Olivier Wintenberger
email: wintenberger@ceremade.dauphine.fr

Sixiang Cai
email: sixiang.cai@u-cergy.fr

Parametric inference and forecasting for continuously invertible volatility models

Keywords: AMS 2000 subject classifications: Primary 62F12; secondary 60H25, 62F10, 62M20, 62M10, 91B84 Invertibility, volatility models, parametric estimation, strong consistency, asymptotic normality, asymmetric GARCH, exponential GARCH, stochastic recurrence equation, stationarity

.

Introduction

Since the seminal paper of [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation[END_REF] and [START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF], the General Autoregressive Conditional Heteroskedasticity (GARCH) type models have been successfully applied to volatility modeling. [START_REF] Nelson | Conditional Heteroskedasticity in Asset Returns : A New Approach[END_REF] is the first attempt to introduce non linearity into volatility models with the Exponential-GARCH(1,1) type models. Since then, many other volatility models have been introduced: APGARCH of Ding, [START_REF] Ding | A long memory property of stock market returns and a new model[END_REF], GJR-GARCH of [START_REF] Glosten | On the relation between the expected value and the volatility of the nominal excess return on stocks[END_REF], TGARCH of [START_REF] Zakoïan | Threshold heteroscedastic models[END_REF], etc. Non linear volatility models have been used extensively in empirical researches (see [START_REF] Brandt | Volatility forecasting with rangebased EGARCH models[END_REF] among many others) and financial industry. Not surprisingly, theoretical investigations of EGARCH has attracted constant attention, see [START_REF] He | Moment structure of a family of first-order exponential GARCH models[END_REF], [START_REF] Harvey | Exponential conditional volatility models[END_REF] and [START_REF] Rodriguez | GARCH models with leverage effect: differences and similarities[END_REF]. However, the validity of the estimation procedures used empirically in [START_REF] Nelson | Conditional Heteroskedasticity in Asset Returns : A New Approach[END_REF] was not proved.

Our study provides the first satisfactory answer to this open question for non linear volatility models including the EGARCH(1,1) model. We give sufficient conditions for the estimator to be strongly consistent and asymptotically normal. Our approach is based on the natural notion of continuous invertibility that we introduce in a very general setting and thus will be applied in other models in future works.

Consider a general volatility model of the form X t = Σ 1/2 t • Z t where Σ t is the volatility and where the innovations Z t are normalized, centered independent identical distributed (iid) random vectors. The natural filtration F t is generated by the past innovations (Z t , Z t-1 , . . .). It is assumed that a transformation of the volatility satisfies some (possibly non-linear) SRE, i.e. there exist a function h and some F t-1 measurable random function ψ t such that the following relation

(h(Σ k )) k≤t = ψ t ((h(Σ k )) k≤t-1 , θ 0 ) (1)
holds. It is the case of all classical models of GARCH and EGARCH types, extensions of the simplest GARCH(1,1) and EGARCH(1,1) univariate models. Let us illustrate our propose in this introduction on these models (in the univariate case, the volatility is denoted σ 2 t ):

GARCH(1,1):

σ 2 t = α 0 + β 0 σ 2 t-1 + γ 0 X 2 t-1 , (2)
EGARCH(1,1): log(σ 2 t ) = α 0 + β 0 log(σ 2 t-1 ) + (γ

0 Z t-1 + δ 0 |Z t-1 |). (3)
One can rewrite (2) as an SRE driven by the innovations:

σ 2 t = α 0 + (β 0 + γ 0 Z 2 t-1 )σ 2 t-1 , i.e. ψ t (x, θ) = α + (β + γZ 2 t-1 )
x. This SRE is used by [START_REF] Nelson | Stationarity and persistence in the GARCH(1,1) model[END_REF] to obtain the Lyapunov condition E[log(β 0 + γ 0 Z 2 0 )] < 0, necessary and sufficient for the stationarity. In general, the functional process (ψ t ) driving the SRE (1) is assumed to be a stationary ergodic process of Lipschitz functions. Such a SRE is said to be convergent when its solution is unique, non anticipative (i.e. function of F t at any time t) and its law does not depend on the initial values. This last property, also called the "stability" of the SRE, ensures the existence of the stationary process (X t ). It coincides with the ergodicity for Markov chains. Sufficient conditions (also necessary in the linear Markov case) for the convergence are the negativity of a Lyapunov coefficient and the existence of logarithmic moments, see [START_REF] Elton | A multiplicative ergodic theorem for Lipschitz maps[END_REF], [START_REF] Bougerol | Stationarity of GARCH processes and of some nonnegative time series[END_REF] and [START_REF] Bougerol | Kalman filtering with random coefficients and contractions[END_REF].

Assume that the model have a non anticipative, stationary solution with invertible volatility matrices Σ t . Using the relation Z t = Σ -1 t • X t in ψ t , it is possible to study a new SRE driven by the observations X t (h(Σ k )) k≤t = φ t ((h(Σ k )) k≤t-1 , θ 0 ).

(4)

Here φ t is an ergodic stationary process generated by (G t-1 ), the sigma-field of the past values (X t-1 , X t-2 , • • • ). The convergence of this new SRE is closely related with the notion of invertibility (i.e. the existence of a non linear AR representation of the observations) see [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] and [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]. As previously, sufficient conditions for the convergence are the negativity of a Lyapunov coefficient and the existence of logarithmic moments; These conditions are expressed this time on φ 1 and not on ψ 1 . For instance, we obtain from (2) that φ t (x, θ) = α + βx + γX 2 t-1 and the GARCH(1,1) model is invertible as soon as 0 ≤ β 0 < 1. For the EGARCH(1,1) model, we obtain from (3) that

φ t (x, θ) = α + βx + (γX t-1 + δ|X t-1 |) exp(-x/2). ( 5
)
This SRE is less stable than in the GARCH(1,1) model because of the exponential function that explodes for large values. However, a sufficient condition for invertibility is given in [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF] under restrictions on the parameters (α 0 , β 0 , γ 0 , δ 0 ), see ( 14) below.

In practice the parameter θ 0 is unknown and is estimated using the SRE as follow. An approximation of the SRE (h(g k )) k≤t = φ t ((h(g k )) k≤t-1 , θ) generates recursively a forecast ĝt (θ) of the volatility Σ t using only the past observations and the model at the point θ. Using Quasi-LIKelihood (QLIK) criteria to quantify the error of the volatility forecasting:

n Ŝn (θ) = n ∑ t=1 ŝt (θ) = n ∑ t=1 2 -1 X T t ( ĝt (θ)) -1 X t + log(det( ( ĝt (θ))) (6)
the best forecast corresponds to the M-estimator θn satisfying θn = argmin θ∈Θ Ŝn (θ).

