Asymptotic behavior of the Laplacian quasi-maximum likelihood estimator of affine causal processes

0.25 MB·

by  Jean-Marc Bardet

Asymptotic behavior of the Laplacian quasi-maximum likelihood
estimator of affine causal processes
Jean-Marc Bardet1 , Yakoub Boularouk2 and Khedidja Djaballah2
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1 1 S.A.M.M., Universit´e Panth´eon-Sorbonne, 90, rue de Tolbiac, 75634, Paris, France.
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2 2 Universit´e des Sciences et de la Technologie Houari Boum´ediene, Alger, Alg´erie

n
a
January 5, 2016
J

2

]
T
S
Abstract: We prove the consistency and asymptotic normality of the Laplacian Quasi-Maximum
.
h
Likelihood Estimator (QMLE) for a general class of causal time series including ARMA, AR( ),
at ∞
GARCH, ARCH( ), ARMA-GARCH, APARCH, ARMA-APARCH,..., processes. We notably ex-
m
∞
[ hibit the advantages (moment order and robustness) of this estimator compared to the classical

1 Gaussian QMLE. Numerical simulations confirms the accuracy of this estimator.
v
5
5 2000 AMS Classification Numbers: 62M10 (primary), 62F12 (secondary)
1
0 Key words: Laplacian Quasi-Maximum Likelihood Estimator; Strong consistency; Asymptotic
0
. normality; ARMA-ARCH processes.
1
0
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1 1 Introduction
:
v
i This paper is devoted to establish the consistency and the asymptotic normality of a parametric
X
r estimator for a general class of time series. This class was already defined and studied in Doukhan
a
and Wintenberger (2007), Bardet and Wintenberger (2009) and Bardet et al. (2012). Hence, we
willconsideranobservedsample(X1, ,Xn)where(Xt)t∈Z isasolutionofthefollowingequation:
···
X = M (X ,X , )ζ +f (X ,X , ), t Z, (1.1)
t θ0 t−1 t−2 ··· t θ0 t−1 t−2 ··· ∈
where
θ Θ Rd, d N∗, is an unknown vector of parameters, also called the ”true” parameters;
0
• ∈ ⊂ ∈
(ζt)t∈Z isasequenceofcentredindependentidenticallydistributedrandomvariables(i.i.d.r.v.)
•
with symmetric probability distribution, i.e. ζ =L ζ , satisfying E[ζ r] < with r 1
0 0 0
− | | ∞ ≥
and E[ζ ] = 1. If r 2, denote σ2 = Var(ζ );
| 0| ≥ ζ 0
(θ,(xn)n∈N) Mθ((xn)n∈N) (0, ) and (θ,(xn)n∈N) fθ((xn)n∈N) R are two known
• → ∈ ∞ → ∈
applications.
1
For instance, if M (X ,X , ) = 1 and f (X ,X , ) = α X with α < 1 then
θ0 t−1 t−2 ··· θ0 t−1 t−2 ··· 0 t−1 | 0|
(X ) is a causal AR(1) process. In Doukhan and Wintenberger (2007) and Bardet and Winten-
t
berger (2009), it was proved that all the most famous stationary time series used in econometrics,
such as ARMA, AR( ), GARCH, ARCH( ), TARCH, ARMA-GARCH processes can be written
∞ ∞
as a causal stationary solution of (1.1).
In Bardet and Wintenberger (2009), it was also established that under several conditions on M , f
θ θ
and if E[ζ r] with r 2, the usual Gaussian Quasi-Maximum Likelihood Estimator (QMLE) of θ
0
| | ≥
is strongly consistent and when r 4 it is asymptotically normal. This estimator was first defined
≥
by Weiss (1986) for ARCH processes, and the asymptotic study of this estimator was first obtained
by Lumsdaine (1996) for GARCH(1,1) processes, Berkes et al. (2003) for GARCH(p,q) processes,
Francq and Zakoian (2004) for ARMA-GARCH processes, Mikosch and Straumann (2006) for gen-
eral heteroskedastic models, and Robinson and Zaffaroni (2006) for ARCH( ) processes. The
∞
results of Bardet and Wintenberger (2009) devoted to processes satisfying almost everywhere (1.1)
as well as its multivariate generalisation, provide a general and unified framework for studying the
asymptotic properties of the Gaussian QMLE.
