Stability of random coefficient ARCH models and aggregation schemes

Journal of Econometrics 120 (2004) 139–158www.elsevier.com/locate/econbase

Stability of random coe#cient ARCH models andaggregation schemes

Vytautas Kazakevi.ciusa, Remigijus Leipusa;b;∗;1, Marie-Claude Vianoc;1

aDepartment of Mathematics and Informatics, Vilnius University, Naugarduko 24,Vilnius 2600, Lithuania

bInstitute of Mathematics and Informatics, Akademijos 4, Vilnius 2600, LithuaniacLaboratoire de Statistique et Probabilit'es, Bat. M2, Universit'e des Sciences et Technologies de Lille,

F-59655 Villeneuve d’Ascq Cedex, France

Accepted 30 June 2003

Abstract

In this paper, we consider ARCH(∞) models with nonnegative random coe#cients. Necessaryand su#cient conditions for the existence of 8rst and second moments are established. The ARCHmodels with deterministic coe#cients are shown to be always short memory processes. The e:ectof small perturbations of the parameters is investigated. The results obtained are applied to someaggregation schemes.c© 2003 Elsevier B.V. All rights reserved.

JEL classi5cation: C22; C43; C65

Keywords: ARCH(∞) model; Aggregation; Random coe#cients; Long memory

1. Introduction

It is known that long memory time series can be obtained by aggregating random para-meter short memory autoregressive processes. Since the pioneering paper of Granger(1980), this question has attracted considerable attention in the econometric and timeseries literature, see Gon@calves and GouriAeroux (1988), Lippi and Za:aroni (1999),and Oppenheim and Viano (1999) among others.

∗ Corresponding author. Department of Mathematics and Informatics, Vilnius University, Naugarduko 24,Vilnius 2600, Lithuania.

E-mail address: [email protected] (R. Leipus).1 Supported by a cooperation agreement CNRS/LITHUANIA (4714).

0304-4076/$ - see front matter c© 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0304-4076(03)00209-4

mailto:[email protected]

140 V. Kazakevi:cius et al. / Journal of Econometrics 120 (2004) 139–158

The basic scheme is always the same, starting from n elementary individuals(X (1)

t ) ; : : : ; (X (n)t ), where each individual evolves according to a random parametric

short memory dynamics. The aggregated process is the limit, as n → ∞, of the nor-malized sums

∑nk=1 X (k)

t , where the normalizing constant is n−1 in the case of commoninput noise, and n−1=2 in the case of independent inputs.For autoregressive conditionally heteroscedastic (ARCH) models, the aggregation

can be performed in di:erent ways. Leipus and Viano (2002) studied the aggrega-tion procedure based on averaging the squares of returns described by the ARCHprocess. Slightly di:erent aggregation scheme modelling the long memory propertywas proposed by Ding and Granger (1996). They discussed the aggregation procedurebased on averaging the volatilities of GARCH(1,1) processes, and their n-componentGARCH(1,1) model is of the form

rt = tt ; 2t =n∑

i=1

w(n)i 2i; t ; (1.1)

2i; t = 2(1− i − �i) + ir2t−1 + �i2i; t−1; (1.2)

where t are i.i.d. random variables with zero mean and unit variance, and the weightsw(n)

i satisfy∑n

i=1 w(n)i = 1. Here i and �i are random coe#cients. Eq. (1.2) can be

formally rewritten as

2i; t = 21− i − �i

1− �i+

∞∑k=1

i�k−1i r2t−k :

Hence, for 8xed n, r2t =: (r(n)t )2 has a representation in the ARCH(∞) form (1.4) below

with bj =: b(n)j =

∑ni=1 w(n)

i i�j−1i and �t=2t . If i, �i, and the weights w(n)

i are properlychosen, then the ARCH(∞) representation of r2t has the limit form (as n → ∞)

r2t = 2t

(∫1− − �1− �

dF( ; �) +∞∑k=1

r2t−k

∫ �k−1 dF( ; �)

); (1.3)

where F( ; �) is some distribution function. In the present paper, this passage to thelimit is given with a rigorous justi8cation. Ding and Granger (1996) showed that,taking a speci8c form of F( ; �), the coe#cients in the ARCH(∞) expansion (1.3)decay hyperbolically. On the basis of this fact, they conjectured that the limiting processhas long memory.In our paper, we study aggregation schemes of general ARCH-type processes with

random coe#cients. These schemes include the n-component GARCH(1,1) model as aspecial case. Our conclusion is directly opposite to the conjecture of Ding and Granger(1996): we prove that such an aggregation scheme does not lead to long memoryprocesses (see Theorem 2.2). The proof easily follows from our general results: 8rst weshow that each process satisfying ARCH equations with deterministic coe#cients is ashort memory process; then we prove that the suitably aggregated (as in Ding-Granger’sscheme) ARCH processes converge to a process which evolves according to someARCH model with nonrandom coe#cients.

V. Kazakevi:cius et al. / Journal of Econometrics 120 (2004) 139–158 141

To prove that an ARCH process with deterministic coe#cients has short memory, weneed necessary and su#cient conditions for the square-integrability of classical ARCHprocesses. By classical ARCH processes we mean the models where the returns rtadmit a representation of the form rt = tt , where t are i.i.d. random variables withzero mean and 8nite variance, and 2t are linear combinations of the squares of pastreturns rt−1, rt−2; : : : . Denoting Xt = r2t and �t = 2t , we represent the ARCH model forsquares r2t in the ARCH(∞) form proposed by Robinson (1991). For convenience, werecall the de8nition.