Under conditions ensuring the asymptotic regularity of Ŝn (•), θn is a good candidate for estimating θ 0 . By construction, the asymptotic regularity of Ŝn (•) depends on the regularity of g t (•) and on the stability of the corresponding SRE. As θ 0 is unknown, [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF] imposed the uniform invertibility over the compact set Θ: The SRE is assumed to be stable in the Banach space of continuous function on Θ (with the sup-norm). The consistency and the asymptotic normality of θn follows, see [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF] and [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]. However, the notion of uniform invertibility is too restrictive; the asymptotic normality of the EGARCH(1,1) model is proved in the degenerate case

β 0 = 0 only; there log(σ 2 t ) = α 0 + (γ 0 Z t-1 + δ 0 |Z t-1 |) which is not realistic.
In this paper, we introduce the notion of continuous invertibility and apply it successfully to the EGARCH(1, 1) model: the asymptotic normality of the procedure described above and used since [START_REF] Nelson | Stationarity and persistence in the GARCH(1,1) model[END_REF] is proved for the first time. The notion of continuous invertibility is a very natural one: the SRE is assumed to be stable at each point θ ∈ Θ and the functional solution (g t (•)) is assumed to be continuous. By definition, uniform invertibility implies the weaker notion of continuous invertibility. We give sufficient conditions (CI) for continuous invertibility and check it on volatility model that are non necessarily uniformly invertible. We prove under (CI) and the identifiability of the model that θn is a strongly consistent estimator of θ 0 and that the natural forecast ĝt ( θt ) of the volatility Σ t is also strongly consistent. We prove that θn is asymptotically normal if moreover the limiting variance exists. The proofs are very general and also valid in the multidimensional case. In absence of uniform invertibility, we use under (CI) new arguments following the ones of [START_REF] Jeantheau | Modèles autorégressifs à erreur conditionellement hétéroscédastique[END_REF] for the strong consistency and of [START_REF] Bardet | Asymptotic normality of the quasi maximum likelihood estimator for multidimensional causal processes[END_REF] for the asymptotic normality. One crucial step in the proof is Theorem 2 below which asserts the logarithmic moments properties of solutions of SRE. This results, by its generality, is of independent interest for the study of probabilistic properties of solutions of SRE (also called Iterated Random Functions).

The commonly used statistical procedure described above is only valid under (CI). Fortunately condition (CI) is automatically satisfied for all invertible GARCH and EGARCH type models known by the authors. The GARCH(1,1) model satisfied (CI) on all compact sets of [0, ∞[ 3 satisfying β < 1 where it is also uniformly invertible. The EGARCH(1,1) model satisfies (CI) on all compact sets of points that satisfy the invertibility condition (13). It is not uniformly invertible there but the statistical inference is still valid. Applying our approach to other models, we recover the results of [START_REF] Berkes | GARCH processes: structure and estimation[END_REF] and [START_REF] Francq | Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes[END_REF] for GARCH(p,q) models, we recover the results of [START_REF] Francq | QML estimation of a class of multivariate asymmetric GARCH models[END_REF] for CCC-GARCH(p,q) models and for AGARCH(p,q) models we refine the results of [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]. On the contrary, it is shown in [START_REF] Sorokin | Non-invertibility in some heteroscedastic models[END_REF] that forecasting the volatility with SRE may be inconsistent when the SRE is unstable. Thus, if the model is not continuously invertible on Θ the minimization (7) is unstable and the estimation procedure is not valid. To sum up, one can think of the following "equivalences":

Stability of the SRE generated by ⇐⇒ stationarity, ergodicity the innovations (Z t , Z t-1 , . . .) (A) and log-moments Stability of the SRE generated by ⇐⇒ invertibility, forecasting the observations (X t , X t-1 , . . .) (B) and statistical inference

The equivalence (A) is crucial when studying existence of stationary solutions of volatility models. We want to emphasize the importance of the second equivalence (B) for the volatility forecast and the statistical inference using the QLIK criteria. For non uniformly invertible models, it is crucial to infer the model only in the domain of invertibility otherwise the whole procedure can fail. The consequences of this work on empirical study is huge as the equivalence (B) has been negligible in most existing works, see [START_REF] Cai | The EGARCH(1,1) process: structure, estimation and prediction[END_REF] for the EGARCH(1,1) case (applications on other classical models are also in progress). Finally, notice that the statistical inference of θ 0 is possible without assuming (CI): it has been done by [START_REF] Zaffaroni | Whittle estimation of EGARCH and other exponential volatility models[END_REF] using Whittle's estimator.

An outline of the paper can be given as follows. In Section 2, we discuss the standard notions of invertibility and introduce the continuous invertibility and its sufficient condition (CI). In Section 3 our main results on the statistical inference based on the SRE are stated. We apply this results in some GARCH type models and in the EGARCH(1,1) model in Section 4. The Appendix contains the technical computation of the necessary and sufficient condition for the existence of the asymptotic variance of θn in the EGARCH(1,1) model.

Continuously invertible volatility models

The general volatility model

In this paper, (Z t ) is a stationary ergodic sequence of real vectors called the innovations. Let us denote F t the filtration generated by (Z t , Z t-1 , . . .). Consider the general volatility model

X t = Σ 1/2 t • Z t where (1) is satisfied: (h(Σ k )) k≤t = ψ t ((h(Σ k )) k≤t-1 , θ 0 )
. The function h is injective from the space of real matrices of size k × k to an auxiliary separable metric space F. The random function ψ t (•, θ 0 ) is a F t-1 adapted random function from the space of the sequences of elements in the image of h to itself. Let us denote the inverse of h (from the image of h to the space of real matrices of size k × k) and call it the link function.

Convergent SRE and stationarity

A first question regarding this very general model is wether or not a stationary solution exists. As the sequence of the transformed volatilities (h(Σ k )) k≤t is a solution of a fixed point problem, we recall the following result due to [START_REF] Elton | A multiplicative ergodic theorem for Lipschitz maps[END_REF] and [START_REF] Bougerol | Kalman filtering with random coefficients and contractions[END_REF]. Let (E, d) be a complete separable metric space. 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. For any sequence of random element in (E, d), (X t ) is said to be exponential almost sure convergence to 0 X t e.a.s.

---→ 0 as t → ∞ if for X t = o(e -Ct ) a.s. for some C > 0.

Theorem 1. Let (Ψ t ) be a stationary ergodic sequence of Lipschitz maps from E to E. Suppose that

E[log + (d(Ψ 0 (x), x))] < ∞ for some x ∈ E, that E[log + Λ(Ψ 0 )] < ∞ and that for some integer r ≥ 1, E[log Λ(Ψ (r) 0 )] = E[log Λ(Ψ 0 • • • • • Ψ -r+1 )] < 0.
Then the SRE X t = Ψ t (X t-1 ) for all t ∈ Z is convergent: 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( Ỹt , Y t ) e.a.s. ---→ 0, t → ∞ such that Ỹt = Ψ t ( Ỹt-1 ) for all t > 0.
The sufficient Lyapunov assumptions E[log Λ(Ψ (r) 0 )] < 0 is also necessary in the linear case, see [START_REF] Bougerol | Stationarity of GARCH processes and of some nonnegative time series[END_REF].

The results of Theorem 1 does not guaranty any moment property on the stationary solution. The following result ensures the existence of logarithmic moments of the stationary solution under natural assumptions. It extends the result of [START_REF] Alsmeyer | Limit theorems for iterated ran-dom functions by regenerative methods[END_REF] to a much more general context.

