Marcel Aloy
email: marcel.aloy@univ-amu.fr

Gilles De Truchis
email: gilles.de-truchis@univ-amu.fr

Estimation and Testing for Fractional Cointegration

Keywords: JEL classification: C32, C15, C53, C58 Fractional cointegration, Long memory, Monte Carlo experiment, Cointegration test

 to estimate jointly β and d, and possibly other nuisance parameters, for a wide range of integration orders when regressors are I(1). The finite sample properties of this estimator are compared with various popular estimation methods of parameters β (LSE, ADL, DOLS, FMLS, GLS, MLE, NBLS, FMNBLS), and d (LPE,LWE,LPM,FML) through a Monte Carlo experiment. We also investigate the crucial question of testing for fractional cointegration (that is, d < 1). The simulation results suggest that the one-step methodology generally outperforms others methods, both in terms of estimation precision and reliability of statistical inferences. Finally we apply this methodology by studying the long-run relationship between stock prices and dividends in the US case.

INTRODUCTION

This paper deals with estimation of a generalized form of the standard triangular cointegration model:

(1 -L) d (y t -βx t ) = ε 1t , (1 -L) δ x t = ε 2t , t = 1, ..., n (1)
In equation ( 1), we assume that t = [ε 1t , ε 2t ] is a vector of I(0) variables, ω t = [y t , x t ] a vector of I(δ) variables and we define (1 -L) d by its binomial expansion

1 (1 -L) d = 1 -dL - d(1 -d) 2! L 2 - d(1 -d)(2 -d) 3! L 3 -... = +∞ k=0 Γ(k -d) Γ(k + 1)Γ(-d) L k (2) Γ(z) = +∞ 0 t z-1 e -t dt,
where L is the lag operator and (1 -L) d the fractional difference operator (denoted ∆ d ). Equation (1) extends the traditional cointegration framework since we allow d ∈ (0, 1) and δ ∈ (0, 1) to be real numbers, while d = 0 and δ = 1 corresponds to the standard cointegration model. According to Granger (1986) and [START_REF] Engle | Cointegration and error correction: representation, estimation and testing[END_REF], cointegration arises when d < δ. The case where d < δ implies a dimensionality reduction, revealing a near-zero-frequency correlation between the two series. Denoting λ = δd the reduction parameter and C(δ, λ) the cointegration relationship we can identify five cases: i) strong fractional cointegration C(δ, λ): 1/2 < δ ≤ 1, 0 ≤ d < 1/2 and λ > 1/2;

ii) weak fractional cointegration C(δ, λ): 0 < δ ≤ 1, 0 < d < δ and λ < 1/2;

iii) stationary fractional cointegration C(δ, λ): 0 < δ < 1/2, 0 ≤ d < δ and λ < 1/2; iv) standard cointegration C(1, 1): δ = 1, d = 0 and λ = 1; v) spurious regression: 1/2 ≤ δ ≤ 1, 1/2 ≤ d ≤ δ and λ = 0.

The concepts of stationary and weak fractional cointegration result from properties of fractionally integrated process demonstrated by [START_REF] Granger | An introduction to long memory time series models and fractional differencing[END_REF] and Hosking (1981): an I(d) process is covariance stationary when d < 0.5 and covariance non-stationary when d ≥ 0.5. As long a d < 1 the process is mean reverting. Consequently , in equation (1), d < δ ≤ 1 implies the existence of a reversion mechanism toward the long run equilibrium.

In the standard cointegration framework the long run coefficient β is the only parameter of interest and the literature provides many studies dealing with estimation of β in this specific case. [START_REF] Panopoulou | A comparison of autoregressive distributed lag and dynamic LSE cointegration estimators in the case of a serially correlated cointegration error[END_REF] lead a finite sample comparison of the most commonly used estimators. In the more general case of fractional cointegration, parameters of interest in equation (1) are β, d and δ. Consequently, the joint estimation of parameters in equation ( 1) is tedious and existing studies focuse on only specific cases. The seminal paper of [START_REF] Cheung | A fractional cointegration analysis of purchasing power parity[END_REF] suggests a two-step methodology carrying out the Least Squares Estimator (LSE) and the Log-Periodogram regression Estimator (LPE) of d, developed by [START_REF] Geweke | The estimation and application of long memory time series models[END_REF]. It consists in estimating β in a first step and d, upon collected residuals, in a second step. Their approach cover the fractional cointegration case but they restrict δ = 1. [START_REF] Marinucci | Semiparametric fractional cointegration[END_REF] provide a good overview of fractional cointegration and investigate asymptotic and finite sample properties of the Narrow-Band Least Squares estimator of β (introduced by Robinson (1994)) for different parameter regions. Nielsen (2007) suggests a quasi-maximum likelihood estimator in order to estimate jointly β, d and δ. This local Whittle estimator operates in frequency domain and is consistent and asymptotically normal for the entire stationary region of δ and d (so it is restricted to the stationary fractional cointegration case). Finally, [START_REF] Hualde | Root-n-consistent estimation of weak fractional cointegration[END_REF] present a time domain maximum likelihood estimator of β, d and δ. [START_REF] Hualde | Root-n-consistent estimation of weak fractional cointegration[END_REF] demonstrate consistency and asymptotic normality of this estimator under assumptions of weak fractional cointegration. However their methodology operates in two non-linear optimizations.

The aim of this paper is to analyse various estimation strategies of bivariate fractional cointegration models under the restriction δ = 1. This specific restriction is relevant to a wide range of financial and macroeconomics applications. For instance, [START_REF] Cheung | A fractional cointegration analysis of purchasing power parity[END_REF] bring out some evidence of cointegration at orders C(1, 0.4) exploring the purchasing power parity relationship between United States and United Kingdom (see [START_REF] Henry | The long range dependence paradigm for macroeconomics and finance[END_REF] for a survey of applications in macroeconomics and finance in presence of long-memory). [START_REF] Marinucci | Semiparametric fractional cointegration[END_REF] analyze the consumption and the GNP in the U.S. case and find them integrated at order 1. However, they get inconclusive results concerning the integration order of cointegrating errors. [START_REF] Caporale | Unemployment and input prices: a fractional cointegration approach[END_REF] test the cointegration hypothesis between unemployment and input prices and find some evidence of cointegration at order C(1, λ), where λ depends on the residuals specification (AR(1), AR(2) or white noise). [START_REF] Baillie | Cointegration, fractional cointegration and exchange rate dynamics[END_REF] investigate dynamic relationships between different exchange rate series and conclude to cointegration at orders C(1, 0.11). Using the same times series, Nielsen (2004) tests the C(1, 1) cointegration hypothesis against the alternative of fractional cointegration. This paper leads up to the crucial question of testing cointegration that constitutes our second point of interest. [START_REF] Cheung | A fractional cointegration analysis of purchasing power parity[END_REF] provide a finite sample analysis of a fractional cointegration test based upon the log-periodogram estimator. The empirical distribution of their test statistic under the null appears to be negatively skewed. As a consequence this cointegration test rejects the null hypothesis of no cointegration too often. [START_REF] Lobato | Efficient Wald tests for fractional unit roots[END_REF] develop a fractional Wald test of fractional integration and suggest that this test can also be applied for testing fractional cointegration. Since they provide neither asymptotic nor finite sample properties of their test in the case of fractional cointegration, we include it in our simulations.

Under the assumption δ = 1, there are two free parameter of interest: d and β. In the two-step methodology initiated by [START_REF] Cheung | A fractional cointegration analysis of purchasing power parity[END_REF], there is a wide range of candidate to estimate d. The main drawback of the semi-parametric frequency approach of [START_REF] Geweke | The estimation and application of long memory time series models[END_REF] used by [START_REF] Cheung | A fractional cointegration analysis of purchasing power parity[END_REF] is a negative bias in small sample. [START_REF] Andrews | A bias-reduced logperiodogram regression estimator for the long memory parameter[END_REF] suggest a Log-Periodogram (LPM) Modified estimator in order to correct the bias. In comparison with the LPE and the LPM, the Local Whittle semi-parametric Estimator (LWE) of Künsch (1987) and Robinson (1995) has convenient asymptotic properties, but requires a numerical optimization. In a different spirit, a parametric frequency maximum likelihood (FML) approach is proposed by [START_REF] Fox | Large-sample properties of parameter estimates for strongly dependent stationary gaussian series[END_REF]. Finally, we can can also mention the time domain maximum likelihood estimator of Sowell (1992). An exhaustive list of fractional integration estimator is considered in the Monte-Carlo experiment performed by [START_REF] Nielsen | Finite Sample Comparison of Parametric, Semi-parametric, And Wavelet Estimators of Fractional Integration[END_REF]. Estimators we have mentioned above exhibit best finite sample properties according to their study (except Sowell (1992)).

