Mohamed Boutahar

Rabeh Khalfaoui

Estimation of the long memory parameter in non stationary models: A Simulation Study

Keywords: Wavelets, long memory, tapering, non-stationarity, volatility

In this paper we perform a Monte Carlo study based on three well-known semiparametric estimates for the long memory fractional parameter. We study the efficiency of Geweke and Porter-Hudak, Gaussian semiparametric and wavelet Ordinary Least-Square estimates in both stationary and non stationary models. We consider an adequate data tapers to compute non stationary estimates. The Monte Carlo simulation study is based on different sample size. We show that for d ∈ [1/4, 1.25) the Haar estimate performs the others with respect to the mean squared error. The estimation methods are applied to energy data set for an empirical illustration.

In recent years, studies about long memory have received the attention of statisticians and mathematicians. This phenomenon has grown rapidly and can be found in many fields such as hydrology, chimistry, physics, economic and finance. For instance, the studies of Diebold and Redebusch (1989) and Sowell (1992b) for real gross national product, [START_REF] Shea | Uncertainty and implied variance bounds in long-memory of the interest term structure[END_REF] for interest rates, [START_REF] Lo | Long-term memory in stock market prices[END_REF] and [START_REF] Taqqu | Estimators of long-range dependence: an empirical study[END_REF] for Ethernet traffic, [START_REF] Boutahar | Estimation Methods of the Long Memory Parameter: Monte Carlo Analysis and Application[END_REF] for US inflation rate and [START_REF] Tsay | Estimating long memory time-series-cross-section data[END_REF] for political time series. Moreover, [START_REF] Bollerslev | Modeling and pricing long memory in stock market volatility[END_REF] proposed the FIGARCH model for modelling the persistence in the volatility in financial time series. The concept of long memory describes the property that many time series models possess, despite being stationary, higher persistence than short memory models, such as ARMA models.

Usually a long memory model X t can be characterized by a single memory parameter d ∈ (0, 1/2) called the degree of memory of the model, which controls the shape of the spectrum near zero frequency and the hyperbolic decay of its autocorrelation function. More precisely, the spectral density, f X (λ ), of the long memory model is approximated in the neighborhood of the zero frequency by f X (λ ) ∼ cλ -2d as λ → 0 + , 0 < c < ∞.

(1.1) thus, f X (λ ) → ∞ as λ → 0 + . Under additional regularity assumptions of f X (λ ), the autocorrelation function ρ (k) of the long memory model has the following asymptotic behavior:

ρ (k) ∼ ck 2d-1 as k → ∞. (1.2)
As a consequence, for 0

< d < 1/2, ∑ +∞ -∞ |ρ (k)| = ∞.
This property of absolute nonsummability of autocorrelations is often considered as definition of long memory and is satisfied by the ARFIMA models [START_REF] Granger | Long memory relationships and the aggregation of dynamic models[END_REF][START_REF] Granger | An introduction to long-memory time series models and fractional differencing[END_REF][START_REF] Hosking | Fractional Differencing[END_REF]. The properties of the model X t depend on the parameter value d. Several estimation techniques have been proposed in the literature for detecting the long memory phenomenon, in both time and frequency domains [START_REF] Sowell | Maximum Likelihood Estimation of Stationary Univariate Fractionally Integrated Time Series Models[END_REF][START_REF] Beran | Statistics for long memory processes[END_REF]. These methods are grouped into three categories: the heuristic, the semiparametric and the parametric method. In the first group, we can cite for example [START_REF] Hurst | Long-term storage capacity of reservoirs[END_REF], [START_REF] Higuchi | Approach to an irregular time series on the basis of the fractal theory[END_REF] and [START_REF] Lo | Long-term memory in stock market prices[END_REF]. The second method focuses on the frequency domain estimation, in this context the most popular method is the one developed by [START_REF] Geweke | The Estimation and Application of Long Memory Time Series Models[END_REF]. See also [START_REF] Reisen | Estimation of fractional difference parameter in the ARFIMA(p,d,q) model using the smoothed periodogram[END_REF], [START_REF] Chen | Lag window estimation of the degree of differencing in fractionally integrated time series models[END_REF], Robinson (1995a) and [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF] among others. The third method based on the maximum likelihood function is proposed by [START_REF] Whittle | Hypothesis testing in time series analysis[END_REF], [START_REF] Fox | Large sample properties of parameter estimates for strongly dependent stationary time series[END_REF], [START_REF] Dahlhaus | Efficient parameter estimation for self-similar processes[END_REF] and [START_REF] Sowell | Maximum Likelihood Estimation of Stationary Univariate Fractionally Integrated Time Series Models[END_REF]. In this class we estimate simultaneously all the parameters such as the long memory parameter d, the short-run AR, the MA parameters and the scale parameter. Another semiparametric method for estimating the long memory parameter is the so-called wavelet method proposed by [START_REF] Mccoy | Wavelet analysis and synthesis of stationary long memory processes[END_REF] and Jensen (1999a). Some recent simulation studies comparing different techniques of estimation in stationary long memory models can be found in [START_REF] Taqqu | Estimators of long-range dependence: an empirical study[END_REF], [START_REF] Reisen | Estimation of seasonal fractionally integrated processes[END_REF], [START_REF] Boutahar | Estimation Methods of the Long Memory Parameter: Monte Carlo Analysis and Application[END_REF] and [START_REF] Tsay | Estimating long memory time-series-cross-section data[END_REF]. We propose here a Monte Carlo study to compare the GPH, Gaussian semiparametric and wavelet OLS estimation method in both stationary and non stationary models. More specifically, our main objective is to determine the best estimation method of the long memory. The rest of the paper is organized as follows. Section 2 describes the importance of fractional differencing. Section 3 briefly recalls definitions of long memory models. In section 4 we provide a brief theoretical background of wavelets. 3 Section 5 describes the GPH, the Local Whittle and the wavelet estimation methods. Section 6 proposes a simulation study. An empirical application will be presented in section 7. Section 8 concludes.

Fractional differencing

Importance of fractional differencing

Most financial and economic time series are non-stationary, with their means and covariance fluctuating in time. Therefore, how to transform a non-stationary time series into a stationary one became an important problem in the field of time series analysis. The point of fractional differencing models is not merely to allow for the memory measure to be fractional, but to allow it to be unknown, and estimated from data. Let X t be a time series, we can obtain a new series Y t by differencing X t (i.e. Y t = (1 -L) d X t ). Let f Y (λ ) be the spectral density of Y t . The spectral density of X t is given by

f X (λ ) = 1 -e iλ -2d
f Y (λ ).

(2.1)

Meaning of fractional differencing

The fractional differencing method proposed in this paper illustrates the essence of long-term memory. It shows the connection between differencing parameter d and long-term memory.

Consider the ARFIMA(0,d,0) model often called fractional white noise which can be expressed as

(1 -L) d X t = ε t where (1 -L) d = ∑ ∞ k=0 δ k (d)L k and δ k (d) = Γ (k -d) Γ (k + 1) Γ(-d) , (2.2)
where Γ(.) denotes the gamma function. When d = 0, X t is merely a white noise, and its autocorrelation function is equal to 0. When d = 1, X t is a random walk, whose value of autocorrelation function is 1, and it can be regarded as a white noise after the first-order differencing.

When

d is non-integer, X t = -∑ +∞ k=1 δ k (d)X t-k + ε t ,
and hence X t is influenced by all historical data (X t-1 , X t-2 , . . . ), this is just the characteristic of long-term memory.

Long memory

Researchers in several fields had noticed that the correlation between observations sometimes decayed at a slower rate than for data following classical ARMA models. Later on, as a direct result of the pioneer research of [START_REF] Mandelbrot | Fractional Brownian motions, fractional noises and applications[END_REF], self-similar and long memory models were introduced in the fields of statistics as a basis for inferences. Since then, this field is experiencing a considerable growth in the number of research results (for instance, see [START_REF] Franco | Bootstrap techniques in semiparametric estimation methods for ARFIMA models: a comparison study[END_REF]; [START_REF] Boutahar | Estimation Methods of the Long Memory Parameter: Monte Carlo Analysis and Application[END_REF]; [START_REF] Reisen | Estimation of seasonal fractionally integrated processes[END_REF] and the references therein). The ARFIMA models are extension of ARMA models (short memory), thus we introduce a long memory characteristic by the fractional integration.

Definition and characteristics

A process X t , t ≥ 0, is an ARFIMA(0,d,0) or I(d) model if:

(1 -L) d (X t -µ) = ε t , (3.1)
where ε t is a white noise model and d is a real number such that |d| < 1/2. This model is stationary for d < 1/2 and invertible for d > -1/2. Its Wold representation is given by:

X t = µ + +∞ ∑ k=1 δ k (-d)ε t-k . (3.2)
where δ k (d) is given by (2.2). For 0 < d < 1/2, the autocorrelations decay hyperbolically like k 2d-1 as k → +∞. Similarly, the spectral density of I(d) models behaves like λ -2d as λ → 0. More generally, an ARFIMA(p,d,q) model is defined as:

Φ(L) (1 -L) d (X t -µ) = Θ(L)ε t , (3.3) where ε t ∼ i.i.d(0, σ 2 ε ) is a white noise process, Φ(L) = 1 -φ 1 L -. . . -φ p L p and Θ(L) = 1 + θ 1 L + . . . + θ q L q . The spectral density of the model {X t } t=1,...,T is f (λ ) = σ 2 ε 2π Θ(e iλ ) Φ(e iλ ) 2 1 -e iλ -2d
, forπ < λ < π.

(3.4) f (λ ) is continuously differentiable for all non-zero frequencies. For d > 0, f (λ ) is discontinuous and unbounded at zero frequency. The principal properties of an ARFIMA(p,d,q) are as follows:

• if d > -1/2, X t is invertible, • if d < 1/2, X t is stationary, • if -1/2 < d < 0
, the autocorrelation function ρ(k) decreases more quickly than the case 0 < d < 1 2 . There is a stronger mean reversion, and X t is called anti-persistent in Mandelbrot's terminology.

• if 0 < d < 1
2 , X t is a stationary long memory model. The autocorrelation function decays hyperbolically to zero and we have for

|k| → ∞, γ(k) ≡ C γ (d, φ , θ ) |k| 2d-1 where C γ (d, φ , θ ) = σ 2 ε π Θ(1) Φ(1) 2 Γ(1 -2d) sin(πd), (3.5)
and then for |k| → ∞,

ρ(k) = γ(k) γ(0) ≡ C γ (d, φ , θ ) π -π f (λ )dλ |k| 2d-1 . (3.6) • if d = 1/2
, the spectral density is unbounded at zero frequency.