However, the definition of the Gaussian QMLE is explicitly obtained with the assumption that (ζ )
t
is a Gaussian sequence and even if it could be applied when the probability distribution of (ζ ) is
t
non-Gaussian, it keeps some drawbacks of this initial assumption. Indeed, the computation of this
estimators requires the minimization of a least squares contrast (typically n M−2(X f )2)
t=1 θ t − θ
and this induces that r =2 is required for the consistency and r = 4 for the asymptotic normality
P
(and therefore confidenceintervals or tests). For numerousreal data such requirementis sometimes
too strong (for instance the kurtosis of economic data is frequently considered as infinite). More-
over, such estimator is notrobustto eventual outliers. Hence, the referenceprobability distribution
of (ζ ) could be a Laplace one and this allows to avoid both these drawbacks. Roughly speaking
t
this choice implies to minimize a least absolute deviations contrast (typically n M−1 X f )
t=1 θ | t− θ|
instead of the previous least squares contrast. And therefore, r = 1 will be sufficient for insuring
P
the strong consistency of this Laplacian-QMLE, while only r = 2 is required for the asymptotic
normality (see below).
SuchprobabilitydistributionchoiceisnotnewsincethisleadstoaLeastAbsoluteDeviations(LAD)
estimation. Hence, for ARMA processes, Davis and Dunsmuir (1997) proved the consistency and
asymptotic normality of the LAD estimator. For ARCH or GARCH processes, the same results
concerning the LAD estimator were already established by Peng and Yao (2003), while Berkes
and Horva´th (2004) proved the consistency and asymptotic normality of the Laplacian-QMLE.
Newey and Steigerwald (1997) considered also the estimator for other conditional heteroskedastic-
ity models. Recently, Francq et al. (2011) proposed a two-stage non-Gaussian-QML estimation
for GARCH models and Francq and Zakoian (2013) proposed an alternative one-step procedure,
based on an appropriate non-Gaussian-QML estimator, the asymptotic properties of both these
approaches were studied.
In this paper we unified all these studies of the Laplacian-QMLE in a simple framework, i.e. causal
stationary solutionsof(1.1). Thisnotablyallows toobtainknownresultsonARMAorGARCHbut
also to establish for the first time the consistency and the asymptotic normality of the Laplacian-
QMLE for APARCH, ARMA-GARCH, ARMA-ARCH( ) and ARMA-APARCH processes.
∞
2
NumericalMonte-Carloexperimentswererealizedtoillustratethetheoreticalresults. Andthereare
convincing, especially when the accuracy of Laplacian-QMLE is compared with Gaussian-QMLE:
except for Gaussian distribution of (ζ ), the Laplacian-QMLE provides a sharper estimation than
t
Gaussian-QMLE for all the other probability distributions we considered. This is notably the case,
and this is not a surprise, for a Gaussian mixing which mimics the presence of outliers.
The following Section 2 will be devoted to provide the definitions and assumptions. In Section
3 the main results are stated with numerous examples of application, while Section 4 presents the
results of Monte-Carlo experiments and Section 5 contains the proofs.
2 Definition and assumptions
2.1 Definition of the estimator
Let (X , ,X ) be an observed trajectory of X which is an a.s. solution of (1.1) where θ Θ
1 n
··· ∈ ⊂
Rd is unknown. For estimating θ we consider the log-likelihood of (X , ,X ) conditionally to
1 n
···
(X ,X , ). If g is the probability density (with respect to Lebesgue measure) of ζ , then, from
0 −1 0
···
the affine causal definition of X, this log-likelihood can be written:
n X ft
log L (X , ,X ) = log g t− θ
θ 1 ··· n Mt
(cid:0) (cid:1) Xt=1 (cid:16) (cid:16) θ (cid:17)(cid:17)
where Mt := M (X ,X , ) and ft := f (X ,X , ), with the assumption that Mt > 0.
θ θ t−1 t−2 ··· θ θ t−1 t−2 ··· θ
However, Mt and ft are generally not computable since X ,X ,... are unknown. Thus, a quasi-
θ θ 0 −1
log-likelihood is considered instead of the log-likelihood and it is defined by:
n X ft
log QL (X , ,X ) = log g t− θ ,
θ 1 n
··· Mt
(cid:0) (cid:1) Xt=1 (cid:16) (cid:16) θ b (cid:17)(cid:17)
with ft := f (X ,...,X ,u) and Mt := M (X ,...,X ,u) , where u = (u ) is a finitely
θ θ t−1 1 θ θ t−1 1 c n n∈N
non-zero sequence (u ) . The choice of (u ) does not have any consequences on the asymp-
n n∈N n n∈N
b c
totic behaviour of L , and (u ) could typically be chosen as a sequence of zeros. Finally, a Quasi-
n n
Maximum Likelihood Estimator (QMLE) is defined by:
θ := Argmax log QL (X , ,X ) .
n θ∈Θ θ 1 ··· n
Usually, the Gaussian distribution is considered,(cid:0)with g(x) = (2π)−(cid:1)1/2e−x2/2 for x R and this
e
∈
provides the Gaussian-QMLE of θ.
