Mohsen Bahmani-Oskooee, Tsangyao Chang, Tsung-Hsien Chen & Han-Wen Tzeng

Transcription

1 Revisiting purchasing power parity in Eastern European countries: quantile unit root tests Mohsen Bahmani-Oskooee, Tsangyao Chang, Tsung-Hsien Chen & Han-Wen Tzeng Empirical Economics Journal of the Institute for Advanced Studies, Vienna, Austria ISSN Volume 52 Number 2 Empir Econ (2017) 52: DOI /s z 1 23

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3 Empir Econ (2017) 52: DOI /s z Revisiting purchasing power parity in Eastern European countries: quantile unit root tests Mohsen Bahmani-Oskooee 1 Tsangyao Chang 2 Tsung-Hsien Chen 3 Han-Wen Tzeng 4 Received: 28 June 2015 / Accepted: 6 April 2016 / Published online: 8 June 2016 Springer-Verlag Berlin Heidelberg 2016 Abstract This study applies quantile unit root test proposed by Koenker and Xiao (J Am Stat Assoc 99(467): , 2004) and Galvao (J Econom 152: , 2009) to revisit the purchasing power parity in 7 transition countries: Bulgaria, Czech Republic, Hungary, Lithuania, Poland, Romania, and Russia over 1998M1 to 2015M3. While traditional unit root tests fail to reject unit root hypothesis, the two quantile unit root tests did reject unit root null hypothesis in all countries, providing support for the PPP and solving the PPP puzzle. The estimated half-life based on quantile autoregressive model is about months (1 2 year). Keywords Purchasing power parity Quantile unit root test Transition countries Valuable comments of two anonymous referees are very much appreciated. Remaining errors, however, are our own. B Mohsen Bahmani-Oskooee bahmani@uwm.edu Tsangyao Chang tychang@mail.fcu.edu.tw Tsung-Hsien Chen cheng523@cyut.edu.tw Han-Wen Tzeng hann@ocu.edu.tw 1 The Center for Research on International Economics and Department of Economics, University of Wisconsin-Milwaukee, Milwaukee, WI, USA 2 Department of Finance, Feng Chia University, Taichung, Taiwan 3 Department of Insurance, Chaoyang University of Technology, Taichung, Taiwan 4 Department of Finance, Overseas Chinese University, Taichung, Taiwan

4 464 M. Bahmani-Oskooee et al. JEL Classification C22 F31 1 Introduction Purchasing power parity (hereafter, PPP) remains an interesting research topic in international macroeconomics. Validating the PPP is critical not only for empirical researchers but also for policymakers. PPP states that due to arbitrage activities in the international commodities market, the real exchange rates that combine the movements of relative prices with nominal exchange rates are expected to return to a constant equilibrium value in the long run. This means real exchange rate will be a mean-reverting process stationary in the long run. In particular, a non-stationary real exchange rate indicates that there is no long-run relationship between nominal exchange rate and domestic and foreign prices, thereby invalidating the PPP hypothesis (Lin et al. 2011). As such, PPP cannot be used to determine the equilibrium exchange rate, and an invalid PPP also disqualifies the monetary approach to exchange rate determination, which requires PPP to hold true. Empirical evidence on the stationarity of real exchange rates is abundant but inconclusive thus far. Previous studies by MacDonald and Taylor (1992), Taylor (1995), Taylor (2006), Peel and Venetis (2003), Lothian and Taylor (2000), Sarno and Taylor (2002), Taylor and Taylor (2004), Sjolander (2007), Bahmani-Oskooee et al. (2008), Bahmani-Oskooee and Hegerty (2009), Chang and Tzeng (2013), He and Chang (2013), He et al. (2014) and Bahmani-Oskooee et al. (2014)have provided in-depth information on its theoretical and empirical aspects. However, the above tests usually focus on the average behavior of real exchange rate without considering the influence of various sizes of shocks on real exchange rate. In other words, the speed of adjustment in real exchange rate toward its equilibrium is usually assumed to be constant, no matter how big or what sign the shock is. As a result, the commonly used conventional unit root tests possibly lead to a widespread failure in the rejection of unit root null hypothesis in real exchange rates. This paper intends to deal with the above deficiency by employing two newly developed quantile unit root tests by Koenker and Xiao (2004) and quantile unit root test with stationary covariates by Galvao (2009) to enhance estimation accuracy. In this study, we use quantile unit root tests to reinvestigate validity of the PPP in seven transition countries of Bulgaria, Czech Republic, Hungary, Lithuania, Poland, Romania, and Russia over 1998M1 to 2015M3. According to Hosseinkouchack and Wolters (2013), there are several advantages of these new tests. First, a quantile-based unit root test can allow for the possibility that shocks of different sign and magnitude to have different impacts on real exchange rate. Second, this approach is not restricted to a specific number of regimes. Indeed, it allows generally for differences in the transmission of all kinds of different shocks. Third, the tests avoid the estimation of additional regime parameters and therefore reduces estimation uncertainty. Fourth, the tests have higher powers than conventional unit root tests as shown by Koenker and Xiao (2004). Finally, the quantile-based unit root test is superior to standard unit root tests in case of departure from Gaussian residuals The transition countries in our sample have recently moved from centrally planned economies toward market-driven economies that motivate us to investigate the behav-

