Table Of ContentA nonlinear model for long memory conditional
heteroscedasticity
Paul Doukhan1, Ieva Grublyte˙1,2 and Donatas Surgailis2
5
1
October 20, 2015
0
2
1
Universit´e de Cergy Pontoise, D´epartament de Math´ematiques, 95302 Cedex, France
t
c 2 Vilnius University, Institute of Mathematics and Informatics, 08663 Vilnius, Lithuania
O
8
1
Abstract. Wediscussaclassofconditionallyheteroscedastictimeseriesmodels
]
T satisfying the equation rt = ζtσt, where ζt are standardized i.i.d. r.v.’s and the
S conditional standard deviation σ is a nonlinear function Q of inhomogeneous
t
.
h linear combination of past values r ,s < t with coefficients b . The existence of
s j
t
a stationarysolutionr withfinitepthmoment,0 < p < isobtainedundersome
t
m ∞
conditions on Q,b and the pth moment of ζ . Weak dependence properties of
[ j 0
2 rt are studied, including the invariance principle for partial sums of Lipschitz
v functions of r . In the case whenQ is the squareroot of a quadratic polynomial,
t
5
we prove that r can exhibit a leverage effect and long memory, in the sensethat
9 t
0 the squared process r2 has long memory autocorrelation and its normalized
0 t
0 partial sums process converges to a fractional Brownian motion.
.
2
0 Keywords: ARCH model, leverage, long memory, Donsker’s invariance princi-
5
ple
1
:
v
i
X 1 Introduction
r
a
A stationary time series r ,t Z is said conditionally heteroscedastic if its conditional
t
{ ∈ }
variance σ2 = Var[r r ,s < t] is a non-constant random process. A class of conditionally
t t| s
heteroscedastic ARCH-type processes is defined from a standardized i.i.d. sequence ζ ,t
t
{ ∈
Z as solutions of stochastic equation
}
r = ζ σ , σ = V(r ,s < t), (1.1)
t t t t s
where V(x ,x , ) is some function of x ,x , .
1 2 1 2
··· ···
The ARCH( ) model corresponds to V(x ,x , ) = a+ ∞ b x2 1/2, or
∞ 1 2 ··· j=1 j j
∞ (cid:0) P (cid:1)
σ2 = a+ b r2 , (1.2)
t j t−j
j=1
X
1
where a 0,b 0 are coefficients.
j
≥ ≥
TheARCH( )modelincludesthewell-knownARCH(p)andGARCH(p,q)modelsofEngle
∞
[13] and Bollerslev [6]. However, despite their tremendous success, the GARCH models are
not able to capture some empirical features of asset returns, in particularly, the asymmetric
or leverage effect discovered by Black [5], and the long memory decay in autocorrelation
of squares r2 . Giraitis and Surgailis [16] proved that the squared stationary solution
{ t}
of the ARCH( ) model in (1.2) with a > 0 always has short memory, in the sense that
∞
∞ Cov(r2,r2)< . (However, for integrated ARCH( ) models with ∞ b = 1,b
j=0 0 j ∞ ∞ j=1 j j ≥
0 and a = 0 the situation is different; see [19].)
P P
The above shortcomings of the ARCH( ) model motivated numerous studies proposing
∞
alternative forms of the conditional variance and the function V(x ,x , ) in (1.1). In
1 2
···
particular, stochastic volatility models can display both long memory and leverage except
that in their case, the conditional variance is not a function of r ,s < t alone and therefore
s
it is more difficult to estimate from real data in comparison with the ARCH models; see
Shephard and Andersen [31]. Sentana [30] discussed a class of Quadratic ARCH (QARCH)
models with σ2 beinga general quadratic form in lagged variables r , , r . Sentana’s
t t−1 ··· t−p
specification of σ2 encompasses a variety of ARCH modelsincludingthe asymmetric ARCH
t
modelofEngle[14]andthe‘linearstandarddeviation’modelofRobinson[27]. Thelimiting
case (when p = ) of the last model is the LARCH model discussed in [15] (see also [16],
∞
[4], [17], [33]) and corresponding to V(x ,x , ) = a+ ∞ b x , or
1 2 ··· j=1 j j
∞ P
σ = a+ b r , (1.3)
t j t−j
j=1
X
where a R,b R are real-valued coefficients. [15] proved that the squared stationary
j
∈ ∈
solution r2 of the LARCH model with b decaying as jd−1,0 < d < 1/2 may have long
{ t} j
memory autocorrelations. The leverage effect in the LARCH model was discussed in detail
in [17]. On the other hand, volatility σ (1.3) of the LARCH model may assume negative
t
values, lacking some of the usual volatility interpretation.
