Olivier Wintenberger

CONTINUOUS INVERTIBILITY AND STABLE QML ESTIMATION OF THE EGARCH(1,1) MODEL

Keywords: AMS 2000 subject classifications: Primary 62F12; Secondary 60H25, 62F10, 62M20, 62M10, 91B84 Invertible models, volatility models, quasi maximum likelihood, strong consistency, asymptotic normality, exponential GARCH, stochastic recurrence equation

Continuous invertibility and stable QML estimation of the EGARCH(1,1) model

Introduction

Since the seminal papers [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation[END_REF][START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF], the General Autoregressive Conditional Heteroskedasticity (GARCH) type models have been successfully applied to financial time series modeling. One of the stylized facts observed on the data is the asymmetry with respect to (wrt) shocks [START_REF] Cont | Empirical properties of asset returns: stylized facts and statistical issues[END_REF]: a negative past observation impacts the present volatility more importantly than a positive one. Nelson introduced in [START_REF] Nelson | Conditional Heteroskedasticity in Asset Returns : A New Approach[END_REF] the Exponential-GARCH (EGARCH) model that reproduces this asymmetric effect. Not surprisingly, theoretical investigations of EGARCH has attracted lot of attention since then, see for example [START_REF] He | Moment structure of a family of first-order exponential GARCH models[END_REF][START_REF] Mikosch | Stochastic volatility models with possible extremal clustering[END_REF]. However, the properties of the Quasi Maximum Likelihood Estimator (QMLE) used empirically in [START_REF] Nelson | Conditional Heteroskedasticity in Asset Returns : A New Approach[END_REF] was not proved except in some degenerate case, see [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]. We give in this paper some sufficient conditions for the strong consistency and the asymptotic normality of the QMLE in the EGARCH(1,1) model. Our approach of the strong consistency is based on the natural notion of continuous invertibility that we introduce in the general setting of volatility model solutions of a Stochastic Recurrent Equation (SRE).

Consider a real valued volatility model of the form X t = σ t Z t where σ t is the volatility and where the innovations Z t are normalized, centered independent identical distributed (iid) random vectors. It is assumed that a transformation of the volatility satisfies some parametric SRE (also called Iterated Random Function): there exist a function h and some ψ t measurable wrt Z t such that the following relations [START_REF] Alsmeyer | On the Harris recurrence and iterated random Lipschitz fonctions and related convergence rate results[END_REF] h(σ 2 t+1 ) = ψ t (h(σ 2 t ), θ 0 ), ∀t ∈ Z hold. Classical examples are the GARCH(1,1) and EGARCH(1,1) models:

GARCH(1,1):

σ 2 t+1 = α 0 + β 0 σ 2 t + γ 0 X 2 t , (2)
EGARCH [START_REF] Alsmeyer | On the Harris recurrence and iterated random Lipschitz fonctions and related convergence rate results[END_REF][START_REF] Alsmeyer | On the Harris recurrence and iterated random Lipschitz fonctions and related convergence rate results[END_REF]: log(σ 2 t+1 ) = α 0 + β 0 log(σ 2 t ) + (γ 0 Z t + δ 0 |Z t |). (3)

In practice, the innovations Z t are not observed. Writing Z t = X t /σ t in the expression of ψ t wrt Z t , we invert the model, i.e we consider a new SRE driven by a function φ t of the observation X t , [START_REF] Berkes | The efficiency of the estimators of the parameters in GARCH processes[END_REF] h(σ 2 t+1 ) = φ t (h(σ 2 t ), θ 0 ), t ∈ Z.

For instance, we obtain from (2) the inverted model φ t (x, θ) = α + βx + γX 2 t for the GARCH(1,1) model. For the EGARCH(1,1) model, we obtain from (3) the inverted model [START_REF] Berkes | GARCH processes: structure and estimation[END_REF] φ t (x, θ) = α + βx + (γX t + δ|X t |) exp(-x/2).

In accordance with the notions of invertibility given in [START_REF] Granger | On the invertibility on time series models[END_REF][START_REF] Tong | Non-Linear Time Series, A Dynamical System Approach[END_REF][START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF][START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF], we will say that the model is invertible if the SRE (4) is stable. Then, as the functions φ t are observed, the volatility is efficiently forecasted by using recursively the relation [START_REF] Billingsley | Convergence of Probability Measures[END_REF] h t+1 = φ t (h t , θ 0 ), t ≥ 0, from an arbitrary initial value h 0 . Sufficient conditions for the convergence of this SRE are the negativity of a Lyapunov coefficient and the existence of logarithmic moments, see [START_REF] Bougerol | Kalman filtering with random coefficients and contractions[END_REF]. So the GARCH(1,1) model is invertible as soon as 0 ≤ β 0 < 1. The invertibility of the EGARCH(1,1) model is more complicated to assert due to the exponential function in [START_REF] Berkes | GARCH processes: structure and estimation[END_REF]. The recursive relation on h t+1 can explode to -∞ for small negative values of h t and negative values of γ 0 X t + δ 0 |X t |. However, assuming that δ 0 ≥ |γ 0 |, the relation γ 0 X t + δ 0 |X t | > 0 holds and conditions for invertibility of the EGARCH(1,1), denoted hereafter INV(θ 0 ), are obtained in [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF][START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]:

INV(θ 0 ): δ 0 ≥ |γ 0 | and (7)
E[log(max{β 0 , 2 -1 (γ 0 X 0 + δ 0 |X 0 |) exp(-2 -1 α 0 /(1 -β 0 )) -β 0 })] < 0. On the oppposite, Sorokin introduces in [START_REF] Sorokin | Non-invertibility in some heteroscedastic models[END_REF] sufficient conditions on θ 0 for the EGARCH(1,1) model to be non-invertible. Then the SRE (5) is completely chaotic for any possible choice of the initial value h 0 and the volatility forecasting procedure based on the model ( 4) is not reliable.

In practice, the value θ 0 = (α 0 , β 0 , γ 0 , δ 0 ) of the Data Generating Process (DGP) EGARCH(1,1) is unknown. Nelson proposed in [START_REF] Nelson | Conditional Heteroskedasticity in Asset Returns : A New Approach[END_REF] to estimate θ 0 with the Quasi Maximum Likelihood Estimator (QMLE). Let us recall the definition of this classical estimator that estimates efficiently many GARCH models, see [START_REF] Berkes | GARCH processes: structure and estimation[END_REF] for the GARCH case and [START_REF] Francq | Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes[END_REF] for the ARMA-GARCH case. To construct the QMLE one approximates the volatility using the observed SRE (6) at any θ. Assume that the SREs driven by φ t (•, θ) are stable for any θ. Let us consider the functions ĝt (θ) defined for any θ as the recursive solutions of the SRE [START_REF] Bougerol | Kalman filtering with random coefficients and contractions[END_REF] ĝt+1 (θ) = φ t (ĝ t (θ), θ), t ≥ 0, for some arbitrary initial value ĝ0 (θ). Assume that h is a bijective function of inverse > 0. Then (ĝ t (θ 0 )) has a limiting law that coincides with the one of σ 2 t . The Quasi Likelihood (QL) criteria is defined as [START_REF] Bougerol | Stationarity of GARCH processes and of some nonnegative time series[END_REF] 2n Ln (θ) =

n t=1 lt (θ) = n t=1 X 2 t / (ĝ t (θ)) 2 + log( (ĝ t (θ)).
The associated M -estimator is the QMLE θn defined by optimizing the QL on some compact set Θ θn = argmin θ∈Θ Ln (θ).

This estimator has been used since the seminal paper of Nelson [START_REF] Nelson | Conditional Heteroskedasticity in Asset Returns : A New Approach[END_REF] for estimating the EGARCH(1,1) model without any theoretical justification, see for example [START_REF] Brandt | Volatility forecasting with range-based EGARCH models[END_REF]. The inverted EGARCH(1,1) model is driven by the SRE (8) that expresses as (denoting

(ĝ t ) = σ2 t ) (10) log(σ 2 t+1 (θ)) = α + β log(σ 2 t (θ)) + (γX t + δ|X t |) exp(-log(σ 2 t (θ))/2).
The consistency and the asymptotic normality are not proved except in the degenerate case β = 0 for all θ ∈ Θ in [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF]. The problem of the procedure (and of any volatility forecast) is that the inverted EGARCH(1,1) model ( 10) is stable only for some values of θ. Thus, contrary to other GARCH models, the QML estimation procedure is not always reliable for the EGARCH model, see the discussion in [START_REF] Harvey | Beta-t-(E)GARCH[END_REF]. Thus, other estimation procedure has been investigated such as the bayesian, bias correction and the Whittle procedure in [START_REF] Vrontos | Full Bayesian Inference for GARCH and EGARCH Models[END_REF][START_REF] Demos | Bias correction of ML and QML estimators in the EGARCH(1,1) model[END_REF][START_REF] Zaffaroni | Whittle estimation of EGARCH and other exponential volatility models[END_REF] respectively. Another approach is to introduce models that behave like the EGARCH(1,1) model but where the QMLE could be more reliable, see [START_REF] Harvey | Beta-t-(E)GARCH[END_REF][START_REF] Sucarrat | The power log-GARCH model[END_REF][START_REF] Francq | GARCH models without positivity constraints: Exponential or Log GARCH?[END_REF].

We prove the strong consistency of the QMLE for the general model (1) when the maximization procedure is done on a continuously invertible domain. We give sufficient conditions called the continuous invertibility of the model such that the QMLE is strongly consistent. More precisely we assume that the SRE (8) produces continuous functions ĝt of θ on Θ. The continuous invertibility holds when the limiting law of ĝt corresponds to the law of some continuous function g t on Θ that does not depend on the initial function ĝ0 . The continuous invertibility ensures the stability of the estimation procedure regardless the initial function ĝ0 chosen arbitrarily in practice. Under few other assumptions, we prove that the QMLE is strongly consistent for continuously invertible models on the compact set Θ. The continuous invertiblity should be checked systematically on models before using QMLE. One example of such continuously invertible models with properties similar than the EGARCH model is the Log-GARCH model studied in [START_REF] Francq | GARCH models without positivity constraints: Exponential or Log GARCH?[END_REF].

