Yacouba Boubacar
email: yacouba.boubacar_mainassara@univ-fcomte.fr

Maïnassara Youssef

Esstafa Bruno Saussereau
email: bruno.saussereau@univ-fcomte.fr

Diagnostic checking in FARIMA models with uncorrelated but non-independent error terms

Keywords: AMS 2000 subject classifications: Primary 62M10, 62F03, 62F05, secondary 91B84, 62P05 Nonlinear processes, Long-memory processes, Weak FARIMA models, Least squares estimator, Box-Pierce and Ljung-Box portmanteau tests, Residual autocorrelations, Self-normalization

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Diagnostic checking in FARIMA models with uncorrelated but non-independent error terms

Introduction

To model the long memory phenomenon, a widely used model is the fractional autoregressive integrated moving average (FARIMA, for short) model (see for instance [START_REF] Granger | An introduction to long-memory time series models and fractional differencing[END_REF], [START_REF] Fox | Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series[END_REF], [START_REF] Dahlhaus | Efficient parameter estimation for self-similar processes[END_REF], [START_REF] Hosking | Fractional differencing[END_REF], [START_REF] Beran | Long-memory processes[END_REF], [START_REF] Palma | Long-memory time series[END_REF], among others). This model plays an important role in many scientific disciplines and applied fields such as hydrology, climatology, economics, finance, to name a few.

We consider a centered stationary process X := (X t ) t∈Z satisfying a FARIMA(p, d 0 , q) representation of the form a(L)(1 -L)

d 0 X t = b(L) t , (1)
where d 0 is the long memory parameter, L stands for the back-shift operator and a(L) = 1p i=1 a i L i , respectively b(L) = 1 -q i=1 b i L i , is the autoregressive, respectively the moving average, operator. These operators represent the short memory part of the model (by convention a 0 = b 0 = 1). In the standard situation := ( t ) t∈Z is assumed to be a sequence of independent and identically distributed (iid for short) random variables with zero mean and with a common variance. In this standard framework, is said to be a strong white noise and the representation (1) is called a strong FARIMA(p, d 0 , q) process. In contrast with this previous definition, the representation (1) is said to be a weak FARIMA(p, d 0 , q) if the noise process is a weak white noise, that is, if it satisfies (A0): E( t ) = 0, Var ( t ) = σ 2 0 and Cov ( t , t-h ) = 0 for all t ∈ Z and all h = 0. A strong white noise is obviously a weak white noise because independence entails uncorrelatedness. Of course the converse is not true. The strong FARIMA model was introduced by [START_REF] Hosking | Fractional differencing[END_REF]. The particular strong FARIMA(0, d 0 , 0) process was discussed by [START_REF] Granger | An introduction to long-memory time series models and fractional differencing[END_REF]. To ensure the stationarity and the invertibility of the model defined by (1), we assume that -1/2 < d 0 < 1/2 and all roots of a(z)b(z) = 0 are outside the unit disk (see [START_REF] Granger | An introduction to long-memory time series models and fractional differencing[END_REF] and [START_REF] Hosking | Fractional differencing[END_REF] for details). It is also assumed that a(z) and b(z) have no common factors in order to insure unique identifiability of the parameters.

The validity of the different steps of the traditional methodology of Box and Jenkins (identification, estimation and validation) depends on the noise properties. After estimating the FARIMA process, the next important step in the modeling consists in checking if the estimated model fits satisfactorily the data. Thus, under the null hypothesis that the model has been correctly identified, the residuals (ˆ t ) are approximately a white noise. This adequacy checking step allows to validate or invalidate the choice of the orders p and q. The choice of p and q is particularly important because the number of parameters (p + q + 1) quickly increases with p and q, which entails statistical difficulties. In particular, the selection of too large orders p and q may introduce terms that are not necessarily relevant in the model. Conversely, the selection of too small orders p and q causes loss of some information, that can be detected by the correlation of the residuals.

Thus it is important to check the validity of a FARIMA(p, d 0 , q) model, for given orders p and q. Based on the residual empirical autocorrelation, [START_REF] Box | Distribution of residual autocorrelations in autoregressiveintegrated moving average time series models[END_REF] have proposed a goodness-of-fit test, the so-called portmanteau test, for strong ARMA models. The intuition behind these portmanteau tests is that if a given time series model with iid innovation is appropriate for the data at hand, the autocorrelations of the residuals ˆ t should be close to zero, which is the theoretical value of the autocorrelations of t (see Assumption (A0) below). A modification of the test of [START_REF] Box | Distribution of residual autocorrelations in autoregressiveintegrated moving average time series models[END_REF] has been proposed by [START_REF] Ljung | On a measure of lack of fit in time series models[END_REF] which is nowadays one of the most popular diagnostic checking tools in strong ARMA modeling of time series. A modified portmanteau test statistic was proposed by [START_REF] Li | Fractional time series modelling[END_REF] for checking the overall significance of the residual autocorrelations of a strong FARIMA(p, d 0 , q) model. All these above test statistics have been obtained under the iid assumption on the noise and they may be invalid when the series is uncorrelated but dependent (see [START_REF] Romano | Inference for autocorrelations under weak assumptions[END_REF], Francq et al. (2005), Boubacar [START_REF] Boubacar Maïnassara | Diagnostic Checking in Multivariate ARMA Models With Dependent Errors Using Normalized Residual Autocorrelations[END_REF], [START_REF] Zhu | A bootstrapped spectral test for adequacy in weak ARMA models[END_REF], [START_REF] Lobato | Testing for autocorrelation using a modified box-pierce q test[END_REF], [START_REF] Lobato | Testing for zero autocorrelation in the presence of statistical dependence[END_REF], [START_REF] Wang | A simple asymptotically f-distributed portmanteau test for diagnostic checking of time series models with uncorrelated innovations[END_REF], to name a few).

As mentioned above, the works on the portmanteau statistic are generally performed under the assumption that the errors t are independent (see for instance [START_REF] Li | Fractional time series modelling[END_REF]). This independence assumption is often considered too restrictive by practitioners. It precludes conditional heteroscedasticity and/or other forms of nonlinearity (see Francq and Zakoïan (2005) for a review on weak univariate ARMA models) which can not be generated by FARIMA models with iid noises. 1Relaxing this independence assumption allows to cover linear representations of general nonlinear processes and to extend the range of application of the FARIMA models.

This paper is devoted to the problem of the validation step of weak FARIMA processes. For the asymptotic theory of weak FARIMA model validation, recently [START_REF] Shao | Testing for white noise under unknown dependence and its applications to diagnostic checking for time series models[END_REF] studied the diagnostic checking for long memory time series models with nonparametric conditionally heteroscedastic martingale difference errors. This author also generalized the test statistic based on the kernelbased spectral proposed by [START_REF] Hong | Consistent testing for serial correlation of unknown form[END_REF] under weak assumptions on the innovation process. Note also that [START_REF] Ling | On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity[END_REF] have studied the [START_REF] Box | Distribution of residual autocorrelations in autoregressiveintegrated moving average time series models[END_REF] type test for FARIMA-GARCH models by assuming a parametric form for the GARCH model.

To our knowledge, it does not exist any diagnostic checking methodology for FARIMA models when the (possibly dependent) error is subject to unknown conditional heteroscedasticity. We think that this is due to the difficulty that arises when one has to estimate the asymptotic covariance matrix of the parameter estimates. In our paper, thanks to the asymptotic results obtained by Boubacar [START_REF] Boubacar Maïnassara | Estimating FARIMA models with uncorrelated but non-independent error terms[END_REF], we are able to extend for weak FARIMA models the diagnostic checking methodology proposed by Francq et al. (2005) as well as the self-normalized approach proposed by Boubacar [START_REF] Boubacar Maïnassara | Diagnostic Checking in Multivariate ARMA Models With Dependent Errors Using Normalized Residual Autocorrelations[END_REF].

The paper is organized as follows. In Section 2, we recall the results on the least squares estimator asymptotic distribution of weak FARIMA models obtained by Boubacar [START_REF] Boubacar Maïnassara | Estimating FARIMA models with uncorrelated but non-independent error terms[END_REF]. In Section 3, a modified version of the portmanteau test is proposed thanks to the investigation of the asymptotic distribution of the residual autocorrelations. Our first main result is stated in Theorem 2. The second main result of this section is obtained in Theorem 7 by means of a selfnormalized approach. Two examples are also proposed in Section B in order to illustrate our results. Some numerical illustrations are gathered in Section 4. They corroborate our theoretical work. An application to the Standard & Poor's 500 and Nikkei returns also illustrate the practical relevance of our theoretical results. All our proofs are given in Section A and figures and tables are brought together in Section 5.

Assumptions and estimation procedure

In this section, we recall the results on the least squares estimator asymptotic distribution of weak FARIMA models obtained by Boubacar [START_REF] Boubacar Maïnassara | Estimating FARIMA models with uncorrelated but non-independent error terms[END_REF] in order to have a self-containing paper.

Let Θ * be the parameter space

Θ * := (θ 1 , θ 2 , . . . , θ p+q ) ∈ R p+q , where a θ (z) = 1 - p i=1 θ i z i , and b θ (z) = 1 - q j=1
θ p+j z j have all their zeros outside the unit disk .

Denote by Θ the cartesian product

Θ * ×[d 1 , d 2 ], where [d 1 , d 2 ] ⊂ ]-1/2, 1/2[ with d 1 -d 0 > -1/2.
The unknown parameter of interest θ 0 = (a 1 , a 2 , . . . , a p , b 1 , b 2 , . . . , b q , d 0 ) is supposed to belong to the parameter space Θ.

The fractional difference operator (1 -L) d 0 is defined, using the generalized binomial series, by

(1 -L) d 0 = j≥0 α j (d 0 )L j ,
the smooth transition autoregressive (STAR), the exponential autoregressive (EXPAR), the bilinear, the random coefficient autoregressive (RCA), the functional autoregressive (FAR) (see [START_REF] Tong | Non-linear time series: a dynamical system approach[END_REF] and [START_REF] Fan | Nonlinear time series: nonparametric and parametric methods[END_REF], for references on these nonlinear time series models).

where for all j ≥ 0, α j (d 0 ) = Γ (jd 0 )/ {Γ (j + 1)Γ (-d 0 )} and Γ (•) is the Gamma function. Using the Stirling formula we obtain that for large j, α j (d 0 ) ∼ j -d 0 -1 /Γ (-d 0 ) (one refers to [START_REF] Beran | Long-memory processes[END_REF] for further details). For all θ ∈ Θ we define ( t (θ)) t∈Z as the second order stationary process which is the solution of

t (θ) = j≥0 α j (d)X t-j - p i=1 θ i j≥0 α j (d)X t-i-j + q j=1
θ p+j t-j (θ).

(2)

Observe that, for all t ∈ Z, t (θ 0 ) = t a.s. Given a realization X 1 , . . . , X n of length n, t (θ) can be approximated, for 0 < t ≤ n, by ˜ t (θ) defined recursively by

˜ t (θ) = t-1 j=0 α j (d)X t-j - p i=1 θ i t-i-1 j=0 α j (d)X t-i-j + q j=1 θ p+j ˜ t-j (θ), (3)
with ˜ t (θ) = X t = 0 if t ≤ 0.
As shown in of Boubacar [START_REF] Boubacar Maïnassara | Estimating FARIMA models with uncorrelated but non-independent error terms[END_REF], these initial values are asymptotically negligible and in particular it holds that t (θ) -˜ t (θ) → 0 in L 2 as t → ∞. Thus the choice of the initial values has no influence on the asymptotic properties of the model parameters estimator. Let Θ * δ denotes the compact set Θ * δ = θ ∈ R p+q ; the roots of the polynomials a θ (z) and b θ (z) have modulus ≥ 1 + δ . We define the set Θ δ as the cartesian product of

Θ * δ by [d 1 , d 2 ], i.e. Θ δ = Θ * δ × [d 1 , d 2 ],
where δ is a positive constant chosen such that θ 0 belongs to Θ δ .

The least squares estimator is defined, almost-surely, by

θn = argmin θ∈Θ δ Q n (θ), where Q n (θ) = 1 n n t=1 ˜ 2 t (θ). (4)
The asymptotic properties of this estimator are well known when the innovation process ( t ) t∈Z is a strong or a semi-strong white noise (see for instance [START_REF] Hualde | Gaussian pseudo-maximum likelihood estimation of fractional time series models[END_REF], [START_REF] Nielsen | Asymptotics for the conditional-sum-of-squares estimator in multivariate fractional time-series models[END_REF] and [START_REF] Cavaliere | Quasi-maximum likelihood estimation and bootstrap inference in fractional time series models with heteroskedasticity of unknown form[END_REF] who have considered the problem of conditional sum-of squares estimation with d 0 allowed to lie in an arbitrary large compact set). To ensure the consistency of the least squares estimator in our context, we assume as in Boubacar Maïnassara et al. ( 2019) that the parametrization satisfies the following condition.

(A1):

The process ( t ) t∈Z is strictly stationary and ergodic.

The consistency of the estimator is obtained under the assumptions (A0) and (A1). Additional assumptions are required in order to establish the asymptotic normality of the least squares estimator. We assume that θ 0 is not on the boundary of the parameter space Θ δ .

(A2):

We have θ 0 ∈

• Θ δ , where

• Θ δ denotes the interior of Θ δ .
The stationary process is not supposed to be an independent sequence. So one needs to control its dependency by means of its strong mixing coefficients {α (h)} h∈N defined by

α (h) = sup A∈F t -∞ ,B∈F ∞ t+h |P (A ∩ B) -P(A)P(B)| , where F t -∞ = σ( u , u ≤ t) and F ∞ t+h = σ( u , u ≥ t + h).
We shall need an integrability assumption on the moment of the noise and a summability condition on the strong mixing coefficients (α (h)) h≥0 .

(A3):

There exists an integer τ such that for some ν ∈]0, 1], we have

E| t | τ +ν < ∞ and ∞ h=0 (h + 1) k-2 {α (h)} ν k+ν < ∞ for k = 1, . . . , τ .
Note that (A3) implies the following weak assumption on the joint cumulants of the innovation process (see [START_REF] Doukhan | Cumulants for stationary mixing random sequences and applications to empirical spectral density[END_REF], for more details).

(A3'):

There exists an integer τ ≥ 2 such that

C τ := i 1 ,...,i τ -1 ∈Z |cum( 0 , i 1 , . . . , i τ -1 )| < ∞ .
In the above expression, cum( 0 , i 1 , . . . , i τ -1 ) denotes the τ -th order joint cumulant of the stationary process . Due to the fact that the t 's are centered, we notice that for fixed (i, j, k)

cum( 0 , i , j , k ) = E [ 0 i j k ] -E [ 0 i ] E [ j k ] -E [ 0 j ] E [ i k ] -E [ 0 k ] E [ i j ] .
Assumption (A3) is a usual technical hypothesis which is useful when one proves the asymptotic normality (see [START_REF] Francq | Estimating linear representations of nonlinear processes[END_REF] for example). Let us notice however that we impose a stronger convergence speed for the mixing coefficients than in the works on weak ARMA processes. This is due to the fact that the coefficients in the infinite AR or MA representation of t (θ) have no more exponential decay because of the fractional operator (see Subsection 6.1 in Boubacar Maïnassara et al. ( 2019) for details and comments).

As mentioned before, Hypothesis (A3) implies (A3') which is also a technical assumption usually used in the fractional ARIMA processes framework (see for instance [START_REF] Shao | A self-normalized approach to confidence interval construction in time series[END_REF][START_REF] Shao | Testing for white noise under unknown dependence and its applications to diagnostic checking for time series models[END_REF]) or even in an ARMA context (see [START_REF] Francq | HAC estimation and strong linearity testing in weak ARMA models[END_REF]; [START_REF] Zhu | A bootstrapped spectral test for adequacy in weak ARMA models[END_REF]).

For all t ∈ Z, let

H t (θ) = 2 t (θ) ∂ ∂θ t (θ) = 2 t (θ) ∂ ∂θ 1 t (θ), . . . , 2 t (θ) ∂ ∂θ p+q+1 t (θ) .
Remind that the sequence ( t (θ)) t∈Z is given by (2). Under the assumptions (A0), (A1), (A2) and (A3) with τ = 4, Boubacar Maïnassara et al. (2019) showed that θn → θ 0 in probability as n → ∞ and √ n( θn -θ 0 ) is asymptotically normal with mean 0 and covariance matrix Σ θ := J -1 I J -1 , where J = J(θ 0 ) and I = I (θ 0 ), with

I (θ) = +∞ h=-∞ Cov (H t (θ), H t-h (θ)) and J(θ) = 2E ∂ ∂θ t (θ) ∂ ∂θ t (θ) a.s.

