Jean-Marc Bardet
email: jean-marc.bardet@univ-paris1.fr

William Kengne

Monitoring procedure for parameter change in causal time series

Keywords: Sequential change detection, Change-point, Causal processes, Quasi-maximum likelihood estimator, Weak convergence

We propose a new sequential procedure to detect change in the parameters of a process X = (X t ) t∈Z belonging to a large class of causal models (such as AR(∞), ARCH(∞), TARCH(∞), ARMA-GARCH processes). The procedure is based on a difference between the historical parameter estimator and the updated parameter estimator, where both these estimators are based on a quasi-likelihood of the model. Unlike classical recursive fluctuation test, the updated estimator is computed without the historical observations. The asymptotic behavior of the test is studied and the consistency in power as well as an upper bound of the detection delay are obtained. Some simulation results are reported with comparisons to some other existing procedures exhibiting the accuracy of our new procedure. The procedure is also applied to the daily closing values of the Nikkei 225, S&P 500 and FTSE 100 stock index. We show in this real-data applications how the procedure can be used to solve off-line multiple breaks detection.

Introduction

In statistical inference, many authors have pointed out the danger of omitting the existence of changes in data.

Many papers have been devoted to the problem of test for parameter changes in time series models when all data are available, see for instance Horváth [START_REF] Horváth | Change in autoregressive processes[END_REF], Inclan and Tiao [START_REF] Inclan | Use of cumulative sums of squares for retrospective detection of changes of variance[END_REF], Kokoszka and Leipus [START_REF] Kokoszka | Leipus Testing for parameter changes in ARCH models[END_REF], Kim et al. [START_REF] Kim | On the CUSUM test for parameter changes in GARCH(1, 1) models[END_REF],

Horváth and Shao [START_REF] Horváth | Limit theorems for permutations of empirical processes with applications to change point analysis Stochastic Processes and their[END_REF], Aue et al. [START_REF] Aue | Delay times of sequential procedures for multiple time series regression models[END_REF] or Kengne [START_REF] Kengne | Testing for parameter constancy in general causal time-series models[END_REF]. These papers consider "retrospective" (off-line) changes i.e. changes in parameters when all data are available. Another point of view is the change detection when new data arrive; this is the sequential change-point problem. For instance, consider the following sequential problems.

Example 1.1 (Industrial quality control). Consider an industrial system producing electronic objects. To know in real time the quality of production, some devices have been installed to rely the informations about the functioning. The performance is evaluated for each production and the system is automatically stopped if a disorder is detected. After an investigation of the root cause of the problem and a possible maintenance, the system is resetting. IR d . Let T ⊂ Z, and for any θ ∈ Θ, define Class M T (M θ , f θ ): The process X = (X t ) t∈Z belongs to M T (M θ , f θ ) if it satisfies the relation:

X t+1 = M θ (X t-i ) i∈I N ξ t + f θ (X t-i ) i∈I N for all t ∈ T . (1)
The existence and properties of this general class of causal and affine processes were studied in Bardet and Wintenberger [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]. Numerous classical time series (such as AR(∞), ARCH(∞), TARCH(∞), ARMA-GARCH or bilinear processes) are included in M Z (M, f ). The off-line change detection for such class of models has already been studied in Bardet et al. [START_REF] Bardet | Detecting multiple change-points in general causal time series using penalized quasi-likelihood[END_REF] and Kengne [START_REF] Kengne | Testing for parameter constancy in general causal time-series models[END_REF].

Suppose now that we have observed X 1 , • • • , X n which are available historical data such that there exists

θ * 0 ∈ Θ such as (X 1 , • • • , X n ) belongs to M {1,••• ,n} (M θ * 0 , f θ * 0 )
. Then, we observe new data X n+1 , X n+2 • • • , X k , • • • : the monitoring scheme starts. For each new observation, we would like to know if a change occurs in the parameter θ * 0 . More precisely, we consider the following test :

H 0 : θ * 0 is constant over the observation X 1 , • • • , X n , X n+1 , • • • i.e. the observations X 1 , • • • , X n , X n+1 , • • • belong to M I N (M θ * 0 , f θ * 0 ); H 1 : there exist k * > n, θ * 1 ∈ Θ such that X 1 , • • • , X n , X n+1 , • • • , X k * , X k * +1 , • • • belongs to M {1,••• ,k * } (M θ * 0 , f θ * 0 ) M {k * +1,••• } (M θ * 1 , f θ * 1 )
.

When new data arrive, Chu et al. [START_REF] Chu | Monitoring structural change[END_REF] proposed to compute an estimator of the parameter based on all the observations and to compare it to an estimator based on historical data. A large distance between both these estimators means that new data come from a model with different parameters. Then the null hypothesis H 0 is rejected and the new parameter is considered; otherwise, the monitoring scheme continues. In their procedure, Leisch et al. [START_REF] Leisch | Monitoring structural changes with the generalized fluctuation test[END_REF] suggested to compute the recursive estimators on a moving window with a fixed width. They fixed a monitoring horizon so that, the procedure will stop after a fixed number of steps even if no change is detected. As Chu et al. [START_REF] Chu | Monitoring structural change[END_REF], the recursive estimators computed by Na et al. [START_REF] Na | Monitoring parameter change in time series models[END_REF] are based on all the observations. As we will see in the next sections, their procedure cannot be effective in terms of detection delay or if a small change in the parameter occurs.

For any k ≥ 1, ℓ, ℓ ′ ∈ {1, • • • , k} (with ℓ ≤ ℓ ′ ) let θ(X ℓ , • • • , X ℓ ′ ) be the quasi-maximum likelihood estimator (QMLE in the sequel) of the parameter computed on {ℓ, • • • , ℓ ′ } as it is defined in [START_REF] Basseville | Detection of Abrupt Changes: Theory and Application[END_REF]. When new data arrive at time k ≥ n, we explore the segment {ℓ, ℓ + 1,

• • • , k} with ℓ ∈ {n -v n , n -v n + 1 • • • k -v n } (where (v n ) n∈I N is a fixed sequence of integer numbers) that the distance between θ(X ℓ , • • • , X k ) and θ(X 1 , • • • , X n ) is the largest. If the norm θ(X ℓ , • • • , X k ) -θ(X 1 , • • • , X n ) is
greater than a suitable critical value, then H 0 is rejected and a model with a new parameter is considered; otherwise, the monitoring scheme continues. More precisely, we construct a detector that takes into account the distance between θ(X ℓ , • • • , X k ) and θ(X 1 , • • • , X n ). It is shown that this detector is almost surely finite under the null hypothesis and almost surely diverges to infinity under the alternative. Hence, the consistency of our procedure follows. Finally, Monte-Carlo experiments have been done, comparing our procedure to the ones of Horváth et al. [START_REF] Horváth | Monitoring changes in linear models[END_REF] (see also Aue et al. [START_REF] Aue | Change point monitoring in linear models[END_REF]) and Na et al. [START_REF] Na | Monitoring parameter change in time series models[END_REF]. It appears that our procedure outperforms these other procedures in terms of test power and detection delay in different frames. An application to financial data (Nikkei 225, S&P 500 and FTSE 100 stock index) allows to detect changes in these data in accordance with historical and economic events.

In the forthcoming Section 2 the assumptions and the definition of the quasi-likelihood estimator are provided. In Section 3 we present the monitoring procedure and the asymptotic results. Section 4 is devoted to a simulation study for AR [START_REF] Aue | Break detection in the covariance structure of multivariate time series models[END_REF] and GARCH(1, 1) processes. In Section 5 we apply our procedure to famous financial data. The proofs of the main results are provided in Section 6.

2 Assumptions and definition of the quasi-likelihood estimator 2.1 Assumptions on the class of models M Z (f θ , M θ )

Let θ ∈ IR d and M θ and f θ be numerical functions such that for all (x i ) i∈I N ∈ IR I N , M θ (x i ) i∈I N = 0 and

f θ (x i ) i∈I N ∈ IR. Denote h θ := M 2
θ . We will use the following classical notations: 1. • applied to a vector denotes the Euclidean norm of the vector; 2. for any compact set K ⊆ IR d and for any g :

K -→ IR d ′ , g K = sup θ∈K ( g(θ) ); 3. for any set K ⊆ IR d , • K denotes the interior of K.
Throughout the sequel, we will assume that the functions θ → M θ and θ → f θ are twice continuously differentiable on Θ. Let Ψ θ = f θ , M θ and i = 0, 1, 2, then for any compact set K ⊂ Θ define Assumption A i (Ψ θ , K): Assume that ∂ i Ψ θ (0)/∂θ i Θ < ∞ and there exists a sequence of non-negative real numbers (α

(i) j (Ψ θ , K)) j≥1 such that ∞ j=1 α (i) j (Ψ θ , |) < ∞ and ∂ i Ψ θ (x) ∂θ i - ∂ i Ψ θ (y) ∂θ i K ≤ ∞ j=1 α (i) j (Ψ θ , K)|x j -y j | for all x, y ∈ IR I N .
In the sequel we refer to the particular case called "ARCH-type process" if f θ = 0 and if the following assumption holds with h θ = M 2 θ :

Assumption A i (h θ , K): Assume that ∂ i h θ (0)/∂θ i Θ <
∞ and there exists a sequence of non-negative real numbers (α

(i) j (h θ , K)) j≥1 such as ∞ j=1 α (i) j (h θ , K) < ∞ and ∂ i h θ (x) ∂θ i - ∂ i h θ (y) ∂θ i K ≤ ∞ j=1 α (i) j (h θ , K)|x 2 j -y 2 j | for all x, y ∈ IR I N .
The Lipschitz-type hypothesis A i (Ψ θ , K) are classical when studying the existence of solutions of the general model (see for instance [START_REF] Doukhan | Weakly dependent chains with infinite memory[END_REF]). Using a result of [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF], for each model M Z (M θ , f θ ) it is interesting to define the following set:

Θ(r) := θ ∈ Θ, A 0 (f θ , {θ}) and A 0 (M θ , {θ}) hold with j≥1 α (0) j (f θ , {θ})+(E |ξ 0 | r ) 1/r j≥1 α (0) j (M θ , {θ}) < 1 θ ∈ Θ, f θ = 0 and A 0 (h θ , {θ}) holds with (E |ξ 0 | r ) 2/r j≥1 α (0) j (h θ , {θ}) < 1 .
Then, if θ ∈ Θ(r) the existence of a unique causal, stationary and ergodic solution

X = (X t ) t∈Z ∈ M Z (f θ , M θ )
is ensured (see more details in [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]). The subset Θ(r) is defined as a union to consider accurately general causal models and ARCH-type models simultaneously.

Here there are assumptions required for studying QMLE asymptotic properties:

Assumption D(Θ): ∃h > 0 such that inf θ∈Θ (|h θ (x)|) ≥ h for all x ∈ IR I N . Assumption Id(Θ): For all (θ, θ ′ ) ∈ Θ 2 , f θ (X 0 , X -1 , • • • ) = f θ ′ (X 0 , X -1 , • • • ) and h θ (X 0 , X -1 , • • • ) = h θ ′ (X 0 , X -1 , • • • ) a.s. ⇒ θ = θ ′ .
Assumption Var(Θ): For all θ ∈ Θ, one of the families

∂f θ ∂θ i (X 0 , X -1 , • • • ) 1≤i≤d or ∂h θ ∂θ i (X 0 , X -1 , • • • ) 1≤i≤d is a.s. linearly independent. Assumption K(f θ , M θ , Θ): for i= 0, 1, 2, A i (f θ , Θ) and A i (M θ , Θ) (or A i (h θ , Θ)) hold and there exists ℓ > 2 such that α (i) j (f θ , Θ) + α (i) j (M θ , Θ) + α (i) j (h θ , Θ) = O(j -ℓ ) for j ∈ IN .
Note that in this last assumption, as in [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF], we use the convention that if A i (M θ , Θ) holds then α

(i) ℓ (h θ , Θ) = 0 and if A i (h θ , Θ) holds then α (i) ℓ (M θ , Θ) = 0.

Two first examples

1. ARMA(p, q) processes.

Consider the ARMA(p, q) process defined by:

X t + p i=1 a * i X t-i = q j=0 b * j ξ t-j , t ∈ Z (2) with b * 0 = 0, θ * 0 = (a * 1 , • • • , a * p , b * 0 , • • • , b * q ) ∈ Θ ⊂ IR p+q+1
and (ξ t ) a white noise such as E (ξ 2 0 ) = 1. When q j=0 b * j X j = 0 and 1 + p i=0 a * i X i = 0 for all |X| ≤ 1, this process can be also written as:

X t = b * 0 ξ t + ∞ j=1 φ j (θ * 0 ) X t-i , t ∈ Z
where θ ∈ Θ → φ j (θ) are functions only depending on θ and decreasing exponentially fast to 0 (j → ∞). The process (2) belongs to the class

M Z (M θ * 0 , f θ * 0 ) where f θ (x 1 , • • • ) = j≥1 φ j (θ)x j and M θ ≡ b * 0 for all θ ∈ Θ. Then Assumptions D(Θ), A 0 (f θ , Θ), A 0 (M θ , Θ) hold with h = |b * 0 | > 0 and α (0) j (f θ , Θ) = φ j (θ) Θ while α (0) j (M θ , Θ) = 0 for j ∈ IN * . Assumption K(f θ , M θ , Θ)
holds since there exists c > 0 and C > 0 such as |φ j | ≤ C e -cj for j ∈ IN . Moreover, if (ξ t ) is a sequence of non-degenerate random variables (i.e. ξ t is not equal to a constant), Assumptions Id(Θ) and Var(Θ) hold. Finally, for any r ≥ 1 such that

E |ξ 0 | r < ∞, then Θ(r) = θ ∈ IR p+q+1 , j≥1 |φ j (θ)| < 1 .
Note that if θ ∈ Θ(r) with r ≥ 1 then the previous conditions of stationarity q j=0 b j X j = 0 and 1 + p i=0 a i X i = 0 for all |X| ≤ 1 are satisfied.

