Abstract

Introduction
11
He and Terasvirta (1999a) defined the following general class of GARCH(1, 1) model:
where Pr{hξ > 0} = 1, δ > 0, zt is a sequence of i.i.d. random variables with mean zero and variance 1, and g(x) and z(x) are nonegative functions. As indicated by these authors, this family of GARCH processes includes the GARCH(1,1) model of Engle (1982) and Bollerslev(1986), the absolute value GARCH (1,1) model of Taylor (1986) and Schwert (1989), the asymmetric GJR-GARCH (1,1) model of Glosten et al. (1993) (which seems to be the most widely used specification for asymmet- ric time-varying volatility), the nonlinear GARCH(1,1) model of Engle (1990), the volatility switching GARCH (1,1) model of Fornari and Mele (1997), the thresh- old GARCH (1,1) model of Zakoian (1994), the fourth-order nonlinear generalized moving-average conditional heteroskedasticity (4NLGMACH(1,1)) model of Yang and Bewley (1995), and the generalized quadratic ARCH (1,1) model of Sentana (1995). It was argued by He and Terasvirta (1999a) that the necessary and sufficient condition for the mδth unconditional moment of model (1.1) to exist is
where m is a positive integer and δ = 1 or 2.  As in Engle (1982) and much of the subsequent literature, He and Terasvirta's (1999a) result in their Theorem 1 was derived under the following assumption:
Even though He and Terasvirta (1999a) restrict attention to the two integers δ = 1, 2, their proofs actually hold for any δ > 0. Furthermore, He and Terasvirta (1999b) derived the fourth moment condition for the higher-order GARCH (p, q) model of 11 Bollerslev (1996) (see also Karanasos (1999) for a related result).  Ling and McAleer (1999) established the necessity and sufficiency of the condition for the existence of the fourth and higher moments for the GARCH (p, q) model. The results in He and Terasvirta (1999a, b) are useful. However, two problems in their family of GARCH(1, 1) processes have not yet been resolved.  First, the condition under which model (1.1) is strictly stationary has not been established. Second, assumption (1.3) is impossible to check in practice, so that it is an axiom rather than an assumption. The purpose of this paper is to resolve these two prob- lems. Without using assumption (1.3), we provide a simple sufficient condition for the strict stationarity of model (1.1), and establish the necessary and sufficient con- dition for the existence of its moments. These conditions are simple to check and should prove useful in practice.
2
Main Results
Theorem 2.1.  If E|z | ∞  < ∞ , E[g(z t)] α < ∞ and E[c(z t)] α < 1 for some α ∈ (0,1], then there exists a unique αδ-order stationary solution to (1.1). The solution is strictly stationary and ergodic, and has the following causal expansion:
where the infinite sum converges almost surely (a.s.). Remark. Theorem 2.1 makes clear that the model given in (1.1) did, in fact, start infinitely many periods ago. However, this is a consequence of the existence of the unique stationary solution and is not an assumption. When g( z –1) ≡ α 0 > 0 and c( z –1) ＝ α z ঙ –1 +β , namely Bollerslev’s GARCH(1,1) model, the condition E [c( z )] α < 1 for some α ∈ (0, 1] is equivalent to ln[c( z )] < 0, which is the necessary and sufficient condition given in Nelson (1990) and Bougerol and Picard (1992). Theorem 2.2.  If E|zt|™8 < ∞ and E[g(zt)]m8 < ∞, then the necessary and sufficient condition for the existence of the mδth moment of the solution {ε } in Theorem 2.1 is
where m is a positive integer. Note that, without strict stationarity, Ehkm = Ehkm is used in (A.3) in He and Terasvirta (1999a), which means that the authors proved the necessity of the condition but not sufficiency.  He and Terasvirta’s (1999a) proof of necessity uses assumption (1.3), whereas the proof of necessity in this paper does not require this assumption.  The proof of sufficiency in this paper uses the concept of Markov chains and Tweedie’s (1988) criterion. In general, the results in Theorem 2.2 cannot be easily extended to higher orders of the family of GARCH processes given in (1.1). However, for some special cases, such as the GARCH (p,q) model in Bollerslev (1986), the sufficient condition for the existence of moments is established in Chen and An (1998) and Ling (1999) (see Ling and McAleer (1999) for the proof of necessity). Ling and McAleer (1999) also established the necessary and sufficient conditions for the existence of moments for an asymmetric power GARCH (p,q) model in Ding et al. (1993), a special case of which is the widely used asymmetric GARCH (1,1) model of Glosten et al. (1993) (this special case of model (1.1) is considered in Example 2.1 below). In the following, we give two examples of the second and fourth moment condi- tions. Example 2.1. In model (1.1), let δ = 2, g( z t − 1) ≡ α 0, and c( z − 1) = β + (α + ωI ( z t −1)) z 2–1, with I ( z t −1) = 1 if z t −1 < 0, and I ( z t −1) = 0 otherwise, which is the asymmetric GJR-GARCH(1,1) model of Glosten et al. (1993). (i) For this model, E[c( z −1)] = β + α + ωb, where b = E[I( z −1) z −1]. When z t follows a symmetric distribution, b = 1/2 by direct calculation. In this case, the second moment condition for the asymmetric GJR-GARCH(1,1) model, which is simple to check, is given by
When ω = 0, condition (2.1) reduces to the well-known second moment condition for the GARCH(1,1) model.  Condition (2.1) means that the admissible region of (α, β ) for second-order stationarity of the asymmetric GJR-GARCH(1,1) model is smaller than that for the symmetric GARCH(1,1) model, as the asymmetry of the model increases its uncertainty. (ii) Furthermore, if z t ～ N (0,1), the fourth moment condition for the asymmetric GJR-GARCH(1,1) model is given by
When ω = 0, condition (2.2) reduces to the well-known fourth moment condition for the GARCH(1,1) model. (iii) If z 〜 t(ν) with ν ≥ 5, the fourth moment condition is given by
where s = 3(ν − 2)/( ν − 4).  When ν → ∞, condition (2.3) reduces to condition (2.2). Example 2.2.  In model (1.1), let δ = 2, g( z –1) ≡ α 0, and c( z –1) = β + α[1 − 2ηsgn( z − 1) + η 2 ] z − 1, namely the nonlinear GARCH(1,1) model of Engle (1990), which is a special case of the asymmetric power GARCH model of Ding et al. (1993). (i) For this model, E[c(z –1)] = β + α + αη² – 2αηb, where b = E[sgn(z –1)² –1]. When z t follows a symmetric distribution, b = 0 and, in this case, the second moment condition for the nonlinear GARCH(1,1) model is given by
When η = 0, condition (2.4) reduces to that for GARCH (1,1). 44 (ii) Furthermore, if z t ～ N (0,1), the fourth moment condition for the nonlinear GARCH(1,1) model is given by
When η = 0, condition (2.5) reduces to the fourth moment condition for GARCH(1,1). (iii) If z t ～ t ( ν ) with ν ≥ 5, the fourth moment condition is given by
where s = 3 ν 2 /[( ν − 4)(ν − 2)].  When ν → ∞, condition (2.6) reduces to condi- tion (2.5).  The comments in Example 2.1 relating to the reason for the increased uncertainty arising from asymmetry remain valid for the nonlinear GARCH(1,1) model.
