diagnostic checking for the adequacy of nonlinear time series models

57 Pages·2003·0.7 MB·English

EconometricTheory,19,2003,1065–1121+PrintedintheUnitedStatesofAmerica+
DOI:10+10170S0266466603196089
DIAGNOSTIC CHECKING FOR
THE ADEQUACY OF NONLINEAR
TIME SERIES MODELS
YOOONNNGGGMMMIIIAAAOOO HOOONNNGGG
Cornell University
TAAAEEE-HWWWYYY LEEEEEE
University of California, Riverside
We propose a new diagnostic test for linear and nonlinear time series models,
using a generalized spectral approach+ Under a wide class of time series models
thatincludesautoregressiveconditionalheteroskedasticity~ARCH!andautoregres-
sive conditional duration ~ACD! models, the proposed test enjoys the appealing
“nuisance-parameter-free”propertyinthesensethatmodelparameterestimation
uncertainty has no impact on the limit distribution of the test statistic+ It is con-
sistentagainstanytypeofpairwiseserialdependenceinthemodelstandardized
residuals and allows the choice of a proper lag order via data-driven methods+
Moreover, the new test is asymptotically more efficient than the correlation
integral–based test of Brock, Hsieh, and LeBaron ~1991, Nonlinear Dynamics,
Chaos, and Instability: Statistical Theory and Economic Evidence! and Brock,
Dechert, Scheinkman, and LeBaron ~1996, Econometric Reviews 15, 197–235!,
the well-known BDS test, against a class of plausible local alternatives ~not in-
cludingARCH!+Asimulation study compares the finite-sample performance of
the proposed test and the tests of BDS, Box and Pierce ~1970, Journal of the
American Statistical Association 65, 1509–1527!, Ljung and Box ~1978, Bio-
metrika65,297–303!,McLeodandLi~1983,JournalofTimeSeriesAnalysis4,
269–273!,andLiandMak~1994,JournalofTimeSeriesAnalysis15,627–636!+
The new test has good power against a wide variety of stochastic and chaotic
alternatives to the null models for conditional mean and conditional variance+ It
can play a valuable role in evaluating adequacy of linear and nonlinear time se-
riesmodels+AnempiricalapplicationtothedailyS&P500priceindexhighlights
themeritsofourapproach+
Wethanktheco-editor~DonAndrews!andtworefereesforcarefulandconstructivecommentsthathaveleadto
significantimprovementoveranearlierversion+WealsothankC+W+J+Granger,D+Tjøstheim,andZ+Xiaofor
helpfulcomments+Hong’sparticipationissupportedbytheNationalScienceFoundationviaNSFgrantSES–
0111769+LeethankstheUCRAcademicSenateforresearchsupport+Addresscorrespondenceto:Yongmiao
Hong,DepartmentofEconomicsandDepartmentofStatisticalScience,CornellUniversity,Ithaca,NY14853-
7601,USA;e-mail:yh20@cornell+edu;ortoTae-HwyLee,DepartmentofEconomics,UniversityofCalifornia,
Riverside,Riverside,CA92521-0427,USA;e-mail:tae+lee@ucr+edu+
© 2003 Cambridge University Press 0266-4666003 $12+00 1065
1066 YONGMIAO HONG AND TAE-HWY LEE
1. INTRODUCTION
The development of nonlinear time series analysis has been advancing rapidly
~e+g+, Subba Rao and Gabr, 1984; Priestley, 1988; Tong, 1990; Brock, Hsieh,
andLeBaron,1991;GrangerandTeräsvirta,1993;Tjøstheim,1994;Teräsvirta,
Tjøstheim, and Granger, 1994; Terdik, 1999!+ In modern time series analysis,
one often considers the data generating process
Yt 5g0~It21!1h0~It21!«t, t50,61,+++, (1.1)
where It21 is the information set available at time t21 and $«t% is a sequence
of independently and identically distributed ~i+i+d+! innovations+ Often, $« %
t
has mean 0 and variance 1 such that g0~It21! 5 E~Yt6It21! and h02~It21! 5
var~Yt6It21! almostsurely~a.s.!.Forvariousexamplesthatbelongtoclass~1+1!,
see, e+g+, Tong ~1990! and Granger and Teräsvirta ~1993!+ It is also possible
that $« % is an i+i+d+ nonnegative sequence with E~« ! 51+ An example is the
t t
autoregressive conditional duration ~ACD! process introduced in Engle and
Russell ~1998! for irregularly spaced time series, where g0~It21! 5 0 and
h0~It21!5E~Yt6It21! is the conditional duration of the nonnegative time dura-
tion process $Y%+
t
Various models for g ~{! and h ~{! have been proposed in the literature+
0 0
Consider a model
Yt 5g~It21,u!1h~It21,u!et~u!, (1.2)
where g~{,u! and h~{,u! are some known parametric specifications for g ~{!