2. GARCH(p, q) processes.

Consider the GARCH(p, q) process defined by:

X t = σ t ξ t , σ 2 t = α * 0 + p j=1 α * j X 2 t-j + q j=1 β * j σ 2 t-j , t ∈ Z (3) with E (ξ 2 0 ) = 1 and θ * 0 := (α * 0 , • • • , α * p , β * 1 , • • • , β * q ) ∈ Θ where Θ is a compact subset of ]0, ∞[×[0, ∞[ p+q such that p j=1 α j + q j=1 β j < 1 for all θ ∈ Θ.
Then there exists (see Bollerslev [START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF] or Nelson and Cao [START_REF] Nelson | Inequality Constraints in the Univariate GARCH Model[END_REF]) a nonnegative sequence (ψ j (θ * 0 )) j≥0 such that

σ 2 t = ψ 0 (θ * 0 ) + j≥1 ψ j (θ * 0 )X 2 t-j with ψ 0 (θ * 0 ) = α * 0 /(1 - q j=1 β * j ). This process belongs to the class M Z (M θ * 0 , f θ * 0 ) where f θ ≡ 0 and M θ (x 1 , • • • ) = ψ 0 (θ) + j≥1 ψ j (θ)x 2 j
for all θ ∈ Θ. Assumption D(Θ) holds with h = inf θ∈Θ (ψ 0 (θ)) > 0. If there exists 0 < ρ 0 < 1 such that for any θ ∈ Θ, q j=1 α j + p j=1 β j ≤ ρ 0 then the sequences ( ψ j (θ) Θ ) j≥1 , ( ψ ′ j (θ) Θ ) j≥1 and ( ψ ′′ j (θ) Θ ) j≥1 decay exponentially fast (see Berkes et al. [START_REF] Berkes | GARCH processes: structure and estimation[END_REF]) and Assumption K(f θ , M θ , Θ) holds. Moreover, (ξ 2 t ) is a sequence of non-degenerate random variables (i.e. ξ 2 t is not equal to a constant), Assumptions Id(Θ) and Var(Θ) hold. Finally for r ≥ 2 we obtain

Θ(r) = θ ∈ Θ ; (E |ξ 0 | r ) 2/r ∞ j=1 φ j (θ) < 1 .

The quasi-maximum likelihood estimator

Let k ≥ n ≥ 2, if (X 1 , • • • , X k ) ∈ M {1,••• ,k} (M θ , f θ ), then for T ⊂ {1, • • • ,
k}, the conditional quasi-(log)likelihood computed on T is given by:

L(T, θ) := -θ(T ) := argmax θ∈Θ ( L(T, θ)). (6)
In Bardet and Wintenberger [4] it was established that if

(X 1 , • • • , X n ) is an observed trajectory of X ⊂ M Z (f θ * 0 , M θ * 0 ) with θ * 0 ∈ • Θ(4) and if Θ is a compact set such as Assumptions A i (f θ , M θ , Θ) (or A i (h θ , Θ))
hold for i = 0, 1, 2 and under Assumptions D(Θ), Id(Θ), Var(Θ), K(f θ , M θ , Θ), then

√ n θ(T 1,n ) -θ * 0 D -→ n→∞ N 0 , F G -1 F , (7)
with

G := E ∂q 0 (θ * 0 ) ∂θ ∂q 0 (θ * 0 ) ∂θ ′ and F := E ∂ 2 q 0 (θ * 0 ) ∂θ∂θ ′ , (8)
where ′ denotes the transpose and with q 0 defined in [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]. Note that under assumptions D(Θ) and Var(Θ), G is symmetric positive definite (see [START_REF] Kengne | Testing for parameter constancy in general causal time-series models[END_REF]) and F is non-singular (see [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]). Also define the matrix

G(T ) := 1 Card(T ) t∈T ∂ q t ( θ(T )) ∂θ ∂ q t ( θ(T )) ∂θ ′ and F (T ) := - 2 Card(T ) ∂ 2 L m (T, θ(T )) ∂θ∂θ ′ . ( 9
)
Under the previous assumptions, G(T 1,n ) and F (T 1,n ) converge almost surely to G and F respectively. Hence,

√ n G(T 1,n ) -1/2 F (T 1,n ) θ(T 1,n ) -θ * 0 D -→ n→∞ N 0 , I d (10)
with I d the identity matrix. This result will be the starting point of the following monitoring procedure.

3 The monitoring procedure and asymptotic results

The monitoring procedure

In the sequel, (X 1 , • • • , X n ) is supposed to be the historical available observations belonging to the class

M {1,••• ,n} (f θ * 0 , M θ * 0 ). For 1 ≤ ℓ ≤ ℓ ′ , denote T ℓ,ℓ ′ := {ℓ, ℓ + 1, • • • , ℓ ′ }.
At a monitoring instant k, our procedure evaluates the difference between θ(T ℓ,k ) and θ(T 1,n ) for any ℓ = n, • • • , k. More precisely, from [START_REF] Billingsley | Convergence of Probability Measures[END_REF], for any k > n define the statistic (called the detector)

C k,ℓ := √ n k -ℓ k G(T 1,n ) -1/2 F (T 1,n ) θ(T ℓ,k ) -θ(T 1,n ) for ℓ = n, • • • , k. Since the matrix G(T 1,n
) is asymptotically symmetric and positive definite (see [START_REF] Kengne | Testing for parameter constancy in general causal time-series models[END_REF]), G(T 1,n ) -1/2 exists for n large enough and C k,ℓ is well defined. At the beginning of the monitoring scheme and when ℓ is close to k, the length of T ℓ,k is too small, therefore the numerical algorithm used to compute θ(T ℓ,k ) cannot converge. This can introduce a large distortion in the procedure. To avoid this, we introduce a sequence of integer numbers (v n ) n∈I N with v n << n and compute

C k,ℓ for ℓ ∈ {n -v n , n -v n + 1, • • • , k -v n }. Thus, for any k > n denote Π n,k := {n -v n , n -v n + 1, • • • , k -v n }.
For technical reasons, assume that,

v n → ∞ and v n / √ n → 0 (n → ∞).
According to Remark 1 of [START_REF] Kengne | Testing for parameter constancy in general causal time-series models[END_REF], we can choose

v n = [(log n) δ ] with δ > 1.
Note that, if change does not occur at time k > n, for any ℓ ∈ Π n,k , the two estimators θ(T ℓ,k ) and θ(T 1,n ) are close and the detector C k,ℓ is not too large. Hence, the monitoring scheme rejects H 0 at the first time k > n where there exists ℓ ∈ Π n,k satisfying C k,ℓ > c for a fixed constant c > 0. To be more general, we will use a b : (0, ∞) → (0, ∞), called a boundary function satisfying:

Assumption B: b : (0, ∞) → (0, ∞) is a non-increasing and continuous function such as Inf 0<t<∞ b(t) > 0.
Then the monitoring scheme rejects H 0 at the first time k > n such as there exists ℓ ∈ Π n,k satisfying

C k,ℓ > b((k -ℓ)/n).
Hence define the stopping time:

τ (n) := Inf k > n / ∃ℓ ∈ Π n,k , C k,ℓ > b((k -ℓ)/n) = Inf k > n / max ℓ∈Π n,k C k,ℓ b((k -ℓ)/n) > 1 .
Therefore, we have

P {τ (n) < ∞} = P max ℓ∈Π n,k C k,ℓ b((k -ℓ)/n) > 1 for some k > n = P sup k>n max ℓ∈Π n,k C k,ℓ b((k -ℓ)/n) > 1 . ( 11
)
The challenge is to choose a suitable boundary function b(•) such as for some given α ∈ (0, 1)

lim n→∞ P H0 {τ (n) < ∞} = α and lim n→∞ P H1 {τ (n) < ∞} = 1
where the hypothesis H 0 and H 1 are specified in Section 1.

In the case where b(•) is a constant positive value, b ≡ c with c > 0, these conditions lead to compute a threshold c = c α depending on α. If change is detected under

H 1 i.e. τ (n) < ∞ and τ (n) > k * ,
then the detection delay is defined by

d n = τ (n) -k * .
Using the previous notations, Na et al. [START_REF] Na | Monitoring parameter change in time series models[END_REF] used the following detector

D k := √ n G(T 1,n ) -1/2 F (T 1,n ) θ(T 1,k ) -θ(T 1,n ) .
At the step k of the monitoring scheme, their recursive estimator is based on

X 1 , • • • , X n , • • • , X k .
One can see that this estimator is highly influenced by the historical data. Assume that a change occurs at time k * ≤ k, in the sequel of the procedure, the recursive estimator contents the observations

X 1 , • • • , X n , • • • , X k * -1 which de- pends on θ * 0 . Then, one must wait longer before the difference between θ(X 1 , • • • , X n ) and θ(X 1 , • • • , X n , • • • , X k
) becomes significant at a step k > k * . Therefore, their procedure cannot be effective in terms of detection delay.

Moreover, if n tends to infinity, it is not almost sure that this change will be detected. These are confirmed by the results of simulations (see Section 4). [START_REF] Berkes | Sequential change-point detection in GARCH(p,q) models[END_REF] considered an estimator based on historical data to compute the quasi-likelihood scores. They used the fact that the partial derivatives applied to a vector u is equal to 0 if and only if u is the true parameter of the model. So, when change occurs, their detector growths asymptotically to infinity. Therefore, their procedure is consistent. They proved this result for GARCH(p,q) models.

Asymptotic behaviour under the null hypothesis

Under H 0 , the parameter θ * 0 does not change over the new observations. Thus we have the result [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]. Under the null hypothesis H 0 , then

Theorem 3.1. Assume D(Θ), Id(Θ), Var(Θ), K(f θ , M θ , Θ), B and θ * 0 ∈ • Θ (
lim n→∞ P {τ (n) < ∞} = P sup t>1 sup 1<s<t W d (s) -sW d (1)) t b(s) > 1
where W d is a d-dimensional standard Brownian motion.

In the simulations, we will use the most "natural" boundary function b(•) = c with c a positive constant since it satisfies the above assumptions imposed to b(•). In such case, the forthcoming corollary indicates that the asymptotic distribution of Theorem 3.1 can be easily computed:

Corollary 3.1. Assume b(t) = c > 0 for t ≥ 0.
Under the assumptions of Theorem 3.1,

lim n→∞ P {τ (n) < ∞} = P sup t>1 sup 1<s<t 1 t W d (s) -sW d (1)) > c = P {U d > c}
where

U d = sup 0<u<1 f (u) W d (u) with f (u) = √ 9 -u + √ 1 -u √ 9 -u + 3 √ 1 -u 2 3 -u + (9 -u)(1 -u) 1/2 . Remark 3.1.

By the law of the iterated logarithm, it comes that

sup 1<s<t 1 t W d (s) -sW d (1)) a.s. -→ t→∞ W d (1)) .
So, the two distributions sup 1<s<t

1 t W d (s) -sW d (1)) as t → ∞ (resp. t → 1) and f (u) W d (u) as u → 1 (resp. u → 0) are equal. It is easy to show (see proof of Corollary 3.1) that sup t>1 sup 1<s<t 1 t W d (s) -sW d (1)) D = sup 0<u<1 f (u) W d (u) .

Under the null hypothesis, it holds that θ(T

-→ n→∞ θ * 0 (see [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]). Thus denote

C (0) k,ℓ := √ n k -ℓ k G(T 1,n ) -1/2 F (T 1,n ) θ(T ℓ,k ) -θ * 0 .
Under the assumptions of Theorem 3.1, one can easily show that

sup k>n max ℓ∈Π n,k 1 b((k -ℓ)/n) C k,ℓ -C (0) k,ℓ = o P (1) as n → ∞.
Thus, the Theorem 3.1 still holds if the stopping time τ (n) is computed by using the detector C (0) k,ℓ . Hence, if the parameter θ * 0 of the historical observations is known, then use the detector

C (0) k,ℓ instead of C k,ℓ . But let us note that this situation is infrequent in practice.
Therefore, at a nominal level α ∈ (0, 1), take c = c(α) be the (1α)-quantile of the distribution of U d = sup 0<u<1 f (u) W d (u) which can be computed through Monte-Carlo simulations. Table 1 shows the (1α)quantile of this distribution for α = 0.01, 0.05, 0. 10 andd

= 1, • • • , 5. d = 1 d = 2 d = 3 d = 4 d = 5 α = 0.

Asymptotic behaviour under the alternative hypothesis

Under the alternative H 1 , the parameter changes from θ * 0 to θ * 1 at k * > n, where θ * 1 ∈ Θ and

θ * 0 = θ * 1 . Then Theorem 3.2. Assume D(Θ), Id(Θ),Var(Θ), K(f θ , M θ , Θ) and B. Under the alternative H 1 , if θ * 1 = θ * 0 and θ * 0 , θ * 1 ∈ • Θ(4) then for k * = k * (n) such as lim sup n→∞ k * (n)/n < ∞ and k n = k * (n) + n δ with δ ∈ (1/2, 1), max ℓ∈Π n,kn C kn,ℓ b((k n -ℓ)/n) a.s. -→ n→∞ ∞.
The forthcoming Corollary 3.2 can be immediately deduced from the relation [START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF].

Corollary 3.2. Under assumptions of Theorem 3.2,

lim n→∞ P {τ (n) < ∞} = 1.
Remark 3.2. We know that the monitoring scheme rejects H 0 at the first time k where

max ℓ∈Π n,k C k,ℓ b((k -ℓ)/n) > 1.
Therefore, it follows from Theorem 3.2 that under the hypothesis H 1 , the detection delay d n of the procedure can be bounded by O P (n 1/2+ε ) for any ε > 0 (or even by O P √ n(log n) a with a > 0 using the same kind of proof ).

