yieldas good a levelas thebestasymptotic
In particular,
approximation.
permutation
givesthe
exactlevel.However,it is impossibleto computethelevelexactlyin practiceforall exceptvery
smallsamplesizes.Hence,Monte Carlo methodsmustbe used to obtainan approximation.
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435
Pairwise Serial Independence
3.
Consistency
conditions.
To stateour consistencytheorem,we impose some additionalregularity
Assumption
2.
Jz12k2(z)dz < oo.
00
0
Assumption3. Put U, = G(X,), where G is the continuousmarginaldistributionof the
strictlystationarymixingprocess {X,} with strongmixingcoefficient
a(j) = 0(yj--6) as
-*
The
of
oo
for
some
6
0.
distribution
has
continuous
>
a
joint
(U,, U,1)
joint density
j
functionon [0, 1]2 thatis bounded by some A < oo, whereA does not dependon j.
Theorem3. Suppose that assumptions1-3 hold, and p = cnVfor some 0 < v <
3
and
0 < c < oo. Let Ma and Mb be definedas in theorem2. Then,withprobability
approaching1
as n -+ oo,
pl2nMD(j)/{2J
(P'/2 /n)Mb
00
k4(z)z
90 E D2(j)
1/2
/fg 00
2
k4(z)dz)
Theorem 3 impliesthat limn,o{P(Ma > C,G) = I for any non-stochasticsequence {Cn=
of Ma againstall pairwisedependencesforcontinuous
o(n/pl/2)1, thusensuringconsistency
randomvariables.It suggeststhat whenmore data become available the testsMa and Mb
Thus, Ma and Mb are useful
largerclass of alternatives.
have poweragainstan increasingly
about possible alternatives.Because negativevalues of
whenwe have no priorinformation
Ma and Mb are oneMa and Mb can occur onlyunderthe null hypothesisasymptotically,
of
sided tests;appropriateupper-tailedcriticalvalues N(0, 1) should be used.
Althoughthe temporalconditionon a(j) is mild,it rulesout strongdependences.However,it is reasonablethat any independencetestshould be expectedto have strongpower
againststrongdependences.In particular,Ma and Mb may be expectedto have good power
againststrongdependencesbecause a long lag is used. We shall investigatethepowerof Ma
and Mb againstlongmemoryprocessesvia simulation.It shouldalso be notedthattheorem3
holds onlyforcontinuousrandomvariables.For discreterandomvariables,Ma and Mb are
notconsistentagainstall pairwisedependences,because D2(j) can be 0 evenifthetimeseries
like otherasymptoticnotions,consistency
against
is not pairwiseindependent.Furthermore,
all pairwisedependencesseems to be mostlyof theoreticalinterest.In practice,p is always
finitegivenany samplesize n. When a kernelwithboundedsupport(i.e. k(z) = 0 if lzl > 1) is
p lags are tested.Whena kernelwith
used,p is thelag truncationnumber,and onlythefirst
unboundedsupportis used,p is not a lag truncationnumber.In thiscase, all n - 1 lags are
fromlags muchlargerthanp are negligible.
used, but thecontributions
An important
forstationary
timeserieswhosedependence
issueis thechoiceofk. Intuitively,
to give
as measuredby statistic(1) decaysto zero as thelag increases,it seemsmoreefficient
moreweightsto lowerorderlags. The asymptotic
analysisbelow showsthatthisis indeedthe
relativeefficiency
betweentwokernels,say k, and k2,we use
case. To comparetheasymptotic
whichis suitableforlargesampletestsunderfixed
Bahadur's(1960) asymptotic
slopecriterion,
alternatives.
Bahadur'sasymptotic
p-valueof thetest
slope is therateat whichtheasymptotic
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436
Y. Hong
statistic
its
goesto 0 as n -* oo. BecauseMa is asymptotically
N(0, 1) underthenullhypothesis,
asymptotic
p-valueis 1 - '1(Ma), where 1 is thecumulativedistribution
functionof N(0, 1).
Define
SA(k) = -21n{1
Given ln{1-?_(z)}
=I
-Iz2{1
+ o(1)} as z
(p/n2)Sn(k)-* 902{
-
-* oo,
(M,)
we have by theorem3 that
D2(j)}
/{2
J
k4(z)dz}
in probability.FollowingBahadur (1960), we call 902{ 5j??D2(j)}2/2 Jo k4(z)dz theasymptotic slope of Ma. Under the conditionsof theorem3, it is straightforward
to show that
Bahadur's asymptoticrelativeefficiency
of k2 to k, is
ARE(k2:kl) =
{
00
dz/
k14(z)
00
o
1/(2-v)
k2(z)dz}
Thus, k2 is moreefficient
thank, if
Jk4(z)dz< J 4(Z)dz
2
o
For example,Bahadur'sasymptotic
efficiency
oftheBartlett
kernel(kB(z) = (1-IZI) (IzI <, 1))
to thetruncated
kernel(kT(z) = (IzOI< 1)) is ARE(kB :kT) = 511(2-v)> 2.23. Followingreasoningthatis analogousto Hong's (1996a,b) local poweranalysisforsomecorrelation
tests,we can
obtainthattheDaniell kernel
k(z) = sin(rzV/3)
rz /3
ZElR
(9)
maximizesthe Bahadur asymptoticslope over theclass of kernels
K(r)
=
{k satisfiesassumptions1, 2, k(2)= r2/2> 0, K(A) > 0 forA E R},
(10)
where
k(r)= lim[{I -
k(z)}/IZIr]
is called the 'characteristic
exponent'of functionk and
K(A) = (2r)-'
J
k(z) exp(-iAz) dz
is the Fouriertransform
of k. This class includesthe Daniell, Parzen and quadraticspectral
kernelsbut rulesout the truncated(r = oo) and Bartlett(r = 1) kernels.Mainly because of
assumption2, K(r) is more restrictive
than a class of kernelsoftenconsideredto derive
optimalkernelsusingappropriatecriteria(see Andrews(1991) and Priestley(1962)) in the
spectralanalysisliterature.
