Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA--GARCH/IGARCH models

0.41 MB·English

by  Ke Zhu

TheAnnalsofStatistics
2011,Vol.39,No.4,2131–2163
DOI:10.1214/11-AOS895
(cid:13)c InstituteofMathematicalStatistics,2011
GLOBAL SELF-WEIGHTED AND LOCAL QUASI-MAXIMUM
EXPONENTIAL LIKELIHOOD ESTIMATORS FOR
2
ARMA–GARCH/IGARCH MODELS
1
0
2 By Ke Zhu and Shiqing Ling1

n
Hong Kong University of Science and Technology
a
J
Thispaperinvestigatestheasymptotictheoryofthequasi-maxi-
0
mumexponentiallikelihoodestimators(QMELE)forARMA–GARCH
3
models. Under only a fractional moment condition, the strong con-

sistency and the asymptotic normality of the global self-weighted
]
T QMELE are obtained. Based on this self-weighted QMELE, the lo-
S cal QMELE is showed to be asymptotically normal for the ARMA
. model with GARCH(finitevariance) andIGARCHerrors. A formal
h
comparison of two estimators is given for some cases. A simulation
t
a study is carried out to assess the performance of these estimators,
m and a real example on the world crudeoil price is given.
[

1 1. Introduction. Assume that y :t=0, 1, 2,... is generated by the
t
v ARMA–GARCH model { ± ± }
6
1 p q
2 (1.1) y =µ+ φ y + ψ ε +ε ,
t i t−i i t−i t
6
i=1 i=1
. X X
1
r s
0 (1.2) ε =η h and h =α + α ε2 + β h ,
2 t t t t 0 i t−i i t−i
1 i=1 i=1
p X X
:
v where α >0,α 0 (i=1,...,r),β 0 (j=1,...,s), and η is a sequence
0 i j t
i ≥ ≥
X of i.i.d. random variables with Eηt=0. As we all know, since Engle (1982)
and Bollerslev (1986), model (1.1)–(1.2) has been widely used in economics
r
a
and finance; see Bollerslev, Chou and Kroner (1992), Bera and Higgins
Received January 2011.
1SupportedinpartbyHongKongResearchGrantsCommissionGrantsHKUST601607
and HKUST602609.
AMS 2000 subject classifications. 62F12, 62M10, 62P20.
Key words and phrases. ARMA–GARCH/IGARCH model, asymptotic normality,
global self-weighted/local quasi-maximum exponential likelihood estimator, strong con-
sistency.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Statistics,
2011,Vol. 39, No. 4, 2131–2163. This reprint differs from the original in
pagination and typographic detail.
1
2 K. ZHU ANDS. LING
(1993), Bollerslev, Engel and Nelson (1994) and Francq and Zako¨ıan (2010).
The asymptotic theory of the quasi-maximum likelihood estimator (QMLE)
was established by Ling and Li (1997) and by Francq and Zako¨ıan (2004)
when Eε4 < . Under the strict stationarity condition, the consistency
t ∞
and the asymptotic normality of the QMLE were obtained by Lee and
Hansen (1994) and Lumsdaine (1996) for the GARCH(1,1) model, and by
Berkes,Horva´thandKokoszka(2003)andFrancqandZako¨ıan(2004)forthe
GARCH(r,s) model. Hall andYao (2003) established theasymptotic theory
of the QMLE for the GARCH model when Eε2 < , including both cases
t ∞
in which Eη4= and Eη4< . Under the geometric ergodicity condition,
t ∞ t ∞
Lang,RahbekandJensen(2011)gavetheasymptoticpropertiesofthemodi-
fiedQMLEforthefirstorderAR–ARCHmodel.Moreover, whenE ε ι<
t
| | ∞
for some ι>0, the asymptotic theory of the global self-weighted QMLE and
the local QMLE was established by Ling (2007) for model (1.1)–(1.2).
