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Purchasing power parity with nonlinear threshold unitroot testTsangyao Chang a , Chi-Wei Su b c & Yu-Shao Liu a ba Department of Finance, Feng Chia University, Taichung, Taiwanb Department of Finance, Xiamen University, Xiamen, PR Chinac Department of International Business, Tamkang University, Taipei, TaiwanPublished online: 15 Sep 2011.

To cite this article: Tsangyao Chang , Chi-Wei Su & Yu-Shao Liu (2012) Purchasing power parity with nonlinear threshold unitroot test, Applied Economics Letters, 19:9, 839-842, DOI: 10.1080/13504851.2011.607110

To link to this article: http://dx.doi.org/10.1080/13504851.2011.607110

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Purchasing power parity with

nonlinear threshold unit root test

Tsangyao Changa, Chi-Wei Sub,c,* and Yu-Shao Liua,b

aDepartment of Finance, Feng Chia University, Taichung, TaiwanbDepartment of Finance, Xiamen University, Xiamen, PR ChinacDepartment of International Business, Tamkang University, Taipei, Taiwan

This study applies a simple and powerful nonlinear threshold unit root testproposed by Caner and Hansen (2001) to test the validity of long-runPurchasing Power Parity (PPP) in a sample of nine East Asian countries.The empirical results indicate that PPP holds true for more than half ofthese nine East Asian countries under study, and the adjustment towardsPPP is found to be nonlinear.

Keywords: purchasing power parity; East Asian countries; nonlinearthreshold unit root test

JEL Classification: C22; F31

I. Introduction

Purchasing Power Parity (PPP) is a cornerstone ofmany theoretical models in international finance.

PPP states that the exchange rates between currenciesare in equilibrium when their purchasing power is the

same in each of the two countries. This means that theexchange rate between any two countries should equal

the ratio of two currencies’ price level of a fixed basketof goods and services. The basic idea behind the PPP

hypothesis is that since any international goods-market arbitrage should be traded away over time,

we should expect the Real Exchange Rate (RER) toreturn to a constant equilibrium value in the long run.

Studies on this issue are critical not only for empiricalresearchers but also for policymakers. In particular, a

nonstationary RER indicates that there is no long-runrelationship between nominal exchange rate and

domestic and foreign prices, thereby invalidating thePPP. As such, PPP cannot be used to determine the

equilibrium exchange rate, and an invalid PPP alsodisqualifies the monetary approach from exchange

rate determination, which requires PPP to hold true.Empirical evidence on the stationarity of RER is

abundant but inconclusive thus far. For details onprevious studies, please refer to the works of Taylor(1995), Rogoff (1996), MacDonald and Taylor (1992),

Taylor and Sarno (1998), Sarno and Taylor (2002),Taylor and Taylor (2004) and Lothian and Taylor(2000, 2008), who have provided in-depth informationon the theoretical and empirical aspects of PPP and

RER. Recently, there has been a growing consensusthat the RER exhibits nonlinearities, and conse-quently, conventional unit root tests such as the

Augmented Dickey–Fuller (ADF, 1981) test havelow power in detecting themean reversion of exchangerate. A number of studies have provided empiricalevidence on the nonlinear adjustment of exchange

rate.1 However, the finding of nonlinear adjustmentdoes not necessarily imply nonlinear mean reversion

*Corresponding author. E-mail: cwsu@mail.tku.edu.tw1 Reasons for the nonlinear adjustment are the presence of transaction costs that inhibit international goods arbitrage andofficial intervention in the foreign exchange market may be such that nominal exchange rate movements are asymmetric (seeTaylor and Peel, 2000; Taylor, 2004; Juvenal and Taylor, 2008). Kilian and Taylor (2003) also suggested that nonlinearity mayarise from the heterogeneity of opinion in the foreign exchangemarket concerning the equilibrium level of the nominal exchangerate: as the nominal rate takes on more extreme values, a great degree of consensus develops concerning the appropriatedirection of exchange rate moves, and traders act accordingly.

