The purpose of this paper is to study the self-similar properties of discrete-time long memory processes. We apply our results to specific processes such as GARMA processes and GIGARCH processes, heteroscedastic models and the processes with switches and jumps.Covariance stationary, Long memory processes, short memory processes, self-similar, asymptotically second-order self-similar, autocorrelation function.

Dominique Guegan

Zhiping Lu

English

Research Papers in Economics

Documents de Travail du
Centre d’Economie de la Sorbonne
Maison des Sciences Économiques, 106-112 boulevard de L'Hôpital, 75647  Paris Cedex 13
http://ces.univ-paris1.fr/cesdp/CES-docs.htm
ISSN : 1955-611X
A note on self-similarity for discrete time series
Dominique GUEGAN, Zhiping LU
2007.55A Note On Self-Similarity For Discrete Time
Series.
Dominique Guégana, Zhiping LUb, ∗
a Centre d’Economie de la Sorbonne, UMR 8147, Université Paris 1 Panthéon Sorbonne, 75647
Paris. E-mail: dguegan@univ-paris1.fr
b Department of Mathematics, East China Normal University, 3663, Zhongshan Road North,
Shanghai, China, 200062; Centre d’Economie de la Sorbonne Antenne de Cachan, Ecole Normale
Supérieure de Cachan, 61 avenue du président Wilson, 94235, Cachan, France. E-mail:
camping9@gmail.com
————————————————————————————————————————
Abstract
The purpose of this paper is to study the self-similar properties of discrete-time long memory
processes. We apply our results to speciﬁc processes such as GARMA processes and GIGARCH
processes, heteroscedastic models and the processes with switches and jumps.
Keywords : Long Memory Processes, Short Memory Processes, Self-similar, Asymptotically
Second-order Self-similar.
———————————————————————————————————————–
1 Introduction
In the past decades there has been a growing interest in studying self-similar processes
and asymptotically self-similar processes, which has been a subject in literature (eg. Beran
1994, Samorodnitsky and Taqqu 1994). However, most of the works concern continuous-time
processes. In this paper, we are devoted to study the relationship between long memory and
self-similarity for discrete-time series and we present some new results. We prove that if a
∗Corresponding author: E-mail Adress: camping9@gmail.com (Z.P. LU), Centre d’Economie de la Sorbonne
Antenne de Cachan, Ecole Normale Supérieure de Cachan, 61 avenue du président Wilson, 94235, Cachan,
France.
1process is both covariance stationary and long memory, then it is asymptotically second-order
self-similar. On the other hand, if a process is covariance stationary and short memory, then
it is not asymptotically self-similar. We apply these results to the k-factor GARMA process
and the k-factor GIGARCH process, the ARMA process, the GARCH process, the model
with switches and the model with breaks, etc. What’s more, for the processes with breaks
and jumps, although they are theoretically short memory, they are empirically long memory
processes. Consequently, theoretically they are not asymptotically second-order self-similar,
while empirically they are asymptotically second-order self-similar. We propose in this paper a
new notion to describe this kind of property which is the “spurious asymptotically second-order
self-similar" behavior.
The paper is organized as follows. In section two, we recall some concepts and give some
new results. Section three is devoted to the study of the self-similar properties for discrete-time
series.
2 Concepts and main results
In this part, we recall some deﬁnitions and give some new results on self-similarity.
2.1 Concepts of long memory and short memory
Deﬁnition 2.1 A covariance stationary process (Xt)t is called short memory process, if its
autocorrelation function ρ(k) satisﬁes Σ∞
k=0ρ(k) < ∞.
Deﬁnition 2.2 A covariance stationary process (Xt)t is called long memory process if it has an
autocorrelation function ρ(k) which behaves like a power function decaying to zero hyperbolically
as
ρ(k) ∼ Cρ(k) · k
−α, as k → ∞, 0 < α < 1, (1)
where ∼ represents the asymptotic equivalence, and Cρ(k) is a function which changes slowly
to inﬁnity, i.e. for all a ∈ R, Cρ(k)(ax)/Cρ(k)(x) → 1, when x → ∞ (or x → 0).
These concepts have been studied a lot in literature, for example, by Beran (1994), Guégan
(2005).
2.2 Concepts of self-similarity
Self-similarity provides an elegant explanation and interpretation for an empirical law that
is commonly referred as the Hurst’s eﬀect. Let (Xt)t be a covariance stationary time series.
Denote
X
(m)
t = 1/m
tm X
k=(t−1)m+1
Xk, k = 1,2,... (2)
2the corresponding aggregated sequence with level of aggregation m (> 1). Denote ρ(m)(k) the