Examples

AR(∞) processes

Consider the generalization of ARMA(p, q) processes defined in (2) i.e. a AR(∞) processes defined by:

X t = φ 0 (θ * 0 ) + j≥1 φ j (θ * 0 )X t-j + ξ t , t ∈ Z (12)
with θ * 0 ∈

• Θ, where we can chose Θ as a compact subset of Θ(4) ⊂ IR d where

Θ(4) = θ ∈ IR d ; j≥1 |φ j (θ)| < 1 .
This process belongs to the class

M Z (M θ * 0 , f θ * 0 ) where f θ (x 1 , • • • ) = j≥1 φ j (θ)x j and M θ ≡ φ 0 (θ) for all θ ∈ Θ and therefore α (0) j (f θ , Θ) = φ j (θ) Θ and α (0) j (M θ , Θ) = 0 for j ∈ IN * . Then • Assumption D(Θ) holds if h = inf θ∈Θ (|φ 0 (θ)|) > 0; • Assumption K(f θ , M θ , Θ) holds if there exists ℓ > 2 and and if θ → φ j (θ) are twice differentiable functions satisfying max ψ j (θ) Θ , φ ′ j (θ) Θ , φ ′′ j (θ) Θ = O(j -ℓ ) for j ∈ IN .
• if (ξ t ) is a sequence of non-degenerate random variables, Assumptions Id(Θ) and Var(Θ) hold.

Case of AR(p) process

Assume that

X t = φ * 0 + p j=1 φ * j X t-j + ξ t with p ∈ IN * .
The true parameter of the model is denoted by

θ * 0 = (φ * 0 , φ * 1 , • • • , φ * p ) ∈ Θ where Θ = {θ = (φ 0 , φ 1 , • • • , φ p ) ∈ IR p+1 / p j=1 |φ j | < 1}. Then, Θ(r) = Θ for any r ≥ 1. Assume that a trajectory (X 1 , • • • , X k ) has been observed, for any t = 1, • • • , k and θ ∈ Θ we have, q t (θ) = X t -φ 0 - p j=1 φ j X t-j 2 , ∂ q t (θ) ∂θ = -2 X t -φ 0 - p j=1 φ j X t-j • (1, X t-1 , X t-2 , • • • , X t-p ). Moreover, ∂ 2 q t (θ) ∂φ 0 ∂φ 0 = 2, for j = 1, • • • , p, ∂ 2 q t (θ) ∂φ 0 ∂φ j = 2X t-j and for 1 ≤ i, j ≤ p, ∂ 2 q t (θ) ∂φ i ∂φ j = 2X t-i X t-j .

ARCH(∞) processes

Consider the generalization GARCH(p, q) processes defined in (3) i.e. a ARCH(∞) processes defined by:

X t = σ t ξ t and σ 2 t = ψ 0 (θ * 0 ) + ∞ j=1 ψ j (θ * 0 )X 2 t-j , t ∈ Z (13)
with θ * 0 ∈

• Θ, where we can chose Θ as a compact subset of Θ(4) ⊂ IR d where

Θ(4) = θ ∈ IR d ; (E |ξ 0 | 4 ) 1/2 ∞ j=1 |φ j (θ)| < 1 .
This process, introduced by Robinson [START_REF] Robinson | Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression[END_REF], belongs to the class

M Z (f θ * 0 , M θ * 0 , ) where f θ (x 1 , • • • ) ≡ 0 and M 2 θ (x 1 , • • • ) = ψ 0 (θ) + j≥1 ψ j (θ)x 2 j for all θ ∈ Θ and therefore α (0) j (f θ , Θ) = 0 and α (0) j (h θ , Θ) = φ j (θ) Θ for j ∈ IN * (X is of course a ARCH-type process). Then • Assumption D(Θ) holds if h = inf θ∈Θ (ψ 0 (θ)) > 0;
• Assumption K(f θ , M θ , Θ) holds if there exists ℓ > 2 and and if θ → φ j (θ) are twice differentiable functions satisfying max

ψ j (θ) Θ , ψ ′ j (θ) Θ , ψ ′′ j (θ) Θ = O(j -ℓ ) for j ∈ IN . • if (ξ 2 t
) is a sequence of non-degenerate random variables, Assumptions Id(Θ) and Var(Θ) hold.

Case of GARCH(1, 1) process Assume that

X t = σ t ξ t with σ 2 t = α * 0 + α * 1 X 2 t-1 + β * 1 σ 2 t-1 with θ * 0 = (α * 0 , α * 1 , β * 1 ) ∈ Θ ⊂]0, ∞[×[0, ∞[ 2 and satisfying α * 1 + β * 1 < 1. The ARCH(∞) representation is σ 2 t = α * 0 /(1 -β * 1 ) + α * 1 j≥1 (β * 1 ) j-1 X 2 t-j . If a trajectory (X 1 , • • • , X k ) has been observed, for any t = 1, • • • , k and θ ∈ Θ we have, h t θ = α 0 /(1 -β 1 ) + α 1 X 2 t-1 + α 1 t j=2 β j-1 1 X 2 t-j and q t (θ) = X 2 t / h t θ + log( h t θ ).
Thus, it follows that

∂ q t (θ) ∂θ = 1 h t θ 1 - X 2 t h t θ ∂ h t θ ∂α 0 , ∂ h t θ ∂α 1 , ∂ h t θ ∂β 1 with ∂ h t θ ∂α 1 = X 2 t-1 + t j=2 β j-1 1 X 2 t-j ∂ h t θ ∂α 0 = 1/(1 -β 1 )
, and

∂ h t θ ∂β 1 = α 0 /(1 -β 1 ) 2 + α 1 X 2 t-2 + α 1 t j=3 (j -1)β j-2 1 X 2 t-k . Let θ = (α 0 , α 1 , β 1 ) = (θ 1 , θ 2 , θ 3 ) ∈ Θ, for 1 ≤ i, j ≤ 3, we have ∂ 2 q t (θ) ∂θ i ∂θ j = 1 ( h t θ ) 2 2X 2 t h t θ -1 ∂ h t θ ∂θ i ∂ h t θ ∂θ j + 1 h t θ 1 - X 2 t h t θ ∂ 2 h t θ ∂θ i ∂θ j with ∂ 2 h t θ ∂α 2 0 = 0, ∂ 2 h t θ ∂α 0 ∂α 1 = 0, ∂ 2 h t θ ∂α 2 1 = 0, ∂ 2 h t θ ∂α 1 ∂β 1 = X 2 t-2 + t j=3 (j -1)β j-2 1 X 2 t-j , ∂ 2 h t θ ∂α 0 ∂β 1 = 1/(1 -β 1 ) 2 and ∂ h t θ ∂β 2 1 = 2α 0 /(1 -β 1 ) 3 + 2α 1 X 2 t-3 + α 1 t j=4 (j -1)(j -2)β j-3 1 X 2 t-j .

TARCH(∞) processes

The process X is called Threshold ARCH(∞) (TARCH(∞) in the sequel) if it satisfies

X t = σ t ξ t and σ t = b 0 (θ * 0 ) + ∞ j=1 b + j (θ * 0 ) max(X t-j , 0) -b - j (θ * 0 ) min(X t-j , 0) , t ∈ Z ( 14
)
where the parameters b 0 (θ), b + j (θ) and b - j (θ) are assumed to be non negative real numbers and θ ∈

• Θ where Θ is a compact subset of Θ(4) where

Θ(4) = θ ∈ IR d E |ξ 0 | 4 1/4 ∞ j=1 max b - j (θ), b + j (θ) < 1 since α (0) j (M, {θ}) = max b - j (θ), b + j (θ)
. This class of processes is a generalization of the class of TGARCH(p,q) processes (introduced by Rabemananjara and Zakoïan [START_REF] Rabemananjara | Threshold ARCH models and asymmetries in volatility[END_REF]). Then,

• Assumption D(Θ) holds if h = inf θ∈Θ b 0 (θ) > 0;
• Assumption K(f θ , M θ , Θ) holds if there exists ℓ > 2 and and if θ → b - j (θ) and θ → b + j (θ) are twice differentiable functions satisfying

max b - j (θ) Θ , b + j (θ) Θ , ∂ ∂θ b - j (θ) Θ , ∂ ∂θ b + j (θ) Θ , ∂ 2 ∂θ 2 b - j (θ) Θ , ∂ 2 ∂θ 2 b + j (θ) Θ = O(j -ℓ ) for j ∈ IN.
Unfortunately, for TARCH(∞) it is not possible to provide simple conditions for obtaining Assumptions Id(Θ)

and Var(Θ) as for AR(∞) or ARCH(∞) processes.

Some simulation and numerical experiments

First remark that, at a time k > n, we need to compute C k,ℓ for all ℓ ∈ Π n,k to test whether change occurs or not. On can see that, the computational time is very long and increases with k. To reduce it, we introduce an integer sequence (u n ) (satisfying

u n / √ n → 0 as n → ∞; typically u n = [ln(n)]) and compute C k,ℓ only for ℓ ∈ Π 0 n,k := {n -v n , n -v n + u n , n -v n + 2u n , • • • , k -v n }.
We have Π 0 n,k ⊂ Π n,k and for any t = ℓ n with ℓ ∈ Π n,k , we can find an integer j ℓ such that n -

v n + j ℓ u n ∈ Π 0 n,k
and nv n + j ℓ u n ≤ ℓ ≤ nv n + (j ℓ + 1)u n . This implies that n-vn+j ℓ un n ≤ t ≤ n-vn+j ℓ un n + un n . Thus, we have asymptotically (as n → ∞), t ∼ n-vn+j ℓ un n . It shows that the previous asymptotic results still hold by computing C k,ℓ for ℓ ∈ Π 0 n,k . The condition u n / √ n → 0 ensures that the Theorem 3.2 still holds by choosing k n = k * (n) + n δ with δ ∈ (1/2, 1). In practice, the use of Π 0 n,k can introduce a distortion in the detection delay. But, the new detection delay must be between d n and d n + u n (where d n is the detection delay obtained by using Π n,k ). In the sequel, we use

u n = [ln(n)].
Moreover, if b ≡ c > 0 is a constant function, according to [START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF], we have

P {τ (n) < ∞} = P sup k>n max ℓ∈Π 0 n,k C k,ℓ > c . (15)
Thus, denote

C k = max ℓ∈Π 0 n,k C k,ℓ for any k > n.
The procedure is monitored from

k = n + 1 to k = n + 500. The set {n + 1, • • • , n + 500} is called monitoring period. According to the Remark 1 of [20], v n = [(log n) δ ] (with 1 ≤ δ ≤ 3) is chosen.
We evaluated the performance of the procedure with

v n = [log n], [(log n) 3/2 ], [(log n) 2 ], [(log n) 3

An illustration

We consider a GARCH(1,1) process :

X t = σ t ξ t with σ 2 t = α * 0 + α * 1 X 2 t-1 + β * 1 σ 2 t-1 . Thus, the parameter of the model is θ * 0 = (α * 0 , α * 1 , β * 1 )
. The historical available data are X 1 , • • • , X 500 (therefore n = 500) and the monitoring period is {501, • • • , 1000}. At the nominal level α = 0.05, the critical values of the procedure is C α = 2.760. The Figure 1 is a typical realization of the statistic ( C k ) 500<k≤1000 . We consider a scenario without change (Figure 1 Figure 1 a-) shows that the detector C k is under the horizontal line which represents the limit of the critical region. On Figure 1 b-) we can see that, before change occurs, C k is under the horizontal line and increases with a high speed after change. Such growth over a long period indicates that something happening in the model.

Monitoring mean shift in times series

Let (X 1 , • • • , X n ) be an (historical) observation of a process X = (X t ) t∈Z . We assume that X satisfy

X t = µ 0 + ǫ t for 1 ≤ t ≤ k * X t = µ 1 + ǫ t for t > k *
with k * > n, µ 0 = µ 1 and (ǫ t ) a zero mean stationary time series belongs to a class M Z (f θ , M θ ). Under H 0 , k * = ∞. The monitoring procedure start at k = n + 1 and the aim is to test mean shift over the new observations X n+1 , X n+2 , • • • . This problem can be seen as monitoring changes in linear model (see Horváth et al. [START_REF] Horváth | Monitoring changes in linear models[END_REF], Aue et al. [START_REF] Aue | Change point monitoring in linear models[END_REF]) with constant regressor. The empirical mean X n = 1 n n i=1 X i is a consistent estimator of µ 0 and the recursive residuals are defined by ǫ k = X k -X n ; for k > n. Horváth et al. [START_REF] Horváth | Monitoring changes in linear models[END_REF] and Aue et al. [START_REF] Aue | Change point monitoring in linear models[END_REF] proposed the

CUSUM detector

Q k = 1 σ n 1 c √ n( k n )(1 -n k ) γ k i=n+1 ǫ i k > n, c > 0, 0 ≤ γ < 1/2, (16)
where σ 2 n is a consistent estimator of the long-run variance σ 2 = lim n→∞

1 n Var( n i=1 ǫ i ). If the process (ǫ t )
are uncorrelated (for instance GARCH-type model), empirical variance of the historical data can be used as estimator of σ 2 . If (ǫ t ) are correlated, the popular Bartlett estimator (see [START_REF] Berkes | Almost sure convergence of the Bartlett estimator[END_REF]) can be used. Under some regular conditions, it hold that (see [START_REF] Horváth | Monitoring changes in linear models[END_REF] and [START_REF] Aue | Change point monitoring in linear models[END_REF])

lim n→∞ P {τ (n) < ∞} = P sup 0<s<1 |W 1 (s)| s γ > c .
Hence, at a nominal level α = 0.05, the critical value of the test is the (1α)-quantile of the distribution of sup 0<s<1 W 1 (s) /s γ . When γ = 0, these quantiles are known (see Table 1 of [START_REF] Na | Monitoring parameter change in time series models[END_REF] for values obtained through a Monte Carlo simulation).