Diagnostic checking in weak FARIMA models

After the estimation phase, the next important step consists in checking if the estimated model fits satisfactorily the data. In this section we derive the limiting distribution of the residual autocorrelations and that of the portmanteau statistics (based on the standard and the self-normalized approaches) in the framework of weak FARIMA models. For t ≥ 1, let êt = ˜ t ( θn ) be the least squares residuals. By (3) we notice that êt = 0 for t ≤ 0 and t > n. By (1) it holds that

êt = t-1 j=0 α j ( d) Xt-j - p i=1 θi t-i-1 j=0 α j ( d) Xt-i-j + q j=1
θp+j êt-j , for t = 1, . . . , n, with Xt = 0 for t ≤ 0 and Xt = X t for t ≥ 1.

For a fixed integer m ≥ 1 consider the vector of residual autocovariances γm = (γ(1), . . . , γ(m)) where γ

(h) = 1 n n t=h+1 êt êt-h for 0 ≤ h < n.
In the sequel we will also need the vector of the first m sample autocorrelations ρm = (ρ(1), . . . , ρ(m)) where ρ(h) = γ(h)/γ(0).

Since the papers by [START_REF] Box | Distribution of residual autocorrelations in autoregressiveintegrated moving average time series models[END_REF] and [START_REF] Ljung | On a measure of lack of fit in time series models[END_REF], portmanteau tests have been popular diagnostic checking tools in the ARMA modeling of time series. Based on the residual empirical autocorrelations, their test statistics are defined respectively by

Q bp m = n m h=1 ρ2 (h) and Q lb m = n(n + 2) m h=1 ρ2 (h) n -h . ( 5
)
These statistics are usually used to test the null hypothesis (H0): (X t ) t∈Z satisfies a FARIMA(p, d 0 , q) representation;

against the alternative (H1): (X t ) t∈Z does not admit a FARIMA representation or admits a FARIMA(p , d 0 , q ) representation with p > p or q > q.

These tests are very useful tools to check the global significance of the residual autocorrelations.

Asymptotic distribution of the residual autocorrelations

First of all, the mixing assumption (A3) will entail the asymptotic normality of the "empirical" autocovariances

γ m = (γ(1), . . . , γ(m)) where γ(h) = 1 n n t=h+1 t t-h for 0 ≤ h < n. (6)
It should be noted that γ(h) is not a computable statistic because it depends on the unobserved innovations t = t (θ 0 ). They are introduced as a device to facilitate future derivations. Let Ψ m be the m × (p + q + 1) matrix defined by

Ψ m = E         t-1 . . . t-m    ∂ t ∂θ      . ( 7
)
By a Taylor expansion of √ nγ m , one should prove that (see Section A.3)

√ nγ m = √ nγ m + Ψ m √ n θn -θ 0 + o P (1), (8)
where Ψ m is given in (7). We shall also prove (see Section A.3 again) that

√ nρ m = √ n γm σ 2 + o P (1). (9)
Thus from (9) the asymptotic distribution of the residual autocorrelations √ nρ m depends on the distribution of γm . In view of (8) the asymptotic distribution of the residual autocovariances √ nγ m will be obtained from the joint asymptotic behavior of √ n( θ n -θ 0 , γ m ) .

In view of Theorem 1 in Boubacar Maïnassara et al. ( 2019) and (A2), we have θn → θ 0 ∈

• Θ in probability. Thus ∂Q n ( θn )/∂θ = 0 for sufficiently large n and a Taylor expansion gives

√ n ∂ ∂θ O n (θ 0 ) + J(θ 0 ) √ n( θn -θ 0 ) = o P (1), (10)
where

O n (θ) = n -1 n t=1 2 t (θ)
and the sequence ( t (θ)) t∈Z is given by (2). The equation ( 10) is proved in Boubacar Maïnassara et al. ( 2019) (see the proof of Theorem 2). Consequently from (10) we have

√ n( θn -θ 0 ) = - 2 √ n n t=1 J -1 (θ 0 ) t (θ 0 ) ∂ t (θ 0 ) ∂θ + o P (1) . ( 11
)
For integers m, m ≥ 1, one needs the matrix Γ m,m = [Γ ( , )] 1≤ ≤m,1≤ ≤m where

Γ ( , ) = ∞ h=-∞ E t t-t-h t-h-.
The existence of Γ ( , ) will be justified in Lemma 3 of the appendix.

Proposition 1. Under the assumptions (A0), (A1), (A2) and (A3) with τ = 4, the random vector √ n θn -θ 0 , γ m has a limiting centered normal distribution with covariance matrix

Ξ =   Σ θ Σ θ,γm Σ θ,γm Γ m,m   = ∞ h=-∞ E U t U t-h , (12)
where from (6) and (11) we have

U t = U 1t U 2t = -2J -1 (θ 0 ) t (θ 0 ) ∂ ∂θ t (θ 0 ) ( t-1 , . . . , t-m ) t . ( 13
)
The proof of the proposition is given in Subsection A.2 of the appendix.

The following theorem which is an extension of the result given in Francq et al. (2005) provides the limit distribution of the residual autocovariances and autocorrelations of weak FARIMA models.

Theorem 2. Under the assumptions of Proposition 1, we have

√ nγ m D -→ n→∞ N (0, Σ γm ) where Σ γm = Γ m,m + Ψ m Σ θΨ m + Ψ m Σ θ,γm + Σ θ,γm Ψ m (14) and √ nρ m D -→ n→∞ N (0, Σ ρm ) where Σ ρm = 1 σ 4 Σ γm . (15)
The detailed proof of this result is postponed to the Subsection A.3 of Appendix.

Remark 1. It is clear from Theorem 2 that for a given FARIMA(p, d 0 , q) model, the asymptotic distribution of the residual autocorrelations depends only on the noise distribution through the quantities Γ ( , ) (which depends on the fourth-order structure of the noise). It is also worth noting that this asymptotic distribution depends on the asymptotic normality of the least squares estimator of the FARIMA(p, d 0 , q) only through the matrix Σ θ.

Remark 2. In the standard strong FARIMA case, i.e. when (A1) is replaced by the assumption that ( t ) t∈Z is iid, Boubacar Maïnassara et al. ( 2019) have showed in Remark 2 that I (θ 0 ) = 2σ 2 J(θ 0 ). Thus the asymptotic covariance matrix is then reduced as Σ θ = 2σ 2 J -1 (θ 0 ). In the strong case, we also have:

Γ ( , ) = 0 when = and Γ ( , ) = σ 4 . Thus Γ m,m is reduced as Γ m,m = σ 4 I m ,
where I m denotes the m × m identity matrix. Because Σ θ = 2σ 2 J -1 (θ 0 ) we obtain that

Σ θ,γm = -2 ∞ h=-∞ E t J -1 (θ 0 ) ∂ t (θ 0 ) ∂θ         t-1-h . . . t-m-h    t-h      = -2σ 2 J -1 (θ 0 )      E       t-1 . . . t-m    ∂ t (θ 0 ) ∂θ         = -Σ θΨ m .
We denote by Σ s γm and Σ s ρm the asymptotic variances obtained respectively in ( 14) and (15) for the strong FARIMA case. Thus we obtain, in the strong case, the following simpler expressions

Σ s γm = σ 4 I m -2σ 2 Ψ m J -1 (θ 0 )Ψ m and Σ s ρm = I m - 2 σ 2 Ψ m J -1 (θ 0 )Ψ m ,
which are the matrices obtained by [START_REF] Li | Fractional time series modelling[END_REF].

To validate a FARIMA(p, d 0 , q) model, the most basic technique is to examine the autocorrelation function of the residuals. Theorem 2 can be used to obtain asymptotic significance limits for the residual autocorrelations. However, the asymptotic variance matrices Σ γm and Σ ρm depend on the unknown matrices Ξ , Ψ m and the positive scalar σ 2 which need to be estimated. This is the purpose of the following discussion.

Modified version of the portmanteau test

From Theorem 2 we can deduce the following result, which gives the limiting distribution of the standard portmanteau statistics (5) under general assumptions on the innovation process of the fitted FARIMA(p, d 0 , q) model. Theorem 3. Under the assumptions of Theorem 2 and (H0), the statistics Q bp m and Q lb m defined by (5) converge in distribution, as n → ∞, to

Z m (ξ m ) = m k=1 ξ k,m Z 2 k ,
where ξ m = (ξ 1,m , . . . , ξ m,m ) is the vector of the eigenvalues of the matrix Σ ρm = σ -4 Σ γm and Z 1 , . . . , Z m are independent N (0, 1) variables.

It is possible to evaluate the distribution of a quadratic form of a Gaussian vector by means of the Imhof algorithm (see [START_REF] Imhof | Computing the distribution of quadratic forms in normal variables[END_REF]).

Remark 3. In view of remark 2 when m is large, Σ s ρm I m -2σ -2 Ψ m J -1 (θ 0 )Ψ m is close to a projection matrix. Its eigenvalues are therefore equal to 0 and 1. The number of eigenvalues equal to 1 is Tr(I m -2σ -2 Ψ m J -1 (θ 0 )Ψ m ) = Tr(I m-(p+q+1) ) = m-(p+q+1) and p+q+1 eigenvalues equal to 0, Tr(•) denotes the trace of a matrix. Therefore we retrieve the well-known result obtained by [START_REF] Li | Fractional time series modelling[END_REF]. More precisely, under (H0) and in the strong FARIMA case, the asymptotic distributions of the statistics Q bp m and Q lb m are approximated by a X 2 m-(p+q+1) , where m > p +q +1 and X 2 k denotes the chi-squared distribution with k degrees of freedom. Theorem 3 shows that this approximation is no longer valid in the framework of weak FARIMA(p, d, q) models and that the asymptotic null distributions of the statistics Q bp m and Q lb m are more complicated. The limit distribution Z m (ξ m ) depends on the nuisance parameter σ 2 , the matrix Ψ m and the elements of Ξ . Therefore, the asymptotic distribution of the portmanteau statistics (5), under weak assumptions on the noise, requires a computation of a consistent estimator of the asymptotic covariance matrix Σ ρm . The m × (p + q + 1) matrix Ψ m and the noise variance σ 2 can be estimated by its empirical counterpart. Thus we may use

Ψm = 1 n n t=1 (ê t-1 , . . . , êt-m ) ∂ê t ∂θ and σ2 = γ(0) = 1 n n t=1 ê2 t .
A consistent estimator of Ξ is obtained by means of an autoregressive spectral estimator, as in Boubacar Maïnassara et al. ( 2019) (see also [START_REF] Berk | Consistent autoregressive spectral estimates[END_REF][START_REF] Boubacar Mainassara | Computing and estimating information matrices of weak ARMA models[END_REF] and den [START_REF] Den Haan | A practitioner's guide to robust covariance matrix estimation[END_REF], to name a few for a more comprehensive exposition of this method).

The stationary process (U t ) t∈Z admits the Wold decomposition

U t = v t + ∞ i=1 i v t-i ,
where (v t ) t∈Z is a (p + q + 1 + m)-variate weak white noise. Assume that the covariance matrix

Σ v := Var(v t ) is non-singular, that ∞ i=1 i < ∞
, where • denotes any norm on the space of the real (p + q + 1 + m) × (p + q + 1 + m) matrices, and that det I p+q+1+m + ∞ i=1 i z i = 0 if |z| ≤ 1. Then (U t ) t∈Z admits an AR(∞) representation (see [START_REF] Akutowicz | On an explicit formula in linear least squares prediction[END_REF]) of the form

∆(L)U t := U t - ∞ i=1 ∆ i U t-i = v t , ( 16
) such that ∞ i=1 ∆ i < ∞ and det {∆(z)} = 0 if |z| ≤ 1.
In view of ( 12), the matrix Ξ can be interpreted as 2π times the spectral density of the stationary process (U t ) t∈Z = ((U 1t , U 2t ) ) t∈Z evaluated at frequency 0 (see p. 459 of [START_REF] Brockwell | Time series: theory and methods[END_REF]). We then obtain that

Ξ = ∆ -1 (1)Σ v ∆ -1 (1)
Since U t is unobservable, we introduce Ût ∈ R p+q+1+m obtained by replacing t (θ 0 ) by ˜ t ( θn ) and J(θ 0 ) by its empirical or observable counterpart Ĵn in (13). Let ∆r (z) = I p+q+1+m -r k=1 ∆r,k z k , where ∆r,1 , . . . , ∆r,r denote the coefficients of the least squares regression of Ût on Ût-1 , . . . , Ût-r . Let vr,t be the residuals of this regression, and let Σvr be the empirical variance of vr,1 , . . . , vr,n . We are now able to state Theorem 4 which is an extension of a result given in Boubacar [START_REF] Boubacar Mainassara | Computing and estimating information matrices of weak ARMA models[END_REF].

Theorem 4. We assume (A0), (A1), (A2) and Assumption (A3') with τ = 8. In addition, we assume that the innovation process ( t ) t∈Z of the FARIMA(p, d 0 , q) model (1) is such that the process (U t ) t∈Z defined in (13) admits a multivariate AR(∞) representation (16), where ∆ i = o(i -2 ) as i → ∞, the roots of det(∆(z)) = 0 are outside the unit disk, and Σ v = Var(v t ) is non-singular. Then the spectral estimator of Ξ satisfies

Ξ SP n := ∆-1 r (1) Σvr ∆ -1 r (1) - → Ξ = ∆ -1 (1)Σ v ∆ -1 (1) in probability when r = r (n) → ∞ and r 5 (n)/n 1-2(d 0 -d 1 ) → 0 as n → ∞ (remind that d 0 ∈ [d 1 ,d 2 ] ⊂] -1/2,1/2[).
The proof of this theorem is similar to the proof of Theorem 3 in Boubacar Maïnassara et al. ( 2019) and it is omitted.

We are now in a position to define the modified versions of the Box-Pierce (BP) and Ljung-Box (LB) goodness-of-fit portmanteau tests. The standard versions of the portmanteau tests are useful tools to detect if the orders p and q of a FARIMA(p, d 0 , q) model are well chosen, provided the error terms ( t ) t∈Z of the FARIMA(p, d 0 , q) equation be a strong white noise and provided the number m of residual autocorrelations is not too small (see Remark 3). Now we define the modified versions which are aimed to detect if the orders p and q of a weak FARIMA(p, d 0 , q) model are well chosen. These tests are also asymptotically valid for strong FARIMA(p, d 0 , q) even for small m. The modified versions of the portmanteau tests will be denoted by BP w and LB w , the subscript w referring to the term weak.

Let Σρm be the matrix obtained by replacing Ξ by Ξ and σ 2 by σ2 in Σ ρm . Denote by ξm = ( ξ1,m , . . . , ξm,m ) the vector of the eigenvalues of Σρm . At the asymptotic level α, it holds under the assumptions of Theorem 2 and (H0) that

lim n→∞ P (Q bp m > S m (1 -α)) = lim n→∞ P (Q lb m > S m (1 -α)) = α,
where S m (1-α) is such that P(Z m ( ξm ) > S m (1-α)) = α. We emphasize the fact that the proposed modified versions of the Box-Pierce and Ljung-Box statistics are more difficult to implement because their critical values have to be computed from the data while the critical values of the standard method are simply deduced from a χ 2 -table. We shall evaluate the p-values

P Z m ( ξm ) > Q bp m and P Z m ( ξm ) > Q lb m , with Z m ( ξm ) = m i=1 ξi,m Z 2 i ,
by means of the Imhof algorithm (see [START_REF] Imhof | Computing the distribution of quadratic forms in normal variables[END_REF]).

A second method avoiding the estimation of the asymptotic matrix is proposed in the next Subsection.

Self-normalized asymptotic distribution of the residual autocorrelations

In view of Theorem 3, the asymptotic distributions of the statistics defined in (5) are a mixture of chi-squared distributions, weighted by eigenvalues of the asymptotic covariance matrix Σ ρm of the vector of autocorrelations obtained in Theorem 2. However, this asymptotic variance matrix depends on the unknown matrices Ξ , Ψ m and the noise variance σ 2 . Consequently, in order to obtain a consistent estimator of the asymptotic covariance matrix Σ ρm of the residual autocorrelations vector we have used an autoregressive spectral estimator of the spectral density of the stationary process (U t ) t∈Z to get a consistency estimator of the matrix Ξ (see Theorem 4). However, this approach presents the problem of choosing the truncation parameter. Indeed this method is based on an infinite autoregressive representation of the stationary process (U t ) t∈Z (see ( 16)). So the choice of the order of truncation is crucial and difficult.

