Montserrat Ferre
email: montserrat.ferre@urv.cat

Julian Ramajo

Julián Ramajo

Montserrat Ferré

EVIDENCE FROM OLD AND NEW TESTS FOR AN OECD PANEL

Keywords: C23 -Models with Panel Data &lt, C2 -Econometric Methods: Single Equation Models &lt, C -Mathematical purchasing power parity, exchange rates, OECD C23, F31, G15 purchasing power parity, exchange rates, OECD

published or not. The documents may come

Introduction

Purchasing Power Parity (PPP) is one of the most thoroughly studied propositions, but even though it has received such attention by the literature, there is not yet clear agreement between the scholars about its empirical validity. In particular, the debate on the validity of long-run PPP for the floating period that followed the collapse of the Bretton Woods system in 1973 has not been concluded. 1 It is now a good time, after more than three decades of debate about PPP, for contributing to this debate and take advantage of the econometric techniques developed in the last 15 years. These techniques have allowed for an increase in the power of nonstationarity tests and to complement them with tests based on nonlinear regressions that are theoretically motivated. For this reason, in this article we will carry out a comprehensive study of long-run PPP in the post-Bretton Woods era applying a large battery of tests, univariate and multivariate, single-equation and panel, linear and nonlinear, using data for 21 OECD countries.

Overview

Much previous research has assessed the validity of long-run PPP by analysing whether the real exchange rate is stationary. The real exchange rate ( t q ) is defined as the nominal exchange rate ( t s ) minus the difference between the domestic price index ( t p ) and the foreign price index ( * t p ) as follows:

* t t t t p p s q + (1)
where t s is defined as units of domestic currency per foreign currency and all variables are in logs. If PPP holds in the long-run, then the log of the real exchange rate, t q , should be zero, that is, the log of the nominal exchange rate, t s , should be equal to the difference in the log price levels (a less strict version postulates that q t may have nonzero mean but it has to be a realization of a stationary process). Therefore, a necessary condition for PPP to hold in the long run is that the real exchange rate is mean reverting or, in the terminology of time series analysis, that it does not contain a unit root.

At the end of the 80s, following on from the development of techniques specifically designed to test for unit roots, a substantial number of studies tested -and 2 failed to reject-whether t q contained a unit root (for a study, see [START_REF] Taylor | An empirical examination of long-run purchasing power parity using cointegration techniques[END_REF], and for a survey, see [START_REF] Taylor | Purchasing Power Parity[END_REF]. However, as shown by a number of papers, the power of these tests when using a reduced number of years was low, and so authors started to look at other avenues of analysis of the long-run behaviour of the real exchange rate.

One such avenue has complemented the univariate analysis of unit roots with panel data tests given that [START_REF] Banerjee | Panel Data Unit Roots and Cointegration: an Overview[END_REF] and [START_REF] Baltagi | Nonstationary panels, cointegration in panels and dynamic panels: A survey[END_REF], among others,

show that unit root tests based on panel data are more powerful than those based on individual data. Initially, many of the studies that applied panel unit root tests to a number of real exchange rates series over the recent float rejected the unit root hypothesis. Nonetheless, as pointed by [START_REF] Taylor | The behaviour of real exchange rates during the post-Bretton Woods period[END_REF], the null hypothesis of these tests is generally that all the series are generated by unit-root processes, and therefore the probability of rejection of the null hypothesis may be quite high, as one only needs that one of the series is stationary to reject the null. Taking into account this criticism, [START_REF] Sarno | Real Exchange Rates under the Recent Float: Unequivocal Evidence of Mean Reversion[END_REF] and [START_REF] Coakley | Is there a base currency effect in long run PPP?[END_REF] find support for the long-run PPP using panel unit root tests.

Another avenue of study of PPP has been to examine whether the real exchange rate does in fact present mean reversion, but in a non-linear way or with high persistence. For instance, Michael et al. (2001), [START_REF] Taylor | Nonlinear mean-reversion in real exchange rates: Towards a solution to the purchasing power parity puzzles[END_REF] and [START_REF] Kapetanios | Testing for a unit root in the nonlinear STAR framework[END_REF] apply smooth-transition autoregressive models to real bilateral exchange rates and show that they are well characterized by nonlinearly mean reverting processes. On the other hand, [START_REF] Cheung | A fractional cointegration analysis of purchasing power parity[END_REF] and [START_REF] Gil-Alana | Estimation and testing of ARFIMA models in the real exchange rate[END_REF] show that real exchange rates are mean reverting but they exhibit significant persistence in the short run.

Due to the common factor restrictions implicit in the analysis of real exchange rates [START_REF] Sarno | Deviations from Purchasing Power Parity Under Different Exchange Rate Regimes: Do They Revert and, If So, How[END_REF], long-run PPP has also been analysed by looking at the presence of a cointegrating relationship between the nominal exchange rate and the prices. For instance, by using the panel cointegration methods that allow to test the null of no cointegration without imposing homogeneity of the cointegrating vector, [START_REF] Canzoneri | Relative labour productivity and the real exchange rate in the long-run: Evidence for a panel of OECD countries[END_REF] and [START_REF] Pedroni | Panel cointegration: asymptotic and finite sample properties of pooled time series tests with an application to the PPP hypothesis[END_REF] find support for the weak version of PPP.

On the other hand, [START_REF] Pedroni | Purchasing power parity tests in cointegrated panels[END_REF] tests directly that the cointegrating vector is homogeneous and equal to one (strong version of PPP) and it is clearly rejected.

Questions addressed and methodology

As indicated before, the objective of this article is to study long-run PPP for 21 OECD countries from the end of the Bretton Woods era by applying a wide range of the econometric techniques available. This will allow us to present a comprehensive review of PPP, covering the weak and strong versions of PPP and individual and panel analysis, including the absence or presence of cross-dependency, the linear or nonlinear behaviour of the real exchange rates, and the degree of persistence. 2

We will start the analysis by studying the stationarity of the real exchange rates of our sample. Due to the well-documented low power of conventional unit root tests we will use powerful tests recently proposed in the econometric literature [START_REF] Elliott | Efficient tests for an autoregressive unit root[END_REF][START_REF] Ng | Lag length selection and the construction of unit root tests with good size and power[END_REF]. We will also benefit from the latest developments in the analysis of unit roots by applying a variety of panel tests, such as those developed by [START_REF] Levin | Unit roots tests in panel data: Asymptotic and finite-sample properties[END_REF], [START_REF] Breitung | The local power of some unit roots tests for panel data[END_REF], [START_REF] Hadri | Testing for stationarity in heteogeneous panel data[END_REF], [START_REF] Im | Testing for unit roots in heterogeneous panels[END_REF], [START_REF] Maddala | A comparative study of unit root tests with panel data and a new simple test[END_REF][START_REF] Pesaran | A simple panel unit root test in the presence of cross section dependence[END_REF], and the PANIC/PASIC decomposition proposed by Bai and Ng (2004a,b).

We will also complement the non-stationary analysis by considering, first, if there is non-linearity and second, if there is persistence in the real exchange rates' behaviour. To that end, we will first use non-linear techniques like the smooth transition regressions (STR). In order to provide a general analysis, and also to take advantage of the fact that the test will, according to the data, choose whether there is a symmetric or an asymmetric behaviour, we will choose a logistic STR model. Second, we will also analyse the persistence of the series by applying ARFIMA models.