The present paper discusses a class of conditionally heteroscedastic models (1.1) with V
of the form
∞
V(x ,x , )= Q(a+ b x ), (1.4)
1 2 j j
···
j=1
X
where Q(x),x R is a (nonlinear) function of a single real variable x R which may
∈ ∈
be separated from zero by a positive constant: Q(x) c > 0, x R. Linear Q(x) = x
≥ ∈
corresponds to the LARCH model (1.3). Probably, the most interesting nonlinear case of
Q in (1.4) is
Q(x)= c2+x2,
p
where c 0 is a parameter. In the latter case, the model is described by equations
≥
2
r = ζ σ , σ = c2+ a+ b r . (1.5)
t t t t t−s s
s (cid:16) Xs<t (cid:17)
2
Note σ c 0 in (1.5) is nonnegative and separated from 0 if c > 0. Particular cases of
t
≥ ≥
volatility forms in (1.5) are:
σ = c2+(a+br )2 (Engle’s [14] asymmetric ARCH(1)), (1.6)
t t−1
p b p
2
σ = c2+ a+ r , (1.7)
t t−j
v p
u j=1
u (cid:0) X (cid:1)
t ∞
σ = a+ b r (Q(x) = x ), (1.8)
t j t−j
| |
j=1
(cid:12) X (cid:12)
(cid:12) (cid:12)
σ = c2+(a+b((1 L)−d 1)r )2. (1.9)
t t
− −
q
In (1.6)-(1.9), a,b,c are real parameters, p 1 an integer, Lx = x is the backward
t t−1
≥
shift, and (1 L)−dx = ∞ ϕ x , ϕ = Γ(d+j)/Γ(d)Γ(j +1),ϕ = 1 is the fractional
− t j=0 j t−j j 0
integration operator, 0 < d < 1/2. The squared volatility (conditional variance) σ2 in
P t
(1.6)-(1.9)and (1.5)is aquadratic form in lagged returnsr ,r , and hencerepresent
t−1 t−2
···
particular cases of Sentana’s [30] Quadratic ARCH (QARCH) model with p = . It
∞
should be noted, however, that the first two conditional moments do not determine the
unconditional distribution. Particularly, (1.1) with (1.5) generally is a different process
from Sentana’s [30] QARCH process, the latter being defined as a solution of a linear
random-coefficient equation for r in contrast to the nonlinear equation in (1.1). See also
t
{ }
Example 2 below.
Let us describe the main results of this paper. Section 2 obtains sufficient conditions
on Q,b and µ := E ζ p for the existence of stationary solution of (1.1)-(1.4) with finite
j p 0
| | | |
moment E r p < , p > 0. We use the fact that the above equations can be reduced to
t
| | ∞
the ‘nonlinear moving-average’ equation
X = b ζ Q(a+X )
t t−s s s
s<t
X
for linear form X := b r in (1.4), and vice-versa. Section 3 aims at providing weak
t s<t t−s s
dependence properties of (1.1) with V in (1.4), in particular, the invariance principle for
P
Lipschitzfunctionsof r and X , undertheassumptionthatb aresummableanddecay
t t j
{ } { }
asj−γ withγ > 1. Section4discusseslongmemorypropertyofthequadraticmodelin(1.5).