0
andh ~{!,u[Qisanunknownfinite-dimensionalparametervector,and$e~u!%
0 t
isanunobservableseries+Specification~1+2!coversmostcommonlyusedlinear
and nonlinear time series models+ Examples includeACD, autoregressive con-
ditional heteroskedasticity ~ARCH!, autoregressive moving average ~ARMA!,
bilinear, nonlinear moving average, Markov regime-switching, smooth transi-
tion,exponential,andthresholdautoregressivemodels+Wheng~{,u!andh~{,u!
are correctly specified for g ~{! and h ~{!, i+e+, when there exists some u [ Q
0 0 0
such that g~{,u ! 5 g ~{! and h~{,u ! 5 h ~{! a.s., the model standardized
0 0 0 0
errorseries$e~u !% coincideswiththetrueinnovationseries$« % andtherefore
t 0 t
is i+i+d+ In contrast, if g~{,u! is inadequate for g ~{! and0or h~{,u! is inade-
0
quate for h ~{!, i+e+, if there exists no u [ Q such that g~{,u!5g ~{! and0or
0 0
h~{,u! 5 h ~{! a+s+, $e~u!% will be serially dependent for all u [ Q+ Conse-
0 t
quently, to test adequacy of model ~1+2!, one can check whether there exists
some u [ Q such that $e~u !% is i+i+d+ Tests for i+i+d+ rather than for white
0 t 0
noisearemoresuitableandusefulinnonlineartimeseriesanalysis,inparticu-
lar when higher order conditional moments or the conditional probability dis-
tribution are of interest ~e+g+, Brock et al+, 1991; Brock, Dechert, Scheinkman,
and LeBaron, 1996, Christoffersen, 1998; Diebold, Gunther, and Tay, 1998;
Kim, Shephard, and Chib, 1998; Clement and Smith, 2000; Elerian, Chib, and
Shephard, 2001!+ See also an empirical application in Section 8+
DIAGNOSTIC CHECKING FOR NONLINEAR TIME SERIES MODELS 1067
Because $e~u !% is unobservable, one has to use the standardized estimated
t 0
residual
e[t 5@Yt2g~IZt21,uZ!#0h~IZt21,uZ!, t51,+++,n, (1.3)
where uZ is a consistent estimator of u0 based on a random sample $Yt%tn51 of
size n and IZ is the observed information set available at period t that may in-
t
volve certain initial values+ To construct an asymptotically valid test proce-
dure,itisimportanttoexaminewhetherandhowtheuseof$e[t%tn51ratherthan
$et[et~u0!%tn51affectsthelimitdistributionofateststatistic,besidesitspower
property ~seeTjøstheim, 1996!+
Often, the information set It21 consists of lagged variables $Yt2j,j . 0%+
When g0~It21! is linear in It21, Yt is called linear in conditional mean on It21
~Lee,White,andGranger,1993!+Ifinadditionh0~It21!5sa.s.,$Yt% iscalled
completely linear in It21 ~Granger, 2001; Granger and Lee, 1999!+ Assuming
h0~It21! 5 s a+s+ in an ARMA framework, Box and Pierce ~1970! and Ljung
and Box ~1978! propose a diagnostic test for anARMA~p ,q ! model:
0 0
p
BPL~p!5n~n12!j(51~n2j!21r[2~j! d&&xp22~p01q0!, p.p01q0,
(1.4)
where r[~j! is the sample autocorrelation function of $e[t%tn51, e[t 5 Yt 2
g~IZt21,uZ!, and g~IZt21,uZ! is an estimated ARMA~p0,q0! model+ Here, $e[t% is
the usual estimated residuals because no conditional variance estimation is