Theorem 2. Under the assumptions of Theorem (1) and E[(log

+ d(Ψ 0 (x), x)) 2 ] < ∞ the unique stationary solution satisfies E[log + (d(Y 0 , y))] < ∞ for all y ∈ E.
Proof. The basic inequality log(1 + y + z) ≤ log(1 + y) + log(1 + z) will be used several time. Notice that for any r.v. X ≥ 0 we have the equivalence

E[log(1 + X)] < ∞ iff E[log + (X)] < ∞. Thus E[log(1 + d(Y 0 , y))] < ∞ for all y ∈ E iff E[log(1 + d(Y 0 , x))] < ∞ since d(Y 0 , y)) ≤ d(Y 0 , x)) + d(x, y). We denote Ψ (-m) = Ψ 0 • • • • • Ψ -m (x), w = d(y, Ψ (1-m) (x)) ≥ 0 and z = Λ(Ψ (1-m) )d(Ψ -m (x), x) ≥ 0. From the triangular inequality d(x, Ψ (-m) (x)) ≤ w + z we obtain that log(1 + d(x, Ψ (-m) (x))) ≤ log(1 + w + z) ≤ log(1 + w) + log(1 + z).
Eqn. 27 in [START_REF] Bougerol | Stationarity of GARCH processes and of some nonnegative time series[END_REF] asserts the existence of 0 < ρ < 1 and > 0 satisfying

lim m→∞ 1 m log(Λ(Ψ (-m) )) ≤ log(ρ) - a.s.
Thus there exists a r.v.

M ∈ N * such that Λ(Ψ (-m) ) ≤ ρ m for all m ≥ M. Writing v m = log(1 + d(x, Ψ (-m) (x))
), for all m ≥ M we have:

v m ≤ v m-1 + log(1 + ρ m-1 d(Ψ -m (x), x)).
A straightforward recurrence leads to the following upper bound of all

(v m ) m≥M v m ≤ v M + ∞ ∑ j=1 log(1 + ρ j-1 d(Ψ -j-M (x), x)).
As log(1

+ d(x, Y 0 )) = lim m→∞ v m a.
s., it remains to prove that the series in the upper bound is summable and that the upper bound is integrable to conclude by the dominated convergence Theorem. Using the stationarity of (v m ),

we know that E[log(1 + ρ j-1 d(Ψ -j-M (x), x))] = E[log(1 + ρ j-1 d(Ψ 0 (x), x))] does not depend on M and j. Thus E[v M ] = E[log(1 + d(x, Ψ 0 (x)))] < ∞ by assumption.
We conclude the proof by comparing the series with an integral:

∑ j≥1 E[log(1 + ρ j-1 d(Ψ 0 (x), x))] ≤ 1 1 -ρ 1 0 E[log(1 + ud(Ψ 0 (x), x))] u du.
Let us prove that his integral converges as soon as E[(log

+ d(Ψ 0 (x), x)) 2 ] < ∞. Using that E[log(1 + ud(Ψ 0 (x), x))] = ∞ 0 P(log(1 + ud(Ψ 0 (x), x)) ≥ t)
dt and denoting v = (e t -1)/u the integral becomes:

1 0 ∞ 0 P(log(1 + ud(Ψ 0 (x), x)) ≥ t) u dtdu = 1 0 ∞ 0 P(d(Ψ 0 (x), x) ≥ v) 1 + uv dvdu.
Using Fubini's theorem and

1 0 (1 + uv) -1 du = log(v + 1)/v for all v ≥ 0 we get an upper bound in term of ∞ 0 log(1 + v)P(d(Ψ 0 (x), x) ≥ v) v dv.
This integral converges in +∞ as

∞ 0 log(1 + v)P(d(Ψ 0 (x), x) ≥ v) 1 + v dv = E[log(1 + d(x, Ψ(x)) 2 ]
and the desired result follows.

In order to apply Theorem 1 in our case, let us denote by E the separable metric space of the sequences of elements in the image of h. Equipped with the metric ∑ j≥1 2 -j d(x j , y j )/(1 + d(x j , y j )), the space E is complete. A sufficient condition for stationarity of (X t ) is that the SRE driven by (ψ t ) converges in E. It simply expresses as the Lyapunov condition E[log Λ(ψ (r) 0 )] < 0 for some integer r ≥ 1 and some logarithmic moments. This assumption of stationarity is sufficient but not optimal in many cases:

Remark 1. The state space of the SRE (1), denoted E, in its most general form, is a space of infinite sequences. However in all classical models we can find a lag p such that (h(Σ k )) t-p+1≤k≤t = ψ t ((h(Σ k )) t-p≤k≤t-1 , θ 0 ). The state space E is now the finite product of p spaces. It can be equipped by unbounded metrics such that p -1 ∑ p j=1 d(x j , y j ) or ∑ p j=1 d 2 (x j , y j ). The product metric has to be carefully chosen as it changes the value of the Lipschitz coefficients of the φ t . Yet, even if the products spaces are embedded, the smallest possible lag p in the SRE yields the sharpest Lyapunov condition. Finally, if E has a finite dimension and if the condition of convergence of the SRE expresses in term of the top Lyapunov coefficient, one can choose any metric induced by any norm, see [START_REF] Bougerol | Kalman filtering with random coefficients and contractions[END_REF] for details.

In view of Remark 1, instead of choosing a specific metric space (E, d) we prefer to work under the less explicit assumption (ST) The process (X t ) satisfying (1) exists. It is a stationary, non anticipative and ergodic process with finite logarithmic moments.

In view of Theorem 2, it is reasonable to require that the solution has finite logarithmic moments. It is very useful when considering the invertibility of the general model, see Proposition 2 below.

The invertibility and the observable invertibility

Now that under (ST) the process (X t ) is stationary and ergodic, we investigate the question of invertibility of the general model (1). We want to emphasize that the classical notions of invertibility are related with convergences of SRE and implied by Lyapunov conditions. 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. The model is invertible if the sequence of the volatilities (Σ t ) is adapted to the filtration (G t-1 ) generated by (X t-1 , X t-2 , • • • ).
It is natural to assume invertibility to be able to forecast the volatility and to ensure the so-called "predictability" of the model. This notion of invertibility is very weak and consists in restricting the underlying filtration (F t ) of the SRE to (G t-1 ). Indeed, under (ST) the filtration G t ⊆ F t is well defined. If the volatility matrices are invertible, using Z t = Σ -1 t • X t in ψ t we can express (1) as ( 4

): (h(Σ k )) k≤t = φ t ((h(Σ k )) k≤t-1 , θ 0 ), a SRE
driven by the whole past of the observations. The sequence of random functions (φ t ) is an ergodic and stationary process adapted to (G t-1 ). Using the sufficient conditions of convergence of SREs given in Theorem 1, the invertibility follows if the φ t (•, θ 0 ) are Lipschitz maps such that for some x ∈ E and r > 0,

E[log + (d(x, φ 0 (x, θ 0 )))] < ∞, E[log + Λ(φ 0 (•, θ 0 ))] < ∞ and E[log Λ(φ 0 (•, θ 0 ) (r) )] < 0. ( 8
)
The Remark 1 also holds for the SRE driven by (φ t ): the metric space (E, d) must be chosen carefully. The conditions (8) (with the optimal metric space (E, d)) are called the conditions of invertibility.