Fractional gaussian noise

The fractional Gaussian noise (fGn) model was independently developed by [START_REF] Granger | Long memory relationships and the aggregation of dynamic models[END_REF], [START_REF] Granger | Long memory relationships and the aggregation of dynamic models[END_REF][START_REF] Granger | An introduction to long-memory time series models and fractional differencing[END_REF][START_REF] Hosking | Fractional Differencing[END_REF]. The best way to introduce the fGn model is to do it from the fractional Brownian motion {B H (t), t ≥ 0}. 4 The fGn model {X t , t ≥ 0} is the increment of the fractional Brownian motion

X t = B H (t) -B H (t -1), t ≥ 1. (3.7)
X t is a stationary Gaussian with autocovariance function

γ(k) = E (X t X t+k ) = 1 2 |k + 1| 2H -2k 2H + |k -1| 2H , k ≥ 0. γ(k) ∼ H(2H -1)k 2H-2 , for k → ∞ and H = 1 2 . (3.8)
with H is the self similarity parameter or Hurst coefficient, 0 < H < 1.

The fGn model reduces to a white noise when

H = 1/2, in this case γ (k) = 0 for all k ≥ 1. When 1/2 < H < 1, X t is a long memory model.
The spectral density of the fGn model is given by

f (λ ) = C H 2 sin λ 2 2 +∞ ∑ k=-∞ 1 |k + 2πk| 2H+1
(3.9)

∼ C H |λ | 1-2H as λ → 0,
C H is a constant.

Wavelets

Wavelets are mathematical tools for analyzing time series. This method has been adopted in the areas of computational economics (see, for example, [START_REF] Davidson | Wavelet Analysis of Commodity Price Behavior[END_REF], then McCoy and 4 Regular Brownian motion is a continuous time stochastic model, B(s), composed of independent Gaussian increments. [START_REF] Mandelbrot | Fractional Brownian motions, fractional noises and applications[END_REF] also note that in a sense fractional Brownian motion, B H (s), can be regarded as the approximation (0.5 -H) fractional derivative of regular Brownian motion:

B H (s) = 1 Γ(H + 1 2 ) s 0 (s -t) H-1 2 dB(t) for s ∈ (0, 1) .
where Γ(.) is the gamma function, B(t) is regular Brownian motion with unit variance and H is the Hurst coefficient, originally due to [START_REF] Hurst | Long-term storage capacity of reservoirs[END_REF]. The autocovariance of fractional Brownian motion is given by:

E(B H (s)B H (t)) = 1 2 s 2H + t 2H -|s -t| 2H .
Walden (1996) suggested an approximate maximum likelihood estimation method for estimating the fractional differencing parameter of a fractional white noise model. [START_REF] Johnstone | Wavelet threshold estimators for data with correlated noise[END_REF] extended the method to an ARFIMA model. The series were decomposed (filtered) using wavelet transform. This analysis provides a time-frequency representation of a time series. It was developed to overcome the short coming of the Time Fourier Transform (TFT), which can be used to analyze non-stationary time series. The Fourier transform can only provide frequency information that comprise the time series. It gives no direct information about when an oscillation occurred. However, wavelets can keep track of time and frequency information. The basic idea behind wavelet analysis is to decompose a time series into a number of components, each component can be associated with a particular scale at a particular time.

Generally, the wavelet refers to a set of functions of the form

ψ sτ (t) = |s| -1/2 ψ t -τ s , (4.1)
where s is the dilation parameter (scaling parameter) which dilates or compresses a time series, It is defined as |1/frequency| and corresponds to frequency information, and τ is the translation parameter which simply moves the wavelet through the time domain. Any function f ∈ L 2 (R) can be expanded into a wavelet series

f (t) = +∞ ∑ j=-∞ +∞ ∑ k=-∞ ω jk ψ jk (t), (4.2)
where ψ jk ∈ L 2 (R) denotes the dilated and translated wavelet defined by

ψ jk (t) = 2 j/2 ψ 2 j t -k . (4.3)
for j, k ∈ Z. The term j gives the scale corresponding to a dilation by 2 j (the octave), and k ∈ Z is the position or translation.

There are several wavelet families such as Haar, Daubechies, Shannon and so on. [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF] constructed the hierarchy of the compactly supported and orthogonal wavelets with a desired degree of smoothness. This smoothness can be achieved by the number of vanishing moments (the wavelet function ψ (t) has M vanishing moments if x r ψ (t) dt = 0, for r = 0, . . . , M -1). Daubechies defined in her famous book Ten lectures on wavelets many different types of wavelet transform; the continuous wavelet transform, the discrete wavelet transform,... . In statistical setting researchs are more usually concerned with discret rather than continuous functions. In the following subsection we present a brief definition of discrete wavelet transform.

The Discrete Wavelet Transform

The wavelet transform is a function of scale in contrast to the Fourier transform which is a function of frequency. The scale is inversely proportional to a frequency interval. If the scale parameter increases, then the wavelet basis is stretched in the time domain, shrunk in the frequency domain, and shifted toward lower frequencies. Conversely, a decrease in the scale parameter reduces the time support, increases the number of frequencies captured, and shifts toward higher frequencies. The wavelet method is essentially a transformation of time series data based on the two types of filters called a wavelet filter and scaling filter denoted by h and g, respectively. The wavelet filter h and the scaling filter g are used to construct the Discrete Wavelet Transform (DW T ) matrix. The wavelet filter h of support L ∈ N (L is the length of the filter), is defined so as to satisfy the following three properties

L ∑ i=1 h i = 0, L ∑ i=1 h 2 i = 1, L ∑ i=1 h i h i+2n = ∞ ∑ i=-∞ h i h i+2n = 0, ∀ n ∈ N * .
The scaling filter of support L similarly satisfies the following three conditions :

L ∑ i=1 g i = √ 2, L ∑ i=1 g 2 i = 1, L ∑ i=1 g i g i+2n = ∞ ∑ i=-∞ g i g i+2n = 0, ∀ n ∈ N * .
The scaling filter g i , i = 1, . . . , L, is the quadrature mirror filter corresponding to the wavelet filter h i , i = 1, . . . , L by the relationship

g i = (-1) i+1 h L-i-1 . (4.4)
In addition, the following relation holds between the wavelet filter h and the scaling filter g L ∑ i=1 g i h i+2n = 0. (4.5)

Let X = (X 1 , X 2 , . . . , X T ) be an T -vector of observations. Usually, the discrete wavelet transformations are used with equally spaced observations with a simple size equal to an integer power of 2. Thus, we assume T = 2 J for some positive integer J. Using the wavelet filter and the scaling filter, the original time series X t , t = 1, . . . , T is transformed to the new series ω j,t and ν j,t , called the wavelet and scaling coefficients, respectively. Then we can write

ω = W X. (4.6)
where ω is a T-vector wavelet coefficients and W is T × T real valued matrix called wavelet transform matrix.5

Wavelet coefficients

The DW T of X = (X 1 , X 2 , . . . , X T ) can be performed using the so-called pyramid algorithm (Mallat 1989b).

• At first, the time series X t , t = 1, . . . , T with T = 2 J is filtered using the wavelet and scaling filters.

We obtain two T /2-vectors of coefficients, ω 1 (wavelet coefficients) and ν 1 (scaling coefficients).

The wavelet coefficients can be obtained by

ω 1,t = L ∑ i=1 h i X 2t+1-i mod T , t = 1, . . . , T 2 . (4.7)
and the scaling coefficients can be obtained by

ν 1,t = L ∑ i=1 g i X 2t+1-i mod T , t = 1, . . . , T 2 . (4.8)
• Second, the vector of scaling coefficients is filtered with both wavelet and scaling filter to obtain a vector of wavelet and scaling coefficients ω 2 and ν 2 each with length T /4. Thus, we filter ν 1,t separately with h i , i = 1, . . . , L and g i , i = 1, . . . , L and subsample to produce two new series, namely

ω 2,t = L ∑ i=1 h i ν 1,2t+1-i mod T 2 , t = 1, . . . , T 4 . ν 2,t = L ∑ i=1 g i ν 1,2t+1-i mod T 2 , t = 1, . . . , T 4 .
• Third, repeating this process recursively, after J applications of DW T to get a wavelet and scaling coefficients given by

ω j,t = L ∑ i=1 h i ν ( j-1), (2t-1) mod ( T 2 j-1 ) +1
.

(4.9)

ν j,t = L ∑ i=1 g i ν ( j-1), (2t-1) mod ( T 2 j-1 ) +1 . (4.10)
After a brief overview of long memory phenomenon and wavelet analysis of time series, we present in the following section some estimation methods for the long memory parameter.

Estimation procedure

Stationary case

We deal with some well known estimation methods of the long memory parameter d. The first one is the semiparametric method based on an approximated regression equation obtained from the logarithm of the spectral density function of a model. This method is proposed by [START_REF] Geweke | The Estimation and Application of Long Memory Time Series Models[END_REF]. The second is the Gaussian semiparametric method developed by [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF]. The third is the wavelet method based on the discrete wavelet coefficients proposed by [START_REF] Abry | Wavelet analysis of long-range-dependence traffic[END_REF] and Jensen (1999a).

GPH estimate

The GPH estimation procedure is a two-step procedure, which begins with the estimation of d. This method is based on least squares regression in the spectral domain, exploits the sample form of the pole of the spectral density at the origin: f X (λ ) ∼ λ -2d , λ → 0. To illustrate this method, we can write the spectral density function of a stationary model X t , t = 1, . . . , T as

f X (λ ) = 4 sin 2 ( λ 2 ) -d f ε (λ ). (5.1)
where f ε (λ ) is the spectral density of ε t , assumed to be a finite and continuous function on the interval [-π, π]; taking the logarithm of the spectral density function f X (λ ), the log-spectral density can be expressed as

log { f X (λ )} = log { f ε (0)} -d log 4 sin 2 ( λ 2 ) + log f ε (λ ) f ε (0) . (5.2)
Let I X (λ j ) be the periodogram evaluated at the Fourier frequencies λ j = 2π j/T, j = 1, 2, . . . , m, T is the number of observations and m is the number of considered Fourier frequencies, that is the number of periodogram ordinates,6 which will be used in the regression

log I X (λ j ) = log { f ε (0)} -d log 4 sin 2 ( λ j 2 ) + log f ε (λ j ) f ε (0) + log I X (λ j ) f X (λ j ) . (5.3)
where log { f ε (0)} is a constant, log 4 sin 2 (λ j /2) is the exogenous variable and log I X (λ j )/ f X (λ j ) is a disturbance error. The GPH estimate requires two major assumptions related to asymptotic behavior of the equation ( 5.3): H1: for low frequencies, we suppose that log f ε (λ j )/ f ε (0) is negligible. H2: the random variables log I X (λ j )/ f X (λ j ) , j = 1, . . . , m are asymptotically i.i.d.