Hong (1996a, b) showedthattheDaniell kernel(9) maximizesthe
local powerof some correlationtestsovera class of kernelsslightly
moregeneralthanK(T).
The resultobtainedheresuggeststhattheoptimalityof theDaniell kernel(9) carriesoverto
thenon-linearHoeffding
measure(1). However,theoptimality
of theDaniell kernel(9) seems
to be only of theoreticalinterest.Simulationstudieslater show that commonlyused non-
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PairwiseSerialIndependence
437
uniformkernelshave similarpowersforMa and Mb, but theyare all morepowerfulthanthe
truncatedkernelin mostcases.
One advantageof Skaug and Tj0stheim's(1993a) testis thatit does not requirechoosing
any kernelfunctionand p as a functionof n. For Ma and Mb, p behaveslike a smoothing
parameteras we requirep -+ oo as n -+oo to ensureasymptoticnull normalityand confromsmoothed
sistencyagainstall pairwisedependences.Of course,Ma and Mb are different
estimationfor
testsbecauseMa and Mb do notrequiresmoothednonparametric
density-based
testwould also need to
each lag. To testall pairwisedependences,a smootheddensity-based
letp -> oo as n -+ oo, in additionto theoriginalsmoothingparameterfordensityestimation.
In thepresentcontext,the optimalp forMa and Mb dependson thealternativedependence
structure.Skaug and Tj0stheim(1993a) noted that choosingp too large will decrease the
powerof statistic(3). The same is trueof Ma and Mb as wellin general.Skaug and Tj0stheim
lag includedinthemodel.
choosingp largerthanthesmallestsignificant
(1993a) recommended
It is reasonableto use a 'ruleof thumb'consistingof takingp equal to thelargestsignificant
lag includedin the model. Of course,thisrule may not make sense forcertainalternatives
of thechoiceofp on bothsize
theeffect
dependentprocesses.We investigate
suchas strongly
and powervia simulation.Our simulationfindsthatthisrule of thumbapplies well to the
kernelsmaximalpoweris oftenachievedforp larger
truncatedkernel,but fornon-uniform
lag includedin themodel.In addition,theuse of thenon-uniform
thanthelargestsignificant
kernelalleviatesthe loss fromchoosingp too largebecause the kerneldiscountstheloss of
degreesoffreedomforhigherorderlags; thismakesthepowerless sensitiveto thechoiceofp.
We deferthiscomplicatedissue
theissueofchoosingan optimalp is important.
Nevertheless,
to otherwork.
4. MonteCarlo evidence
of thetestsproposed.In additionto an identically
We now studyfinitesampleperformances
AR(l),
distributed
and independently
process,we also considerfollowingalternatives:
X, = 0.2X,_1+,El;
ARFIMA(0, d, 0),
X, =(1-L)
026,,
LX, =X,;
ARCH(l),
X, = ,(l ?0 5X2
)1/2;
GARCH(1, 1),
h, = 1 +O.1X,2_ + 0.8h,l1;
X, = c,h1/2,
TAR(l),
X, =
(-0.5X,_
if X,1 > 1,
+ c,
if X,_1 < 1;
0.4X,t1+c,,
NMA,
X, = 0.8,1E,-2
+ El;
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438
Y. Hong
NAR(3)
X, = 0.25X,2
-
0.4E,_1X,I + 0.2ct-2Xt-2 + 0.5Eu.3X,_3 +
E,;
NAR(5)
XI
=
O.3(ct,1Xt-1-Et
-2 +
c,.3X,3 -c,4X,.4
-c,_5X,.5)
+
l,;
EXP(3)
3
X,=0.2 Z X1j exp(- IX2 )+E,;
i=l
EXP(1 0),
10
X,= 0.8 E X,< exp(- -X2 ) + ,.
j=,
Here, AR(1) is a first-order
autoregressive
process,ARFIMA(0, d, 0) is a fractionally
differencedintegratedprocess,a popular long memorymodel (see Robinson (199ib, 1994)),
ARCH(1) is a first-order
autoregressive
conditionalheteroscedastic
process,GARCH(1, 1) is
a generalizedARCH processof order(1, 1), TAR(1) is a first-order
thresholdautoregressive
process, NMA is a non-linearmoving average process, NAR(3) and NAR(5) are nonlinearautoregressive
processesof orders3 and 5 respectively,
and EXP(3) and EXP(10) are
non-linearexponentialautoregressive
processesof orders3 and 10 respectively.
