Goodness-of-fit tests for extended Log-GARCH models

0.51 MB·

by  Christian Francq

Goodness-of-fit tests for log and exponential GARCH models
Christian Francq, Olivier Wintenberger and Jean-Michel Zakoïan
∗ † ‡
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2 Abstract

n
u This paper studies goodness of fit tests and specification tests for an extension of the Log-
J
GARCH model which is both asymmetric and stable by scaling. A Lagrange-Multiplier test is
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derivedfortestingtheextendedLog-GARCHagainstmoregeneralformulationstakingtheform
]
T
ofcombinationsofLog-GARCHandExponentialGARCH(EGARCH). The nullassumptionof
S
. an EGARCH is also tested. Portmanteau goodness-of-fit tests are developed for the extended
h
t Log-GARCH. An application to real financial data is proposed.
a
m
Keywords: EGARCH,LMtests,Invertibilityoftimeseriesmodels,log-GARCH,Portmanteau
[
tests, Quasi-Maximum Likelihood

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v Mathematical Subject Classifications: 62M10; 62P20
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It is now widely accepted that, to model the dynamics of daily financial returns, volatility models
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5
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0 havetoincorporatetheso-calledleverageeffect. AmongthevariousasymmetricGARCHprocesses
.
1 introduced in the econometric literature, E(xponential)GARCH and Log-GARCH models share
0
6 the property of specifying the dynamics of the log-volatility, rather than the volatility, as a linear
1
:
v combination ofpastvariables. Oneadvantage ofsuchspecifications istoavoid positivityconstraints
i
X on the parameters, which complicate statistical inference of standard GARCH formulations. A
r
a class of (asymmetric) Log-GARCH(p,q) models was recently studied by Francq, Wintenberger and
Zakoïan (2013) (FWZ). In this class, originally introduced by Geweke (1986), Pantula (1986) and
Milhøj (1987) (see Sucarrat, Grønneberg and Escribano (2015) for a more recent reference), the
∗CREST and University Lille 3 (EQUIPPE), BP 60149, 59653 Villeneuve d’Ascq cedex, France. E-Mail:
[email protected]
†Universities of Paris 6 and Copenhagen, LSTA 4 Place Jussieu, 75005 Paris, France. E-Mail:
[email protected]
‡Corresponding author: Jean-Michel Zakoïan, EQUIPPE (University Lille 3) and CREST, 15 boulevard Gabriel
Péri, 92245 Malakoff Cedex,France. E-mail: [email protected], Phone number: 33.1.41.17.77.25.
1
This effect, typically observed on most stock returns series, means that negative returns have more impact on
thevolatility than positive returnsof the same magnitude.
1
dynamics is defined by
ǫ = σ η ,
t t t
 logσ2 = ω+ q α 1 +α 1 logǫ2 (0.1)
 t i=1 i+ {ǫt−i>0} i− {ǫt−i<0} t−i
 + Ppj=1βj(cid:0)logσt2 j (cid:1)
−


 P
where σ > 0 and(η ) is a sequence of independent and identically distributed (iid) variables such
t t
that Eη2 = 1.
1
One drawback of this model is that it is generally not stable by scaling. Indeed, if (ǫ ) is a
t
solution of Model (0.1), the process (ǫ ) defined by ǫ = cǫ with c > 0 satisfies ǫ = σ η with
∗t ∗t t ∗t t∗ t
σ 2 = ω + q α 1 +α 1 logǫ 2 + p β logσ 2 where
t∗ t∗−1 i=1 i+ {ǫ∗t−i>0} i− {ǫ∗t−i<0} ∗t−i j=1 j t∗−j
(cid:16) (cid:17)
P P
p q
ω = logc2 1 β α 1 +α 1
t∗−1  − j − i+ {ǫ∗t−i>0} i− {ǫ∗t−i<0} 
Xj=1 Xi=1(cid:16) (cid:17)
 
