The Kapetanios, Shin, and Snell (KSS, 2003) test for a nonlinear unit root is used to study purchasing power parity using Taylor's extensive data set, d to include recent exchange rate and price level data. The results i) indicate that PPP holds with respect to the US dollar for most countries and ii) are more supportive of PPP than those from standard linear unit root tests.

Frederick Wallace

English

Research Papers in Economics

Nonlinear unit root tests of PPP using long-horizon data 
Frederick Wallace
Universidad de Quintana Roo
Abstract
The Kapetanios, Shin, and Snell (KSS, 2003) test for a nonlinear unit root is used to study
purchasing power parity using Taylor’s extensive data set, updated to include recent
exchange rate and price level data. The results i) indicate that PPP holds with respect to the
US dollar for most countries and ii) are more supportive of PPP than those from standard
linear unit root tests.
I thank Alan Taylor for providing the data from his study. Errors are mine, of course. Financial support from the National
Council for Science and Technology (CONACYT) of Mexico is gratefully acknowledged.
Citation: Wallace, Frederick, (2008) "Nonlinear unit root tests of PPP using long-horizon data." Economics Bulletin, Vol. 6,
No. 33 pp. 1-8
Submitted: July 17, 2008.  Accepted: August 25, 2008.
URL: http://economicsbulletin.vanderbilt.edu/2008/volume6/EB-08F30066A.pdf1. Introduction 
As a long run equilibrium condition of open economy macro models, the 
hypothesis of purchasing power parity (PPP) has been the focus of much empirical 
work. The absolute version of PPP indicates that the price of a market basket of (traded) 
goods is the same everywhere in terms of a common currency. Deviations from PPP are 
corrected by goods arbitrage. A partial list of techniques used in such empirical work 
includes single equation unit root tests, variance ratio tests, cointegration studies, and 
panel unit root tests. The voluminous empirical evidence is mixed. Sarno and Taylor 
(2002) provide a thorough review of the PPP literature.  
In an influential paper, Taylor (2002) concludes that long run purchasing power 
parity held during the past century in the sample of countries he studied and suggests 
that it is time to focus attention on short-run, real exchange rate adjustment rather than 
PPP. His findings are particularly important because his data set includes more than one 
hundred years of annual observations ending in 1996, hence provides a long horizon 
perspective lacking in most other studies. Subsequently, numerous researchers have 
applied different econometric methods to Taylor’s data set. Lopez, Murray, and Papell 
(LMP, 2005) criticize Taylor’s approach for the use of suboptimal lag length selection 
in the unit root tests. Employing unit root tests with optimally selected lags to Taylor’s 
data, LMP find support for PPP in the data for just nine of sixteen developed countries.
1  
Bahmani-Oskooee, Kutan, and Zhou (BKZ, 2007) apply the Kwiatkowski, 
Phillips, Schmidt, Shin (KPSS, 1992) test for stationarity to Taylor’s data and conclude 
that the evidence supports PPP for eighteen of the twenty countries in the data set. They 
base their conclusions on two different versions of the KPSS test, one in which the null 
is a stationary real exchange rate (RER) and the other in which the RER is trend 
stationary. Furthermore they use these two versions with two different real rates, one 
with respect to the US dollar and the other against a world basket of currencies. Thus 
data for each country are subject to four different tests. If any one of these produces 
results indicating a stationary series, BKZ cite this as evidence supportive of PPP. 
Indeed, for five of the eighteen countries they mention as displaying results supportive 
of PPP, only one of the four test versions supports PPP. A skeptic of purchasing power 
parity using the same criterion could interpret their evidence as not indicating PPP 
because stationarity is rejected for fifteen countries at least one of the four empirical 
specifications. 
In other work with the Taylor data Bahmani-Oskooee, Kutan, and Zhou (2006) 
apply the nonlinear unit root test of Kapetanios, Shin, and Snell (2003) to the Taylor 
data. They find rejections of the unit root null, that is, evidence in favor of purchasing 
power parity, for sixteen of nineteen countries when the US dollar is the numeraire 
currency. Instead of relying on unit root tests Wallace and Shelley (2006) apply the 
Fisher-Seater test with bootstrapped errors to the Taylor data and conclude that PPP 
holds for at least twelve of nineteen countries with respect to the United States. 
In this study, as in that of BKZ (2006) the Kapetanios, Shin, and Snell (KSS) 
nonlinear unit root test is used to study purchasing power parity in the Taylor data. But 
there are three important differences from the BKZ study. First, the Taylor data are 
updated to 2007 (2006 in the case of Argentina). The BKZ study is through 1996 as in 
Taylor’s original study. Second, Taylor’s data set actually includes information for three 
additional countries, Chile, Greece, and New Zealand although results for these 
countries were not published in his study, presumably because there are fewer than one 
                                                 