Proposition 1. Under (ST) and (8), the general model (4) is invertible.

Another notion of invertibility is introduced in Straumann ( 2005). We call it observable invertibility. As φ t (x, θ) = φ θ,x (X t , X t-1 , . . .) where φ θ,x is measurable for any x, θ, denote φt (x, θ) = φ θ,x (X t , . . . , X 1 , u) where u is an arbitrary deterministic sequence of E N .

Definition 2. The model is observably invertible if and only if the

SRE (h( Σk )) k≤t = φt ((h( Σk )) k≤t-1 , θ 0 ) t ≥ 1 (9)
is convergent for any arbitrary initial values h( Σk )) k≤0 and such that Σt -Σ t → 0 in probability as t → ∞.

Notice that in general the approximative SRE does not fit the conditions of Theorem 1 and in particular ( φt ) is not necessarily stationary and ergodic. However, the Proposition below gives sufficient conditions for observable invertibility. It is a very useful result for the sequel of the paper, see Remark 2. Notice that logarithmic moments are assumed and Theorem 2 is very useful to check this condition.

Proposition 2. If (ST) and (8) hold, if the link function is continuous and it exists

x ∈ E such that d( φt (x), φ t (x))

e.a.s.

---→ 0 and Λ( φt (•, θ 0 )φ t (•, θ 0 )) e.a.s.

---→ 0 as t → ∞, then the model is observably invertible.

Proof. One can extend 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] written for Banach spaces to the case of the complete separable met-

ric space (E, d). That d((h(Σ k )) k≤t ), (h( Σk )) k≤t ))
e.a.s.

---→ 0 follows from 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] under the following assumptions:

S1 E[log + (d(x, φ 0 (x, θ 0 )))] < ∞ for some x ∈ E, S2 E[log Λ(φ 0 (•, θ 0 ))] < ∞ and E[log Λ(φ 0 (•, θ 0 ) (r) )] < 0 for some r > 0, S2' E[log + (d(y, (h(Σ k )) k≤t ))] < ∞ for all y ∈ E, S3 d( φt (x), φ t (x))
e.a.s.

---→ 0 and Λ( φt (•, θ 0 )φ t (•, θ 0 )) e.a.s.

---→ 0 as t → ∞.

The conditions S1-S2 are equivalent to the invertibility conditions (8). S3 holds from the assumptions in Proposition 2 and S2' follows from (ST). Finally, using the continuity of the projection on the first coordinate and the one of the link function , the desired result follows.

Remark 2. Classical models such that GARCH(p,q) or EGARCH(p,q) models satisfy an SRE for finite p lags (h(Σ k )) t-p+1≤k≤t = φ t ((h(Σ k )) t-p≤k≤t-1 , θ 0 ) and for some φ t generated by only a finite of past observation (X t-1 , . . . , X t-q ). In this context, the approximative SRE coincides with the initial ones, i.e. one can choose φt = φ t for t > q. Therefore, conditions of Proposition 2 hold systematically; invertibility and observable invertibility are equivalent, i.e. they are induced by the same Lyapunov condition. As for any initial values of φt (for 0 ≤ t ≤ q) the conditions of Proposition 2 are satisfied, seeking simplicity we work in the sequel with φt = φ t for all t ≥ 1.

The continuous invertibility

Now we are interested in inferring the unknown parameter θ 0 . To that ends, we extend the notions of invertibility at the point θ 0 to the compact set Θ used in the definition (7) of our estimator. For the sake of simplicity, we assume in all the sequel that we are in the framework of Remark 2, i.e. φ t is observable for t > q and φt = φ t . For any θ ∈ Θ, let us consider from the functional SRE of the form

( ĝk (θ)) t-p+1≤k≤t = φ t (( ĝk (θ)) t-p≤k≤t-1 , θ), ∀t ≥ 1, (10)
with arbitrary initial values ( ĝk (θ)) 1-p≤k≤0 . If the model is invertible for all θ ∈ Θ, the function ĝt (•) is well defined and converges to g t (•), the unique stationary solution of (10). For statistical inference, the regularity of g t (•) is required. We call uniform integrability the notion used in [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF] and [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]:

Definition 3. The model is uniformly invertible on Θ if and only if the SRE (10) is convergent when considering it on the Banach space of continuous functions C(Θ).

This notion is too restrictive and we introduce the weaker notion of continuous invertibility as follows Definition 4. The model is continuously invertible on Θ if and only if the SRE (10) is convergent for all θ ∈ Θ and the stationary solution g t (•) is continuous.

We have seen that sufficient conditions for pointwise invertibility are classically expressed in term of Lyapunov conditions. Let us consider models with parametric functions having continuous Lipschitz coefficients:

(CL) For any metric spaces X , Y and Z, a function f : X × Y → Z satisfies (CL) if there exists a continuous function

Λ f : Y → R + such that Λ( f (•, y)) ≤ Λ f (y) for all y ∈ Y.
In this context, the uniform invertibility holds under the Lyapunov condition

E[log sup Θ Λ (r)
φ 0 (θ)] < 0. This condition is too restrictive to handle EGARCH type models because of the supremum inside the expectation. We introduce a sufficient condition (CI) for continuous invertibility in term of a weaker Lyapounov condition than the preceding one:

(CI) Assume that the SRE (10) holds with φ t satisfying (CL) for stationary (Λ φ t ) such that conditions there exists an positve integr r such that E[log Λ (r)

φ 0 (θ)] < 0 on the compact set Θ. Assume moreover that E[sup Θ log + Λ (r) φ 0 (θ)] < ∞ and that there exists y ∈ E such that E[sup Θ log + (d(φ 0 (y, θ), y))] < ∞.
The condition (CI) implies the convergence of the SRE (10) for all θ ∈ Θ. If θ 0 ∈ Θ it implies the invertibility of the model as described in Subsection 2.3. It also implies the local uniform regularity of g t (•) and thus the continuous invertibility:

Theorem 3. Assume that (ST) and (CI) hold. Then the functions g t (•) are continuous for all θ ∈ Θ and all t ∈ Z. Moreover, for any θ ∈ Θ there exists an > 0 such that ĝt (θ) satisfying (10) satisfies lim sup θ ∈B(θ, )∩Θ d( ĝt (θ ), g t (θ )) e.a.s.

---→ 0.

