Paolo Zaffaroni

Whittle estimation of EGARCH and other exponential volatility models

Keywords: EGARCH, GJR, Stochastic Volatility, Whittle estimation, asymptotics

, are also comprised by our assumptions. The variable of interest might not have finite fractional moment of any order and so, in particular, finite variance is not imposed. We allow for a wide range of degrees of persistence of shocks to conditional variance, allowing for both short and long memory.

Introduction

Consider an observable satisfying

x t = z t e 0.5 h t , t ∈ Z, (1)
h t = ω 0 + ∞ k=0 ψ 0k t-k-1 almost surely (a.s.), ∞ j=0 ψ 2 0j < ∞, (2)
where Z = {t : t = 0, ±1, ...}. The {z t , t } form a sequence of independently identically distributed (i.i.d.) unobservable bivariate random variable, although we allow z t and s to be cross-correlated for t = s. We require, at minimum, E 0 = 0, 0 < E 2 0 < ∞ and Elogz 2 0 < ∞ although existence of the moments of the z t is not required.

When z t and t are mutually independent and normally distributed, model ( 1)-( 2) becomes the well-known exponential stochastic volatility (SV) model of [START_REF] Taylor | Modelling Financial Time Series[END_REF]. Simple yet successful estimation of such SV models can be carried out by noting that

logx 2 t = logz 2 t + h t , (3)
represents an example of a linear signal-plus-noise model. For parameterizations of the ψ 0j that ensure Markovianity, the Kalman filter could be successfully applied to SV models (see [START_REF] Nelson | The time series behaviour of stock market volatility and returns[END_REF] and [START_REF] Harvey | Multivariate Stochastic Variance Models[END_REF], where (3) represents the measurement equation and h t is the state unobserved variable. An alternative estimation approach for (3) is the Whittle estimator, obtained maximizing the frequency domain approximation of the Gaussian likelihood, so-called Whittle function (see [START_REF] Whittle | Gaussian estimation in stationary time series[END_REF]). This paper is prompted by considering that another popular class of exponential volatility models, where t = (z t ) for some instantaneous transformation (•), also belongs to the class (1)-(2). Such one-shock models, the most important case of which is the exponential generalized autoregressive conditional heteroskedasticity (EGARCH) model of [START_REF] Nelson | Conditional heteroscedasticity in asset pricing: a new approach[END_REF], can be seen a SV models for which a singularity occurs with respect to the joint distribution of the z t and t . For instance EGARCH requires that

t = (z t ) = θ 0 (z t -µ z ) + δ 0 (| z t | -µ |z| ), (4)
for constant parameters θ 0 , δ 0 with θ 0 δ 0 = 0 where µ y = Ey for any random variable y with finite first moment. Another one-shock model, popular among practitioners, is the so-called GJR volatility of Glosten, Jaganathan, and where 1 (•) denotes the indicator function. Note that (•) is an odd function in both ( 4) and ( 5) yielding asymmetric models. Since (1)-( 2) holds, the signalplus-noise representation (3) can be obtained for one-shock models although now the signal h t and the noise logz 2 s are correlated for some t, s. It turns out that all the statistical literature of Whittle estimation of linear signal-plus-noise models requires uncorrelated components (see in particular [START_REF] Hosoya | A limit theory for long-range dependence and statistical inference on related models[END_REF], [START_REF] Dunsmuir | A central limit theorem for parameter estimation in stationary vector time series and its applications to models for a signal observed with noise[END_REF] and [START_REF] Hosoya | A central limit theorem for stationary processes and the parameter estimation of linear processes[END_REF]) and, indeed, the Whittle estimator has bee successfully applied to estimation of SV models by [START_REF] Harvey | Long memory in stochastic volatility[END_REF], [START_REF] Breidt | The detection and estimation of long memory in stochastic volatility[END_REF] and [START_REF] Deo | Forecasting realized volatility using a long memory stochastic volatility model: estimation, prediction and seasonal adjustment[END_REF]. However, for one-shock exponential models such as the EGARCH or the GJR, one cannot use these results since, by (4) or (5), log z 2 t and h s could be correlated for some t, s. The difficulty in estimating the signal-plus-noise model ( 3) is due to the fact that, despite linearity, the spectral density of the observable logx 2 t is not easily factored meaning that it cannot be expressed as m 2 0 /(2π) | 1 + ∞ k=1 n 0k e ikλ | 2 with m 0 and the n 0k , k = 0, 1, ... being respectively function of two disjoint sets of the model parameters (see [START_REF] Hannan | The asymptotic theory of linear time series models[END_REF]). An alternative approach to this problem is proposed in [START_REF] Robinson | Rates of convergence and optimal bandwidth for longrange dependence[END_REF], who presents various cases where the model spectral density is not easily factored, other than the signal-plus-noise model. Linearity is nowhere assumed but long memory, except for a mild form, is ruled out and, due to its generality, some assumptions appear un-primitive.

The appeal of the exponential model ( 1)-( 2) is based on the fact that it solves many of the drawbacks characterizing the structure of the ARCH model of [START_REF] Engle | Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom[END_REF]. In particular, one needs not to impose non-negativity of the ω 0 , ψ 0k (k ≥ 0). Second, asymmetric effects, whereby volatility tends to rise in response to 'bad' news and to fall in response to 'bad news', are easy to parameterize. Third, there is no ambiguity on the interpretation of the persistence of shocks to conditional variance. These were in fact the chief motivations that led in fact [START_REF] Nelson | Conditional heteroscedasticity in asset pricing: a new approach[END_REF] to introduce the EGARCH model. Fourth, as the model is observed at a finer and finer time interval, it has a diffusion limit which belongs to the class of continuous time processes frequently used in continuous time mathematical finance. This property holds for both one and two-shock models (see Ghysels, Harvey, and Renault (1995, Section 4.1 and 4.3)). Not surprisingly, EGARCH models are attracting a constantly increasing attention in theoretical (see [START_REF] Duan | Approximatin the GJR-GARCH and the EGARCH option pricing models analytically[END_REF]) and empirical finance (see [START_REF] Brandt | Volatility Forecasting With Range-Based EGARCH Models[END_REF] among many others).

The main contributions of this paper can be synthesized as follows:

First, we extend the statistical theory of Whittle estimation to cover correlated signal-plus-noise models, providing a formal asymptotic distribution theory specifically tailored for parameter estimation of the exponential model ( 1)-( 2), both for the two-shock as well as for the one-shock version. This is relevant since, with the exception of the low-order EGARCH result of [START_REF] Straumann | Estimation in conditionally heteroskedastic time series models[END_REF], for general EGARCH, GJR and any other exponential one-shock model, no other estimation approach exists for which we have a complete, formal, understanding of its asymptotic statistical properties.

Second, our theory covers both cases of summable and non-summable | ψ 0j |. Important examples of the latter case, which implies long memory in h t , is the fractionally integrated EGARCH (FIEGARCH) of [START_REF] Bollerlsev | Modeling and pricing long memory in stock market volatility[END_REF], when considering one-shock models, and the long memory SV of [START_REF] Harvey | Long memory in stochastic volatility[END_REF] and [START_REF] Breidt | The detection and estimation of long memory in stochastic volatility[END_REF], when considering twoshock models. Note that the statistical literature does cover the case of linear-plus-signal model with long memory (see [START_REF] Hosoya | A limit theory for long-range dependence and statistical inference on related models[END_REF]) but, again, the case of correlated signal and noise is ruled out. Third, our asymptotic results are based on a set of regularity conditions, easily verifiable with respect to any given choice of the ψ 0k . Our result covers the situation of uncorrelated signal and noise, as a special case. Therefore, even for the case of SV models, one could use our results. This is highly desirable since the statistical literature of Whittle estimation typically defines regularity conditions in terms of smoothness of the model spectral density and their higher-order mixed derivatives, and checking such conditions can become an arduous task. (See for instance Assumptions C and D of Hosoya (1997).) Fourth, we present a unified theory which depends on a set of regularity conditions designed to apply to both one and two-shock models that applies to a class of models wider than EGARCH and SV. [START_REF] Meddahi | Temporal aggregation of volatility models[END_REF] firstly pointed out that for many purposes the difference between oneshock GARCH-type and two-shock SV-type models is only apparent rather than substantial. Our results provide a case where this analogy holds with respect to Whittle estimation.

Practical estimation of the exponential model ( 1)-( 2) requires to finiteparameterize the

ψ 0k = ψ k (ζ 0 ) for a known set of functions ψ k (•) and unknown p × 1 parameter ζ 0 , where p < ∞. Next, let α 0 = α(φ 0 ) = var(log z 2 0 ), β 0 = β(φ 0 ) = var( 0 ), γ 0 = γ(φ 0 ) = cov(log z 2 0 , 0 ) for known functions α(•), β(•), γ(•) of a q × 1
unknown vector φ 0 with q < ∞. This includes both the case of a known parametric specification for the joint distribution of the {z t , t }, depending on the unknown φ 0 , as well as the case of an unspecified distribution in which case φ 0 = (α 0 , β 0 , γ 0 ) with q = 3. We wish to estimate the (p + q) × 1 vector ϑ 0 = (ζ 0 , φ 0 ) , on the basis of a sample (x 1 , ..., x T ) of observations. Denote by ϑ = (ζ , φ ) any admissible value to which corresponds the function

f (λ; ϑ) = α(φ) 2π + β(φ) 2π ψ(e iλ ; ζ) 2 + γ(φ) 2π e iλ ψ(e iλ ; ζ) + e -iλ ψ(e -iλ ; ζ) , -π ≤ λ < π (6) with ψ(z; ζ) = ∞ j=0 ψ j (ζ)z j , | z |≤ 1. ( 7
)
Note that no truncation of the transfer function ψ(z; ζ) is needed here. It can be easily seen that f (λ) = f (λ; ϑ 0 ) is the spectrum of the log x 2 t and, thus, f (λ; ϑ) denotes the model-spectrum. Let Θ be a prescribed compact subset of R p+q . The Whittle estimator of ϑ 0 is

θT = arg min ϑ∈Θ Q T (ϑ),
where the discrete Whittle function is

Q T (ϑ) = 1 T T -1 t=1 log( f (λ t ; ϑ))+ I T (λ t ) f (λ t ; ϑ) , λ t = 2πt T . ( 8
)
Hereafter

I T (λ) = (2πT ) -1 T t=1 log x 2 t e iλt 2
, -π ≤ λ < π, is the periodogram based on T consecutive observations of the log x 2 t where we can avoid mean correction since I T (λ) is evaluated at the Fourier frequencies.