is not constant (except in the symmetric case where α = α for all i). It is important that
i+ i
−
2
a volatility model be stable by scaling. The standard log-GARCH has the stability by scaling
property, but is not able to capture the leverage effect.
In this paper, we will consider an extension of Model (0.1) which is both stable by scaling and
asymmetric. Our main foci concern specification tests of this model and the comparison with the
EGARCH model. The latter formulation, introduced by Nelson (1991), appears as a widely used
competitor of the Log-GARCH in applications. As we will see, the two models display very similar
properties and their volatility dynamics may coincide. However, the Log-GARCH and EGARCH
models are not equivalent from a statistical point of view. In particular, it is obvious to invert
the Log-GARCH model, i.e. to express the volatility as an explicit function of the past returns,
whereas the EGARCH(1,1) is invertible only under strong restrictions on the parameters. This is
a major drawback for the statistical inference of the second specification, see Wintenberger (2013)
and FWZ. However, the two models are not compatible for a same series and one has to discuss if
one specification is more likely to fit the data at hand than the other. It is therefore of interest to
develop testing procedures for one specification against the other. This constitutes the main aim of
the present paper.
2
Indeed, as remarked by a referee, a practitioner is essentially faced by three choices: (a) leave returns untrans-
formed, i.e. set c=1, (b) expressreturnsin terms of percentages, i.e. set c=100, or (c) express returnsin terms of
basis points, i.e. set c=10,000. Clearly, it is desirable that the dynamics of thevolatility model be not affected by
thechoice of c.
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The remainder of the paper is organized as follows. Section 1 introduces the extended Log-
GARCH model and discusses its similarities with the EGARCH. It also provides strict stationarity
conditions. Section 2 studies the asymptotic properties of the quasi-maximum likelihood (QML)
estimator. Section 3 considers testing the null assumption of a Log-GARCH against more general
formulations including the EGARCH. Section 4 considers the reverse problem, in which the null
assumption is the EGARCH model. In Section 5, Portmanteau goodness-of-fit tests are developed
for the Log-GARCH. Section 6 compares the Log-GARCH and EGARCH models for series of
exchange rates.
1 Extended Log-GARCH model
Consider the Asymmetric and stable by Scaling Log-GARCH (AS-Log-GARCH) model of order
(p,q), defined by
ǫ = σ η ,
t t t
 logσ2 = ω+ q ω 1 + p β logσ2 (1.1)
 t i=1 i− {ǫt−i<0} j=1 j t−j
 + Pqi=1 αi+1{ǫt−i>0}+αPi−1{ǫt−i<0} logǫ2t−i,