For b βjd−1, j , 0< d < 1/2 as in (1.9), we prove that the squaredprocess r2 has
j ∼ → ∞ { t}
long memory autocorrelations and its normalized partial sums process tend to a fractional
Brownian motion with Hurst parameter H = d + 1/2 (Theorem 10). Finally Section 5
establishes the leverage effect in spirit of [17], viz., the fact that the ‘leverage function’
h := Cov(σ2,r ),j 1 of model (1.5) takes negative values provided the coefficients a
j t t−j ≥
and b have opposite signs. All proofs are collected in Section 6 (Appendix).
j
Notation. In what follows, C,C(...) denote generic constants, possibly dependent on
the variables in brackets, which may be different at different locations. a b (t ) is
t t
∼ → ∞
equivalent to lim a /b = 1.
t→∞ t t
3
2 Stationary solution
This section discusses the existence of a stationary solution of (1.1) with V of (1.4), viz.,
r = ζ Q a+ b r , t Z. (2.10)
t t t−s s
∈
s<t
(cid:0) X (cid:1)
Denote
X := b r . (2.11)
t t−s s
s<t
X
Then r in (2.10) can be written as r = ζ Q(a + X ) where (2.11) formally satisfies the
t t t t
following equation:
X = b ζ Q(a+X ). (2.12)
t t−s s s
s<t
X
Below we give rigorous definitions of solutions of (2.10) and (2.12) and a statement (Propo-
sition 3) justifying (2.12) and the equivalence of (2.10) and (2.12).
In this section we consider a general case of (2.10)-(2.12) when the innovations may have
infinite variance. More precisely, we assume that ζ ,t Z are i.i.d. r.v.’s with finite
t
{ ∈ }
moment µ := E ζ p < , p > 0. We use the following moment inequality.
p t
| | | | ∞
Proposition 1 Let Y ,j 1 be a sequence of r.v.’s such that E Y p < for some p > 0
j j
{ ≥ } | | ∞
and the sum on the r.h.s. of (2.13) converges. If p > 1 we additionally assume that Y
j
{ }
is a martingale difference sequence: E[Y Y , ,Y ] = 0, j = 2,3, . Then there exists
j 1 j−1
| ··· ···
a constant K depending only on p and such that
p
∞ ∞ E Y p, 0 < p 2,
E Y p K j=1 | j| ≤ (2.13)
j p
(cid:12)Xj=1 (cid:12) ≤ P ∞j=1(E|Yj|p)2/p p/2, p > 2.
(cid:12) (cid:12)
(cid:0)P (cid:1)
Remark 1 In the sequel, we shallrefer to K in (2.13) as the Rosenthal constant. For
p
0 < p 1 and p = 2, inequality (2.13) holds with K = 1, and for 1 < p < 2, it is known as
p
≤
von Bahr and Ess´een inequality, see [34], which holds with K = 2. For p > 2, inequality
p
(2.13) is a consequence of the Burkholder and Rosenthal inequality (see [7], [29], also [18],
Lemma 2.5.2). Ose¸kowski [26] proved that K1/p 2(3/2)+(1/p)(p +1)1/p 1+ p , in
p ≤ 4 log(p/2)
1/4
particular, K 27.083. See also [23]. (cid:0) (cid:1)
4 ≤
Let us give some formal definitions. Let = σ(ζ ,s t),t Z be the sigma-field gener-
t s
F ≤ ∈
ated by ζ ,s t. A random process u ,t Z is called adapted (respectively, predictable)
s t
≤ { ∈ }
if u is -measurable for each t Z (respectively, u is -measurable for each t Z).
t t t t−1
F ∈ F ∈
Define
∞ b p, 0< p < 2,
B := j=1| j| (2.14)
p
P ∞ b2 p/2, p 2.
j=1 j ≥
(cid:0)P (cid:1)
Definition 2 Let p > 0 be arbitrary.
(i) By Lp-solution of (2.10) we mean an adapted process r ,t Z with E r p < such
t t
{ ∈ } | | ∞
that for any t Z the series b r converges in Lp and (2.10) holds.