De�nition 1.1. A random sequence (Xt; t ∈Z) is said to satisfy ARCH(∞) equationsif there exist a sequence of i.i.d. nonnegative random variables � = (�t ; t ∈Z) and asequence of nonnegative numbers b= (bi; i¿ 0) such that

Xt =

(b0 +

∞∑i=1

biXt−i

)�t : (1.4)

This model was later studied by Giraitis et al. (2000). In particular, they proved thatthe condition

(E�20)1=2

∞∑i=0

bi ¡∞ (1.5)

is su#cient for the existence of a unique second-order stationary solution to (1.4),which is given by

Xt = b0�t

∞∑k=0

∞∑i1 ; ::: ; ik=1

bi1 · · · bik �t−i1 · · · �t−i1−···−ik : (1.6)

Moreover, under (1.5), the covariance Cov(Xk; X0) is nonnegative and summable.Kazakevi.cius and Leipus (2002, Section 3) proved that this solution is minimal inthe sense that every other strictly stationary solution to (1.4), say X , satis8es theinequality

Xt6 X t a:s: for all t: (1.7)

We denote this minimal solution X by m(b; �).In Section 2, we establish necessary and su#cient conditions for the existence of a

second-order stationary solution to (1.4) and prove (see Theorem 2.2 below) that allsquare-integrable processes X satisfying (1.4) have a summable covariance function,i.e., are short memory processes.Necessary and su#cient conditions for the existence of an integrable or square-

integrable random coe#cient ARCH (RC ARCH) process are easily obtained from thecorresponding conditions in the nonrandom coe#cient case. The class of RC ARCH(∞)processes is de8ned as follows.

De�nition 1.2. A random sequence {Xt; t ∈Z} is said to satisfy random coe#cientARCH(∞) equations if there exist a sequence of i.i.d. nonnegative random variables


� = (�k ; k ∈Z) and a sequence of nonnegative random variables b = (bi; i¿ 0)independent of � such that (1.4) holds almost surely.

Note that the random coe#cient ARCH(∞) model (1.4) is a nonergodic processwith covariance function which does not tend to zero as lag increases (see Section 3).Convergence of aggregated processes is a special case of general stability problem:

do small perturbations of the coe#cients bi cause small changes of the solution m(b; �)?Here m(b; �) is the minimal solution given by (1.6). More precisely, let b(n) be asequence of random sequences such that, as n → ∞,

b(n)iP→ bi ∀i∈Z; (1.8)

where P→ denotes the convergence in probability, and bi are some random variables.Does this imply that the processes X (n) =m(b(n); �) tend to X =m(b; �) in some sense?We consider this problem in Section 4 and then, in Section 5, we apply the obtainedresults to the aggregating scheme of ARCH processes. The generalization of stabilityresults is provided for general nonlinear models of the form

Xt = �t

∞∑k=0

∞∑i1 ; ::: ; ik=1

�(i1; : : : ; ik)�t−i1 · · · �t−i1−···−ik (1.9)

with random coe#cients �. In the case of deterministic coe#cients, (1.9) is a Volterraexpansion, some properties of which are presented by Priestley (1988). Assuming thatmodel (1.9) is de8ned in the L1 or L2-sense, the stability problem can be well-statedas well (see Section 4.3). This easily leads to necessary and su#cient conditions forthe convergence in the L1 or L2-sense of 8nitely aggregated sequences to their limit inthe scheme based on a simple averaging of n random coe#cient ARCH(∞) models(1.6) (see Leipus and Viano (2002) and Remark 4.1 below).The paper is organized as follows. In Section 2, we consider the nonrandom co-

e#cient ARCH(∞) model. We establish necessary and su#cient conditions for theexistence of a square integrable solution and give an expression of the second momentEX 2

t . Section 3 is devoted to the random coe#cient ARCH(∞) models. The problemof stability of the ARCH(∞) model and of the extended model (1.9) is studied inSection 4. Finally, in Section 5, we apply the results obtained to the aggregation ofrandom coe#cient ARCH(∞) models.

2. Second-order properties of ARCH(∞) model with nonrandom coe!cients

In this section, we consider the ARCH(∞) model (1.4) with nonrandom coe#cients.An integrable solution to (1.4) exists if and only if the minimal solution (1.6) satis8esEXt ¡∞ (see Proposition 3.1 below). Set �1 := E�0, B :=

∑∞i=1 bi, and

�1(b) = b0�1∞∑k=0

∞∑i1 ; ::: ; ik=1

bi1 · · · bik �k1 = b0�1

∞∑k=0

(B�1)k : (2.1)

If

B�1 ¡ 1; (2.2)


then the series on the right-hand side converges, and we have

�1(b) =b0�1

1− B�1: (2.3)

Otherwise, �1(b)=∞. It is easily seen that �1(b)=EXt . Therefore, condition B�1 ¡ 1is necessary and su#cient for the existence of an integrable solution to (1.4).In order to derive necessary and su#cient conditions for the existence of a square

integrable solution to (1.4) and to obtain a simple expression for

�2(b) = EX 2t ;

we have to establish a condition of 8niteness of the sum (cf. equality (2.11) of Giraitiset al., 2000)

�2(b) = b20�2∞∑

k;l=0

∞∑i1 ; ::: ; ik=1

∞∑j1 ; ::: ; jl=1

bi1 · · · bik bj1 · · · bjl�k+l1

(�2�21

)�(i1 ; ::: ;ik ; j1 ; ::: ; jl);

(2.4)

where �2 := E�20 and

�(i1; : : : ; ik ; j1; : : : ; jl)

