weighted least absolute deviations estimation for arma models with infinite variance

28 Pages·2007·0.44 MB·English

EconometricTheory,23,2007,852–879+PrintedintheUnitedStatesofAmerica+
DOI:10+10170S0266466607070363
WEIGHTED LEAST ABSOLUTE
DEVIATIONS ESTIMATION FOR ARMA
MODELS WITH INFINITE VARIANCE
JIIIAAAZZZHHHUUU PAAANNN
Peking University
and
London School of Economics
HUUUIII WAAANNNGGG
Peking University
QIIIWWWEEEIII YAAAOOO
Peking University
and
London School of Economics
Forautoregressivemovingaverage~ARMA!modelswithinfinitevarianceinno-
vations,quasi-likelihood-basedestimators~suchasWhittleestimators!sufferfrom
complexasymptoticdistributionsdependingonunknowntailindices+Thismakes
statisticalinferenceforsuchmodelsdifficult+Incontrast,theleastabsolutedevi-
ations estimators ~LADE! are more appealing in dealing with heavy tailed pro-
cesses+ In this paper, we propose a weighted least absolute deviations estimator
~WLADE!forARMAmodels+WeshowthattheproposedWLADEisasymptot-
icallynormal,isunbiased,andhasthestandardroot-nconvergencerateevenwhen
thevarianceofinnovationsisinfinity+Thispavesthewayforstatisticalinference
based on asymptotic normality for heavy-tailedARMAprocesses+ For relatively
smallsamplesnumericalresultsillustratethattheWLADEwithappropriateweight
is more accurate than theWhittle estimator, the quasi-maximum-likelihood esti-
mator ~QMLE!, and the Gauss–Newton estimator when the innovation variance
is infinite and that the efficiency loss due to the use of weights in estimation is
notsubstantial+
1. INTRODUCTION
Let $y % be a stationary autoregressive moving average ~ARMA! time series
t
generated by the equation
y (cid:1)f y (cid:3){{{(cid:3)f y (cid:3)« (cid:3)u « (cid:3){{{(cid:3)u « , (1.1)
t 1 t(cid:2)1 p t(cid:2)p t 1 t(cid:2)1 q t(cid:2)q
Theauthorsthankthetworefereesfortheirvaluablesuggestions+TheworkwaspartiallysupportedbyanEPSRC
researchgrant~UK!andtheNaturalScienceFoundationofChina~grant10471005!+Addresscorrespondenceto
JiazhuPan,SchoolofMathematicalSciences,PekingUniversity,Beijing100871,China;e-mail:jzpan@math+
pku+edu+cn+
852 ©2007CambridgeUniversityPress 0266-4666007$15+00
WLADE ESTIMATION FOR IVARMA MODELS 853
where the innovation process $« % is a sequence of independent and identi-
t
cally distributed ~i+i+d+! random variables and b (cid:1) ~f ,+++,f ,u ,+++,u !'
1 p 1 q
is an unknown parameter vector+ When E«2 (cid:4) `, it is well known that vari-
t
ous estimators such as the maximum likelihood estimator ~MLE!, Whittle
estimators, and least squares estimators ~LSE! for b are all asymptotically
normalandunbiased~BrockwellandDavis,1991!+Furthermore,theleastabso-
lute deviations estimator ~LADE! is also asymptotically normal; see Duns-
muir and Spencer ~1991! and Davis and Dunsmuir ~1997!+ When E«2 (cid:1) `,
t
model ~1+1! is called the infinite variance ARMA ~IVARMA! model, which
defines a heavy-tailed process $y %+ The IVARMA models are pertinent in
t
modeling heavy-tailed time series data often encountered in, for example,
economics and finance ~Koedijk, Schafgans, and De Vries, 1990; Jansen
and de Vries, 1991!+ For further references on statistical modeling for heavy-
tailedphenomena,werefertoResnick~1997!andAdler,Feldaman,andTaqqu
~1997!+
Statistical inference for IVARMA models has not been well explored yet+
Most available results concern infinite variance autoregressive ~IVAR! models
~i+e+,q(cid:1)0in~1+1!!+GrossandSteiger~1979!andAnandChen~1982!obtained