Here, we chose the Laplacian distribution, i.e.,
1
g(x) = e−|x| for x R, (2.1)
2 ∈
and this implies E ζ = 1.
0
| |
Therefore, we respectively define the Laplacian-likelihood and Laplacian-quasi-likelihood by:
(cid:2) (cid:3)
n
L (Θ) = q (Θ) with q (Θ) = log Mt + Mt −1 X ft (2.2)
n − t t | θ| | θ| | t− θ|
t=1
X
n
L (θ) = q (θ) with q (θ):= log Mt + Mt −1 X ft . (2.3)
n − t t | θ| | θ| | t− θ|
t=1
X
b b b c c b
3
A Laplacian-QMLE θ is a maximizer of L :
n n
b θ b:= argmaxL (θ).
n n
θ∈Θ
We restrict the set Θ in such a way thabt a stationarybsolution (X ) of order 1 or 2 of (1.1) exists.
t
Additional conditions are also required for insuring the consistency and the asymptotic normality
of θ . More details are given now.
n
b
2.2 Existence and stationarity
As it was already donein Doukhan and Wintenberger (2007) and Bardet andWintenberger (2009),
several Lipshitz-type inequalities on f and M are required for obtaining the existence and r-order
θ θ
stationarity of an ergodic causal solution of (1.1). First, denote g = sup g with m N∗
k θkΘ θ∈Θk θk ∈
and the usual Euclidian norm. Now, let us introduce the generic symbol K for any of the
k ·k
functions f or M and for g : RN Rm or (R), For k = 0, 1, 2 and some subset Θ of Rd,
θ m
7→ M
define a Lipshitzian assumption on function K :
θ
Assumption (A (K,Θ)) x R∞, θ Θ K (x) k(Θ) and∂kK satisfies ∂kK (0) <
k ∀ ∈ ∈ 7→ θ ∈ C θ θ θ θ Θ ∞
and there exists a sequence α(k)(K,Θ) of nonnegative numbers such that x, y RN
j j ∀ (cid:13)(cid:13) ∈ (cid:13)(cid:13)
(cid:0) (cid:1)
∞ ∞
∂kK (x) ∂kK (y) α(k)(K,Θ)x y , with α(k)(K,Θ) <
θ θ − θ θ Θ ≤ j | j − j| j ∞
j=1 j=1
(cid:13) (cid:13) X X
(cid:13) (cid:13)
For ensuring a stationary r-order solution of (1.1), for r 1, define the set
≥
Θ(r):= θ Rd, (A (f, θ )) and (A (M, θ )) hold,
0 0
∈ { } { }
n ∞ ∞
α(0)(f, θ )+(E[ζ r])1/r α(0)(M, θ ) < 1 .
j { } | 0| j { }
Xj=1 Xj=1 o
Then, from Doukhan and Wintenberger (2007), we obtain:
Proposition 2.1. If θ Θ(r) for some r 1, then there exists a unique causal (X is independent
0 t
∈ ≥
of (ζ ) for t Z) solution X of (1.1), which is stationary, ergodic and satisfies E X r < .
i i>t 0
∈ | | ∞
(cid:2) (cid:3)
Finally, the following lemma insures that if a process X satisfies Proposition 2.1, a causal pre-
dictable ARMA process with X as innovation also satisfies Proposition 2.1. We first provide the
classical following notion for a sequence (un)n∈N of real numbers:
(un)n∈N is an exponentially decreasing sequence (EDS)
⇐⇒
there exists ρ [0,1[ such as u = (ρn) when n .
n
∈ O → ∞
Lemma 2.1. Let X be a.s. a causal stationary solution of (1.1) for θ Rd. Let X be such
0
∈
as X = Λ (L) X for t Z with Λ (L) = P−1(L)Q (L) where (P ,Q ) are the coprime
t β t ∈ β0 β0 β0 β0 β0 e
polynomials of a causal invertible ARMA(p,q) processes with a vector of parameters β Rp+q.