= |{i1; i1 + i2; : : : ; i1 + · · ·+ ik} ∩ {j1; j1 + j2; : : : ; j1 + · · ·+ jl}| (2.5)

(|A| stands for the number of elements of a set A).In the sequel, we abbreviate �1(b) and �2(b) to �1 and �2. For k; l¿ 1, de8ne

�kl =

{bk+l if k6 l;

bk+l + bk−l if k ¿ l:

Let l∞ be the Banach space of all bounded sequences endowed with the norm ‖x‖=supk¿1|xk |. Let � and b′ be respectively the linear operator and linear form on l∞de8ned by

(�x)k =∞∑l=1

�klxl; b′x =∞∑l=1

blxl;

where x = (xk ; k¿ 1)∈ l∞. Then

‖�‖= supk¿1

∞∑l=1

�kl6B; ‖b′‖=∞∑l=1

bl = B:

Together with the above inequality, B�1 ¡ 1 implies that the operator I − �1� is in-vertible. In this case, we denote

G = b′(I − �1�)−1b: (2.6)

Let !k = Cov(Xt; Xt−k), "k = !k=!0, and " = ("k ; k¿ 1). The following theoremprovides necessary and su#cient conditions for the 8niteness of the second moment�2 and establishes a simple formula for �2 and the correlations "k .


Theorem 2.1. �2 ¡∞ if and only if

B�1 ¡ 1 and G�2 ¡ 1: (2.7)

If this condition holds, then

�2 =b20�2

(1− B�1)2· 1− G�211− G�2

(2.8)

and

"= �1(I − �1�)−1b: (2.9)

The proof is given in appendix.Note that, for computational purposes, (2.6) can be rewritten as

G =∞∑j=0

�j1

∞∑k;l=1

∞∑i1 ;:::; ij=1

bk�k; i1 · · ·�ij;lbl: (2.10)

Expression (2.9) shows that the correlation sequence " does not depend on �2. Usingthis fact, one can easily derive that the covariance function of an ARCH(∞) processis always summable.

Theorem 2.2. Every square-integrable ARCH(∞) process X = m(b; �) with nonran-dom coeDcients is a short memory process, i.e.,∑

k¿0

Cov(Xk; X0)¡∞: (2.11)

Proof. By Giraitis et al. (2000, Lemma 2.1), the covariances Cov(Xk; X0) are positive.Introduce the new process X = m(b; �), where � = (�t ; t ∈Z) is a sequence of i.i.d.random variables with expectation E�t = E�t = �1 and the second moment E�2t = �2satisfying the inequality B

√�2 ¡ 1 (it is possible, since �2 can be chosen arbitrar-

ily close to �21 and, by Theorem 2.1, B�1 ¡ 1). Then, by Proposition 3.1 of Giraitiset al. (2000),

∑k¿0 Corr(X k ; X 0)¡∞. On the other hand, by (2.9), the correlations "k

depend on �1 and b only. Therefore, Corr(X k ; X 0)=Corr(Xk; X0), and (2.11) holds.

Remark 2.1. In the ARCH(p) case, bi = 0 for i¿p. Then � and b in (2.6) can bereplaced by the matrices

� =

�11 · · · �1p

.... . .

...

�p1 · · · �pp

; b=

b1

...

bp

:

In the case of Gaussian noise, if �1=1 and �2=3, the second condition of Theorem 2.1becomes

3b′(I − �)−1b¡ 1;

which coincides with the condition of Theorem 3 of MilhHj (1985).


Remark 2.2. In the GARCH(p; q) case, the expressions for B and G can be obtaineddirectly. For instance, consider the GARCH(1,1) model de8ned by

Xt = ht�t ; ht = 0 + 1Xt−1 + �1ht−1:

Then, taking expectations of both sides of the equations hit =( 0 + ht−1( 1�t−1 +�1))i,

i = 1; 2, direct calculations lead to

�1 = 0�11− –1 ; �2 =

20�2(1 + –1)(1− –1)(1− –2) ; (2.12)

where –i := E( 1�0 + �1)i, i = 1; 2. Comparison of (2.12) with (2.3) and (2.8), yields

B= 1

1− �1(2.13)

and

G = 21

1− �21 − 2 1�1�1: (2.14)

Hence, since –216 –2, condition (2.7) is equivalent to 21�2 + 2 1�1�1 + �21 ¡ 1.Note that, in a number of particular cases of the GARCH model, Karanasos (1999)

obtained necessary conditions for the 8niteness of �2. Our results allow one easily toconclude that these conditions also are su#cient. The su#ciency also follows fromHe and TerRasvirta (1999); however, the proof of these authors is rather complicatedand uses the assumption that the GARCH model begins at some 8nite value in8nitelymany periods ago (see Ling and McAleer, 2002).

Remark 2.3. An alternative form of �2 can be obtained using a martingale represen-tation of the stationary solution (1.6) of model (1.4). Rewrite equations (1.4) as

Xt = ht�t ; ht = b0 +∞∑i=1

biXt−i =: b0 + �−11 (1− (L))Xt; (2.15)

where, by (2.2), (z) ≡ 1− �1∑∞

i=1 bizi does not vanish for |z|6 1. Eqs. (2.15) canbe rewritten as

(L)(Xt − EXt) = �t ; (2.16)

where �t = Xt − ht�1, t ∈Z, is a stationary sequence of martingale di:erences, i.e.,E(�t |Ft−1)=0, where Ft=(�j; j6 t) is the -8eld generated by the random variables�t ; �t−1; : : : : Since �2 ¡∞, �t also have 8nite second moment, and it is easy to verifythat E�20 =�−1

