Is the PPP valid for the EA-11 countries? New evidence from nonlinear unit root tests

Abstract In the empirical literature there is a prevalent view that real exchange rates tend to converge towards levels predicted by the Purchasing Power Parity (PPP) only in the long-run and that short-run deviations from the PPP relationship are frequently sizable. The progressing of European monetary integration and the forming of monetary union spurred the interest of researchers to assess the relevance of the PPP theory in the case of the single European currency. Our paper therefore examines this exchange rate theory by testing a dataset of monthly real exchange rates for a sample of 11 eurozone members with respect to different benchmark currencies. Because of the documented drawbacks of linear specifications in examining this exchange rate theory, we utilise a nonlinear unit root test based on the ESTAR model proposed by Kapetanios, Shin, and Snell (2003). The results of unit root tests for the US dollar-based real exchange rate series as well as for Japanese yen-based series suggest that the PPP proposition does not hold in the case of eurozone countries. The absence of real exchange rates’ nonlinear reversion reported in this study thus confirms the thesis of Wu and Lin (2011) regarding the PPP relationship since the inception of the euro.


Introduction
The theory of Purchasing Power Parity (PPP) remains one of the most thoroughly investigated topics in the field of empirical international economics. In recent years researchers have been analysing various aspects of this exchange rate theory by considering a range of different methodological approaches, estimation techniques and datasets. The application of otherwise popular linear specifications by PPP testing is criticised in cases when the speed of exchange rate adjustment critically depends on the degree of deviations from the long-run equilibrium level (Taylor & Taylor, 2004). A number of factors that can be held responsible for the asymmetric adjustment of real exchange rates were summarised, for example, by Taylor (2006) and include: presence of trade barriers and transaction costs, heterogeneous interactions of traders in the foreign exchange market concerning the expected exchange rate adjustment, and the coordination effects of official intervention by monetary authorities on the main traders in the foreign exchange market.
The aim of this paper is to implement, in addition to the conventional augmented Dickey-Fuller (ADF) test, the Kapetanios et al. (2003) (hereafter KSS) unit root test that accounts for possible nonlinear adjustment of real exchange rates for 11 countries of the euro area . The empirical literature on the PPP theory for the original euro area economies is rather limited, while the results of explicit tests are surprisingly mixed (Chang, Chang, & Su, 2013;Christidou & Panagiotidis, 2010;Wu & Lin, 2011;Zhou & Kutan, 2011). From the perspective of European integration processes it is especially intriguing to observe their effects on the stationarity characteristics of real exchange rates and on the overall price convergence in this region. A second important avenue of research is to evaluate the impact of the single European currency on the functioning of PPP in the international environment. Our study focuses on these questions by testing the PPP relationship in the post-euro period with respect to different numeraire currencies.
The paper is structured as follows: Section 2 provides a brief overview of the relevant empirical literature on PPP testing for euro countries. Section 3 describes the methodology of the KSS test. The dataset employed and the empirical results are described in Section 4. Section 5 presents the implications of the study.

An overview of the literature
In the academic literature on PPP as well as among experts (Rogoff, 1996;Taylor, 2006) there is a dominant view that real exchange rates tend to converge on levels predicted by PPP only in the long-run and that short-run deviations from the PPP relationship are usually substantial and variable. Despite accumulating a volume of empirical research, the debate on the validity of the PPP concept is far from being settled (Taylor, 2006). The progressing of European monetary integration and the forming of monetary union spurred the interest of researchers to assess the relevance of this exchange rate theory in the case of the single European currency. Koedijk, Tims, and van Dijk (2004) analysed the relationship between the synthetic euro and eight major currencies within the SURADF model for the period 1979-2003 and gained ambiguous outcomes. Initially, they find support for the theory in the full panel of real exchange rates, but after considering the heterogeneous mean reversion in the selected sample, PPP holds only for the euro-Swiss franc series. Lopez and Papell (2007) stress that within the euro area and between the euro area and other industrial EU countries the rejection of the unit root hypothesis can be detected in the period 1996-1999. Giannellis and Papadopoulos (2010) examine national exchange rates per euro for 12 eurozone countries and conclude that between 1980 and 2000 the majority of tested real exchange rates followed an adjustment path towards PPP. For eight out of 15 EU economies, Emirmahmutoglu and Omay (2014) provided evidence of nonlinear stationarity of real exchange rates against the US dollar over the period 1988-2013 by using a sequential panel selection method in ESTAR tests.