0
e ∈
4
Denote Λ−1(x) = Q−1(x)P (x) = 1+ ∞ ψ (β )xj. Then X is a.s. a causal stationary solution
β0 β0 β0 j=1 j 0
of the equation
P
e
Xt = Mθe0 (Xt−i)i≥1 ζt+fθe0 (Xt−i)i≥1) for t ∈Z,
where f and M are given in (cid:0)(5.1) and θ(cid:1) = (θ ,(cid:0)β ). Moreov(cid:1)er, for i = 0, 1, 2 and with K = f
θ0 θ0 e f e 0 e0 0e
or M and K = f or M,
e f e
if α(i)(K, θ ) = O(j−β) and β > 1, then α(i)(K, θ ) = O(j−β);
• j e {e0} f j { 0}
(i) (i)
• if αj (K,{θ0}) is EDS, then αj (K,{θ0}) is EDeS. e
e e
2.3 Assumptions required for the convergence of the Laplacian-QMLE
The Laplacian-QMLE could converge and be asymptotically Gaussian but this requires some ad-
ditional assumptions on Θ and functions f and M :
θ θ
Condition C1 (Compactness) Θ is a compact set.
•
Condition C2 (Lower bound of the conditional variance) There exists a deterministic con-
•
stant M > 0 such that for all θ Θ, then M (x) > M for all x RN.
θ
∈ ∈
Condition C3 (Identifiability) The functions M and f are such that: for all θ , θ Θ,
θ θ 1 2
• ∈
then M = M and f = f implies that θ = θ .
θ1 θ2 θ1 θ2 1 2
3 Consistency and asymptotic normality of the estimator
3.1 Consistency and asymptotic normality
First we prove the strong consistency of a sequence of Laplacian-QMLE for a solution of (1.1). The
proof of this theorem, is postponed in Section 5, as well as the other proofs.
Theorem 3.1. Assume Conditions C1, C2 and C3 hold and θ Θ(r) Θ with r 1. Let X be
0
∈ ∩ ≥
the stationary solution of (1.1). If (A (f,Θ)) and (A (M,Θ)) hold with
0 0
2
α(0)(f,Θ)+α(0)(M,Θ) = (j−ℓ) for some ℓ > (3.1)
j j O min(r, 2)
a.s.
then a sequence of Laplacian-QMLE (θ ) strongly converges, that is θ θ .
n n n 0
n−→→∞
Of course, the conditions required for tbhis strong consistency of a sequebnce of Laplacian-QMLE are
almost thesamethan theones requiredfor thestrongconsistency of a sequenceof Gaussian-QMLE
except that r [1,2) is proved to be possible in Theorem 3.1 and not in case of Gaussian-QMLE
∈
(see Bardet and Wintenberger, 2009). Moreover, if r = 2, the condition (3.1) on Lipshitzian co-
efficients is weaker for Laplacian-QMLE than for Gaussian-QMLE. As we will see below, many
usual time series can satisfy the assumptions of Theorem 3.1; for example, an AR( ) process can
∞
be defined for satisfying the strong consistency of Laplacian-QMLE while the conditions given in
Bardet and Wintenberger (2009) do not ensure the strong consistency of Gaussian-QMLE.
Now we state an extension of Theorem 1 proved in Davis and Dunsmuir (1997) which will be
an essential step of the proof of the asymptotic normality of the estimator.
5
Theorem 3.2. Let (Zt)t∈Z be a sequence of i.i.d.r.v such as Var(Z0) = σ2 < , with common
∞
distribution function which is symmetric (F( x) = 1 F(x) for x R) and is continuously
− − ∈
differentiable inaneighborhood of0withderivativef(0)in0. Denote = σ(Z ,Z , )fort Z
t t t−1
F ··· ∈
and let (Yt)t∈Z and (Vt)t∈Z two stationary processes adapted to (Ft)t and such as E Y02V02 < ∞.
