Some crucial time series of market data, such as electricity spot prices, exhibit long memory, in the sense of slowly-decaying correlations combined with heteroscedasticity. To e able to model such a behaviour, we consider the k-factor GIGARCH process and we propose two methods to address the related parameter estimation problem. For each method, we develop the asymptotic theory for this estimation.GIGARCH process – estimation theory – Electricity spot prices

Abdou Kâ Diongue

Dominique Guegan

English

Research Papers in Economics

Estimating parameters of a k-factor GIGARCH process.
Estimation des paramŁtres d’un processus GIGARCH ￿ k
facteurs.
Abdou K￿ DIONGUE a,b, Dominique GUEGAN a
aENS Cachan IDHE-MORA, UMR CNRS 8533, 61, avenue du prØsident Wilson 94231 Cachan cedex.
Tel: 01-47-40-55-75; Fax: 01-47-40-24-60.
bEDF R&D, 1 avenue du gØnØral de gaulle 92141 Clamart cedex.
Tel: 01-47-65-53-61; Fax: 01-47-65-30-37.
Abstract
Some crucial time series of market data, such as electricity spot prices, exhibit long-memory, in the sense of slowly-
decaying correlations combined with heteroskedasticity. To be able to modelized such a behaviour, we consider in
this Note the k-factor GIGARCH process and we propose two methods to address the related parameter estimation
problem. For each method, we develop the asymptotic theory for estimation. To cite this article: A.K. Diongue,
D. GuØgan C. R. Acad. Sci. Paris, Ser. I 336 (2003).
RØsumØ
Plusieurs donnØes de marchØ, telles que les prix spot de l’ØlectricitØ, prØsentent de la longue mØmoire, au sens de la
dØcroissance hyperbolique des autocorrØlations combinØe avec un phØnomŁne d’hØtØroskØdasticitØ. Pour modØliser
de tels comportements, nous considØrons dans cette Note les processus GIGARCH ￿ k facteurs et nous proposons
deux mØthodes d’estimation des paramŁtres de ce modŁle. En￿n, nous dØveloppons les propriØtØs asymptotiques
de ces estimateurs. Pour citer cet article : A.K. Diongue, D. GuØgan C. R. Acad. Sci. Paris, Ser. I 336 (2003).
Version fran￿aise abrØgØe
Dans cette Note, nous nous intØressons ￿ l’estimation des paramŁtres d’un processus GIGARCH ￿ k
facteurs par la mØthode des moindres carrØs conditionnels (CSS) et la mØthode du maximum de vraisem-
blance de Whittle. Ce processus dØ￿ni par les Øquations (1)-(2) a ØtØ introduit et ØtudiØ dans les articles
Email addresses: abdou-ka.diongue@edf.fr (Abdou K￿ DIONGUE), guegan@ecogest.ens-cachan.fr (Dominique
GUEGAN).
Preprint submitted to Elsevier Science 15 juin 2004de GuØgan [7], [8]. Soit {Xt}
T
t=1 un processus GIGARCH ￿ k facteurs stationnaire de￿ni par les equa-
tions (1)-(2). Posons γ = (φ1,···,φp,θ1,···,θq,d1,···,dk), δ = (a0,a1,···,ar,b1,···,bs) et ω = (γ,δ).
Supposons que ω0 = (γ0,δ0) soit la vraie valeur du paramŁtre ω et se trouve ￿ l’intØrieur du compact
Θ ⊆ Rp+q+k+r+s+1. Nous supposons, dans toute la suite, que toutes les G-frØquences sont connues. Dans
le ThØorŁme 2.1, nous donnons les propriØtØs asymptotiques de l’estimateur des paramŁtres par la mØ-
thode CSS. Les propriØtØs asymptotiques des estimateurs des paramŁtres par la mØthode de Whittle sont
fournies dans le ThØorŁme 2.2 pour les paramŁtres de mØmoire longue et de mØmoire courte homoscØdas-
tiques et dans le ThØorŁme 2.3 pour les paramŁtres hØtØroscØdastiques.
ThØorŁme 2.1 Soit {Xt}
T
t=1 un processus generØ par les Øquations (1)-(2). Supposons que a0 > 0,
a1,···,ar,b1,···,bs ≥ 0,
r X
i=1
ai+
s X
i=1
bi < 1, E
 