(11)

Proof. Without loss of generality, one can assume that E[log Λ (r) ρ) ∩ Θ}, where B(θ, ρ) stands for the closed ball centered at θ with radius ρ. Noticing that

φ 0 (θ)] > -∞ such that lim K→∞ E[log Λ (r) φ 0 (θ) ∨ (-K)] = E[log Λ (r) φ 0 (θ)] for any θ ∈ Θ. For any ρ > 0, let us write Λ (r) * (θ, ρ) = sup{Λ (r) φ 0 (θ ), θ ∈ B(θ,
E[sup Θ | log Λ (r) φ 0 (θ) ∨ (-K)|] < ∞, using 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 lim ρ→0 log Λ (r) * (θ, ρ) = log Λ (r) φ 0 (θ) and for sufficiently large K lim ρ→0 E(log Λ (r) * (θ, ρ) ∨ (-K)) = E(log Λ (r) φ 0 (θ) ∨ (-K)) < 0.
Thus, there exists an > 0 such that

E(log Λ (r) * (θ, )) ≤ E(log Λ (r) * (θ, ) ∨ (-K)) < 0.
Let us now work on C(B(θ, ) ∩ Θ), the complete metric space of continuous functions from B(θ, ) ∩ Θ to R equipped with the supremum norm d ∞ = sup B(θ, )∩Θ d. In this setting ( ĝt ) satisfy a functional SRE ( ĝk ) k≤t = φt (( ĝk ) k≤t-1 ) with Lipschitz constants satisfying

Λ ∞ ( φ(r) t (•)) ≤ sup s 1 ,s 2 ∈C(B(θ, )∩Θ) d ∞ ( φ(r) t (s 1 ), φ(r) t (s 2 )) d ∞ (s 1 , s 2 ) ≤ sup s 1 ,s 2 ∈C(B(θ, )∩Θ) sup B(θ, )∩Θ d(φ (r) t (s 1 (θ ), θ ), φ (r) t (s 2 (θ ), θ ) d ∞ (s 1 , s 2 ) ≤ sup s 1 ,s 2 ∈C(B(θ, )∩Θ) sup B(θ, )∩Θ Λ(φ (r) t (•, θ ))d(s 1 (θ )s 2 (θ )) d ∞ (s 1 , s 2 ) ≤ sup s 1 ,s 2 ∈C(B(θ, )∩Θ) sup B(θ, )∩Θ Λ(φ (r) t (•, θ ))d ∞ (s 1 , s 2 ) d ∞ (s 1 , s 2 ) ≤ Λ (r) * (θ, ).
As E[sup B(θ, )∩Θ log + (d(φ t (y, θ ), y))] ≤ E[sup Θ log + (d(φ t (y, θ), y))] is finite and E(log Λ (r) φ 0 (θ)) < 0 we apply Theorem 1. By recurrence

φ t • • • • • φ t-m (ζ 0 ) ∈ C(B(θ, ) ∩ Θ)
is continuous in θ and so is g t as the convergence holds uniformly on B(θ, ) ∩ Θ. It is true for any θ ∈ Θ and the result follows.

Statistical inference under continuous invertibility

Consider θn = argmin θ∈Θ Ŝn (θ) the M-estimator associated with the QLIK criteria ( 6) where ( ĝt ) is obtained from the approximative SRE (10).

Remark 3. The statistical procedure described here does not coincide with the Quasi Maximum Likelihood for non uniformly invertible models. For a detailed discussion in the EGARCH(1,1) case see [START_REF] Cai | The EGARCH(1,1) process: structure, estimation and prediction[END_REF].

Strong consistency of the parametric inference

From now on, we assume that the innovations process (Z t ) is iid:

(IN) The Z t are iid variables such that E[Z T 0 Z 0 ]
is the identity matrix. The next assumption implies that the volatility matrices are invertible and that the link function is continuous:

(IV) The functions -1 and log(det( )) are Lipschitz satisfying det( (g 0 (θ))) ≥ C(θ) for some continuous function C : Θ → (0, ∞).

Remark 4. The SRE criteria converges to the possibly degenerate limit

S(θ) = E[s 0 (θ)] = 2 -1 E X T 0 (g 0 (θ)) -1 X 0 + log(det[ (g 0 (θ))]) Notice that S(θ 0 ) = 2 -1 E[Z T 0 Z 0 + log(det(Σ 0
))] is finite under (ST), (IN) and (IV) because h(Σ 0 ) has logarithmic moments and log(det( )) is Lipschitz. It is a considerable advantage of the QLIK criteria: it does not need moments of any order to be defined in θ 0 . Even if S(θ) may be equal to +∞ for θ = θ 0 , it does not disturb the statistical procedure that defines θn as a minimizer.

If the model is identifiable, the estimator θn is strongly consistent: Theorem 4. Assume that (ST) and (CI) are satisfied on the compact set Θ. If (IN) and (IV) are satisfied and the model is identifiable, i.e. g 0 (θ) = h(Σ 0 ) iff θ = θ 0 , then θn → θ 0 a.s. for any θ 0 ∈ Θ.

Proof. With no loss of generality we restrict our propose to θ ∈ Θ satisfying the relation det( ( φt (•, θ))) ≥ C(θ) for all t ≤ 0. We adapt the proof of [START_REF] Jeantheau | Modèles autorégressifs à erreur conditionellement hétéroscédastique[END_REF] and its notation s * t (θ, ρ) = inf{s t (θ ), θ ∈ B(θ, ρ)} and ŝ * t (θ, ρ) = inf{ ŝt (θ ), θ ∈ B(θ, ρ)}. Let us recall Theorem 5.1 in [START_REF] Jeantheau | Modèles autorégressifs à erreur conditionellement hétéroscédastique[END_REF] : The M-estimator associated with the loss ( 6) is strongly consistent under the hypothesis H1-H6: H1 Θ is compact. H2 Ŝn (θ) → S(θ) a.s. under the stationary law P θ 0 . H3 S(θ) admits a unique minimum for θ = θ 0 in Θ. Moreover for any θ 1 = θ 0 we have: lim inf θ→θ 1 S(θ) > S(θ 1 ).

H4 ∀θ ∈ Θ and sufficiently small ρ > 0 the process (ŝ * t (θ, ρ))) t is ergodic.

H5 ∀θ ∈ Θ, E θ 0 [s * 1 (θ, ρ)] > -∞. H6 lim ρ→0 E θ 0 [s * 1 (θ, ρ)] = E[s * 1 (θ)].
Let us check H1-H6 in our case. H1 is satisfied by assumption. H2 is verified in two steps. First, by the e.a.s. convergence given by Theorem 3, arguments of [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF] and the Lipschitz properties of -1 and log(det( )) we obtain 1

n n ∑ t=1 sup B(θ, ) |ŝ t (θ ) -s t (θ )| → 0 P θ 0 -a.s.
Second we use that (s t ) is an ergodic sequence and Proposition 1.1 of [START_REF] Jeantheau | Modèles autorégressifs à erreur conditionellement hétéroscédastique[END_REF]. As the s t are bounded from below n -1 ∑ n t=1 s t (θ) converges P θ 0 -a.s. to S(θ) (taking values in R ∪ {+∞})

1 n n ∑ t=1 |s t (θ) -S(θ)| → 0 P θ 0 -a.s.
Combining this two steps leads to H2. The first part of H3 is checked similarly than in (ii) p.2474 of [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF] and with the help of the Remark 4. Notice that S has a unique minimum iff E[Tr(Σ 0 • (g 0 (θ)) -1 )log(det(Σ 0 • (g 0 (θ)) -1 ))] has a unique minimum. As this criteria is the integrand of a sum of the λ ilog(λ i ) where the λ i are positive eigenvalues, we conclude under the identifiability condition from the property xlog(x) ≥ 1 for all x > 0 with equality iff x = 1. The second part is checked using the fact that lim inf

θ→θ 1 S(θ) ≥ E[lim inf θ→θ 1 s 0 (θ)] = E[s 0 (θ 1 )] = S(θ 1 )
where the first inequality was already used for proving Theorem 3 and the first equality comes from the local continuity of g 0 and . H4 is satisfied from the ergodicity of (ŝ t ). H5 and H6 follows from Theorem 3 that ensures the continuity of the function s * 1 and by the lower bounded assumption on det( ), see Proposition 1.3 of [START_REF] Jeantheau | Modèles autorégressifs à erreur conditionellement hétéroscédastique[END_REF].