1 They eliminate Argentina, Brazil and Mexico (included in Taylor) from their study.   
 
 
2
hundred years of observations.
2 Findings for these three countries are reported in this 
paper. Finally, the results of the KSS tests are compared to those from three linear unit 
root tests; the augmented Dickey-Fuller (ADF), the Elliott, Rothenberg, and Stock 
(ERS, 1996) test which Taylor applied, and the KPSS test as used in BKZ. 
2. Data and Methodology 
The Taylor data set consists of annual observations on logged nominal exchange 
rates and logged price indexes (consumer price index) for the twenty-three countries 
listed in Table I. For most countries the data set is updated through 2007 using IMF, 
OECD, and Federal Reserve Board data. Recent data for Chile are from the country’s 
Central Bank. In the case of Argentina, the data for 1997-2006 from the country’s 
statistical agency are added to the data set.  The nominal exchange rate is measured as 
units of foreign currency per US dollar. For each country except Chile, Greece, and 
New Zealand the data span more than 100 years.  
KSS begin with a univariate, smooth transition autoregressive model of order one 
in which the transition function is exponential. Using a Taylor series approximation, 
they derive their test equation given by 
  t
j
i
i t i t t u y y y + Δ + = Δ ∑
=
− −
1
3
1 φ δ  (1) 
In this study, yt is the demeaned (or demeaned and detrended) log real exchange rate. 
The coefficient of interest for the unit root test is δ. As shown in equation (1) lags of the 
dependent variable may be included to eliminate serial correlation. Under the null of a 
(nonlinear) unit root, δ=0 while the alternative of a stationary series is δ<0. As with 
linear unit root tests, failure to reject the null implies a nonstationary real exchange rate 
thus does not support PPP. Rejection of the null suggests a stationary real exchange rate 
and evidence in favor of PPP.
3 
It is fairly common to conduct one set of unit root tests on the real exchange rate 
including only a constant in the estimations and then another including a constant and 
trend in the estimations. The possibility of Balassa-Samuelson effect is usually cited as 
justification for the inclusion of a trend. For comparability to other studies this practice 
is followed in this paper although a theoretical justification for why the Balassa-
Samuelson effect would yield a trended real exchange rate is lacking in the literature. 
3. Results 
To address serial correlation in the ADF and ERS tests lags of Δyt are added (if 
needed) with the Schwarz information criterion used to determine lag length. In the 
nonlinear versions Lagrange multiplier tests are used to fix the lag length. Each 
nonlinear unit root test is estimated with zero to eight lags of the dependent variable. 
The shortest lag length which yields an obs.R
2 statistic having a marginal significance 
level of .2 or better is selected as the number of lags to use in the estimation. For a few 
countries this criterion was not satisfied and the nonlinear tests were conducted with 
additional, nine to twelve lags. The added lags eliminated autocorrelation in the case of 
Norway with nine lags used in the nonlinear test on the demeaned real exchange rate.  
However, in the case of tests on both the demeaned and demeaned-detrended real 
exchange rate for Germany, not a single specification using from zero to twelve lags 
                                                 
2 Table I lists the countries and the period for which data are available. Although the earliest observations 
are for England in 1850, use of the dollar as numeraire implies that all empirical estimations begin no 
earlier than 1870. 
3 Strictly speaking rejection of the null implies that the real exchange rate is mean reverting, evidence 
supportive of the relative version of PPP but not sufficient for absolute PPP. In the absence of models in 
which relative, but not absolute, PPP holds in the long run, most macroeconomists, however, probably 
regard evidence of stationarity as indicative of absolute PPP.  
 
