1. INTRODUCTION AND MOTIVATION
He and Teräsvirta (1999a) considered a general class of first-order generalized autoregressive conditional heteroskedasticity (GARCH) models and examined the moment structure within this family. In their paper, the sequence of random variables {εt}t=−∞∞ belongs to this general class of GARCH(1,1) processes if
where {zt} is a sequence of independent and identically distributed random variables with zero mean, k equals 1 or 2, and gt = g(zt) and ct = c(zt) are well-defined functions of zt. Furthermore, they assume that Pr{htk > 0} = 1.
Many GARCH(1,1) models are included in this family. For example, the choices k = 2, gt−1 = α0, and ct−1 = β + α1 zt−12 yield the linear GARCH model of Bollerslev (1986). When k = 1, gt−1 = α0, and ct−1 = β + α1|zt−1| the absolute value GARCH model of Taylor (1986) and Schwert (1989) is obtained. For k = 2, gt−1 = α0, and ct−1 = β + (α1 + ωI(zt−1))zt−12 (where I(zt−1) = 1 if zt−1 < 0 and I(zt−1) = 0 otherwise) the model reduces to the GJR–GARCH model of Glosten, Jaganathan, and Runkle (1993). For a more extensive list, see He and Teräsvirta (1999a).
He and Teräsvirta (1999a) give conditions for the existence of moments of arbitrary order for the GARCH process (1)-(2). In particular, they show that the conditions E [|zt|km] < ∞ and E [ctm] < 1 are necessary and sufficient for the existence of the kmth absolute moment of εt, E [|εt|km]. They also give an explicit formula for this moment under the stated conditions. He and Teräsvirta (1999b) consider a special case of the model (1)-(2), where it is assumed that gt−1 = α0 and ct−1 has a particular parametric form. The exponent k is no longer restricted to take one of the values 1 and 2 but is only assumed to be a positive real number. They show that the conditions E [|zt|2k] < ∞ and E [ct2] < 1 are necessary and sufficient for the existence of E [|εt|2k].
Ling and McAleer (2002) also considered the model (1)-(2) and complemented the results given in He and Teräsvirta (1999a, 1999b). Ling and McAleer (2002) assume that k is a positive real number and show that if E [|zt|km] < ∞, E [gtm] < ∞, and E [ctm] < 1 for some m ∈ (0,1], then there exists a unique kmth-order stationary solution to (1)-(2), which is also strictly stationary and ergodic. Furthermore, under the conditions E [|zt|km] < ∞ and E [gtm] < ∞ and assuming now that m is a positive integer, they show that the necessary and sufficient condition for the existence of the kmth absolute moment of εt, E [|εt|km], is E [ctm] < 1 (in Ling and McAleer, 2002, Thm. 2.2, the exponent of gt and ct is km instead of m, but this appears to be a typographical error; the exponent appearing in their proof is m).
A yet unresolved issue is the necessary and sufficient condition for the existence of a strictly stationary solution to (1)-(2). In the case of a linear GARCH(1,1) model of Bollerslev (1986), this condition was derived in Nelson (1990). Bougerol and Picard (1992) extended this result to the linear GARCH(p,q) process. Duan (1997) introduced the augmented GARCH(p,q) model, which overlaps with the model (1)-(2), and gave a sufficient condition for the existence of a strictly stationary solution; in a number of cases the condition is also necessary. In all of these papers the condition is derived, in principle, using the theory of random matrices and is formulated using the so-called Lyapunov exponent.
In this short note we demonstrate that the same approach can be used to prove the necessity and sufficiency of a similar condition also in the case of the model (1)-(2). In fact, this readily follows from the results already given in Bougerol and Picard (1992) in the context of generalized autoregressive equations.
2. RESULT AND DISCUSSION
We consider the case in which the exponent k is assumed to be a positive real number (it plays no role in the following proof). First note that {gt} and {ct} are sequences of independent and identically distributed random variables because {zt} is. We make the following assumption.
Assumption 1. For all t, gt and ct are nonnegative, and gt is strictly positive with nonzero probability.
This assumption is not very restrictive compared with the requirement Pr{htk > 0} = 1 made in He and Teräsvirta (1999a). From a practical point of view, most of the models listed as special cases in He and Teräsvirta (1999a) also satisfy this condition.
Rewriting (2) as
makes it clear that the process {htk} follows an autoregressive equation in
with independent and identically distributed nonnegative coefficients gt and ct. Conditions for strict stationarity in such a situation are discussed in Bougerol and Picard (1992). The following result gives necessary and sufficient conditions for the existence of a strictly stationary solution of (3). We use the notation ln+(x) = max{ln(x),0}.
THEOREM 1 (Corollary of Theorem 3.2 of Bougerol and Picard, 1992). Suppose that Assumption 1 is satisfied and that E [ln+(ct)] is finite. If (3) has a strictly stationary nonnegative solution, then E [ln(ct)] < 0. Conversely, if E [ln+(gt)] is finite and E [ln(ct)] < 0, then for all
, the series
converges a.s., and the process
is the unique strictly stationary solution of (3).
Proof. Assumption 1 ensures that condition (C) of Bougerol and Picard (1992) is satisfied (see Bougerol and Picard, 1992, pp. 122–123). In Theorem 3.2 of Bougerol and Picard (1992), the key condition for the existence of a strictly stationary solution is that the so-called top Lyapunov exponent associated with a sequence of certain matrices is negative. In the present (univariate) case the top Lyapunov exponent equals
The stated result now follows from Theorem 3.2 of Bougerol and Picard (1992). █
The strict stationarity of the process {εt} follows from that of {zt} and {htk}. In the case of the linear GARCH(1,1) process, gt is a constant, and the condition for ct has the form E [ln(β + αzt2)] < 0, a condition already derived in Nelson (1990). The conditions for other members of the family of GARCH processes (1)-(2) are easily derived from Theorem 1.
Nelson (1990, Thm. 6) derived an explicit expression for the moment E [ln(β + αzt2)] in the case of standard normal or Cauchy errors using functions standard in the mathematical literature yet rather exotic in the econometric one. This derivation relies on the existence of explicit integral formulas for the logarithm of a polynomial (see the references in Nelson, 1990). Unfortunately, however, similar formulas for the moment E [ln(ct)] do not seem to be available without making stringent assumptions about the functional form of c(·).
