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

Béchir Dola
email: bechir.dola@univ-paris1.fr

Semiparametric stationarity and fractional unit roots tests based on data-driven multidimensional increment ratio statistics

Keywords: Gaussian fractionally integrated processes, semiparametric estimators of the memory parameter, test of long-memory, stationarity test, fractional unit roots test

Semiparametric stationarity and fractional unit roots tests based on data-driven multidimensional increment ratio statistics

Introduction

The set I(d) of fractionally integrated stochastic process X = (X k ) k∈Z was defined and used in many articles (see for instance, [START_REF] Granger | An Introduction to Long-memory Time Series and Fractional Differencing[END_REF]). Here we consider the following spectral version of this set for -0.5 < d < 1.5:

Set I(d): X = (X t ) t∈Z is a stochastic process and there exists a continuous function f * : [-π, π] → [0, ∞[ satisfying:

1. if -0.5 < d < 0.5, X is a stationary process with a spectral density f satisfying f (λ) = |λ| -2d f * (λ) for all λ ∈ (-π, 0) ∪ (0, π), with f * (0) > 0.

2. if 0.5 ≤ d < 1.5, U = (U t ) t∈Z = (X t -X t-1 ) t∈Z is a stationary process with a spectral density f satisfying f (λ) = |λ| 2-2d f * (λ) for all λ ∈ (-π, 0) ∪ (0, π), with f * (0) > 0.

The case d ∈ (0, 0.5) is the case of long-memory processes, while -0.5 < d ≤ 0 corresponds to short-memory processes while 0.5 ≤ d < 1.5 corresponds to non-stationary processes having stationary increments. ARFIMA(p, d, q)

1 processes (which are linear processes), as well fractional Gaussian noises (with parameter H = d + 1/2 ∈ (0, 1)) or fractional Brownian motions (with parameter H = d -1/2 ∈ (0, 1)) are famous examples of processes satisfying Assumption I(d). The purpose of this paper is twofold: firstly, we establish the consistency of an adaptive data-driven semiparametric estimator of d for any d ∈ (-0.5, 1.25). Secondly, we use this estimator to build new stationarity and fractional unit roots semiparametric tests.

Numerous articles have been devoted to the estimation of d in the case d ∈ (-0.5, 0.5) only. The books of [START_REF] Beran | Statistics for Long-Memory Processes[END_REF] and [START_REF] Doukhan | Theory and applications of long-range dependence[END_REF] provide large surveys of such parametric estimators (as maximum likelihood or Whittle estimators) or semiparametric estimators (as local Whittle, log-periodogram or wavelet based estimators). Here we will focus on the case of semiparametric estimators of processes satisfying Assumption I(d).

Even if first versions of local Whittle, log-periodogram and wavelet based estimators are considered in the case d < 0.5 only (see for instance Robinson, 1995aand 1995b[START_REF] Veitch | On the Automatic Selection of the Onset of Scaling[END_REF] , new extensions have been provided to estimate d when d ≥ 0.5 also (see for instance Hurvich andRay, 1995, Velasco, 1999a [START_REF] Bardet | Adaptive semiparametric wavelet estimator and goodness-of-fit test for long-memory linear processes[END_REF]. The first objective of this paper is to propose for the first time an adaptive data-driven estimator of d satisfying a CLT, providing confidence intervals or tests, that is valid for d < 0.5 but also for d ≥ 0.5. This objective is achieved by using Multidimensional Increment Ratio (MIR) statistics.

The original version of the Increment Ratio (IR) statistic was defined in [START_REF] Surgailis | The increment ratio statistic[END_REF] from an observed trajectory (X 1 , . . . , X N ) of a process X satisfying I(d) and for any ℓ ∈ N * as:

IR N (ℓ) := 1 N -3ℓ N -3ℓ-1 k=0 k+ℓ t=k+1 X t+ℓ - k+ℓ t=k+1 X t + k+2ℓ t=k+ℓ+1 X t+ℓ - k+2ℓ t=k+ℓ+1 X t k+ℓ t=k+1 X t+ℓ - k+ℓ t=k+1 X t + k+2ℓ t=k+ℓ+1 X t+ℓ - k+2ℓ t=k+ℓ+1 X t . (1.1)
Under conditions on X, if ℓ → ∞ and N/ℓ → ∞, it is proved that the statistics IR N (ℓ) converges to a deterministic monotone function Λ 0 (d) on (-0.5, 1.5) and a CLT is also established for d ∈ (-0.5, 0.5) ∪ (0.5, 1.25) when ℓ is large enough with respect to N . As a consequence of this CLT and using the Delta-method, the estimator d N (ℓ) = Λ -1 0 (IR N (ℓ)), where d → Λ 0 (d) is a smooth and increasing function defined in (2.6), is a consistent estimator of d satisfying also a CLT (see more details below). However this new estimator was not totally satisfying. Firstly, it requires the knowledge of the second order behavior of the spectral density, which is clearly unknown in practice, to select ℓ. Secondly, its numerical accuracy is reasonable but clearly lower than those of local Whittle or log-periodogram estimators. As a consequence, in [START_REF] Bardet | Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic[END_REF], we built a data-driven Multidimensional IR (MIR) estimator d (MIR) N computed from d N (ℓ 1 ), • • • , d N (ℓ p ) (see its precise definition in (3.2)) improving both these points but only for -0.5 < d < 0.5. This is an adaptive data-driven semiparametric estimator of d achieving the minimax convergence rate (up to a multiplicative logarithm factor) and requiring no regulation of any auxiliary parameter (as bandwidth or trimming parameters). Moreover, its numerical performances are comparable to the ones of local Whittle, log-periodogram or wavelet based estimators. Here we extend this previous work to the case 0.5 ≤ d < 1.25. Hence we obtain a CLT satisfied by d (MIR) N for all d ∈ (-0.5, 1.25) with an explicit asymptotic variance depending on d only. This especially allows to obtain confidence intervals of d using Slutsky Lemma. The case d = 0.5 is now studied and this offers new perspectives: our data-driven estimator can be used for building a stationarity (or non-stationarity) test since 0.5 is the "border number" between stationarity and non-stationarity. The case d = 1 is also now studied and it provides another application of d (MIR) N to test fractional unit roots, that is to decide between d = 1 and d < 1.

There exist several famous stationarity (or non-stationarity) tests. We may cite parametric tests defined by [START_REF] Elliott | Efficient Tests for an Autoregressive Unit Roots[END_REF] or Ng andPerron (1996, 2001). For non parametric stationarity tests we may cite the LMC test (see Leybourne and McCabe, 2000) and the KPSS (Kwiatkowski, Phillips, Schmidt, Shin) test (see [START_REF] Kwiatkowski | Testing the null hypothesis of stationarity against the alternative of a unit root[END_REF], improved by the V/S test (see [START_REF] Giraitis | Rescaled variance and related tests for long-memory in volatility and levels[END_REF]. For non-stationarity tests we may cite the Augmented Dickey-Fuller test (see [START_REF] Said | Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown Order[END_REF] and the Philipps and Perron test (PP test in the sequel, see Philipps and Perron, 1988). All these tests are unit roots tests (except the V/S test which is also a short-memory test), which are, roughly speaking, tests based on the model X t = ρ X t-1 + ε t with |ρ| ≤ 1. A right-tailed test d ≥ 0.5 for a process satisfying Assumption I(d) is therefore a refinement of a basic unit roots test since the case ρ = 1 is a particular case of I(1) and the case |ρ| < 1 a particular case of I(0). Thus, a stationarity (or non-stationarity test) based on the estimator of d provides a useful complementary test to usual unit roots tests. This principle of stationarity test linked to d has been already investigated in many articles. We can cite [START_REF] Robinson | Efficient Tests of non-stationary Hypotheses[END_REF], [START_REF] Tanaka | The non-stationary Fractional Unit Root[END_REF], [START_REF] Ling | Asymptotic Inference for non-stationary fractionally integrated autoregressive moving-average models[END_REF], [START_REF] Ling | Adaptive estimators and tests of stationary and non-stationary short and long memory ARIMA-GARCH models[END_REF] or [START_REF] Nielsen | Efficient Likelihood Inference in non-stationary Univariate Models[END_REF]. It also be used to define fractional unit roots tests, like the Fractional Dickey-Fuller test defined by [START_REF] Dolado | A Fractional Dickey-Fuller Test For Unit Roots[END_REF] or the cointegration rank test defined by [START_REF] Breitung | Inference on the Cointegration Rank in Fractionally Integrated Processes[END_REF]. However, all these papers provide parametric tests, with a specified model (for instance ARFIMA or ARFIMA-GARCH processes). Extensions proposed by Lobato an [START_REF] Lobato | Efficient Wald tests for fractional unit roots[END_REF] are convincing: this estimator is accurate and robust for all the considered processes and is globally as efficient as local Whittle, log-periodogram or wavelet based estimators. Secondly, the new stationarity S N and non-stationarity T N tests are compared to the most famous unit roots tests (KPSS, V/S, ADF and PP tests) for numerous I(d) processes. And the results are quite surprising: even on AR [START_REF] Abadir | non-stationarity-extended local Whittle estimation[END_REF] or ARIMA(1, 1, 0) processes, S N and T N tests provide convincing results which are comparable to those obtained with ADF and PP tests while those tests are especially built for these specific processes. For long-memory processes (such as ARFIMA processes), the results are clear: S N and T N tests are accurate tests of (non)stationarity while ADF and PP tests are only helpful when d is close to 0 or 1. Concerning the new MIR fractional unit roots test F N , it provides satisfying results for all considered processes, while fractional unit roots tests such as the fractional Dickey-Fuller test developed by [START_REF] Dolado | A Fractional Dickey-Fuller Test For Unit Roots[END_REF] or the efficient Wald test introduced by Lobato and Velasco (2007) are respectively only performing for ARFIMA(0, d, 0) processes or a class of long-memory processes containing ARFIMA(p, d, 0) processes but not ARFIMA(p, d, q) processes with q ≥ 1.

The forthcoming Section 2 is devoted to the definition and asymptotic behavior of MIR estimators of d and Section 3 studies an adaptive MIR estimator. The stationarity and non-stationarity tests are presented in Section 4 while Section 5 deals with the results of simulations, Section 6 provides conclusive remarks and Section 7 contains all the proofs.

The Multidimensional Increment Ratio statistic

Now we consider a semiparametric class IG(d, β) which is a refinement of the general class I(d). For -0.5 < d < 1.5 and β > 0 define: Assumption IG(d, β): X = (X t ) t∈Z is a Gaussian process such that there exist ǫ > 0, c 0 > 0, c ′ 0 > 0 and c 1 ∈ R satisfying:

1. if d < 0.5, X is a stationary process with a spectral density f satisfying for all λ ∈ (-π, 0) ∪ (0, π)

f (λ) = c 0 |λ| -2d + c 1 |λ| -2d+β + O |λ| -2d+β+ǫ and |f ′ (λ)| ≤ c ′ 0 λ -2d-1 . (2.1) 2. if 0.5 ≤ d < 1.5, U = (U t ) t∈Z = (X t -X t-1
) t∈Z is a stationary process with a spectral density f satisfying for all λ ∈ (-π, 0) ∪ (0, π)

f (λ) = c 0 |λ| 2-2d + c 1 |λ| 2-2d+β + O |λ| 2-2d+β+ǫ and |f ′ (λ)| ≤ c ′ 0 λ -2d+1 . (2.2)
Note that Assumption IG(d, β) is a particular (but still general) case of the set I(d) defined above.