The following section lists our assumptions, with discussion. Section 3 presents the main results, namely strong consistency and asymptotic normality of θT under conditions that cover a wide variety of parametric specifications, comprising both exponentially and hyperbolically decaying coefficients ψ j (ζ). Section 4 illustrates how the main results apply to EGARCH and clarifies the advantages and disadvantages of the Whittle estimation approach with respect to other methods. A number of extensions are introduced in Section 5 such as nonstationary logx 2 t , arising from non square-summability of the ψ 0j , and filtering and forecasting. Concluding remarks make Section 6. The proofs are reported in the final appendix.

Assumptions

Denote by K a generic finite constant, not always the same. Let k 1 ≥ 0, k 2 ∈ {0, 1} and l a non-negative integer. Let ∼ denote asymptotic equivalence:

a(x) ∼ b(x) as x → x 0 when a(x)/b(x) → 1. Assumption A(k 1 ). The {z t , t } are i.i.d. variates with E 0 = 0 and E | log z 2 0 | k 1 < ∞, E | 0 | k 1 < ∞.
Assumption B. Θ = Z × Φ where Z is a compact subspace of R p and Φ is a compact subspace of R q with s = p + q < ∞. ϑ 0 is an interior point of Θ. Assumption D(k 2 , l). For any ϑ ∈ Θ, α(φ), β(φ), γ(φ) and all the ψ j (ζ) have continuous lth derivative and

Assumption C(k 2 , l). For any ϑ ∈ Θ, | ψ k (ζ) |≤ K | ψ j (ζ) | for 1 ≤ j ≤ k, all k ≥ 1,
∂ r ψ j (ζ) ∂ζ i 1 ...∂ζ i r - ∂ r ψ j+1 (ζ) ∂ζ i 1 ...∂ζ i r ≤ K k 2 + | 1 -k 2 | j -1 ∂ r ψ j (ζ) ∂ζ i 1 ...∂ζ i r for any j > J,
for some constant J < ∞ and all i h = 1, ..., s, h = 1..., r, r = 0, ..., l.

Assumption E. For any ϑ ∈ Θ, α(φ), β(φ), γ(φ) and all the ψ j (ζ) are continuously differentiable and there exist integers j i (ϑ), i = 2, ..., s, such that 1 ≤ j 2 (ϑ) < ... < j s (ϑ) < ∞ and the s × s matrix

∂ ∂ϑ c 0 (ϑ), ∂ ∂ϑ c j 2 (ϑ), . . . , ∂ ∂ϑ c j s (ϑ) (9)
has full rank, setting

c u (ϑ) = 1 (u=0) α(φ)+β(φ) ∞ j=0 ψ j (ζ)ψ j+u (ζ)+1 (u =0) γ(φ)ψ |u|-1 (ζ), u = 0, ±1, ... ( 10
) Assumption F . For any ϑ ∈ Θ, | γ(φ) |< (α(φ) β(φ)) 1 2 < ∞.
Remarks.

(i). We require Ass. A(2) for consistency and A(4) for asymptotic normality of θT . For two-shock models the conditional expectation E(x 2δ t | F t-1 ), δ > 0, might not be bounded, where F t-1 defines the sigma-algebra induced by the {z s , s ; s ≤ t -1}. Instead, for one-shock model a stronger moment condition for the z t is implied, since E(x 2 t | F t-1 ) < ∞ when (4) holds. However, the unconditional moment E | x t | δ need not to be bounded for any δ > 0. Important examples of this case are when the z t have a Studentt distribution with ν > 2 degrees of freedom or a generalized exponential distribution (henceforth GED) with tail thickness parameter ν ≤ 1. See Nelson (1991, p.453).

(ii). The x t are strictly stationary and ergodic under Ass. A(2) and square summability of the ψ 0j whereas | log x 2 t |= ∞ a.s. when the ψ 0j are not square summable .

(iii). Ass. B implies that there exist constant 0

< α L < α U < ∞, 0 < β L < β U < ∞, -1 < e L < e U < 1 and -∞ < d L < d U < 1/2 such that α L ≤ α(φ) ≤ α U , β L ≤ β(φ) ≤ β U and e L ≤ e(ζ) ≤ e U , d L ≤ d(ζ) ≤ d U for any ϑ ∈ Θ.
(iv). The parameter ω 0 is not identified by the Whittle function, since enters linearly in logx 2 t and it disappears when calculating the empirical autocovariances of the log x 2 t . Nevertheless alternative estimation methods do exist and will be discussed in the sequel.

(v). We are concerned here with two cases: exponentially decaying (case k 2 = 1) and hyperbolically decaying (case k 2 = 0) coefficients ψ j (ζ). The functions E r (•), D r (•) arise as a result of differentiation in most cases of interest. For example, typically |D r (j; ζ)|< K(log(j + 1)) r . When d(ζ) = 0 one has to distinguish the two cases of summable and non-summable | D r (j; ζ) | j -1 . In the latter case the model spectral density is still unbounded at zero frequency, although it diverges very slowly as the zero frequency is approached. In the former case we allow for the possibility that D r (j; ζ) = 0 when d(ζ) = 0.

(vi). Ass. B and C(k 2 , 0) imply that for any ϑ ∈ Θ sup ϑ∈Θ

ψ j (ζ) ≤ K j d U -1 , j ≥ 1, (11)
and thus sup ϑ∈Θ

∞ j=0 ψ 2 j (ζ) ≤ ∞ j=0 sup ϑ∈Θ ψ 2 j (ζ) < ∞.
(vii). Ass. D(k 2 , l) implies that the ψ j (ζ) and its derivatives converge toward zero in a sufficiently smooth manner, implying quasi monotonic convergence toward zero and pure bounded variation (see Yong (1974, Definitions I-2 and I-4)). As we will see, these, together with the exact rate condition of Ass. C(k 2 , l), define unambiguously the behaviour near the origin of the model spectral density and its derivatives, as well as a form of uniform continuity away from zero frequency. When k 2 = 1 this is already implied by Ass.

C(1, l), which imparts absolute summability of ψ j (ζ) and of their derivatives.

(viii). Ass. E is a rank identification assumption. It is easy to see that c u = c u (ϑ 0 ), u = 0, ±1, ... defines the autocovariance function of the log x 2 t . By simple calculations, the left hand side of ( 9) can be expressed

       e 1 ∂α(φ) ∂φ +      ∆ 0 (ζ) ∆ j 2 (ζ) . . . ∆ js (ζ)      ∂β(φ) ∂φ +      0 ψ j 2 -1 (ζ)
. . .

ψ js-1 (ζ)      ∂γ(φ) ∂φ . . . . . . . . . . . . e 1 Φ 0 (ϑ)+e 2 Φ j 2 (ϑ) + . . . e s Φ js (ϑ)        (12) setting ∆ u (ζ) = ∞ j=0 ψ j (ζ)ψ j+u (ζ) u = 0, ±1, ..., Φ 0 (ϑ) = β(φ) ∂∆ 0 (ζ) ∂ζ , Φ j (ϑ) = β(φ) ∂∆ j (ζ) ∂ζ + γ(φ) ∂ψ j-1 (ζ) ∂ζ j ≥ 1
, and e j is the s × 1 vector with all zeros but 1 in the jth entry. Simple inspection of (12) shows that a necessary order condition for identification is q ≤ 3. Moreover, when q = 3 the identification condition must include ∂c 0 (ζ)/∂ϑ for otherwise φ is not identified.

(ix). Ass. F is a sufficient condition for strict positivity of the model spectral density at all frequencies. This is required for asymptotic normality of θT although not for consistency. When {z t , t } have a parametric distribution depending on φ 0 , then we can assume that for any ϑ ∈ Θ, there exists a collection of i.i.d. bivariate stochastic processes {z t (φ), t (φ)}, indexed by φ, such that

α(φ) = var(logz 2 0 (φ)), β(φ) = var( 0 (φ)), γ(φ) = cov(logz 0 (φ), 0 (φ)). (13)
Ass. F is violated whenever the Schwarz inequality holds with the equality sign: either perfectly collinear log z 2 t (φ) and t (φ) or, alternatively, degenerate log z 2 t (φ) or t (φ). For instance, perfect collinearity arises for a version of the EGARCH model where ( 4) is substituted by t = (z t ) = θ 0 (logz 2 tµ logz 2 ) where δ 0 = 0.