where ω and thecomponents of Pthe v(cid:0)ectors ω = (ω ,...,ω (cid:1)), α = (α ,...,α ),
 1 q ′ + 1+ q+ ′
− − −
α = (α ,...,α ), and β = (β ,...,β ) are real coefficients, which are not a priori subject to
1 q ′ 1 p ′
− − −
positivity constraints, under the same assumptions on (η ) as in Model (0.1). The main features
t
of the asymmetric Log-GARCH(p,q) model - volatility which is not bounded below, persistence of
small values, power-aggregation - continue to hold in this extended version. We refer the reader
to FWZ for details. Contrary to Model (0.1), the extended formulation (1.1) is stable by scaling.
Moreover, this model leads to a different interpretation of the usual leverage effect.
1.1 News Impact Curves
Compared to model (0.1), the AS-Log-GARCH model (1.1) contains additional asymmetry pa-
rameters. Through the introduction of the coefficients ω , Model (1.1) allows for an asymmetric
i
−
impactofthepastpositiveandnegative returns onthelog-volatility whichdoesnotdependontheir
magnitudes. For instance, consider the AS-Log-ARCH(1) model with α = α = α. We have
1+ 1
−
σ2 = eω+ω1−1{ǫt−1<0}(ǫ2 )α.
t t 1
−
If ω > 0, a decrease of the price, whatever its amplitude, will increase the volatility by a scaling
1
−
factor eω1−. In the limit case where α = 0, the volatility takes only two values depending only on
3
6 6
t=0 w −=0,t=0
t=0.2 w −=0.5,t=0
5 t=0.4 5 w −=0,t=0.5
t=0.6 w −=0.5,t=0.5
4 t=0.8 4
2t 3 2t 3
s s
2 2
1 1
0 0
−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3
et-1 et-1
6
w −=0
8 5 w −=0.2
a =0.2 w −=0.4
a =0.4 w −=0.6
6 a =0.6 4 w −=0.8
a =0.8
2t a =1 2t 3
s 4 s
2
2
1
0 0
−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3
et-1 et-1
Figure 1: News Impact Curves: σt as a function of ǫt−1 in (1.2). The parameter ω is set to 0. The top
graphs are obtained for τ = 0, the left graphs for ω− = 0, the right graphs and the bottom left graph for
α=0.8.
the sign (not the size) of the past return. Now we turn to the second leverage effect. If α = α
1+
and α = α+τ with τ > 0, we have
1
−
σ2 = eω+ω1−1{ǫt−1<0}(ǫ2 )α(ǫ2 )τ1{ǫt−1<0}. (1.2)
t t 1 t 1
− −
The effect of a large negative return (ǫ < 1) is an increase of volatility, but the effect may
t 1
− −
be reversed for very small returns. For small but not too small returns, this effect is balanced by
the presence of the scaling factor eω1−. To summarize, the AS-Log-GARCH is in fact capable of
detecting two types of leverage: one type where the leverage effect depends on the magnitude of
negative return, and one type in which it does not. The so-called News Impact Curves, displaying
σ as a function of ǫ , are provided in Figure 1.
t t 1
−
1.2 Similarities with the EGARCH dynamics
The dynamics of the logarithm of the volatility of the EGARCH(p,ℓ) model is provided by the
recursion
p ℓ
logσ2 = ω˜ + β˜ logσ2 + γ η˜+ +γ η˜ , (1.3)
t j t j k+ t k k t−k
− − − −
j=1 k=1
X X
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where the innovations η˜ are iid random variables such that Eη˜2 = 1, with the notation x+ =
t 1
max x,0 and x = max x,0 . If one substitutes logσ2 + logη2 for logǫ2 in (1.1), the
{ } − {− } t i t i t i
− − −
probabilistic structures of the two classes of models seem similar. More precisely, we have the
following result.
Proposition 1.1 (i) For any EGARCH process ˜ǫ = σ η˜ satisfying (1.3) with Ees0η˜1 < for
t t t |
| ∞
some s > 0, there exists a AS-Log-GARCH process ǫ = σ η satisfying (1.1), with the same
0 t t t
volatility process σ and η measurable with respect to η˜.
t t t