involved+The degrees of freedom of the Box–Pierce–Ljung~BPL! test depend
on p 1q , the number of the estimated parameters+ Hong ~1996! and Paparo-
0 0
ditis ~2000a, 2000b! propose spectrum-based diagnostic tests that generalize
the BPL~p! test+
It has been pointed out for a long time ~e+g+, Granger and Anderson, 1978;
Granger, 1983! that the Box–Pierce–Ljung test has no power against nonlinear
dependencies with zero autocorrelation, such as some bilinear and nonlinear
moving-average~MA!processes+Usingthesampleautocorrelationfunctionof
squared residuals, McLeod and Li ~1983! suggest a test for linearity against
unspecified nonlinearity+ For a nullARMA~p ,q ! model, the McLeod and Li
0 0
~1983! test statistic is
p
ML~p!5n~n12!(~n2j!21r[22~j! d&&xp2, (1.5)
j51
where r[2~j! is the sample autocorrelation function of $e[t2%tn51 and e[t 5 Yt 2
g~IZt21,uZ!+ This test has good power against ARCH+ It also has power against
departures from linearity that have apparent ARCH structures+ The null limit
distributionoftheteststatisticisax2distribution;thedegreesoffreedomneed
p
not be adjusted when only an ARMA model is estimated+ As pointed out in
1068 YONGMIAO HONG AND TAE-HWY LEE
Granger and Teräsvirta ~1993!, ML~p! is asymptotically equivalent to the La-
grange multiplier test forARCH of Engle ~1982!+
When testing the adequacy of anARCH0generalized autoregressive moving
average ~GARCH! model, many researchers have applied the McLeod–Li test
tothesquaresoftheestimatedstandardizedresiduals+1LiandMak~1994!show
thatthisprocedureismisleadingbecauseitsasymptoticdistributionisnotax2
distribution if ML~p! is applied to the residuals standardized by estimated
ARCH0GARCH models+ Li and Mak ~1994! propose corrected statistics+ In
fact, their test is asymptotically equivalent to the Lagrange multiplier test pre-
sented in Lundbergh and Teräsvirta ~1998!, which is a test of the standardized
errors being i+i+d+ against the alternative that they follow anARCH model+ Li
and Mak ~1994! also provide a simpler statistic when the fitted conditional
variance model is ARCH~r!+ For a null ARMA~p ,q !-ARCH~r! model, the
0 0
simpler version of Li and Mak’s test statistic can be written as
p
LM~p,r!5n~n12! ( ~n2j!21r[22~j! d&&xp22r, p.r, (1.6)
j5r11
where r[2~j! is the sample autocorrelation function of $e[t2%tn51, e[t 5
@Yt 2 g~IZt21,uZ!#0h~IZt21,uZ!, and h~IZt21,uZ! is an estimated ARCH~r! model+
The null limit distribution depends on r, the order of theARCH model+ When
other conditional variance models are estimated, the test statistic itself has to
be modified as suggested by Li and Mak ~1994! or by Lundbergh and Teräs-
virta ~2002!+
Brock et al+ ~1991, 1996! propose a diagnostic test for model ~1+2!, using
chaos theory:
BDS~m,d!5n102@CZ ~d!2CZ ~d!m#0VZ 102, (1.7)
m 1 m
where the sample correlation integral ~cf+ Grassberger and Procaccia, 1983!
2 n t21 m21
CZm~d!5 n~nF21! t5(m11s(5m j)50 1~6e[t2Gj2e[s2j6,d!
