Explaining the European exchange rates deviations: Long memory or non-linear adjustment?

GREQAM Groupement de Recherche en Economie

Quantitative d'Aix-Marseille - UMR-CNRS 6579 Ecole des Hautes Etudes en Sciences Sociales

Universités d'Aix-Marseille II et III

Document de Travail n°2006-34

EXPLAINING THE EUROPEAN EXCHANGE RATES DEVIATIONS: LONG MEMORY OR

NONLINEAR ADJUSTMENT?

Gilles DUFRENOT Sandrine LARDIC

Laurent MATHIEU Valérie MIGNON

Anne PEGUIN-FEISSOLLE

Septembre 2006

Explaining the European exchange ratesdeviations: long memory or nonlinear

adjustment?�

Gilles DUFRENOTy Sandrine LARDICz

Laurent MATHIEUx Valérie MIGNON{

Anne PEGUIN-FEISSOLLEk

Abstract

The standard macroeconomic view links the equilibrium level of foreign ex-change rates to the state of the macroeconomic fundamentals. Any deviationfrom the equilibrium level is viewed as temporary since there are forces ensuringquickly mean-reverting dynamics. The aim of this article is to investigate whetherthe empirical observation of the real exchange rate misalignments in �ve Euro-pean countries over the period 1979-1999 was consistent with the hypothesis oftemporary deviations from the fundamentals, or whether they must be associatedwith signi�cant persistent dynamics. We depart from the traditional framework oflinear cointegration by using fractional cointegration or nonlinear cointegration.Therefore, we will try to discriminate between linear long memory dynamics andnonlinear short memory dynamics.

This version: July 27, 2006J.E.L. Classi�cation: C22, F31.Keywords: exchange rates, nonlinear cointegration, fractional cointegration.

�Corresponding author: Valérie MIGNON, University of Paris 10, EconomiX-CNRS, 200 avenue dela République, 92001 Nanterre Cedex, France. Tel: +33.1.40.97.58.60. Fax: +33.1.40.97.59.73. Email:[email protected]

yGREQAM and ERUDITE, University of Paris 12, France.zEconomiX-CNRS, University of Paris 10, France.xC3ED, University of Saint-Quentin en Yvelines and EconomiX, University of Paris 10, France.{EconomiX-CNRS, University of Paris 10, and CEPII, Paris, France.kGREQAM-CNRS, France.

1

1. Introduction

The standard macroeconomic view links the equilibrium level of foreign exchange

rates to the state of the macroeconomic fundamentals. Any deviation from the equi-

librium level is viewed as temporary since there are forces ensuring quickly mean-

reverting dynamics. However, in many countries, the experience of real exchange

rates over the last two decades has been characterized by substantial misalignments,

with time lengths much higher than suggested by the theoretical models.

The perception that exchange rates can spend long periods away from their funda-

mental values implied a revival of interest in the study of exchange rate misalignments.

Indeed, policy implications are really important. In the case of temporary disequilib-

rium, there might be some room for the policy-makers to prevent further disequilib-

rium through policy measures that correct the imbalance and promote fast convergence

towards the equilibrium level. On the contrary, in situations of prolonged deviations

from the equilibrium level, efforts towards more rigorous stance in macroeconomic

policies and increased convergence would prove futile.

Our aim is to investigate whether the empirical observation of the real exchange

rate misalignments in the European countries over the period 1979-1999 was consis-

tent with the hypothesis of temporary deviations from the fundamentals, or whether

they must be associated with signi�cant persistent dynamics. Indeed, the case of the

European countries is a salient example of a controlled exchange rate regime. The

stylized facts suggest that the European currencies, over this period, were not charac-

2

terized by quick realignments and mean-reversion dynamics.

We derive the long-run value of the real exchange rates from a Behavioral Equilib-

rium Exchange Rate model (BEER) as proposed by Clark andMcDonald (1999, 2000).