As the continuous invertibility is an abstract assumption, we provide sufficient conditions for continuous invertibility collected in the assumption (CI) below. These conditions ensure the invertibility of the model at any point θ of the compact set Θ and some regularity of the model with respect to the parameter θ. As the inverted EGARCH(1,1) model ( 10) is a regular function of θ, it satisfies (CI) on any Θ such that the invertibility condition INV(θ) is satisfied for any θ ∈ Θ. Thus we prove the strong consistency of the QMLE for the invertible EGARCH(1,1) model when INV(θ) is satisfied for any θ ∈ Θ. It is a serious advantage of our approach based on the continuous invertibility condition (CI) compared with the approach of [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF]. Based on uniform Lipschitz coefficients this last approach is more restrictive than our when applied to the EGARCH(1,1) model. Moreover, we also prove the strong consistency of the natural volatility forecasting σ2 n = (ĝ n ( θn )) of σ 2 n+1 under (CI). Continuous invertibility seems to be well suited to assert volatility forecasting because σ2 n expresses as functions ĝn evaluated at points θn = θ 0 . To infer in practice the EGARCH(1,1) model, we propose to stabilize the QMLE. We constrain the QMLE on some compact set satisfying the empirical version of the condition INV(θ 0 ). This new estimator θS n called Stable QMLE (SQMLE) produces only reliable volatility forecasting such that σ2 n = (ĝ n ( θS n )) does not depend asymptotically of the initial value σ2 0 = (ĝ 0 ). It is not the case of the classical QMLE the continuous invertibility condition IN V (θ) is not observed in practice. Thus INV(θ) might not be satisfied for any θ in the compact set Θ of the maximization procedure. And the whole procedure might have some chaotic behavior with respect to any initial value σ2 0 used in the inverted model.

The asymptotic normality of the SQMLE in the EGARCH(1,1) model is proved under the additional assumption (MM). The moment conditions in (MM) are sufficient for the existence of the asymptotic covariance matrix. No uniform moment condition on the score vector is assumed. The proof is based on functional SREs as [START_REF] Elton | A multiplicative ergodic theorem for Lipschitz maps[END_REF] and their perturbations. As for the strong consistency, we need to refine arguments from [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]. As we do not apply any uniform Strong Law of Large Number, we develop in the proof new arguments based on SREs satisfied by differences such as | Ln ( θn )-Ln (θ 0 )|.

We consider conditions of moments (MM) only at the point θ 0 where the expression of the score vector simplifies. In the EGARCH(1,1) model, the conditions (MM) take the simple form

E[Z 4 0 ] < ∞ and E[(β 0 -2 -1 (γ 0 Z 0 + δ 0 |Z 0 |) 2
] < 1 and can be checked in practice by estimating the innovations. We believe that this new approach gives sharp conditions for asymptotic normality for other models.

The paper is organized as follows. In Section 2, we discuss the standard notions of invertibility and introduce the continuous invertibility and its sufficient condition (CI). We prove the strong consistency of the QMLE for general continuously invertible models in Section 4.1. The consistency of the volatility forecasting is also proved under the sufficient condition (CI). We apply this results in the EGARCH (

] = 0 and E[Z 2 0 ] = 1. Consider the general DGP X t = σ t Z t satisfying h(σ 2 t+1 ) = ψ t (h(σ 2 t ), θ 0 ) for all t ∈ Z.
The function h is a bijection from some subset R + to some subset of R of inverse called the link function. A first question regarding such general SRE is the existence of the model, i.e. wether or not a stationary solution exists. Hereafter, we work under the general assumption (ST): The SRE (1) admits a unique stationary solution denoted (σ 2 t ) that is non anticipative, i.e. σ 2 t is independent of (Z t , Z t+1 , Z t+2 , . . .) for all t ∈ Z, and has finite log-moments: E log + σ 2 0 < ∞. The GARCH(1,1) model (2) satisfies the condition (ST) if and only if (iff) E[log(β 0 + γ 0 Z 2 0 )] < 0, see [START_REF] Nelson | Stationarity and persistence in the GARCH(1,1) model[END_REF] for the existence of the stationary solution and [START_REF] Berkes | GARCH processes: structure and estimation[END_REF] for the existence of log moments. The EGARCH(1,1) model (3) satisfies the condition (ST) iff |β 0 | < 1, see [START_REF] Nelson | Conditional Heteroskedasticity in Asset Returns : A New Approach[END_REF]. In this case, the model has nice ergodic properties: any process recursively defined by the SRE from an arbitrary initial value approximates exponentially fast a.s. the original process (σ 2 t ). In the sequel, we say that the sequence of non negative r.v. (W t ) converges exponentially almost surely to 0, W t e.a.s.

---→ 0 as t → ∞, if W t = o(e -Ct ) a.s. for some r.v. C > 0. We will also use the notation x + for the positive part of x, i.e. x + = x ∨ 0 for any x ∈ R.

2.2.

Invertible models. Under (ST) the process (X t ) is stationary, non anticipative and thus ergodic as a Bernoulli shift of an ergodic sequence (Z t ), see [START_REF] Krengel | Ergodic Theorems[END_REF]. Let us now investigate the question of invertibility of the general model [START_REF] Alsmeyer | On the Harris recurrence and iterated random Lipschitz fonctions and related convergence rate results[END_REF]. The classical notions of invertibility are related with convergences of SRE and thus are implied by Lyapunov conditions of Theorem 3.1 in [START_REF] Bougerol | Kalman filtering with random coefficients and contractions[END_REF]. Following [START_REF] Tong | Non-Linear Time Series, A Dynamical System Approach[END_REF], we say that a volatility model is invertible if the volatility can be expressed as a function of the past observed values: Definition 1. Under (ST), the model is invertible if the sequence of the volatilities (σ 2 t ) is adapted to the filtration generated by

(X t-1 , X t-2 , • • • ).
Using the relation Z t = X t / (h(σ t )) in the expression of ψ t yields the new SRE (4): h(σ 2 t+1 ) = φ t (h(σ 2 t ), θ 0 ). Now the random functions φ t (•, θ 0 ) depends only on X t . As (X t ) is an ergodic and stationary process, it is also the case of the sequence of parametrized maps (φ t (•, θ 0 )). Using Theorem 3.1 in [START_REF] Bougerol | Kalman filtering with random coefficients and contractions[END_REF], the invertibility of the model follows if the φ t (•, θ 0 ) are Lipschitz maps such that there exists r > 0 satisfying [START_REF] Cont | Empirical properties of asset returns: stylized facts and statistical issues[END_REF] inv

(θ 0 )E[log + |φ 0 (x, θ 0 )|] < ∞ for some x ∈ E, E[log + Λ(φ 0 (•, θ 0 ))] < ∞ and E[log Λ(φ 0 (•, θ 0 ) (r) )] < 0.
Here Λ(f ) denotes the Lipschitz coefficient of any function f defined by the relation (in the case where f is real valued)

Λ(f ) = sup x =y |f (x) -f (y)| |x -y| f (r) t denotes the iterate f t o f t-1 o • • • o f t-r
for any sequence of function (f t ). The conditions (11) are called the conditions of invertibility in [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF] and is proved there that Proposition 1. Under (ST) and [START_REF] Cont | Empirical properties of asset returns: stylized facts and statistical issues[END_REF], the general model (4) is invertible.

The GARCH(1,1) model ( 2) is invertible as soon as 0 ≤ β 0 < 1.

The invertibility of the EGARCH(1,1) model is more difficult to assert due to the exponential function in the SRE [START_REF] Berkes | GARCH processes: structure and estimation[END_REF]. Let us describe the sufficient condition of invertibility INV(θ 0 ) of the EGARCH(1,1) model given in [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF][START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]. It expresses as a Lyapunov condition (7) on the coefficients θ 0 = (α 0 , β 0 , γ 0 , δ 0 ). This condition does not depend on α 0 when the DGP (X t ) is itself the stationary solution of the EGARCH(1,1) model for θ 0 . Indeed, (log

σ 2 t ) admits a MA(∞) representation log σ 2 t = α 0 (1 -β 0 ) -1 + ∞ k=1 β k-1 0 (γ 0 Z t-k + δ 0 |Z t-k |).
Plugging in this MA(∞) representation into [START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF], we obtain the equivalent sufficient condition

(12) E log max β 0 , 2 -1 exp 2 -1 ∞ k=0 β k 0 (γ 0 Z -k-1 + δ 0 |Z -k-1 |) (γ 0 Z 0 + δ 0 |Z 0 |) -β 0 < 0.
Using the Monte Carlo algorithm and assuming that Z 0 is N (0, 1)-distributed, we report in Figure 1 the largest values of β 0 that satisfies the condition (7) on a grid of values of (γ 0 , δ 0 ). The constraint on β 0 is always stronger than the stationary constraint |β 0 | < 1. It exists stationary EGARCH(1,1) models that are not invertible, i.e. the inverted model log σ2

t+1 = α 0 + β 0 log σ 2 t + (γ 0 X t + δ 0 |X t |) exp(-log σ 2 t+1 /2), t ≥ 0,
is not stable wrt any possible choice of initial value log σ2 0 . On the opposite, Sorokin exhibits in [START_REF] Sorokin | Non-invertibility in some heteroscedastic models[END_REF] sufficient conditions for some chaotic behaviour of the inverted EGARCH(1,1) model under

|β 0 | < 1.
To emphasize the danger to work with non invertible EGARCH(1,1) models, we report in Figure 2 the convergence criterion N t=1 (log σ 2 t -log σ2 t ) wrt arbitrary initial values for two different values of θ 0 (N = 10 000). The first picture represents a stable case where INV(θ 0 ) holds. The second picture represents a chaotic case where the condition of non invertibility given in [START_REF] Sorokin | Non-invertibility in some heteroscedastic models[END_REF] is satisfied. The stationary constraint |β 0 | < 1 is satisfied in both cases. It is interesting to note that the convergence criterion does not explode in the chaotic case. It is an important difference between the non invertible GARCH(1,1) and EGARCH(1,1) models: the non invertible GARCH(1,1) is always explosive because it is also non stationary. The inference by QMLE remains stable due to this very specific behaviour, see [START_REF] Jensen | Asymptotic Inference for Nonstationary GARCH[END_REF]. On the opposite, we have driven numerical experiments and we are convinced that the QMLE procedure is not stable when the EGARCH(1,1) model is non invertible. It is not surprising as it is not possible to recover the volatility process from the inverted model, even when the parameter θ 0 of the DGP is known. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -100 -50 0 50 100 5e-17 6e-17 7e-17 8e-17 9e-17 1e-16

Stability of the SRE (Good Parameter)

Initial values of the SRE MAD between real log-volatility and the SREs q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -100 -50 0 50 100 8.5 4) for an unknown value θ 0 . Consider the inference of the QMLE θn defined as θn = argmin θ∈Θ Ln (θ) for some compact set Θ. Here Ln is the QL criteria defined in ( 9) and based on the approximation ĝt (θ) of the volatility: ĝt+1 (θ) = φ t (ĝ t (θ), θ), ∀t ≥ 0, ∀θ ∈ Θ starting at an arbitrary initial value ĝ0 (θ), for any θ ∈ Θ. The invertibility of the model is not sufficient to assert the consistency of the inference: as the parameter θ 0 is unknown, the conditions of invertibility [START_REF] Cont | Empirical properties of asset returns: stylized facts and statistical issues[END_REF] do not provide the stability of the approximation ĝt (θ) wrt the initial value ĝ0 (θ) when θ = θ 0 . satisfied for (φ t (•, θ)) for any θ ∈ Θ then the unique stationary solution (g t ) exists and is defined by the relation [START_REF] Elton | A multiplicative ergodic theorem for Lipschitz maps[END_REF] g

t+1 (θ) = φ t (g t (θ), θ), ∀t ∈ Z, ∀θ ∈ Θ.
Assume there exists some subset K of R such that the random functions (x, θ) → φ t (x, θ) restricted on K × Θ take values in K for. Consider that the initial values θ → ĝ0 (θ) constitutes a continuous function on Θ that takes its values in K. Denote • Θ the uniform norm and C Θ the space of continuous functions from Θ to K. By a recursive argument, it is obvious that the random functions ĝt belong to C Θ for all t ≥ 0.