 
3
resolved the problem. In this instance, inspection of the graphs of residuals from 
estimating equation (1) with different lags revealed exceptionally large (absolute) errors 
during the years immediately following World War I. During this period, of course, 
Germany experienced several years of hyperinflation. Restricting the sample period to 
1928-2007 and again estimating equation (1) with zero to eight lags produces a much 
better behaved specification that satisfies the criterion described above. For Germany, 
this means one lag of the dependent variable eliminates serial correlation in the 
nonlinear unit root test. The same problem occurs in tests using the demeaned and 
detrended real exchange rates for Denmark, Japan, and Norway. Inspection of residual 
graphs is used to select restricted sample periods and the above described criterion again 
applied to search for a specification free of serial correlation. In the case of Japan, the 
period from 1941-1948 (World War II and its immediate aftermath) display relatively 
large (in absolute terms) residuals. No specification free of serial correlation could be 
found which included the post WW II period. Consequently, only results for 1885-1940 
are reported for Japan.
4 
Table II reports the t-statistics for the nonlinear tests on demeaned real exchange 
rates, that is estimation of equation (1) for each country with sufficient lags to eliminate 
autocorrelation. For comparison, the results from the ADF, ERS, and KPSS linear tests 
are also provided. The final row of the table shows the number of statistics indicating 
stationarity, thus evidence of PPP, of the real exchange rates. Critical values for the 
nonlinear tests are from Table I of KSS. As can be observed from the table, the 
nonlinear test results are supportive of purchasing power parity for eighteen of the 
twenty-two countries in the study. The linear test results show fewer significant t-stats 
(insignificant t-stats in the case of the KPSS test) thus indicate less support for PPP. In 
some cases, the linear tests present conflicting results. For example, in the case of 
Sweden both the ADF and ERS tests indicate rejection of the unit root null hence are 
supportive of PPP but the KPSS test statistic indicates rejection of the stationarity null, 
thus is not supportive of PPP. 
Results are not notably different when the nonlinear tests are carried out on 
demeaned and detrended data (Table III). The only differences are for Denmark and 
Switzerland which do not display evidence of PPP with demeaned data but do for 
demeaned and detrended data; and Germany and Japan whose real exchange rates 
appear stationary when demeaned but not so when demeaned and detrended. But, note 
that the sample periods for Japan are different in the two versions of the nonlinear tests 
as previously explained. Again, the linear tests which include a constant and trend in the 
estimation equations present mixed results and weaker evidence for PPP than observed 
in the nonlinear estimations. 
4. Conclusions 
The Taylor data are the most extensive long horizon data available on exchange 
rates and price levels with more than a century of annual observations for most included 
countries. Long span data are important for testing long run propositions such as PPP, 
particularly if mean reversion is slow as discussed in Rogoff (1996). When applied to 
the real exchange rates constructed from the Taylor data, the linear unit root tests yield 
mixed results but the nonlinear, KSS tests produce strong evidence in favor of 
purchasing power parity. The results of this study reinforce Taylor’s conclusion; PPP 
has generally held over the twentieth century for the countries in the sample. 
                                                 
4 As mentioned previously, there is no theoretical justification for including a trend in PPP tests. Absent 
such a rationale, one should probably have greater confidence in the results with demeaned data.  
 
 
4
Table I-Countries and Period Coverage 
Country Taylor  data-years 
of coverage 
Updated to 
Argentina 1884-1996  2006 
Australia 1870-1996  2007 
Belgium 1870-1996  2007 
Brazil 1880-1996 2007 
Canada 1870-1996  2007 
Denmark 1880-1996  2007 
Finland 1881-1996  2007 
France 1880-1996 2007 
Germany 1880-1996  2007 
Italy 1880-1996  2007 
Japan 1885-1996 2007 
Mexico 1886-1996  2007 
Netherlands 1870-1996  2007 
Norway 1870-1996  2007 
Portugal 1890-1996  2007 
Spain 1880-1996 2007 
Sweden 1880-1996  2007 
Switzerland 1892-1996  2007 
UK 1850-1996  2007 
US 1870-1996  2007 
Additional countries in the data set but for which 
results are not reported in Taylor (2002) 
Chile 1913-1996 2007 
Greece 1948-1996 2007 
New Zealand  1948-1996  2007 
  