3
Proofs of the Main Results
Proof of Theorem 2.1. Let S n = g(z −1) + ∑ k =0 Π k =0 c(zt−1− j )g(zt−1− k ). Then
so that S n converges a.s.  under the given assumptions.  Denote the limit of S n by hç , that is,
It is easy to verify that h t is a solution of (1.1) and Eh t δ < ∞ . From (3.1), h t is a fixed functional of a sequence of i.i.d. random variables, and hence is also strictly stationary and ergodic, in which case so is ε t. To prove uniqueness, let h 1 t be another αδ th-order stationary solution of model (1.1).  Then
● as k → ∞. Thus, ht = h 1 t a.s.. This completes the proof. Proof of the necessity of Theorem 2.2 . It is easy to show that
Since Eh ε m < ∞, the above inequality holds only if E [ g ( z − 1 )] m < ∞ and E [ c ( z − 1 ) ● ]m < 1. This completes the proof. The proof of the sufficiency of Theorem 2.2 needs the concept of Markov chains and Tweedie's (1988) criterion. Let { X t ; t = 1, 2, . . } be a temporally homogeneous Markov chain with a locally compact completely separable metric state space ( S, B ). The transition probability is P(x,A) = Pr(Xn E A|Xn-1 = x), where x ∈ S and A ∈ B . Tweedie's criterion is given in the following lemma. Lemma 3.1. (Tweedie, 1988, Theorem 2) Suppose that { X i } is a Feller chain. (1) If there exists, for some compact set A ∈ B, a non-negative function f  and ε > 0 satisfying
then there exists a σ-finite invariant measure μ for P with 0 < μ(A) < ∞; (2)Fur- thermore, if
then μ is finite, and hence π = μ/μ(S) is an invariant probability; (3) Furthermore, if
then μ admits a finite g-moment, that is,
It is clear that any solution hể of model (1.1) is a Markov chain with state space R 0 = (0, ∞). Tweedie's criterion can be used to find the sufficient condition for the existence of a stationary and finite g –moment solution of model (1.1).  Now we give a lemma that is used in the proof of the sufficiency of Theorem 2.2. Lemma 3.2.  There exists an interval A = (0,K) such that the function f defined by
satisfies
and
where Ac = [K, ∞), △ is a positive constant, and τ and K are constants not de- pending on x. Proof. From the given conditions, we can show that
where τ 1 = 1 − E[c( z −1)] m , and b = ∑ m −1 ( m  ) E m−i[g( z −1)] E i[c( z −1)]. Note that π ∈ (0,1), and x π /(1 + x π ) and b(1 + x ) π − 1 /(1 + x π ) converge to 1 and 0, respectively, as x → ∞. It is obvious that there exists a constant K such that, when x ≥  K, [π x m + b(1 + x)m−1]/(1 + x m ) >  π /2 − π /4 > 0.  Denote A = (0, K). Let τ = τ 1 /4 and Δ = 1 + K m . It follows that
77
This completes the proof. ● Proof of the sufficiency of Theorem 2.2. First, note that { h } is a Markov chain with state space R 0 . It is straightforward to prove that, for each bounded continuous function f on R 0 , E [ f ( h f ) | h f_1 = x ] is continuous in x , that is, { h f } is a Feller chain. From Lemma 3.2, there exists a constant K > 0, A = (0,K) and τ > 0 such that the function defined by f(x) = 1 + x m satisfies
and
where τ and △ are positive constants. By (3.7)-(3.8), it is not difficult to show that the function f ( x ) satisfies conditions (3.2) and (3.3) in Lemma 3.1 and, hence, there is an invariant probability measure π for the process {h }. Let g(x) = τf(x). By (3.8) and Lemma 3.1 (3), we know that E πf(h f) is finite. Thus, Eπh fm < ∞. By Hölder’s inequality, E π h σ δ ≤ ( E π h σ m )α/m < ∞ , and hence π is the αδ th-order stationary distribution of h t . Thus, ε t =  z h t is the αδ th-order stationary solution of model (1.1). By Theorem 2.1, the solution is unique. Thereforce, the solution {ε } □ in Theorem 2.1 has finite mth-order moment.  This completes the proof.
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