2. AN ARMA REPRESENTATION OF THE GARCH(1,1) MODEL

We consider the GARCH(1,1) process given by

We assume that

, where

is the σ-field generated by {z_t,z_t−1,…}. We can write x_t ≡ y_t² as

where ε_t = x_t − σ_t² is a martingale difference sequence with respect to

. From this expression, we see that x_t is a (heteroskedastic) ARMA(1,1) process with parameters φ and θ. We shall throughout assume that φ < 1, implying that E [x_t] < ∞ (cf. Bollerslev, 1988). We introduce the covariance function of the process

Assuming that {x_t} is stationary with second moment, γ(·) is well defined. Using standard results, we then have that the autocorrelation function, ρ(k) = γ(k)/γ(0), solves the following set of Yule–Walker equations:

(cf. Harvey, 1993, Ch. 2, eqns. (4.13a) and (4.13b); these equations were also derived in Bollerslev, 1988, and He and Teräsvirta, 1999). We can express (4) as a quadratic equation in θ,

Observe that b is only well defined if φ ≠ ρ(1). It is easily checked that ρ(1) ≥ φ with equality if and only if φ² = 1 or θ = 0. The first case is ruled out by our assumption that φ < 1, whereas in the following discussion we assume β > 0.¹

There is a second root that is reciprocal to the one stated here; however this has |θ| = β > 1, which is ruled out by φ < 1.²

This last expression is utilized by Engle and Sheppard (2001) to profile out ω in their estimation procedure.

The expressions (4), (5), and (6) can now be used to obtain estimators of the parameters α, β, and ω. First, we can estimate φ by

, where

is the sample autocorrelation function of x_t,

with

Substituting the estimator of φ into (5), we obtain an estimator of θ,

assuming that

. This leads to the following estimators of λ = (α,β,ω)^{[top ]}:

In practice, this method may lead to

. To deal with this problem, the estimator can be Winsorized (censored) at zero and one or at ε and 1 − ε for small positive ε.

Note that

for any w_j sequence with

so that a more general class of estimators can be defined based on this relationship. It can be expected that for a sufficiently general class of weights one can obtain the same efficiency as the Whittle estimator of Giraitis and Robinson (2001). We do not pursue this approach to achieving efficiency because a more standard approach is available; see the discussion that follows. Nevertheless, some smoothing of the ratio of autocorrelations based on (10) may be desirable in practice.

3. ASYMPTOTIC PROPERTIES OF THE ESTIMATOR

We derive the asymptotic properties of our estimator

under the following set of assumptions:

A.1. The error process {z_t} is independent and identically distributed (i.i.d.) with marginal distribution given by a lower semicontinuous density with support

. It satisfies E [z_t] = 0 and E [z_t²] = 1.

A.2. The observed process {y_t} is strictly stationary with E [(β + αz_t²)²] < 1.

A.3. E [(β + αz_t²)⁴] < 1.

Under (A.1), the moment condition in (A.2) is necessary and sufficient for the GARCH model to have a strictly stationary solution with a fourth moment. This solution will be β-mixing with geometrically decreasing mixing coefficients (cf. Meitz and Saikkonen, 2004, Thm. 1). We then assume that this is the one we have observed. Assumption (A.3) is a strengthening of (A.2) implying that also the eighth moment of y_t exists.

In some of the results stated later, the i.i.d. assumption in (A.1) can be weakened to {z_t} being a stationary martingale difference sequence; this is more realistic, allowing for dependence in the rescaled errors. In (A.2) we assume that we have observed the stationary version of the process. This is merely for technical convenience and can most likely be removed.

Under (A.1)–(A.2), we prove consistency of the estimator and derive its asymptotic distribution (toward which it converges with rate slower than T^−1/2). If additionally (A.3) holds, the estimator is shown to be

-asymptotically normally distributed.

We first state a lemma that shows that the basic building blocks of our estimator are consistent and gives their asymptotic distribution. Under (A.2), E [y_t⁴] < ∞, but the eighth moment does not necessarily exist. One can show for ARMA processes with homoskedastic errors that the sample autocorrelation function is asymptotically normally distributed with only a second moment of the errors. In our case however, the errors, ε_t, are heteroskedastic, and to obtain asymptotic normality the fourth moment, E [ε_t⁴] < ∞, seems to be needed; see, for example, Hannan and Heyde (1972) for results in both the homoskedastic and heteroskedastic cases. The fourth moment of ε_t translates into the eighth moment of y_t.