∈ s<t t−s s
P
4
(ii) By Lp-solution of (2.12) we mean an predictable process X ,t Z with E X p <
t t
{ ∈ } | | ∞
such that for any t Z the series b ζ Q(a+X ) converges in Lp and (2.12) holds.
∈ s<t t−s s s
P
Let Q(x),x R be a Lipschitz function, i.e., there exists Lip > 0 such that
∈ Q
Q(x) Q(y) Lip x y , x,y R. (2.15)
| − | ≤ Q| − | ∈
Note (2.15) implies the bound
Q2(x) c2+c2x2, x R, (2.16)
≤ 1 2 ∈
where c 0, c Lip and c can bechosen arbitrarily close to Lip ; in particular, (2.16)
1 ≥ 2 ≥ Q 2 Q
holds with c2 = (1+ǫ2)Lip2,c2 = Q2(0)(1+ǫ−2), where ǫ > 0 is arbitrarily small.
2 Q 1
Proposition 3 Let Q be a measurable function satisfying (2.16) with some c ,c 0 and
1 2
≥
ζ be an i.i.d. sequence with µ = E ζ p < and satisfying Eζ = 0 for p > 1. In
t p 0 0
{ } | | | | ∞
addition, assume B < .
p
∞
(i) Let X be a stationary Lp-solution of (2.12). Then r := ζ Q(a+X ) is a stationary
t t t t
{ } { }
Lp-solution of (2.10) and
E r p C(1+E X p). (2.17)
t t
| | ≤ | |
Moreover, for p > 1, r , ,t Z is a martingale difference sequence with
t t
{ F ∈ }
E[r ]= 0, E[r p ] = µ Q(a+ b r )p. (2.18)
t t−1 t t−1 p t−s s
|F | | |F | | |
s<t
(cid:12) X
(cid:12)
(ii) Let r be a stationary Lp-solution of (2.10). Then X in (2.11) is a stationary
t t
{ } { }
Lp-solution of (2.12) such that
E X p CE r p. (2.19)
t t
| | ≤ | |
Moreover, for p 2
≥
∞
E[X X ] = Er2 b b , t = 0,1,.... (2.20)
t 0 0 t+s s
s=1
X
Remark 2 Let p 2 and µ < , then by inequality (2.13), r being a stationary
p t
≥ | | ∞ { }
Lp-solution of (2.10) is equivalent to r being a stationary L2-solution of (2.10) with
t
{ }
E r p < . Similarly, if Q and ζ satisfy the conditions of Proposition 3 and p 2,
0 t
| | ∞ { } ≥
then X being a stationary Lp-solution of (2.12) is equivalent to X being a stationary
t t
{ } { }
L2-solution of (2.12) with E X p < .
0
| | ∞
Thefollowingtheoremobtainsasufficientconditionin(2.21)fortheexistenceofastation-
ary Lp-solution of equations (2.10) and (2.12). Condition (2.21) involves the pth moment of
innovations, the Lipschitz constant Lip , the sum B in (2.14) and the Rosenthal constant
Q p
K in (2.13). Part (ii) of Theorem 4 shows that for p = 2, condition (2.21) is close to
p
optimal, being necessary in the case of quadratic Q2(x) = c2+c2x2.
1 2
5
Theorem 4 Let the conditions of Proposition 3 be satisfied, p > 0 is arbitrary. In addition,
assume that Q satisfies the Lipschitz condition in (2.15).
(i) Let
p
K µ Lip B < 1. (2.21)
p| |p Q p
Then there exists a unique stationary Lp-solution X of (2.12) and
t
{ }
C(p,Q)µ B
E X p | |p p , (2.22)
t p
| | ≤ 1 K µ Lip B
− p| |p Q p
where C(p,Q)< depends only on p and c ,c in (2.16).
1 2
∞
(ii) Assume, in addition, that Q2(x) = c2+c2x2, where c 0,i = 1,2, and µ = Eζ2 = 1.