Then
(cid:2) (cid:3)
n
V Z n−1/2Y Z D f(0)E V Y2 , E V2Y2 (3.2)
t−1 | t− t−1|−| t| n−→→∞ N 0 0 0 0
Xt=1 (cid:0) (cid:1) (cid:16) (cid:2) (cid:3) (cid:2) (cid:3)(cid:17)
Then, the asymptotic normality of the Laplacian-QMLE can be established using additional as-
sumptions:
◦ ◦
Theorem 3.3. Assume that θ0 Θ Θ(r) where r 2 and Θ denotes the interior of Θ. Let X be
∈ ∩ ≥
the stationary solution of the equation (1.1). Assume that the conditions of Theorem 3.1 hold and
for i = 1,2, assume (A (f,Θ)) and (A (M,Θ)) hold. Then, if the cumulative probability function
i i
of ζ is continuously differentiable in a neighborhood of 0 with derivative g(0) in 0 and if matrix
0
Γ or Γ , defined in (5.21), are definite positive symmetric matrix, then
F M
√n θ θ D 0 , Γ +2g(0)Γ −1 σ2 1 Γ +Γ Γ +2g(0)Γ −1 . (3.3)
n− 0 n−→→∞ Nd M F ζ − M F M F
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0)(cid:0) (cid:1) (cid:1)(cid:0) (cid:1)
As it wabs already proved for the median estimator (see van der Vaart, 1996) or for least absolute
deviations estimator ofARMAprocess(seeDavis andDunsmuir,1997), itisnotsurprisingthatthe
probability density function g of the white noise (ζ ) impacts the asymptotic covariance of (3.3).
i i
However, when f = 0, this is not such the case and this is what happens for GARCH processes
θ
(see Francq et al., 2011, or Francq and Zakoian, 2015) where g does not appear in the asymptotic
covariance.
3.2 Examples
In this section, several examples of time series satisfying the conditions of previous results are con-
sidered. Like it could be boring to state the results for all sufficiently famous processes, we refer,
mutatis mutandis, to Bardet and Wintenberger (2009) and Bardet et al. (2012) for ARCH( ) and
∞
TARCH( ).
∞
1/ APARCH processes. APARCH(δ,p,q) model has been introduced (see Ding et al, 1993)
as the solution of equations
X = σ ζ ,
t t t
(3.4)
 σδ = ω+ p α (X γ X )δ + q β σδ ,
 t i=1 i | t−i|− i t−i j=1 j t−j
where δ 1, ω > 0, 1 < γ < 1 aPnd α 0 for i = 1,...,p, βP 0 for j = 1,...,q with α ,β
≥ − i i ≥ j ≥ p q
q
strictlypositiveand β < 1. Hence,wedenotehereθ = δ,ω,α ,...,α ,γ ,...,γ ,β ,...,β .
j=1 j 1 p 1 p 1 q
Using L the usual backward operator such as LX = X , 1 q β Lj −1 exists and simple
P t t−1 (cid:0) − j=1 j (cid:1)
computations imply for t Z:
∈ (cid:0) P (cid:1)
q p
σδ = 1 β Lj −1 ω+ α (1 γ )δ(max(X ,0))δ +α (1+γ )δ( min(X ,0))δ
t − j i − i t−i i i − t−i
(cid:0) Xj=1 (cid:1) h Xi=1 i
= b + b+(max(X ,0))δ + b−(max( X ,0))δ.
0 i t−i i − t−i
i≥1 i≥1
X X
6
where b = w(1 q β )−1 and the coefficients (b+,b−) are defined by the recursion relations
0 − j=1 j i i i≥1
P
b+ = q β b+ +α (1 γ )δ with α (1 γ )= 0 for i> p
i k=1 k i−k i − i i − i (3.5)
b−i = Pqk=1βkb−i−k +αi(1+γi)δ with αi(1+γi)= 0 for i> p
with b+ = b− =0 for iP0. As a consequence, for APARCH model, ft 0 and Mt = σ . It is clear
i i ≤ θ ≡ θ t
thatα(0)(f,Θ)= 0andsimplecomputationsimplyα(0)(M,Θ) = sup max b+(θ)1/δ, b−(θ)1/δ .
j j θ∈Θ | j | | j |
q q
ThereforeA (f,Θ)holdsand β < 1impliesthatasequencedefinedbyu = β u for
0 j=1 j (cid:0)n k=1 k n−k (cid:1)
n large enough is such as (un)Pn∈N is an exponentially decreasing sequence and thePrefore A0(M,Θ)
holds. Thus for r 1, the stationarity set Θ(r) is defined by
≥
∞
Θ(r) = θ R2p+q+2 E ζ r 1/r max b+ 1/δ, b− 1/δ < 1 . (3.6)
∈ | 0| | j | | j |
n . (cid:0) (cid:2) (cid:3)(cid:1) Xj=1 (cid:0) (cid:1) o
NowthestrongconsistencyandasymptoticnormalityoftheLaplacian-QMLEforAPARCHmodels
can be established (see the proof in Section 5):
Proposition 3.1. Assume that X is a stationary solution of (3.4) with θ Θ where Θ is a
0
∈
compact subset of Θ(r) defined in (3.6). Then,
a.s.
