2. STATIONARITY OF ARCH(∞)

The minimal condition for covariance stationarity of the ε_t, is κ = E(z_t²) < ∞, by (1). Following Robinson (1991), who considers the case κ = 1, setting

, we can rewrite (3) as

setting ν_t := ε_t² − κσ_t². By (1) and the i.i.d.-ness of the z_t,

, where

is the σ-field of events induced by the ε_s (s ≤ t) (for the definition of conditional expectations when the corresponding unconditional expectations may not exist, see Loève, 1978, Sect. 27.2).

Assume that, for complex-valued z, the following “invertibility” condition holds:

A sufficient condition for (6) is

implying ψ(1) > 0, but we want to allow for the possibility that ψ(1) = 0 such as when the ψ_j are the AR(∞) coefficients of the autoregressive fractionally integrated moving average (ARFIMA) filter:

L being the lag operator (Lz_t = z_t−1) with d > 0 (by summability of the ψ_j) and where the finite-order polynomials a(L), b(L) have zeros outside the unit circle in the complex plane. Indeed, note that (6) does not imply absolute summability of the δ_j.

Given (6), (5) can be rewritten as

These simple manipulations suggest that τ, κ < ∞, and

imply covariance stationarity of the ε_t. Indeed, it turns out that these are the necessary and sufficient conditions for E(ε_t²) < ∞. This is developed in Theorem 1, where the necessary and sufficient conditions for covariance stationarity of the squares ε_t² are also established.

THEOREM 1. Assume that condition (6) holds.

(i) The necessary and sufficient conditions for E(σ_t²) < ∞ are

Under these conditions

(ii) The necessary and sufficient conditions for E(σ_t⁴) < ∞ are

setting

, for any square summable sequence c_i.

Under these conditions the ε_t² are covariance stationary with ACF

Remark 1.1. The ACF of the squares ε_t² for GARCH(p,q) can be obtained as follows. We first establish the ARCH(∞) representation of GARCH(p,q). Exploiting the ARMA(max(p,q), p) representation of GARCH(p,q), by standard arguments (see Nelson and Cao, 1992) one obtains

where α(L) := α₁ L + ··· + α_q L^q, β(L) := β₁ L + ··· + β_p L^p, ρ_i (i = 1,…, p) define the inverse of the roots of |1 − β(z)| = 0, assumed all outside the unit circle in the complex plane and distinct, and

where A_p = A₁ = 1 for p = 1. The ARCH(∞) coefficients ψ_j of GARCH(p,q) are then

with μ_j := A₁(ρ₁)^j + ··· + A_p(ρ_p)^j, j ≥ 0. Based on the previous definitions, Theorem 1 yields the ACF of GARCH(p,q) ε_t² with

deriving the

and the χ_δ(u), and thus E(ν_t²), based on (15) and deriving the δ_i from (6).

Remark 1.2. Necessary and sufficient conditions for covariance stationarity of GARCH(p,q) ε_t² have been independently established by He and Teräsvirta (1999) and Karanasos (1999). When considering GARCH(p,q), these results and (12)–(14) are equivalent, but formal derivation of their equivalence is very cumbersome. He and Teräsvirta (1999) is closer to our result, because their work considers the slightly more general case of nonstandardized z_t, though from a computational point of view our conditions (12)–(14) seem more appealing because they permit immediate use of known results for the theoretical ACF of ARMA(p,q).

As an example, for GARCH(1,1), setting κ = 1 and π₁ := α₁ + β₁, i.e., the “persistence” parameter, we get

When the z_t have zero fourth-order cumulant (such as for Gaussian z_t) then θ = 2. Under this condition, by simple manipulations, (14) yields the well-known covariance stationarity conditions for ε_t² (cf. Bollerslev, 1986, Sect. 3)

which, in turn, implies (11): E(z_t²)α(1) + β(1) < 1.

For the ARCH(2), namely, ψ_k = 0, k ≥ 3, setting (1 − ξ₁⁻¹L)(1 − ξ₂⁻¹L) := (1 − ψ₁ L − ψ₂ L²) and assuming that the roots ξ₁, ξ₂ are greater than one in modulus (cf. (6)) and satisfy, for simplicity's sake, ξ₁ ≠ ξ₂, one gets

when κ = 1 (see Brockwell and Davis, 1987, Example 3.3.5). Using

and by simple manipulations, from (14) one gets the covariance stationarity condition for ARCH(2) ε_t² (cf. Milhøj, 1985, Theorem 3; He and Teräsvirta, 1999, equation (12)), setting θ = 2,

Remark 1.3. When

Giraitis et al. (2000, Theorem 2.1) show that E(σ_t⁴) < ∞. However, (17) is more restrictive than (14). For instance, for GARCH(1,1) with θ = 2κ = 2, their condition becomes

strictly implying (16), unless β₁ = 0, the ARCH(1).

Remark 1.4. Hyperbolically decaying specifications of the ψ_i and hence of the δ_i are allowed for. Note, however, that from (11),

and thus κ = 1 is ruled out when

. Thus, unless κ < 1, (7) is not compatible with covariance stationary ε_t.