ε4
t

< ∞, 0 < di < 1
2 si |νi| < 1, ou 0 < di < 1
4 si |νi| = 1
pour i = 1,···,k et toutes les racines de φ(B) et θ(B) soient en dehors du cercle unitØ. Si les frØquences
νi sont connues alors
(i) Il existe un estimateur CSS ˆ ωT satisfaisant
∂L(ω)
∂ω = 0 et ˆ ωT
P → ω0 quand T → ∞.
(ii)
√
T (ˆ ωT − ω0)
D → N
 
0,Ω
−1
0

quand T → ∞, avec Ω0 = diag (Ωγ0,Ωδ0) .
(iii) De plus, les estimateurs consistants des matrices d’information Ωγ et Ωδ sont donnØs en (5).
ThØorŁme 2.2 Soit {Xt}
T
t=1 un processus dØ￿ni par les Øquations (1)-(2). Supposons vØri￿Øes les hypo-
thŁses du ThØorŁme (2.1). Alors, les estimateurs de Whittle des paramŁtres γ sont tels que :
(i) ˆ γT
p.s
→ γ0 quand T → ∞.
(ii)
√
T (ˆ αT − α0)
D → N

0,4πV (α0)
−1

, quand T → ∞, oø α = (φ1,···,φp,θ1,···,θq), et oø V (α)ij
est dØ￿ni par l’Øquation (6).
(iii)
√
T

ˆ dT − d

D → N

0,4πV (d)
−1

, oø V (d)ij est donnØe en (7)
ThØorŁme 2.3 Soit {Xt}
T
t=1 un processus dØ￿ni par les Øquations (1)-(2). Supposons vØri￿Øes les hypo-
thŁses du ThØorŁme (2.1). Alors
(i) Sous (H0)(J = 4) et (H1), on a ˆ δT
P → δ0, quand T → ∞.
(ii) Sous (H0)(J = 8),(H1) et (H2), on a
√
T

ˆ δT − δ0

D → N
 
0,2W−1 + W−1V W−1
, quand T → ∞,
oø V est donnØ en (9)
1. Introduction
Assume that (ξt)t∈Z is a white noise process with unit variance and let the polynomials φ(B) and θ(B)
denote the ARMA operators. Let B denote the backshift operator and 0 < di < 1
2 if |νi| < 1 or 0 < di < 1
4
if |νi| = 1 for i = 1,···,k. We de￿ne a centered k-factor GIGARCH process (Xt)t∈Z by, ∀ t
φ(B)
k Y
i=1
 
I − 2νiB + B2di Xt = θ(B)εt, (1)
2where
εt =
p
htξt, with ht = a0 +
r X
i=1
aiεt−i +
s X
i=1
biht−i. (2)
For i = 1,···,k, the frequencies λi = arccos(νi) are called the Gegenbauer frequencies (or G-frequencies).
The process de￿ned by the equations (1)-(2) was introduced by GuØgan (see [7] and [8]). In the following
section, we provide some results related to the asymptotic properties of the k-factor GIGARCH process
estimators, obtained by two methods: the conditional sum of squares and the Whittle approach.
2. Asymptotic theory for estimation
Given a stationary k-factor GIGARCH process {Xt}
T
t=1 de￿ned by the equations (1)-(2). We denote
γ = (φ1,···,φp,θ1,···,θq,d1,···,dk), δ = (a0,a1,···,ar,b1,···,bs) and ω = (γ,δ) its parameters. We
assume that ω0 = (γ0,δ0) is the true value of ω and is in the interior of the compact set Θ ⊆ Rp+q+k+r+s+1.
Let us assume that all the G-frequencies are known.
2.1. Conditional Sum of Squares estimation
The conditional sum of squares estimator ˆ ωT of ω in Θ maximizes the conditional logarithmic likelihood
L(ω) on F0, where Ft is the σ-algebra generated by (Xs,s ≤ t). If we assume that the innovations (εt)t∈Z
have a conditional Gaussian distribution then the conditional log-likelihood is de￿ned by:
L(ω) =
1
T
T X
t=1
`t, `t = −
1
2
log(ht) −
ε2
t
2ht
. (3)
Now, if we assume that the innovations (εt)t∈Z have a conditional Student distribution with l degrees of
freedom, then the CSS estimator ˆ ωT maximizes the likelihood function L(ω) de￿ned by
L(ω) =T