1. If r = 1, then θ θ .
n 0
n−→→∞
2. If r = 2, and ifbΓ defined in (5.21) is a definite positive symmetric matrix, then
M
√n θ θ D 0 , (σ2 1) Γ−1 .
n− 0 n−→→∞ N2p+q+2 ζ − M
(cid:0) (cid:1) (cid:0) (cid:1)
b
To our knowledge, this is the first statement the asymptotic properties of Laplacian-QMLE for
APARCH processes.
2/ ARMA-GARCH processes. ARMA(p,q)-GARCH(p′,q′) processes have been introduced
by Ding et al. (1993) and Ling and McAleer (2003) as the solution of the system of equations
P (L) X = Q (L) ε ,
θ t θ t
(3.7)
 ε =σ ζ , with σ2 = c + p′ c ε2 + q′ d σ2
 t t t t 0 i=1 i t−i i=1 i t−i
P P
where 
• c0 > 0, ci ≥0 for i = 1,...,p′, di ≥ 0 for i= 1,...,q′, qi=′ 1di < 1 and cp′,dq′ positive;
P (x) = 1 a x a xp and Q (x) = 1 b x P b xq are coprime polynomials with
θ 1 p θ 1 q
• − −···− − −···−
p p
a < 1 and b < 1.
i=1| i| i=1| i|
P P
Let θ = (c0,c1,...,cp′,d1,...,dq′,a1,...,ap,b1,...,bq). We are going to use Lemma 2.1. Since
(ε ) is supposed to be a GARCH(p′,q′), then fε = 0 and Mε = 1 q′ d Lj −1 c +c ε2 +
t θ θ − j=1 j 0 1 t−1
···+cp′ε2t−p′ 1/2 and direct computations imply that the Lipsh(cid:0)i(cid:0)tz coPefficients o(cid:1)f (ε(cid:0)t) are such as
α(0)(fε, θ ) = 0 and α(0)(Mε, θ ) = β with 1+ ∞ β xj 1 q′ d xj = p′ c xj.
j { 0}(cid:1) j { 0} | j| j=1 j − j=1 j j=0 j
Therefore (α(0)(fε, θ )) and (α(0)(Mε, θ )) are EDS (see for instance Berkes and Horvath,
j { 0} j j { 0} j (cid:0) P (cid:1)(cid:0) P (cid:1) P
7
2004). Thus (A (fε, θ )) and (A (Mε, θ )) hold.
0 0 0 0
{ } { }
Considering the ARMA part and denoting (ψ ) such as 1+ ∞ ψ xj 1 ∞ a xj = 1
j j=1 j − j=1 j −
∞ b xj , then from Lemma 2.1 we deduce that:
j=1 j (cid:0) P (cid:1)(cid:0) P (cid:1) (cid:0)
P (cid:1)
(0)
α (f, θ ) = ψ
j { 0} | j| .
(0) j
( αj (M,{θ0}) ≤ k=1|ψk|×|βj−k|
P
(0) (0)
Thenwededucethat(α (f, θ )) and(α (M, θ )) areEDS,(A (f, θ ))and(A (M, θ ))
j { 0} j j { 0} j 0 { 0} 0 { 0}
hold, and X is a.s. a solution of (1.1) for θ included in the r-order stationarity set Θ(r) defined by
∞ ∞ j
Θ(r) = θ Rp+q+p′+q′+1 ψ (θ) + E ζ r 1/r ψ β < 1 . (3.8)
i 0 k j−k
∈ | | | | | |×| |
n . Xi=1 (cid:0) (cid:2) (cid:3)(cid:1) Xj=1Xk=1 o
Now the strong consistency and asymptotic normality of the Laplacian-QMLE for ARMA-GARCH
processes can be established:
Proposition 3.2. Assume that X is a stationary solution of (3.7) with θ Θ where Θ is a
0
∈
compact subset of Θ(r) defined in (3.8). Then,
a.s.