Applied Economics Letters ISSN 1350–4851 print/ISSN 1466–4291 online # 2012 Taylor & Francishttp://www.informaworld.com

DOI: 10.1080/13504851.2011.607110

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Applied Economics Letters, 2012, 19, 839–842

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(stationarity). As such, stationarity tests based on anonlinear framework must be applied.This empirical study contributes to this line of

research by determining whether PPP holds true in asample of nine East Asian countries, and whether theadjustment process towards its equilibrium is in anonlinear way. We test the nonstationarity of RERfor nine East Asian countries using a simple andpowerful nonlinear threshold unit root test of Canerand Hansen (2001). The major advantage of thisapproach is that it allows us to simultaneously inves-tigate nonstationarity and nonlinearity. With this, thecurrent research hopes to fill the existing gap in theliterature. To the best of our knowledge, this study isthe first, to date, that utilizes the nonlinear ThresholdAutoregressive (TAR) unit root test in nine East AsianRERs. We find that the TAR unit test strongly rejectsthe unit root process for more than half of these EastAsian countries examined, while the traditional unitroot tests such as the ADF, Phillips and Perron (PP,1988) and Kwiatkowski et al. (KPSS, 1992) did notlead to rejection. Furthermore, the adjustment processtowards its equilibrium for most of these East Asiancountries is nonlinear.This article is organized as follows: Section II pre-

sents the data used in our study. Section III brieflydescribes the TAR test and our empirical results.Section IV concludes this article.

II. Data

Our empirical analysis covers a sample of nine EastAsian countries: China, Hong Kong, Indonesia,Japan, Korea, Malaysia, Philippines, Singapore andThailand. Monthly data are employed in this study,and the time span is from January 1986 to October2009. All Consumer Price Indices (CPIs; based on2000 = 100) and nominal exchange rates relative tothe US dollar data are taken from the InternationalMonetary Fund’s International Financial StatisticsCD-ROM. Testing for PPP against the United Statesis based on the argument that internal foreign exchangemarkets are mostly dollar dominated. In addition, theUnited States is themajor trading partner for these nineEast Asian countries.

III. Methodology and Empirical Results

Caner and Hansen’s (2001) threshold unit root test

Following the work of Caner and Hansen (2001), weadopt a two-regime TAR(k) model with an autore-gressive unit root as follows:

�rt ¼ �01xt�1I Z�lf g þ �02xt�1I Zt>lf g þ et;

t ¼ 1; . . . ;T ð1Þ

where rt is the RER for t ¼ 1, 2,:::,T, xt�1 ¼ ðrt�1,v0t, �rt�1, . . . , �rt�kÞ0; I �f g is the indicator function; etis an i.i.d. disturbance; Zt�1 ¼ rt�1 � rt�m is the

threshold variable; m represents the delay parameter;

and 1 � m � k, vt is a vector of exogenous variables

including an intercept and possibly a linear time trend.

The threshold value l is unknown and takes the values

in the compact interval l 2 � ¼ ½l1, l2�, where l1 andl2 are selected according to PðZt � l1Þ ¼ 0:15 and

PðZt � l2Þ ¼ 0:85, respectively. The components of

�1 and �2 can be partitioned as follows:

�1 ¼r1b1a1

0@

1A; �2 ¼

r2b2a2

0@

1A ð2Þ

where r1 and r2 are scalar terms; b1 and b2 have the

same dimensions as vt; and a1 and a2 are k-vectors.

Thus ðr1, r2Þ are the slope coefficients on rt�1;ðb1, b2Þ are the slopes on the deterministic compo-

nents; and ða1, a2Þ are the slope coefficients on

ð�rt�1, . . . , �rt�kÞ in the two regimes.The threshold effect in Equation 1 has the null

hypothesis of H0: �1 ¼ �2, which is tested using the

familiar Wald statistic:WT ¼ WTðl̂Þ ¼ supl2�WTðlÞ.The stationarity of the process rt can be established in

two ways. First, when there is a unit root in both

regimes. Here the null hypothesis is of the form

H0: r1 ¼ r2 ¼ 0, which is tested against the unrest-

ricted alternative r1� 0 or r2� 0 using the Wald sta-

tistic. The parameters of r1 and r2 from Equation 1

will control the regime-dependent unit root process of

the RER. If r1 ¼ r2 ¼ 0 holds, the RER has a unit

root that can be described as a rejection of PPP. This

statistic is

R2T ¼ t21 þ t22 ð3Þ

where t1 and t2 are the t ratios for r̂1 and r̂2, respec-tively, from the ordinary least squares estimation.

