Discussion Paper No. 534

NECESSARY AND SUFFICIENT MOMENTCONDITIONS FOR THE GARCH(r,s) AND
ASYMMETRIC POWER GARCH(r,s) MODELS*

Shiqing Ling
andMichael McAleer

April 2001

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan* To appear in Econometric Theory.

Necessary and Suﬃcient Moment Conditions for
the GARCH(r, s) and Asymmetric PowerGARCH(r, s) Models∗
Shiqing Ling
Department of Mathematics
Hong Kong University of Science and Technology
Michael McAleer
Department of Economics
University of Western Australia
Revised: February 2001

Abstract
Although econometricians have been using Bollerslev’s (1986) GARCH (r,s) model for over a decade, the higher-order moment structure of the model
remains unresolved. The suﬃcient condition for the existence of the higherorder moments of the GARCH (r, s) model was given by Ling (1999a). This
paper shows that Ling’s condition is also necessary. As an extension, thenecessary and suﬃcient moment conditions are established for Ding, Granger
and Engle’s (1993) asymmetric power GARCH (r, s) model.
∗

The authors wish to acknowledge the helpful comments of the Co-Editor, Bruce Hansen, anda referee. The first author wishes to acknowledge the financial support of the Australian ResearchCouncil. The second author wishes to acknowledge the financial support of the Australian ResearchCouncil and the Institute of Social and Economic Research at Osaka University.

1

Introduction

A process εt is said to follow Bollerslev’s (1986) general autoregressive conditional
heteroskedasticity (GARCH (r, s)) model if it satisfies the equations:qεt = ηt ht ,ht = α0 +

r
X

(1.1)
αi ε2t−i +

i=1

s
X

βi ht−i ,

(1.2)

i=1

where α0 > 0, αi ≥ 0 (i = 1, · · · , r) with at least one αi > 0, and βi ≥ 0 (i = 1, · · · , s).
When s = 0, the GARCH(r,s) model (1.1)-(1.2) reduces to Engle’s (1982) autoregressive conditional heteroskedasticity (ARCH (r)) model. Both the ARCH andGARCH models have been applied widely in the econometric and finance literature
to model volatility (see Bollerslev, Chou and Kroner (1992), Bollerslev, Engle andNelson (1994), and Li, Ling and McAleer (1999) for recent reviews).
Bollerslev (1986) showed that the necessary and suﬃcient condition for thesecond-order stationarity of model (1.1)-(1.2) is
r
X
i=1

αi +

s
X

βi < 1.

(1.3)

i=1

Bougerol and Picard (1992) provided the necessary and suﬃcient condition for thestrict stationarity and ergodicity of model (1.1)-(1.2). Ling and Li (1997) proved
that, under (1.3), there exists a unique Ft−1 -measurable and second-order stationary
solution to model (1.1)-(1.2), and that the solution is strictly stationary and ergodic,
where Ft is a σ−field generated by {ηt , ηt−1 , · · ·}. Thus, the second-order moment
structure of model (1.1)-(1.2) is now complete.
However, the higher-order moment structure of model (1.1)-(1.2) remains unresolved. When s = 0, Milhφj (1985) gave the necessary and suﬃcient condition for
the existence of the 2m-th moment of the ARCH model. Bollerslev (1986) provided
the necessary and suﬃcient condition for the existence of the 2m-th moment of the
GARCH(1,1) model, and the necessary and suﬃcient condition for the fourth-ordermoments of the GARCH(1,2) and GARCH(2,1) models. Karanasos (1999) and He1