These models
are a fairlyrepresentative
selectionof both linearand non-lineartimeseries.Exceptforthe
ARFIMA(0, d, 0) process,all have been used in a varietyof existingsimulationstudiesfor
independencetests.We considertwo typesof c,: normaland log-normalwithzero mean and
unit variance,forn = 100 and n = 200. The process ARFIMA(0, d, 0) is generatedusing
Davies and Harte's(1987) fastFouriertransform
algorithm.ExceptfortheARFIMA(0, d, 0)
process,we generatedn + 100 observationsforeach n and thendiscardedthefirst100 to reduce the effectof initialvalues. For the ARFIMA(O, d, 0) process,we generatedn + 156
= 28observationsand discardedthefirst156forn = 100,and generatedn + 312 = 29observations and discardedthe first312 forn = 200. The simulationexperiments
were conducted
usingthe GAUSS 386 randomnumbergeneratoron a Cyrixpersonalcomputer.
To examineeffects
ofvariousweightson thetestsproposed,we consideredfivekernels:the
truncated,Bartlett,Daniell, Parzen and quadratic spectralkernels.The last threekernels
belongto theclass K(er/V/3)
in expression(10). To investigate
theeffectof choosingdifferent
values of p, we consideredp from1 to 20; this covers a sufficiently
large rangeof p given
n = 100 and n = 200. We also studied Skaug and Tj0stheim's(1993a) test STia and its
variantsSTib, ST2a and ST2b, as well as Skaug and Tj0stheim's(1996) smootheddensitybased test
p
J= J,i(),
i=l
where
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Pairwise Serial Independence
.
0.30
.
.,..
0.30
0.27
0.27
0.24
0.24
ST1a
ST1b,
0.21
0.21
,Ma-Dan
0.18
STa
0.18
Ma-Trun
STbb
01
0.15
0.06
439
SMb-Trun
0ST2c,2
0.03
0.09
P
P
(a)
0.16
0.20
.
'.
.
.
' . 'M'D
/
0.20
0.18
0.18
0.14
0.14
STia
STlb
0.12
/
0.12
Ma-Dan
0.10
(b)
0.16
'Ma-Trun
o.1o
0.08
ST1aa
Mb-Dan
STIb'
,Mb-Trun
?a
0
Ma-Trun
0.06~~~~~~~~~~~~S2
0.04
,=1 o
0o0
1
b
.0
0.0
" Mb-Trun
ST2b,
0.02
0.2
0.00
o
ST2aoS
3
5
J-Test
0.02
00
7
9
I1
1 13
15
17
19
0.00
1
3
5
7
9
1 13
I1
p
p
(c)
(d)
15
17
19
Fig. 1. Size underthe normalprocess: (a) empiricalsize at the 10% level; (b) bootstrapsize at the 10% level; (c)
empiricalsize at the 5% level; (d) bootstrapsize at the 5% level
= (n - })f,1Qj)
S
ii
{Jj(1)(X1,
X,..1)-J,()(X,)A,..
X,..) = {(n -j)hn}-lt=j+l~~~~~~~t
fJ(,)(X,,
E
K{Xs=j+I
f(l)(
=
s$I
{ (n - I)hn }
E
s=I,sot
J)(X,..J)},
X)/h,1} K{ (X,..1-X)/,}
K{ (X,-Xs)/hn
}
withK: R -R+ a kernelfunctionand hna bandwidth.Here, 'leave-one-out'bivariateand
are used. As in Skaug and Tj0stheim(1996),we firststandardized
marginaldensityestimators
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440
Y. Hong
10
r
.0
0.8
0.8
0.6
,Mo-Don
0.6
,,Mo-Trun
,Mo-Dan
,
,Mo-Trun
0.404
o.2
0.2
3
1
5
7
11
9
i3
15
17
1
19
3
5
7
9
p
11
13
15
17
__ _Mo-
Don
19
p
(a)
(b)
1.0
1.....
Don
~~~~~Mo-
>
0.404
o.o
.T2
..
.
3
5
7
.
.
11
13
o.
.