m21
p&&P ) 1~6et2j2es2j6,d! [Cm~d!, (1.8)
j50
1~{! is an indicator function, m is the so-called embedding dimension, d is a
distance parameter, and VZ is an asymptotic variance estimator+ The statistic
m
CZm~d! measures the fraction of pairs of histories$e[t2j,e[s2j%jm5201 that are within
distance d of each other+ If e[ and e[ are close in value, so will be subsequent
t s
pairs for a chaotic process but not for an i+i+d+ sequence+ Thus, BDS~m,d! is
expectedtohavegoodpoweragainstchaos+Inaddition,italsohaspoweragainst
a wide range of stochastic dependent processes+To see this, observe that when
$e % is i+i+d+, we have
t
DIAGNOSTIC CHECKING FOR NONLINEAR TIME SERIES MODELS 1069
C ~d!5C ~d!m (1.9)
m 1
for all positive integers m and all distances d . 0+ In other words, the correla-
tion integral C ~d! behaves like the characteristic function of a serial string in
m
the sense that the correlation integral of a serial string is the product of corre-
lation integrals of component substrings+ If C ~d!(cid:222)C ~d!m, there is evidence
m 1
against the i+i+d+ hypothesis, and BDS will gain power+2
As shown in Brock et al+ ~1991, Ch+ 2 andAppendix D!, BDS~m,d! has the
appealing “nuisance-parameter-free” property that any n102-consistent param-
eter estimator uZ has no impact on its null limit distribution under a class of
conditionalmeanmodelsg~{,u!+This,togetherwithgoodpoweragainstawide
rangeofdependentalternatives,hasmadeBDS~m,d!aconvenientandpower-
ful diagnostic tool for nonlinear time series models+ It has been recommended
byBrocketal+~1991!asaportmanteaulackoffittestfornonlineartimeseries
models in the same spirit as Box and Jenkins ~1970, p+ 29! recommend Box–
Pierce–Ljung’s test for linear time series models+
Nevertheless, BDS~m,d! has certain features one might consider undesir-
able+ First, the “nuisance-parameter-free” property holds only under condi-
tional mean models but not under ARCH models ~cf+ Brock et al+, 1991,
Appendix D!+ More generally, when conditional variance estimation is in-
volved, the limit distribution of BDS~m,d! depends on the nature of estimator
uZ, and how to modify the test statistic is unknown+3 This is troublesome in
practice+ Second, although serial independence implies ~1+8!, the converse is
not true ~Brock et al+, 1991, p+ 47!+ There are examples in which $e % is not
t
i+i+d+ but ~1+8! holds+ For such alternatives, BDS~m,d! may have no power+
Also,BDS~m,d!involvesthechoiceoftwoparameters—mandd+Bothmand
d are fixed but arbitrary+ Because m21 is actually the largest lag order used,
BDS~m,d! has no power against alternatives for which serial dependence in
$e % occurs only at the lag orders equal to or larger than m+ Ideally, a proper
t
choiceofmshoulddependonthealternative,which,however,isunknownwhen
serialdependenceof$e % isofunknownform+Similarly,somechoiceofdmay
t
render BDS~m,d! inconsistent against certain alternatives+ There exists no
rule guided by chaos theory for choosing parameters m and d, although Brock
et al+ ~1991! have recommended a simple rule of thumb based on their simula-
tionstudy+Moreover,asshowninSection5,BDS~m,d!hassuboptimalpower
against some local alternatives+ For example, it can detect a local ARCH~1!-
type alternative with parametric rate n2102 but a local MA~1! alternative with
rate n2104 only+
In this paper, we propose a new diagnostic test for time series model ~1+2!,
using a generalized spectrum proposed in Hong ~1999!+ The test enjoys the
“nuisance-parameter-free”propertyoftheBDStestunderawiderclassoftime
seriesmodels,whichincludebutarenotrestrictedtoARCHandACDmodels+