Finally, following a cointegrating vector autoregresssion (VAR) approach -that do not impose any a priori common factor restriction as single-equation and panel unit root test do-we will use the tests recently developed by [START_REF] Pedroni | Critical values for cointegration tests in heterogeneous panels with multiple regressors[END_REF][START_REF] Pedroni | Panel cointegration: asymptotic and finite sample properties of pooled time series tests with an application to the PPP hypothesis[END_REF], [START_REF] Mccoskey | A residual-based test of the null of cointegration in panel data[END_REF], Westerlund (2005a,b,c) and [START_REF] Larsson | Likelihood-based cointegration tests in heterogeneous panels[END_REF] to look for the presence of cointegration among the nominal exchange rate and the prices. When cointegration is found, there will be evidence in favour of the weak 2 We should point out several relevant caveats. First, our analysis is inevitably incomplete because all types of tests are not included, particularly the nonparametric and the Bayesian approaches. We are aware of this limitation but, similarly to the bulk of the literature, we rely on the parametric or semiparametric approaches in our empirical work. Secondly, most of the assumptions underpinning these tests are not mutually independent. For instance, non-linearity and temporal aggregation reinforce each other, fractionally integrated processes may have some cross-sectional dependence or cointegration may happen in the presence of non-linearities. Due to the complex nature of these questions we do not attempt to resolve the econometric problems surrounding the above issues leaving such ambitious aim for future work. version of PPP. In the case that there is cointegration, we will also test for the fulfilment of the strong version of PPP.

Main results and interpretation

Our results provide overall support for the validity of long-run PPP in the Post Bretton Woods era. In the first place, there seems to be evidence of mean reversion of the real exchange rates, which would give support to the strong version of PPP. and ARFIMA models. The STR models point towards the presence of nonlinearities in some of the series, and the analysis of highly persistent behaviour through ARFIMA models provides evidence of stationarity in all the series even though with a considerable degree of temporal dependence in the evolution of real exchange rates.

Organization

The article is structured as follows. In section 2 we report the unit root analysis through univariate and multivariate tests. Section 3 presents the non-linearity and persistence analysis of real exchange rates. In section 4 the panel cointegration tests are discussed and, finally, section 5 concludes.

Unit root analysis

In this section we will test the stationarity of the real exchange rates considered in our sample. To calculate the real exchange rate as in (1), we will use quarterly nominal exchange rates (end of period and expressed as domestic currency per US dollar) and consumer price indexes for 21 industrialised OECD countries and the United States. The period of analysis will be the 31 post-Bretton Woods years since 1973Q1 to 2004Q4. All data is available from the International Financial Statistics Online Service of the International Monetary Fund.

In Table 1a we present the results of different individual unit root tests.

Columns 1 and 2 of Table 1a report the well known augmented Dickey-Fuller -ADFtest, whose null hypothesis is the existence of a unit root, and the Kwiatkowski-Phillips-Schmidt-Shin -KPSS-test, whose null hypothesis is the stationarity of the variable analysed. The reason for using both tests lies, as pointed by [START_REF] Maddala | Unit Roots, Cointegration, and Structural Change[END_REF], in the well-known low power of the ADF test in small samples and in the tendency of the KPSS test for over-rejection in this type of samples. Columns 3 to 5 of Table 1a show the results from three types of tests that, according to the literature [START_REF] Elliott | Efficient tests for an autoregressive unit root[END_REF][START_REF] Ng | Lag length selection and the construction of unit root tests with good size and power[END_REF], are more powerful than the ADF in the analysis of non-stationarity: the augmented Dickey-Fuller (DF GLS ), the modified Sargan-Bhargava (MSB GLS ) and the modified Elliot-Rothenberg-Stock (MP T GLS ) tests under GLS detrending. It can be seen that, even though the ADF test only rejects the presence of a unit root in 4 cases (and only at the 90% significance level in 3 of them), all the other tests point towards the stationarity of the majority of the real exchange rates. Thus, the KPSS indicates the presence of only 5 non-stationary variables, the DF GLS test rejects the presence of unit roots in 12 variables, and for the MSB GLS and MP T GLS tests the rejection of the unit root hypothesis is fairly generalised (in 18 of the 21 variables).

In order to complement the univariate study, we have also carried out a panel analysis of the real exchange rate series. The extension to panel analysis is justified by the results from recent studies (see [START_REF] Banerjee | Panel Data Unit Roots and Cointegration: an Overview[END_REF], [START_REF] Baltagi | Nonstationary panels, cointegration in panels and dynamic panels: A survey[END_REF] or [START_REF] Breitung | Unit roots and cointegration in panels[END_REF] (2000) analyse, through a Monte Carlo simulation, the power of some of the unit root tests for panel data used here and conclude that for panels with a considerable temporal dimension (T>100) there is a risk of over-rejecting nonstationarity, whereas the opposite is true for panels with a small temporal dimension.

To minimise this risk, they propose a simultaneous analysis, as the one carried out in this article, of the individual and panel tests. Before we present the results in Table 1b, we would like to point out a few methodological notes about these tests.

All panel tests to be used are based on the null hypothesis of the presence of a unit root in the series, with the exception of [START_REF] Hadri | Testing for stationarity in heteogeneous panel data[END_REF] test, whose hypothesis is that the series are stationary. The tests differ from each other on the restrictions imposed on the autoregressive process of each of the panel series. Thus, the tests of [START_REF] Levin | Unit roots tests in panel data: Asymptotic and finite-sample properties[END_REF], [START_REF] Breitung | The local power of some unit roots tests for panel data[END_REF] and Hadri impose a common persistence parameter to all the series -therefore, if the null is rejected, the alternative would be that all the series are simultaneously stationary for the first two tests and non-stationary for the latter. On the other hand, the tests of [START_REF] Im | Testing for unit roots in heterogeneous panels[END_REF], the Fisher-type tests suggested by [START_REF] Maddala | A comparative study of unit root tests with panel data and a new simple test[END_REF], and the [START_REF] Pesaran | A simple panel unit root test in the presence of cross section dependence[END_REF] CADF test allow for the autoregressive parameter to change freely among the different cross-sectional variables under consideration. Therefore, the alternative hypothesis in these cases is the presence of a non-null proportion of stationary series of the total. The latter set of tests seem more adequate from an empirical point of view as they impose less restrictions on the data generating process.

All the above mentioned panel tests, with the exception of Pesaran's, assume that there are no short-run or long-run cross-correlations among the autoregressive processes that govern the behaviour of each time series. In particular, all these tests are based in the absence of cross-correlation or cointegration among the variables of the panel. However, O'Connell (1998) and [START_REF] Banerjee | Some cautions on the use of panel methods for integrated series of macro-economic data[END_REF][START_REF] Banerjee | Testing for PPP: Should we use panel methods?[END_REF] have demonstrated that all tests are affected when this property is missing. This will lead to less reliability as the null hypothesis will be rejected more often than it should be according to the confidence level prefixed. Nevertheless, it is most likely that in practice there are significant cross dependencies among the real exchange rates of the different countries given the presence of common components. For example, when using the US dollar as the base currency, not only independent changes in the dollar value and in the US price index will be included in the real exchange rate, but also any other type of variable or global shock that is common to all or some of the countries from the sample. For the 21 real exchange rates of our sample, the cross-correlations oscillate from a minimum value of -0.317 to a maximum of 0.996, which reveals the relevance of the crossdependency problem. This fact brings about a potentially important bias in the standard tests, which we have tried to lessen in two ways.

In the first place, we have extracted a specific time effect which would collect the contemporaneous factors that are common to all exchange rates. In practice this implies working with the time-demeaned real exchange rates, which, as shown by [START_REF] Luintel | Heterogeneous panel unit root tests and purchasing power parity[END_REF], does not eliminate all the present correlation, but it does reduce it considerably. Second, we have dealt with the cross-dependency problem through the implementation of two tests that take into account the presence of cross-correlation and/or cointegration: the cross-dependence modified ADF test (CADF) suggested by [START_REF] Pesaran | A simple panel unit root test in the presence of cross section dependence[END_REF] and the decomposition procedure put forward by Bai and Ng (2004 a,b). 3 Levin-Lin-Chu imposes strong parametric restrictions which imply that under the alternative hypothesis all series must be stationary and have the same autoregressive parameter. Hadri's test, on the other hand, has a tendency to over-reject the null hypothesis as it is based on KPSS tests and, as shown by [START_REF] Caner | Size distortions of tests of the null hypothesis of stationarity: Evidence and implications for the PPP debate[END_REF], the KPSS statistics tend to reject the stationarity hypothesis more often than they should at the specified significance level. Therefore, we can conclude that there is evidence that at least a significant proportion of the 21 real exchange rates are stationary.