and Terasvirta (1999b) gave conditions for the existence of the fourth moment ofmodel (1.1)-(1.2). He and Terasvirta (1999b) state that their condition is necessary
and suﬃcient. From the proof in Karanasos (1999), it can be seen that his conditionis necessary, but it is not clear whether the condition is also suﬃcient.
Based on Theorem 2.1 in Ling and Li (1997) and Theorem 2 in Tweedie (1988),
Ling (1999a) provided a suﬃcient condition for the existence of the 2m-th momentof model (1.1)-(1.2). Ling’s result does not need to assume that the GARCH (r,s) process starts infinitely far in the past with finite 2m-th moment, as is requiredin Bollerslev (1986) and He and Terasvirta (1999b), and has a far simpler form ascompared with Milhφj (1985), Karanasos (1999), and He and Terasvirta (1999b).
In this paper, it is shown that the suﬃcient condition in Ling (1999a) is also
necessary. As an extension, the necessary and suﬃcient moment condition is established for the asymmetric power GARCH (r, s) model proposed by Ding, Granger
and Engle (1993).

2

Main Results

Denote A⊗m = A ⊗ A ⊗ · · · ⊗ A (m factors), where ⊗ is the Kronecker product. Our
result for the GARCH (r,s) model is as follows.
Theorem 2.1. The necessary and suﬃcient condition for Eε2m
< ∞ is ρ[E(A⊗mt )] <t
1, where ρ(A) = min{|eigenvalues of a matrix A|}, and

At = α1 ηt2···I(r−1)×(r−1)α1···
O(s−1)×r

αr ηt2β1 ηt2O(r−1)×1
αr

β1

···

βs ηt2O(r−1)×s···βsI(s−1)×(s−1) O(s−1)×1





,



in which Ir×r is the r × r identity matrix.
The suﬃciency comes from Theorem 6.1 of Ling (1999a). The proof of necessity
is given in Section 4. Ling (1999a) showed that, when r = s = 1, the condition in
Theorem 2.1 is equivalent to that in Bollerslev (1986).

2

Recently, He and Terasvirta (1999b) investigated the necessary and suﬃcientcondition for the fourth moment of the GARCH (r, s) model. It is instructive to
examine their condition carefully. By assuming that Eh2t < ∞, He and Terasvirta(1999b) derive their conditions by the following equation:
Eh2t = α20 + 2α0 γ1 Eht + γ2 Eh2t + 2

X

E(cl,t−l cm,t−m ht−l ht−m ),

(2.1)

l<m
2
where γ1 and γ2 are some suitable constants and ci,t−i = αi ηt−i
+ βi . Because of

the assumption that Eh2t < ∞, their condition is clearly necessary rather thansuﬃcient1 . Karanasos (1999) used an equation similar to (2.1), and hence his condition is also necessary rather than suﬃcient.
As in Engle (1982) and Bollerslev (1986), He and Terasvirta (1999b) assumed
that the GARCH (r, s) process starts infinitely far in the past with finite 2m-th
moment. In fact, this assumption is unnecessary. It is interesting to clarify thispoint. For simplicity, we consider only the case m = 2 and the ARCH(1) model, i.e.
√εt = ηt ht and ht = α0 + α1 ε2t−1 .

(2.2)

Ling and Li (1997) proved that, when 0 < α1 < 1, there exists a unique second-order
stationary solution to model (2.2), and the solution has the following expansion inmean square:
j
∞ Y
√
X
2
(α1 ηt−i).
εt = ηt ht and ht = α0 + α0

(2.3)

j=1 i=1

The structure of this solution actually is a transformation of a series of i.i.d. randomvariables ηt , and does not involve the initial value. The suﬃcient condition in Ling
1

Moreover, the proof of necessity in He and Terasvirta (1999b) is incomplete as some steps intheir proof, such as the one given below, do not hold in general, but require additional
P∞restrictionsiwhich were notestablishedbytheauthors.
Note
that
it
cannotbeclaimedthat
i=1 A < ∞P∞ ifollows from i=1 A ξ < ∞, where A and ξ are some suitable matrix and vector, respectively,
except for some special A and ξ as in Ling (1999b) and in this paper. It is not clear whether
He and Terasvirta’s (1999b) necessary condition for the existence of the fourth moment, namelyλ(Γ) < 1, holds generally because, for example, without other arguments, their (A.21) converging
Pk−1
does not ensure that i=m−l+1 Γi−(m−l+1) converges and hence does not ensure that λ(Γ) < 1. A
possible solution is to find a vector with all elements positive and to use the ideas established inthis paper to prove that λ(Γ) < 1. However, in He and Terasvirta (1999b), ep−1 in (A.21) includes
some zero elements, and Γ is quite complicated. Thus, such a vector with all elements positive
would not be easy to establish, even if it were to exist.

