1. No category

A New Test for ARCH Effects and Its Finite-Sample Performance
Author(s): Yongmiao Hong and Ramsey D. Shehadeh
Source: Journal of Business & Economic Statistics, Vol. 17, No. 1 (Jan., 1999), pp. 91-108
Published by: American Statistical Association
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A
New
Test
for
ARCH
Effects
and
Its
Performance
Finite-Sample
Yongmiao HONGand Ramsey D. SHEHADEH
Departmentof Economics,CornellUniversity,Ithaca,NY 14853-7601 ([email protected])
([email protected])
We proposea test for autoregressiveconditionalheteroscedasticity
based on a weightedsum of
the squaredsampleautocorrelations
of squaredresidualsfrom a regression,typicallywith greater
weight given to lower-orderlags. The tests of Engle, Box and Pierce, and Ljungand Box are
equivalentto the test with equalweighting.Ourtest does not requireformulationof an alternative
and permitschoice of the lag numbervia data-drivenmethods.Simulationstudiesshow that the
new test performsreasonablywell in finitesamplesespeciallywith greaterweighton lower-order
lags. We applythe test in two empiricalexamples.
KEY WORDS: Cross-validation;
Efficiency;Frequencydomain;Monte Carlo;Spectraldensity;
Weighting.
Since Engle (1982) introducedthe autoregressivecon- also asymptoticallydistributedas chi-squaredrandomvariditionalheteroscedasticity(ARCH)model, there has been ables with q df. Grangerand Terdisvirta
(1993, pp. 93-94)
considerableinterest in estimationof and testing for dy- showed that they are asymptoticallyequivalentto Engle's
namic conditionalheteroscedasticityof the regressiondis- (1982) LM test.
turbance.The ARCHmodel andits variousgeneralizations Detection of ARCH effects has attracted significant
[e.g., Bollerslev's (1986) generalized ARCH (GARCH)] recent attention from researchers.Contributionsinclude
have provedquite useful in modelingthe disturbancebe- those of Bera and Higgins (1992), Brock, Dechert, and
haviorof regressionmodelsof economicandfinancialtime Scheinkman(BDS, 1987),Gregory(1989),Lee (1991),Lee
series. See Bera and Higgins (1993), Bollerslev,Chou,and and King (1993), and Robinson(1991a).In this article,we
Kroner(1992),andBollerslev,Engle,andNelson (1994)for proposea new test for the presenceof ARCHin the residuals from a possibly nonlinearregressionmodel. The test
recent surveysof this literature.
From the perspectiveof econometricinference,neglect- is basedon an extensionof Hong's (1996) spectraldensity
ing ARCH effects may lead to arbitrarilylarge loss in approachand results for testing serial correlationof unasymptoticefficiency(Engle 1982) and cause overrejection known form in conditionalmean.Specifically,the null hyof standardtests for serial correlationin conditionalmean pothesisof no ARCHimplies that the normalizedspectral
(Taylor1984; Milhoj 1985; Diebold 1987; Domowitz and densityof the squaredresidualsfrom a regressionmodelis
Hakkio 1987). In the contextof autoregressivemovingav- the uniformdensity on the frequencyinterval[-7r,7r].Alerage(ARMA)modeling,Weiss (1984) pointedout thatig- ternatively,if ARCHexists, the normalizedspectraldensity
noringthe ARCHeffect will resultin overparameterizationof the squaredresidualsis nonuniformin general.Therefore, we can construct a test for ARCH by comparing
of an ARMA model.
a
kernel-basedspectraldensity estimatorof the squared
In practice,the most populartest for ARCH is Engle's
residuals
to the uniformdensity via an appropriatediver(1982) Lagrangemultiplier(LM)test for ARCH(q)undera
measure.
When a quadraticnorm is used, the new
gence
two-sidedalternativeformulation.When the null hypothetest
turns
out
to
be a properlystandardizedversionof the
sis of no ARCHis true,this statisticis asymptoticallydissum
of
squaresof all n - 1 sample autocorrelatributedas a chi-squaredrandomvariablewith q degrees weighted
tions
of
the
squaredresiduals,with weighting depending
of freedom.It is simple to calculateand is asymptotically
on
the
kernel
function.Most commonlyused kernelstyplocally most powerfulif the truealternativeis ARCH(q),a
characteristicit shareswith the likelihoodratio and Wald ically give greaterweight to lower-orderlags. The exceptests (Engle 1982) undera two-sided alternativeformula- tion is the truncatedkernel, which gives equal weight to
tion. [In the ARCHcontext,it is also possible to construct each lag.
Interestingly,when using the truncatedkernel (i.e., unia one-sidedtest. Withthe one-sidedalternative,the LM test
form
weighting),our approachdeliversa generalizedBox
has a standardchi-squareddistribution.On the otherhand,
and
Pierce
(1970)type of test, whichalso is asymptotically
the likelihood ratio and Wald tests may have an asympto
totic mixture of chi-squareddistributions.See Bera, Ra, equivalent a generalizedversion of Engle's (1982) LM
and Sarkar(1998).] An alternativeapproachis to subject test for ARCH. (See Theorem3 following.)Because ecothe squaredresidualsto such standardtests for serialcorre- nomic agentsnormallydiscountpast information,the older
lationas the portmanteau
tests of Box andPierce(1970)and
and
Box
which
are based on the sum of the
(1978),
Ljung
? 1999 American Statistical Association
firstq-squaredsampleautocorrelations
of the squaredresidJournal of Business & Economic Statistics
uals. These tests, as shown by McLeodand Li (1983), are
January 1999, Vol. 17, No. 1
91
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92
Journalof Business & EconomicStatistics,January1999
the information,the less effect it has on currentvolatility.
Therefore,when a relativelylarge q is used, it seems that
uniformweightingtests are likely not fully efficient;a better test should put greaterweight on recent information.
Indeed,it can be shownthat,when a large q is used, many
nonuniformkernels deliver more powerfultests than the
truncatedkernel.Therefore,our approachmight be more
powerfulthanthe Box-Pierce-LjungandLM tests in many
practicalsituations.In fact, a linearlydecliningweighting
schemewith a relativelylong lag oftenhasbeenusedin testing and estimatingARCHmodels (e.g., Engle 1982, 1983;
EngleandKraft1983;Engle,Lilien,andRobins1987;Bera
and Higgins 1993).Amongotherthings,such a schemeincreasespower,althoughit has been criticizedby Bollerslev
(1986)andothersas an adhoc approach.In comparison,our
kernel-basedweighting scheme arises endogenouslyfrom
the frequency-domainapproach.For persistentARCH efinferencesuggeststhatit is better
fects, frequency-domain
to employ a long lag. Therefore,it is desirableto allow
growthof the numberof lags with the samplesize for such
alternatives.
In addition to its relative efficiency gain, our test is
also relativelyconvenientto implement.It is based on a
weighted sum of squaredautocorrelationsof the squared
residualsand has a null, asymptoticallyone-sidednormal
distribution.Furthermore,we do not requireformulation
of the alternativemodel. Ourasymptotictheoryallows the
lag numberto be chosen via such data-drivenmethodsas
cross-validation.Such methodsmay reveal some information about the true alternative.With a data-drivenq, one
can obtain a nonparametricspectraldensity estimate for
the squaredresiduals.This estimate usually enjoys some
optimalitypropertiesand containsinformationon autocorrelationsin squaredresiduals-that is, volatilityclustering.
This maybe appealingwhenno priorinformationaboutthe
true alternativeis available,as is common in practice.In
applicationsof the existing ARCHtests, one usuallymust
calculatestatisticsfor several,possiblymany,valuesof q. It
is commonthatsome statisticsare significantand some insignificant.Consequently,drawingconclusionsfrom sucha
sequencecan be a delicatebusinessbecausethese statistics
are not independent.
In Section 1, we introducea class of new ARCHtests. In
Section2, we discusschoice of the lag numberfor our test
via data-dependentmethods,particularlycross-validation.
In Section3, we discussthe relationshipsbetweenour tests
andsome importantexistingtests. In Section4, we conduct
a simulationstudy to comparefinite-sampleperformances
of the new tests andthose of Engle (1982), Box andPierce
(1970), Ljung and Box (1978), Lee and King (1993), and
BDS. We also presenttwo empiricalexamplesin Section
5. In Section6, we concludethe article.All proofs are collected in the Appendix.
1. A NEW TEST FOR ARCH
Considerthe data-generating
process (DGP)
Yt = g(Xt, bo) + Et,
t = 1,..., n,
(1)
with ARCHerror
Et =
tth/2,
(2)
where Yt is the dependentvariable;Xt is a d x 1 vector containingexogenous variablesand lagged dependent
variables;bo is an 1 x 1 unknown true parametervector in IRl;g(Xt,b) is a given, possibly nonlinear,function such that, for each b,g(, b) is measurablewith respect to Il-1, the informationset available at period
t - 1; and g(Xt, -) is twice differentiablewith respect to
b in an open neighborhoodN(bo) of bo almost surely,
with limn• nr1 t=1 ESUPbEN(bo) flVbg(Xt, b)l4 < 0c
and limn-+ n-' •-•n ESUPbEN(bo) V2g(Xt, b)l2 <
where1I-IIis the Euclideannormin I1. [Forlinearmodelsi.e., g(Xt, b) = XIb--these dominanceconditionsreduceto
limn-+on- 1jE
EjjXt114<
00.]
The functionht is a positive,time-varying,and measurable functionwith respectto tl-. We assumethat?t is iid
with E(?t) = 0, E(t2) = 1, and E(?t8)< 00. In particular,
we do not requirethat•t be N(O,1), which may be too restrictivefor manyhigh-frequencyfinancialdata.In addition,
we assumethat?t andX, aremutuallyindependentfor t >
s. By definition,Etis seriallyuncorrelatedwith E(Et) = 0.
= ht, maychange
But, its conditionalvariance,E(et
Ctl)
overtime.Forexample,if ht follows an ARCH(qo) process,
we have h = to -+ i=1 a2-i, whereao > 0 and ai 0
to ensurepositivityof ht. If ht follows a GARCH(po,qo)
E?
Po2
qo
process,
h = ao ++
process, ht
-i +
jht-j, where
+j=l
i=1
o
a0o > 0.