We compare our procedure to this CUSUM one (see [START_REF] Horváth | Change in autoregressive processes[END_REF] with γ = 0) in two situations 1. (ǫ t ) is an AR(1) process;

ǫ t = φ * 1 ǫ t-1 + ξ t with φ * 1 = 0.2; 2. (ǫ t ) is a GARCH(1, 1) process; ǫ t = σ t ξ t with σ 2 t = α * 0 + α * 1 ǫ 2 t-1 + β * 1 σ 2 t-1 and (α * 0 , α * 1 , β * 1 ) = (0.01, 0.3, 0.2).
The historical sample size are n = 500 and n = 1000. These procedures are evaluated at times k = n + 100, n + 200, n + 300, n + 400, n + 500, while the change occurs at k * = n + 50 or k * = n + 250. Tables 2 and3 indicate the empirical levels and the empirical powers based of 200 replications. The elementary statistics of the empirical detection delay are reported in Tables 4.

The results of Table 2 and Table 3 show that both the procedures based on detectors C k and Q k are conservative. One can also see that the larger n (length of historical data) the smaller the distortion size of these procedures. This is due to the fact that the length of monitoring period is fixed and does not increase with n. Under H 1 , the change has been detected before the monitoring time k = n + 500. But, as we mentioned above, the challenge of this problem is to minimize the detection delay. For this criteria, it can be seen in Table 4 that in the case of the mean shift in AR process, our procedure works well as Horváth et al.'s procedure when the change occurs at the beginning of the monitoring (k * = n + 50); but our procedure is a little more accurate when the change occurs a long time after the beginning of the monitoring (k * = n + 250). For the case of the mean shift in GARCH process, our test procedure outperforms the Horváth et al.'s test in terms of mean and quantiles of the detection delay.

Monitoring parameter changes in AR(1) and GARCH(1,1) processes

In this subsection, we present some simulations results for monitoring parameter changes in AR(1) and GARCH(1,1) models and compare our procedure to the one proposed by Na et al. [START_REF] Na | Monitoring parameter change in time series models[END_REF]. If the boundary

d n Mean SD Min Q 1 Med Q 3 Max AR (
D k > c = P sup 0<s<1 W d (s) > c ,
where

D k := √ n G(T 1,n ) -1/2 F (T 1,n ) θ(T 1,k ) -θ(T 1,n ) .
Hence, at a nominal level α, the critical value of their procedure is the (

= (α * 0 , α * 1 , β * 1 ) = (0.01, 0.3, 0.
2) is constant (hypothesis H 0 ) and the empirical powers are computed when θ 0 = (0.01, 0.3, 0.2) changes to θ 1 = (0.05, 0.5, 0.2) (hypothesis H 1 ). this mean is not estimated. For AR model, it appears in Table 5 that both procedures based on detector C k and D k are conservative. This is not the case for GARCH model (Table 6). The high size distortions when n = 500 is due to the difficulty to estimate the parameter of GARCH model. This size distortion decreases when n increases and Corollary 3.1 ensures that with infinite monitoring period, the empirical level tends to the nominal one as n → ∞.

For both the cases of AR and GARCH processes, the procedure based on detector C k,ℓ detects the change before the monitoring time k = n + 500. Unlike Na et al. [START_REF] Na | Monitoring parameter change in time series models[END_REF], we consider a scenario of GARCH model with moderate change in parameter and it can be seen in Table 6 that the procedure based on detector D k provides unsatisfactory results. At the monitoring time k = n + 500, it is not sure that the change must be detected even when k * = n + 50. This is not surprising according to the comment of subsection 3.1.

Table 7 indicates the distribution of the detection delay d n . We can see in Table 7 (even in Table 4) that for our procedure, the relation d 1000 ≤ 1000/500 d 500 is globally satisfied (from Theorem 3.2, we deduced that d n = O P n 1/2 log n when n is large enough). Moreover, elementary statistics (mean and quantiles) show that the detection delay using our procedure is shorter than using the Na et al.'s one. The results of Table 5, 6 and 7 show that, our test is uniformly better and the procedure based on detector C k could be recommended in this frame.

Real-Data Applications

We consider the returns of the daily closing values of the Nikkei 225 stock index (from January 2, 1995 to October 19, 1998), S&P 500 and FTSE 100 (from January 2, 2004 to June 11, 2012). These data are available on Yahoo! Finance at http://finance.yahoo.com/. They are represented on Figure 2 and Figure 6. These series are known to represent ARCH effect and GARCH(1,2) (resp. GARCH(1,1)) can be used to capture it in returns of Nikkei 225 (resp. S&P 500 and FTSE 100), see the book of Francq and Zakoïan 2010.

d n Mean SD Min Q 1 Med Q 3 Max AR (
Consider the observations going from January 2, 1995 to December 31, 1996 (resp. January 2, 2004 to December 30, 2005) as the historical data for the Nikkei 225 (resp. S&P 500 and FTSE 100) stock index.

These periods are known to be stable in the financial community. To verify it, we apply three procedures to test for parameter change in the historical observations. The null hypothesis is that the parameter is constant over the observations against the parameter changes alternative.

• The first test is proposed by Kengne [START_REF] Kengne | Testing for parameter constancy in general causal time-series models[END_REF]. Define the asymptotic covariance matrix (which take into account the change possibility) of the estimator

θ n (T 1,n ) by Σ n,k := k n F n (T 1,k ) G n (T 1,k ) -1 F n (T 1,k )1 det( Gn(T 1,k )) =0 + n -k n F n (T k,n ) G n (T k,n ) -1 F n (T k,n )1 det( Gn(T k,n )) =0 .
The test is based on the statistic

Q n := max Q (1) n , Q (2)
n where

Q (1) n := max vn≤k≤n-vn Q (1) n,k with Q (1)
n,k :=

k 2 n θ n (T 1,k ) -θ n (T n ) ′ Σ n,k θ n (T k ) -θ n (T n ) , Q (2) n := max vn≤k≤n-vn Q (2) n,k with Q (2) n,k :
= (n -k) 2 n θ n (T k,n ) -θ n (T 1,n ) ′ Σ n,k θ n (T k,n ) -θ n (T k,n ) .
This test is applied with v n = (log n) δ where 2 ≤ δ ≤ 5/2.

• The second test (see Lee and Song [START_REF] Lee | Test parameter change in ARMA models with GARCH innovations[END_REF]) is based on the statistic

Q (0) n := max vn≤k≤n-vn k 2 n θ n (T 1,k ) -θ n (T 1,n ) ′ Σ n,n θ n (T 1,k ) -θ n (T 1,n ) with v n = (log n) 2 .
• The third procedure is the CUSUM test see Kulperger and Yu [START_REF] Kulperger | High moment partial sum processes of residuals in GARCH models and their applications[END_REF].

At a nominal level α ∈ (0, 1), each of these procedure rejects null hypothesis if the test statistic is greater than a critical value C α . Table 8 provides the results of these tests to the historical data that we have chosen. Note that, these series are very closed to a nonstationary process, in the sense that q j=1 α j + p j=1 β j ≃ 1 (see [START_REF] Aue | Delay times of sequential procedures for multiple time series regression models[END_REF]). Therefore, it would be difficult to compute the estimator θ n (T k,l ) (1 ≤ k < l ≤ n). For the statistics Q n and Q (0) n , we consider only the time k that the computation of θ n (T 1,k ) and θ n (T k,n ) converges. This 1. For the Nikkei 225, from January 2, 1995 to October 19, 1998 ; break is detected at • t N ≃ 17 September 1997 which correspond to the turmoil period of the Asian financial crisis (1997)(1998).

Q n Q 0 n CUSUM Nikkei 225 3.
See also Figure 2.

2. For the S&P 500 and FTSE 100, from January 2, 2004 to June 11, 2012 ; break are detected at ( t S,i and t F,i are referred to the breakpoint in the S&P 500 and FTSE 100 respectively) See also Figure 6.

• t S,

Breaks detection in the S&P 500

Returns S&P 500 -0. Recall that (X

1 , • • • , X n ) is an observed trajectory of a process M Z (M θ * 0 , f θ * 0 ). Let k ≥ n ≥ 2 and T 1,n = {1, • • • , n}, T ℓ,k = {ℓ, ℓ + 1, • • • , k} with ℓ ∈ Π n,k = {v n , v n + 1, • • • , k -v n }, and define C k,ℓ := √ n k -ℓ k G -1/2 F • θ(T ℓ,k ) -θ(T 1,n ) ,
with θ defined in [START_REF] Basseville | Detection of Abrupt Changes: Theory and Application[END_REF].

Lemma 6.1. Under the assumptions of Theorem 3.1,

sup k>n max ℓ∈Π n,k 1 b((k -ℓ)/n) C k,ℓ -C k,ℓ = o P (1) as n → ∞.
Proof. For any n ≥ 1, we have sup

k>n max ℓ∈Π n,k 1 b((k -ℓ)/n) C k,ℓ -C k,ℓ = 1 inf s>0 b(s) sup k>n max ℓ∈Π n,k C k,ℓ -C k,ℓ .
Now, proceed similarly as in the proof of Lemma 3 of [START_REF] Kengne | Testing for parameter constancy in general causal time-series models[END_REF].

Lemma 6.2. Under the assumptions of Theorem 3.1

sup k>n max ℓ∈Π n,k 1 b((k -ℓ)/n) √ n k (k -ℓ)F • θ(T ℓ,k ) -θ(T 1,n ) -2 ∂ ∂θ L(T ℓ,k , θ * 0 ) - k -ℓ n ∂ ∂θ L(T 1,n , θ * 0 ) = o P (1) as n → ∞.
Proof. Let k ≥ n and T ⊂ {1, • • • , k}. By applying the Taylor expansion to the coordinates of ∂ L(T, •)/∂θ, and using the fact that ∂ L(T, θ(T ))/∂θ = 0 we have 2 Card(T )

∂ ∂θ L(T, θ * 0 ) = F (T ) • ( θ(T ) -θ * 0 ) where F (T ) = -2 1 Card(T ) ∂ 2 L(T, θ i (T )) ∂θ∂θ i 1≤i≤d
for some θ i (T ) between θ(T ) and θ * 0 . Hence for any ℓ ∈ Π n,k

F ( θ(T ℓ,k ) -θ * 0 ) = 2 k -ℓ ∂ ∂θ L(T ℓ,k , θ * 0 ) + F -F (T ℓ,k ) θ(T ℓ,k ) -θ * 0 + 2 k -ℓ ∂ ∂θ L(T ℓ,k , θ * 0 ) - ∂ ∂θ L(T ℓ,k , θ * 0 ) . and
F ( θ(T 1,n ) -θ * 0 ) = 2 n ∂ ∂θ L(T 1,n , θ * 0 ) + F -F (T 1,n ) θ(T 1,n ) -θ * 0 + 2 n ∂ ∂θ L(T 1,n , θ * 0 ) - ∂ ∂θ L(T 1,n , θ * 0 ) . Therefore, for any ℓ ∈ Π n,k √ n k (k -ℓ)F θ(T ℓ,k ) -θ(T 1,n ) -2 ∂ ∂θ L(T ℓ,k , θ * 0 ) - k -ℓ n ∂ ∂θ L(T 1,n , θ * 0 ) = √ n k -ℓ k F -F (T ℓ,k ) θ(T ℓ,k ) -θ * 0 + 2 √ n k ∂ ∂θ L(T ℓ,k , θ * 0 ) - ∂ ∂θ L(T ℓ,k , θ * 0 ) - √ n k -ℓ k F -F (T 1,n ) θ(T 1,n ) -θ * 0 -2 k -ℓ k 1 √ n ∂ ∂θ L(T 1,n , θ * 0 ) - ∂ ∂θ L(T 1,n , θ * 0 ) . ( 17
)
For k > n and with some ℓ k ∈ Π n,k , we have max

ℓ∈Π n,k 1 b((k -ℓ)/n) √ n k -ℓ k F -F (T ℓ,k ) θ(T ℓ,k ) -θ * 0 ≤ 1 Inf s>0 b(s) k -ℓ k F -F (T ℓ k ,k ) θ(T ℓ k ,k ) -θ * 0 .
According to [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF] and [START_REF] Bardet | Detecting multiple change-points in general causal time series using penalized quasi-likelihood[END_REF],

F -F (T ℓ k ,k ) = o P (1) and θ(T ℓ k ,k ) -θ * 0 = O P (1/ √ k -ℓ k ) as k -ℓ k → ∞. Hence sup k>n max ℓ∈Π n,k 1 b((k -ℓ)/n) √ n k -ℓ k F -F (T ℓ,k ) θ(T ℓ,k ) -θ * 0 = o P (1) as n → ∞. (18)
Similar arguments imply that sup

k>n max ℓ∈Π n,k 1 b((k -ℓ)/n) √ n k -ℓ k F -F (T 1,n ) θ(T 1,n ) -θ * 0 = o P (1) as n → ∞. (19)
For k > n and for some ℓ k ∈ Π n,k , we have max

ℓ∈Π n,k 1 b((k -ℓ)/n) √ n k ∂ ∂θ L(T ℓ,k , θ * 0 ) - ∂ ∂θ L(T ℓ,k , θ * 0 ) ≤ 1 Inf s>0 b(s) 1 √ k -ℓ k ∂ ∂θ L(T ℓ k ,k , θ * 0 ) - ∂ ∂θ L(T ℓ k ,k , θ * 0 ) . According to [4], 1 √ k -ℓ k ∂ ∂θ L(T ℓ k ,k , •) - ∂ ∂θ L(T ℓ k ,k , •) Θ = o P (1) as k -ℓ k → ∞. Hence sup k>n max ℓ∈Π n,k 1 b((k -ℓ)/n) √ n k ∂ ∂θ L(T ℓ,k , θ * 0 ) - ∂ ∂θ L(T ℓ,k , θ * 0 ) = o P (1) as n → ∞. (20)
Similar arguments show that sup

k>n max ℓ∈Π n,k 1 b((k -ℓ)/n) k -ℓ k 1 √ n ∂ ∂θ L(T 1,n , θ * 0 ) - ∂ ∂θ L(T 1,n , θ * 0 ) = o P (1) as n → ∞. (21)
Thus, Lemma 6.2 follows from ( 17), ( 18), ( 19), ( 20) and ( 21).