2 (�2 −�21)EX20 . From (2.16) we obtain the representation (see Lemma 4.1

in Giraitis et al. 2000)

Xt − EXt = −1(L)�t : (2.17)

The coe#cients in the expansion −1(z) = 1 +∑∞

i=1 izi are nonnegative and∑i i ¡∞. By (2.17), if t¿ 0, then

Cov(Xt; X0) = E�20

∞∑i=0

i t+i with 0 := 1: (2.18)


Thus, denoting ‖ ‖22 :=∞∑i=1

2i , the variance of Xt is �2 − �21 = E�20(1 + ‖ ‖22): This,

together with (2.3), leads to

�2 =b20�2

(1− B�1)2· 1

1− (�2 − �21)�−21 ‖ ‖22

: (2.19)

Hence, by (2.8), we have

G =‖ ‖22

�21(1 + ‖ ‖22)and condition G�2 ¡ 1 becomes (�2 − �21)�

−21 ‖ ‖22 ¡ 1.

A formula for the covariance Cov(Xt; X0) can be derived from (2.18) and (2.19),taking into account that E�20 = �−1

2 (�2 − �21)�2.We complete this remark by noting that if B(z) :=

∑∞j=1 bjzj, then

‖ ‖22 =12(

∫ (

−(

∣∣∣∣∣∣∞∑j=1

jeij�

∣∣∣∣∣∣2

d�=12(

∫ (

−(

∣∣∣∣ �1B(ei�)1− �1B(ei�)

∣∣∣∣2

d�

=12(

∫ (

−(

∣∣∣∣∣∣∞∑j=1

(�1B(ei�)) j

∣∣∣∣∣∣2

d�=12(

∞∑j; j′=1

�j+j′

1

∫ (

−(Bj(ei�)Bj′

(e−i�) d�

=∞∑

j; j′=1

�j+j′

1

∑k1+···+kj=l1+···+lj′

bk1 · · · bkj bl1 · · · blj′ :

This gives another explicit expression for G (cf. (2.10)) and �2.For an alternative approach, based on the orthogonal representation of ARCH(∞)

model and leading to the same result, see Giraitis and Surgailis (2002).

3. First and second-order properties of ARCH(∞) model with random coe!cients

Consider the general random coe#cient ARCH model given in De8nition 1.2. Thefollowing simple result is vital for our investigation.

Proposition 3.1. Let Xt be the minimal solution (1.6). Then condition EXt ¡∞ (re-spectively, EX 2

t ¡∞) is necessary and suDcient for the existence of an integrable(respectively, square-integrable) solution to (1.4).

Proof. According to Theorem 2.1 of Kazakevi.cius and Leipus (2002) (which alsoholds in the random coe#cient case), X given by (1.6) satis8es (1.7). If EXt ¡∞,then X is an integrable solution to (1.4). Conversely, if X is some integrable solution,then (1.7) implies that EXt6EX t ¡∞.The square integrability can be treated similarly.


Obviously, �1(b) (respectively, �2(b)) is the conditional expectation of Xt (respec-tively, X 2

t ) given b and, therefore, EXt = E�1(b) (respectively, EX 2t = E�2(b)). Thus,

Proposition 3.1 implies the following result.

Theorem 3.1. (i) Conditions

B�1 ¡ 1 a:s: and E[

b01− B�1

]¡∞ (3.1)

are necessary and suDcient for the existence of an integrable solution to (1.4).(ii) Conditions

B�1 ¡ 1 a:s:; G�2 ¡ 1 a:s:; and E[

b20(1− B�1)2(1− G�2)

]¡∞ (3.2)

are necessary and suDcient for the existence of a square-integrable solution to (1.4).

Proof. The 8rst assertion follows from equality (2.1). To prove (ii), note that, underconditions B�1 ¡ 1 and G�2 ¡ 1,

�2(b) =b20�2(1− G�21)

(1− B�1)2(1− G�2):

Therefore, the proof follows from Theorem 2.1 and from the inequalities �2 − �21 ¡�2× (1− G�21)¡�2.

Note that, under condition (3.1) (respectively, (3.2)), the solution X is unique in acertain class of integrable (respectively, square-integrable) processes (see Leipus andViano (2002) for the exact formulation and the proof of this statement).Concerning su#cient conditions for the 8niteness of E�2(b) note that, since �16

√�2,

from (2.4) we obtain the estimate

�2(b)6 b20�2

[ ∞∑k=0

(B√

�2)k]2

(3.3)

and, hence, the conditions

B√

�2 ¡ 1 a:s: and E[

b01− B

√�2

]2¡∞ (3.4)

imply the existence of a square-integrable solution to (1.4).We complete this section with a remark on the dependence structure of ARCH(∞)

models with random coe#cients. Let EbXt denote the conditional expectation of Xt

given b. Di:erently from the nonrandom coe#cient case, under the assumptionVar EbX0 ¿ 0, any square-integrable random coe#cient ARCH(∞) process X exhibitsthe long memory property (see Leipus and Viano, 2002):∑

k

Cov(Xk; X0) =∞:

This result follows from the equality

Cov(Xk+h; Xk) = ECovb(Xk+h; Xk) + Var EbXk ; (3.5)


which clearly shows that the covariance function Cov(Xk; X0) does not tend to zeroat in8nity, except the deterministic coe#cient case, and that the process X has nospectral density. More precisely, by (2.11), Covb (Xh; X0) → 0 (h → ∞) a.s. andCovb (Xh; X0)6Varb X0, where EVarbX06EX 2

0 ¡∞. Thus, by the Dominated Con-vergence Theorem, the 8rst term on the right-hand side of equality (3.5) tends to zeroimplying that

Cov(Xh; X0) → Var EbX0 ¿ 0 as h → ∞: (3.6)

From (3.6) it is easy to see that such random coe#cient ARCH models are nonergodic,di:erently from the deterministic coe#cient case. The statistical inference for such typemodels is more complicated. However, the fact that their covariance does not vanishat in8nity may provide an explanation to empirical 8ndings showing that the sampleautocorrelations of some 8nancial data remain signi8cantly nonzero even at large lags(see, e.g., Ding et al., 1993; Andersen et al., 2001).