The nullhypothesisof no ARCHeffectis Ho:ht = a2 almost surelyfor some 0 < a < 00 andall t = 1, 2, ..This
is equivalentto conditionalhomoscedasticity.Define ut =
E
0o - E(E2). Then,underHo,ut = 1t2
t
Et/g2
ao0_ 1,, where
whrt g2
a white-noiseprocess.The white-noisehypothesisimplies
a uniformnormalizedspectraldensityor distributionfunction.WhenARCHexists,the spectraldensityor distribution
function will not be uniformin general.Therefore,a test
for Ho can be basedon the shapeof the spectraldensityor
distributionfunction.Researchershaveemployedthe shape
of the spectraldistributionfunctionto test varioushypotheses on serial dependence(e.g., white noise). These include
Anderson(1993), Bartlett(1955), Durbin(1967), Durlauf
(1991), and Grenanderand Rosenblatt(1953, 1957). See
Priestley(1981) and Anderson(1993) for surveys.Extending these works to the ARCHcontext shouldbe possible,
thoughwe have not seen such resultsin the literature.
Relatively few tests are based on the spectraldensity
function.Althoughthe periodogramis not consistentfor
the spectraldensity,one can use Parzen's(1957) smoothedkernelspectraldensityestimatorto test the white-noisehypothesis, as did Hong (1996). This approachinvolves the
choiceof a smoothingparameter,a sometimesdelicatebusiness. But, maximalpoweris often attainableif this parameter is chosenvia an appropriate
data-drivenmethod.In this
article,we extendHong's (1996) spectraldensityapproach
to the ARCHcontext.Ourasymptotictheoryhere permits
a data-dependent
smoothingparameter.
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Hong and Shehadeh:A New Test forARCHEffects
93
Let f(w) be the normalizedspectraldensityfunctionof
where the secondequalityfollows by Parseval'sidentity,
ut. Consequently, f(w) = fo(w) - 1/21r for all frequencies
w e [-7r, wr]underHo. In contrast,f(w) fo(w) in general
when ARCH exists. It follows that a test for Ho can be
based on the L2 norm
L2(f; fo)
n-1
Cn(k)
=
E
(1 - j/n)k2(j/q)
j=1
and
n-2
=
2
7r(f(w)
- fo(w))2
dw(3)
Dn(k)
=
>
(1 - j/n)(1 - (j + 1)/n)k4(j/q).
j=1
where f is a consistent spectraldensity estimatorfor f.
As will be seen, tests based on (3) can be rathersimpleto
compute;no integrationis required.
We now constructan estimatorfor f. Let b be an n1/2consistentestimatorfor bo.For example,b can be the nonlinearleast squaresestimator
b = argmin n'
b
(Y - g(Xt, b))2.
t=1
We note that Cn(k) and Dn(k) are approximatelythe mean
andvarianceof n En-. k2(j/q)132(j).The factors(1-j/n)
and (1 - j/n)(1 - (j + 1)/n) canbe viewed as finite-sample
correctionsandare asymptoticallynegligible.Althoughwe
motivateour test using (3), Q(q) involvesneithernumerical
mointegrationnor estimationof f. This frequency-domain
tivationleads to an appropriatechoice of q via data-driven
methods,as will be seen.
Our first result shows that Q(q) is asymptoticallystandardnormalunderHo.
Such an estimatordoes not take into accountthe possible
Theorem 1. Let q --+ o, q/n -+ 0. Then under the stated
ARCHeffects but consistentlyestimatesbo underboth the
conditionsand H0, Q(q) -+ N(O,1) in distribution.
null and alternative.The estimatedresidualof (1) is Et =
The test is one-sidedbecausein generalQ(q) divergesto
=
Yt - g(Xt, b). Put Ut = t2 1, where t = ?t/8n and
r t=> 2. Then the sampleautocorrelationfunctionof positive infinitywhen ARCH exists. Consequently,appron-1
{ut} is (j) = i(j)/r(O),j = O,+1,..., +(n - 1), where priateupper-tailedcriticalvalues of N(O, 1) must be used.
For example,the criticalvalue at the 5% significancelevel
i(j) = n-1 Z=iil+ utut-IjI. A kernelestimatorfor f is is 1.645.
given by
For largeq, q-Dn (k) -+ D(k) = fo7 k4(z) dz. Thus,one
n-1
can replaceDn(k) by qD(k) withoutaffectingthe asympf(w) = (2r)-1
totic distributionof Q(q). Under some additionalcondik(j/q)1(j)cos(jw)
j=1-n
tions on k and/or q, q-1Cn(k) = C(k) + o(q-1/2), where
fo k2(z) dz; we also can replaceCn(k) by qC(k).
for w E [-ir, wr].Here, the bandwidthq - q(n) -+ oc and C(k) _a more
Thus,
compact,asymptoticallyequivalenttest statisq/n -+ 0. The kernel function k: R -- [-1, 1] is symmetric, tic is
continuousat 0, and all but a finite numberof points,with
"n-1
=
k(0) = 1 and f_0Vk2(z) dz < 00. This implies that, for j
(2qD(k))1/2
n
qC(k)
smallrelativeto n, the weightgivento p(j) is close to unity, Q*(q)
k2(j/q)2(j)
j=1
the maximumweight.The squareintegrabilityof k implies
00.
as
that k(z)
0
Thus, eventually, less weight is
jzj -+
Because Q(q) is basedon the whole-sampleautocorrelagiven to p(j) as j increases.
tion function 3(j) of the squared residual fit, it can detect
Examples of k include the Bartlett, Daniell, general the whole class of lineardependenciesof ht-that is, auto(QS), and truncatedker- correlationin the squaredresiduals.Hence, it may be powTukey,Parzen,quadratic-spectral
nels. (See, e.g., Priestley 1981, p. 441.) Of these, the erful against ARCH and GARCH alternatives,especially
Bartlett,generalTukey,Parzen,and truncatedkernels are stronglypersistentones when a long lag is used. The test
of compact support--k(z) = 0 for z > 1. For these kernels, q is the lag truncation number because lags of order j > q receive zero weight. In contrast, the Daniell
and QS kernels are of unbounded support. Here, q is not
a truncation point but determines the degree of smoothing for f(w). When the kernel is of unbounded support, all
n- 1 sample autocorrelations of the squared residual are
used.
Following Hong (1996), we define our test statistic
Q(q)
=
(2(f;
-(
1n-1-
fo)
- Cn(k)) /(2Dn(k))1/2
cannot, however, detect all deviations from conditional homoscedasticity. For example, it has no power if ht follows
a tent map, a typical nonlinearity from chaos theory. Of
course, like the LM test, the test has power against many
types of nonlinearity.
When the regression model (1) is misspecified, the Q(q)
test may falsely reject the null hypothesis Ho. This is not peculiar to Q(q). The same is true of all existing ARCH tests;
all, explicitly or implicitly, assume correct specification of
the regression model. In the present context, Equations (1)
and (2) with existence of a consistent estimator b for bo (cf.
Assumptions A.1-A.5 in the Appendix) imply correct specification of regression model (1). Such an assumption makes
sense because one is usually interested in ARCH only when
the regression model is correctly specified.
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94
Journalof Business & EconomicStatistics,January1999
The Q(q) test is based on the squareddeviationof f (w) because r = 00. Consequently,the conditionq/q - 1 =
from fo(w), weighted equally for all frequenciesw over op(q-((3/2)v-1)) is ratherweak.FortheDaniellkernel,any
[-7r,ir]. This is appropriatewhen no prior informationis v > 3/2 is allowed because 7 = 1. In this case, the rate
available.One can directpower towardspecific directions condition
q/q - 1 = op(q-((3/2)v-1)) is also easyto satisfy.
of interestby giving differentweightsto differentfrequen- For suchmethodsas Andrews's(1991)parametric"plug-in"
cies. For example,if one is interestedin detectingstrongly bandwidth,q/q - 1 vanishesin probabilityat the parametric
persistentARCHeffects,greaterweightcanbe givento fre- rate n-1/2 with q oc n1/5 for most kernels,but for such
quenciesnear0. Whennonuniformweightingfor frequen- methodsas Newey andWest's(1994)nonparametric
"plugcies is used, an L2-norm-based
test will involve numerical in" bandwidth,q/q - 1 vanishesat a rate slower than the
integration.(We emphasizethat weightingfor frequencies parametricrate with q oc n1/3 for the Bartlettkernel. In
is differentfrom weightingfor lags.)
general, our conditionon q allows for a variety of dataAlternatively,tests can be based on other appropriate dependentmethods.
One appropriatedata-drivenprocedureis the crossglobal divergencemeasures,such as the supremumover
validationproposedby BeltraoandBloomfield(1987).This
[0,7r]of the absolutevalue of the deviationf(w) - fo(w):
delivers an automaticq by minimizinga cross-validated
frequency-domainlikelihood function. Robinson (1991b)
(q) =()1/2 2
sup
showed that, asymptotically,such chosen q minimizes a
wE[o,Gr]If(w)-fo(w)
weightedintegratedmean squarederrorof f with suitable
n-1
weights dependingon the true spectraldensity f(w). This
71/2 sup
k(j/q)1(j)xV/cos(jw)
global procedureis more appropriatein the presentconwE[,0-r]j=1
text than the narrow-bandmethodsused in estimatingan
covariancematrix(e.g., Andrews
autocorrelation-consistent
This test is computationallymore complicatedthan Q(q). 1991;
Newey andWest 1994) becauseQ(q) is basedon all
Following reasoninganalogousto, but more complicated frequenciesover [-w, 7r]ratherthanon frequency0 only.
than,thatof Anderson(1993) and Durlauf(1991), one can
The Beltrdo-Bloomfieldprocedurecan be describedas
null asymptoticdistribu- follows. Define
expectthatQ(q)has a nonstandard
tion thatis the supremumof a Gaussianprocesswith mean
n-1-2
0 and covariancedependingon both w and k. For analytic
n,
t exp(-iAt))
cE(-oo, oo).
I(A)
E
simplicity,we focus on Q(q) here, but we studythe power
t=o
of Q(q) in our simulations.