(ii) Conversely, there exist AS-Log-GARCH processes ǫ = σ η for which there is no EGARCH
t t t
process ǫ˜ = σ η˜ with the same volatility process σ and η˜ measurable with respect to η .
t t t t t t
Proof: Let us prove (i). For simplicity of notation, we assume that ǫ˜ = σ η˜ follows the first
t t t
order EGARCH(1,1) model, and we drop the indexes i, j and k. Let the Log-GARCH(1,1) process
ǫ = σ η satisfying (1.1) with the parameters α:= α+ = α = 0, ω+αc = ω˜, ω = α(c c ),
t t t + +
− 6 − − −−
and α+β = β˜, and the noise ηt = ec2+eγ2+α|η˜t|1η˜t≥0 −ec2−eγ2−α|η˜t|1η˜t<0, with constants c+ and c− to
be chosen later. The Log-GARCH volatility then satisfies
logσ2 = ω+ω 1 +αlogη2 +(α+β)logσ2
t ηt−1<0 t 1 t 1
− − −
= ω˜ +(γ 1 +γ 1 )η˜ +β˜logσ2 ,
+ η˜t−1>0 − η˜t−1<0 | t−1| t−1
which is the equation satisfied by the volatility of the EGARCH(1,1) model. It then suffices to
choose α such that γ /α < s and γ /α < s , and then c and c such that Eη2 = 1.
+ 0 0 + t
− −
Now we turn to (ii). Let (ǫ ) denote any AS-Log-GARCH process satisfying (1.1), with α =
t 1+
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α ,andsufficientlygeneralsothatthesupportofthelawoflogσ2 containsatleastthreedifferent
1 t 1
− −
values. Alsoassumethat logη2 has afinite variance. We proceed by contradiction. Suppose there
t 1
−
exists an EGARCH process satisfying ǫ˜ = σ η˜ with η˜ = f(η ) for some measurable function f.
t t t t t
We thus have
logσ2 = ω+ω 1 + α 1 +α 1 logη2
t − ηt−1<0 1+ {ǫt−1>0} 1− {ǫt−1<0} t−1
p
(cid:0) (cid:1)
+ α 1 +α 1 logσ2 + β logσ2
1+ {ǫt−1>0} 1− {ǫt−1<0} t−1 j t−j
j=1
(cid:0) (cid:1) X
= ω˜ +(γ 1 +γ 1 )η˜ +β˜logσ2
+ η˜t−1>0 − η˜t−1<0 | t−1| t−1
p ℓ
+ β˜ logσ2 + γ η˜+ +γ η˜ ,
j t j k+ t k k t−k
− − − −
j=1 k=2
X X
which entails
a(η ) = b +c(η )logσ2
t 1 t 2 t 1 t 1
− − − −
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where b denotes a variable belonging to σ-field generated by the η with j 0. We
t 2 t 2 t 2 j
− F− − − ≥
have
0 = var a(η ) b c(η )logσ2
{ t−1 − t−2− t−1 t−1|Ft−2}
= var a(η ) +log2σ2 var c(η ) 2logσ2 cov a(η ),c(η ) ,
{ t−1 } t−1 { t−1 }− t−1 { t−1 t−1 }
fromwhichitfollowsthatlogσ2 takesatmosttwovalues. Thiscontradictstheaboveassumptions.
t 1
−
✷
This proposition allows to complete the interpretation of the two types of leverage effects in
the AS-Log-GARCH. The coefficients ω produce the leverage effect of the EGARCH volatility,
0,i
−
i.e. an asymmetry depending on the amplitude of the innovations η˜ . On the opposite, the
t i
−
EGARCH model cannot capture the asymmetric effect induced by the coefficients α ,α and
0,i 0,i+
−
the amplitude of the returns ǫ . Thus, the class of the Log-GARCH models generates a richer
t i
−
class of volatilities than the EGARCH.
1.3 Strict stationarity
We now show that the introduction of a time varying intercept in the log-volatility of Model (1.1)
does not modify the strict stationarity conditions of the Log-GARCH model. The study being
very similar to that of the Log-GARCH model (0.1) in FWZ, details are omitted. Let ω = ω +
t
q
ω 1 . Because coefficients equal to zero can always be added, it is not restrictive to
i=1 i− {ǫt−i<0}
Passume p > 1 and q > 1. Let the vectors
ǫ+ = (1 logǫ2,...,1 logǫ2 ) Rq,
t,q {ǫt>0} t {ǫt−q+1>0} t−q+1 ′ ∈
ǫ = (1 logǫ2,...,1 logǫ2 ) Rq,
−t,q {ǫt<0} t {ǫt−q+1<0} t−q+1 ′ ∈
z = (ǫ+ ,ǫ ,logσ2,...,logσ2 ) R2q+p,
t t,q −t,q t t−p+1 ′ ∈
bt = (ωt+logηt2)1{ηt>0},0′q−1,(ωt+logηt2)1{ηt<0},0′q−1,ωt,0′p−1 ′ ∈ R2q+p,
(cid:0) (cid:1)
and the matrix
1 α 1 α 1 β
ηt>0 ′+ ηt>0 ′ ηt>0 ′
{ } { } − { }
 