In this section, we propose an alternative method where we do not estimate an asymptotic covariance matrix which is an extension to the results obtained by Boubacar [START_REF] Boubacar Maïnassara | Diagnostic Checking in Multivariate ARMA Models With Dependent Errors Using Normalized Residual Autocorrelations[END_REF]. It is based on a self-normalization approach to construct a test-statistic which is asymptotically distribution-free under the null hypothesis. This approach has been studied by Boubacar [START_REF] Boubacar Maïnassara | Diagnostic Checking in Multivariate ARMA Models With Dependent Errors Using Normalized Residual Autocorrelations[END_REF] in the weak ARMA case, by proposing new portmanteau statistics. In this case the critical values are not computed from the data since they are tabulated by [START_REF] Lobato | Testing that a dependent process is uncorrelated[END_REF]. In some sense this method is finally closer to the standard method in which the critical values are simply deduced from a X 2 -table. The idea comes from [START_REF] Lobato | Testing that a dependent process is uncorrelated[END_REF] and has been already extended by Boubacar Maïnassara and Saussereau (2018), [START_REF] Kuan | Robust M tests without consistent estimation of the asymptotic covariance matrix[END_REF], [START_REF] Shao | A self-normalized approach to confidence interval construction in time series[END_REF], Shao (2010a) and [START_REF] Shao | Parametric inference in stationary time series models with dependent errors[END_REF] to name a few in more general frameworks. See also [START_REF] Shao | Self-normalization for time series: a review of recent developments[END_REF] for a review on some recent developments on the inference of time series data using the self-normalized approach.

Other alternative methods that avoid the estimation of the covariance of the parameter estimates by directly eliminating the estimation effect of the test statistics can be found in [START_REF] Delgado | An asymptotically pivotal transform of the residuals sample autocorrelations with application to model checking[END_REF] or [START_REF] Velasco | A joint portmanteau test for conditional mean and variance time-series models[END_REF]. [START_REF] Delgado | An asymptotically pivotal transform of the residuals sample autocorrelations with application to model checking[END_REF] developed an asymptotically distribution-free transform of the sample autocorrelations of residuals in general parametric linear time-series models and shown that the proposed Box-Pierce-type test statistic based on the transformed autocorrelation is not affected by the estimation effect. [START_REF] Velasco | A joint portmanteau test for conditional mean and variance time-series models[END_REF] proposed an asymptotic simultaneous distribution-free transform of the sample autocorrelations of standardized residuals and their squares, which extended the approach developed by [START_REF] Delgado | An asymptotically pivotal transform of the residuals sample autocorrelations with application to model checking[END_REF] to the conditional mean and variance models diagnosis.

We denote by Λ the block matrix of R m×(p+q+1+m) defined by Λ = (Ψ m |I m ). In view of ( 8) and ( 11) we deduce that

√ nγ m = 1 √ n n t=1 ΛU t + o P (1).
At this stage, we do not rely on the classical method that would consist in estimating the asymptotic covariance matrix Ξ . We rather try to apply Lemma 1 in [START_REF] Lobato | Testing that a dependent process is uncorrelated[END_REF]. So we need to check that a functional central limit theorem holds for the process U := (U t ) t≥1 . For that sake, we define the normalization matrix C m of R m×m by

C m = 1 n 2 n t=1 S t S t where S t = t j=1 (ΛU j -γ m ) .
To ensure the invertibility of the normalization matrix C m (it is the result stated in the next proposition), we need the following technical assumption on the distribution of t .

(A4):

The process ( t ) t∈Z has a positive density on some neighbourhood of zero.

Proposition 5. Under the assumptions of Theorem 2 and (A4), the matrix C m is almost surely non singular.

The proof of this proposition is given in Subsection A.4 of the appendix.

Let (B K (r )) r ≥0 be a K -dimensional Brownian motion starting from 0. For K ≥ 1, we denote by U K the random variable defined by:

U K = B K (1)V -1 K B K (1), (17)
where

V K = 1 0 (B K (r ) -r B K (1)) (B K (r ) -r B K (1)) dr. ( 18
)
The critical values of U K have been tabulated by [START_REF] Lobato | Testing that a dependent process is uncorrelated[END_REF].

The following theorem states the asymptotic distributions of the sample autocovariances and autocorrelations.

Theorem 6. Under the assumptions of Theorem 2, (A4) and under the null hypothesis (H0) we have

nγ m C -1 m γm in law ---→ n→∞ U m and nσ 4 ρ m C -1 m ρm in law ---→ n→∞ U m .
The proof of this theorem is given in Subsection A.5 of Appendix.

Of course, the above theorem is useless for practical purpose because the normalization matrix C m and the nuisance parameter σ 2 are not observable. This gap will be fixed below (see Theorem 7) when one replaces the matrix C m and the scalar σ 2 by their empirical or observable counterparts. Then we denote

Ĉm = 1 n 2 n t=1 Ŝt Ŝ t where Ŝt = t j=1 ΛÛ j -γm , with Λ = ( Ψm |I m )
and where Ût and σ2 are defined in Subsection 3.2. The above quantities are all observable and the following result is the applicable counterpart of Theorem 6.

Theorem 7. Under the assumptions of Theorem 6, we have

nγ m Ĉ -1 m γm in law ---→ n→∞ U m and Q sn m = nσ 4 ρ m Ĉ -1 m ρm in law ---→ n→∞ U m .
The proof of this result is postponed in Subsection A.6 of Appendix. Based on the above result, we propose a modified version of the Ljung-Box statistic when one uses the statistic

Qsn m = nσ 4 ρ m D 1/2 n,m Ĉ -1 m D 1/2 n,m ρm , where D n,m ∈ R m×m is diagonal with (n + 2)/(n -1), . . . , (n + 2)/(n -m) as diagonal terms.
These modified versions of the portmanteau tests will be denoted by BP sn and LB sn , the subscript sn referring to the term self-normalized.

Numerical illustrations

In this section, by means of Monte Carlo experiments, we investigate the finite sample properties of the asymptotic results that we introduced in this work. The numerical illustrations of this section are made with the open source statistical software R (see http://cran.r-project.org/).

Simulation studies and empirical sizes

We study numerically the behavior of the least squares estimator for FARIMA models of the form

(1 -L) d 0 (X t -aX t-1 ) = t -b t-1 , (19)
where the unknown parameter is θ 0 = (a, b, d 0 ). First we assume that in (19) the innovation process ( t ) t∈Z is an iid centered Gaussian process with common variance 1 which corresponds to the strong FARIMA case. For the weak FARIMA case, we consider that in ( 19) the innovation process ( t ) t∈Z follows firstly a GARCH(1, 1) process given by the model

t = σ t η t σ 2 t = ω + α 1 2 t-1 + β 1 σ 2 t-1 , (20)
with ω > 0, α 1 ≥ 0 and where (η t ) t∈Z is a sequence of iid centered Gaussian random variables with variance 1. Secondly we consider that in (19) a noise defined by

t = η 2 t η t-1 . (21)
The example ( 21) is an extension of a noise process in [START_REF] Romano | Inference for autocorrelations under weak assumptions[END_REF]. Contrary to the GARCH(1, 1) process, the noise defined in Equation ( 21) is not a martingale difference sequence for which the limit theory is more classical.

We simulate N = 1, 000 independent trajectories of size n = 10, 000 of models ( 19). The same series is partitioned as three series of sizes n = 1, 000, n = 5, 000 and n = 10, 000. For each of these N replications, we use the least squares estimation method to estimate the coefficient θ 0 and we apply portmanteau tests to the residuals for different values of m ∈ {1, 2, 3, 6, 12, 15}, where m is the number of autocorrelations used in the portmanteau test statistic. For the nominal level α = 5%, the empirical size over the N independent replications should vary between the significant limits 3.6% and 6.4% with probability 95%. When the relative rejection frequencies are outside the 95% significant limits, they are displayed in bold type in Tables 1, 2 and3. For the standard Box-Pierce test, the model is therefore rejected when the statistic Q bp m or Q lb m is larger than χ 2 (m-p-q-1) (0.95) in a FARIMA(p, d 0 , q) case (see [START_REF] Li | Fractional time series modelling[END_REF]). Consequently the empirical size is not available (n.a.) for the statistic Q bp m or Q lb m because they are not applicable for m ≤ p + q + 1. For the proposed self-normalized test BP sn or LB sn , the model is rejected when the statistic Q sn m or Qsn m is larger than U m (0.95), where the critical values U K (0.95) (for K = 1, . . . , 20) are tabulated in Lobato (see Table 1 in Lobato (2001)).

Table 1 displays the relative rejection frequencies of the null hypothesis (H0) that the data generating process (DGP for short) follows a strong FARIMA(0, d 0 , 0) model ( 19), over the N independent replications. For all tests, the percentages of rejection belong globally to the confident interval with probabilities 95%, except for LB s and BP s (see Table 8). Now, we repeat the same experiments on two weak FARIMA models. As expected Tables 2 and3 show that the standard LB s or BP s test poorly performs in assessing the adequacy of these particular weak FARIMA models. Indeed, we observe that the observed relative rejection frequencies of LB s and BP s are definitely outside the significant limits. Thus we draw the conclusion that the error of the first kind is globally well controlled by all the tests in the strong case, but only by the proposed tests in the weak cases.

Empirical power

In this section, we repeat the same experiments as in Section 4.1 to examine the power of the tests for the null hypothesis of Model ( 19) with a = b = 0 (i.e. a FARIMA(0, d 0 , 0)) against the FARIMA(0, d 0 , 1) alternative defined by Model ( 19) with θ 0 = (0, b, d 0 ) and where the innovation process ( t ) t∈Z follows the two weak white noises introduced in Section 4.1.

For each of these N replications we fit a FARIMA(0, d 0 , 0) model ( 19) and perform standard andmodified tests based on m = 1, 2, 3, 6, 12 and15 residual autocorrelations. Tables 4 and5 compare the empirical powers of Model ( 19) with θ 0 = (0, 0.2, d 0 ) over the N independent replications. For these particular weak FARIMA models, we notice that the standard BP s and LB s and our proposed tests have very similar powers except for BP sn and LB sn when n = 5, 000.

In these Monte Carlo experiments, we illustrate that the proposed test statistics have reasonable finite sample performance. Under nonindependent errors, it appears that the standard test statistics are generally non reliable, overrejecting severely, while the proposed tests statistics offer satisfactory levels. Even for independent errors, they seem preferable to the standard ones when the number m of autocorrelations is small. Moreover, the error of first kind is well controlled. Contrarily to the standard tests based on BP s or LB s , the proposed tests can be used safely for m small.For all these above reasons, we think that the modified versions that we propose in this paper are preferable to the standard ones for diagnosing FARIMA models under nonindependent errors.

Illustrative example

We now consider an application to the daily log returns (also simply called the returns) of the Nikkei and Standard & Poor's 500 indices (S&P 500, for short). The returns are defined by r t = 100 log(p t /p t-1 ) where p t denotes the price index of the S&P 500 index at time t. The observations of the S&P 500 (resp. the Nikkei) index cover the period from January 3, 1950 to to February 14, 2019(resp. from January 5, 1965to February 14, 2019). The length of the series is n = 17, 391 (resp. n = 13, 319) for the S&P 500 (resp. the Nikkei) index. The data can be downloaded from the website Yahoo Finance: http://fr.finance.yahoo.com/.

In Financial Econometrics the returns are often assumed to be a white noise. In view of the so-called volatility clustering, it is well known that the strong white noise model is not adequate for these series (see for instance A long-range memory property of the stock market returns series was largely investigated by [START_REF] Ding | A long memory property of stock market returns and a new model[END_REF] which shown that there are more correlation beetwen power transformation of the absolute return |r t | v (v > 0) than returns themselves (see also [START_REF] Beran | Long-memory processes[END_REF], [START_REF] Palma | Long-memory time series[END_REF], [START_REF] Baillie | Analysing inflation by the fractionally integrated ARFIMA-GARCH model[END_REF] and [START_REF] Ling | On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity[END_REF]). We choose here the case where v = 2 which corresponds to the squared returns (r 2 t ) t≥1 process. The mean and the standard deviation of (r 2 t ) t≥1 are 0, 9347 and 5, 0036 (resp. 1, 6167 and 5, 4759) for the S&P 500 (resp. the Nikkei) index. Following a similar way as in [START_REF] Ling | Adaptive estimators and tests of stationary and nonstationary short-and longmemory ARFIMA-GARCH models[END_REF] we denote by (X t ) t≥1 the centered series of the squared returns, that is, X t = r 2 t -0, 9347 (resp. X t = r 2 t -1, 6167) for the S&P 500 (resp. the Nikkei) index. Figure 1 (resp. Figure 3) plots the returns and the sample autocorrelations of (X t ) t≥1 of the S&P 500 (resp. of the Nikkei). The centered squared returns (X ) t≥1 have significant positive autocorrelations at least up to lag 80 (see Figure 1 and Figure 3) which confirm the claim that stock market returns have long-term memory (see for instance [START_REF] Ding | A long memory property of stock market returns and a new model[END_REF], for more details).

We first fit a FARIMA(1, d 0 , 1) model defined in (19) to the process (X ) t≥1 of the S&P 500 and the Nikkei returns. Let θSP500

where the estimated asymptotic standard errors obtained from Σ θ := J -1 I J -1 (respectively the p-values), of the estimated parameters (first column), are given into brackets (respectively in parentheses). Note that for these series, the estimated coefficients |â n | and | bn | are smaller than one. This is in accordance with the assumptions that the power series a -1 θ and b -1 θ are well defined (remind that the moving average polynomial is denoted b θ and the autoregressive polynomials a θ ). We also observe that the estimated long-range dependence coefficients dn is significant for any reasonable asymptotic level and is inside ] -0.5, 0.5[. So we think that the assumption (A2) is satisfied and thus our asymptotic normality theorem on the residual autocorrelations can be applied.

Concerning the S&P 500, the estimators of the parameters a and b are significant whereas it is not the case for the Nikkei (see ( 22)). In the Nikkei case, the coefficients could reasonably be set to zero. So we adjust a FARIMA(0, d 0 , 0) for the squares of Nikkei returns and ( 22) is reduced as θNikkei n = 0.2132 [0.0259] (0.0000) and σ2 = 25.9793 × 10 -8 .

We thus apply portmanteau tests to the residuals of FARIMA(1, d 0 , 1) (resp. FARIMA(0, d 0 , 0)) model for the process (X ) t≥1 of S&P 500 (resp. of Nikkei). Table 6 (resp. Table 7) displays the statistics and the p-values of the standard and modified versions of BP and LB tests of model ( 19). From Tables 6 and7, we draw the conclusion that the strong FARIMA(1, 0.2338, 1) and FARIMA(0, 0.2132, 0) models are rejected but the weak FARIMA(1, 0.2338, 1) and FARIMA(0, 0.2132, 0) models are not rejected.

Figure 2 (resp. Figure 4) displays the residual autocorrelations and their 5% significance limits under the strong FARIMA and weak FARIMA assumptions. In view of Figures 2 and4, the diagnostic checking of residuals does not indicate any inadequacy for the proposed tests. All of the sample autocorrelations should lie between the bands (at 95%) shown as dashed lines (green color) and solid lines (red color) for the modified tests, while the horizontal dotted (blue color) for standard test indicate that strong FARIMA is not adequate. Figure 2 (resp. Figure 4) confirms the conclusions drawn from Table 6 (resp. Table 7). The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7.

Figures and tables

Table 1 Empirical size (in %) of the modified and standard versions of the LB and BP tests in the case of a strong FARIMA(0, d 0 , 0) defined by (19) with θ 0 = (0, 0, d 0 ). The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000. Empirical size (in %) of the modified and standard versions of the LB and BP tests in the case of a weak FARIMA(0, d 0 , 0) defined by (19) with θ 0 = (0, 0, d 0 ) and where ω = 0.4, α 1 = 0.3 and β 1 = 0.3 in (21).