Our conclusion is given further support when we use the methodology suggested by Bai and Ng (2004 a,b). Bai and Ng put forward an approach, which they call PANIC/PASIC (Panel Analysis of Non-Stationarity/Stationarity in Idiosyncratic and Common Components), which consists of decomposing a time series panel in two 3 The most recent literature labels the panel tests that take into account the cross-dependency problem as "second generation tests" (see the surveys of, among others, [START_REF] Hurlin | Second generation panel unit root tests[END_REF] and [START_REF] Breitung | Unit roots and cointegration in panels[END_REF]). Among the second generation tests, we have decided to use those suggested by Pesaran and Bai-Ng based on the results of [START_REF] Gengenbach | Panel unit root tests in the presence of cross-sectional dependences: Comparison and implication for modelling[END_REF], [START_REF] Baltagi | Panel Unit Root Tests and Spatial Dependence[END_REF], [START_REF] Gutierrez | Panel unit root tests for cross-sectionally correlated panels: a Monte Carlo comparison[END_REF] and [START_REF] Moon | An empirical analysis of nonstationarity in panels of exchange rates and interest rates with factors[END_REF] components, a part that is common to all the series and a part with idiosyncratic components. Then, unit root and stationarity tests are carried out separately on each component. The advantage of using this methodology lies in the fact that the decomposition allows for the construction of panel tests that verify the crossindependence hypothesis, so that the tests applied to the estimated components will display better statistical properties than those based on the original observed series.

To be more precise, the PANIC/PASIC methodology assumes that the real exchange rate observed series, it q , can be decomposed in the form

it t i i it e F q + + = ' ,
where i is the deterministic component of the series, t F is a k-vector of common factors and it e is a specific term. t F and it e are unobserved elements that must be estimated using the information from the complete panel. This factor model makes clear that for a real exchange rate to be stationary, the common and idiosyncratic components must also be stationary. Non-stationarity can arise from the presence of a unit root in the common factors or in the specific component of each series.

We have applied the principal components method put forward by Bai and Ng to our empirical analysis, estimating the factor model

it t i i it e F q + + = '
. We selected one single common factor using the ) ( 1 K IC information criterion suggested by [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF], which accounted for over 71% of the total variation of the data (the second factor accounted for only an additional 5.7% of the variation, a similar percentage to that of the third factor). Next we estimated the idiosyncratic component of each series and, finally, we applied the unit root and/or stationarity tests to the two estimated components. The results from this analysis are shown in Table 2a. In the first place, it can be seen that four out of the five tests applied to the common factor t F ˆ point toward its stationarity. On the other hand, four panel tests (with the exception, again, of the Levin-Lin-Chu and Hadri tests) accept the alternative hypothesis for the idiosyncratic errors, and therefore there is evidence of the stationarity of a non-null proportion of these components. Taken together, both results suggest -given the additive nature of the factor model used-that at least a significant part of the original real exchange rate series presents mean reversion and so PPP is verified for this significant part.

In Table 2b we study the relative weight of each of the estimated components (note that the selection applied depends on the cutting point chosen and thus the following statements should be taken with caution). It can be seen that the specific component dominates only for the real exchange rates of Canada, Japan, Iceland, Australia and New Zealand, whereas in the remaining cases the common factor explains the majority of the variations. This result indicates an additional value added from the PANIC/PASIC approach, as it shows that for the majority of the real exchange rates analysed the time dynamics are dominated by a common factor with a clear European profile.

The set of individual and panel tests carried out in this section provide, taken as a whole, evidence in favour of the stationarity of (at least) some of the 21 real exchange rates analysed, and thus they are indicating that the long-run PPP proposition is verified.

Non-linearity and persistence analysis

To complement the analysis of individual and panel stationarity, we have also studied the non-linearity, time-dependence and persistence properties of the real exchange rates, which have recently appeared as alternative or complementary ways of analysing long-run PPP in the literature.

In the first place, we have used non-linear techniques to assess the validity of the conclusions from recent theoretical models which predict a non-linear adjustment of the real exchange rates towards their long-run equilibrium values (see, for instance, the discussions of [START_REF] Taylor | Nonlinear mean-reversion in real exchange rates: Towards a solution to the purchasing power parity puzzles[END_REF] and [START_REF] Taylor | Purchasing Power Parity[END_REF]). In other words, these studies point towards the presence of non-linear stationary autoregressive processes for real exchange rates. In contrast with non-linear stationary models, the maintained hypothesis of linear stationary models is that the adjustment towards equilibrium in exchange rates happens continuously and at constant speed, not taking into account the deviations from equilibrium at each point in time.

Non-linear behaviour can be characterised through Smooth Transition Regression (STR) models. 4 In particular, we have used a general class of logistic models of the type , where the function G has its minimum value. Note that when 0 = , in both models the transition function is constant and thus both models become a linear autoregressive specification. On the other hand, when , the LSTR1 and LSTR2 models evolve into a standard two-regime and three-regime selfexciting autoregressive (SETAR) model, respectively.

t t t t t u s c G w w q + + = ) , , ( , where ) , , , 1 ( 1 '
= p t t t q q w K is
The results from the non-linearity tests of the 21 real exchange rates analysed are reported in column 6 of Table 1a. In the STR modelling process we have followed a successive application of the specification, estimation and evaluation stages and concluded afterwards whether to apply a linear model or one of the two non-linear models LSTR1 or LSTR2. There is evidence in favour of the non-linear adjustment towards equilibrium hypothesis for more than a third of the sample, in particular for 8 of the 21 real exchange rates analysed, and in two of them (Finland and New Zealand) the adjustment is asymmetric (LSTR1). This result raises two interesting issues. First, it demonstrates the difficulty for conventional unit root tests to detect mean reversion in exchange rates, as these tests are based on linear processes for the variables which, as shown, are not always linear. Second, it shows the importance of recent theories of the PPP which predict different behaviours for the exchange rates depending on the size of the deviation from their long-run equilibrium positions.

Another approach proposed by the literature to analyse whether real exchange rates have a unit root or are mean reverting has been the extension of the standard linear autoregressive models (ARIMA) to include more general specifications. For instance, the ARFIMA models are used for fractionally integrated variables, whose highly persistent behaviour make standard ARIMA models -and the stationarity tests based on them-inadequate. For our analysis, these models allow for a higher degree of persistence in the temporal dynamics of exchange rates and thus, for less restrictive processes of mean reversion. and the speed of adjustment would be higher or lower depending on whether the estimated value is smaller or bigger than 0.5.

F

We have estimated the 21 ARFIMA(p,d,q) models for the real exchange rate series, and we report the estimates of the parameter d in column 7 of Table 1a. 5 It can be seen that the estimated parameter d ˆ is statistically different from zero at a 90% significance level in only four cases (Sweden, Canada, Finland and Spain), but in all four cases the value is significantly lower than 0.5 (the null hypothesis of a value above 0.5 was rejected through a Wald statistic). This would imply that all the series analysed are stationary: 17 of them would follow stationary ARMA processes and 4 of them would (marginally) follow stationary ARFIMA processes. To complete the analysis, the last 2 columns of Table 1a present the estimations of the autoregressive and moving average parameters of the ARMA(1,1) models of the 17 series for which the hypothesis 0 = d could not be rejected, and the estimations of the ARFIMA(1,d,0) models for the remaining 4 series. In all cases the estimated parameter of the autoregressive component of the model is statistically significant and has a high value. This would indicate a high degree of temporal dependence in the evolution of exchange rates, and further, would prove the difficulty of standard unit root tests to detect the non- 5 We started with a maximum value of 2 for p and q and we chose the final model according to the Akaike (AIC) and Schwarz (SBC) information criteria. In all cases we obtained models of the type ARFIMA(1,d,1) or inferior, which justifies that in columns 8 and 9 we only present the results of the estimation of the first order model for the exchange rates. All estimations are available from the authors upon request. To summarise, the results obtained from the analysis of stationarity of the real exchange rates through a variety of techniques applied in section II and III indicate that there is strong evidence that the 21 real exchange rates considered in this article are stationary and therefore this provides support for the existence of PPP in its strong version.