Contrary to the above-mentioned studies, testing results for the stationarity of real exchange rates appear to be distinctly heterogeneous in the research presented by Zhou and Kutan (2011). In their cases of PPP evidence for countries in the euro area, the adjustment of real exchange rates frequently obeys nonlinear dynamics. Furthermore, Christidou and Panagiotidis (2010) could not find support for the PPP hypothesis for the original members of the Eurozone, neither in the complete span of 1973-2009 nor in the subperiod after the introduction of the single European currency. This finding is additionally corroborated by Wu and Lin (2011) and by Huang and Yang (2015). Both studies applied Pesaran's (2007) panel unit root test allowing for cross-sectional dependencies and confirmed the mean-reversion of real exchange rates against the US dollar before the adoption of the euro, but failed to reject the null of a unit root under the euro regime. Instead of relying on bilateral exchange rates, the study by Su, Cheung, and Roca (2014) involves monthly data on real effective exchange rates for 61 countries over the period 1994 to 2012. The authors demonstrated that after considering heteroscedasticity and nonlinearity, the real effective exchange rates for the euro area as well for the majority of other countries in the sample are nonstationary.
Our overview of studies examining the long-run purchasing power parity between euro members and new EU countries shows fairly mixed results, but at the same time reflects the degree of price convergence and overall economic integration of European economies around the euro. Among papers that produced solid evidence on PPP for the new EU countries, all using euro-based exchange rates, are those written by Caporale, Ciferri, and Girardi (2011), Giannellis and Papadopoulos (2010), and by Maican and Sweeney (2013). While Caporale et al. (2011) focused on three Baltic countries testing the theory within a vector error correction model, Giannellis and Papadopoulos (2010) operated with a cluster of countries consisting of four new eurozone members and six possible eurozone candidates. In a similar way, Giannellis and Papadopoulos (2010), and Maican and Sweeney (2013) also investigated the exchange rate parity proposition by employing a range of nonlinear models, concentrating on a group of ten Central and Eastern European economies. All three studies quoted use observations that generally cover the period 1993-2005.
Some PPP-related econometric exercises provide no conclusive evidence of the empirical fulfilment of this theory. Koukouritakis (2009), for example, scrutinised the validity of PPP for 12 new EU countries using Johansen cointegration methodology. The author concludes that the long-run PPP vis-à-vis the eurozone holds only for Bulgaria, Cyprus, Romania and Slovenia. Cuestas (2009), after elaborating two sources of nonlinearities in real exchange rates against the euro, found evidence in favour of PPP for five of the eight Central and Eastern European (CEE) economies, whereas Boršič, Baharumshah, and Bekő (2012) were able to reject the unit root hypothesis by applying the panel SURADF test merely for five out of 12 euro rates of CEE countries. In addition, Karabulut, Bilgin, and Gozgor (2013) addressed the behaviour of real exchange rates in the Czech Republic, Poland and Hungary by analysing the currencies of their five largest trading partners and found weak evidence of mean reversion solely among the currencies of Hungary's main trading partners.
This paper extends the list of PPP literature given above by investigating different currencies as a numeraire and by applying the KSS test procedure on a group of 11 EU economies that were the first to officially adopt the single European currency.

PPP theory and econometric methodology
The absolute version of PPP theory states that a basket of goods should cost the same in two countries when the value of the basket is declared in the same currency. The relative version of purchasing power parity theory holds if the movements of relative prices of goods are offset by the movements of the prices of currencies in two observed countries in the long run. The basic model for empirical tests of the relative PPP is presented in the following form: where e t stands for nominal exchange rates, defined as the price of foreign currency in the units of domestic currency, p t denotes domestic price level and p t * foreign price level, while ξ t is the error term showing deviations from PPP. All the variables are used in the logarithmic form. The strict version of relative PPP contains two types of restrictions imposed on the parameters. Under α 0 =0, the symmetry restriction suggests that α 1 and α 2 are equal in absolute terms, whereas the requirement of α 1 =1 and α 2 = -1 is referred to as the proportionality restriction.