autocorrelation function of the process (X
(m)
t )t.
Deﬁnition 2.3 A strictly stationary stochastic process (Xt)t is exactly self-similar (or asymp-
totically self-similar) if for all t,
Xt =
d m
1−HX
(m)
t
holds for all m (or as m → ∞), where (X
(m)
t )t is deﬁned as in (2) and 1/2 < H < 1.
Deﬁnition 2.4 Let (Xt)t be a covariance stationary process,
(1) The process (Xt)t is called exactly second-order self-similar, or s.o.s.s, if m1−HX
(m)
t has the
same autocorrelation as X, for all m and all t. Thus we have V ar(X(m)) = V ar(X)m2H−2 and
ρ(m)(k) = ρ(k) where 1/2 < H < 1, m > 1, k = 0,1,2,···, and ρ(k) ∼ Ck2H−2, as k → ∞.
(2) The process (Xt)t is called asymptotically second-order self-similar, or a.s.o.s.s, if
lim
m→∞ρ
(m)(k) =
1
2
[(k + 1)
2H − 2k
2H + (k − 1)
2H], ∀ k > 0. (3)
So the autocorrelation function of the process (Xt)t is such that :
ρ(k) =
1
2
[(k + 1)
2H − 2k
2H + (k − 1)
2H], ∀k > 0, (4)
We can also use the relationship(3) and (4) as the deﬁnition of second-order self-similarity
processes. The notion of exactly (asymptotically) self-similar concerns all the ﬁnite-dimensional
distributions of a strict stationary process, while the notion of exact (asymptotical) second-
order self-similarity concerns only the variance and autocorrelation function of a covariance
stationary process.
2.3 Main Results
Lemma 2.5 Let (Xt)t be a covariance stationary process, if this process is short memory as
is deﬁned by deﬁnition 2.1, then it is not asymptotically second-order self-similar.
Proof : For a short memory process (Xt)t, its autocorrelation function decays exponentially
to zero, so it does not satisfy the equation (4), which means that the process (Xt)t is not
asymptotically second-order self-similar. ￿
Lemma 2.6 Let (Xt)t be a covariance stationary long memory process with 1
2 < H < 1, then
this process is asymptotically second-order self-similar. Furthermore, under Gaussianity, the
process is asymptotically self-similar.
3Proof : For a covariance stationary process (Xt)t, its autocorrelation function decays hyperbo-
lically, i.e. limk→∞
ρ(k)
k2H−2 = c, 1
2 < H < 1. According to the results of Tsybakov and Georganas
(1997), we deduce that limm→∞ ρ(m)(k) = 1
2[(k + 1)2H − 2k2H + (k − 1)2H], ∀ k = 1,2,··· .
Thus the process is asymptotically second-order self-similar following the deﬁnition 2.4. ￿
From the lemma 2.6, the following result is straightforward :
Lemma 2.7 Let (Xt)t be a covariance stationary long memory process with 1
2 < H < 1, if
the spectral density of the process blows up at the origin, then the process is asymptotically
second-order self-similar. Under Gaussianity, it is asymptotically self-similar.
3 Examples of self-similar processes
In this part, we investigate the classical discrete-time processes and their self-similar pro-
perties. The study concerns the long memory processes under stationarity, for example, fGn,
k−factor GARMA process, k−factor GIGARCH process, and some short memory processes
such as processes with switches, processes with breaks, and also processes with threshold.
3.1 Fractional Gaussian noise (fGn)
Deﬁnition 3.1 A process (Xt)t∈Z is called a fractional Gaussian noise, or fGn, if it satisﬁes,
for all t ∈ Z, Xt = BH(t) − BH(t − 1), where {BH(t)}t∈R is a fractional Brownian motion.
The fractional Gaussian noise (fGn) is the unique stationary Gaussian process which is
exactly self-similar process with zero mean, which has been studied, for instance, by Beran
(1994), Samorodnitsky and Taqqu (1994).
3.2 k-factor GARMA process
Deﬁnition 3.2 A stationary process (Xt)t is called a k-factor GARMA process, if it has the
following representation
φ(B)
k Y
i=1
(1 − 2uiB + B
2)
diXt = θ(B)εt (5)
where k is a ﬁnite integer, |ui| ≤ 1 for all i = 1,··· ,k, (εt)t is a white noise with mean zero
and variance σ2
ε, and φ(B) and θ(B) are polynomials of order p and q respectively, di ∈ R, B
is the backshift operator satisfying BXt = Xt−1.
Woodward, Cheng and Gray (1998) proved that a stationary k-factor GARMA process
exhibits long memory behavior, if ui are distinct, all the roots of the polynomials φ(B) and
θ(B) are distinct and lie outside the unit circle and if (i) 0 < di < 1
2 and |ui| < 1 or if (ii)
0 < di < 1
4 and |ui| = 1, for i = 1,··· ,k.
4Proposition 3.3 Let (Xt)t be a covariance stationary and long memory k-factor GARMA
process, then it is asymptotically second-order self-similar. Furthermore, under Gaussianity, it
is also asymptotically self-similar.
Proof : Due to lemma 2.6, a covariance stationary and long memory k-factor GARMA process
is asymptotically second-order self-similar. The conclusion is the same for particular k-factor