Lemma 6.3. Under the assumptions of Theorem 3.1

sup k>n max ℓ∈Π n,k 1 b((k -ℓ)/n) √ n k -ℓ k F • θ(T ℓ,k ) -θ(T 1,n ) D -→ n→∞ sup t>1 sup 0<s<t W G (s) -sW G (1) t b(s) where W G is a d-dimensional Gaussian centered process with covariance matrix E (W G (s)W G (τ ) ′ ) = min(s, τ )G.
Proof. We are going to apply Lemma 6.2 for specifying the asymptotic behaviour of θ(T ℓ,k )θ(T 1,n ).

For k > n and ℓ ∈ Π n,k , we have

2 √ n k ∂ ∂θ L(T ℓ,k , θ * 0 ) - k -ℓ n ∂ ∂θ L(T 1,n , θ * 0 ) = - n k 1 √ n k i=ℓ+1 ∂q i (θ * 0 ) ∂θ - k -ℓ n n i=1 ∂q i (θ * 0 ) ∂θ .
Now we are going to proceed in two steps.

Step 1. Let T > 1. We have

max n<k<nT max ℓ∈Π n,k 1 b((k -ℓ)/n) 2 √ n k ∂ ∂θ L(T ℓ,k , θ * 0 ) - k -ℓ n ∂ ∂θ L(T 1,n , θ * 0 ) = max n<k<nT max ℓ∈Π n,k 1 b((k -ℓ)/n) √ n k k i=ℓ ∂q i (θ * 0 ) ∂θ - k -ℓ n n i=1 ∂q i (θ * 0 ) ∂θ = max t∈{1,1+ 1 n ,••• ,T } max s∈{1-vn n ,2-vn n ,••• ,t-vn n } 1 b(([nt] -[ns])/n) n [nt] 1 √ n [nt] i=[ns]+1 ∂q i (θ * 0 ) ∂θ - [nt] -[ns] n n i=1 ∂q i (θ * 0 ) ∂θ . Define the set S := {(t, s) ∈ [1, T ] × [1, T ]/ s < t}.
According to [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF], ∂q i (θ * 0 ) ∂θ t∈Z is a stationary ergodic martingale difference sequence with covariance matrix G. By Cramér-Wold device (see [START_REF] Billingsley | Convergence of Probability Measures[END_REF]

p. 206), it holds that 1 √ n [nt] i=[ns]+1 ∂q i (θ * 0 ) ∂θ D(S) -→ n→∞ W G (t -s).
√ n [nt] i=[ns]+1 ∂q i (θ * 0 ) ∂θ - [nt] -[ns] n n i=1 ∂q i (θ * 0 ) ∂θ D(S) -→ n→∞ W G (t -s) -(t -s)W G (1). Therefore max n<k<nT max ℓ∈Π n,k 1 b((k -ℓ)/n) 2 √ n k ∂ ∂θ L(T ℓ,k , θ * 0 ) - k -ℓ n ∂ ∂θ L(T 1,n , θ * 0 ) D -→ n→∞ sup 1<t<T sup 1<s<t W G (t -s) -(t -s)W G (1) t b(t -s) D -→ n→∞ sup 1<t<T sup 1<s<t W G (s) -s W G (1) t b(s) . (22)
Step 2. We will show that the limit distribution (as n, T → ∞) of

sup k>nT max ℓ∈Π n,k 1 b((k -ℓ)/n) 2 √ n k ∂ ∂θ L(T ℓ,k , θ * 0 ) - k -ℓ n ∂ ∂θ L(T 1,n , θ * 0 )
exists and is equal to the limit distribution (as

T → ∞) of sup t>T sup 1<s<t W G (s) -s W G (1) t b(s) . Let k > nT . We have max ℓ∈Π n,k 1 b((k -ℓ)/n) √ n k ∂ ∂θ L(T ℓ,k , θ * 0 ) ≤ 1 Inf s>0 b(s) √ n k k i=ℓ k +1 ∂q i (θ * 0 ) ∂θ for some ℓ k ∈ Π n,k .
It comes from the Hájek-Rényi-Chow inequality (see [START_REF] Chow | A martingale inequality and the law of large numbers[END_REF]) that, for any ε > 0 lim

T →∞ lim n→∞ P sup k>nT √ n k k i=ℓ k +1 ∂q i (θ * 0 ) ∂θ > ε = 0. Hence sup k>nT max ℓ∈Π n,k 1 b((k -ℓ)/n) 2 √ n k ∂ ∂θ L(T ℓ,k , θ * 0 ) = o P (1) as T, n → ∞. (23)
Moreover, since the function b(•) is non-increasing, for any n, T > 1, we have:

sup k>nT max ℓ∈Π n,k 1 b((k -ℓ)/n) 2 √ n k k -ℓ n ∂ ∂θ L(T 1,n , θ * 0 ) = 1 √ n n i=1 ∂q i (θ * 0 ) ∂θ × sup k>nT max ℓ∈Π n,k 1 b((k -ℓ)/n) k -ℓ k = 1 √ n n i=1 ∂q i (θ * 0 ) ∂θ × sup k>nT 1 b((k -v n )/n) k -v n k = 1 Inf s>0 b(s) 1 √ n n i=1 ∂q i (θ * 0 ) ∂θ D -→ n→∞ 1 Inf s>0 b(s) W G (1) , (24)
using again the Cramèr-Wold device. It comes from ( 23) and ( 24

It comes from ( 26) and ( 27) that the limit of ( 22) satisfies when T → ∞, . Therefore with u = u ′ v,

sup 0<v<1 sup 0<u<v 1 -v 1 -u W d (u) = sup 0<v<1 sup 0<u ′ <1 (1 -v)v 1/2 1 -u ′ v W d (u ′ ) .
D = sup 0<u ′ <1 f (u ′ ) W d (u ′ ) with f (u ′ ) = √ 9 -u ′ + √ 1 -u ′ √ 9 -u ′ + 3 √ 1 -u ′
1 b((k n -ℓ)/n) √ n k n -ℓ k n G(T 1,n ) -1/2 F (T 1,n ) • θ kn (T ℓ,kn ) -θ(T 1,n ) ≥ 1 b((k n -k * )/n) √ n k n -k * k n G(T 1,n ) -1/2 F (T 1,n ) • θ kn (T k * ,kn ) -θ(T 1,n ) ≥ c √ n n δ k * + n δ G(T 1,n ) -1/2 F (T 1,n ) θ kn (T k * ,kn ) -θ(T 1,n ) ≥ c n 1/2+δ c 0 n + n δ G(T 1,n ) -1/2 F (T 1,n ) θ kn (T k * ,kn ) -θ(T 1,n ) ≥ c n δ-1/2 (c 0 + 1) G(T 1,n ) -1/2 F (T 1,n ) θ kn (T k * ,kn ) -θ(T 1,n ) . (30)
According to [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF] and [START_REF] Kengne | Testing for parameter constancy in general causal time-series models[END_REF], G(T

  ] and we recommend to use v n = [(log n) 3/2 ] for linear model and v n = [(log n) 2 ] for ARCH-type model. The nominal level used in the sequel is α = 0.05.

  a-)) and a scenario with change at k * = n + 250 = 750 (Figure 1 b-)).

Figure 1 :

 1 Figure 1: Typical realization of the statistics C k for GARCH(1,1), n = 500 and k = 501, • • • , 1000. a-) The parameter θ * 0 = (0.01, 0.3, 0.2) is constant ; b-) the parameter θ * 0 = (0.01, 0.3, 0.2) changes to θ * 1 = (0.05, 0.5, 0.2) at k * = 750. The horizontal solid line represents the limit of the critical region, the vertical dotted line indicates where the change occurs and the vertical solid line indicates the time where the monitoring procedure detecting the change.

Figure 2 :

 2 Figure 2: The top figure is a realization of the statistics C k with k going from January 2, 1995 to October 19, 1998 for Nikkei 225 data; the historical data considered are the series going from January 2, 1995 to December 31, 1996. The horizontal solid line represents the limit of the critical region, the vertical dotted line indicates the date of the beginning of the Asian financial crisis (1997-1998) and the vertical solid line indicates the time where the monitoring procedure will stop. The bottom figure is the returns of Nikkei 225 data from January 2, 1995 to October 19, 1998; the vertical solid line indicates the date where break have been detected using retrospective test after the monitoring stops.

Figure 4 :

 4 Figure 4: Realization of the statistics C k for S&P 500 and FTSE 100 data; the historical data are the series going from June 18, 2007 to November 16, 2007 (for S&P 500 data) and July 6, 2007 to September 4, 2007 (for FTSE 100 data). The horizontal solid line represents the limit of the critical region, the vertical dotted line indicates the date of the Lehman Brothers Crisis and the vertical solid line indicates the time where the monitoring procedure stopped.

Figure 6 :

 6 Figure 6: Break detection in the returns of S&P 500 and FTSE 100 data using monitoring procedure based on C k . The verticals lines indicate the dates where breaks have been detected.

  convergence on the Skorohod space D(S). Hence 1

  the coordinates of W G are Brownian motions, by the law of the iterated logarithm there exists t 0 > exp(1) such ass > t 0 ⇒ W G (s) ≤ √ s log(s) almost surely.Thus, for any t > t 0 , we obtain almost surelysup 1<s<t W G (s) ≤ sup 1<s<t0 W G (s) + √ t log(t).Therefore, for T large enough, we have sup b(•) is non-increasing, for any T > 1, we have sup

From Step 1 andW 2 .G

 12 Step 2 (the relations[START_REF] Kokoszka | Leipus Testing for parameter changes in ARCH models[END_REF],[START_REF] Lai | Efficient Recursive Algorithms for Detection of Abrupt Changes in Signals and Control Systems[END_REF] and[START_REF] Mei | Sequential change-point detection when unknown parameters are present in the pre-change distribution[END_REF]), it comes that sup k>nT maxℓ∈Π n,k 1 b((kℓ)/n) 2 √ n k ∂ ∂θ L(T ℓ,k , θ * 0 ) -kℓ n ∂ ∂θ L(T 1,n , θ * 0 ) G (s) -sW G (1) t b(s) .Hence, Lemma 6.3 follows from Lemma 6.Proof of Theorem 3.1We know thatP {τ (n) < ∞} = P sup 1,n ) -1/2 F (T 1,n ) • θ(T ℓ,k )θ(T 1,n ) > 1 . Since G(T 1,n ) 1,n ) -1/2 F (T 1,n ) • θ(T ℓ,k )θ(T 1,n ) -1/2 (W G (s) -sW G (1)) t b(s) .Since the covariance matrix of {W G (s) ; s ≥ 0}, is min(s, τ )G, the covariance matrix of{G -1/2 W G (s) ; s ≥ 0}is min(s, τ )I d (where I d is the d-dimensional identity matrix). Hence Theorem 3.1 follows. Proof of Corollary 3.1 Since b ≡ c a positive constant, it follows immediately from Theorem 3.1 that lim n→∞ P {τ (n) < ∞} = P sup t>1 sup 1<s<t 1 t W d (s) -sW d (1)) > c . Now, it suffices to show that sup

It remains to compute sup 0<v<1 ( 1 -

 0<v<11 v)v 1/2 1u ′ v. Classical computations show that this supremum is obtained byv = 2 3u ′ + (9u ′ )(1u ′ ) s) -sW d (1))

2 3 - 2 . 2

 322 u ′ + (9u ′ )(1u ′ ) 1/Denote k n = k * +n δ for δ ∈ (1/2, 1). For n large enough, we have v n < n δ and thus k n -v n = k * +n δ -v n ≥ k * . Moreover, since k * > n then k * ∈ Π n,k for n large enough.In addition, since k * = k * (n) ≥ n and lim sup n→∞ k * (n)/n < ∞, there exists c 0 > 1 such that k * ≤ c 0 n for n large enough. Hence, according to assumption B, there exists a constant c > 0 such that max ℓ∈Π n,knC kn,ℓ b((k nℓ)/n) = max ℓ∈Π n,kn

1 .

 1 θ kn (T k * ,kn) ) Since G is symmetric positive definite, F is invertible, θ * 0 = θ * 1 and δ > 1/2 , then(30)implies that max ℓ∈Π n,kn C kn,ℓ b((k nℓ)/n)

Table 1 :

 1 Empirical (1 -α)-quantile of the distribution of U

	01 2.583 3.035 3.335 3.631 3.914
	α = 0.05 1.954 2.432 2.760 3.073 3.334
	α = 0.10 1.652 2.156 2.486 2.784 3.028

d , for d = 1, • • • , 5.

