Mohamed Boutahar

Imène Mootamri
email: imene.mootamri@etumel.univmed.fr

Anne Péguin-Feissolle

A fractionally integrated exponential STAR model applied to the US real effective exchange rate *

Keywords: Fractional integration, Nonlinearity, STAR models, Long memory, Real effective exchange rate, Forecasting. JEL classification: C22, C51, C52, F31

 in the case when the transition function is an exponential function and we develop an estimation procedure. Indeed, these models can take into account processes characterized by several distinct dynamic regimes and persistence phenomena.

namics of monthly US unemployment rates. [START_REF] Smallwood | Joint tests for non-linearity and long memory: the case of purchasing power parity[END_REF] extends these results to the FISTAR model with an exponential transition function, and applies this model to the purchasing power parity puzzle by considering the real exchange rate processes for twenty countries against the United States.

In this paper, we study this class of models because these FISTAR models, indeed, make it possible to generate nonlinearity, since they are defined by several distinct modes in dynamics, and to take into account the persistence phenomenon. We consider the case of an exponential transition function and propose a two-step estimation method: in the first step, we estimate the long memory parameter, then, in the second step, the STAR model parameters via nonlinear least squares estimation.

The remainder of this paper is organized as follows. In Section 2, we present the FISTAR model with an exponential transition function and the two-step estimation procedure; we describe also the out-of-sample forecasting. In Section 3, we analyze the monthly US real effective exchange rate in order to illustrate the various elements of the modelling cycle. Finally, Section 4 concludes.

The Econometric Specification 2.1 The model

Let us consider a process y t that satisfies the following long memory scheme:

(1 -L) d y t = x t ( 1
)
where L is the lag operator, d is the long memory parameter and x t is a covariance-stationary I(0) process. The parameter d is possibly noninteger, in which case the time series y t is called fractionally integrated (F I) (see, among others, [START_REF] Granger | An introduction to long-range time series models and fractional differencing[END_REF] and [START_REF] Hosking | Fractional differencing[END_REF]). If -0.5 < d < 0.5, y t is covariance stationary and invertible process. For 0 < d < 0.5, y t is a stationary long memory process in the sense that autocorrelations are not absolutely summable and decay hyperbolically to zero. Finally, if 0.5 ≤ d < 1, y t is nonstationary and the shocks do not have permanent effects.

To capture the nonlinear feature of time series, a wide variety of models can be used (see [START_REF] Franses | Nonlinear Time Series Models in Empirical Finance[END_REF]). In this paper, we consider the fractionally integrated STAR (FISTAR) 1 model introduced by van Dijk et al. (2002) given by: ½ (1 -L) d y t = x t x t = (ϕ 10 + P p i=1 ϕ 1i x t-i ) + (ϕ 20 + P p i=1 ϕ 2i x t-i ) F (s t , γ, c)

+ ε t (2)
where ε t is a martingale difference sequence with

E [ε t | Ω t-1 ] = 0 and E £ ε 2 t ¯Ωt-1 ¤ = σ 2
and Ω t is the information set available at time t. γ is the transition parameter (γ > 0) and c is the threshold parameter;s t ,the transition variable2 , is generally the lagged endogenous variable, i.e. s t = y t-m for certain integer m > 0 where m is the delay parameter. In most applications, the transition function F (s t , γ, c) is an exponential function or a logistic function. The FISTAR model can be also be written as follows:

½ (1 -L) d y t = x t x t = π 0 1 w t + π 0 2 w t F (s t , γ, c) + ε t (3)
where

w t = (1, x t-1 , ..., x t-p ) 0 , π i = (π i0 , π i1 , ..., π ip ) 0 and π i (L) = ϕ i (L) (1 -L) d for i = 1, 2.
The fractional parameter d and the autoregressive parameters make the FISTAR model potentially useful for capturing both nonlinear and long memory features of the time series y t . Indeed, the long-run properties of y t are restricted to be constant and these are determined by the fractional differencing parameter, however, the short-run dynamics are determined by autoregressive parameters.