It is well known that the asymptotic normality of the QMLE requires
Eη4< and this property is lost when Eη4= ; see Hall and Yao (2003).
t ∞ t ∞
Usually, the least absolute deviation (LAD) approach can be used to reduce
the moment condition of η and provide a robust estimator. The local LAD
t
estimator was studied by Peng and Yao (2003) and Li and Li (2005) for the
pure GARCH model, Chan and Peng (2005) for the double AR(1) model,
and Li and Li (2008) for the ARFIMA–GARCH model. The global LAD
estimator was studied by Horva´th and Liese (2004) for the pure ARCH
model and by Berkes and Horva´th (2004) for the pure GARCH model, and
by Zhu and Ling (2011a) for the double AR(p) model. Except for the AR
models studied by Davis, Knight and Liu (1992) and Ling (2005) [see also
Knight(1987,1998)],thenondifferentiableandnonconvexobjectivefunction
appears when one studies the LAD estimator for the ARMA model with
i.i.d. errors. By assuming the existence of a √n-consistent estimator, the
asymptotic normality of the LAD estimator is established for the ARMA
modelwithi.i.d. errorsby Davis andDunsmuir(1997)forthefinitevariance
case and by Pan, Wang and Yao (2007) for the infinite variance case; see
also Wu and Davis (2010) for the noncausal or noninvertible ARMA model.
Recently, Zhu and Ling (2011b) proved the asymptotic normality of the
global LAD estimator for the finite/infinite variance ARMA model with
i.i.d. errors.
In this paper, we investigate the self-weighted quasi-maximum exponen-
tial likelihood estimator (QMELE) for model (1.1)–(1.2). Under only a frac-
tional moment condition of ε with Eη2 < , the strong consistency and
t t ∞
the asymptotic normality of the global self-weighted QMELE are obtained
by using the bracketing method in Pollard (1985). Based on this global self-
weighted QMELE,the local QMELE is showed to beasymptotically normal
for the ARMA–GARCH (finite variance) and –IGARCH models. A formal
comparison of two estimators is given for some cases.
QMELE FOR ARMA–GARCH/IGARCHMODELS 3
Fig. 1. The Hill estimators {αˆη(k)} for {ηˆt2}.
To motivate our estimation procedure,we revisit the GNP deflator exam-
ple of Bollerslev (1986), in which the GARCH model was proposed for the
first time. The model he specified is an AR(4)–GARCH(1,1) model for the
quarterly data from 1948.2 to 1983.4 with a total of 143 observations. We
use this data set and his fitted model to obtain the residuals ηˆ . The tail
t
{ }
index of η2 is estimated by Hill’s estimator αˆ (k) with the largest k data
{ t} η
of ηˆ2 , that is,
{ t}
k
αˆ (k)= ,
η k (logη˜ logη˜ )
j=1 143−j − 143−k
where η˜ is the jth order sPtatistic of ηˆ2 . The plot of αˆ (k) 70 is given
j { t} { η }k=1
in Figure 1. From this figure, we can see that αˆ (k)>2 when k 20, and
η
≤
αˆ (k)<2whenk>20.NotethatHill’s estimator isnotsoreliablewhenk is
η
too small. Thus, the tail of η2 is most likely less than 2, that is, Eη4= .
{ t} t ∞
Thus, the setup that η has a finite forth moment may not be suitable, and
t
hence the standard QMLE procedure may not be reliable in this case. The
estimation procedure in this paper only requires Eη2 < . It may provide
t ∞
a more reliable alternative to practitioners. To further illustrate this advan-
tage, a simulation study is carried out to compare the performance of our
estimators andtheself-weighted/local QMLEinLing(2007),andanewreal
example on the world crude oil price is given in this paper.
This paper is organized as follows. Section 2 gives our results on the
global self-weighted QMELE. Section 3 proposes a local QMELE estimator
4 K. ZHU ANDS. LING
and gives its limiting distribution. The simulation results are reported in
Section 4. A real example is given in Section 5. The proofs of two techni-
cal lemmas are provided in Section 6. Concluding remarks are offered in
Section 7. The remaining proofs are given in the Appendix.