Table 4 :

 4 Elementary statistics of the empirical detection delay for monitoring mean shift in AR(1) and GARCH(1,1). function b(•) ≡ c > 0 with a real number c > 0, Na et al. show that under H 0 , lim

	1)	n = 500 ; k * = n + 50	C k 54.74 14.95	18	44	54	64	103
			Q k 53.78 14.72	16	43	54	63	102
		n = 500 ; k * = n + 250	C k 63.14 23.18	12	45	61	77	135
			Q k 72.70 21.47	7	56 71.5 90	139
		n = 1000 ; k * = n + 50	C k 75.84 14.19	37	66	75	83	114
			Q k 72.60 13.23	41	63	73	82	111
		n = 1000 ; k * = n + 250	C k 76.24 19.15	23	60	76	89	140
			Q k 86.82 22.57	27	70	85	100 151
	GARCH(1,1)	n = 500 ; k * = n + 50	C k 20.21	6.15	1	16	20	24	35
			Q k 27.06	4.52	16	24	27	30	44
		n = 500 ; k * = n + 250	C k 25.53	8.04	3	20	25	31	50
			Q k 35.40 10.01	13	28	35	41	62
		n = 1000 ; k * = n + 50	C k 28.43	7.41	6	24	28	33	51
			Q k 36.98	5.09	21	33	37	40	48
		n = 1000 ; k * = n + 250	C k 31.16	8.52	4	26	33	39	53
			Q k 44.35 10.04	14	37	45	50	71

n→∞ P {τ (n) < ∞} = lim n→∞ P sup k>n

Table 6 :

 6 Empirical levels and powers for monitoring parameter change in GARCH(1,1) process. The empirical levels are computed when θ 0

	1 -α)-quantile of the distribution of

Table 7 :

 7 Elementary statistics of the empirical detection delay for monitoring parameter change in AR(1) and GARCH(1,1).

	1)	n = 500 ; k * = n + 50	C k	55.36 18.75	9	42	56	67	121
			D k	71.54 38.44	2	52.75	69	89	167
		n = 500 ; k * = n + 250	C k	66.81 25.27	5	49	65	83	149
			D k	97.80 39.42	21	68	89	123 222
		n = 1000 ; k * = n + 50	C k	75.13 19.87	24	62	74	90	147
			D k	87.70 28.72	14	66	85	109 195
		n = 1000 ; k * = n + 250	C k	76.89 26.16	15	56	77	96	172
			D k 101.20 37.97	20	75	96	129 245
	GARCH(1,1)	n = 500 ; k * = n + 50	C k	29.41 15.84	4	22	31	40	98
			D k	86.05 90.50	2	36	61	99	416
		n = 500 ; k * = n + 250	C k	38.02 19.33	5	27	37	44	113
			D k	87.72 50.96	1	49.25	79	112 236
		n = 1000 ; k * = n + 50	C k	41.96 13.93	3	32	41	48	94
			D k	71.29 37.12	6	46	66	88	287
		n = 1000 ; k * = n + 250	C k	44.99 17.16	5	35	41	52	117
			D k	75.78 35.10	7	52	71	95	198

Table 8 :

 8 Results of test for parameter changes in the historical data of Nikkei 225 (from January 2, 1995 to December 31, 1996), S&P 500 and FTSE 100 (from January 2, 2004 to December 30, 2005). Figures in brackets the critical values of the procedure at the nominal level α = 0.05.

		35 (3.98) 2.31 (3.45) 0.98 (1.36)
	S&P 500	2.13 (3.47) 2.01 (3.06) 0.93 (1.36)
	FTSE 100 1.95 (3.47) 2.31 (3.06) 1.13 (1.36)

  1 ≃ 18 June 2007 and t F,1 ≃ 6 July 2007 which correspond to the beginning of the Subprime Crisis in US; • t S,2 ≃ 14 August 2008 and t F,2 ≃ 17 September 2008 which correspond to the Lehman Brothers Bankruptcy; • t S,3 ≃ 5 January 2009 and t F,3 ≃ 29 December 2008 which correspond worldwide governments intervention to solve the financial crisis; • t S,4 ≃ 26 June 2009, t S,5 ≃ 5 April 2010, t S,6 ≃ 27 September 2010, t S,7 ≃ 19 July 2011, t S,8 ≃ 11 January 2012 and t F,4 ≃ 30 June 2009, t F,5 ≃ 27 July 2011, t F,6 ≃ 21 December 2011. These breaks indicates the turmoils periods in the 2010 -2012 + Greece and European debt crisis.

Acknowledgements

The authors are grateful to the anonymous referee for its report which helped to improve this paper.

Table 3: Empirical levels and powers for monitoring means shift in GARCH(1,1) with (α * 0 , α * 1 , β * 1 ) = (0.01, 0.3, 0.2). The empirical levels are computed when µ 0 = 0 and the empirical powers are computed when the mean µ 0 = 0 changes to µ 1 = 0.3. W d (s) which can be found in Table 1 of [START_REF] Na | Monitoring parameter change in time series models[END_REF].

The comparisons between their procedure based on D k and ours based on C k,ℓ are made in the following situations:

1. For AR(1) model :

The sizes of historical samples are n = 500 and n = 1000. The procedures are evaluated at times k = n + 100, n + 200, n + 300, n + 400, n + 500, while the change occurs at k * = n + 50 or k * = n + 250. Tables 5 and6

2 is constant and the empirical powers are computed when θ 0 = 0.2 changes to θ 1 = -0.5.

The considered AR and GARCH processes have zero mean. Contrary to the mean shift studied above, certainly introduces distortions on these tests. On the other hand, the CUSUM procedure needs to compute only the estimator θ n (T 1,n ) which convergence is obtained. According to these results, we conclude that the parameter does not change over these historical observations.

For Nikkei 225 data, monitoring starting at January 2, 1997. Figure 2 shows the realization of the sequence ( C k ) for k going from January 2, 1997 to October 19, 1998. Monitoring procedure stops at October 27, 1997.

Recall that, the monitoring scheme can be used as an alarm system. When it triggered, we need to apply retrospective test to estimate the breakpoint. According to Kengne [START_REF] Kengne | Testing for parameter constancy in general causal time-series models[END_REF], the test based on Q n and Q (0) n are more powerful than the CUSUM test. Thus, in the retrospective procedure, we applied these two tests and considered the one which provides more significant result (in terms of p-value).

Retrospective procedure is applied to the observations going from January 2, 1995 to October 27, 1997 and break is detected at t N ≃ September 17, 1997; see Figure 2. This change corresponds to the Asian financial crisis (1997)(1998) where the turmoil period started at July 1997.

We are going to see for S&P 500 and FTSE 100 data how multiple changes can be monitored. For these series, monitoring starts at January 2, 2006. Figure 3 After monitored the first change, we need to update the procedure. The new historical data are the series going from t S,1 to November 16, 2007 (for S&P 500 data) and the series going from t F,1 to September 4, 2007 (for FTSE 100 data). Therefore, monitoring continues at November 19, 2007 (for S&P 500 data) and at September 5, 2007 (for FTSE 100 data). Figure 4 shows the curve of the sequence ( C k ). The monitoring stops at October 17, 2008 and November 10, 2008 for S&P 500 and FTSE 100 data respectively. The retrospective test is applied and the break point estimation are t S,2 ≃ August 14, 2008 and t F,2 ≃ September 17, 2008 respectively for these two series; see Figure 6. These breaks correspond to the Lehman Brothers Bankruptcy which affects the worldwide financial system.

After that, the procedure is updated and monitoring continues at October 20, 2008 andNovember 11, 2008 for S&P 500 and FTSE 100 data. Figure 5 shows the sequence ( C k ).

Summary of the real-data applications

Both monitoring procedure (based on detector C k ) and retrospective test have been applied to detect breaks in the Nikkei 225, S&P 500 and FTSE 100 stock index. The following results are obtained :


H Y Yg

G A Ar

R C Ch

Mohamed Chikhi
email: chikhi@lameta.univ-montp1.fr

Anne Péguin-Feissolle

Michel Terraza
email: mterraza@lameta.univ-montp1.fr

MODELING OF DOW JONES RETURN PERSISTENCE

Keywords: SEMIFARMA model, HYGARCH model, nonparametric deterministic trend, kernel methodology, long memory JEL Classification: C14, C22, C58, G17

This paper analyzes the cyclical behavior of Dow Jones by testing the existence of long memory through a new class of semiparametric ARFIMA models with HYGARCH errors (SEMIFARMA-HYGARCH); this class includes nonparametric deterministic trend, stochastic trend, short-range and long-range dependence and long memory heteroscedastic errors. We study the daily returns of the Dow Jones from 1896 to 2006. We estimate several models and we find that the coefficients of the SEMIFARMA-HYGARCH model, including long memory coefficients for the equations of the mean and the conditional variance, are highly significant. The forecasting results show that the informational shocks have permanent effects on volatility and the SEMIFARMA-HYGARCH model has better performance over some other models for long and/or short horizons. The predictions from this model are also better than the predictions of the random walk model; accordingly, the weak efficiency assumption of financial markets seems violated for Dow Jones returns studied over a long period.

Introduction

Stock prices have specific statistical properties whose consideration is fundamental to address the problem of modeling. Thus, the presence of long memory in financial series has important implications especially concerning the property of weak efficiency of markets. Indeed, if a series exhibits long memory, this implies significant autocorrelations between observations that, although they are distant in time, can help predict future returns, which violates the assumption of market efficiency (see, among others, [START_REF] Fama | Random Walks in Stock Market Prices[END_REF][START_REF] Fama | Efficient Capital Markets: A Review of Theory and Empirical Work[END_REF][START_REF] Fama | Market efficiency, long-term returns, and behavioral finance[END_REF][START_REF] Grossman | On the Efficiency of Competitive Stock Markets Where Trades Have Diverse Information[END_REF][START_REF] Jensen | Some anomalous evidence regarding market efficiency[END_REF][START_REF] Lillo | The Long Memory of the Efficient Market[END_REF][START_REF] Christodoulou-Volos | Long Range Dependence in Stock Market Returns[END_REF][START_REF] Barkoulas | Long memory and forecasting in Euroyen deposit rates[END_REF][START_REF] Gursakal | Detecting Long Memory in Bulls and Bears Markets: Evidence from Turkey[END_REF]. Therefore, modeling persistence in financial time series has been a high priority in the field of economic research. Moreover, time series often exhibit deterministic or stochastic trends. Thus, [START_REF] Beran | SEMIFAR models-a semiparametric framework for modelling trends, long-range dependence and nonstationarity[END_REF] proposed a SEMIFAR (Semiparametric fractional autoregressive) model; this model is interesting because it takes into account both short-term behavior of the series through autoregressive parameters, longterm behavior through the parameter of fractional integration and also the nonparametric deterministic trend. [START_REF] Beran | SEMIFAR models-a semiparametric framework for modelling trends, long-range dependence and nonstationarity[END_REF]Feng (2002a, 2002b) then introduce the moving average part into the SEMIFAR model, calling it SEMIFARMA (SEMIFAR Moving Average) model. The SEMIFARMA model thus includes the ARIMA model and the fractionally autoregressive process [START_REF] Granger | An introduction to long-memory time series models and fractional differencing[END_REF][START_REF] Hosking | Fractional differencing[END_REF]. The assumption of white noise on the SEMIFARMA model residuals ignores the presence of conditional heteroskedasticity; however, the financial series are generally characterized by a time-varying volatility that can be modeled by ARCH-type models [START_REF] Engle | Autoregressive conditional heteroskedasticity with estimation of U.K. inflation[END_REF][START_REF] Bollerslev | Generalized Autoregressive Conditional Heteroskedasticity[END_REF]). The SEMIFARMA-GARCH model proposed by [START_REF] Feng | Modelling financial time series with SEMIFAR-GARCH model[END_REF] corresponds to a specific representation of nonlinearity allowing for simplified modeling of uncertainty. We will extend this model by using the HYGARCH (Hyperbolic GARCH) model [START_REF] Davidson | Moments and memory properties of linear conditional heteroskedasticity models and a new model[END_REF]) that provides a direct measure of persistence through the fractional integration parameter.

The present article belongs to the field of the above mentioned research work. It is applied to analysis of the persistence of informational shocks and to the search for a possible long memory in Dow Jones returns. By studying the daily returns of this stock price over a long period, from 1896 to 2006 (30292 observations), we show that the SEMIFARMA-HYGARCH model has a significant predictive superiority over all other proposed models for long horizons. The predictions from this model are also better than the predictions of the random walk model, either short term or long term; accordingly, the weak efficiency assumption of financial markets seems violated for Dow Jones returns studied over a long period.

The rest of the paper is organized as follows. Section 2 focuses on presentation of the SEMIFARMA-HYGARCH model. Section 3 is devoted to empirical study of the daily series of Dow Jones; we compare the predictive quality of SEMIFARMA-GARCH, SEMIFARMA-FIGARCH and SEMIFARMA-HYGARCH models with that of a random walk. We give some conclusions in Section 4.