the strong consistency and the convergence rates for LADE for IVAR models+
Davis and Resnick ~1985, 1986! derived the limiting distributions of the LSE
for IVAR models+ A more comprehensive asymptotic theory of M-estimators
forautoregressive~AR!modelswasderivedbyDavis,Knight,andLiu~1992!+
ForIVARMAmodels,theasymptoticpropertiesforseveralestimatorshavebeen
derived when the innovation distribution is in the domain of attraction of a
stablelawdistributionwithindexbetween0and2+Forexample,boththeWhittle
estimator proposed by Mikosch, Gadrich, and Adler ~1995! and the Gauss–
Newton estimator proposed by Davis ~1996! converge in distribution to some
functions of a sequence of stable random variables+ Furthermore, it has been
provedthattheM-estimatorconvergesindistributiontotheminimizerofasto-
chastic process; see Davis ~1996! and Calder and Davis ~1998!+ Kokoszka and
Taqqu~1996!extendedtheresultsofMikoschetal+~1995!tofractionalARMA
modelswithinfinitevariance+However,alltheprecedinglimitingdistributions
arecomplicatedanddependintimatelyontheunknowntailindicesoftheunder-
lying processes+ This makes it difficult to develop asymptotic approximations
for the purpose of statistical inference+ This paper provides a remedy for this
problem+
The difficulty of the asymptotic theory for LADE for IVARMA processes
may at least partially result from the fact that the residual of a linear predic-
tion for y based on its lagged values depends on b nonlinearly, whereas
t
such a dependence is completely linear for pure AR processes+ Note that
this linearity implies that the objective function for least absolute devia-
tions estimation is convex and therefore the asymptotic normality of LADE
may be readily derived from the convex lemma ~Hjort and Pollard, 1993!+
One way to deal with a nonconvex objective function is to adopt a local
854 JIAZHU PAN ET AL.
linearization around the true value of the parameter, which enables one to
establish asymptotic properties of a local estimator defined as a local mini-
mizer around the true value of the parameter+ This is the line taken in
the paper by Davis and Dunsmuir ~1997!, which dealt with LADE for
ARMA models with E«2 (cid:4) `+ On the other hand, Ling ~2005! proposed a
t
weighted least absolute deviations estimator ~WLADE! for IVAR models+The
key idea of the WLADE is to weigh down the observations that are exces-
sively large, either positively or negatively+ Ling ~2005! showed that the
WLADEisasymptoticallynormal+Theideaofweighingdownthelargeobser-
vations has also been used in estimation for autoregressive conditionally het-
eroskedastic ~ARCH! models with heavy tailed innovations by Horvath and
Liese ~2004!+
In this paper, we deal with the WLADE for IVARMA models+ By adopt-
ing the idea of local linearization mentioned previously, we show that a
local WLADE is asymptotic normal and unbiased under the condition that
E6« 6d (cid:4) (cid:3)` for some d (cid:5) 0 and the density function of « and its deriva-
t t
tive are bounded+ This facilitates statistical inference for IVARMA models
~even when E6« 6 (cid:1) `! in a conventional fashion+ For example, a Wald
t
test for a linear hypothesis can be constructed; see Section 2+ For relatively
small samples a simulation study indicates that the proposed WLADE is
more accurate than the Whittle estimator, the quasi-maximum-likelihood es-
timator ~QMLE!, and the Gauss–Newton estimator when Var~« ! (cid:1) `+ Fur-
t
thermore, the efficiency loss of the WLADE with respect to the ~unweighted!
