Mohsen Bahmani-Oskooeea, Ali M. Kutanb and Su Zhouc
Testing PPP in the non-linear STAR framework
ABSTRACT
Testing for stationarity of the real exchange rates is a common practice to validate the
purchasing power parity (PPP) theory. Recent studies that have applied the unit root tests to the
real exchange rates have shown that comparing the standard ADF test to the one that
incorporates non-linearity in the rates, the latter provides more support for PPP. Unlike the
previous research that applied the tests to the real bilateral exchange rates, we apply the two tests
to the real effective exchange rates of 23 OECD countries and confirm the findings of the
previous research.
JEL Classification: F31
Key Words: PPP, Non-linearity, Unit Roots.
a. The Center for Research on International Economics and the Department of Economics,
The University of Wisconsin-Milwaukee, Milwaukee, WI 53201. e-mail:
bahmani@uwm.edu
b. Department of Economics, Southern Illinois University, Edwardsville , IL 62026-1102.
email:akutan@siue.edu
c. Department of Economics, University of Texas at San Antonio, San Antonio, TX.
Email:szhou@utsa.edu
1. Introduction
The literature on the behavior of real exchange rates (RERs) has been an important
branch of the international finance and economics literature. Movements in RER are considered
vital for the competitiveness and hence the stability of trade flows, as well as testing the validity
of the purchasing power parity (PPP) hypothesis, an important cornerstone of many openeconomy macroeconomic models. Given the failure of linear models to confirm the validity of
PPP (for a recent survey, see Sarno, 2005), a number of recent theoretical studies that emphasize
the role of transaction costs have turned to nonlinear dynamic adjustment models to explain the
behavior of real bilateral exchange rates and hence to test PPP (e.g., Michael, Nobay and Peel,
1997; Taylor, Peel and Sarno, 2001; Sarno, Taylor and Chowdhury; 2004). A general finding
from these studies, based on the post-1973 floating period data and a battery of bilateral real
exchange rates, is that some selected bilateral rates can be characterized by nonlinear mean
reversion.
Our focus in this paper is on the real effective exchange rates (REERs), rather than real
bilateral exchange rates. Previous studies focus on and report nonlinear mean reversion in some
selected real bilateral exchange rates. We ask a similar but fundamentally different question: Do
REERs follow a nonlinear pattern? REER movements are critical for studies of trade flows. If
REERs are characterized by nonlinear mean reversion, this would suggest that real devaluations
will affect trade flows in a nonlinear fashion in that the REER would adjust to its long-run
equilibrium nonlinearly. In addition, a finding of stationary REERs would suggest that PPP hold
not only with respect to a country’s bilateral trading partners but also with respect to its many
trading partners. Hence, testing whether REERs follow non-stationary mean reverting behavior
is also a test of the multi-country version of PPP, rather than that of PPP based on bilateral
1
trading partners. Unit root tests on REERs therefore reflect a much stronger condition for PPP to
hold.1
To test the nonlinearity of REERs, this paper employs a new testing procedure suggested
by Kapetanios, Shin and Snell (hereafter, KSS) (2003). They have developed a new technique
for the null hypothesis of a unit root against an alternative of nonlinear stationary smooth
transition autoregressive (STAR) process. Assuming REERs follow nonlinear stationary
processes, the alternative hypothesis of the augmented Dickey–Fuller (ADF) unit root tests based
on the linear model will be misspecified. KSS (2003) have illustrated that their tests are more
powerful than the ADF tests. Using quarterly bilateral real exchange rates with the US dollar and
real interest rates for the 1957-2000 period for some selected OECD countries, they have shown
some applications of their tests. Chortareas and Kapetanios (2004), Hasan (2004) and Venus et.
al (2004) have recently applied the KSS tests to the bilateral real exchange rates of Japan,
Turkey, India and Asia, respectively.
In this paper, we apply the new tests to REERs of 23 developed economies, which
includes 15 founding members2 of the European Union (EU) and 8 other developed economies
(Australia, Canada, Iceland, Japan, New Zealand, Norway, Switzerland, and the US). Our sample
focuses on the post Breton Woods floating period. We use monthly data from 1973:01 through
2005:09. We also carry out our analysis for some sub-samples to capture the impact of the
Maastricht Treaty and the introduction of the euro on real REER behavior. Furthermore, since
the KSS test is an extension of the ADF test to account for non-linearity, the null is still the unit
1
For testing PPP in the context of REER, see Corbae and Ouliaris (1991), Bahmani-Oskooee and Rhee (1992) and
Bahmani-Oskooee (1995).