The traditional way to estimate a BEER model is by using OLS on a static equation

relating the real exchange rate to its fundamentals and by applying a cointegration test

to the residuals. When testing for cointegration, one assumes that the deviations are

captured by linear processes. Although the BEER can be thought as a steady state

attractor for the actual rate, the adjustment process towards the equilibrium is however

not appropriately represented by linear equations. Firstly, the deviations at a given

date are not unaffected by the magnitude of previous misalignments. The dependence

of exchange rates to historical paths has received several theoretical interpretations in

the recent literature. For instance, some authors have proposed explanations based

on heterogeneous behaviors in exchange rate markets (De Long et al. (1990)), or on

target zone models (Tronzano, Psaradakis and Sola (2003)). Temporal dependence

implies that the adjustment is likely not to operate at a constant rate as assumed in

linear models. Further, although the standard models of exchange rate determination

often assume that prices move quickly, in practice the latter seem more in�exible. This

means that, when a disequilibrium appears, prices do not quickly and painlessly move

to bring the actual real exchange rate to its equilibrium level. In this context, using

an empirical framework based on nonlinear cointegration is more appropriate than the

standard linear cointegration regressions. A great deal of research has been done on

the empirical implications of nonlinear cointegration in economics (for an overview,

3

see Dufrénot and Mignon (2002)). In line with the idea that the misalignments can

switch in different states, we proposed a parametric model based on smooth transition

autoregressive processes (STAR).

A second reason why we do not use the linear cointegration framework is that it is

not adequate to describe the dynamic impact of persistent shocks on future misalign-

ments. The observation of persistent dynamics in the exchange rate movements has

theoretical counterparts in the literature. For instance, it has been shown that the de-

viations of the exchange rates from their fundamental value are described by a long

memory process, when these deviations come from market sentiments, possibly un-

correlated with the macroeconomic fundamentals (see, among others, Kirman and

Teyssière (2002)). In this context, it is more natural to study the real exchange rate

misalignments using fractional cointegration models.

In summary, our goal is to provide a better understanding of the adjustment dy-

namics that underlies the misalignments of the European exchange rates, by exploring

the type of cointegration between exchange rates and their fundamentals.

The paper is organized as follows. Section 2 describes the empirical model, the

data and cointegration tests. In Section 3, long memory and exponential STAR models

are compared. Section 4 concludes the paper.

4

2. The econometric model, data and cointegration tests

2.1. The model

The equilibrium relationship between the real exchange rates and the macroeconomic

fundamentals is modelled using the following speci�cation:

log(qt) = �0 + �1 log(TOTt) + �2 log(TNTt) + �3NFAt + �4�t (1)

+�5FISCALt + �6(rt � r�t ) + "t

The dependent variable q is the real effective exchange rate. It is the consumer

price index (CPI)-based multilateral real exchange rate. It is de�ned as the ratio of

the domestic CPI to the foreign CPI. The term �effective� means that it is constructed

as the trade weighted average of the exchange rates, where the weights are based on

bilateral trade share.

We consider the following fundamentals for the equilibrium exchange rates1:

� TOT is an indicator of the terms of trade, de�ned as the ratio of export price

index to import price index;

� TNT is the ratio of the prices of non traded goods over the prices of domestic

traded goods;

� NFA is a variable of net foreign assets; we use data on net income from abroad

as a proxy;1All the variables are de�ned in relative terms: we consider a country's variable over a weighted

average of the same variable for the other countries.

5

� � is the ratio of the government's de�cit over GDP;

� FISCAL is an indicator of �scal wedge. It is measured as the difference be-

tween the real wages and salaries paid by the employers and the real wages

and salaries earned by the employees. This variable captures the in�uence of

the wage-price loop on the real exchange rate. The �scal wedge is introduced to

evaluate the impact of tax reforms undertaken on the labour markets. Indeed, the

�scal wedge affects the unit labour costs, which in turn have some implications

on the unit prices of both the non traded and traded goods and thus indirectly

affects the real exchange rate;

� (rt � r�t ) denotes the interest rate differential (10 year government bond yield);

rt is the domestic interest rate and the interest rate r�t is de�ned as a weighted

mean of the interest rates of the other countries.

"t is a white noise process.