The characteristics of real exchange rates, namely the strict version of equation (1), are most commonly explored by the empirical analysis of PPP. In accordance with the relative PPP, the changes in the nominal exchange rates should indemnify for price level shifts. Thus, a stationary time series of the real exchange rate implies that the relative version of PPP theory should hold. The nominal exchange rate and relative price levels in two observed economies determine the real exchange rate: where y t stands for the real exchange rate. Real exchange rates and the validity of PPP are an integral part of new open economy macroeconomics developments, which gives the PPP approach an important role in the dynamic stochastic general equilibrium models (DSGE). Among others, Ahmad, Lo, and Mykhaylova (2013) analyse the real exchange rate dynamics in the setting of DSGE models, while Clarida (2014) applies the PPP concept to optimal monetary policy outcomes in the DSGE framework.
There are several reasons that support the assumption that the adjustment of real exchange rates is nonlinear and asymmetric. As stated in Enders and Dibooglu (2001) another source of nonlinearity might arise from the well-known rigidities of national price levels in the short-run, when monetary policy changes might generate PPP deviations since prices do not react as quickly as the exchange rates. Taking into account the downward price rigidity it cannot be presumed that the real exchange rate adjustment is symmetric. Being aware of the shortcomings of linear-type empirical techniques, this paper implements a nonlinear unit root test with respect to the set of real exchange rates series in the selected euro area countries. Kapetanios et al. (2003), also described in Bekő, Kavkler, and Boršič (2012), developed a test for the null hypothesis of the unit root against the alternative hypothesis of a nonlinear stationary smooth transition autoregressive (STAR) model. The authors attempted to distinguish between the nonstationary linear processes and the stationary nonlinear ones. The motivation for the development of the new test lies in the persistent failure of the standard ADF test to reject the null of a unit root. Consequently, two alternative frameworks for unit root testing were proposed in recent years. The first approach utilises panel tests and their higher power in comparison with standard unit root tests. The second approach incorporates stationary models other than the simple AR or ARMA under the alternative hypothesis, including nonlinear transition dynamics. Kapetanios et al. (2003) extended the last framework by analysing a particular kind of nonlinear dynamics, namely exponential smooth transition autoregressive (ESTAR) models.
The smooth transition autoregressive (STAR) model of order 1 is given by the equation (3) y t = y t−1 + * y t−1 G( , c;y t−d ) + t , t = 1, 2, ..., T d ≥ 1 where β and β * are unknown parameters and ɛ t is a sequence of independent identically distributed errors. Initially, y t is assumed to be a zero-mean process, but the framework can easily be extended to include more general processes with non-zero mean and time trend. G represents a continuous transition function bounded between 0 and 1. The slope parameter γ is an indicator of the speed of transition between 0 and 1, whereas the threshold parameter c points to where the transition takes place. y t-d is the transition variable and stands for the variable y lagged d times. The most popular functional forms are the Logistic Smooth Transition Autoregressive (LSTAR) form with logistic transition function and ESTAR with exponential transition function. The LSTAR transition function is monotonously increasing, while ESTAR is U-shaped around c and thus enables re-switching. The ESTAR functional form can be defined as Kapetanios et al. (2003) applied the ESTAR transition function with c equal to zero. By substituting G in equation (3) with the ESTAR transition function from equation (4), we obtain the ESTAR model The null hypothesis of unit root implies β = 1 and γ = 0, since G(0; y t-d ) = 0. Model (5) postulates the nonstationary linear First-Order Autoregressive (AR(1)) model under the null hypothesis and a stationary model (with γ > 0) under the alternative. When y t-d is close to zero, model (7) resembles a unit root process, since G(γ; 0) = 0. For large values of y t-d , on the other hand, we obtain an approximation of the linear AR(1) with the root equal to β + β * . We assume that -1 < β + β * < 1 (i.e. -2 < β * < 0), as this condition implies stable roots and a stationary AR(1) model.