2
15 EU members include 12 Euro-zone countries, namely, Austria, Belgium, Denmark, Finland, France, Germany,
Greece, Ireland, Italy, Luxembourg, the Netherlands, Portugal, Spain, and Denmark, Sweden, and the UK. We also
report the results for EU15 and Euro-zone as two individual groups.
2
root versus an alternative of stationarity. To this end, we briefly explain the KSS test in Section 2
and report the results in Section 3. Section 4 concludes.
2. The Methods
Since the introduction of cointegration and error-correction modeling techniques,
researches have been required to establish the degree of integration of every time-series variable.
Perhaps, the most common test is the ADF test in which the null is non-stationarity of a variable
against an alternative of stationarity. Recently, Kapetanios, Shin, and Snell (2003) expand the
standard ADF test by keeping the null hypothesis as nonstationarity in a time series variable
against the alternative of a nonlinear but globally stationary process. They demonstrate that the
new test could be based on the following exponential smooth transition autoregressive (ESTAR)
specification:
yt   yt 1[1 - exp(- yt21 )] + t
(1)
where yt is the de-meaned or de-trended series of interest and  t is an i.i.d. error with zero mean
and constant variance.
The null hypothesis of nonstationarity with the KSS test procedure is H 0:  = 0 against
the alternative of H1:  > 0. Because  in (1) is not identified under the null, it is not feasible to
directly test the null hypothesis. KSS thus reparameterize (1) by computing a first-order Taylor
series approximation to specification (1) to obtain the auxiliary regression specified by (2)
below:
yt   yt31 + error
(2)
For a more general case where the errors in (2) are serially correlated, regression (2) is extended
to
3
yt 
p
  j yt  j   yt31  error
j 1
(3)
with the p augmentations in order to correct for serially correlated errors. The null hypothesis to
be tested with either (2) or (3) is H0:  = 0 against the alternative of H1:  < 0. KSS show that the
t-statistic for  = 0 against  < 0, i.e., tNL, does not have an asymptotic standard normal
distribution and they tabulate the asymptotic critical values of the tNL statistics via stochastic
simulations.
We estimate the tNL statistics with regression (3) and refer them to as tNL11 and tNL12,
which are the KSS test statistics for the de-meaned and the de-trended data, respectively.
Similarly, tADF1 and tADF2 are the standard ADF test statistics for the null of nonstationarity without
and with a trend, respectively, using the model with augmentations.3 The de-meaned or detrended data are obtained by first regressing each series on a constant or on both a constant and a
time trend, respectively, and then saving the residuals.
3. The Results
We report the results of the KSS test along with the ADF statistic in Table 1 and 2 for EU
and non-EU countries, respectively. In Table 1, the tests are applied to 17 EU economies
(including 15 EU founding countries, 15 EU countries together as a group, and 12 Euro-zone
countries as another group). Table 2 reports the results for 8 non-EU industrial countries.4
Tables 1 and 2 go about here
3
Following the suggestion of KSS (2003, p. 365), the number of augmentations p for either the ADF tests or the
KSS tests is selected using the significance testing procedure in Ng and Perron (1995). The maximum number of p
was set to be 24 mostly because of the data being monthly.
4
The real effective exchange rate data for all countries come from the OECD Economic Indicators.
4
As indicated before, four statistics are reported based on the models with augmentations.
tNL11 and tNL12 are the KSS test statistics for the de-meaned and the de-trended data, respectively,
while tADF1 and tADF2 are the standard ADF test statistics for the null of nonstationarity of the
variable without and with a trend, respectively. The results show that the KSS tests reject the null
of nonstationarity more often than the ADF tests. For the full sample period 1973:1 – 2005:9,
the ADF tests reject the null in seven cases (EU and Non-EU combined) while the KSS tests
reject the null for 12 out of 25 cases at the 10% level of significance. There is more evidence for
the stationarity of the REERs for longer sample period that includes data from more recent years
as compared to the shorter periods. For period 1973:1-1998:12, the ADF and the KSS tests reject
the null for 5 and 8 cases, respectively. However, for period 1973:1-1991:12 they reject the null
of unit root in only 2 and 7 out of 25 cases respectively. Again, even for the shorter period the
KSS rejects the null more often than the ADF.
The evidence for the convergence toward PPP for the EU countries seems to be stronger
for the samples including the years when the Maastricht Treaty is in effect.
The null of
nonstationary of REER is rejected by the ADF or KSS tests for 7 and 9 EU countries for the
periods 1973:1-1991:12 and 1973:1-1998:12, respectively. Both ADF and KSS tests reject the
null hypothesis for the EU15 or Euro-zone group for the full sample period but none of the two
tests was able to do so for period 1973:1-1991:12.