2. Global self-weighted QMELE. Let θ =(γ′,δ′)′ be the unknown pa-
rameterofmodel(1.1)–(1.2)anditstruevaluebeθ ,whereγ=(µ,φ ,...,φ ,
0 1 p
ψ ,...,ψ )′ and δ=(α ,...,α ,β ,...,β )′. Given the observations y ,...,
1 q 0 r 1 s n
{
y and the initial values Y y ,y ,... , we can rewrite the parametric
1 0 0 −1
} ≡{ }
model (1.1)–(1.2) as
p q
(2.1) ε (γ)=y µ φ y ψ ε (γ),
t t i t−i i t−i
− − −
i=1 i=1
X X
η (θ)=ε (γ)/ h (θ) and
t t t
(2.2)
pr s
h (θ)=α + α ε2 (γ)+ β h (θ).
t 0 i t−i i t−i
i=1 i=1
X X
Here, η (θ )=η , ε (γ )=ε and h (θ )=h . The parameter space is Θ=
t 0 t t 0 t t 0 t
Θ Θ , where Θ Rp+q+1, Θ Rr+s+1, R=( , ) and R =[0, ).
γ× δ γ ⊂ δ ⊂ 0 −∞ ∞ 0 ∞
Assume that Θ and Θ are compact and θ is an interior point in Θ.
γ δ 0
Denote α(z)= r α zi, β(z)=1 s β zi, φ(z)=1 p φ zi and
ψ(z)=1+ q ψi=z1i. Wi e introduce−the fio=l1lowiing assumpti−ons:i=1 i
i=1 i
P P P
AssumptPion 2.1. For each θ Θ, φ(z)=0 and ψ(z)=0 when z 1,
∈ 6 6 | |≤
and φ(z) and ψ(z) have no common root with φ =0 or ψ =0.
p q
6 6
Assumption 2.2. For each θ Θ, α(z) and β(z) have no common root,
α(1)=1,α +β =0 and s β ∈<1.
6 r s6 i=1 i
Assumption 2.3. η2 Phas a nondegenerate distribution with Eη2< .
t t ∞
Assumption 2.1 implies the stationarity, invertibility and identifiability
of model (1.1), and Assumption 2.2 is the identifiability condition for mo-
del (1.2). Assumption 2.3 is necessary to ensure that η2 is not almost surely
t
(a.s.) a constant. When η follows the standard double exponential distri-
t
bution, the weighted log-likelihood function (ignoring a constant) can be
written as follows:
n
1 ε (γ)
t
(2.3) L (θ)= w l (θ) and l (θ)=log h (θ)+ | | ,
sn t t t t
n h (θ)
t=1 t
X p
where w =w(y ,y ,...) and w is a measurable, positipve and bounded
t t−1 t−2
function on RZ0 with Z = 0,1,2,... . We look for the minimizer, θˆ =
0 sn
{ }
QMELE FOR ARMA–GARCH/IGARCHMODELS 5
(γˆ′ ,δˆ′ )′, of L (θ) on Θ, that is,
sn sn sn
θˆ =argminL (θ).
sn sn
Θ
Sincetheweight w onlydependson y itself andwedonotassumethatη
t t t
{ }
follows the standard double exponential distribution, θˆ is called the self-
sn
weighted quasi-maximum exponential likelihood estimator (QMELE) of θ .
0
When h is a constant, the self-weighted QMELE reduces to the weighted
t
LAD estimator of the ARMA model in Pan, Wang and Yao (2007) and Zhu
and Ling (2011b).
The weight w is to reduce the moment condition of ε [see more discus-
t t
sions in Ling (2007)], and it satisfies the following assumption:
Assumption 2.4. E[(w +w2)ξ3 ]< for any ρ (0,1), where ξ =
t t ρt−1 ∞ ∈ ρt
1+ ∞ ρi y .
i=0 | t−i|
WPhen w 1, the θˆ is the global QMELE and it needs the moment
t sn
≡
condition E ε 3< for its asymptotic normality, which is weaker than the
t
| | ∞
moment condition Eε4< as for the QMLE of θ in Francq and Zako¨ıan
t ∞ 0
(2004). It is well known that the higher is the moment condition of ε , the
t
smaller is the parameter space. Figure 2 gives the strict stationarity region
and regions for E ε 2ι < of the GARCH(1,1) model: ε = η √h and
t t t t
| | ∞
h =α +α ε2 +β h , where η Laplace(0,1). From Figure 2, we can
t 0 1 t−1 1 t−1 t∼
see that the region for E ε 0.1 < is very close to the region for strict
t
| | ∞
stationarity of ε , and is much bigger than the region for Eε4< .