0.202
1
9
1S
17
19
1
3
5
7
9
11
p
p
(c)
(d)
13
15
17
19
power, AR(1),
(a) size-corrected
d, 0) processes:
Power at the 5% level under AR(1) and ARFIMA(O,
power, AR(1), log-normal error; (d)
error; (b) bootstrap power, AR(1), normal error; (c) size-corrected
d, 0), normal error; (f) bootpower, ARFIMA(O,
bootstrap
power, AR(1), log-normal error; (e) size-corrected
d, 0), log-normal error; (h)
power, ARFIMA(O,
d, 0), normal error; (g) size-corrected
strap power, ARFIMA(O,
d, 0), log-normal error
bootstrap power, ARFIMA(O,
Fig. 2.
normal
thedata by thesampledeviationand used h = n-116withthe Gaussian kernel.This testhas
been provento be morepowerfulthanor comparablewithmanyexistingsmootheddensitybased testsagainst a varietyof non-linearprocesses(see Skaug and Tj0stheim(1993a, b,
as n -+ oo forsome a 2
-+ N(O, 42) in distribution
(n/p)l12J
1996)). Under independence,
we also used a bootstrapprocedureforall theteststo
In additionto asymptoticinference,
studybootstrapsize and power.To describethe procedure,we firstconsiderMa. Given a
sample {X,},n1, we generate m bootstrap samples {X*jL'1, 1= 1, . . ., m. For each bootstrap
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Pairwise Serial Independence
t.0
,
,
.
.
0.8
.
.
.Ma- Don
1.0
I
0.8
M -Dan
,Mo-Trun
, -Ma-Trun
0.6
--
,
441
I
STlo
,'ST1oaST
T
Test
0.404
0.2
0.2
0.0
1
3
.
7
5
11
9
13
15
17
19
0.0
1
3
5
7
9
13
15
17
19
17
19
(f)
(e)
,
.
1.0
11
p
p
,
1.0
0.8
,
,.
.
0.60.
"
\Mo-Trun
\STlao
0.4
0.0
*1
"0ST2
0ST2.4
//J-Test
0.2
'Mo-Trun
ST a
-JTs
/0.2
3
5
7
11
9
13
15
17
19
.
*
3
5
7
11
9
p
p
(g)
(h)
13
t5
Fig.2. (continued)
ofthe
function
Mt*.To estimatethedistribution
sample{X*/},1,wecomputea bootstrapstatistic
if
of
function
distribution
the
empirical
{MII}I
bootstrapstatisticM*a we could use
I m is
a
=
50. For
we
choose
m
is
rather
simulation
extensive,
experiment
large.Becauseour
sufficiently
the
estimate
to
section
and
bootstrap
3.3,
Tj0stheim(1996),
sucha smallm,we followHjellvik
kernelestimator
function
distribution
F*(z) = P(M* < z) by a smoothednonparametric
ft(z) =
rsni
J [rn-ijE {h,n,\/(2ir)F-'
exp{-(y
m
FW=-00
/= 1/1hl
- M*
)2/2h, dy
In
= M-1E (D{ (z -M*)Ih
l=l
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442
Y. Hong
,
1.0
,
.
1.0
0.8
0.8
0.8
0.6
~~~~,
\
,' ,Mo~~-Don
\
0.404
o.*
\
\,J-Test
\,,M-Don
J-Test
,'
,
\
\<
,',~~Mo-Trun .
,',',ST1o
' ',, T20X
o
,,','
ST2o
0.2
0.2
p
p
(a)
(b)
1.0
,
1.0
0.8
0.8
0.6
0.6
, J-Test
\
0.4
,J-Test
Mo-Don
0.4
, ' , ',Mo-Trun
\
,
\
3
S
7
9
ll
,',
,' ,
',',ST1 a/
13
15
17
19
,
/ /
0.2
02
'
0.2
*
\,,'Mo-Trun
\,,',Stlo
.
1
3
5
7
/ / / ',
11
9
p
p
(c)
(d)
'
13
Mo-Don
Mo-Trun
STio
ST2a
15
17
19
Fig. 3. Power at the 5% level underARCH(1) and GARCH(1, 1) processes: (a) size-correctedpower,ARCH(1),
normalerror;(b) bootstrappower,ARCH(1), normalerror;(c) size-correctedpower,ARCH(1), log-normalerror;
(d) bootstrap power, ARCH(1), log-normalerror;(e) size-correctedpower, GARCH(1, 1), normal error; (f)
bootstrappower,GARCH(1, 1), normalerror;(g) size-correctedpower,GARCH(1, 1), log-normalerror;(h) bootstrap power,GARCH(1, 1), log-normalerror
of N(O,1), h,,,= Sn,,m"5is thebandwidth
function
distribution
where'1 is thecumulative
For moredetails,see Hjellvikand
and S,A,
is the samplestandarddeviationof {Ma*!}72=
if
ofindependence
Tj0stheim
(1996).HavingobtainedPn,wecan rejectthenullhypothesis
based on the
leveland Ma is theteststatistic
F*t(Ma) > 1 - a, whereaois thesignificance
is appliedto theothertests.
The sameprocedure
originalseries{XJ,},1.
criticalvalues.For STia,
and bootstrap
both
asymptotic
sizes
using
consider
by
first
We
of
values
10000simulations
critical
using
their
tabulate
we
asymptotic
and
ST2b,
STIb,ST2a
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Pairwise Serial Independence
1.0
-
1fX--rl.0
- - -s r---
0.8
0.8
0.6
0.6
0.4
,
, J-Test
,',Mo-Don
,',Mc-Trun
, J-Test
,Mo-Dan
Ma-Trun
0.,
ST2a
0.2
__:
0.0
1
_
5
3
~~~~ST2a
7
9
11
0.2
13
15
17
19
.