LADE is not significant with appropriately selected weights+ Because the
WLADE converges at a slower rate than theWhittle estimator and the Gauss–
Newton estimator, we also studied the large-sample properties of theWLADE
numerically+
Although we only deal with IVARMA models in this paper, the basic idea
of combining a weighted objective function with local linearization of the
residuals may apply to other infinite variance time series models, such as
the infinite variance autoregressive integrated moving average ~ARIMA! and
the integrated generalized autoregressive conditionally heteroskedastic
~IGARCH!, which are popular in financial econometrics+ Another open prob-
lem is to develop appropriate methods for choosing weight functions; see
Remark 3 in Section 2+2+
The rest of paper is organized as follows+ The WLADE and the associated
asymptotic properties are presented in Section 2+ In addition to showing the
asymptotic normality of the local WLADE, we also show that a ~global! esti-
matorsharingthesameasymptoticpropertycouldbeobtainedbyminimizinga
convexobjectivefunctionifthereisavailableaninitialestimatorwithinroot-n
distance from the true value; seeTheorem 2 in Section 2+2+ Section 3 gives all
thetheoreticalproofsoftheresultsinSection2+Section4reportssomenumer-
ical results from a simulation study+
WLADE ESTIMATION FOR IVARMA MODELS 855
2. WLADE AND ITS ASYMPTOTIC PROPERTIES
2.1. Weighted Least Absolute Deviations Estimators
Denote by Q (cid:1) Rp(cid:3)q the parameter space that contains the true value b (cid:1)
0
~f0,+++,f0,u0,+++,u0!' oftheparameterbasaninnerpoint+Forb(cid:1)~f ,+++,
1 p 1 q 1
f ,u ,+++,u !', put
p 1 q
(cid:1)y (cid:2)f y (cid:2){{{(cid:2)f y (cid:2)u « ~b!(cid:2){{{(cid:2)u « ~b!,
t 1 t(cid:2)1 p t(cid:2)p 1 t(cid:2)1 q t(cid:2)q
« ~b!(cid:1) ift(cid:5)0, (2.1)
t
0, otherwise,
where y [0 for all t(cid:1)0+ Note that « (cid:6)«~b ! because of this truncation+
t t t 0
We define the objective function as
n
Wn~b!(cid:1) ( wK t6«t~b!6 (2.2)
t(cid:1)u(cid:3)1
and theWLADE as
bZ (cid:1)argminWn~b!, (2.3)
b
where u(cid:1)u~n! is a positive integer and the weight function, depending on a
constant a (cid:5) 2, is defined as
(cid:2) (cid:3)
t(cid:2)1 (cid:2)2
wK t (cid:1) 1(cid:3) ( k(cid:2)a6yt(cid:2)k6 + (2.4)
k(cid:1)1
2.2. Asymptotic Normality of WLADE
To state the asymptotic normality of bZ, we introduce some notation first+ Let
v(cid:1)~v ,+++,v !'(cid:1)Mn~b(cid:2)b !+
1 p(cid:3)q 0
It is easy to see that bZ (cid:1)b0(cid:3)v[0Mn, where v[ is a minimizer of
n
Tn~v!(cid:1) ( wK t~6«t~b0(cid:3)n(cid:2)102v!6(cid:2)6«t~b0!6!+
t(cid:1)u(cid:3)1
Denote At~b! (cid:1) ~At,1~b!,+++,At,p(cid:3)q~b!!', where At,i~b! (cid:1) (cid:2)]«t~b!0]bi+ By
~8+11+9! of Brockwell and Davis ~1991!, it holds for t(cid:2)max~p,q! that
(cid:4)
u~B!A ~b!(cid:1)y , i(cid:1)1,+++,p
t,i t(cid:2)i
(2.5)
u~B!A ~b!(cid:1)« ~b!, i(cid:1)1,+++,q,
t,i(cid:3)p t(cid:2)i
where B is the backshift operator+
856 JIAZHU PAN ET AL.
For t(cid:1)0,61,62,+++, define
p q
U (cid:2)(f0U (cid:1)« , V (cid:3)(u0V (cid:1)« + (2.6)
t i t(cid:2)i t t j t(cid:2)j t
i(cid:1)1 j(cid:1)1
Put Qt(cid:1)~Ut(cid:2)1,+++,Ut(cid:2)p,Vt(cid:2)1,+++,Vt(cid:2)q!', wt[~1(cid:3)(k`(cid:1)1k(cid:2)a6yt(cid:2)k6!(cid:2)2, and
S(cid:1)E~w Q Q'!, V(cid:1)E~w2Q Q'!+ (2.7)
t t t t t t
We denote by 7v7 the euclidean norm for a vector v+
Some regularity conditions are now in order+
A1+ For b (cid:2) Q, the polynomials u~z!(cid:1)1(cid:3)u z(cid:3){{{(cid:3)u zq and f~z!(cid:1)
1 q
1(cid:2)f z(cid:2){{{(cid:2)f zp have no common zeros, and all roots of f~z! and u~z!