The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000. Empirical size (in %) of the modified and standard versions of the LB and BP tests in the case of weak FARIMA(0, d 0 , 0) defined by ( 19)-( 21) with θ 0 = (0, 0, d 0 ). The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000. Empirical power (in %) of the modified and standard versions of the LB and BP tests in the case of a weak FARIMA(0, d 0 , 1) defined by (19) with θ 0 = (0., 0.2, d 0 ) and where ω = 0.4, α 1 = 0.3 and β 1 = 0.3 in (20). The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000. Empirical power (in %) of the modified and standard versions of the LB and BP tests in the case of a weak FARIMA(0, d 0 , 1) defined by ( 19)-( 21) with θ 0 = (0., 0.2, d 0 ). The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000. 99.9 99.9 99.9 99.9 0.05 n = 5, 000 3 69.1 69.1 99.9 99.9 99.9 99.9 6 75.9 75.9 99.9 99.9 99.9 99.9 12 71.9 71.4 99.9 99.9 99.9 99.9 15 68.5 68.0 99.9 99.9 99.9 99.9 The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7.

1
Appendix A: Supplemental material: Proofs

The following proofs are quite technical and are adaptations of the arguments used in Francq and Zakoïan (1998), Francq et al. (2005) and Boubacar Maïnassara and Saussereau (2018). The results of Boubacar Maïnassara et al. (2019) which will be needed for all the proofs are collected in the following Subsection A.1 in order to have a self-containing paper.

In all our proofs, K is a positive constant that may vary from line to line.

A.1. Preliminary results

In this subsection, we shall give some results on estimations of the coefficients of formal power series that will arise in our study. We begin by recalling the following properties on power series. If for |z| ≤ R, the power series f (z) = i≥0 a i z i and g (z) = i≥0 b i z i are well defined, then one has (f g )(z) = i≥0 c i z i is also well defined for |z| ≤ R with the sequence (c i ) i≥0 which is given by c = a * b where * denotes the convolution product between a and b defined by

c i = i k=0 a k b i-k = i k=0 a i-k b k .
We will make use of the Young inequality that states that if the sequence a ∈ r 1 and b ∈ r 2 and such that 1

r 1 + 1 r 2 = 1 + 1 r with 1 ≤ r 1 , r 2 , r ≤ ∞, then a * b r ≤ a r 1 × b r 2 .
Now we come back to the power series that arise in our context. Remind that for the true value of the parameter,

a θ 0 (L)(1 -L) d 0 X t = b θ 0 (L) t . (23)
Thanks to the assumptions on the moving average polynomials b θ and the autoregressive polynomials a θ , the power series a -1 θ and b -1 θ are well defined. Thus the functions t (θ) defined in (2) can be written as

t (θ) = b -1 θ (L)a θ (L)(1 -L) d X t (24) = b -1 θ (L)a θ (L)(1 -L) d-d 0 a -1 θ 0 (L)b θ 0 (L) t (25)
and if we denote γ(θ) = (γ i (θ)) i≥0 the sequence of coefficients of the power series b -1 θ (z)a θ (z)(1z) d , we may write for all t ∈ Z:

t (θ) = i≥0 γ i (θ)X t-i . (26)
In the same way, by (24) one has

X t = (1 -L) -d a -1 θ (L)b θ (L) t (θ)
and if we denote η(θ) = (η i (θ)) i≥0 the coefficients of the power series

(1 -z) -d a -1 θ (z)b θ (z) one has X t = i≥0 η i (θ) t-i (θ) . ( 27
)
We strength the fact that γ 0 (θ) = η 0 (θ) = 1 for all θ.

For large j, [START_REF] Hallin | Local asymptotic normality for regression models with long-memory disturbance[END_REF] have shown that uniformly in θ the sequences γ(θ) and η(θ)

satisfy ∂ k γ j (θ) ∂θ i 1 • • • ∂θ i k = O j -1-d {log(j)} k , for k = 0, 1, 2, 3, (28)
and

∂ k η j (θ) ∂θ i 1 • • • ∂θ i k = O j -1+d {log(j)} k , for k = 0, 1, 2, 3. (29)
One difficulty that has to be addressed is that (26) includes the infinite past (X t-i ) i≥0 whereas only a finite number of observations (X t ) 1≤t≤n are available to compute the estimators defined in (4). The simplest solution is truncation which amounts to setting all unobserved values equal to zero. Thus, for all θ ∈ Θ and 1 ≤ t ≤ n one defines

˜ t (θ) = t-1 i=0 γ i (θ)X t-i = i≥0 γ t i (θ)X t-i (30)
where the truncated sequence γ t (θ) = (γ t i (θ)) i≥0 is defined by

γ t i (θ) = γ i (θ) if 0 ≤ i ≤ t -1 , 0 otherwise.
Since our assumptions are made on the noise in (1), it will be useful to express the random variables t (θ) and its partial derivatives with respect to θ, as a function of ( t-i ) i≥0 . From ( 25), there exists a sequence λ(θ) = (λ i (θ)) i≥0 such that

t (θ) = ∞ i=0 λ i (θ) t-i (31)
where the sequence λ(θ) is given by the sequence of the coefficients of the power series b

-1 θ (z)a θ (z)(1- z) d-d 0 a -1 θ 0 (z)b θ 0 (z). Consequently λ(θ) = γ(θ) * η(θ 0 ) or, equivalently, λ i (θ) = i j=0 γ j (θ)η i-j (θ 0 ). (32)
As in [START_REF] Hualde | Gaussian pseudo-maximum likelihood estimation of fractional time series models[END_REF], it can be shown using Stirling's approximation that there exists a positive constant K such that

sup θ∈Θ δ |λ i (θ)| ≤ K sup d∈[d 1 ,d 2 ] i -1-(d-d 0 ) ≤ K i -1-(d 1 -d 0 ) . (33)
Equation ( 31) and Inequality (33) imply that for all θ ∈ Θ the random variable t (θ) belongs to L 2 , that the sequence ( t (θ)) t is an ergodic sequence and that for all t ∈ Z the function t (•) is a continuous function. We proceed in the same way as regard to the derivatives of t (θ). More precisely, for any θ ∈ Θ, t ∈ Z and 1 ≤ k, l ≤ p +q +1 there exists sequences

. λ k (θ) = ( . λ i,k (θ)) i≥1 and .. λ k,l (θ) = ( .. λ i,k,l (θ)) i≥1 such that ∂ t (θ) ∂θ k = ∞ i=1 . λ i,k (θ) t-i (34) ∂ 2 t (θ) ∂θ k ∂θ l = ∞ i=1 .. λ i,k,l (θ) t-i . (35)
Of course it holds that

. λ k (θ) = ∂γ(θ) ∂θ k * η(θ 0 ) and .. λ k,l (θ) = ∂ 2 γ(θ) ∂θ k ∂θ l * η(θ 0 ). Similarly we have ˜ t (θ) = ∞ i=0 λ t i (θ) t-i , ( 36
) ∂˜ t (θ) ∂θ k = ∞ i=1 . λ t i,k (θ) t-i , (37)
∂ 2 ˜ t (θ) ∂θ k ∂θ l = ∞ i=1 .. λ t i,k,l (θ) t-i , (38)
where

λ t (θ) = γ t (θ) * η(θ 0 ), . λ t k (θ) = ∂γ t (θ) ∂θ k * η(θ 0 ) and .. λ t k,l (θ) = ∂ 2 γ t (θ)
∂θ k ∂θ l * η(θ 0 ). In order to handle the truncation error t (θ)-˜ t (θ), one needs some information on the sequence λ(θ) -λ t (θ). In Boubacar Maïnassara et al. ( 2019) the following two lemmas are proved.

Lemma 1. For 2 ≤ r ≤ ∞ and 1 ≤ k, l ≤ p + q + 1, we have

λ (θ) -λ t (θ) r = O t -1+ 1 r -(d-max(d 0 ,0)) , . λ k (θ) - . λ t k (θ) r = O t -1+ 1 r -(d-max(d 0 ,0)) and .. λ k,l (θ) - .. λ t k,l (θ) r = O t -1+ 1 r -(d-max(d 0 ,0))
for any θ ∈ Θ δ if d 0 ≤ 0 and for θ with non-negative memory parameter d if d 0 > 0.

Remark 4. The above lemma implies that the sequence

. λ k (θ 0 ) - . λ t k (θ 0
) is bounded and more precisely there exists K such that

sup j≥1 . λ j,k (θ 0 ) - . λ t j,k (θ 0 ) ≤ K t 1+min(d 0 ,0) (39)
for any t ≥ 1 and any 1 ≤ k ≤ p + q + 1.

Remark 5. In order to prove our asymptotic results, it will be convenient to give an upper bound for the norms of the sequences introduced in Lemma 1 valid for any θ ∈ Θ δ . Since d 1 -d 0 > -1/2, Estimation (33) entails that for any r ≥ 2,

λ (θ) -λ t (θ) r = O t -1+ 1 r -(d 1 -d 0 ) , ∀θ ∈ Θ δ .
This can easily be seen since λ(θ)

-λ t (θ) r ≤ K ( i≥t i -r -r (d 1 -d 0 ) ) 1/r ≤ K t -1+1/r -(d 1 -d 0 ) .
As in [START_REF] Hallin | Local asymptotic normality for regression models with long-memory disturbance[END_REF], the coefficients . λ j,k (θ) and .. λ j,k,l (θ) are O(j -1-(d-d 0 )+ζ ) for any small enough ζ > 0, so we have

. λ k (θ) - . λ t k (θ) r = O t -1+ 1 r -(d 1 -d 0 )+ζ and .. λ k,l (θ) - .. λ t k,l (θ) r = O t -1+ 1 r -(d 1 -d 0 )+ζ
for any r ≥ 2, any 1 ≤ k, l ≤ p + q + 1 and all θ ∈ Θ δ .

Lemma 2. For any 2 ≤ r ≤ ∞, 1 ≤ k ≤ p + q + 1 and θ ∈ Θ, there exists a constant K such that we have

λ t k (θ) r ≤ K and . λ t k (θ) r ≤ K .

A.2. Proof of Proposition 1

First we remark that the asymptotic normality of the joint distribution of √ n( θ n -θ 0 , γ m ) can be established along the same lines as the proof of Theorem 2 in Boubacar Maïnassara et al. ( 2019). The detailed proof is omitted. From ( 6) and ( 11) we have

√ n θn -θ 0 γ m = 1 √ n n t=1 -2J -1 (θ 0 ) t ∂ ∂θ t (θ 0 ) ( t-1 , . . . , t-m ) t + o P (1) 0 m = 1 √ n n t=1 U t + o P (1),
where 0 m is the vector of R m×1 with zero components. It is clear that U t is a measurable function of t , t-1 , . . . Thus by using the same arguments as in Boubacar Maïnassara et al. ( 2019) (see proof of Theorem 2), the central limit theorem (CLT) for strongly mixing processes (U t ) t∈Z of [START_REF] Herrndorf | A functional central limit theorem for weakly dependent sequences of random variables[END_REF] implies that (1/ √ n) n t=1 U t has a limiting normal distribution with mean 0 and covariance matrix Ξ .

For i ≥ 1, we denote Λ i (θ 0 ) = ( . λ i,1 (θ 0 ), . . . , . λ i,p+q+1 (θ 0 )) . From (34) we deduce that

∂ t (θ 0 ) ∂θ = ∞ i=1 Λ i (θ 0 ) t-i . (40)
In view of ( 11) and ( 40), by applying the CLT for mixing processes we directly obtain

Σ θ = lim n→∞ Var 2J -1 1 √ n n t=1 t ∂ ∂θ t (θ 0 ) := J -1 I J -1 = 4J -1 ∞ , =1 Λ (θ 0 ) Λ (θ 0 ) ∞ h=-∞ E ( t t-t-h t--h ) J -1 = 4J -1 ∞ , =1 Λ (θ 0 ) Λ (θ 0 ) Γ ( , )J -1 ,
which gives the first block of the asymptotic covariance matrix of Proposition 1. By the stationarity of ( t ) t∈Z and Lebesgue's dominated convergence theorem, we obtain the ( , )-th entry of the matrix Γ m,m :

lim n→∞ Cov( √ nγ( ), √ nγ( )) = lim n→∞ 1 n n t= +1 n s= +1 E t t-s s- = ∞ h=-∞ E t t-t-h t-h-:= Γ ( , ).
We thus have Γ m,m = [Γ ( , )] 1≤ , ≤m . Finally, by the stationarity of ( t ) t∈Z and ( t ∂ t (θ 0 )/∂θ) t∈Z we have

Cov -2J -1 1 √ n n t=1 t ∂ ∂θ t (θ 0 ), √ nγ( ) = -2J -1 1 n n t=1 n t = +1 Cov t ∂ ∂θ t (θ 0 ), t t - = -2J -1 1 n n-1 h=-n+1 (n -|h|)Cov t ∂ t (θ 0 ) ∂θ , t-h t--h .
By the dominated convergence theorem and from ( 40), it follows that

lim n→∞ Cov -2J -1 1 √ n n t=1 t ∂ ∂θ t (θ 0 ), √ nγ( ) = -2J -1 ∞ h=-∞ Cov t ∂ ∂θ t (θ 0 ), t-h t--h = -2J -1 j≥1 Λ j (θ 0 ) ∞ h=-∞ E t t-j t-h t--h = -2J -1 j≥1 Λ j (θ 0 ) Γ (j, ) := Σ θ,γm (•, ).
It is clear that the existence of the above matrices is ensured by the existence of Γ ( , ) and

∞ , =1 Λ (θ 0 )Λ (θ 0 )Γ ( , )
. The proof will thus follow from Lemma 3 below. We now justify the existence of the Γ ( , ) and ∞ , =1 Λ (θ 0 )Λ (θ 0 )Γ ( , ) in the following result.

Lemma 3. Under the assumptions (A0) and (A3') with τ = 4, we have for ( , ) = (0, 0)

Γ ( , ) = ∞ h=-∞ E t t-t-h t-h-< ∞ (41) and ∞ , =1 Λ (θ 0 ) Λ (θ 0 ) Γ ( , ) < ∞. ( 42
)
Proof. Note that, for all h ∈ Z and all ( , ) = (0, 0) we have

E t t-t-h t-h- ≤ cum t , t-, t-h , t-h- + |E [ t t-]| E t-h t-h- + |E [ t t-h ]| E t-t-h- + E t t-h- |E [ t-t-h ]| .
Then, using the stationarity of ( t ) t∈Z , and under the assumptions (A0) and (A3') with τ = 4 it follows that

Γ ( , ) ≤ E 2 t 2 + ∞ h=-∞ cum 0 , -, -h , -h- ≤ K
which proves (41). Similarly, we obtain

∞ , =1 Λ (θ 0 ) Λ (θ 0 ) Γ ( , ) ≤ ∞ h=-∞ ∞ , =1 cum 0 , -, -h , -h- + E 2 t 2 ∞ =1 Λ (θ 0 ) 2 ≤ K
where we have used Lemma 2. The conclusion follows.

A.3. Proof of Theorem 2

The proof is divided in two steps.

A.3.1.

Step 1: Taylor's expansion of √ nγ m and √ nρ m

The aim of this step is to prove ( 8) and ( 9). First we prove that for h = 1, . . . , m

√ nγ(h) = √ nγ(h) + E t-h ∂ ∂θ t (θ 0 ) √ n θn -θ 0 + o P (1). ( 43
) A Taylor expansion of (1/ √ n) n t=1+h ˜ t (•)˜ t-h (•) around θ 0 gives √ nγ(h) = 1 √ n n t=1+h ˜ t (θ 0 )˜ t-h (θ 0 ) + 1 n n t=1+h Dt (θ * n ) √ n θn -θ 0 = √ nγ(h) + (E [D t (θ 0 )]) √ n θn -θ 0 + R n,h,1 + R n,h,2 + R n,h,3 ,
where

Dt (θ) = ∂˜ t (θ) ∂θ ˜ t-h (θ) + ˜ t (θ) ∂˜ t-h (θ) ∂θ , D t (θ 0 ) = ∂ t (θ 0 ) ∂θ t-h + t ∂ t-h (θ 0 ) ∂θ , R n,h,1 = 1 √ n n t=1+h {˜ t (θ 0 )˜ t-h (θ 0 ) -t (θ 0 ) t-h (θ 0 )} , R n,h,2 = 1 n n t=1+h Dt (θ * n ) -D t (θ 0 ) √ n θn -θ 0 , R n,h,3 = 1 n n t=1+h D t (θ 0 ) -E [D t (θ 0 )] √ n θn -θ 0 ,
and where θ * n is between θn and θ 0 . Using the orthogonality between t and any linear combination of the past values of t (in particular ∂ t-h /∂θ), we have

√ nγ(h) = √ nγ(h) + E t-h ∂ ∂θ t (θ 0 ) √ n θn -θ 0 + R n,h,1 + R n,h,2 + R n,h,3 . (44)
Thus, to obtain (43), we just need to prove that in (44) the sequences of random variables (R n,h,1 ) n≥1 , (R n,h,2 ) n≥1 and (R n,h,3 ) n≥1 converge in probability to 0. One of the three above term is easy to handle. Indeed, by the ergodic theorem, we have

n -1 n t=1+h D t (θ 0 ) -E [D t (θ 0 )] → 0 almost-surely as n → ∞.
Thus using the tightness of the sequence ( √ n( θn -θ 0 )) n , we deduce that R n,h,3 = o P (1). The proof of (43) will thus follow from Lemmas 4 and 5 in which the two others terms R n,h,1 and R n,h,2 are discussed. These lemmas are stated and proved hereafter (see subsections A.3.3 and A.3.4).