The second parameter of interest is β. In the standard cointegration case, [START_REF] Stock | Asymptotic properties of least squares estimators of cointegrating vector[END_REF] demonstrates that the LSE is still consistent but super-convergent at the rate O(n) rather than the optimal rate O(n 1/2 ). It implies that the LSE will be biased in small sample. This result is extended by [START_REF] Robinson | Narrow-Band Analysis of Nonstationary Processes[END_REF] to the fractional cointegration case: when δ = 1, it can be shown that the LSE converges at the rate n λ for λ > 1/2, (n/ log(n)) 1/2 for λ = 1/2, and n 1/2 for 0 < λ < 1/2. Hence, it easy to recover the standard results, for instance the spurious regression case when λ = 0, and the traditional cointegration case when λ = 1. A second issue is the inconsistency of the LSE estimator of β when cointegrating errors and regressors both have long memory and have non-zero coherence at the zero frequency. Consider the equation y t = βx t + υ t . One sufficient condition of consistency of the LSE estimator of β is that n t=1 υ 2 t / n t=1 x 2 t converges stochastically to zero when n → ∞. This condition is satisfied when δ > 1/2 and d < 1/2, ore more generally when the residuals are "less nonstationary" than y t and x t [START_REF] Robinson | Narrow-Band Analysis of Nonstationary Processes[END_REF]). However, the LSE are generally inconsistent when x t and υ t are correlated and d < δ < 1/2: in this case, n t=1 x 2 t does not dominate n t=1 υ 2 t . Focusing on the stationary case, Robinson (1994) has developed a semi-parametric Narrow-Band Least Squares estimator (NBLS) in the frequency domain. NBLS exploits the dominance of the spectral density of x t on υ t at low frequencies (since cointegration implies δ > d). However the NBLS estimator is confined to the case δ < 1/2 (where y t , x t and z t have finite variances) and in practice it is difficult to ensure that we are in the stationary regions of δ. In order to solve this drawback, [START_REF] Nielsen | Fully modified narrowband least squares estimation of weak fractional cointegration[END_REF] suggest a Fully Modified Narrow-Band Least Squares (FMNBLS) to take into account bias that appears in the limiting distribution of the NBLS in presence of a non-zero long run coherence between regressors and cointegrating errors, in the weak fractional cointegration case (λ < 1/2). Their approach is equivalent to the so called time domain Fully-Modified Least Squares (FMLS) estimator of [START_REF] Phillips | Statistical inference in instrumental regressions with I(1) processes[END_REF]. [START_REF] Panopoulou | A comparison of autoregressive distributed lag and dynamic LSE cointegration estimators in the case of a serially correlated cointegration error[END_REF] propose to exploit an Autoregressive Distributed Lags structure of the equation (1) to estimate β. They demonstrate the good finite sample properties of the LSE in the ADL form throughout simulations. They also investigate the Dynamic Ordinary Least Squares of [START_REF] Stock | A simple estimator of cointegrating vectors in higher-order integrated systems[END_REF] and show this estimator suffer from truncation bias in finite sample. An other candidate to estimate β is the time domain Generalized Least Squares (GLS) estimator. A frequency domain version of this parametric estimator is given by [START_REF] Robinson | Time series regression with long-range dependence[END_REF]: they demonstrate root-n-consistency and asymptotic normality of this estimator when the true specification of errors is known. More recently, they adapt it to unknown specification of disturbance (see [START_REF] Hidalgo | Adapting to Unknown Disturbance Autocorrelation in Regression With Long Memory[END_REF]). Finally, we can mention the well-known Maximum Likelihood Estimator (MLE) of Johansen (1988): since this estimator is not designed to estimate fractional cointegrated models, we can expect mixed results in non-stationary regions of cointegrating errors.

All estimators presented above are relevant either in estimation of β or in estimation of d. In the spirit of Nielsen (2007) and [START_REF] Hualde | Root-n-consistent estimation of weak fractional cointegration[END_REF] we are interested in the joint estimation of parameters of interest, which are limited to β and d since we assume δ = 1. We are also interested in estimating those parameters for a wide range of integration orders. Since we adopt a parametric approach, we cannot avoid issues of serial correlation and the lack of orthogonality between errors and regressors. In consequence, we suggest an adaptation of the model of [START_REF] Hualde | Root-n-consistent estimation of weak fractional cointegration[END_REF] that is still relevant with the Maximum Likelihood Estimator (MLE). We also advise a selection procedure of the parametric form of this model. We investigate the finite sample properties of the MLE of β and d. The question of testing cointegration is also performed using the asymptotic variance, derived by Tanaka (1999), in the framework of a Wald test. We compare our one step methodology with most of estimators mentioned previously. The outline of the paper is as follows. The section 2 presents some technical aspects of the fractional cointegration framework and the model we consider, and then details various estimators used in our simulation. In the section 3 we comment our simulations and the results. We finally lead an empirical study on the present value model in the section 4. The section 5 concludes.

SIMULATION DESIGN

The fractional cointegration model

We consider now a generalized form of equation ( 1) that allows for serial correlation of errors and weak exogeneity of regressors. Let ω t and ζ t be two bivariate processes where ω t = [y t , x t ] is a vector of I(1) variables and ζ t = [υ t , z t ] a vector of their residuals. We assume that [∆ d υ t , z t ] is a two dimensional vector of I(0) variables with a VAR(1) structure driven by t = [ε 1t , ε 2t ] and the data generating process (DGP) for y t is given by the following system,

y t = βx t + υ t
(3)

x t = x t-1 + z t (4) (1 -L) d υ t z t = α 11 α 12 α 21 α 22 (1 -L) d υ t-1 z t-1 + ε 1,t ε 2,t (5)
ε 1t ε 2t ∼ i.i.d N 0 0 , σ 11 σ 12 σ 21 σ 22 (6)
Our analysis will be restricted to the case where α 21 = α 12 = α 22 = 0: if the two last restrictions are only simplifying assumptions, the first one (α 21 =0) is crucial as it insures that x t is weakly exogenous for β, since x t does not adjust to past cointegration errors. In the standard I(1)/I(0) framework, Johansen (1992) shows that under weak exogeneity of x t , the single equation modelling is equivalent to the ML estimation of the full system. We assume further that eigenvalues of the matrix A=[α i j ], i, j = 1, 2 are less than one in modulus: under the restriction α 21 = 0, this hypothesis implies α 11 < 1 (and α 22 < 1 if one would relax our simplifying assumption α 22 = 0).

The contemporaneous correlation between the elements of e t leads us to suggest a triangular representation in which we assume

ε 1t = αε 2t + ε t (7)
where,

α = σ 12 /σ 22 , V(ε) = σ 11 -(σ 2 12 /σ 22 ) = σ 2 ε , ε t ∼ i.i.d N(0, σ 2 ε )
The process for the cointegration errors can thus be written more compactly as it follows

∆ d υ t = α 11 ∆ d υ t-1 + αε 2t + ε t (8)
Under our set of assumptions, it results that: x t is a weakly exogenous I(1) process for β and follows a simple random walk; -assuming β 0, x t and y t are two I(1) fractional cointegrated processes provided that d < 1. These two processes are not cointegrated if d = 1, and are cointegrated under the usual I(1)/I(0) framework if d = 0; -if x t and y t are two I(1) processes and are cointegrated under the I(1)/I(0) framework, the long-run multiplier between x t and y t is given by the parameter β while the short-run multiplier is given by the sum β + α. It is thus important to consider the case where α 0 since it corresponds to a short-run undershooting (α < 0) or overshooting (α > 0) following a shock on x t . It is also important to consider the case where α 11 0: if the two series are cointegrated under the I(1)/I(0) framework, the cointegrating errors will follow a simple and unrealistic white noise if α 11 = 0.

Restricting some coefficients of this model, we define four specifications which will be used later in our Monte Carlo study.

A 0 0 0 0 0 B 1 0 > 0 0 0 0 B 2 < 0 0 0 0 0 B 3 < 0 > 0 0 0 0
The first model, denoted A, describes a bivariate relationship between y t and x t in line with [START_REF] Cheung | A fractional cointegration analysis of purchasing power parity[END_REF]. Indeed, innovations are a simple fractional white noise integrated at order d while x t is defined as a random walk with drift. The model B 1 corresponds to the case where α = 0 and α 11 > 0: the cointegration residuals follows thus an ARFIMA(1,d,0) process with a positive AR coefficient (since α 11 > 0). Conversely, the model B 2 corresponds to the case of α 11 = 0 and α < 0. In other terms, we introduce here the second order bias since α < 0 breaks the orthogonality condition: E[υ t |x t ] 0. In the specifications B 3 , the model faces simultaneously serial correlation and engoneity in the cointegration residuals (α 11 > 0 and α < 0) : innovations follow an ARFIMA(1,d,0) process with a positive AR coefficient and are adversely influenced by current shocks on x t (since α < 0).

Estimators

We will examine eight different estimators of the long run coefficient β: the ordinary least squares (LSE), the generalized least squares (GLS), the fully modified least squares (FMLS), the dynamic LSE (DOLS), the autoregressive distributed lags (ADL), the maximum likelihood estimator (MLE) of Johansen (1988) (although the methodology of Johansen (1988) is clearly limited to the standard I(1)/I(0) case, we considered that it was interesting to evaluate its perfor-mances in the case of fractional cointegration), the narrow band least squares (NBLS) and the fully modified narrow band least squares (FMNBLS).