This is the periodogramof it = E^2
/-2 1. Note thatI is pe2. CROSS-VALIDATION
riodic in A with periodicity27. Put the Fourierfrequencies
for j = 0, 1,..., n- 1. Then definethe crossAn importantpracticalissue with Q(q) or Q*(q) is the Aj = 27j/n
estimatorat the Fourier
validated
spectral-density-function
choice of q, which may have significantimpacton power
as
in finite samples.Clearly,the optimalchoice of q depends frequencies
on the unknownalternative.It is, therefore,desirableto
fJ(AJ;q) =j(q)-1
W(qAl)I(Aj -A•),
choose q via data-drivenmethods.Althoughwe need not
ION(n,j)
computef to computeQ(q), our approachsuggeststhat it
is appropriateto choose q using an optimalitycriterionfor whereaj (q) = lgN(n,j) W(qA1),W(A)= (27)-1 f_c k(z)
the estimationof f. Before we discuss some data-driven exp(iAz)dz is the Fouriertransformof a kernelfunctionk,
methodsin detail,we extendTheorem1, which assumesa and N(n,j) = {0,+n,...} U {2j, 2j n,...} is the set of
nonstochasticq, to allow for a data-dependentbandwidth indicesj for which I(A• - Al) = I(Aj). Note thatwhen W
> 7, the
q. For this purpose,we furtherrestrictthe class of kernels is of compactsupportsuch that W(A) = 0 for AlA
k such that lk(zi) - k(z2)1 _ All1 - z21 for any z1,z2 and effective summationis only over 1 for Il1< n/2q.
The cross-validatedq• solves
some 0 < A < co, and Ik(z)l ? alzl- for all z E R and
for some 7 > •. The Lipschitz condition on k rules out
the truncatedkernel but allows for most commonlyused
nonuniformkernels.
[n/2-1]
qc= argmin
qE[an,bn]
Theorem2. Suppose the data-dependentbandwidth
where v > (27 satisfies O/q - 1 = op(q-((3/2)v-1)),
)/(2r - 1) and q is a nonstochastic bandwidth such that
q -+ oo, q"/n -+ 0. Then, under the stated conditions and
-+ N(0, 1) in distribuHo, Q(q) - Q(q) = op(1) and Q(Q)
tion.
5
{ln(Aj;q) +I(Aj)/fj(Aj;q)},
(4)
j=l
where [n/2 - 1] is the integer part of n/2 - 1 and
the interval[an,b,] is predetermined.The objectivefunction In j (A•j;q) + I(Aj)/l (Aj; q) is the well-knownWhittle approximationfor the negativelog-likelihoodfunction
of {it}.
Therefore, t• approximately maximizes the log-
likelihoodof {St }. The parameter0 canbe real-valued,but
aninteger-valued
on
which
in
turn
The range of admissible0 depends v,
• is moreconvenienthere.Thenthe probcan
lem
be
solved
k.
the
is
the
on
or
The
smaller
is
(4)
v,
using grid search. By letting b, -+ 00
larger
range
depends r
of admissibleratesfor q. Forkernelswith boundedsupport as n -+ 00, 0 always will tend to infinity under the altersuchas the BartlettandParzenkernels,anyv > 1 is allowed natives with nonuniformf(w). Under the null hypothesis
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Hong and Shehadeh:A New Test forARCHEffects
95
of no ARCH,q will convergeto the lower bound a, as the use of the truncatedkernel-that is, uniformweighting.
n -+ 00 because the spectrum is flat. Consequently, we also Becausemanynonuniformkernelsdeliverbetterpowerthan
let an -+ oo to satisfythe conditionsof Theorem2; namely, the truncatedkernelwhenq is large,we expectthatourtests
q, -+ o0. As long as an diverges at a slow rate and bn di- may have greaterpowerthanthese tests for large q.
verges at a fast rate, the theoreticallyoptimalqc will fall
Suppose that k is the truncated kernel, k(z) = 1 for Izl <
in this interval,and q^ will convergeto it (cf. Robinson 1 and 0 for IzI> 1;Q(q) becomes
1991b).For example,with the Daniell kernel,q = Cn1/5
= (BP(q) - q)/(2q)1/2
QTRUN(q)
for the alternativesfor which f"(w) exists, where C dependson f(w). Thenqc will convergeto q, underthe alter- given q3/2/n -+ 0, where
native as n -+ oo if an/n1/5 -+ 0 and bn/n1/5 -+ 00o. We
q
note thatthe choices of an and bn are to some degreearbiBP(q) = n PZ2(j)
trary,butthesechoicesareof secondaryimportancebecause
j=1
qc will convergeto qc for large n. This is similarin spirit
to Newey and West's (1994) method.Oursimulationstud- is a Box and Pierce (1970) type of statisticfor the squared
ies, like those of BeltrdoandBloomfield(1987) andRobin- residuals.The conditionq3/2/n -+ 0 is strongerthanq/n -+
son (1991b),reveal thatqc is variablein finite samples.It 0. This ensures q-1Cn(k) = 1 + o(q-1/2) and q-1Dn(k) =
gives better power,however,than simple "rule-of-thumb" 2 + o(1) for the truncatedkernel.The test BP(q), as shown
choices of q, suggestinggains from its use. The procedure by McLeodand Li (1983), is asymptoticallyx2 underHo.
can be implementedefficientlyusing the fast Fouriertrans- Intuitively,when q is large, we can transformBP(q) into
an N(0, 1) by first subtractingthe mean q and dividingby
form (FFT).
We now summarizethe proceduresto implementthe the standarddeviation (2q)1/2. Therefore QTRUN(q)can be
viewed as a generalizedBP(q) test for largeq. In practice,a
test:
modifiedbut asymptoticallyequivalentstatistic,originally
1. Obtainany nl/2-consistent estimatorb and save the
proposedby Ljungand Box (1978),
=
residuals Et
Yt
g(Xt, b).
q
functionp(j) of
2. Constructthe sampleautocorrelation
Et=
a
S
LB(q) = n2 E
1.
3. Choose a kernel k. For example,we can choose the
2(j)/(n - j),
j=1
Daniell kernel k(z) = sin(rz)/rz, z e (-oo, 00oo).Its Fourier often is used for testing ARCH. The weights n/(n - j)
transform W(A) = 7-11[I1AI< 7r].
are introducedto improvesize performanceand do not af4. Choose qc by the Beltrdo-Bloomfield(1987) cross- fect asymptoticpower. These weights are fundamentally
validationprocedure.
differentfrom those of our Q(q) test, which affects the
5. Computethe statistic Q(qc) and compare it to the asymptoticpower.The test LB(q) is asymptoticallyequivupper-tailedcriticalvalueC, of N(0, 1) at significancelevel alent to BP(q). It follows that QTRUN(q) is also asymptotrq.If Q(qc) > C,, then one rejects Ho at level qj.
ically equivalentto a generalizedversion of LB(q) under
Step 4 is not always necessary.If some q is preferreda Ho; namely,
= Op(l).
priori,one can omit this step. A GAUSS programimpleQTRUN(q) - (LB(q) - q)/(2q)1/2
from
us.
is
available
these
steps
menting
We now relate a generalizedversion of Engle's (1982)
Althoughthe cross-validatedq, is asymptoticallyoptimal for the estimationof f(w) in terms of an integrated LM test to QTRUN(q). Put U = (Ui,...,un)' and Z =
Then the
(1,2_ ,...,t2_q)'.
weightedmean squarederrorcriterion,it may not be opti- (Z',... ,Z)', where Zj
mal for power of Q(q). For hypothesistesting, a sensible LM test
criterionis to choose q so as to maximizepower and/or
LM(q) = nU'Z(Z'Z)-1Z'/UU/U'
minimize size distortion.It can be expectedthat the optimal q chosen this way will generally be different from ic,
although they may be of the same order of magnitude. This
is a theoretically interesting but technically complicated issue that deserves further investigation; we defer it to future
work. Nevertheless, our simulation and application suggest
that • often delivers maximal or reasonably good power
for Q(q) in finite samples.
3.
RELATIONSHIPTO SOME EXISTINGTESTS
We now discuss the relationship of our test to some important existing tests for ARCH. We will show that, when q
is large, a generalized version of Engle's (1982) LM test, as
well as those of Box and Pierce (1970) and Ljung and Box
(1978), can be viewed as a special case of our approachwith
- nR2 ,
where R2 is the uncentered squared multicorrelation coefficient from the ARCH regression
4
O+
at
1 +
+2'2
+
'..
+ Qq
+tq,
t= 1,...,n,
with the initial values g2 = 0 for t = -q + 1,...,0. This
test is asymptotically x2 under Ho and is asymptotically
locally most powerful if the true alternative is ARCH(q)
with q fixed (cf. Engle 1982, 1984). Lee (1991) showed that
a modified LM(q) test for GARCH(p, q) is the same as the
LM(q) test for ARCH(q). Granger and Teriisvirta (1993)
showed that the LM(q) test is asymptotically equivalent to
BP(q) and LB(q) for fixed q.
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Journalof Business & EconomicStatistics,January1999
96
Forlargeq it is naturalto considerthe generalizedversion kT(Z) = 1[ zI _ 1] becauseEFF(kB : kT) > 51/2. As done
of Engle'stest,
by Hong (1996),withina suitableclass of kernelfunctions,
the Daniellkernelmaximizesthe powerof Q(q) underHan.
= (LM(q) - q)/(2q)1/2
The relativeefficiencyover the LM test does not contraQREG(q)
the well-knownresultthat the LM test is locally most
dict
=
(nR2 - q)/(2q)/2
powerfulwhen the true alternativeis ARCH(q)with fixed
The following theorem shows that QTRUN(q) is asymptotically equivalent to QREG(q).
Theorem 3. Let q - oo, q5/n
O0.Then, under
and
the stated conditions
Ho, QTRUN(q) - QREG(q)
in
-?
N(0, 1) distribution.
op(l), QREG(q)
Like the LM test, QREG(q) can be viewed as a test for
the hypothesisthatall q coefficientsof an ARCH(q)model
arejointly equalto 0, where q grows with the samplesize
n. Intuitively,if the data are homoscedastic,then the variance cannotbe predictedby any past squaredresiduals.If
a large variancefor it can be predictedby large values of
the past squaredresiduals,however,thenARCHeffects are
present.Any stationaryinvertibleprocessin Et with continwell by a truncatedARCH
uous f(w) can be approximated
model of sufficiently high order. Therefore, QREG(q) even-
q; here we consider a different regime; namely, q -
00o.
Many earlier applications(e.g., Engle 1982, 1983; Engle
and Kraft1983;Engle et al. 1987;Bera andHiggins 1993)
used linearlydecliningweighting schemes with relatively
long lags. Such weightingschemes also serve to discount
past information.But, as pointedout by Bollerslev(1986),
these weightingschemesare somewhatad hoc. In comparison, ourweightingdependson the kernelfunctionk, which
arisesendogenouslyfrom the kernelspectralestimation.
