International Journal of Arts & Sciences, CD-ROM. ISSN: 1944-6934 :: 4(22):215–220 (2011) c 2011 by InternationalJournal.org Copyright

OLD WINE IN NEW BOTTLES: TESTING PPP WITH A NEW TESTING STRATEGY Nilgun Cil Yavuz and Veli Yilanci Istanbul University, Turkey This study examines the Purchasing Power Parity hypothesis for G7 countries from January 1972 to December 1998 using a nonlinear strategy. We, first employ Escribano and Jorda (2001)’s two step nonlinear test. At first step in this step, we test whether the real exchange rates are linear or nonlinear and at second step, we determine the type of nonlinearity; whether they are logistic smooth transition autoregressive (LSTAR) or exponential smooth transition autoregressive (ESTAR) process. For the linear series, we applied augmented Dickey-Fuller unit root test, we used modified KSS unit root test for the real exchange rates (RER)s which follow ESTAR process, for the LSTAR RER we used LSTAR unit root test. In conclusion, the results show that only the RER of United Kingdom is stationary which show PPP is valid only for United Kingdom. Keywords: Purchasing power parity, Nonlinearity.

1. INTRODUCTION The purchasing power parity (PPP), which shows a unit of one country’s currency will have the same purchasing power in a foreign country (Taylor and Taylor, 2004) in the case of its validity, is one of the most attracting and popular theories in economics which also has not been resolved yet. The underlying reason for its popularity can be ordered as follows: First; the PPP can be used to determine whether a currency is over or under-valued. Second; the PPP used in many exchange rate theories construction as expressed by Holmes (2001). Third; understanding of its behavior can be helpful for policy design since the measurement of economic integration and external competitiveness can be made by employing real exchange (Wei and Parsley, 1995). There are numerous studies in the literature to test the PPP hypothesis which use different date intervals, different countries and different econometric methodologies so have different conclusions. This study based on unit root tests to examine the PPP. For this purpose, we use the real exchange rate of G7 countries which was computed using following formulae: qt = et + pt − pt*

where et is the nominal exchange rate, pt is the domestic price level (we used consumer prices index) and pt* foreign price level. All variables transformed into natural logarithms. If the real exchange rate is found to be stationary, deviations from equilibrium will be temporary, and

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so we can conclude that the PPP is valid for the relevant country, on the other hand the nonstationary characteristic of the real exchange rate implies the invalidity of PPP. In this study, we analyze the real exchange rates in a nonlinear smooth transition autoregressive model framework since the most of empirical studies found the rates as STAR process (see Chortareas et al.(2002), Liew et al. (2004), Bec et al. (2004) among others). This study organized as follows: In the next section we review the econometric methodology, in the section 3 we present the data and empirical results. Finally, section 4 concludes the study. ECONOMETRIC METHODOLOGY The nonlinearity test introduced by Escribano and Jorda (2001) contains two stages. At first step, we test the linearity against the nonlinearity. In the case of rejection of linearity, two different F tests applied to choose the type of nonlinearity. Let’s consider the following smooth transition autoregressive model:

yt = π ′xt + F ( zt −d , γ , c ) Θ′xt + ut

(1)

Where xt = (1, yt −1 ,..., yt − p )′ = (1, xɶt′ )′ ,

π ′ = (π 0 , π 1 ,..., π p ) = (π 0 , πɶ ′ ) ,

ɶ ′ ) and 1 ≤ d ≤ p . The value of z generally choosen as y but, it Θ′ = ( Θ 0 , Θ1 ,..., Θ p ) = ( Θ0 , Θ t −d t −d can be also another given parameter or an exogenous variable. Exponential STAR (ESTAR) and Logistic STAR (LSTAR) models have the following transition functions respectively: 2 F ( zt − d , γ , c ) = 1 − exp −γ ( zt − d − c )   

{

}

1 −1  F ( zt − d , γ , c ) = 1 + exp −γ ( zt − d − c ) −  2 

{

}

Linearity ( H 0 : Θ′ = 0 ) is tested against the STAR type nonlinearity in the Model 1. But there are unidentified parameters under the null, so Escribano and Jorda (2001) suggest to use the following auxiliary regression in which a Taylor series approximation used instead of the

F ( zt − d , γ , c ) function to overcome the unidentified parameters problem.

yt = δ 0 + δ1′xɶt + β1′xɶt zt − d + β 2′ xɶt zt2− d + β 3′xɶt zt3− d + β 4′ xɶt zt4− d + et

