2. ASYMPTOTIC THEORY

We consider the GARCH(1,1) sequence {y_n, −∞ < n < ∞} defined by the equations

where θ = (ω,α,β) is the parameter of the process, ω > 0,α ≥ 0,β ≥ 0, and

Nelson (1990) shows that (2.1) and (2.2) have a unique stationary solution if and only if

The goal of this paper is to find an n^−1/2 consistent estimator for the number of moments of y₀.

Let

We are interested in the estimation of ν₀ satisfying

because then by the corollary on p. 322 of Nelson (1990) we have

Clearly, if Eε₀^2ν₀ < ∞, then (2.6) implies (1.1).

If Z_n(t) is an estimator for Z(t), then the natural estimator for ν₀ is ν_n, the smallest positive random variable satisfying

The random variables {ε_i², −∞ < i < ∞} cannot be observed and must be estimated from the sample. Using the recursion in (2.2) one can easily verify (cf. also Lee and Hansen, 1994; Lumsdaine, 1996) that

Let

be an estimator for θ. We estimate σ_k² with

, where

In light of (2.1),

can be used as an estimator for ε_k². We show that

is a uniformly, weakly consistent estimator for Z(t), where

. We assume that

is weakly consistent with order n^−γ, i.e.,

where ∥·∥ denotes the maximum norm of vectors. We note that (2.8) holds with γ = ½ in case of quasi (pseudo) -likelihood estimators (cf. Lumsdaine, 1996; Lee and Hansen, 1994; Berkes, Horváth, and Kokoszka, 2002) and in case of likelihood estimators under weak assumptions on the distribution of the innovations and model parameters (cf. Berkes and Horváth, 2001).

THEOREM 2.1. If (2.1)–(2.4) and (2.8) hold and 0 < ν₀ < ∞,

and

then there is a unique ν₀ satisfying (2.5) and

The main result of this paper is the asymptotic normality of n^1/2(ν_n − ν₀), which will require stronger conditions than (2.8). We postulate a representation of the form

where

with some functions [ell ]_i and h_i satisfying

Representation (2.11) was established by Horváth, Kokoszka, and Teyssière (2001 in the context of ARCH(p) processes for several classes of estimators with the functions [ell ]_i and h_i explicitly identified. It was extended to GARCH models in Berkes et al. (2002)) and Berkes and Horváth (201).

We now define the quantities needed to define the asymptotic variance. Let

We note that conditions (2.9) and (2.2) imply that p₁ < ∞, whereas the existence of p₂ will follow from Lemma 4.2 in Section 4.

The asymptotic variance of n^1/2(ν_n − ν₀) is given by

THEOREM 2.2. We assume that (2.1)–(2.4), (2.9), (2.11), and (2.12) hold, 0 < ν₀ < ∞, and

Then

where N(0,d₀²) stands for a normal random variable with mean 0 and variance d₀².

3. SMALL SAMPLE PROPERTIES

In this section we describe the results of a small simulation study that we conducted to assess the performance of the estimator in finite samples. We worked with time series of length n = 1,000, which may be considered small for applications to financial data; Mikosch and Stărică (2000) consider a sample of 70,000 observations, and samples of this length are not unusual when analyzing intradaily trading of speculative assets. The goal of our study was merely to gain some insight into the behavior of the estimator, and we do not view the results presented in this section as a guide for practitioners; for this a more extensive study focusing on a specific application at hand would be required.

We considered several GARCH(1,1) models with standard normal innovations ε_k. Roughly speaking, the smaller the parameters α and β in (2.2), and consequently the higher the maximal moment exponent, the worse the performance of the estimator becomes. This is intuitively clear because if the parameters α and β are small, then the observations y_k look like the noise sequence ε_k, which has finite moments of any order. This results in long right tails of the sampling distribution of the moment estimator. In Figures 1a and 2a we plotted the estimated sampling densities of the estimator ν_n. We also studied the effect of parameter estimation on the moment estimator. Figures 1b and 2b show the estimated sampling densities of ν_n when the estimators

in the definition of Z_n(t) are replaced by the true parameter values β and α. (We used the residuals as Mikosch and Stărică, 2000, not the innovations.) It is seen that the variability of the estimators

has a significant impact on the estimator ν_n and should be taken into account in empirical studies. This variability is described by (2.11) and the definitions leading to Theorem 2.2.