However, Caner and Hansen (2001) claimed that this

two-sided Wald statistic may have less power than a

one-sided version of the test. As a result, they pro-

posed the following one-sided Wald statistic as

follows:

R1T ¼ t21I r̂1<0f g þ t22I r̂1<0f g ð4Þ

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R1T tests H0 against the one-sided alternative r1< 0or r2< 0. Caner and Hansen (2001) showed that bothtests R1T and R2T will have power against bothalternatives.2

Empirical results

For the sake of comparison, we also incorporate theADF, PP (1988) and KPSS (1992) tests into our study.The results of these three conventional unit root testsfor RERs using ADF, PP and KPSS tests (notreported here but are available upon request) indicatethat the RERs are nonstationary for these nine EastAsian countries. As stated earlier, there is a growingconsensus that the RER exhibits nonlinearities, andconsequently, conventional unit root tests such asADF test have low power in detecting the mean rever-sion of exchange rate. A number of studies have alsoprovided empirical evidence on the nonlinear adjust-ment of exchange rate. Therefore, we proceed to testthe RER by using Caner and Hansen’s (2001) non-linear TAR unit root tests.First, we use Wald test WT to examine whether we

can reject the linear autoregressive model in favour ofa threshold model. The results of Wald test and alsothe bootstrap critical values generated at conventionallevels of significance are reported in Table 1. Thebootstrap p-value for threshold variables of the formZt�1 ¼ rt�1 � rt�m for delay parameters m is rangedfrom 1 to 12. The parameters m are generallyunknown; there is no reason to think that the optimaldelay parameter will be the same across countries. Tocircumvent this, Caner and Hansen (2001) suggestedmaking m endogenous by selecting the least squaresestimate of m that minimizes the residual variance.

This amounts to selecting m at a value that maximizesthe WT statistic. We find that WT statistic is maxi-mized for China when m ¼ 1; for Hong Kong, Japanand Indonesia when m ¼ 4; and for Malaysia,Philippines, Singapore, South Korea and Thailandwhen m ¼ 2. Taken together, these results implystrong statistical evidence against the null hypothesisof linearity at least at the 10% significance level inthese nine East Asian countries indicating that simplelinear models are inappropriate, except for Japan.Next, we explore the threshold unit root properties of

RER based onR1T statistic for each delay parameterm,ranging from 1 to 12, paying particular attention to theresults obtained for our preferred model. The R1T testresults, together with the bootstrap critical value at theconventional levels of significance and the bootstrap p-value, are reported in Table 2. We are able to reject theunit root null hypothesis for more than half of thesenine Ease Asian countries at least at the 10% signifi-cance level. However, we are unable to reject thethreshold unit root hypothesis for Japan, Philippinesand Singapore. Taken together, our results providestrong support for PPP for most of the East Asiancountries and point that the RERs of these countriesare nonlinear stationary, implying that deviations ofexchange rate are mean reverting towards the PPPequilibrium. As we know trade barriers, transactioncosts, as well as interventions in the exchange marketcould be behind this nonlinear behaviour.The major policy implication that emerges from our

study is that PPP can be used to determine the equili-brium exchange rates for most of these East Asiancountries and the unbounded gains from arbitrage intraded goods are not possible among them with excep-tion of Japan, Philippines and Singapore.