Our empirical results show that the fractionally integrated exponential STAR (F IEST AR) model is more appropriate for modelling real exchange rate dynamics than the FISTAR model with the logistic function (F ILST AR). Then, the simple transition function suggested by [START_REF] Teräsvirta | Characterizing nonlinearities in business cycles using smooth transition autoregressive models[END_REF] and [START_REF] Teräsvirta | Specification, estimation, and evaluation of smooth transition autoregressive models[END_REF], which is particularly attractive in the present context, is the exponential function3 that takes the following form:

F (s t , γ, c) = 1 -exp µ - γ σ 2 s t (s t -c) 2 ¶ (4)
where σ st is the standard deviation of s t . We present the main steps of the specification procedure for FISTAR models, such as it is proposed by van Dijk et al. ( 2002):

• Specify a linear ARFI(p) model by selecting the autoregressive order p by means of information criteria 4 (Akaike (1974) or [START_REF] Schwarz | Estimating the dimension of a model[END_REF]).

• Test the null hypothesis of linearity against the alternative of a FISTAR model. If linearity is rejected, select the appropriate transition variable.

• Estimate the parameters in the FISTAR model.

• Evaluate the estimated model using misspecification tests (no remaining nonlinearity, parameter constancy, no residual autocorrelation, among others).

Linearity tests

Teräsvirta (1994) developed the procedure of testing linearity against STAR models; he pointed out that this procedure is complicated by the presence of unidentified nuisance parameters under the null hypothesis. To overcome this problem, [START_REF] Luukkonen | Testing linearity against smooth transition autoregressive models[END_REF] propose to replace the transition function F (s t , γ, c) with a suitable Taylor series approximation about γ = 0. In the reparametrized equation, the identification problem is no longer present, and linearity can be tested by means of a Lagrange multiplier (LM) statistic.

For an extensive presentation of the test when the alternative is a FISTAR model, the reader is referred to van Dijk et al. (2002) and [START_REF] Smallwood | Joint tests for non-linearity and long memory: the case of purchasing power parity[END_REF].

The, we consider the model given by ( 3) and (4); the LM-type test statistic can be computed in a few steps as follows:

• Estimate an ARF I(p), obtain the set of residuals b ε t . The sum of squared errors, denoted SSR 0 , is then constructed from the residuals b ε t , SSR 0 = P T t=1 b ε 2 t .

• Regress b ε t on w t , -P t-1 j=1 b ε t-j j and w t s i t , i = 1, 2, and compute the sum of squared residuals SSR 1 under the alternative hypothesis.

• The χ 2 version of the LM test statistic is calculated as:

LM χ 2 = T (SSR 0 -SSR 1 ) SSR 0 (5)
and is distributed as χ 2 (2 (p + 1)) under the null hypothesis of linearity (T denotes the sample size).

Estimation of the FISTAR model

S (λ) = T X t=1 ε 2 t (λ) , (6)
where λ = (π 0 1 , π 0 2 , d, γ, c) denotes the parameters of the FISTAR model (3). The residuals ε t (λ) are calculated as follows:

ε t (λ) = (1 -L) d y t - ³ π 10 + P t+p-1 j=1 π 1,j y t-j - ³ π 20 + P t+p-1 j=1 π 2,j y t-j ´F (s t , γ, c) (7)
where F (s t , γ, c) is given by (4). Thus, conditional upon d, γ and c, van Dijk et al. (2002) remark that the FISTAR model is linear in the remaining parameters; estimates of π 1 and π 2 can be thus obtained by ordinary least squares as:

b µ (d, γ, c) 0 = Ã T X t=1 w t (d, γ, c) w t (d, γ, c) 0 ! -1 Ã T X t=1 w t (d, γ, c) y t ! , (8)
where w t (d, γ, c) = (w 0 t , w 0 t F (s t , γ, c)) 0 . Therefore, the sum of squares function can be obtained by :

S (d, γ, c) = T X t=1 ¡ y t -b µ (d, γ, c) 0 w t (d, γ, c) ¢ 2 . (9)
According to van Dijk et al. (2002), it can be difficult to estimate the model parameters jointly. In particular, accurate estimation of the smoothness parameter γ is quite difficult when this parameter is large. They proposed an algorithm that is based on a grid search over d, γ and c in order to obtain starting values for the nonlinear least squares procedure.