Concerning the fractional differencing parameter (d) of the cointegrating errors, we will consider four estimators: the log-periodogram regression estimator (LPE), the modified logperiodogram estimator (LPM), the gaussian semi-parametric estimator (LWE) and the approximate Whittle Frequency domain Maximum Likelihood estimator (FML). Notice that these estimators of d have not been developed to be applied to estimated residuals. Moreover, some of them are semi-parametric and thus we can expect some biases in presence of short run dynamics in residuals. Alternatively, some of them are parametric and they potentially outperform semi-parametric estimators, provided that we are able to select the true underlying data generation process. In addition we suggest a parametric model, based on Tanaka (1999) and [START_REF] Hualde | Root-n-consistent estimation of weak fractional cointegration[END_REF], that allows to estimate simultaneously β and d. It clearly appears as an alternative convenient way to address issues raised by the two-step methods. We denote it FCMLE.

Estimators of β

In the fractional cointegration case, the least squares estimates of the cointegrating parameter is consistent and converge in probability at the rate O(n min(2δ-1,λ) ) [START_REF] Robinson | Narrow-Band Analysis of Nonstationary Processes[END_REF] and [START_REF] Robinson | Cointegration in fractional systems with unkown integration orders[END_REF]). Thus, the case where δ = 1 and λ = 1 corresponds to the I(1)/I(0) framework and we find back the convergence results demonstrated by [START_REF] Stock | Asymptotic properties of least squares estimators of cointegrating vector[END_REF]. When δ = 1 and λ = 0, least squares estimates does not converge and the cointegrating regression is spurious. Since we focus on the case δ = 1, we don't cover the stationary case considered by Robinson (1994).

The fractional cointegration framework we consider involves serially dependent errors and thus non-spherical disturbances. Therefore, least squares estimator is no longer efficient and a convenient way to deal with this issue is to use GLS estimators. Since the covariance matrix Ω υ is unknown, we can apply feasible GLS (FGLS). For instance, the [START_REF] Prais | Trend Estimators and Serial Correlation[END_REF] method can be used to build the unknown covariance Ω(ς) υ , which is assumed to be dependent upon several unobservables parameters ς:

Ω-1 υ (ς) = 1 σ 2 υ                             1 -α11 0 0 • • • 0 -α11 1 + α2 11 -α11 0 • • • 0 0 -α11 1 + α2 11 -α11 • • • 0 . . . . . . . . . . . . 0 • • • 0 -α11 1 + α2 11 -α11 0 • • • 0 0 -α11 1                            
The FGLS estimator of β is thus:

βGLS =        n t=1 x t Ω(ς) -1 υ x t        -1        n t=1 x t Ω(ς) -1 υ y t        (9)
When α 0, the orthogonality condition is no longer satisfied since Cov(υ t |x t ) 0. [START_REF] Phillips | Statistical inference in instrumental regressions with I(1) processes[END_REF] suggest a fully modify least square estimator (FMLS), applying non-parametric corrections to the covariance matrix to take into account endogeneity and serial correlation. The FMLS estimator of β is given by βFMLS

=        n t=1 x t x t        -1        n t=1 x t (y t -λ+ ) -T δ+        , ( 10
)
where λ+ is the correction term associated with the correction for the endogeneity bias while δ+ eliminates the non-centrality bias. Assuming that the long-run variance of residuals is positive definite and can be expressed as Ω = Λ + Ξ = Σ + Ξ + Ξ , we can partition Ω and Λ such as

Ω = ω 11 ω 12 ω 21 Ω 22 , Λ = λ 11 λ 12 λ 21 Λ 22 ,
where

Σ = lim n→∞ 1/n n t=1 E(υ t υ t ) and Ξ = lim n→∞ 1/n n-1 j=1 n-j t=1 E(υ t υ t+ j ). It results that λ+ = y t -ω12 Ω-1 22 z t and δ+ = [0, λ21 -Λ22 Ω-1 22 ω21 ] ,
where z t refers to the error term of x t . Alternatively, [START_REF] Stock | A simple estimator of cointegrating vectors in higher-order integrated systems[END_REF] consider a parametric correction based upon an augmented regression equation, in order to provide an estimator robust to endogeneity. The dynamic LSE (DOLS) estimator of

β is βDOLS =         n-k t=k+1 ũt ũ t         -1         n-k t=k+1 ũt ỹt         , (11)
where, u t = [1, x t ] and ũt and ỹt are regression residuals of u t and y t on ζ 2t = (z t+k , ..., z t-k ) , respectively (for a detailed review of these estimators, see [START_REF] Kurozumi | Asymptotic properties of the efficient estimators for cointegrating regression models with serially dependent errors[END_REF]). Johansen (1988) adopts a vectorial approach of the cointegration theory in a I(1)/I(0) framework. Starting from the long run equation, the vectorial error correction model is

(1 -L)y t = µ + Πy t-1 + p-1 j=1 Θ j ∆y t-j + v t
where y t = [y t , x t ] and v t is an k = 2 × 1 vector of innovations. This equation is clearly a combination of I(1) and I(0) variables. However, if the coefficient matrix Π has reduced rank r < k, then we can decompose Π as k × r matrix α and ξ such that Π = αξ . It appears now that ξ y t is stationary since ξ is the matrix of cointegrating vectors, while α is the matrix of adjustment coefficients.

Let u 1t be the residuals in the regressions of ∆y t on z t = {∆y t-1 , ∆y t-2 , ..., ∆y t-p+1 } and u 2t the residuals in the regressions of y t-p on z t . With respect to parameters Θ 1 , ..., Θ p-1 , µ, the maximum likelihood function is the following,

⇒ L -2/n max (r) = |S 00 | r j=1 (1 -λ j ) (12)
where the eigenvalue λ j is the jth largest canonical correlation of (1 -L)y t with y t-1 after correcting for lag differences and S i j the cross-correlation matrix between variables in set i and set j for i, j = u 1t , u 2t1 . Notice that in a bivariate cointegration model, the Π matrix contains at most one cointegrating vector.

When λ = 1, the model defined by equations ( 3), ( 4), ( 5), and ( 6), can also be parametrized as an autoregressive distributed lag (ADL) model (see [START_REF] Panopoulou | A comparison of autoregressive distributed lag and dynamic LSE cointegration estimators in the case of a serially correlated cointegration error[END_REF]). Hence the conditional density of y t , with an ADL(1,1), can be expressed as

D(y t |x t , ω t-1 , θ) = N( βx t + c 1 y t-1 + c 2 x t-1 , σ 2 e ), θ ≡ (c 0 , c 1 , c 2 , σ 2 e ), with β = β + σ 12 σ 22 c 1 = α 11 -α 21 σ 12 σ 22 = α 11 c 2 = α 12 -(α 22 + 1 -α 21 β) σ 12 σ 22 -α 11 β = -α 11 β σ 2 u = σ 11 - σ 2 12 σ 22
Hence, the ADL equation is given by

y t = βx t + c 1 y t-1 + c 2 x t-1 + e t , (13)
and the equation of β by

β = β + c 2 1 -c 1 , (14)
where error term e t in this ADL representation is orthogonal to z t . [START_REF] Panopoulou | A comparison of autoregressive distributed lag and dynamic LSE cointegration estimators in the case of a serially correlated cointegration error[END_REF] show that the LSE of ( 13) can be employed to obtain an efficient estimator of β using ( 14). In a seminal paper, Robinson (1994) provides a semi-parametric frequency domain narrowband least squares (NBLS) estimator of β. Robinson (1994) restricts his analysis to the stationary region of δ and d, and deals with long-run non-zero coherence between υ t and x t . Since we impose δ = 1, our simulation is not theoretically relevant with this framework. However, we are still interested in the finite sample performance of the NBLS when δ ≥ 1/2. Assume any non-negative real γ which transforms the following hypothetical model,

(1 -L) γ y t = β(1 -L) γ x t + (1 -L) γ υ t ,
into a stationary model. In the special case investigated by Robinson (1994), γ = 0 since it is assumed that variables of the model are covariance stationary. Then, the NBLS estimator of β is defined by:

βNBLS (m, γ) = F-1 xx (γ, 1, m) Fxy (γ, 1, m), (15)
where the average co-periodogram is

Fq,r (γ, k, l) = 2π n l j=k Re(I qr (γ, λ j )), 0 ≤ k ≤ l ≤ n -1, (16)
and the co-periodogram is

Îq,r (γ, λ) = 1 2πn n t=1 n s=1 ((1 -L) γ q t )((1 -L) γ r s ) e -i(t-s)λ (17)
Nielsen and Frederiksen (2011) extend the NBLS estimator to the case where γ > 0 (in this case, the time series are covariance non-stationary) and propose a fully modified NBLS (FMNBLS) estimator in order to correct the bias that appears in presence of non-zero coherence between errors and regressors. The FMNBLS estimator of β is expressed as

βFMNBLS (m 3 , γ) = βNBLS (m 3 , γ) -λ - d m 3 λ δ m 3 λ d m 2 λ - δ m 2 Γm 2 (γ), ( 18
)
where λ j is the angular frequency. The estimators of the bias is given by

Γ(m 2 , γ) = F-1 xx (γ, m 0 + 1, m 2 ) Fxυ (γ, m 0 + 1, m 2 ), ( 19
)
where the modified average co-periodogram Fq,r is

Fq,r (γ, k, l) = 2π n l j=k Re(e iλ j (d q -d r )/2 I qr (γ, λ j )), 0 ≤ k ≤ l ≤ n -1 (20)
Bandwidths can be chosen such as m 0 = m 3 = 0.3, m 1 > 0.675 and m 2 ∈ (0.675, 0.85), when λ min = 0.4 (λ refers here to the reduction parameter). [START_REF] Nielsen | Fully modified narrowband least squares estimation of weak fractional cointegration[END_REF] use the local Whittle estimator of Robinson (1995) to estimate δ with bandwidth m 1 . [START_REF] Nielsen | Fully modified narrowband least squares estimation of weak fractional cointegration[END_REF] demonstrate the asymptotic normality of this estimator, albeit δ is estimated beforehand.

