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

Bertrand Vignal

Dominique Guegan

English

Research Papers in Economics

A k-factor GIGARCH process: Estimation and
Application on electricity market spot prices.
Abdou Kâ Diongue∗, Dominique Guégan†, and Bertrand Vignal‡
∗Ecole Normale Supérieure de Cachan
IDHE-MORA, 61, avenue du président Wilson 94235 Cachan cedex
Email: abdou-ka.diongue@edf.fr
†Ecole Normale Supérieure De Cachan
IDHE-MORA, 61, avenue du président Wilson 94235 Cachan cedex
Email: guegan@ecogest.ens-cachan.fr
‡EDF R&D, 1 avenue du général de gaulle 92141 Clamart cedex
Email: bertrand.vignal@edf.fr
Abstract—Some crucial time series of market data, such as
electricity spot price, exhibit long-memory, in the sense of slowly-
decaying correlations, combined with heteroskedasticity. To be
able to modelized such a behaviour, we consider the k-factor
GIGARCH process. The related parameter estimation problem is
addressed using an extension of Whittle estimation. We develop
the corresponding asymptotic theory for estimation. We apply
this approach to electricity prices (spot prices) from the German
energy market (European Energy eXchange). For these data, we
propose two models of k-factor GIGARCH process. To conclude,
the forecasting performances of these models are analysed in
detail.
I. INTRODUCTION
However, Long-range dependence is now a phenomenon
which have attracted the attention of more and more
statisticians and economists since the works of Granger and
Joyeux (1980), Hosking (1981). The long memory parameter
d of the FARIMA(p,d,q) model is generally estimated in
the spectral domain by considering a least squares procedure
based on the periodogram, see for instance Geweke and
Porter-Hudak (1983), Robinson (1994, 1995) or Giraitis and
al. (1998) or frequency-domain maximum likelihood (ML)
estimation procedure proposed by Fox and Taqqu (1986),
Sowell (1992) or Cheung and Diebold (1993). For a review
of estimation techniques in long memory models, we refer to
Beran (1994), Guégan (1994), Bisaglia and Guégan (1998)
or Bisaglia (1998).
many empirical time series often possess a persistent periodic
behaviour, especially when dealing with hourly, daily or
monthly frequencies. For intance, the hourly spot prices
of Germany electricity market shows a strong seasonality
of daily and weekly. Unfortunately, the FARIMA process
does not allow to take into account a cyclical or periodic
behaviour. Therefore, many authors proposed different
long memory processes, fo which the common point is to
have a spectral density with one or more singularities, not
restrict at the origin, somewhere on the interval [0,π]. For
instance, we refer to the works of Gray and al. (1989),
Chung (1996(a), 1996(b)) or Yajima (1996) for models
with periodogram having a single peak, see the paper of
Giraitis and Leipus (1995), Hosoya (1997) or Woodward
and al. (1998) for models with periodogram having more
than one peak. Some authors proposed a two-steps procedure
to estimate the parameter the Gegenbauer autoregressive
moving-average (GARMA) model. In the ﬁrst step, they
estimate the frequency in which the periodogram becomes
unbounded by using a grid-search procedure, see Gray and
al. (1989) or Chung (1996(a), 1996(b)) for more details, or
by taking the maximum of the periodogram (Yajima (1996)).
Then, in a second step, the memory parameter is estimated
by using a classical parametric or semi-parametric methods
of the long memory domain. Later, a simultaneous Whittle’s
method is proposed. For more details, we refer to Férrara
and Guégan (2001), Férrara (2000) or Giraistis and Leipus
(1995). All of the aforementioned works assume that the
conditional variance of time series is a constant over time,
however.
Time series models with a time-varying conditional variance
(ARCH) was ﬁrst propsed by Engle (1982). This class of
models has important applications, particulary in ﬁnance
and economics, see Bollerslev (1986), Bollerslev, Engle and
Woodridge (1988). An extension of this model, considering
the long memory behaviour, has been introduced by Baillie
and al. (1996): it is the so-called FIGARCH models.