It is consistent against any type of pairwise serial dependence across various
lags in the model standardized residuals, a property not attainable by the BDS
1070 YONGMIAO HONG AND TAE-HWY LEE
test+ It can detect a class of local alternatives with a rate slightly slower than
the parametric rate n2102 but much faster than n2104+ This class includes both
MAandARCH-typelocalalternatives+Finally,generalizedspectralsmoothing
allows one to choose a lag order via data-driven methods, which are more ob-
jectivethananarbitrarychoiceora“ruleofthumb”andthusgivemorerobust
power+ A simulation study compares the proposed test and the tests of BDS,
Box–Pierce–Ljung, and McLeod–Li0Li–Mak in finite samples+ The new test
has reasonable power against a wide variety of stochastic and chaotic alterna-
tivestothenullmodels+Itisausefuladditiontotheexistingdiagnostictoolkit
for time series models ~see Barnett, Gallant, Hinich, Jungeilges, Kaplan, and
Jensen, 1997!+An empirical application to the daily S&P500 index highlights
the merits of the proposed test+ We emphasize, however, that our procedure is
best viewed as a complement rather than a substitute to the BDS test, which is
motivated from an interesting chaotic theory+4
It should be pointed out that there are a variety of nonparametric tests for
serial dependence in the literature+ These include the tests of Chan and Tran
~1992!, Cameron and Trivedi ~1993!, Delgado ~1996!, Hong ~1998!, Pinkse
~1998!, Skaug and Tjøstheim ~1993a, 1993b, 1996!, and Robinson ~1991!+All
ofthesetestsarebasedonobservedrawdataratherthanonestimatedstandard-
izedresiduals+Whetherandhowthelimitdistributionsofthesetestswillchange
when applied to estimated standardized residuals has not been investigated+ In
this paper, we do not consider how to adapt these tests to estimated standard-
ized residuals $e[ %+
t
2. A NEW DIAGNOSTIC TEST
Hong~1999!proposesageneralizedspectrumasananalytictoolforlinearand
nonlinear time series+ Suppose the time series $e % is strictly stationary+ The
t
basicideaofHong~1999!istoconsiderthespectrumofthetransformedseries
$eiuet%, where u [ R5~2‘,‘!+ Define
sj~u,v!5cov~eiuet,eivet2j!, i5M21,j50,61,+++, (2.1)
the covariance between eiuet and eivet2j+ Straightforward algebra yields
s~u,v!5w ~u,v!2w~u!w~v!, (2.2)
j j
where wj~u,v! 5 E@ei~uet1vet2j!# and w~u! 5 E~eiuet! are the joint and
marginal characteristic functions of ~et,et2j!+5 Thus, sj~u,v! 5 0 for all
~u,v! [ R2 if and only if et and et2j are independent+ Suppose
sup~u,v![R2(j‘52‘6sj~u,v!6 , ‘, which holds when, for example, $et% is a
stationary a-mixing process with the mixing coefficients $a~j!% satisfying
(j‘50a~j!~n21!0n , ‘ for some n . 1+ Then the Fourier transform of sj~u,v!
exists:
DIAGNOSTIC CHECKING FOR NONLINEAR TIME SERIES MODELS 1071
1 ‘
f~v,u,v!5 ( s~u,v!e2ijv, v[@2p,p#+ (2.3)
2p j52‘ j
No moment condition on $e % is required+ When var~e ! exists, however, the
t t
negative partial derivative of f~v,u,v! with respect to ~u,v! at ~0,0! yields the
conventional spectral density:
]2f~v,u,v!* 1 ‘
2 5 ( R~j!e2ijv,
]u]v ~0,0! 2p j52‘
where R~j!5cov~et,et2j!+ For this reason f~v,u,v! is called in Hong ~1999!
a “generalized spectral density” of $e %+ The introduction of parameters ~u,v!
t
offers much flexibility in capturing serial dependence in $e %+ The generalized
t
spectrumf~v,u,v!cancaptureanytypeofpairwisedependenceacrossvarious
lags in $e %, including those with zero autocorrelations+ Searching over the do-
t
main of ~u,v!, for example, one can find the “maximal dependence” of $e % at
t
each frequency v, as given by
s~v!5 sup 6f~v,u,v!6, v[@2p,p#,
~u,v![R2