We are now ready to define the notion of continuous invertibility:

Definition 2. The model is continuously invertible on Θ iff ĝt (θ) -g t (θ) Θ → 0 a.s. when t → ∞.
This continuous invertibility notion is crucial for the strong consistency of the QMLE.

3.2.

Strong consistency of the QMLE constrained to a continuously invertible domain.

Let us assume the regularity and boundedness from below on the link function :

(LB): The maps x → 1/ (x) and log( (x)) are Lipschitz functions on K and there exists m > 0 such that (x) ≥ m for all x ∈ K. Classically, we also assume the identifiability of the model (ID): (g 0 (θ)) = σ 2 0 a.s. for some θ ∈ Θ iff θ = θ 0 . The consistency of the QMLE follows from the continuous invertibility notion: Theorem 1. Assume that (ST), (LB) and (ID) for a volatility model that is continuously invertible on the compact set Θ. Then the QMLE on Θ is strongly consistent, θn → θ 0 a.s., when θ 0 ∈ Θ.

Proof. First, note that from the continuous invertibility of the model on Θ, the SRE (13) admits a stationary solution in E = C Θ that is a separable complete metric space (equipped with the metric d(x, y) = x -y Θ ). Let us denote Φ t the mapping acting on C Θ and satisfying

g t+1 = Φ t (g t ) iff g t+1 (θ) = φ t (g t (θ), θ), ∀θ ∈ Θ.
We will apply the principle of Letac [START_REF] Letac | A contraction principle for certain Markov chains and its applications. random matrices and their applications[END_REF] extended to the stationary sequences as in [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]: the existence of a unique stationary non anticipative solution follows from the convergence a.s. of the backward equation for any initial value g ∈ C Θ :

Z t (g) := Φ 0 o Φ -1 o • • • o Φ -t (g) → t→∞ Z
where Z does not depend on g. By stationarity, Z t (g) is distributed as ĝt when ĝ0 = g. Thus, we have P( lim t→∞ Z t (g) -g 0 Θ = 0) = P( lim t→∞ ĝt -g t Θ = 0) where g 0 (θ) is well defined for each θ ∈ Θ as the solution of the SRE [START_REF] Elton | A multiplicative ergodic theorem for Lipschitz maps[END_REF]. Thus, under continuous invertibility, an application of Letac's principle leads to the existence of a unique stationary solution distributed denoted also (g t ) that coincides with g t (θ) at any point θ ∈ Θ. In particular, g t belongs to C Θ and

2nL n (θ) = n t=1 l t (θ) = n t=1 X 2 t / (g t (θ)) 2 + log( (g t (θ)).
is a continuous function on Θ.

Let us turn to the proof of the strong consistency based on standard arguments, see for example the book of Francq and Zakoïan [START_REF] Francq | GARCH Models -Structure, Estimation and Finance Applications[END_REF]. As the model satisfies the identifiability condition (ID), the strong consistency follows the intermediate results

(a) lim n→∞ sup θ∈Θ |L n (θ) -Ln (θ)| = 0 a.s. (b) E|l 0 (θ 0 )| < ∞ and if θ = θ 0 then E[l 0 (θ)] > E[l 0 (θ 0 )]. (c) Any θ = θ 0 has a neighborhood V (θ) such that lim inf n→∞ inf θ * ∈V (θ) Ln (θ * ) > El t (θ 0 ) a.s.
Let us prove that (a) is satisfied when the model is continuously invertible and the condition (LB) holds. As 1/ and log( ) are Lipschitz continuous functions there exists some constant C > 0 such that [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation[END_REF] |L n (θ) -

Ln (θ)| ≤ C 1 n n t=1 |ĝ t (θ) -g t (θ)|.
The desired convergence to 0 of the upper bound that is a Cesaro mean follows from the definition of the continuous invertibility.

The first assertion of (b) follows from the identity l 0 (θ 0 ) = log(σ 2 0 ) + Z 2 0 . Thus, as σ 0 = (g 0 (θ 0 )) ≥ m from (LB) and E log + σ 2 0 < ∞ under (ST), we have that log(σ 2 0 ) is integrable. Moreover, EZ 2 0 = 1 by assumption and the assertion E|l 0 (θ 0 )| < ∞ is proved. To prove the second assertion, note that θ → E[l 0 (θ)] has a unique minimum iff E[σ 2 0 / (g 0 (θ)) -log(σ 2 0 / (g 0 (θ))] has a unique minimum. Under the identifiability condition, as x -log(x) ≥ 1 for all x > 0 with equality iff x = 1, we deduce that for any θ = θ 0 we have

E[l 0 (θ)] > E[l 0 (θ 0 )].
Finally, let us prove (c) under continuous invertibility, (LB) and (ID). First, under continuous invertibility, we have lim inf

θ * →θ Ln (θ * ) = lim inf n→∞ inf θ * ∈V (θ) L n (θ * ) + lim inf n→∞ inf θ * ∈V (θ) ( Ln (θ * ) -L n (θ * )) = lim inf n→∞ inf θ * ∈V (θ)
L n (θ * ) by using the convergence to 0 of the Cesaro mean [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation[END_REF]. Second, by ergodicity of the stationary solution (g t ) we have the ergodicity of the sequence (inf θ * ∈V (θ) l t (θ * )). Moreover, under the condition (LB), we have that inf θ * ∈V (θ) l t (θ * ) ≥ log(m) > -∞ a.s. Then, for any K > 0, the sequence inf θ * ∈V (θ) l t (θ * ) ∧ K is integrable. We use the classical SLLN and obtain

lim n→∞ inf θ * ∈V (θ) L n (θ * ) ∧ K = E inf θ * ∈V (θ) l 0 (θ * ) ∧ K a.s. Letting K → -∞ we obtain that lim n→∞ inf θ * ∈V (θ) L n (θ * ) = E inf θ * ∈V (θ) l 0 (θ * ) ∈ R ∪ {+∞}.
Finally, remark that by continuity of θ → g t (θ) the function l 0 is continuous. Thus, for any ε > 0 we can find a neighborhood V (θ) such that

E inf θ * ∈V (θ) l 0 (θ * ) ≥ E[l 0 (θ)] + ε.
From the second assertion of (b) we choose ε > 0 such that E[l 0 (θ)] + ε > E[l 0 (θ 0 )] and (c) is proved.

The end of the proof of the strong consistency is based on a classical compact argument and thus is omitted.

3.3. Sufficient conditions for continuous invertibility. The definition of continuous invertibility does not give any explicit condition. In order to apply Theorem 3.1 in [START_REF] Bougerol | Kalman filtering with random coefficients and contractions[END_REF], one classically assumes some uniform Lipschitz conditions on the SRE [START_REF] Elton | A multiplicative ergodic theorem for Lipschitz maps[END_REF]. Hereafter we will use a continuity argument instead to obtain a tractable sufficient condition of continuous invertibility for general models.

Consider some generic function f : K × Θ → K. Assume that there exists a continuous function Λ f on Θ such that for each x, y ∈ E we have

|f (x, θ) -f (y, θ)| ≤ Λ f (θ)|x -y|.
The approach followed by Straumann and Mikosch in [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF] is to consider the SRE (8) in the complete metric space of continuous functions C Θ on Θ with values in K equipped with d(x, y) = x -y Θ . Straightforward conditions for continuous invertibility are the following ones

(15) E log + ( φ 0 (y, •) Θ ) < ∞ for some y ∈ K, E log + ( Λ φ0 Θ ) < ∞ and E log Λ φ (r) 0 Θ < 0.
The EGARCH(1,1) model satisfies this condition under restrictive assumptions on Θ, for example when β = 0 for any θ ∈ Θ, see [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF].

We collect more general sufficient conditions for continuous invertibility in the assumption (CI)

(CI): E log + ( φ 0 (y, •) Θ ) < ∞ for some y ∈ K, E log + ( Λ φ0 Θ ) < ∞ and E log Λ φ (r) 0 (θ) < 0 for any θ ∈ Θ.
The difference with conditions [START_REF] Francq | GARCH models without positivity constraints: Exponential or Log GARCH?[END_REF] is that the Lyapunov condition holds pointwisely and not necessarily uniformly on Θ. Of course (CI) is weaker than the uniform Lyapunov condition [START_REF] Francq | GARCH models without positivity constraints: Exponential or Log GARCH?[END_REF]. Due to the regularity of classical models wrt to θ, (CI) is satisfied as soon as the model is invertible on Θ (see Section 4.1 for the EGARCH(1,1) case). Next Theorem proves that the continuity argument is sufficient to assert continuous invertibility:

φ0 (θ)∨ K|] < ∞ because E[sup Θ log + Λ φ (r) 0 (θ)] < ∞ under (CI). Applying the dominated convergence theorem we obtain lim ρ→0 E[log Λ (r) * (θ, ρ) ∨ K] = E[lim ρ→0 log Λ (r) * (θ, ρ) ∨ K]. By continuity we have lim ρ→0 log Λ (r) * (θ, ρ) = log Λ φ (r) 0 (θ) and for sufficiently small K < 0 lim ρ→0 E[log Λ (r) * (θ, ρ) ∨ K] = E[log Λ φ (r) 0 (θ) ∨ K] < 0.
Thus, there exists an > 0 such that

E[log Λ (r) * (θ, )] ≤ E[log Λ (r) * (θ, ) ∨ K] < 0.
Denote V(θ) the compact neighborhood B(θ, ) ∩ Θ of θ. Let us now work on C V(θ) , the complete metric space of continuous functions from V(θ) to K equipped with the supremum norm • V(θ) .