1 p
are outside the unit circle+
A2+ Innovation« haszeromedianandadifferentiabledensityfunctionf~x!
t
satisfyingtheconditionsf~0!(cid:5)0,supx(cid:2)R6f~x!6(cid:4)B1(cid:4)`,andsupx(cid:2)R6f'~x!6(cid:4)
B (cid:4) `+ Furthermore, E6« 6d (cid:4) (cid:3)` for some d (cid:5) 0, and a (cid:5) max$2,20d%+
2 t
A3+ As n r `, u r ` and u0n r 0+
The following proposition indicates that model ~1+1! has a unique strictly
stationary and ergodic solution under conditionsA1 and A2+
Proposition 1+ Suppose that conditionA1 holds and E6« 6d (cid:4) (cid:3)` for some
t
d (cid:5) 0. Then model (1.1) defines a unique strictly stationary and ergodic pro-
cess $y %.
t
Proof+ ConditionA1 implies that
`
f(cid:2)1~z!u~z!(cid:1) (c zj,
j
j(cid:1)0
where6cj6(cid:1)Crj forsomeconstantsC(cid:5)0and0(cid:4)r(cid:4)1+Let dD (cid:1)min$d,1%+
Then
`
(6cj6dDE6«t(cid:2)j6dD (cid:4)`+
j(cid:1)0
The same argument as for Proposition 13+3+2 of Brockwell and Davis ~1991!
(cid:1)
yields the result+
Remark 1+ Condition A3 eliminates asymptotically the bias in the estima-
tion resulting from the lack of observations y for t(cid:1)0+
t
Remark 2+ Condition A2 does not rule out the possibility that E6« 6 (cid:1) `+
t
Thepurposeofintroducingweightsw istoweighdownexcessivelylargeobser-
Kt
vationsthatreflecttheheavy-tailedinnovationdistribution+Thereforetheasymp-
toticcovariancematrixofthenormalizedWLADE,dependingonSandVgiven
in ~2+7!, is a well-defined ~finite! matrix+ Note that wK t (cid:2) ~0,1#+ ConditionsA1
andA2 imply that for dD (cid:1)min$d,1%,
WLADE ESTIMATION FOR IVARMA MODELS 857
(cid:2) (cid:3)
` dD `
E ( k(cid:2)a6yt(cid:2)k6 (cid:1) ( k(cid:2)adDE6yt(cid:2)k6dD (cid:4)(cid:3)`+
k(cid:1)1 k(cid:1)1
Hence(` k(cid:2)a6y 6(cid:4)`withprobabilityone,whichensuresthatw iswell
k(cid:1)1 t(cid:2)k t
defined+ Note that w is stationary and ergodic under condition A1 and it is
t
asymptotically equivalent to wK t for t (cid:5) u ~see A3!+
We are now ready to state our main results+
THEOREM 1+ Let conditions A1–A3 hold. For any given positive random
variableMwithP~0(cid:4)M (cid:4)`!(cid:1)1,thereexistsalocalminimizerbZ ofWn~b!
that lies in the random region $b:7b(cid:2)b (cid:2)j0Mn7(cid:1)M0Mn% for which
(cid:2) (cid:3)0
1
Mn~bZ (cid:2)b0!rLN 0,4f2~0! S(cid:2)1VS(cid:2)1 ,
where j is a normal random vector with mean 0 and covariance matrix
~104f2~0!!S(cid:2)1VS(cid:2)1.
Notice that the lack of convexity for the objective function W ~b! compli-
n
cates the search for its minimizer+ As in Davis and Dunsmuir ~1997!, W ~b!
n
may be linearized in a neighborhood of a good initial estimate bZ 0 as follows:
n
WGn~b!(cid:1) ( wK t6«t~bZ 0!(cid:2)A't~bZ 0!~b(cid:2)bZ 0!6+
t(cid:1)u(cid:3)1
The resulting estimator b(cid:1)argmin W ~b! shares the same asymptotic prop-
D b Gn
erty as the localWLADE+ SeeTheorem 2, which follows+
THEOREM 2+ Let conditionsA1–A3 hold. Then
(cid:2) (cid:3)
1
Mn~bD (cid:2)b0!rLN 0,4f2~0! S(cid:2)1VS(cid:2)1 ,
provided that b0(cid:1)b (cid:3)O ~n(cid:2)102!.