Considering the behaviour of very complex data, such that
the electricity spot prices, it seems natural to consider time
series both taking into account persistence with quasi periodic
behaviour and heteroskedasticity. Guégan (2000) introduced
the k-factor GIGARCH process and studied later in (2003) the
properties of stationarity and invertibility of the process. Our
aim is to study in this article, the parameters estimation of the
k-factor GIGARCH process and the asymptotic properties of
the estimators. Two estimations procedures were proposed: the
ﬁrst is based on the conditional sum of squares method (CSS)
and the second approach is an extension of Whittle estimation
procedure. The article is organized as follow. Section 2 givesthe models deﬁnition. Section 3 discusses the two estimations
procedures and the asymptotic properties of the estimators.
Section 4 applies the k-factor GIGARCH process to the hourly
spot prices of German electricity market (August, 15th 2000
to December, 31th 2002).
November 18, 2002
II. GIGARCH PROCESSES
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 deﬁne a centered k-factor GIGARCH
process (Xt)t∈Z by, ∀ t
φ(B)
k Y
i=1
￿
I − 2νiB + B2￿di Xt = θ(B)εt, (1)
where
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 deﬁned by the equations (??)-(??) was introduced
by Guégan (2000, 2003) generalizing fractionally integrated
generalized autoregressive conditional heteroskedasticity
process (FIGARCH) introduced by Baillie, Bollerslev and
Mikkelsen (1996).
We recall that the Gegenbauer polynomials, often used
in applied mathematics because of their orthogonality and
recursion properties, are deﬁned by:
￿
I − 2νz + z2￿−d
=
X
j≥0
Cj (d,ν)zj, (3)
where |z| ≤ 1 and |ν| ≤ 1.
The coefﬁcients (Cj (d,ν))j∈Z of this development can
be computed in many different ways. For example, Rainville
(1960) shows that:
Cj (d,ν) =
[
j
2] X
k=0
(−1)
k Γ(d + j − k)(2ν)
j−2k
Γ(d)Γ(k + 1)Γ(j − 2k + 1)
, (4)
where
￿
j
2
￿
is the integer part of
j
2
. An easiest way to
compute the Gegenbauer polynomials (Cj (d,ν))j∈Z is based
on the following recursion formula ∀ j > 1:
Cj (d,ν) = 2ν
￿
d − 1
j
+ 1
￿
Cj−1 (d,ν)
−
￿
2
d − 1
j
+ 1
￿
Cj−2 (d,ν), (5)
where C0 (d,ν) = 1 and C1 (d,ν) = 2dν.
The properties of stationarity and invertibility of a k-factor
GIGARCH process are established and proved by Guégan
(2003).
In this section, we provide some results related to the
asymptotic properties of the k-factor GIGARCH process es-
timators, obtained by two different methods: the conditional
sum of squares (CSS) and the Whittle approach.
III. ESTIMATION OF GIGARCH MODELS
Given a stationary k-factor GIGARCH process
{Xt}
T
t=1 deﬁned by the equations (??)-(??). 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.
A. 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
deﬁned by:
L(ω) =
1
T
T X
t=1
ℓt, ℓt = −
1
2
log(ht) −
ε2
t
2ht
. (6)
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) deﬁned 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)
)
￿￿
. (7)
In the following Theorem, L(ω) represents the log likelihood
introduced in (??) or in (??).
Theorem 3.1: Suppose that the process (Xt)t∈Zis
generated by equations (??)-(??). Assume that
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 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. If we assume that
all the G-frequencies are known. Then
1) There exists a CSS estimator ˆ ωT that satisﬁes
∂L(ω)
∂ω = 0
and ˆ ωT
P → ω0 as T → ∞.
2)
√
T (ˆ ωT − ω0)
D → N
￿
0,Ω
−1
0
￿
as T → ∞, where
P → and
D → denotes respectively the convergence in probabilityand 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) 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
￿
, (8)
and
ˆ Ωδ =
1
T
T X
t=1
￿￿
1
2h2
t
∂ht
∂δ
∂ht
∂δT
￿￿
. (9)
The proof is given in for the Gaussian case (Diongue and
al. (2003)). 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
previous case.