Although the results imply increased convergence toward PPP for the EU group in recent
years, they do not show more evidence of stationarity of the REER for EU countries than for
non-EU countries. The test statistics in Tables 1 and 2 for the full sample period indicate that the
null of nonstationary of REER is rejected for 8 or 5 out of 15 (i.e., 53% or 33% of) EU founding
members at the 10% or 5% significance level, respectively. Yet, it can be rejected for five out of
5
eight (i.e., 63% of) non-EU industrial countries (including three European countries: Iceland,
Norway, and Switzerland, and two Pacific Rim countries: Australia and New Zealand) at the 5%
significance level.
4. Summary and conclusion
When a new test for unit root is introduced, its application to real exchange rates as a way
of testing the well-known PPP theory becomes a common practice. Previous studies that applied
the standard ADF test to detect stationarity in real exchange rates were not successful in
supporting PPP. A recent test that extends the standard ADF test by introducing non-linearity in
a time-series variable has gained momentum in providing relatively more support for PPP.
However, these studies have applied the new test to the real bilateral exchange rates. Since a
country normally has more than one trading partner, adjustment of the prices to changes in
bilateral exchange rates of most trading partners may lubricate the adjustment toward long-run
equilibrium between prices and exchange rates. Thus, using the real effective exchange rate in
testing PPP has more appeal than using the bilateral rate.
In this paper we apply the standard ADF test as well as the new test which allows nonlinearity in the testing procedure to the real effective exchange rate of 23 industrial countries.
The new test provides support for PPP in more countries than the standard ADF test, a finding
that is in line with previous research. The results also suggest that, while more REERs of
industrial countries tend to be stationary when the sample period is extended to the recent years
along with enhanced economic integration among the countries in the study, whether or not
being a member of an economic union does not make a notable difference for the behavior of a
country’s real effective exchange rate.
6
References
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Parity: Experiences of 19 Industrial Countries. Economic Notes 24, 239-250.
Bahmani-Oskooee, Mohsen and Hyun-Jae Rhee, 1992. Testing for Long-Run Purchasing Power
parity: Examination of Korean Won. International Economic Journal 6, 93-103.
Chortareas, G and Kapetanios, G., 2004. The Yen Real Exchange Rate may be Stationary after
all: Evidence from Non-linear Unit-root Tests. Oxford Bulletin of Economics and
Statistics 66(1), 113-131.
Corbae, Dean and Sam Ouliaris, 1991. A Test of Long-run Purchasing Power Parity Allowing for
Structural Breaks. The Economic Record 67, 26-33.
Hasan, M.S., 2004. Univariate time series behaviour of the real exchange rate: Evidence from
Colonial India, Economics Letters 84(1), 75-80.
Kapetanios, G., Shin, Y., Snell, A., 2003. Testing for a unit root in the nonlinear STAR
framework, Journal of Econometrics 112(2), 359-379.
Michael, P., Nobay, A.R. and Peel, D.A., 1997. Transaction Costs and Nonlinear Adjustment in
Real Exchange Rates: An Empirical Investigation. Journal of Political Economy 105(4),
862–879.
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methods for the selection of truncation lag. Journal of American Statistical Association
90(429), 268-281.
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Sarno Lucio, Taylor, Mark P. and Chowdhury, Ibrahim, 2004. Nonlinear dynamics in deviations
from the law of one price: a broad-based empirical study. Journal of International Money
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Toward a solution to the Purchasing Power Parity puzzles. International Economics
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7
Table 1.