3

(1999a), i.e. the suﬃcient condition in Theorem 2.1, is the moment condition forsolution (2.3). Since the solution is unique, the εt from model (2.2) is almost surely
the same as (2.3) if Eε2t < ∞. The method of proof in Ling (1999a) used the drift
criterion in Tweedie (1988). This method has been a common tool in nonlinear timeseries [see Chan and Tong (1985), Feigin and Tweedie (1985), and Ling (1999a)]. In
the next section, we will use this method to analyze the asymmetric power GARCH(r,s) model proposed by Ding, Granger and Engle (1993).

3

Asymmetric GARCH (r,s) modelA special asymmetric power GARCH (r,s) model in Ding, Granger and Engle (1993)[see also He and Terasvirta (1999a)] is given by:qεt = ηt ht ,δ/2

ht

= α0 +

(3.1)
r
X
i=1

αi (|εt−i | − γεt−i )δ +

s
X

δ/2

βi ht−i ,

(3.2)

i=1

where α0 > 0, αi ≥ 0 (i = 1, · · · , r) with at least one αi > 0, βi ≥ 0 (i = 1, · · · , s),δ ≥ 0 and |γ| < 1. The primary feature of the asymmetric power GARCH (r,s) model (3.1)-(3.2) is the presence of a Box-Cox power transformation of theconditional variances and the asymmetric absolute errors. It can be seen thatmodel (3.1)-(3.2) is a general version of GARCH. Ding, Granger and Engle (1993)
showed that the asymmetric power GARCH model includes as special cases sevenother ARCH-type models, including ARCH, GARCH, Higgins and Bera’s (1990)
NARCH, Geweke’s (1986) and Pantula’s (1986) log-ARCH, and simple asymmetricand threshold GARCH models. In the following, Theorem 3.1 gives the suﬃcientcondition for the stationarity, ergodicity and δ−order stationarity of model (3.1)(3.2). Theorem 3.2 provides the necessary and suﬃcient condition for the existenceof the higher-order moments.
Theorem 3.1. Suppose that α0 > 0, αi , βi ≥ 0, δ ≥ 0, and |γ| < 1. Denote
Zt = (|ηt | − γηt )δ . ThenPr

i=1 αi EZt +Ps

j=1 βj4< 1 is a necessary and suﬃcientcondition for the existence of a unique Ft -measurable δ-order stationary solution{εt } to model (3.1)-(3.2). The solution {εt } has the following causal representation:
j
∞
X
Y
2
1/2
0
εt = ηt ht and ht = [α0 +c(Aδt−i ) ξδt−j ] δ
j=1
i=1

a.s.,

(3.3)

where ξδt = (α0 Zt , 0, · · · , 0, α0 , 0, · · · , 0)(r+s)×1 , that is, the first component is α0 Zt
and the (r + 1)-th component is α0 , c = (α1 , · · · , αr , β1 , · · · , βs )0 , andAδt = 



α1 Zt

···

I(r−1)×(r−1)
α1

···

αr Zt
β1 Zt
O(r−1)×1
αr

β1

O(s−1)×r

···

βs Zt

···

βs

O(r−1)×sI(s−1)×(s−1) O(s−1)×1

Hence, {εt } is strictly stationary and ergodic.





.