B. Whittle estimation
In this paragraph, we investigate the Whittle’s method to
estimate all parameters of the k-factor GIGARCH process
deﬁned by equations (??)-(??). The ﬁrst step consists to
estimate the long-memory parameters d = (d1,    ,dk) and
the ARMA(p,q) parameters α = (φ1,    ,φp,θ1,    ,θq)
using the Whittle’s approach (for more details, see
Chung (1996(a), 1996(b))) or Férrara and Guégan
(2001)). 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 (2001) for more details).
Theorem 3.2: Let {Xt}
T
t=1 be a k-factor GIGARCH pro-
cess deﬁned by equations (??)-(??). Let us assume that the
same hypothesis given in Theorem (??) are veriﬁed. Then
1) ˆ γT
a.s → γ0 as T → ∞.
2) Furthermore:
√
T (ˆ αT − α0)
D →
N
￿
0,4πV (α0)
−1
￿
, as T → ∞, Where,
V (α)ij =
Z π
−π
g2 (λ,ω)
∂g−1 (λ,ω)
∂αi
∂g−1 (λ,ω)
∂αj
dλ.
(10)
Here g (λ,ω) denotes the spectral density of the process
(Xt)t∈Z.
3) Moreover
√
T
￿
ˆ dT − d
￿
D → N
￿
0,4πV (d)
−1
￿
, with
V (d)ij =
Z π
−π
log
￿ ￿ ￿
￿4sin
￿
(λ − λi)
2
￿
sin
￿
(λ + λi)
2
￿
×log
￿ ￿ ￿
￿4sin
￿
(λ − λj)
2
￿
sin
￿
(λ + λj)
2
￿￿ ￿ ￿
￿dλ.
(12)
The Theorem ?? follows from the proof of Hosoya’s Theorem
2.3 (1997).
To estimate the GARCH(r,s) parameters δ, we consider the
process
￿
ε2
t
￿
t∈Z in its ARMA representation. This means
that we can rewritte (??) 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 deﬁned 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 (??), is strictly stationary, ergodic with ﬁnite
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).
1)
R π
−π logf (λ,δ)dλ = 0, for all δ, with f (λ,δ) the
spectral density of the process (εt)t∈Z.
2) f (λ,δ)
−1 is continuous in (λ,δ) ∈ [−π,π] × Λ, where
Λ ⊂ Rr+s+1 is a compact.
3)  L ({λ;f (λ,δ)  = f (λ,δ0)}) ≥ 0, for δ ∈ Λ with  L
the Lebesque measure.
(H2).
1) 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.
2)
∂f(λ,δ0)
−1
∂δ is K-Lipchitzienne with K > 1
2.
3) The matrix W given by
W =
1
2π
Z π
−π
∂ logf (λ,δ0)
∂δ
∂ logf (λ,δ0)
∂δT dλ (13)
is non singular.
Theorem 3.3: Let (Xt)t∈Z be a stationary, causal and in-
vertible process deﬁned by the equations (??)-(??).
1) Under (H0) with J = 4 and (H1), ˆ δT
P → δ0, as T → ∞.
2) Under (H0) with J = 8, (H1)
and (H2),
√
T
￿
ˆ δT − δ0
￿
→
N
￿
0,2W−1 + W−1V W−1￿
, as T → ∞. Here
V is given by
V =
2π
σ2
Z π
−π
Z π
−π
∂f (λ,δ0)
−1
∂δ
∂f (ω,δ0)
−1
∂δT h(λ,−ω,ω)dλdω,
(14)TABLE I
GENERAL DESCRIPTIVE STATISTICS.
Statistics Value
Mean 2.97
Variance 0.375
Skewness -2.6
Kurtosis 29.67
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 ?? is similar to the proofs of Theorem
2.1 and Theorem 2.2 given in Giraitis and Robinson’s (2001).