Old Wine in New Bottles: Testing PPP with a New Testing Strategy

217

Where the null hypothesis of linearity H 0 : β1′ = β 3′ = β 3′ = β 4′ = 0 can be tested employing LM test-statistic. After rejection of the null, EJ (2001) suggest to use following selection procedure to determine the type of nonlinear process; that is the series under investigate are LSTAR or ESTAR:

1-Test the null H 0 L : β 2′ = β 4′ = 0 with an F-test (FL) and 2- Test the null H 0 E : β1′ = β3′ = 0 with an F-test (FE), 3- Compare the p-values of the F tests and select the process whose p-values bigger. After determining the appropriate model, we will test if the true process linear unit root or stationary nonlinear (ESTAR/LSTAR) process by using modified Kapetonious et al. (2003) test of Su et al. (2010) and Pascalau (2007) unit root tests. Kapetonious et al. (2003) use a Taylor approximation for the Exponential function and use the following auxiliary model to test the unit root hypothesis: p

∆yt = α1 yt3−1 + ∑ α i ∆yt −i

(2)

i =1

Where yt is de-meaned/de-trended/raw series of interest. We test the null of linear unit root ( α1 = 0 ) against the stationary exponential smooth transition autoregressive model ( α1 < 0 ) using Dickey-Fuller type test for α1 . Su et al. (2010) modified this test suggesting a recursive detrending method following So and Shin (1999) and Shin and So (2001). Since we will use the demaned series in this study, following Su et al. (2010) we will use the following demeaned series instead of yt and yt −1 in Model 2 respectively: yt −

1 t −1 ∑ yi t − 1 i =1

yt −1 −

1 t −1 ∑ yi t − 1 i =1

In their paper Su et al. (2010) approved this method improves the power of the Kapetonious et al. (2003)’s test only when mean needs to be removed. Critical values of this modified KSS test is tabulated in Table 1 in Su et al. (2010). Pascalau (2007) introduced a test to test the null of unit root against the logistic smooth transition autoregressive model employing following auxiliary model which he obtained by using a third-order Taylor series approximation of logistic function: p

∆yt = α1 yt2−1 + α 2 yt3−1 + α 3 yt4−1 + ∑ p j ∆yt − j + et j =1

Where

yt

is demeaned series. The null of unit root can be tested employing

H 0 : α1 = α 2 = α 3 = 0 using a F-test. The relevant critical values are tabulated in Table 1 in

Pascalau(2007).

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We can summarize the testing strategy as in Figure 1.

Figure 1: Testing Strategy.

DATA AND EMPIRICAL RESULTS In this study, we will investigate the mean reversion of the real exchange rates of G7 countries by using both official market exchange rates which we obtained from the study of Reinhart and Rogoff (2004). We obtained the CPI data from OECD-Stat. The dataset used in this study is monthly and from January 1972 to December 1998. In the first step of the analysis we perform the EJ nonlinearity test. Before applying this test, we must determine the optimal lag length and the delay parameter. Following Tsay (1989), we first select the optimal lag of length then the delay parameter. We select the optimal lag length using AIC criteria and the delay parameter using the value minimizing the p value of the linearity test. Table 1: Results of Nonlinearity Test.

Country

Lag Length model

in

AR Delay Parameter: p-value EJ Test d

Choice

Canada

2

2

0.2216

Linear

France

2

6

0.2874

Linear

Germany

3

4

0.2559

Linear

Italia

10

1

0.0372 ESTAR

Japan

2

10

Uni. King.

5

4

0.1119

Linear

0.0459 LSTAR

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219

As Table 1 shows we found only the real exchange rates of Italia and United Kingdom as nonlinear. According to test results the real exchange rates of these countries follow ESTAR and LSTAR process respectively. On the other hand, remaining countries, namely Canada, France, Germany and Japan follow linear process. On the second step of the analysis, we test the stationarity of the exchange rates, for the linear series we used ADF unit root test, for the ESTAR series used Su et al. (2010) test, for the LSTAR series used Pascalau(2007) test. Table 2 shows the results of these tests. Table 2. Results of Unit Root Tests.