Panel cointegration tests

In this section we will study the weak version of long-run PPP, which relaxes the hypotheses of symmetry and proportionality that underlie the analysis of real exchange rates. In particular, we will look at expressions of the type:

t i t i i USA t i i t i p p s , , , 2 , 1 , 0 , + + + = , (2)
that do not impose the restrictions ).

The analysis to be developed next will be based on testing for the cointegration hypothesis and we will apply the econometric techniques developed in the recent literature. These techniques exploit the panel dimension of the data, considerably improving the statistical properties of standard cointegration tests used in the analysis of individual time series and allow for a higher degree of heterogeneity in the parameters and in the time dynamics of the series. 6Before we proceed with the cointegration analysis, we will look at whether the nominal exchange rates and the domestic and foreign prices are unit root processes.

Table 3 shows the results from the ADF and KPSS tests. The ADF test (first column of we can conclude that there is general evidence in favour of the presence of a unit root both in the nominal exchange rates and in the consumer price indexes considered. 7

We will apply four groups of tests that have been proposed, respectively, by [START_REF] Pedroni | Critical values for cointegration tests in heterogeneous panels with multiple regressors[END_REF][START_REF] Pedroni | Panel cointegration: asymptotic and finite sample properties of pooled time series tests with an application to the PPP hypothesis[END_REF] Pedroni has developed seven different cointegration statistics, all of them based on the least squares residuals of (2), and on the null hypothesis of no cointegration.

Four of these tests have the panel test feature (within dimension), as they are constructed adding separately the numerator and the denominator over the crosssectional dimension of the panel. At the same time, each of these tests can be constructed weighted -using an estimation of the long-run variances as weights-or nonweighted, so that actually there are 8 different tests. The remaining three tests are group average tests (between dimension), constructed dividing first each numerator and denominator and afterwards adding over the cross-sectional dimension of the panel. In any case, the standardised distributions of the panel and group statistics are given by ( )

) 1 , 0 ( N N k k k µ !
, where k ! is the corresponding statistic, and k µ is their 7 We also investigated the presence of a second unit root in the series and in all cases it was rejected. Further, we also applied alternative tests, such as those of Table 1a for real exchange rates, and in all cases the conclusions were similar to those of the ADF and KPSS. These complementary results are available from the authors upon request. (  )

) 1 , 0 ( * N LM N LM = " " µ ,
where " µ and 2 " are, respectively, the expected mean and variance of the LM statistic.

Table 5 reports the individual statistics and the (standardised and nonstandardised) panel statistics proposed by McCoskey and Kao, the estimated residuals being obtained from the dynamic generalised least squares (DGLS) method suggested by [START_REF] Stock | A simple estimator of cointegrating vectors in higher order integrated systems[END_REF] applied to the regressions (2). This method includes not only the lead and lagged explanatory variables to correct for their endogeneity, but it also includes an autoregressive process for the errors of the model in order to obtain autocorrelation-free residuals. In our case, we started with a general model for each equation with 4 lags and 4 leads of the variables USA p and i p and an AR(2) process for the errors i , and have simplified according to the statistical significance of the parameters. It can be seen from Table 5 that, at the individual level, the null hypothesis is only marginally rejected in two cases (Belgium and the Netherlands), whereas the The CUSUM test is an extension for panel data of the test proposed by [START_REF] Xiao | A residual test for the null hypothesis of cointegration[END_REF] and [START_REF] Xiao | A CUSUM test for cointegration using regression residuals[END_REF] for individual time series and tries to measure the extent of the fluctuations in the estimated residuals it ˆ through a statistic, PC , that is asymptotically free of nuisance parameters, so that it verifies asymptotically that ( ) ( )

) 1 , 0 ( N PC N µ
, where µ and 2 are the first and second order moments of the PC statistic, which have been tabulated by Westerlund (2005a). Amongst the non-cointegration tests, [START_REF] Westerlund | New simple tests for panel cointegration[END_REF] proposes two non-parametric tests that do not require any hypothesis about the time dynamics of the errors it , although they assume cross-independency of the errors. These two variance ratio tests are a within group (panel type) statistic, P VR , and a between group (group average) statistic, G VR , and both have an asymptotic normal standard distribution. Finally, [START_REF] Westerlund | Panel cointegration tests of the Fisher hypothesis[END_REF] proposes two cointegration tests for panel data that do not impose cross-independence among the units of the panel. To model the cross-dependencies, a factorial approach like that of Bai and Ng (2004a,b) ). To avoid this problem, [START_REF] Larsson | Likelihood-based cointegration tests in heterogeneous panels[END_REF] have developed a panel statistic based in the multivariate approach of [START_REF] Johansen | Statistical analysis of cointegrating vectors[END_REF][START_REF] Johansen | Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models[END_REF], which allows for the presence of multiple cointegration relationships amongst the variables. 9

The

)

) 1 , 0 ( ) ( ) ( )] ( / ) ( [ ) ( / ) ( N Z Var Z E p H r H LR N p H r H k k NT LR = &
, where . On the other hand, some of the individual trace statistics and the panel statistics LR & reject the hypothesis of one and even two cointegrating vectors. It is worth pointing out that a similar result has been obtained in other studies, like those of [START_REF] Coakley | Is there a base currency effect in long run PPP?[END_REF], [START_REF] Cerrato | Symmetry, proportionality and the purchasing power parity: Evidence from panel cointegration tests[END_REF] or [START_REF] Caporale | Panel data tests of PPP. A critical overview[END_REF]. Nonetheless, this last result should be taken with caution as the Larsson-Lyhagen-Löthgren test is based on the individual Johansen statistics, which tend to overestimate the number of cointegrating vectors and, further, are very sensitive to the inclusion of different deterministic components, to the error 9 Recently, [START_REF] Breitung | A parametric approach to the estimation of cointegration vectors in panel data[END_REF] has extended the approach of [START_REF] Larsson | Likelihood-based cointegration tests in heterogeneous panels[END_REF] to more general cases. In particular, deterministic components in the underlying VAR model are allowed (this is why in our empirical analysis we use the tabulated critical values given in Breitung's article), and a new estimator for the cointegrating vector(s) is proposed, which can be modified in case of contemporaneous crosscorrelation. distribution, to the number of lags chosen and to the size of the time series used [START_REF] Maddala | Unit Roots, Cointegration, and Structural Change[END_REF].

) ( k Z E
The evidence from all the range of individual and panel cointegration tests applied in this section clearly points towards the presence of a long-run equilibrium relationship between nominal exchange rates and domestic and foreign prices according to equation ( 2). Therefore, the next stage in our analysis is the estimation of the parameters of each of these relationships, and of the parameters of an average function for the complete panel of 21 exchange rates.

We have taken into account two issues when deciding the estimator to be used.

A first issue, already mentioned before, refers to the problems that arise from the Johansen method, whose estimators are generally not very robust to changes in the initial VAR model used (further, some works have pointed out the weaknesses of the statistical properties of the maximum likelihood estimators for small samples). Second, the study of [START_REF] Maddala | Unit Roots, Cointegration, and Structural Change[END_REF] reveals that amongst the alternative methods proposed by the literature for the estimation of cointegration equations, the DGLS method offer the best results in finite samples with respect to other estimators asymptotically more efficient.

Thus, our estimation of the long-run relations for the nominal exchange rates is based on DGLS estimators and the results for our sample of 21 countries are reported in Table 8. It can be seen that there is a high variability in the significance and in the coefficients' estimates, both for the foreign price ( USA p ) and for the domestic price (p).