Two steps estimation

The properties of the process y t depend on the value of the parameter d.

Many researchers have proposed methods for estimating the long memory parameter d. These methods can be summarized in three classes: the heuristic methods [START_REF] Hurst | Long-term storage capacity of reservoirs[END_REF], [START_REF] Higuchi | Approach to an irregular time series on the basis of the fractal theory[END_REF], [START_REF] Lo | Long-term memory in stock market prices[END_REF]...), the semiparametric methods (Geweke and Porter-Hudak (1983), [START_REF] Robinson | Semiparametric analysis of long-memory time series[END_REF]Robinson ( , 1995a and b), [START_REF] Lobato | Averaged periodogram estimation of long memory[END_REF]...) and the maximum likelihood methods [START_REF] Dahlhaus | Efficient parameter estimation for self-similar processes[END_REF], [START_REF] Fox | Large sample properties of parameter estimates for strongly dependent stationary time series[END_REF], [START_REF] Sowell | Maximum likelihood estimation of stationary univariate fractionally integrated time series models[END_REF] ...). In the first two classes, we can estimate only the long memory parameter d, and in the last, we estimate simultaneously all the parameters, see [START_REF] Boutahar | Estimation Methods of The Long Memory Parameter: Monte Carlo Analysis and Application[END_REF] for more details.

The estimation method of the FISTAR model we propose proceeds in two steps:

• In the first step, we estimate the long memory parameter d in the simple model (1) using the heuristic method via the R/S statistic proposed by [START_REF] Hurst | Long-term storage capacity of reservoirs[END_REF] and modified by [START_REF] Lo | Long-term memory in stock market prices[END_REF]. The R/S statistic is the range of partial sums of deviations of a time series from its mean, rescaled by its standard deviation. Specifically, the R/S statistic is defined as:

Q T = 1 S T Ã max 1≤k≤T k X j=1 (y j -y) -min 1≤k≤T k X j=1 (y j -y) ! (10)
where y = 1 T P T i=1 y i is the empirical mean and 2 is the empirical variance. [START_REF] Lo | Long-term memory in stock market prices[END_REF] modified the R/S statistic as follows:

S 2 T = 1 T P T i=1 (y i -y)
e Q T = 1 Sq(T ) Ã max 1≤k≤T k X j=1 (y j -y) -min 1≤k≤T k X j=1 (y j -y) ! (11)
where

Sq(T ) = " S 2 T + 2 T q X j=1 w j (q) Ã T X i=j+1 (y i -y) (y i-j -y)
!# 1/2 ; w j (q) = 1 -j q+1 are the weights proposed by Newey and West (1987), with j = 1, ...., q. There is no optimal choice of the parameter q. Lo and MacKinlay (1988) and [START_REF] Andrews | Heteroskedasticity and autocorrelation consistent covariance matrix estimation[END_REF] showed by a Monte Carlo study that, when q is relatively large compared to the sample size, then the estimator is skewed and thus q must be relatively small. By default, for obtaining the long run variance, q is chosen to be h 4 (T/100) 1/4 i , where T is the sample size, and [x] denotes integer part of x. However, when the stationary process y t has long memory, [START_REF] Mandelbrot | Statistical methodology for non-periodic cycles: from the covariance to R/S analysis[END_REF] showed that the R/S statistic converges to a random variable at rate T H , where H is the Hurst coefficient. The link between the parameter H and the ARFI parameter d is that H = d+ 1 2 (Boutahar et al.( 2007)).

• Once we obtain b d R/S , in the second step, we filter out the long memory component and we estimate the STAR model parameters via nonlinear least squares estimation.