5 Revisiting purchasing power parity in Eastern European 465 ior of real exchange rate. Determining the behavior of exchange rates in such countries would provide information not only to policy makers in developing sound exchange rate and monetary policies but also to traders and global investors in their trade and investment strategies. Even though a number of studies have tested the PPP hypothesis in the transition economies (see, among others, Baharumshah and Borsic 2008; Lin et al. 2011; Solakoglu 2006; Cuestas 2009; Telatar and Hasanov 2009), this study differs from those studies by applying the quantile-based unit root tests for the first time. The remainder of this paper is organized as follows. Section 2 presents the data used in our study. Section 3 first briefly describes the quantile unit root tests proposed by Koenker and Xiao (2004) and Galvao (2009). Empirical results in support of PPP are reported in Sect. 4. Finally, a summary and conclusion appears in Sect. 5. 2Data This empirical analysis covers seven transition countries: Bulgaria, the Czech Republic, Hungary, Lithuania, Poland, Romanian, and Russia. We employ monthly data in our empirical study, and the time span is from January 1998 to March Due to data availability, our study can only focus on these seven transition countries. Some transition countries are excluded because they joined the European community and no longer have its own currency such as Estonia and Latvia. All consumer price indices, CPI (based on 2000 = 100), and nominal exchange rates relative to the USA dollar data, respectively, are taken from the Datastream. Each of the consumer price index and nominal exchange rate series was transformed into natural logarithms before performing the econometric analysis. Testing for the PPP against the USA is based on the argument that internal foreign exchange markets are mostly dollar dominated. In addition, funds for economic reconstructions are being provided by US-sponsored institutions (Bahmani-Oskoee et al. 2015). We define the bilateral real exchange rate, RE, as: ( P ) RE t = NE t (1) P where NE t is the nominal exchange rate defined as number of units of local currency per US dollar; P t and Pt are the domestic and US price levels, respectively, measured by consumer price indices (CPI). Taking log from both sides of (1) and denoting the log of each variable by small letter corresponding to that variable yield: t re t = ne t + p t P t (2) From a statistical point of view, the validity of the purchasing power parity (PPP) hypothesis reduces to a unit root test of re t. The presence of a unit root in the real 1 The period prior to 1998 was eliminated from the analysis because changes in overall inflation during the early years of the transition process (especially appreciation of the real exchange rate) were dominated by firm-level restructuring involving massive lay-offs, the adjustment of distorted relative prices from the Communist era and pegged exchange rate regimes motivated by concerns for macroeconomic stabilization.