Under the hypotheses H1 and H2, we can write the linear regression (5.4) where e j ∼ i.i.d(-c, π 2 /6). 7 Let Y j =log 4 sin 2 (λ j /2) , the GPH estimator is the OLS estimate of the regression log I X (λ j ) on the constant α and Y j . The estimate of d, say dGPH is

log I X (λ j ) = α -d log 4 sin 2 ( λ j 2 ) + e j ,
dGPH = ∑ m j=1 Y j -Ȳ log I X (λ j ) ∑ m j=1 Y j -Ȳ 2 . (5.5) where Ȳ = m -1 ∑ m j=1 Y j and m = g(T ) with lim T →∞ g(T ) = ∞ and lim T →∞ g(T )/T = 0.
Geweke and Porter-Hudak (1983) showed that, if T → ∞ and |d| < 1/2 we have

√ m dGPH -d ∼ N   0, π 2 6 m ∑ j=1 Y j -Ȳ 2 -1   .
(5.6)

Porter-Hudak (1990), [START_REF] Crato | Long range dependence in the conditional variance of stock returns[END_REF], showed that the parameter m must be selected so that m = T υ , with υ = 0.5, 0.6, 0.7. Robinson (1995a), [START_REF] Hurvich | The Mean Squared Error of Geweke and Porter-Hudak's Estimator of the Memory Parameter of a Long Memory Time Series[END_REF], [START_REF] Tanaka | The nonstationary fractional unit root[END_REF] and [START_REF] Lieberman | Valid Edgeworth expansion for the sample autocorrelation function under long range dependence[END_REF] have analyzed the GPH estimate dGPH in great detail. Under the assumption of normality for X t , it has been proved that the estimate is consistent and asymptotically normal, so that the estimated standard error of dGPH can be used for inference. An alternative semiparametric estimator has been proposed by [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF].

Gaussian semiparametric estimate

The Gaussian semiparametric estimate called Local Whittle (LW) estimate was originally developed by [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF] under the assumption that X t , t = 1, . . . , T is a stationary model and its spectral density behaves at low frequencies like Gλ -2d . 8 Robinson (1995b) studied the Gaussian semiparametric estimate of d based on minimization of a local Whittle frequency domain log-likelihood,

m (d, G) = 1 m m ∑ j=1 log Gλ -2d j + I X (λ j ) Gλ -2d j ,
(5.7) 7 c is Euler constant equal to 0.57721... 8 At low frequencies, the spectral density

f X (λ ) ∼ G X |λ | -2d as λ → 0. for some finite constant G X > 0.
where m 9 is some integer less than T controlling the number of frequencies included in the local likelihood. Given the interval of admissible estimate of d by, (5.9)

D = [∇ 1 , ∇ 2 ], where ∇ 1 and ∇ 2 are numbers such that -1/2 < ∇ 1 < ∇ 2 < 1/2,
where

R m (d) = log G LW m (d) -2d 1 m m ∑ j=1 log λ j , G LW m (d) = 1 m m ∑ j=1 λ 2d j I X (λ j ).
(5.10)

For linear time series with homoskedastic martingale difference innovations, with spectral density satisfying the same regularity conditions as for log-periodogram estimation based in GPH estimate, [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF] proved that dLW m is consistent and asymptotically normal

√ m dLW m -d d → N 0, 1 4 .
(5.11)

Wavelet OLS estimate

Let a time series X t , t = 1, . . . , T follows a long memory model and let ψ(t) a wavelet function satisfying ψ(t)dt = 0, (5.12) t r ψ(t)dt = 0, r = 0, . . . , M -1.

(5.13)

Consider the family of dilations and translations of the wavelet function ψ(t) defined by equation (4.3). It can be shown that

ψ 2 jk (t)dt = ψ 2 (t)dt.
(5.14)

The DW T of the model X t , t = 1, . . . , T is then defined by 9 The bandwith m satisfies

1 m + m 5 log 2 m T 4 → 0. if the approximation f X (λ ) ∼ G X |λ | -2d has error O |λ | 2-2d as λ → 0.
ω jk = X t ψ jk dt, for j, k ∈ Z.

(5.15) if ψ jk are orthonormal basis,10 we obtain the following representation of the model X t :

X t = +∞ ∑ j=-∞ +∞ ∑ k=-∞ ω jk ψ jk (t).
(5.16)

Consider the DW T coefficients ω jk discussed in subsection (4.2) and define the statistics

μ j = 1 n j n j ∑ k=1 ω 2 jk , (5.17)
where n j is the number of coefficients at octave j available to be computed. As shown by [START_REF] Veitch | A wavelet-based estimator of the parameters of long rangedependence[END_REF]. μ j ∼ z j n j χ 2 n j .

(5.18)

where z j = 2 2d j c, c > 0, and χ 2 n j is Chi-squared random variable with n j degrees of freedom. Thus, by taking logarithms we may write11

log 2 μ j ∼ 2d j + log 2 c + log χ 2 n j log 2 -log 2 n j . (5.19)
Recall that the expected value and the variance of the random variable log χ 2 n are given by

E log χ 2 n = ψ n 2 + log 2, (5.20) Var log χ 2 n = ζ 2, n 2 .
where ψ (z) is the psi function, ψ (z) = d/dz log {Γ (z)}, and ζ (2, n/2) is the Riemann zeta function

ζ (x) = 1 Γ(x) ∞ 0 u x-1 e u -1 du = 1 1 -2 1-s ∞ ∑ n=1 (-1) n-1 n s .
(5.21)

The equation (5.19) can be written as (5.22) where y j = log 2 μ jg j , α = log c, β = 2d, x j = log 2 (2 j ) = j and ε j = log 2 log χ n jlog 2 n jg j , g j = ψ n j /2log n j /2 . We conclude that ε j satisfies E ε j ≈ 0,

y j = α + β x j + ε j ,
(5.23)

Var ε j = ζ 2, n j 2 (log 2) 2 ≈ 2n j log 2 2 -1 .
In order to obtain a wavelet estimate, an OLS regression is fitted to points x j , y j for j = 1, . . . , J with weights κ j = n j . Thus, once the estimate β is obtained, an estimate for long memory parameter d is given by d = β /2. Furthermore, an estimate of the variance of d is provided by the estimate of the variance of β , Var d = 1/4Var β .

It is well known that under regularity conditions, the wavelet OLS estimate of d is efficient and consistent.

Non-stationary case

It is necessary to extend the concept of long memory to the non-stationary framework, d ≥ 1/2. [START_REF] Hurvich | Estimation of the memory parameter for nonstationary or noinvertible fractionally integrated processes[END_REF] proposed a general model for possibly non-stationary integrated vector models with components X t , t = 1, . . . , T , each with memory parameter d > -1/2. So that, we say

X t has memory parameter d > -1/2 if Y t = (1 -L) D X t , D = d + 1/2 , is stationary with mean µ and spectral density f Y (λ ) behaving as Gλ -2(d-D) around the origin, -1/2 ≤ d -D ≤ 1/2.
To estimate the memory parameter of an integrated model beyond the stationary regime (d > 1/2), it has been suggested to apply a data taper either to the time series X t or to its D-th order difference Y t .

The idea of tapering in improving a Fourier approximation has a long history dating back to 1900 (see [START_REF] Brillinger | Time Series: Data analysis and Theory[END_REF]. [START_REF] Tukey | An introduction to the calculations of numerical spectrum analysis[END_REF] introduced this technique to time series for reducing the periodogram bias due to the strong peaks and throughs in spectral density. 12Velasco (1999a, b) A taper is a sequence h t ,t = 1, . . . , T . Given a series of T observations, X t ,t = 1, . . . , T , the tapered discrete Fourier transform and the periodogram are defined as

D X (λ j ) = 2π T ∑ t=1 h 2 t -1 2 T ∑ t=1
h t X t e iλ j t , (5.24)

I X (λ j ) = D X (λ j ) 2 .
(5.25)

The expectation of the periodogram is

E I X (λ j ) = π -π f X (λ )K(λ -λ j )dλ . (5.26)
where f X (λ ) is the spectral density of X t defined in (1.1) and

K(λ ) = (2πT ) -1 ∑ T 1 exp(iλt)
2 is the Fejér kernel. Velasco (1999a) showed that when X t is non-stationary, f X (λ ) plays exactly the same role as a spectral density in the asymptotics for the discrete Fourier transform at frequencies λ j , j = 0 mod T and he showed that the periodogram is (asymptotically) unbiased for λ if j is growing slowly with T and d < 1. 13 Velasco (1999a, b) and [START_REF] Velasco | Whittle pseudo-maximum likelihood estimation for nonstationary time series[END_REF] have considered several tapering schemes such as the cosine bell and Zurbenko-Kolmogorov tapers [START_REF] Zurbenko | On the efficiency of estimates of a spectral density[END_REF]. These tapers h t ,t = 1, . . . , T have the property of being orthogonal to polynomials up to given order, for a subset of Fourier frequencies,