1. If r = 1, then θ θ .
n 0
n−→→∞
2. If r = 2, and ifbΓ and Γ defined in (5.21) are definite positive symmetric matrix, then the
f M
asymptotic normality (3.3) of θ holds.
n
This result is a new one and extendbs the previous results already obtained with Gaussian-QMLE
for such processes (see for instance, Ling and McAleer, 2003, and Bardet and Wintenberger, 2009).
3/ ARMA-ARCH( ) processes. ARMA(p,q)-ARCH( ) processes are a natural extension
∞ ∞
of ARMA-GARCH processes. They are the solution of the system of equations
P (L) X = Q (L) ε ,
θ t θ t
(3.9)
 ε = σ ζ , with σ2 = c + ∞ c ε2
 t t t t 0 i=1 i t−i
P
where 
c > 0, c 0 for i 1;
0 i
• ≥ ≥
P (x) = 1 a x a xp and Q (x) = 1 b x b xq are coprime polynomials with
θ 1 p θ 1 q
• − −···− − −···−
p p
a < 1 and b < 1.
i=1| i| i=1| i|
P P
ARCH( )processeswereintroducedbyRobinson(1991)andtheasymptoticpropertiesofGaussian-
∞
QMLE were studied in Robinson and Zaffaroni (2006), Straumann and Mikosch (2006) or Bardet
and Wintenberger (2009). Hence, we assume that there exists β = (β , ,β ) such as for
1 m
···
all i N, c = c(i,β), with c() a known function. Let θ = (β,a ,...,a ,b ,...,b ). We
i 1 p 1 q
∈ ·
are going to use Lemma 2.1. Since (ε ) is supposed to be an ARCH( ), then fε = 0 and
t ∞ θ
Mε = c(0,β)+ ∞ c(i,β)ε2 1/2 and direct computations imply that the Lipshitz coefficients
θ i=1 t−i
(cid:0) P (cid:1)
8
of (ε ) are such as α(0)(fε, θ ) = 0 and α(0)(Mε, θ ) = c(j,β ). Therefore we assume that there
t j { 0} j { 0} 0
exists ℓ > 1 such as
c(j,β )= j−ℓ) when j . (3.10)
0
O → ∞
(cid:0)
Thus (A (fε, θ )) and (A (Mε, θ )) hold.
0 0 0 0
{ } { }
Considering the ARMA part and denoting (ψ ) such as 1+ ∞ ψ xj 1 ∞ a xj = 1
j j=1 j − j=1 j −
∞ b xj , then from Lemma 2.1 we deduce that:
j=1 j (cid:0) P (cid:1)(cid:0) P (cid:1) (cid:0)
P (cid:1)
(0)
α (f, θ ) = ψ
j { 0} | j| .
(0) j
( αj (M,{θ0}) ≤ k=1|ψk|×c(j,β0)
P
Then we deduce that (α(0)(f, θ )) is EDS and (α(0)(M, θ )) = j−ℓ). Then (A (f, θ ))
j { 0} j j { 0} j O 0 { 0}
and (A (M, θ )) hold, and X is a.s. a solution of (1.1) for θ included in the r-order stationarity
0 0 (cid:0)
{ }
set Θ(r) defined by
∞ ∞ j
Θ(r)= θ Rp+q+m ψ (θ) + E ζ r 1/r ψ c(j,β ) < 1 . (3.11)
i 0 k 0
∈ | | | | | |×
n . Xi=1 (cid:0) (cid:2) (cid:3)(cid:1) Xj=1Xk=1 o
NowthestrongconsistencyandasymptoticnormalityoftheLaplacian-QMLEforARMA-ARCH( )
∞
processes can be established:
Proposition 3.3. Assume that X is a stationary solution of (3.9) where (3.10) holds and with
θ Θ where Θ is a compact subset of Θ(r) defined in (3.11). Then,
0
∈
a.s.
1. If r 1 and ℓ 2/min(r,2), then θ θ .
n 0
≥ ≥ n−→→∞
2. If r = 2, ℓ > 1 and if ∂ic(j,β) = b j−ℓ for i = 1, 2, and if Γ and Γ defined in (5.21)
β O f M
are definite positive symmetric matrix, then the asymptotic normality (3.3) of θ holds.
(cid:0) (cid:1) n
This result is a new one. Note that ℓ > 1 and r = 2 is required for the asymptoticbnormality of
Laplacian-QMLE while r = 4 and ℓ > 2 is required for Gaussian-QMLE for such processes (see for
instance Bardet and Wintenberger, 2009). This confers a clear advantage to Laplacian-QMLE.