Estimators of d

In this section we review some estimators of d that we consider in the Monte Carlo study2 . Fist we recall some useful properties of long memory process. Let υ t an autoregressive fractionally integrated moving average model, ARFIMA(p,d,q), defined by

Φ(L)(1 -L) d υ t = Ψ(L)e t , d ∈ (0, 1/2) ( 21
)
where

Φ(L) = 1 - p j=1 φ j L j , Ψ(L) = 1 + q j=1 ψ j L j ,
are polynomials of orders p and q in the lag operator, with roots outside the unit circle.

The autocorrelation function of the process in ( 21) decays at an hyperbolic rate (2d -1) and thus satisfies

ρ k ∼ c ρ k 2d-1 , 0 < c ρ < ∞, as k → ∞ (22)
The autocovariance function of υ t is defined by,

γ(k) = 1 2π π -π f υ (λ)e -ikλ dλ, π ≥ λ ≥ -π (23)
where f υ (λ) denotes the spectral density function of υ t :

f υ (λ) = σ 2 2π |1 -e iλ | -2d Ψ(e iλ ) Φ(e iλ ) 2 , ( 24
)
with λ j = 2π j n the angular frequencies. Under some additional conditions it can be shown that equation ( 22) is equivalent to the following approximation of ( 24), near zero frequency,

f υ (λ) ∼ g|λ| -2d , 0 < g < ∞, as k → 0 (25)
Exploiting those properties, [START_REF] Fox | Large-sample properties of parameter estimates for strongly dependent stationary gaussian series[END_REF] suggest a convenient parametric frequency maximum likelihood estimator (FML) based upon the spectral density of υ t and the log-periodogram I(λ),

I(λ) = 1 2πT n t=1 υ t e itλ 2 , (26)
The Whittle form of the log-likelihood function they obtain is:

L(d, Φ, Ψ, σ 2 ) = - n/2 j=1 ln f y (λ j ) + I(λ j ) f y (λ j ) , (27)
Assuming gaussianity and d ∈ {-1/2, 1/2}, the FML estimator verifies asymptotically

√ n( dFML -d) → N 0, 4π Ω(ς) ,
where Ω(ς) is a p × p matrix, depending on unknown parameters ς and with j, kth entry such that,

ω j,k (ς) = π -π f (υ, ς) ∂ 2 ∂ς j ∂ς k f -1 (υ, ς)dυ
The so called log-periodogram least square regression (LPE) of Geweke and Porter-Hudak (1983) is one of the most commonly applied estimator. While such estimator does not depend on shortrun dynamics parameters, it is less efficient since only √ m-consistency is achieved. From equation (25), [START_REF] Geweke | The estimation and application of long memory time series models[END_REF] suggest to estimate the following equation using the LSE:

ln I(λ j ) = c -d ln(4 sin 2 (λ j /2)) + e(λ j ), j = 1, ..., m (28)
The bandwidth number for the log-periodogram is arbitrarily fixed to m = √ n. The LSE of d ∈ (-1/2, 1/2) is asymptotically normally distributed such as:

√ m( dLPE -d) → N 0, π 2 24
Andrews and Guggenberger ( 2003) investigate the bias source of the LPE estimator and suggest to replace the constant term c in (28) by the polynomial R r=0 c τ λ 2r j , leading to the new regression model:

ln I(λ j ) = R r=0 c τ λ 2r j -d ln(4 sin 2 (λ j /2)) + ε, (29)
Thus, assuming that d ∈ {-1/2, 1/2}, the asymptotic distribution of this modified log-periodogram estimator (LPM) is

√ m( dLPM -d) → N 0, π 2 24 c R , c R=1 = 2.25, d ∈ {-1/2, 1/2}
Another semi-parametric approach is suggested by Künsch (1987) using a local Whittle approach and revisited by Robinson (1995). The Whittle form log-likelihood function of the local Whittle estimator (LWE) is defined as follows:

L(g, d) = - 1 m m j=1         ln(gλ -2d j ) + I(λ j ) gλ -2d j         , (30)
where gλ -2d corresponds to the approximation of the spectral density in (25). Robinson (1995) also demonstrates that the asymptotic distribution of the LWE is normal:

√ m( dLWE -d) → N (0, 1/4) , d ∈ {-1/2, 1/2}
Various others long memory estimators have been developed during the last three decades (see [START_REF] Nielsen | Finite Sample Comparison of Parametric, Semi-parametric, And Wavelet Estimators of Fractional Integration[END_REF] for a Monte Carlo comparison of numerous ARFIMA(p,d,q) estimators). However, we limit ourselves to the most diffused and efficient.

A simultaneous estimator of β and d

Most of the time, the two-step methodology (e.g. [START_REF] Cheung | A fractional cointegration analysis of purchasing power parity[END_REF]) uses semi-parametric methods to estimate fractional integration parameters and parametric methods to estimate longrun parameters. This approach involves a loss of efficiency since the estimators are globally less than √ n-consistent. According to [START_REF] Hualde | Root-n-consistent estimation of weak fractional cointegration[END_REF] these parameters ought to be jointly estimated parametrically to achieve √ n-consistency and they advise to use the Gaussian Maximum Likelihood Estimator in order to estimate δ, β and d. Outside the cointegration context, Tanaka (1999) proves that the time domain Gaussian maximum likelihood estimator of d is asymptotically efficient and normally distributed. He shows that √ n( dd) → N(0, 6/π 2 ) when υ t is a fractional white noise, and that

√ n( d -d) → N(0, ω -2 ),
where ω is detailed in (39), when υ t follows a process similar to (21). Moreover, Tanaka (1999) proposes a two-step parametric cointegration methodology based on the LSE of β and the MLE of d. We suggest to estimate jointly β and d by MLE, assuming δ = 1 and Gaussian errors. First we rewrite the model (8) using the following reparametrization

ε t = ∆ dy t + ∆ d (y t-1 -βx t-1 ) -β∆ d x t - p j=1 α j 11 (∆ d y t-j -β∆ d x t-j ) - m l=1 α l ∆ d x t-l+1 + α l (∆x t-l+1 -∆ d x t-l+1 ) , ( 31
)
where d = d + δ = d + 1 and ∆ d is the binomial expansion defined in (2). The Gaussian MLE of (31), denoted FCMLE, is defined as

θ = arg min θ f θ (x), d ∈ D, (32)
where θ = (d, β) is the vector of parameters of interest, D ⊆ (0, 1) and the log likelihood function of f θ (x) is given by

L(θ | y) = - 1 2 n t=1 ln(2π) + ln |Ω| + ε t Ω -1 ε t (33)
Proving consistency and limiting distribution of the Gaussian MLE of ( 31) is outside the scope of this paper. Difficulties in studying asymptotic theory when δ > 1/2 are briefly discussed in [START_REF] Hualde | Root-n-consistent estimation of weak fractional cointegration[END_REF] and motivates a Monte-Carlo experiment (the use of Monte Carlo simulations when appropriate asymptotic theory is not sufficiently developed is advocated by da Silva and [START_REF] Da | Finite Sample Performance in Cointegration Analysis of Nonlinear Time Series with Long Memory[END_REF]). However, the Gaussian MLE of (31) raises some important issues. For instance, our estimator is subject to the uniform convergence problem described in Saikkonen (1995) : in cointegrated systems, different rates of convergence can apply in different directions of the parameter space. Consequently, we are not sure that the √ n-consistency is achieved for all parameters of interest. One solution to overcome this problem is to concentrate out β (Saikkonen (1995), [START_REF] Robinson | Cointegration in fractional systems with unkown integration orders[END_REF]). Nevertheless, we choose to not concentrate β out and the numerical simulations presented in section 3 suggest that, at least empirically, this issue does not result in major consequences. In addition, our simulation results permit the interpretation that the Gaussian MLE of β(δ, d) is root-n-consistent, since [START_REF] Robinson | Cointegration in fractional systems with unkown integration orders[END_REF] show that the asymptotic distributions of β(δ, d) and β(δ, d) are equivalent as long as d is root-n-consistent.

This model is fully parametric: in order to select the most appropriate model, we suggest to perform an unconstrained estimation and then to proceed to a sequential model selection of lag structure using the t-statistics. Since the true parametric model is unknown, this procedure can generate some selection bias that will be assessed in our Monte Carlo study.

Fractional cointegration tests

Given the estimation methods of the key parameters β and d discussed above, it is an interesting point to test for fractional cointegration hypothesis. Since both series, y t and x t , are assumed to be I(1), the question is to test for the hypothesis:

H 0 : d = 1 (y t and x t are not cointegrated) H 1 : d < 1 (y t and x t are cointegrated) In the standard cointegration framework cointegrating errors are assumed to be I(1) or I(0). Unfortunately, usual unit root test have low power against the fractional alternative (see Dittman (2000) for a comparison of some residuals-based tests for fractional cointegration).