Presentation of the SEMIFARMA-HYGARCH model

Some authors have used the HYGARCH model or the SEMIFAR model to study financial time series, but always separately. Thus, concerning the HYGARCH model, [START_REF] Davidson | Moments and memory properties of linear conditional heteroskedasticity models and a new model[END_REF] examines the volatility dynamics of three Asian currencies. [START_REF] Tang | Long memory in stock index future markets: a value-at-risk approach[END_REF] analyze the value-at-risk in the case of three stock indexes (S&P 500, Dow Jones and Nasdaq 100). [START_REF] Cardamone | HYGARCH Approach to Estimating Interest Rate and Exchange Rate Sensitivity of a Large Sample of U[END_REF] study the sensitivity of U.S. interest rates and exchange rates. [START_REF] Härdle | Value at risk and expected short fall when there is long range dependence[END_REF] focus on the value-at-risk of some stock indexes. [START_REF] Mcmillan | Are Risk Metrics forecasts good enough? Evidence from 31 stock markets[END_REF] compare the predictions of the HYGARCH model and other models for 31 stock indexes. [START_REF] Kasman | Dual long memory property in returns and volatility: Evidence from the CEE countries' stock markets[END_REF] investigate the presence of long memory in eight stock indexes of countries from Central Europe and Eastern Europe. [START_REF] Aloui | Value-at-risk estimations of energy commodities via longmemory, asymmetry and fat-tailed GARCH models[END_REF] consider the value-at-risk for oil and gas. [START_REF] Conrad | Non-negativity conditions for the hyperbolic GARCH model[END_REF] derives the conditions for non-negativity of the conditional variance in a HYGARCH model and presents two empirical applications, one for the daily FTSE, the other for the DAX30. [START_REF] Wei | Forecasting crude oil market volatility: Further evidence using GARCH-class models[END_REF] are concerned with the volatility of the oil market. Finally, [START_REF] Kwan | On the estimation and diagnostic checking of the ARFIMA-HYGARCH Model[END_REF] study some theoretical properties of the HYGARCH model and give an empirical illustration devoted to the exchange rates of the Korean currency. Regarding the SEMIFAR modeling, Beran and Ocker (1999a) apply SEMIFAR to model some European and Asian exchange rates; [START_REF] Beran | Volatility of stock market indices -an analysis based on SEMIFAR models[END_REF] and [START_REF] Feng | Modelling financial time series with SEMIFAR-GARCH model[END_REF] study different stock series.

The SEMIFARMA-HYGARCH model we use is as follows (see Beran and Feng (2002a) for the SEMIFAR part and [START_REF] Davidson | Moments and memory properties of linear conditional heteroskedasticity models and a new model[END_REF] for the HYGARCH part). { } t Y is a fractional semiparametric process with hyperbolic GARCH error, called SEMIFARMA-HYGARCH, if it verifies the following relationship:

{ } t t t d d B x g Y B B B ε θ φ ) ( ) ( ) 1 ( ) 1 )( ( 1 2 = - - - (1) with ) , ~iid( , u , σ σ u ε t t t t t 1 0 0 > = (2)
and

∑ ∞ = + = + = 1 2 * 2 * 2 ) ( i t t i i t B B ε λ ω ε λ ω σ (3) where { } ] 1 ) 1 [( 1 ) ( )] ( 1 [ 1 ) ( 1 - - + - - = - d B B B B α ψ β λ (4) and 1 * )) ( 1 ( - - = B β ω ω (5) with ( ) ∑ ∑ ∞ = ∞ = - = + - Γ + Γ + Γ - = - 1 0 ) ( 1 ) 1 ( ) 1 ( ) 1 ( 1 ) 1 ( k k k k k k d B d c B k d k d B ;moreover, , ) ( 1 d d c = ), 1 ( 2 1 ) ( 2 d d d c - = and
(.)

Γ is the gamma function. Furthermore, we have

p p B B B B φ φ φ φ - - - - = .... 1 ) ( 2 2 1 , q q B B B B θ θ θ θ - - - - = .... 1 ) ( 2 2 1 and r r B B B β β β - - - = ... 1 ) ( 1 ; s s B B B B ψ ψ ψ ψ + + + + = ... 1 ) ( 2 2 1
and ) (B λ denote polynomials in the lag operator with degrees given by p, q, r, s and ∞ respectively, with roots outside the unit circle. B is the lag operator, 1 d is an integer:

{ } 1 , 0 1 ∈ d , t x t =
is the trend and g : [0, 1] → R is the smoothing function. Finally, the process is stationary and invertible,

2 1 2 1 2 < < - d
, and t u is an i.i.n.

process. [START_REF] Davidson | Moments and memory properties of linear conditional heteroskedasticity models and a new model[END_REF] shows that the HYGARCH process generalizes the FIGARCH model [START_REF] Baillie | Fractionally integrated generalized autoregressive conditional heteroskedasticity[END_REF]. If . The long memory intervenes in the mean equation (1) (through the parameter 2 d ) and in the variance equation (3) (through the parameter d). It is important to note that the definition of long memory is an asymptotic definition; in other words, what matters is how the autocorrelations converge to zero when the lag increases. To summarize, the distinction, in terms of autocorrelations, between a short memory process and a long memory process results from the speed of geometric or hyperbolic convergence of the autocorrelations toward zero.

Finally, the

) ( t x g
function is a nonlinear deterministic trend. The kernel estimation in the case of nonparametric regression has been studied especially by [START_REF] Hall | Nonparametric regression with long-range dependence[END_REF], [START_REF] Ray | Bandwidth selection for kernel regression with long-range dependence[END_REF] and [START_REF] Beran | SEMIFAR models-a semiparametric framework for modelling trends, long-range dependence and nonstationarity[END_REF] in the case of long memory errors. We estimate g by the kernel method using the following model:

t t t d X x g Y B + = - ) ( ) 1 ( 1 (6) It is equivalent to t t t X x g Y + = ) ( if 0 1 = d and to t t t t X x g Y Y + = -- ) ( 1 if 1 1 = d
, where t X is a process of long memory stationary errors if 2 d >0 and short memory errors if 2 d =0 ;

the case 2 d <0 is sometimes called "anti-persistence". We consider the polynomial kernel defined by:

∑ = = τ α 0 2 ) ( l l l x x K with 1 ≤ x (7) and 0 ) ( = x K if x >1.
Here we have ∈ τ {0, 1, 2, …} and the ι α coefficients are such that

∫ - = 1 1 1 ) ( dx x K
(see Beran and Feng (2002a and b) for details on the estimation method).

Empirical analysis of the Dow Jones

The empirical study focuses on the logarithmic series of daily Dow Jones covering a historical period from May 26, 1896to August 17, 2006 (n = 30292). Unit root tests results (Dickey and Fuller 1981;[START_REF] Phillips | Testing for Unit Roots in Time Series Regression[END_REF][START_REF] Schmidt | LM Test for a Unit Root in the Presence of Deterministic Trends[END_REF][START_REF] Kwiatkowski | Testing the Null Hypothesis of Stationary Against the Alternative of a Unit Root: How Sure are we that Economic Time Series have a Unit Root?[END_REF][START_REF] Elliott | Efficient Tests for an Autoregressive Unit Root[END_REF] show that this series is characterized by a unit root (see Table 1). The series is finally differentiated to obtain the returns (see Figure 1).

[Insert Table 1 here] [Insert Fig. 1

here]

The assumption of normality of returns is clearly rejected (see Table 2 andFigure 2). The observed asymmetry may indicate the presence of nonlinearities in the evolution process of returns. The scatter plot of the series (Figure 3) does not appear in the form of a regular ellipsoid, and confirms nonlinearity. In addition, the series is heteroscedastic according to the results of White and Breusch-Pagan tests reported in Table 3, since the null hypothesis of homoscedasticity is rejected at 5%. The conditional heteroskedasticity test result shows that Dow Jones returns are characterized by the presence of an ARCH effect frequently encountered in financial time series [ ) 2 ( 1114.3201

2 05 . 0 2 χ > = nR
].

[Insert Table 2 here] [Insert Fig. 2 here]

[Insert Table 3 here] [Insert Fig. 3 here]

In view of Table 4, the random walk hypothesis is clearly rejected. Indeed, Mizrach [START_REF] Mizrach | A Simple Nonparametric Test for Independence[END_REF] and BDS [START_REF] Brock | A test for independence based on the correlation dimension[END_REF]) statistics, which test the presence of linear or nonlinear dependences, are strictly greater than the critical value at 5%.

[Insert Table 4 here]

These first tests generally highlight the presence of significant non-zero autocorrelations in the short term; even if these autocorrelations are significant from a statistical point of view, they are not significant from an economic point-of-view in that it is impossible to exploit these autocorrelations to establish speculative rules leading to abnormal profits. Tests lead us to reject the null hypothesis of no autocorrelation, but do not detect the presence of a structure of long-term dependence. Given this situation, we analyze the cyclical behavior of returns by considering longer horizons. By plotting the periodogram of this series (see Figure 4) (with Tuckey window), we note that the spectral density is concentrated around low frequencies and tends to infinity when the frequency tends to zero. This is a sign of long memory that we have to verify with statistical tests.

[Insert Fig. 4 here]

For this reason, we choose different values for the periodogram ordinates around the square root of the number of observations. This choice aims to examine the stability of the estimators when the number of the periodogram ordinates varies. From Table 5, it is obvious that the series of Dow Jones returns is generated by a long memory process. Some values of the Student statistic (with a power of 0.8) are strictly greater than the critical value at 5%. In addition, the memory parameter estimated by the [START_REF] Andrews | A Bias-Reduced Log-Periodogram Regression Estimator for the Long-memory Parameter[END_REF] method is positive and significant. The estimation result is very close to those found with the GPH [START_REF] Geweke | The estimation and application of long-memory time series models[END_REF] method. The returns are long-term predictable: the presence of a long memory indicates that agents can anticipate their returns to a sufficiently long time horizon. Indeed, the observed movements appear as the result of lasting exogenous shocks which affect the New York market, i.e. the return will not come back to its fundamental value.

[Insert Table 5 here]

The results of the SEMIFARMA model estimation by the exact maximum likelihood method are shown in Table 6. After estimating the deterministic trend, the optimal window and the cross-validation criteria by the kernel method based on the methodology of Nadaraya-Watson [START_REF] Nadaraya | On estimating regression[END_REF][START_REF] Watson | Smooth regression analysis[END_REF], the results indicate that the Dow Jones series is characterized by a long memory: the estimated fractional integration parameter is significantly different from zero. This result is consistent with the different spectral method estimators.

[Insert Table 6 here]

We note that the residuals (Figure 5) are not characterized by a Gaussian distribution (Table 7) and are leptokurtic (Figure 6). The asymmetry may indicate the presence of nonlinearities in the residuals. However, these residuals can be modeled by GARCH models because the presence of an ARCH effect is confirmed by the result of the ARCH-LM test on residuals (

)

1 ( 1646 . 20 2 2 χ > = nR
).Thus, the spectrum of squared residuals (see Figure 7) is concentrated around low frequencies and tends to infinity as the frequency tends to zero. It is likely that the conditional variance has a persistent long memory structure. The financial asset prices often exhibit heteroscedastic behavior with persistence. For this reason, we will study the conditional variance of Dow Jones returns, to consider the possibility of lasting shocks on volatility.

[Insert Fig. 5 here] [Insert Table 7 here] [Insert Fig. 6 here] [Insert Fig. 7

here]

The process followed by Dow Jones returns is then estimated by the simultaneous maximization of the mean and variance processes. The tests allow the adequacy of the model to be appraised, firstly, concerning the absence of autocorrelation and of heteroscedasticity, and secondly, concerning the existence of long memory and the good modeling of the ARCH effect. In practical terms, we estimate several models for different lags: a SEMIFARMA (p, d, q) jointly with a GARCH, FIGARCH and HYGARCH model. For each model, we calculate both [START_REF] Akaike | Statistical Predictor Identification[END_REF] and [START_REF] Schwarz | Estimating the dimension of a Model[END_REF] information criteria. The estimation of different models is only based on 30032 observations, in order to make further comparisons with the predictions of the 260 remaining observations.

In view of Table 8, we find that the information criteria are minimum for the SEMIFARMA-HYGARCH model and the coefficients of this model are highly significant. In addition, long memory coefficients for the equations of the mean and the conditional variance are also significant. The residuals of the models are characterized by the absence of conditional heteroskedasticity: the ARCH-LM statistics are strictly less than the critical value of 2 2 χ at 5%. It should be noted that the normality assumption of residuals of the models is clearly rejected because the Jarque-Bera statistics are strictly greater than the critical value of 2 2 χ at 5%. In view of Table 9, the series of the SEMIFARMA-HYGARCH residuals show no dependence structure where the BDS statistics are strictly less than the critical value 1.96.

[Insert Table 8 here] [Insert Table 9 here]

To compare the forecasting performance of the proposed models and the random walk model, two criteria are used: the mean squared error (MSE) and the mean absolute error (MAE) given by

∑ = + - + - - - = H h h H n h H n Y Y H MSE 1 2 1 ) ( (8)
∑ = + - + - - - = H h h H n h H n Y Y H MAE 1 1 ˆ (9)
where h is the forecasting horizon and H is the total number of predictions for the horizon h over the forecast period.

Table 10 contains the results of in-sample predictions provided by the different models. MSE and MAE criteria generally give the same results. We note that, whatever the forecast horizon, the random walk model is beaten by all the other models. We generally find good predictive results from the SEMIFARMA-FIGARCH model and especially the SEMIFARMA-HYGARCH model with a horizon of 90, 180 and 260 days. As shown in Table 11, which gives the statistical comparisons of out-of-sample forecasts, all the models beat also the random walk model and we find good predictive results from the SEMIFARMA-HYGARCH with a horizon of 30, 90 and 180 days. Indeed, the random walk takes into account only the short-term memory of the series and therefore completely neglects the longterm memory. MSE and MAE criteria give nearly the same results. Consequently, as shown in Tables 10 and11, the long memory models of conditional variance, such as the SEMIFARMA-HYGARCH model, provide superior quality forecasts over a long horizon. Taking into account the long memory implies that, for a long-term forecast (180 or 260 days), not only the last observed value but also the entire weighted history of this series should be used.

[Insert Table 10 here] [Insert Table 11 here]

Given that the Dow Jones returns are characterized by the presence of long-term dynamics in the equations of the mean and conditional variance and by heteroscedasticity, the SEMIFARMA-HYGARCH model allows computation of better short-term and long term forecasts than the random walk model. This model is also clearly superior to all other models for long horizons.