We now remark that in Equation ( 43), E[ t-h (∂ t (θ 0 )/∂θ )] is the line h of the matrix Ψ m ∈ R m×(p+q+1) defined by ( 7). So as h = 1, . . . , m, Equation ( 43) becomes

√ nγ m = √ nγ(1), . . . , √ nγ(m) = √ nγ m + Ψ m √ n θn -θ 0 + o P (1).
Therefore the Taylor expansion (8) of γm is proved. Now, it is clear that the asymptotic distribution of the residual autocovariances √ nγ m is related to the asymptotic behavior of √ n( θ n -θ 0 , γ m ) obtained in Subsection A.2. We come back to the vector ρm = (ρ(1), . . . , ρ(m)) . Note that from (43), we have √ n(γ(0) -γ(0)) = o P (1). Applying the CLT for mixing processes (see [START_REF] Herrndorf | A functional central limit theorem for weakly dependent sequences of random variables[END_REF]) to the process ( 2 t ) t∈Z , we obtain

√ n σ2 -σ 2 = 1 √ n n t=1 2 t -E[ 2 t ] + o P (1) in law ---→ n→∞ N 0, ∞ h=-∞ Cov 2 t , 2 t-h . So we have √ n(σ 2 -σ 2 ) = O P (1) and √ n(γ(0) -σ 2 ) = O P (1)
. Now, using ( 14) and the ergodic theorem, we have The joint asymptotic distribution of √ nγ m and √ n( θn -θ 0 ) shows that √ nγ m has a limiting normal distribution with mean zero and covariance matrix

n γ(h) γ(0) - γ(h) σ 2 = √ nγ(h) √ n σ 2 -γ(0) σ 2 γ(0) = O P (1), which means √ nρ(h) = √ nγ(h)/σ 2 + O P (n -1/2
lim n→∞ Var √ nγ m = lim n→∞ Var √ nγ m + Ψ m lim n→∞ Var √ n( θn -θ 0 ) Ψ m + Ψ m lim n→∞ Cov √ n( θn -θ 0 ), √ nγ m + lim n→∞ Cov √ nγ m , √ n( θn -θ 0 ) Ψ m = Γ m,m + Ψ m Σ θΨ m + Ψ m Σ θ,γm + Σ θ,γm Ψ m .
Consequently, we have

lim n→∞ Var √ nρ m = lim n→∞ Var √ n γm σ 2 = 1 σ 4 Σ γm .
This ends our second step and the proof is completed.

In the following, we justify the convergence of R n,h,1 and R n,h,2 .

A.3.3.

Step 3: convergence of R n,h,1

Lemma 4. Under the assumptions of Theorem 2, the sequence of random variables

R n,h,1 = 1 √ n n t=1+h {˜ t (θ 0 )˜ t-h (θ 0 ) -t (θ 0 ) t-h (θ 0 )} (45)
converges in probability to zero as n → ∞.

Proof. Throughout this proof, θ = (θ 1 , . . . , θ p+q , d) ∈ Θ δ is such that max(d 0 , 0) < d ≤ d 2
where d 2 is the upper bound of the support of the long-range parameter

d 0 . Let R 1 n,h,1 = 1 √ n n t=1+h {˜ t (θ 0 ) -t (θ 0 )} ˜ t-h (θ 0 ) (46) and R 2 n,h,1 = 1 √ n n t=1+h t (θ 0 ) {˜ t-h (θ 0 ) -t-h (θ 0 )} . ( 47
)
The lemma will be proved as soon as we show that R 1 n,h,1 and R 2 n,h,1 tend to zero in probability when n → ∞.

Proof of the convergence in probability

of R 1 n,h,1
The arguments follow the one of Lemma 4 in Boubacar [START_REF] Boubacar Maïnassara | Estimating FARIMA models with uncorrelated but non-independent error terms[END_REF] in a simpler context. The proof is quite long so we divide it in four steps.

Step 1: preliminaries. We have

R 1 n,h,1 = 1 √ n n t=1+h {˜ t (θ 0 ) -˜ t (θ)} ˜ t-h (θ 0 ) + 1 √ n n t=1+h {˜ t (θ) -t (θ)} ˜ t-h (θ 0 ) + 1 √ n n t=1+h { t (θ) -t (θ 0 )} ˜ t-h (θ 0 ) = ω n,h,1 (θ) + ω n,h,2 (θ) + ω n,h,3 (θ),
where

ω n,h,1 (θ) = 1 √ n n t=1+h {˜ t (θ 0 ) -˜ t (θ)} ˜ t-h (θ 0 ), ω n,h,2 (θ) = 1 √ n n t=1+h {˜ t (θ) -t (θ)} ˜ t-h (θ 0 ) and ω n,h,3 (θ) = 1 √ n n t=1+h { t (θ) -t (θ 0 )} ˜ t-h (θ 0 ).
Therefore, if we prove that the two sequences of random variables (ω n,h,2 (θ)) n≥1 and (ω n,h,1 (θ) + ω n,h,3 (θ)) n≥1 converge in probability to 0, then the convergence in probability of R 1 n,h,1 to zero will be true.

Step 2: convergence in probability of (ω n,h,2 (θ)) n≥1 to 0 For all β > 0, we have

P (|ω n,h,2 | ≥ β) ≤ 1 √ nβ n t=1+h E [|˜ t (θ) -t (θ)| |˜ t-h (θ 0 )|] ≤ 1 √ nβ n t=1+h ˜ t (θ) -t (θ) L 2 ˜ t-h (θ 0 ) L 2 .
First, from (36) and using Lemma 2, we have

˜ t-h (θ 0 ) 2 L 2 = E   ∞ i=0 λ t-h i (θ 0 ) t-i-h 2   = ∞ i=1 ∞ j=1 λ t-h i (θ 0 ) λ t-h j (θ 0 ) E [ t-i-h t-j-h ] + σ 2 λ t-h 0 (θ 0 ) 2 = σ 2 ∞ i=1 λ t-h i (θ 0 ) 2 + σ 2 ≤ K . ( 48
)
In view of ( 31), ( 36) and ( 48), we may write

P (|ω n,h,2 (θ)| ≥ β) ≤ K β √ n n t=1+h E (˜ t (θ) -t (θ)) 2 1/2 ≤ K β √ n n t=1   i≥0 j≥0 λ t i (θ) -λ i (θ) λ t j (θ) -λ j (θ) E [ t-i t-j ]   1/2 ≤ σ K β √ n n t=1   i≥0 λ t i (θ) -λ i (θ) 2   1/2 ≤ σ K β √ n n t=1 λ(θ) -λ t (θ) 2 .
We use Lemma 1, the fact that d > max(d 0 , 0) and the fractional version of Cesàro's Lemma 2 to obtain

P (|ω n,h,2 (θ)| ≥ β) ≤ σ K β 1 √ n n t=1 1 t 1/2+(d-max(d 0 ,0)) ---→ n→∞ 0.
This proves the expected convergence in probability.

2 Recall that the fractional version of Cesàro's Lemma states that for (ht )t a sequence of positive real numbers, κ > 0 and c ≥ 0 we have

lim t→∞ ht t 1-κ = |κ| c ⇒ lim n→∞ 1 n κ n t=0 ht = c.
Step 3: convergence in probability of (ω n,h,1 (θ) + ω n,h,3 (θ)) n≥1 Note now that, for all n ≥ 1, we have

ω n,h,1 (θ) + ω n,h,3 (θ) = 1 √ n n t=1+h ( t (θ) -˜ t (θ)) -( t (θ 0 ) -˜ t (θ 0 )) ˜ t-h (θ 0 ).
A Taylor expansion of the function

( t -˜ t )(•) around θ 0 gives ( t (θ) -˜ t (θ)) -( t (θ 0 ) -˜ t (θ 0 )) ≤ ∂( t -˜ t ) ∂θ (θ ) R p+q+1 θ -θ 0 R p+q+1 ( 49
)
where θ is between θ 0 and θ. Following the same method as in the previous step we obtain

E ( t (θ) -˜ t (θ)) -( t (θ 0 ) -˜ t (θ 0 )) 2 ≤ K θ -θ 0 2 R p+q+1 p+q+1 k=1 E ∂( t -˜ t ) ∂θ k (θ ) 2 ≤ K θ -θ 0 2 R p+q+1 p+q+1 k=1 σ 2 ( . λ k - . λ k t )(θ ) 2 2 .
As in [START_REF] Hallin | Local asymptotic normality for regression models with long-memory disturbance[END_REF], it can be shown using Stirling's approximation and the fact that d > d 0 that

( . λ k - . λ k t )(θ ) 2 ≤ K 1 t 1/2+(d -d 0 )-ζ
for any small enough ζ > 0. We then deduce that

( t (θ) -˜ t (θ)) -( t (θ 0 ) -˜ t (θ 0 )) L 2 ≤ K θ -θ 0 R p+q+1 1 t 1/2+(d -d 0 )-ζ . ( 50
)
The expected convergence in probability follows from ( 48), ( 50) and the fractional version of Cesàro's Lemma.

Proof of the convergence in probability of R 2 n,h,1

Under Assumption (A3) with τ = 2 it follows that t (θ 0 ) belongs to L 2 . Thus the proof of the convergence in probability of R 2 n,h,1 to zero is shown in the same way as the proof of the convergence in probability of R 1 n,h,1 to 0.

Conclusion : convergence in probability of R n,h,1

The conclusion is a consequence of the above convergences.

A.3.4. Step 4: convergence of R n,h,2
Lemma 5. Under the assumptions of Theorem 2, the sequence of random variables

R n,h,2 = 1 n n t=1+h Dt (θ * n ) -D t (θ 0 ) √ n θn -θ 0 (51)
tends to zero in probability as n → ∞ and where θ * n is between θn and θ 0 .

Proof. Since ( √ n( θn -θ 0 )) n is a tight sequence, we have √ n( θn -θ 0 ) = O P (1). Hence, to prove the convergence in probability of (R n,h,2 ) n to 0, it suffices to show that

1 n n t=1+h Dt (θ * n ) -D t (θ 0 ) = o P (1). (52)
This will be proved using Remark 5 and Cesàro's Lemma. Nevertheless, the proof is quite long so we divide it in four steps.

Step 1: preliminaries. We have

1 n n t=1+h Dt (θ * n ) -D t (θ 0 ) = T n,h,1 (θ * n ) + T n,h,2 (θ * n ) + T n,h,3 (θ * n ) + T n,h,4 (θ * n ) + T n,h,5 (θ * n ),
where

T n,h,1 (θ) = 1 n n t=1+h ∂˜ t (θ) ∂θ (˜ t-h (θ) -t-h (θ)) , T n,h,2 (θ) = 1 n n t=1+h (˜ t (θ) -t (θ)) ∂˜ t-h (θ) ∂θ , T n,h,3 (θ) = 1 n n t=1+h ∂˜ t (θ) ∂θ - ∂ t (θ) ∂θ t-h (θ), T n,h,4 (θ) = 1 n n t=1+h t (θ) ∂˜ t-h (θ) ∂θ - ∂ t-h (θ) ∂θ and T n,h,5 (θ) = 1 n n t=1+h (D t (θ) -D t (θ 0 )) .
Therefore, if we prove that the five sequences of random variables (T n,h,i (θ * n )) n (for i = 1, . . . , 5) converge in probability to 0, then (52) will be true.

Step 2: convergence in probability of (T n,h,1 (θ * n )) n to 0 For all β > 0, we have

P ( T n,h,1 (θ * n ) ≥ β) ≤ 1 nβ n t=1+h E ∂˜ t (θ * n ) ∂θ |˜ t-h (θ * n ) -t-h (θ * n )| ≤ 1 nβ n t=1+h ˜ t-h (θ * n ) -t-h (θ * n ) L 2 ∂˜ t (θ * n ) ∂θ R p+q+1 L 2 .
First, from (36) and using Lemma 2 we have

∂ ∂θ ˜ t (θ * n ) R p+q+1 2 L 2 ≤ K p+q+1 k=1 E   ∞ i=1 . λ t i,k (θ * n ) t-i 2   ≤ K p+q+1 k=1 sup θ∈Θ δ E   ∞ i=1 . λ t i,k (θ) t-i 2   ≤ K σ 2 p+q+1 k=1 sup θ∈Θ δ ∞ i=1 . λ t i,k (θ) 2 (53) ≤ K , (54)
where we have used the fact that the function

θ → E ∂˜ t ) ∂θ k (θ)
2 is bounded and continuous. In view of ( 31), ( 36), ( 54) and following the same way as the step 2 of Lemma 4 we have

P (|T n,h,1 (θ * n )| ≥ β) ≤ K βn n t=1+h E (˜ t-h (θ * n ) -t-h (θ * n )) 2 1/2 ≤ K βn n t=1+h sup θ∈Θ δ E (˜ t-h (θ) -t-h (θ)) 2 1/2 ≤ K βn n-h t=1 sup θ∈Θ δ   i≥0 j≥0 λ t i (θ) -λ i (θ) λ t j (θ) -λ j (θ) E [ t-i t-j ]   1/2 ≤ σ K βn n t=1 sup θ∈Θ δ   i≥0 λ t i (θ) -λ i (θ) 2   1/2 ≤ σ K βn n t=1 sup θ∈Θ δ λ(θ) -λ t (θ) 2 .
We use Remark 5, the fact that d 1 -d 0 > -1/2 and Cesàro's Lemma to obtain

P (|T n,h,1 (θ * n )| ≥ β) ≤ σ K β 1 n n t=1 1 t 1/2+(d 1 -d 0 ) ---→ n→∞ 0.
This proves the expected convergence in probability of T n,h,1 (θ * n ). The same calculations holds for the sequences of random variables

(T n,h,2 (θ * n )) n , (T n,h,3 (θ * n )) n and (T n,h,4 (θ * n )) n .
Step 3: convergence in probability of (T n,h,5 (θ * n )) n to 0

For 1 ≤ i, j ≤ p + q + 1, t, s ∈ Z and θ * * n between θ * n and θ 0 , one has in view of (34) and Remark 5

E ∂ ∂θ i t (θ * * n ) ∂ ∂θ j s (θ * * n ) ≤ sup θ∈Θ δ   E   k≥1 . λ k,i (θ) t-k 2     1/2 sup θ∈Θ δ   E   k≥1 . λ k,i (θ) s-k 2     1/2 ≤ K σ 2 sup θ∈Θ δ . λ k (θ) 2 2 ≤ K . (55)
Similar calculation can be done to obtain

E t (θ * * n ) ∂ 2 ∂θ i ∂θ j s (θ * * n ) < ∞. (56)
A Taylor expansion of D t (•) around θ 0 implies that

T n,h,5 (θ * n ) ≤ 1 n n t=1 ∂ ∂θ D t (θ * * n ) θ * n -θ 0 ,
for some θ * * n between θ * n and θ 0 . From ( 55) and ( 56), it follows that

E ∂ ∂θ D t (θ * * n ) = E t-h (θ * * n ) ∂ 2 ∂θ∂θ t (θ * * n ) + ∂ ∂θ t-h (θ * * n ) ∂ ∂θ t (θ * * n ) + ∂ ∂θ t (θ * * n ) ∂ ∂θ t-h (θ * * n ) + t (θ * * n ) ∂ 2 ∂θ∂θ t-h (θ * * n ) ≤ K . (57)
We use Equation ( 57), the ergodic theorem and the convergence in probability of ( θn -θ 0 ) n to 0 to deduce that T n,h,5 (θ) converges in probability to 0.

Step 4: end of the proof of the convergence in probability of R n,h,2 to zero. By Step 2 and 3 we deduce that R n,h,2 = o P (1)

and the convergence in probability is proved. The proof of the lemma is completed.

A.4. Proof of Proposition 5

The following proofs are quite technical and are adaptations of the arguments used in Boubacar Maïnassara and Saussereau (2018).