I 0 0 0
q 1 q 1 (q 1) q (q 1) p
− − − × − ×
 
 1 α 1 α 1 β 
Ct =  {ηt<0} ′+ {ηt<0} ′− {ηt<0} ′ .
 0 I 0 0 
 (q 1) q q 1 q 1 (q 1) p 
 − × − − − × 
 α α β 
 ′+ ′ ′ 
 − 
 0(p 1) q 0(p 1) q Ip 1 0p 1 
 − × − × − − 
 
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Model (0.1) is rewritten in matrix form as
z = C z +b .
t t t 1 t
−
Let γ(C) be the top Lyapunov exponent of the sequence C = C ,t Z ,
t
{ ∈ }
1 1
γ(C)= lim E(log C C ...C ) = inf E(log C C ...C ).
t t 1 1 t t 1 1
t t k − k t 1 t k − k
→∞ ≥
It can be noted that the sequence (C ,b ) is only strictly stationary and ergodic (not iid) but this
t t
property suffices to extend the proof of Theorem 2.1 in FWZ.
Theorem 1.1 Assume that Elog+ logη2 < . A sufficient condition for the existence of a
| 0| ∞
strictly stationary solution to the AS-Log-GARCH model (1.1) is γ(C)< 0. When γ(C)< 0, there
exists only one stationary solution, which is non anticipative and ergodic.
It follows that the presence of the coefficients ω does not modify the stationarity condition.
i
−
2 QML estimation of the AS-Log-GARCH model
We turn to the inference of the AS-Log-GARCH model. Let ǫ ,...,ǫ be observations of the sta-
1 n
tionary solution of (1.1), where θ = (ω,ω ,α ,α ,β ) is equal to an unknown value θ belonging
′ ′+ ′ ′ ′ 0
− −
to some parameter space Θ Rd, with d= 3q+p+1. A QMLE of θ is defined as any measurable
0
⊂
solution θ of
n
θ = arg minQ (θ), (2.1)
b n n
θ Θ
∈
with b e
n ǫ2
Q (θ) = n 1 ℓ (θ), ℓ (θ) = t +logσ2(θ),
n − t t σ2(θ) t
t=Xr0+1 t
where r is a fixede integer and logeσ2(θ) iserecursively definede by logσ2(θ) = ω +
0 t e t
q α logǫ2 1 +(ω +α logǫ2 )1 + p β logσ2 (θ), fort = 1,2,...,n,
i=1 i+ t−i {ǫt−i>0} i− i− e t−i {ǫt−i<0} j=1 j t−j e
Pusing(cid:0)initial values for ǫ0,...,ǫ1 q,σ02(θ),...,,σ12 p(θ).(cid:1)WePassume that these initial values are
− − e
such that there exists a real random variable K independent of n satisfying
e e
sup logσ2(θ) logσ2(θ) < K, a.s. for t = q p+1,...,q, (2.2)
t − t −
θ Θ
∈
(cid:12) (cid:12)
where σ2(θ) is defined(cid:12)by e (cid:12)
t
(B)logσ2(θ) = ω+ (B)1 + +(B)1 logǫ2
Bθ t Oθ− {ǫt<0} Aθ {ǫt>0} t
+ (B)1 logǫ2, (2.3)
A−θ {ǫt<0} t
7
where B is the the lag operator and, for any θ Θ, +(z) = q α zi, (z) = q α zi,
∈ Aθ i=1 i,+ A−θ i=1 i,−
and Bθ(z) = 1− pj=1βjzj and Oθ−(z) = qi=1ωi−zi. By convPention, A+θ(z) = 0, A−θP(z) = 0 and
Oθ−(z) = 0 if q =P0, and Bθ(z) = 1 if p = 0P. Theorem 1.1 shows that a strict stationarity condition
of the Log-GARCH can be obtained from the behaviour of the sequence C. As in FWZ, it can be
shown that moment conditions can be obtained by constraining the matrix
µ (η ) ... µ (η ) µ (η )
1 t 1 r 1 t r+1 r t r
A = − − − − , (2.4)
t
 