2.2. Data and unit root tests

We use quarterly data over the period 1979-1999 for the following �ve countries:

France, Germany, the UK, the Netherlands and Portugal. The data were obtained from

the OECD main economic indicators and Eurostat database. The weights on bilateral

trades were obtained from IMF.

As a �rst step to our analysis, we investigate the order of integration of the variables

by applying three unit root tests: the Augmented Dickey-Fuller (ADF), the Phillips-

6

Perron (PP) and the Kwiatkowski et al. (KPSS) tests. As expected, the results2 indicate

that the unit root hypothesis is generally not rejected for the series.

2.3. Cointegration tests

2.3.1. Linear cointegration tests

We �rst test for linear cointegration using residual-based tests: we apply the ADF, PP

and KPSS tests to estimated residuals from equation (1). Recall that ADF and PP test

the null hypothesis of non cointegration, while KPSS is based on the null of cointe-

gration. Our results3 indicate that, neither the unit root hypothesis, nor the assumption

of stationarity can be rejected at the 5% signi�cance level. The exception is Portugal

for which the three tests lead to the same conclusion, i.e. no linear cointegration for

this country. Thus, for most countries, opposite null hypotheses are not being rejected.

In this case, there is no evidence of whether there is cointegration or not. The oppo-

site conclusions lead us to examine more deeply the type of cointegration inherent to

the estimated residuals and look beyond linearity. The �rst issue of concern will be

fractional cointegration and the second issue nonlinear cointegration.

2.3.2. Fractional cointegration and long memory dynamics

In order to test for the hypothesis of a slow mean-reverting dynamics, we use long

memory models such as ARFIMA(p; d; q) processes (Auto Regressive Fractionally

Integrated Moving Average):

�p (B) (1�B)d "t = �q (B) �t (2)2The results are available upon request to authors.3The results are available upon request to authors.

7

where �t is a strong white noise with �nite variance, B is the backward shift operator,

�p (B) and �q (B) are two lag polynomials of respective orders p and q: �p (B) =

1� �1B � �2B2 � :::� �pBp and �q (B) = 1� �1B � �2B2 � :::� �qBq.

The fractional cointegration test consists in testing the null hypothesis that the real

exchange rate and the fundamentals are not cointegrated, e.g. "t � I(1) against the al-

ternative that the real exchange rate and the fundamentals are fractionally cointegrated,

e.g. "t � ARFIMA(p; d; q), d < 1.

The results of the preceding section suggested that the estimated residuals of the

long-run relationship were possibly I(1). Instead of testing the d = 1 hypothesis on

the levels of the residuals, we thus test the hypothesis d0 = 0 on the differenced series

(since the procedures we use to estimate the fractional parameter work on stationary

time series), with d0 = d � 1. The test can thus be re-stated as follows: H 00 : d

0 = 0

against H 01 : d

0 < 0.

We consider three estimators for d0: the Lo (1991) modi�ed R=S statistic, the

Geweke and Porter-Hudak (1983, GPH) log-periodogrammethod and the Sowell (1992)

exact maximum likelihood estimation. Table 1 reports the results relating to the modi-

�ed R=S statistic. We see that the null is rejected for the UK and the Netherlands.

[insert table 1]

From the GPH procedure (table 2), we conclude that the estimated residuals for

Germany, the UK and Portugal may be considered as I(1). In the cases of France and

the Netherlands, the residuals seem characterized by a long memory dynamics.

8

[insert table 2]

The results based on the R=S analysis are usually more powerful than those ob-

tained with the GPH estimator, when one considers residual-based analysis (see the

Monte Carlo results by Dittmann (2000)). So, in regard to the contradictory results

with the GPH analysis, we are inclined to prefer the conclusions based on the mod-

i�ed R=S statistic. Therefore, we may expect a fractional cointegration relationship

between the real exchange rate and its fundamentals for the UK and the Netherlands.

We �nally apply the Sowell (1992) exact maximum likelihood approach to esti-

mate the value of d with autoregressive and moving average components. The results

reported in table 3 seem to be in line with those of the GPH analysis. Indeed, no

long-term dependence is found for Germany, the UK and Portugal, while the residual-

based analysis yields to �nd ARFIMA models with values of d that are statistically

signi�cant for France and the Netherlands.