In this setting (ĝ t ) satisfies a functional SRE ĝt+1 = Φ t (ĝ t ) with Lipschitz coefficients satisfying

Λ(Φ (r) t ) ≤ sup s1,s2∈C(V(θ)) Φ (r) t (s 1 ) -Φ (r) t (s 2 ) V(θ) s 1 -s 2 V(θ) ≤ sup s1,s2∈C(V(θ)) sup θ * ∈V(θ) φ (r) t (s 1 (θ * ), θ * ) -φ (r) t (s 2 (θ * ), θ * ) s 1 -s 2 V(θ) ≤ sup s1,s2∈C(V(θ)) sup θ * ∈V(θ) Λ(φ (r) t (•, θ * )) s 1 (θ * ) -s 2 (θ * ) s 1 -s 2 V(θ) ≤ sup s1,s2∈C(V(θ)) sup θ * ∈V(θ) Λ(φ (r) t (•, θ * )) s 1 -s 2 V(θ) s 1 -s 2 V(θ) ≤ Λ (r) * (θ, ).
As

E[log + ( Φ 0 (y) V(θ) )] ≤ E[log + ( φ 0 (y, θ) Θ )] < ∞, E[log + (Λ(Φ 0 ))] ≤ E[log + ( Λ φ0 (θ) Θ )] < ∞ under (CI) and E(log Λ(Φ (r) 0
)) < 0 we can apply Theorem 3.1 of [START_REF] Bougerol | Kalman filtering with random coefficients and contractions[END_REF]. The unique stationary solution (g t ) exists and satisfies [START_REF] Francq | Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes[END_REF] ĝt

-g t V(θ)
e.a.s.

---→ 0.

Now, let us remark that Θ = ∪ θ∈Θ V(θ). As Θ is a compact set, there exists a finite number N such that Θ = ∪ N k=1 V(θ k ) and we obtain ĝt -g t V(θ k ) e.a.s.

---→ 0 ∀1 ≤ k ≤ N.

The desired result follows from the indentity --→ 0 as n → ∞.

• Θ = ∨ N k=1 • V(θ k ) .
Proof. Theorems 1 and 2 assert the strong consistency of θn under the assumption of Theorem 3. By continuity of and continuous invertibility on Θ, | (ĝ t ( θn ))-(g t ( θn ))| a.s.

--→ 0. Thus, using again the continuity of , the result is proved if |g t ( θn ) -g t (θ 0 )| a.s.

--→ 0. Keeping the notation used in the proofs of Theorems 1 and 2, we have θn ∈ V(θ 0 ) for n sufficiently large. The Lyapunov condition

E[log Λ(Φ (r) 0 )] < 0 is satisfied for the functional SRE (Φ t ) restricted on V(θ 0 ). For any θ ∈ V(θ 0 ), t ∈ Z we have |g (t+1)r (θ) -g (t+1)r (θ 0 )| ≤ Λ(Φ (r) t )|g t (θ) -g t (θ 0 )| + w t (θ) where w t (θ) = |φ t (g t (θ 0 ), θ) -φ t (g t (θ 0 ), θ 0 )|. Thus |g t (θ) -g t (θ 0 )| is bounded by a linear SRE.
Applying the Borel-Cantelli Lemma as in [START_REF] Berkes | GARCH processes: structure and estimation[END_REF][START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF], the convergence of the series

g tr -g tr (θ 0 ) V(θ0) ≤ ∞ i=0 Λ(Φ (r) tr ) • • • Λ(Φ (r) (t-i+1)r ) w (t-i)r V(θ0) < ∞ follows from the fact that Λ(Φ (r) tr ) • • • Λ(Φ (r) (t-i+1)r )
e.a.s.

---→ 0 when i → ∞ and E log + w t-i V(θ0) < ∞ Thus the difference |g tr (θ) -g tr (θ 0 )| is bounding by an a.s. normally convergent function on C V(θ0) that we denote a(θ). Moreover, as lim θ→θ0 w t (θ) = 0 for any t ∈ Z a.s., we also have lim θ→θ0 a t (θ) = 0 a.s. from the normal convergence. Finally, using the strong consistency of θn , we obtain that [START_REF] Francq | GARCH Models -Structure, Estimation and Finance Applications[END_REF] |g t ( θn ) -g t (θ 0 )| ≤ a( θn ) a.s.

--→ 0 ∀n ≥ t when t → ∞.

= α 0 + β 0 log σ 2 t-1 + γ 0 Z t + δ 0 |Z t |.
The volatility process (σ 2 t ) exists, is stationary and ergodic as soon as |β 0 | < 1 with no other constraint on the coefficients. and (ST) holds. However, it does not necessarily have finite moment of any order. The model is identifiable as soon as the distribution of Z 0 is not concentrated on two points, see [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]. Let us assume in the rest of the paper these classical conditions satisfied such that assumptions (ST), (LB) and (ID) automatically hold.

The continuous invertibility of the stationary solution of the EGARCH(1, 1) model does not hold on any compact set Θ. Recall that the inverted model is driven by the SRE [START_REF] Berkes | GARCH processes: structure and estimation[END_REF]. Under the constraint δ ≥ |γ|, we have that γX t + δ|X t | ≥ 0 a.s.. By a straightforward monotonicity argument, we can consider the restriction of the SRE (5) on the intervall K × Θ where K = [α/(1 -β), ∞) and δ ≥ |γ| for any θ ∈ Θ. By regularity of φ t wrt x, the Lipschitz coefficients are computed using the first partial derivative:

Λ(φ t (•, θ)) ≤ max{β, 2 -1 (γX t + δ|X t |) exp(-2 -1 α/(1 -β)) -β}.
This Lipschitz coefficient is continuous in θ and thus coincides with Λ φt (θ). The assumption (CI) is satisfied if INV(θ) is satisfied for any θ ∈ Θ as the uniform log moments exists by continuity and because E[log + X 0 ] < ∞. As (ST) is also satisfied, an application of Theorem 2 asserts the continuous invertibility on any compact sets Θ such that INV(θ) is satisfied for any θ ∈ Θ.

Remark that the domain of invertibility represented in Figure 1 does not coincide with the domain of continuous invertibility, i.e. the set of θ ∈ R 4 satisfying INV(θ) for an EGARCH(1,1) DGP with θ 0 ∈ R 4 . This situation is more complicated to represent as the domain INV(θ) depends on the fixed value θ 0 and on the parameter α when θ = θ 0 . However, for any θ 0 in the invertibility domain represented in Figure 1, there exists some compact neighborhood Θ of continuous invertibility. For the QMLE constrained to such compact set Θ, an application of Theorems 1, 2 and 3 gives directly Theorem 4. Consider the EGARCH(1,1) model. If INV(θ) is satisfied for any θ ∈ Θ and θ 0 ∈ Θ then θn → θ 0 and σ2 n -σ 2 n → 0 a.s. as n → ∞ with σ2 n = exp(ĝ n ( θn )) for any initial value σ2 0 . Remark we extend the result of [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF][START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF] under the sufficient condition [START_REF] Francq | GARCH models without positivity constraints: Exponential or Log GARCH?[END_REF] that expresses in the EGARCH(1,1) model as

E[sup Θ log(max{β, 2 -1 (γX t-1 + δ|X t-1 |) exp(-2 -1 α/(1 -β)) -β})] < 0.
This more restrictive condition is less explicit than [START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF] because it is a Lyapunov condition uniform on Θ. It is difficult to check in practice when β = 0 for some θ ∈ Θ.

The Stable QMLE (SQMLE).

Noe that the procedure is valid only if the invertibility condition INV(θ) is satisfies for any θ ∈ Θ. Thus, we want to constrain the optimization of the QL on a continuously invertible domain. However, the condition (7) depends on the distribution of X 0 and on the unknown parameter θ 0 that drives the DGP. We propose to constrain the optimization of the QL under the empirical constraint

INV(θ): δ ≥ |γ| and n t=1 log(max{β, 2 -1 (γX t + δ|X t |) exp(-2 -1 α/(1 -β)) -β}) ≤ -ε.
We introduce artificially ε > 0 (as small as we want) such that INV(θ) defines a closed set on R 4 . This set is non empty because the constraint is satisfied for θ ∈ R 4 such that β = 0 and the parameters δ ≥ |γ| and α are sufficiently small. Definition 3. The SQMLE is the M -estimator

θS n = argmin θ∈Θ S n t=1 2 -1 X 2 t exp(-ĝ t (θ)/2) + ĝt (θ)
where Θ S = {θ ∈ Θ satisfying INV(θ)} for any compact set Θ.

Consider in the sequel that Θ S is non empty. It is always the case in practice where we use a steepest descent algorithm on the constraints INV(θ) and some maximum number of iterations. The following theorem gives the strong consistency of the SQMLE for the EGARCH(1,1) model if INV(θ 0 ) is satisfied and θ 0 ∈ Θ. It also shows that the volatility forecasting using the SQMLE does not depend on the arbitrary choice of the initial value even when INV(θ 0 ) is not satisfied. Thus the SQMLE is more reliable than the QMLE for which chaotic behavior of the volatility forecasting described in [START_REF] Sorokin | Non-invertibility in some heteroscedastic models[END_REF] can occur if INV(θ) is not satisfied for some θ ∈ Θ. --→ θ 0 . If INV(θ 0 ) is not satisfied, the asymptotic law of σ2 t = exp(ĝ t ( θS n )) still does not depend on the initial value σ2

0 . Proof. Denote Λ n (θ) = max{β, 2 -1 (γX n + δ|X n |) exp(-2 -1 α/(1 -β)) -β}. We will prove that n -1 log Λ n (θ) -E log Λ 0 (θ) Θ a.s.
--→ 0 as n → ∞. Then, if INV(θ 0 ) is satisfied and θ 0 ∈ Θ, Θ S coincides asymptotically a.s. to a compact (because bounded and close) continuously invertible domain containing θ 0 . An application of Theorem 4 yields the strong consistency of the SQMLE. The second assertion follows from the fact that asymptotically INV( θS

n ) is satisfied even if θ 0 / ∈ Θ S .
Let us prove that n -1 log Λ n (θ) -E log Λ 0 (θ) Θ a.s.