Thus, the foreign price is significant in the UK, Belgium, the Netherlands, Japan, Iceland, Ireland and Portugal (a third of the total panel), with the correct expected sign in all cases except for Japan. The domestic price, on the other hand, is significant in 14 out of the 21 countries (the UK, Austria, Belgium, Denmark, France, Italy, the Netherlands, Finland, Greece, Iceland, Ireland, Portugal, Australia and New Zealand), presenting the correct expected sign in all these cases. In the last column we present the Wald statistics to test the validity for each country of the joint symmetry and proportionality restrictions, 1

,

1 = i and 1 , 2 = i .
We reject the null for 5 countries (Austria, Denmark, France, Canada and Australia) at the 5% level and for 2 countries (United Kingdom and Japan) at the 1% level, and thus, for the remaining 14 countries we do not reject the null and so we cannot reject the strong version of PPP for them.

Hence, the evidence on the rejection of the symmetry and proportionality conditions at the individual country-level is in line with similar results obtained by other studies (see, for example, [START_REF] Cheung | A fractional cointegration analysis of purchasing power parity[END_REF] or [START_REF] Cerrato | Symmetry, proportionality and the purchasing power parity: Evidence from panel cointegration tests[END_REF], where the null is also rejected for some, not all, of the countries).

The last row of Table 8 presents a pooled estimation for the complete panel in order to establish a basis for comparison. We have used a fixed effect for each country and assumed homogeneity in the slopes of the exchange rate equation, that is, we followed a specification of type (2) but imposing the restrictions

Conclusions

This article has carried out a detailed empirical study of long-run PPP in the post-Bretton Woods period for 21 OECD countries. In doing so, we have reviewed the current status quo of the empirical analysis of PPP. We have analysed the statistical properties of the real exchange rates, which is equivalent to testing for PPP in its strong version. We have also examined the relationship between nominal exchange rates and domestic and foreign prices for each country, which implies the analysis of PPP in its weak version.

Overall, the results obtained through the analysis implemented in this article point in favour of the validity of PPP. Thus, when analysing real exchange rates through individual and panel unit root tests, through non-linear stationarity models and through high persistence ARFIMA models, evidence indicates that a considerable number of the 21 real exchange rates examined are stationary. Further, the group of cointegration tests applied clearly point towards the presence of a long-run equilibrium 10 It might seem very restrictive to impose homogeneity of the effect of the domestic and foreign prices amongst all the members of the panel, but for our data, both the individual likelihood ratio tests for the domestic and foreign prices ( 44 . ) do not reject the corresponding null hypothesis. 11 We obtain similar results using the dynamic GLS estimation procedure for SUR cointegration regression models recently proposed in [START_REF] Moon | Efficient Estimation of the Seemingly Unrelated Regression Cointegration Model and Testing for Purchasing Power Parity[END_REF]. On the other hand, our estimates under the homogeneity hypothesis are very similar to those obtained by [START_REF] Breitung | A parametric approach to the estimation of cointegration vectors in panel data[END_REF]

+ =
, where p USA is the US aggregate log-price level; 2) ADF is the augmented Dickey-Fuller unit root t test (only with intercept) and the number between parenthesis is the lag order of the corresponding regression (based on modified Akaike information criterion -MAICusing a step-down procedure starting from K=12); 3) KPSS is the Kwiatkowski-Phillips-Schmidt-Shin unit root test; the computed bandwidth was 9 (using the Newey-West approach and a Barlett kernel as spectral estimation method); 4) DF GLS , MSB GLS [START_REF] Pedroni | Critical values for cointegration tests in heterogeneous panels with multiple regressors[END_REF][START_REF] Pedroni | Panel cointegration: asymptotic and finite sample properties of pooled time series tests with an application to the PPP hypothesis[END_REF]. 3) An * (**) [***] indicates rejection of the null hypothesis at the 10% (5%) [1%] significance level based on the appropriate critical values (1.28, 1.64 and 2.33, respectively). 3) For the semiparametric PP tests we have used the Newey-West (1994) rule for truncating the lag length for the kernel bandwidth, and for the parametric ADF tests we have used a step-down procedure starting from K=12; 4) Panel and group mean time-demeaned statistics have been demeaned with respect to common time effects to accommodate for some forms of cross-sectional dependency.; 5) The residuals have been estimated using the least squares estimator. 3) The residuals have been estimated using the generalized dynamic least squares (DGLS) estimator proposed by [START_REF] Stock | A simple estimator of cointegrating vectors in higher order integrated systems[END_REF]. .28, 1.64 and 2.33, respectively); 3) The residuals for the CUSUM test have been estimated using the fully modified least squares estimator (FM-OLS) [the CUSUM test based on DOLS estimates was 0.996, with p-value of 0.16]; 4) The residuals for the VR tests have been estimated using the ordinary least squares (OLS) estimator; 4) For the DH tests the number of factors have been estimated using the IC 1 (K) criterion with the maximum number of factors set equal to five. et al. (1999) p-values; 3) The optimal lag length of the VAR model for each country has been chosen according to the Akaike (AIC) and Schwarz (SBC) information criteria, and for the individual trace tests the probabilities of [START_REF] Mckinnon | Numerical distribution functions of likelihood ratio tests for cointegration[END_REF] have been used. NOTES: 1) The numbers within parentheses (below coefficients) are t values; 2) AR(p) denotes the order of the autoregressive model used in the estimation (a step-down procedure starting from p=2 has been used); 3) LL(q) denotes the order of the leads-lags terms used in the estimation (a step-down procedure starting from Q=4 has been used); 4)

  parameter that controls the slope of the function, s is the transition variable, which in our application is given by d t y , where d is the lag parameter of the transition function. The most common choices for K are K=1 and K=2, which generate the so-called LSTR1 and LSTR2 models. In the LSTR1 model, the parameters ) asymmetrically) from the initial values to the final values + , whereas in the LSTR2 model, those parameters change symmetrically around

  series, given the low power of these tests when the autoregressive parameter is near unity.

  ,[START_REF] Mccoskey | A residual-based test of the null of cointegration in panel data[END_REF],Westerlund (2005a,b,c) and[START_REF] Larsson | Likelihood-based cointegration tests in heterogeneous panels[END_REF].8 The tests of Pedroni, McCoskey-Kao and Westerlund are based on the residual estimates of the individual cointegration relationships, whereas the Larsson-Lyhagen-Löthgren test is based on the analysis of multiple cointegrating vectors. All these tests allow for a high degree of individual heterogeneity, so that the coefficients of each cointegrating relationship can vary freely for each exchange rate. It is interesting to note that the tests of Pedroni, Larsson-Lyhagen-Löthgren and two of the tests proposed by Westerlund have a null hypothesis of absence of cointegration among the variables of each equation whereas in the LM tests of McCoskey-Kao and the CUSUM tests of Westerlund the null hypothesis is stationarity of the residuals -and so the presence of cointegration among the variables. Finally, of all these tests, only the Durbin-Hausman-type tests proposed by Westerlund allow explicitly for the presence of dependency among the panel data.

  in the non-rejection area of the null hypothesis of cointegration. Therefore, these results give further support to the conclusion obtained with Pedroni's tests, in the sense that there are stable long-run equilibrium relationships between the nominal exchange rate and the domestic and foreign (US) prices for each country.Next we study the cointegration tests put forward by Westerlund. These tests are all applied to the estimated residuals of the potential equilibrium relationship, even though the null and the alternative hypotheses differ according to the version of the test.Thus, the CUSUM test for panel data tests the null of cointegration, whereas the variance ratio tests -VR-and the Durbin-Hausman tests have the null of absence of cointegration amongst the variables.