[insert table 3]

We �nally note that the estimated value of d0 obtained for the Netherlands is out-

side the range where the process is invertible and stationary. One reason may be the

presence of structural changes in the exchange rate dynamics. As shown by Diebold

and Inoue (2001), structural changes can be easily confused with long memory in em-

pirical applications.

9

2.3.3. Nonlinear cointegration

The mixed results of the preceding cointegration tests may come from the presence of

nonlinearities in the adjustment process towards the equilibrium value. Various mod-

elling approaches have been proposed, among which the Smooth Transition Autore-

gressive models (STAR). Generally two types of STAR models are considered. If one

assumes that the current deviation of the real exchange rates responds symmetrically

to either positive or negative past deviations, then the exponential smooth transition

autoregressive models (ESTAR) are convenient. If the adjustment towards the funda-

mental value in response to positive or negative shocks is instead asymmetric, logistic

STAR models (LSTAR) are more appropriate. In this paper, we considered both types

of adjustment: the nonlinearity tests lead us to reject the case of an asymmetric adjust-

ment of a logistic type (see below). We therefore use the ESTAR models.

Let "t be the deviation of the real exchange rate to its fundamental value. We

consider the following nonlinear speci�cation:

�"t = �0+�1"t�1+

pXi=1

��i�"t�i+

"�0 + �1"t�1 +

pXj=1

��j�"t�j

#F (xt�d; ; c)+�t;

(3)

where �t is a white noise process with a �nite variance. F (xt�d; ; c) is a transition

function with xt�d as the transition variable and ( ; c) are two parameters. Two transi-

tion functions are usually considered:

� the logistic function:

F (xt�d; ; c) = [1 + exp (� (xt�d � c))]�1 (4)

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� the exponential function:

F (xt�d; ; c) = 1� exp�� (xt�d � c)2

�(5)

xt�d is a transition variable, that we consider to be either "t�d or �"t�d, where d

is a lag. xt�d = "t�d means that it is the magnitude of past deviations that in�uence

the current misalignment. xt�d = �"t�d means that the speed of past adjustments

determines the current deviation. The slope parameter ( > 0) indicates how rapidly

the transition between the two extreme regimes is. c is the location parameter, which

indicates where the transition occurs.

To �nd an appropriate speci�cation for the European exchange rate deviations, we

use a sequential procedure (see Escribano and Jorda (1999) for instance). Concerning

the transition function, the choice is based on a linearity test when the null of a linear

model is tested against the alternatives of STAR and ESTAR models. The optimal

choice for p, d and xt (either xt = "t or xt = �"t) in (3) is based on the information

criteria (Bayesian, Akaike and Schwarz criteria) and the speci�cation tests on the es-

timated residuals. Depending upon the results of the linearity tests, either a LSTAR

or an ESTAR model is estimated with the transition variable that was �nally retained.

We use the log-likelihood estimator, although the method is costly (this implies many

iterations to obtain a global maximum).

[insert table 4]

The application of the linearity tests yielded us to reject a logistic STAR speci�ca-

11

tion4. Table 4 thus presents the estimates based on ESTAR speci�cations. The transi-

tion variables are those selected from the linearity tests. There is a clear indication of

a nonlinear mean-reverting dynamics for the Netherlands, the UK and the Portugal, in

spite of the presence of a �linear� unit root. We indeed see that �1 is statistically not

signi�cant, but �1 is statistically signi�cant and negative. The case of Germany illus-

trates a situation where the linear component contains an explosive root but the model

is globally stable (�1 < 0 and �1+ �1 < 0). This can be explained by a higher volatil-

ity of the German real exchange rate during the EMS period in comparison with the

other countries. France is the only country for which there is no evidence of nonlinear

cointegration. Not only �1 is signi�cantly negative, but we also have j�1j > j�1j with

�1+ �1 < 0. This means that the misalignment is characterized by a mean-reverting

dynamics, mainly due to the linear component.