Imposing κ = 1, a choice compatible with covariance stationary levels is obtained using the autoregressive coefficients of the ARFIMA filter as follows. Set (cf. (7))

where d > 0 and a(L) and b(L) are finite-order polynomials, all of whose roots are outside the unit circle in the complex plane. Assuming that the nonnegativity constraints on the coefficients hold, i.e., ψ_i ≥ 0, set ψ_i = ψ_i (i ≥ 2) and ψ₁ = ψ₁ε for some given 0 < ε < 1. Condition (14) is more involved, but again by suitable modification of the first n (say) coefficients ψ_j (j = 1,…, n), a feasible sequence of coefficients can be obtained from (18).

Remark 1.5. An alternative, frequency domain characterization of (14) is

setting

where κ²f_ψ(λ) = 1 + f_δ⁻¹(λ) − 2Re(δ⁻¹(e^iλ)) (−π ≤ λ < π) and Re(·) denotes the real part of its argument. This frequency domain specification seems easier to compute than (14). For instance, when considering parameterization (7) of the ψ_j (as for GARCH(p,q)) this allows us to exploit the much greater computational simplicity of rational power spectra compared with their Fourier transforms. This might be relevant when imposing covariance stationarity of the ε_t² in practical estimation, e.g., when estimating ARCH(∞) using the Whittle estimator (see Giraitis and Robinson, 2001).

The equivalent (19) representation of (14) could be used to show that (14) strictly implies (11). In fact, setting θ = 2κ = 2 for simplicity's sake, (19) can be rewritten as

However, f_δ(λ) ≥ 0 (−π ≤ λ < π), and it is arbitrary, necessarily requiring

Given the nonnegativity of the ψ_i and cos(jλ) ≤ 1, with the equality achieved for λ = 0 (mod. 2π), this is equivalent to

strictly implying ψ(1) > 0.

Finally, exploiting the relation between the ψ_i (i ≥ 1) and the δ_j (j ≥ 0) (cf. (6)), (19) allows us to express the time domain characterization (14) more simply as

Remark 1.6. The original formulation of ARCH(∞), analogous to Robinson (1991) but allowing κ ≠ 1, is

for some

. The reparameterization (20) is clearly permitted only for covariance stationary ε_t, i.e., when ψ(1) > 0, given that

. In fact, assume that one starts directly from (20) rather than from (3). By the nonnegativity constraint

because ψ(1) < 0 is not allowed. Imposing ψ(1) = 0 and assuming (6), the linear moving average representation (8) for the ε_t² would then be

for any constant ζ, given that ψ(L)ε_t² = ψ(L)ε_t² − ψ(1)ζ = ψ(L)(ε_t² − ζ), which is meaningless.

Remark 1.7. (9)–(11) and (12)–(14) are the necessary and sufficient conditions for E(ε_t²) < ∞ and for E(ε_t⁴) < ∞, respectively.

3. MEMORY OF ARCH(∞)

Giraitis et al. (2000, Proposition 3.1) show that (17) implies absolute summability of the ACF for the ε_t², ruling out long memory. However, considering that (17) is more restrictive than effectively required to obtain covariance stationary ε_t², it is important to assess the impact of the weaker condition (14) on the memory of the ε_t².

Insights can be obtained by looking at the linear representation (8) for ε_t². In fact, it follows that the memory of the ε_t² is expressed by the asymptotic behavior of δ_j as j → ∞. Surprisingly, it turns out that even the much weaker covariance stationarity condition (11) for the levels ε_t rules out long memory in the ε_t². In fact, from (6) and (11),

Expression (14) ensures that the uncorrelated ν_t have finite variance but the rate of decay of the δ_j, imposed by (11), is already quick enough to imply their absolute summability. We summarize our results on the memory of the ε_t² as follows.

THEOREM 2. Assume that conditions (6) and (11) hold. Then

where

When the ψ_i decay toward zero more slowly than exponentially, namely, ψ_i /ζⁱ → ∞ as i → ∞ for any 0 < ζ < 1, (21) implies that, as u → ∞,

for some 0 < C < ∞, with c(x) ∼ d(x) as x → x₀, meaning that c(x)/d(x) → 1.

Remark 2.1. When ψ_i ∼ ci^−δ as i → ∞ for 0 < c, δ − 1 < ∞, as for the parameterization described in Remark 1.4, E(ε_t²) < ∞ implies

for some 0 < C < ∞, ruling out long memory in the ε_t². Under the same assumptions on the asymptotic behavior of the ψ_i, the exact rate in (24) was also obtained in Giraitis et al. (2000, Proposition 3.2), although they impose (17), a sufficient condition for E(ε_t⁴) < ∞.

Remark 2.2. Theorem 2 makes it clear that whereas the bounded second moment conditions (9)–(11) impart the degree of memory of the ε_t², the stronger bounded fourth moment conditions (12)–(14) ensure that the martingale difference sequence ν_t (the innovations in the linear representation (8) of the squares) are square integrable, without changing the memory implications of the model. This double role of the coefficients is simply a by-product of ARCH(∞) nonlinearity.

Remark 2.3. In the case of exponentially decaying ψ_i, i.e., when there does exist 0 < ζ < 1 such that ψ_i /ζⁱ → c as i → ∞ for some 0 < c < ∞, it clearly follows that

for some 0 < δ < 1, 0 < C < ∞, as in the GARCH(p,q) case.