logΓ

(l + 1)
2

− logΓ

l
2

−
1
2
log(l − 2)

−
1
2
T X
t=1

log(ht) + (l + 1)

log(1 +
ε2
t
ht(l − 2)
)

. (4)
In the following Theorem, L(ω) represents the log likelihood introduced in (3) or in (4).
Theorem 2.1 Suppose that the process (Xt)t∈Zis generated by equations (1)-(2). Assume that a0 >
0,a1,···,ar,b1,···,bs ≥ 0,
Pr
i=1 ai +
Ps
i=1 bi < 1, E
 
ε4
t

< ∞, 0 < di < 1
2 if |νi| < 1 or 0 < di < 1
4
if |νi| = 1 for i = 1,···,k and all roots of the polynomials φ(B) and θ(B) lie outside the unit circle. Then
(i) There exists a CSS estimator ˆ ωT that satis￿es
∂L(ω)
∂ω = 0 and ˆ ωT
P → ω0 as T → ∞.
(ii)
√
T (ˆ ωT − ω0)
D → N
 
0,Ω
−1
0

as T → ∞, where
P → and
D → denotes respectively the convergence in
probability and in distribution. Furthermore, Ω0 = diag (Ωγ0,Ωδ0) and Ωγ0 and Ωδ0 are values of
Ωγ and Ωδ at ω = ω0, with Ωγ = E
h
1
ht
∂εt
∂γ
∂εt
∂γT + 1
2h2
t
∂ht
∂γ
∂ht
∂γT
i
and Ωδ = E
h
1
2h2
t
∂ht
∂δ
∂ht
∂δT
i
.
3(iii) The information matrices Ωγ and Ωδ can be estimated consistently by
ˆ Ωγ =
1
T
T X
t=1

1
ht
∂εt
∂γ
∂εt
∂γT +
1
2h2
t
∂ht
∂γ
∂ht
∂γT

and ˆ Ωδ =
1
T
T X
t=1

1
2h2
t
∂ht
∂δ
∂ht
∂δT

. (5)
The proof is given in Section 3 for the Gaussian case and details can be found in Diongue, GuØgan and
Vignal [4]. Note that, if the innovations (εt)t∈Z have a conditional Student distribution with l degrees of
freedom, then the proof can easily done using the same steps as in Section 3.
2.2. Whittle estimation
In this paragraph, we investigate the Whittle’s method to estimate all parameters of the k-factor
GIGARCH process de￿ned by equations (1)-(2). The ￿rst step consists to estimate the long-memory pa-
rameters d = (d1,···,dk) and the ARMA(p,q) parameters α = (φ1,···,φp,θ1,···,θq) using the Whittle’s
approach (for more details, see Chung [2], [3] and Ferrara and GuØgan chapter 8 of [5]). In the second
step, the GARCH(r,s) parameters δ = (a0,a1,···,ar,b1,···,bs) are estimated using Whittle’s method
applied to the residuals of the long-memory process (see Giraitis and Robinson [6] for more details).
Theorem 2.2 Let {Xt}
T
t=1 be a k-factor GIGARCH process de￿ned by equations (1)-(2). Let us assume
that the same hypothesis given in Theorem (2.1) are veri￿ed. Then
(i) ˆ γT
a.s → γ0 as T → ∞.
(ii) Furthermore:
√
T (ˆ αT − α0)
D → N