Out-of-sample forecasting performance

Unlike the linear model, forecasting with nonlinear models is more complicated, especially for several steps ahead (see, for instance, [START_REF] Granger | Modelling Nonlinear Economic Relationships[END_REF]). Let us consider the FIESTAR model given by (3) and (4) which can be written as:

     (1 -L) d y t = x t x t = G (w t , ω) + ε t F (s t , γ, c) = 1 -exp ³ -γ σ 2 s t (s t -c) 2
´ [START_REF] Clarida | Nonlinear permanenttemporary decomposition, with applications in macroeconomics and finance[END_REF] where G (w t , ω) = π 0 1 w t + π 0 2 w t F (s t , γ, c) and ω = (π 0 1 , π 0 2 , γ, c) 0 . The optimal one-step ahead forecast of x t is given by:

x t+1 |t = E (x t+1 |Ω t ) = G (w t+1 , ω) ; (13)
this forecast can be achieved with no difficulty and can be estimated by

b x t+1 |t = G (w t+1 , b ω) ( 14
)
where b ω is the parameter estimate. However, when the forecast horizon is larger than one period, things become more complicated because the dimension of the integral grows with the forecast horizon. For example, the two-step ahead forecast of x t is given by:

b x t+2|t = E ¡ G ¡ b w t+2|t , ω ¢ |Ω t ¢ = Z ∞ -∞ G ¡ b w t+2|t , b ω ¢ f (ε)dε (15) with b w t+2/t = ¡ 1, b x t+1 |t + ε t+1 , x t , ..., x t+2-p ¢ 0 .
The analytic expression for the integral ( 15) is not available. We thus need to approximate it using integration techniques. Several methods obtaining forecasts to avoid numerical integration have been developed (see [START_REF] Granger | Modelling Nonlinear Economic Relationships[END_REF]). In this paper, we use a bootstrap method suggested by Lundberg and [START_REF] Lundbergh | Forecasting with smooth transition autoregressive models[END_REF]. This approach is based on the approximation of

E ¡ G ¡ b w t+2|t , ω ¢ |Ω t ¢
, the optimal point forecast is given by :

b x t+2|t = 1 k k X i=1 G ³ b w (i) t+2|t , b ω ´, ( 16
)
where k is some large number and the values of ε t+1 in b w

(i)
t+2|t are drawn with replacement from the residuals from the estimated model b ε t .

In general, forecasts are evaluated using the mean squared prediction error (MSPE) and the root mean squared prediction (RMSE), where m is the number of steps-ahead forecasts. Models with smaller MSPE have a better forecast performance. Further, in order to assess the accuracy of forecasts derived from two different models, the Diebold and Mariano (1995) test is likely to be widely used in empirical evaluation studies, and is considerably more versatile than any alternative test of equality of forecast performance.

Let y 1 t+h/t and y 2 t+h/t denote two competing forecasts of y t+h from FIES-TAR and ARFI models, respectively, based on Ω t , where Ω t = {y t , y t-1 , ..} is the information set available at time t. The forecast errors from the two models are given by e i t+h/t = y t+hy i t+h/t , i = 1, 2. The accuracy of each forecast is measured by a particular loss function:

g ¡ y t+h , y i t+h/t ¢ = g ¡ e i t+h/t ¢ , i = 1, 2.
To determine if a model predicts better than the other one, we may test the null hypothesis of equality of expected forecast performance:

   H 0 : E ³ g ³ e 1 t+h/t ´´= E ³ g ³ e 2 t+h/t ´H 1 : E ³ g ³ e 1 t+h/t ´´6 = E ³ g ³ e 2
t+h/t

´´.

The Diebold-Mariano test is based on the loss differential:

d t = g ¡ e 1 t+h/t ¢ -g ¡ e 2 t+h/t ¢ . ( 17
)
The null of equal predictive accuracy is then: H 0 : E (d t ) = 0. The Diebold-Mariano test statistic is:

S 1 = d q 2π b f d (0) T , ( 18
)
where T is the sample size, d = 1

T P T t=1 d t is the sample mean of d t and b f d (0) = 1 2π P ∞ τ =-∞ b γ d (τ )
is a consistent estimate of the spectral density of the loss differential function at frequency zero,