t t ∞
Under Assumption 2.4, we only need a fractional moment condition for
the asymptotic property of θˆ as follows:
sn
Assumption 2.5. E ε 2ι< for some ι>0.
t
| | ∞
The sufficient and necessary condition of Assumption 2.5 is given in The-
orem 2.1 of Ling (2007). In practice, we can use Hill’s estimator to estimate
the tail index of y and its estimator may provide some useful guidelines
t
{ }
for the choice of ι. For instance, the quantity 2ι can be any value less than
the tail index y . However, so far we do not know how to choose the op-
t
{ }
timal ι. As in Ling (2007) and Pan, Wang and Yao (2007), we choose the
weight function w according to ι. When ι=1/2 (i.e., E ε < ), we can
t t
| | ∞
choose the weight function as
∞ −4
1
(2.4) w = max 1,C−1 y I y >C ,
t k9| t−k| {| t−k| }
( )!
k=1
X
where C >0 is a constant. In practice, it works well when we select C as
the 90% quantile of data y ,...,y . When q=s=0 (AR–ARCH model),
1 n
{ }
6 K. ZHU ANDS. LING
Fig. 2. The regions bounded by the indicated curves are for the strict stationarity and
for E|εt|2ι<∞ with ι=0.05,0.5,1,1.5 and 2, respectively.
for any ι>0, the weight can be selected as
p+r −4
1
w = max 1,C−1 y I y >C .
t k9| t−k| {| t−k| }
( )!
k=1
X
Whenι (0,1/2) andq>0ors>0,theweightfunctionneedtobemodified
∈
as follows:
∞ −4
1
w = max 1,C−1 y I y >C .
t ( k1+8/ι| t−k| {| t−k| })!
k=1
X
Obviously, these weight functions satisfy Assumptions 2.4 and 2.7.For more
choices of w , we refer to Ling (2005) and Pan, Wang and Yao (2007). We
t
QMELE FOR ARMA–GARCH/IGARCHMODELS 7
first state the strong convergence of θˆ in the following theorem and its
sn
proof is given in the Appendix.
Theorem 2.1. Suppose η has a median zero with E η =1. If Assump-
t t
| |
tions 2.1–2.5 hold, then
θˆ θ a.s., as n .
sn 0
→ →∞
To study the rate of convergence of θˆ , we reparameterize the weighted
sn
log-likelihood function (2.3) as follows:
L (u) nL (θ +u) nL (θ ),
n sn 0 sn 0
≡ −
where u Λ u=(u′,u′)′:u+θ Θ . Let uˆ =θˆ θ . Then, uˆ is the
∈ ≡{ 1 2 0∈ } n sn− 0 n
minimizer of L (u) on Λ. Furthermore, we have
n
n n n
(2.5) L (u)= w A (u)+ w B (u)+ w C (u),
n t t t t t t
t=1 t=1 t=1
X X X
where
1
A (u)= [ε (γ +u ) ε (γ )],
t t 0 1 t 0
h (θ ) | |−| |
t 0
p εt(γ0) εt(γ0)
B (u)=log h (θ +u) log h (θ )+ | | | | ,
t t 0 t 0
− h (θ +u) − h (θ )
t 0 t 0
p p
1 1 p p
C (u)= [ε (γ +u ) ε (γ )].
t t 0 1 t 0
h (θ +u) − h (θ ) | |−| |
(cid:20) t 0 t 0 (cid:21)
Let I() be tphe indicator funpction. Using the identity
·
x y x = y[I(x>0) I(x<0)]
(2.6) | − |−| | − −
y
+2 [I(x s) I(x 0)]ds
≤ − ≤
Z0
for x=0, we can show that
6
−qt(u)
(2.7) A (u)=q (u)[I(η >0) I(η <0)]+2 X (s)ds,
t t t t t
−
Z0
where X (s)=I(η s) I(η 0), q (u)=q (u)+q (u) with
t t t t 1t 2t
≤ − ≤
u′ ∂ε (γ ) u′ ∂2ε (ξ∗)
t 0 t
q (u)= and q (u)= u,
1t h (θ ) ∂θ 2t 2 h (θ ) ∂θ∂θ′
t 0 t 0
and ξ∗ lies betwpeen γ0 and γ0+u1. Moreover, letpt=σ ηk :k t and
F { ≤ }
−q1t(u)
ξ (u)=2w X (s)ds.