 
 
5
Table II-Unit Root Tests for Mean Stationary Real Exchange Rates 
 
Country KSS 
demeaned 
ADF 
constant 
ERS 
constant 
KPSS 
constant 
Argentina -8.242**  5.035**  -4.900**  .107 
Australia -1.804  -2.027  -2.143*  .745** 
Belgium -10.274**  -4.745**  -2.810*  .793** 
Brazil -5.223**  -2.637  -2.248*  .154 
Canada -1.907  -2.531  -1.733  .798** 
Denmark -1.420  -2.148  -2.066  .695** 
Finland  -7.587**  -6.259** -6.109**  .164 
France  -3.564**  -4.048** -2.028*  .896** 
Germany
a  -3.324*  -2.892 -2.489*  .173 
Italy  -4.843**  -4.116** -4.130**  .077 
Japan  -3.225*  -2.405 -1.006  1.127** 
Mexico  -12.592**  -6.021** -3.148**  .868** 
Netherlands  -3.162*  -2.862 -2.501* .501* 
Norway  -3.581**  -3.632** -2.271**  .509* 
Portugal  -6.020**  -2.976* -1.897  .362 
Spain  -4.276**  -3.342* -2.906**  .281 
Sweden  -5.098**  -4.008** -3.165**  .551* 
Switzerland  -2.887  -1.632 -0.802 .981** 
UK  -4.279**  -3.243* -2.800**  .335 
Additional Countries 
Chile  -3.073*  -1.735 -1.076  1.092** 
Greece  -8.808**  -7.665** -0.585  .218 
New Zealand  -4.301**  -3.827** -3.941**  .596* 
# stationary  18  11 14  9 
* Significant at 5% 
** Significant at 1% 
 
  
 
 
6
Table III-Unit Root Tests for Trend Stationary Real Exchange Rates 
 
Country KSS 
detrended 
ADF constant, 
trend 
ERS constant, 
trend 
KPSS 
constant, trend 
Argentina  -8.262**  -5.014** -5.010**  .105 
Australia  -2.636  -3.080 -2.851 .174* 
Belgium  -9.705**  -5.603** -5.641**  .070 
Brazil  -4.245**  -2.695 -2.704  .091 
Canada  -2.569  -3.606* -2.067  .211* 
Denmark
a  -3.411*  -3.495* -2.926  .160* 
Finland  -7.531**  -6.357** -6.408**  .056 
France  -7.123**  -4.989** -4.924**  .212* 
Germany
b  -3.067  -2.916 -2.895  .119 
Italy  -4.824**  -4.120** -4.143**  .062 
Japan
c  -3.049  -2.502 -2.556 .204* 
Mexico  -11.676**  -7.038** -7.014**  .065 
Netherlands  -3.583*  -3.441 -3.481* .176* 
Norway
a  -4.755**  -3.349 -3.325 .193* 
Portugal  -5.689**  -2.923 -2.811 .251** 
Spain  -4.471**  -3.273 -3.232* .219** 
Sweden  -4.905**  -4.546** -4.562**  .064 
Switzerland  -5.056**  -3.911* -3.912**  .130 
UK  -4.620**  -3.328 -3.013*  .130 
Additional Countries 
Chile  -4.039**  -3.459* -3.416*  .185* 
Greece  -8.736**  -7.436** -0.858  .161* 
New Zealand  -3.645*  -4.519** -4.512**  .035 
# stationary  18  13 13 11 
* Significant at 5% 
** Significant at 1% 
a Estimation period 1915-2007 
b Estimation period 1928-2007 
c Estimation period 1885-1940  
 
 
7
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A century of purchasing power parity: Further evidence,

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Are Asian real exchange rates stationary?

Efficient tests for an autoregressive unit root,

State of the art unit root tests and purchasing power parity,

Stationarity of Asian-Pacific real exchange rates.

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Testing the null hypothesis of stationarity against the alternative of a unit-root,

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Nonlinear unit root tests of PPP using long-horizon data

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Nonlinear unit root tests of PPP using long-horizon data

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