(a) Estimated sampling density of the estimator νn with n = 1,000. The true data generating process is GARCH(1,1) with α = 0.3,β = 0.3, and standard normal innovations εk. The 2.5, 25, 50, 75, and 97.5 percentiles of the sampling distribution are, respectively, 2.06, 2.68, 3.12, 3.7, and 5.62. The approximate true value of ν is 3.436. (b) Estimated sampling density of the estimator νn with n = 1,000 but with the estimators replaced by the true values β and α. The estimated densities are based on 1,000 replications of νn smoothed by a Gaussian kernel.

(a) Estimated sampling density of the estimator νn with n = 1,000. The true data generating process is GARCH(1,1) with α = 0.1,β = 0.8, and standard normal innovations εk. The 2.5, 25, 50, 75, and 97.5 percentiles of the sampling distribution are, respectively, 3.4, 4.82, 6.04, 7.78, and 14.04. The approximate true value of ν is 6.263. (b) Estimated sampling density of the estimator νn with n = 1,000 but with the estimators replaced by the true values β and α. The estimated densities are based on 1,000 replications of νn smoothed by a Gaussian kernel.

The true value of ν cannot be found analytically even when the values of α and β are known. The “true values” reported in the captions in Figures 1 and 2 were obtained by approximating Z(t) by the mean of one million values of (β + αε_i²)^t for i.i.d. standard normal ε_i.

We also investigated the effect of model misspecification on the sampling distribution of the estimator. As can be expected, the estimator is biased in such situations, and the bias is probably not asymptotically negligible. In samples of length n = 1,000 it appears however that for time series with heavy-tailed marginal distributions such that the dependence between the observations decays exponentially, the extremal behavior of the series can to some extent be approximated by fitting a GARCH(1,1) model. Figure 3 shows the sampling density of ν_n when the true DGP (data generating process) is AR(1) with stable innovations. It is well known (see, e.g., Embrechts et al., 1997, Ch. 7), that for linear processes the extremal moment index of the observations is equal to the index of stability of the innovations, so in this case we know the true value of ν.

Estimated sampling density of the estimator νn with n = 1,000. The true data generating process is AR(1) with the autoregressive parameters equal to 0.5 and i.i.d. symmetric stable innovations with the index of stability equal to 1.75. The 2.5, 25, 50, 75, and 97.5 percentiles of the sampling distribution are, respectively, 0.63, 0.86, 1.05, 1.25, and 1.51. The exact true value of ν is 0.875. The estimated densities are based on 1,000 replications of νn smoothed by a Gaussian kernel.

4. PROOFS OF THEOREMS 2.1 AND 2.2

The following lemma will be used in the proofs of Theorems 2.1 and 2.2.

LEMMA 4.1. Let 0 < a < b < 1 and d > 0 and define

If

then

and

for all t ≥ 1, where κ = ν − ν(log(bb_*))/log(b/a) with any 1 < b_* < 1/b and C does not depend on k.

Proof. For any N ≥ 1 we have (x ∧ y = min(x,y))

By (4.1) and the Markov inequality we have

It is easy to see that

Assume that t > 1 and let N = c₁ log t with c₁ = 1/log(b/a). Now (4.4) yields

and by (4.5) we have

completing the proof of (4.2).The proof of (4.3) is similar and hence is omitted. █

The following result is an immediate consequence of Lemma 4.1.

LEMMA 4.2. Let 0 < a < 1 and d > 0 and define

If (4.1) holds with some ν > 0, then

and

for all k ≥ 1 and μ ≥ 0, where C₁(μ) and C₂(μ) do not depend on k.