Remark 1.

• The extension of the definition from d ∈ (-0.5, 0.5) to d ∈ [0.5, 1.5) is classical since the conditions on the process is replaced by conditions on the process' increments.

• The condition on the derivative f ′ is not really usual. However, this is not a very restrictive condition since it is satisfied by all the classical long-range dependent processes.

• In the literature, all the theoretical results concerning the IR statistic for time series have been obtained under Gaussian assumptions. In [START_REF] Surgailis | The increment ratio statistic[END_REF] and [START_REF] Bardet | Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic[END_REF], simulations exhibited that the obtained limit theorems should be also valid for linear processes. However a theoretical proof of such result would require limit theorems for functionals of multidimensional linear processes difficult to be established, even if numerical experiments seem to show that this assumption could be replaced by the assumption that X is a linear process having a fourth-moment order like it was done in Giraitis and [START_REF] Giraitis | A central limit theorem for quadratic forms in strongly dependent linear variables and its applications to the asymptotic normality of Whittle estimate[END_REF].

In this section, under Assumption IG(d, β), we establish central limit theorems which extend to the case d ∈ [0.5, 1.25) those already obtained in [START_REF] Bardet | Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic[END_REF] for d ∈ (-0.5, 0.5). Let X = (X k ) k∈N be a process satisfying Assumption IG(d, β) and (X 1 , • • • , X N ) be a path of X. The statistic IR N (see its definition in (1.1)) was first defined in [START_REF] Surgailis | The increment ratio statistic[END_REF] as a way to estimate the memory parameter. In [START_REF] Bardet | Measuring the roughness of random paths by increment ratios[END_REF] a simple version of IR-statistic was also introduced to measure the roughness of continuous time processes, and its connection with level crossing index by geometrical arguments. The main interest of such a statistic is to be very robust to additional or multiplicative trends.

As in [START_REF] Bardet | Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic[END_REF], let m j = j m, j = 1

d (τ ), • • • , Z (p)
d (τ ) such as for τ ∈ R,

Z (j) d (τ ) :=        2d(2d + 1) |4 d+0.5 -4| 1 0 B d-0.5 (τ + s + j) -B d-0.5 (τ + s) ds if d ∈ (0.5, 1.25) 1 |4 d+0.5 -4| B d+0.5 (τ + 2j) -2 B d+0.5 (τ + j) + B d+0.5 (τ ) if d ∈ (-0.5, 0.5) . (2.3)
Using a continuous extension when d → 0.5 of the covariance of Z (j) d (τ ), we also define the stationary multidimensional centered Gaussian processes Z

(i) 0.5 (0), Z (j) 0.5 (τ ) := 1 4 log 2 -h(τ + i -j) + h(τ + i) + h(τ -j) -h(τ ) for τ ∈ R,
where h(x) = 1 2 |x -1| 2 log |x -1| + |x + 1| 2 log |x + 1| -2|x| 2 log |x| for x ∈ R, using the convention 0 × log 0 = 0. Now, we establish a multidimensional CLT satisfied by (IR N (j m)) 1≤j≤p for all d ∈ (-0.5, 1.25):

Proposition 1. Assume that Assumption IG(d, β) holds with -0.5 ≤ d < 1.25 and β > 0. Then

N m IR N (j m) -E IR N (j m) 1≤j≤p L -→ [N/m]∧m→∞ N (0, Γ p (d)) (2.4)
with Γ p (d) = (σ i,j (d)) 1≤i,j≤p where for i, j ∈ {1, . . . , p},

σ i,j (d) : = ∞ -∞ Cov |Z (i) d (0) + Z (i) d (i)| |Z (i) d (0)| + |Z (i) d (i)| , |Z (j)
d (τ ) + Z (j) d (τ + j)| |Z (j) d (τ )| + |Z (j) d (τ + j)| dτ.
(2.5)

The proof of this proposition as well as all the other proofs can be found in Section 7.

In the sequel, we will assume that Γ p (d) is a positive definite matrix for all d ∈ (-0.5, 1.25). Extensive numerical experiments seem to give strong evidence of such a property. Now, the CLT (2.4) can be used for estimating d.

To begin with,

2 π arctan 1 + r 1 -r + 1 π 1 + r 1 -r log( 2 1 + r ) for |r| ≤ 1. (2.7)
Therefore by choosing m and N such as N/m m -β log m → 0 when m, N → ∞, the term E IR(jm) can be replaced by Λ 0 (d) in Proposition 1. Then, using the Delta-method with the function (

x i ) 1≤i≤p → (Λ -1 0 (x i )) 1≤i≤p (the function d ∈ (-0.5, 1.5) → Λ 0 (d) is a C ∞ increasing function), we obtain: Theorem 1. Let d N (j m) := Λ -1 0 IR N (j m) for 1 ≤ j ≤ p. Assume that Assumption IG(d, β) holds with 0.5 ≤ d < 1.25 and 0 < β ≤ 2. Then if m ∼ C N α with C > 0 and (1 + 2β) -1 < α < 1, N m d N (j m) -d 1≤j≤p L -→ N →∞ N 0, (Λ ′ 0 (d)) -2 Γ p (d) . (2.8)
This result is an extension to the case 0.5 ≤ d ≤ 1.25 from the case -0.5 < d < 0.5 already obtained in [START_REF] Bardet | Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic[END_REF]. Note that the consistency of d N (j m) is ensured when 1.25 ≤ d < 1.5 but the previous CLT does not hold (the asymptotic variance of N m d N (j m) diverges to ∞ when d > 1.25, see [START_REF] Surgailis | The increment ratio statistic[END_REF].

Now define Σ N (m) := (Λ ′ 0 ( d N (m)) -2 Γ p ( d N (m)).
(2.9)

The function d ∈ (-0.5, 1.5) → σ(d)/Λ ′ (d) is C ∞ and therefore, under assumptions of Theorem 1,

Σ N (m) P -→ N →∞ (Λ ′ 0 (d)) -2 Γ p (d).
Thus, a pseudo-generalized least square estimation (PGLSE) of d can be defined by

d N (m) := J ⊺ p Σ N (m) -1 J p -1 J ⊺ p Σ N (m) -1 d N (m i ) 1≤i≤p
with J p := (1) 1≤j≤p and denoting J ⊺ p its transpose. From a Gauss-Markov Theorem type (see again [START_REF] Bardet | Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic[END_REF], the asymptotic variance of d N (m) is smaller than the one of any d N (jm), j = 1, . . . , p. Hence, we obtain under the assumptions of Theorem 1:

N m d N (m) -d L -→ N →∞ N 0 , Λ ′ 0 (d) -2 J ⊺ p Γ -1 p (d)J p -1 .
(2.10)

3 The adaptive data-driven version of the estimator Theorem 1 and CLT (2.10) require the knowledge of β to be applied. But in practice β is unknown. The procedure defined in [START_REF] Bardet | Adaptive semiparametric wavelet estimator and goodness-of-fit test for long-memory linear processes[END_REF] or [START_REF] Bardet | Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic[END_REF] can be used for obtaining a data-driven selection of an optimal sequence ( m N ) derived from an estimation of β. Since the case d ∈ (-0.5, 0.5) was studied in [START_REF] Bardet | Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic[END_REF] we consider here d ∈ [0.5, 1.25) and for α ∈ (0, 1), define

Q N (α, d) := d N (j N α ) -d N (N α ) ⊺ 1≤j≤p Σ N (N α ) -1 d N (j N α ) -d N (N α ) 1≤j≤p , (3.1)
which corresponds to the sum of the pseudo-generalized squared distance between the points ( d N (j N α )) j and PGLSE of d. Note that by the previous convention,

d N (j N α ) = d N (j [N α ]) and d N (N α ) = d N ([N α ]
). Then Q N (α) can be minimized on a discretization of (0, 1) and define:

α N := Argmin α∈AN Q N (α) with A N = 2 log N , 3 log N , . . . , log[N/p] log N .
Remark 2. The choice of the set of discretization A N is implied by our proof of convergence of α N . If the interval (0, 1) is stepped in N c points, with c > 0, the used proof cannot attest this convergence. However log N may be replaced in the previous expression of A N by any negligible function of N compared to functions N c with c > 0 (for instance, (log N ) a or a log N with a > 0 ).

From the central limit theorem (2.8) one deduces the following limit theorem:

Proposition 2. Assume that Assumption IG(d, β) holds with 0.5 ≤ d < 1.25 and 0 < β ≤ 2. Then,

α N P -→ N →∞ α * = 1 (1 + 2β) .
Finally define

m N := N αN with α N := α N + 6 α N (p -2)(1 -α N ) • log log N log N .
and the estimator

d (MIR) N := d N ( m N ) = d N (N αN ). (3.2)
(the definition and use of α N instead of α N are explained just before Theorem 2 in [START_REF] Bardet | Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic[END_REF]. The following theorem provides the asymptotic behavior of the estimator d

(MIR) N : Theorem 2. Under assumptions of Proposition 2, N N αN d (MIR) N -d L -→ N →∞ N 0 ; Λ ′ 0 (d) -2 J ⊺ p Γ -1 p (d)J p -1 . (3.3) Moreover, ∀ρ > 2(1 + 3β) (p -2)β , N β 1+2β (log N ) ρ • d (MIR) N -d P -→ N →∞ 0.
The convergence rate of d

Stationarity, non-stationarity and fractional unit roots tests

Assume that (X 1 , . . . , X N ) is an observed trajectory of a process X = (X k ) k∈Z . We define here new stationarity, non-stationarity and fractional unit roots tests for X based on d (MIR) N .

A stationarity test

There exist many stationarity and non-stationarity tests. The most famous stationarity tests are certainly the following unit roots tests:

• The KPSS (Kwiatkowski, Phillips, Schmidt, Shin) test (see Kwiatkowsli et al., 1992);

• The V/S test (see its presentation in [START_REF] Giraitis | Testing for long-memory in the presence of a general trend[END_REF] which was first defined for testing the presence of long-memory versus short-memory. As it was already notified in [START_REF] Giraitis | Rescaled variance and related tests for long-memory in volatility and levels[END_REF][START_REF] Giraitis | A test for stationarity versus trends and unit roots for a wide class of dependent errors[END_REF], the V/S test is also more powerful than the KPSS test for testing the stationarity.

• A test based on unidimensional IR statistic and developed in [START_REF] Surgailis | The increment ratio statistic[END_REF].

More precisely, we consider here the following statistical hypothesis test:

• Hypothesis H 0 (stationarity): (X t ) t∈Z is a process satisfying Assumption IG(d, β) with d ∈ (-0.5, 0.5) and 0 < β ≤ 2.