GARMA process like GARMA process, Gegenbauer process and FARIMA process. ￿
3.3 Heteroscedastic Processes
3.3.1 k-factor GIGARCH processes
Deﬁnition 3.4 A process (Xt)t is called a k-factor GIGARCH process, if it has the following
representation
φ(B)Π
k
i=1(I − 2uiB + B
2)
di(Xt − µ) = θ(B)εt (6)
where εt = ξtσt with (ξt)t∈Z a white noise process with unit variance and mean zero, σ2
t =
a0+Σr
i=1aiε2
t−i+Σs
j=1bjσt−j, µ the mean of the process (Xt)t∈Z, φ(B) and θ(B) polynomials in
B of order p and q respectively, B the backshift operator satisfying BXt = Xt−1, d = (d1,...,dk)
the memory parameters and u = (u1,...,uk) the frequency location parameters, di ∈ R, |ui| ≤ 1,
i = 1,··· ,k.
Guégan (2000, 2003) proposed this model and has given out the corresponding covariance
stationary condition and long memory conditions.
Proposition 3.5 Let (Xt)t be a covariance stationary and long memory k-factor GIGARCH
process, then it is asymptotically second-order self-similar. Furthermore, under Gaussianity, it
is also asymptotically self-similar.
Proof : For a covariance stationary and long memory k-factor GIGARCH process, according
to lemma 2.6, a k-factor GIGARCH process is asymptotically second-order self-similar. ￿
3.3.2 GARCH processes and related processes
Deﬁnition 3.6 A process (Xt)t is a generalized autoregressive conditional heteroscedastic pro-
cess with order p and q respectively, or a GARCH(p,q) process, if it has the following repre-
sentation Xt = σtεt and
σ
2
t = α0 + Σ
q
i=1αiX
2
t−i + Σ
p
j=1βjσ
2
t−j = α0 + a(B)X
2
t + b(B)σ
2
t (7)
where α0 > 0, αi ≥ 0,i = 1,··· ,q, βj ≥ 0,j = 1,··· ,p, {εt} ∼ IID(0,1), and εt is inde-
pendent of {Xt−k,k ≥ 1} for all t, B is the backshift operator, a(B) and b(B) are polynomials
in B of order q and p respectively.
5This model has been introduced by Bollerslev (1986) and has several particular cases : if
βj = 0 in (7), we get an ARCH process (Engle 1982). If
Pq
i=1 αi +
Pp
j=1 βj = 1, we get an
IGARCH process (Bollerslev 1988).
Proposition 3.7 Let (Xt)t the covariance stationary GARCH process introduced in (7), then
it is not asymptotically second-order self-similar.
Proof : If the GARCH model is covariance stationary, then it is short memory process. Thus,
following the lemma 2.5, it is not asymptotically second-order self-similar. The conclusion is
the same for particular GARCH models like ARCH and IGARCH models. ￿
3.4 Processes with switches and jumps
Structural breaks have been observed in many economic and ﬁnancial time series. A lot of
models have been proposed in order to capture the existence of structural changes and complex
dynamic patterns. Let (Xt)t be a process whose recursive scheme is
Xt = µst + εt (8)
where (µst)t is a process we specify below and (εt)t is a strong white noise, independent of
(µst)t. With respect to the process (µst)t, we distinguish two cases :
1. (µst)t depends on a hidden ergodic Markov chain (st)t,
2. If (µst)t = µt, then we will assume that this process depends on a probability p.
The ﬁrst class of models considered above includes “models with switches" and the second
class includes “models with breaks". For most of these processes, they are covariance stationary
and short memory. Nevertheless, some processes exhibit empirically a kind of long memory
behavior when we observe the behavior of the sample autocorrelation function.
3.4.1 Processes with switches
The processes with switches have been studied in literature by Hamilton (1988), Diebold
and Inoue (2001), Guégan (2003, 2006), etc. We consider a two-state Markov Switching model
(Xt)t deﬁned by the following equations :
Xt = µst + φstXt−1 + σstεt, (9)
where µst, φst and σst (i = 1,2) are real parameters, σi are positive and (εt)t is a strong white
noise with mean m ∈ R and variance σ2 ∈ R∗
+, and (εt)t is independent of the hidden ergodic
Markov chain (st)t, which is characterized by its transition probabilities pij, deﬁned by :
P[st = j|st−1 = i] = pij (10)
with 0 ≤ pij ≤ 1 and
P2
j=1 pij = 1, i = 1,2.
6Proposition 3.8 Let {Xt}t be a process deﬁned by (9) and (10), then if maxi=1,2{pi1|φ1|2 +
pi2|φ2|2} < 1, the model is not asymptotically second-order self-similar.
Proof : Yang (2000) proved that under the above conditions, the process {Xt}t is covariance
stationary . Under stationarity, this model is known to be theoretically short memory. Thus,
following lemma 2.5, it is not asymptotically second-order self-similar. ￿
Actually, there are many other interesting models with switches contained in equation (8),
for example : the mean switching model(Xt = µst + εt), the mean variance switching model
(Xt = µst + σstεt), the sign model (Xt = sign(Xt−1) + εt, where εt ∼i.i.d N(0,σ2)). For all