Recently,Lee and King (1993) proposeda locally most
meanpowerfulscored-based(LBS)test for ARCHthatexploits the one-sidednatureof the ARCHalternative.Their
is
test statisticfor ARCH,whichis robustto nonnormality,
given by
((n--
LBS(q)
q)
(2 /
_- 1) Eq1
q)
2-i
-1Zt-i)
t- _
tuallywill capturea wide rangeof alternativesas morelags
[Et(^21
2_ \1)2]
Et
Eq
2 )
of g2 are includedwhenn increases.This test is convenient
- (Et Eql 2-i)2}1/2
to implement,butit cannotbe expectedto be fully efficient,
(5)
as we show next.
The preceding discussions show that the tests of
UnderHo, LBS(q)is asymptoticallyone-sidedN(O,1). ObQTRUN(q), QREG(q), BP(q), LB(q), and LM(q) put unialso puts uniformweight on all q sample
form,or roughlyuniform,weightson all q sampleautocor- viously, LBS(q)
of
the squaredresiduals.Consequently,it
autocorrelations
relations.Intuitively,this mightnot be the optimalweightin detectingthe ARCH alternabe
not
efficient
fully
may
ing scheme.For most stationaryARCHprocesses,the au- tives
whose
autocorrelation
p(j) decays to 0 as the lag j
tocorrelationdecays to 0 as the lag increases.Intuitively,a
increases.
we
Therefore,
expect that Q(q) may be competbetter test should put greaterweight on lower-orderlags.
in
with
some
itive
cases, as illustratedin our folLBS(q)
We thus expect that tests based on nonuniformweightsimulations
and
empiricalapplications.Comparing
ing may deliver better power than BP(q), LB(q), LM(q), lowing
and fo(w) at frequency0, Hong (1997) constructeda
f(w)
Pitman's
relative
and
efficiency
QREG(q). Using
QTRUN(q),
test thatexploitsthe one-sidednatureof the ARCHalternacriterion,it can be shownthatthis is indeedthe case. Contive and uses a flexible weightingscheme.AlthoughQ(q)
siderthe followingclass of local ARCHalternatives:
does not exploitthis, it is asymptoticallymoreefficientthan
Hong's (1997) test becauseQ(q) can detecta class of local
an
alternativesof O(ql/4/nl1/2). Hong's (1997) test only can
Han: ht
2 1+?(ql/4/nl/2)Eaj
detect a class of local alternativesof O(ql/2/nl1/2). The
j=1
weighting schemes of Q(q) and Hong's (1997) test also
where aj 2 0 and -jl, aj < o00, which implies aj -+ differ.
0 as j -+ 00. Without loss of generality, we assume
1
a•j < for all n to ensure positivity of ht.
Following reasoning analogous to (but more tedious than)
that of Hong (1996), it can be shown that, if q2/n -+ 0
and q -+ co, Q(q) -+ N(p, 1) in distribution under Han with
Note
noncentrality parameter p = (Z7= aJ)2/{D(k)}1/2.
that, whenever ARCH exists-that is, at least some aj > 0
always has nontrivial power because p > 0.
exist-Q(q)
With q = CnY, where 0 < y < 1 and 0 < C < o00, the relative efficiency of the Q(q) test using kernel k2 with respect
to using kernel kl under Han is given by
(q1/4 /n1/2)
-'=1
EFF(k2 : kl)= (D(k2)/D(kl))1/(2-Y )
Therefore, the Bartlett kernel kBg(z) = (1 - zl)1[zl _ 1]
is about 120% more efficient than the truncated kernel
4.
MONTE CARLO EVIDENCE
We now study the finite-sample performances of our
tests in comparison to a variety of existing ARCH tests.
Consider the following DGP: Yt = X'bo + t, t = tt/2
where Et - NID(0, 1) and Xt = (1,mt)' with mt =
and vt
Amt-1 +
-vt
NID(0, o-). This model was first
used by Engle, Hendry, and Trumble (1985). We consider
four processes for ht--(1) ht = w, (2) ht = w + a12_l,
i 2t- /i2, and (4) ht =
(3) ht = w + a(E 1 1/i2)-1
w + aEt-1 + pht_1.
Under (1), ARCH is not present. This permits us to examine the sizes. Alternative (2) is an ARCH(1) process often examined in existing simulation studies (e.g., Engle et
al. 1985; Diebold and Pauly 1989; Luukkonen, Saikkonen,
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Hong and Shehadeh:A New TestforARCHEffects
97
and Terasvirta1988;Bollerslevand Wooldridge1992;Lee
andKing 1993).Alternative(3) is an ARCH(8)processwith
weights decliningat a geometricrate as the lag increases.
Finally, alternative(4) is a GARCH(1, 1) process. The
GARCHmodel has been the workhorsein the literature.
We set b, = (1, 1)' and w = 1. For the exogenousvari-
25
S.........BDS1
20
able mt, we set A = .8 and ao2= 4. As done by Engle et
al. (1985), mt is generatedfor each experimentand then
held fixed from iterationto iteration.For alternatives(2)
and (3), we considertwo values of a, .3 and .95. For alternative(4), we choose threecombinationsof (a, r0)-(.3, .2),
(.5, .2), and (.3, .65). We considersamplesizes of n = 2"
for m = 6, 7, 8, and 9. These samplesizes are chosen for
conveniencebecausewe employthe Cooley-TukeyFFT algorithmin our cross-validationprocedure.(Ourprocedure
is applicablewhenn is not a powerof 2, thoughat some sacrifice of computationalefficiency.This sacrificeis costly in
simulation.)We set Ec = 0 for t < 0 and ho = 1. To reduce
the possible effects of these initialconditions,we generate
n + 100 observationsand then discardthe first 100. The
iterationnumberis 10,000 for (1), and 1,000 for (2)-(4). A
GAUSS-386random-number
generatoris used.
We choose the Daniell kernel for our tests,
QDAN
(q) and
QDAN(q), and also considertheir cross-validatedversions,
Qcv = QDAN (c) and Qcv = QDAN(qc). We only study the
powers of Qcv and QDAN(q) becausetheir null limit distributionsare unknown.We compareour tests to Engle's
LM(q), Box and Pierce's BP(q), Ljung and Box's LB(q),
Lee and King's LBS(q), the truncatedkernel-basedtest
QTRUN(q), the generalized LM test QREG(q), and BDS, (q)
for q = 1,..., 20. We include BDS.y(q) because it has been
popularin the literaturepartly for its capacity to detect
ARCH alternatives.This test involves choice of a distance
parametery E (0, 1) in terms of the data spread.To study
the impactof y on size andpower,we considertwo values,
7 = 1/4 and 1/2. (We use the BDS code of D. Dechert.)
For economy, we reportresults for n = 128 in detail
and briefly mention results for other sample sizes. Figure 1 shows the sizes at the 5% level. The QDAN(q) test
performswell for all q and overrejectsonly slightly. For
large q, BP(q), QTRUN(q), LM(q), and QREG(q) all tend
to underreject,most seriouslyfor LM(q). The LB(q) test
underrejectsfor small q andperformswell for mediumand
large q. The LBS(q) test has reasonable size for small q,
underrejects for medium q, and overrejects for large q. The
BDS,(q) test significantly overrejects for the two choices
of 7. Finally, the cross-validated Qcv gives a rejection rate
of 7.79%, illustrating some overrejection at the 5% level.
(To facilitate comparison, the result for Qcv is reported in
each figure as a horizontal line, the same value for each q.
Since 4• is chosen endogenously it may vary for each iteration of the simulation.) Comparison of QDAN(q) and Qcv
indicates that this overrejection arises from the additional
noise of cross-validation. (Here, the mean of j0 is 3.8 with a
standarddeviation of 3.7; the empirical distribution of q0 is
left-skewed.) As will be seen, the gain in power from using
0• may compensate for this moderate overrejection. At the
10% and 1% levels, the size of LBS(q) exhibits a U-shaped
"BDSI
1at5%
4
2the
3
Level.
8
5
7
6
The
simulated
11is 12
9
10
model
yLB=13 114 + 15mt 16+
LBS
2
3
4
5
20
19
t, 18where
LM
0
1
17
6
7
I
8
I
9
I
I
I
10 11 12 13 14 15 16 17 18 19 20
I
I
I
Lag(q)
Figure 1. EmpiricalSize: RejectionRates of Tests Underthe Null
at the 5% Level. The simulated model is Yt = 1 + mt + Et, where Et
- , 1), mt
+
=
=
+ vt,
t
ht =
.8, and vt N(O,
- ,thtllP
th112,ht
N(O, i), mt == Amt-l
vt, A == .8,
N(O,
1- - N(_1, t
Imt
I and"vt
4). Calculationsare based on a sample size of 128 and 10,000 replications. TheresultforQcv (forwhich;Tcis chosen viacross-validation
and,
therefore,may change foreach iteration)is includedforcomparisonas
a horizontalline.
curve, performingbest for small q. At the 10%level, the
finite-samplecorrectionagainimprovesthe sizes of LB(q)
versusBP(q),while it inducessome overrejectionfor LB(q)
in the 1%level. Cross-validationfor Qcv deliversa rejection rate of 11.6%at the 10%level, and a strongrejection
rate of 4.2% at the 1%level. The LM(q) test underrejects
at bothlevels. The sizes of the tests improve,to varyingdegrees, in largersamples.For n = 512 andthe 5% level, the
Q tests have sizes between5%and6%;the othertests have
sizes between 3% and 5%. The cross-validatedQcv test
still exhibitssome overrejection,7%at the 5%level. At the
10%level, the sizes of all of the tests are similarandrange
between7%and 11%.At the 1%level, LBS(q),LB(q),and
BP(q) performwell, close to 1%.The Q tests show some
overrejectionbut never exceed 2.5%;Qcv remains3.5%.
The LM(q) test exhibitssome underrejection.