0fX(_.0._.
OO_
'
3
5
7
/
9
11
p
0.83
0.6
0.6
A-Test
I
'
,'
/,Mc04
/
zSTlo/
5
7
9
STia
0.2
* 1
5
Mo-Trun
STlo
,
3
19
, J-Test
,'MoDo
ST2o~~~~~~~/
0.0
17
1.0 .
0.81
0.2 .
15
(f)
,
040g
13
p
(e)
1.0
0
443
71
9
0.
/
3
a5
p
13
17
15
17
1
P
(g)
(h)
Fig.3. (continued)
200 200
EE
i=l j~=]
-
1)-I X2J(P)'
a truncated
versionofthelimitdistribution
ofSTia. For theJ-test,
we onlystudyitsbootstrapsize.Sizes forasymptotic
and bootstrap
criticalvaluesare based on 5000and 1000
replications
respectively.
Fig. I reports
sizesofMa, Mb, STI a, ST Ib, ST2a,ST2bandtheJ-test
underthenullhypothesis
ofindependence
at the10% and5% levels,forn = 100andnormal
innovations.
We onlyreporttheDanielland truncated
kernels
forMa and Mb,becausethe
fournon-uniform
kernels
perform
We havethefollowing
similarly.
observations.
In
(a)
terms
ofasymptotic
critical
values,Ma andMb havereasonable
sizes,although
they
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Y. Hong
444
.0
.
1.0
0.8
0.8
0.6
0.6
,Mo-Don
,
0.2
0.0
5
3
1
|
e
i\
-
(
,'/,S2
15
I
17
19
0.0
t
3
5
7
J-Test
11
9
p
/
13
''S2
17
15
19
JTs
(b)
p
s
a
0. \\0
0.6
0.48STla
0.6
13
11
9
7
,,'~~ p
(a)
,l
.e
Mo-Trun
Mo-Trun
'
'
i
p
-
,',ST2o
0.8 _
nm1.0
M
,MoMa-Da
.
b
e
p
M
norma
i
.SW
0.6
(c)~~~~~~~~~~~~~~~~~~~~~~S
I(d)
error
power,NMA,log-normal
error;(h) bootstrap
power,NMA,log-normal
error;(g) size-corrected
at the 5%/level.Both theDaniell and thetruncatedkernels
show slightoverrejections
have similarsizes.Sizes are robustto thechoiceofp. The leave-one-outversionMb has
slightlybettersizes than Ma has, suggestingsome gain fromusing the leave-one-out
statistics.The testsSTi a' ST2a, STi b and ST2b have precisesizes forsmallp. For large
but themodifiedstatisticsST2a and
p, STi a and STi b tendto givesome overrejection,
ST2b continueto have good sizes. The leave-one-outversionST2b has slightlybetter
sizes thanST2a.
(b) All the testshave good bootstrapsizes. In particular,bootstrapsizes are betterthan
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Pairwise Serial Independence
1.0
1.0
0.8
0.8
~~~~~~~~~~~~~~~~0.8
0.6
A
, ~~~J-Test
0-6 t/\
,,~~~~Mo-Don
\
Mo-Trun
0.4
,J-Test
'MoDon
Mo-Trun
0.4
0.2
0.2
0.0
445
1
5
3
13
11
9
7
15
17
19
0.0
1.0
1.0
0.8
0.8
\
Test
,J-
_--~~
\
\
0 4
\5
,
__- '
-
S
o'
'
3
5
7
9
11
13
15
17
19
(0
(e)
0.6 o.
1
p
p
0.6
0.6
,STl1o
,J-Test
,,M-Don
,', Mo-Trun
,
,Mo-Don
,
' Mo-Trun
o.
',,STl1a
0.2
0.2
oo~~~~~~~~~~~~~~~~~~
P~~~~~~~~~~~~~~~.
o.o
0.0
Fig. 4.
PL
p
p
(g)
(h)
(continued)
at the5% levelforall thetests.
approximation
thosegivenbytheasymptotic
gives
approximation
We also considersizesat the1% level.At thislevel,theasymptotic
to variousdegreesforall thetestsin mostcases. The bootstrapsizesare
overrejections
havingthebestsizeundernormalinnovawiththeJ-test
andreasonable,
better
significantly
sizesat all the
haveinvariant
function
distribution
tions.The testsbasedon theempirical
For n = 200,we studiedsizes
innovations.
threelevelsunderbothnormaland log-normal
for
forall thetests,especially
values.The sizesareimproved
critical
onlyusingasymptotic
and
ST2a.
STIa
powerat the5% levelforn = 100.To comparethetestson an equalbasis,
Figs2-6 report
of
underthenullhypothesis
from5000replications
valuessimulated
critical
weuseempirical
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Y. Hong
446
,
,
,
,
,
1.0
1.0
,
, .,
0.8
0.81
0.6
- - ---J-Test
SMa-Dan
\
-
,
\
\,,
7
5
x
. P
1a
15 Ma17
19
t
error;(b) bootstrap
5%
l
, N
u
+
0.8- t
3
<
<
7
5
11 /
9
ST2a15
13
17
19
~~~~~~~~~~~~~~~~0.6
(b)
(a)
a
0.2
o4t
p
p
0.4
1.