Main Results

We present the asymptotic results for the Whittle estimator θT .

Theorem 1 Under Assumptions A(2), B, C(k 2 , 0), D(k 2 , 0), E, F , as T → ∞, θT → a.s. ϑ 0 . Theorem 2 Under Assumptions A(4), B, C(k 2 , 2), D(k 2 , 2), E, F ,as T → ∞, T 1 2 ( θT -ϑ 0 ) → d N p+q 0, M -1 V M -1 , where M (ϑ) = 1 2π π -π N (λ; ϑ)N (λ; ϑ) dλ, M = M (ϑ 0 ), N (λ; ϑ) = ∂ ln f (λ; ϑ) ∂ϑ , N (λ) = N (λ; ϑ 0 ), and
V = 1 π π -π N (ω)N (ω)dω + 1 2π π -π π -π N (ω 1 ) f (ω 1 ) N (ω 2 ) f (ω 2 ) Q(-ω 1 , ω 2 , -ω 2 )dω 1 dω 2 , (14)
Q(ω 1 , ω 2 , ω 3 ) denoting the trispectrum of the log x 2 t (the Fourier transform of the fourth-order cumulants of the log x 2 t ).

Under Assumptions A(4), B, C(k 2 , 3), D(k 2 , 3), F as T → ∞, M T ( θT ), V T ( θT ),
defined respectively in ( 31) and ( 32), are consistent estimates of M, V .

Remarks.

(i) Both the rate of convergence and asymptotic normality do not depend on whether d(ζ) is zero or not. This result represents one of the finest feature of the Whittle estimator, due to the automatic compensation, characterizing the Whittle function, for possible lack of square integrability of the model spectral density, occurring when 1/4 ≤ d(ζ) < 1/2. On the other hand, obviously, the asymptotic covariance matrix depends on the assumed parametric choice made for ψ j (ζ). But the Whittle estimator displays other advantages. For one, the discrete Whittle function does not require estimation of the mean which might otherwise affects the small sample performance of the Whittle estimator (see [START_REF] Diebold | Maximum Likelihood Estimation of Fractionally Integrated Noise with Unknown Mean[END_REF]). Second, the Whittle function does not require any truncation, such as substituting log x 2 t by logz 2 t + ω 0 + t-2 k=0 ψ 0k t-k-1 . This is because the spectral density ( 6) is a function of all the ψ 0k , k = 0, 1, ..., through the transfer function ( 15). This contrasts with the maximum likelihood estimator (henceforth MLE) where typically one needs to distinguish the observable pseudo log likelihood, function of a sample (x 1 , x 2 , ..., x T ), from the unobservable pseudo log likelihood, function of all {x s ; t = ..., -1, 0, 1, .., T }. Such truncation is usually asymptotically negligible but might not be so for long memory parameterizations, and can induce an asymptotic bias, such as for the pseudo MLE (henceforth PMLE) of ARCH(∞), as shown by [START_REF] Robinson | Pseudo maximum likelihood estimation of ARCH(∞) models[END_REF].

(ii) The mean parameter ω 0 is not identifiable by the autocorrelation function, and thus it cannot be estimated by the Whittle estimator. A simple estimate is based on the sample mean log

x 2 T = 1/T T t=1 log x 2 t , which is a consistent estimate of Elog x 2 0 = ω 0 + Elogz 2 0 under Ass. A(2). Thus, a √
T -consistent estimate of ω 0 is obtained by subtracting the Whittle estimate

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT
of Elog z 2 0 , which depends on φT , from log x 2 T .

(iii) For practical use of the asymptotic distribution result, a consistent estimate of asymptotic covariance matrix is required. Here we prove consistency of plug-in estimators of M and V , respectively in Lemma 8 and 9. Alternatively, for V one can use [START_REF] Taniguchi | On estimation of the integrals of the fourth order cumulant spectral density[END_REF], although it is unclear whether Taniguchi's result extends to case d(ζ) ∈ (0, 1/2).

Implications for EGARCH

We first check that the regularity conditions requested by Theorem 1 and 2 are satisfied by EGARCH. For the same model, we discuss how the properties of the Whittle estimator relates to the ones of the MLE. Define the generating function

ψ(z; ζ) = ∞ j=0 ψ j (ζ)z j , | z |≤ 1, (15)
and consider the class of functions

ψ(z; ζ) = a(z; ζ) b(z; ζ) (1 -z) -d(ζ) (16
a(z; ζ) = 1 + m j=1 ζ j z j for m ≥ 0, a(z; ζ) = 0, |z| ≤ 1, (17) b(z; ζ) = 1 - n j=1 ζ j+m z j for n ≥ 0, b(z; ζ) = 0, |z| ≤ 1, (18)
setting n j=1 c j = 0 when n < 1 and with ζ i denoting the ith element of ζ. Hereafter assume that E(z 0 ) = 0, var(z 0 ) = 1. Then, the EGARCH(m, n) model is defined by ( 4) and ( 16)-( 17 The function E r (j; ζ) could either be monotonically non decreasing as well as a trigonometric function, the latter case arising in case of dominant complex conjugates roots. Concerning the parameter φ, (4) implies that the distribution of the z t can at most depend on one parameter φ 03 due to Ass. E, since q ≤ 3, and imposing Ez 0 = 0 and var(z 0 ) = 1 one gets

α(φ 0 ) = var(logz 2 0 ), β(φ 0 ) = φ 01 + φ 2 02 (1 -µ 2 |z| ), γ(φ 0 ) = φ 02 cov(logz 2 0 , | z 0 |),
where φ 01 = θ 2 0 , φ 02 = δ 0 . The sign of θ 0 is not identifiable through the (univariate) model spectral density although it will be using a bivariate extension of the Whittle estimator, as discussed below. Identification requires that φ 03 = var(logz 2 0 ) only can be left as a free parameter, and one must then set 'a priori' both µ |z| and cov(logz 2 0 , | z 0 |) in such a way that Ass. F is satisfied. This represents the most general (semi-parametric) specification of the EGARCH model, in terms of the distribution of the z t . Considering parametric specifications, meaning that the distribution of the z t depends on an unknown parameter φ 03 , then var(logz 2 0 ), µ |z| and cov(logz 2 0 , | z 0 |) are jointly determined. Ass. F will be automatically satisfied, whenever the z t have a non-degenerate (parametric) distribution. For instance, when z t are i.i.d GED with tail thickness parameter 0 < φ 03 ≤ ∞

α 0 = ( 2 φ 03 ) 2 Ψ( 1 φ 03 ), β 0 = φ 01 +φ 2 02 (1-µ 2 |z| ), γ 0 = 2φ 02 φ 03 µ |z| ψ( 2 φ 03 ) -ψ( 1 φ 03 ) , ( 19
) with µ |z| = Γ(2/φ 03 )/( Γ(3/φ 03 )Γ(1/φ 03 ))
where ψ(z) is the digamma function (the derivative of logΓ(z) with Γ(z) being the Gamma function) and Ψ(z) is the trigamma function (the derivative of ψ(z)); see Gradshteyn and Ryzhik (1994, sections 6.3 and 6.4). A distribution with fatter tails than the normal is obtained whenever φ 03 < 2 and with thinner tails whenever φ 03 > 2 (see Nelson (1991, p.353)). The GED nests the normal distribution, for φ 03 = 2, and the uniform distribution, for φ 03 → ∞. Let us discuss now Ass. E more in depth. Consider first the left hand side sub matrix of ( 12), made up of the first q columns. It is evident that linear independence of the columns of this sub matrix requires ∂α(φ)/∂φ, ∂β(φ)/∂φ, ∂γ(φ)/∂φ to be linearly independent. Since for EGARCH ∂α(φ)/∂φ 1 = ∂α(φ)/∂φ 2 = ∂γ(φ)/∂φ 1 = 0 it is easy to see that this is achieved when

∂α(φ) ∂φ 3 ∂β(φ) ∂φ 1 ∂γ(φ) ∂φ 2 = 0. ( 20
)
For the normal case ( 20) is not satisfied since α(φ) is independent from φ and, indeed, we can identify only φ 01 = φ 01 φ 03 and φ 02 = φ 02 √ φ 03 , implying q = 2. However, both for the GED and Students't case (20) is satisfied. Under this circumstance, full rank of the left hand side sub matrix of ( 12) is then guaranteed if also ∆ 0 (ζ) ∆ j a (ζ) = 0 for at least one j a , a ∈ {2, ..., s}.

Let us now consider the right hand sub matrix made up by the last p columns of ( 12), and consider for sake of simplicity the EGARCH(1, 1) case, yielding p = 2 and ζ = (ζ 1 , ζ 2 ) . It is well known that for this case

ψ 0 = 1, ∆ 0 (ζ) = (1 + ζ 2 1 + 2ζ 1 ζ 2 )/(1 -ζ 2 2 ) and ψ u = ζ u-1 2 (ζ 1 + ζ 2 ), ∆ u (ζ) = ζ u-1 2 (ζ 1 + ζ 2 )(1 + ζ 1 ζ 2 )/(1 -ζ 2 2 ), u ≥ 1. Simple yet tedious calculations yield Φ 0 (ϑ) = 2β(φ) ζ 1 + ζ 2 1 -ζ 2 2   1 1+ζ 1 ζ 2 1-ζ 2 2   , Φ 1 (ϑ) = β(φ) 1 -ζ 2 2    1 + 2ζ 1 ζ 2 + ζ 2 2 (1+ζ 1 ζ 2 ) 2 +(ζ 1 +ζ 2 ) 2 1-ζ 2 2    Φ u (ϑ) = ζ u-2 2 β(φ) ζ 2 ∂∆ 1 (ζ) ∂ζ + γ(φ) ∂ψ 1 (ζ) ∂ζ +ζ u-3 2 0 (u -1) ζ 2 β(φ)∆ 1 (ζ) + (u -2) γ(φ)ψ 1 (ζ) , u ≥ 2, which form a nonsingular basis in R 2 whenever | ζ 1 |< 1, | ζ 2 |< 1, ζ 1 +ζ 2 = 0.
Under the same conditions, full rank of matrix ( 12) follows since each of the first q column vector is linearly independent from the last p column vectors.