6 466 M. Bahmani-Oskooee et al. Table 1 Summary statistics of real exchange rate Bulgaria Czech Hungary Lithuania Poland Romanian Russia Mean Median Maximum Minimum SD Skewness Kurtosis Jarque Bera Probability Observations exchange rate series would imply that PPP does not hold in the long run. If PPP holds, it implies that nominal exchange rate is corrected for inflation differentials, as shown by (2). Chang et al. (2011) note that non-stationarity in real exchange rates has many macroeconomic implications. For example, Dornbusch (1987) has argued that if real exchange rate depreciates, it could bring a gain in international competitiveness, which in turn could shift the employment toward the country whose currency depreciates. Therefore, it is important to establish the empirical validity of the purchasing power parity theory. Another important implication of non-stationary in real exchange rate is that unbounded gains from arbitrage in traded goods are possible. In fact, Parikh and Williams (1998) have mentioned that a non-stationary real exchange rate can cause severe macroeconomic disequilibrium that would lead to real exchange rate devaluation in order to correct for external imbalance. Table 1 reports summary statistics of data, and we find that all data series are non-normal. As pointed by Koenker and Xiao (2004), the quantile-based unit root test has higher power than conventional unit root tests in case of departure from Gaussian residuals, and these further confirm the use of quantile unit root test in this paper. 3 The methods: quantile autoregressive unit root tests Let re t denote the log of real exchanger rate in our case and ε t a serially uncorrelated error term. An AR(q) process for re t with drift a is given by: re t = a + q γ i re t 1 + ε t, t = q + 1, q + 2,...,n. (3) i=1 The sum of the autoregressive coefficients is α = q i=1 γ i a measure of persistence that we will focus on in our study. We can rewrite Eq. (3) as:

7 Revisiting purchasing power parity in Eastern European 467 q 1 re t = αre t 1 + a + φ i re t i + ε t (4) Here we can run the usual unit root test. If α = 1, then the real exchange rate has a unit root, and therefore, shocks have permanent effects on real exchange rate. If we have α<1, then real exchange rate is stationary. In this case, shocks have only temporary effects on real exchange rate and this means PPP holds in this case. To gain more detailed estimates to analyze persistence, we can focus not only on the conditional mean, but also in the tails of the conditional distribution of re t, and here we can estimate Eq. (4) using quantile autoregression methods. The τth conditional quantile is defined as the value Q τ (re t ret 1,...,re t q ) such that the probability that output conditional on its recent and past history will be less than Q τ (re t ret 1,...,re t q ) is τ. For example, if exchange rate is very high (low) relative to recent exchange rate level, this means that a large positive (negative) shock has occurred and that re t is located above (below) the mean conditional on past observations re t 1,...,re t q somewhere in the upper (lower) conditional quantiles. The AR(q) process of real exchange rate at quantile τ can be written as: i=1 ( ) q 1 Q τ ret ret 1,...,re t q = α(τ)ret 1 + a(τ) + φ i (τ) re t i. (5) The test has been extended by Galvao (2009) to include deterministic components which is essential for unit root tests of drifting time series like real exchange rate in our case. Following Galvao (2009), we can extend Eq. (5) to include stationary covariates for unit root testing. Galvao (2009) has proved that his proposed model has more power than standard quantile unit test model. We can express his model as: ( ) q 1 Q τ ret ret 1,...,re t q = α(τ)ret 1 + a(τ) + φ i (τ) re t i + q2 I = q1 i=1 i=1 γ I (τ) X t I + F 1 u (τ) (6) where F u denotes the common distribution function of errors. Let a(τ) = a + Fu 1 (τ) define Z = (1, re t 1, re t 2,..., re t q+1, X t q2..., X t+q1 ) T. Here we can define X t I as stationary covariates (such as difference nominal exchange rate, stock returns, and inflation rate in our study). By estimating Eqs. (5) and (6) at different quantiles τ (0, 1), we can get a set of estimates of the persistence measure as α(τ). We can test α(τ) = 1 at different values of τ to analyze the persistence of the exchange rate impact of positive and negative shocks and shocks of different magnitude using the quantile autoregression-based unit root test proposed by Koenker and Xiao (2004). Let α(τ) be the quantile regression estimator. To test H 0 : α(τ) = 1, we use the t-stat for α(τ) proposed by Koenker and Xiao (2004) which can be written as