4. Stability of ARCH(∞) model with random coe!cients

In this section, we study the problem formulated in the introduction: given a sequenceof random coe#cient processes X (n)=m(b(n); �) such that, for all i¿ 0, b(n)i

P→ bi, whichadditional conditions are necessary and su#cient for the convergence of X (n)

t to Xt inthe L1 or L2-sense?The results of this section are based on the following well-known fact from proba-

bility theory (see, e.g., Theorem 13 in Section 7.5 of Galambos, 1988).

Theorem 4.1. Let (Yn) be a sequence of nonnegative random variables fromLp (16p¡∞), such that Yn

P→Y as n → ∞. Then the following three statementsare equivalent:

(1) YnLp→Y as n → ∞;

(2) the sequence (Ypn ) is uniformly integrable;

(3) EYpn → EYp as n → ∞.

4.1. L1-stability

The main result of this subsection is the following theorem.

Theorem 4.2. Suppose that X (n) =m(b(n); �) and X =m(b; �) are integrable processesand that (1.8) holds. Then X (n)

tL1→Xt if and only if E�1(b(n)) → E�1(b) as n → ∞.

Proof. Let {p(i1; : : : ; ik)} be a family of positive numbers such that∞∑k=0

∞∑i1 ; ::: ; ik=1

p(i1; : : : ; ik) = 1 (4.1)


and let K; I1; I2; : : : be random variables independent of b and � and such that

P{K = k; I1 = i1; : : : ; Ik = ik}= p(i1; : : : ; ik)

for all k¿ 0 and i1; : : : ; ik ¿ 1. Set

Yn = b(n)0 b(n)I1 · · · b(n)IK �K+11 =p(I1; : : : ; IK);

Y = b0bI1 · · · bIK �K+11 =p(I1; : : : ; IK): (4.2)

Then

|EYn − EY |= |EX (n)t − EXt |6E|X (n)

t − Xt |6E|Yn − Y |: (4.3)

By assumption (1.8), b(n)0 b(n)i1 · · · b(n)ikP→ b0bi1 · · · bik . Applying the Dominated Conver-

gence Theorem, we get that

P{|Yn − Y |¿ j}

=∞∑k=0

∞∑i1 ; ::: ; ik=1

p(i1; : : : ; ik)

×P{�k+11 |b(n)0 b(n)i1 · · · b(n)ik − b0bi1 · · · bik |¿p(i1; : : : ; ik) j}

tends to 0 as n → ∞. Therefore, YnP→Y .

By Theorem 4.1 with p=1, the outer terms in (4.3) simultaneously tend to 0. Hence,the same is true for the inner terms.

4.2. L2-stability

The following theorem provides a necessary and su#cient condition for the L2-convergence of random coe#cient ARCH(∞) processes.

Theorem 4.3. Suppose that X (n) = m(b(n); �) and X = m(b; �) are square-integrableprocesses and that (1.8) holds. Then X (n)

tL2→Xt if and only if E�2(b(n)) tends to

E�2(b) as n → ∞.

Proof. Suppose that X (n)t

L2→Xt . Then

|[E�2(b(n))]1=2 − [E�2(b)]1=2|= ‖|X (n)t ‖L2 − ‖Xt‖L2 |6 ‖X (n)

t − Xt‖L2 → 0:

To prove the su#ciency, suppose that E�2(b(n)) → E�2(b). Let p(i1; : : : ; ik), K ,I1; I2; : : :, Yn, Y denote the same objects as in the proof of Theorem 4.2, and let(L; J1; J2; : : :) be an independent copy of (K; I1; I2; : : :) (both independent of b).De8ne

Zn = b(n)0 b(n)J1 · · · b(n)JL �L+11 (�2=�21)

�(I1 ; ::: ; IK ; J1 ; ::: ; JL)+1=p(J1; : : : ; JL)

and

Z = b0bJ1 : : : bJL�L+11 (�2=�21)

�(I1 ; ::: ; IK ; J1 ; ::: ; JL)+1=p(J1; : : : ; JL):


Then

E[YnZn] = E�2(b(n)) → E�2(b) = E[YZ]: (4.4)

Since YnZnP→YZ , Theorem 4.1 and (4.4) yield the uniform integrability of (YnZn).