The asymptotic covariance matrix of θT involve the trispectrum Q(ω 1 , ω 2 , ω 3 ), defined in (33), which in turn depends on various mixed fourth-order cu-mulants for the logz 2 t , t . Since cum(a, b, c, d) = E(abcd) -E(ab)E(cd) -E(ac)E(bd)-E(ad)E(bc), assuming symmetric z t , for EGARCH one gets:

κ logz 2 logz 2 logz 2 = δ 0 E(((logz 2 0 ) 3 | z 0 |) -3µ (logz 2 ) 2 Elogz 2 0 | z 0 |) , κ logz 2 logz 2 = (θ 2 0 + δ 2 0 )E(logz 2 0 ) 2 z 2 0 -(θ 2 0 + δ 2 0 )µ (logz 2 ) 2 -2δ 2 0 (Elogz 2 0 | z 0 |) 2 , κ logz 2 = (δ 3 0 + 3θ 2 0 δ 0 )Elogz 2 0 | z 0 | 3 -3δ 0 (θ 2 0 + δ 2 0 )Elogz 2 0 | z 0 |, κ = (θ 4 0 + δ 4 0 + 6θ 2 0 δ 2 0 )µ z 4 -3(θ 2 0 + δ 2 0 ) 2 .
We now discuss the analogies of our results with respect to the MLE of EGARCH. Nelson (1991, p.93) proposed estimation of the EGARCH by ML based on the GED with tail thickness parameter ν, but its asymptotic properties remain unknown even when ν is assumed known (ν = 2 yields the Gaussian likelihood). ML estimation requires invertibility of the model, that is the possibility to express z t as a convergent (in some norm) function of all the x s (s ≤ t). Invertibility is necessary for the 'observed' likelihood, function of a sample (x 1 , ..., x T ) of size T , to be well-behaved asymptotically without exploding nor converging toward zero for any admissible parameter value. Establishing invertibility is typically a formidable task to be achieved for nonlinear moving average models (see [START_REF] Granger | On the invertibility of time series models[END_REF]). Recently Straumann and Mikosch (2006, eq.(3.18) and Remark 3.13) provide a sufficient condition for invertibility of EGARCH(0, 1) but also suggest that such condition is practically infeasible, except when ζ 01 = 0, that is for the EGARCH(0, 0) which, in turn, implies ψ 00 = 1, ψ 0j = 0, j ≥ 1. This means, for instance, that h t is i.i.d. and x t is independent of x t-j for all j > 1. Based on this invertibility condition, the asymptotics properties of the MLE of the EGARCH(0, 0) are then established (see Straumann (2005, Theorem 5.7.9).

However, it is easy to establish the central limit theorem (CLT) for the first derivative of the log likelihood function evaluated at the true parameter value for a general EGARCH(m, n). This represents a necessary step to establish the asymptotic distribution of the PMLE which, however, does not require invertibility of the model. It turns out that Ez 2 0 + E | z 0 | 2ν < ∞ is required, when adopting a GED log likelihood for z t , with tail thickness parameter 0 < ν < ∞. (Obviously, when z t is truly GED distributed then all its moments exist.) Since ν > 1 is likely to be required (see Nelson (1991, last paragraph of p.106)), this simplifies to E | z 0 | 2ν < ∞, which equals Ez 4 0 < ∞ when adopting a Gaussian likelihood. Concerning the Whittle estimator, Theorem 2 simply requires Ez 4 0 < ∞ and E(log z 2 0 ) 2 < ∞ .(The latter is implied by the forme for most distributions of interest.) Therefore, our moment condition equals the one required by Gaussian PMLE and will only be slightly stronger for GED PMLE with ν < 2. This contrasts with estimation of ARCH(∞) model whereas Whittle estimation requires Ex 8 0 < ∞ (see [START_REF] Giraitis | Whittle estimation of ARCH models[END_REF]). Conditions required for asymptotic normality of Gaussian PMLE of ARCH(∞) are instead much weaker, since finite variance of the x t is not even required (see [START_REF] Robinson | Pseudo maximum likelihood estimation of ARCH(∞) models[END_REF]). [START_REF] Linton | Estimating semiparametric ARCH(∞) models by kernel smoothing methods[END_REF] consider semiparametric estimation of a class of asymmetric ARCH(∞), with a nonparametric specification of the effect of past squares on the conditional variance. Their asymptotic results require E|x 0 | δ < ∞, δ > 4. [START_REF] Zaffaroni | Gaussian inference on certain long-range dependent volatility models[END_REF] considers Whittle estimation of a class of nonlinear moving average processes, which however does not admit a signal-plus-noise representation, treating x 2 t as the observable. A bounded fourth moment condition is postulated.

Extensions

We now present a number of relevant extensions, the additional proofs of which follow by suitably adapting the corresponding proofs of Theorem 1 and 2, and are not reported for sake of simplicity. Further extensions that lead to efficiency improvements and applicability to high-frequency return and duration data are discussed in [START_REF] Zaffaroni | Gaussian inference on certain long-range dependent volatility models[END_REF].

Non-stationarity

The very first empirical application of EGARCH, based on the value-weighted market index from the CRSP tapes, indicates a great deal of persistence, with the largest estimated autoregressive roots approximately equal to 0.99962, yielding a t statistic for testing the unit root hypothesis of -0.448 Nelson (1991, Table 5.2 and p.99). As another example, Bollerlsev and Mikkelsen (1996, Table 5) 0k < ∞ might be too strong in certain circumstances. Let us consider for instance the case where ψ 0j ∼ c j d 0 -1 as j → ∞, for 1/2 ≤ d < 3/2. Then square summability of the ψ 0j clearly fails but ∞ k=1 (ψ 0kψ 0k-1 ) 2 < ∞. Several approaches can be considered in such circumstance. First, one can adapt the idea of using data tapering used by [START_REF] Velasco | Whittle pseudo-maximum likelihood estimation for nonstationary time series[END_REF] for parametric Whittle estimation of nonstationary linear time models. Second, one can simply differentiate logx 2 t in order to achieve stationarity. For the case just described, this implies considering as observable the series xt =

x t x t-1 = z t z t-1 e 0.5(ht-h t-1 ) = zt e 0.5(ht-h t-1 ) , setting zt = z t /z t-1 , and then consider Whittle estimation based on the log x2 t . The differenced process will satisfy the conditions listed in Section 2, and the corresponding spectral density will be |1e iλ | 2 f (λ) where f (λ) is the (pseudo) spectral density of the log x 2 t . The Whittle estimator of the model parameters will then be strongly consistent and asymptotically normal but the limit covariance matrix will have a different form from the un-differenced case reported in Theorem 2, since a different model spectral density is considered. Moreover, an efficiency loss is likely to occur. In fact, logx 2 t = logz 2 t + (h th t-1 ), and the noise term logz 2 t has a larger variance, in fact double, than logz 2 t . Note that logz 2 t is autocorrelated and moreover correlated with h th t-1 , except for the simple case of mutually uncorrelated log z 2 t , t . For Gaussian z t , the zt will be Cauchy distributed so the first moment

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

does not exist and the first absolute moment is unbounded. This suggests that applying the same idea to PMLE could be problematic, stressing instead the great flexibility enhanced by the Whittle estimator. The log-difference transformation mainly affects the proof of the CLT (cf. Lemma 7 in the appendix), which nevertheless could be generalized to the case where the observable is the log-square of

m-1 j=0 z c j t-j e 0.5(ω 0 + P ∞ k=0 ψ 0k t-k-1 )
, with known c j = 0, ±1, ... and square summable ψ 0j . This implies that any finite number of log-square differenciations is permitted in order to achieve stationarity. By differencing one might obtain that the spectral density of the differenced series is O(| λ | δ ), λ → 0 for some δ > 0. This affects the proof of asymptotic normality, unless δ < 1, but not of consistency. Therefore, in order to construct confidence intervals, two estimations are required with the first one aiming at finding the suitable degree of differentiation to possibly ensure δ < 1.

Filtering and forecasting

Consider one-shock models such as the EGARCH and the GJR models. If the (nonlinear) autoregressive representation of x t was available, it could be used for both forecasting h T +s ,s ≥ 0, based on a sample of x 1 , ..., x T -1 or, alternatively, to back out the within-sample (yet unobserved) volatility h s , 1 ≤ s ≤ T -1. Establishing the existence of such representation is analog to establish invertibility of the x t , which appears a formidable task to be achieved, as discussed before. However, the frequency domain set up allows us to make use of the Wiener-Kolgomorov (WK) theory of forecasting and signal extraction for linear models (see [START_REF] Whittle | Gaussian estimation in stationary time series[END_REF]). [START_REF] Harvey | Long memory in stochastic volatility[END_REF] proposed to use the WK filter for long memory SV models when γ 0 = 0. It is relatively less known, though, that the WK theory naturally allows for correlated signal and noise and it can then be applied to one-shock models, once a consistent parameters estimate is available. Therefore, in what follows, we simply describe how to implement the WK theory within our framework, when correlation between signal and noise is allowed for.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

Assume for sake of simplicity that Elogx 2 0 = Elogz 2 0 + ω 0 = 0. In the more realistic case of non-zero mean, a consistent estimate of the mean (see Remark (ii) of Section 3) will be subtracted from the data prior to filtering and forecasting yielding asymptotically equivalent results.