of these processes, if they are covariance stationary, then they are theoretically short memory
processes, although empirically they exhibit long memory behavior. Likewise, they are not
asymptotically second-order self-similar theoretically.
3.4.2 Processes with breaks
For processes with breaks, many people have investigated their properties, for example,
Engle and Smith (1999), Granger and Hyung (2004), Hyung and Franses (2005), Guégan
(2003, 2006). Assume the process (Xt)t is deﬁned by
Xt = µt + εt (11)
where the process (µst)t = µt depends on a probability p. Diﬀerent dynamics of the process
(µt)t correspond to diﬀerent break models, for example : the Binomial model, the random walk
model with a Bernouilli process, the STOPBREAK model, the stationary random level shift
model, the mean-plus-noise model, etc.
Under stationarity, these models are short memory and cannot be asymptotically second-
order self-similar theoretically.
3.4.3 Processes with threshold
Consider the general form of the processes with threshold :
Xt = f(Xt−1)(1 − G(Xt−d,γ,c)) + g(Xt−1)G(Xt−d,γ,c) + εt (12)
where the function f and g can be any linear or nonlinear functions of the past values of Xt or
εt. The process (εt)t is a strong white noise and G is an indicator function or some continuous
function. For a given threshold c and the position of the random variable Xt−d with respect to
this threshold c, the process (Xt)t contains diﬀerent models, for example, the SETAR model,
the STAR model ( Tong 1990).
Proposition 3.9 For the process with threshold deﬁned as in (12), if the functions f and g
correspond to short memory process, then it is not asymptotically second-order self-similar.
7Proof : If the functions f and g correspond to short memory process, then the process is short
memory. Thus, following lemma 2.5, it is not asymptotically second-order self-similar.￿
Now we consider the long memory SETAR model, deﬁned as follows :
Xt = (1 − B)
−dε
(1)
t It(Xt−d ≤ c) + ε
(2)
t [1 − It(Xt−d ≤ c)], (13)
and the assumptions :
(H5) : the process (ε
(i)
t )t (i = 1,2) is a sequence of independent identically distributed random
variables.
(H6) : the long memory parameter d is such that 0 < d < 1/2.
Proposition 3.10 Under the assumptions (H5) and (H6), the process (Xt)t deﬁned by the
relation(13) is globally stationary, and then it is asymptotically second-order self-similar. Fur-
thermore, under Gaussianity, it is asymptotically self-similar.
Proof : Under the assumptions (H5) and (H6), the stationary model deﬁned in (13) is long
memory in the covariance sense. According to lemma 2.6, it is asymptotically second-order
self-similar. ￿
3.4.4 A New Concept
For the models with switching and models with breaks introduced above, theoretically
they are not asymptotically second-order self-similar. However, empirically, they exhibit long
memory behavior with the sample autocorrelation function which decreases in an hyperbolic
way towards zero. Since we often call this long memory behavior the “Spurious long memory"
behavior, we can also call this self-similar behavior the “spurious asymptotical second-order
self-similar" behavior.
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9

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A new model : the k-factor GIGARCH process,

A prospective study of the k-factor Gegenbauer with heteroscedastic errors and applications to inﬂation t rates,

Forecasting time series with long memory and level shifts,

How can we deﬁne the concept of long memory? An econometric survey,

Is network traﬃc self-similar or multifractal,

Les processus longues mémoires, Note de Cours, http ://www.ces.ens-cachan.fr, ENS-Cachan,

Long memory and regime switching,

Nonlinear Time Series,

Occasional structural breaks and long memory with application to the S&P500 absolute stock returns,

On the corrrelation structure for the generalized autoregressive conditonal heteroskedastic process,

On the self-similar traﬃc in ATM queues : deﬁnitions, overﬂow probablility bound, and cell delay distribution,

Rational expectations econometric analysis of change in regime : an investigation of the term structure of interest rates,

Some properties of vector autocoregressive processes with Markov-Switching coeﬃcients,

Stable Non-Gaussian Random Processes Stochastic Models With Inﬁnite Variance, Chapman and Hall,

Statistics For Long Memory Processes, Chapman and Hall,

Stochastic permanent breaks,

A note on self-similarity for discrete time series

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A note on self-similarity for discrete time series

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