T ∑ 1 1 + t + . . . + t D-1 h t e itλ j = 0, j ∈ ℑ D,T ⊂ 1, . . . , T , (5.27)
where T = (T -1)/2 . The usual discrete Fourier transform is obtained setting h t ≡ 1,t = 1, . . . , T , while the full cosine bell taper is given by h t = 1/2 (1cos[2πt/T ]), and ∑ h 2 t = 3T /8. For sample size T = 4N, where N is an integer, the weights h P t of the Parzen window 14 are

h P t =    2 1 -2t-T T 3 , 1 ≤ t ≤ N or 3N ≤ t ≤ 4N, 1 -6 2t-T T 2 -2t-T T 3 , N < T < 3N.
(5.28) [START_REF] Zurbenko | On the efficiency of estimates of a spectral density[END_REF] used a class of data tapers h Z t , t = 1, . . . , T suggested by Kolmogorov, indexed by order p = 1, 2, . . . , assuming N = T /p integer. For p = 3, Zurbenko's weights are similar to the cosine window, and when p = 4, h Z t are very close to h P t . If p = 2, Zurbenko taper is equal to Bartlett's triangular window 15 and when p = 1 they are constant. Hurvich and Chen (2000) [START_REF] Hurvich | An efficient taper for potentially overdifferenced longmemory time series[END_REF] have defined a family of data tapers. This family depends on a single 13 See theorem 1 in Velasco (1999a) for more details. 14 The Parzen window is a piecewise curve window obtained by the convolution of two triangles of half length or four rectangles of one-four length.

w(n) =      1 -6 n-N/2 N/2 2 + 6 |n-N/2| N/2 3 , 0 ≤ n -N 2 ≤ N 4 2 1 -|n-N/2| N/2 3 , N 4 ≤ n -N 2 ≤ n 2 n = 0, 1, 2, . . . , N -1,
where N is the length of the window. 15 The Bartlett window of width N also known as triangular window is defined as

w(n) =    1 -2n N , 0 ≤ n ≤ N 2 1 + 2n N , -N 2 ≤ n ≤ 0 0, otherwise
parameter p, referred to as the taper order. Set h t = 1exp(2iπt/T ) and for any integer p ≥ 0, define the tapered discrete Fourier transform of order p of the sequence X t , t ∈ Z as follows:

D p X (λ ) = (2πTa p ) -1/2 T ∑ t=1 h p t X t exp(itλ ),
(5.29)

I p X (λ ) = D p X (λ ) 2 .
(5.30)

where a p = n -1 ∑ T t=1 |h t | 2p is a normalization factor. As shown in Hurvich and Chen (2000, Lemma 0), the decay of the discrete Fourier transform of the taper of order p is given by

(2πTa p ) -1/2 T ∑ t=1 h p t exp(itλ ) ≤ C T (1 + T |λ |) p , λ ∈ (-π, π).
(5.31)

This property means that higher-order tapers control the leakage more effectively. The Fourier transform of the taper may be expressed as a finite sum of shifted Dirichlet kernels,

T ∑ t=1 h p t e itλ = p ∑ k=0 T ∑ t=1 e it(λ +λ k ) .
(5.32)

Chen ( 2001) defined a new class of tapers. He proposed a linear combinations of [START_REF] Hurvich | An efficient taper for potentially overdifferenced longmemory time series[END_REF] tapers with different phase shifts, h(x) = 1e i2πx p a 0 + a 1 e i2πx + . . . + a q-1 e i2(q-1)πx .

(5.33)

GPH estimate

Under some smoothness conditions on the short-memory density function f * , and by assuming that the taper order p is larger than D, the ratios of the pooled periodogram of Y t divided by its spectral density ĪY τ,p ( λ j )/ f Y ( λ j ) are i.i.d.16 τ is the pooling order of the periodogram. where

ĪY τ,p ( λ j ) = (p+τ)(l-1)+p ∑ j=(p+τ)(l-1)+1 I X τ (λ j ),
(5.34) and

λ j = p -1 (p+τ)l ∑ (p+τ)(l-1)+1 λ j = [2(p + τ)(l -1) + p + τ + 1] π T .
(5.35)

The parameter I X τ (λ j ) is the periodogram of X evaluated at the frequencies λ j = 2π j/T, j = 1, . . . , m. The log-periodogram estimate of [START_REF] Geweke | The Estimation and Application of Long Memory Time Series Models[END_REF] is defined as the least squares estimate in the linear regression model (5.36) where Y j = -2 log 1e i λ j and u j ∼ = log ĪY τ,p ( λ j )/ f X ( λ j ) . The GPH estimate is (5.42) Velasco (1999a) showed that the LW estimate with τ = 0 and D = 0 is consistent for d ∈ (-1/2, 1) and asymptotically N(0, 1/4) for d ∈ (-1/2, 3/4). He showed that if d > 1 and p ≥ d + 1/2 + 1 the estimate is consistent. In the next section, we report the results of a Monte Carlo experiment based on the three estimation methods defined above.

log ĪY τ,p ( λ j ) = log f * (0) + (d -D)Y j + u j , 1 ≤ j ≤ m,
dGPH (m) = ∑ m j=1 Y j -Ȳ ∑ m j=1 Y j -Ȳ 2 × log ĪY

Simulation study

In this section, a simulation study is used to assess the performance of the three estimation procedures. For this purpose we consider four models: ARFIMA(0,d,0), ARFIMA(1,d,0), ARFIMA(0,d,1) and ARFIMA(1,d,1), d = 0.05, 0.15, 0.25, 0.35, 0.45, 0.50, 0.75, 1.00, 1.25, 1.50. 1000 replications of sample size T = 2 8 , 2 9 , 2 10 are generated. The GPH method was applied using truncation value m = T 0.5 . The realizations of ARFIMA(1,d,1) model were generated for same sample size and of AR coefficient φ = 0.2 and MA coefficient θ = 0.1. Wavelet analysis is performed with the Haar or D(2) 18 and Daubechies filters with 2 and 4 vanishing moments. In the non stationary case, for all estimators we have used the taper of order p = 2 and p = 3 of Bartlett and Zhurbenko Kolmogorov, respectively. Our aim is divided in three points: first, to show the efficiency of the estimates, we computed the Bias and the Root Mean Squared Error of the estimates of long memory parameter d. Second, to show the effect of choosing the wavelet filter in estimating the long memory parameter for stationary and non stationary times series. Third, to show the importance of data tapers in non stationary case.

To compare the different estimators we considered the Bias given by Bias = dd, where d = 1 n ∑ n i=1 di and the Root Mean-Squared Error value, denoted hereafter by RMSE , i.e, RMSE =

1 n ∑ n i=1 di -d 2 .
The boxplots for the ARFIMA(0,d,0) and ARFIMA(1,d,1) models with d = 0.35 (stationary case) and d = 1.25 (non stationary case) are given in figures 6.3. The RMSE and Bias for the GPH, LW and wavelet OLS estimates over 1000 simulated realizations are provided in table 6.1, 6.2, 6.3 and 6.4. Some general comments are as follows :

• In stationary case, for the four models the LW estimate performs the GPH estimate (small RMSE). Such results were pointed out in [START_REF] Boutahar | Estimation Methods of the Long Memory Parameter: Monte Carlo Analysis and Application[END_REF]. In non stationary case, we have found the same results.

• Generally, for the four models used in our simulation and for d ≤ 0.2, the LW estimate is better than the wavelet OLS one.

• Increasing the number of vanishing moments, we showed that the LW estimate is better. For instance, for the LA(8) estimate (the number of vanishing moments is 4) the RMSE is larger than those of all estimates. 18 The Haar filter corresponds to the Daubechies filter with one vanishing moments. q q q q q q q q q q (a)

q GPH LW Haar D( 4) LA( 8 RMSE q q q q q q q q q q (d) RMSE q q q q q q q q q q (b) RMSE q q q q q q q q q q (e) RMSE q q q q q q q q q q (c) RMSE q q q q q q q q q q (f) q q q q q q q q q q (g)

q GPH LW Haar D (

RMSE

q q q q q q q q q q (j) RMSE q q q q q q q q q q (h)

RMSE

q q q q q q q q q q (k)

RMSE

q q q q q q q q q q (i) RMSE q q q q q q q q q q (l)

q GPH LW Haar D(4) LA( 8 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 q q q q q q q q q q q q q q q q q q q q q q 0.35 4) D( 8) 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 q 1.25 1.25 1.25 1.25 1.25 4) D(8) 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 q q q q q q q q q q q q q q q 0.35 4) D(8) 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 1.25 1.25 1.25 1.25 1.25 • For 0.5 ≤ d < 1.5, we have used the Bartlett taper of order p = 2. We found that the Haar filter performs better results than the GPH, LW, D(4) and LA(8) estimates. But, for d ≥ 1.5 we have used the Zhurbenko Kolmogorov taper of order p = 3. Using the Zhurbenko Kolmogorov taper the D(4) is better than the GPH, Haar and LA(8) estimates. Hence, we have small RMSE.

• In non stationary case, for d ≥ 1.5 the LW estimate is better than all other estimates. We concluded that the RMSE of the LW estimate are smaller than those of GPH and wavelet OLS estimates for d ≤ 0.2 and d ≥ 1.5. Thus, the LW estimation method has a good performance. Moreover, the performance of the Haar estimate is better for 0.3 ≤ d < 1.5. is the previous week's price.

In the literature, the oil price has been analyzed by many authors, see for example [START_REF] Chang | Oil price fluctuations and Singapore economy[END_REF] for oil price fluctuations in Singapore economy, [START_REF] Sadorsky | Modeling and forecasting petroleum futures volatility[END_REF] for forecasting and modeling petroleum volatility, [START_REF] Lardic | The impact of oil prices on GDP in European countries: an empirical investigation based on asymmetric cointegration[END_REF], [START_REF] Lardic | Oil prices and economic activity: An asymmetric cointegration approach[END_REF] for oil prices and economic activities, [START_REF] Chen | Oil prices and real exchange rates[END_REF] for financial assets, [START_REF] Narayan | Modeling oil price volatility[END_REF] for modeling and forecasting crude oil market volatility and [START_REF] Kang | Forecasting volatility of crude oil markets[END_REF] for forecasting volatility of daily crude oil markets. that the two series are stationary models. Hence, the volatility crude oil price series are stationary. Thus, we can examine evidence for any possible long-range dependence phenomenon.

To implement the tapering on non stationarity crude oil data, we have concentrated on the Bartlett and Zhurbenko Kolmogorov tapers. We plotted these data tapers for p = 2 and p = 3 for sample size T = 1024 on the first row of Fig. 7.3. The tapered crude oil series h t p t are plotted in the second and fourth rows. In the third and fifth rows of the pictures we plotted the log-periodogram of tapered crude oil series.