8 468 M. Bahmani-Oskooee et al. t n (τ) = f (F 1 (τ)) τ (1 τ) ( er 1 M Z er 1 ) 1/2 (α(τ) 1), (7) where f (u) and F(u) are the probability and cumulative density functions of ε t,re 1 is the vector of lagged log-real exchange rate, and M z is the projection matrix onto the space orthogonal to Z = (1, re t 1, re t 2,..., re t q+1 ) T for Koenker and Xiao (2004) approach or Z = (1, re t 1, re t 2,..., re t q+1, X t q2..., X t+q1 ) T for Galvao (2009) approach. 2 We use the results derived by Koenker and Xiao (2004) and Galvao (2009) to find the critical values of t n (τ) for different quantile levels. We can estimate f (F 1 (τ)) following the rule given in Koenker and Xiao (2004) and Galvao (2009). Besides allowing for asymmetric effects of shocks on real exchange rate, an important advantage of QAR-based unit root tests over standard unit root tests is that they have more power (Koenker and Xiao 2004; Galvao 2009). In contrast, a more complete inference of the unit root process based on the quantile approach involves exploring the unit root property across a range of quantiles. To this end, both Koenker and Xiao (2004) and Galvao (2009) suggest the Quantile Kolmogorov Smirnov (QKS) test, which is given as QKS = sup t n (τ) (8) τ Ɣ where t n (τ) is given by Eq. (7) and Ɣ = (0.1, 0.2,...0.9) in our later applications. In other words, we first calculate t n (τ) for all τ s in Ɣ and then construct the QKS test statistic by selecting the maximum value across Ɣ. While the limiting distributions of both t n (τ) and QKS tests are nonstandard, both Koenker and Xiao (2004) and Galvao (2009) also suggest the use of a resampling (number of bootstrap = in our case) procedure to approximate their small-sample distributions. 3 4 Empirical results We begin our empirics by first applying three conventional unit root tests ADF, PP, and KPSS tests. The results in Table 2 clearly indicate that both the ADF and the PP tests fail to reject the null of non-stationary real exchange rate for all 7 transition countries, with the exception of Romania. KPSS test get similar results. These results are consistent with those of Baharumshah and Borsic (2008), Liu et al. (2012), Chang and Tzeng (2013), and Bahmani-Oskoee et al. (2015), indicating that PPP does not hold in most transition countries. As mentioned before, due to deficiencies associated with conventional unit root tests, we now consider the quantile unit root of Koenker and Xiao (2004) first.here we test the null of α(τ) = 1forτ = 0.1, 0.2, 0.3, 04,...,0.9 byusingt statistic (t n (τ)) based on Eq. (7). Table 3 show the point estimates, the t statistics, the critical 2 Details about the selection of bandwidth, of kernel, and of truncation parameters, we follow the suggestion of Koenker and Xiao (2004) and Galvao (2009). Interested reader can refer to Koenker and Xiao (2004) and Galvao (2009). 3 For more details, see Koenker and Xiao (2004)andGalvao (2009).

9 Revisiting purchasing power parity in Eastern European 469 Table 2 Univariate unit root tests Level 1st Difference ADF PP KPSS ADF PP KPSS Bulgarian 1.253(0) 1.261[1] 1.587(11) (0) [0] 0.278(0) Czech 1.523(0) 1.514[3] 1.506(11) (0) [3] 0.271(2) Hungarian 1.529(0) 1.516[3] 1.346(11) (0) [2] 0.203(2) Lithuanian 1.382(0) 1.384[3] 1.581(11) (0) [3] 0.286(3) Poland 1.868(0) 1.930[4] 1.312(11) (0) [3] 0.(3) Romanian 4.516(0) 4.084[7] 1.204(11) (0) [8] 0.585(8) Russian 1.247(1) 1.234[6] 1.478(11) (0) [4] 0.187(6),,and indicate significance at the 0.01, 0.05, and 0.1 level, respectively. Bold values represent the significant relation to the critical values. The number in parentheses indicates the lag order selected based on the recursive t statistic, as suggested by Perron (1989). The number in the brackets indicates the truncation for the Bartlett Kernel, as suggested by the Newey West test (1987) values, half-life of a shock, and QKS for each country. We find that H 0 : α(τ) = 1 can be rejected at the 10 % significance level over the whole conditional real exchanger rate distribution based on Quantile Kolmogorov Smirnov test (QKS) for 3 out of 7 countries (i.e., Bulgaria, Lithuania, and Romania). The results confirm that all types of shocks to real exchange rate lead to temporary effects in these 3 transition countries. This means that PPP holds in these 3 transition countries (i.e., Bulgaria, Lithuania, and Romania). Table 3 also shows the persistent estimates of α(τ)for τ = 0.1, 0.2, 0.3,...,0.9 in each transition country. The persistence parameter estimates are close to one for all the quantiles considered in Czech Republic, Hungary, Poland, and Russia. The persistent point estimate is slightly above one at the upper tail quantile for Bulgaria, Czech Republic, Hungary, and Poland and lower tail quantile for Lithuania. Overall, the parameter estimates are relatively homogeneous over the conditional exchange rate distribution. 4 Because PPP holds in Bulgaria, Lithuania, and Romania, Table 3 also calculate half-life of a shock for those 3 countries. We find that the estimated half-life based on quantile autoregressive model is about months (1 2 year). Can we improve the results by shifting to Galvao (2009) test? Galvao (2009) points out that his proposed quantile unit root test with stationary covariates has more power than that of Koenker and Xiao (2004). We follow Elliott and Pesavento (2006), Su et al. (2012) and Liu and Chang (2013) to choose the stationary covariates according to economic theory. In our study, we use the first differences in nominal exchange rates, inflation, and stock returns. Results for these three stationary covariates are reported in Tables 4, 5, and 6, respectively. Based on results from Table 4, we can see that PPP now holds in all seven transition countries when difference nominal exchange rates serve as stationary covariates, based on QKS statistics. Table 5 reports our empirical results when we use stock returns to serve as our stationary covariates, and we find that PPP holds in four of these seven transition 4 Significant results in certain quantiles indicate asymmetric adjustment of real exchange rate process.