3. Comparing long memory and nonlinear models

We now examine how the different models (ARFIMA and ESTAR) perform in fore-

casting using Diebold and Mariano (1995) procedures. The latter amount to test

whether the models have equal accuracy in the predictions on the whole sample. Re-

sults in table 5 correspond to the case where the ARFIMA processes are estimated

by the exact maximum likelihood method and selected by the Schwarz information

criterion. We do not report the results corresponding to the GPH method since the

conclusions are identical. There is no signi�cant difference between the predictive4The results are available upon request to the authors.

12

methods for the UK, the Netherlands and Portugal. On the contrary, for France and

Germany, ARFIMA and ESTAR models give signi�cantly different performance. In-

deed, for these two countries, the ARFIMA models seem to outperform the ESTAR

models (there is a high fraction of the residuals of the ARFIMA models that are less

than those of the ESTAR models). However, this conclusion must be considered with

caution, given that nonlinear models sometimes perform quite poorly in forecasting.

[insert table 5]

4. Conclusion

The aim of this paper was to study the adjustment dynamics of the real exchange rates

towards the fundamentals for �ve European countries. After an analysis in terms of

cointegration tests, we were interested in explaining their con�icting results. In par-

ticular, we tried to discriminate between dynamics of short memory nonlinear adjust-

ment (STAR processes) and dynamics of long memory linear adjustment (ARFIMA

processes). We conclude that the adjustment dynamics of the real exchange rate for

France can be well represented by long memory process, showing the persistency of

the real exchange rate deviations from its fundamentals. In the case of the United King-

dom and the Netherlands, the tests for comparing the predictive accuracy do not detect

signi�cant difference between a smooth transition nonlinear process and a long mem-

ory process. This last remark illustrates the dif�culty to distinguish long memory from

switching regime processes. Finally, concerning Germany, the tests for comparing the

predictive accuracy show that the long memory model is better than the ESTARmodel,

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even if the tests for long memory do not detect a signi�cant persistency phenomena.

The �nding of both long-memory and persistent nonlinear dynamics for the real ex-

change rate has some implications for policymaking and market practitioners. Firstly,

a persistent dynamics of a long-memory type can be interpreted as the consequence

of trade barriers and sunk costs of international arbitrage. So, traders must wait for

suf�ciently large arbitrage opportunities before entering the markets. They may be

reluctant to very frequent interventions since the marginal cost of buying and selling

at high frequencies exceeds the marginal bene�ts. Secondly, a long-memory dynam-

ics may results from real-side shocks, for instance technological shocks, that affect

the real exchange rate through their impact on the terms of trade or the ratio of the

prices of non traded goods over those of traded goods. In this case, to correct for an

overvalued exchange rate it may be more indicated to adopt supply-side policies rather

than demand-side policies based on monetary innovations. Thirdly, nonlinear cointe-

gration is very useful to evaluate the exact degree of misalignment (see, Dufrénot et al.

(2006)).

References

Clark, P.B., MacDonald, R., 1999. Exchange rates and economic fundamentals: amethodological comparison of BEERs and FEERs. In: MacDonald, R., Stein, J.L.(Eds.), Equilibrium Exchange Rates. Kluwer Academic Press: London, U.K.

Clark, P.B., MacDonald, R., 2000. Filtering the BEER: a permanent and transitorydecomposition. Working paper 00144, IMF.

De Long, J.B., Shleifer, A., Summers, L.H., Waldmann, R.J., 1990. Noise trader riskin �nancial markets. Journal of Political Economy 98, 703-738.

Diebold, F.X., Inoue, A., 2001. Long memory and regime switching. Journal ofEconometrics 105, 131-159.

Diebold, F.X., Mariano, R., 1995. Comparing predictive accuracy. Journal of Business

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and Economic Statistics 13, 253-263.Dittmann, I., 2000. Residual-based tests for fractional cointegration: a Monte Carlostudy. Journal of Time Series Analysis 21(6), 615-647.

Dubois, E., Lardic, S., Mignon, V., 2004. The exact maximum likelihood based-test for fractional cointegration: critical values, power and size. ComputationalEconomics 24(3), 239-255.