,

  have used a DGLS type estimator, similar to the one used for the individual regressions, adding four leads and lags of the explanatory variables ( USA p and p ) and an AR(2) model for the errors of the model. It can be seen that in this specification the two parameters are highly significant and their values are Prob=0.77) from the theoretical values needed for the PPP to hold in its strong version (

  using the FM-OLS exchange rates and domestic and foreign prices for each country, which gives support to the validity of the weak version of PPP. There is also moderate evidence in favour of the strong version of PPP when tested through cointegration techniques.method and the new two-stage estimator proposed in his work. Further, his Wald statistics do not reject in both cases the null hypothesis of 1

  and MP T GLS are the augmented Dickey-Fuller, modified Sargan-Bhargava and modified Elliot-Rothenberg-Stock tests under GLS detrending (based on MAIC criterion using a step-down procedure starting from K=12); 5) STR is the Smooth Transition Regression model (with a Logistic transition function) estimated for each variable to analyze nonlinear real exchange rate behaviour; 6) ARFIMA is the estimate of the d parameter (and of the standard error between parentheses) of a Fractionally Integrated ARMA model estimated for each variable to analyze long-run dependence in real exchange rate behaviour; 7) ARMA ROOTS are the estimated AR and MA parameters of an ARMA/ARFIMA adjusted for each series; 8) An * (**) [***] indicates rejection of the null hypothesis at the 10% (5%) [1%] significance level based on the appropriate critical values.

  The individual and panel stationarity tests carried out together with the PANIC/PASIC approach indicate a tendency to reject non-stationarity for the series of real exchange rates analysed. The problem of conventional unit root tests to detect non-linear mean reversion is highlighted with the findings through Smooth Transition Regression (STR)

Table 1b

 1b

reports the panel tests results. With the exception of the Pesaran test, under each standard panel test we report the corresponding version time demeaned, which will be less affected by the cross-dependency problem. The tests reported in Table 1b present, on balance, evidence in favour of PPP. The null hypothesis of nonstationarity is rejected in all tests with the exception of those of Levin-Lin-Chu and Hadri. Nonetheless, Levin-Lin-Chu and Hadri´s tests both have some limitations.

Table 3 )

 3 does not reject the unit root hypothesis for any of the nominal exchange rates, whereas the KPSS test (second column) accepts the unit root hypothesis for only 12 of On the other hand, the ADF test accepts the stationarity of 5 of the price indexes considered (Denmark, the Netherlands, Japan, Finland and Ireland), whereas the KPSS test rejects in all cases the stationarity hypothesis. From these results

	F o r
	P
	e
	e r
	R e v
	i e
	w

Table 4

 4 reports the estimations of the Pedroni's cointegration tests. We present the results for the original series and also for the time-demeaned series, with the aim to address the cross-dependency problem mentioned above. For the original series, the

	panel statistics	"	stat	and	ADF	stat	clearly reject the null hypothesis of non-
	cointegration, both in the weighted and non-weighted versions; of the group tests, only
	the	ADF	stat	rejects the null hypothesis. Further, the cross-dependency corrected
	series all strongly reject the absence of cointegration, for any type and version of the
	tests applied. These results clearly point towards the presence of a cointegration
	relationship among the nominal exchange rate and price variables where there could be
	a long-run equilibrium relationship like (2) for a non-null proportion of the exchange
	rates analysed.

McCoskey-Kao´s test is a panel version of the Lagrange multipliers -LM-tests

proposed by

[START_REF] Harris | A test of the null hypothesis of cointegration[END_REF]

and

[START_REF] Shin | A residual based test of the null of cointegration against the alternative of no cointegration[END_REF]

for the individual analysis of cointegration. The proposed LM test is a weighted mean of the individual Lagrange tests of each equation, and McCoskey and Kao (1998) have shown that the standardised version of such average statistic is given by ( )

  is considered, but it is applied to the long-run regressions' errors it . The two statistics are constructed following the Durbin-

	Hausman principle, obtaining a panel test,	P DH , and a group average test,	G DH ,
	which, when normalised, have normal standard limit distributions.

Table 6

 6 The tests of Pedroni, McCoskey and Kao and Westerlund all share a weakness due to the fact that they all assume that there is only one cointegrating vector among

	shows the estimations of the 3 groups of tests put forward by
	Westerlund. Interestingly, they all indicate that there is heterogeneous cointegration

between nominal exchange rates and domestic and foreign (US) prices, which provides further evidence in favour of PPP in the weak sense.

  , where p is the number of variables in the model. The null hypothesis of the test is that all countries of the panel have a maximum common rank of r cointegrating vectors, even though it is allowed that each country has its own r i number of stable equilibrium relationships.Larsson et al. demonstrate that the asymptotic distribution of the standardised version of

	Larsson-Lyhagen-Löthgren's test,	LR NT	[	H	(	r	)	/	H	(	p	)]	, is the mean of the trace
	iT LR statistics proposed by Johansen to test the hypothesis	H	0	:	rank	(	i = # )	r i	r
	against the alternative	H	1	:	rank	(	i = # )	p	for each countryNT LR is
	given by	[				]

Table 1a :

 1a Individual unit root tests for the log-level of the real exchangerates, 1973:1-2004:4

		1. ADF	2. KPSS	3. DF GLS	4. MSB GLS	5. MPT GLS			6. STR	7. ARFIMA	8. ARMA ROOTS
	United	-2.02	0.36 *	-0.88	0.14 ***	1.60 ***		LSTR2,	-0.08	0.89 ***	-0.25 ***
	Kingdom	(2)		(9)					d=2	(0.15)
	Austria	-2.28	0.12	-1.33	0.17 ***	1.91 **		LINEAR	0.07	0.91 ***	-0.14
		(0)		(4)							(0.10)
	Belgium	-2.36	0.19	-2.11	0.19 **	1.88 **		LINEAR	0.11	0.93 ***	-0.19 **
		(3)		(3) **							(0.09)
	Denmark	-2.58	0.09	-1.50	0.19 **	2.18 **		LSTR2,	0.10	0.90 ***	-0.22 **
		(3) *		(3)					d=4	(0.10)
	France	-2.01	0.17	-1.84	0.26 *	3.41 *		LSTR2,	0.11	0.91 ***	-0.18 **
		(0)		(0) *					d=4	(0.10)
	Germany	-2.05	0.15	-2.09	0.19 **	1.84 **		LINEAR	0.07	0.92 ***	-0.15
		(0)		(3) **							(0.10)
	Italy	-1.96	0.11	-1.98	0.24 **	3.37 *		LINEAR	0.14	0.92 ***	-0.15
		(0)		(0) **							(0.11)
	Netherlands	-2.14	0.14	-1.82	0.18 **	1.98 **		LINEAR	0.08	0.90 ***	-0.18 *
		(0)		(3) *							(0.10)
	Norway	-2.36	0.18	-1.52	0.23 **	3.10 **		LSTR2,	0.04	0.89 ***	-0.15
		(0)		(0)					d=2	(0.10)
	Sweden	-2.28	0.49 **	-2.24	0.22 **	2.53 **		LSTR2,	0.17	0.92 ***	---
		(3)		(3) **					d=1	(0.10) *
	Switzerland	-2.86	0.22	-0.83	0.30	6.00		LINEAR	0.04	0.88 ***	-0.10
		(0) **		(0)							(0.10)
	Canada	-2.19	0.86 ***	-1.78	0.29	4.17 *		LINEAR	0.14	0.96 ***	---
		(6)		(6) *							(0.08) *
	Japan	-2.38	0.73 **	-0.56	0.38	7.91		LINEAR	0.13	0.92 ***	-0.19 **
		(0)		(5)							(0.12)
	Finland	-2.67	0.29	-1.58	0.20 **	2.07 **		LSTR1,	0.17	0.90 ***	---
		(3) *		(5)					d=4	(0.10) *
	Greece	-1.75	0.15	-1.95	0.16 ***	2.08 **		LINEAR	0.02	0.95 ***	0.05
		(0)		(5) **							(0.09)
	Iceland	-2.63	0.15	-1.68	0.24 *	3.92 *		LINEAR	0.07	0.89 ***	-0.04
		(0) *		(0) *							(0.12)
	Ireland	-1.98	0.20	-2.10	0.18 **	2.55 **		LSTR2,	0.06	0.89 ***	-0.15
		(2)		(3) **					d=4	(0.11)
	Portugal	-1.54	0.26	-1.43	0.18 **	2.47 **		LINEAR	0.05	0.95 ***	-0.05
		(0)		(4)							(0.09)
	Spain	-1.80	0.13	-1.27	0.20 **	2.80 **		LINEAR	0.17	0.91 ***	---
		(0)		(3)							(0.09) *
	Australia	-1.71	0.83 ***	-1.64	0.38 **	7.21		LINEAR	0.06	0.95 ***	-0.01
		(0)		(0) *							(0.09)
	New Zealand	-1.88	0.07	-1.79	0.26 *	3.80 *		LSTR1,	0.15	0.93 ***	-0.08
		(0)		(0) *					d=4	(0.09)
	NOTES: 1) The logarithm of the real exchange rate was computed as	q	s	p	USA	p