Proof. Let μ be given and choose b > a so near to a that κ > μ + 2. Let R be the smallest integer such that iaⁱ ≤ bⁱ for R ≤ i < ∞. With this choice of b and R we have that

and by Lemma 4.1 we have that Eξ_k^μ < ∞. On the other hand,

completing the proof of (4.6). Similar arguments give (4.7). █

The proofs of Theorems 1.1 and 1.2 will use the following decomposition:

Using the definition of

we write

where

Let

be arbitrary constants satisfying

and define the event B_n on which

. Condition (2.8) yields that

We continue with a technical lemma. █

LEMMA 4.3. If (2.1)–(2.4), (2.8), and (2.9) hold and

then

and

Proof. By (4.10) we can assume that B_n holds. On the event B_n

So using (2.8) we get

and

Condition (4.11) yields

and therefore (4.17), (4.18), and δ > 1/γ imply (4.12) when i = 1, 2, 5, and 6. On the event B_n we have

By Lemma 2.3 in Berkes et al. (2002), there is ν* such that E|y₀|^ν* < ∞. We can choose any β < β to have (4.10) so we can choose β < β so close to β such that

Hence by Lemma 4.1 we have

Again using (2.8), (4.19), and (4.20) we conclude

Using the mean-value theorem on the event B_n we have

so following the proof of (4.21) one can easily verify that

The estimates in (4.13) are immediate consequences of (4.10), (4.16), and (4.19).

Following the proofs of (4.12) and (4.13) one can easily verify (3.13) and (4.15). █

The next result shows that Z_n(t) and Z_n′(t) are uniformly consistent estimators for Z(t) and Z′(t). █

LEMMA 4.4. If the conditions of Lemma 4.3 are satisfied, then

and

Proof. Let η₁ > 0 and η₂ > 0. Putting together (4.8), (4.9), and Lemma 4.3 we get that there is an integer N = N(η₁,η₂) satisfying

Since the convergence of monotone functions to a limit is uniform over finite intervals, by the strong law of large numbers we have

Thus by (4.25) and (4.26) we have

Since

and

relation (4.22) is proved. Similar arguments give (4.23). █

Proof of Theorem 2.1. Let us observe first that Z(0) = 1,Z′(t) and Z′′(t) exist on [0,δ]. In (2.10) we assumed that ν₀ < δ. By (2.4) we have that Z′(0) < 0. Also Z′′(t) > 0, and thus Z(t) is convex on [0,δ] . The corollary on p. 322 in Nelson (1990) yields that

for any

. Since Z(0) = 1 and Z′(0) < 0 there is μ > 0 such that Z(t) < 1 for all 0 < t ≤ μ. By

we conclude that there is a ν₀,

such that Z(ν₀) = 1. The observation that Z′′(t) > 0 gives that Z′(t) is strictly increasing. By Z(0) = Z(ν₀) = 1 and 0 < ν₀, Z(t) reaches its minimum at a unique point ν_* satisfying 0 < ν_* < ν₀. Of course, Z′(ν_*) = 0, and by the strict increase of Z′ we obtain that Z′(t) > 0 for all t > ν_* and therefore ν₀ is unique.

Next we show that

Let

. Then Z(ζ) > 1 and by Lemma 4.4 we have

establishing (4.28).

There are μ* > 0 and η* > 0 such that Z′(t) < −μ* on

. Hence by Lemma 4.4 we obtain that

Since Z′(ν_*) = 0, we get that Z(t) < 1 for all 0 < t ≤ ν_*. So by Lemma 4.4 we conclude that

Since Z_n(t) is uniformly close to Z(t) and ν₀ is the only point in [ν_*,η*] where Z(t) = 1, standard arguments give Theorem 2.1.