For forecasting purposes, the minimum mean-square linear forecast of logx 2

T +h ,h = 0, 1, ... based on the infinite past of the series

{logx 2 T -j , j = 1, ..} is logx 2 T +h = - h j=1 a j logx 2 T +h-j - ∞ j=1 a j+h logx 2 T -j , h = 0, 1, ... ( 21
)
Forecast of h T +h-1 easily follow by ( 3). The coefficients a j are obtained by means of the canonical factorization of f (λ) (see Whittle (1983, p.26)) which exists by Lemma 4, part (ii). A consistent estimate of the a j is âu =

(2N ) -1 N j=-N +1 Âj e iuµ j , u = 0, 1, .., N -1, where Âu = exp -N -1 j=1 ĉj e -ijµ u = Ā-u , ĉu = 1 N N -1 k=1 log(f (µ k ; θT )) cos(uµ k ) for µ n = πn/N, n = 0, ±1, ..., ±N , N = [T 1 2 /4], [•]
is the integer part of its argument and ā is the conjugate of the complex number a. Computation of the âu implies that ( 21) is replaced by the finite sum logx

2 * T +h = -h j=1 âj logx 2 * T +h-j -N -h-1 j=1
âj+h logx 2 * T -j , h = 0, 1, .... Hereafter {x * t-j , j = 1, 2, ..., N -1} represents a sample of the x t not used in the estimation of ϑ 0 .

Signal extraction of h s , 1 ≤ s ≤ T -1 based on the infinite sequence log x 2 t , t = 0, ±1, ... can be carried out evaluating the best linear predictor

ĥt = ∞ j=-∞ g j logx 2 t-j (22)
where the

g j = (2π) -1 π -π e -ijλ f -1 (λ) (f hh (λ) + f hlogz 2 (λ)) dλ and f hh (λ) = β 0 /(2π) ∞ u=-∞ ∆ |u| e iuλ , f hlogz 2 (λ) = γ 0 /(2π) ∞ u=1 ψ 0u-1 e iuλ , -π ≤ λ < π, setting ∆ u = ∆ u (ζ 0 ) = ∞ j=0 ψ 0j ψ 0j+u . Similarly to before, a consistent esti- mate of g j is ĝj = 1 2N N -1 p=-N +1 f -1 (µ p ; θT ) f hh (µ p ; θT ) + f hlogz 2 (µ p ;
θT ) e -ijµp , j = 0, ±1, ... Truncating suitably ( 22) and plugging in the ĝj yields ĥ * t = t-1 j=t-T ĝj logx 2 * t-j , t = 1, ..., T . Theoretical justification for logx 2 * T +h and ĥ * t , as T → ∞, can be obtained adapting [START_REF] Hidalgo | Prediction and signal extraction of strongly dependent processes in the frequency domain[END_REF] and [START_REF] Bhansali | Asymptotic properties of the Wiener-Kolmogorov predictior.I[END_REF].

We have established the asymptotic distribution theory of the Whittle estimate of a class of exponential volatility models (1)-( 2) the most notable element of which is the EGARCH model of [START_REF] Nelson | Conditional heteroscedasticity in asset pricing: a new approach[END_REF] and the FIE-GARCH of [START_REF] Harvey | Long memory in stochastic volatility[END_REF] and [START_REF] Breidt | The detection and estimation of long memory in stochastic volatility[END_REF]. [START_REF] Perez | Finite Sample Properties of Maximum Likelihood and Whittle Estimators in EGARCH and FIEGARCH Models[END_REF] present a finite-sample comparison of the Whittle estimator with MLE for EGARCH and FIEGARCH models. We have discussed generalizations, in particular when dealing with nonstationarity as well as with filtering and forecasting. Other generalizations are of interest. The frequency domain set-up easily allows to estimate seasonality effects with various degrees dependence. Second, we focused on estimation of the conditional variance parameters by assuming martingale difference observations, but one can consider simultaneous estimation of both conditional mean and conditional variance parameters. Third, multivariate extensions of the exponential model, along the lines of [START_REF] Harvey | Multivariate Stochastic Variance Models[END_REF] but also considering one-shock models, can be developed and estimated based on a multivariate version of the Whittle estimator. Fourth, the exponential model can be generalized by leaving the news impact curve (z t ) unspecified, similarly to the semiparametric ARCH(∞) model of [START_REF] Linton | Estimating semiparametric ARCH(∞) models by kernel smoothing methods[END_REF]. Similarly to them, the estimation procedure of this semiparametric exponential model will then combine aspects of parametric and nonparametric estimation.

Appendix: mathematical proofs K defines a non zero constant, not always the same, and K 0 a constant that could be eventually zero. → a.s. , → d define a.s. convergence and convergence in distribution respectively. For any i h = 1, ..., p, h = 0, ..., l, let

U i 1 ,i 2 ,..,i l (λ; ζ) = ∂ l ∂ζ i 1 ...∂ζ i l ψ(e iλ ; ζ) 2 ,V i 1 ,i 2 ,..,i l (λ; ζ) = ∂ l ∂ζ i 1 ...∂ζ i l e iλ ψ(e iλ ; ζ) + e -iλ ψ(e -iλ ; ζ)
and for any j h = 1, ..., s, h = 0, ..., l, let f j 1 ,j 2 ,..,j l (λ; ϑ) = ∂ l ∂ϑ j 1 ...∂ϑ j l f (λ; ϑ), any -π ≤ λ < π and ϑ ∈ Θ. When l = 0 we mean that no differentiation occurs.

Lemma 1 Under Assumptions C(0, l) and D(0, l):

(i) As λ → 0 + V i 1 ,i 2 ,..,i r (λ; ζ) ∼        K D r (λ -1 ; ζ) λ -d(ζ) , d(ζ) ∈ (0, 1/2), K D r (λ -1 ; ζ), d(ζ) = 0 and non summable | D r (j; ζ) | j -1 , K 0 , d(ζ) = 0 and summable | D r (j; ζ) | j -1 , K 0 , d(ζ) < 0, U i 1 ,i 2 ,..,i r (λ; ζ) ∼        K D r (λ -1 ; ζ) λ -2d(ζ) , d(ζ) ∈ (0, 1/2), K D r (λ -1 ; ζ), d(ζ) = 0 and non summable | D r (j; ζ) | j -1 , K 0 , d(ζ) = 0 and summable | D r (j; ζ) | j -1 , K 0 , d(ζ) < 0,
for all i h = 1, ..., s, h = 1...,
| ∆ u (ζ) -∆ u+1 (ζ) ≤| K | ∆ u (ζ) | u as u → ∞. (23)
This implies that the c u (ϑ) are quasi-monotonically convergent to zero and also satisfy the pure-bounded variation condition: Robinson (1994, Lemma 11) when

∞ k=u |∆ k (ζ) -∆ k+1 (ζ)|= O(| ∆ u (ζ) |) as u → ∞. See, for example,
d ∈ (0, 1/2). Moreover ∆ u (ζ) = u j=0 ψ j (ζ)ψ j+u (ζ) + ∞ j=u+1 ψ j (ζ)ψ j+u (ζ) ∼ ψ u u j=0 ψ j + ∞ j=u+1 ψ 2 j (ζ) ∼ D 0 (j; ζ)u 2d(ζ)-1 , as u → ∞, setting D 0 (j; ζ) = D 2 0 (j; ζ)
, where the last ∼ relationship follows by Yong (1974, Lemma I-11, (1-32') and (1-32") and Lemma I-16). It easily follows that same properties apply to the c u (ϑ). When

d(ζ) = 0, ∆ u (ζ) = u j=0 ψ j (ζ)ψ j+u (ζ) +
Proof. (i) We proceed by contradiction. Let us assume that there exist ϑ ∈ Θ such that ϑ = ϑ 0 and f (λ; ϑ) = f (λ; ϑ 0 ) = f (λ) for -π ≤ λ < π. However, since f (λ; ϑ) is uniquely identified by its Fourier transform, it also follows that c u (ϑ) = c u (ϑ 0 ), u = 0, ±1, .... By Assumption E and the mean value theorem

     c 0 (ϑ) c j 2 (ϑ) . . . c j s (ϑ)      =      c 0 (ϑ 0 ) c j 2 (ϑ 0 ) . . . c j s (ϑ 0 )      +      ∂ ∂ϑ c 0 ( θ) ∂ ∂ϑ c j 2 ( θ) . . . ∂ ∂ϑ c js ( θ)      (ϑ -ϑ 0 ),
where θϑ ≤ ϑ 0ϑ and • is the Euclidian norm. But this implies ϑ = ϑ 0 since the s × s matrix of derivatives is full rank.