10 470 M. Bahmani-Oskooee et al. Table 3 Empirical results of quantile estimation and unit root tests for each quantile (without taking into stationary covariates) Koenker and Xiao (2004) τ Bulgaria α1(τ) Half-lives QKS for quantiles of %: (0.009) Czech Republic α1(τ) Half-lives QKS for quantiles of %: Hungary α1(τ) Half-lives QKS for quantiles of %: Lithuania α1(τ) Half-lives QKS for quantiles of %: (0.012) Poland α1(τ) Half-lives QKS for quantiles of %: 1.368

11 Revisiting purchasing power parity in Eastern European 471 Table 3 continued τ Romania α1(τ) Half-lives QKS for quantiles of %: (0.022) Russia α1(τ) Half-lives QKS for quantiles of %: and denotes significance at 5 and 1 % levels, respectively. Bold values represent the significant relation to the critical values. Numbers in parentheses denote bootstrap p values with the bootstrap replications set to be 10,000. For α1(τ), the unit root null is examined with the tn(τ) statistic. The lag length q is selected based on robust Schwarz information criterion as suggested by Galvao (2009) with a maximum lag set to be 12. The number in parentheses is p value.

12 472 M. Bahmani-Oskooee et al. Table 4 Empirical results of quantile estimation and unit root tests for each quantile (taking into account stationary covariates first difference of nominal exchange rate) Galvao (2009) τ Bulgaria α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.000) [8, 1, 1] Czech Republic α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.000) [1, 1, 1] Hungary α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.000) [1, 1, 1] Lithuania α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.000) [1, 1, 1]

13 Revisiting purchasing power parity in Eastern European 473 Table 4 continued τ Poland α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.000) [8, 1, 1] Romania α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.000) [1, 2, 1] Russia α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.000) [11, 1, 2] and denotes significance at 5 and 1 % levels, respectively. Bold values represent the significant relation to the critical values. Numbers in parentheses denote bootstrap p values with the bootstrap replications set to be 10,000. For α1(τ), the unit root null is examined with the tn(τ) statistic. The lag lengths p and q in the bracket are selected based on robust Schwarz information criterion as suggested by Galvao (2009) with a maximum lag set to be 12. The number in parentheses is p value

14 474 M. Bahmani-Oskooee et al. Table 5 Empirical results of quantile estimation and unit root tests for each quantile (taking into account stationary covariates Stock Return) Galvao (2009) τ Bulgaria α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.006) [1, 1, 1] Czech Republic α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.009) [1, 1, 1] Hungary α1(τ) ˆδ Half-lives QKS for quantiles of %: [1, 1, 1] Lithuania α1(τ) ˆδ Half-lives QKS for quantiles of %: [1, 1, 1] Poland α1(τ) ˆδ Half-lives