• Hypothesis H 1 (non-stationarity): (X t ) t∈Z is a process satisfying Assumption IG(d, β) with d ∈ [0.5, 1.25) and 0 < β ≤ 2.

We use a test based on d

(MIR) N
for deciding between both these hypothesis. Hence from the previous CLT (3.3) and with a significance level α, define

S N := 1 d (M IR) N >0.5+σp(0.5) q1-α N ( α N -1)/2 , (4.1)
where σ p (0.5

) = Λ ′ 0 (0.5) -2 J ⊺ p Γ -1 p (0.5)J p -1 1/2
(see (3.3)) and q 1-α is the (1α) quantile of a standard Gaussian random variable N (0, 1).

Then we define the following rules of decision: "H 0 (stationarity) is accepted when S N = 0 and rejected when S N = 1." Remark 3. In fact, the previous stationarity test S N defined in (4.1) can also be seen as a semiparametric test d < d 0 versus d ≥ d 0 with d 0 = 0.5. It is obviously possible to extend it to any value d 0 ∈ (-0.5, 1.25) by defining

S (d0) N := 1 d (M IR) N >d0+σp(d0) q1-α N ( α N -1)/2
. The particular case d 0 = 1 will be considered thereafter as a fractional unit roots test.

From previous results, it is clear that: Property 1. Under Hypothesis H 0 , the asymptotic type I error of the test S N is α and under Hypothesis H 1 , the test power tends to 1.

Moreover, this test can be used as a unit roots (UR) test. Indeed, define the following typical problem of UR test. Let X t = at + b + ε t , with (a, b) ∈ R 2 , and ε t an ARIMA(p, d, q) with d = 0 or d = 1. Then, a (simplified) problem of a UR test is to decide between:

• H UR 0 : d = 0 and (ε t ) is a stationary ARMA(p ′ , q ′ ) process. • H UR 1 : d = 1 and (ε t -ε t-1 ) t is a stationary ARMA(p ′ , q ′ ) process.
Then, Property 2. Under Hypothesis H UR 0 , the type I error of this unit roots test problem using S N decreases to 0 when N → ∞ and under Hypothesis H UR 1 , the test power tends to 1.

A non-stationarity test

Unit roots tests are also often used as non-stationarity test. Hence, between the most famous non-stationarity tests and in a nonparametric framework, consider

• The Augmented Dickey-Fuller (ADF) test (see [START_REF] Said | Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown Order[END_REF]);

• The Philipps and Perron (PP) test (see for instance [START_REF] Phillips | Testing for a Unit Root in Time Series Regression[END_REF].

Using the statistic d

(MIR) N
we propose a new non-stationarity test T N for deciding between:

• Hypothesis H ′ 0 (non-stationarity): (X t ) t∈Z is a process satisfying Assumption IG(d, β) with d ∈ [0.5, 1.25) and β ∈ (0, 2]. • Hypothesis H ′ 1 (stationarity): (X t ) t∈Z is a process satisfying Assumption IG(d, β) with -0.5 < d < 1/2 and β ∈ (0, 2].
Then, the decision rule of the test under the significance level α is the following: "Hypothesis H ′ 0 is accepted when T N = 1 and rejected when T N = 0"

where

T N := 1 d (M IR) N <0.5-σp(0.5) q1-α N ( α N -1)/2 . (4.2)
Then, Property 3. Under Hypothesis H ′ 0 , the asymptotic type I error of the test T N is α and under Hypothesis H ′ 1 the test power tends to 1.

As previously, this test can also be used as a unit roots test where X t = at + b + ε t , with (a, b) ∈ R 2 , and ε t an ARIMA(p, d, q) with d = 0 or d = 1. We consider here a "second" simplified problem of unit roots test which is to decide between:

• H UR ′ 0 : d = 1 and (ε t -ε t-1 ) t is a stationary ARMA(p ′ , q ′ ) process. • H UR ′ 1 : d = 0 and (ε t ) t is a stationary ARMA(p ′ , q ′ ) process.
Then, Property 4. Under Hypothesis H UR ′ 0 , the type I error of the unit roots test problem using T N decreases to 0 when N → ∞ and under Hypothesis H UR ′ 1 the test power tends to 1.

A fractional unit roots test

Fractional unit roots tests have also been defined for specifying the eventual long-memory property of the process in a unit roots test. In our Gaussian framework, they consist on testing

• Hypothesis H F UR 0 : (X t ) t∈Z is a "random walk"-type process such as:

X t = X t-1 + u t (4.3)
with (u t ) t a process satisfying Assumption IG(0, β) with 0 < β ≤ 2. Therefore (X t ) is a process satisfying Assumption IG(1, β).

• Hypothesis H F UR 1 : (X t ) t∈Z is a process satisfying the following relation:

X t = X t-1 + φ ∆ d1 X t-1 + u t (4.4)
where (u t ) t is a process satisfying Assumption IG(0, β) with 0 < β ≤ 2, φ < 0, and ∆ d1 is the fractional integration operator of order 0

< d 1 < 1, i.e. ∆ d1 X t-1 = t-1 i=0 π i (d 1 )X t-1-i and π i (d 1 ) = Γ(i -d 1 ) Γ(i + 1)Γ(-d 1 ) -1 .
After computations, it follows that if X satisfies (4.4), then X satisfies Assumption IG(d First of all, softwares used in this Section are available on http://samm.univ-paris1.fr/-Jean-Marc-Bardet with a free access on (in Matlab language). The concrete procedure for applying the MIR-test of stationarity is the following:

1. using additional simulations (performed on ARMA, ARFIMA, FGN processes and not presented here in order to avoid overloading the paper), we have observed that the value of the parameter p is not really important with respect to the accuracy of the test (there are less than 10% of fluctuations on the value of d (MIR) N when p varies). However, for optimizing our procedure (in the sense of minimizing from simulation the mean square error of the d estimation) we chose p as a stepwise function of N :

p = 5 × 1 {N <120} + 10 × 1 {120≤N <800} + 15 × 1 {800≤N <10000} + 20 × 1 {N ≥10000} .
2. as the values of σ p (0.5) and σ p (1) are essential for computing the thresholds of the tests, we have estimated them and obtained:

• σ 5 (0.5) ≃ 0.9082, σ 10 (0.5) ≃ 0.8289, σ 15 (0.5) ≃ 0.8016 and σ 20 (0.5) ≃ 0.7861.

• σ 5 (1) ≃ 0.8381, σ 10 (1) ≃ 0.8102, σ 15 (1) ≃ 0.8082 and σ 20 (1) ≃ 0.7929.

3. then after computing m N presented in Section 3, the adaptive estimator d

(MIR) N defined in (3.
2), the test statistics S N defined in (4.1), T N defined in (4.2) and F N defined in (4.5) are computed.

Monte-Carlo experiments on several time series

In the sequel the results are obtained from 1000 generated independent trajectories of each process defined below. The concrete procedures of generation of these processes are obtained from the circulant matrix method, as detailed in [START_REF] Doukhan | Theory and applications of long-range dependence[END_REF]. The simulations are realized for different values of d and N and processes which satisfy Assumption IG(d, β):

1. the usual ARIMA(p ′ , d, q ′ ) processes with respectively d = 0 or d = 1 and an innovation process which is a Gaussian white noise. Such processes satisfy Assumption IG(0, 2) or IG(1, 2) (respectively);

2. the ARFIMA(p ′ , d, q ′ ) processes with parameter d such that d ∈ (-0.5, 1.25) and an innovation process which is a Gaussian white noise. Such ARFIMA(p ′ , d, q ′ ) processes satisfy Assumption IG(d, 2) (note that ARIMA processes are particular cases of ARFIMA processes).

3. the Gaussian stationary processes X (d,c1,d) with the spectral density

f 3 (λ) = 1 |λ| 2d (1 + c 1 |λ| β ) for λ ∈ [-π, 0) ∪ (0, π], (5.1)
with d ∈ (-0.5, 1.5), c 1 > 0 and β ∈ (0, ∞). Therefore the spectral density f 3 implies that Assumption IG(d, β) holds. In the sequel we will first use c 1 = 1 and β = 0.1, implying that the second order term of the spectral density is "less negligible" than in case of ARFIMA processes, and c 1 = 0, implying that the second order term of the spectral density is "more negligible" than in case of ARFIMA processes.

4. the Gaussian stationary processes X (d,log) , such as its spectral density is

f 4 (λ) = 1 |λ| 2d (1 + | log(λ)| |λ|) for λ ∈ [-π, 0) ∪ (0, π], (5.2)
with d ∈ (-0.5, 1.5). Therefore the spectral density f 4 implies that Assumption I(d) holds, but not IG(d, β) stricto sensu.

5. the Gaussian non-stationary process X (trend) which can be written as X (trend) t = a n (t)+σ n (t)×ARF IM A(0, d, 0), where the additive and multiplicative trends are respectively a n (t) = sin(2πt/n) and σ n (t) = 2t/n (for us we chose a non-polynomial but smooth additive trend). are not data-driven adaptive among the 5 estimators. Table 1 provides the results of simulations for ARIMA(1, d, 0) (N = 500, N = 5000 and N = 50000). For ARFIMA(0, d, 0), ARFIMA(1, d, 1), X (d,1,1) , X (d,0,1) , X (d,log) and X (trend) processes and several values of d, the results of simulations are presented for N = 500 (Table 2), N = 5000 (Table 3) and N = 50000 (Table 4).

Comparison of d

Conclusions of simulations: Even if the estimator d (ADG) N

often provides the more accurate estimation of d for stationary processes, it is not more accurate anymore than d (MIR) N in case of trended time series. Moreover since this is not a data-driven estimator, with a bandwidth m fixed to be N 0.65 , it is not a consistent estimator when β is small enough: this is such the case for X (d,log) where we observe that the MSE is globally larger for N = 50000 than for N = 5000. The estimator d ). Finally the other data-driven estimators d (MS) and d (W AV ) provide correct results but are often less efficient than d (MIR) N .

Comparison of MIR tests S N and T N with other stationarity or non-stationarity tests

Monte-Carlo experiments were done for evaluating the performances of new tests S N and T N and for comparing them to most famous stationarity tests (KPSS and V/S) or non-stationarity (ADF and PP) tests (see more details on these tests in the previous section). As it is suggested for the corresponding R-software commands (see also [START_REF] Banerjee | Cointegration, Error Correction, and the Econometric Analysis of Non-Stationary Data[END_REF], we chose the following trimming parameters for the classical tests:

• k = 3 13 √
n for KPSS test;

• k = √ N for V/S test;

• k = (N -1) 1/3 for ADF test;

• k = 4 N 100 1/4 for PP test;
The results of these simulations with a type I error classically chosen to 0.05 are provided in Tables 5,6, 7 and 8. provide comparable and convincing results. Note also that KPSS and V/S provide reasonable results but less efficient than the other tests. In case of processes with d ∈ (0, 1), the tests computed from d (MIR) N obtain clearly better performances than classical non-stationarity tests ADF or PP which accept the non-stationarity assumption H ′ 0 even if the processes are stationary when 0 < d < 0.5 for instance. The case of the V/S test is different since this test is built for distinguishing between short and long-memory processes. However, as it was already established in [START_REF] Giraitis | Rescaled variance and related tests for long-memory in volatility and levels[END_REF], V/S test is slightly more accurate than KPSS for testing the stationarity. Note also that a renormalized version of this test has been defined in [START_REF] Giraitis | A test for stationarity versus trends and unit roots for a wide class of dependent errors[END_REF] for taking into account the value of d.