Z 0 p
Remark 3+ Although we only deal with the weight function defined in ~2+4!
explicitly, the preceding theorem holds for general weight function g [
t
g~yt(cid:2)1,yt(cid:2)2,+++! provided
E$~g (cid:3)g2!~j2(cid:3)j3!%(cid:4)`, E$~g (cid:3)g2!~j2(cid:3)j3(cid:3)j4!%(cid:4)`,
t t t t t t t
where j (cid:1) (` ri6y 6, j (cid:1) C (` ri6y 6, 0 (cid:4) r (cid:4) 1, and C (cid:5) 0 are
i(cid:1)0 (cid:2)i t 0 i(cid:1)t t(cid:2)i 0
constants+ For example, we may use weights of more general form:
(cid:2) (cid:3)
t(cid:2)1 (cid:2)g
g (cid:1) 1(cid:3) ( k(cid:2)a~logk!d6y 6 , a(cid:5)2, g(cid:2)2, d(cid:2)0+ (2.8)
t t(cid:2)k
k(cid:1)1
ThenumericalexamplesinSection3indicatethattheaccuracyoftheWLADE
is not sensitive with respect to the value of a+ However the choice of g (cid:1) 2
typically leads to a better estimator than those with g (cid:5) 2, at least for model
858 JIAZHU PAN ET AL.
~4+1!+ Furthermore, it seems that d(cid:1)2,3,4 behave almost equally well+ How-
everhowtochooseaweightfunctioningeneralsuchthattheresultingestima-
tor is of certain optimality remains an open question+
2.3. A Wald Test for Linear Hypotheses
The asymptotic normality of the estimator b stated in Theorem 1 facilitates
Z
inference for model ~1+1!+ For example, we may consider a general form of
linear null hypothesis:
H :Gb (cid:1)k,
0 0
where G is an s(cid:7)~p(cid:3)q! constant matrix with rank s and k is an s(cid:7)1 con-
stant vector+AWald test statistic may be defined as
(cid:4) (cid:5)
1 (cid:2)1
Zn(cid:1)~GbZ (cid:2)k!' G 4nf2~0! SZ (cid:2)1VZ SZ (cid:2)1G' ~GbZ (cid:2)k!,
D
and we reject H for large values of Z + In the preceding expression,
0 n
1 n 1 n
SZ (cid:1) n(cid:2)u t(cid:1)(u(cid:3)1wK tQZtQZt', VZ (cid:1) n(cid:2)u t(cid:1)(u(cid:3)1~wK t2QZtQZt'!+ (2.9)
Here QZt is defined in the same manner as Qt but with b0 replaced by bZ, «t
replaced by «t~bZ !, and yt(cid:1)0 for all t(cid:1)0 ~see ~2+6! and ~2+7!!, and fD~0! is an
estimate for f~0! defined as
(cid:2) (cid:3)(cid:6)
1 n « ~b! n
fD~0!(cid:1) bn t(cid:1)(u(cid:3)1wK tK tbnZ t(cid:1)(u(cid:3)1wK t, (2.10)
where K~{! is a kernel function on R and b (cid:5) 0 is a bandwidth+ Theorem 3,
n
whichfollows,showsthattheasymptoticnulldistributionofZ isx2+Itinfact
n s
still holds if bZ in the definition of Zn is replaced by bD+
THEOREM 3+ Suppose conditions A1–A3 hold. Let kernel function K
be bounded, Lipschitz continuous, and of finite first moment. Let b r 0 and
n
nb4 r ` as n r `. Then Z r x2 under H .
n n L s 0
3. PROOFS
We use the same notation as in Section 2+ For any fixed v (cid:2) Rp(cid:3)q, put
n
S ~v!(cid:1) ( w ~6« ~b (cid:3)n(cid:2)102v!6(cid:2)6« ~b !6!,
n t t 0 t 0
t(cid:1)u(cid:3)1
n
S(cid:3)~v!(cid:1) ( w ~6« (cid:2)n(cid:2)102Q'v6(cid:2)6« 6!,
n t t t t
t(cid:1)u(cid:3)1
n
S*~v!(cid:1) ( w ~6« ~b !(cid:2)n(cid:2)102A'~b !v6(cid:2)6« ~b !6!+
n t t 0 t 0 t 0
t(cid:1)u(cid:3)1
WLADE ESTIMATION FOR IVARMA MODELS 859
We linearize «~b! around b ; that is, «~b! is approximated by
t 0 t
« ~b !(cid:2)A'~b !~b(cid:2)b !,
t 0 t 0 0
where A ~b! is defined in ~2+5!+
t
We denote by rL the convergence in distribution and by P&& the conver-
genceinprobability+LetC~Rs!bethespaceofthereal-valuedcontinuousfunc-
tionsonRs~Rudin,1991!+ForprobabilitymeasuresP andPonC~Rs!,wesay
n
that P converges weakly to P in C~Rs! if *fdP r *fdP for any bounded and
n n
continuousfunctionfdefinedonC~Rs!+ForrandomfunctionsS ,Sdefinedon
n
C~Rs!,S r SifthedistributionofS convergesweaklytothatofSinC~Rs!