IV. APPLICATION
In this section, we provide an application of k-factor GI-
GARCH process, which points out the interest of such model
to analyze and to forecast time series with long-range de-
pendence, especially when the forecast horizon is wide. We
consider here the hourly series {St}
20856
t=1 of the spot prices
of German electricity market from August, 15th 2000 to
December, 31th 2003, and we note Yt = log(St) (??) this
series of 20856 points. As shows in Figure ??, the serie studied
seems to be stationary in mean but we observe the presence
of peaks around December 2001 and July 2001. Let us look
at the ACF (AutoCorrelation Function), represented in Figure
??. The ACF of the series Yt seems to exhibit a slow decay
pattern for the 2500 lags and a strong seasonality of twenty
four hours and one hundred sixty eight hours is observed.
The periodogram (Figure ??) of the series is unbounded for
relatively three frequencies. Note that the frequencies coincid
respectively to the period of one week, one day and one half-
day.
Summary statistics of the data over the sample period are
presented in Table ??. The skewness is non positive, indicating
that the series have long left hand side tails, and the kurtosis
statistic is greater than 3, signifying that the series have fat
tails.
Our aim is to forecast from January, 1th 2003 to January,
30th 2003. To reach our goal, we consider two different
approachs of modelling and we compare them according to
their predictive ability. Note that all the approachs are long
memory approach. We assess the predictive ability of each
model by considering the RMSE forecast criterion deﬁned by:
RMSE (f) =
v u
u t1
h
h X
k=1
￿
Xt+k − ˆ Xt (k)
￿2
,
where h is the forecast horizon and ˆ Xt (k) is the predict
value of Xt+k.
First approach:
When dealing with time series having a strong seasonality,
TABLE II
PARAMETER ESTIMATES FOR THE 1-FACTOR GIGARCH PROCESS WITH
CONDITIONAL STUDENT DISTRIBUTION WITH 3 DEGREES OF FREEDOM.
Parameter Estimate St. dev.
d 0.38 0.017
φ24 0.874 0.342
θ24 0.706 0.342
θ168 0.851 0.303
a0 0.0167 3.36 10−4
a1 0.594 0.072
TABLE III
PARAMETER ESTIMATES FOR THE 3-FACTOR GIGARCH PROCESS WITH
CONDITIONAL STUDENT DISTRIBUTION WITH 3 DEGREES OF FREEDOM.
Parameter Estimate St. dev.
d1 0.135 0.009
d2 0.214 0.006
d3 0.141 0.015
φ24 0.863 0.179
φ168 0.951 0.221
θ24 0.720 0.331
θ168 0.794 0.298
a0 0.0141 3.39 10−4
a1 0.756 0.365
denote S, a ﬁrst idea is to remove this seasonality by using
the
￿
I − BS￿
ﬁlter. Let us denoted (Zt)t the seasonally ﬁlterd
series such that:
￿
I − B168￿￿
Yt − ¯ Y
￿
= Zt.
The ACF of the series (Zt)t (Figure ??) seems to exhibit a
relatively slow decay pattern for the 2500 ﬁrst lags. Moreover,
the periodogram of the series (Zt)t (Figure ??) clearly pos-
sesses a peak for a frequency located very close, but not equal,
to zero. We assume here that the spectral density of this series
is unbounded for only one frequency. Furthermore, we have
detect great variability in the variance of this series (Figure
??).
The parameter estimation is done using the Whittle’s
method and the values are resumed in Table ??. Using Yajima
(1996), the periodogram is unbounded at the frequency λ =
0.001.
Second approach:
In this approach, we assume that the periodogram of the series
(Yt)t is unbounded for three frequencies. Therefore, we ﬁt a
3-factor GIGARCH process to the series (Yt), considering the
frequencies λ1 = 0.2658, λ2 = 0.0374, λ3 = 0.5236. The
parameter estimation is resumed in Table ??.
V. CONCLUSION
The conclusion goes here.[Evolution] [ACF] [Periodogram]
Fig. 1. Evolution (top), estimation of the ACF (center) and the spectral density (buttom) of the hourly log spot prices of German elctricity market
[Evolution] [ACF] [Periodogram]
Fig. 2. Evolution (top), estimation of the ACF (center) and the spectral density (buttom) of the hourly log spot prices of German elctricity market
ACKNOWLEDGMENT
The authors would like to thank...
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A k- factor GIGARCH process : estimation and application to electricity market spot prices,

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