As suggestedby asymptotictheory,the sizes of Qcv improve as the samplesize n increases,but very slowly. In a
finitesample,one can use bootstrapto obtainmoreaccurate
sizes for Qcvy. Givena sampleof size n, we firstobtainthe
ordinary least squares (OLS) residuals ut = Yt- X/b, where
b is the OLS estimator for b0. We then draw a sample of
residuals {'24} of size n, with replacement, from the empirical distribution function of {it}. Next, we obtain a sample
i4. On each sample
{Yt*} of size n, where Y1* = Xb
{1Y*}, we perform the regression described previously and
submit the regression residuals to each test. We repeat this
for 1,000 iterations and obtain 1,000 bootstrap test statistics, which are used to form the bootstrap empirical distribution function for each test. We then compare the actual
test statistics to the bootstrap empirical distribution functions to find the bootstrap p values. The bootstrap sizes of
Qcv at the 10%, 5%, and 1% levels are 12.2%, 5.3%, and
1.5%, respectively. These are significantly better than the
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98
Journalof Business & EconomicStatistics,January1999
sizes using asymptoticcriticalvalues,especiallyat the 1%
level.
Figure 2 shows powers against ARCH(1) using the
5% empirical critical values. The empirical critical values are obtained from the iterations under (1). First,
LBS(1) is slightly more powerfulthan LM(1), LB(1), and
QDAN(1), suggesting some gain from exploitingthe onesided nature of the alternative.The tests LM(1), LB(1),
QDAN(1),
QTRUN (1), and QREG (1) have the same power,
slightlybetter than QDAN(1) and BDS1/4(1). Next, the powers of all the tests except QDAN(q) decreaserapidlyas q
increases,most dramaticallyfor LBS(q) and BDS1/4(q).
50
40
BDS
30
LBS
.
..
QCV
20
BDS
BP andQMI
LB
f7
LM and QI
10
-
80
I
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Lag(q)
(a)
120
------------------------- -....
"100
SBP
andQ
"80
E?
--
--"
.--
BDS
.
-.
-
-
LMandQ ,
40
L
BDSBDS
(a)
120
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
100
--
-
.
QD-
----80
IBP
andQ..
rates are based on the size-correctedempiricalcriticalvalues for the
5% level. The resultforQcv (forwhich?, is chosen via cross-validation
and, therefore,maychange foreach iteration)is includedforcomparison
as a horizontalline. (a) a = .3; (b) a = .95.
SBDS
._..
-.._
BDS
20
0
..
--------
I
I
LBS
The tests LM(q), LB(q), BP(q),QTRUN(q),
QREG(q), and
For
>
q 1, QDAN(q)is the most
QDAN(q) perform similarly.
-.-
I
I
Figure 3. EmpiricalPower: Rejection Rates of Tests Under the
ARCH(8)Alternativeat the 5% Level. The simulatedmodel is yt = 1
LB
LMandQ
40
(b)
I
I
I
I
I
I
I
I
I
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Lag(q)
(b)
powerfultest. Its power loss as q increasesis smallerthan
the losses of the other tests because QDAN
(q) discounts
The
of
is
higher-orderlags.
power BDS1/4(q) comparable
to that of LBS(q) except for small q: BDS1/2(q) has low
power for small q but performs better than BDS 1/4(q) for
q > 8. (We report y = 1/4 becausethis choice appearsto
maximizethe powerof BDS for n 128 in most cases. We
know of no theoreticalbasis for choosing
For large
critica.)
,forthe
as in Figure2(b), the power of QDAN
increases
slightly
(q)
Figure 2. EmpiricalPower: Rejection Rates of Tests Under the
ARCH(1)Alternativeat the 5% Level. The simulated model is yt =
1 + mt + Et, whereEt - 5th112,ht = 1 + ar2t1, Jt N N(O,1),
mt = Amt_1 + vt, A = .8, and v t N(O, 4). Calculations are based on a
sample size of 128 and 1,000 replications.Rejectionrates are based on
the size-correctedempiricalcriticalvalues for the 5% level. The result
forQcv (forwhichqc is chosen via cross-validationand, therefore,may
and then decreases as q increases. This is not surprising;
change foreach iteration)is includedforcomparisonas a horizontalline.
(a) a = .3; (b) a = .95.
the spectraldensityof an ARCH(1)process includescon-
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99
Hongand Shehadeh:A New Test forARCHEffects
tributions from q > 1 that are substantial for large a. Finally, we note that the cross-validation delivers reasonable
power for Qcv, although a little lower than those of LM(1)
and LB(1), which use the correct information that the true
ARCH order qo = 1. The power of Qcv is low, 31% and
80%, respectively. (We do not include Qcv in the figures.)
Next, Figure 3 reports powers against ARCH(8). In Figure 3(a), QDAN (q) has the highest power for all q, including
q = 8. The tests QDAN(8) and BDS1/4(8) perform similarly.
The Qcv test has roughly the same or better power than
QDAN(8);
CV only has power 27%. The power of QDAN(q)
increases slightly and then decreases as q increases. Although LBS(8), LM(8), BP(8), and LB(8) are asymptotically locally most powerful against ARCH(8), none of
them achieves its maximum power at q = 8. The powers
of LBS(q), LM(q), LB(q), QTRUN(q),
80
LBS
70
-..V
60
50
BPand
30
20
10
1
2
4
3
5
6
8
7
9 10 11 12 13 14 15 16 17 18 19 20
and QREG(q) attain
their maxima at q = 1 and then decrease as q increases.
Again, LBS(1) has better power than the related tests at
q = 1. The tests LM(1), LB(1) and QDAN(1) have similar power. As q increases, the power of LBS(q) drops dramatically. The tests LM(q), BP(q), LB(q), QTRUN(q), and
QREG(q)perform similarly. BDS1/4(q) performs quite well
for small q but is dominated by BDS1/2(q) for large q.
Again, the power of QDAN(q) declines slowly. In Figure
3(b), BDS1/4(q) dominates QDAN(q) for q < 6.
Figure 4 reports powers against GARCH(1, 1). First, we
examine Figure 4, (a) and (b), in which a changes but /
is fixed. For (a, /) = (.3, .2), the power patterns and ranking of the tests are similar to those under ARCH(1). In
particular, all the tests have maximum power at q = 1, except QDAN(q) and QDAN(q), which have maximum power
at q = 2 and 3, respectively. LBS(q) is the most powerful at q = 1, and the other tests have similar power. [As
pointed out by Lee (1991) and Lee and King (1993), LM(1)
and LBS(1) are the appropriateLM and LBS tests for both
ARCH(1) and GARCH(1, 1).] For q > 2, QDAN(q)has the
highest power. The power of Qcv is close to QDAN(1).For
(a, /) = (.5, .2), we note two primary differences from the
preceding. First, the power curves of BDS, (q) and QDAN (q)
are concave and each achieves its maximum for q > 5. Second, both the advantage of LBS(1) over the other tests and
the cost of using a cross-validated q, are smaller. Last, Figure 4(c) contains results for (a, p) = (.3, .65), where a +/3
is close to 1, a characteristic common in empirical research.
All of the power curves now become concave; none of the
tests achieves its maximum power at q = 1. Indeed, it is
reasonable that LM(1) and LBS(1) cannot be expected to
be optimal here. For persistent GARCH, it is better to include more lags in the tests. Here, the power of QDAN(q)
decreases very slightly even for large q and achieves its
maximum power at q as large as 7. As /3increases, the relative power ranking of the tests remains much the same,
though the concavity is more pronounced. Finally, we note
that for most GARCH(1, 1) processes, cross-validation delivers Qcv with reasonable power, performing better than
LM(1), LB(1), and BP(1) in every case, while tcv has low
power.
[We also examined power against several other parameter
values and variance processes including ARCH(4) and
Q
S40
Lag (q)
(a)
100
ioo
-
---
Q-
-..
•
BP and Q
.DA
LB
60
LMand QRED
.
40
20
BDS
LBS
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Lag (q)
(b)
100
80
BP and Q
.Tn
QDA
60
BDS"
"LBS
40
BDS,
20
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Lag (q)
(c)
Figure 4. Empirical Power: Rejection Rates of Tests Under the
GARCH(1, 1) Alternative at the 5% Level. The simulated model is Yt =
+ pht-1, Ct N N(0, 1),
1+ mt + et, where et = 5th1/2, ht =1 + e
w
mt = Amt_1 + vt, A = .8, and vt
N(O, 4). Calculations are based on a
sample size of 128 and 1,000 replications.Rejectionrates are based on
the size-correctedempiricalcriticalvalues for the 5% level. The result
forQcv (forwhichqc is chosen via cross-validationand, therefore,may
change foreach iteration)is includedforcomparisonas a horizontalline.
(a) (0, P) = (.3, .2); (b) (0, 3) = (.5, .2); (c) (0, 3) = (.3, .65)
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Journalof Business & EconomicStatistics,January1999
100
ARCH(12),as well as the Student'st and gamma distributionsfor ?t. The resultsare largelysimilarto those presentedpreviously.For some of these additionalresults,see
Hong and Shehadeh(1996).]
In all of our simulations,the mean of q^increaseswith
the sample size and ARCH persistence.For Figure 2(a)
(a = .3), the mean of q^is 5.8 with a standarddeviation
of 3.9. For n = 512, the mean and standarddeviationare
7.7 and 4.7. For Figure4(c) [(a, 0) = (.3, .65)], the means
and standarddeviationsof q^are 11.1 and 5.0, respectively,
for n = 128, and 17.2 and 3.6 for n = 512. The distributions of q^areskewedleft for weakpersistenceandbecome
increasinglysymmetricunderstrongerpersistence.
Summing up, we can draw the following conclusions
from our simulation:
1. Both QDAN(q) and LB(q) have reasonablesize and
performbetter than the uniform-weighttests for most q.
The QREG(q)test has bettersize performancethan LM(q)
for all q;QTRUN(q)has bettersize performancethanBP(q).
BDS, (q) has poor size performance,overrejectingthe null
hypothesisin most cases.
2. For each q, our test with nonuniform weights,
QDAN(q), has power as good or better than the uniformweightingtest, QTRUN(q),againstall the alternativesunderconsideration.LM(q),BP(q),LB(q),andQREG(q)have
powerssimilarto QTRUN(q).
3. Lee and King's LBS(q) test often has the best power
for q = 1. For large q, however,LBS(q) is dominatedby
all of the other tests, independentof the true alternative.
The supremum-normtest QDAN(q) and BDS1/4(q) have
the best power for persistentARCH alternativessuch as
ARCH(8)but performless well againstGARCHalternatives. For large q, QDAN(q) is often the most powerful.
4. ForpersistentGARCH(1,1) processes,the LM(1)and
LBS(1)tests do not havethe best power.Instead,thesetests
have betterpower when more lags are employed.On the
otherhand,the powerof QDAN(q) is only slightlyaffected
by the truncationlag numberq over ratherwide rangesof
q and often is the greatestof all the tests.