4/ ARMA-APARCH processes. The ARMA(p,q)-APARCH(p′,q′) processes have been also
introduced by Ding et al. (1993) as the solutions of the equations
P (L)X = Q (L)ε ,
θ t θ t
(3.12)
 ε = σ ζ , with σδ = ω+ p′ α (ε γ ε )δ + q′ β σδ
 t t t t i=1 i | t−i|− i t−i j=1 j t−j
P P
where: 
δ 1, ω > 0, 1< γi < 1 and αi 0 for i= 1,...,p′ 1, βj 0 for j = 1,...,q′ 1, αp′, βq′
• ≥ − p′ ≥ − ≥ −
positive real numbers and α < 1;
j=1 j
P (x) = 1 a x a xPp and Ψ (x) = 1 b x b xq are coprime polynomials with
θ 1 p θ 1 q
• − −···− − −···−
p q
a < 1 and b < 1 .
i=1| i| i=1| i|
P P
9
Letθ = (δ,ω,α1,...,αp′,γ1,...,γp′,β1,...,βq′,a1,...,ap,b1,...,bq). Then,asforARMA-GARCH
processes, we are going to use Lemma 2.1. Thanks to the computations realized for APARCH
processes, we obtain α(0)(fε, θ ) = 0 and α(0)(Mε, θ ) = max(b+ 1/δ, b− 1/δ) with (b+,b−)
j { 0} j { 0} | j | | j | i i i≥1
defined in (3.5).
Then, we have
(0)
α (f, θ ) ψ
j { 0} ≤ | j| .
( α(j0)(M,{θ0}) ≤ jk=1|ψk|×max |b+j−k|1/δ, |b−j−k|1/δ
(ψ ) such as 1+ ∞ ψ xj 1 ∞ aPxj = 1 ∞ (cid:0)b xj . From Lemm(cid:1)a 2.1, (A (f,Θ)) and
j j=1 j − j=1 j − j=1 j 0
(A (M,Θ)) hold since (α(0)(fε, θ )) = 0 and (α(0)(Mε, θ )) are EDS. As a consequence, for
0 (cid:0) P j (cid:1)(cid:0) {P0} j (cid:1) (cid:0) j P { 0}(cid:1) j
r 1, the stationarity set Θ(r) is defined by
≥
∞ ∞ j
Θ(r)= θ Rp+q+p′+q′+2 ψ + E ζ r 1/r ψ max b+ 1/δ, b− 1/δ < 1 .
∈ | j| 0| | k|× | j−k| | j−k|
n . Xj=1 (cid:0) (cid:12) (cid:3)(cid:1) Xj=1Xk=1 (cid:0) (cid:1) o
(cid:12)
Now, we are able to provide the asymptotic properties of QMLE for ARMA-APARCH models.
Proposition 3.4. Assume that X is a stationary solution of (3.12) with θ Θ where Θ is a
0
∈
compact subset of Θ(r) defined in (3.8). Then,
a.s.
1. If r = 1, then θ θ .
n 0
n−→→∞
2. If r = 2, and ifbΓ and Γ defined in (5.21) are definite positive symmetric matrix, then the
f M
asymptotic normality (3.3) of θ holds.
n
ThisresultisstatedforthefirsttimeforLaplacian-QMLE.ThecaseofGaussian-QMLEforARMA-
b
APARCH could be also obtained following the previous decomposition and the paper Bardet and
Wintenberger (2009). Once again, the asymptotic normality of Laplacian-QMLE only requires
r = 2 while this requires r = 4 for Gaussian-QMLE.
4 Numerical Results
To illustrate the asymptotic results stated previously, we realized Monte-Carlo experiments on
LQL
the bevarior of Laplacian-QMLE (denoted θ ) for several time series models, sample sizes and
n
probability distributions. A comparison with the results obtained by Gaussian QMLE (denoted
GQL b
θ ) is also proposed.
n
More precisely, the considered probability distributions of (ζ ) are:
t
b
Centred Gaussian distribution denoted ;
• N
Centred Laplacian distribution denoted ;
• L
Centred Uniform distribution denoted ;
• U
Centred Student distribution with 3 freedom degrees, denoted t ;
3
•
Normalized centred Gaussian mixture with probability distribution 0.05 ( 2,0.16) +
• ∗ N −
(0,1)+0.05 (2,0.16) and denoted .
N ∗N M
10