Dufrénot, G., Mathieu, L., Mignon, V., Péguin-Feissolle, A., 2006. Persistent mis-alignments of the European exchange rates: some evidence from nonlinear coin-tegration. Applied Economics 38, 203-229.

Dufrénot, G., Mignon, V., 2002. Recent Developments in Nonlinear Cointegrationwith Applications toMacroeconomics and Finance. Kluwer Academic Press: Dor-drecht.

Escribano, A., Jorda, O., 1999. Improved testing speci�cation of smooth transitionregression models. In: Rothman, P. (Ed.), Nonlinear Time Series Analysis ofEconomic and Financial Data. Kluwer Academic Press: Boston.

Geweke, J., Porter-Hudak, S., 1983. The estimation and application of long memorytime series models. Journal of Time Series Analysis 4, 221-238.

Hurvich, C.M., Tsai, C.L., 1989. Regression and time series model selection in smallsamples. Biometrika 76, 297-307.

Kirman, A., Teyssière, G., 2002. Bubbles and long range dependence in asset pricesvolatilities. In: Hommes, C., Ramer, R., Withagen, C. (Eds.), Equilibrium, Mar-kets and Dynamics. Springer Verlag.

Lo, A.W., 1991. Long-term memory in stock market prices. Econometrica 5, 1279-1313.

Sowell, F., 1992. Maximum likelihood estimation of stationary univariate fractionallyintegrated time series models. Journal of Econometrics, 165-188.

Tronzano, M., Psaradakis, Z., Sola, M., 2003. Target zones and economic fundamen-tals. Economic Modelling 20, 2003, 791-807.

15

Table 1. Modi�ed R/S analysis on the residuals

V d0

France 0.7696 -0.0598Germany 0.9713 -0.0066UK 0.5551** -0.1343Netherlands 0.6863** -0.0859Portugal 1.1247 0.0268

Note: This table reports the value of the V statistic developed by Lo (1991) which is de�nedas the ratio of the modi�ed R/S statistic to the square root of the number of observations; moreover,d0 = d � 1. The test is based on the null hypothesis of short memory against the alternative oflong memory. It is applied to residual in �rst differences.**: rejection of the null hypothesis at the 5%signi�cance level (critical values in Dittmann (2000)).

Table 2. ARFIMA estimation by Gewekeand Porter-Hudak method

T 0:45 T 0:5 T 0:55 T 0:8

France d0td0

�0:85��2:21

�0:73��2:10

�0:63��2:29

�0:35��2:61

Germany d0td0

�0:40�1:05

�0:29�0:83

�0:37�1:34

�0:01�0:10

UK d0td0

�0:49�1:28

�0:32�0:94

�0:24�0:89

�0:06�0:44

Netherlands d0td0

�1:39��3:62

�1:05��3:03

�0:62��2:27

�0:16�1:15

Portugal d0td0

0:070:18

0:220:63

0:100:37

�0:02�0:16

Note: This table reports the estimated value of the fractional differencing parameter d0 accordingto the GPH procedure applied to the residuals in �rst differences. Four numbers of frequencies areused, corresponding to four power transformations of the number of observations T . td0 is the t-statisticassociated with the estimated fractional integration parameter d0. *: rejection of the null hypothesis atthe 5% signi�cance level (critical values in Dittmann (2000)).

16

Table 3. ARFIMA estimation by exactmaximum likelihood method

AICc SICARFIMA(p; d0; q) LV ARFIMA(p; d0; q) LV

France (1;�0:5940; 0) 208:84 (0;�0:3562; 0) 207:23td0 = �4:0704� td0 = �3:0887��

Germany (1;�0:6489; 0) 212:16 (1;�0:6489; 0) 212:16td0 = �2:1495 td0 = �2:1495

UK (1;�0:5581; 0) 140:52 (0;�0:1239; 0) 138:94td0 = �1:2626 td0 = �1:2484

Netherlands (2;�1:8429; 1) 240:25 (2;�1:8429; 1) 240:25td0 = �11:2386��� td0 = �11:2386���

Portugal (0;�0:1847; 0) 171:59 (0;�0:1847; 0) 171:59td0 = �1:9722 td0 = �1:9722

Note: This table reports the estimated ARFIMA(p,d',q) processes using the exact maximum like-lihood procedure applied to �rst differenced residuals. The displayed models have been selected usingtwo information criteria: the Akaike information criterion corrected by Hurvich and Tsai (1989), de-noted as AICc, and the Schwarz information criterion (SIC). td0 is the t-statistic associated with theestimated fractional integration parameter d0. *: rejection of the null hypothesis at the 10% signi�cancelevel, **: at the 5% signi�cance level, ***: at the 1% signi�cance level (critical values are tabulated inDubois et al. (2004)). LV is the value of the log-likelihood at the optimum.

17

Table 4. Estimation of ESTAR models

Model�"t = �0 + �1"t�1 + �2�"t�1 + [�0 + �1"t�1

+�2�"t�1]F (xt�d; ; c) + "tCountry France Germany Netherlands UK Portugalxt�d "t�2 "t�11 �"t�3 �"t�4 �"t�1�0 � � 0:002

(0:86)0:0058(0:722)

�0:002(�0:212)

�1 �2:56(�4:59)

0:744(2:80)

�0:004(�0:033)

�0:0145(�0:107)

0:114(1:078)

�2 � � 0:425(1:732)

0:109(0:508)

0:143(0:389)

�0 � � �0:032(�1:417)

�0:085(�0:824)

0:007(0:086)

�1 2:30(3:97)

�1:00(�3:61)

�2:141(�1:862)

�4:185(�2:218)

�1:204(�2:158)

�2 � � �3:031(�1:936)

�1:867(�0:647)

�1:179(�1:876)

365:0 80:0 80:0 5:00 95:0c �0:0467 �0:057 0:0369 0:112 0:04GB 0:918 0:975 0:606 0:863 0:819JB 0:2� 10�4 0:947 0:210 0:912 0:201NL 0:10 0:66 0:517 0:33 0:118

CONST 0:006 0:91 0:624 0:59 0:344

Note: This table reports the estimation of Exponential Smooth Transition AutoRegressive (ESTAR)models for the deviations of the real exchange rate to its equilibrium value. A method based on thequasi maximum likelihood has been used. Student t-statistics are in parentheses. GB is the probabilityassociated to the Godfrey-Breusch test statistic of order-p serial correlation (results are reported forp = 4). JB is the probability associated to the Jarque-Bera normality test. NL and CONST are theprobabilities associated respectively to the remaining nonlinearity and parameter constancy tests.

Table 5. Tests for comparing predictive accuracy of ESTARand ARFIMA models (p-values). Case where the ARFIMAmodels are estimated by the exact maximum likelihood method

France Germany UK Netherlands PortugalAsymptotic test 0.028 0.003 0.405 0.667 0.283Sign test 0.001 0.000 1.000 0.568 0.910Wilcoxon's test 0.000 0.000 0.868 0.848 0.856Naive benchmark test 0.000 0.000 0.699 0.530 0.750Morgan-Granger-Newbold's test 0.000 0.000 0.269 0.880 0.270Meese-Rogoff's test 0.004 0.000 0.115 0.870 0.313

Note: This table reports the p-values associated with the six tests of predictive accuracy describedin Diebold and Mariano (1995). The null hypothesis is the hypothesis of equal accuracy of ESTAR

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and ARFIMA models. ARFIMA processes are estimated by the exact maximum likelihood procedureand have been selected by the Schwarz information criterion. The loss function is quadratic. The teststatistics follows asymptotically different distributions: N(0; 1) for the asymptotic test, the sign test,the Wilcoxon's test, the Meese-Rogoff's test, F (T; T ) for the Naive benchmark test and a tT�1 forthe Morgan-Granger-Newbold's test. The Meese-Rogoff test statistic is computed with the Diebold-Rudebusch covariance matrix estimator. The truncation lag is 2 for the asymptotic test and is given bythe integer part of T 4=5 for the Meese-Rogoff's test.

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