γ d (τ ) = E [(d t -µ) (d t-τ -µ)]
is the autocovariance of the loss differential at rate τ, and µ is the population mean loss differential. Under the null hypothesis of equal forecasts, the statistic S 1 has an asymptotic standard normal distribution. [START_REF] Harvey | Testing the equality of prediction mean squared errors[END_REF] noted that the Diebold-Mariano test statistic could be seriously over-sized as the prediction horizon increases, and therefore provide a modified Diebold-Mariano test statistic. [START_REF] Harvey | Testing the equality of prediction mean squared errors[END_REF] and [START_REF] Clark | Tests of equal forecast accuracy and encompassing for nested models[END_REF] show that this modified test statistic performs better than the Diebold-Mariano test statistic, and also that the power of the test is improved when the pvalues are computed with a Student distribution with (T -1) degrees of freedom, rather than from the standard normal distribution. Thus, the modified Diebold-Mariano statistic is given by:

S * 1 = µ T + 1 -2h + T -1 h (h -1) T ¶ 1/2 S 1 (19)
where S 1 is the original Diebold and Mariano statistic [START_REF] Diebold | Comparing predictive accuracy[END_REF].

Empirical results

The fractionally integrated models 5 [START_REF] Smallwood | Joint tests for non-linearity and long memory: the case of purchasing power parity[END_REF] to the case of purchasing power parity. In this paper, we study the behaviour of exchange rates and compare the forecast performances of the FIESTAR modelling to some other models.

The data

We use monthly data of the seasonally adjusted US real effective exchange rate covering the period June 1978 until April 2002, these data were obtained from the IMF International Financial Statistics. The series is expressed in logarithm. The use of monthly data provides us with a reasonably large sample and hence meets the requirement of the linearity tests for many degrees of freedom. The series is shown in Figure 1, which demonstrates a real appreciation of the dollar during the beginning of the 1980's followed by depreciation in 1985. As noted by [START_REF] Smallwood | Joint tests for non-linearity and long memory: the case of purchasing power parity[END_REF], consistently with the theoretical foundation of [START_REF] Sercu | The exchange rate in the presence of transactions costs: implications for tests of purchasing power parity[END_REF], we observe four periods after 1987 in which the dollar steadily appreciates and then rapidly depreciates after reaching approximately the same value. This provides some support for the use of nonlinear models.

Linearity tests results

Application of the linearity tests models requires stationary time series. The unit root tests 6 of Phillips and Perron (1988), Kwiatkowski et al. (1992) 7and Dickey-Fuller Augmented (1979) for the levels and first differences of the real effective exchange rates, measured in logarithms, are shown in Table 1. These results indicate that the time series are integrated of order 1, at both 5% and 1% significance levels.

The selection of the maximum lag p, of the linear ARFI model was made using the AIC and BIC criteria under the non autocorrelation hypothesis. We allow for a maximum autoregressive order of p = 6. Both AIC and BIC indicate that an ARFI model with p = 4 is adequate.

The linearity tests are displayed in Table 2. In carrying out linearity tests, we have considered values for the delay parameter m over the range [START_REF] Akaike | A new look at the statistical model identification[END_REF][START_REF] Beran | Maximum likelihood estimation of the differencing parameter for invertible short and long memory autoregressive integrated moving average models[END_REF], and calculated the pvalues for the linearity test in each case, the estimate of m is chosen by the lowest pvalue. Using 5% as a threshold pvalue, the test classifies the US real effective exchange rates as nonlinear. Although the pvalue is slightly higher than 5%, we show thereafter that a nonlinear model describes the features of a time series better than a linear model 8 . Then the lowest pvalue corresponds to m = 4 (m ≤ p).