From �2¿ �21 we get that Zn¿Y ′n, where

Y ′n = b(n)0 b(n)J1 · · · b(n)JL �L+1

1 =p(J1; : : : ; JL):

Therefore, (YnY ′n) also is uniformly integrable. Thus, since YnY ′

nP→YY ′, where

Y ′ = b0bJ1 · · · bJL�L+11 =p(J1; : : : ; JL);

we get

YnY ′nL1→YY ′: (4.5)

Let B denote the -8eld generated by the sequences b; b(1); b(2); : : :, and let EB bethe corresponding conditional expectation. By the inequality

E|EB[YnY ′n]− EB[YY ′]|6EEB|YnY ′

n − YY ′|= E|YnY ′n − YY ′|

relation (4.5) yields

EB[YnY ′n]

L1→EB[YY ′]: (4.6)

Since the vectors (Yn; Y ) and (Y ′n; Y

′) are conditionally independent, we getEB[YnY ′

n] = [EBYn]2 and EB[YY ′] = [EBY ]2. Hence, (4.6) becomes

[EBYn]2L1→ [EBY ]2: (4.7)

We will show that

EBYnL1→EBY: (4.8)

Set n := E|[EBYn]2−[EBY ]2|. By (4.7), n → 0. Then, since Yn and Y are nonnegativevariables, we have

E|EBYn − EBY |=E|EBYn − EBY |1{EBYn+EBY¿1=2n }

+E|EBYn − EBY |1{EBYn+EBY≤1=2n }

6 −1=2n E|[EBYn]2 − [EBY ]2|+ 1=2n = 21=2n → 0:

Hence, (4.8) follows and, therefore,

E�1(b(n)) = EEBYn → EEBY = E�1(b):


By Theorem 4.2, X (n)t

L1→Xt , which implies X (n)t

P→Xt . From the assumptions of the

theorem and from Theorem 4.1 (with p= 2) we obtain that X (n)t

L2→Xt .

Denote by U�2(b) the right-hand side of (3.3), i.e.,

U�2(b) = b20�2

( ∞∑k=0

(B√

�2)k)2

:

In the following theorem, we establish a su#cient condition for the L2-convergence interms of U�2(b).

Theorem 4.4. Let X (n)=m(b(n); �) and X =m(b; �), and let condition (3.4) be satis5edfor both X (n) and X . Assume that (1.8) holds and

E U�2(b(n)) → E U�2(b): (4.9)

Then X (n)t

L2→Xt .

Proof. The same arguments as in the proof of Theorem 4.3 imply the uniform inte-grability of the sequence

b(n)0 b(n)I1 · · · b(n)IK b(n)J1 · · · b(n)JL (√

�2)K+L; n¿ 1:

Since �16√�2, the sequence

b(n)0 b(n)I1 · · · b(n)IK b(n)J1 · · · b(n)JL �K+L1 (�2=�21)

�(I1 ; :::; IK ; J1 ; :::; JL)

also is uniformly integrable. Applying Theorem 4.1, we get E�2(b(n)) → E�2(b). By

Theorem 4.3, X (n)t

L2→Xt .

4.3. Extension to general nonlinear sequences

The results established in the previous two subsections are restricted to specialmodels. In particular they rule out the aggregation schemes which do not lead toa limiting process having an ARCH(∞) structure.These results can be easily generalized to more general nonlinear time series with

random coe#cients. More precisely, a slight change in the proof of Theorems 4.2 and4.3 shows that the same results hold for processes of the form

Xt = �t

∞∑k=0

∞∑i1 ; ::: ; ik=1

�(i1; : : : ; ik)�t−i1 · · · �t−i1−···−ik ; (4.10)

where �(i1; : : : ; ik) for all k, and (i1; i2; : : : ; ik)∈Nk are nonnegative random variablesindependent of the sequence (�t). Denote by M (�; �) the process X =(Xt; t ∈Z) givenin (4.10), and let M1(�) and M2(�) be the corresponding conditional expectations of


Xt and X 2t :

M1(�) =∞∑k=0

∞∑i1 ; ::: ; ik=1

�(i1; : : : ; ik)�k+11 ;

M2(�) = �2∞∑

k;l=0

∞∑i1 ; ::: ; ik=1

∞∑j1 ; ::: ; jl=1

�(i1; : : : ik)�( j1; : : : ; jl)�k+l1

(�2�21

)�(i1 ; ::: ;ik ; j1 ; ::: ; jl):

Then we have the following result.

Theorem 4.5. Suppose that X (n) =M (�n; �) and X =M (�; �) are integrable (respec-tively, square-integrable) processes and that convergence

�n(i1; : : : ; ik)P→ �(i1; : : : ; ik) as n → ∞

holds for all k and (i1; i2; : : : ; ik)∈Nk . Then X (n)t

L1→Xt (respectively, X(n)t

L2→Xt) if andonly if EM1(�n) → EM1(�) (respectively, EM2(�n) → EM2(�)) as n → ∞.

An analogous su#cient condition as in Theorem 4.4 can be also established.

Remark 4.1. A particular case of model (4.10) is the model of Leipus and Viano(2002) obtained by considering the following aggregation scheme. Let X = m(b; �),X (i) = m(b(i); �), i = 1; 2; : : :, be a sequence of random coe#cient ARCH(∞) pro-cesses, where the sequences b(1), b(2); : : : are independent copies of the random se-quence b= (b0; b1; : : :), and all are independent of the noise �. Consider the sequenceZ (n) = (Z (n)

k ; k ∈Z) of normalized sums Z (n)k = n−1∑n

i=1 X(i)k . Then, for

�n(i1; : : : ; ik) =1n

n∑j=1

b( j)0 b( j)i1 · · · b( j)ik ;

by the Strong Law of Large Numbers we have for all k and i1; : : : ; ik ,

�n(i1; : : : ; ik) → E[b0bi1 · · · bik ] = : �(i1; : : : ; ik) a:s:

Therefore, applying Theorem 4.5, we get that if Z := M (�; �), then Z (n)t

L1→Zt (respec-

tively, Z (n)t

L2→Zt) if and only if EXt ¡∞ (respectively, EX 2t ¡∞), see Leipus and

Viano (2002). Obviously, the sequences Z (n) and the limit process Z are no longerARCH(∞) processes.Another aggregation procedure generalizing the scheme of Ding and Granger (1996)

mentioned in the introduction, is studied in the next section.