5. For our test Qcv, cross-validationexhibitssome size
overrejectionbut alwaysdeliversgood power.To some extent, cross-validationrevealsinformationaboutthe truealternative.The costs associatedwith misspecificationof an
unknowntrue alternativeappearhigh for all of the other
tests, thoughlower for QDAN(q). As opposedto Qcv, tcv
performspoorlyrelativeto QDAN(q) in each experiment.
5.
EMPIRICALAPPLICATIONS
We now applyour tests to two datasets,the U.S. implicit
pricedeflatorfor grossnationalproduct(GNP)andthe daily
nominalU.S. dollar/Deutschemark
exchangerate.
5.1 U.S. GNP Deflator
We considerinflationas measuredby the log changein
the quarterlyimplicit price deflatorof U.S. GNP, 7lt =
In(GDt/GDti). (These deflatordata are reportedby the
U.S. Bureauof Economic Analysis.) Among others, En-
gle and Kraft(1983) and Bollerslev (1986) examinedthis
series in detail. These researchersreportedthat the conditionalmean E(7rtlt-1) appearswell specifiedby an
autoregressive(AR) model.Standardunivariatetime series
methodsleadto identificationof the followingAR(4)model
for 7rt:
7rt= .148 +
(.076)
.355 7rt-1 + .265
(.090)
(.094)
+
.199
(.093)
Wt-3+
7rt-2
.059
(.088)
rt-4
+ Et
The model is estimatedon quarterlydata from the second quarterof 1952 to the firstquarterof 1984, a total of
128 observations.OLS standarderrorsappearin parentheses andthe R2 = .648.The p valuesof the LM,BP,andLB
tests for serialcorrelationin the meanare .80 or greaterfor
lags between 1 and 20, suggestingan absenceof autocorrelationin the residuals.[Inthe presenceof ARCHeffects,
these standardtests for serialcorrelationin meanwill tend
to overrejectthe null hypothesisof no serial correlation.
This fact does not changethe conclusionhere. But, for an
earlierperiodthe BP,LB, andLM tests for serialcorrelation
in meanareall significantat the 10%level for a lag of 5. As
Bollerslev(1986, p. 323) noted,"[f]romthe late 1940suntil
the mid-1950s the inflation rate was ... hard to predict."Al-
thoughthe heteroscedasticityin this periodis striking,the
mean is difficultto specify. We thereforeomit this period
and begin our analysisat the earliestdate at which serial
correlationin meanis not evidentwiththese tests.]Wenote
thatthe dynamicregressionmodelhereis differentfromthe
static regressionmodels with exogenoustime series variables used in the simulationexperiment.Nevertheless,one
can expect that ARCH tests will performsimilarlyunder
both the regressionmodels with the same ARCH alternative because ARCH tests, based on the squaresof residuals, are not very sensitive to the estimationeffect even
in moderatelysmall samples.The estimationeffect has an
asymptoticallynegligible impact on both size and power
of ARCH tests, whetherthe regressionmodel is static or
dynamic.
We apply our tests, as well as those of Engle (1982),
LB, BP, BDS, and Lee and King (1993) to the estimated
squaredresidualsfrom the precedingregression.In Table
1, we report the p values for the tests at different q. With
use of the Daniell kernel, cross-validation delivers •? = 6.
Our test Qcv = QDAN(gc) = 1.901, giving a p value of
.028 using the asymptotic critical value. Thus, our test suggests that ARCH effects exist at the 5% significance level.
Engle's LM test statistic LM(O•) = 11.00. This gives a p
value of .09; the LM statistic is just significant at the 10%
level. Lee and King's test LBS(O•) = -2.75. This has a p
value equal to .997, delivering an opposite conclusion. This
results from the fact that the sample autocorrelationsof the
squared residuals are negative at some lags, although these
negative autocorrelations are not significant for any given
lag at reasonable levels. Likewise, BDS1/4(4c) = -.12
which fails to reject the null. As can be seen in Table 1,
the choices of q = 4, 8, and 12 deliver significant p val-
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Hong and Shehadeh: A New Test for ARCH Effects
101
Table2. TestStatisticsforARCHEffectsin the Residualsof the
MA(1)Regressionfor the DM/U.S.DollarExchangeRate
Table1. p ValuesFromTests forARCHEffectsin the Residuals
of the AR(4)Regressionof the U.S. GNP Deflator
q
1
4
6
8
12
16
BP
.442
(.439)
.046
(.039)
.104
(.077)
.172
(.101)
.394
(.263)
.544
(.361)
LB
.437
(.439)
.040
(.039)
.091
(.077)
.150
(.106)
.351
(.281)
.479
(.395)
LM
.511
(.515)
.076
(.055)
.088
(.064)
.154
(.104)
.367
(.282)
.340
(.234)
LBS
.253
(.225)
.912
(.863)
.997
(.989)
.999
(.983)
1.000
(.987)
1.000
(.945)
QTRUN QDAN
.614
(.439)
.022
(.039)
.096
(.077)
.187
(.101)
.446
(.263)
.589
(.361)
.547
(.362)
.052
(.048)
.028
(.037)
.032
(.041)
.070
(.062)
.138
(.094)
BDS1/4
q
BP
LB
LM
LBS
QTRUN QDAN
.792
(.832)
.516
(.587)
.905
(.937)
.724
(.768)
.422
(.619)
.007
(.075)
10
20
29
30
40
50
155.50
188.66
191.88
193.69
205.09
208.58
155.92
189.33
192.60
194.44
206.05
209.63
111.45
120.30
125.61
126.30
140.78
144.11
8.11
7.43
6.29
6.32
5.38
4.55
32.53
26.67
21.39
21.13
18.46
15.86
40.44
37.04
34.00
33.62
30.49
27.91
BDS1/4
7.44
9.43
9.72
9.74
9.76
9.87
NOTE: Cross-validation
generatedqc = 29. UnderH0, BP,LB,and LMare Xq;BDS1/4,LBS,
are N(O, 1).
QTRUN,and OQDAN
mated a GARCH(3,3) and appliedthe tests to these new
standardizedsquaredresiduals.The GARCH parameters
=
were significant at any reasonablelevel, and all of the
NOTE:The bootstrapp values appearin parentheses.Cross-validation
generatedqc 6.
tests failed to reject the null of homoscedasticityat any
ues (smallerthan 10%) for QDAN(q), but the choices of conventionallevel.] Beyond this, we see that QDAN(q) is
q = 1 and 16 deliver insignificantp values (largerthan much largerthan Lee and King's LBS(q) and BDS1/4(q).
10%).This suggeststhe importanceof a properchoice of q. Moreover,our QDAN(q)is always larger than the trunWe see thatthe cross-validatedq^gives the smallestp value cated kernel-based test QTRUN(q)for all q, which can be
viewed as a normal approximationof Box and Pierce's
here.
We also performa bootstrapprocedureto bettercompare BP(q) test andhas approximatelythe samepoweras BP(q)
the tests in this empiricalsetting.The bootstrapresultsare and LB(q).
Of special interesthere is the long lag, q^ = 29, chosen
largelysimilarto those using the asymptoticcriticalvalues
andarereportedin parenthesesin Table1. In particular,the by cross-validation.This resultis consistentwith otherevidence of strongpersistencein the conditionalvolatilityof
relativerankingof the tests is unchanged.
we esti- high-frequencydata.We have Qcv = QDAN(qc) = 34.00,
[Toconfirmthe sourceof the heteroscedasticity,
matedan ARCH(3)model, in which only the interceptand significantat any level. In particular,as we allow the samcoefficienton q = 3 were significant.We appliedthe ARCH ple size to grow, q, grows as well. For sample sizes of
tests to the new standardizedsquaredresidualsand failed n = 2m with m = 6,7,..., 11 and a common start date,
to reject the null of homoscedasticityat reasonablelevels cross-validationyieldedq^= 1, 12, 8, 11, 19, and29, respecfor q < 20. To check robustnessof the obtainedresults,we tively.Thistrendis consistentwith ourrequirementsfor the
appliedthe tests to some later samples of the same size. theoreticalresults.
The conclusionsremainlargelyunchanged.]
Again, we performeda bootstrapof each test. The conclusion using our test is the same based on the boot5.2 Exchange-RateData
strapcriticalvalues.In threeothercases, the conclusionis
We now considera high-frequencyseries,the dailynomi- changed.For lags greaterthan20, the LM(q)and QREG(q)
nal Deutschemark/U.S.dollarexchangerate.The data,ob- tests arenow significantonly at the 5%level. Withthe boottained from Chicago MercantileExchange,cover the pe- strapcriticalvalues, LBS(q) is significantat the 1%level
riod June 8, 1983, to July 24, 1991-2,048 observations. for q < 16, at the 5% level for q < 20, and at the 10%
As noted by Bollerslev et al. (1994), most researchers level for q < 25 and q > 35. This is consistentwith the
have reacheda consensusthat nominalexchangerates are U-shapedsize curvewe observedfor LBS(q)in the Monte
modeledbest as integratedprocesses.We thereforemodel Carlosection.
st = 100 I1n(DMt/DMt_1)as a first-ordermoving average [MA(1)], where DMt is the spot exchange rate on
day t. Quasi-maximum likelihood estimation of this process
yields
6.
CONCLUSIONS
We propose a new ARCH test based on the sum of
weighted squares of the sample autocorrelations, with the
weights depending on a kernel function. Typically, the
st = .0190 - .0149
et-1 iet.+
kernel function gives greater weight to lower-order lags.
(.0149)
(.0210)
When a relatively large lag is employed, Engle's (1982)
We note that neither the constant nor the MA(1) parameter LM test, as well as the tests of Box and Pierce (1970)
is significant, consistent with the integration hypothesis.
and Ljung and Box (1978), is equivalent to the trunWe test for ARCH effects as in the previous example cated kernel-based test that imposes a uniform weighting
using the estimated squared residuals. A summary of our scheme and is less efficient than nonuniform kernel-based
results appears in Table 2. All of the tests in Table 2 are tests. Our method permits choice of an appropriate lag
significant at any reasonable level for q = 10, 20,..., 50 us- via data-drivenmethods-for example, Beltrio and Blooming the asymptotic critical values. Therefore, all the tests field's (1987) cross-validation. Simulation studies show that
under consideration deliver the same conclusion. [Rejec- the new test performs reasonably well in finite samples.
tion here is not surprising given the sample size. We esti- The test also is applied in two empirical examples. These
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102
Journalof Business & EconomicStatistics,January1999
n-1
empiricalexamples illustratesome merits of the present
test.