Volatility forecasting

From the inference of θ 0 , we deduce a natural forecast of the volatility Σt = ( ĝt ( θt )). It is strongly consistent:

Theorem 5. Under the conditions of Theorem 4 then Σt -Σ t → 0 a.s. as t → ∞.

Proof. It is a consequence of Theorems 3 and 4 that assert the a.s. convergence of θt toward θ 0 and the local uniform convergence of ĝt toward g t . Notice that for t sufficiently large such that θt ∈ B(θ, ), a ball where the uniform Lyapunov condition E[log Λ ∞ (φ t (•))] < 0 is satisfied. Thus ĝt ( θt )g t (θ t ) → 0 a.s. and by continuity of and g t and from the identification Σ t = (g t (θ 0 )) the result follows if g t ( θt ) converges to g t (θ 0 ). For proving it, we use

d(g t ( θt ), g t (θ 0 )) ≤ Λ ∞ (φ t (•))d(g t-1 ( θt ), g t-1 (θ 0 )) + w t ( θt ) where w t ( θt ) = d(φ t (g t-1 (θ 0 ), θt ), φ t (g t-1 (θ 0 ), θ 0 )). The quantity d(g t ( θt ), g t (θ 0 )) is upper bounded by v t satisfying the SRE of linear stationary maps v t = Λ ∞ (φ t (•))v t-1 + w t ( θt ). We apply Theorem 1 as E[log(Λ ∞ (φ t (•)))] < 0 and E log + (w 0 ( θt )) ≤ E[sup Θ log + (2d(φ t (y, θ), y))] < ∞. We get d(g t ( θt ), g t (θ 0 )) ≤ ∞ ∑ i=0 Λ ∞ (φ t (•)) • • • Λ ∞ (φ t-i+1 (•))w t-i ( θt ).
Conditioning on ( θt ), the upper bound is a stationary normally convergent series of functions and

P ∞ ∑ i=0 Λ ∞ (φ t (•)) • • • Λ ∞ (φ t-i+1 (•))w t-i ( θt ) → 0 = E P ∞ ∑ i=0 Λ ∞ (φ t (•)) • • • Λ ∞ (φ t-i+1 (•))w t-i ( θt ) → 0 | ( θt ) = E P ∞ ∑ i=0 Λ ∞ (φ 0 (•)) • • • Λ ∞ (φ -i+1 (•))w -i ( θt ) → 0 | ( θt ) = E[1] = 1,
the last inequalities following from the continuity of normally convergent series of functions, θt → θ 0 and w i ( θt ) → w i (θ 0 ) = 0 for all i a.s. as t → ∞.

Asymptotic normality of the parametric inference

Classical computations show that if the M-estimator θn is asymptotically normal then the asymptotic variance is given by the expression

V = P -1 QP -1 with P = E[Hs 0 (θ 0 )] and Q = E[∇s 0 (θ 0 )∇s 0 (θ 0 ) T ],
where Hs 0 (θ 0 ) and ∇s 0 are the Hessian matrix and the gradient vector of s 0 (θ 0 ).

(AV) Assume that E( Z 0 Z T 0 2 ) < ∞ and that the functions and φ t are 2-times continuously differentiable on the compact set Θ that coincides with the closure of its interior.

The following moments assumptions ensure the existence of Q and P:

(MM) Assume that E[ ∇s 0 (θ 0 ) 2 ] < ∞ and E[ Hs 0 (θ 0 ) ] < ∞.
These moments assumptions holds only for θ = θ 0 ; they are simpler to verify than for the moment conditions for θ = θ 0 due to the specific form of the derivatives of the SRE criteria at θ 0 , see Remark 4 for details. The next assumption is classical and ensures to the existence of P -1 :

(LI) The components of the vector ∇g 0 (θ 0 ) are linearly independent.

Let V = B(θ 0 , ) ⊂ Θ with θ 0 ∈ • Θ and > 0 chosen in accordance with Theorem 3, i.e. such that E[log(sup V Λ φ 0 )] < 0. The two next assumptions are specific to the SRE approach. They ensure that ∇ŝ t (θ) is a good approximation of ∇s t (θ) uniformly on the neighborhood V of θ 0 :

(DL) The partial derivatives Φ t = D x (φ t ), = D θ (φ t ), = D 2 x 2 (φ 0 ), D 2 θ,x (φ 0 ) or D 2 θ 2 (φ 0 ) satisfy (CL) for stationary (Λ Φ t ) with E[sup V log(Λ Φ 0 )] < ∞. Assume there exists y ∈ E such that E[sup V (log + (d(φ 0 (y, θ), y))) 2 ] < ∞.
(LM) Assume that y → ∇ -1 (y) and y → ∇ log(det( (y))) are Lipschitz functions.

Now that V is well defined in terms of derivatives of s 0 that are well approximated by derivatives of the SRE, the procedure is asymptotically normal:

Theorem 6. Under the assumptions of Theorem 4, (AV), (MM), (LI), (DL) and (LM) then the asymptotic variance V is a well defined invertible matrix and

√ n( θn -θ0 ) d → N (0, V) if θ 0 ∈ • Θ .
Proof. Under (CI) and (LM) we apply Theorem 2 to (sup V g t (θ)). We obtain

E[log + (sup V g 0 (θ))] < ∞.
From the Lipschitz condition in (DL) we have that

E[sup V log + ( Φ 0 (θ) )] < ∞ for Φ 0 (θ) = D θ (φ 0 )(g 0 (θ), θ) or D 2 x 2 (φ 0 )(g 0 (θ), θ) or D 2 θ,x (φ 0 )(g 0 (θ), θ) or D 2 θ 2 (φ 0 )(g 0 (θ), θ).
Using the existence of these logarithmic moments and the relation E[log(sup V Λ φ 0 )] < 0, we apply recursively the Theorem 1 and prove the existence of continuous first and second derivatives of (g t (θ)) on V as solutions of functional SRE. The asymptotic normality follows from the Taylor development of Section 5 of [START_REF] Bardet | Asymptotic normality of the quasi maximum likelihood estimator for multidimensional causal processes[END_REF] on the first partial derivatives ∇ i of S n

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

1. n -1/2 ∇S n (θ 0 ) → N (0, Q), 2.
n -1 HS n ( θn ) -P converges a.s. to 0 for any sequence ( θn ) converging a.s. to θ 0 and P is invertible, 3. n -1/2 ∇ Ŝn ( θn ) -∇S n ( θn ) converges a.s. to 0.