1 2 i ≥ 2 0
Then c2B < 1 is a necessary and sufficient condition for the existence of a stationary
2 2
L2-solution X of (2.12) with a = 0.
t
{ } 6
Remark 3 Condition (2.21) agrees with the contraction condition for the operator defined
by the r.h.s. of (2.12) and acting in a suitable space of predictable processes with values
in Lp. For the LARCH model, explicit conditions for finiteness of the pth moment were
obtained in [15], [17] using a specific diagram approach for multiple Volterra series. For
larger values of p> 2, condition (2.21) is preferable to the corresponding condition
(2p p 1)1/2 µ 1/pB1/p < 1, p = 2,4,6, , (2.23)
− − | |p p ···
in([15],(2.12))fortheLARCHmodel,sincethecoefficient(2p p 1)1/2growsexponentially
− −
1/p
with p in contrast to the bound on K in Remark 1. See also ([20], sec. 4.3). On
p
the other hand for p = 4 (2.23) becomes √11 µ 1/4B1/2 < 1 while (2.21) is satisfied if
| |4 2
1/4 1/4 1/2 1/4 1/2
K µ B 27.083µ B <1, see Remark 1, which is worse than (2.23).
4 | |4 2 ≤ | |4 2
Example 1 (The LARCH model) Let Q(x) = x and ζ be a standardized i.i.d. se-
t
{ }
quence with zero mean and unit variance. Then (2.12) becomes the bilinear equation
X = b ζ (a+X ). (2.24)
t t−s s s
s<t
X
Thecorrespondingconditionallyheteroscedasticprocess r = ζ (a+X ) inProposition3(i)
t t t
{ }
is the LARCH model discussed in [15], [17] and elsewhere. As shown in ([15], Thm.2.1),
equation (2.24) admits a covariance stationary predictable solution if and only if B =
2
∞ b2 < 1. Note the last result agrees with Theorem 4 (ii). A crucial role in the study
j=1 j
of the LARCH model is played by the fact that its solution can be written in terms of the
P
convergent orthogonal Volterra series
∞
X = a b b ζ ζ .
t t−s1··· sk−1−sk s1··· sk
Xk=1sk<·X··<s1<t
Except forQ(x) = x, in other cases of (2.12)includingtheQARCHmodelin(1.5), Volterra
series expansions are unknown and their usefulness is doubtful.
6
Example 2 (Asymmetric ARCH(1)) Consider the model (1.1) with σ in (1.6), viz.
t
r = ζ c2+(a+br )2 1/2, (2.25)
t t t−1
(cid:0) (cid:1)
where ζ are standardized i.i.d. r.v.’s. By Theorem 4 (ii), equation (2.25) has a unique
t
{ }
stationary solution with finite variance Er2 =(a2 +c2)/(1 b2) if and only if b2 < 1.
t −
In parallel, consider the random-coefficient AR(1) equation
r = κε +bη r , (2.26)
t t t t−1
where (εt,ηt) are i.i.d. random vecetors with zeroemean Eεt = Eηt = 0 and unit variances
{ }
E[ε2] = E[η2]= 1 and κ,b are real coefficients. As shown in Sentana [30] (see also Surgailis
t t
[32]), equation(2.26)hasastationarysolutionwithfinitevarianceunderthesamecondition
b2 < 1as(2.25). Moreover, ifthecoefficientsκandρ:= E[ε η ] [ 1,1]in(2.26)arerelated
t t
∈ −
to the coefficients a,c in (2.25) as
κρ = a, κ2 = a2+c2, (2.27)
then the processes in (2.25) and (2.26) have the same volatility forms since
σ2 := E[r2 r ,s < t] = κ2+2κbρ+b2r2
t t| s t−1
= c2+(a+br )2
t−1
e e e e
agrees with the corresponding expression σt2 = c2+(a+brt−e1)2 in the case of (1.6).