Estimation results

Estimation results for the ARFI and FIESTAR models are shown in Table 3. The second column gives the ARFI model estimation, the estimate of d is -0.484, showing that the process y t is stationary and invertible. The results of the second model are based on the specification (3) where y t is the first difference of the US real effective exchange rates. The third column of Table 3 contains simultaneous estimation results of the parameters. In particular, the estimate of d is equal to -0.169 and belongs thus to the interval ]-0.5, 0.5[, suggesting that the process is stationary and invertible. The autocorrelation function decreases more quickly than in the case where 0 < d < 0.5: y t is an anti-persistent process. It is also interesting to note, in the last column corresponding to the two-step estimation, that the degree of persistence measured by the differentiation parameter increases. The Lo's (1991) estimator using the modified R/S statistic is b d R/S = 0.221, then, the process is stationary and invertible, the autocorrelation function decays hyperbolically to zero and y t is a long memory process. The modified R/S statistic 1.896 is significant at 5%. The ratio of the standard errors for the nonlinear and linear models for the simultaneous estimation of the FIESTAR model is equal to 0.840, it's higher than for the two-step estimation 0.670. We can thus confirm that the nonlinear model improves the modelling of the exchange rate process, as shown by both estimation methods. It is worthwhile noting here the relative small value of the estimation of γ for the second estimation (2.547 compared to 12.655 for simultaneous estimation), suggesting that the transition from one regime to the other is rather slow, contrary to first estimation which assumes a slightly sharp switch. The parameter c indicates the halfway point between the different phases of the exchange rate. The value of c is negative for the first case, and not significantly different from zero in the other. These values belong to the neighborhood of the sample mean for the first difference exchange rates. Figures 2 and3 show the curves of the exponential transition function corresponding to the estimation of the FIESTAR model, the first one using the simultaneous estimation method and the second one the two-step method.

Table 4 gives summary statistics and misspecification tests for ARFI and FIESTAR models. In particular, the hypothesis of no residual autocorrelation, no conditional heteroscedasticity, and normality are not rejected in the residuals for both models at 5% level of significance. From the skewness and kurtosis of the series, it is evident that the US real effective exchange rate is symmetric and the frequency curve is normal, this is confirmed by the Jarque-Bera test for normality. Moreover, the null hypothesis of parameter constancy against the alternative of smoothly changing parameters for s t = t, and the null of no remaining non linearity are not rejected, following the LM test statistics LM NL and LM C for the FIESTAR model.

Forecasting performance of estimated models

The final two years of data from January 2002 to April 2004 for US real effective exchange rate are used to evaluate the forecast performance of the estimated linear ARFI and FIESTAR models. For each point, we compute 1-12-step-ahead forecasts of real exchange rates. To obtain the forecasts from nonlinear model, we use the bootstrap method exposed in section 2.4.

The results of forecasting performance are reported in Table 5. Forecast accuracy is evaluated using mean squared prediction error (MPSE) criterion. The forecasts produced by the FIESTAR are compared to the forecasts generated by a random walk and linear ARFI models. Further, in order to assess the accuracy of forecasts derived from two different models, we employ the modified Diebold and Mariano test statistic proposed by [START_REF] Harvey | Testing the equality of prediction mean squared errors[END_REF] discussed in Section 2.4 for which the null hypothesis is the hypothesis of equal accuracy of different predictive methods.

The results successfully provide evidence in favour of the predictive superiority of the FIESTAR model against the random walk and ARFI models using MPSE: the MPSE of the linear model and a random walk is actually greater than the MPSE of the FIESTAR model. Comparing our results to those obtained in the previous literature we can see that the FIESTAR model gives very much more accurate forecasts and outperform random walk and linear ARFI models in out-of-sample forecasting performances for all forecast horizons. The statistical significance of this result is confirmed executing the modified Diebold and Mariano test: there is a statistically significant difference in predictive accuracy for the FIESTAR model over the random walk and ARFI specifications. We can thus conclude that the forecasts of the FIESTAR modelling are significantly better than those of the other models. The same conclusion is given by [START_REF] Chung | The out-of-sample forecasts of nonlinear longmemory models of the real exchange rate[END_REF] who finds clear evidence in favour of the nonlinear long-memory model over some other estimated models for the real exchange rates of Germany, France, Italy, UK, Japan, and Switzerland.

Conclusion

The aim of this paper was to study the dynamic modelling of the US real effective exchange rates covering the period June 1978 until April 2002. We considered the FISTAR model, as proposed by van Dijk et al. (2002), that can describe long memory and nonlinearity simultaneously and be used to produce out-of-sample forecasts. We used their model to the case of an exponential transition function. To this end, we employ two modelling approaches corresponding to two different estimations (simultaneous estimation or twostep estimation) of a FIESTAR model. The estimated FIESTAR model seem to provide a satisfactory description of the nonlinearity and persistency found in the US real effective exchange rates. With regards to the out-of-sample forecasting performance for US exchange rate, the tests for comparing the predictive accuracy show that the FIESTAR model seems better that the random walk and linear models. Note: The table presents selected diagnostic and misspecification tests statistics for the estimated FIESTAR on two step and ARFI models for the US real effective exchange rate; the numbers in parentheses are p-values. SK is skewness, K r is kurtosis, JB is the Jarque-Bera test of normality of the residuals, ARCH(r) is the LM test of no autoregressive conditional heteroscedasticity up to order r, LM SI (q) denotes the LM test of no serial correlation in the residuals up to order q, LM NL is the LM test of no remaining non linearity, and LM C is the LM test of parameter constancy.

