Minimum Hellinger Distance Estimation of a Univariate GARCH Process

General Autoregressive conditionally heteroscedastic (GARCH) models were pioneered by Engle(1982) and Bollerslev(1986), and have ever since been widely used to analyze financial time series. Parameters of GARCH models are usually estimated by the quasi-maximum likelihood estimator (QMLE) (Berkes, Horváth, & Kokoszka, 2003) and (Francq & Zakoïan, 2004). The QML estimator is well-known for its efficiency asymptotic properties under regular conditions, however it has very bad robustess properties.


Introduction
General Autoregressive conditionally heteroscedastic (GARCH) models were pioneered by Engle(1982) and Bollerslev(1986), and have ever since been widely used to analyze financial time series.Parameters of GARCH models are usually estimated by the quasi-maximum likelihood estimator (QMLE) (Berkes, Horváth, & Kokoszka, 2003) and (Francq & Zakoïan, 2004).The QML estimator is well-known for its efficiency asymptotic properties under regular conditions, however it has very bad robustess properties.
In this paper we estimate the parameters of GARCH process using the minimum Hellinger distance (MHD) method, under uniform mixing (or ϕ-mixing) condition.
The interest for this method of parametric estimation is that the minimum Hellinger distance estimation method gives efficient and robust estimators (Beran, 1977).The minimum Hellinger distance estimators have been used in parameter estimation for independent observations (Beran, 1977), for nonlinear time series models (Hili, 1995) and recently for univariate long memory linear processes (Bitty & Hili, 2010), for nonlinear univariate and multivariate gaussian process (N'dri & Hili, 2011(N'dri & Hili, , 2013)), for parameter estimation of one-dimensional diffusion process (Apala & Hili, 2013).
The paper is organized as follows.In section 2 we give the definition and some properties of the GARCH model.Section 3 contains the definition of the estimator and some assumptions.Sections 4 and 5 are the main results of the paper.They respectively establish the consistency and the asymptotic normality of the estimator θ n .In section 6 we did some numerical simulations.In section 7 we apply MHD method to a financial time series.In section 8 we open problem.

Definition and Some
The α i and β i are nonnegative constants and ω is a (strictly) positive constant.
then, The GARCH(p, q) process (X t ) t∈Z defined in (1.2) admits a unique strictly stationary solution.
For the following properties see Davis & Mikosch (2008).

Definition of the Estimator and Some Assumptions
Let X 1 , ... , X n be an observed sequence of GARCH processes with the density belonging to a specified parametric family { f θ } θ∈Θ where Θ is the parameter espace, a compact set of R p+q+1 .Note that in our study the form of the density is not explicit.
Let f n be a nonparametric estimator of the density f θ defined as where K (.) is a kernel function and (b n ) is a sequence of bandwidths.
The Minimum Hellinger Distance estimator θ n of θ 0 is the value in the parameter espace Θ which minimizes the Hellinger distance (denoted H 2 ) between f n and f θ defined by: To establish the asymptotic properties of the estimator θ n , we need the following assumptions.

Assumption A2
For each θ ∈ Θ, the density f θ of X t is positive over all R and twice continuously differentiable.

Assumption A4
We chose b n such that lim

Assumption A5
The continuous function K is symmetric positive, bounded function with compact support such that : } is a set of positive Lebesgue measure.
Proof of theorem 4.1.
Let F denote the set of all densities with respect to the Lebesgue measure on R.
Define the functional U : provided such minimum exists.In cas U (g) is multiple-value, the notation U (g) will represent one of possible values chosen arbitrarily.
We have By lemmas 4.2 and 4.3 Thus f n (x) −→ f θ 0 (x) a.s when n −→ ∞ in the Hellinger topology.
We have Using assumption (A5) and Jensen's inequality, we get where K 0 is a constant.
Using assumption (A5) and the Taylor's expansion in a neighbourhood of x, we have where Then, by assumption (A3) and ( A5) where C 1 is a constant.

Asymptotic Normality of the Estimator θ n
For the following theorem, denote by .
Proof of lemma 5.2.
We have ) . Thus, )) Step 1: we prove that where u ∈supp(K) the support of kernel density K (.) a compact set.
On other hand, for all u ∈supp(K) and for all n ∈ N we obtain ∫ Therefore, by the dominated convergence theorem Step 2: we prove that Let ψ be the function defined by: for all u ∈supp(K) a compact set and for all x ∈ R ψ is a continuous function.Therefore ψ (X i ) is ϕ-mixing.
We have Thus, there exists a constant C 5 such that On the other hand, we know that where C > 0 and 0 < ρ < 1. Therefore, Let χ be the decreasing function defined by: for all k ∈ N )) .
Proof of lemma 5.4.
We have We have )) and )) Using the same arguments as in the lemma 5.3, we can prove that where χ is defined in (5.3).
We conclued that, To confirm the performance of the estimator, we use the sample bias and the root means quare error (RMSE) defined as follows: Table 1 shows the consistency of the MHD estimator.The above results show the good performance of the MHD method because all estimations biases are close to 0. Also, the table 1 shows that the estimations biases and the RMSE of the MHD method are small or almost equal to the estimation biases and the RMSE of QMLE.The MHD estimator seems better performed than the QML estimator.
To illustrate the robustness of the MHD estimator, we proceed in this manner: Let and δ[0, 1] the uniform density on the interval [0, 1].We vary λ between 0 and 1 and consider the MHD associated estimator.In each case, we replace f n by f n,λ .This gives the following table with nine values of λ.

Application in Finance
The data used for our empirical study are daily returns of S&P500 index of 1272 observations.The study period is from 21-01-2009 to 02-01-2004 (cf figure 1 page 22).
We define the daily returns r t of S&P500 index as follows: where p t is the price at the end of trading day t.One can adjust a GARCH(1,1) model to the series (r t ).Supposing that we are looking for the return at the date t.Two informations are particulaly interesting: the average and the variance.Particulary, we hope modeling the average and the conditional variance of r t .Then, for the GARCH modeling, it amounts to write: where X t = √ h t η t n t i.i.dN (0, 1) with µ and h t are respectily the conditional average and conditional variance of r t .
To estimate the parameters (w, α, β) , we will use the MHD method.For that, we use the method described in the paragraph 6 with 50 replications of the model (7.1) and the bandwidth b n = n −0.23 .We obtain w = 1.30022275, α = 0.06422973, β = 0.92449759.and µ = −0.2311027 Finaly we can write, r t = −0.2311027+ X t

Table 1 .
Consistency of the MHD estimator and the comparison between MHDE and QMLE