(ii) By Assumption F we can always find a sufficiently small δ(φ

) > 0 such that | γ(φ) |< (α (φ)) 1 2 (β(φ)) 1 2 with α (φ) = α(φ) -δ(φ). Then f (λ; ϑ) = δ(φ) 2π + α (φ) 2π + β(φ) 2π ψ(e iλ ; ζ) 2 + γ(φ) 2π e iλ ψ(e iλ ; ζ) + e -iλ ψ(e -iλ ; ζ) ,
and the term in brackets on the right hand side is nonnegative for any -π ≤ λ < π, since it represents a well-defined model spectral density. When (13) holds, one can take δ(φ) = var(log z 2 0 (φ) -E(log z 2 0 (φ) | 0 (φ))). Finally take K = inf ϑ∈Θ δ(φ)/(2π).

Lemma 5 Under Assumption A(2) and (2) the x t satisfy | x t |< ∞ a.s. and are ergodic and strictly stationary.

Proof. Almost sure boundedness, ergodicity and strict stationarity follow easily adapting Nelson (1991, Theorem 2.1).

Lemma 6 (i) Under Assumptions

A(2), C(k 2 , 0), D(k 2 , 0), F , Q T (ϑ) → a.s. Q(ϑ) as T → ∞, uniformly in ϑ ∈ Θ, where Q(ϑ) = 1 2π π -π log(f (λ; ϑ))dλ+ 1 2π π -π f (λ)f -1 (λ; ϑ)dλ. (ii) Under Assumption E, F , Q(ϑ) ≥ Q(ϑ 0 ) for any ϑ ∈ Θ.
Proof. (i) All the convergences below hold as T → ∞. Uniform convergence of 1/T T -1 t=1 I T (λ t )/f (λ t ; ϑ) is obtained by Hannan (1973, Lemma 1). Let us consider the non random term 1/T T -1 t=1 log( f (λ t ; ϑ)). When f (λ; ϑ) is continuous for all λ ∈ [-π, π) then uniform convergence also follows adapting Hannan (1973, Lemma 1) since, uniformly on Θ, |log( f (λ; ϑ))g M (λ; ϑ)| < δ, for some arbitrary δ > 0 taking large enough M , where g M (λ; ϑ) is the M -terms Cesaro sum of the Fourier series of log( f (λ; ϑ)). When f (λ; ϑ) is not continuous at zero frequency the result follows by adapting Zaffaroni (2003, Lemma 10) bearing in mind that sup ϑ∈Θ f (λ; ϑ)

≤ K | λ | -2d U , inf ϑ∈Θ f (λ; ϑ) ≥ K > 0, λ ∈ [-π, π), with d U < 1/2.
(ii) By Assumption F one gets f (λ)/f (λ; ϑ)-1 ≥ -log(f (λ; ϑ)/f (λ)), equality holding only for ϑ = ϑ 0 by Lemma 4-(i).

Lemma 7 Under Assumptions

A(4), B, C(k 2 , 1), D(k 2 , 1), F, T 1 2 π -π g(λ)(I T (λ) -EI T (λ))dλ → d N (0, Ṽ ), as T → ∞, (24)
where g(λ) = g(λ; ϑ 0 ), g(λ; ϑ) = ∂ ∂ϑ f -1 (λ; ϑ), -π ≤ λ < π, and Ṽ defines an s × s positive semi definite matrix.

Proof. At first we characterize the local behaviour of g(λ; ϑ) near the zero frequency. As λ → 0 + by Lemma 1

f -2 (λ; ϑ) ∂f (λ; ϑ) ∂φ i ∼            K β φ i D 0 (λ -1 ;ζ) λ 2d(ζ) , d(ζ) ∈ (0, 1/2), K β φ i D 0 (λ -1 ;ζ) , d(ζ) = 0 and non summable | D 0 (j; ζ) | j -1 , K 0 , d(ζ) = 0 and summable | D 0 (j; ζ) | j -1 , K 0 , d(ζ) < 0,
for i = 1, ..., q, and

f -2 (λ; ϑ) ∂f (λ; ϑ) ∂ζ j ∼            K D 1 (λ -1 ;ζ) D 2 0 (λ -1 ;ζ) λ 2d(ζ) , d(ζ) ∈ (0, 1/2), K D 1 (λ -1 ;ζ) D 2 0 (λ -1 ;ζ) , d(ζ) = 0 and non summable | D 1 (j; ζ) | j -1 , K 0 , d(ζ) = 0 and summable | D 1 (j; ζ) | j -1 , K 0 , d(ζ) < 0,
for j = 1, ..., p. Just considering the first case d(ζ) > 0, it easily follows since 2d(ζ) . The other cases can be obtained in the same manner, noting that the terms that involve V (λ, ζ) are dominated by terms in U (λ, ζ).

f 2 (λ; ϑ) ∼ KD 2 0 (λ -1 ; ζ)λ -4d(ζ) and ∂ ∂φ i f (λ; ϑ) ∼ K β φ i D 0 (λ -1 ; ζ)λ -
All the convergences below hold as T → ∞. Let h u (ϑ) = (2π) -1 π -π g(λ; ϑ)e iuλ dλ, u = 0, ±1, ±2, ... be the Fourier coefficients of g(λ; ϑ) and write h u (ϑ) = (h u φ 1 (ϑ), ..., h u φq (ϑ), h u ζ 1 (ϑ), ..., h u ζp (ϑ)) . Set h u = h u (ϑ 0 ), h u ϑ i = h u ϑ i (ϑ 0 ), i = 1, ..., s and y t = log x 2 tµ log x 2 , where µ log x 2 = ω 0 + µ log z 2 . We follow the approach put forward by [START_REF] Giraitis | A central limit theorem for quadratic forms in strongly dependent linear variabels and its application to asymptotically normality of Whittle's estimate[END_REF] and approximate

π -π g(λ)I T (λ)dλ = 1 4π 2 T T t,s=1 h |t-s| (log x 2 t -μlogx 2 )(log x 2 s -μlogx 2 ) (here μlogx 2 = T t=1 log x 2 t /T
) by another quadratic form, which shares the same asymptotic distribution but is nevertheless much easier to handle. First note that by Fox and Taqqu (1987, Lemma 8.1) no change in the asymptotic distribution of the previous quadratic form occurs when substituting μlog x 2 with the population mean µ log x 2 . Next, set

P T = T t,s=1 h |t-s| y t y s , P T (N ) = T t,s=1 h |t-s| y t (N )y s (N ) with y t (N ) = logz 2 t -µ log z 2 + N j=0 ψ 0j t-j-1 , 0 < N < ∞.
The main part of the proof is devoted to establish var(P T -P T (N )) = O(T δ N ) for a sequence of positive terms satisfying δ N → 0 as N → ∞.

We first show that var(P T ) = O(T ). In fact

var(P T ) = T t 1 ,s 1 ,t 2 ,s 2 =1 h |t 1 -s 1 | h |t 2 -s 2 | c |t 1 -t 2 | c |s 1 -s 2 | + c |s 1 -t 2 | c |t 1 -s 2 | (25) + T t 1 ,s 1 ,t 2 ,s 2 =1 h |t 1 -s 1 | h |t 2 -s 2 | cum (y t 1 , y t 2 , y s 1 , y s 2 ) , (26)
where cum(•, •, •, •) defines the fourth-order cumulant operator. Setting Q T (f ), Q T (g) equal to the T × T Toeplitz matrix based on the Fourier transforms of f (λ) and g(λ) respectively, the two terms on the right hand side of (25) can be written as tr(Q

T (f )Q T (g)Q T (f )Q T (g)). When | f (λ) |= O(λ -a-δ ), | g(λ) |= O(λ -b-δ
) as λ → 0, any δ > 0, and satisfying the regularity condition of Fox and Taqqu (1987, p.215), then tr(Q Fox and Taqqu (1987, Theorem 1). Since any slowly varying function satisfies | L(u) |= O(u δ ), u → ∞, any δ > 0, and for all possible of our cases -b ≥ a ≥ 0 the result follows. For (26), it can be easily seen that cum (y t 1 , y t 2 , y s 1 , y s 2 ) is made by the sum of sixteen terms of the form:

T (f )Q T (g)Q T (f )Q T (g)) = O(T ) when 2(a + b) < 1 by
κ logz 2 logz 2 logz 2 logz 2 1 t 1 =t 2 =s 1 =s 2 (one term), κ logz 2 logz 2 logz 2 ψ s 2 -t 1 -1 1 t 1 =t 2 =s 1 (four terms), κ logz 2 logz 2 1 t 1 =t 2 ψ s 1 -t 2 -1 ψ s 2 -t 1 -1 (six terms), κ logz 2 ψ t 2 -t 1 -1 ψ s 1 -t 1 -1 ψ s 2 -t 1 -1 (four terms), κ ∞ k=0 ψ k ψ k+t 2 -t 1 ψ k+s 1 -t 1 ψ k+s 2 -t 1 (one term), (27)
where we set κ abcd = cum(a 0 , b 0 , c 0 , d 0 ) for i.i.d. {a t , b t , c t , d t } with bounded fourth moment. Consider the last term. Since ψ j = 0, j < 0 then 0 ≤ s