--→ 0. Note that log Λ n (θ) is a random element in the Banach space C(Θ). The desired result is a consequence of the ergodic theorem if

E log Λ 0 Θ < ∞. Since log Λ 0 Θ ≤ max{ log |β| Θ , log |2 -1 (γX 0 + δ|X 0 |) exp(-2 -1 α/(1 -β)) -β| Θ }
and Θ is a compact set, the desired result follows by continuity and the dominated convergence theorem as

E log |X 0 | = E log σ 0 + E log |Z 0 | < ∞.
The advantage of the SQMLE θS n is that the procedure is stable wrt the choice of the initial value, whereas the QMLE is not stable if there is one θ ∈ Θ satisfiying the non invertibility condition of [START_REF] Sorokin | Non-invertibility in some heteroscedastic models[END_REF]. The stabilization procedure relies on the explicit continuous invertibility condition (CI). The expression of the sufficient constraint INV(θ) is specific to the EGARCH(1,1) model. We believe that this constraint is not sharp; it should be possible to improve the invertibility condition INV(θ) of [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF], thus to improve (CI), to extend the constraint INV(θ) and finally to obtain more general SQMLE. Remark that such stabilization of the QMLE should be done before using this classical estimator on models that are not everywhere invertible nor explosive.

Asymptotic normality of the SQMLE in the EGARCH(1,1) model

In this section we extend the result of Theorem 5.7.9 of [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF] to non-degenerate cases when β 0 = 0. We obtain the asymptotic normality of the SQMLE without assuming uniform moment on the compact set Θ for the likelihood and its derivatives. However, we assume the additional condition (MX): The EGARCH(1,1) volatility DGP (σ 2 t ) is geometrically ergodic.

The geometric ergodicity is a classical assumption in the context of Markov chains, see [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF]. Condition (MX) is satisfied if Z 0 is nonsingular w.r.t. the Lebesgue measure on R, see [START_REF] Alsmeyer | On the Harris recurrence and iterated random Lipschitz fonctions and related convergence rate results[END_REF]. Geometric ergodicity is equivalent to the strong mixing property of the DGP with a geometric rate of decrease of the coefficients. The notion of geometric strongly mixing is not restricted to the case of Markov chains. As we will use it in the proof, let us recall that it is equivalent to the existence of 0 < a < 1 and b > 0 such that α r ≤ ba r for all r ≥ 1. Here the α r , r ≥ 1, are the strong mixing coefficients defined by [START_REF] Rosenblatt | A central limit Theorem and a strong mixing condition[END_REF] as

α r = sup A∈σ(...,X-1,X0) ,B∈σ(Xr,Xr+1,...) |P(A ∩ B) -P(A) P(B)| .
Let us also assume the finite moments assumption which is necessary and sufficient for the existence of the asymptotic covariance matrix, see Lemma 1 below:

(MM): E[Z 4 0 ] < ∞ and E[(β 0 -2 -1 (γ 0 Z 0 + δ 0 |Z 0 |)) 2 ] < 1.
Theorem 6. Assume that the conditions of Theorem 4 are satisfied, that θ 0 ∈

• Θ and that (MX) and (MM) hold. Then

√ n( θS n -θ 0 ) d -→ N (0, Σ)
where Σ is an invertible matrix.

Proof. As the SQMLE is asymptotically a.s. equivalent to the QMLE under the assumptions of Theorem 4 we will prove the result only for this last estimator. We first prove that Assumption (MM) yields the existence of the asymptotic variance

Σ = J -1 IJ -1 with J = E[H 0 (θ 0 )] and I = E[∇ 0 (θ 0 )∇ 0 (θ 0 ) T ]
, where H 0 (θ 0 ) and ∇ 0 (θ 0 ) are the Hessian matrix and the gradient vector of 0 at the point θ 0 ∈ • Θ.

Lemma 1. Under condition (MM) the covariance matrix Σ exist and it is an invertible matrix.

The proof of this lemma is given in Section 6. Then we prove that the functions ĝt and g t are twice continuously differentiable refining the arguments developed in [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF] Lemma 2. Under the assumptions of Theorem 4 the functions ĝt and g t are twice continuously differentiable on a compact neighborhood V(θ 0 ) of θ 0 ∈ • Θ and ∇ĝ t -∇g t V(θ0) + Hĝ t -Hg t V(θ0) e.a.s.

---→ 0.

The proof of this lemma is given in Section 6. The asymptotic normality follows from the Taylor development used in Section 5 of Bardet and Wintenberger [START_REF] Bardet | Asymptotic normality of the quasi maximum likelihood estimator for multidimensional causal processes[END_REF] on the partial derivatives ∇ i of the real valued function L n :

∇ i L n ( θn ) -∇ i L n (θ 0 ) = HL n ( θn,i )( θn -θ 0 ) for all 1 ≤ i ≤ d.
Then the asymptotic normality follows from the following sufficient conditions:

(a) n -1/2 ∇L n (θ 0 ) → N (0, I), (b) n -1 HL n ( θn ) -J a.s.
--→ 0 for any sequence ( θn ) converging a.s. to θ 0 and J is invertible, (c) n -1/2 ∇ Ln ( θn ) -∇L n ( θn ) converges a.s. to 0. Note that

∇L n (θ 0 ) = n i=t 4 -1 ∇g t (θ 0 )(1 -Z 2 t )
is a martingale as ∇g t (θ 0 ) is independent of Z t for all t ≥ 1 and E[1 -Z 2 t )] = 0 by assumption. Under (MM), this martingale has finite moments of order 2 due to Lemma 1. An application of the CLT for differences of martingales of [START_REF] Billingsley | Convergence of Probability Measures[END_REF] yields (a).

The following Lemma is used to prove (b) without uniform moment assumption on the compact set Θ for the likelihood and its derivatives Lemma 3. Under the assumptions of Theorem 6 we have HL n ( θn ) -HL n (θ 0 ) a.s.

--→ 0 for any sequence ( θn ) converging a.s. to θ 0 when n → ∞.

The proof of this lemma is given in Section 6. Applying Lemma 3, it is sufficient to prove that n -1 HL n (θ 0 ) -J a.s.

--→ 0 to obtain that n -1 HL n ( θn ) -J a.s.

--→ 0. The ergodic Theorem applied to the process (Hl t (θ 0 )) (integrable under (MM)) yields n -1 HL n ( θn ) -J a.s.

--→ 0. Thus the first assertion of (b) is proved. The fact that J is an invertible matrix is already known, see Lemma 1.

Finally (c) is obtained by using the exponential decrease of the approximation ( 16) that holds uniformly on some compact neighborhood V(θ 0 ) of θ 0 and the identity

∇L n = n i=t 4 -1 ∇g t (1 -X 2 t exp(-g t )).
Note that here we use the fact that exp(-x) is a Lipschitz function on [c, ∞) where c := min V(θ0) α/(1β).

6. Proofs of the technical lemmas

6.1. Proof of Lemma 1. Let us denote U t = (1, log σ 2 t , Z t , |Z t |) and V t = β 0 -2 -1 (γ 0 Z t + δ 0 |Z t |). Then (∇g t (θ 0 )) is the solution of the linear SRE ∇g t+1 (θ 0 ) = U t + V t ∇g t (θ 0 ), ∀t ∈ Z.
Let us consider the process Y t = (∇g t (θ 0 ), log(σ 2 t )) ∈ R 5 . It satisfies the relation

Y t+1 =       V t 0 0 0 0 0 V t 0 0 1 0 0 V t 0 0 0 0 0 V t 0 0 0 0 0 β 0       Y t +       1 0 Z t |Z t | α 0 + γ 0 Z t + δ 0 |Z t |       =: Γ t Y t + R t , ∀t ∈ Z.
Thus (Y t ) is a random coefficients autoregressive processsatisfying the assumptions of Theorem 4 (a) of [START_REF] Tweedie | Invariant measures for Markov chains with no irreducibility Assumptions[END_REF] iff EV 2 t < 1. By a direct application of the Theorem 4 (a) of [START_REF] Tweedie | Invariant measures for Markov chains with no irreducibility Assumptions[END_REF] we obtain that the process (Y t ) is second order stationary and thus the existence of the matrix

B = E[∇g t (θ 0 )(∇g t (θ 0 )) T ].
Let us prove that B is invertible. By classical arguments, it is sufficient to prove that the components of the vector ∇g 0 (θ 0 ) are linearly independent. It is the case in the AGARCH(1,1) model as soon as the density of Z 0 is not concentrated on two points, se Lemma 8.2 of [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]. Thus B is an invertible matrix.

Finally, we have the identity I = 2 -1 B as

I = 2 -1 E (∇g t (θ 0 )(∇g t (θ 0 )) T Z 2 0 + Hg t (θ 0 )(1 -Z 2 0 ) = 2 -1 E[∇g t (θ 0 )(∇g t (θ 0 )) T ] = 2 -1 B.
We also have the identity J = 4 -1 (EZ 4 0 -1)B because

J = E 4 -1 E ∇g t (θ 0 )(∇g t (θ 0 )) T (1 -Z 2 t ) 2 |F t-1 = 4 -1 E[(1 -Z 2 0 ) 2 ]E[∇g t (θ 0 )(∇g t (θ 0 )) T ] = 4 -1 (EZ 4 0 -1)
B. Thus, using the identity Σ = (EZ 4 0 -1)B -1 , this matrix exists and is invertible. The lemma is proved.

6.2. Proof of Lemma 2. The proof of the existence of the derivatives of the process (g t ) requires a refinement of Theorem 3.1 of [START_REF] Bougerol | Kalman filtering with random coefficients and contractions[END_REF]. We give this new result in full generality when the SRE is on a Polish space (E, d). A map f : E → E is a Lipschitz map if Λ(f ) = sup (x,y)∈E 2 d(f (x), f (y))/d(x, y) is finite. The regularity property we study is the following one: let (U t ) t≥0 be a sequence of non negative r.v.

(EAS): For any non negative sequence W t e.a.s.

---→ 0 the series (W t U t ) converges a.s.

Remark that (EAS) is implied for stationary sequences by a condition of log-moment of order 1 as in [START_REF] Bougerol | Kalman filtering with random coefficients and contractions[END_REF] (it is a straightforward application of the Borel Cantelli Lemma also used in [START_REF] Berkes | GARCH processes: structure and estimation[END_REF][START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]). More generally, (EAS) is satisfied for any non necessarily stationary sequence (Y t ) such that the series (P(U t ≥ ρ t )) = (P (log + U t ≥ tε)) converge for any 0 < ρ < 1 and ε > 0. Remark also that (EAS) is also automatically satisfied for any sequence (U t ) such that U t e.a.s.

---→ 0. The property (EAS) is very useful in our context as it also satisfied for any solution of a convergent SRE under minimal additional assumptions: finite log-moments of order p > 2 and the geometric strongly mixing condition Theorem 7. Let (Ψ t ) be a stationary ergodic sequence of Lipschitz maps from E to E that is also strongly mixing with geometric rate. Assume that (d(Ψ t (x), x)) satisfies (EAS) for some x ∈ E, E[(log + Λ(Ψ 0 )) p ] < ∞ for some p > 2 and

(18) E[log Λ(Ψ (r) 0 )] = E[log Λ(Ψ 0 • • • • • Ψ -r+1 )] < 0 for some r ≥ 1.
Then the SRE Y t+1 = Ψ t (Y t ), t ∈ Z, converges: it admits a unique stationary solution (Y t ) t∈Z which is ergodic and for any y ∈ E

Y t = lim m→∞ Ψ t • • • • • Ψ t-m (y), t ∈ Z.
The Y t are measurable with respect to the σ(Ψ t-k , k ≥ 0) and d( Ŷt , Y t ) e.a.s.