n L n
~Billingsley, 1999!+The term C denotes a positive constant that may be differ-
ent at different places+
3.1. Proof of Theorem 1
BeforeweproveTheorem1,wefirstintroduceapropositionthatisofindepen-
dent interest+ Its proof is divided into several lemmas+ We always assume that
conditionsA1–A3 hold+
PROPOSITION 2+ As n r `, it holds that T ~v! r T~v! on C~Rp(cid:3)q!,
n L
where T~v!(cid:1)f~0!v'Sv(cid:3)v'N and N denotes an N~0,V! random vector.
LEMMA 1+ It holds that 6« (cid:2)«~b !6 (cid:1) j and 6A ~b !(cid:2)Q 6 (cid:1) j for
t t 0 t t,i 0 t,i t
i(cid:1)1,+++,p(cid:3)q, where j (cid:1)C (` rj6y 6, 0 (cid:4) r (cid:4) 1, C is a positive con-
t 0 j(cid:1)t t(cid:2)j 0
stant, and Q is the ith component of Q .
t,i t
Proof+ See Brockwell and Davis ~1991, pp+ 265–268!+
LEMMA2+ S(cid:3)~v! r T~v! on C~Rp(cid:3)q!.
n L
Proof+ We first prove the convergence for any fixed v+ Using the identity
6z(cid:2)y6(cid:2)6z6(cid:1)(cid:2)ysgn~z!(cid:3)2~y(cid:2)z!$I~0(cid:4)z(cid:4)y!(cid:2)I~y(cid:4)z(cid:4)0!%,
which holds for z(cid:6)0, we have
n
S(cid:3)~v!(cid:1)(cid:2)n(cid:2)102 ( w Q'vsgn~« !
n t t t
t(cid:1)u(cid:3)1
n
(cid:3)2 ( w ~n(cid:2)102Q'v(cid:2)« !
t t t
t(cid:1)u(cid:3)1
(cid:7)@I~0(cid:4)« (cid:4)n(cid:2)102Q'v!(cid:2)I~n(cid:2)102Q'v(cid:4)« (cid:4)0!#
t t t t
(cid:1):A (cid:3)B +
n n
860 JIAZHU PAN ET AL.
Notice that, by Lemma 1, we have 6Q 6(cid:1)C(` rj6y 6 and
t,i j(cid:1)1 t(cid:2)j
`
C(rj6y 6
t(cid:2)j `
6w102Q 6(cid:1) j(cid:1)1 (cid:1)C ( kark+ (3.1)
t t,i `
1(cid:3) ( k(cid:2)a6y 6 k(cid:1)1
t(cid:2)k
k(cid:1)1
Then, Ew2~Q'v!2 (cid:4) (cid:3)`+ But, from conditionsA1 andA2, $w Q'vsgn~« !% is
t t t t t
a stationary martingale difference sequence+ Therefore, applying a martingale
central limit theorem ~Hall and Heyde, 1980!, we obtain A r v'N+
n L
For B , let
n
W (cid:1)w ~n(cid:2)102Q'v(cid:2)« !I~0(cid:4)« (cid:4)n(cid:2)102Q'v!
nt t t t t t
and Ft(cid:2)1(cid:1)s~«j,j(cid:1)t(cid:2)1!+ Then
nEW2(cid:1)nE~E~W26F !!
nt nt t(cid:2)1
(cid:2) (cid:7)(cid:8)
(cid:1)nE w2 n(cid:2)102Qt'v~n(cid:2)102Q'v(cid:2)z!2~f~z!(cid:2)f~0!!dz
t t
0 (cid:8) (cid:9)(cid:3)
(cid:3) n(cid:2)102Qt'v~n(cid:2)102Q'v(cid:2)z!2f~0!dz
t
0
(cid:1)nE~w2B n(cid:2)2~Q'v!4(cid:3)w2B n(cid:2)302~Q'v!3!+
t 2 t t 1 t
Similarly to ~3+1!, we can obtain
Ew2~Q'v!4 (cid:4)(cid:3)`, Ew2~Q'v!3 (cid:4)(cid:3)`+
t t t t
Therefore, we have proved that
limsupnEW2(cid:1)0+ (3.2)
nt
nr`
On the other hand, on the set $Q'v (cid:5) 0%, we may show that
t
n f~0!