To deal with this issue, [START_REF] Lobato | Efficient Wald tests for fractional unit roots[END_REF] 3 propose a fractional integration Wald test. We apply this test on cointegration residuals in order to test the fractional cointegration hypothesis [START_REF] Lobato | Efficient Wald tests for fractional unit roots[END_REF] mention this possibility in the conclusion of their article). Considering the simplest case where the cointegrating errors follow an fractional white noise, we can rewrite (1

-L) d υ t = ε t as (1 -L)υ t = (1 -(1 -L) d-1 )(1 -L)υ t + ε t and thus, (1 -L)υ t = ϕ 2 ((1 -L) d-1 -1)(1 -L)υ t + ε t , (34)
where ϕ 2 = 0 under the null hypothesis and ϕ 2 = -1 under the alternative hypothesis. To make the regressor continuous when d = 1, the equation ( 34) must be re-written:

(

1 -L)υ t = ϕ 2 v t-1 (d) + ε t , v t-1 (d) = (1 -L) d-1 -1 1 -d (1 -L)υ t (35)
Lobato and Velasco (2007) also consider the case where (1 -L) d is serially correlated. The testing equation is in this case

(1 -L)υ t = ϕ 2 (Φ(L)v t-1 (d)) + p j=1 α j (1 -L)υ t-j + ε t , (36)
where v t-1 (d) is given in equation ( 35). [START_REF] Lobato | Efficient Wald tests for fractional unit roots[END_REF] show that estimation of α is consistent and under the null hypothesis, a simple t-test can be used. We also consider LPE, LWE and FML residual-based test of fractional cointegration (for all semi-parametric tests we select a bandwidth equal to n 0.5 ). Following the methodology of [START_REF] Cheung | A fractional cointegration analysis of purchasing power parity[END_REF], we lead the LPE residual-based test using the following statistic under the null hypothesis of no cointegration:

T LPE = ( d -1) σ , σ2 = mπ 2 6 det XX , X k = 1, -2 ln 2 sin λ k 2
Since residuals are estimated, critical values are non-standard. We use here critical values tabulated by Sephton (2002). The LWE et FML residual-based tests follow the same methodology, the only difference being that we use critical values from a Student distribution in order to test (1 -dυ ) = 0 under the null hypothesis.

In the case of the the Gaussian MLE, we apply the fractional integration Wald test introduced in Tanaka (1999). The test statistic depends on whether errors are serially dependent or not. When errors follow a fractional white noise (that is, (1 -L) d υ t = ε t ) the test statistic is,

W FCMLE = √ n × ( d -d H 0 :d=1 ) 6/π 2 (37)
Conversely, in the case when errors are autocorrelated the test statistic is,

W FCMLE = √ n × ω × ( d -d H 0 :d=1 ), ( 38
)
where ω depends on the Fisher information matrix and the ARMA structure of (31). ω can be easily computed in the simplest case of one AR lag:

ω2 = π 2 6 - 1 -α 2 11 α 2 11 (log(1 -α 11 )) 2 (39)

MONTE-CARLO SIMULATIONS

Several Monte Carlo studies concerning estimators of β and/or d and/or δ are given in the literature. For instance, [START_REF] Robinson | Cointegration in fractional systems with unkown integration orders[END_REF] investigate the finite sample properties of a GLS estimator of β when δd > 1/2. [START_REF] Hualde | Root-n-consistent estimation of weak fractional cointegration[END_REF] explore the finite sample behavior of a MLE of β, d and δ when δd < 1/2. However they restrict their experiment to the comparison with the LSE of β and they do not provide any comparison with other long memory parameters estimators. da Silva and [START_REF] Da | Finite Sample Performance in Cointegration Analysis of Nonlinear Time Series with Long Memory[END_REF] focus on nonlinearity and compare two narrow band estimators of β and three frequency estimators of d, without relaxing orthogonality conditions. They find out that optimal choice depends on the specification: for instance they show that the weighted narrow band least squares estimator has generally both lowest bias and lowest root mean squared error (RMSE). However, in presence of nonlinearity, the simple narrow band least squares estimator of Robinson (1994) is shown to be optimal. Concerning the fractional integration parameter of cointegrating residuals, it appears that the modified local Whittle estimator is not systematically the optimal choice, considering the finite sample performance of the simple local Whittle approach. [START_REF] Marinucci | Semiparametric fractional cointegration[END_REF] compare the NBLS with the LSE and conclude to the superiority of the NBLS in terms of bias and MSE. Another interesting approach is led by [START_REF] Kurozumi | Asymptotic properties of the efficient estimators for cointegrating regression models with serially dependent errors[END_REF], considering only the I(1)/I(0) case. They analytically explain the poor finite sample performance of three efficient estimators (i.e. [START_REF] Phillips | Statistical inference in instrumental regressions with I(1) processes[END_REF], [START_REF] Stock | A simple estimator of cointegrating vectors in higher-order integrated systems[END_REF] and the canonical cointegrating regression method) with a moderate serial correlation. They finally explain why finite samples performance of these estimators are clearly different while they are asymptotically equivalent. Remaining in the I(1)/I(0) paradigm, [START_REF] Panopoulou | A comparison of autoregressive distributed lag and dynamic LSE cointegration estimators in the case of a serially correlated cointegration error[END_REF] compare several time domain estimators of β.

Parametrization

We suggest a Monte Carlo study in order to investigate finite sample properties of both first and second step estimators described above, plus the one step FCMLE. As in [START_REF] Nielsen | Finite Sample Comparison of Parametric, Semi-parametric, And Wavelet Estimators of Fractional Integration[END_REF], we generate 1000 replications of artificial time series, with 512 observations, from the data generation process (DGP) given by equations ( 3), ( 4), ( 5), and ( 6) where β = 1.0. We consider seven values of d which represent different long range dependence behaviors: d = {0.0, 0.2, 0.4, 0.6, 0.8, 0.9, 1.0}. The different values of the nuisance parameters α and α 11 are detailed below:

A 0 0 0 0 0 B 1 0 0.8 0 0 0 B 2 -0.5 0 0 0 0 B 3 -0.5 0.8 0 0 0
For each estimators, we compute the root mean squared error (RMSE), defined by RMSE= The critical value at 5% used for LPE residual-based is -2.25 (see Sephton (2002)). The test of [START_REF] Lobato | Efficient Wald tests for fractional unit roots[END_REF] is based on the FML estimator of d. Bandwidths are arbitrary fixed to n 0.5 for all semi-parametric frequency estimators of d. The optimal lags of ADL, MLE and FML, are selected using the bayesian information criterion (BIC) of Schwarz (1978). Others information criterion have also been envisaged, however, there is no particular changes in the results. Regarding the two-step methodologies, estimation of d are performed using the LSE cointegrating residuals. Concerning the nuisance parameters of in the FCMLE, the model selection is based on sequential t-tests. In order to extract an approximation of the performance of our selection procedure, we also perform the FCMLE under the hypothesis that the true DGP is known (FCMLE k ) and compare it with the selected model FCMLE u . β LSE 0,000 0,006 0,000 0,013 0,000 0,030 -0,002 0,079 0,005 0,197 -0,006 0,344 -0,008 0,571 ADL 0,000 0,006 0,000 0,013 0,000 0,032 -0,003 0,085 0,004 0,209 -0,010 0,364 -0,013 0,599 DOLS 0,000 0,006 0,000 0,013 0,000 0,031 -0,002 0,080 0,005 0,200 -0,006 0,350 -0,008 0,580 FMLS 0,000 0,006 0,000 0,013 0,000 0,030 -0,002 0,080 0,005 0,199 -0,006 0,348 -0,008 0,577 GLS 0,000 0,006 0,000 0,012 0,000 0,028 -0,001 0,055 0,002 0,049 0,001 0,044 0,001 0,042 MLE 0,000 0,006 0,000 0,013 0,000 0,034 0,136 4,766 -0,275 9,573 -0,142 4,945 -0,597 25,959 NBLS 0,001 0,020 0,000 0,021 0,001 0,032 0,001 0,054 0,001 0,096 0,006 0,167 -0,004 0,244 FMNBLS 0,001 0,020 0,000 0,019 -0,001 0,020 0,000 0,034 0,003 0,081 -0,005 0,188 -0,002 0,360 FCMLE u 0,000 0,006 0,001 0,014 0,000 0,019 0,001 0,030 -0,002 0,059 -0,011 0,120 -0,017 0,224 FCMLE k 0,000 0,006 0,000 0,014 0,000 0,018 0,002 0,028 0,002 0,042 0,001 0,044 0,001 0,043 dυ LPE -0,042 0,184 -0,038 0,180 -0,044 0,191 -0,043 0,194 -0,038 0,190 -0,052 0,190 -0,071 0,191 LWE -0,057 0,159 -0,049 0,157 -0,047 0,173 -0,033 0,172 -0,028 0,161 -0,037 0,155 -0,049 0,156 LPM -0,076 0,318 -0,073 0,327 -0,087 0,334 -0,097 0,346 -0,102 0,347 -0,112 0,343 -0,133 0,336 FML -0,019 0,057 -0,019 0,056 -0,016 0,060 -0,010 0,065 -0,002 0,052 -0,003 0,053 -0,022 0,053 FCMLE u -0,019 0,061 -0,030 0,075 -0,017 0,061 -0,014 0,060 -0,009 0,051 -0,002 0,059 -0,011 0,051 FCMLE k -0,014 0,038 -0,016 0,048 -0,010 0,038 -0,010 0,038 -0,006 0,040 0,004 0,040 -0,007 0,032 LSE residual-based tests 0,0 0,2 0,4 0,6 0,8 0,9 1,0 H0 H1 H0 H1 H0 H1 H0 H1 H0 H1 H0 H1 H0 H1 LPE 0,999 0,001 0,987 0,013 0,898 0,102 0,637 0,363 0,245 0,755 0,131 0,869 0,069 0,931 LWE 1,000 0,000 1,000 0,000 0,993 0,007 0,910 0,090 0,521 0,479 0,288 0,712 0,129 0,871 LV 1,000 0,000 1,000 0,000 1,000 0,000 1,000 0,000 0,971 0,029 0,553 0,447 0,125 0,875 FML 1,000 0,000 1,000 0,000 1,000 0,000 1,000 0,000 0,815 0,185 0,219 0,781 0,028 0,972 FCMLE u 1,000 0,000 1,000 0,000 1,000 0,000 1,000 0,000 0,992 0,008 0,855 0,145 0,074 0,926 FCMLE k 1,000 0,000 1,000 0,000 1,000 0,000 1,000 0,000 1,000 0,000 0,839 0,161 0,056 0,944 β LSE 0,000 0,027 0,001 0,061 0,001 0,149 -0,008 0,393 0,024 0,983 -0,028 1,713 0,082 2,780 ADL -0,001 0,029 0,001 0,065 -0,003 0,160 -0,017 0,419 0,029 1,039 -0,027 1,801 0,106 2,931 DOLS 0,000 0,028 0,001 0,063 0,001 0,152 -0,009 0,400 0,025 1,000 -0,029 1,743 0,085 2,824 FMLS 0,000 0,028 0,002 0,062 0,001 0,151 -0,008 0,398 0,025 0,994 -0,028 1,734 0,084 2,809 GLS 0,000 0,023 0,001 0,035 0,000 0,043 0,001 0,047 0,002 0,053 0,001 0,061 0,001 0,069 MLE -0,001 0,032 0,003 0,076 0,001 0,371 0,337 12,998 0,667 12,965 2,935 88,529 0,985 75,996 NBLS 0,004 0,071 0,001 0,088 0,004 0,144 0,006 0,261 0,006 0,472 0,026 0,821 -0,025 1,214 FMNBLS 0,002 0,075 0,002 0,081 0,003 0,169 0,008 0,520 0,003 0,962 0,027 1,573 -0,017 2,194 FCMLE u 0,000 0,030 -0,002 0,048 -0,006 0,074 -0,005 0,056 -0,001 0,043 -0,003 0,045 -0,004 0,063 FCMLE k 0,000 0,030 0,001 0,041 0,000 0,042 0,000 0,040 0,002 0,035 0,000 0,035 0,001 0,032 dυ LPE 0,109 0,209 0,114 0,210 0,109 0,214 0,101 0,214 0,088 0,204 0,059 0,188 0,021 0,184 LWE 0,104 0,183 0,117 0,196 0,123 0,211 0,127 0,205 0,126 0,198 0,101 0,181 0,084 0,175 LPM -0,050 0,308 -0,048 0,320 -0,056 0,326 -0,074 0,338 -0,083 0,340 -0,088 0,329 -0,114 0,343 FML 0,024 0,147 0,038 0,177 0,017 0,180 0,014 0,278 0,249 0,389 0,252 0,319 0,147 0,208 FCMLE u 0,019 0,143 0,013 0,184 0,012 0,149 0,042 0,167 0,062 0,164 0,040 0,111 -0,005 0,078 FCMLE k 0,017 0,134 0,010 0,176 0,012 0,148 0,041 0,163 0,058 0,154 0,036 0,105 -0,008 0,070 LSE residual-based tests 0,0 0,2 0,4 0,6 0,8 0,9 1,0 H0 H1 H0 H1 H0 H1 H0 H1 H0 H1 H0 H1 H0 H1 LPE 0,993 0,007 0,937 0,063 0,702 0,298 0,323 0,677 0,097 0,903 0,044 0,956 0,025 0,975 LWE 1,000 0,000 0,998 0,002 0,949 0,051 0,656 0,344 0,173 0,827 0,080 0,920 0,040 0,960 LV 1,000 0,000 0,996 0,004 0,952 0,048 0,696 0,304 0,142 0,858 0,036 0,964 0,030 0,970 FML 1,000 0,000 0,972 0,028 0,927 0,073 0,762 0,238 0,226 0,774 0,068 0,932 0,023 0,977 FCMLE u 1,000 0,000 0,994 0,006 0,974 0,026 0,852 0,148 0,449 0,551 0,194 0,806 0,050 0,950 FCMLE k 1,000 0,000 0,999 0,001 0,973 0,027 0,850 0,150 0,450 0,550 0,197 0,803 0,050 0,950 β LSE -0,005 0,009 -0,011 0,018 -0,028 0,044 -0,070 0,111 -0,168 0,267 -0,270 0,447 -0,414 0,709 ADL 0,000 0,006 -0,004 0,014 -0,019 0,038 -0,060 0,108 -0,160 0,272 -0,269 0,462 -0,412 0,732 DOLS 0,000 0,006 -0,006 0,014 -0,022 0,039 -0,064 0,107 -0,164 0,266 -0,267 0,450 -0,413 0,716 FMLS 0,000 0,006 -0,006 0,015 -0,023 0,040 -0,065 0,108 -0,165 0,266 -0,268 0,449 -0,413 0,714 GLS -0,005 0,009 -0,014 0,021 -0,055 0,086 -0,390 0,413 -0,475 0,478 -0,493 0,495 -0,497 0,498 MLE 0,000 0,006 -0,004 0,014 -0,015 0,038 -0,049 0,310 -0,176 2,948 -0,085 13,876 -0,238 25,164 NBLS 0,000 0,022 -0,005 0,025 -0,018 0,041 -0,045 0,086 -0,116 0,193 -0,160 0,294 -0,256 0,551 FMNBLS 0,001 0,026 -0,001 0,021 -0,003 0,026 -0,008 0,045 -0,024 0,131 -0,014 0,260 -0,180 0,608 FCMLE u 0,000 0,006 -0,001 0,016 -0,001 0,023 0,001 0,046 -0,074 0,206 -0,259 0,377 -0,465 0,539 FCMLE k 0,000 0,006 0,000 0,016 -0,001 0,023 0,001 0,046 0,003 0,132 0,019 0,849 0,362 3,041 dυ LPE -0,024 0,179 -0,018 0,168 -0,038 0,189 -0,045 0,186 -0,036 0,187 -0,054 0,191 -0,066 0,188 LWE -0,038 0,150 -0,033 0,145 -0,042 0,167 -0,041 0,171 -0,029 0,156 -0,040 0,155 -0,042 0,153 LPM -0,038 0,304 -0,039 0,304 -0,073 0,327 -0,086 0,328 -0,089 0,336 -0,112 0,347 -0,131 0,352 FML -0,014 0,045 -0,016 0,049 -0,017 0,059 -0,016 0,052 -0,009 0,057 -0,006 0,053 -0,021 0,048 FCMLE u -0,015 0,054 -0,023 0,065 -0,014 0,064 -0,011 0,047 -0,001 0,056 0,019 0,065 -0,009 0,055 FCMLE k -0,011 0,032 -0,012 0,041 -0,008 0,035 -0,009 0,036 -0,008 0,046 -0,005 0,047 -0,013 0,055 LSE residual-based tests 0,0 0,2 0,4 0,6 0,8 0,9 1,0 H0 H1 H0 H1 H0 H1 H0 H1 H0 H1 H0 H1 H0 H1 LPE 1,000 0,000 0,987 0,013 0,907 0,093 0,642 0,358 0,242 0,758 0,136 0,864 0,058 0,942 LWE 1,000 0,000 0,999 0,001 0,988 0,012 0,931 0,069 0,533 0,467 0,300 0,700 0,128 0,872 LV 1,000 0,000 1,000 0,000 1,000 0,000 1,000 0,000 0,972 0,028 0,566 0,434 0,104 0,896 FML 1,000 0,000 1,000 0,000 1,000 0,000 1,000 0,000 0,850 0,150 0,237 0,763 0,032 0,968 FCMLE u 1,000 0,000 1,000 0,000 1,000 0,000 1,000 0,000 0,988 0,012 0,611 0,389 0,041 0,959 FCMLE k 1,000 0,000 1,000 0,000 1,000 0,000 1,000 0,000 1,000 0,000 0,911 0,089 0,083 0,917

The spurious regression case

The case where d = 1 corresponds to the no-cointegration case, introducing the well-known spurious regression issue. Consequently, estimates of β are highly significant but don't fit with any econometric inference. In our Monte-Carlo experiment, it leads to dramatic increase in variances for all estimators of β (tables 3 -4). Moreover, in our framework, equation ( 8) can be reformulated as follows,

∆y t = (α + β)∆x t + α 11 ∆y t-1 -α 11 β∆x t-1 + ε t (40)
This equation highlights an additional problem: the identification of parameter β. When α 11 = 0 there is no way to identify separately α and β and all estimators are severely biased (table 5).