I 0
r 1 r 1
− −
 
where r = max(p,q) and µ (η )= α 1 +α 1 +β with the convention α = α = 0
i t i+ {ηt>0} i− {ηt<0} i i+ i−
for i > p and β = 0 for i > q. The spectral radius of a square matrix A is denoted by ρ(A). For
i
any vector or matrix A, we denote by Abs(A) the matrix whose elements are the absolute values
of the corresponding elements of A.
The following assumptions will be used to establish the strong consistency and asymptotic
normality of the QMLE.
A1: θ Θ and Θ is compact.
0
∈
A2: γ C < 0 and θ Θ, (z) = 0 z > 1.
θ
{ } ∀ ∈ |B | ⇒ | |
A3: the support of η contains at least two positive values and two negative values, Eη2 = 1
0 0
and E logη2 s0 < for some s > 0.
| 0| ∞ 0
A4: If p > 0 and q > 1, there is no common root to the polynomials (z), + (z), (z)
Oθ−0 Aθ0 A−θ0
and (z). Moreover (ω ,α ,α ) = 0 and ω α α + β = 0 if p > 0.
Bθ0 0− 0+ 0− 6 | 0q−|| 0q+|| 0q−| | 0p|6
A5: E logǫ2 < .
t ∞
(cid:12) (cid:12)
A6: θ0(cid:12)∈Θ◦ an(cid:12)d κ4 := E(η04) < ∞.
A7: There exists some s > 0 such that Eexp(s logη2 ) < and ρ esssupAbs(A ) < 1,
0 0| 0| ∞ { 1 }
where A is defined by (2.4).
1
In the case p = q = 1, omitting the index i, Assumption A2 simplifies to the conditions α +
0+
|
β a α +β 1 a < 1, where a = P(η > 0), and β < 1, θ Θ (see FWZ, Example 2.1).
0 0 0 − 0
| | − | | | ∀ ∈
Let Q = ( Q,..., Q) and HQ = (H Q,...,H Q) be the vector and matrix of the
1 d ′ 1. ′ d. ′ ′
∇ ∇ ∇
first-order and second-order partial derivatives of a function Q :Θ R.
→
8
Theorem 2.1 (Asymptotic properties of the QMLE) Let (θ ) be a sequence of QMLE satis-
n
fying (2.1), where (ǫ ) is the stationary solution of the AS-Log-GARCH model (1.1) with parameter
t b
θ . Under the assumptions (2.2) and A1-A5, θ θ a.s. as n . If, moreover, A6-A7 hold
0 n 0
→ → ∞
we have √n(θ θ ) d (0,(κ 1)J 1) as n , where J = E[ logσ2(θ ) logσ2(θ )] is a
n− 0 → N 4− − b→ ∞ ∇ t 0 ∇ t 0 ′
d
positive definite matrix and denotes convergence in distribution.
b →
Proof: The proof is similar to those of Theorems 4.1-4.2 of FWZ. We will only show the identifia-
bility of the extended model, that is,
σ2(θ) = σ2(θ ) a.s. θ = θ .
1 1 0 ⇒ 0
Note that if the left-hand side holds, by stationarity we have logσ2(θ)= logσ2(θ ) for all t. From
t t 0
the equality (2.3) we then have, almost surely,
Oθ−(B) Oθ−0(B) 1 + A+θ(B) A+θ0(B) 1 logǫ2
(Bθ(B) − Bθ0(B)) {ǫt<0} (Bθ(B) − Bθ0(B)) {ǫt>0} t
+ A−θ(B) A−θ0(B) 1 logǫ2 = ω0 ω .
(Bθ(B) − Bθ0(B)) {ǫt<0} t Bθ0(1) − Bθ(1)
Throughout thepaperlet R denote any generic random variable, whosevaluecan bemodified from
t
one line to the other, which is measurable with respect to σ( η ,u t ). If
u
{ ≤ }
Oθ−(B) = Oθ−0(B) or A+θ(B) = A+θ0(B) or A−θ(B) = A−θ0(B), (2.5)
(B) 6 (B) (B) 6 (B) (B) 6 (B)
Bθ Bθ0 Bθ Bθ0 Bθ Bθ0
there exists a non null (c ,c ,d ) R3, such that
+
− − ∈
d 1 +c 1 logǫ2+c 1 logǫ2+R = 0 a.s.
− ηt<0 + {ηt>0} t − {ηt<0} t t−1
This is equivalent to the two equations
c logη2+c logσ2+R 1 = 0 a.s.
+ t + t t−1 {ηt>0}
(cid:0) (cid:1)
and
d +c logη2+c logσ2+R 1 = 0 a.s.
− − t − t t−1 {ηt<0}
(cid:0) (cid:1)
Note that if an equation of the form alogx21 + b1 = 0 admits two positive solutions
x>0 x>0
{ } { }
then a = 0. This result, A3, and the independence between η and (σ2,R ) imply that c = 0
t t t 1 +
−
and R = 0. Similarly we obtain c = 0. Plugging c = c = 0 in the equations above yields
t 1 +
− − −
c = c = d = 0 that is a contradiction. We conclude that (2.5) cannot hold true, and the
+
− −
conclusion follows from A4. ✷
9
3 Test of AS-Log-GARCH
In this section, we are interested in testing the AS-Log-GARCH specification against more general
formulations, including both the Log-GARCH and the EGARCH models. For our testing problem,
we therefore introduce the general model
ǫ = σ η ,
t t t
 logσ2 = ω + q ω 1
 t 0 i=1 0,i− {ǫt−i<0} (3.1)
 + qi=P1 α0,i+1{ǫt−i>0}+α0,i−1{ǫt−i<0} logǫ2t−i
 +Ppj=1(cid:0)β0jlogσt2−j + ℓk=1γ0,k+ηt+−k +(cid:1)γ0,k−ηt−−k.