( E~W 6F !r E@w ~Q'v!2I~Q'v(cid:5)0!#
t(cid:1)u(cid:3)1 nt t(cid:2)1 2 t t t
and
(cid:2) (cid:3)
n
Var ( ~W (cid:2)E~W 6F !! r0;
nt nt t(cid:2)1
t(cid:1)u(cid:3)1
see Davis and Dunsmuir ~1997!+ Therefore,
n f~0!
( W r E@w ~Q'v!2I~Q'v(cid:5)0!#+ (3.3)
t(cid:1)u(cid:3)1 nt 2 t t t
WLADE ESTIMATION FOR IVARMA MODELS 861
Using the same argument for the second indicator in the summands of B ,
n
we obtain that
Bn P&&f~0!v'Sv, (3.4)
so that the finite-dimensional distributions of S(cid:3) converge to those of T+ But
n
because S(cid:3) has convex sample paths, this implies that the convergence is in
fact on C~nRp(cid:3)q! ~see Davis and Dunsmuir, 1997, proof of Prop+ 1!+ (cid:1)
LEMMA3+ Sn*~v!(cid:2)Sn(cid:3)~v! P&& 0 uniformly on compact sets.
Proof+ Notice that
S*~v!(cid:2)S(cid:3)~v!
n n
n
(cid:1) ( w @~6« ~b !(cid:2)n(cid:2)102A'~b !v6(cid:2)6« ~b !6!(cid:2)~6« (cid:2)n(cid:2)102Q'v6(cid:2)6« 6!#
t t 0 t 0 t 0 t t t
t(cid:1)u(cid:3)1
n n
(cid:1)(cid:2)n(cid:2)102 ( w A'~b !vsgn~« ~b !!(cid:3)n(cid:2)102 ( w Q'vsgn~« !
t t 0 t 0 t t t
t(cid:1)u(cid:3)1 t(cid:1)u(cid:3)1
n
(cid:3)2 ( w ~n(cid:2)102A'~b !v(cid:2)« ~b !!@I~0(cid:4)« ~b !(cid:4)n(cid:2)102A'~b !v!
t t 0 t 0 t 0 t 0
t(cid:1)u(cid:3)1
(cid:2)I~n(cid:2)102A'~b !v(cid:4)« ~b !(cid:4)0!#
t 0 t 0
n
(cid:2)2 ( v ~n(cid:2)102Q'v(cid:2)« !
t t t
t(cid:1)u(cid:3)1
(cid:7)@I~0(cid:4)« (cid:4)n(cid:2)102Q'v!(cid:2)I~n(cid:2)102Q'v(cid:4)« (cid:4)0!#+
t t t t
First, we consider
n
L (cid:1)n(cid:2)102 ( w @Q'vsgn~« !(cid:2)A'~b !vsgn~« ~b !!#
1 t t t t 0 t 0
t(cid:1)u(cid:3)1
n
(cid:1)n(cid:2)102 ( w Q'v~sgn~« !(cid:2)sgn~« ~b !!!
t t t t 0
t(cid:1)u(cid:3)1
n
(cid:3)n(cid:2)102 ( w ~Q'v(cid:2)A'~b !v!sgn~« ~b !!
t t t 0 t 0
t(cid:1)u(cid:3)1
(cid:1):K (cid:3)K +
1 2
By Lemma 1 and the proof of ~3+1!, we know that
6w ~Q'(cid:2)A'~b !!v6(cid:1)C7v7j +
t t t 0 t
Then
n ` n `
K2(cid:1)n(cid:2)102C7v7 ( (rj6yt(cid:2)j6(cid:1)n(cid:2)102C7v7 ( rt ( rh6y(cid:2)h6 P&&0
t(cid:1)u(cid:3)1j(cid:1)t t(cid:1)u(cid:3)1 h(cid:1)0