11
9
_J-Test
, Ma-Don
- , -Mo-Trun
,
- /'STla
2
_
~
_--
\
,Mo-Trun
,',STla
4
<
)
0o2t
0.6
N
) nom-al
N
; () s
1.0 p
,
(
e
p
r
, n
J-Test
~~~~~~~~~~~~~~~~0.2
0.0
(ecess:()
lognoma
ne
h %leeerror;
pi.5 ower,a NAR(3),
normalero;f)btsapow,NA()
power,
NAR(5),
size-corrected
error
log-normal
power,NAR(5),
error;(h) bootstrap
log-normal
power,NAR(5),
error;(g) size-corrected
normal
independence.We also considerpower by using bootstrapcriticalvalues. Powers by using
empiricaland bootstrapcriticalvalues are based on 500 and 200 replicationsrespectively.
a werealso studied,and similarrankingpatternswerefound.
Powersat the10% and 1 levels
Both Ma and Mb have similarpowers.Also, STia and ST2a have powersimilarto STib and
For clarity,we only reportresultsfor M STi ST2 and the J-test.
ST2b respectively.
and thetruncatedkernelsforMa are reported,because thefournonDaniell
the
only
Again,
powers. (In some cases, the Bartlettkernelis slightlymore
similar
have
uniformkernels
kernels,but this is not inconsistentwith the
non-uniform
three
powerfulthan the other
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Pairwise Serial Independence
1.0
I
.0
447
I.
,J-Test
, ,Ma-Dan
1,J-Test
,,',a-Dan
0.4~~~~~~~~~0
o.o
1
ST.
..
.ST..l.o.
3
5
7
9
11
13
15
17
1
19
3
5
7
.
,
os
*
*
*
,
.
.
,M-TDon
?
|
.
.
.
.
Fig. 5.
19
13
1S
17
19
,.
|
oo0.6
.
~~~~~~~~~~~~~~~~0.4
0.4
1
17
*
1.0.
~ ~ Ma- Dan
<
.o
o.o
__-
k
15
(f)
X
~\
13
p
(e)
,
1.0
11
9
p
3
5
7
9
11
13
15
17
19
*1
3
5
7
9
11
p
p
(g)
(h)
(continued)
of theDaniell kernelbecause theBartlettkernelis outsidetheclass of kernelsover
optimality
whichtheDaniell kernelis optimal.)We can make thefollowingobservations.
(a) For Ma, the Daniell kernelis more powerfulthan or comparablewiththe truncated
weighting.For the
kernelin mostcases, suggestingthe gains fromusingnon-uniform
alternativesNAR(3), NAR(5), EXP(3) and EXP(10), the truncatedkernelis more
powerfulthan or comparablewiththe Daniell kernelfor smallp, but as p becomes
largetheDaniell kernelbecomesdominant.The testsSTIa and ST2a have roughlythe
same power as the truncatedkernel-basedMa-test.(The truncatedkernel-basedMatest and ST2a have identicalpower because thereis an exact relationshipbetween
them.)
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448
Y. Hong
1.0
.
.
.
.
1.0
0.8
0.8
, Ma-Dan
, ,Ma-Trun
0.6
,
,Ma -Dan
Ma -Trun
13
17
0.6
0.404
0.202
1
3
.
1.0
os5
5
.
h
7
9
I
I11
13
is
17
19
5
*1
9
p
p
(a)
(b)
,
.
*
*
1.0
*
010.8
\ %
\Ma-Don
\
0.6
,Mo-Don
',Ma-Trun
\Ma-Trun
0.0
*1
1 o
/ ',',ST2o
0.6
% Ma0
0.4
0.4
0.2
21
. 20
0,
3
5
7
II
9
13
15
17
19
.
*1
,
_
5
9
13
p
p
(c)
(d)
17
21
Fig.6. Powerat the5% levelunderEXP(3)and EXP(10)processes:(a) size-corrected
power,EXP(3),normal
error;
(b) bootstrap
power,EXP(3),normal
error;
(c) size-corrected
power,EXP(3),log-normal
error;
(d) bootstrap
power,EXP(3),log-normal
error;
(e) size-corrected
power,EXP(10),normal
error;
(f)bootstrap
power,EXP(10),
normal
error;
(g) size-corrected
power,EXP(10), log-normal
error;
(h) bootstrap
power,EXP(10), log-normal
error
(b) For the alternatives
AR(1), TAR(1), NAR(3), NAR(5), EXP(3) and EXP(10), STia,
ST2a and thetruncatedkernel-basedMa-testachievemaximalpowerswhenp is equal
to thelargestsignificant
lag includedin themodel. This is consistentwiththe findings
of Skaug and Tj0stheim(1993a). For the ARFIMA(O, d, 0) process,STia, ST2a and
the truncatedkernel-basedMa-testachieve maximal powers at p = 1 under normal
innovations,but theirpowersgrowwithp underlog-normalinnovations.In contrast,
the Daniell kernelgenerallyachieves the maximalpower whenp is largerthan the
largestsignificant
lag. This is trueevenfortheAR(1) process.A possiblereasonis that
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PairwiseSerialIndependence
0.8
0.8
0.6
0.6
0.4
04MD
\XSTl a
\ST2a
\XSTl a
\ST2a
5
3
7
..