Conclusions of simulations

Comparison of MIR Fractional

T DGM = N t=2 (X t -X t-1 )∆ d1 X t-1 N t=2 ∆ d1 X t-1 2 × 1 N N t=2 X t -X t-1 -φ∆ d1 X t-1 2 1/2 , (5.3
)

with φ = N t=2 (Xt-Xt-1)∆ d 1 Xt-1 N t=2 ∆ d 1 Xt-1 2
and d 1 is the minimum between an ordinary least square estimator of d 1 and 1c with c > 0 small enough (typically c = 0.02). This is an extension in a fractional framework of the Dickey-Fuller test.

2. The efficient Wald test statistic defined in Lobato and Velasco ( 2007) is based on a two-step student test of a regression coefficient, i.e. T LV is the student test of the (z t ) t coefficient for the regression of

X t -X t-1 onto variables z t -α 1 z t-1 -• • • -α p z t-p , X t-1 -X t-2 , • • • , X t-p -X t-p-1 for t = p + 1, • • • , n
, where z t is defined as

z t = ∆ d X t -(X t -X t-1 ) 1 -d , (5.4)
and where ( α 1 , • • • , α p ) are obtained as a minimizer of

n k=p ∆ d X t -α 1 ∆ d X t-1 -• • • -α p ∆ d X t-p 2 and d
is a semi-parametric local Whittle type estimator of d. Note that T LV is depending on p and in the sequel we will chose p = 1 and p = 10, defining respectively T LV 1 and T LV 10 .

We applied the fractional unit roots tests F N , T LV 1 and T LV 10 to several fractional processes and displayed the results in Table 9. As it is a test specially devoted to FARIMA(0, d, 0) processes, we only applied the fractional unit roots test T DGM to those processes. Finally, note that we also consider the additional sample size N = 200 to N = 500 and N = 5000 used in other simulations because this could help to better evaluate the power of the several tests (since for N = 500 and N = 5000 the test powers are often 1).

Conclusions of simulations:

If T DGM and T LV 1 provide extremely convincing results for ARFIMA(0, d, 0) processes, T LV 1 is still very accurate for ARFIMA(1, d, 0) processes. From its definition, the fractional unit roots T LV 1 can not be used fruitfully for ARFIMA(1, d, 1) or X (d,0,1) processes but it can clearly be applied to a more general class of processes than T DGM .

The same for T LV 10 which can be applied likely to a more general class of processes than T LV 1 . However, if the results obtained here for ARFIMA(1, d, 1) or X (d,0,1) are satisfying and indicate that this test could be applied when N = 500 and N = 5000, this is not the case for N = 200 for all the considered processes because this test requires the estimation of too many parameters. Moreover, T LV 10 can not theoretically be applied to ARFIMA(1, d, 1) or X (d,0,1) processes and additional simulations realized with N = 500000 indicate a rejection probability ≃ 0.16 for ARFIMA(1, d, 1) processes when d = 1 (instead of 0.05). However, when N = 500000, a user could probably chose T LV 20 or T LV 50 which would provide satisfying results. The fractional unit roots test F N constructed from d (MIR) N

does not have these drawbacks: this is a data-driven test and it can be applied to a large family of fractional processes. Hence, the results of simulations obtained with F N are satisfying (even if they are less efficient for specific processes than those obtained with T LV 10 which requires the knowledge of the AR component). Even if this is a semiparametric test, the results obtained for N = 200 are reasonable. However, note that the rejection probability of F N for d = 0.9 is much bigger than for d = 1 when N = 200 and N = 500. Hence one could size adjust this test to get a better performance.

Conclusion

The adaptive data-driven memory parameter estimator d

(MIR) N
proposed in this paper has a lot of advantages. Firstly, for any process belonging to the set IG(d, β) -0.5<d<1.25, 0<β≤2 , it follows a CLT with a convergence rate reaching the minimax convergence rate (up to a multiplicative logarithm term) and this CLT is obtained without any choice of bandwidth or trimming parameter. Secondly, the numerical performances of this estimator are often better than those of the most accurate semiparametric memory parameter estimators, especially in case of trended processes (the robustness of IR estimator was already established in [START_REF] Surgailis | The increment ratio statistic[END_REF]. Finally, datadriven stationarity and fractional unit roots tests are constructed from this estimator and they provide accurate competitive results with respect to classical unit roots or fractional unit roots tests. Improving the performance of those tests could be an interesting task.

An asymptotic study of these new estimator and tests for linear processes could be an interesting extension of this paper. However, this requires to proof a multidimensional CLT theorem for a non-polynomial function of a multidimensional linear process which is a difficult result to be established.

Proofs

Two technical lemmas are first established:

Lemma 7.1. For all λ > 0 1. For a ∈ (0, 2), 2 |λ| a-1 ∞ 0 sin(λx) x a dx = 4a 2 a |λ| a ∞ 0 sin 2 (λx) x a+1 dx = π Γ(a) sin( aπ 2 ) ; 2. For b ∈ (-1, 1), 1 2 1-b -1 ∞ 0 sin 4 (λx) x 4-b dx = 16 -15 + 6 • 2 3-b -3 3-b × ∞ 0 sin 6 (λx) x 4-b dx = 2 3-b |λ| 3-b π 4 Γ(4 -b) sin( (1-b)π 2 ) ; 3. For b ∈ (1, 3), 1 1 -2 1-b ∞ 0 sin 4 (λx) x 4-b dx = 16 15 6 • 2 3-b + 3 3-b × ∞ 0 sin 6 (λx) x 4-b dx = 2 3-b |λ| 3-b π 4 Γ(4 -b) sin( (3-b)π 2 ) .
Proof. These equations are given or deduced (using decompositions of sin j (•) and integration by parts) from (see Doukhan et al., p. 31).

Lemma 7.2. For j = 4, 6, denote

J j (a, m) := π 0 x a sin j ( mx 2 ) sin 4 ( x 2 )
dx.

Then, we have the following expansions when m → ∞:

J j (a, m) =                C j1 (a) m 3-a + O m 1-a if -1 < a < 1 C ′ j1 (1) m 3-a + O log(m) if a = 1 C ′ j1 (a) m 3-a + O 1 if 1 < a < 3 C ′ j2 (3) log(m) + O 1 if a = 3 C ′′ j1 (a) + O m -((a-3)∧2) ) if a > 3 (7.2)
with the following real constants (which do not vanish for any a on the corresponding set): dx.

• C 41 (a) := 4 π(1 -2 3-a 4 ) (3 -a)Γ(3 -a) sin( (3-a)π
Proof. The proof of these expansions follows the steps than those of Lemma 5.1 in [START_REF] Bardet | Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic[END_REF]. Hence we write for j = 4, 6,

J j (a, m) = J j (a, m) + π 0 x a sin j ( mx 2 ) 1 ( x 2 ) 4 dx + π 0 x a sin j ( mx 2 ) 2 3 1 ( x 2 ) 2 dx (7.3) with J j (a, m) := π 0 x a sin j ( mx 2 ) 1 sin 4 ( x 2 ) - 1 ( x 2 ) 4 - 2 3 1 ( x
2 ) 2 dx. The expansions when m → ∞ of both the right hand sided integrals in (7.3) are obtained from Lemma 7.1. It remains to obtain the expansion of J j (a, m). Then, using classical trigonometric and Taylor expansions:

∀j ≥ 1, |γ m (j)| = E Y m (j)Y m (0) = 2 V 2 m π 0 cos(jx) x c 0 + O(x β ) sin 4 ( mx 2 ) sin 4 ( x 2 )
dx . q=1 I 2 (q) with |I 2 (q)| ≤ Cm 4 j -1 when 1 ≤ q ≤ j/m and |I 2 (q)| ≤ Cq -4 j 3 when j/m ≤ q ≤ j (see details p. 536-537 of [START_REF] Surgailis | The increment ratio statistic[END_REF]. Then, the inequality (5.41) remains true for d = 0.5 and since we consider here j ≥ m,

Denote γ m (j) = ρ m (j) = 2
|I 2 | ≤ C m 3 j -1 =⇒ |I 1 + I 2 | ≤ C m 3 j -1 =⇒ |γ m (j)| = |ρ m (j)| ≤ 2 V 2 m |I 1 + I 2 | ≤ C m j .
N m IR N (m) -E IR N (m) L -→ [N/m]∧m→∞ N 0, σ 2 (0.5) , (7.4)
with σ 2 (0.5) ≃ (0.2524) 2 .

To establish the multidimensional CLT for 0.

E IR N (m) = E |Y 0 + Y 1 | |Y 0 | + |Y 1 | = Λ( R m V 2 m ) with R m V 2 m := 1 -2 π 0 f (x) sin 6 ( mx 2 ) sin 2 ( x 2 ) dx π 0 f (x) sin 4 ( mx 2 ) sin 2 ( x 2 ) dx . Therefore an expansion of R m /V 2
m provides an expansion of E IR N (m) when m → ∞.

Step 1 Let f satisfy Assumption IG(d, β). Then we are going to establish that there exist positive real numbers C 1 , C 2 and C 3 specified in (7.5), (7.6) and (7.7) such that for 0.5 ≤ d < 1.5 and with ρ(d) defined in (2.6),

1. if β < 2d -1, R m V 2 m = ρ(d) + C 1 (2 -2d, β)m -β + O m -2 + m -2β ; 2. if β = 2d -1, R m V 2 m = ρ(d) + C 2 (2 -2d, β)m -β + O m -2 + m -2-β log(m) + m -2β ; 3. if 2d -1 < β < 2d + 1, R m V 2 m = ρ(d) + C 3 (2 -2d, β)m -β + O m -β-ǫ + m -2d-1 log(m) + m -2β ; 4. if β = 2d + 1, R m V 2 m = ρ(d) + O m -2d-1 log(m) + m -2 .
Under Assumption IG(d, β) and with J j (a, m) defined in (7.2) in Lemma 7.2, it is clear that,

R m V 2 m = 1 -2 J 6 (2 -2d, m) + c1 c0 J 6 (2 -2d + β, m) + O(J 6 (2 -2d + β + ε)) J 4 (2 -2d, m) + c1 c0 J 4 (2 -2d + β, m) + O(J 4 (2 -2d + β + ε)) , since π 0 O(x 2-2d+β+ε ) sin j ( mx 2 ) sin 2 ( x 2 ) dx = O(J j (2 -2d + β + ε))
. Now using the results of Lemma 7.2 and constants

C jℓ , C ′ jℓ and C ′′ jℓ , j = 4, 6, ℓ = 1, 2 defined in Lemma 7.2, 1. Let 0 < β < 2d -1 < 2, i.e. -1 < 2 -2d + β < 1. Then R m V 2 m = 1-2 C 61 (2 -2d) m 1+2d +O m 2d-1 + c1 c0 C 61 (2 -2d + β)m 1+2d-β +O m 2d-1-β C 41 (2 -2d)m 1+2d +O m 2d-1 + c1 c0 C 41 (2 -2d + β)m 1+2d-β +O m 2d-1-β = 1- 2 C 41 (2 -2d) C 61 (2 -2d)+ c 1 c 0 C 61 (2 -2d + β)m -β 1- c 1 c 0 C 41 (2 -2d + β) C 41 (2 -2d) m -β +O m -2 = 1- 2C 61 (2 -2d) C 41 (2 -2d) +2 c 1 c 0 C 61 (2 -2d)C 41 (2 -2d + β) C 41 (2 -2d)C 41 (2 -2d) - C 61 (2 -2d + β) C 41 (2 -2d) m -β +O m -2 + m -2β .
As a consequence,,

R m V 2 m = ρ(d) + C 1 (2 -2d, β) m -β + O m -2 + m -2β (m → ∞), with 0 < β < 2d -1 < 2 and C 1 (2 -2d, β) := 2 c 1 c 0 1 C 2 41 (2 -2d) C 61 (2 -2d)C 41 (2 -2d + β) -C 61 (2 -2d + β)C 41 (2 -2d) , (7.5)
and numerical experiments proves that C 1 (2 -2d, β)/c 1 is negative for any d ∈ (0.5, 1.5) and β > 0.