2 -t 1 + k implies s 1 -s 2 ≤ s 1 -t 1 + k. Likewise s 2 -s 1 ≤ s 2 -t 1 + k, yielding | ψ k+s 1 -t 1 ψ k+s 2 -t 1 |= 1 s 2 -s 1 >0 | ψ k+s 1 -t 1 ψ k+s 2 -t 1 | +1 s 1 -s 2 ≥0 | ψ k+s 1 -t 1 ψ k+s 2 -t 1 |≤ 1 s 2 -s 1 >0 K | ψ s 2 -s 1 | +1 s 1 -s 2 ≥0 K | ψ s 1 -s 2 |≤ K | ψ |s 2 -s 1 | | which in turn implies κ ∞ k=0 | ψ k ψ k+t 2 -t 1 ψ k+s 1 -t 1 ψ k+s 2 -t 1 |≤ K | ψ |s 2 -s 1 | | ∞ k=0 | ψ k ψ k+|t 2 -t 1 | |≤ K ∞ k=0 | ψ k ψ k+|s 2 -s 1 | | ∞ k=0 | ψ k ψ k+|t 2 -t 1 | |.
Therefore we can apply Fox and Taqqu (1987, Theorem 1) precisely as we have done above, yielding that the term of (26) involving κ ∞ k=0 ψ k ψ k+t 2 -t 1 ψ k+s 1 -t 1 ψ k+s 2 -t 1 is O(T ). Along the same lines, one can easily show that all the other terms of (26) are O(T ), implying var(P T ) = O(T ). Write var(P T -P T

(N )) = A 1 +A 2 + A 3 where A 1 = var T t,s=1 h |t-s| y t (y s -y s (N )) , A 2 = var T t,s=1 h |t-s| y s (N ) (y t -y t (N )) , A 3 = 2cov T t,s=1 h |t-s| y t (y s -y s (N )), T t,s=1 h ϑ i ,|t-s| y s (N )(y t -y t (N ))
. The same bound apply to A 1 and A 2 and therefore, by Schwarz inequality, to A 3 as well, so we just consider A 1 . By the cumulants theorem (see [START_REF] Leonov | On a method of calculation of semi-invariants[END_REF]

) one obtains A 1 = B 1 + B 2 + B 3 with B 1 = T t 1 ,s 1 ,t 2 ,s 2 =1 h |t 1 -s 1 | h |t 2 -s 2 | cov (y t 1 , y t 2 ) cov (y s 1 -y s 1 (N ), y s 2 -y s 2 (N )) B 2 = T t 1 ,s 1 ,t 2 ,s 2 =1 h |t 1 -s 1 | h |t 2 -s 2 | cov (y t 1 , y s 2 -y s 2 (N )) cov (y t 2 , y s 1 -y s 1 (N )) , B 3 = T t 1 ,s 1 ,t 2 ,s 2 =1 h |t 1 -s 1 | h |t 2 -s 2 | cum (y t 1 , y t 2 , y s 1 -y s 1 (N ), y s 2 -y s 2 (N )). By Assumption A(2), cov (y t , y s ) = α 0 1 t=s + β 0 ∆ |t-s| (ζ 0 ) + γ 0 1 t =s ψ 0|s-t|-1 , cov (y t -y t (N ), y s -y s (N )) = β 0 ∞ j=N +1 ψ 0j ψ 0j+|t-s| , cov (y t , y s -y s (N )) = γ 0 ψ 0s-t-1 1 (s-t>N +2) + β 0 ∞ j=N +1 ψ 0j ψ 0j+|t-s| ,

and by Assumption

A(4) cum (y t 1 , y t 2 , y s 1 -y s 1 (N ), y s 2 -y s 2 (N )) = (28) κ logz 2 logz 2 ψ s 1 -t 2 -1 ψ s 2 -t 1 -1 1 t 1 =t 2 1 s 1 -t 2 ≥N +1 1 s 2 -t 1 ≥N +1 + κ logz 2 ψ |t 2 -t 1 |-1 ψ s 1 -t 1 -1 ψ s 2 -t 1 -1 1 s 1 -t 1 ≥N +1 1 s 2 -t 1 ≥N +1 + κ logz 2 ψ |t 1 -t 2 |-1 ψ s 1 -t 2 -1 ψ s 2 -t 2 -1 1 s 1 -t 2 ≥N +1 1 s 2 -t 2 ≥N +1 + κ ∞ k=0 1 k≥max{0,N +1-s 1 +t 1 ,N +1-s 2 +t 1 } ψ k ψ k+t 2 -t 1 ψ k+s 1 -t 1 ψ k+s 2 -t 1 .
For B 1 , for some 0 < η < 1

| cov (y t , y s ) |≤ K ∞ j=0 | ψ 0j | 1-η/2 | ψ 0j+|t-s| | 1-η/2 , ( 29
) | cov (y t -y t (N ), y s -y s (N )) |≤ β 0 | ∞ j=N +1 | ψ 0j | η/2 | ψ 0j | 1-η/2 | ψ 0j+|t-s| | η/2 | ψ 0j+|t-s| | 1-η/2 ≤ K | ψ 0N | η ∞ j=0 | ψ 0j | 1-η/2 | ψ 0j+|t-s| | 1-η/2 . (30)
Consider Ass. C(0, 1) and d = d(θ 0 ) ∈ (0, 1/2) and set D r (u) = D r (u; ϑ 0 ), r = 0, ±1, .... As u → ∞, -1) . Take j large enough so that ψ i > 0, j ≤ i < ∞. The same applies when ψ i < 0, j ≤ i < ∞. This is because D 0 (•) is slowly varying at infinity and thus does not change sign asymptotically. Then for any η > 0

∞ k=0 | ψ k | 1-η/2 | ψ k+u | 1-η/2 ∼| ψ u | 1-η/2 u k=0 ψ k + ∞ k=u+1 | ψ k | 2-η ∼ K | D 0 (u) | 2-η u (1-η/2)(2d
, | (ψ η 0j -ψ η 0j+1 )ψ 1-η 0j |≤| (ψ η 0j - ψ η 0j+1 )ψ 1-η 0j + ψ η 0j+1 (ψ 1-η 0j -ψ 1-η 0j+1 ) |=| ψ η 0j ψ 1-η 0j -ψ η 0j+1 ψ 1-η 0j+1 |=| ψ 0j -ψ 0j+1 |≤ Kψ 0j /j implying that ψ η
0j is quasi-monotonically convergent to zero, any η > 0. Therefore, by Yong (1974, Theorem III-23 and III-33 (ii)) the Fourier transform of the terms in { }-brackets on left hand side of ( 29) and ( 30 Fox and Taqqu (1987, Theorem 1) as long as 2(-2d + 1 + (1η/2)(2d -1)) < 1 implied by η < 1/(1 -2d). The same arguments carry through to show that B 2 = O(| ψ 0N | η T ). For the cumulant term B 3 , consider the last term on the left hand side of (28). When both s 1t

) is O(λ -(1+(1-η/2)(2d-1))-δ ), any η, δ > 0, as λ → 0. Therefore B 1 = O(| ψ 0N | η T ) by
1 > N + 1, s 2 -t 1 > N + 1 this bounded by K | ψ 0N | η ∞ k=0 | ψ k ψ k+|s 2 -s 1 | | 1-η/2 ∞ k=0 | ψ k ψ k+|t 2 -t 1 | | 1-η/2
by following the same arguments used for the corresponding term of (26). When either s 1t 1 > N + 1 or s 2t 1 > N + 1 then the bound is

K | ψ 0N | η/2 ∞ k=0 | ψ k ψ k+|s 2 -s 1 | | 1-η/2 ∞ k=0 | ψ k ψ k+|t 2 -t 1 | | 1-η/2 . Finally when both s 1 -t 1 ≤ N + 1, s 2 -t 1 ≤ N + 1 the bound is K | ψ 0N | ∞ k=0 | ψ k ψ k+|s 2 -s 1 | | ∞ k=0 | ψ k ψ k+|t 2 -t 1 | |.
A tighter bound applies to the other terms of (28) yielding B 3 = O(| ψ 0N | η/2 T ). Cases C(0, 1), D(0, 1), with d ≤ 0, and C(1, 1) follow along the same lines, with an even simpler proof.

The proof ends considering that P T (N )/ √ T is a quadratic form in Ndependent variates. This implies that it is φ-mixing with arbitrarily fast decreasing mixing coefficients and Ibragimov and Linnik (1971, Theorem 18.5.1) applies. A 4 (ϑ) = (π) -1 π -π f (λ)/f 3 (λ; ϑ)( ∂ ∂ϑ f (λ; ϑ))( ∂ ∂ϑ f (λ; ϑ))dλ. When d(ζ) ∈ [0, 1/2), much in the same way as in Zaffaroni (2003, Lemma 14), A 1T (ϑ) → a.s. A 1 (ϑ) = (2π) -1 π -π f -1 (λ; ϑ)( ∂ 2 ∂ϑ∂ϑ f (λ; ϑ))dλ, A 2T (ϑ) → a.s. A 2 (ϑ) = -(2π) -1 π -π f -2 (λ; ϑ)( ∂ ∂ϑ f (λ; ϑ))( ∂ ∂ϑ f (λ; ϑ))dλ, whereas when d(ζ) < 0 we adapt Hannan (1973, Lemma 1) idea, using the decomposition | 1/T T t=1 g(λ t )-1/(2π) π -π g(λ)dλ |≤| 1/T T t=1 g(λ t )-1/T T t=1 g T (λ t ) | + | 1/T T t=1 g T (λ t )-1/(2π) π -π g(λ)dλ | for a generic Lip-continuous function g(λ) with Cesaro sum g T (λ). It remains to show that the following equicontinuity property holds: sup θ: θ-ϑ < A iT ( θ) -A iT (ϑ) → 0 as → 0, and that A i (ϑ) are continuous, for i = 1, 2. These are implied by ∂A iT (ϑ) ∂ϑ + ∂A i (ϑ) ∂ϑ < ∞, for i = 1, 2, and the latter easily follow by the same arguments used to establish pointwise convergence, given the smoothness of the third-order derivatives of f (λ; ϑ) away from zero frequency, and making use of Lemma 1 regarding their local behaviour around zero frequency. T -1

t 1 =1
T -1

t 2 =1
N (λ t 1 ; ϑ) f (λ t 1 ; ϑ) N (λ t 2 ; ϑ) f (λ t 2 ; ϑ) Q(-λ t 1 , λ t 2 , -λ t 2 ; ϑ), (32)