Long memory of crude oil spot price volatility

Baillie et al. (1996), introduced the Fractionally Integrated GARCH (FIGARCH) model as generalization of the GARCH and Integrated GARCH (IGARCH) specifications, see [START_REF] Beine | Accounting for conditional leptokurtosis and closing days effects in FIGARCH models of daily exchange rates[END_REF], [START_REF] Giraitis | LARCH, leverage and long memory[END_REF], [START_REF] Lardic | Term premium and long-range dependence in volatility: a FIGARCH-M estimation on some Asian countries[END_REF], [START_REF] Lee | Forecasting conditional volatility of returns by using the relationship among returns, trading volume, and open interest in commodity futures markets[END_REF] and [START_REF] Baillie | Long memory and FIGARCH models for daily and high frequency commodity prices[END_REF] for a few recent applications. They suggested the FIGARCH model because is able to distinguish between short and long memory in conditional variance behavior. Let the conditional variance h t = Var(ε t |Ω t-1 ) where Ω t-1 is the information set in period t -1. The GARCH model is written Notes: The value between parentheses are standard deviation. he mean µ was clearly nonsignicant for the tow processes. Notes: The columns correspond to data tapers of orders p=2,3, respectively. In the first row, the plots correspond to the weights h t . In second and fourth rows we plot the tapered series h t p t . In the third and fifth rows appear the log-periodogram of the above series plotted against frequency.

Conclusion

In this paper we considered a simulation study to evaluate the procedure for estimating the long memory parameter in stationary and non stationary time series. We have compared three semiparametric methods for the estimation of the fractional parameter d in four models: ARFIMA(0,d,0), ARFIMA(1,d,0), ARFIMA(0,d,1) and ARFIMA(1,d,1). These estimates are the Geweke and Porter-Haduk (1983), the Local Whittle [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF] and the wavelet Ordinary Least-Square [START_REF] Jensen | Using wavelets to obtain a consistent ordinary least squares estimator of the long-memory parameter[END_REF]. We have used the Bartlett and Zhurbenko Kolmogorov tapers with orders p = 2 and p = 3, respectively, in non stationary models. According to the simulation results we have showed that the order p of taper has a strong impact on the performance of estimates. For d ≤ 0.25 and for d ≥ 1.5 the Local Whittle estimate possesses a small mean squared error than the GPH and wavelet OLS estimates for small and large sample sizes and for different values of long memory parameter. But, for d ∈ [0.25, 1.5) the performance of Haar estimate [START_REF] Jensen | Using wavelets to obtain a consistent ordinary least squares estimator of the long-memory parameter[END_REF] usually is better than the two other estimates.

The estimation results showed that the volatility of crude oil prices exhibit strong evidence of long memory which can be captured by FIGARCH model.

  the LW estimates are defined by dLW m , ĜLW m = arg min d∈D, 0<G<∞ m (d, G) . (5.8) Concentrating equation (5.7) with respect to G, we obtain dLW m = arg min d∈D R m (d).

  Let p (G, d) be the objective function 17 p (G, d) j ) is defined in (5.25). Define the lower and upper bound of the admissible estimates of d by ∇ 1 and ∇ 2 . ∇ 1 and ∇ 2 are numbers such that -1/2 < ∇ 1 < ∇ 2 < d * , where d * is the maximum value of d we can estimate with tapers of order p. The estimates of (G p , d p ) are defined as Ĝp , ,...,m λ 2d j I(λ j ).

Figure 6 . 1 .

 61 Figure 6.1. RMSE as a function of d for ARFIMA(0,d,0) and ARFIMA(1,d,0) models.

Figure 6 . 2 .

 62 Figure 6.2. RMSE as a function of d for ARFIMA(0,d,1) and ARFIMA(1,d,1) models.

  RMSE as a function of long memory parameter using GPH, LW and wavelet OLS (Haar, D(4) and LA(8) ) estimation methods. (g), (h), (i) for an ARFIMA(0,d,1); (j), (k) and (l) for an ARFIMA(1,d,1) models, with T = 256, T = 512 and T = 1024.

Figure 6

 6 Figure 6.3. Box-plots of GPH, LW and wavelet OLS estimators.

  results are based on 1000 realizations with sample size T = 1024 for an ARFIMA(0,d,0) and ARFIMA(1,d,1) model. The dotted blue lines indicate 0.35 and 1.25 ordinates, respectively. The first row of the picture corresponds to the ARFIMA(0,d,0) model and the second row corresponds to the ARFIMA(1,d,1) model.

Figure ( 7

 7 .1) shows the dynamics of crude oil spot prices and return oil price of the two sample WTI 20 and Europe Brent 21 and figure (7.2) depicts a plot of volatility (i.e. of the absolute returns |r t |) for the two crude oils and corresponding correlogram and spectrum function. We observe a slow decay of correlogram indicating the long memory behavior of the volatility of the weekly energy price.

Figure

  Figure 7.1. Oil price time series.

Figure

  Figure 7.2. Energy price analysis.

Figure 7 . 3 .

 73 Figure 7.3. Data tapers, tapered oil series and log-periodogram of tapered series.

Table 6 .

 6 1. Bias and RMSE of the GPH, LW and wavelet OLS estimator of the long memory parameter for an ARFIMA(0,d,0).

				GPH	LW			Haar		D(4)	LA(8)
	T	d	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE
	256 0.05 -0.0043 0.2101 0.0018 0.0084 -0.0401 0.1643 -0.0426 0.2022 -0.0428 0.3039
		0.15 0.0021 0.2072 -0.0759 0.0851 -0.0559 0.1664 -0.0392 0.1956 -0.0508 0.3150
		0.25 -0.0047 0.2152 -0.1181 0.1410 -0.0610 0.1672 -0.0485 0.1970 -0.0661 0.3056
		0.35 0.0083 0.2136 -0.1348 0.1695 -0.0627 0.1794 -0.0518 0.1983 -0.0450 0.3037
		0.45 0.0118 0.2105 -0.1399 0.1853 -0.0664 0.1674 -0.0557 0.2011 -0.0550 0.2967
		0.50 0.0952 0.2626 0.0158 0.2116 -0.0001 0.1837 0.0702 0.2312 0.1255 0.3453
		0.75 0.0870 0.2651 0.0471 0.2149 0.0210 0.1812 0.0647 0.2285 0.1088 0.3484
		1.00 0.1385 0.2820 0.0372 0.2077 0.0180 0.1873 0.0575 0.2370 0.1100 0.3583
		1.25 0.2058 0.3330 0.0779 0.2152 0.0644 0.2066 0.0571 0.2463 0.0643 0.3658
		1.50 0.3191 0.4058 0.0109 0.2608 0.2158 0.3264 0.1027 0.2779 0.1507 0.4117
	512 0.05 -0.0039 0.1725 0.0009 0.0051 -0.0412 0.1135 -0.0307 0.1202 -0.0204 0.1415
		0.15 0.0095 0.1686 -0.0755 0.0833 -0.0370 0.1138 -0.0290 0.1156 -0.0225 0.1371
		0.25 0.0116 0.1769 -0.1117 0.1338 -0.0384 0.1130 -0.0305 0.1188 -0.0326 0.1407
		0.35 0.0140 0.1684 -0.1267 0.1611 -0.0482 0.1265 -0.0347 0.1195 -0.0241 0.1418
		0.45 0.0193 0.1725 -0.1313 0.1768 -0.0462 0.1185 -0.0301 0.1176 -0.0411 0.1478
		0.50 0.0834 0.2229 -0.0111 0.1491 0.0161 0.1250 0.0570 0.1420 0.1023 0.1845
		0.75 0.0701 0.2137 -0.0035 0.1447 0.0133 0.1275 0.0516 0.1410 0.0920 0.1868
		1.00 0.1157 0.2380 0.0026 0.1548 0.0173 0.1316 0.0424 0.1463 0.0863 0.1767
		1.25 0.1796 0.2788 0.0232 0.1500 0.0562 0.1494 0.0569 0.1513 0.0808 0.1950
		1.50 0.2903 0.3540 0.0071 0.2021 0.1708 0.2484 0.1099 0.2021 0.1164 0.2270
	1024 0.05 0.0097 0.1333 0.0005 0.0031 -0.0241 0.0797 -0.0174 0.0800 -0.0166 0.0851
		0.15 -0.0026 0.1343 -0.0749 0.0825 -0.0304 0.0866 -0.0258 0.0812 -0.0126 0.0817
		0.25 0.0022 0.1366 -0.1152 0.1356 -0.0322 0.0847 -0.0233 0.0774 -0.0173 0.0828
		0.35 0.0085 0.1266 -0.1330 0.1648 -0.0340 0.0857 -0.0279 0.0824 -0.0195 0.0820
		0.45 0.0120 0.1351 -0.1424 0.1849 -0.0396 0.0872 -0.0267 0.0834 -0.0260 0.0847
		0.50 0.0663 0.1744 0.0098 0.1137 0.0099 0.0851 0.0491 0.0990 0.0784 0.1194
		0.75 0.0460 0.1719 0.0266 0.1185 0.0100 0.0936 0.0419 0.0967 0.0774 0.1189
		1.00 0.0848 0.1805 0.0316 0.1200 0.0158 0.0969 0.0383 0.0979 0.0681 0.1185
		1.25 0.1452 0.2283 0.0438 0.1234 0.0413 0.1148 0.0439 0.1114 0.0638 0.1177
		1.50 0.2446 0.2985 0.0114 0.1473 0.1541 0.2169 0.0894 0.1532 0.0982 0.1513

Table 6 .

 6 2. Bias and RMSE of the GPH, LW and wavelet OLS estimator of the long memory parameter for an ARFIMA(1,d,0).