A natural question is whether the above stationary solutions r and r of (2.25)
t t
{ } { }
and (2.26), with parameters related as in (2.27), have the same (unconditional) finite-
dimensional distributions? As shown in ([32], Corollary 2.1), the answer is poesitive in the
case when ζ and (ε ,η ) are Gaussian sequences. However, the conditionally Gaussian
t t t
{ } { }
case seems tobetheonlyexception andingeneral theprocesses r and r have different
t t
{ } { }
distributions. This can be seen by considering the 3rd conditional moment of (2.25)
e
E[r3 r ]= µ c2+(a+br )2 3/2 (2.28)
t| t−1 3 t−1
(cid:0) (cid:1)
which is an irrational function of r (unless µ = Eζ3 = 0 or b = 0), while a similar
t−1 3 0
moment of (2.26)
E[r3 r ] = κ3ν +3bκ2ν r +3b2κν r2 +b3ν r3 (2.29)
t| t−1 3,0 2,1 t−1 1,2 t−1 0,3 t−1
is a cubic polynomeiael in rt−1, where νi,j :=eE[εi0η0j]. Moreoever, (2.28) heas a constant sign
independentof r while the sign of the cubic polynomial in (2.29) changes with r ranging
t−1 t
from to if the leadeing coefficient b3ν =0.
0,3
∞ −∞ 6
Using the last observation we can prove that the bivariate distributions of (r ,er ) and
t t−1
(r ,r ) are different under general conditions on the innovations and the parameters of
t t−1
the two equations. The argument is as follows. Let b > 0,c > 0,µ > 0,ν = Eη3 > 0.
3 0,3 0
Aessueme that ζ has a bounded strictly positive density function 0 < f(x) < C,x R and
0
∈
7
(ε ,η ) has a bounded strictly positive density function 0 < g(x,y) < C,(x,y) R2. The
0 0
∈
above assumptions imply that the distributions of r and r have infinite support. Indeed,
t t
by (2.25) and the above assumptions we have that P(r > K) = P(c2 +(a+br )2 >
t R t−1
(K/y)2)f(y)dy > 0 for any K > 0 since lim P(c2 +e (a + br )2 > (K/y)2) = 1.
y→∞ Rt−1
Similarly, P(r > K) = P(r > (K κx)/by)g(x,y)dxdy > 0 and P(r < K) >
t R2 t−1 t
− −
0,P(r < K) > 0 for any K > 0. Since h(x) := µ c2+(a+bx)2 3/2 1 for all x > K
t R 3
− ≥ | |
and any sufficeiently large K > 0e, from (2.28) we obtain that for any K > 0
(cid:0) (cid:1)
e
Er31(r > K) = Eh(r )1(r > K)> 0 and
t t−1 t−1 t−1
Er31(r < K) = Eh(r )1(r < K)> 0. (2.30)
t t−1 − t−1 t−1 −
Ontheotherhand,sinceh(x) := κ3ν +3bκ2ν x+3b2κν x2+b3ν x3 1forx > K and
3,0 2,1 1,2 0,3
≥
h(x) 1 for x < K and K large enough, from (2.29) we obtain that for all sufficiently
≤ − − e
large K > 0
e
Er31(r > K) = Eh(r )1(r > K)> 0 and
t t−1 t−1 t−1
Er31(r < K) = Eh(r )1(r < K)< 0. (2.31)
te t−e1 − e et−1 et−1 −
Clearly, (2.30) ande(2.31e) imply that the beivaeriate deistributions of (rt,rt−1) and (rt,rt−1)
are different under the stated assumptions.
e e
Formodels(2.25)and(2.26),wecanexplicitlycomputecovariancesρ(t) = cov(r2,r2),ρ(t) =
t 0
cov(r2,r2) and some other joint moment functions, as follows.
t 0
Let µ = Eζ3 = 0,µ = Eζ4 < and m := Er2, m (t) := Er2r , m (t) := Er2r2,t e0.