Unit root test results for EU countries
tADF1
tNL1
tADF2
tNL2
tADF1
tNL1
tADF2
tNL2
tADF1
tNL1
tADF2
tNL2
Sample period:
1973M1 – 2005M9
1973M1 – 1991M12
1973M1 – 1998M12
Austria
-2.02
-2.46
-1.91
-2.49
-1.38
-1.24
-2.32
-2.83
-1.62
-2.09
-2.49 -3.13*
Belgium
-2.61* -2.20
-2.78 -3.17*
-2.18 -2.67* -2.49 -3.46**
-2.43
-2.39
-2.47 -3.16*
Denmark
-1.83
-1.86
-2.80
-3.00
-2.11 -3.12** -2.24
-2.72
-1.78
-2.28
-2.43
-2.55
Finland
-1.56
-1.85
-2.48
-1.36
-2.46
-1.63
-2.75
-2.80
-2.60 -2.76* -2.82
-2.19
France
-2.86** -2.40 -3.80** -4.07**
-2.62* -2.43 -3.18* -3.36*
-3.22** -2.56 -3.66** -3.26
Germany
-2.86** -3.84** -3.02 -3.37*
-2.07
-2.47
-1.91
-2.67
-2.31 -3.16** -2.20
-2.77
Greece
-1.33
-2.23
-1.88
-1.49
-1.83
-1.73
-1.31
-2.25
-1.48
-1.16
-1.36
-1.08
Ireland
-1.86 -3.62** -2.18
-3.13
-0.92
-1.97
-2.83
-2.68
-1.51 -2.66* -1.55
-2.49
Italy
-2.16 -3.04** -2.15
-3.03
0.53
-0.46
-2.62
-2.79
-1.87 -2.87* -1.87
-3.08
Luxembourg
-2.01
-1.76
-2.00
-2.69
-1.23
-1.84
-1.53 -3.33*
-1.86
-2.05
-2.04
-2.90
Netherlands
-2.54
-2.45
-2.94 -3.24*
-2.19
-2.64
-3.22 -3.17*
-2.30
-2.18 -3.22* -3.21*
Portugal
-0.97
-1.54
-2.44 -4.00**
-2.16 -3.23** -1.39
-2.55
-1.18
-2.63
-1.95 -3.74**
Spain
-2.50
-1.92
-2.62
-1.80
-1.09
-0.48
-1.68
-1.76
-2.28
-1.61
-2.33
-1.90
Sweden
-1.64
-2.00 -3.55** -2.60
-1.87
-2.22
-1.44
-1.50
-1.95
-2.14
-2.86
-0.25
UK
-2.37
-2.56
-3.05
-2.28
-2.03
-2.19
-2.31
-1.89
-2.87** -2.93
-2.99
-2.45
EU 15
-2.71* -3.03** -2.74 -3.29*
-1.69
-1.67
-1.38
-1.86
-2.34
-2.34
-2.31
-2.52
Euro zone
-2.76* -2.82* -2.89 -3.34*
-2.00
-1.93
-1.73
-2.25
-2.63* -2.53
-2.59
-2.38
Notes: tADF1 and tADF2 are the standard ADF test statistics for the null of nonstationarity of the variable in the study without and
with a trend, respectively, in the model for testing. tNL1 and tNL2 are the KSS test statistics for the de-meaned and the detrended data, respectively, using the models with augmentations. The 10% and 5% asymptotic critical values are -2.57 and
-2.86 for tADF1 respectively, and are -3.12 and -3.41 for tADF2 respectively. The 10% and 5% asymptotic critical values for
tNL1 are -2.66 and -2.93 respectively and those for tNL2 are -3.13 and -3.40 respectively, taken from Kapetanios et al. (2003, p.
364). * and ** denote rejection of the null hypothesis at the 10% and 5% significance levels, respectively.
Table 2.
Unit root test results for non-EU countries
tADF1
tNL1
tADF2
tNL2
Sample period:
1973M1 – 2005M9
-2.11 -3.26** -2.16
-1.76
Australia
-1.79
-1.92
-1.81
-2.16
Canada
-2.98** -3.92** -3.17* -2.68
Iceland
-1.94
-2.59
-1.73
-2.78
Japan
New Zealand -4.09** -4.22** -3.97** -3.94**
-2.56 -4.88** -2.68 -3.99**
Norway
Switzerland -2.97** -3.05** -3.55** -4.20**
-2.34
-2.33
-2.33
-2.30
US
See notes to Table 1.
tADF1
tNL1
tADF2
tNL2
1973M1 – 1991M12
tADF1
tNL1
tADF2
tNL2
1973M1 – 1998M12
-1.64
-2.08
-2.59
-2.45
-1.52
-2.81*
-3.24*
-2.73
-1.84
-2.16
-1.64
-2.68
-0.59
-0.36
-1.67
-2.23
-4.85** -4.00** -4.84** -4.11**
-1.21
-1.43
-3.09** -3.32**
-2.22
-1.88
-1.93
-3.66**
-3.08
-3.41**
-2.23
-2.12
-2.73* -3.42** -3.76** -3.62**
-1.74
-2.49
-2.85
-4.86**
-3.90** -4.36** -3.88** -4.23**
-2.56
-2.69*
-2.15
-2.64
-2.68* -4.28** -3.58** -3.68**
-2.77* -2.99** -3.78** -4.09**
-1.52
-1.92
-1.92
-1.31
-1.93
-1.99
-2.02
-2.03