where 6{6 is the Euclidean norm+ This maximal spectral dependence may be
contributed from linear or nonlinear serial dependence in $e %+ A generalized
t
spectral peak at some frequency will indicate a cycle, seasonality, or periodic-
ityduetononlineardependence~e+g+,volatilityclustering!when$e % isawhite
t
noise+6
The generalized spectrum f~v,u,v! differs from the well-known higher
order spectra, which are the Fourier transforms of higher order cumulants ~cf+
Brillinger and Rosenblatt, 1967a, 1967b; Subba Rao and Gabr, 1980, 1984;
Terdik, 1999!+ It does not require any moment condition on e + This is appeal-
t
ing because, for example, it has been argued that many high-frequency eco-
nomic and financial time series have infinite variances ~e+g+, Fama and Roll,
1968; Pagan and Schwert, 1990!+ It can effectively capture any pairwise
serially dependent processes, includingARCH with zero third cumulants+ For
such ARCH processes, the bispectrum—the Fourier transform of third-order
cumulants—will miss them+ We note, however, that f~v,u,v! cannot capture
dependent processes that are pairwise serially independent ~i+e+, et and et2j are
independent for any nonzero j but $e % is serially dependent!, which may or
t
may not be captured by the bispectrum+ It would be interesting to compare the
generalizedspectrumandthebispectrumthoroughly,butthisisbeyondthescope
of this paper and should be pursued elsewhere+
When $e % is i+i+d+, f~v,u,v! becomes a flat generalized spectrum
t
1
f ~v,u,v!5 s ~u,v!, v[@2p,p#+ (2.4)
0 2p 0
1072 YONGMIAO HONG AND TAE-HWY LEE
Any deviation of f~v,u,v! from f ~v,u,v! is evidence of serial dependence of
0
$e %+ To test the i+i+d+ hypothesis for $e %, Hong ~1999! suggests that one com-
t t
paretwoconsistentestimatorsoff~v,u,v!andf ~v,u,v!viaanL -norm+Define
0 2
s[ ~u,v!5w[ ~u,v!2w[ ~u,0!w[ ~0,v!, j50,61,+++,6~n21!, (2.5)
j j j j
where
5
n
~n2j!21 ( ei~ue[t1ve[t2j! ifj$0,
w[ ~u,v!5 t511j (2.6)
j n
~n1j!21 ( ei~ue[t1j1ve[t! ifj,0+
t512j
Notethatw[j~u,v!5w[2j~v,u!+Akernelestimatorforf~v,u,v!canbedefinedas
1 n21
fZ ~v,u,v!5 ( ~126j60n!102k~j0p!s[ ~u,v!e2ijv, (2.7)
n 2p j512n j
where k:R r @21,1# is a symmetric kernel and p[p is a bandwidth ~or lag
n
order! such that p r ‘, p0n r 0 as n r ‘+ Examples of k~{! include the
Bartlett,Daniell,quadratic-spectral,andtruncatedkernels~e+g+,Priestley,1981,
p+ 441!+ The factor ~12 6j60n!102 is a finite-sample correction factor that de-
livers a better approximation to the finite-sample distribution+ We also have a
consistent estimator for f ~v,u,v!:
0
1
fZ ~v,u,v!5 s[ ~u,v!, v[@2p,p#+ (2.8)
0 2p 0
Let W:R r R1 be a nondecreasing function such that W’~u!5w~v! exists
and is symmetric about 0, with *dW~u!5*w~u!du , ‘+ Examples of W~{!
are the cumulative distribution functions of N~0,1!, double exponential, and
uniformdistributions+7 Thenatestforthei+i+d+hypothesisof$e % canbebased
t
on a properly standardized L -norm:
2
EE
p n21
np 6fZ ~v,u,v!2fZ ~v,u,v!62dvdW~u!dW~v!2CZ ( k2~j0p!
n 0 0
MZ ~p!5 2p F G j51
n22 102
2DZ ( k4~j0p!
0
j51
E
n21 n21
( k2~j0p!~n2j!6s[ ~u,v!62dW~u!dW~v!2CZ ( k2~j0p!
j 0
5 j51 F G j51
,
n22 102
2DZ ( k4~j0p!
0
j51
(2.9)
DIAGNOSTIC CHECKING FOR NONLINEAR TIME SERIES MODELS 1073
wherethesecondequalityfollowsfromParseval’sidentity,s[j~u,v!5s[2j~v,u!,
symmetry of weighting functions k~{! and w~{!+ Moreover, the centering and
scaling factors have the following values:
F G
E
2
CZ 5 s[ ~u,2u!dW~u! , (2.10)
0 0
F G
E
2
DZ 5 6s[ ~u,v!62dW~u!dW~v! + (2.11)
0 0
Throughout, unspecified integrals are taken over the entire Euclidean space of
proper dimension+ The test statistic MZ ~p! involves one- and two-dimensional