3 0 4 0 ∞ 2 0 3 t 0 4 t 0 ≥
Thene e
m = (a2+c2)/(1 b2), m (0) = 0,
2 3
−
m (1) = E[((a2 +c2)+2abr +b2r2)r ] = 2abm +b2m (0) = 2abm ,
3 0 0 0 2 3 2
m (t) = E[((a2 +c2)+2abr +b2r2 )r ] = b2m (t 1) = = b2(t−1)m (1)
3 t−1 t−1 0 3 − ··· 3
2ab(a2 +c2)
= b2(t−1), t 1. (2.32)
1 b2 ≥
−
Similarly,
m (0) = µ E[((a2+c2)+2abr +b2r2 )2]
4 4 −1 −1
= µ (a2+c2)2+(2ab)2m +b4m (0)+2b2(a2+c2)m ,
4 2 4 2
{ }
m (t) = E[((a2+c2)+2abr +b2r2 )r2] = (a2+c2)m +b2m (t 1), t 1
4 t−1 t−1 0 2 4 − ≥
resulting in
µ ((a2+c2)2+((2ab)2 +2(a2+c2)b2)m )
4 2
m (0) = , (2.33)
4 1 µ b4
4
−
1 b2t
m (t) = m (a2+c2) − +b2tm (0), t 1,
4 2 · 1 b2 4 ≥
−
and
ρ(t) = (m (0) m2)b2t, t 0. (2.34)
4 − 2 ≥
8
In a similar way, when the distribution of ζ is symmetric one can write recursive linear
0
equations for joint even moments E[r2p(0)r2p(t)] of arbitrary order p = 1,2,... involving
E[r2l(0)r2p(t)],1 l p 1 and m (0) = E[r2k(0)],1 k 2p. These equations can be
2k
≤ ≤ − ≤ ≤
explicitly solved in terms of a,b,c and µ ,1 k 2p.
2k
≤ ≤
A similar approach can be applied to find joint moments of the random-coefficient AR(1)
processin(2.26),withthedifferencethatsymmetryof(ε ,η )isnotneeded. Letm := Er2,
0 0 2 t
m (t) := E[r2r ], m (t) := E[r2r2] and ρ(t) := Cov(r2,r2), ν := E[εiηj]. Then
3 t 0 4 t 0 t 0 i,i 0 0
e e
e m2 = eκe2/(1e b2), e e e e e
−
m (0) = E[(κε +bη r )3] = κ3ν +3κb2ν m +b3ν m (0),
3 0 0 −1 3,0 1,2 2 0,3 3
e
m (1) = E[(κ+2κρbr +b2r2)r ] = 2κρbm +b2m (0),
3 0 0 0 2 3
e e e e
m (t) = E[(κ2+2κρbr˜ +b2r˜2 )r˜ ] = b2m (t 1) = = b2(t−1)m (1), t 2,
e 3 e t−1 e et−1 0 e 3 −e ··· 3 ≥
aned e e
m (0) = E[(κε +bη r )4]
4 0 0 −1
= κ4ν +6κ2b2ν m +4κb3ν m (0)+b4ν m (0),
4,0 2,2 2 1,3 3 0,4 4
e e
m (1) = E[(κε +bη r )2r2] = κ2m +2κρbm (0)+b2m (0)
4 t t 0 0 2 3 4
e e e
m (t) = E[(κε +bη r )2r2] = κ2m +b2m (t 1), t 2,
e 4 t tet−1e 0 e 2 e 4 − e ≥
leading to e e e e e
κ3ν +3κb2ν m
3,0 1,2 2
m (0) = ,
3 1 ν b3
0,3
−
m (t) = b2(t−1)(2κρbm +eb2m (0)) t 1.
e 3 2 3
≥
κ4ν +6κ2b2ν m +4κb3ν m (0)
4,0 2,2 2 1,3 3
m (0) = , (2.35)
e4 e 1 ν eb4
0,4
−
1 b2t e e
me (t) = m κ2 − +b2t(m (0)+2κρm (0)/b), t 1,
4 2 1 b2 4 3 ≥
(cid:16) − (cid:17)
and e e e e
ρ (t) = b2(t−1)ρ (1), t 1,
4 4
≥
ρ (1) = 2ρκbm (0)+b2(m (0) m2).