Conversely our fully-parametric FCMLE allows to identify separately β and α 11 and thus reduces the bias quite considerably.

Synthesis

Overall, β estimators can be gathered into four categories.

i) LSE, ADL, DOLS and FMLS perform similarly in all cases. According to our simulations there is no significant benefit to the use of methods ADL, DOLS or FMLS compared to OLS to correct the endogeneity or autocorrelation bias in the fractional cointegration case. These biases appears to be closely related to the long memory parameter value. However, they are negligible as long as d < 0.4. ii) GLS estimator is heavily biased when residuals face both serial correlation and endogeneity (table 6) and is not robust to endogeneity when residuals are non-stationary (table 5).

iii) NBLS perform generally better than the estimators discussed above, although theoretically, it should not be applied in the parameter space of δ that we consider. FMNBLS appear to be less robust to autocorrelation than NBLS. Nevertheless this problem is confined to weak fractional cointegration and spurious regression cases. In line with the theory, FMNBLS outperform other estimators when residuals are subject to endogeneity bias (table 5).

iv) In comparison with other estimators, the FCMLE is less affected by endogeneity and autocorrelation biases when d increases. Overall, this estimator clearly outperforms other methods both in terms of bias and variance.

Concerning d estimators we can identify two categories.

i) Regarding the non-parametric estimators of d, our simulations do not reveal large differences between LPR and LWE whatever the cases studied. In most cases LWE exhibit a slight advantage in terms of RMSE. In contrast and in line with the theory, the LPM estimator is characterized by a significant bias reduction at the cost of increased variance. However its performance deteriorates in the spurious regression case.

ii) The behavior of FML and FCMLE are closely related and globally these estimators outperforms the non-parametric estimators. The FCMLE is less sensitive to serial correlation bias that affect FML when d is close to 1 (table 4 and6).

Fractional cointegration tests: powers and sizes

Overall, most tests are satisfactory in terms of power, since they reject H 0 when H 0 is false with a probability of 100%, as long as residuals are stationary. As we can expect, the power of the different tests decreases when d increases. Moreover, all tests are sensitive to autocorrelation, which causes a dramatic deterioration of powers. This behavior is much more pronounced in the case of the LPE test (figure 3) and FCMLE clearly exhibits the best results. Size are globally satisfactory expect in some cases for the test of [START_REF] Lobato | Efficient Wald tests for fractional unit roots[END_REF] and the LWE test (table 3 and5).

EMPIRICAL ILLUSTRATION

The present value model

We present an empirical investigation of the present value model to the S&P 500 stock prices [START_REF] Campbell | Cointegration and tests of present value models[END_REF]). Under the efficient markets hypothesis (EMH), an asset price y t depends linearly on the present discounted value of expected dividends x t ,

y t = ∞ j=1 1 (1 + r) j × E[x t+ j |I t ] (41)
Solving this equation leads to a cointegration relationship between y t and x t . Numerous papers have investigated this relation from the restrictive point of view of the I(1)/I(0) paradigm. Most studies have led to inconclusive results, or have concluded against the EMH because "spread between stock prices and dividends moves too much and deviations from the present value model are quite persistent" (see [START_REF] Campbell | Cointegration and tests of present value models[END_REF]). Recently, some studies have explored the possibility of fractionally integrated deviations from equilibrium. [START_REF] Marinucci | Semiparametric fractional cointegration[END_REF] find little evidence in favor of weak cointegration (that is, δd < 1/2) and they cannot reject the null hypothesis of no cointegration. In the same spirit, [START_REF] Caporale | Fractional cointegration and tests of present value models[END_REF] reveal strong evidence of in favor weak fractional cointegration with d ∈ [0.6, 0.7] and conclude to the rejection of the null hypothesis.

Estimations results

In order to test the EMH, we use the updated monthly database of [START_REF] Campbell | Cointegration and tests of present value models[END_REF] 5 : y t refers to the Standard and Poor's composite stock price index from January 1871 to December 2011 and x t refers to the associated dividends.

Firstly, we lead several integration tests upon y t and x t including the fractional Wald test of [START_REF] Lobato | Efficient Wald tests for fractional unit roots[END_REF]. Results support the hypothesis of non-stationary I(1) process for both y t and x t . The LV test confirms that processes are both I(1), against the alternative hypothesis of I(d) processes. Secondly, we lead several estimations corresponding to different sample sizes. We consider first the original dataset of [START_REF] Campbell | Cointegration and tests of present value models[END_REF]. Next, we consider the complete sample. Finally we consider the complete sample excluding the Cowles (1939Cowles ( )'s data (i.e. from 1871Cowles ( to 1926)), since calculation methods are different. We perform estimations using the FCMLE and we also report the [START_REF] Cheung | A fractional cointegration analysis of purchasing power parity[END_REF] methodology and the combination {LSE,FML}. Results are reported in the table (8).

Results depend heavily on the period and the estimation method. For instance, the methodology of [START_REF] Cheung | A fractional cointegration analysis of purchasing power parity[END_REF] cannot reject at 5% the null hypothesis of no cointegration (T LPE = -1.43 > -2.25) as well as our one step methodology, when we consider the complete sample. Conversely, the LSE/FML methodology reject the null hypothesis accepting cointegration (T F ML = -3.79 < -1.645). Removing the Cowles (1939)'s data, the conclusion is the same for the LSE/LPE methodology (T LPE = -1.07 > -2.25), whereas the LSE/FML and the FCMLE both lead to reject the null hypothesis (T F ML = -2.87 < -1.645 and W FCMLE = -2.40 < -1.645). According to the data set used by [START_REF] Campbell | Cointegration and tests of present value models[END_REF], the LSE/LPE and the LSE/FML reject H 0 (T F ML = -4.21 < -1.645 and T LPE = -3.28 < -1.645) while

FINAL COMMENTS

In this paper we have compared through Monte Carlo simulations the finite sample properties of different estimation methods of a fractional cointegration model for a wide range of the integration order of residuals, in the case where the integration order of regressors is known and equal to 1. To improve efficiency, we have proposed to estimate jointly all parameters of interest, using the Gaussian maximum likelihood estimator. Our approach complements the contributions of [START_REF] Hualde | Root-n-consistent estimation of weak fractional cointegration[END_REF], which is restricted to weak fractional cointegration (that is, δd < 1/2), and Nielsen (2007) solely concerned with the stationary fractional cointegration case (that is, 0 ≤ d < δ < 1/2). We have studied the finite sample properties of the Gaussian maximum likelihood estimator and several estimators that operate in two steps. The Monte Carlo experiment shows that our one-step parametric time domain approach compares favorably with other estimators of β and d and sometimes performs better, even when cointegrating errors face endogeneity and serial correlation. We have also suggested to use the fractional integration Wald test of Tanaka (1999), which has good power and size according to our simulation. Finally, we have revisited the test of the present value model of [START_REF] Campbell | Cointegration and tests of present value models[END_REF] and found little evidence in favor of fractional cointegration.

  d id]) ≡ var( d) + (Bias[ d|d]) 2 , and the bias in order to compare finite sample performances 4 . Values in bold indicate the smallest RMSE. Concerning fractional cointegration tests, results are reported in percentage of rejection of the null hypothesis at a threshold of 5%.

Figure 1 :

 1 Figure 1: Finite sample distributions of and d (model A, d = 0.2 and n = 512)

Figure 2 :

 2 Figure 2: Finite sample distributions of β and d (model A, d = 0.6 and n = 512)

Figure 3 :

 3 Figure 3: Power and size of various tests at 5%, for the model B 2 with n = 512.

Table 1 :

 1 Some parameters combinations Model α α 11 α 12 α 21 α 22

Table 2 :

 2 The different parametrizations Model α α 11 α 12 α 21 α 22

Table 3 :

 3 Residuals face neither serial correlation nor endogeneity (α 11 = α = 0)

		Model A	I(1)/I(0)	Strong fractional		Weak fractional		Spurious
		d	0,0	0,2	0,4	0,6	0,8	0,9	1,0
	n	θ Est.	Bias RMSE	Bias RMSE	Bias RMSE	Bias RMSE	Bias RMSE	Bias RMSE	Bias RMSE
	512

Table 4 :

 4 Residuals face serial correlation (α 11 = 0.8, α = 0)

		Model B 1	I(1)/I(0)	Strong fractional		Weak fractional		Spurious
		d	0,0	0,2	0,4	0,6	0,8	0,9	1,0
	n	θ Est.	Bias RMSE	Bias RMSE	Bias RMSE	Bias RMSE	Bias RMSE	Bias RMSE	Bias RMSE
	512

Table 5 :

 5 Residuals face non-zero coherence with regressors (α 11 = 0, α = -0.5)

		Model B 2	I(1)/I(0)	Strong fractional		Weak fractional		Spurious
		d	0,0	0,2	0,4	0,6	0,8	0,9	1,0
	n	θ Est.	Bias RMSE	Bias RMSE	Bias RMSE	Bias RMSE	Bias RMSE	Bias RMSE	Bias RMSE
	512