The estimator

has a well-defined asymptotic distribution without requiring the eighth moment of the GARCH process to exist however, but the limit will not be Gaussian. This result has been established in Mikosch and Stărică (2000), see also Basrak, Davis, and Mikosch (2002) and Davis and Mikosch (1998). They show that

converges in distribution toward a so-called stable, regularly varying distribution with index κ ∈ (1,2); see Samorodnitsky and Taqqu (1994) and Resnick (1987) for an introduction. The index κ is shown to be the solution to E [(αz_t² + β)^κ] = 1. The convergence takes place with rate Ta_T⁻¹ where a_T = T^1/κl(T) and l(T) is a slowly varying function. The sequence a_T satisfies TP(y_t⁴ > a_T) → 1, such that the index κ is a measure of the tail thickness.

LEMMA 1. Under (A.1)–(A.2), the estimators

in (7) and (8) satisfy

. Furthermore,

where

and

for some κ ∈ (1,2), where W(m) = (Z_i − ρ(i)Z₀)_i=1,…,m and (Z_i)_i=0,…,m has a κ-stable distribution. If additionally (A.3) holds, then

where V_ρ(m) is the m × m matrix given by

where Y_t is an m × 1 vector with Y_t,i = (y_t² − σ²)(y_t+i² − σ²), i = 1,…,m.

Mikosch and Stărică (2000) establish a weak convergence result without the fourth moment condition (A.2). But in this case, one does not have any consistency result because the law of large numbers does not hold. The preceding weak convergence result of

under (A.1)–(A.2) has the flaw that the limit distribution is not in explicit form, which makes it difficult in practice to carry out any inference.

The consistency results and the weak convergence results in (11) and (13) in the preceding lemma can be proved without the i.i.d. assumption on {z_t} in (A.1) by using martingale limit theory. This can be done by utilizing the ARMA structure of {x_t} and applying the results of Hannan and Heyde (1972). It is not clear whether the more general weak convergence result in (12) can be extended to a non-i.i.d. setting however.

Observe that our estimator of λ can be expressed in terms of

. A simple application of the continuous mapping theorem therefore yields the following result.

THEOREM 2. Under (A.1)–(A.2), the estimator

given in (9) is consistent,

, and

where W(2) and κ are as in Lemma 1 and the function D is given subsequently in equations (A.1)–(A.3). If additionally Assumption (A.3) holds, then

An estimator of the covariance matrix V in (16) can be obtained by first estimating V_σ² and V_ρ(2) using heteroskedasticity and autocorrelation consistent (HAC) variance estimators (see Robinson and Velasco, 1997) and then substituting

into ∂D(σ²,ρ(1),ρ(2))/∂(σ²,ρ(1),ρ(2)). One can alternatively use the analytic expressions of ρ(k) to obtain an estimator of V_σ².

4. EFFICIENCY ISSUES

We here give a brief discussion of how one may improve on the efficiency of the closed-form estimator, in terms of both convergence rate and asymptotic variance. The basic idea is to perform a number of NR iterations using either the Whittle objective function or the Gaussian quasi-likelihood. Given closed-form estimates, one may wish to proceed to the QMLE or the Whittle estimator,³

with the initial value being the closed-form estimator,

the criterion function.

We here assume that

is invertible; Robinson (1988) discusses these issues in detail. Also, the step length is kept constant; it may however be a good idea in practice to use a line search in each iteration.

, one can show that

In general, the NR estimator will satisfy

where

is the actual M estimator (cf. Robinson, 1988, Thm. 2). Under regularity conditions, the M estimator will satisfy

where H = E [H_T(λ₀)] and Σ = E [S_T(λ₀)S_T(λ₀)^{[top ]}] (see, e.g., Lee and Hansen, 1994, Thm. 2; Giraitis and Robinson, 2001, Thm. 1). Combining (18) and (19), we obtain the following result.

THEOREM 3. Assume that (19) holds together with (A.1)–(A.2). Then, with κ as in Lemma 1, for k ≥ 2 + [log(κ/2(κ − 1))/log(2)],

If additionally (A.3) is satisfied, the preceding result holds for k ≥ 1.