numerical integrations with respect to ~u,v!, which can be implemented using,
e+g+,Gauss–Legendrequadratures+Notethat MZ ~p!involvesnonumericalinte-
gration over frequency v, which has been integrated out as a result of the use
of the L -norm+ Divergence measures rather than the L -norm could be used,
2 2
but they would generally involve numerical integrations over v and also over
~u,v!, and the distribution theory might be different also+ A GAUSS code for
computing MZ ~p! with p chosen via a data-driven method is available from the
authors+
3. ASYMPTOTIC DISTRIBUTION
We now derive the null limit distribution of MZ ~p! and establish its “nuisance-
parameter-free” property under a wide class of time series models+ Following
are regularity conditions+
Assumption A+1+ $Y % is a strictly stationary a-mixing process with
t
(j‘50a~j!~n21!0n , ‘ for some n . 1+
Assumption A+2+ n102~uZ 2u !5O ~1!, where u 5plim~uZ!+
0 P 0
Assumption A+3+ Let I be the pseudo information set from time t
t
to the infinite past and let Q be a small convex neighborhood of u +
0 0
The functions g~I ,{! and h~I ,{! are twice continuously differentiable
t t
bwEEyisstuuhsoppmuur[[eeQQs00pc77oehhcn22tst11a~~tnoIIttt,,Cuuu!![~~]][02~0]0]u,Qu‘!2h0!!~,hIawt~,.Ishut+,e,!u7re4!w,7e2itt,~hEua!sEnu5dpsuu@[YpEQtu[s20u7Qhp0g7u2~[h1IQt2~20I11t@~,,eIuut4t!,!~#~uu]0!!h2#~0]~]I0uta]2lu21l!!,ggub~~!oII+ttu,,nuud!!77e42d,,
Assumption A+4+ Let IZ be the observed information set available at time t
t
that may involve certain initial values+ Then
1074 YONGMIAO HONG AND TAE-HWY LEE
n *h~I ,u!2h~IZ ,u!*
lim (E sup t t #C
nr‘ t51 u[Q0 h~IZt,u!
and
n *g~I ,u!2g~IZ ,u!*
lim (E sup t t #C+
nr‘ t51 u[Q0 h~IZt,u!
Assumption A+5+ k:R r @21,1# is symmetric about 0 and is continuous
at 0 and all points in R except a finite number of points, with k~0! 5 1,
*‘k2~z!dz , ‘, 6k~z!6#C6z62b as z r ‘ for some b . 1_ and C [ ~0,‘!+
0 2
Assumption A+6+ W:R r R1 is nondecreasing such that the derivative
W’~u! 5 w~u! exists and is symmetric about 0, with *‘ dW~u! , ‘ and
2‘
*‘ u4dW~u! , ‘+
2‘
Assumption A+7+ D [@*6s ~u,v!62dW~u!dW~v!#2 . 0+
0 0
These are conditions on the data generating process ~DGP! $Y%, model pa-
t
rameter estimator uZ, initial value conditions, models g~{,u! and h~{,u!, and
weight functions k~{! and W~{!+ In Assumption A+1, we permit but do not re-
quirevar~Y!,‘+Anexamplewithvar~Y!5‘istheintegratedGARCH~1,1!
t t
process~EngleandBollerslev,1986!+InAssumptionA+2,wepermitbutdonot
requireuZ tobeaquasi–maximumlikelihoodestimator~LeeandHansen,1994;
Lumsdaine,1996!+Anyn102-consistentestimatoruZ suffices+AssumptionA+3is
astandardconditionontheconditionalmeanandconditionalvariancemodels+
We require that the fourth moment of the standardized error e exist+
t
Assumption A+4 is a start-up value condition+ It ensures that the impact of
initialvalues~ifany!assumedinIZ isasymptoticallynegligible+Thiscondition
t
easilyholdsformanytimeseriesmodels+Toillustratethis,wefirstconsideran
invertibleMA~1!modelYt5aut211ut,whereut5set,$et% isi+i+d+~0,1!,and
6a6,1+Here,wehaveg~It21,u!5aut21andh~It21,u!5s,whereu5~a,s!’+
Furthermore, we have It21 5 $Yt21,+++,Y1,Y0,+++% and IZt21 5 $Yt21,+++,Y1,u[0%,
where u[ is some assumed value ~e+g+, u[ 50! for u + The condition on h~{,{!
0 0 0
holds trivially, so we focus on the condition on g~{,{!+ By recursive substitu-
tion, we obtain
g~It21,uF!2g~IZt21,u! G
t21 t21
5 (~21!j21ajYt2j1atu02(~21!j21ajYt2j2atu[0 +
j51 j51
It follows that
F G
(n E sup*g~It21,u!2g~IZt21,u!*# (‘ E sup 6a6t~6u0616u[06! #C
t51 u[Q0 h~IZt21,u! t51 u[Q0 s
provided 6a6 , 1, 0 , s , ‘, E6e 6 , ‘, and E6u[ 6 , ‘+
t 0