9
- J T s
Te t0.2
------------
0.2
0.0
*
.I
13
II
I __
.0
17
t5
I
19
1
3
5
7
9
p
*
,
*
*
*
.
17
19
1.0
.
0.8
0.6
0.6
0.6 ;
=
0.40.4
\ s s\"Mo-Dan
s
\\ Ma-Trun
\\STl1a
|
[
.
.
04S2
-J-Test
0.2
Fig. 6.
I
15
(f)
*
0.8
0.0
0.03
13
II
p
(e)
1.0
449
S
(continued)
7
11
9
13
15
~~~~~~~~~~~,',Mo-Trun
,
,
'STlo
''S2
',J -Test
0.2
17
19
0.C,
0.03
5
7
tl
9
p
p
(g)
(h)
13
15
17
19
D2(j) may not vanish whenj is largerthan the largestsignificant
lag, althoughits
magnitudemay be smaller.Thus, to include some extra termsbeyond the largest
significant
lag may enhancethepowerifD2(j) is sufficiently
largeto offsetthe loss of
additionaldegreesof freedom.For the ARFIMA(O, d, 0) process,the Daniell kernel
has the maximalpower whenp = 3 under normal innovations,and its power also
growswithp underlog-normalinnovations.Note thattheDaniell kerneloftenmakes
power less sensitiveto the choice of p, because it discountshigherorderlags, which
usuallycontributeless in power.
(c) None of the testsuniformly
dominatesthe othersagainst all the alternativesunder
study.The J-testis morepowerfulagainstARCH(1), GARCH(1) and NMA processes
withnormalinnovations,whereasthetestsbased on theempiricaldistribution
function
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450
Y. Hong
are more powerfulthan the J-testagainst the AR(1), ARFIMA(O, d, 0), TAR(l),
EXP(3) and EXP(10) processes.For NAR(3), NAR(5) and NMA with log-normal
innovations,theJ-testis morepowerfulwhenp is small,whereasthetestsbased on the
functionare more powerfulforlargep. It seems thatthe tests
empiricaldistribution
functionare morepowerfulagainstthealternatives
based on theempiricaldistribution
whosedependencesare mostlycapturedby theconditionalmean,whereastheJ-testis
whose dependencesare mostlycapturedby the
morepowerfulagainstthe alternatives
powers
conditionalsecond moments.Almost all the testshave bettersize-corrected
under log-normalinnovationsthan under normal innovations;only the J-testhas
betterpowers against the ARFIMA(O, d, 0) and ARCH(1) processesunder normal
innovationsthanunderlog-normalinnovations.
(d) The bootstrappowersforall thetestsare similarto thoseusingempiricalcriticalvalues
is small.
undernormalinnovations.In thiscase, theloss ofpowerdue to bootstrapping
For log-normalinnovations,however,the bootstrappowersare substantiallysmaller
powersin quite a fewcases. An exceptionis the J-testagainst
thanthe size-corrected
the ARFIMA(O, d, 0) process,whose bootstrappower is much betterthan its sizecorrectedpower.
powerswereconsidered.All the testshave betterpowers
For n = 200, onlysize-corrected
to variousdegrees,buttherelativerankingsremainunchanged.The
againstall thealternatives
functionstillhave relativelylow powersagainstthe
testsbased on theempiricaldistribution
ARCH(1), GARCH(1, 1) and NMA processes.However,thisdoes notmeanthattheyshould
it is
If interestindeedis in detectingthesealternatives,
not be used to detectsuchalternatives.
functionto the square of the
sensibleto apply the testsbased on the empiricaldistribution
originalseries.This can be expectedto yieldgood poweragainstARCH-type alternatives.
Acknowledgements
I would like to thanktwo referees,the JointEditor,N. Kieferand T. Lyons for helpful
of thispaper.
improvement
commentsand suggestionsthatled to a significant
Appendix A
constant
finite
0 < A < oo denotesa generic
1-3 here.Throughout,
We sketchtheproofoftheorems
in different
places.
thatmaydiffer
A.1. Proofof theorem1
1-5
1is basedon propositions
Theproofoftheorem
WeproveforMa only;theproofforMb is similar.
areavailablefromtheauthoron request.
statedbelow.Theproofsofthesepropositions
1 showsthattheweighted
sumofvonMisesstatistics
V, in equation(4) can be approxProposition
relatedto variouslags.
V-statistics
sumofthird-order
imatedbya weighted
Proposition1. Put h(x, y) = I (x < y) - G(y). For 0 < j
fI(j)
=
(n -])'
<
n, define
E Af](XI,
XIj),
I=j+l
where
11j(x,y) = (n -])'
n
E
t=j+l
h(X,,x) h(X,1, y).