				GPH	LW			Haar		D(4)	LA(8)
	T	d	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE
	256 0.05 0.0048 0.2047 0.0005 0.0005 0.0305 0.1543 0.0718 0.2030 0.1173 0.3179
		0.15 0.0109 0.2093 -0.0974 0.0979 0.0183 0.1532 0.0635 0.2107 0.1093 0.3184
		0.25 0.0198 0.2129 -0.1612 0.1671 0.0274 0.1611 0.0569 0.2000 0.1140 0.3132
		0.35 0.0304 0.2177 -0.1888 0.2017 0.0083 0.1580 0.0460 0.2003 0.0965 0.3269
		0.45 0.0250 0.2229 -0.1986 0.2188 0.0079 0.1623 0.0404 0.1978 0.1039 0.3289
		0.50 0.1110 0.2714 0.0576 0.2228 0.0747 0.2002 0.1573 0.2685 0.2840 0.4403
		0.75 0.1033 0.2696 0.0689 0.2189 0.0736 0.2013 0.1573 0.2757 0.2609 0.4257
		1.00 0.1537 0.3118 0.0722 0.2255 0.0713 0.2063 0.1485 0.2656 0.2404 0.4269
		1.25 0.2369 0.3493 0.0888 0.2318 0.1173 0.2343 0.1298 0.2691 0.2213 0.4202
		1.50 0.3379 0.4108 0.0394 0.2773 0.2379 0.3324 0.1870 0.3135 0.2506 0.4574
	512 0.05 0.0088 0.1689 0.0005 0.0005 0.0235 0.1119 0.0545 0.1283 0.0890 0.1623
		0.15 0.0052 0.1647 -0.0983 0.0985 0.0171 0.1127 0.0414 0.1228 0.0833 0.1599
		0.25 0.0105 0.1715 -0.1565 0.1613 0.0078 0.1098 0.0360 0.1236 0.0758 0.1571
		0.35 0.0168 0.1732 -0.1783 0.1903 0.0045 0.1091 0.0353 0.1205 0.0766 0.1584
		0.45 0.0205 0.1679 -0.1916 0.2091 0.0011 0.1087 0.0290 0.1232 0.0590 0.1541
		0.50 0.0767 0.2190 0.0038 0.1484 0.0639 0.1350 0.1242 0.1803 0.2084 0.2558
		0.75 0.0913 0.2175 0.0056 0.1430 0.0581 0.1390 0.1240 0.1807 0.1902 0.2454
		1.00 0.1092 0.2357 0.0172 0.1487 0.0567 0.1395 0.0995 0.1722 0.1814 0.2450
		1.25 0.1841 0.2785 0.0359 0.1559 0.0822 0.1624 0.1115 0.1776 0.1685 0.2419
		1.50 0.2880 0.3533 0.0397 0.1972 0.2047 0.2705 0.1606 0.2261 0.1888 0.2651
	1024 0.05 0.0008 0.1332 0.0005 0.0005 0.0170 0.0809 0.0327 0.0823 0.0582 0.0995
		0.15 0.0020 0.1342 -0.0990 0.0991 0.0099 0.0810 0.0314 0.0812 0.0593 0.0986
		0.25 0.0018 0.1379 -0.1536 0.1578 0.0092 0.0800 0.0296 0.0860 0.0552 0.0950
		0.35 0.0132 0.1295 -0.1763 0.1873 0.0024 0.0773 0.0283 0.0813 0.0490 0.0945
		0.45 0.0110 0.1351 -0.1891 0.2060 -0.0033 0.0826 0.0198 0.0803 0.0484 0.0958
		0.50 0.0624 0.1789 0.0224 0.1103 0.0485 0.1007 0.0925 0.1268 0.1486 0.1736
		0.75 0.0674 0.1721 0.0232 0.1187 0.0424 0.1038 0.0877 0.1254 0.1369 0.1657
		1.00 0.0825 0.1804 0.0384 0.1189 0.0416 0.1033 0.0810 0.1230 0.1322 0.1640
		1.25 0.1466 0.2254 0.0447 0.1258 0.0666 0.1247 0.0803 0.1262 0.1287 0.1613
		1.50 0.2426 0.2942 0.0179 0.1504 0.1686 0.2241 0.1207 0.1726 0.1443 0.1818

Table 6 .

 6 3. Bias and RMSE of the GPH, LW and wavelet OLS estimator of the long memory parameter for an ARFIMA(0,d,1).In this section, the proposed methodology is applied to a real example for illustration. The data consist of weekly crude oil spot prices (in US dollars per barrel) during the period from Jun 15, Table6.4. Bias and RMSE of the GPH, LW and wavelet OLS estimator of the long memory parameter for an ARFIMA(1,d,1), φ = 0.2, θ = 0.1 The data are from the U.S. Energy Information Administration.19 We consider a weekly nominal percentage return for crude oil series, i.e., r t = ln (P t /P t-1 ) * 100 for t = 1, 2, . . . , T , where r t is the return for crude oils at time t, P t is the weekly current price and P t-1

				GPH	LW			Haar		D(4)	LA(8)
	T	d	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE
	256 0.05 -0.0015 0.2096 0.0051 0.0165 -0.0788 0.1751 -0.0960 0.2179 -0.1131 0.3176
		0.15 -0.0106 0.2116 -0.0646 0.0833 -0.0907 0.1820 -0.0783 0.2048 -0.1114 0.3155
		0.25 -0.0023 0.2139 -0.1119 0.1401 -0.0909 0.1901 -0.1056 0.2261 -0.1176 0.3212
		0.35 0.0106 0.2125 -0.1207 0.1690 -0.0948 0.1887 -0.1003 0.2228 -0.1285 0.3136
		0.45 0.0180 0.2016 -0.1314 0.1893 -0.1009 0.1920 -0.1038 0.2211 -0.1360 0.3192
		0.50 0.1129 0.2682 0.0437 0.2181 -0.0125 0.1753 0.0240 0.2187 0.0678 0.3309
		0.75 0.0854 0.2624 0.0488 0.2100 -0.0147 0.1916 0.0174 0.2156 0.0551 0.3367
		1.00 0.1325 0.2792 0.0635 0.2250 0.0093 0.1893 0.0188 0.2318 0.0336 0.3572
		1.25 0.2242 0.3525 0.0713 0.2213 0.0374 0.2000 0.0257 0.2283 0.0396 0.3558
		1.50 0.3285 0.4044 0.0258 0.2784 0.2095 0.3137 0.0934 0.2808 0.0760 0.3780
	512 0.05 -0.0158 0.1719 0.0031 0.0123 -0.0606 0.1223 -0.0600 0.1302 -0.0661 0.1543
		0.15 -0.0061 0.1657 -0.0678 0.0816 -0.0697 0.1294 -0.0640 0.1301 -0.0696 0.1554
		0.25 0.0002 0.1662 -0.1102 0.1368 -0.0652 0.1293 -0.0697 0.1348 -0.0782 0.1579
		0.35 0.0011 0.1664 -0.1148 0.1602 -0.0707 0.1321 -0.0685 0.1315 -0.0803 0.1613
		0.45 0.0146 0.1748 -0.1189 0.1744 -0.0734 0.1314 -0.0829 0.1467 -0.0795 0.1615
		0.50 0.0991 0.2257 -0.0081 0.1478 -0.0109 0.1203 0.0270 0.1289 0.0563 0.1625
		0.75 0.0884 0.2193 0.0007 0.1424 -0.0095 0.1250 0.0163 0.1280 0.0415 0.1568
		1.00 0.1205 0.2322 0.0146 0.1522 0.0040 0.1285 0.0175 0.1402 0.0380 0.1633
		1.25 0.1805 0.2758 0.0273 0.1460 0.0323 0.1413 0.0209 0.1492 0.0457 0.1717
		1.50 0.2930 0.3557 0.0188 0.1910 0.1735 0.2528 0.0880 0.1915 0.0818 0.1986
	1024 0.05 0.0025 0.1331 0.0015 0.0065 -0.0478 0.0918 -0.0461 0.0916 -0.0472 0.0942
		0.15 -0.0009 0.1360 -0.0728 0.0833 -0.0459 0.0886 -0.0457 0.0855 -0.0508 0.0939
		0.25 -0.0013 0.1353 -0.1107 0.1368 -0.0494 0.0924 -0.0466 0.0871 -0.0542 0.0996
		0.35 0.0057 0.1360 -0.1315 0.1708 -0.0535 0.0963 -0.0471 0.0902 -0.0550 0.0996
		0.45 0.0074 0.1403 -0.1379 0.1867 -0.0547 0.0977 -0.0544 0.0951 -0.0517 0.0947
		0.50 0.0563 0.1655 0.0202 0.1152 -0.0038 0.0863 0.0197 0.0932 0.0528 0.1037
		0.75 0.0529 0.1665 0.0199 0.1163 -0.0121 0.0926 0.0193 0.0901 0.0379 0.1012
		1.00 0.0876 0.1831 0.0344 0.1196 0.0060 0.0979 0.0220 0.0962 0.0466 0.1061
		1.25 0.1485 0.2258 0.0483 0.1239 0.0276 0.1083 0.0241 0.0963 0.0331 0.1034
		1.50 0.2454 0.2986 0.0218 0.1477 0.1493 0.2095 0.0824 0.1483 0.0735 0.1369
	7. Empirical application
	7.1. Data

Table 7 .

 7 1. Summary statistics and Unit Root tests of energy price volatility.

		Descriptive statistics		Unit Root tests
		Mean Std.dev Skewness Kurtosis	J-B	PP	KPSS	ADF
	WTI	3.321 3.014	2.202	7.870	3487.88	-28.202 0.182 -17.440
	t-value				0.000 *	0.000 * 0.022 * * 0.000 *
	BRENT 3.352 3.005	2.159	7.990	3536.99	-28.008 0.197 -17.315
	t-value				0.000 *	0.000

* 0.017 * * 0.000 * Notes: Sample period, weekly data: Juin 15, 1990 to January 29, 2010. J-B is the Jarque-Bera statistic for null hypothesis of normality in sample volatility distribution. PP and ADF are the Phillips-Perron and Augmented-Dickey-Fuller adjusted t-statistics of lagged dependent variable in a regression with intercept and trend (H 0 : Data are I(1)). KPSS is the

Kiwatkoski et al. (1992)

test statistic based on residuals from regression with intercept and trend (H 0 : Data are I(0)). The critical values for PP,

, respectively, at 1% significance level. * and ** indicate significance at 1% and 5% level, respectively.

Table ( 7

 ( .1) displays descriptive statistics and unit root tests of weekly volatility for both WTI and Brent crude oil spot prices. The sample mean and variance of volatility are around 3 and skewness and kurtosis statistics indicate that the volatility distribution is not normally distributed. We apply three standard unit root tests on each individual transformed series, PP (Phillips-Perron), KPSS (Kwiatkoski, Phillips, Schmidt and Shin) and Augmented-Dickey-Fuller in order to show the stationarity behavior for weekly crude oil data. Results are given in table (7.1). For the PP and ADF tests, large negative values for the statistics support the rejection of the null hypothesis of unit root at the 1% significance level. For KPSS test, the statistics are significant at 5% level implying

Table 7 .