Figure 1 : 20 Figure 2 :

 1202 Figure 1: Monthly US real effective exchange rate (Log)

Figure 3 :

 3 Figure 3: Exponential transition function (two-step estimation)

  Therefore, the long memory models, such as the FISTAR, are not only able to study the persistence but also to capture nonlinearity features such as thresholds or asymmetries. They can be applied in various economic and financial fields, in particular the stock indexes, the exchange rates and the interest rates. van Dijk et al.(2002) apply the FISTAR models to US unemployment and

have been already applied in economics and finance, for instance to exchange rates

[START_REF] Diebold | Real exchange rates under the Gold Standard[END_REF]

,

[START_REF] Cheung | Long memory and nonlinear mean reversion in Japanese yen-based real exchange rates[END_REF]

,

[START_REF] Baillie | Cointegration, fractional cointegration, and exchange rate dynamics[END_REF]

), inflation

[START_REF] Hassler | Long memory in inflation rates: international evidence[END_REF]

,

[START_REF] Baillie | Analysing inflation by the fractionally integrated ARFIMA-GARCH model[END_REF]

) and unemployment modelling

[START_REF] Diebold | Long memory and persistence in aggregate output[END_REF]

,

[START_REF] Tschernig | Illusive persistence in German unemployment[END_REF]

,

[START_REF] Koustas | Unemployment hysteresis in Canada: an approach based on long-memory time series models[END_REF]

,

[START_REF] Crato | Measuring hysteresis in unemployment rates with long memory models[END_REF]

).

Table 1 .

 1 Unit root tests : The unit root tests are Phillips and Perron (PP), Kwiatkowski, Phillips, Schmidt and Shin (KPSS) and Dickey-Fuller Augmented (ADF) tests. For ADF test, the 1%, and 5% critical values are -3.455 and -2.871, respectively. For KPSS test, the 1%, and 5% critical values are 0.739 and 0.463, respectively.

		Level First difference
	ADF	-1.118	-7.287
	PP	-1.106	-12.281
	KPSS	3.090	0.251

Note

Table 2 .

 2 Linearity tests (pvalues)

	m	1	2	3	4	5	6
	LM-test 0.868 0.346 0.087 0.073 0.251 0.171

Table 3 .

 3 Estimation of the different models : The standard errors are displayed in parentheses. * : Lo's (1991) estimator based on first difference; the value of modified R/S statistic for long memory test is in parentheses. S E is the ratio of residual variance for the nonlinear and linear models.

			FIESTAR	FIESTAR
		ARFI	(simultaneous	(two-step
			estimation)	estimation)
	π 10 0.850 (0.284)	-0.063 (0.026)	-0.003 (0.014)
	π 11 -0.183 (0.125)	0.665 (0.295)	-0.175 (0.142)
	π 12 0.179 (0.079)	-0.167 (0.335)	0.156 (0.144)
	π 13 -0.059 (0.083)	0.194 (0.394)	0.345 (0.163)
	π 14 0.103 (0.065)	-0.683 (0.290)	0.217 (0.126)
	π 20		-0.001 (0.001)	-0.004 (0.015)
	π 21		1.256 (0.078)	0.470 (0.115)
	π 22		-0.458 (0.122)	-0.238 (0.120)
	π 23		0.172 (0.119)	0.121 (0.103)
	π 24		0.035 (0.075)	-0.195 (0.121)
	d	-0.484 (0.282)	-0.169 (0.007) 0.221 * (1.896)
	γ		12.655 (8.648)	2.574 (1.190)
	c		-0.101 (0.003)	0.022 (0.020)
	S E		0.840	0.670

Note

Table 4 .