Let

β = 2d -1, i.e. 2 -2d + β = 1. Then, R m V 2 m = 1-2 C 61 (2 -2d) m 1+2d +O m 2d-1 + c1 c0 C ′ 61 (1)m 1-2d +O log(m) C 41 (2 -2d)m 1+2d +O m 2d-1 + c1 c0 C ′ 41 (1)m 1-2d +O log(m) = 1- 2 C 41 (2 -2d) C 61 (2 -2d)+ c 1 c 0 C ′ 61 (1)m 1-2d 1- c 1 c 0 C ′ 41 (1) C 41 (2 -2d) m 1-2d +O m -2 + m -2d-1 log(m) = 1- 2C 61 (2 -2d) C 41 (2 -2d) +2 c 1 c 0 C 61 (2 -2d)C ′ 41 (1) C 41 (2 -2d)C 41 (2 -2d) - C ′ 61 (1) C 41 (2 -2d) m 1-2d +O m -2 + m -2d-1 log(m) + m 2-4d .
As a consequence,

R m V 2 m = ρ(d) + C 2 (2-2d, β) m -β + O m -2 +m -2-β log(m)+m -2β (m → ∞), with 0 < β = 2d -1 < 2 and C 2 (2 -2d, β) := 2 c 1 c 0 1 C 2 41 (2 -2d) C 61 (2 -2d)C ′ 41 (1) -C ′ 61 (1)C 41 (2 -2d) , (7.6)
and numerical experiments proves that C 2 (2 -2d, β)/c 1 is negative for any d ∈ [0.5, 1.5) and β > 0.

Let 2d

-1 < β < 2d + 1, i.e. 1 < 2 -2d + β < 3. Then, R m V 2 m = 1-2 C 61 (2 -2d)m 1+2d + c1 c0 C ′ 61 (2 -2d + β)m 1+2d-β +O m 1+2d-β-ǫ + log(m) C 41 (2 -2d)m 1+2d + c1 c0 C ′ 41 (2 -2d + β)m 1+2d-β +O m 1+2d-β-ǫ + m -2d-1 log(m) = 1- 2 C 41 (2 -2d) C 61 (2 -2d)+ c 1 c 0 C ′ 61 (2 -2d + β)m -β 1- c 1 c 0 C ′ 41 (2 -2d + β) C 41 (2 -2d) m -β +O m -β-ǫ + m -2d-1 log(m) = 1- 2C 61 (2 -2d) C 41 (2 -2d) +2 c 1 c 0 C 61 (2 -2d)C ′ 41 (2 -2d + β) C 41 (2 -2d)C 41 (2 -2d) - C ′ 61 (2 -2d + β) C 41 (2 -2d) m -β +O m -β-ǫ + m -2d-1 log(m) .
As a consequence,

R m V 2 m = ρ(d) + C 3 (2 -2d, β) m -β + O m -β-ǫ + m -2d-1 log(m) + m -2β (m → ∞), and
C 3 (2 -2d, β) := 2 c 1 c 0 1 C 2 41 (2 -2d) C 61 (2 -2d)C ′ 41 (2 -2d + β) -C ′ 61 (2 -2d + β)C 41 (2 -2d) , (7.7)
and numerical experiments proves that C 3 (2 -2d, β)/c 1 is negative for any d ∈ [0.5, 1.5) and β > 0.

4. Let β = 2d + 1. Then, Once again with Lemma 7.2:

R m V 2 m = 1-2 C 61 (2 -2d) m 1+2d +O m 2d-1 + c1 c0 C ′ 62 (3) log(m)+O 1 C 41 (2 -2d)m 1+2d +O m 2d-1 + c1 c0 C ′ 42 (3) log(m)+O 1 = 1- 2 C 41 (2 -2d) C 61 (2 -2d)+ c 1 c 0 C ′ 62 (3)m -β log(m) 1- c 1 c 0 C ′ 42 (3) C 41 (2 -2d) m -β log(m) +O m -2 + m -2d-1 = 1- 2C 61 (2 -2d) C 41 (2 -2d) +2 c 1 c 0 C 61 (2 -2d)C ′ 42 (3) C 41 (2 -2d)C 41 (2 -2d) - C ′ 62 (3) C 41 (2 -2d) m -β log(m)+O m -2 .
As a consequence,

R m V 2 m = ρ(d) + O m -2d-1 log(m) + m -2 (m → ∞), with 2 < β = 2d + 1 < 4. (7.8)
Step 2: A Taylor expansion of Λ(•) around ρ(d) provides:

Λ R m V 2 m ≃ Λ ρ(d) + ∂Λ ∂ρ (ρ(d)) R m V 2 m -ρ(d) + 1 2 ∂ 2 Λ ∂ρ 2 (ρ(d)) R m V 2 m -ρ(d) 2 .
Note that numerical experiments show that ∂Λ ∂ρ (ρ) > 0.2 for any ρ ∈ (-1, 1). As a consequence, using the

previous expansions of R m /V 2 m obtained in Step 1 and since E IR N (m) = Λ R m /V 2 m , then for all 0 < β ≤ 2: E IR N (m) = Λ 0 (d) +          c 1 C ′ 1 (d, β) m -β + O m -2 + m -2β if β < 2d -1 c 1 C ′ 2 (d, β) m -β + O m -2 + m -2-β log m + m -2β if β = 2d -1 c 1 C ′ 3 (d, β) m -β + O m -β-ǫ + m -2d-1 log m + m -2β if 2d -1 < β < 2d + 1 O m -2d-1 log m + m -2 if β = 1 + 2d with C ′ ℓ (d, β) = ∂Λ ∂ρ (ρ(d)) C ℓ (2 -2d, β) for ℓ = 1, 2, 3
and C ℓ defined in (7.5), (7.6) and (7.7).

Proof of Theorem 1.

Using Property 2.1, if m ≃ C N α with C > 0 and (1+2β) -1 < α < 1 then N/m E IR N (m) - Λ 0 (d) -→ N →∞
0 and it implies that the multidimensional CLT (2.4) can be replaced by

N m IR N (m j ) -Λ 0 (d) 1≤j≤p L -→ N →∞ N (0, Γ p (d)). (7.9)
It remains to apply the Delta-method with the function Λ -1 0 to CLT (7.9). This is possible since the function

d → Λ 0 (d) is an increasing function such that Λ ′ 0 (d) > 0 and Λ -1 0 ) ′ (Λ 0 (d)) = 1/Λ ′ 0 (d) > 0 for all d ∈ (-0.5, 1.

5). It achieves the proof of Theorem 1.

Proof of Proposition 2. We use the proof of Proposition 2 in [START_REF] Bardet | Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic[END_REF]. Indeed, this proof is only based on the definitions of d N (jN α ), d N (N α ), Q N (α) and α N which are exactly the same here, and on the CLT satisfied by d N (jN α ) 1≤j≤p The only difference is that we suppose here 0 < β ≤ 2 and 0.5 ≤ d < 1.25 instead of 0 < β and -0.5 < d < 0.5 in [START_REF] Bardet | Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic[END_REF]. Then we consider only the case where β ≤ 2d + 1 and

α * = 1 (1+2β)∧(4d+3) = 1 (1+2β) in our framework.
Proof of Theorem 2. This proof is exactly the same as the proof of Theorem 2 in [START_REF] Bardet | Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic[END_REF].

N = 500 d = 0 d = 0 d = 0 d = 0 d = 0 d = 1 d = 1 d = 1 d = 1 d = 1 ARIMA(1, d, 0)
φ=-0.1 φ=-0.3 φ=-0.5 φ=-0.7 φ=-0.9 φ=-0.1 φ=-0.3 φ=-0.5 φ=-0.7 φ=-0.9

√ M SE d (M IR) N 0.
N = 5000 d = 0 d = 0 d = 0 d = 0 d = 0 d = 1 d = 1 d = 1 d = 1 d = 1 ARIMA(1, d, 0)
φ=-0.1 φ=-0.3 φ=-0.5 φ=-0.7 φ=-0.9 φ=-0.1 φ=-0.3 φ=-0.5 φ=-0.7 φ=-0.9

√ M SE d (M IR) N 0.
N = 50000 d = 0 d = 0 d = 0 d = 0 d = 0 d = 1 d = 1 d = 1 d = 1 d = 1 ARIMA(1, d, 0)
φ=-0.1 φ=-0.3 φ=-0.5 φ=-0.7 φ=-0.9 φ=-0.1 φ=-0.3 φ=-0.5 φ=-0.7 φ=-0.9 , d ) applied to fractionally integrated processes for N = 50000, several values of d ∈ (-0.5, 1.25) and 1000 independent replications.