5. Application to aggregation of ARCH(∞) processes

Consider the model

X (n)t = Uh(n)t �t ; Uh(n)t = n−1

n∑k=1

h(k)t ; (5.1)


h(k)t = a(k)0 +∞∑i=1

a(k)i X (n)t−i ; (5.2)

where a(k) := (a(k)i ; i¿ 0) are independent copies of a random sequence a=(ai; i¿ 0)(a(k) and a are independent of �). Then X (n) satis8es equations (1.4) with bi

replaced by

b(n)i =1n

n∑k=1

a(k)i : (5.3)

If Eai ¡∞, then, by the Strong Law of Large Numbers,

b(n)i → Eai a:s:

and the limit process becomes a nonrandom coe#cient process.The following two theorems establish the L1 and L2-convergence of the process X (n)

tunder conditions which only involve the disaggregated process Y = m(a; �).

Theorem 5.1. Let X (n) be given by (5.1) and (5.2), A =∑∞

i=1 ai, and X = m(b; �),where bi = Eai ¡∞. If the conditions

A�1 ¡ 1 a:s: and E[1 + a01− A�1

]¡∞

are satis5ed, then X (n)t ∈L1, Xt ∈L1, and X (n)

tL1→Xt .

Proof. Set

Ak =∞∑i=1

a(k)i and Bn =∞∑i=1

b(n)i =1n

n∑k=1

Ak:

By the concavity of (1 − B�1)−1 (as a function of B) we get

b(n)0

1− Bn�16

b(n)0

n

n∑k=1

11− Ak�1

: (5.4)

The Strong Law of Large Numbers implies that the right-hand side of (5.4) convergesa.s. to b0E(1− A�1)−1. On the other hand,

E

[b(n)0

n

n∑k=1

11− Ak�1

]=

1n2

n∑k=1

n∑l=1

E

[a(l)0

1− Ak�1

]

=1nE[

a01− A�1

]+

n − 1n

b0E[

11− A�1

]→ b0E

[1

1− A�1

]:

By Theorem 4.1, the right-hand side of (5.4) is uniformly integrable, which impliesthe same property of the left-hand side. Therefore, E�1(b(n))¡∞, E�1(b)¡∞, and

E�1(b(n)) → E�1(b). By Theorem 4.2, X (n)t

L1→Xt .

In the L2-case the following result holds (the proof is similar to that of Theorem 5.1and is omitted).


Theorem 5.2. Let X (n) be given by (5.1) and (5.2), A =∑∞

i=1 ai, and X = m(b; �),where bi = Eai ¡∞. If conditions

A√

�2 ¡ 1 a:s: and E[

1 + a01− A

√�2

]2¡∞

are satis5ed, then E U�2(b(n))¡∞, E U�2(b)¡∞, and X (n)t

L2→Xt .

Example 5.1. A particular case of (5.1) and (5.2) is the model considered by Dingand Granger (1996). These authors assume that �t = 2t , where t are i.i.d. randomvariables with zero mean and unit variance and that the coe#cients a(k)i in (5.2) aregiven by the equalities

a(k)0 = 2(1− ∗k ) and a(k)i = ∗

k (1− �k)�i−1k ;

where �k are i.i.d. random variables on [0; 1] having the Beta distribution B(p; q), and ∗k are i.i.d. random variables on [0; 1] independent of the �k such that E ∗

k = �¿ 0.It is easily seen that h(k)t satis8es the following recursive GARCH(1,1)-type equation

h(k)t = 2(1− ∗k )(1− �k) + ∗

k (1− �k)X(n)t−1 + �kh

(k)t−1:

The disaggregated process Y=m(a; �) satis8es the following GARCH(1,1) equations:

Yt = ht�t ;

ht = 2(1− ∗)(1− �) + ∗(1− �)Yt−1 + �ht−1

( ∗ and � denote independent random variables with the same distribution as ∗k

and �k). By (2.13), A = ∗ and, since �1 = 1, the assumptions of Theorem 5.1 areequivalent to

∗ ¡ 1 a:s: and E[

11− ∗

]¡∞:

Similarly, if

∗√�2 ¡ 1 a:s: and E[

11− ∗√�2

]2¡∞; (5.5)

then the conditions of Theorem 5.2 are satis8ed and X (n)t

L2→Xt . For instance, if ∗ isnonrandom and equals �, then the second inequality in (5.5) is automatically satis8edand condition (5.5) simply becomes �

√�2 ¡ 1.

Note that the condition

∗√�26 1 a:s: (5.6)

is necessary for the square-integrability of Yt . Indeed, by (2.14), the quantity G corre-sponding to the process Yt equals

( ∗)2(1− �)2

1− �2 − 2 ∗(1− �)�=

( ∗)2(1− �)1 + � − 2 ∗�

: (5.7)

The right-hand side of (5.7) tends to ( ∗)2 as � → 0. Therefore, if G�2 ¡ 1 a.s., then(5.6) holds.


Remark 5.1. If p+ q= 1 and 0¡q¡ 1=2, the limit process X is

Xt = Uht�t ;

Uht = 2(1− �) + �(1− (1− L)q)Xt;

where L stands for the backshift operator (LiXt = Xt−i). Suggested by the form of X ,Ding and Granger (1996) conjectured that the limit process is long-range dependentin the sense that its covariance function is not absolutely summable. However, byTheorem 2.2, the conjecture fails.

Acknowledgements

The authors are grateful to the referees for many helpful comments.