(3.4)

For a more general asymmetric power GARCH (r,s) model, Ding, Granger andEngle (1993) provided a condition for E|εt |δ < ∞, which is the same as that in
Theorem 3.1 for model (3.1)-(3.3). When r = s = 1, γ = 0 and δ = 1, thecondition in Theorem 3.1 is also the same as the condition for the absolute-value
GARCH model in Taylor (1986). The strict stationarity and ergodicity condition
for the asymmetric power model is a new result. The uniqueness of second-order
stationarity and the expansion in (3.3) are also useful results.
Theorem 3.2. The necessary and suﬃcient condition for E|εt |mδ < ∞ in model(3.1)-(3.2) is ρ[E(A⊗mδt )] < 1, where Aδt is defined by (3.4).
To establish the suﬃcient moment condition, usually there are two drift criteria
to be employed, the first being that in Tweedie (1983). The results derived from thiscriterion will ensure that there is a unique strictly stationary and (geometric) ergodic
solution to the underlying model with finite moment, as in Chan and Tong (1985)
and Feigin and Tweedie (1985). However, this criterion usually needs to assume thatthe density function of ηt is lower semi-continuous or positive in a neighbourhoodof the original point to prove the irreducibility condition.
The second criterion is that in Tweedie (1988), but does not need any irreducibility condition for the moment condition. However, the results from this criterion5ensure only that there exists a strictly stationary solution for the underlying modelwith finite moment. Such a result is not especially helpful in practice since, without
uniqueness, one cannot guarantee that the process from the underlying model has
finite moment. Our method here uses the criterion in Tweedie (1988) to avoid theassumption on the density of ηt , and then uses Theorem 3.1 to establish uniqueness.
Thus, the result in Theorem 3.2 should be very useful in practice.

4

Proofs of the Main Results

Proof of Theorem 2.1. Multiplying (1.2) by ηt2 yields
ε2t = α0 ηt2 +rX

αi ε2t−i ηt2 +

i=1

s
X

βi ht−i ηt2 .

(4.1)

i=1

Now rewrite (1.2) and (4.1) in vector form as
Xt = At Xt−1 + ζt ,(4.2)

where Xt = (ε2t , · · · , ε2t−r , ht , · · · , ht−s )0 and ζt = α0 (ηt2 , 0, · · · , 0, 1, 0, · · · , 0)0 , of which⊗mthe first element is ηt2 and the (r+1)-th element is 1. If E(ε2m)<
t ) < ∞, then E(Xt∞, where “a vector A < ∞” means that all the elements of A are finite.
Note that all the elements of At , Xt and ξt are non-negative. We haveE(Xt⊗m ) ≥ E(At Xt−1 )⊗m + E(ξt⊗m )⊗m⊗m= E(A⊗mt )E(Xt−1 ) + C1 R1

≥ C1

k
X

i ⊗m
[E(A⊗mt )] R1 ,

(4.3)

i=0

where C1 = min{all the positive elements of E(ξt⊗m )}, R1 = (1, 0, · · · , 0, 1, 0, · · · , 0)0 ,and “a vector A > a vector B” means that each element of A exceeds the corresponding element of B. Similarly, define A ≥ B. If k tends to infinity, from (4.3)
we have
∞
X
i=0

i ⊗m
[E(A⊗m< ∞.
t )] R1

6(4.4)

As discussed in Section 2, we cannot claim directly that ρ(E(A⊗mt )) < 1 from (4.4).
Our proof here makes full use of the advantage of the non-negativity of the elements⊗mof E(A⊗mt ) and R1 .

Denote r? = max{r, s}. In the following, we first show that?