√ M SE d (M IR) N 0.
N = 500 d = 0 d = 0 d = 0 d = 0 d = 0 d = 1 d = 1 d = 1 d = 1 d = 1 ARIMA(1, d, 0) φ=0.
N = 5000 d = 0 d = 0 d = 0 d = 0 d = 0 d = 1 d = 1 d = 1 d = 1 d = 1 ARIMA(1, d, 0) φ=0.
N = 5000 d = 0 d = 0 d = 0 d = 0 d = 0 d = 1 d = 1 d = 1 d = 1 d = 1 ARIMA(1, d, 0) φ=0.
(M S) N , d (
N = 500 d = 0 d = 0 d = 0 d = 0 d = 0 d = 1 d = 1 d = 1 d = 1 d = 1 ARIMA(1, d, 0)
φ=-0.1 φ=-0.3 φ=-0.5 φ=-0.7 φ=-0.9 φ=-0.1 φ=-0.3 φ=-0.5 φ=-0.7 φ=-0.9

N = 5000 d = 0 d = 0 d = 0 d = 0 d = 0 d = 1 d = 1 d = 1 d = 1 d = 1 ARIMA(1, d, 0)
φ=-0.1 φ=-0.3 φ=-0.5 φ=-0.7 φ=-0.9 φ=-0.1 φ=-0.3 φ=-0.5 φ=-0.7 φ=-0.9

N = 500 d = 0 d = 0 d = 0 d = 0 d = 0 d = 1 d = 1 d = 1 d = 1 d = 1 ARIMA(1, d, 0) φ=0
N = 5000 d = 0 d = 0 d = 0 d = 0 d = 0 d = 1 d = 1 d = 1 d = 1 d = 1 ARIMA(1, d, 0) φ=0

  and Dolado et al. (2008) allow to extend these tests to I(d) processes with ARMA component but requiring the knowledge of the order of this component. Several papers have been recently devoted to the construction of semiparametric tests, see for instance Giraitis et al. (2006), Abadir et al. (2007) or Surgailis et al.(2008). But these semiparametric tests require the knowledge of the second-order expansion of the spectral density at the zero frequency for adjusting a trimming or a bandwidth parameter; an a priori choice of this parameter always implies a bias of the estimator and therefore of the test when this asymptotic expansion is not smooth enough. The MIR estimator d (MIR) N does not present this drawback. It converges to d following a CLT with minimax convergence rate without any a priori choice of a parameter. This result is established for time series belonging to the Gaussian semiparametric class IG(d, β) defined below (see the beginning of Section 2) which is a restriction of the general set I(d). As a consequence, we construct a stationarity test S N which accepts the stationarity assumption when d (MIR) N ≤ 0.5 + s with s a threshold only depending on the type I error test, d (MIR) N and N . A non-stationarity test T N accepting the non-stationarity assumption when d (MIR) N ≥ 0.5s is also proposed. By the same principle, d (MIR) N also provides a fractional unit roots test F N for deciding between d = 1 and d < 1, i.e. whether F N ≥ 1s ′ or not, where s ′ is a threshold depending on the type I error test. İn Section 5, numerous simulations are realized on several models of time series (short and long-memory processes). First, the new MIR estimator d (MIR) N is compared to the most efficient and famous semiparametric estimators for several values of d ∈ (-0.5, 1.25). The performances of d (MIR) N

Property 2 . 1 .

 21 Let X satisfy Assumption IG(d, β) with 0.5 ≤ d < 1.5 and 0 < β ≤ 2. Then, there exists a non-vanishing constant K(d, β) depending only on d and β such that for m large enough, E IR N (m) = Λ 0 (d) + K(d, β) × m -β 1 + o(1) if β < 1 + 2d Λ 0 (d) + K(0.5, β) × m -2 log m 1 + o(1) if β = 2 and d = 0.5 with Λ 0 (d) := Λ(ρ(d)) where ρ(d)

  the same (up to a multiplicative logarithm factor) than the one of minimax estimator of d in this semiparametric framework (see[START_REF] Giraitis | Rate optimal semi-parametric estimation of the memory parameter of the Gaussian time series with long range dependence[END_REF]. As it was already established in[START_REF] Surgailis | The increment ratio statistic[END_REF], the use of IR statistics confers a robustness of d (MIR) N to smooth additive or multiplicative trends (see also the results of simulations thereafter). The additional advantage of d (MIR) N with respect to other adaptive estimators of d (see Moulines and Soulier, 2003, for an overview over frequency domain estimators of d) is the central limit theorem (3.3) satisfied by d (MIR) N . This central limit theorem provides asymptotic confidence intervals on d which are unobtainable for instance with FEXP or local periodogram adaptive estimator (see respectively Iouditsky et al., 2001, and Giraitis et al., 2000 or Henry, 2007). Moreover d (MIR) N can be used for d ∈ (-0.5, 1.25), i.e. as well for stationary and non-stationary processes, without modifications in its definition. Both these advantages allow to define stationarity and fractional unit roots tests based on d (MIR) N .

  (MIR) N to 1. Hence, the decision rule of the test under the significance level α is the following:"Hypothesis H F UR 0 is accepted when F N = 1 and rejected when F N = 0" whereF N := 1 d (M IR) N >1-σp(1) q1-α N ( α N -1)/2 . (4.5) Then as previously Property 5. Under Hypothesis H F UR 0 , the asymptotic type I error of the test F N is α and under Hypothesis H F UR 1 the test power tends to 1. 5 Results of simulations 5.1 Numerical procedure for computing the estimator and tests

  (MIR) N with other semiparametric estimators of d Here we first compare the performance of the data-driven MIR estimator d (MIR) Nwith other famous semiparametric estimators of d:• d (IR) Nis the original version of the IR based estimator defined in[START_REF] Surgailis | The increment ratio statistic[END_REF]. As it was recommended in that article, we chose m = 10.• d (MS) Nis the global log-periodogram estimator introduced by[START_REF] Moulines | Semiparametric spectral estimation for fractional processes[END_REF], also called FEXP estimator, with bias-variance balance parameter κ = 2. Such an estimator was shown to be consistent for d ∈ (-0.5, 1). This semiparametric estimator is an adaptive data-driven estimator of d.• d (ADG) Nis the extended local Whittle estimator defined by Abadir, Distaso and Giraitis (2007) which is consistent for d > -3/2. It is a generalization of the local Whittle estimator introduced by Robinson (1995b), consistent for d < 0.75, following a first extension proposed byPhillips (1999) and[START_REF] Shimotsu | The Exact Local Whittle Estimation of Fractional Integration[END_REF]. This estimator avoids the tapering used for instance inVelasco (1999b) or[START_REF] Hurvich | An Efficient Taper for Potentially Over differenced Long-Memory Time Series[END_REF]. The trimming parameter is chosen as m = N 0.65 (this is not an adaptive data-driven estimator) following the numerical recommendations of[START_REF] Abadir | non-stationarity-extended local Whittle estimation[END_REF].• d (W AV ) Nis an adaptive data-driven wavelet based estimator introduced in Bardet and Bibi (2012) using a Lemarie-Meyer type wavelet (another similar choice could be the adaptive wavelet estimator introduced in Veitch et al., 2003, using a Daubechie's wavelet, but its robustness property are slightly less interesting). The asymptotic normality of such estimator is established for d > -0.5 (when the number of vanishing moments of the wavelet function is large enough).Note that only d

  good trade-off with always one of the smallest √ M SE among the 5 semiparametric estimators and almost never bad results (except perhaps for X(trend) , N = 500 and d < 0.5). Moreover, the larger N the more efficient d (MIR) N with respect to the other estimators. Note also that the use of a data-driven multidimensional version of IR statistics (i.e. the estimator d (MIR) N ) considerably improves the quality of the estimation with respect to the original estimator based on unidimensional IR statistics (the estimator d (IR) N

:

  As it is well known, from their constructions, KPSS , V/S, ADF and PP tests should asymptotically decide the stationarity hypothesis when d = 0, and the non-stationarity hypothesis when d > 0. It was exactly what we observe in these simulations. For ARIMA(p, d, 0) processes with d = 0 or d = 1 (i.e. AR(1) process when d = 0), ADF and PP tests are more efficient tests than our adaptive MIR tests when N = 500. However, note that all stationarity tests do not control the size for φ = -0.9. But when N = 5000 the tests computed from d (MIR) N

2

 2

) 2 )

 2 and C 61 (a) := π(15 -6 • 2 3-a + 3 3-a ) 4(3a)Γ(3a) sin( (3-a)π a -15 cos(y) + 6 cos(2y)cos(3y) dy • C ′ 42 (a) := 6 • 1 {a=3} + 1 {a=1} and C ′ 62 (a) := 5 • 1 {a=3}

  J j (a, m) can be obtained. Numerical experiments show that C ′′ 41 (a) = 0, C ′′ 61 (a) = 0, C ′′ 42 (a) = 0 and C ′′ 62 (a) = 0. Proof of Proposition 1. This proposition is based on results of Surgailis et al. (2008) and was already proved in Bardet et Dola (2012) in the case -0.5 < d < 0.5. Mutatis mutandis, the case 0.5 < d < 1.25 be treated exactly following the same steps. The only new proof which has to be established concerns the case d = 0.5 since Surgailis et al. (2008) do not provide a CLT satisfied by the (unidimensional) statistic IR N (m) in this case. Let Y m (j) the standardized process defined Surgailis et al. (2008). Then, for d = 0.5,

V 2 mI 1 +

 21 I 2 as in (5.39) of Surgailis et al. (2008). The expansion (2.20) of Surgailis et al. (2008) remains true for d = 0.5 and therefore V 2 m ∼ c 0 V (0.5)m 2 when m → ∞. The same for the inequality (5.42) when d = 0.5 and thus |I 1 | ≤ C m 4 j -2 . Finally, when d = 0.5, we still have I 2 = j -1 j/2

j 2 ,

 2 Now let η m (j) := |Ym(j)+Ym(j+m)| |Ym(j)|+|Ym(j+m)| := ψ Y m (j), Y m (j +m) . The Hermite rank of the function ψ is 2 and therefore the equation (5.23) of Surgailis et al. (2008) obtained from Lemma 1 of Arcones (1994) remains valid. Hence: Cov(η m (0), η m (j)) ≤ C m 2 and then the equations (5.28-5.31) of Surgailis et al. (2008) remain valid for all d ∈ [0.5, 1.25). Then for d = 0.5,

  ARIMA(1, d, 0) process (defined by Xt + φXt-1 = εt for d = 0 and (Xt -Xt-1) + φ(Xt-1 -Xt-2) = εt for d = 1) for several values of φ and N and 1000 independent replications.

of ARIMA( 1 ,

 1 d, 0) processes (defined by Xt + φXt-1 = εt for d = 0 and (Xt -Xt-1) + φ(Xt-1 -Xt-2) = εt for d = 1) for several values of φ and N . The accuracy of tests is measured by the rejection probabilities.