t t t
Z0
8 K. ZHU ANDS. LING
Then, from (2.7), we have
n
(2.8) w A (u)=u′T +Π (u)+Π (u)+Π (u),
t t 1n 1n 2n 3n
t=1
X
where
n
w ∂ε (γ )
t t 0
T = [I(η >0) I(η <0)],
1n t t
h (θ ) ∂θ −
t=1 t 0
X
n p
Π (u)= ξ (u) E[ξ (u) ] ,
1n t t t−1
{ − |F }
t=1
X
n
Π (u)= E[ξ (u) ],
2n t t−1
|F
t=1
X
n
Π (u)= w q (u)[I(η >0) I(η <0)]
3n t 2t t t
−
t=1
X
n −qt(u)
+2 w X (s)ds.
t t
t=1 Z−q1t(u)
X
By Taylor’s expansion, we can see that
n
(2.9) w B (u)=u′T +Π (u)+Π (u),
t t 2n 4n 5n
t=1
X
where
n
w ∂h (θ )
t t 0
T = (1 η ),
2n t
2h (θ ) ∂θ −| |
t 0
t=1
X
n 3 ε (γ ) 1 1 ∂h (ζ∗) ∂h (ζ∗)
Π (u)=u′ w t 0 t t u,
4n t 8 h (ζ∗) − 4 h2(ζ∗) ∂θ ∂θ′
t=1 (cid:18) (cid:12) t (cid:12) (cid:19) t
X (cid:12) (cid:12)
Π (u)=u′ n w 1(cid:12)(cid:12)p1 εt(γ(cid:12)(cid:12)0) 1 ∂2ht(ζ∗)u,
5n t=1 t(cid:18)4 − 4(cid:12) ht(ζ∗)(cid:12)(cid:19)ht(ζ∗) ∂θ∂θ′
X (cid:12) (cid:12)
and ζ∗ lies between θ0 and θ0+(cid:12)(cid:12)up. (cid:12)(cid:12)
We further need one assumption and three lemmas. The first lemma is
directly from the central limit theorem for a martingale difference sequence.
The second- and third-lemmas give the expansions of Π (u) for i=1,...,5
in
and n C (u). The key technical argument is for the second lemma for
t=1 t
which we use the bracketing method in Pollard (1985).
P
QMELE FOR ARMA–GARCH/IGARCHMODELS 9
Assumption 2.6. η has zero median with E η =1 and a continuous
t t
| |
density function g(x) satisfying g(0)>0 and sup g(x)< .
x∈R ∞
Lemma 2.1. Let T =T +T . If Assumptions 2.1–2.6 hold, then
n 1n 2n
1
T N(0,Ω ) as n ,
n d 0
√n → →∞
where denotes the convergence in distribution and
d
→
w2 ∂ε (γ ) ∂ε (γ ) Eη2 1 w2 ∂h (θ ) ∂h (θ )
Ω =E t t 0 t 0 + t − E t t 0 t 0 .
0 h (θ ) ∂θ ∂θ′ 4 h2(θ ) ∂θ ∂θ′
(cid:18) t 0 (cid:19) (cid:18) t 0 (cid:19)
Lemma 2.2. If Assumptions 2.1–2.6 hold, then for any sequence of ran-
dom variables u such that u =o (1), it follows that
n n p
Π (u )=o (√n u +n u 2),
1n n p n n
k k k k
where o () 0 in probability as n .
p
· → →∞
Lemma 2.3. If Assumptions 2.1–2.6 hold, then for any sequence of ran-
dom variables u such that u =o (1), it follows that:
n n p
(i) Π (u )=(√nu )′Σ (√nu )+o (n u 2),
2n n n 1 n p n
k k
(ii) Π (u )=o (n u 2),
3n n p n
k k
(iii) Π (u )=(√nu )′Σ (√nu )+o (n u 2),
4n n n 2 n p n
k k
(iv) Π (u )=o (n u 2),
5n n p n
k k
n
(v) C (u )=o (n u 2),
t n p n
k k
t=1
X
where
w ∂ε (γ ) ∂ε (γ )
t t 0 t 0
Σ =g(0)E
1 h (θ ) ∂θ ∂θ′
(cid:18) t 0 (cid:19)
and
1 w ∂h (θ ) ∂h (θ )
t t 0 t 0
Σ = E .