Z k2(j/q)-2(j)
Some directionsfor furtherresearchmay be pursued.As
j=1
n-1
is well known, it is importantto check the adequacyof
= k2(j/q)2(j)/r2
an estimatedGARCHmodel. For example,misspecifica(0)
tion of a GARCH-Mmodel will lead to inconsistentquasi
j=1
n-1
maximumlikelihoodestimation.Our approachcan be ex_
+ {f-2(0)tendedto checkadequacyof a GARCHmodel.If the model
T2((0)}
k2(j/q)r2(j)
is correctlyspecified,then the standardizedsquaredresidj=1
n-1
ual is a white-noiseprocess;otherwise,thereis evidenceof
+ -2(0) k2(j/q)2(j) - f2(j)}. (A.1)
misspecificationfor GARCH.We conjecturethat the proj=1
posed test also will be asymptoticallyN(O,1) undercorrect
specificationof the GARCH,but the technicalityinvolved
(q/n)
appearsnontrivial.These considerationsare left for subse- By Markov'sinequalityand n-1k2(j/q)Ef2(j)
where
quentwork.
r2(0)
{q-1E k2(j/q)}= O(q/n),
q- j:lnk2
= Op
(j/q) --+ f k2(z) dz, we have •?1k2(j/q)i2(j)
ACKNOWLEDGMENTS
(q/n). Moreover,i(0) - r(0) = Op(n-1/2). It follows that
We thanka coeditor,an associateeditor,three referees, the secondtermin (A.1) is Op(q/n3/2) = op(ql/2/n) given
and seminarparticipantsat CornellUniversityfor helpful q/n -+ 0. The last termis also op(ql/2/n) by LemmaA.1.
commentsandsuggestions.Any remainingerrorsare solely Therefore,we have
attributable
to the authors.We also thankJau-LianJengfor n--1
us
providing with data.
(j)
Sk2(j/q)2
APPENDIX:MATHEMATICAL
PROOFS
j=1
n-1
For rigor and completeness,we state explicitlythe con= S
(A.2)
k2(j/q)i2(j)/r2(0) + op(ql/2/n).
ditionsfor the theorems.
j=1
=
is
with
AssumptionA.1. (a) {(t} iid,
E(?t) 0, E(t2) =
1, and E(t8) < 00. (b) ?t is independentof X, for s By Hong (1996, theoremA.2), AssumptionsA.1 and A.4,
< t.
and q -+ oco,q/n -+ 0, we have
R1, where
AssumptionA.2. (a) For each b E B
n-1
1 E N, g(., b) is a measurablefunction, and (b) g(Xt,.)
2
- Cn
n )ik2(j/q)
is twice differentiablewith respect to b in an open con(k)
(j)/r2(0)
vex neighborhoodN(bo) of bo E I, with lim,,, n-1 Et1
ESUPbEN(b,) IVbg(Xt,b)ll4 < 00 and limno n-1 Etn=
+ (2Dn(k))1/2 _d N(0, 1). (A.3)
EsupbEN(bo) IVg(Xt, b)!2 < 00, where Vb and V2 are
the gradientand Hessianoperators,respectively.
Both (A.2) and (A.3), togetherwith q-1Dn(k) -+ fOO
k4
Assumption A.3. n1/2(b - bo) = Op(l).
AssumptionA.4. k: R - [-1, 1] is a symmetricfunction (z) dz, implythatQ(q) _+dN(0, 1). The proof will be comcontinuousat 0 and all but a finite numberof points,with pleted providedLemmaA.1 is proven.
Proof of LemmaA.1. Noting ri2(j) - 2(j) = (?(j) k(0) = 1 and fo k2(u)du < 00.
<
A.5.
For
IR,
f(j))2 + 2f?(j)(^(j)- f(j)), we write
Assumption
any z1,z2 E !k(zi) k(z2)l
- z21 for some 0 < A < oo andIk(z)l? Alzl-' for
Alzl
n-1
all z gR
E and some 7 > .
5 k2(j/q){C2(j)
. - i2(j)}
Theorem1. Suppose AssumptionsA.1-A.4 hold. Let
j=l
q
-
oo, q/n -+ O.Then under Ho,
n--1
=
Q(q) _d N(0, 1).
To proveTheorem1, we first state a lemma.
Lemma A.1. For j _ 0, define f(j) = n-1
1 (2_j - 1). Then
+ 2
-=j+l(s2 _
n-1
Sk2(j/q){2(j
j=1
_ 2(j)} =
5
j=l
op(ql/2/n).
Proof of Theorem 1. Put r(j) = Ef(j). We can write
k2(j/q)(?(j) - e(j))2
S
j=l
k2(j/q)i?(j)(?(j) - i(j)).
(A.4)
We now considerthe firsttermof (A.4).By straightforward
algebra,we have
i(j) - i(j)
t=j+l
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103
Hongand Shehadeh:A New Test forARCHEffects
given A.1-A.4, where q-1 n-1 k2(j/q)
and n-1 E• 1 IIVbg(Xt,b)114= Op(1).
n
j)
(t2 - 1)( t-j -_t-
n-1
k2(z) dz
Next, noting t - Et = (bo - b)'Vbg(Xt, b) +
b)'V g(Xt, b)(bo b) for some b, we have
t=j+1
n
t=j+l
(bo
-
n-1
^2
( (_?2)(
+-1
f
-+
5
2j
k2(j/q)22(j)
j=1
t=j+1
= At (j) + A2(j)
(A.5)
say.
A3(j),
-1
=
-
utEt-(t-j
k2(j/q)
j=1
t-j)
t=j+l
n-1
Recall that it = it/&7n, t = et/co, and ut =-2
have
5
- 1. We
- j112 k2(j/q)
< 411bo
j=1
2
n
2-1
1+lj
(/)t2- 1•2
x n-1
• j
E
t=ij+l
utEt-jVbg(Xt-j,bo)
n--1
t=j+l
Z
- b114
+ 2 jlbo
.(_1W)
tE an
0-1
E-2
k2(j/q)
S2
j=1
t=j+l
+ ( .2
t=
X 7-1 u•uEt_ g(Xt-,b)
3t-j3.
(o').-1
-
t=j+l
t=j+
n
2
n-(-1)t=j+l
A,
(A.8)
= Op(q/n2)
j)
_•-_j
by Markov'sinequalityand Elln-1ZEnj+1Utt-jVbg
t=j+l
(Xt-j,Ibo)ll2
+ (& 0•o)n-1
UtEt-j•
n
= O(n-1) given E(utlLt-1) = 0, where Lt-1
is the informationset availableat periodt -1. We also have
/(2
t=j+l
+ (
1(j) + 222B12(j)
-2 _
=
ao2)!Bl3(j), say.
(A.6)
t=ij+
j=1
j=1
-2
=
UtEt-j
n-1
2(jq)
=
Sk2(j/q)B3(i)
(A.9)
Op(q/n)
by Markov's inequality and E(n-1 E
j+l UtE•j)2
O(n-1) by AssumptionA.1.
=
By the Cauchy-Schwarz inequality and noting ?t - et
jb- bol <IlIb- bo ,
(bo - b)'Vbg(Xt,b) for some b, where
we have
Combining (A.6)-(A.9) with an
Op(n-1/2),
2
we
obtain
n-1
Sk2(j/q)A(j)
= Op(q/n2).
(A.10)
j=1
7n-1
j=1
Similarly, we have
Sk2 k21(j)
=-1
n--1
5 k2(j/q)A
0/n_-+
<^bo211
We now turn to Ij=l-n 1k2(j/q)A(j). By the CauchySchwarz inequality and noting t - t = a•2(?t - t) +
(&2 - -2)et,
we have
2u(-1
k2 J
t=l
j=1
x
n-1l
= Op(q/n2),
I
(A.11)
j=1
t=j+l
j=1
(j) = Op(q/n2).
n
sup la3(j)12
_n-l
I
Vbg(Xt, b)
l<j<n-1
4
(A.7)
= •Zn-1
t)2
t=l
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(I
-
(• - )2
t=l
104
Journalof Business & EconomicStatistics,January1999
n
-2)2 n-1 E
+ 2(-2
Proof of Lemma A.2. Let integer d = [q"], where [x]
denotes the integer part of x. Then d/n
d/q -- co.
O0,
4
t=1
Write
= Op(n-2)
d
given AssumptionsA.1-A.3. It follows that
Cn((k) = Cni(k)+ E
Sk2(j/q) A3(j) = Op(q/n2).
k2(j/q)}
n-1
(A.12)
+
j=1
n-1
n-1
- f(j))2 = Op(q/n2).
k2(j/q)((J)
((1-j/n)k2(j/l)
j=d+l
Hence, from (A.5) and (A.10)-(A.12),we have
E
j=1
-
(1 - j/n){k2(j/l)
j=1
n-1
(A.13)
S
(1-j/n)k2(j/q)
j=d+l
= Cn (k) + C
+ 02n - 3n, say.
(A.15)
= OF(q/n),
(j/q)
2(j) =
- j=l k2(jq)f2(j)
On the otherhand,becauseEn-1
Op(q/n),
We show that the last three terms are op(ql/2). First, we
we have
considerC2n. Given jk(z)j < Alzl-r in AssumptionA.5
n-1
and O/q--+P0, we have
- (j))f(j) = op(ql/2/n)
(A.14)
E k2(j/q)((J)(
j=1
n-1
?
by the Cauchy-Schwarzinequality,(A.13), and q/n -+ 0.
The lemma then follows from (A.4), (A.13)-(A.14), and
j-27
IC2Tn[ A2q2r(/q)27
q/n -+ 0. This completes the proof.
=
j=d+l
Op(q2r/d2T-1)
= Op(ql/2),
(A.16)
Theorem2. Suppose AssumptionsA.1-A.5 hold. Let
where the last equality follows from d = [q"]for v > (27 q be a data-dependentbandwidthsuch that q/q - 1 =
- 1). Similarly,
where *)/(27
op(q-((3/2)v-1))
q -- oo,q'q/n
--
for some v > (27
)/(2,r- 1),
0. Then, under H0, Q(q) - Q(q) = op(l),
and
C3n
= o(qi/2).