⊗mr[E(A⊗m> 0.
t )] R1(4.5)

We will prove (4.5) for the case α1 > 0. For other cases, (4.5) can be similarly⊗mproved. First, E(A⊗m= E(At R1 )⊗m , where At R1 = (α1 ηt2 +β1 , 1, 0, · · · , 0, α1 +t )R1β1 , 1, 0, · · · , 0)0 . Let C2 = min{all the positive elements of E(At R1 )⊗m }, and R2 =(1, 1, · · · , 0, 1, 1, 0, · · · , 0)0 . It follows that⊗m
≥ C2 R2⊗m .
E(A⊗m
t )R1

(4.6)

⊗m
2 ⊗m
[E(A⊗m≥ C2 E(A⊗m= C2 E(At R2 )⊗m .
t )] R1
t )R2

(4.7)

From (4.6), we haveNow, At R2 = (α1 ηt2 +β1 +α2 ηt2 +β2 , 1, 1, 0, · · · , 0, α1 +β1 +α2 +β2 , 1, 1, 0, · · · , 0). Let
C3 = min{all the positive elements of E(At R2 )⊗m }, and R3 = (1, 1, 1, 0, · · · , 0, 1,
1, 1, 0, · · · , 0)0 . From (4.7), we have
2 ⊗m[E(A⊗m≥ C2 C3 R3⊗m .
t )] R1(4.8)

Repeating the above procedure r? times, we can show thatr?
Y
⊗m r? ⊗m
[E(At )] R1 ≥ ( Ci )R⊗m
r? ,
i=2

(4.9)

0

where Ci > 0 and Rr? = (1, 1, · · · , 1)(r+s)×1 . Thus, (4.5) holds. From (4.4)-(4.5), we
have
∞
X
i=0

?

⊗m r
⊗m
i
[E(A⊗m< ∞.
t )] [E(At )] R1(i)i
Let akj be the (k, j)-th element of [E(A⊗mt )] . From (4.10), we know that∞ for all, k, j = 1, · · · , (r + s)m , that is,
∞
X
i=0

i
[E(A⊗mt )] < ∞,7(4.10)
P∞(i)
i=0 akj <and hence ρ[E(A⊗mt )] < 1. This completes the proof.

2

Proof of Theorem 3.1. Multiplying (3.2) by Zt yields
δ(|εt | − γεt ) = α0 Zt +

r
X
i=1

δ

αi (|εt−i | − γεt−i ) Zt +

s
X

δ/2

βi ht−i Zt .

(4.11)

i=1

Rewrite (3.2) and (4.11) in vector form as
Xt = Aδt Xt−1 + ζδt ,(4.12)
δ/2

δ/2

where Xt = [(|εt | − γεt )δ , · · · , (|εt−r+1 | − γεt−r+1 )δ , ht , · · · , ht−s ]0 and ζδt = α0 (Zt , 0,
· · · , 0, 1, 0, · · · , 0), of which the first element is Zt and (r + 1)-th element is 1. By(4.12), following exactly the steps of Theorem 2.1 in Ling and Li (1997), we canprove that there is a δ−order solution to (3.1)-(3.2), and the solution has expansion(3.4), and hence it is strictly stationary and ergodic.
For necessity, since E|εt |δ = Ehtδ/2 = a constant < ∞, we have
δ/2

Eht

= α0 +rX

δ/2

αi EZi Eht

+

i=1

s
X

δ/2

βi Eht ,i=1that is,(1 −δ/2

Since 0 < Eht
2

r
X
i=1

< ∞, we have

αi EZi −

Pr

s
X

δ/2

βi )Eht= α0 .

i=1

i=1 αi EZi +Ps

i=1 βi < 1. This completes the proof.

Proof of Theorem 3.2. Using (4.12), the drift criterion in Tweedie (1988), andfollowing the steps of Theorem 4.2 in Ling (1999a), we can show that there exists a
strictly stationary solution to model (3.1)-(3.2) with E|εt |mδ < ∞.
By Hölder’s inequality, E|εt |δ ≤ (E|εt |mδ )1/m < ∞, i.e. {εt } is δ−order stationary. By Theorem 3.1, the solution is unique ergodic. This means that a process εtsatisfying (3.1)-(3.2) has finite mδ−th moment if ρ(E(A⊗mδt )) < 1.
In a similar manner to the proof of Theorem 2.1, it can be proved that thenecessity of Theorem 3.2 holds. This completes the proof.
82

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11