Table 6 :

 6 Residuals face both serial correlation and endogeneity (α 11 = 0.8, α = -0.5)

		Model B 3	I(1)/I(0)	strong fractional		weak fractional		Spurious
		d	0,0	0,2	0,4	0,6	0,8	0,9	1,0
	n	θ Est.	Bias RMSE	Bias RMSE	Bias RMSE	Bias RMSE	Bias RMSE

Table 7 :

 7 Unit root tests

		S&P500		Dividend
	Sample (T )	1871-2010 (1667)		1871-2010 (1667)
	(µ)	KPSS ADF	PP	LV	KPSS ADF	PP	LV
	Level	30.14 1.085 0.730 0.898 31.64 3.964 1.709 0.16
	Difference	0.266 -30.32 -29.98	-	0.674 -9.903 -9.709	-
	Crit. value at 5% 0.463 -2.864 -2.864 -1.96 0.463 -2.864 -2.864 -1.96
	(µ + δ t )	KPSS ADF	PP	LV	KPSS ADF	PP	LV
	Level	6.559 -1.639 -1.878	-	6.340 -2.153 -1.900	-
	Difference	5.789 -74.28 -62.88	-	4.084 -152.6 -88.16	-
	Crit. value at 5% 0.146 -3.415 -3.415	-	0.146 -3.415 -3.415	-

Table 8 :

 8 Estimation of β, d and tests FCMLE Wald test cannot reject H 0 . However, with respect to power of tests, we cannot eliminate any doubt of type I error and results have to be interpreted carefully.

	(n)		1871-2011 (1680)	1926-2011 (1032)		1871-1986 (1380)
	θ	FCMLE LSE/LPE LSE/FML FCMLE	LSE/LPE LSE/FML FCMLE LSE/LPE LSE/FML
	d	1.0061	0.7959	0.9086	0.8611	0.8812	0.8510	0.9984	0.6549	0.7913
		(0.0318) (0.1158)	(0.0424)	(0,10172) (0.1346)	(0.0517)	(0.0329) (0.1232)	(0.0495)
	β	0.8973	1.1819	1.1819	1.0298	1.2572	1.2572	0.8588	1.0638	1.0638
		(0.1379) (0.0046)	(0.0046)	(0.1967)	(0.0073)	(0.0073)	(0.1416) (0.0058)	(0.0058)
	α 1 11	0.2531	-	0.3618	0.3786	-	0.3899	0.2575	-	0.4309
		(0.0512) -	(0.0479)	(0.1214)	-	(0.0586)	(0.0557) -	(0.0543)
	α 2 11	-0.0567 -	-0.1056	-	-	-	-0.0498 -	-
		(0.0316) -	(0.0257)	-	-	-	(0.0336) -	-
	R 2	0.3874	0.9750	0.9750	0.3139	0.9661	0.9661	0.3837	0.9605	0.9605

the

Lasak (2010) suggests to extend the Johansen MLE estimator to fractional processes. However, we do not consider this estimator since the asymptotic theory is still a work in progress.

The time domain exact maximum likelihood estimator (EML) ofSowell (1992) is not considered here. According to the Monte-Carlo study of[START_REF] Nielsen | Finite Sample Comparison of Parametric, Semi-parametric, And Wavelet Estimators of Fractional Integration[END_REF], finite sample performances of EML estimator strongly depend on specifications and has poor finite sample properties relatively to other parametric estimators like FML. Notice also that[START_REF] Dubois | The Exact Maximum Likelihood-Based Test for Fractional Cointegration: Critical Values, Power and Size[END_REF] investigate the EML-based fractional cointegration test and point out that the power of this test decreases when the sample size increases. They also bring out that the size of this test strongly depends on the short run coefficients and perform badly in some cases.

Notice that[START_REF] Dolado | A Fractional DickeyFuller Test for Unit Roots[END_REF] suggest a fractional Dickey-Fuller test, but[START_REF] Lobato | Efficient Wald tests for fractional unit roots[END_REF] show that this test is inefficient. Moreover, their tests are locally asymptotically equivalent to the optimal Lagrange multiplier tests ofRobinson (1991).

All computations are performed using RATS 8.01. RATS codes and unreported results are available upon request.

0,217 0,133 0,216 0,117 0,218 0,104 0,207 0,087 0,200 0,047 0,189 0,007 0,185 LWE 0,121 0,192 0,132 0,200 0,127 0,209 0,126 0,204 0,123 0,194 0,091 0,175 0,063 0,170 LPM -0,011 0,296 -0,013 0,302 -0,042 0,321 -0,056 0,327 -0,066 0,330 -0,085 0,332 -0,109 0,344 FML 0,031 0,154 0,045 0,178 0,028 0,182 0,029 0,270 0,247 0,382 0,223 0,298 0,119 0,186 FCMLE u 0,016 0,128 0,007 0,156 0,008 0,142 0,030 0,157 0,057 0,149 0,047 0,111 -0,004 0,084 FCMLE k 0,015 0,122 0,004 0,145 0,008 0,130 0,025 0,144 0,056 0,148 0,043 0,104 -0,008 0,065

http://www.econ.yale.edu/ shiller/

Bias RMSE Bias RMSE 512 β LSE -0,022 0,041 -0,053 0,089 -0,135 0,214 -0,335 0,541 -0,809 1,312 -1,295 2,195 -1,985 3,498 ADL -0,008 0,031 -0,034 0,077 -0,112 0,206 -0,323 0,553 -0,791 1,344 -1,290 2,268 -1,985 3,633 DOLS -0,014 0,034 -0,046 0,083 -0,128 0,210 -0,332 0,545 -0,814 1,329 -1,311 2,231 -2,011 3,550 FMLS -0,016 0,036 -0,048 0,085 -0,130 0,212 -0,333 0,544 -0,814 1,324 -1,308 2,220 -2,006 3,534 GLS -0,458 0,463 -0,475 0,477 -0,489 0,492 -0,496 0,499 -0,499 0,502 -0,498 0,502 -0,501 0,507 MLE -0,001 0,032 -0,011 0,080 -0,044 0,239 -0,282 2,749 -0,682 17,404 -3,484 84,141 2,306 82,262 NBLS

-0,005 0,079 -0,038 0,110 -0,103 0,205 -0,240 0,439 -0,582 0,966 -0,795 1,457 -1,250 2,728 FMNBLS 0,005 0,088 -0,016 0,102 -0,006 0,209 -0,239 0,653 -1,048 1,557 -1,668 2,396 -2,684 3,949 FCMLE u 0,001 0,042 -0,013 0,120 -0,059 0,211 -0,053 0,207 -0,008 0,103 -0,008 0,101 -0,001 0,145 FCMLE k 0,001 0,042 0,004 0,109 0,005 0,122 0,002 0,133 -0,006 0,111 -0,012 0,129 -0,012 0,154 dυ LPE 0,128

Simulation Results

The I(1)/(0) case

The first columns of tables 3 to 6 correspond to the traditional cointegration case (δ = 1 and d = 0). When cointegrating errors follow a simple white noise (table 3), most estimators of β (LSE, ADL, DOLS, FMLS, GLS, MLE and FCMLE) perform well with similar bias and RMSE. Adding short-run noise (table 4), the RMSE is deteriorated, but these estimation methods seem fairly robust to endogeneity (table 5). The worst performances are observed when residuals face both autocorrelation and endogeneity (table 6).

Concerning semi-parametric estimators of d based on cointegrating residuals, it is well known that in the presence of short-run dynamics, a smaller bandwidth is more appropriate to avoid biased results. Since we have selected a bandwidth equal to T 0.5 , these methods are moderately impacted by autocorrelation (table 4 and6). Overall, FML estimator and FCMLE exhibit lowest bias and RMSE.

The fractional cointegration case with stationary residuals

In this case, most estimators of β (LSE, ADL, DOLS, FMLS, GLS, MLE and FCMLE) exhibit similar performances, both in terms of bias and RMSE. It may be noted that the RMSE of these estimators increases gradually as d increase. When residuals face endogeneity, n t=1 y 2 t / n t=1 x 2 t no longer dominates asymptotically n t=1 υ 2 t / n t=1 x 2 t and biases of most of estimators of β increase as long as d goes up. The behavior of the estimators of d is very similar to the I(1)/I(0) case. Figure 1

The weak fractional cointegration case

When cointegrating errors are not stationary (that is, d ≥ 0.5), the MLE estimator of Johansen (1988) is not convergent and clearly inappropriate. GLS estimator outperform other estimators of β in model A and B 1 but is no longer convergent when residuals face endogeneity (models B 2 and B 3 ): referring to our notation, this estimator cannot distinguish between β and β + α when d is high and α 0. The problem of endogeneity also affect the behavior of other estimation methods (LSE, ADL, DOLS, FMLS and FCMLE). Without endogeneity, the bias increases moderately with d while the variance increases sharply (table 3 and4). In the presence of non-zero coherence between errors and regressors both bias and variances are strongly impacted when d increase. In this context, the FMNBLS estimator behaves correctly in the presence of endogeneity (table 5) but its performance deteriorates when α 11 0 (tables 4 and 6). Concerning the estimators of d, the simulations show that the bias increases slightly when d goes up, while the RMSEs remain broadly similar.