To prove the invertibility of the normalized matrix C m , we need to introduce the following notation.

Let S t (i) be the i-th component of the vector S t = t j=1 (ΛU j -γ m ) ∈ R m . We remark that

S t-1 (i) = S t (i) - p+q+1 k=1 δ i,k t ∂ ∂θ k t (θ 0 ) -t t-i + γ(i), (58)
where δ i,k is the (i, k)-th entry of the m × (p + q + 1) matrix ∆ := -2Ψ m J -1 . If the matrix C m is not invertible, there exists some real constants c 1 , . . . , c m not all equal to zero, such that we have

m i=1 m j=1 c j C m (j, i)c i = 1 n 2 n t=1 m i=1 m j=1 c j S t (j)S t (i)c i = 1 n 2 n t=1 m i=1 c i S t (i) 2 = 0,
which implies that m i=1 c i S t (i) = 0 for all t ≥ 1. Then by ( 58), it would imply that

m i=1 p+q+1 k=1 c i δ i,k t ∂ ∂θ k t (θ 0 ) + m i=1 c i t t-i = m i=1 c i γ(i). (59)
By the ergodic Theorem, we also have m i=1 c i γ(i) → 0 almost-surely as n goes to infinity. Consequently replacing this convergence in (59) implies that for all t ≥ 1

m i=1 p+q+1 k=1 c i δ i,k t ∂ ∂θ k t (θ 0 ) + m i=1 c i t t-i = 0, a.s.
Using (31), it yields that

t    ≥1 m i=1 p+q+1 k=1 c i δ i,k . λ ,k (θ 0 ) t-+ m =1 c t-    = 0, a.s. Or equivalently, t    m =1 m i=1 c i p+q+1 k=1 δ i,k . λ ,k (θ 0 ) + c t-+ ≥m+1 m i=1 c i p+q+1 k=1 δ i,k . λ ,k (θ 0 ) t-    = 0, a.s.
Thanks to Assumption (A4), t has a positive density in some neighborhood of zero and then

t = 0 almost-surely. Hence we obtain m =1 m i=1 c i p+q+1 k=1 δ i,k . λ ,k (θ 0 ) + c t-+ ≥m+1 m i=1 c i p+q+1 k=1 δ i,k . λ ,k (θ 0 ) t-= 0, a.s.
Since the variance of the linear innovation process in not equal to zero, we deduce that

     m i=1 c i p+q+1 k=1 δ i,k . λ ,k (θ 0 ) + c = 0 for all ∈ {1, . . . , m} m i=1 c i p+q+1 k=1 δ i,k . λ ,k (θ 0 ) = 0 for all ∈ {m + 1, . . . } .
Then we would have c 1 = • • • = c m = 0 which is impossible. Thus we have a contradiction and the matrix C m ∈ R m×m is non singular.

A.5. Proof of Theorem 6

We recall that the Skorokhod space D [0,1] is the set of R -valued functions on [0,1] which are right-continuous and have left limits everywhere. It is endowed with the Skorokhod topology and the weak convergence on D [0,1] is mentioned by D -→. The integer part of x will be denoted by x .

The proof is divided in two steps.

A.5.1. Functional central limit theorem for (ΛU t ) t≥1

In view of ( 8) and ( 13), we deduce that

√ nγ m = √ nγ m + √ nΨ m θn -θ 0 + o P (1) = 1 √ n n t=1 U 2t + Ψ m 1 √ n n t=1 U 1t + o P (1) + o P (1) = 1 √ n n t=1 ΛU t + o P (1). (60)
Now, it is clear that the asymptotic behaviour of γm is related to the limit distribution of U t = (U 1t , U 2t ) . Our first goal is to show that there exists a lower triangular matrix Π with nonnegative diagonal entries such that

1 √ n nr t=1 ΛU t D m -→ n→∞ ΠΠ 1/2 B m (r ), (61)
where (B m (r )) r ≥0 is a m-dimensional standard Brownian motion. Using (31), U t can be rewritten as

U t = -2 ∞ i=1 . λ i,1 (θ 0 ) t t-i , . . . , ∞ i=1 . λ i,p+q+1 (θ 0 ) t t-i J -1 , t t-1 , . . . , t t-m .
The non-correlation between t 's implies that the process (U t ) t∈Z of R p+q+1+m is centered. In order to apply the functional central limit theorem for strongly mixing process (see [START_REF] Herrndorf | A functional central limit theorem for weakly dependent sequences of random variables[END_REF]), we need to identify the asymptotic covariance matrix in the classical central limit theorem for the sequence

(U t ) t∈Z . It is proved in Proposition 1 that 1 √ n n t=1 U t in law ---→ n→∞ N (0, Ξ := 2πf U (0)) ,
where f U (0) is the spectral density of the stationary process (U t ) t∈Z evaluated at frequency 0. The existence of the matrix Ξ has already been discussed in Lemma 3. Since the matrix Ξ is positive definite, it can be factorized as Ξ = ΥΥ , where the (p + q + 1 + m) × (p + q + 1 + m) lower triangular matrix Υ has nonnegative diagonal entries. Therefore, we have

1 √ n n t=1 ΛU t in law ---→ n→∞ N 0, ΛΞ Λ ,
and the new variance matrix can also been factorized as ΛΞ Λ = (ΛΥ)(ΛΥ) := ΠΠ , where Π ∈ R m×(p+q+1) . Thus

n -1/2 n t=1 (ΠΠ ) -1/2 ΛU t in law -→ n→∞ N (0, I m ),
where (ΠΠ ) -1/2 is the Moore-Penrose inverse (see [START_REF] Magnus | Matrix differential calculus with applications in statistics and econometrics[END_REF], p. 36) of (ΠΠ ) 1/2 . Using the same arguments as in the proof of Theorem 2 in Boubacar Maïnassara et al. ( 2019), the asymptotic distribution of n -1/2 n t=1 U t when n tends to infinity is obtained by introducing the random vector U k t defined for any positive integer k by

U k t = -2 k i=1 . λ i,1 (θ 0 ) t t-i , . . . , k i=1 . λ i,p+q+1 (θ 0 ) t t-i J -1 , t t-1 , . . . , t t-m .
Since U k t depends on a finite number of values of the noise-process ( t ) t∈Z , it also satisfies a mixing property (see Theorem 14.1 in Davidson (1994), p. 210). Then applying the central limit theorem for strongly mixing process of [START_REF] Herrndorf | A functional central limit theorem for weakly dependent sequences of random variables[END_REF] shows that its asymptotic distribution is normal with zero mean and variance matrix Ξ k that converges when k tends to infinity to Ξ . More precisely we have

1 √ n n t=1 U k t in law ---→ n→∞ N (0, Ξ k ) .
The above arguments also apply to matrix Ξ k with some matrix Π k which is defined analogously as Π. Consequently we obtain

1 √ n n t=1 ΛU k t in law ---→ n→∞ N 0, ΛΞ k Λ
and we also have

n -1/2 n t=1 (Π k Π k ) -1/2 ΛU k t in law ---→ n→∞ N (0, I m ).
Now we are able to apply the functional central limit theorem (see [START_REF] Herrndorf | A functional central limit theorem for weakly dependent sequences of random variables[END_REF]) and we obtain that

1 √ n nr t=1 (Π k Π k ) -1/2 ΛU k t D m -→ n→∞ B m (r ).
Since for all t ∈ {1, . . . , nr } we write

(ΠΠ ) -1/2 ΛU k t = (ΠΠ ) -1/2 -(Π k Π k ) -1/2 ΛU k t + (Π k Π k ) -1/2 ΛU k t ,
we obtain the following weak convergence on D m [0, 1]:

1 √ n nr t=1 (ΠΠ ) -1/2 ΛU k t D m -→ n→∞ B m (r ).
In order to conclude that ( 61) is true, it remains to observe that uniformly with respect to n

Y k n (r ) := 1 √ n nr t=1 (ΠΠ ) -1/2 ΛZ k t D m -→ k→∞ 0, (62)
where

Z k t = -2 ∞ i=k+1 . λ i,1 (θ 0 ) t t-i , . . . , ∞ i=k+1 . λ i,p+q+1 (θ 0 ) t t-i J -1 , t t-1 , . . . , t t-m .
Using the same arguments as those used in the proof of Theorem 2 in Boubacar Maïnassara et al.

(2019), we have

sup n Var 1 √ n n t=1 Z k t -→ k→∞ 0 and since nr ≤ n, sup 0≤r ≤1 sup n Y k n (r ) -→ k→∞ 0.
Thus ( 62) is true and the proof of ( 61) is achieved.

A.5.2. Limit theorem

To conclude the prove of Theorem 6, we follow the arguments developed in Boubacar [START_REF] Boubacar Maïnassara | Diagnostic Checking in Multivariate ARMA Models With Dependent Errors Using Normalized Residual Autocorrelations[END_REF]. Note that the previous step ensures us that Assumption 1 in Lobato ( 2001) is satisfied for the sequence (ΛU t ) t≥1 . Firstly from (61) we deduce that

1 √ n S nr = 1 √ n nr t=1 ΛU t - nr n 1 √ n n t=1 ΛU t D m -→ n→∞ (ΠΠ ) 1/2 B m (r ) -r (ΠΠ ) 1/2 B m (1). ( 63
)
Observe now that the continuous mapping theorem implies

C m = 1 n n t=1 1 √ n S t 1 √ n S t D m -→ n→∞ (ΠΠ ) 1/2 1 0 {B m (r ) -r B m (1)} {B m (r ) -r B m (1)} dr (ΠΠ ) 1/2 = (ΠΠ ) 1/2 V m (ΠΠ ) 1/2 .
Using ( 60), ( 63) and again the continuous mapping theorem on the Skorokhod space, one finally obtains

nγ m C -1 m γm D m -→ n→∞ (ΠΠ ) 1/2 B m (1) (ΠΠ ) 1/2 V m (ΠΠ ) 1/2 -1 (ΠΠ ) 1/2 B m (1) = B m (1)V -1 m B m (1) := U m .
Consequently, from (9) it follows that

nσ 4 ρ m C -1 m ρm D m -→ n→∞ U m ,
which completes the proof of Theorem 6.

A.6. Proof of Theorem 7

The proof follows the same line than in the proof of Theorem 2 in Boubacar Maïnassara and Saussereau (2018) (see also the proof of in Boubacar Maïnassara et al. ( 2019)).

Appendix B: Example of explicit calculation of Σ ρm and C m

The results of the previous subsections 3.2 and 3.3 are particularized in the FARIMA(1, d 0 , 0) and FARIMA(0, d 0 , 1) cases. First we consider the case of a FARIMA(1, d 0 , 0) model of the form

(1 -L) d 0 (X t -aX t-1 ) = t , (64)
where the unknown parameter is θ 0 = (a, d 0 ). We assume that in (64) the innovation process ( t ) t∈Z is a GARCH(1, 1) process given by ( 20). We also assume that in (20):

α 2 1 κ + β 2 1 + 2α 1 β 1 < 1, 3 where κ := Eη 4
1 and we assume that κ > 1. For the sake of simplicity we assume that the variables (η t ) t∈Z involved in (20) have a symmetric distribution. More precisely, we have the following symmetry assumption

E[ t 1 t 2 t 3 t 4 ] = 0 when t 1 = t 2 , t 1 = t 3 and t 1 = t 4 , (65)
made in [START_REF] Francq | Bartlett's formula for a general class of nonlinear processes[END_REF][START_REF] Boubacar Mainassara | Computing and estimating information matrices of weak ARMA models[END_REF]. For this particular GARCH(1, 1) model with fourth-order moments and symmetric innovations satisfying (65), it can be shown that

E [ t t-t-h t-h-] =    E 2 t 2 t- if h = 0 and = 0 otherwise. ( 66
)
Now we need to compute the autocovariance structure of ( 2 t ) t∈Z . We will use the fact that the GARCH process ( t ) t∈Z is fourth-order stationary, then ( 2 t ) t∈Z is a solution of the following ARMA(1, 1) model

2 t = ω + (α 1 + β 1 ) 2 t-1 + ν t -β 1 ν t-1 , t ∈ Z (67)
where ν t = 2 t -σ 2 t is the innovation of ( 2 t ) t∈Z . From (67) the autocovariances of ( 2 t ) t∈Z take the form

γ 2 ( ) := Cov( 2 t , 2 t-) = γ 2 (1)(α 1 + β 1 ) -1 , ≥ 1, (68)
where

γ 2 (1) = (κ -1)(α 1 -α 1 β 2 1 -α 2 1 β 1 ) 1 -β 2 1 -2α 1 β 1 -α 2 1 κ σ 4 , γ 2 (0) := Var( 2 t ) = (κ -1)(1 -β 2 1 -2α 1 β 1 ) 1 -β 2 1 -2α 1 β 1 -α 2 1 κ σ 4 ,
and

σ 2 := ω 1 -α 1 -β 1 .
From ( 66) and ( 68) we deduce that for any ≥ 1

Γ ( , ) = E 2 t 2 t-= Cov( 2 t , 2 t-) + E 2 t E 2 t- = 1 + 1 σ 4 γ 2 (1)(α 1 + β 1 ) -1 σ 4 . (69)

B.1. Examples of analytic and numerical computations of Σ ρm

As mentioned before, the subject of this subsection is to give an explicit expression of the asymptotic variance of residual autocorrelations Σ ρm defined in (15) in the particular case of model ( 64). For that sake, we need the following additional expressions. It is classical that the noise derivatives (∂ t (θ 0 )/∂a, ∂ t (θ 0 )/∂d) in ( 64) can be represented as

  ∂ t (θ 0 ) ∂a ∂ t (θ 0 ) ∂d   = - j≥1 a j-1 1 j t-j . (70)
We compute the information matrices J(θ 0 ) and I (θ 0 ) by using ( 70). Then we have

J(θ 0 ) = 2σ 2   1 1-a 2 -ln(1-a) a -ln(1-a) a π 2 6   . (71)
A simple calculation implies that

J -1 (θ 0 ) = 1 2σ 2 c(a)   π 2 6 ln(1-a) a ln(1-a) a 1 1-a 2   , (72)
where

c(a) = π 2 6(1 -a 2 ) - ln(1 -a) a 2 . (73)
We now investigate a similar tractable expression for I (θ 0 ). Using ( 70) and (65) we have

I (θ 0 ) = 2σ 2 J(θ 0 ) + 4σ 4 (κ -1)(α 1 -α 1 β 2 1 -α 2 1 β 1 ) 1 -β 2 1 -2α 1 β 1 -α 2 1 κ   1 1-a 2 (α 1 +β 1 )
ln[1-a(α 1 +β 1 )] a(α 1 +β 1 )

ln[1-a(α 1 +β 1 )] a(α 1 +β 1 ) Li 2 (α 1 +β 1 )

α 1 +β 1   , ( 74
)
where Li 2 is the Spence function defined by Li 2 (z) = ∞ k=1 z k k -2 . Note that we retrieve the well know result: I (θ 0 ) = 2σ 2 J(θ 0 ) in the strong FARIMA case (i.e. when α 1 = β 1 = 0 in (74)).

The matrix defined in (7) can be rewritten as

Ψ m = -σ 2 1 a . . . a m-1 1 1 2 . . . 1 m . ( 75
)
Using ( 69) and under the symmetry assumption (65), the matrix Γ m,m takes the simple following diagonal form

Γ m,m = σ 4 I m + σ 4 (κ -1)(α 1 -α 1 β 2 1 -α 2 1 β 1 ) 1 -β 2 1 -2α 1 β 1 -α 2 1 κ diag(1, (α 1 + β 1 ), . . . , (α 1 + β 1 ) m-1 ). (76)
Using ( 65), ( 70) and ( 72), the matrix Σ θ,γm is given by

Σ θ,γm = 1 σ 2 c(a)         π 2 6 + ln(1-a) a Γ m,m (1, 1) 1 1-a 2 + ln(1-a) a Γ m,m (1, 1) a π 2 6 + ln(1-a) 2a Γ m,m (2, 2) 1 2(1-a 2 ) + ln(1 -a) Γ m,m (2, 2) . . . . . . a m-1 π 2 6 + ln(1-a) ma Γ m,m (m, m) 1 m(1-a 2 ) + a m-2 ln(1 -a) Γ m,m (m, m)         , (77)
where for any 1 ≤ i, j ≤ m, Γ m,m (i, j) is given by ( 76).