Table 1b :

 1b Panel unit root tests for the log-level of the real exchangerates, 1973:1-2004:4

		Statistic	Prob.
	Null: Unit root (assumes common unit root process)
	Levin-Lin-Chu
	Standard:	0.37	0.64
	Time demeaned:	-0.10	0.46
	Breitung
	Standard:	-4.92	0.00 ***
	Time demeaned:	-3.19	0.00 ***
	Null: Unit root (assumes individual unit root process)
	Im-Pesaran-Shin
	Standard:	-3.39	0.00 ***
	Time demeaned:	-3.84	0.00 ***
	Maddala-Wu ADF-Fisher Standard: Time demeaned: Maddala-Wu PP-Fisher Standard: Time demeaned: F o r	66.01 84.03 82.38 98.90	0.01 *** 0.00 *** 0.00 *** 0.00 ***
	Pesaran (allows for cross-sectional dependence) CADF: -2.51 P e Null: No unit root (assumes common unit root process) Hadri Standard: 4.19 e r	(1% Critical value: -2.36) *** 0.00 ***
	Time demeaned: NOTES: 1) The probabilities for the Fisher tests have been computed using an asymptotic Chi-square distribution (all 18.24 0.00 *** the other tests assume asymptotic normality); 2) Time-demeaned statistics have been demeaned with respect to common time effects to accommodate for some forms of cross-sectional dependency; 3) An * (**) [***] indicates rejection of the null hypothesis at the 10% (5%) [1%] significance level. R e v
		i e
			w

Table 2a :

 2a PANIC/PASIC approach to panel testing of real exchange rates: global results The probabilities for the Fisher tests have been computed using an asymptotic Chi-square distribution (all the other panel tests assume asymptotic normality); 2) To test the (non)stationarity of the idiosyncratic components, the Levin-Lin-Chu, Breitung and the Maddala tests have been computed in a model with no deterministic term; 3) An * (**) [***] indicates rejection of the null hypothesis at the 10% (5%) [1%] significance level.

		Statistic	Prob.
		COMMON FACTOR ( F ˆ)
	ADF (Null: Unit root)	-1.78	---
	KPSS (Null: No unit root)	0.09	---
	DF GLS (Null: Unit root)	-1.81 *	---
	MSB GLS (Null: Unit root)	0.26 *	---
	MP T GLS (Null: Unit root)	3.89 *	---
	IDIOSYNCRATIC COMPONENTS ( i e ˆ)
		Panel unit root tests
	Null: Unit root (assumes common unit root process)
	Levin-Lin-Chu	-2.82	0.00 ***
	Breitung	-1.30	0.09 *
	Null: Unit root (assumes individual unit root process) Im-Pesaran-Shin -2.86 Maddala-Wu ADF-Fisher 57.15 Maddala-Wu PP-Fisher 75.00 Null: No unit root (assumes common unit root process) Hadri 18.92 o r P NOTES: 1) F e	0.00 *** 0.06 * 0.00 *** 0.00 ***
		e r
		R e v
		i e
		w

Table 2b :

 2b PANIC/PASIC approach to panel testing of real exchange rates: individual results for the idiosyncratic components

		Var	( ) ( ) i i q Var e ' 'ˆ( ) ( ) i e F ' i	Dominant component
	United			0.44	1.09	Common
	Kingdom
	Austria			0.06	4.14	Common
	Belgium			0.07	2.76	Common
	Denmark			0.07	5.95	Common
	France			0.09	3.96	Common
	Germany			0.07	3.52	Common
	Italy			0.28	1.94	Common
	Netherlands			0.06	3.45	Common
	Norway			0.19	2.96	Common
	Sweden Switzerland Canada Japan Finland F o r	0.32 0.20 0.98 0.61 0.26	1.13 2.26 0.10 0.62 1.28	Common Common Idiosyncratic Idiosyncratic Common
	Greece Iceland Ireland Portugal Spain Australia	0.33 0.65 P 0.15 0.23 e 0.28 0.88 e r	1.87 1.56 1.86 1.45 1.85 0.34	Common Idiosyncratic Common Common Common Idiosyncratic
	New Zealand ( ) ( ) i i q Var e ' 'ˆ is the ratio of the standard deviation of the idiosyncratic component to the standard 0.70 0.95 Idiosyncratic deviation of the differenced data; 2) ( ) ( ) NOTES: 1) Var i i e F ' is the ratio of the standard deviation of the common factor to the idiosyncratic component; 3) An Idiosyncratic dominant component has benn selected if the ( ) ( ) i i q Var e Var ' ' ˆstatistic R e v exceeds 0.6 and a Common dominant component otherwise. i e
					w

Table 3 :

 3 Unit root tests for the log-level of the originalvariables, 1973:1-2004:4

		s		p
		ADF	KPSS	ADF	KPSS
	United Kingdom	-1.83 (0)	0.15 **	-2.81 (5)	0.32 ***
	Austria	-2.42 (0)	0.11	-2.89 (4)	0.32 ***
	Belgium	-2.43 (0)	0.09	-3.00 (2)	0.33 ***
	Denmark	-1.60 (0)	0.12 *	-3.33 (0) *	0.34 ***
	France	-1.45 (0)	0.14 *	-2.33 (3)	0.34 ***
	Germany	-2.41 (0)	0.11	-2.07 (4)	0.27 ***
	Italy	-1.21 (0)	0.17 **	-1.81 (1)	0.34 ***
	Netherlands	-2.23 (0)	0.10	-3.54 (4) **	0.25 ***
	Norway	-2.14 (0)	0.09	-1.11 (4)	0.35 ***
	Sweden	-2.45 (3)	0.10	0.05 (1)	0.35 ***
	Switzerland	-2.99 (0)	0.13	-2.50 (4)	0.29 ***
	Canada	-0.67 (0)	0.10	-2.36 (1)	0.34 ***
	Japan	-2.03 (0)	0.17 **	-6.39 (5) ***	0.30 ***
	Finland	-2.76 (3)	0.07	-3.80 (6) **	0.34 ***
	Greece	0.56 (0)	0.25 ***	1.06 (9)	0.33 ***
	Iceland	-0.09 (2)	0.33 ***	-1.02 (4)	0.35 ***
	Ireland	-1.53 (0)	0.15 **	-3.19 (4) *	0.33 ***
	Portugal	0.01 (0)	0.30 ***	-0.31 (1)	0.35 ***
	Spain	-0.90 (0)	0.16 **	-3.11 (4)	0.34 ***
	Australia	-1.33 (0)	0.17 **	-2.68 (2)	0.35 ***
	New Zealand	-0.72 (0)	0.23 ***	-1.25 (2)	0.35 ***
	United States	---	---	-3.01 (3)	0.32 ***
	(numeraire country)

NOTES: 1) s is the logarithm of the nominal exchange rate (relative to the US dollar) and p is the logarithm of the aggregate price level (CPI); 2) ADF is the augmented Dickey-Fuller unit root t test (with intercept and time trend) and the number between parentheses is the lag order of the corresponding regression (based on MAIC criterion using a stepdown procedure starting from K=12); 3) KPSS is the Kwiatkowski-Phillips-Schmidt-Shin unit root test; the computed bandwidth was 9 (using the Newey-West approach and a Barlett kernel as spectral estimation method); 4) An * (**) [***] indicates rejection of the null hypothesis at the 10% (5%) [1%] significance level based on the appropriate critical values.