  , Velasco and Robinson, 2000, Moulines and Soulier, 2003, Shimotsu and Phillips, 2005, Giraitis et al., 2003, 2006, Abadir et al., 2007 or Moulines et al., 2007). Moreover, adaptive data-driven versions of these estimators have been defined to avoid any trimming or bandwidth parameters, generally required by these methods (see for instance Giraitis et al., 2000, Moulines and Soulier, 2003, Veitch et al., 2003, or

  , • • • , p with p ∈ N * and m ∈ N * , and define the random vector (IR N (m j )) 1≤j≤p . In the sequel we naturally extend the results obtained for m ∈ N

	modifications of Surgailis et al. (2008), for d ∈ (-0.5, 1.25) and p ∈ N * , define the stationary multidimensional centered Gaussian processes Z (1)
	to m ∈ (0, ∞) by the convention: (IR N (j m)) 1≤j≤p = (IR N (j [m])) 1≤j≤p (which does not change the asymptotic results).
	For H ∈ (0, 1), let B H = (B H (t)) t∈R be a standard fractional Brownian motion, i.e. a centered Gaussian process having stationary increments and such as Cov B H (t) , B H (s) = 1 2 |t| 2H + |s| 2H -|t -s| 2H . Now, using obvious

*

  Unit Roots test F N and Dolado et al. and Lobato and Velasco Fractional Unit Roots tests Monte-Carlo experiments were also done for evaluating the performances of new Fractional Unit Root test F N and for comparing it to the Fractional Unit Roots tests defined in Dolado et al. (2002) and in Lobato and Velasco (2007). 1. The student-type test statistic defined in Dolado, Gonzalo and Mayoral (2002) is such as:

  5 ≤ d ≤ 1.25 from (7.4) and unidimensional CLT of Surgailis et al (2008) for 0.5 < d ≤ 1.25, we can reproduce exactly the steps 1 and 2 of the proof in Proposition 2.1 of Bardet and Dola (2012). of Property 2.1. As in Surgailis et al (2008), we can write:

	Proof

Table 1 :

 1 : Comparison between d (M IR)

		1	φ=0.3	φ=0.5	φ=0.7	φ=0.9	φ=0.1	φ=0.3	φ=0.5	φ=0.7	φ=0.9
	√ √ M SE d (M IR) N M SE d (IR) N √ M SE d (M S) N √ M SE d (ADG) N √ M SE d (W AV ) N	0.0161 0.0217 0.0146 0.0158 0.0114	0.0280 0.0407 0.0150 0.0151 0.0147	0.0435 0.0662 0.0153 0.0144 0.0174	0.0672 0.1130 0.0162 0.0147 0.0213	0.0727 0.0880 0.0261 0.0136 0.0258	0.0143 0.0148 0.0132 0.0133 0.0185	0.0152 0.0160 0.0136 0.0142 0.0260	0.0163 0.0189 0.0137 0.0132 0.0446	0.0162 0.0176 0.0148 0.0140 0.0194	0.0191 0.0193 0.0209 0.0150 0.0260
			N	and other famous semiparametric estimators of d ( d	(IR) N

Table 2 :

 2 : Comparison between d (M IR)

			(M IR) N M SE d √ (IR) N √ M SE d (M S) N √ M SE d (ADG) N √ M SE d (W AV ) N	0.0911 0.1900 0.1405 0.0764 0.0716	0.0968 0.2156 0.1441 0.0803 0.0795	0.0988 0.2229 0.1432 0.0787 0.0849	0.0949 0.2081 0.1523 0.0838 0.0865	0.1018 0.2008 0.1681 0.0778 0.0808	0.1022 0.1806 0.1765 0.0785 0.0848	0.0973 0.1622 0.1703 0.0800 0.0701	0.1055 0.1432 0.1643 0.0758 0.0707
			ARFIMA(1,d,1)
			√ √ M SE d (M IR) N M SE d (IR) N √ M SE d (M S) N √ M SE d (ADG) N √ M SE d (W AV ) N	0.1527 0.2255 0.1393 0.0939 0.1728	0.1363 0.2328 0.1448 0.0914 0.1625	0.1315 0.2217 0.1446 0.0925 0.1591	0.1173 0.2205 0.1521 0.1012 0.1424	0.1212 0.2080 0.1668 0.0933 0.1373	0.1099 0.1773 0.1744 0.0887 0.1210	0.1129 0.1592 0.1688 0.0897 0.1026	0.1098 0.1353 0.1625 0.0872 0.0922
			X (d,1,0.1)
			√ √ M SE d (M IR) N M SE d (IR) N √ M SE d (M S) N √ M SE d (ADG) N √ M SE d (W AV ) N	0.0892 0.1803 0.1418 0.0808	0.1003 0.2081 0.1438 0.0836 0.0871	0.1010 0.2186 0.1425 0.0804 0.0891	0.1093 0.2062 0.1472 0.0864 0.0856	0.1168 0.2043 0.1538 0.0817 0.0772	0.1126 0.1840 0.1680 0.0812 0.0757	0.1158 0.1700 0.1677 0.0842 0.0798	0.1271 0.1569 0.1697 0.0817 0.0856
			X (d,0,1)
			√ √ M SE d (M IR) N M SE d (IR) N √ M SE d (M S) N √ M SE d (ADG) N √ M SE d (W AV ) N	0.0915 0.1839 0.1393 0.0746 0.0756	0.0950 0.2141 0.1437 0.0790 0.0786	0.0962 0.2094 0.1446 0.0750 0.0767	0.1018 0.2179 0.1447 0.0808 0.0750	0.1043 0.2010 0.1524 0.0778 0.0667	0.1111 0.1827 0.1709 0.0779 0.0724	0.1104 0.1625 0.1721 0.0754 0.0789	0.1205 0.1411 0.1708 0.0780 0.0846
			X (d,log)
			√ √ M SE d (M IR) N M SE d (IR) N √ M SE d (M S) N √ M SE d (ADG) N √ M SE d (W AV ) N	0.0836 0.1810 0.1500 0.0822 0.0974	0.1064 0.2100 0.1529 0.0864 0.1087	0.1089 0.2089 0.1564 0.0844 0.0996	0.1161 0.2009 0.1677 0.0901 0.1068	0.1138 0.1853 0.1649 0.0827 0.1007	0.1197 0.1819 0.1654 0.0797 0.1031	0.1252 0.1666 0.1660 0.0852 0.0967	0.1380 0.1542 0.1578 0.0846 0.0829
			X (trend)
			√ √ M SE d (M IR) N M SE d (IR) N √ M SE d (M S) N √ M SE d (ADG) N √ M SE d (W AV ) N	0.4684 0.2793 0.9077 0.5674 0.0961	0.2922 0.2192 0.6067 0.3564 0.0908	0.1633 0.2048 0.3444 0.1787 0.0886	0.1051 0.2029 0.2150 0.1009 0.0913	0.1027 0.1964 0.2024 0.0845 0.0917	0.1176 0.1824 0.1994 0.0901 0.0907	0.1176 0.1616 0.1683 0.0880 0.0804	0.1279 0.1443 0.1471 0.0878 0.0896
				N	and other famous semiparametric estimators of d ( d	(IR) N	, d	(M S) N	, d	(ADG) N	and
	d	(W AV ) N	) applied to fractionally integrated processes for N = 500, several values of d ∈ (-0.5, 1.25) and 1000 independent
	replications.

Table 3 :

 3 : Comparison between d (M IR)

	(IR) N	, d	(M S) N	, d	(ADG) N	and

N

and other famous semiparametric estimators of d ( d

Table 4 :

 4 : Comparison between d (M IR)

	(IR) N	, d	(M S) N	, d	(ADG) N	and

N

and other famous semiparametric estimators of d ( d

Table 5 :

 5 Comparisons of stationarity and non-stationarity tests from 1000 independent Monte Carlo experiment replications

Table 6 :

 6 Comparisons of stationarity and non-stationarity tests from 1000 independent Monte Carlo experiment replications of ARFIMA(0, d, 0) processes for several values of d and N . The accuracy of tests is measured by the rejection probabilities.

	S N : Rejected H 0	0	0	0	0.003	0.276	0.917	0.998	0.999
	KP SS: Rejected H 0	0	0.059	0.395	0.771	0.946	0.989	0.999	0.999
	V /S : Rejected H 0	0	0.052	0.446	0.847	0.970	0.993	0.998	1.000
	T N : Rejected H ′ 0	1.000	1.000	0.965	0.421	0.017	0	0	0
	ADF: Rejected H ′ 0	1.000	1.000	1.000	0.977	0.615	0.233	0.065	0.005
	PP : Rejected H ′ 0	1.000	1.000	1.000	1.000	0.919	0.447	0.065	0.002
	N = 5000
	ARFIMA(0, d, 0)	d = -0.2 d = 0	d = 0.2	d = 0.4	d = 0.6	d = 0.8	d = 1	d = 1.2
	S N : Rejected H 0	0	0	0	0	0.912	1.000	1.000	1.000
	KP SS: Rejected H 0	0	0.042	0.674	0.996	1.000	1.000	1.000	1.000
	V /S : Rejected H 0	0		0.694	0.992	1.000	1.000	1.000	1.000
	T N : Rejected H ′ 0	1.000	1.000	1.000	0.946	0	0	0	0
	ADF: Rejected H ′ 0	1.000	1.000	1.000	1.000	0.946	0.448	0.050	0.004
	: Rejected H ′ 0	1.000	1.000	1.000	1.000	1.000	0.705	0.042	0

Table 7 :

 7 Comparisons of stationarity and non-stationarity tests from 1000 independent Monte Carlo experiment replications of ARFIMA(1, d, 1) processes with φ = -0.3 and θ = 0.7 for several values of d and N . The accuracy of tests is measured by the rejection probabilities.

	.015	0.442	0.898	0.987	0.999

Table 9 :

 9 Comparisons of Fractional Unit Roots tests, as 1000 independent Monte Carlo experiment replications, of processes for several values of d and N . Note that the AR parameter of the ARFIMA(1, d, 0) is 0.5 and the AR and MA parameters of ARFIMA(1, d, 1) are respectively -0.3 and 0.7. The accuracy of tests is measured by the rejection probabilities.

Acknowledgments. The authors are grateful to both the referees for their very careful reading and many relevant suggestions and corrections that strongly improve the content and the form of the paper.


Mariacristina Picchio
email: mariacristina.picchio@istc.cnr.it

Susanna Mayer
email: susanna.mayer@istc.cnr.it

The socialization of children of migrant parents during their first year in crèche

Introduction

Recent policy documents stated that the attendance of children at high quality ECEC services supports their social inclusion and has positive effects on their long run school success. The quality of this experience matters particularly in the case of children from migrant families (European Commission, Communication 66/2011; Recommendation 112/2013).

As several studies have pointed out, the presence of children with different cultural backgrounds requires that ECEC everyday practices would take into account the plural identity of these children as well as their families' linguistic and cultural diversity [START_REF] Almodóvar | Children Crossing Borders: Immigrant Parent and Teacher Perspectives on Preschool[END_REF]Brougère, Guénif-Souilamas & Rayna, 2008;[START_REF] Favaro | Une crèche pour apprendre à vivre ensemble[END_REF][START_REF] Vandenbroeck | Éduquer nos enfants à la diversité: sociale culturelle, ethnique, familiale[END_REF][START_REF] Vandenbroeck | Immigrant mothers crossing borders: nomadic identities and multiplies belongings in early childhood education[END_REF].

Also in Italy, we see teachers' increasing demand of professional support in questioning their practices according to children's and families' needs.

Which experiences do Italian ECEC service offer to children from migrant families? Which practices can be realized in order to support their social inclusion?

We faced these issues in a previous study, in which we analysed the first experience of two young children from migrant families in an Italian municipal nido (crèche) [START_REF] Picchio | Participer et communiquer: l'expérience de deux enfants de parents immigrants dans une crèche italienne[END_REF].

Aim of the study

In this contribution we present the main results of an action-research initiative aimed at better understanding how children of migrant parents cope with the transition from the home to the nido, which difficulties they encounter, how their experience in the new context changes over a whole academic year and which conditions support or impede their participation in activities and in social interactions with peers and teachers. This initiative was carried out in collaboration with the Department of Education of the Municipality of Pistoia, which over the years developed attentive policies aimed at support continuous professional development of ECEC professionals [START_REF] Musatti | Origins and evolution of professionalism in the context of municipal ECEC institutions[END_REF][START_REF] Musatti | How to construct a curriculum in an Italian nido[END_REF].