2 8 h2(θ ) ∂θ ∂θ′
(cid:18) t 0 (cid:19)
The proofs of Lemmas 2.2 and 2.3 are given in Section 6. We now can
state one main result as follows:
Theorem 2.2. If Assumptions 2.1–2.6 hold, then:
(i) √n(θˆ θ ) = O (1),
sn 0 p
−
(ii) √n(θˆ θ ) N(0,1Σ−1Ω Σ−1) as n ,
sn− 0 →d 4 0 0 0 →∞
where Σ =Σ +Σ .
0 1 2
10 K. ZHU ANDS. LING
Proof. (i) First, we have uˆ = o (1) by Theorem 2.1. Furthermore,
n p
by (2.5), (2.8) and (2.9) and Lemmas 2.2 and 2.3, we have
(2.10) L (uˆ )=uˆ′ T +(√nuˆ )′Σ (√nuˆ )+o (√n uˆ +n uˆ 2).
n n n n n 0 n p k nk k nk
Let λ >0 be the minimum eigenvalue of Σ . Then
min 0
1
L (uˆ ) √nuˆ T +o (1) +n uˆ 2[λ +o (1)].
n n n n p n min p
≥−k k √n k k
(cid:20)(cid:13) (cid:13) (cid:21)
(cid:13) (cid:13)
Note that Ln(uˆn) 0. By t(cid:13)he previ(cid:13)ous inequality, it follows that
≤ (cid:13) (cid:13)
1
(2.11) √n uˆ [λ +o (1)]−1 T +o (1) =O (1),
n min p n p p
k k≤ √n
(cid:20)(cid:13) (cid:13) (cid:21)
(cid:13) (cid:13)
where the last step holds by Lemma 2(cid:13).1. Thu(cid:13)s, (i) holds.
(ii) Let u∗ = Σ−1T /2n. Then, by(cid:13)Lemma(cid:13)2.1, we have
n − 0 n
√nu∗ N(0,1Σ−1Ω Σ−1) as n .
n→d 4 0 0 0 →∞
Hence,itissufficienttoshowthat√nuˆ √nu∗ =o (1).By(2.10)and(2.11),
n− n p
we have
1
L (uˆ )=(√nuˆ )′ T +(√nuˆ )′Σ (√nuˆ )+o (1)
n n n n n 0 n p
√n
=(√nuˆ )′Σ (√nuˆ ) 2(√nuˆ )′Σ (√nu∗)+o (1).
n 0 n − n 0 n p
Note that (2.10) still holds when uˆ is replaced by u∗. Thus,
n n
1
L (u∗)=(√nu∗)′ T +(√nu∗)′Σ (√nu∗)+o (1)
n n n √n n n 0 n p
= (√nu∗)′Σ (√nu∗)+o (1).
− n 0 n p
By the previous two equations, it follows that
L (uˆ ) L (u∗)=(√nuˆ √nu∗)′Σ (√nuˆ √nu∗)+o (1)
n n − n n n− n 0 n− n p
(2.12)
λ √nuˆ √nu∗ 2+o (1).
≥ mink n− nk p
Since L (uˆ ) L (u∗)=n[L (θ +uˆ ) L (θ +u∗)] 0 a.s., by (2.12),
we haven √nnuˆ− n√nnu∗ =osn(1).0Thisnco−mpslnete0s thenpro≤of. (cid:3)
k n− nk p
Remark 2.1. When w 1, the limiting distribution in Theorem 2.2 is
t
≡
the same as that in Li and Li (2008). When r=s=0 (ARMA model), it
reducestothecaseinPan,WangandYao(2007)andZhuandLing(2011b).
In general, it is not easy to compare the asymptotic efficiency of the self-
weighted QMELE and the self-weight QMLE in Ling (2007). However, for
the pureARCH model, a formal comparison of these two estimators is given
in Section 3. For the general ARMA–GARCH model, a comparison based
on simulation is given in Section 4.