 4 Diagnostic tests ) 7.605 (0.107) 4.276 (0.370) LM SI (2) 0.765 (0.467) 1.764 (0.175) LM SI (4) 1.174 (0.325) 2.179 (0.075) LM SI (6) 1.280 (0.271) 2.111 (0.057) LM SI (8) 1.118 (0.355) 1.690 (0.106) LM SI (31) 0.746 (0.817) 0.965 (0.529)

		ARFI	FISTAR
	AIC	-8.195	-8.181
	BIC	-7.846	-0.132
	SK	-0.166	-0.133
	K r	3.297	3.006
	JB	1.313 (0.518) 0.463 (0.793)
	ARCH(1) 0.981 (0.321) 0.714 (0.398)
	ARCH(2) 1.778 (0.411) 1.292 (0.524)
	ARCH(3) 5.634 (0.130) 2.933 (0.402)
	ARCH(4LM NL	-	0.937 (0.521)
	LM C	-	0.701 (0.778)

Table 5 .

 5 Out-of-sample MPSE and modified Diebold-Mariano statistics from random walk (Rw), ARFI and FIESTAR models : Columns 2-4 report the MPSE for the random walk and ARFI models, and columns 5-6 report the modified DM test statistics with p-values in parentheses.

	h	Rw	ARF I F IEST AR ARF I&F IEST AR Rw&F IEST AR
	1 0.0079 0.0053	0.0019	8.71 (0.000)	8.98 (0.000)
	2 0.0176 0.0098	0.0085	7.52 (0.000)	8.11 (0.000)
	3 0.0292 0.0218	0.0192	6.39 (0.000)	7.13 (0.000)
	4 0.0495 0.0438	0.0346	6.22 (0.000)	6.87 (0.000)
	5 0.0799 0.0748	0.0543	5.49 (0.000)	6.22 (0.000)
	6 0.1314 0.1044	0.0775	5.36 (0.000)	5.74 (0.000)
	7 0.1670 0.1565	0.1059	5.22 (0.000)	5.49 (0.000)
	8 0.2045 0.1926	0.1363	4.54 (0.000)	4.72 (0.000)
	9 0.2449 0.2398	0.1728	4.44 (0.000)	4.59 (0.000)
	10 0.3298 0.2896	0.2129	3.71 (0.001)	4.10 (0.000)
	11 0.3770 0.3595	0.2583	3.68 (0.002)	3.88 (0.001)
	12 0.7021 0.6812	0.3087	3.07 (0.007)	3.58 (0.002)

Note

See also[START_REF] Smallwood | Joint tests for non-linearity and long memory: the case of purchasing power parity[END_REF].

The transition variable can also be assumed an exogenous variable, or a possibly nonlinear function of lagged endogenous variables. See[START_REF] Teräsvirta | Specification, estimation, and evaluation of smooth transition autoregressive models[END_REF] for more details.

[START_REF] Paya | Temporal aggregation of an ESTAR process: some implications for purchasing power parity adjustment[END_REF], Michael, Nobay and Peel (1997), Taylor, Peel and Sarno (2001), and Sarantis (1999) applied the ESTAR models to exchange rates for differents countries.

[START_REF] Beran | On unified model selection for stationary and nonstationary short-and long-memory autoregressive processes[END_REF] proposed versions of the AIC, BIC and the HQ (Hannan and Quinn, 1979) which are suitable for fractional autoregressions, but do not consider moving average components.

For a survey on long memory models and their application in economics and finance, see[START_REF] Baillie | Long memory processes and fractional integration in econometrics[END_REF],[START_REF] Robinson | Time series with long memory[END_REF] among others.

For other unit root tests see[START_REF] Elliot | Efficient tets of an autoregressive unit root[END_REF], among others.

Contrary to ADF test, the KPSS test considers the stationnarity under the null hypothesis, and the alternative hypothesis is the presence of unit root.

[START_REF] Boutahar | Estimation Methods of The Long Memory Parameter: Monte Carlo Analysis and Application[END_REF] This result is also found in[START_REF] Sarantis | Modelling non-linearities in real effective exchange rates[END_REF].