Due to its specific expression and Assumption (IN) the process (∇S n (θ 0 )) is a martingale. Under (MM), the CLT for martingales applied to (∇S n (θ 0 )) leads to the first condition. The first part of the second condition is derived from similar arguments than in the proof of Theorem 5 and an application of the Cesaro mean theorem ensuring that n -1 HS n ( θn ) -∑ n t=1 Hs t (θ 0 ) → 0 a.s. The ergodic Theorem on (Hs t (θ 0 )) with (MM) leads to n -1 HS n ( θn ) -P → 0 a.s. The fact that P is invertible follows from (LI), see [START_REF] Bardet | Asymptotic normality of the quasi maximum likelihood estimator for multidimensional causal processes[END_REF] for detailed computations. Finally the third condition is obtained by applying Theorem 2.10 of [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF] to the SRE satisfied by (∇g t ) and its approximative SRE satisfied by (∇ ĝt ) uniformly on V. Thus sup V ∇ ĝt -∇g t e.a.s.

---→ 0 as t → ∞ and Lipschitz conditions on ∇ -1 and ∇ log(det( )) in (LM) and arguments similar than in [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF] leads to the desired result.

Applications to GARCH and EGARCH type models

Some applications to GARCH type models

In GARCH type models, the stationarity assumption (ST) is crucial, whereas the continuous invertibility condition (CI) is automatically satisfied on well chosen compact sets Θ due to the form of the model (they are also uniformly invertible). The asymptotic properties of the QMLE (that coincides with the procedure described here in these cases, see Remark 3) follow from Theorems 4 and 6. Thus, we recover and slightly refine existing results in the AGARCH and CCC-GARCH models (we refer the reader to [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF] and [START_REF] Francq | QML estimation of a class of multivariate asymmetric GARCH models[END_REF] respectively for details). Moreover, we prove also for the first time the strong consistency of the natural volatility forecast in both cases under the conditions of strong consistency of θn .

First, let us detail the case of the univariate APGARCH(p, q) model introduced in [START_REF] Ding | A long memory property of stock market returns and a new model[END_REF], [START_REF] Zakoïan | Threshold heteroscedastic models[END_REF] and studied in [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF]:

σ 2 t = α 0 + p ∑ i=1 α i (|X t-i | -γX t-i ) 2 + q ∑ j=1 β j σ 2 t-j , t ∈ Z,
where α 0 > 0, α i , β j ≥ 0 and |γ| ≤ 1 (it coincides with the GARCH(p, q) model if γ = 0. Then we derive the strong consistency and the asymptotic normality directly from our Theorems 4 and 6. The conditions we obtained coincides with these of Theorem 5.5 and Theorem 8.1 of [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF] except their useless condition (8.1) as one does not need moment of any order.

Second, let us consider the multivariate CCC-GARCH(p, q) model introduced by [START_REF] Bollerslev | Modeling the coherence in short run nominal exchange rates: a multivariate generalized ARCH approach[END_REF], first studied in [START_REF] Jeantheau | Strong consistency of estimation for multivariate ARCH models[END_REF] and refined in [START_REF] Francq | QML estimation of a class of multivariate asymmetric GARCH models[END_REF]

Diag(Σ 2 t ) = A 0 + q ∑ i=1 A i Diag(X t-i X T t-1 ) + p ∑ i=1 B i Diag(Σ 2 t-i )
and (Σ 2 t ) i,j = ρ i,j (Σ 2 t ) i,i (Σ 2 t ) j,j ) for all (i, j), where Diag(M) is the vector of the diagonal elements of M. A necessary and sufficient conditions for (ST) is given in term of top Lyapunov condition in [START_REF] Francq | QML estimation of a class of multivariate asymmetric GARCH models[END_REF]. We recover the strong consistency and the asymptotic normality of [START_REF] Francq | QML estimation of a class of multivariate asymmetric GARCH models[END_REF] directly from our Theorems 4 and 6.

Application to the EGARCH(1, 1) model

Let (Z t ) be an iid sequence of random variables not concentrated on two points such that E(Z 2 0 ) = 1. The EGARCH(1, 1) model introduced by Nelson ( 1991) is an AR(1) model for log σ 2 t ,

X t = σ t Z t with log σ 2 t = α 0 + β 0 log σ 2 t-1 + W t-1 (θ 0 )
where W t (θ 0 ) = γ 0 Z t + δ 0 |Z t | are the innovations of this AR(1) model. Let θ 0 = (α 0 , β 0 , γ 0 , δ 0 ) be the unknown parameter. Assume that |β 0 | < 1 such that there exists a stationary solution having a MA(∞) representation:

log σ 2 t = α 0 (1 -β 0 ) -1 + ∞ ∑ k=1 β k-1 0 W t-k (θ 0 ). (12)
The moments assumptions on Z t ensures that the process (log σ 2 t ) is ergodic, strongly and weakly stationary. Then the volatilities process (σ 2 t ) is also ergodic and strongly stationary and (ST) holds. However, it does not necessarily have finite moment of any order.

The invertibility of the stationary solution of the EGARCH(1, 1) model does not hold in general. A sufficient condition for invertibility is given in [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]. As (log

σ 2 t ) satisfies the SRE log σ 2 t = α 0 + β 0 log σ 2 t-1 + (δ 0 X t-1 + γ 0 |X t-1 |) exp(-log σ 2 t-1 /2),
if it has a non anticipative solution the model is invertible. Keeping the notation of Section 2, the function h is now the logarithmic function and the SRE (9) holds with (φ t ) defined by

φ t (•; θ) : s → α + βs + (γX t-1 + δ |X t-1 |) exp(-s/2)
We check that the φ t are random functions generated by G t-1 . For any θ ∈

R × R + × {γ ≥ |δ|} we restrict φ t (•; θ) on the complete separable metric space [α/(1 -β), ∞) equipped with d(x, y) = |x -y|. The process (φ t (•; θ)) is a sta- tionary ergodic sequence of Lipschitz maps from [α/(1 -β), ∞) to [α/(1 - β), ∞) with Lipschitz coefficients Λ(φ t (•, θ 0 )) ≤ max{β, 2 -1 (γX t-1 + δ|X t-1 |) exp(-2 -1 α/(1 -β)) -β}.
The technical smoothness assumption (CL) is automatically satisfied as

(Λ φ t (θ)) t = (max(β, 2 -1 (γX t-1 + δ|X t-1 |) exp(-2 -1 α/(1 -β)) -β)) t is a stationary process of continuous functions of θ. The EGARCH(1,1) is con- tinuously invertible on any compact set Θ ⊂ R × R + × {γ ≥ |δ|} such that E[log(max{β, 2 -1 (γX 0 + δ|X 0 |) exp(-2 -1 α/(1 -β)) -β})] < 0. ( 13
)
This sufficient condition for continuous invertibility depends on the distribution of the observations (X t ). Notice that if θ 0 ∈ Θ then it satisfies the condition of stationarity β 0 < 1 and, from the MA(∞) representation ( 12) of log σ 2 t , our condition (13) expressed at θ 0 implies that