(A.17)
For the secondtermin (A.15), we decompose
Q() __d N(0, 1).
To show Theorem2, we first state two lemmas.
d
and
Lemma A.2. Let Cn(k) = E=(1
j/n)k2(j/q)
- j/n)(1 - (j + 1)/n)k4(j/^).
Then
D(k) = EI(1
and q-iDn(k) =
q-lCn(k) = q-Cn(k) + op(q-1/2),
+
op(l).
q-'1D(k)
Lemma A.3. EZl {k2(j/q) - k2(j/q)2(j)
= Op
Cln
<
{k(j/) )-
k(j/q)}2
j=1
d
J{k(j/l) - k(j/q)}k(j/q)l.
+ 2
j=1
Given the Lipschitzconditionon k (in AssumptionA.5),
of
we
can
From
the
definition
2.
Theorem
=
- =
Q(q),
Proofof
q/q 1 op(q-((3/2)v-1)), and d [qv], we have
write
(q1/2/nr).
d
Q(q)-Q(q)
=
E {k(j/q)- k(j/q)}2
j=i
-- Cn(k)}
{Dn(k)}-1/2{Cn(k)
+
Q(q){(Dn(k)/Dn(k))1/2
+ {Dni(k)}-1/2
n
-
d
< A2-2(/q)2(t/q
1}
(k2(j/O) - k2(i/q))2(J)
--
1)2 5j2
= Op(1).
j=i
}.
Here, the first term vanishes in probability by Lemma A.2
and {D,(k)}-1/2
= Op(q-1/2), the latter follows from
Lemma A.2, and q-iD,(k) -- fo k4(z) dz. Next, the second term vanishes in probability because D! (k) /D, (k) --+P
1 by Lemma A.2 and Q(q) = Op(l) by Theorem 1. Finally,
the last term vanishes in probability by Lemma A.3 and
=
{Djn(k)}-1/2 = Op(q-1/2). Consequently, Q(O)- Q(q)
so
Q(j) --+dN(0, 1) given Q(q) _+dN(0, 1) by TheOp(l),
orem 1. The proof is completed provided Lemmas A.2-A.3
are proven.
This, together with Ed=l
k2(j/q) = O(q) and the Cauchy-
Schwarz inequality, implies
Z
{k(j/4)-
j=1
k(j/q)}k(j/q)-
op(ql/2).
It follows that
Cl,
= op(ql/2).
(A.18)
Collecting (A.15)-(A.18) yields q-l(C'n(k) = q-Cn(k)+
(k) = qiD (k) + op(1) can
op(q-1/2). The result q-1
be shown analogously.
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Hongand Shehadeh:A New Test forARCHEffects
105
Proof of Lemma A.3. Recalling the definition of f(j) in
have
LemmaA.1, we can write
1/2
d
n-1
(D11)1/2
JD12nj
1
E {k2(j/q) - k2(j/q)}P2(j)
j=1
k2(j/q)f2(j)
j=1
n-1
(A.25)
op(q1/2/n).
{k(2(j/) - k2(j/q)} 2(j)
=-2(0)
It follows from (A.23)-(A.25)that
j=1
n-1
+
Din = op(q1/2/n).
{k2(j/l) - k2(j/q)}
-2(0) E
(A.26)
j=1
x {12(j)
_
(A.19)
2(j)}.
For the firsttermof (A.19), we have
Combining(A.20)-(A.22) and (A.26), we obtain that, for
the firsttermof (A.19),
n-1
n-1
S {k2(j/q) - k2(j/q)}f2(j) = op(q1/2/n).
{k2(j/l) - k2(j/q)} 2(j)
E
j=1
n-1
d
=
I{k(2(j/4)
-
>
k2(j/q)} 2(j) +
k2(j/q)f2(j)
j=d+l
j=1
n-1
Next, we considerthe second term of (A.19). Because
_ 2(j)} = op(ql/2/n) by Lemma
j= k2(j/q){2(
_
f2(j)}
A.1, it suffices to show --•1k2(j/){ri2(j)
op(ql/2/n). Put
- E3k2(j/q)i2(j)
n-1
j=d+l
= Din + D2n - D3n, say.
A2q27r(/q)27
Wn = (n/q1/2) E k2(j/q)lr2(j)
(A.20)
WefirstconsiderD2n. GivenIk(z)l< Alzj-', we have
n-i
D2n
(A.27)
j=1
-
2(j)[.
j=1
We shall show WI,= op(l). For any given q > 0, 6 > 0,
<
P(Wtn> q) P(WTr,> rl,Iq/q- 11< 6) +P(li q 11> 6),
-2()}j=dl
j=d+1
(A.21) where the last probabilityconvergesto 0 as n -+ oc, given
l/q - 1 = op(1). Thus, it remainsto show that the first
givenq/q -+P1 andd = [q"]for v > (27--)/(27 - probabilityalso vanishes.Put
(j) = Op(dl-2r/n) by Markov's
1), where E -jf-27
if I < zo
=
1
and
Ef2(j)
inequality
O(n-1). Similarly,we have
k(z) = { Ajzj- if 1z, > zo,
= op(ql/2/n),
D3n
= op(ql/2/n).
(A.22)
Next, we considerthe firsttermin (A.20). We write
d
Din
{k(j/l) - k(j/q)}2f2(j)
-
d
{k(j/)
Then [k(z)l k(z), and k(z) is monotonicallydecreasing
on IR+ = [0,00]; that is, k(zi) > k(z2) if 0 < z1 < Z2.
Furthermore, we have fo k2(z) dz < 00, given 7 > 2. It
follows that [k(j/q^)l
k(j/^) <_ k{j/(1 - 6)q} if
2>
(1- 6)q.Whence,the firstprobabilitywill vanishas n -+ o00
j=1
+ 2
where 0 < zo < oc and A, 7 are given in Assumption A.5.
- k(j/q)}k(j/q)f2(j)
if
j=1
n-1
= Dlln + 2D12n, say.
(A.23)
Given the Lipschitz condition on k (Assumption A.5), we
have
D11, < A2q-2q(q/c)2( /q - 1)2
j2r2(j)
j=1
= A2 -2op(q-(3v-2))Op(d3/n)
=
Op(n-1),
(n/(1
- 6)q1/2)
x
j=1
c2{j/(1 - 6)q}li2(j)
_
•2(J)l = Op(1)
for any given small 5 > 0, which follows by a reasoning
analogous to the proof of Lemma A.1 and the fact that
k
q-1 Ej=I k2(j/q)
2(Z) dZ < 00 as q -+ o00. Therefore, we have Wn = op(1), or equivalently,
n-1
(A.24)
where c/q - 1 = op(q-((3/2)V-1)) and Zj]lJ2f2(J)
Op(d3/n) by Markov's inequality. Next, by the CauchySchwarzinequalityand E
k2(J/q) 2(j) = Op(q), we
> k2(j/q)r2(j)
-
i2(j)[ = op(ql/2/n).
(A.28)
j=1
Combining (A.19), (A.27)-(A.28), and Lemma A.1 then
yields the desired result.
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Journalof Business & EconomicStatistics,January1999
106
Theorem3. Suppose AssumptionsA.1-A.3 hold. Let For the firstterm of (A.30), we have
- QREG(q)=
q -+ oo, q5/n -+ 0. ThenunderHo, QTRUN(q)
q
op(1) and
02 n-1
Bl, = sup
(AQREG(q)-+d N(0,1).
t=l
Oeeq j=l
Proof of Theorem3. Put Z = (Z1,I...I, )', Zt =
n
(
^2
(1,
2/2
7
6_tq)
_1,...,
= (^2/^2
6
( 12/ 1,...,O-
1•
2/n
Rq = (R(1),.. .,R(q))', where R(j)
n- Et+
=
1
- 1), and
n
4)
_ n11
t
(-4_j_4)
n
t=j+1
- 3). Then by definition(e.g., see Engle 1982),
an)(2_j
2
we have
0 ( n-1
sup
OEeq j=1
q
R2= U'Z(Z'Z)-1Z'U/(U'U).
n
5
t=n-j+l
By straightforward
algebra,we haveR2 = (nR2(0))2RA'-1
Rq,,whereA is a q x q symmetricmatrixwith the (i, j) element
2
-n
/2
+ j)(n -- 3
n
-
Aij
_ -2
2)
(^2
• _-
E
t)
S
qn
1
t=max(i,j)+
3
and 3 =n-1 E
2. BecausenE=l2
R22(j)R (0), it follows that
<supj=11 .0 n-1t=n-j+l4
Eeq
=1
()=n
q
(&+2
Sup
q
nR2 -n
8EEqj=l
P2(j)
(
-1 5
-
n
t=j+l
j=1
=
nri-2(0)(R(O)RIA-lRq
1-
< n-
Rq)
(A.29)
= nR2[r-2(0)Oq(R(0)Iq
- A)Oq,
is a q x 1 unit vector in Eq =
where , = A-1/2q/V
{0 E R•:0'0-=
n
1}.
t=n-q+1
= sup
ISnl
n
2)6
(
n-1
OEeq i=1 j=1
t=1
q j-1
E
(
2
2
-
T
Gr
<
5
GEeq
q
j=1
n
-1
q
j-1
n
j=2
i=1
t=j+l
-2)(-2-j
(-i
E
i=1
q
(n
(A.31)
t=j+1
E
nz-1 n1 C-i-j
O,
t=max(i,j)+1
sup
12
t=n-q+l
j=2
n
-2
2
For the secondtermof (A.30), we have
B2n <
q
-4 4
E
= Op(q/n).
Put Sn =supoeO'(i(O)Iq- A)O.We first show Sn =
Op(q2/n1/2). For this, we write
q
n
t +23n&n-• t
5
1
(?2&2)2
n
j-1
--1 - 2_ -
i=1
j=2
t=j+1
t=l
q
j-1
j=2
i=1
+
n
n-
t=j+1 E
(t-i - ct- )
-
t=j+1
+2 sup
Z
Geeq
j=2
?•xi
--)•
Zn-1
i=1
3
= B1, + 2B2,, say.
q
j-1
n
j=2
i=1
t=j+1
j=2
i=1
t=j+1
t=j+l
-
),
(A.30)
= CI
+ C2• + C3n + C4n = Op(q2/nZ/2).
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(A.32)
Hong and Shehadeh:A New Test forARCHEffects
107
Here,
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