e4 e3 4 − 2
Then if ν3,0 = ν1,2 = 0 weehave m3(0) = 0eand ρ4(t) =e(m4(0)−e m22)b2t; moreover, m2 = m2
in view of (2.27). Then ρ (t) = ρ (t) is equivalent to m (0) = m (0), which follows from
4 4 4 4
e e e e e
µ4e= ν0,4 = ν4,0 and 6ν2,2 =eµ4(4ν12,1 +2), (2.36)
see (2.34), (2.33), (2.35). Note that (2.36) hold for centered Gaussian distribution (ε ,η )
0 0
with unit variances Eε2 = Eη2 = 1.
0 0
9
3 Weak dependence
Various measures of weak dependence for stationary processes y = y ,t Z have been
t t
{ } { ∈ }
introducedintheliterature,seee.g. [9]. Usually,thedependencebetweenthepresent(t 0)
≥
and the past (t n) values of y is measured by some dependence coefficients decaying
t
≤ − { }
to 0 as n . Thedecay rate of these coefficients plays a crucial role in establishing many
→ ∞
asymptotic results. The classical problem is proving Donsker’s invariance principle:
[nτ]
1
(y Ey ) σB(τ), in the Skorohod space D[0,1], (3.37)
t
√n − →
t=1
X
where B = B(τ),τ [0,1] is a standard Brownian motion. The above result is useful
{ ∈ }
in change-point analysis (Cso¨rgo˝ and Horva´th [8]), financial mathematics and many other
areas. Further applications of weak dependence coefficients include empirical processes
[12] and the asymptotic behavior of various statistics, including the maximum likelihood
estimators. See Ibragimov and Linnik [24] and the application to GARCH estimation in
[25].
The present Section discusses two measures of weak dependence - the projective weak
dependencecoefficients ofWu[35]andtheτ-dependencecoefficients introducedinDedecker
and Prieur [10], [11] - for stationary solutions r , X of equations (2.10), (2.12). We
t t
{ } { }
show that the decay rate of the above weak dependence coefficients is determined by the
decay rate of the moving average coefficients b .
j
3.1 Projective weak dependence coefficients
Let us introduce some notation. For r.v. ξ, write ξ := E1/p ξ p, p 1. Let y ,t Z
p t
k k | | ≥ { ∈ }
be a stationary causal Bernoulli shift in i.i.d. sequence ζ , in other words,
t
{ }
y = f(ζ ,s t), t Z,
t s
≤ ∈
where f : RN R is a measurable function. We also assume Ey = 0, y 2 = Ey2 < .
→ 0 k 0k2 0 ∞
Introduce the projective weak dependence coefficients
ω (i; y ) := f (ξ ) f (ξ′) , δ (i; y ) := f(ξ ) f(ξ′) , (3.38)
2 { t} k i 0 − i 0 k2 2 { t} k i − i k2
where ξ := ( ,ζ ,ζ ,ζ , ,ζ ), ξ′ := ( ,ζ ,ζ′,ζ , ,ζ ), f (ξ ) := E[f(ξ )ξ ] =
i ··· −1 0 1 ··· i i ··· −1 0 1 ··· i i 0 i | 0
E[y ] is the conditional expectation and ζ′,ζ ,t Z are i.i.d. r.v.s. Note the i.i.d.
i|F0 { 0 t ∈ }
sequences ξ and ξ′ coincide except for a single entry. Then ω (i; y ) δ (i; y ),i 0
i 2 { t} ≤ 2 { t} ≥
and condition
∞
ω (k; y ) < (3.39)
2 t
{ } ∞
k=0
X
guarantees the weak invariance principle in (3.37) with σ2 := Cov(y ,y ), see Wu
j∈Z 0 j
[35]. The last series absolutely converges in view of (3.39) and the bound in ([35], Thm. 1),
P
implying Cov(y ,y ) ∞ ω (k; y )ω (k+j; y ),j 0.
| 0 j | ≤ k=0 2 { t} 2 { t} ≥
P
10