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Pairwise Serial Independence
Then
n-I
Z k2(j/p)(n -j)
j=l
DJ2(j)=
n-I
?,k2(j/p)(n -j)
j=lI
451
fIt(i) + Op(p/n'12).
The nextresultis a projectiontheoryfordependentdegenerateV-statistics.
Proposition2. For 0 < j < n, define
y) dG(x) dG(y),
ftj2(x,
=
fi2(j)
wherefIn(j)(x,y) is definedin proposition1. Then
n-I
n-I
E k2(j/p)(n -j) fI2(j) = E k2(j/p)(n -j) FI2(j) +
Op(pln'2).
J
To statesubsequentresults,put
A,,(j) =
h(X,,x) h(Xs,x) h(X,1, y) h(X51, y) dG(x) dG(y).
Note thatA,s(j) = As,(j). In addition,E{A,s(j)[F,_} = 0 almostsurelyforall t > s and allj > 0, where
and hereafter{F,} is the sequenceof a-fieldsconsistingof {X,, -r< t}. We can now write
n-1
jp
Zk2(]/p)(n _j)f2(j)=
j=l
n-2
n
2jl
k2(ip)(n-j)1
-j)-< E A1(j) +2
n-1 jp
Zk2(/p)(n
I
j=l I=j+1
n 1-1
t=j+I s=j+l
As(j)
=An + 4n,say.
term.
Proposition3 showsthatAuican be approximatedby a non-stochastic
Proposition3. LetA^nbe definedas above, and define
AO=
[JG(x) { 1-G(x)
} dG(x)1
Then
n-I
An =-AOE k(j/p)
j=l
The statistic4nis a weightedsum of degeneratesecond-orderU-statisticswith mean 0. By rearrangingthe summationindices,we can write
n f -1 s-1
(n _j) AI,(j)
-2k
Bn= Z
I
1=3 tvs=2j=1
{
sequence.
Because E{A,s(j) JF,_
I} = 0 forall t > s and all j > 0, Anis a sum of a martingaledifference
By applyingBrown's(1971) martingalelimittheorem,we can showthata properlystandardizedversion
to N(O, 1), as statedbelow.
of Bn convergesin distribution
Proposition4. Define
Bo = (|[G {min(x,y)} - G(x) G(y)]2dG(x) dG(y))
Then
{2Bo
n-2
k4(j/p)}
~~~1/2
4n
N(O, 1).
Both Ao and Bo mustbe estimatedif we do not know whetherG(x) is continuousor discrete.The
followingresultshowsthatAoand Bo definedin theorem1 are consistentforAo and Bo.
Proposition5. LetAOand Bo be definedas in theorem1. Then
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452
Y. Hong
AO- AO = Op(n
1/2),
ho- Bo = op(l)
Now, combiningpropositions1-3 yields
n-I
n-I
j=l
j=1
E k2(j/p)(n-j) DJn(j)= Ao E k2(j/p) + Bn+ Op(p/n'12)
n-I
=Ao E k2(j/p) + B? + Op(p/n'12),
j=l
wherethe secondequalityfollowsfromproposition5 and p -+ oo. It followsthat
2Bo k4(E j}] /p)
=
{(n-j) Dn(j)-Ao)
1=1-I
j=1
2Bo
=
k4( /p)}
n + Op(p2/nl2
N(O
)
by proposition4 and p = cn' for0 < v < 1, wherewe have also made use of the factthat
n-2
k4( j/p)
?.
j=l
00
=p
o
k4(z) dz{I1+ o(l)}.
By replacingBo by A0,we have
d
-*
Ma
by Slutsky'stheoremand ho -B
N(O, 1)
= op(l). This completestheproofforMa.
D
A.2. Proofof theorem2
on (0, 1). This factsuggeststhatwe can
For a continuousdistribution
distributed
G, G(X,) is uniformly
computeAo and Bo directly:
Ao={
u(l-u) du}=
and
Bo=
[JJ{min(ul,u2)-uiu2}2du2duu,=9d.
Hence, we need not useAOand Bo. The desiredresultfollowsimmediately.
A.3. Proofof theorem3
O
The consistencyof Ma followsfromthe factthat
E kr(j/p)= p
j=lI
f
kr(z)dz{I+ o(l)}
forr > 2,p = cn' for0 < v < 3, and twopropositionsstatedbelow.The proofsof thesepropositionsare
available fromtheauthoron request.
Proposition6. Put Dj(x, y) = Fj(x, y) - G(x) G(y). Define
D2( j) =n_
2
Dj(XI,
XIs)
j)_,
1=j+l
Then
n-I
n-I
n-I
j=I
j=1
Ek2(j/p)(n _j)J25(j) = n- Ek2(j/p)(n _j)D2(j)
+ op(l).
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PairwiseSerialIndependence
453
Proposition7. Let D2(j) be definedas in equation(1) and D2(j) be definedas in proposition6. Then
-I E k2(j/p)(n_j)
j=l
D2(j) = E D2(j) + op(l).
j=l
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