15 Revisiting purchasing power parity in Eastern European 475 Table 5 continued τ QKS for quantiles of %: [1, 1, 1] Romania α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.075) [1, 1, 3] Russia α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.035) [8, 1, 5] and denotes significance at 5 and 1 % levels, respectively. Bold values represent the significant relation to the critical values. Numbers in parentheses denote bootstrap p values with the bootstrap replications set to be 10,000. For α1(τ), the unit root null is examined with the tn(τ) statistic. The lag lengths p and q in the bracket are selected based on robust Schwarz information criterion as suggested by Galvao (2009) with a maximum lag set to be 12. The number in parentheses is p value

16 476 M. Bahmani-Oskooee et al. Table 6 Empirical results of quantile estimation and unit root tests for each quantile (taking into account stationary covariates Inflation) Galvao (2009) τ Bulgaria α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.046) [1, 1, 1] Czech Republic α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.049) [1, 1, 1] Hungary α1(τ) ˆδ Half-lives QKS for quantiles of %: [1, 1, 1] Lithuania α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.008) [1, 1, 2] Poland α1(τ) ˆδ Half-lives

17 Revisiting purchasing power parity in Eastern European 477 Table 6 continued τ QKS for quantiles of %: [1, 1, 1] Romania α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.08) [1, 1, 2] Russia α1(τ) ˆδ Half-lives QKS for quantiles of %: (0.047)[1, 12, 4] and denotes significance at 5 and 1 % levels, respectively. Bold values represent the significant relation to the critical values. Numbers in parentheses denote bootstrap p values with the bootstrap replications set to be 10,000. For α1(τ), the unit root null is examined with the tn(τ) statistic. The lag lengths p and q in the bracket are selected based on robust Schwarz information criterion as suggested by Galvao (2009) with a maximum lag set to be 12. The number in parentheses is p value

18 478 M. Bahmani-Oskooee et al. countries (i.e., Bulgaria, Czech Republic, Romania, and Russia). On the other hand, Table 6 reports our empirical results when we use inflation rates to serve as our stationary covariates, and we find that PPP holds in five of these seven transition countries (i.e., Bulgaria, Czech Republic, Lithuania, Romania, and Russia). Based on these findings, we can conclude that quantile unit root test with stationary covariates proposed by Galvao (2009) provides more evidence in support of the PPP. Perron (1989) has pointed out that failure to account for structural breaks in testing for unit root could result in misleading findings. We, therefore, extend quantile-based unit root test model by incorporating both sharp shifts and smooth breaks proposed by Bahmani-Oskooee et al. (2014) to enhance estimation accuracy. We model the mean reversion properties in the real exchange rates series with both sharp shifts and smooth breaks for estimation of a level equation following Bahmani-Oskooee et al. (2014) and specify its function as: m+1 re t = α + θ l DU l,t + l=1 n ( 2πkt γ 1,k sin T k=1 ) + n ( 2πkt γ 2,k cos T k=1 ) + ε t (9) In Eq. (9), re t follows the same definition as before, and t, T, and m are time trend, sample size, and the optimum number of breaks, respectively. The other regressors are defined as: { 1 if TBk 1 < t < TB DU k,t = k 0 otherwise (10) Note that DU is entered into the specification in order to capture sharp shifts. 5 Following Gallant (1981), in order to obtain a global approximation from the smooth transition, Bahmani-Oskooee et al. (2014) use the Fourier approximation and enter both n k=1 γ 1,k sin( 2πkt T ) and n k=1 γ 2,k cos( 2πkt T ) into the model. Here n and k present the number of frequencies that are contained in the approximation where n T 2. Estimation of Eq. (9) involves selecting values for m, n, and k. As noted by Becker et al. (2004), it is reasonable that we restrict n = 1 because if γ 1,k = γ 2,k = 0 can be rejected for one frequency, then the null hypothesis of time invariance is also rejected. Also Enders and Lee (2012) noted that imposing the restriction n = 1 is useful in order to save the degrees of freedom and avoid over-fitting problem. Hence, we re-specify Eq. (9) as follows: m+1 re t = α + θ l DU l,t + γ 1 sin l=1 ( 2πkt T ) + γ 2 cos ( 2πkt T ) + ε t (11) The approach is to remove the impact of possible structural breaks (sharp and smooth) from the real exchange rate once we approximate the break dates. To this end, we 5 Equation (9) is not only an extension of Enders and Holt (2012) but also a combination of Carrión-i- Silvestre et al. (2006)andBecker et al. (2006) tests.