---→ 0, t → ∞ for ( Ỹt ) satisfying Ỹt+1 = Ψ t ( Ŷt ), t ≥ 0, and Ŷ0 = y for any y ∈ E. Moreover d(Y t , y) satisfies (EAS) for any y ∈ E.

Proof. Note that the proof of the existence of the stationary solution is due to Elton [START_REF] Elton | A multiplicative ergodic theorem for Lipschitz maps[END_REF]. That the approximation scheme is e.a.s. convergent is due to Bougerol [START_REF] Bougerol | Kalman filtering with random coefficients and contractions[END_REF]. Both results hold under the assumption that the Lipschitz coefficients have a finite log-moment of order 1. A careful look at the proof of Theorem 3.1 of [START_REF] Bougerol | Kalman filtering with random coefficients and contractions[END_REF] shows that the condition E[log + d(Ψ 0 (x), x)] < ∞ is only used there to assert (EAS) on (d(Ψ t (x), x)). Thus, the first assertions follow the classical arguments developed in [START_REF] Elton | A multiplicative ergodic theorem for Lipschitz maps[END_REF][START_REF] Bougerol | Kalman filtering with random coefficients and contractions[END_REF]. It remain to prove the (EAS) property only. )a -j/r 1 -a 1/r d(Ψ 1 (y), y).

Let us prove that sup 1≤j≤t Λ(Ψ (j-1) j

)a -j/r satisfies (EAS). It is implied by the Borel-Cantelli Lemma from the convergence of the series This series converges by an application of Theorem 1 in [START_REF] Shao | Complete convergence for α-mixing sequences[END_REF] as log(Λ(Ψ t )) has a finite moment of order p > 2 and is strongly mixing with geometric rate. Finally, for any non negative sequence W t e.a.s.

P sup 1≤j≤t log Λ(Ψ (j-1) j ) -j log(a 1/r ) ≥ tε .
---→ 0 the series (W t sup 1≤j≤t Λ(Ψ (j-1) j

)a -j/r ) converges and the desired result follows.

We are now ready to state the following refinement of the theorem 2.10 of [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF] used to prove that the functions ĝt and g t are twice continuously differentiable in a neighborhood of θ 0 : Theorem 8. Let B be a separable Banach space and (Ψ t ) be a stationary ergodic sequence of Lipschitz maps from B into B that is strongly mixing with geometric rate. Assume that S:

( Ψ t (0) ) satisfies (EAS), E[(log + Λ(Ψ 0 )) p ] < ∞ for some p > 2 and E[log Λ(Ψ (r) 0 )] < 0 for some r ≥ 1.
Let ( Ψt ) t∈N be a sequence of Lipschitz maps such that S': Ψt (0) -Ψ t (0) e.a.s.

---→ 0 and Λ( Ψt -Ψ t ) e.a.s.

---→ 0 as t → ∞. Then the unique stationary solution Y t of the SRE Y t+1 = Ψ t (Y t ), t ∈ Z, exists, ( Y t ) satisfies (EAS) and for every solution ( Ŷt ) t∈N of the perturbed SRE Ŷt+1 = Ψt ( Ŷt ), t ≥ 0, we have Ŷt -Y t e.a.s.

---→ 0 regardless the initial value Ŷ0 ∈ B.

Proof of Theorem 8. We assume the same conditions than in the theorem 2.10 of [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF] except that the conditions E[log + Ψ 0 (0) ] < ∞ and E[log + Y 0 ] < ∞ do not hold. Remark that we introduce this new approach based on the (EAS) property due to the difficulty to check this last condition on the derivatives of the SRE [START_REF] Berkes | GARCH processes: structure and estimation[END_REF]. A careful look at the proof of Theorem 2.10 in [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF] shows that these conditions are used for the convergence of series of the form (W t Ψ t (0) ) and (W t Y t ) converge for some (W t ) and (W t ) such that W t , W t e.a.s.

---→ 0. The first series converges by assumption, the second one converges as ( Y t ) satisfies (EAS) from the last assertion of Theorem 7.

Let us come back to the proof of Lemma 2. First note that as (σ 2 t ) is strongly mixing with geometric rate it is also the case of the process

(φ t (x, θ)) = (α + βx + (γX t + δ|X t |) exp(-x/2)).
Deriving the SRE (3), we obtain the new linear SRE [START_REF] Harvey | Beta-t-(E)GARCH[END_REF] ∇ĝ

t+1 (θ) = φ t (ĝ t (θ), θ)∇ĝ t (θ) + ∇ θ φ t (ĝ t (θ), θ), t ≥ 0 where φ t (x, θ) = β -2 -1 (γX t + δ|X t |) exp(-x/2), ∇ θ φ t (x, θ) = (1, x, X t exp(-x/2), |X t | exp(-x/2)) .
Note that both functions φ t (x, θ) and ∇ θ φ t (x, θ) are continuous in (x, θ). Under (CI), we derive from an application of Theorem 2 the existence of a compact neighborhood V(θ 0 ) ⊂

• Θ of θ 0 such that E[log Λ(φ (r) 0 ) V(θ0)
] < 0 for some r ≥ 1. The SRE ( 19) is a linear perturbed SRE ∇ĝ t+1 = Φ t (∇ĝ t ) on the Banach space of continuous functions on V(θ 0 ). The sequence ( Φ t ) is not stationary but it is well approximated by the stationary sequences (Φ t ) defined by the relation

Φ t (h) := φ t (g t , •)h + ∇ θ φ t (g t , •), t ∈ Z.
We will apply Theorem 8 on the SREs driven by ( Φ t ) and (Φ t ) to assert the existence of the first derivatives (∇g t ). Let us check the first condition of the assumption S of Theorem 8. The series ( Φ t (0) V(θ0) ) = ( ∇ θ φ t (g t , •) V(θ0) ) satisfies (EAS) because E log + |X t | < ∞ and ( ĝt V(θ0) ) satisfies (EAS) by an application of Theorem 7 on (g t ). By definition of V(θ 0 ), as φ t (x, θ) ≤ Λ(φ 0 , θ), we check the last two conditions of the assumption S by the continuous invertibility of the model and because E(log + X 0 ) p < ∞ for p = 8 as EZ 4 0 < ∞.

Let us check the first condition of the assumption S' of the theorem 8. As φ t is twice continuously differentiable, we have

Φ t (0) -Φ t (0) = ∇ θ φ t (g t , •) -∇ θ φ t (ĝ t , •) V(θ0) ≤ sup x≥c ∇ θ φ t V(θ0) g t -ĝt V(θ0)
where

∇ θ φ t (x, θ) = (0, 1, X t /2 exp(-x/2), |X t |/2 exp(-x/2)) and c := min V(θ0) α/(1 -β). Remark that E[log + sup x≥c ∇ θ φ t V(θ0) ] < ∞ because E[log + |X t |] < ∞. Thus ∞ t=0 P sup x≥c ∇ θ φ t V(θ0) ≥ ρ -t < ∞
for any ρ > 1 and then Φ t (0) -Φ t (0) e.a.s.

---→ 0 by an application of the Borel-Cantelli Lemma because g t -ĝt V(θ0) e.a.s.

---→ 0.

To check the second condition of the assumption S', we remark that as φ t is twice continuously differentiable, we also have

Λ(Φ t -Φ t ) ≤ φ t (ĝ t , •) -φ t (g t , •) V(θ0) ≤ sup x≥c φ t V(θ0) g t -ĝt V(θ0)
where φ t (x, θ) = 4 -1 (γX t +δ|X t |) exp(-x/2). That Φ t (0)-Φ t (0) e.a.s.

---→ 0 follows again by an application of the Borel-Cantelli Lemma because g t -ĝ t V(θ0) e.a.s.

---→ 0 and E[log

+ sup x≥c φ t V(θ0) ] < ∞ as E[log + |X t |] < ∞.
The existence of the stationary solution (∇g t ) of the SRE driven by (Φ t ) follows by an application of Theorem 8. It also provides that ( ∇g t V(θ0) ) satisfies the property (EAS) that will be useful to derive the existence of the second derivatives below. This stationary solution (∇g t ) is continuous as it is the locally uniform asymptotic law of ∇ĝ t that is continuous by construction.

Deriving a second time and keeping the same notation as above, we obtain another linear SRE satisfied by (Hĝ t+1 (θ)) for any 1 ≤ i, j ≤ d

(20) Hĝ t+1 (θ) = φ t (ĝ t (θ), θ)Hĝ t (θ) + φ t (ĝ t (θ), θ)∇ĝ t (θ)∇ĝ t (θ) T + ∇ θ φ t (ĝ t (θ), θ)∇ĝ t (θ) T + ∇ĝ t (θ)∇ θ φ t (ĝ t (θ), θ) T + H θ φ t (ĝ t (θ), θ).
For the EGARCH ( The details of the proof are omitted as they are similar than those used above on the first derivative. Note that the property (EAS) on ( ∇g t V(θ0) ) is required to check the first condition of the assumption S of Theorems 8 applied to Φ t (0). 6.3. Proof of Lemma 3. Let us fix V(θ 0 ) as in the proof of Lemma 2 such that g t is twice continuously invertible on V(θ 0 ) and such that the uniform log moments on the derivatives exist. Because θt -θ 0 a.s.

--→ 0 there exists some random integer M ≥ 1 such that θt ∈ V(θ 0 ) for any t ≥ M . Consider the SRE ∇g t+1 (θ) = φ t (g t (θ), θ)∇g t (θ) + ∇ θ φ t (g t (θ), θ) ∀t ∈ Z where φ t (x, θ) = β -2 -1 (γZ t + δ|Z t |) exp(-(x -g t (θ 0 ))/2), ∇ θ φ t (x, θ) = (1, x, Z t exp(-(x -g t (θ 0 ))/2), |Z t | exp(-(x -g t (θ 0 ))/2)) .

As x → exp(-x/2) is a Lipschitz continuous function for x ≥ c, as V(θ 0 ) is a compact set there exists some C > 0 such that for all x ∈ K, all θ ∈ V(θ 0 ) we have |φ t (x, θ) -φ t (g t (θ 0 ), θ 0 )| + ∇ θ φ t (x, θ) -∇ θ φ t (g t (θ 0 ), θ 0 ) ≤ C|Z t |( θ -θ 0 + |x -g t (θ 0 )|).