 7 2. GPH, LW and wavelet OLS estimates.Notes: (p t ), (r t ) and |r t | are, respectively, crude oil spot price, return and absolute return. The value between parentheses are p-values. p=1,2 and 3 are the orders of the taper. p=1 means no tapering, p=2 is the Bartlett taper and p=3 is the Zhurbenko Kolmogorov taper.

			WTI			Brent
		p = 1	p = 2	p = 3	p = 1	p = 2	p = 3
	Panel A: GPH (1983)
	Spot price (p t )	0.904 (0.000)	1.142 (0.000)	1.187 (0.000)	0.864 (0.000)	1.117 (0.000)	1.122 (0.000)
	return (r t )	-0.144 (0.282)	0.019 (0.984)	0.082 (0.934)	-0.181 (0.177)	-0.001 (0.998)	0.075 (0.940)
	Absolute return (|r t |)	0.379 (0.004)	0.542 (0.000)	0.515 (0.000)	0.395 (0.003)	0.623 (0.000)	0.626 (0.000)
	Panel B: Local Whittle
	Spot price (p t )	1.008 (0.000)	0.925 (0.000)	0.857 (0.000)	1.008 (0.000)	0.971 (0.000)	0.832 (0.000)
	return (r t )	0.027 (0.079)	0.021 (0.172)	0.027 (0.079)	0.006 (0.658)	0.004 (0.757)	-0.002 (0.876)
	Absolute return (|r t |)	0.623 (0.000)	0.374 (0.000)	0.347 (0.000)	0.518 (0.000)	0.299 (0.000)	0.223 (0.000)
	Panel C: Haar filter
	Spot price (p t )	0.854 (0.001)	1.558 (0.000)	1.541 (0.000)	0.826 (0.001)	1.510 (0.000)	1.497 (0.000)
	return (r t )	-0.055 (0.000)	0.033 (0.000)	0.046 (0.000)	-0.030 (0.000)	0.023 (0.000)	0.032 (0.000)
	Absolute return (|r t |)	0.137 (0.000)	0.207 (0.001)	0.227 (0.002)	0.114 (0.000)	0.208 (0.000)	0.229 (0.000)
	Panel D: D(4) filter
	Spot price (p t )	0.608 (0.001)	1.318 (0.000)	1.359 (0.000)	0.636 (0.002)	1.318 (0.000)	1.344 (0.000)
	return (r t )	-0.117	-0.019	-0.003	-0.126	-0.040	-0.026
		(0.000)	(0.000)	(0.001)	(0.008)	(0.006)	(0.005)
	Absolute return (|r t |)	0.159 (0.004)	0.195 (0.004)	0.190 (0.004)	0.087 (0.003)	0.171 (0.004)	0.180 (0.005)
	Panel E: LA(8) filter
	Spot price (p t )	0.529 (0.012)	1.324 (0.005)	1.400 (0.001)	0.571 (0.016)	1.328 (0.007)	1.398 (0.005)
	return (r t )	-0.070 (0.004)	0.031 (0.007)	0.047 (0.008)	-0.006 (0.036)	0.076 (0.043)	0.089 (0.040)
	Absolute return (|r t |)	0.069 (0.025)	0.139 (0.027)	0.144 (0.031)	-0.018 (0.001)	0.092 (0.011)	0.196 (0.014)

Table 7 .

 7 3. ARMA model estimation for the oil price returns.

		φ0	φ1	φ2	φ3	θ1	θ2	θ3
	WTI	0.1420	0.2767	-0.5153	-0.2293	-0.1677	0.4134	0.4273
		(0.1563)	(0.0312)	(0.0278)	(0.0318)	(0.3456)	(0.00)	(0.1488)
	BRENT 0.1527	-0.3251	0.0893	0.0988	0.5041
		(0.1819)	(0.0312)	(0.0484)	(0.0323)	(0.2603)

Table 7 .

 7 4. Estimation results for Crude oil priceNotes: p-values are in parentheses below corresponding parameter estimates. Q s (24) is the Box-Pierce test statistic for lag 24.

			WTI	BRENT
		GARCH FIGARCH	GARCH FIGARCH
	ω	0.6866 (0.004)	0.3758 (0.022)	0.4427 (0.020)	0.2549 (0.017)
	α	0.0.091		0.0945
		(0.000)		(0.000)
	β	0.871 (0.000)	0.7941 (0.000)	0.8843 (0.000)	0.8732 (0.000)
	Q s (24)	25.621 (0.372)	26.434 (0.331)	26.575 (0.324)	26.884 (0.3098)
	d-FIGARCH		0.7262		0.9477
			(0.000)		(0.000)
	φ		0.1888 (0.000)		0.0286 (0.850)

M. Boutahar GREQAM, Université de la Méditerranée

rue de la charité, 13236, Marseille cedex 02, France e-mail: boutahar@univmed.fr 2 R. Khalfaoui GREQAM, Université de la Méditerranée 2 rue de la charité, 13236, Marseille cedex 02, France e-mail: r.kalfaoui@gmail.com

For a broader view of wavelet theory see[START_REF] Daubechies | Ten Lectures on Wavelets[END_REF]Daubechies ( , 1988)),[START_REF] Mallat | A theory for multiresolution signal decomposition: The wavelet representation[END_REF] or[START_REF] Meyer | Wavelets: Algorithms and Applications[END_REF].

W satisfies W T W = I T . Orthonormality implies that X = W T ω and ω 2 = X 2 , seePerceival and Walden (2000) for more details.

We can note that the chose of m is an important issue since it strongly affects estimation results. On the one hand, m should be sufficiently small in order to consider only near zero frequencies. On the other hand, m should be sufficiently large to ensure convergence of OLS estimation.

The family ψ i j satisfies : ψ i j (t)ψ kl (t) = 0.

see Veitch and Abry (1999) for details.

Tapering was originally used in nonparametric spectral analysis of short memory (d = 0) time series in order to reduce bias due to frequency domain leakage, where part of the spectrum "leaks" into adjacent frequencies. The leakage is due to the discontinuity caused by the finitness of the sample and is reduced by using tapers which smooth this discontinuity.

See Faÿ et al. (2009)

Velasco (1999a).

http://tonto.eia.doe.gov/dnav/pet/pet_pri_spt_s1_w.htm

West Texas Intermidiate or Texas light Sweet is a type of crude oil used in the pricing of US domestic crudes, as well as oil imports into the US.

Brent crude or Brent petroleum is used as a reference for pricing a number of other crude streams. It is produced in North sea region.

as

(1β (L)) h t = ω + α(L)ε 2 t .

(7.1)

where β (L) and α(L) are polynomials of order p and q, respectively. Let ν t = ε 2 th t . The FIGARCH model is defined as follows:

Note that the FIGARCH model reduces to a GARCH model when d = 0 and to an IGARCH model when d = 1. The conditional variance of FIGARCH model may be written as

In our analysis, we first estimate the fractional integration parameter d by the three methods described previously in section 5. The estimates results are provided in table (7.2). These estimates indicate evidence of long memory in WTI and Brent crude oil volatility.

We also assess the null hypothesis of a White noise model for WTI and Brent returns using Box-Pierce test statistics Q(24). The corresponding t-statistics are 70.722 and 69.397 for WTI and Brent returns, with pvalue = 0, respectively. Thus, we find significant evidence of serial dependence in the volatility series. Therefore, this finding implies non-normality and serial correlation in the crude oil volatilities. We used a filter to remove the serial dependence in the return series and the resulting residuals series are re-tested for GARCH and FIGARCH models to detect the long memory phenomenon in the volatility of crude oils spot prices. We used an autoregressive moving average ARMA(p,q) model to take out all the linearity in the return series. The identification of the autoregressive and moving average orders p and q are based on the lowest AIC. The Box-Pierce Q-statistics (not presented here) showed that the residuals of an ARMA(3,3) and ARMA(3,1) are White noise for the WTI and Brent return, respectively. Second, we reported the estimation results of the GARCH and FIGARCH models in table (7.4). We showed that the FIGARCH (1,d,1) describe volatility persistence for the two crude oil prices. The estimates of long memory parameter d are 0.72 and 0.94 for the WTI and Brent, respectively, rejecting the null hypothesis of the GARCH model (d = 0). This result indicates persistence in the conditional variance of the volatility crude oils. Therefore, the FIGARCH model is able to capture persistence in the volatility of crude oils.


Hakiri Akram
email: hakiri@laas.fr

Berthou Pascal
email: berthou@laas.fr

Gayraud Thierry
email: gayraud@laas.fr

Design of Low Cost PC-based Simulators for Education and Training Purpose Using DDS

The use of simulation in training and education enables to prepare personal in realistic environment. But the cost and the complexity to create and reuse simulations often limits their application. In this paper we investigate a low cost and high fidelity PC-based simulator based on Data Distribution Service (DDS) middleware. The main parts of the systems and the architecture, including the hardware and the software are introduced. Real-time networking between distributed simulators is achieved using a reliable distributed communication, which employs publish-subscribe middleware build using OMG-DDS. Result shows these methods could produce low cost, extensible, reliable and distributed simulators.

I. INTRODUCTION

We have seen an inordinate amount of attention on novel 3D input devices, e.g. data gloves, data suits, space balls, "flying mice", trackers, etc. We have also seen the same level of attention on novel 3D output devices, e.g. 3D TV displays, etc. The second stage of this innovation is focusing on more important part of the problem, the cost and the real time communication. Indeed, design of real simulators remains domain specific and highly expensive for small companies, because the high performance of multi-core systems for graphics and capturing the distributed sensors and actuators to emulate a real conditions. Here, we investigate high performance, low cost PC simulators for education and training purpose: a real car driving cab, with a modular configuration to fit every need, from initial to advanced driving training.

The remainder of this paper is organized as follow: after a this introduction,Section II uses two case studies to motivate common requirements of simulation for training purpose. The design and the architecture of the simulators, including hardware and the software are described in Section III. Section IV focalizes on the architectural overview of the DDS infrastructure. Section V show the networking part when connecting several simulators together in distributed environment using publish-subscribe paradigm build using OMG-DDS. Section VI compares our work with related researches. Finally, some concluding remarks and future works are presented in Section VII.

II. SCOPE AND MOTIVATION

Although simulators have become more populated in flight and driving simulation to teach tactics, resources management and strategic thinking, computer simulation is still remarkably complex, high expensive and hard to evolve and extend. What is needed, therefore, is low complexity, high performance and low cost simulators for social interaction.