In order to test the statistical significance of the forecasting improvements obtained with the SEMIFARMA-HYGARCH model over the random-walk model, we can use also a battery of tests based on loss functions: the asymptotic test, the sign tests, then Wilcoxon's test, the Naive benchmark test, the Morgan-Granger-Newbold test and the Meese Rogoff test (all these tests are summarized in [START_REF] Diebold | Comparing Predictive accuracy[END_REF]). The p-values in table 12 clearly indicate that the null hypothesis of equal accuracy of the two models is strongly rejected (the p-values are less than 0.05). So, we accept different predictive accuracy; it means that, in this case, the SEMIFARMA-HYGARCH model beats the random walk process. The evidence is encouraging for the predictive ability of non-linear fractional models for the Dow Jones returns (the same conclusion is given in [START_REF] Barkoulas | Long-memory forecasting of US monetary indices[END_REF] for some US monetary indices).

[Insert Table 12 here]

Although the gap between the MSE and MAE criteria does not appear very important, what is significant in this study is the systematic nature of the models. Indeed, the price movements appear as the result of lasting exogenous shocks which affect the U.S. stock market; in other words, the consequences of a shock will be sustainable, the Dow Jones returns will not come back to their previous fundamental value and the shock will be persistent in the long term. This suggests that, due to the long-term predictability of returns, it will be possible a priori to establish remunerative strategies on the New York stock market.

Conclusion

In this article, we investigated the presence of long memory in the Dow Jones returns. In this context, we proposed a semiparametric long memory model called SEMIFARMA with hyperbolic GARCH errors. We implemented the exact maximum likelihood method to estimate exactly this class of models by taking into account the phenomenon of long-term persistence for the conditional variance. From the results, informational shocks have lasting effects on volatility and the SEMIFARMA-HYGARCH model shows a clear superiority over all the other proposed models for long horizons. Specifically, the forecasts of the long memory model show a clear improvement compared to the random walk model at all horizons. The returns have long term memory structure because distant observations are positive autocorrelated: future predictions are improved by past returns and this put into question the validity of the efficient market hypothesis. It appears that the Dow Jones returns are long-term dependent, suggesting stock market inefficiency. Consequently, low efficiency of financial markets seems violated for the Dow Jones returns studied over a long period. Thus, recent works on volatility modeling through FIGARCH or HYGARCH processes seem particularly promising and may provide new evidence to better understand the dynamics of financial series.

FigureFigure 7 -

 7 Figure 1 -Dow Jones (logarithmic series and returns)

Table 1 -Unit root tests

 1

			Test		Logarithmic series				Returns
		Dickey-Fuller	(I)		-1.6316		(I)		-73.810
							(-3.4127)			(-3.4127)
		Phillips-Perron	(III)		-1.6323		(II)		-167.53
							(-3.4127)			(-3.4127)
					Z(Rho)		-6.4786		Z(Rho)		-27677.6873
	Schmidt-Phillips			(-18.1)				(-18.1)
					Z(Tau)		-1.7970		Z(Tau)		-159.0685
							(-3.02)				(-3.02)
								Spectral estimation method
	Test	Window	Bartlett kernel Logarithmic series	Returns	Quadratic spectral kernel Logarithmic series Returns
					(II)	(III)	(II)		(III)	(II)	(III)	(II)	(III)
	KPSS	Newey-	20.413	3.564	0.165	0.033	72.771	12.631	0.173	0.034
	H	0	:	West	(0.463)	(0.146)	(0.463)	(0.146)	(0.463)	(0.146)	(0.463)	(0.146)
	Stationary series	Andrews	0.7086 (0.463)	0.150 (0.146)	0.177 (0.463)	0.035 (0.146)	2.29 (0.463)	0.159 (0.146)	0.181 (0.463)	0.036 (0.146)
	Elliott-Rothenberg-Stock	Newey-West	0.00272 (3.26)	0.0066 (5.62)	138.388 (3.26)	16.367 (5.62)	0.0028 (3.26)	0.0069 (5.62)	144.822 (3.26)	17.151 (5.62)
	: H unit 0 root	Andrews	0.0029 (3.26)	0.007 (5.62)	147.891 (3.26)	17.527 (5.62)	0.0029 (3.26)	0.0070 (5.62)	151.514 (3.26)	17.959 (5.62)
	(I): model without constant and deterministic trend (5%)
	(II): model with constant and without deterministic trend (5%)
	(III): model with constant and deterministic trend (5%)

Table 2 -Normality tests on distribution of returns

 2

	Skewness	Kurtosis	J.B	A.D	K.S
	-1.1208	38.2164	1571629	1015.4123	0.581 (0.092
	J.B is the Jarque-Bera statistic,
	K.S is the Kolmogorov-Smirnov statistic,
	A.D is the Anderson-Darling statistic.
	The Kolmogorov-Smirnov critical values are given between parentheses at 5%.

Table 3 -Homoscedasticity tests

 3

	Breusch-Pagan statistic	White statistic	LM -ARCH(2) statistic
	1481.7629	1370.9883	1114.3201

Table 4 -BDS and Mizrach test results on the series of returns

 4

		BDS		Mizrach
	m	Fraction of pairs	Standard Deviation	T	≤	1000
	2	41.85412	37.41965	4.3802
	3	53.06906	48.96358	3.1270
	4	60.44449	57.82082	2.1684
	5	67.01797	67.16815	2.1346
	6	73.79956	78.60053	1.1964
	7	80.97002	92.77881	1.7861
	8	88.54294	110.1803	1.8503
	9	97.17348	132.5872	1.9537
	10	106.8643	161.0307	1.5520
	11	117.9473	198.9349			-
	12	131.0207	250.3263			-
	13	146.1977	318.0141			-
	14	163.8683	408.8996			-
	15	184.3967	530.3570			-
	16	208.5077	694.4048			-
	17	237.1340	922.2850			-
	18	271.0007	1242.474			-
	19	311.1570	1689.988			-
	20	359.0700	2325.839			-
	The BDS statistic is computed by two methods with ε = 0.7

Table 5 -Results from the ARFIMA estimation using spectral methods on Dow Jones returns

 5

							Ordinates
	Windows	n	. 0	4	n	5 . 0	n	. 0	6	n	7 . 0	n	. 0	8
	GPH	-0.0489	0.0329	0.0411	0.0305	0.0311
		(-0.5431)	(0.6527)	(1.4095)	(1.7834)	(3.0528)
	Rectangular	-0.0501	0.0114	0.0322	0.0282	0.0293
		(-0.5045)	(0.2061)	(1.0023)	(1.4961)	(2.6160)
	Bartlett	-1.1681	0.0132	0.0304	0.0253	0.0271
		(-0.0670)	(0.4104)	(1.6382)	(2.3253)	(4.1915)
	Daniell	-0.0694	0.0122	0.0306	0.0255	0.0271
		(-0.9878)	(0.3098)	(1.3452)	(1.9079)	(3.4235)
	Tukey	-0.0646	0.0115	0.0309	0.0260	0.0277
		(-1.0326)	(0.3288)	(1.5249)	(2.1866)	(3.9173)
	Parzen	-0.0614	0.5466	0.0307	0.0255	0.0274
		(-1.1900)	(0.0158)	(1.8373)	(2.6063)	(4.7018)
	B-priest	-0.0606	0.0103	0.0317	0.0270	0.0285
		(-0.7878)	(0.2406)	(1.2713)	(1.8500)	(3.2790)
	Andrews-		-			-		-			-	0.0294
	Guggenberger											(3.047)

The values in parentheses are the Student statistics. The statistically significant values are in bold.

Table 6 -Semiparametric estimation by the exact maximum likelihood method

 6

			Nonparametric estimation of the
	Exact maximum likelihood	deterministic trend Optimal CV Criteria	opt h ˆ
	(2, 0.0423, 2) t d =12,5369	0.9382	0.1384
	CV : cross validation criteria and opt h ˆ : optimal window
	Tableau 7 -Main characteristics of residuals
	Skewness	Kurtosis	J.B statistic	ARCH-LM statistic
	-1.02	37.47	1505311.66	20.1646

Table 8 -Maximum likelihood estimation -BHHH algorithm

 8

	Parameters	SEMIFARMA-GARCH	SEMIFARMA-FIGARCH	SEMIFARMA-HYGARCH
	1 φ	(-2.3134) -0.1767	(-2.5513) -0.2036	-
	θ	1	0.2668 (3.7227)	0.5011 (0.7244)	0.1067 (12.3140)
	d	2	0.0326 (3.4389)	0.0231 (2.6932)	0.0226 (2.7058)
	α		0	1.3564 (0.8644)	1.1025 (2.0876)	1.8451 (2.0039)
	α	1	0.0895 (2.1494)	0.0612 (2.1511)	0.0567 (2.1765)
	β	1	0.0949 (4.1676)	0.1032 (3.9842)	0.1256 (2.4783)
	d ˆ	-	0.04311 (4.2745)	0.04420 (4.8655)
	α ˆ	-	-	0.1592 (2.9252)
	opt h ˆ	0.1384	0.1384	0.1384
	IMSE	0.9382	0.9382	0.9382
	Akaike	5.6842	5.6438	5.6395
	JB statistic	3436.18	3407.22	3391.46
	Schwarz	5.6903	5.6422	5.6416
	ARCH(1)	2.8847	2.6512	2.4956

The Student statistics are in parentheses. IMSE: Minimum Integrated Mean Squared Error

Table 9 -BDS test on SEMIFARMA-HYGARCH residuals

 9

	m	BDS statistic
	2	0.2531
	3	0.2844
	4	0.3142
	5	0.3578
	6	0.4812
	7	0.6740
	8	0.8124

Table 10 -Comparison of predictive qualities (in-sample predictions)

 10

		Horizon Criteria	SEMIFARMA-GARCH	SEMIFARMA-FIGARCH	SEMIFARMA-HYGARCH	Random Walk
		1 day	MSE MAE	6.0644 2.1432	6.1632 2.0323	6.1665 2.0342	7.0328 3.2744
		2 days	MSE MAE	6.0648 2.1263	6.0728 2.1344	6.0813 2.1397	7.1214 3.3328
	Conditional mean (Returns)	15 days 30 days	MSE MAE MSE MAE	6.6231 2.6302 6.2003 2.8804	6.6420 2.6481 6.2078 2.8902	6.6422 2.6486 6.2076 2.8869	7.1488 3.3611 7.2201 3.4027
		90 days	MSE MAE	7.0201 2.6327	7.0194 2.6319	7.0072 2.6289	8.1746 3.4726
		180 days	MSE MAE	6.9303 2.4411	6.9211 2.4207	6.9072 2.3995	8.3011 3.5814
		260 days	MSE MAE	6.7012 2.1354	6.6981 2.1349	6.6837 2.1219	8.1233 3.4916
		1 day	MSE MAE	5.0333 2.1112	5.3682 1.0721	5.3684 1.0725	--
		2 days	MSE MAE	5.0232 2.1152	5.0188 2.1084	5.1029 2.0891	--
	Conditional variance (Volatility)	15 days 30 days 90 days	MSE MAE MSE MAE MSE MAE	5.5178 2.1317 5.1002 2.7901 6.0104 1.9346	5.4411 2.1466 5.1287 2.8023 6.0044 1.9222	5.4677 2.1578 5.1366 2.7811 5.4607 1.8344	------
		180 days	MSE MAE	5.8322 1.3208	5.8188 1.2024	5.7702 1.1109	--
		260 days	MSE MAE	5.2019 2.1453	5.1298 2.0933	5.0913 2.0532	--

Table 11 -Comparison of predictive qualities (out-of-sample predictions)

 11

		Horizon Criteria	SEMIFARMA-GARCH	SEMIFARMA-FIGARCH	SEMIFARMA-HYGARCH	Random Walk
		1 day	MSE MAE	6.1232 2.1432	6.1353 2.0323	6.1422 2.0344	7.0632 3.4811
	Conditional mean (Returns)	2 days 15 days 30 days	MSE MAE MSE MAE MSE MAE	6.1271 2.1824 6.6401 2.6902 6.2602 2.9460	6.1452 2.1956 6.6511 2.7105 6.2588 2.9410	6.1488 2.1397 6.6823 2.7113 6.2552 2.9400	7.1568 3.6232 7.2402 3.8600 7.6314 3.5602
		90 days	MSE MAE	7.0911 2.7033	7.0812 2.6963	7.0253 2.6289	8.2679 3.9601
		180 days	MSE MAE	6.9219 2.4903	6.9008 2.4855	6.8814 2.3836	8.7246 3.9901
		1 day	MSE MAE	5.0965 2.1708	5.0836 1.8814	5.0844 1.9203	--
		2 days	MSE MAE	5.0811 2.1758	5.0993 2.1801	5.1036 2.1107	--
	Conditional variance (Volatility)	15 days 30 days	MSE MAE MSE MAE	5.5701 2.1908 5.1685 2.9067	5.5233 2.2012 5.1786 2.8511	5.5420 2.1897 5.1383 2.8423	----
		90 days	MSE MAE	6.0210 1.9458	5.6049 1.8922	5.5417 1.9015	--
		180 days	MSE MAE	5.5589 1.2840	5.4274 1.1871	5.4110 1.1701	--

Table 12 -Comparing predictive accuracy of SEMIFARMA-HYGARCH model over the random walk: Diebold-Mariano test

 12

	Forecast
	observations	1 S	S	2	S	3	Naïve Benchmark test	MGN	MR
	numbers
	30	2.2856 (0.0223)	2.9212 (0.0035)	3.6303 (0.0003)	F = 0.1400 (1.0000)	F = 7.1452 (0.0000)	-7.2450 (0.0000)	-2.3343 (0.0211)
	The P-Values are in parentheses, 1 S : Asymptotic test statistic, 2 S : Sign test statistic, 3 S : Wilkoxon test statistic, MGN: Morgan-
	granger-Newbold test statistic, MR: Meese-Rogoff test statistic. Truncation lag used for asymptotic test = 3, Truncation lag used for
	Meese-Rogoff test = 8.