Table 1. Threshold test

CountriesWaldstatistic

Bootstrapp-value

Optimaldelayparameterm

Thresholdparameter l̂

Number ofobservationsin regime 1(its percentage)

China 36.252 0.054 1 –0.007 41 (15)Hong Kong 52.501 0.000 4 0.003 176 (63)Indonesia 53.357 0.019 4 0.034 235 (84)Japan 12.688 0.720 4 0.032 179 (64)Malaysia 30.713 0.066 2 0.015 238 (85)Philippines 34.840 0.003 2 0.016 213 (76)Singapore 23.650 0.047 2 –0.003 131 (47)South Korea 81.779 0.000 2 0.022 238 (85)Thailand 43.216 0.015 2 0.021 238 (84)

Notes: Following much of the existing empirical literature on monthly Real Exchange Rates (RERs) andPurchasing Power Parity (PPP), we set a maximum lag of 12 and base all our bootstrap tests on 5000 replications.Most of the statistics are significant, which supports the presence of threshold effects.

2 Since R1T has more power than that of R2T, we only report the results of R1T in our study.

PPP with nonlinear threshold unit root test 841

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IV. Conclusions

This study applies a simple and powerful nonlinearTAR unit root test proposed by Caner and Hansen(2001) to test the validity of long-run PPP in a sampleof nine East Asian countries over the period January1986 to October 2009. The empirical results indicatethat PPP holds true for more than half of the EastAsian countries studied, and the adjustment towardsPPP is nonlinear.

Acknowledgements

We are grateful to Bruce Hansen for making availablehis MATLAB codes for the TAR model, which weremodified for this exercise. However, any remainingerrors are my own.

References

Caner, M. and Hansen, B. (2001) Threshold autoregressionwith a unit root, Econometrica, 69, 1555–96.

Dickey, D. A. and Fuller, W. A. (1981) Likelihood ratiostatistics for autoregressive time series with a unitroot, Econometrica, 49, 1057–72.

Juvenal, L. and Taylor, M. P. (2008) Threshold adjustmentof deviations from the law of one price, Studies inNonlinear Dynamics and Econometrics, 12, 1–44.

Kilian, L. and Taylor, M. P. (2003) Why is it so difficult tobeat the random walk forecast of exchange rates?,Journal of International Economics, 60, 85–107.

Kwiatkowski, D., Phillips, P., Schmidt, P. and Shin, Y.(1992) Testing the null hypothesis of stationarityagainst the alternative of a unit root: How sure are wethat economic time series have a unit root?, Journal ofEconometrics, 54, 159–78.

Lothian, J. R. and Taylor, M. P. (2000) Purchasing powerparity over two centuries: strengthening the case forreal exchange rate stability reply to Cuddington andLiang, Journal of International Money and Finance, 19,759–64.

Lothian, J. R. and Taylor, M. P. (2008) Real exchange ratesover the past two centuries: How important is theHarrod–Balassa–Samuelson effect?, Economic Journal,118, 1742–63.

MacDonald, R. and Taylor, M. P. (1992) Exchange rateeconomics: a survey, IMF Staff Papers, 39, 1–57.

Phillips, P. C. B. and Perron, P. (1988) Testing for a unit rootin time series regression, Biometrika, 75, 335–46.

Rogoff, K. (1996) The purchasing power parity puzzle,Journal of Economic Literature, 34, 647–68.

Sarno, L. and Taylor, M. P. (2002) Purchasing power parityand the real exchange rate, IMFStaff Papers, 49, 65–105.

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Table 2. One-sided unit root tests

Bootstrap critical values (%)

CountriesOptimal delayparameter m R1T statistic 10 5 1

Bootstrapp-value

China 1 18.128 16.130 21.197 35.564 0.081Hong Kong 4 18.358 10.105 12.097 16.440 0.006Indonesia 4 27.868 14.073 18.629 33.215 0.017Japan 4 4.472 9.918 12.273 17.771 0.440Malaysia 2 14.008 12.498 16.658 27.225 0.079Philippines 2 9.255 10.194 12.377 18.086 0.133Singapore 2 3.049 9.647 11.565 15.707 0.636South Korea 2 35.411 10.968 13.773 20.303 0.001Thailand 2 13.525 11.749 15.549 26.672 0.073

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