19 Revisiting purchasing power parity in Eastern European 479 Table 7 Empirical results of quantile estimation and unit root tests for each quantile (taking into account both sharp and smooth breaks) τ Bulgaria α 1 (τ) Half-lives QKS for quantiles of %: (0.032) Czech Republic α 1 (τ) Half-lives QKS for quantiles of %: (0.023) Hungary α 1 (τ) Half-lives QKS for quantiles of %: Lithuania α 1 (τ) Half-lives QKS for quantiles of %: (0.019) Poland α 1 (τ) Half-lives ,875 QKS for quantiles of %: (0.034) Romania α 1 (τ) Half-lives QKS for quantiles of %: (0.002) Russia α 1 (τ) Half-lives QKS for quantiles of %: (0.017) denotes significance at 5 % level. Bold values represent the significant relation to the critical values. Numbers in parentheses denote bootstrap p values with the bootstrap replications set to be 10,000. The lag length p and q are selected based on robust Schwarz information criterion as suggested by Galvao (2009) with a maximum lag set to be 12. For α1(τ), the unit root null is examined with the tn (τ) statistic. The number in parentheses is p value follow the procedure adopted by Tsong and Lee (2011) and reconstruct a new time series of the real exchange rate by taking into account both sharp shifts and smooth breaks as: m+1 y t = re t ˆα l=1 ( 2πkt ˆθ l DU l,t ˆγ 1 sin T ) ˆγ 2 cos ( 2πkt T ) + ε t (12)

20 480 M. Bahmani-Oskooee et al. Table 8 Estimation results for the mean-reverting function in Eq. (11) Countries Optimum frequency F stat 90 % 95 % % 99 % Panel A: The results for optimum frequency and the F statistic and its critical values Bulgarian , Czech Rep Hungary Lithuania Poland Romania Russian Countries Break dates Panel B: The results for sharp drift dates in Eq. (11) Bulgarian Czech Rep Hungary Lithuania Poland Romania Russian where y t is real exchange rate adjusted by the effects of possible structural breaks (for both sharp shifts and smooth breaks). 6 We then apply quantile unit root test proposed by Koenker and Xiao (2004) to test for unit root in our adjusted new series y t.we report these empirical results in Table 7. Table 7 also shows the point estimates, t statistics, critical values, half-life of a shock, and QKS for each country. We find that H 0 : α(τ) = 1 can be rejected at the 5 % significance level over the whole conditional real exchanger rate distribution based on Quantile Kolmogorov Smirnov test (QKS) for all seven countries (i.e., Bulgaria, Czech Republic, Lithuania, Hungary, Poland, Romania, and Russia). These results confirm that all types of shocks to real exchange rates lead to temporary effects which implies that PPP holds in these seven transition countries after we take into account both sharp shifts and smooth breaks. From Table 7, we also gather that the estimated half-life based on quantile autoregressive model is about 1 12 months (1 month to 1 year). Our empirical results highlight the importance of modeling both sharp shifts and smooth breaks into quantile-based unit root test model. Table 8 reports the estimation results for the mean-reverting function in Eq. (11) which further confirm our findings. Figure 1 displays the time paths of the real exchange rates where a positive change in the exchange rate indicates real depreciation. We can clearly observe structural 6 For details of how to estimate (11) and(12) refer to Bahmani-Oskooee et al. (2014), Bahmani-Oskoee et al. (2015).