 P P
Let ϑ = (θ ,γ ) where γ = (γ ,γ ,...,γ ) and θ is as in Section 2.
0 ′0 ′0 ′ 0 01,+ 01, 0ℓ, ′ 0
− −
We wish to test the hypothesis that, in (3.1),
Hγ :γ = 0 against Hγ :γ = 0 .
0 0 2ℓ×1 1 0 6 2ℓ×1
In the time series literature, similar testing problems are solved by a standard test, using for
example the Wald, Lagrange-Mutiplier (LM) or Likelihood-Ratio (LR) principle. See among others
Luukkonen, Saikkonen and Teräsvirta (1988), Francq, Horváth and Zakoïan (2010).
A difficulty, in the present framework, is that we do not have a consistent estimator of the
parameter ϑ . Two problems arise to prove that the QMLE is consistent. First, the stationarity
0
conditions of Model (3.1) are unknown. Second, due to the presence of the η ’s, it seems
t k
| − |
extremely difficult to obtain invertibility conditions allowing to write logσ2(ϑ) (where ϑ denotes
t
any parameter value) as a function of the observations.
c
To circumvent these problems, we propose a LM approach. Denote by ϑ the constrained (by
n
Hγ) estimator of ϑ , defined by
0 0 b
c
ϑ = (θ′ ,0 )
n n 1 2ℓ ′
×
where θ is the QMLE of the AS-Log-GAbRCHbparameters defined in (2.1).
n
For any ϑ in Θ R2ℓ, define logσ2(ϑ) recursively, for t = 1,2,...,n, by
b × t
q q
logσ2(ϑ) = ω+ ω e1 + α 1 +α 1 logǫ2
t i− {ǫt−i<0} i+ {ǫt−i>0} i− {ǫt−i<0} t−i
i=1 i=1
X X(cid:0) (cid:1)
e p ℓ
+ βjlogσt2 j(ϑ)+ (γk+ǫ+t k +γk ǫ−t k)e−12logσet2−k(ϑ),
− − − −
j=1 k=1
X X
e
using positive initial values for ǫ2,...,ǫ2 ,σ2(ϑ),...,,σ2 (ϑ). The random vector
0 1 max(q,ℓ) 0 1 max(p,ℓ)
− −
e e
10