We write

We show that the first term is essentially is a linear combination of

. A two-term Taylor expansion gives

where η_k is between

and β + αε_k². It is easy to see that

LEMMA 4.5. If (2.1)–(2.4), (2.9), (2.11), and (2.12) hold and

then

and

Proof. Since for any i = 1,2,3, the random variables [ell ]_i(ε_k²)h_i(y_k−1,y_k−2,…), −∞ < k < ∞ are stationary and uncorrelated with zero mean, we get

so condition (2.11) yields

By the law of large numbers for any fixed t ≤ δ we have

and therefore the monotonicity argument leading to (4.25) yields

Combining (4.31) and (4.32) we obtain (4.29). Similarly to (3.31) we have

proving (4.30) because of (4.31).

Using the definition of

we write

Next we use (4.9) and get

where

LEMMA 4.6. If the conditions of Lemma 4.5 are satisfied, then

Proof. Applying the ergodic theorem to the stationary ergodic sequence t(β + αε_k²)^t−1ε_k²/σ_k² and using the monotonicity argument leading to (4.25), we get

and therefore (4.31) implies Lemma 4.6. █

LEMMA 4.7. If the conditions of Lemma 4.5 are satisfied, then

Proof. By (2.7) we have

and therefore

Since 0 < β < 1, we conclude that

and therefore (4.31) and (4.34) yield Lemma 4.7. █

LEMMA 4.8. If the conditions of Lemma 4.5 are satisfied, then

Proof. Two-term Taylor expansion gives

where η_i is between β and

. For any 0 < Δ < Δ′ < 1 − β there is C = C(Δ,Δ′,β) such that

Since Δ,Δ′ can be as small as we wish, Lemma 4.1 and our previous estimate imply that

Again using the ergodic theorem we conclude that

and therefore

By (4.31) we have for any

, and therefore it is sufficient to show that for some Δ > 0

Following the proof of (4.35) one can derive (4.36) from the ergodic theorem and Lemma 4.1. █

LEMMA 4.9. If the conditions of Lemma 4.5 are satisfied, then

Proof. The proof of Lemma 4.9 is similar to the proofs of Lemmas 4.5–4.8 and hence is omitted. █

LEMMA 4.10. If the conditions of Lemma 4.5 are satisfied,

Proof. Combining (4.31) and Lemmas 4.6–4.9 we get that

and therefore

Thus we have

and therefore it suffices to show that

The arguments leading to (4.37) can be used to establish (4.38), and therefore the details are omitted. █

Next we show that n^1/2(Z_n(t) − Z(t)) converges in

to a continuous Gaussian process. The covariance of the limit is

LEMMA 4.11. If the conditions of Lemma 4.5 are satisfied, then

where Γ(t) is a Gaussian process with EΓ(t) = 0 and EΓ(t)Γ(s) = r(t,s).

Proof. By Lemma 4.10 and condition (2.11) it is enough to prove the weak convergence of R_n(t) = n^−1/2 [sum ]_1≤i≤n z_j(t), where

Let

be the σ-algebra generated by ε_i,−∞ < i ≤ j. Let K be a positive integer, let λ_i,0 ≤ t_i ≤ δ,1 ≤ i ≤ K be constants, and define z_j* = [sum ]_1≤i≤K λ_i z_j(t_i). It is easy to see that z_j* is a stationary and ergodic sequence and

is a martingale.

Using Theorem 23.1 of Billingsley (1968) we conclude the asymptotic normality of [sum ]_1≤j≤n z_j* and therefore the Cramér–Wald device (cf. Billingsley, 1968, p. 49) gives the convergence of the finite dimensional distributions of R_n(t).

Next we show the tightness of R_n(t). It is easy to see that R_n(0) = 0 and

(i = 1,2,3), since the variance of the preceding sum is O(n). Next we note that

with some constant C. Also, by the mean-value theorem

and therefore

with some constant C. Now Theorem 12.3 in Billingsley (1968) gives tightness. █

Proof of Theorem 2.2 Observing that

by the mean-value theorem we have

where η_n is between ν_n and ν₀. Theorem 2.1 and the continuity of Z′ give that

. Theorem 2.1, Lemma 4.11, and the continuity of Γ(t) yield

, completing the proof of Theorem 2.2. █