V (ϑ) = 1 π π -π N (ω; ϑ)N (ω; ϑ)dω + 1 2π π -π π -π N (ω 1 ; ϑ) f (ω 1 ; ϑ)
N (ω 2 ; ϑ) f (ω 2 ; ϑ) Q(-ω 1 , ω 2 , -ω 2 ; ϑ)dω 1 dω 2

Proof of Theorem 1. In view of Lemmas 4-6, the result follows adapting the proof of Hannan (1973, Theorem 1). . Proof of Theorem 2. The result follow by the Delta method, in view of Lemmas 7 and 8, once we show that ). Expression (32) of V follows directly as a by-product from the proof of Lemma 7, for the part just involving the spectral density, whereas for the part involving the trispectrum one can relatively easily adapt Hosoya (1997, Lemma 3.2). A consistent estimate of asymptotic covariance is obtained by Lemma 8 and 9 by plugging θT into M T (ϑ) and V T (ϑ). To obtain the estimator of V , we also require a consistent estimate of the joint cumulants in the log z 2 t and t , listed in (33), which can be easily obtained since the cumulants, as any mixed moment of (z t , t ), are known functions of φ 0 . A consistent estimate is simply obtained by plugging φT into the expression for the theoretical cumulants.

  ) where d(ζ) < 1/2 is a known function of ζ, and a(z; ζ) and b(z; ζ) are polynomials in z of known degrees m and n respectively, whose coefficients are known functions of ζ, which have no zeros in common:

  )-(18) with d(ζ) ≡ 0. The differentiability and the rate of convergence of the ψ j (ζ) follow since the former are wellknown analytic functions of the roots of the polynomials a(z; ζ), b(z; ζ) (see Brockwell and Davis (1987, section 3.6)). Ass. C(1, l) and D(1, l) are then satisfied with e(ζ) equal to the maximum of the inverse modulus of the roots of b(z; ζ).

  report the results of an empirical application of FIEGARCH, based on the Standard & Poor's 500 composite stock index, yielding a point estimate of the long memory parameter d 0 = d(ζ 0 ) equal approximately to 0.633. The t statistic for testing the hypothesis d = 0.5 is 2.111. Both examples suggest that condition ∞ k=0 ψ 2

  r, r ≤ l where D r (x; ζ), D r (x; ζ), D r (x; ζ) are slowly varying functions. (ii) For all ϑ and λ = 0 U i 1 ,i 2 ,..,ir (λ; ζ) and V i 1 ,i 2 ,..,ir (λ; ζ) satisfy an approximate Lipschitz continuous conditions Lip(δ) with δ ≥ min[1, 1 -2d(ζ)] (see Zygmund (1977, Section 3)). Proof. (i) Consider case l = 0 and U (λ; ζ). Case l > 0 easily follows. When d(ζ) < 0 the ψ j (ζ) are summable implying that the c j (ϑ) are summable. Hence f (λ; ϑ) is continuous for all λ ∈ [-π, π) including λ = 0. The same applies when d(ζ) = 0 with summable | D 0 (j; ζ) | j -1 . We discuss case d(ζ) ∈ [(0, 1/2) with non summable | D 0 (j; ζ) | j -1 . It is easy to see that our assumptions imply

Lemma 8

 8 Under Assumptions A(2), B, C(k 2 , 3), D(k 2 , 3), F, for M T (ϑ) = ∂ 2 ∂ϑ∂ϑ Q T (ϑ), sup ϑ∈Θ M T (ϑ) -M (ϑ) → a.s. 0 as T → ∞.Proof. All the convergences below hold as T → ∞. We first establish pointwise convergence of M T (ϑ) to M (ϑ) a.s. for each ϑ ∈ Θ. LetM T (ϑ) = A 1T (ϑ) + A 2T (ϑ) + A 3T (ϑ) + A 4T (ϑλ t ) f 3 (λ t ; ϑ) .By Hannan (1973, Lemma 1), given Lemmas 3 and 4-(ii), uniformly in ϑ ∈ Θ,A 3T (ϑ) → a.s A 3 (ϑ) = -(2π) -1 π -π f (λ)/f 2 (λ; ϑ)( ∂ 2 ∂ϑ∂ϑ f (λ; ϑ))dλ, A 4T (ϑ) → a.s

Lemma 9 N

 9 Under Assumptions A(4), B, C(k 2 , 2), D(k 2 , 2), F, sup ϑ∈Θ V T (ϑ) -V (ϑ) → 0 as T → ∞, (λ t ; ϑ)N (λ t ; ϑ) + 2π T 2

  s relation the left hand side of (34) equalsT -1 u=-T +1 (1u/T )c u π -π g(λ)e iuλ dλ -∞ u=-∞ h u c u ,and the norm of the above expression is bounded byO ∞ u=T -1 h u c u + ∞ u=T -1 u T h u c u = O(| L(T ) | T -1

Acknowledgment: I thank two anonymous referees and the Associate Editor (Miguel A. Delgado) for their comments that led to a considerably improved version of the paper. I also thank Claudia Miani for remarkable research assistance.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

 Yong (1974, Lemma III-7, (i), (ii) and (iii)) where D 0 (•; ζ) = D 0 (u; ζ) u 1 D 0 (t; ζ) t -1 dt is slowly varying. Note that the latter always dominates D 2 0 (•; ζ) in all cases. By the same arguments (23) follows. Therefore, the proof is completed using Yong (1974, Lemma III-12) when d(ζ) ∈ (0, 1/2) and Yong (1974, Lemma III-22

The result follows by Robinson (1994, Lemma 8).

Lemma 2 Under Assumptions C(0, a), D(0, a) for any i m = 1, ..., q and j n = q + 1, ..., q

Proof. Straightforward.

Lemma 3 Under Assumptions C(k 2 , a), D(k 2 , a), for any i m = 1, ..., q, 0 ≤ m ≤ b ≤ a and j n = q + 1, ..., q + p, 0 ≤ n ≤ c ≤ a, f i 1 ,...,i b ,j 1 ,..,j c (λ; ϑ) is continuous for all ϑ ∈ Θ and λ = 0 ( mod. 2π).

Proof. When the c u (ϑ) are absolutely summable then continuity holds for any ϑ ∈ Θ and any λ ∈ (-π, π]. Summability holds for Robinson (1994, Lemma 8) applies, in the second case also using Lemma 1 together with Yong (1974, Lemma III-7, (i), (ii) and (iii)) and Yong (1974, Lemma III-22, (i)).

A C C E P T E D M A N U S C R I P T

ACCEPTED MANUSCRIPT

and where the trispectrum

(e iω 1 ψ(e iω 1 ;ζ)+e iω 2 ψ(e iω 2 ;ζ)+e iω 3 ψ(e iω 3 ;ζ)+e -i(ω 1 +ω 2 +ω 3 ) ψ(e -i(ω 1 +ω 2 +ω 3 ) ;ζ) )

Proof. We first establish (33) evaluated at ϑ 0 . The general case ϑ = ϑ 0 follows by substituting the ψ 0k with the

We need to evaluate the Fourier transform of each of the sixteen terms which make cum(y 0 , y u 1 , y u 2 , y u 3 ), listed in (27). The first term is trivial. Let us consider the four terms which involve κ logz 2 logz 2 logz 2 . These correspond to the four cases u 1 = u 2 = 0, u 1 = u 3 = 0, u 2 = u 3 = 0 and u 1 = u 2 = u 3 . For the first, neglecting constants, ∞ u 1 =u 2 =0, u 3 =-∞ ψ u 3 -1 e i(u 1 ω 1 +u 2 ω 2 +u 3 ω 3 ) = e iω 3 ∞ u 3 =1 ψ 0u 3 -1 e i(u 3 -1)ω 3 = e iω 3 ψ(e iω 3 ; ζ 0 ), and likewise for the second and third term. For the fourth ∞ u 1 =u 2 =u 3 =u, u=-∞ ψ -u-1 e i(u 1 ω 1 +u 2 ω 2 +u 3 ω 3 ) = -1 u=-∞ ψ 0-u-1 e iu(ω 1 +ω 2 +ω 3 ) = e -i(ω 1 +ω 2 +ω 3 ) ψ(e -i(ω 1 +ω 2 +ω 3 ) ; ζ 0 ). Evaluation of the other terms of (33) follows along the same lines and details are skipped for sake of simplicity. Pointwise convergence of V T (ϑ) to V (ϑ) easily follow by the same arguments used to establish convergence for terms A iT (ϑ), i = 1, 2 in Lemma 8, since the factorization of Q(-ω 1 , ω 2 , -ω 2 ; ϑ) permits to evaluate the double sum term of V T (ϑ) by looking at each sum separately. Finally, uniform convergence follows by equicontinuity of V T (ϑ) and continuity of V (ϑ) which in turn hold since by the same arguments ∂V T (ϑ) ∂ϑ + ∂V (ϑ) ∂ϑ < ∞.