The null hypothesis H 0 :γ = 0 needs to be tested against the alternative H 1 :γ > 0. β * is not identified under the null, and testing such a hypothesis is not feasible. To overcome this problem, Kapetanios et al. (2003) used the Taylor series approximation, as interpreted by Granger and Teräsvirta (1993). In the first step, the authors assumed d = 1 (which can be done without loss of generality) and respecified the ESTAR model (5) as After replacing the right-hand side expression with its first-order Taylor approximation, one obtains the following auxiliary regression: Using the t-statistic approach, the Nonlinear Augmented Dickey-Fuller (NLADF) statistic is defined as where ̂ denotes the Ordinary Least Squares (OLS) estimate from auxiliary regression (9) and ̂ its standard error. In a more general framework, when the errors of model (8) are serially correlated, the equation is augmented with lagged differences of the process y t : as first proposed by Dickey and Fuller in the derivation of the ADF test. The number of lags (p) is defined as the minimal number that removes residual autocorrelation. Auxiliary regression augmented with p lagged differences can be given as The NLADF test statistic is calculated from equation (10), as before. Kapetanios et al. (2003) derived the limiting nonstandard distribution of the NLADF statistic that involves Brownian motion.

Data and empirical results
Our sample consisted of the following eurozone countries: Austria, Belgium, Finland, France, Germany, Ireland, Italy, Luxembourg, the Netherlands, Portugal and Spain. The monthly averages of nominal exchange rates and consumer price indices were obtained from the European Central Bank and Eurostat. The real exchange rate series for US dollar rates covered the period January 1998 to February 2012, while for Japanese yen rates we calculated the real exchange rates over the period January 1996 to January 2012. The basic statistical properties of the real exchange rate series are summarised in the Appendix (Tables A1 and A2). For all countries in the sample, the consumer price indices referred to the beginning of the time period.
The results of the KSS test and the ADF test for models with constant and for models with constant and time trend are given for both numeraire currencies in Tables 1 and 2. Estimates from the standard ADF unit root test with and without the trend component make clear that we cannot reject the null hypothesis of nonstationarity of real exchange rates for any country in the sample. This holds true for the USD-based real exchange rate series as well as for JPY-based series.
The validity of the PPP concept gains no support even if we observe the results of the nonlinear unit root test procedure for the group of 11 original eurozone members. At the conventional significance levels, the KSS test fails to reject the null hypothesis of a unit root for any individual country. Our results from testing the nonlinear adjustment of real (10) NLADF =̂̂(

11)
exchange rates for the selected euro area countries appear to be robust with respect to the inclusion of the time trend and to changes in benchmark currencies. In other words, even after considering the trend element and estimating the nonlinear adjustment of euro area economies' real exchange rates by an ESTAR model, the PPP hypothesis could not be confirmed for any of the countries in the sample, irrespective of the chosen base currency. Our empirical outcomes reported here are in line with the findings of Christidou and Panagiotidis (2010), Wu and Lin (2011), and Huang and Yang (2015). However, instead of applying Pesaran's (2007) unit root test on 11 eurozone countries, as the studies of Wu and Lin (2011), and Huang and Yang (2015) did, or testing only for a single benchmark currency (Christidou & Panagiotidis, 2010), in this paper, we carried out the KSS unit root test procedure on USD-based and JPY-based series.  Table 2. Results of unit root test for japanese yen rates.