5. A SIMULATION STUDY

We now examine the quality of our estimator in finite sample through a Monte Carlo study. In particular, we are interested in its behavior when the fourth moment does not exist. We simulate the GARCH process given in (1)–(2) with z_t ∼ i.i.d. N(0,1) for four different sets of parameter values. For each choice of parameter values, we simulated 5,000 data sets with T = 200, 400, 800, and 1,000 observations. For each data set, we obtained the QMLE using the Matlab GARCH Toolbox,

, j = 1,2,3, and w_j = 0, j > 3, and Winsorized at ε and 1 − ε with ε = 0.001. The GLS estimator reported here is based on two iterations. For the two first sets of parameter values, λ = (0.2, 0.15, 0.25)^{[top ]} and λ = (0.2, 0.25, 0.35)^{[top ]}, both the fourth and eighth moments exist; for the third, λ = (0.2, 0.35, 0.45)^{[top ]}, only the fourth moment is well defined; and for the fourth, λ = (0.2, 0.10, 0.89)^{[top ]}, neither the fourth nor the eighth moment exists, but the second does.

The results in terms of bias, variance, and mean squared error (MSE) are reported in Tables 1, 2, 3, and 4 for the four different sets of parameters. For these results, we have discarded a small number of data sets (eight in total) where the Matlab GARCH Toolbox stalled and did not terminate the optimization procedure. We have also set the ARMA estimator equal to zero whenever it returned negative estimates.

MSE of the QMLE and ARMA estimator for λ = (0.2, 0.15, 0.25)

MSE of the QMLE and ARMA estimator for λ = (0.2, 0.25, 0.35)

MSE of the QMLE and ARMA estimator for λ = (0.2, 0.35, 0.45)

MSE of the QMLE and ARMA estimator for λ = (0.2, 0.10, 0.89)

For the first set of parameters, the ARMA and GLS estimators are serious competitors to the QMLE. For small GARCH effects, the closed-form estimators seem to be a good alternative to the QMLE in terms of MSE. For the remaining three sets of parameters, the QMLE has significantly better performance compared to the two other estimators as expected. As α and/or β increases the variances of the ARMA and GLS estimators in general increase; this is consistent with the theory, which predicts that the convergence rate of the estimator deteriorates as α + β increases. However, contrary to the theory, it seems as if the ARMA estimator of α and β remains consistent even if the fourth moment is not well defined; the performance of the closed-form estimator of ω is very poor, though. The GLS estimator based on two iterations yields a large improvement in the precision relative to the initial ARMA estimator, except for the estimation of ω. The improvement might have been even more pronounced if we had done further iterations.

6. CONCLUDING REMARKS

The procedure easily extends to the case where there is a mean process, say, y_t = β^{[top ]}x_t + σ_t z_t for some covariates x_t, by applying the preceding process to the residuals from some preliminary fitting of the mean. Some aspects of the procedure extend to various multivariate cases and to GARCH(p,q). Specifically, in the multivariate case the relationships (3) and (6) continue to be useful and can be used as in Engle and Sheppard (2001) to reduce the dimensionality of the optimization space.

One may also consider other GARCH models that can be represented as an ARMA-like process. For example, suppose that the conditional variance is given as σ_t² = ω + βσ_t−1² + αy_t−1²1(y_t−1² > 0) + δy_t−1²1(y_t−1² ≤ 0). Then for x_t ≡ y_t² we have x_t = ω + φx_t−1 + u_t, where u_t is a martingale difference sequence with respect to

and φ = β + αm₂⁺ + δm₂⁻, where m₂⁺ = E [z_t²1(z_t > 0)] and m₂⁻ = E [z_t²1(z_t ≤ 0)]. Furthermore, σ² = ω/(1 − φ) so that we can identify ω and φ as before from the variance and the second-order covariance. To proceed further one needs to make strong assumptions about the distribution of z_t, for example, symmetry about zero, in which case φ = β + (α + δ)/2.

Our estimator also allows for a simple t-test for no GARCH effect because the asymptotic distribution derived earlier does not require α,β > 0 in contrast to the QMLE, where the Taylor expansion used requires this.