Table 4 :

 4 Pedroni's panel and group cointegration tests for the process (

	p s USA , ,	p	), 1973:1-2004:4

NOTES: 1) All of the panel and group statistics have been standardized by the means and variances given in

[START_REF] Pedroni | Critical values for cointegration tests in heterogeneous panels with multiple regressors[END_REF]

so that all reported values are distributed as

Table 5 :

 5 McKoskey-Kao's panel cointegration test for the process ( ] indicates rejection of the null hypothesis at the 10% (5%) [1%] significance level based on the appropriate critical values (for the LM statistic these values are 1.28, 1.64 and 2.33, respectively; for the LM i statistics they are 0.16, 0.22 and 0.38, respectively).

			p s USA , ,	p	), 1973:1-2004:4
		LM i statistics
	United Kingdom	0.046
	Austria	0.098
	Belgium	0.173 *
	Denmark	0.063
	France	0.080
	Germany	0.123
	Italy	0.087
	Netherlands	0.208 *
	Norway	0.114
	Sweden	0.088
	Switzerland	0.114
	Canada	0.029
	Japan	0.052
	Finland	0.081
	Greece	0.108
	Iceland	0.079
	Ireland	0.064
	Portugal	0.163
	Spain	0.083
	Australia	0.054
	New Zealand	0.100
	LM µ	0.096 0.1219
	2	0.0099
		-1.21
	NOTES: 1) The panel test					mean and
	variance given in McCoskey and Kao, 1998), so the reported value is distributed as	N	(	) 1 , 0	under the null hypothesis of
	cointegration; 2) An * (**) [***

* LM panel test * LM is N times the standardized version of the LM statistic (using the

Table 6 :

 6 Westerlund's tests for panel cointegration of the process ( All of the statistics have been standardized by the means and variances given inWesterlund (2005a,b,c) so that all reported values are distributed as N under the null hypothesis of cointegration or no cointegration; 2) An * (**) [***] indicates rejection of the null hypothesis at the 10% (5%) [1%] significance level based on the appropriate critical values (1

	p s USA , ,	p	), 1973:1-2004:4
	Statistic	Prob.

Table 7 :

 7 Individual(Johansen) and panel(Larsson-Lyhagen-Löthgren) trace cointegration tests for the process ( mean and variance given in[START_REF] Breitung | A parametric approach to the estimation of cointegration vectors in panel data[END_REF] Table B.1/Case 3) so the reported value is distributed as 30 under the null hypothesis of no cointegration; 2) Prob denotes McKinnon

	p s USA , ,	p	), 1973:1-2004:4
			Lag (k i )			LR(r=0)/Prob 1	LR(r=1)/Prob 1	LR(r=2)/Prob 1
	United Kingdom		5				60.06 (0.00)	24.75 (0.00)	5.77 (0.02)
	Austria		5				45.26 (0.00)	20.08 (0.01)	6.17 (0.02)
	Belgium		5				57.44 (0.00)	21.56 (0.01)	4.85 (0.03)
	Denmark		2				64.29 (0.00)	20.39 (0.01)	0.85 (0.36)
	France		2				46.55 (0.00)	18.98 (0.01)	0.13 (0.71)
	Germany		5				30.88 (0.04)	15.83 (0.04)	4.76 (0.03)
	Italy		4				36.09 (0.01)	8.54 (0.41)	0.30 (0.59)
	Netherlands		5				47.53 (0.00)	18.51 (0.02)	6.43 (0.01)
	Norway		4				45.59 (0.00)	13.18 (0.11)	3.67 (0.06)
	Sweden		4				32.07 (0.03)	13.39 (0.10)	0.88 (0.35)
	Switzerland		5				42.82 (0.00)	23.20 (0.00)	9.32 (0.00)
	Canada		2				34.54 (0.01)	3.51 (0.94)	0.39 (0.53)
	Japan		4				72.78 (0.00)	15.82 (0.05)	4.94 (0.03)
	Finland		5				60.10 (0.00)	18.73 (0.02)	7.12 (0.01)
	Greece		5				32.34 (0.02)	12.60 (0.13)	3.80 (0.05)
	Iceland		4				41.64 (0.00)	12.87 (0.12)	0.12 (0.73)
	Ireland		4				49.98 (0.00)	14.41 (0.07)	2.56 (0.11)
	Portugal		2				63.27 (0.00)	21.81 (0.01)	5.14 (0.02)
	Spain		5				35.45 (0.01)	10.15 (0.27)	3.71 (0.05)
	Australia		2				64.13 (0.00)	18.10 (0.02)	3.46 (0.06)
	New Zealand LR NT Var [ ( H / ) r H [ ] k Z E [ ] k Z [ ] ) 3 ( / ) ( H r H LR & panel test 4 ] ) 3 (			32.87 (0.02) 47.41 19.35 31.84 22.79	10.32 (0.26) 16.03 8.27 14.28 9.42	0.62 (0.43) 3.57 0.98 1.91 8.59
	NOTES: 1) The panel test	[ H	/ ( H ) r	(	) 3	]	LR NT	H	/ ( H ) r	(	) 3	statistic
	(using the ) 1 , 0 ( N

LR

&

is N times the standardized version of the [ ]

Table 8 :

 8 Long-run equilibrium nominal exchange rates functions(Stock-Watson DGLS estimates)

		p	*	p	AR(p)	LL(q)	!	2 WALD
	United Kingdom	-1.26 *	0.96 **	2	3	13.11 ***
		(-1.89)	(1.91)			(0.00)
	Austria	0.63	3.68 **	1	2	7.25 **
		(0.54)	(2.26)			(0.03)
	Belgium	-2.20 **	2.87 **	1	3	2.74
		(-2.35)	(2.35)			(0.25)
	Denmark	-0.09	3.15 ***	1	2	6.80 **
		(-0.07)	(2.65)			(0.03)
	France	-1.21	3.25 ***	1	0	7.21 **
		(-1.19)	(3.45)			(0.03)
	Germany	-1.10	1.18	1	3	0.02
		(-1.17)	(0.78)			(0.99)
	Italy	-1.85	2.65 ***	1	3	3.88
		(-1.40)	(3.06)			(0.14)
	Netherlands	-1.65 **	2.33**	1	2	2.29
		(-2.41)	(2.25)			(0.32)
	Norway	-0.38	0.62	1	0	2.46
		(-0.72)	(1.45)			(0.29)
	Sweden	-0.31	0.71	2	0	3.35
		(-0.52)	(1.53)			(0.19)
	Switzerland

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

For a general revision of these models see, among others,[START_REF] Van Dijk | Smooth transition autoregressive models. A survey of recent developments[END_REF] or[START_REF] Teräsvirta | Smooth transition regression modelling[END_REF].

If nonlinear behaviour is present in some time series (as indeed we have shown in section three) the statistical properties (mainly size and power) of the linear tests used in this section will be affected, given that nonlinear tests would be appropriate in this case. The extension of panel tests that have been designed for the linear case to the non-linear case would be an original and interesting contribution to the existing literature, but it is beyond the scope of this article. See the recent work of[START_REF] Sarno | Deviations from Purchasing Power Parity Under Different Exchange Rate Regimes: Do They Revert and, If So, How[END_REF] for an attempt to take into consideration simultaneously the issues of nonlinearity and regime shifts within the cointegrating VAR approach.

See Gutierrez (2003) for a Monte Carlo analysis of the statistical properties of some of the cointegration tests proposed in the literature.

WALD

The authors gratefully acknowledge financial help from the Institute for Fiscal Studies (Treasury Ministry, Spain) and from project SEJ2006-12392.