From Remark 2, in the strong FARIMA case the asymptotic variance of residual autocorrelations takes a simpler form

Σ s ρm = I m - 1 c(a) π 2 6 a i+j-2 + 1 1 -a 2 1 ij + ln(1 -a) a a j-1 i + a i-1 j 1≤i,j≤m
where c(a) is the constant given in (73).

From the above explicit expressions we deduce that the asymptotic variance of residual autocorrelations for this model is in the form

Σ ρm = Σ s ρm + (κ -1)(α 1 -α 1 β 2 1 -α 2 1 β 1 ) 1 -β 2 1 -2α 1 β 1 -α 2 1 κ (α 1 + β 1 ) i-1 1 1 {i=j} + 1 c(a)
M(i, j)

-(α 1 + β 1 ) i-1 + (α 1 + β 1 ) j-1 1 c(a) π 2 6 a i+j-2 + 1 1 -a 2 1 ij + ln(1 -a) a a j-1 i + a i-1 j 1≤i,j≤m
, where

M(i, j) = ln(1 -a) a 1 1 -a 2 (α 1 + β 1 ) - 1 1 -a 2 ln(1 -a(α 1 + β 1 )) a(α 1 + β 1 ) π 2 6 a j-1 i + 1 ij ln(1 -a) a + Li 2 (α 1 + β 1 ) α 1 + β 1 1 1 -a 2 - ln(1 -a) a ln(1 -a(α 1 + β 1 )) a(α 1 + β 1 ) ln(1 -a) a a j-1 i + 1 ij 1 1 -a 2 + π 2 6 1 1 -a 2 (α 1 + β 1 ) - ln(1 -a) a ln(1 -a(α 1 + β 1 )) a(α 1 + β 1 ) π 2 6 a i+j-2 + a i-1 j ln(1 -a) a + Li 2 (α 1 + β 1 ) α 1 + β 1 ln(1 -a) a - π 2 6 ln(1 -a(α 1 + β 1 )) a(α 1 + β 1 ) ln(1 -a) a a i+j-2 + a i-1 j 1 1 -a 2 .
For simplicity, we take in the sequel β 1 = 0 to consider the case of an ARCH(1) model. For instance when m = 3, κ = 3, ω = 1 and a = -0.55 we have

Σ ρ3

Eigenvalues ξ 3 = (ξ 1,3 , ξ 2,3 , ξ 3,3 ) Z 3 (ξ 3 ) It is clear that for α 1 = 0.55, the [START_REF] Li | Fractional time series modelling[END_REF] approximation by a χ 2 1 distribution will be disastrous. The eigenvalues ξ m can be very different from those of strong FARIMA models which are close to 1 or 0 when the lag m is large enough (see Remark 3). More precisely, for instance for α 1 = 0 and m = 12 we obtain ξ 12 = (1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 0.0665, 0.0000) ,

α 1 = 0   0.
In this weak FARIMA(1, d, 0) with α 1 = 0.55 and m = 12 we also obtain ξ 12 = (5.4628, 3.7524, 2.3222, 1.7930, 1.4152, 1.2405, 1.1295, 1.0723, 1.0387, 1.0207, 0.0827, 0.0000) .

The same result holds for FARIMA(0, d, 1) model with a replaced by b in θ 0 .

B.2. Explicit form of the matrix C m

The following example gives an explicit form of the normalization matrix C m for the model given in (64). For reading convenience, we restrict ourselves to the case m = 3. Using the expression of J -1 (θ 0 ) given in (72) and Equation ( 70), we obtain that for all 1 ≤ j ≤ n

-2J -1 (θ 0 ) j   ∂ j (θ 0 ) ∂a ∂ j (θ 0 ) ∂d   =   v (1) j (a) v (2) j (a)   , where v (1) j (a) = 1 σ 2 c(a) k≥1 π 2 6 a k-1 + ln(1 -a) a 1 k j j-k and v (2) j (a) = 1 σ 2 c(a) k≥1 ln(1 -a) a a k-1 + 1 1 -a 2 1 k j j-k .
Thus, the vector ΛU j is given by

ΛU j =      -σ 2 v (1) j (a) -σ 2 v (2) j (a) + j j-1 -σ 2 av (1) j (a) -σ 2 v (2) j (a)/2 + j j-2 -σ 2 a 2 v (1) j (a) -σ 2 v (2) j (a)/3 + j j-3      .
A simple calculation shows that, for any 1 ≤ j 1 , j 2 ≤ n,

(ΛU j 1 ) (ΛU j 2 ) =        K (1) j 1 (a)K (1) j 2 (a) K (1) j 1 (a)K (2) j 2 (a) K (1) j 1 (a)K (3) j 2 (a) K (2) j 1 (a)K (1) j 2 (a) K (2) j 1 (a)K (2) j 2 (a) K (2) j 1 (a)K (3) j 2 (a) K (3) j 1 (a)K (1) j 2 (a) K (3) j 1 (a)K (2) j 2 (a) K (3) j 1 (a)K (3) j 2 (a)        , where K (1) j (a) = -σ 2 v (1) j (a) -σ 2 v (2) j (a) + j j-1 , K (2) j (a) = -σ 2 av (1) j (a) -σ 2 v (2) j (a)/2 + j j-2 and K (3) j (a) = -σ 2 a 2 v (1) j (a) -σ 2 v (2) j (a)/3 + j j-3 .
Therefore we deduce that for all positive integer t

S t = t j=1 (ΛU j -γ 3 ) = t j=1      -σ 2 v (1) j (a) -σ 2 v (2) j (a) + j j-1 -σ 2 av (1) j (a) -σ 2 v (2) j (a)/2 + j j-2 -σ 2 a 2 v (1) j (a) -σ 2 v (2) j (a)/3 + j j-3      - t n     n j=2 j j-1 n j=3 j j-2 n j=4 j j-3     .
The same result holds for FARIMA(0, d 0 , 1) model with a replaced by b in θ 0 . Empirical power (in %) of the modified and standard versions of the LB and BP tests in the case of a strong FARIMA(1, d 0 , 2) defined by (78) with θ 0 = (0.9, 1, -0.2, d 0 ). The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000. 98.9 98.9 100.0 100.0 100.0 100.0 15 99.5 99.4 100.0 100.0 100.0 100.0

Table 12

Empirical power (in %) of the modified and standard versions of the LB and BP tests in the case of a weak FARIMA(1, d 0 , 2) defined by (78) with θ 0 = (0.9, 1, -0.2, d 0 ) and where ω = 0.4, α 1 = 0.3 and β 1 = 0.3 in (20). The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000. Empirical power (in %) of the modified and standard versions of the LB and BP tests in the case of a weak FARIMA(1, d 0 , 2) defined by (78)-( 21) with θ 0 = (0.9, 1, -0.2, d 0 ). The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000.

C.2. Small sample size

The following tables deal with the same numerical experiments that in Section 4 when the sample sizes are less than 500.

C.3. GARCH process with infinite moment

In order the see if the test procedures remain reliable for GARCH process with infinite moment (for α 1 + β 1 ≥ 1), we replicate the numerical experiments made on Model ( 19)-( 20) with ω = 0.04, α 1 = 0.13 and β 1 = 0.88.

As showing in Figures 5,. . . ,10 the results are qualitatively similar to what we observe here in Tables 2,3 9 and10. Figures 5,. . . ,9 display the residual autocorrelations of a realization of size n = 2, 000 for weak FARIMA models ( 19)-( 20) with ω = 0.04, α 1 = 0.13, β 1 = 0.88 and three values of d 0 , and their 5% significance limits under the strong FARIMA and weak FARIMA assumptions. These figures confirm clearly the conclusions drawn in Subsection 4.1. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption. The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7. 19)-( 20) with θ0 = (0.9, 0.2, 0.01) and where ω = 0.04, α1 = 0.13 and β1 = 0.88. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption. The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7. 19)-( 20) with θ0 = (0.9, 0.2, 0.25) and where ω = 0.04, α1 = 0.13 and β1 = 0.88. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption. The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7. 19)-( 20) with θ0 = (0.9, 0.2, 0.49) and where ω = 0.04, α1 = 0.13 and β1 = 0.88. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption. The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7. 19)-( 20) with θ0 = (0, 0, 0.01) and where ω = 0.04, α1 = 0.13 and β1 = 0.88. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption. The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7. 19)-( 20) with θ0 = (0, 0, 0.25) and where ω = 0.04, α1 = 0.13 and β1 = 0.88. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption. The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7. 19)-( 20) with θ0 = (0, 0, 0.49) and where ω = 0.04, α1 = 0.13 and β1 = 0.88. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption. The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7.

  [START_REF] Francq | GARCH Models: Structure, Statistical Inference and Financial Applications[END_REF]; Lobato et al. (2001); Boubacar Mainassara et al. (2012); Boubacar Maïnassara and Saussereau (2018)).

  least squares estimators of the parameter θ 0 = (a, b, d 0 ) for the model (19) in the case of the S&P 500 and the Nikkei. The least squares estimators were obtained as θSP500 3371 [0.1105] (0.0023) -0.1795 [0.0788] (0.0227) 0.2338 [0.0367] (0.0000)   and σ2 = 22.9076 × 10 -8 0217 [0.1105] (0.9528) 0.1579 [0.0788] (0.6050) 0.3217 [0.0367] (0.0000)   and σ2 = 25.6844 × 10 -8 ,

Fig 1 .

 1 Fig 1. Returns and the sample autocorrelations of squared returns of the S&P 500.

Fig 2 .

 2 Fig 2. Autocorrelation of the FARIMA(1, 0.2338, 1) residuals for the squares of the S&P 500 returns. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption.The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7.

Fig 3 .

 3 Fig 3. Returns and the sample autocorrelations of squared returns of the Nikkei.

Fig 4 .

 4 Fig 4. Autocorrelation of the FARIMA(0, 0.2132, 0) residuals for the squares of the Nikkei returns. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption.The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7.

Fig 5 .

 5 Fig 5. Autocorrelation of a realization of size n = 2, 000 for a weak FARIMA(1, 0.01, 1) model (19)-(20) with θ0 = (0.9, 0.2, 0.01) and where ω = 0.04, α1 = 0.13 and β1 = 0.88. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption. The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7.

Fig 6 .

 6 Fig 6. Autocorrelation of a realization of size n = 2, 000 for a weak FARIMA(1, 0.25, 1) model (19)-(20) with θ0 = (0.9, 0.2, 0.25) and where ω = 0.04, α1 = 0.13 and β1 = 0.88. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption. The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7.

Fig 7 .

 7 Fig 7. Autocorrelation of a realization of size n = 2, 000 for a weak FARIMA(1, 0.49, 1) model (19)-(20) with θ0 = (0.9, 0.2, 0.49) and where ω = 0.04, α1 = 0.13 and β1 = 0.88. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption. The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7.

Fig 8 .

 8 Fig 8. Autocorrelation of a realization of size n = 2, 000 for a weak FARIMA(0, 0.01, 0) model (19)-(20) with θ0 = (0, 0, 0.01) and where ω = 0.04, α1 = 0.13 and β1 = 0.88. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption. The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7.

Fig 9 .

 9 Fig 9. Autocorrelation of a realization of size n = 2, 000 for a weak FARIMA(0, 0.25, 0) model (19)-(20) with θ0 = (0, 0, 0.25) and where ω = 0.04, α1 = 0.13 and β1 = 0.88. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption. The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7.

Fig 10 .

 10 Fig 10. Autocorrelation of a realization of size n = 2, 000 for a weak FARIMA(0, 0.49, 0) model (19)-(20) with θ0 = (0, 0, 0.49) and where ω = 0.04, α1 = 0.13 and β1 = 0.88. The horizontal dotted lines (blue color) correspond to the 5% significant limits obtained under the strong FARIMA assumption. The solid lines (red color) and dashed lines (green color) correspond also to the 5% significant limits under the weak FARIMA assumption. The full lines correspond to the asymptotic significance limits for the residual autocorrelations obtained in Theorem 2. The dashed lines (green color) correspond to the self-normalized asymptotic significance limits for the residual autocorrelations obtained in Theorem 7.

Table 2

 2

Table 3

 3

Table 4

 4

Table 6

 6 Modified and standard versions of portmanteau tests to check the null hypothesis that the S&P 500 squared returns follow a FARIMA(1, 0.2338, 1) model (19).

	Lag m	1	2	3	4	5	6	7
	ρ(m)	0.0002 -0.0033 -0.0350 -0.0393 0.0893 -0.0040 -0.0179
	LB sn	0.0653 18.150	41.924	58.057	186.72	313.78	341.38
	BP sn	0.0653 18.146	41.912	58.037	186.64	313.64	341.20
	LB w	0.0008 0.1885	21.445	48.248	186.95	187.23	192.77
	BP w	0.0008 0.1884	21.439	48.232	186.88	187.15	192.67
	p lb w	0.8525 0.6985	0.0916	0.3137	0.0678	0.0717	0.0752
	p bp w	0.8525 0.6986	0.0917	0.3138	0.0679	0.0718	0.0753
	p lb s	n.a.	n.a.	n.a.	0.0000	0.0000	0.0000	0.0000
	p bp s	n.a.	n.a.	n.a.	0.0000	0.0000	0.0000	0.0000
	Lag m	8	9	10	11	12	13	14
	ρ(m)	0.0047 0.0137 -0.0040 0.0295	0.0093 -0.0077 -0.0286
	LB sn	397.27 397.38	415.22	465.52	468.76	567.87	573.02
	BP sn	397.04 397.13	414.93	465.17	468.33	567.38	572.49
	LB w	193.16 196.42	196.69	211.82	213.31	214.34	228.55
	BP w	193.09 196.34	196.61	211.74	213.22	214.25	228.45
	p lb w	0.0758 0.0786	0.0986	0.1053	0.1148	0.1226	0.1047
	p bp w	0.0758 0.0787	0.0987	0.1054	0.1150	0.1228	0.1048
	p lb s	0.0000 0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	p bp s	0.0000 0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	Lag m	15	16	17	18	19	20	21
	ρ(m)	0.0021 0.0086	0.0097	0.0137 -0.0023 0.0016	0.0132
	LB sn	588.61 701.16	738.23	738.58	749.24	778.88	788.01
	BP sn	588.04 700.44	737.42	737.73	748.33	777.90	786.97
	LB w	228.63 229.91	231.54	234.83	234.92	234.97	238.00
	BP w	228.52 229.80	231.44	234.72	234.81	234.86	237.89
	p lb w	0.1079 0.1113	0.2212	0.2138	0.2127	0.2169	0.2324
	p bp w	0.1080 0.1114	0.2214	0.2140	0.2130	0.2171	0.2327
	p lb s	0.0000 0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	p bp s	0.0000 0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

Table 7

 7 Modified and standard versions of portmanteau tests to check the null hypothesis that the Nikkei squared returns follow a FARIMA(0, 0.2132, 0) model as in (19) with a = b = 0.