21 Revisiting purchasing power parity in Eastern European BULGARIA BULGARIAH CZECH CZECHH HUNGARY HUNGARYH LITHUANIA LITHUANIAH POLAND POLANDH ROMANIAN ROMANIANH RUSSIA RUSSIAH Fig. 1 Time series plot of real exchange rates and fitted nonlinear flexible intercept with sharp shifts and smooth breaks for each country shifts in the trend of the data. Accordingly, it appears sensible to allow for both sharp shifts and smooth breaks in testing for a unit root (and/or stationary). The estimated time paths of the time-varying intercepts are also shown in all figures. Because the actual nature of break(s) is generally unknown, there is no specific guide as to where and how many breaks to use in testing for a unit root or stationarity. Using an incorrect specification for the form and number of breaks can be as problematic as ignoring the breaks altogether. A further examination of the figures indicates that both dummy variables (sharp shifts) and Fourier approximations (smooth beaks) seem reasonable and support the notion of long swings in real exchange rates. Accordingly, it appears sensible to allow for both sharp drifts and smooth breaks in our approach. One major policy implication of our study is that the validity of using PPP to determine the equilibrium exchange is unambiguous for all of these seven transition countries. In these countries, the PPP could be used to determine whether a currency is over- or undervalued. Nevertheless, reaping unbounded gains from arbitrage in traded goods is not possible. Furthermore, the findings in support of the PPP imply that deviations in the short run from the PPP are not prolonged for most of the transition countries and there are some forces which are capable of bringing the exchange rate back to its PPP values in the long run. Our results have important policy implication on cross-border agreement for international trade and investment with these countries. Given the goods and services markets appear quite integrated, future liberalization will likely affect financial markets. If we envision this process of integration continuing, in particular in the transition

22 482 M. Bahmani-Oskooee et al. countries, and to the extent that this process requires even more political engagement, we believe the prospects for cooperation along a variety of dimensions are good. It is noteworthy that the results here are in contrast with those of Beko and Borsic (2007), Baharumshah and Borsic (2008), Acaravci and Ozturk (2010) and Liu et al. (2012) who provided weak (or no) support for PPP in various groups of transition countries using traditional unit root tests. 5 Conclusions This study applies quantile unit root test proposed by Koenker and Xiao (2004) and Galvao (2009) to revisit the purchasing power parity (PPP) in seven transition countries: Bulgaria, Czech Republic, Hungary, Lithuania, Poland, Romania, and Russia using monthly data over January 1998 to March While traditional unit root tests fail to support PPP, quantile unit root tests provide strong support for the PPP even after accounting for sharp shifts and smooth breaks in all seven countries. The estimated half-life based on quantile autoregressive model is about months (1 2 year). The transition countries started their liberalization programs in the late 1980s and early 1990s. In some of these countries, this period was characterized by dramatic improvements in budget deficits, debts, and inflation. As these countries became increasingly open to trade (and inflation and growth rates converged to those of developed countries), we would expect to find more favorable evidence of the parity condition using data for recent years (Bahmani-Oskoee et al. 2015). In fact, many of these countries adopted trade policies that mimic those of the European Union (EU), with a view to alignment in readiness for membership. As the reform process (price liberalization and trade opening) intensified, we could expect a reduction in persistent shocks to international parity (Bahmani-Oskoee et al. 2015). Our empirical results seem to support these claims. References Acaravci A, Ozturk I (2010) Testing purchasing power parity in transition countries: evidence from structural breaks. Amfiteatru econ 12(27): Bahmani-Oskooee M, Kutan AM, Zhou Z (2008) Do real exchange rates follow a nonlinear mean reverting process in developing countries. South Econ J 74: Bahmani-Oskooee M, Hegerty S (2009) Purchasing power parity in less-developed and transition economies: a review article. J Econ Surv 23: Bahmani-Oskooee M, Chang T, Wu TP (2014) Revisiting purchasing power parity in African countries: panel stationary test with sharp and smooth breaks. Appl Financ Econ 24: Bahmani-Oskoee M, Chang T, Wu TP (2015) Purchasing power parity in transition countries: panel stationary test with smooth and sharp breaks. Int J Financ Stud 3: Baharumshah AZ, Borsic D (2008) Purchasing power parity in central and Eastern European countries. Econ Bull 6:1 8 Becker R, Enders W, Lee J (2004) A general test for time dependence in parameters. J Appl Econom 19: Becker R, Enders W, Lee J (2006) A stationairy test in the presence of an unknown number of smooth breaks. J Time Ser Anal 27: Beko J, Borsic D (2007) Purchasing power parity in transition economies: does it hold in the Czech Republic, Hungary and Slovenia? Post Commun Econ 19:

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