A. Challenge 1: Reducing the cost of computer simulation Context. simulation is often used in the training of military personnel and in emergency response preparedness [START_REF] Sanjay | An Architecture for integrating Modeling and Simulation For Emergency Response[END_REF]. Simulators are used when it is very expensive or simply dangerous to allow trainees to use real world. Problem. design of real simulators remains domain specific and highly expensive for small companies, because the high performance of multi-core systems for graphics and capturing the distributed sensors and actuators to emulate a real conditions. Solution approach. our approach suggests a high performance, low cost PC-based simulators for save driving in car driving education: a real car driving cab, with a modular configuration to fit every need, from initial to advanced driving training.

B. Challenge 2: Allowing high performance for distributed simulators context. Developing and deploying distributed applications in traditional pragmatic approach imposes significant challenge to most developers. It requires considerable learning curve for experts to be able to develop and deploy large scale distributed applications. Problem. Both client-server and Peer to Peer architecture are limited in their usability when used in distributed realtime data-centric communication. Many details are mandatory for understanding the interaction between components. In the other hand, the existing software solutions are tightly integrated in a particular programming language. Before a distributed system can by implemented, developers should familiarize them self's with the new language syntax and network programming paradigm. Solution approach. The current effort in improving the technology base for Human Interface Systems have been somewhat lacking with respect to distributed middleware. What is needed, therefore, are middleware-guided network QoS provisioning solutions that are not tied to a particular network QoS mechanism. These solutions should ideally operate across wide range of network QoS mechanisms without requiring modifying application code source. Thereby we show that the newly developed technologies lead in favor of DDS-based solutions which provide distributed real-time communication.

III. HARDWARE AND SOFTWARE ARCHITECTURE

The driving simulator, is among the most sophisticated low cost systems: it is high performance, low latency, low cost, highly distributed and fulfills the most relevant QoS requirements of distributed architecture.

A. Hardware configuration

The hardware architecture is shown in Figure 1. It is composed of numerous functionalities embedded in specific modules. Each simulator is composed of the following components: • adjustable seat, steering wheel, seat belt, rugged shifter and pedal assembly with force feedback • steering wheel with motorized force feedback,

• small-size driving cab. Real-time data are sent across field bus network based on CAN [START_REF]Can in Automation[END_REF] to ensure the communication between the sensors and the actuators within the driving simulator. Further, camera is used to capture the movement of each participant and send them to teaching post. The flow issued from each camera is composed of sound and video streams. Both IP-Camera and any other webcam can be used within this system.

B. Modular Software Architecture

The design of the simulator is modular and extensible: several modules are developed separately and plagued within one more general component. Many libraries and databases were included to allow driving in real condition. Roads were designed to be fully compliant with realistic driving applications. Information such as lane topology, traffic attributes, crossroad modeling, road markings, road signs, urban furniture, etc. are integrated. As illustrated in Figure 2, night driving module is also developed to provide the training in many metrologies environment. In order to fit every need and Fig. 2. City and night driving modules to provide a safe driving, an emergency braking and safety distance module (total reaction time test and 2 seconds rule) is developed, including risk observation module with more than 40 situations. A Huge 3D database including city, country, highway and maneuvering area is provided, like shown in Figure 2. In addition, weather conditions and brightness like rain, snow, fog, daylight, dusk and night are added to meet the realistic condition, which every trainer can oppose. Further, several diverse traffics with more than 50 visible automate vehicles at the same time, and also pedestrian and animals can be confronted during the simulation exercise. The scenario information is used for saving the exercise, and replying it when needed. These scenarios data files are containing not only computer data to be processed by the simulation, but also a descriptive text with teacher interpretation of the trainee mistakes. Replay function is provided with an external view and a printable report. An eco-driving module is provided for the trainees and permits accessing to all other modules in replay mode and allows the display of the following information for each driving: the driving time, the covered distance, the total consumption, the total CO2 emission, the average speed, the average consumption, the average CO2 emission, etc. moreover, these information are displayed within a graph with curves indicating the immediate values of consumption/speed/CO2 emission.

IV. KEY CONCEPTS BEHIND DDS DDS [START_REF] Omg | Data Distribution Service for Real-Time Systems Specification[END_REF] is ultra-low latency, real-time, and Data-Centric Publisher/Subscriber (Pub/Sub) middleware [START_REF] Douglas | Addressing the Challenges of Tactical Information Management in Net-Centric Systems With DDS, The journal of Defense software Engineering[END_REF]: real-time means that all the right data should be delivered at the right place on the right time, all the time. Ultra-low latency means that it is able to distribute high volume of data with very low latency. Data centric means that it is able to ensure availability, reliability, safety and integrity [START_REF] Xinjie | QoS-Aware Publish-Subscribe Service for Real-Time Data Acquisition[END_REF]. It is policy driven and allows to configure all the QoS that relay matters when distributing information. DDS supports dynamic discovery (publisher and subscriber can join and leave the simulation at any point of time) and full time decoupling (publisher delivers some kind of data and subscriber is interested is any combination of those in any time).

The DDS specification implements a Global Data Space (GDS) to be be fully distributed, avoid a single point of failure, and share information between participants [START_REF] Lu | A Novel QoS-Enable Real-Time Publish-Subscribe Service[END_REF]. Applications 3, information are exchanged between distributed application using Pub/Sub paradigm with the aid of the following entities: publisher and datawriter on the sending side, subscriber and datareader at the receiving side. Publisher: is the object responsible for sending of data, owning and managing the DataWriter. A DataWriter can be only owned by a single Publisher while a Publisher can own many DataWriters. Subscriber: is the object responsible for the receipt of published data. The subscriber own and manages DataReaders, which can be only owned by a single subscriber while a subscriber can own multiple DataReaders. Topic: represents the unit of information that is produced and/or consumed. A Topic is an association between a Type, a name and QoS policies used to control the properties of this Topic. As described in Figure4, Topic Types are expressed by means of structures and represented by the subset of OMG-IDL [START_REF]Common Object Request Broker Architecture (CORBA/IIOP) v3.1[END_REF] The architecture of the distributed simulation including many simulators in the same live simulation is depicted in Figure 5. The network module allows delivering the data issued from the virtual environment by sending data to all the simulators sharing the same simulation exercise. In order I, the traffic profile is divided into three classes: signal, states (events) and streams: Signals represent data which change continuously, for example data taken from an speed sensor (accelerator). The reliability QoS policy is chosen as Best-Effort (BE) because many samples are sent during the simulation, so that can be lost without the need of retransmitting them. Signal producer indicate the speed at which it can publish by offering an update deadline equal to 50ms. Latency Budget can be chosen in the User QoS profile to meet the requirement of each trainee, without external stress factors, especially for new trainees. Events are in the most time very critical and they should be sent reliably (R). The durability QoS Specifies whether or not the middleware will store and deliver previously published data to new DataReaders, events use Persistent (P) durability. Multimedia stream is less critical because they are defined using a precedence relationship depending on the codec used, but they can be sent/received unreliable (BE) to avoid the network latency.Further, data samples are sent continuously between the distributed simulators, so the QoS history setting is 'KEEP LEAST' for the three flows. Finally, in order to evaluate the performance of the distributed interactive simulation, we measured the network traffic transmitted during a simulation exercise as shown in Figure 7. The scenario used to evaluate the network module is composed of two simulators publishing and subscribing data during the simulation exercise, and we used another computer (for monitoring and debriefing) to collect all the traffic. Each simulator is considered at the same time as publisher and subscriber, and has a webcam for capturing multimedia (video and audio).

To ensure the inter-operability between the distributed simulators, the OMG-RTPS [START_REF]Data Distribution Ser vice Interoperability Wire-Protocol Specification[END_REF] which is a real time wired protocol build upon of UDP is used. Using the DDS middleware provided by RTI [START_REF]Real Time Innovation[END_REF], the throughput at each simulator is up to 2Mbps. Hence simulators are able to deliver high volume data (multimedia), time critical data (signal in the simulation flow) and events without network bottlenecks. The experiments wa scaled to more six simulators and the traffic remains satisfying the QoS of all the participants.

VI. RELATED WORKS

This Section compare our activities on low cost PC-based simulators with related works. Multi-domain and low cost simulation. Earlier work [START_REF] Smailai | New Integrated Modeling and Simulation Techniques for Reasearch and Training Applications[END_REF] investigated development methods to design an integrated fly simulator for research and training purpose. Our solution provides the way to everyone to be trained in safe condition with, among other, a real car driving cab using existing software and hardware. high performance collaborative simulation. High Level Architecture (HLA) [START_REF] Wu | HLA-based Multiaircraft Comabt Simulation System[END_REF] has a wide applicability, across a full range of simulations areas, including education and training, analysis, engineering, and web-based distributed applications.However, HLA focuses on the inter-operability of simulators without giving any features for the QoS management. In our investigation, high performance and rich QoS policies offered by DDS-based architecture are used to ensure both network centric and inter-operability with a minimum development effort.

VII. CONCLUSION

In this paper, together hardware, software and network architecture design for low cost PC-based driving simulator is addressed. The successful implementation leads in favor of DDS, which is ultra-low latency middleware, providing data-centric and decoupled communication with a QoS provision and offers the possibility of the management, the online resources reconfiguration, and address the challenge of information exchange in high performance communication systems. The future works will focus intercontinental PCbased simulation using DDS.

Fig. 1 .

 1 Fig. 1. Hardware architecture of the simulator

Fig. 3 .

 3 Fig. 3. Publisher-Subscriber communication architecture based on DDS

  standard. Domain: It provides a virtual network linking all applications that have joined it (having the same Domain ID). The Domain consists of one or more DomainParticipant which isolate applications into several sub domains. Partition: this entity represents a logical grouping of Topics associated with the Domain. It serves as a container and logical manager of DDS entities.

Fig. 4 .

 4 Fig. 4. IDL definition of the driving car Type

Fig. 5 .

 5 Fig. 5. Distributed simulators interconnected in live simulation

Fig. 6 .

 6 Fig. 6. Organization of Topics of the Simulator

Fig. 7 .

 7 Fig. 7. Throughput at two distributed simulator

ACKNOWLEDGMENT

This research is supported by the French FUI-DGE (Single Inter-Ministerial Fund of the Directorate General for Enterprize) program within the network simulation Platform (PLATSIM).