	Lag m	1	2	3	4	5	6	7
	ρ(m)	-0.0678 0.0400 0.0634 -0.0022 0.0165	0.0320 -0.0158
	LB sn	5.7332 29.005 34.758 34.779	66.692	288.57	324.46
	BP sn	5.7319 28.997 34.745 34.764	66.657	288.40	324.24
	LB w	61.211 82.507 136.13 136.20	139.84	153.46	156.78
	BP w	61.198 82.487 136.09 136.16	139.76	153.41	156.73
	p lb w	0.1086 0.2186 0.1830 0.2551	0.3002	0.3519	0.3609
	p bp w	0.1086 0.2187 0.1831 0.2552	0.3003	0.3521	0.3611
	p lb s	n.a.	0.0000 0.0000 0.0000	0.0000	0.0000	0.0000
	p bp s	n.a.	0.0000 0.0000 0.0000	0.0000	0.0000	0.0000
	Lag m	8	9	10	11	12	13	14
	ρ(m)	0.0295 0.0384 0.0121 0.0133	0.0503	0.0076	0.0068
	LB sn	387.88 512.70 575.09 600.81	791.67	808.20	808.27
	BP sn	387.59 512.28 574.57 600.22	790.83	807.29	807.30
	LB w	168.41 188.08 190.01 192.36	226.12	226.89	227.50
	BP w	168.35 187.10 189.93 192.29	225.10	226.76	227.39
	p lb w	0.3627 0.3757 0.3802 0.3825	0.3320	0.3447	0.3526
	p bp w	0.3629 0.3759 0.3804 0.3827	0.3323	0.3450	0.3529
	p lb s	0.0000 0.0000 0.0000 0.0000	0.0000	0.0000	0.0000
	p bp s	0.0000 0.0000 0.0000 0.0000	0.0000	0.0000	0.0000
	Lag m	15	16	17	18	19	20	21
	ρ(m)	0.0538 0.0073 0.0173 0.0067 -0.0027 -0.0057 0.0153
	LB sn	839.87 842.24 842.31 845.36	885.74	935.70	946.03
	BP sn	838.80 841.10 841.11 844.10	884.35	934.15	944.40
	LB w	266.16 266.88 270.85 271.45	271.56	271.99	275.13
	BP w	265.99 266.71 270.68 271.28	271.38	271.82	274.94
	p lb w	0.3105 0.3163 0.3161 0.3264	0.3289	0.3329	0.3366
	p bp w	0.3108 0.3166 0.3165 0.3268	0.3293	0.3333	0.3369
	p lb s	0.0000 0.0000 0.0000 0.0000	0.0000	0.0000	0.0000
	p bp s	0.0000 0.0000 0.0000 0.0000	0.0000	0.0000	0.0000

Table 9

 9 Empirical size (in %) of the modified and standard versions of the LB and BP tests in the case of a weak FARIMA(1, d 0 , 1) defined by (19) with θ 0 = (0.9, 0.2, d 0 ) and where ω = 0.4, α 1 = 0.3 and β 1 = 0.3 in (20). The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000.

	d 0	Length n	Lag m	LBsn	BPsn	LBw	BPw	LBs	BPs
			1	4.9	4.9	6.7	6.7	n.a.	n.a.
			2	3.8	3.8	6.3	6.3	n.a.	n.a.
	0.05	n = 1, 000	3	3.2	3.2	5.2	5.2	n.a.	n.a.
			6	3.9	3.8	4.9	4.8	18.5	18.3
			12	2.3	2.3	4.1	4.0	10.2	9.7
			15	2.7	2.3	4.4	4.2	9.7	9.3
			1	5.1	5.1	5.6	5.6	n.a.	n.a.
			2	4.9	4.9	5.4	5.4	n.a.	n.a.
	0.05	n = 5, 000	3	2.6	2.6	5.0	5.0	n.a.	n.a.
			6	3.5	3.5	4.4	4.4	19.6	19.6
			12	2.7	2.7	3.3	3.2	11.4	11.4
			15	3.4	3.4	4.2	4.1	10.8	10.7
			1	4.8	4.8	6.9	6.9	n.a.	n.a.
			2	4.8	4.8	6.7	6.7	n.a.	n.a.
	0.05	n = 10, 000	3	4.7	4.7	5.5	5.5	n.a.	n.a.
			6	3.3	3.3	6.4	6.4	20.2	20.2
			12	4.2	4.2	6.3	6.3	12.4	12.3
			15	3.6	3.6	5.5	5.5	11.6	11.6
			1	5.3	5.3	7.8	7.7	n.a.	n.a.
			2	3.6	3.4	5.7	5.7	n.a.	n.a.
	0.20	n = 1, 000	3	3.1	3.1	4.9	4.8	n.a.	n.a.
			6	3.3	3.2	4.5	4.5	17.6	17.4
			12	2.3	2.0	4.1	4.1	9.4	8.9
			15	2.4	2.1	4.4	4.2	9.0	8.1
			1	4.6	4.6	4.3	4.3	n.a.	n.a.
			2	4.3	4.3	4.4	4.4	n.a.	n.a.
	0.20	n = 5, 000	3	3.1	3.1	4.4	4.3	n.a.	n.a.
			6	4.1	4.1	3.9	3.9	19.0	19.0
			12	2.6	2.6	2.9	2.9	10.9	10.6
			15	3.4	3.3	4.0	4.0	10.0	9.9
			1	4.8	4.8	5.1	5.1	n.a.	n.a.
			2	4.7	4.7	5.0	5.0	n.a.	n.a.
	0.20	n = 10, 000	3	4.5	4.5	4.8	4.8	n.a.	n.a.
			6	3.5	3.5	5.6	5.6	19.1	19.1
			12	4.1	4.1	5.9	5.9	12.1	12.1
			15	3.7	3.7	5.3	5.3	11.3	11.3
			1	4.4	4.4	11.1	11.0	n.a.	n.a.
			2	3.4	3.4	5.4	5.3	n.a.	n.a.
	0.45	n = 1, 000	3	3.1	3.1	4.9	4.9	n.a.	n.a.
			6	3.1	2.9	4.5	4.4	15.3	15.1
			12	2.2	2.1	4.0	4.0	7.9	7.5
			15	2.1	2.0	4.4	4.3	7.0	6.5
			1	3.9	3.9	4.2	4.2	n.a.	n.a.
			2	3.4	3.4	4.2	4.2	n.a.	n.a.
	0.45	n = 5, 000	3	2.9	2.9	4.4	4.4	n.a.	n.a.
			6	3.5	3.5	3.9	3.9	18.4	18.4
			12	2.4	2.4	2.8	2.7	9.9	9.8
			15	3.2	3.2	3.9	3.8	9.2	9.2
			1	4.6	4.6	5.3	5.3	n.a.	n.a.
			2	4.3	4.3	5.1	5.0	n.a.	n.a.
	0.45	n = 10, 000	3	3.5	3.5	5.0	5.0	n.a.	n.a.
			6	2.8	2.8	5.3	5.3	19.3	19.3
			12	4.2	4.2	5.5	5.5	12.2	12.2
			15	3.6	3.5	5.5	5.5	11.4	11.4

Table 10

 10 Empirical size (in %) of the modified and standard versions of the LB and BP tests in the case of a weak FARIMA(1, d 0 , 1) defined by (19)-(21) with θ 0 = (0.9, 0.2, d 0 ). The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000.

	d 0	Length n	Lag m	LBsn	BPsn	LBw	BPw	LBs	BPs
			1	5.1	5.1	7.3	7.3	n.a.	n.a.
			2	3.6	3.6	6.9	6.9	n.a.	n.a.
	0.05	n = 1, 000	3	2.9	2.9	4.3	4.1	n.a.	n.a.
			6	2.6	2.5	3.1	3.0	10.3	10.3
			12	0.9	0.9	1.2	1.1	8.7	8.3
			15	0.4	0.4	1.0	0.8	8.0	7.3
			1	3.9	3.9	5.4	5.4	n.a.	n.a.
			2	3.9	3.9	5.9	5.9	n.a.	n.a.
	0.05	n = 5, 000	3	3.9	3.9	5.5	5.5	n.a.	n.a.
			6	3.2	3.1	3.8	3.8	10.6	10.6
			12	2.4	2.4	3.5	3.4	8.3	8.2
			15	2.7	2.7	3.3	3.3	8.4	8.3
			1	5.0	5.0	5.2	5.2	n.a.	n.a.
			2	4.9	4.9	4.5	4.5	n.a.	n.a.
	0.05	n = 10, 000	3	3.8	3.8	5.6	5.6	n.a.	n.a.
			6	3.6	3.6	4.5	4.5	10.4	10.4
			12	3.3	3.3	4.3	4.3	8.5	8.4
			15	4.7	4.7	3.8	3.8	7.7	7.4
			1	5.7	5.6	10.1	10.0	n.a.	n.a.
			2	3.4	3.4	5.5	5.5	n.a.	n.a.
	0.20	n = 1, 000	3	3.7	3.7	4.0	4.0	n.a.	n.a.
			6	2.9	2.8	2.5	2.4	10.2	9.7
			12	0.9	0.9	1.1	1.1	7.9	7.2
			15	0.5	0.5	0.8	0.8	7.5	6.9
			1	3.5	3.5	4.0	3.9	n.a.	n.a.
			2	3.7	3.7	4.3	4.3	n.a.	n.a.
	0.20	n = 5, 000	3	4.1	4.1	5.0	5.0	n.a.	n.a.
			6	3.1	3.1	3.5	3.5	10.0	10.0
			12	2.8	2.8	3.3	3.3	8.2	8.2
			15	2.4	2.4	3.1	3.1	7.9	7.8
			1	5.1	5.1	4.8	4.8	n.a.	n.a.
			2	4.7	4.7	4.2	4.2	n.a.	n.a.
	0.20	n = 10, 000	3	3.8	3.8	4.7	4.7	n.a.	n.a.
			6	3.8	3.8	4.1	4.1	10.1	10.1
			12	3.4	3.4	4.0	4.0	8.0	8.0
			15	4.8	4.8	3.6	3.6	7.5	7.4
			1	3.8	3.8	12.1	12.0	n.a.	n.a.
			2	2.4	2.4	4.4	4.4	n.a.	n.a.
	0.45	n = 1, 000	3	2.7	2.6	3.8	3.7	n.a.	n.a.
			6	3.2	3.0	2.3	2.3	8.3	7.9
			12	1.1	0.9	1.0	0.9	6.4	6.3
			15	0.3	0.3	1.4	1.1	6.8	6.4
			1	3.1	3.1	4.4	4.4	n.a.	n.a.
			2	2.7	2.7	4.5	4.5	n.a.	n.a.
	0.45	n = 5, 000	3	3.2	3.2	4.9	4.9	n.a.	n.a.
			6	3.2	3.1	3.4	3.4	9.7	9.7
			12	3.3	3.3	3.3	3.3	7.3	7.3
			15	2.4	2.4	3.2	3.1	7.2	7.0
			1	5.1	5.1	4.8	4.8	n.a.	n.a.
			2	4.9	4.9	4.3	4.3	n.a.	n.a.
	0.45	n = 10, 000	3	3.6	3.6	4.9	4.9	n.a.	n.a.
			6	3.5	3.5	4.3	4.2	10.2	10.2
			12	3.7	3.7	3.7	3.7	7.7	7.6
			15	4.8	4.8	3.9	3.9	7.2	7.1

Table 11

 11

Table 13

 13

	d 0	Length n	Lag m	LBsn	BPsn	LBw	BPw	LBs	BPs
			1	22.5	22.5	32.8	32.7	n.a.	n.a.
			2	27.3	27.3	41.7	41.8	n.a.	n.a.
	0.05	n = 5, 000	3	32.4	32.3	20.1	20.0	n.a.	n.a.
			6	52.1	52.0	34.0	34.0	55.8	55.7
			12	54.1	54.1	23.5	23.5	34.2	34.1
			15	53.9	53.4	17.1	16.9	31.9	31.8
			1	36.1	36.1	53.2	53.2	n.a.	n.a.
			2	44.9	44.9	64.5	64.5	n.a.	n.a.
	0.05	n = 10, 000	3	56.5	56.5	33.1	33.1	n.a.	n.a.
			6	83.1	83.1	71.2	71.2	86.4	86.2
			12	84.0	83.9	59.0	59.0	70.4	70.2
			15	80.6	80.5	40.1	40.1	67.4	67.2
			1	14.6	14.5	51.0	50.9	n.a.	n.a.
			2	21.8	21.8	67.1	67.1	n.a.	n.a.
	0.20	n = 5, 000	3	22.4	22.3	37.7	37.7	n.a.	n.a.
			6	32.3	32.3	68.3	68.3	81.9	81.9
			12	51.6	51.5	55.9	55.8	68.7	68.5
			15	51.7	51.6	64.2	64.1	64.8	64.6
			1	22.8	22.8	74.1	74.0	n.a.	n.a.
			2	29.6	29.6	86.2	86.2	n.a.	n.a.
	0.20	n = 10, 000	3	32.9	32.9	56.6	56.5	n.a.	n.a.
			6	43.1	43.1	92.3	92.3	97.1	97.1
			12	72.9	72.8	88.3	88.3	93.8	93.8
			15	71.2	71.1	89.1	88.9	92.0	92.0
			1	30.1	30.1	99.8	99.8	n.a.	n.a.
			2	40.1	40.1	100.0 100.0	n.a.	n.a.
	0.45	n = 5, 000	3	57.9	57.9	100.0 100.0	n.a.	n.a.
			6	65.7	65.7	100.0 100.0 100.0 100.0
			12	78.8	78.5	100.0 100.0 100.0 100.0
			15	84.7	84.6	100.0 100.0 100.0 100.0
			1	62.2	62.2	99.9	99.9	n.a.	n.a.
			2	72.2	72.2	100.0	99.9	n.a.	n.a.
	0.45	n = 10, 000	3	84.8	84.8	100.0 100.0	n.a.	n.a.
			6	89.8	89.7	100.0 100.0 100.0 100.0
			12	97.7	97.7	100.0 100.0 100.0 100.0
			15	99.0	99.0	100.0 100.0 100.0 100.0

To cite few examples of nonlinear processes, let us mention: the generalized autoregressive conditional heteroscedastic (GARCH) model (see[START_REF] Francq | GARCH Models: Structure, Statistical Inference and Financial Applications[END_REF]), the self-exciting threshold autoregressive (SETAR),

This is a necessary and sufficient condition for the existence of a nonanticipative stationary solution process ( t ) t∈Z with fourth-order moments (see(Francq and Zakoïan, 2010, Example 2.3)).

Diagnostic checking in FARIMA models with uncorrelated but non-independent error terms: Complementary results that are not submitted for publication

Appendix C: Supplemental material: Additional Monte Carlo experiments

For the nominal level α = 5%, the empirical size over the N independent replications should vary between the significant limits 3.6% and 6.4% with probability 95%. When the relative rejection frequencies are outside the 95% significant limits, they are displayed in bold type in Tables.

C.1. FARIMA models with a = 0 and b = 0

Table 8 displays the relative rejection frequencies of the null hypothesis (H0) that the DGP follows a strong FARIMA model ( 19), over the N independent replications. When p = q = 1 for all tests, the percentages of rejection belong to the confident interval with probabilities 95%, except for LB s and BP s (see Table 8). Consequently all these tests well control the error of first kind.

We draw the conclusion that in these strong FARIMA cases the proposed modified version may be clearly preferable to the standard ones. Now, we repeat the same experiments on two weak FARIMA models. As expected Tables 9 and10 show that the standard LB s or BP s test poorly performs in assessing the adequacy of these particular weak FARIMA models. Indeed, we observe that

• the observed relative rejection frequencies of LB s and BP s are definitely outside the significant limits, • the errors of the first kind are only globally well controlled by the proposed tests when n is large.

We also investigate the case where the GARCH model ( 20) have infinite fourth moments. As showing in Figures 5,. . . ,10 the results are qualitatively similar to what we observe here in Tables 9 and10.

In this section, we repeat the same experiments as in Section 4.1 to examine the power of the tests for the null hypothesis of Model (19) against the following FARIMA alternative defined by

with θ 0 = (a, b 1 , b 2 , d 0 ) and where the innovation process ( t ) t∈Z follows a strong or weak white noise introduced in Section 4.1.

For each of these N replications we fit a FARIMA(1, d, 1) model ( 19) and perform standard and modified tests based on m = 1, 2, 3, 6, 12 and 15 residual autocorrelations.

Tables 11, 12 and 13 compare the empirical powers of Model (78) with θ 0 = (0.9, 1, -0.2, d 0 ) over the N independent replications. For these particular strong and weak FARIMA models, we notice that the standard BP s and LB s and our proposed tests have very similar powers except for BP sn and LB sn when n = 5, 000. Empirical size (in %) of the modified and standard versions of the LB and BP tests in the case of a weak FARIMA(0, d 0 , 0) defined by (19) with θ 0 = (0, 0, d 0 ) with ω = 0.4, α 1 = 0.3 and β 1 = 0.3 in (21). The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000. Empirical size (in %) of the modified and standard versions of the LB and BP tests in the case of a weak FARIMA(0, d 0 , 0) defined by ( 19)-( 21) with θ 0 = (0, 0, d 0 ). The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000. Empirical size (in %) of the modified and standard versions of the LB and BP tests in the case of a weak FARIMA(1, d 0 , 1) defined by ( 19) with θ 0 = (0.9, 0.2, d 0 ) and where ω = 0.4, α 1 = 0.3 and 20). The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000. Empirical size (in %) of the modified and standard versions of the LB and BP tests in the case of a weak FARIMA(1, d 0 , 1) defined by (19) with θ 0 = (0.9, 0.2, d 0 ) and where ω = 0.04, α 1 = 0.12 and β 1 = 0.85 in (20). The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000. Table 20 Empirical size (in %) of the modified and standard versions of the LB and BP tests in the case of a weak FARIMA(1, d 0 , 1) defined by (19) with θ 0 = (0.9, 0.2, d 0 ) and where ω = 0.04, α 1 = 0.12 and β 1 = 0.85 in (20). The nominal asymptotic level of the tests is α = 5%. The number of replications is N = 1, 000.