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. (14)
It is the condition of invertibility of the EGARCH(1,1) model given in [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]. Applying our results of Theorem 4 and Theorem 5, we obtain Theorem 7. For any compact subset Θ of R × R + × {γ ≥ |δ|} satisfying (13) then θn → θ 0 and σ2

n -σ 2 n → 0 a.s. as n → ∞ with σ2 t = exp( ĝt ( θn )) if θ 0 ∈ Θ. Proof. The condition (CI) follows from E[log Λ(φ t (, θ))] < 0 by assumption of Θ and E[sup Θ log Λ(φ t (, θ))] < ∞ since E log |X t-1 | = E(log σ + log |Z t-1 |) < ∞ as log σ 2
t has a MA(∞) representation ( 12) and Z is integrable. Moreover as log + (d(φ 0 (0, θ), 0

)) = log + |α + (γX -1 + δ|X -1 |)| then fixing y = 0 one has E[sup Θ (log + (d(φ 0 (y, θ), y)) 2 ] < ∞.
In the EGARCH(1, 1) model the link function is the exponential function (x) = exp(x) and since we have log

σ 2 t ≥ α/(1 -β), 1/ (x) = exp(-x) is a Lipschitz function (log(det( )) = id is also a Lipschitz function). Moreover the volatility process (σ 2 t ) is bounded from below by C(θ) = exp(α/(1 -β)).
Finally, the identifiability condition g 0 (θ) = h(θ 0 ) iff θ = θ 0 is checked in Section 5.1 of [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF].

Notice that the procedure is valid only if θ ∈ Θ satisfies (13). In practice, we suggest to optimize the QLIK criteria under the empirical constraint

n ∑ t=1 log(β, 2 -1 (γX t + δ|X t |) exp(-2 -1 α/(1 -β)) -β) < 0.
Existing procedures does not constrain the model to be invertible. The corresponding empirical results are not asymptotically valid since the underlying SREs are not stable, see [START_REF] Cai | The EGARCH(1,1) process: structure, estimation and prediction[END_REF] for evidences on simulations. Notice also that the strong consistency of θn has already been proved in [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF] and [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF] under the uniform invertibility condition E[sup Θ log(max{β, 2 -1 (γX t-1 + δ|X t-1 |) exp(-2 -1 α/(1β)) -β})] < 0.

We share the appreciation of the authors that this condition is very restrictive and might not be satisfied for reasonable compact sets Θ with β = 0 and non empty interior.

As a corollary of Theorem 6 we get the asymptotic normality of the statistical inference in the EGARCH(1, 1) model. The only existing result was Theorem 5.7.9 of [START_REF] Straumann | Estimation in Conditionally Heteroscedastic Time Series Models[END_REF] valid only for degenerate EGARCH(1,1) models with β = 0. Our result holds under consistency assumptions and the following necessary and sufficient condition for the existence of V: Proof. By definition, (φ t ) is 2-times continuously differentiable and simple computations give D x (φ t )(x, θ) = β -2 -1 (γX t-1 + δ|X t-1 |) exp(-x/2), D θ (φ t )(x, θ) = (1, x, X t-1 exp(-x/2), |X t-1 | exp(-x/2)) T , D 2 x 2 (φ t )(x, θ) = 4 -1 (γX t-1 + δ|X t-1 |)

exp(-x/2), D 2 x,θ (φ t )(x, θ) = (0, 1, 2 -1 X t-1 exp(-x/2), 2 -1 |X t-1 | exp(-x/2)) T and D 2 θ 2 (φ t )(x, θ) = 0. Moreover, as the link function is (x) = exp(x) is also 2-times continuously differentiable, the last assertion of the condition (AV) of Theorem 6 holds. The fact that (MM) holds iff (MM') holds is technical and postponed to the Appendix. The fact that (LI) holds if Z 0 is not concentrated on two points is classical, see for instance Lemma 8.2 of [START_REF] Straumann | Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach[END_REF]. Assumption (DL) is satisfied from the expressions of the derivatives (that are Lipschitz functions) and as all the logarithmic moments are finite due to E[log(X 2 t-1 )] < ∞. Finally (LM) is automatically satisfied due to the specific expression of the link function. Thus Theorem 6 applies.

Appendix: (MM) holds in the EGARCH(1,1) model iff (MM') holds

To check (MM) is technical as we have to compute explicitly the diagonal terms the matrix B = E[∇g t (θ 0 )(∇g t (θ 0 )) T ]. Similar computations have been done in [START_REF] Demos | Asymptotic expansions of the QMLEs in the EGARCH(1,1) model[END_REF]. Remember that V = P -1 QP -1 with P = E[Hs 0 (θ 0 )] and Q = E[∇s 0 (θ 0 )∇s 0 (θ 0 ) T ]. Let us first prove the three identities P = 2 -1 B, Q = 4 -1 (EZ 4 0 -1)B and thus V = (EZ 4 0 -1)B -1 . For the first identity, we compute

P = 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.
For the second identity, we compute

Q = E 1 4 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 and the third identity follows the first ones. Thus, for checking the assumption (MM), it is enough to check that diagonal coefficients B ii are well defined when E(Z 4 0 ) < ∞. Let us denote W t = γ 0 Z t + δ 0 |Z t |, 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 (θ 0 ) = U t-1 + V t-1 ∇g t-1 (θ 0 ) = ∞ ∑ l=1 U t-l l-1 ∏ k=1 V t-k .
Using the convention ∏ 0 k=1 V t-k = 1, we obtain the expression

∇g t (θ 0 ) = ∞ ∑ l=1 U t-l l-1 ∏ k=1 V t-k .
More precisely, we have the expressions: then

B 11 = E ∂g t (θ 0 ) ∂θ 1 2 = E ∞ ∑ l=1 l-1 ∏ k=1 V t-k 2 , B 22 = E ∂g t (θ 0 ) ∂θ 2 2 = E ∞ ∑ l=1 log σ 2 t-l l-1 ∏ k=1 V t-k 2 , B 33 = E ∂g t (θ 0 ) ∂θ 3 2 = E ∞ ∑ l=1 Z t-l l-1 ∏ k=1 V t-k 2 , B 44 = E ∂g t (θ 0 ) ∂θ i 2 = E ∞ ∑ l=1 |Z t-l | l-1 ∏ k=1 V t-k 2 .
To prove that condition (MM) is satisfied, i.e. that ∑ 4 i=1 B ii < ∞, we use the following Lemma

(

  MM') E[Z 4 0 ] < ∞ and E[(β 0 -2 -1 (γ 0 Z 0 + δ 0 |Z 0 |) 2 ] < 1.Theorem 8. Assume that assumptions of Theorem 7 are satisfied and that (MM')holds. Then √ n( θnθ 0 ) d → N (0, V) where V is an invertible matrix if θ 0 ∈ • Θ.

Acknowledgments

One of the author would like to thank C. Francq and J.-M. Zakoïan for helpful discussions and for having pointing out some mistakes in an earlier version.

Lemma 1. ∑ 4 i=1 B ii < ∞ iff EV 2 0 < 1. Proof. That the first coefficient B 11 is finite comes easily from the identities

For the second coefficient B 22 , it is more complicated. We need some preliminary work. We know that

so, we decompose B 22 into three parts,

That the first term of the sum is finite is already known. For the last term, it is straightforward from

For the second term of the sum, we need an expansion

and in expectation we obtain a bounded term if EV 2 0 < 1:

That the last coefficient is also finite comes form the computation