Appendix A.

Proof of Theorem 2.1. By (3.1), condition (2.2) is necessary for the integrability (and,hence, for the square-integrability) of Xt . We will now prove that, under (2.2),

�2 =b20�2(1− G�21)(1− B�1)2

∞∑n=0

(G�2)n: (A.1)

If G�2 ¡ 1, then the series on the right-hand side converges and, hence, �2 ¡∞. IfG�2¿ 1, then �2 =∞. Therefore, (A.1) proves the theorem.For i¿ 1, set pi = bi=B and denote u = B�1, v = �2=�21. Then u¡ 1,

∑∞i=1 pi = 1

and, by (2.4),

�2 = b20�2(1− u)−2F;

where

F =∞∑

k;l=0

∞∑i1 ; ::: ; ik=1

∞∑j1 ; ::: ; jl=1

pi1 · · ·pikpj1 · · ·pjl(1− u)2uk+lv�(i1 ; ::: ; ik ; j1 ; ::: ; jl) (A.2)

with �(i1; : : : ; ik ; j1; : : : ; jl) de8ned by (2.5).The rest of the proof is based on the following

Lemma A.1. Let F be de5ned by (A.2). Then

F = EvN ; (A.3)

where N is a geometrically distributed random variable, i.e.,

P{N = m}= (1− G)Gm; m¿ 0 (A.4)

with G = P{N¿ 1}.


Proof. It is easily seen that (A.3) holds with N = �(51; : : : ; 5T ; 5′1; : : : ; 5

′T ′), where

T; T ′; 51; 5′1; 52; 5

′2; : : : are independent random variables such that

P{T = k}= P{T ′ = k}= (1− u)uk ; k¿ 0;

P{5j = i}= P{5′j = i}= pi; i; j¿ 1:

Note that

G6P{T¿ 1; T ′¿ 1}= u2 ¡ 1: (A.5)

De8ne S0 = S ′0 = 0 and, for n¿ 1, Sn = 51 + · · ·+ 5n, S ′

n = 5′1 + · · ·+ 5′

n. Let 8m and8′m, m¿ 0, be successive moments of occurence of the event {Sn = S ′

n′}, i.e., 80 = 0,

8m =min{n¿8m−1|Sn = S ′n′ for some n′}

and let the 8′m be de8ned by the equalities S8m = S ′

8′m. Then, for all m¿ 1,

P{N¿m}= P{T¿ 8m; T ′¿ 8′m}

=∞∑

l;l′=1

P{T¿ l; T ′¿ l′}P{8m = l; 8′m = l′}: (A.6)

It is easily seen that, for m¿ 1, 16 k6 l − 1, and 16 k ′6 l′ − 1,

P{8m = l; 8′m = l′|8m−1 = k; 8′

m−1 = k ′}= P{81 = l − k; 8′1 = l′ − k ′}: (A.7)

On the other hand,

P{T¿ l; T ′¿ l′}= ul+l′ = uk+k′ul−k+l′−k′

= P{T¿ k; T ′¿ k ′}P{T¿ l − k; T ′¿ l′ − k ′}: (A.8)

By (A.6)–(A.8) we get

P{N¿m}=∞∑

l;l′=1

l−1∑k=1

l′−1∑k′=1

P{T¿ k; T ′¿ k ′}P{T¿ l − k; T ′¿ l′ − k ′}

×P{8m−1 = k; 8′m−1 = k ′}P{81 = l − k; 8′

1 = l′ − k ′}= P{N¿m − 1}P{N¿ 1};

which is equivalent to (A.4).

By (A.3) and (A.4) we have

F =∞∑m=0

(1− G)(Gv)m;

which implies (A.1) with G replaced by G=�21:

�2 =b20�2(1− G)(1− B�1)2

∞∑n=0

(G�2=�21)n: (A.9)


It remains to show that G = G�21. The quantities G and G only depend on �1 and onthe bj’s. Hence, we can suppose that G�2=�21 ¡ 1. By (A.9), EX 2

t =�2 ¡∞. By (1.4),for k¿ 1, we have

!k = �1E

[(b0 +

∞∑i=1

biXt−i

)Xt−k

]− �21

= �1

(b0�1 +

∞∑i=1

bi(!k−i + �21)

)− �21

= �1∞∑i=1

bi!k−i : (A.10)

Dividing by !0 and applying the de8nition of �kl, we get

"k = �1∞∑i=1

bi"k−i

= �1∞∑l=1

�kl"l + �1bk ; k¿ 1

or, in the vector form,

"= �1�"+ �1b

and, hence (2.9) holds.On the other hand,

�2 = �2E

(b0 +

∞∑i=1

biXt−i

)2

= �2

b20 + 2b0B�1 +

∞∑i; j=1

bibj(!i−j + �21)

;

which, applying (A.10), gives

�2 = �2

(b20 + 2b0B�1 + B2�21 + �−1

1

∞∑i=1

bi!i

)

= �2(b0 + B�1)2 + �2�−11 (�2 − �21)b

′": (A.11)

Substituting (2.9) into (A.11), we obtain

�2 = H�2 + G�2�2 (A.12)

with some H depending on b and �1 only.


By (A.9) and (A.12), we have

H�21− G�2=�21

=H�2

1− G�2; (A.13)

where H does not depend on �2. This equality holds for all �2 in the interval (�21; �21=G),

which is not empty by (A.5). Expanding both sides of (A.13) into power series andcomparing the coe#cients at �2 and �22, we get that H =H and H G=�21 =HG. Hence,G = G�21.

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Stability of random coefficient ARCH models and aggregation schemes