0,4πV (α0)
−1

, as T → ∞, Where,
V (α)ij =
π Z
−π
g2 (λ,ω)
∂g−1 (λ,ω)
∂αi
∂g−1 (λ,ω)
∂αj
dλ. (6)
Here g (λ,ω) denotes the spectral density of the process (Xt)t∈Z.
(iii) Moreover
√
T

ˆ dT − d

D → N

0,4πV (d)
−1

, with
V (d)ij =
π Z
−π
log

 
4sin

(λ − λi)
2

sin

(λ + λi)
2

 
log

 
4sin

(λ − λj)
2

sin

(λ + λj)
2

 
dλ. (7)
The Theorem 2.2 follows from the proof of Hosoya’s Theorem 2.3 ([9]).
To estimate the GARCH(r,s) parameters δ, we consider the process
 
ε2
t

t∈Z in its ARMA represen-
tation. This means that we can rewritte (2) as: ε2
t −
max(r,s) X
i=1
(ai + bi)ε2
t−i = a0 + υt −
s X
j=1
bjυt−j, where
bi = 0 if i ∈ (s,r] and ai = 0 if i ∈ (r,s]. The process (υt)t∈Z de￿ned by υt = ε2
t − ht constitutes a white
noise sequence with mean zero and variance σ2. We introduce now some complementary assumptions to
get the consistency and asymptotic normality of ˆ δT:
(H0). For t = 0,±1,..., the process (ξt)t∈Z introduced in equation (2), is strictly stationary, ergodic
4with ￿nite Jth moment and E(ξt|Ft−1) = 0, E(ξ2
t|Ft−1) = 1, and E(ξ
2j
t |Ft−1) = υ2j almost-surely, with
j = 2,···, J
2, where υ2j are constants such that |υJ|
2
J
 
r X
i=1
ai +
s X
i=1
bi
!
< 1.
(H1).
(i)
R π
−π logf (λ,δ)dλ = 0, for all δ, with f (λ,δ) the spectral density of the process (εt)t∈Z.
(ii) f (λ,δ)
−1 is continuous in (λ,δ) ∈ [−π,π] × Λ, where Λ ⊂ Rr+s+1 is a compact.
(iii) µL ({λ;f (λ,δ) 6= f (λ,δ0)}) ≥ 0, for δ ∈ Λ with µL the Lebesgue measure.
(H2).
(i) δ0 is an interior point of Λ and in a neighborhood of δ0,
∂f(λ,δ)
−1
∂δ and
∂
2f(λ,δ)
−1
∂δ∂δT exist and are
continuous in λ and δ.
(ii)
∂f(λ,δ0)
−1
∂δ is K-Lipchitzienne with K > 1
2.
(iii) The matrix W given by
W =
1
2π
π Z
−π
∂ logf (λ,δ0)
∂δ
∂ logf (λ,δ0)
∂δT dλ (8)
is non singular.
Theorem 2.3 Let (Xt)t∈Z be a stationary, causal and invertible process de￿ned by the equations (1)-(2).
(i) Under (H0) with J = 4 and (H1), ˆ δT
P → δ0, as T → ∞.
(ii) Under (H0) with J = 8, (H1) and (H2),
√
T

ˆ δT − δ0

D → N
 
0,2W−1 + W−1V W−1
, as T → ∞.
Here V is given by
V =
2π
σ2
π Z
−π
π Z
−π
∂f (λ,δ0)
−1
∂δ
∂f (ω,δ0)
−1
∂δT h(λ,−ω,ω)dλdω, (9)
with h(λ,ω,υ) = 1
8π3
+∞ X
j,k,l=−∞
eijλ−ikω−ilυCum(ε0,εj,εk,εl), and Cum is the order four’s cumulant
for the process (εt)t∈Z .
The proof of Theorem 2.3 is similar to the proofs of Theorem 2.1 and Theorem 2.2 given in Giraitis and
Robinson’s ([6]).
3. Proof of the Theorem 2.1
Here, we assume that the process (εt)t∈Z have a conditional Gaussian distribution. We will ￿rst show
(iii). The strict stationarity and ergodicity of the process (Xt)t∈Z and (εt)t∈Z imply the consistency of
the information matrices ˆ Ωγ and ˆ Ωδ.
In order to proof (ii), we need to check the following Basawa’s conditions (see Basawa, Feign and Heyde
[1]):
5￿ 1
T
T X
t=1
∂`t (ω0)
∂ω
P → 0,
￿ there exists a nonrandom positive de￿nite matrix M (ω0) such that for all  > 0,
P
 