Thus, for any n ≥ t ≥ M , denoting v t ( θn ) = ∇g t+1 ( θn ) -∇g t+1 (θ 0 ) we obtain v t+1 ( θn ) ≤|φ t (g t (θ 0 ), θ 0 )|v t ( θn )| + ∇g t ( θn ) |φ t (g t (θ 0 ), θ 0 ) -φ t (ĝ t ( θn ), θn )| + ∇ θ φ t (ĝ t ( θn ), θn ) -∇ θ φ t (g t (θ 0 ), θ 0 ) ≤|φ t (g t (θ 0 ), θ 0 )|v t ( θn ) + ( ∇g t V(θ0) + 1)C|Z t |( θn -θ 0 + |g t ( θn ) -g t (θ 0 )|).

By a recursive argument, we obtain for all n ≥ t ≥ M v t ( θn ) ≤ ---→ 0 follows from the assumption (MM) Eφ i (g i (θ 0 ), θ 0 ) 2 < 1 and by using the subadditive ergodic theorem of [START_REF] Kingman | Subadditive ergodic theory[END_REF] on the logarithms. Thus the last term of the sum converges e.a.s to 0 and the corresponding Cesaro mean c n = n -1 n t=M t-1 i=M |φ i (g i (θ 0 ), θ 0 )|v M also converges e.a.s. to 0. This series converges a.s. because ( ∇g t V(θ0) ) satisfies (EAS) by an application of Theorem 8. Thus we obtain that there exist some random variable a > 0 such that Finally we obtain that n -1 n t=M v t ( θn ) ≤ a θn -θ 0 + b( θn ) + c n . Using this bound and the SRE satisfied by the differences Hg t ( θn )-Hg t (θ 0 ), we obtain following similar arguments than above that ---→ 0. We conclude that, conditionally on any possible value of M = m, we have n -1 n t=m Hg t ( θn ) -Hg t (θ 0 ) a.s.

Let us treat the term

--→ 0. Thus, we obtain that Hg t ( θn ) -Hg t (θ 0 ) = 0 P(M = k).

By continuity of the second derivative Hg t we have that Hg t ( θn ) → Hg t (θ 0 ) a.s. We deduce that for any k ≥ 1

P lim n→∞ 1 n k t=1
Hg t ( θn ) -Hg t (θ 0 ) = 0 = 1.

The desired result follows easily combining these two last equations with [START_REF] Jeantheau | Strong consistency of estimation for multivariate ARCH models[END_REF].

Figure 1 .

 1 Figure 1. Perspective and contour plots of the domain for invertibility.

Figure 2 . 3 . 3 . 1 .

 2331 Figure 2. Stable and non stable inverted EGARCH(1,1) models.

Theorem 2 .

 2 If (ST) and (CI) hold on some compact set Θ then the model is continuously invertible on Θ. Proof. For any ρ > 0, let us write Λ (r) * (θ, ρ) = sup{Λ φ (r) 0 (θ * ), θ * ∈ B(θ, ρ)∩Θ}, where B(θ, ρ) stands for the closed ball centered at θ with radius ρ. Note that for any K > 0 we have E[sup Θ | log Λ (r)

3. 4 .Theorem 3 .

 43 Volatility forecasting. Based on the QMLE θn we deduce a natural forecasting σ2 n+1 = (ĝ n ( θn )) of the volatility σ 2 n+1 . It is strongly consistent: Assume that (ST), (LB), (ID) and (CI) hold for θ 0 ∈ Θ. Then |σ 2 n+1 -σ 2 n+1 | a.s.

Theorem 5 .

 5 If INV(θ 0 ) and θ 0 ∈ Θ then θS n a.s.

First

  notice that (d( Ŷt , Y t )) satisfies (EAS) because d( Ŷt , Y t ) e.a.s. ---→ 0. Let us show that d( Ŷt , y) satisfies (EAS) when Ŷ0 = y. Fix K < 0 such that E[log Λ(Ψ (r) 0 ) ∨ K] ≤ log a for some 0 < a < 1. Then d( Ŷt , y) ≤

  By subadditivity, assuming that t/r ∈ N for convenience, we haveP sup 1≤j≤t log Λ(Ψ (j-1) j) -j log(a 1/r ) ≥ tε ≤ P sup

  i (g i (θ 0 ), θ 0 )|( ∇g t V(θ0) + 1)C|Z t |( θn -θ 0 + |g t ( θn ) -g t (θ 0 )|) + t-1 i=M |φ i (g i (θ 0 ), θ 0 )|v M . That t-1 i=j+1 |φ i (g i (θ 0 ), θ 0 )| e.a.s.

  g i (θ 0 ), θ 0 )|( ∇g t V(θ0) + 1)C|Z t | θn -θ 0 .It is a.s. smaller than θn -g i (θ 0 ), θ 0 )|( ∇g t V(θ0) + 1)C|Z t |.

  |φ i (g i (θ 0 ), θ 0 )|( ∇g t V(θ0) + 1)C|Z t | θn -θ 0 ≤ a θn -θ 0 .Finally, the reminding term of the upper bound is i (gi (θ 0 ), θ 0 )|( ∇g t V(θ0) + 1)C|Z t ||g t ( θn ) -g t (θ 0 )|.We treat it as in the proof of Theorem 3. Its uniform norm converges a.s. by an application of similar arguments than above. Thus there exists a random continuous function b satisfying b(θ0 ) i (g i (θ 0 ), θ 0 )|( ∇g t V(θ0) + 1)C|Z t ||g t ( θn ) -g t (θ 0 )| |φ i (g i (θ 0 ), θ 0 )|( ∇g t V(θ0) + 1)C|Z t ||g t ( θn ) -g t (θ 0 )| ≤ b( θn ).

1 n n t=M

 n Hg t ( θn ) -Hg t (θ 0 ) ≤ a θn -θ 0 + b ( θn ) + c n ,where a is a positive r.v., b is a random continuous function satisfying b (θ 0 ) = 0 and c n e.a.s.

  Hg t ( θn ) -Hg t (θ 0 ) = 0

  The general volatility model. In this paper, the innovations Z t ∈ R are iid random variables (r.v.) such that Z t is centered and normalized, i.e. E[Z 0

	1,1) model in Section 4. For this model, we propose a new method called Stable
	QMLE that produces only reliable volatility forecasting. The asymptotic normality of SQMLE for
	the EGARCH(1,1) model is given in Section 5. The proofs of technical Lemmas are collected in
	Section 6.
	2. Preliminaries
	2.1.

  1,1) model, H θ φ t (x, θ) is identically null. Thus we consider the perturbed SREΦ t (h) := φ t (ĝ t , •)h + φ t (ĝ t , •)∇ĝ t ∇g T t + ∇ θ φ t (ĝ t , •)∇g T t + ∇ĝ t ∇ θ φ t (ĝ t , •) T , t ≥ 0.We can apply Theorem 8 on this perturbed SRE and the corresponding stationary SREΦ t (h) := φ t (g t , •)h + φ t (g t , •)∇g t ∇g T t + ∇ θ φ t (g t , •)∇g T t + ∇g t ∇ θ φ t (g t , •) T , t ∈ Z.

Acknowledgments. I would like to thank C. Francq and J.-M. Zakoïan for helpful discussions and for pointing out some mistakes in an earlier version.


4$'55'"%6(1,-0'7$,-6 8

I. INTRODUCTION

In the last decades, a lot of Image Quality Metrics (IQMs) have been proposed in the literature. Different approaches have been proposed: Full Reference (FR), Reduced Reference (RR) and No Reference (NR) metrics. FR-IQMs are the most developed measures. These metrics use both original image and its degraded version. One of the most used metric is the Peak Signal to Noise Ratio (PSNR). Nevertheless, this metric is not well correlated with the subjective judgment. Hence, some authors propose to integrate some Human Visual System (HVS) models such as VDP [START_REF] Daly | The Visible Differences Predictor: An Algorithm for the Assessment of Image Fidelity[END_REF]. Some others propose to take into account some HVS characteristics implicitly such SSIM [START_REF] Wang | Multi-scale structural similarity for image quality assessment[END_REF].

RR-IQM approach used only some characteristics of the original image. Some features are extracted from both original image and its degraded version. Then, these features are compared to evaluate the quality of these images. Note that only few RR metrics are proposed in the literature [START_REF] Wang | Reduced-reference image quality assessment using a wavelet-domain natural image statistic model[END_REF], [START_REF] Carnec | Objective quality assessment of color images based on a generic perceptual reduced reference[END_REF].

NRs are the most attractive and require only the degraded image. However, these metrics are generally developed only for specific degradations such as blocking effect [5], blur [START_REF] Chetouani | A new free reference image quality index for blur estimation in the frequency domain[END_REF] and ringing effect [START_REF] Sheikh | No-Reference Quality Assessment Using Natural Scene Statistics: JPEG2000[END_REF]. A recent method has been proposed to combine some NR measures to obtain universal NR-IQM [START_REF] Chetouani | A free reference image quality measure using neural networks[END_REF].

In this paper, we present a new NR image quality estimation approach based on classification step. The main objective of this work is to propose a multi-degradations image quality system without reference. This system will be used to estimate the quality of a given degraded image without any assumption, including assumptions about the type of degradation contained in the image usually assumed known. The idea is to first extract some features from the degraded image and then use it as inputs to a Linear Discriminant Analysis (LDA) classifier. The most common degradations and artifacts considered here are: Noise, Blur, blocking and ringing.

The paper is organized as follows: In section 2 the image database used in our experiments and the proposed method are described. The experimental results are presented and discussed in section 3. The final section is dedicated to the conclusions and perspectives. Our goal is to develop a system, which can be used to estimate the quality of a given degraded image without reference whatever any knowledge about the type of the distortion. In other words, the objective is here to overcome one of the major assumptions of this kind of metrics to extend it's the utilization in practical situations. Hence, we propose to characterize and identify the type of degradation contained in a given image before estimating its quality according to the more appropriate NR-IQM. The block-diagram of the proposed system is shown in Fig. 1.

II. Proposed Method

Noise, blur, blocking and ringing compressed artefacts are considered in this study (see Fig. 2). Some known NR-IQMs are then used to capture these features.

The usefulness of this classification process can be easily demonstrated by computing the Pearson and Spearman correlation coefficients obtained using only one NR-IQMs for all the considered degradations (see Table 1) [5]. Note that for blocking artefact high correlation is obtained. Whereas, the correlation is low for all the other considered distortions. The image database, descriptors and the classifier used in this study are described in this section.