notes: the number of lags in the auxiliary regression is defined as the minimal number that removes residual autocorrelation. the 1%, 5% and 10% asymptotic critical values for aDF with intercept are -3.46, -2.88 and -2.57, respectively. the 1%, 5% and 10% asymptotic critical values for aDF with intercept and trend are -4.01, -3.43 and -3.14, respectively. the 1%, 5% and 10% asymptotic critical values for kss with intercept are -3.48, -2.93 and -2.66, respectively. the 1%, 5% and 10% asymptotic critical values for kss with intercept and trend are -3.93, -3.40 and -3.13, respectively. the critical values for kss are taken from kapetanios et al. (2003). A common feature of all the presented empirical results is that they invalidate the PPP for eurozone countries in the monetary-union period. The viability of purchasing power parity requires either an appropriate adjustment of nominal exchange rate through trading in foreign exchange markets or arbitrage in goods markets, which triggers shifts of relative prices of goods and services. Using detailed micro-level data from 1990 onwards for 20 industrial countries (with the exception of Ireland covering all the original eurozone countries) and the US, Bergin, Glick, and Wu (2013) showed that variations in aggregate real exchange rates are mainly due to changes of nominal exchange rates, while relative price shocks play a substantially less important role in rebuilding the real exchange rate equilibrium. The contention that responses of nominal exchange rates to real exchange rate deviations from PPP are empirically greater than the corresponding adjustment of relative prices is further strengthened by Huang and Yang (2015) for 11 eurozone economies. Moreover, the same authors have also shown that relative prices did not adjust to correct the real exchange rates along the parity lines under the euro regime. The failure of international relative prices to restore parity combined with the regime of inflexible nominal exchange rates can be an important source of violation of the long-run PPP relationship in euro area economies after 1998 (see Huang & Yang, 2015).
Nevertheless, the sufficient flexibility of relative prices of goods and services facilitates adjustment towards PPP and it is simultaneously an indicator of true integration of goods markets in the currency union. In the case of European markets, there is some evidence that this price adjustment mechanism might be functioning quite poorly. Thus, Bergin and Glick (2007) argue that a growing trade share of least traded goods, following the European monetary integration after 1998, has been associated with an increase and not decrease of price dispersion. Berka and Devereux (2013) also found no evidence for price convergence in the group of euro countries after 1999 and emphasise significant departures from PPP for nontraded as well as for traded goods. What are the underlying factors for such a development of relative prices in the eurozone markets? According to the study by Bénassy-Quéré and Coulibaly (2014), increasing product market regulations in nontradable sectors and various employment protection measures across these countries can be held responsible for the lack of adjustment in relative prices. This is a topical thesis that calls for continuous scrutiny of the PPP theory in the euro area countries.

Conclusions
In the literature, the presence of nonlinear dynamics in real exchange rate movements is explained with a range of models taking into account transaction costs, heterogeneous market expectations about the equilibrium nominal exchange rate, nonlinearities stemming from the goods-aggregation problem and the effects of official interventions in the foreign exchange market. Following such a line of reasoning, we employed a nonlinear unit root test based on the ESTAR model developed by Kapetanios et al. (2003) to test the PPP concept for 11 countries of the euro area.
The outcomes of the unit root tests applied in this paper for different model specifications and numeraire currencies unequivocally suggest that the PPP hypothesis does not hold in the case of the selected eurozone members after the introduction of the single currency. Our conclusion is in accord with the findings of Christidou and Panagiotidis (2010), Wu and Lin (2011), and Huang and Yang (2015) for US dollar-based series. The absence of real exchange rates nonlinear reversion reported in this study thus confirms the thesis of Wu and Lin (2011) regarding the shift in the PPP relationship primarily since the inception of the euro. An important source of PPP breach might originate from the unresponsiveness of relative prices to readjust the real exchange rate towards its constant equilibrium under the euro regime (see Huang & Yang, 2015). It is noteworthy that some researchers have already hinted at the functioning of the price adjustment mechanism that contradicts the one predicted for the sustainable operation of the eurozone system. Bergin and Glick (2007) stress that a growing trade share of least traded goods, following the European monetary integration after 1998, has been associated with an increase and not decrease of price dispersion. Additionally, Berka and Devereux (2013) also found no evidence for price convergence in the group of euro countries after 1999 and emphasise significant departures from PPP for nontraded as well as for traded goods. Among the main quantitative drivers of relative price dispersion in eurozone markets, Bénassy-Quéré and Coulibaly (2014) identified product market regulations in nontradable sectors and employment protection measures across these countries. To what extent and when greater competition in nontradable parts of the euro economies and in the national labour markets would provoke necessary price adjustment remains, however, an open question. Given the results presented, further empirical investigation is clearly required to solve the PPP debate for the eurozone countries.