Country

Applied Test

Test Result

Result-1

Result-2

Canada

ADF

-2.0010

Unit Root

PPP is not valid

France

ADF

-1.9630

Unit Root

PPP is not valid

Germany

ADF

-1.7688

Unit Root

PPP is not valid

Italy

Modified KSS

-0.9007

Unit Root

PPP is not valid

Japan

ADF

-1.4035

Unit Root

PPP is not valid

United Kingdom

LSTAR Unit Root

3.8405**

LSTAR

PPP is valid

Note: The optimal lag lengths are determined using general to specific t-significance method. ** shows the significance at %5 level.

As can be seen from Table 2, we find only the real exchange rates of United Kingdom as stationary: that is follow s a LSTAR process, on the other hand, the remaining series found to be nonstationary which indicates the Purchasing Power Parity is only valid for the United Kingdom. This result can be interpreted as PPP can be used to determine the equilibrium exchange rate for United Kingdom.

CONCLUSION In this study, we tested the Purchasing Power Parity (PPP) Hypothesis for G7 countries from January 1972 to December 1998 using official exchange rates. First, we tested the linearity of the exchange rates against the alternative logistic or exponential smooth transition autoregressive process, since lots of studies find real exchange rates follow smooth transition autoregressive process. After finding two of real exchange rates as nonlinearity, we determine which of nonlinearity they follow. We find the real exchange rates of Italy and United Kingdom follow ESTAR and LSTAR respectively. Eventually, we tested the stationarity properties of the real exchange rates (RER) by using appropriate unit root tests. For the linear RERs (Canada, France, Germany, Japan), we used linear augmented Dickey-Fuller unit root test, for the RER which follow exponential smooth autoregressive process (Italy), we used modified KSS unit root test of Su et. al (2010) and for the RER which follow logistic smooth autoregressive process (United Kingdom), we used Pazcalau (2007)’s LSTAR unit root test. The findings from these unit root tests show that only RER of United Kingdom is stationary (stationary LSTAR) which indicate

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PPP is valid for only United Kingdom in the sample period and this result imply that PPP can be used to determine the equilibrium exchange rate for United Kingdom.

REFERENCES Bec, F., Salem, M.B. and Carrasco, M. (2004) “Detecting Mean Reversion in Real Exchange Rates from a Multiple Regime STAR Model” RCER Working Papers 509, University of Rochester - Center for Economic Research (RCER). Chortareas, G. E., Kapetanios, G. and Shin, Y. (2002) “Nonlinear mean reversion in real exchange rates”, Economics Letters, 77 (3), 411-417. Escribano, Á. and Jordá, O. (2001) “Testing nonlinearity: Decision rules for selecting between logistic and exponential STAR models”, Spanish Economic Review, 3(3), 193-209. Holmes, M. J. (2001) “New evidence on real exchange stationarity and purchasing power parity in less developed countries”, Journal of Macroeconomics, 23, 601–614. Su, J., Cheung, W. and Choo, A. (2010) “On the power of modified Kapetanios-Snell-Shin (KSS) tests”, Economics Bulletin, 30(3), 2028-2036. Liew, V. K., Baharumshah, A.Z. and Chong, T. T. L. (2004) “Are Asian real exchange rates stationary?”, Economics Letters, 83 (3), 313-316. Pascalau, R. (2007) “Testing for a Unit Root in the Asymmetric Nonlinear Smooth Transition Framework”, http://www.cba.ua.edu/~rpascala/documents/V4URLSTAR.pdf Reinhart, C. M. and Rogoff, K. S. (2004) “The modern history of exchange rate arrangements: a reinterpretation”, Quarterly Journal of Economics, 119(1), 1–48. Shin, D. and So, B. (2001) “Recursive mean adjustment for unit root tests” Journal of Time Series Analysis, 5, 595-612. So, B. and D. Shin (1999) “Recursive mean adjustment in time series inferences” Statistics and Probability Letters, 43, 65-73. Taylor, A. M. and Taylor, M. P. (2004) “The Purchasing Power Parity Debate”, Journal of Economic Perspectives, 18(4), 135-158. Terasvirta, T. (1994) “Specification, Estimation and Evaluation of Smooth Transition Autoregressive Models”, Journal of the American Statistical Association, 89, 208–218. Tsay, R. S. (1989) “Testing and Modeling Threshold Autoregressive Processes”, Journal of the American Statistical Association, 84(405), 231-240. Wei, S-J., Parsley, D.C. (1995) “Purchasing Power Disparity During the Floating Rate Period: Exchange Rate Volatility, Trade Barriers and Other Culprits”, NBER Working Papers 5032, National Bureau of Economic Research.