A. Image database

Different image databases are now available [START_REF] Sheikh | LIVE Image Quality Assessment Database[END_REF][START_REF] Callet | Subjective quality assessment IRCCyN/IVC database[END_REF]. The Tampere Image Database 2008 is used in this study [START_REF] Ponomarenko | Color Image Database for Evaluation of Image Quality Metrics[END_REF]. In this database, [START_REF] Martinez | PCA versus LDA[END_REF] of degradations are considered. Each degradation type is composed by 100 degraded images from 25 reference images (see Fig. 3). For each image the Mean Opinion Score (MOS) is also given in this database.

B. Descriptors : NR-IQMs

As descriptors, we propose here to use some NR-IQMs. The best results have been obtained using 8 NR-IQMs (i.e. 4 for blur, 1 for blocking and ringing artefacts, and 2 for noise). All these descriptors are briefly presented in the following.

Blur Measures. Four Blur measures are considered in this study. The first one is based on wavelet transform [START_REF] Tong | Blur detection for digital images using wavelet transform[END_REF]. An edge map is first derived from the high frequencies coefficients at each decomposition level. The blur measure is then obtained by analyzing the type of the edge contained in the image using some rules.

The second NR blur metric used is based on a subjective observation [START_REF] Crête | Estimer, mesurer et corriger les artefacts de compression pour la télévision[END_REF]. The authors propose to measure the impact of adding blur to the degraded image. The index quality is obtained by analyzing the spatial variations of the degraded image and its blurred version. Based on the same principle, the third measure is performed in the frequency domain [START_REF] Chetouani | A new free reference image quality index for blur estimation in the frequency domain[END_REF]. The index quality is derived from the difference between the radial spectrum of the degraded image and its blurred version.

The last one is based on some subjective tests, where the Just Noticeable Blur (JNB) is measured for different contrast levels [START_REF] Ferzli | A No-Reference Objective Image Sharpness Metric Based on the Notion of Just Noticeable Blur[END_REF]. A blur model is then derived from these tests and used to estimate the image quality.

Degraded

Image

Features

Extraction

Linear

Discriminant Analysis

Classifier Degradation Type Blocking Effect Measure. The block-based measure used here has been proposed in [5]. The vertical and horizontal gradients are first computed. The global index quality is obtained after summing the gradient values along the block boundaries followed by a zero detection process.

Ringing Measure. As NR Ringing measure, we use the metric described in [START_REF] Sheikh | No-Reference Quality Assessment Using Natural Scene Statistics: JPEG2000[END_REF]. This method is based on wavelet transform and Natural Scene Statistics (NSS). Statistical models are derived from the wavelet coefficients and are used to estimate the image quality.

Noise Measures. The noise measures used here are based on local variance estimation. The first one is derived from the eigenvalues [START_REF] Van De Ville | SURE-Based Non-Local Means[END_REF]. The image is first decomposed in different blocks. After vectorizing all blocks, the Eigen values are then computed through the covariance matrix. The variance of the noise is finally estimated from lowest values.

The second metric used here is based on the variance estimation in the DCT domain [START_REF] Buclkey | Fast computation of a discretized thin-plate smoothing spline for image data[END_REF]. The image is first smoothed in DCT domain using a thin-plate smoothing spline model. Then, the generalized cross validation process determines the variance of the noise.

C. Classifier: Linear Discriminant Analysis: NR-IQMs

As degradation identification tool, the Linear Discriminant Analysis (LDA) classifier is used [START_REF] Martinez | PCA versus LDA[END_REF]. The maximum class discrimination of this classifier is given by minimizing the within class distance and maximizing the between class distance simultaneously.

In our context, each type of degradation is considered as a class (i.e. 4 classes), and descriptors are seen as the input vector (i.e. 8 descriptors).

III. EXPERIMENTAL RESULTS

The proposed method is evaluated in terms of classification accuracy using all the degraded images of the considered degradation types (i.e. 400 degraded images, 100 per degradation) through the crossvalidation principle. Hence, the database is divided into three parts without overlapping. At each step, 2/3 of the image database is used for the learning phase and 1/3 for the test step. Note that, images used during the training and the testing steps are different.

The process is, thus, very simple and uses only the degraded image, as summarized in Fig. 5. From a given degraded image, we first extract the selected descriptors (NR-IQMs). After projecting these features onto the optimal space, the type of distortion is then determined by a distance criterion. Here, Mahalannobis distance provides the best results.

Note that blocking and ringing artefacts are generated, respectively, using JPEG and JPEG2000 compression methods. Table 2 shows the confusion matrix. We can see that the high percentage correspond well to the diagonal of this matrix (correct classification) with the less percentage is obtained for blur degradation (86%). The mean percentage of good classification is equal to 93.51%.

Table 2. Confusion matrix of the proposed method (%). There are also some confusions between different classes. The more important confusion is between Blur and JPEG2000 classes. This kind of confusions can be easily explained by the fact that Blur artifact appears also in JPEG2000 compressed images as we can see in Fig. 4. Our results are also compared to the BIQI (Blind Image Quality Index) method [START_REF] Moorthy | A Two-Step Framework for Constructing Blind Image Quality Indices[END_REF], which is based on natural statistics (Table 3). A Support Vector Machine (SVM) tool is used as classifier. Note that the mean percentage of good classification obtained by our method is higher than the BIQI method. The gain is around 13.5%.

Estimated Class

True class

Noise

Table 3. Comparison between the proposed method and the BIQI method [START_REF] Moorthy | A Two-Step Framework for Constructing Blind Image Quality Indices[END_REF]. To better compare, the confusion matrix of the BIQI method is also shown. We can easily see that there are more confusions than our method. One of the most surprising confusion is between Blur and Noise classes. Indeed, these types of degradations are completely different in terms of impact. Both affect the high frequency content: Blur distortion decreases or attenuates high frequencies and Noise artefact adds high frequencies.

Accuracy (%)

There are also some confusions between JPEG2000 and Blur classes. Of course, even if this kind of confusion is relatively acceptable as explained above. However, the confusion rate reached 33%, which very high comparing to the obtained result (10%). Once the degradation classification performed, the most appropriate NR-IQM can be used for estimating the image quality without any assumption on the degradation type and for different kinds of degraded images. Hence, the system permits to estimate the image quality without reference automatically.

IV. CONCLUSION

In this paper, a new approach to estimate the image quality without reference whatever the degradation type. The proposed method provides an efficient tool to solve one of the major limitations of no reference image quality metrics. The more common distortions have been considered (noise, blur, blocking and ringing artefacts). The proposed system aims to permit to evaluate the quality of a given image without the original image and whatever the degradation type. The performance of the proposed method has been efficiently evaluated in terms of classification accuracy and identification.

  a) Ringing (JPEG2000), b) Blocking (JPEG), c) Blur and d) Noise degradations.

Fig. 1 .

 1 Fig. 1. Flowchart of the proposed method.

Fig. 3 .

 3 Fig. 3. Sample of original images from the TID2008 database.

4 .

 4 Degraded Images: a) Blur degradation, b) JPEG2000 compression degradation.

Table 1 .

 1 Correlation coefficients using only a blocking metric for all the considered degradations.

		Pearson	Spearman
		Correlation	Correlation
	NOISE	0.52	0.57
	BLUR	0.34	0.44
	JPEG	0.93	0.91
	JPEG2000	0.40	0.40

Table 4 .

 4 Confusion matrix of the BIQI method[START_REF] Martinez | PCA versus LDA[END_REF] (%).

			Estimated Class
			Noise	Blur	JPEG	JPEG2000
	True class	Noise Blur JPEG	99 11 0	0 92 1	0 3 87	1 4 12
		JPEG2000	2	33	14	51

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Degraded image

Features extraction

NR-IQMs Classifier (Linear Discriminant Analysis)

Noise Estimation Blur Estimation

Blocking Estimation

Ringing Estimation


S Lacroix

Jj Pernelle Cemagref

Differential sthA gene expression between filamentous and isolated forms of Sphaerotilus natans

Background:

The activated sludge is the most commonly used technology for biological wastewater treatment. Filamentous bacteria are essential component of the activated sludge due to the important roles they play in sludge settleability. Excessive growth of filamentous bacteria, known as bulking, leads to a poor settlement of activated sludge. Sphaerotilus natans h a s b e e n r e p o r t e d a s t h e d o m i n a n t filamentous organism in 12% of bulking sludge samples in United-States (Jenkins D., 1984). This bacterium is characterized by a sheathed structure in which long chains of rod-shaped cells are enclosed. S. natans h a s a l s o b e e n i d e n t i f i e d i n s i n g l e -cell form [START_REF] Stokes | Studies on the filamentous sheathed iron bacterium Sphaerotilus natans[END_REF] a s f e w o t h e r s filamentous bacteria species isolated from activated sludge. The growth morphology is determined by the environmental conditions. The molecular mechanisms that induce the filamentous growth from a single-cell population are still widely unknown. Nevertheless a gene essential for the filamentous growth and the sheath formation of S. natans ATCC 15291 has been identified and named sthA [START_REF] Suzuki | Identification of a gene essential for sheathed structure formation in Sphaerotilus natans, a filamentous sheathed bacterium[END_REF]. An inactivation of this gene results in sheathless single-cell form growth. In order to find more efficient treatments against bulking the molecular mechanisms of filamentation have to be better known.

Methods and results:

In this study, the presence of sthA gene in all the S. natans collection strains was demonstrated as its absence from Leptothrix genome, which is the closest phylogenetic neighbour of S. natans. This gene was sequenced. To analyze the expression of sthA gene in filamentous and single-cell form of S. natans, sthA specific primers useable in RT-QPCR were designed. In opposition with Takeda and al. results, we determined that sthA is more expressed in filamentous form than in single cells form. The expression level is also depending of the S. natans strain studied (Fig. 1). We finally determine that sthA expression increase during filamentous growth until it reaches a plateau (Fig. 2).

Time (hour)

Relative gene expression

Relative sthA gene expression

Conclusions:

These findings indicate that sthA gene is highly specific to S. natans as it is not found in Leptothrix genome. This gene is directly implicated into the sheathed structure formation. As its expression vary in function of the morphology adopted and as it seems to be a very specific gene of S. natans, sthA may play a key role into the molecular mechanism of filamentation.

References:

Jenkins D., R. M. G., Daiggert G. T. (1984), Manual of the Causes and Control of Activated Sludge Foaming and Bulking, 1st ed., Lewis Publishers.

Figure 1 :

 1 Figure 1 : Relative sthA gene expression in ATCC 13338 and ATCC 15291 Sphaerotilus strains

Figure 2 :

 2 Figure 2 : Relative sthA gene expression during filamentous growth of Sphaerotilus natans.