Methodology and empirical data

Our research group was composed by 12 teachers, 2 pedagogical coordinators and 2 researchers. In September and in October 2013 researchers spent some days in a municipal nido in Pistoia in order to become familiar with children and adults and the organization of daily activities. Then, from November 2013 to May 2014, they filmed the activities and social behaviour of children and adults during their everyday life using two video cameras simultaneously. The videos were recorded from 8.30 to 13, once per month, for a total of 50 hours.

Teachers produced weekly written reports focused on children's daily experience in the nido [START_REF] Picchio | Documentation and analysis of children's experience: an ongoing collegial activity for early childhood professionals[END_REF]. Their situated perspective, as actors taking part in the nido everyday life, provided further elements for understanding the meaning of events and children's behaviour and grasping their evolution over the months.

Video and written documentation were analysed and discussed by research group in periodical meetings.

The group of children was composed of 26 children (aged from 19 to 34 months), 17 of whom were of migrant parents from different countries (Nigeria, Albania, Estonia, Ukraine, Somalia, Philippines, Romania). Most of the children of migrant parents (13 children) have been attending nido for the first year, 7 of whom before attending nido were exposed only to the family language.

Children's experience during their first period of attendance

During the first months, most of the children expressed discomfort and experienced difficulties in participating in daily activities and social interactions with peers and teachers. Their discomfort emerged more or less noticeably. In the following we report 4 examples of children's reactions to the new experience.

Ex. 1. Marcus is 27 months old. During the first period of attendance, according to the nido procedure, his mother spends several mornings with him in the service in order to allow child to experience brief separations and become gradually familiar with teachers and environment. Nevertheless, as soon as Marcus enters the nido, he starts crying and shouting and has trouble in separating from his mother, who seems to be reluctant to leave him. Often he cries and shouts continuously until breakfast time despite the teacher's efforts to reassure him. However, when the teacher involves him in setting tables for breakfast or lunch he feels comforted. During transitions between daily activities or when the group move to another room, he gets again very uncomfortable; he cries and shouts.

Ex. 2. Isaac (30 months old), lives with his mother in a difficult condition. She is unemployed and hasn't any relatives or friends in the city. Teachers reported that they encounter difficulties in communication with her because of her poor knowledge of the Italian language

and she appears to be uneasy and uncomfortable during exchange with them. Isaac seems to cope with the daily separation from her mother easily but during all over the day he is very upset. Repeatedly he tries to open the door and when he sees the door open, he rushes towards the nido's exit. He throws play materials around the room and spends a lot of time climbing every kind of furniture. He is particularly able to climb. At his first lunch at nido, he is completely upset and lost, he speeds around the room crying and shouting and he doesn't let teachers approach him. Some days later, teachers will know from his mother that he never has regular meals at home seating at the table. In the following days, during breakfast and lunch time, he starts seating at the table with the other children, though for a short time. He leaves his place a lot of times but he comes back quickly when the teacher proposes Italian rhymes and jingles. Although he doesn't understand Italian, he gets involved in this activity happily, having positive and significant interactions with teacher and peers. Ex. 3. Jordan (26 months old) lives with his mother and siblings in a residential home for single mothers from disadvantaged background. Particularly during the moments of free play, Jordan spends a lot of time wandering all around the room; he often seems confused. Sometimes he plays apart, usually browsing through books. Jordan is often excluded by his peers in several situations (for instance when Jordan tries to approach them) and assaulted without reasons (for instance while he is playing apart). During moments of transition between activities, or when the noise in the room became too high, he appears uncomfortable and suffering. He cries, plugging his ears and hiding himself under the table.

We found that other children expressed their effort to cope with the new experience in the nido, in a less impressive and evident way. We report an example.

Ex. 4. James is 21 months old. His family is well integrated in the city. The child copes with the daily separation from his mother quite easily. All over the day, he is very attentive and interested in activities and social interactions between other children and adults, though he doesn't get involved in them. He spends a lot of time observing at a certain distance children's and adults' behaviour in the different daily activities. He studies how children use play materials, how they move in the environment, how they interact, how teachers intervene in the different situations, for instance, during conflicts or in order to make children respect the rules. Although he doesn't understand and speak Italian yet, often he gazes at children who are chatting during meal time attentively. This behaviour reveals James's engagement in understanding norms and habits of the nido and behaviours expected in the new context.

While teachers spent a lot of energies to alleviate Manuel, Isaac and Jordan's unease and to support them to cope with the new experience, we observed that James's behaviour was unnoticed by teachers in many situations. Unlike Isaac, Marcus and Jordan, apparently James was calm all over the day. He expressed his effort to cope with the new experience silently; this fact often made him invisible at adults' and peers' eyes and his need to be facilitated in knowing the new context could be neglected.

During the first period of attendance none of these children used verbal communication: they didn't understand Italian language and the use of their family languages was not encouraged at all. Rarely, we could catch someone of them mumbling in his family language while playing apart. However, most of them used non-verbal communicative modalities competently, such as gestures, facial expressions and body posture. We report an example.

Ex. 5. Isaac, who has banged his head against a door, goes near the teacher, takes her hand and put it on his head in order to be consoled.

Practices that support children's participation

We found that children's participation in the social context of the nido increased over the months. After some months of attendance:

-each child seemed to have a happy time at nido; each one appeared relaxed and at ease throughout the day; -each child had positive relationships with teachers and other children (or at least with some of them); each one had significant interactions and communicative exchanges with them, though she/he didn't master Italian language yet (most of migrant children didn't speak Italian but seemed to understand it); -each child participated actively and with pleasure in both care and play activities; -each child seemed to have developed a feeling of belonging to nido's community.

During meetings, the research group analysed and discussed which educational practices facilitated and supported these changes.

The first aspect discussed was the role of teachers. Teachers have an important role in mediating between each child and the context, interpreting her/his specific difficulties and needs as well as valuing her/his capacities and desires, and, most important, making her/him explicit and visible to other children (for instance giving voice to child's emotions and feelings, commenting her/his actions, echoing her/his non-verbal communications). This kind of mediation contributes to establish a positive relationship not only between child and teacher but also between children, because it helps them to pay attention to their respective needs and moods and develop a mutual respect.

We observed that the role of mediating between child and the context can be played also by a peer. We will present two examples.

Ex. 6. Isaac, during lunch time, after having eaten a plate of pasta seated at the table with the other children, climbs up the windowsill with a piece of bread in his hand. Although Isaac breaks a rule (no getting up during mealtimes), shared and respected by the group, his

behaviour is accepted by teachers and peers, who acknowledge Isaac's effort to be seated, allowing him an alternative. In this situation a little girl values Isaac's behaviour as she says: "Isaac loves eating bread looking out the window!". Ex. 7. Jordan -the child often excluded by his peers during the first months -goes near a small group of children who are drawing. He struggles to find his place at the table, so the teacher, engaged in reading a book to another small group of children not far away, asks Ramona (34 months) to help him to seat down. The girl answers: "Jordan desires to look what we are doing", interpreting and giving voice to Jordan's intention. In fact, Jordan observes children's activities for a long time, then takes a piece of cardboard, but he hasn't any marker pens for drawing and he doesn't ask one to the other children. Ramona notices him and passes him a marker pen. Jordan starts to drawing. Some minutes later, Giulia (29 months) approaches Jordan and tries to wrestle the marker pen out of his hands. Jordan starts to complain, looking for teacher's attention, but immediately Ramona intervenes once again to sustain Jordan, reprimanding Giulia: "Giulia, leave him in pace! He wants to draw!".

We point out that these examples confirm that the organization in stable small groups of children during play and care activities can support children's reciprocal familiarity and understanding of each other's needs and intentions.

The teachers' organization affects the experience of familiarization of each child as well. During play and care activities each teacher had the primary responsibility of a stable small group of children (7-8 children). This organization helped teachers to better focus their attention on the needs of each child and her/his family, adapt their communicative and relational styles, and support child's involvement in activities and interactions within the large group of peers and teachers. Furthermore, a stable organization of physical environment, time and activities can provide children a clear, recognizable and predictable reference framework. We found that this frame supported particularly children who didn't understand the verbal communications by which teachers explained the organization of nido's daily life (for instance announcing the end of an activity or a move from a room to another). In particular, we found that children's participation increased when activities were marked by stable routines and habits (for instance preparing and rearranging together materials after play, preparing tables before lunch) or rituals (for instance singing rhymes and jingles at the end of breakfast in small group). These practices contributed to develop children's familiarity with the organization of nido's everyday life and their feeling of belonging to a community [START_REF] Musatti | How to construct a curriculum in an Italian nido[END_REF].

As in our previous study [START_REF] Picchio | Participer et communiquer: l'expérience de deux enfants de parents immigrants dans une crèche italienne[END_REF], we found that during the process of second language acquisition, some children from migrant families went through a silent phase (Tabordos, 1997). However, we noticed that the ritualized proposal of nursery rhymes or reading, facilitated children's active participation and learning of the new language, as it is also pointed out by Rayna, [START_REF] Sèguret | Lire en chantant des albums de comptines[END_REF].

Discussion

Our analysis has shown that some educational and organizational practices particularly supported a positive evolution of children's social experience over the months. However, it also has highlighted the difficulties that children's of migrant families had to cope during the first period of their attendance. Indeed, when children enter an ECEC service they have to cope with a double transition. As any child, they have to cope with a new experience in a wider and different context from their home where they meet unfamiliar adults and many peers. Moreover, as children from migrant families they make experience of a context which is different from the one they are familiar with, from a cultural point of view (different rules, habits and behaviours) and, most important, because of the language used. This experience requires a great emotional investment as well as an important engagement of children's cognitive and social competences. Analysing children's experience, teachers became aware that children from migrant families could be particularly disoriented during the first encounter with the nido contest. At their eyes the nido appeared to be a mono-cultural place because of its material and symbolic features. Diversity was not represented among the staff because all the professionals were Italian natives. Even linguistic diversity was not considered, as teachers used only Italian language in verbal communications to children and their parents, and the use of family languages of migrant children was not expected or favoured.

Furthermore, teachers became aware that communication with migrant parents was too often brief and poor -especially with migrant parents from disadvantaged background -and despite teachers' efforts, during the first period of children's attendance, they encountered difficulties in promoting parents' involvement by their usual welcoming practices. Teachers acknowledged that these difficulties affected children's experience of familiarization with the nido and that improving quality of children's experience in the ECEC services claimed to rethink procedures and goals of practices not only towards children but towards parents as well [START_REF] Picchio | Welcoming migrant parents in ECEC services in professionals[END_REF].

As [START_REF] Vandenbroeck | Diversity in early childhood services[END_REF] outlined, for many children the enrolment in an ECEC service represents their first step into out-of-home social world and the service reflects how society looks at them and thus how they look at themselves. Therefore, it is essential to construct an educational context that will value their identity, in which cultural diversity will be immediately recognized and made visible to the eyes of all children and their families.