−
1
T
T X
t=1
∂2`t
∂ω∂ωT ≥ M (ω0)
!
> 1 −  for all T > T1 (),
￿ there exists a constant M < ∞ such that E



∂
3`t(ω)
∂ωi∂ωj∂ωk


 < M for all ω ∈ Θ, where ωi is the ith com-
ponent of ω.
From ∂`t
∂γ = 1
2ht

ε
2
t
ht − 1

∂ht
∂γ − εt
ht
∂εt
∂γ and ∂`t
∂δ = 1
2ht

ε
2
t
ht − 1

∂ht
∂δ , we have E
 ∂`t
∂ω

ω=ω0 = 0 and using
the ergodic theorem, Basawa’s ￿rst condition follows.
The matrix Ω0 is de￿nite positive and hence the second Basawa’s condition holds.
Now the last conditions of Basawa is obtained by di￿erentiating ∂
2`t
∂ω∂ωT and using E

∂
3εt
∂di∂dj∂dk
2
< ∞.
Condition (ii) follows.
Using (iii) of Theorem 2.1, we get 1
T
T X
t=1

∂`t
∂ω
∂`t
∂ωT

ω=ω0
a.s → Ω0. Let ST de￿ned by ST =
T X
t=1

b0
∂`t
∂ω

ω=ω0
,
where b0 is an arbitrary constant vector and b0bT
0 6= 0. Then, ST is a martingal with 1
T E
 
S2
T

=
bT
0 Ω0b0 > 0. Now, using the strict stationarity and ergodicity of the process (Xt)t∈Z and (εt)t∈Z, we
get
 1
T E (ST)
−1  1
T E
 
S2
T|Ft−1
 a.s → 1. From the Central Limit Theorem of Stout [10], the asymptotic
normality convergence of the CSS estimators is derived.
Remark 1 : For applications and more details see Diongue, GuØgan and Vignal [4].
References
[1] Basawa, I. V., Feign, P. D. and Heyde, C. C. Asymptotic properties of maximum likelihood estimators for stochastic
processes, Sankhya, Ser. A, 38 (1976), 259-270.
[2] Chung, C.-F. Estimating a generalized long-memory process, Journal of Econometrics, 73 (1996), 237-259.
[3] Chung, C.-F. A generalized fractionally integrated ARMA process, Journal of Time Series Analysis, 17 (1994), 111-140.
[4] Diongue, A.K., GuØgan, D. and Vignal, B. Processus GIGARCH: Estimation et applications aux prix spot de l’ØlectricitØ,
Preprint MORA, 14 (2003).
[5] Ferrara, L. GuØgan, D. Comparison of parameter estimation methods in cyclical long-memory time series, Development
in Forecasts Combination and Portfolio Choice, C. Dunis and J. Timmermann, J. Wiley,(2001).
[6] Giraitis, L. Robinson, P.M. Whittle Estimation of ARCH models, Econometrics Theory, 17 (2001), 608-631.
[7] GuØgan D. A new model: The k-factor GIGARCH process, Journal of Signal Processing, 4 (2000), 265-271.
[8] GuØgan D. A prospective study of the k-factor Gegenbauer process with heteroscedastic errors and an application to
in￿ation rates, Finance India, 17 (2003), 1-20.
[9] Hosoya, Y. A Limit theory for long-range dependence and statistical inference on related models, The Annals of
Statistics, 25, (1997), 105-137.
[10] Stout, W. F. Almost sure convergence, New York: Academic press, (1974).
6

Estimating parameters for a k-GIGARCH process

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