investigating nonlinearities in real exchange rate adjustment: threshold cointegration and the...

Journal of International Money and Finance 29 (2010) 770e790

Contents lists available at ScienceDirect

Journal of International Moneyand Finance

journal homepage: www.elsevier .com/locate/ j imf

Investigating nonlinearities in real exchange rateadjustment: Threshold cointegration and the dynamicsof exchange rates and relative prices

Hironobu Nakagawa*

Department of International Economics, SIPEC, Aoyama Gakuin University, 4-4-25 Shibuya, Tokyo 150-8366, Japan

JEL classification:F3F4

Keywords:Real exchange rateNominal exchange ratePurchasing power parityMean reversionThreshold vector error correction modelThreshold cointegration

* Tel.: þ81 3 3409 8111.E-mail address: [email protected]

0261-5606/$ e see front matter � 2010 Elsevier Ltdoi:10.1016/j.jimonfin.2010.03.002

a b s t r a c t

Motivated by growing evidence of nonlinear mean-revertingbehavior in real exchange rates, this paper investigates theunderlying dynamics in the context of a threshold vector errorcorrection model (TVECM) of nominal exchange rate and relativeprices. Unlike univariate models, our nonlinear multivariateframework takes into explicit account the joint behavior andindividual dynamics of the nominal exchange rate and relativeprices when these two key variables are threshold cointegrated.Our empirical application unravels their relative contribution tomean reversion and underscores the importance of capturing theirinteractions in investigating the nonlinear adjustment towardpurchasing power parity.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

There is a large and ever-growing literature on purchasing power parity (PPP). Empirical studiestesting the validity of PPP have found that deviations from PPP are highly persistent. The real exchangerate adjusts so slowly that robust evidence of mean reversion (parity reversion) appears elusive.Consensus estimates of the half-life of PPP deviations range between 3 and 5 years, suggesting anextremely slow decay rate for international price differentials (Rogoff, 1996). Recently, in a newlyemerging strand of the literature, an attack has beenmade from a slightly different angle to explain thePPP puzzle: the possibility of nonlinear adjustment due to transaction costs is taken into account. Thereis now a growing body of evidence of nonlinear mean reversion in real exchange rates, thereby

d. All rights reserved.

mailto:[email protected]

www.sciencedirect.com/science/journal/02615606

www.elsevier.com/locate/jimf

H. Nakagawa / Journal of International Money and Finance 29 (2010) 770e790 771

confirming convergence to PPP (e.g., Michael et al., 1997; Obstfeld and Taylor,1997; Taylor, 2001; Tayloret al., 2001; Lothian and Taylor, 2008). Details are discussed below.

To gain further insights into nonlinear adjustment toward PPP e a building block upon which thelink between exchange rate and prices is formede this paper extends the existing nonlinear univariatemodels of real exchange rates. We pursue a multivariate extension in conjunction with the idea ofcointegration and propose a threshold vector error correction model (TVECM) of nominal exchangerate and relative prices that jointly drive nonlinear mean reversion of the real exchange rate.1 Our basicintuition is that if the real exchange rate becomesmean revertingwhen the deviation fromPPP exceedsa critical threshold, then in that case we would expect cointegration effects to emerge between thenominal exchange rate and relative prices. As described by Balke and Fomby (1997), this type ofdiscrete adjustment can be characterized in terms of threshold cointegration, making possiblea threshold error-correction representation e in this case a TVECM of the nominal exchange rate andrelative prices. We utilize such a multivariate framework, rather than univariate models, because itaccommodates the appropriate way to capture threshold nonlinearity in real exchange rates throughthe proper consideration of the joint dynamics of the nominal exchange rate and relative prices.

In earlier empirical work investigating real exchange rate behavior (i.e., prior to the advances inempirical research on nonlinearities in real exchange rates), a plethora of studies employed techniquessuch as unit root and cointegration tests and linear vector error correction models (VECMs). (Thisliterature is discussed in section 3 below.) Work along this line used these techniques to see whetherthe real exchange rate is stationary, implying long-run PPP, or whether a long-run relationship indi-cated by cointegration between the nominal exchange rate and relative prices can be uncovered.However, previous studies produced mixed results. Due primarily to the high degree of persistence inreal exchange rates, the power of these tests is generally low.2

More recently, as an alternative line of approach, the emerging theoretical and empirical literature onnonlinear real exchange rate adjustment regards potential nonlinearity in adjustment to PPP asa possible explanation for the empirical failure of PPP. Nonlinearity may arise in the presence of thetransaction costs that preclude goods-market arbitrage; only when price differentials become largeenough to outweigh the costs, will arbitrage operate to eliminate deviations from PPP. Accordingly, therewill be a band of inaction (i.e., no-arbitrage region): inside the band no mean reversion is observed;whereas outside the band the error-correcting force kicks in, drivingmean reversion of the real exchangerate. This kind of nonlinear adjustment is now receiving broad empirical support (see, for example,Michael et al.,1997; Obstfeld and Taylor,1997; Taylor, 2001; Taylor et al., 2001; Lothian and Taylor, 2008).On thewhole, in empirical applications of univariate nonlinearmodeling of real exchange rates, linearityis rejected and results indicate evidence of nonlinear mean-reverting dynamics, suggesting that trans-action costs are integral to the understanding of real exchange rate adjustment.

In light of the recent findings of nonlinearities in real exchange rate adjustment, we considera threshold vector error correction model (TVECM) of the nominal exchange rate and relative prices.The reason is twofold: the mounting evidence of nonlinear behavior of the real exchange rate suggeststhat dynamics of the nominal exchange rate and relative prices e the driving forces underlying realexchange rate adjustment e warrant investigation by way of extending the existing univariateapproach. Second, the previous empirical difficulty in establishing cointegration between the nominal

1 After several drafts of this paper, we became aware of the paper e written independently of our work e by Lo and Zivot(2001), who consider multivariate threshold cointegration models when examining nonlinearity in price adjustment. Theirwork differs from ours in that: (i) they focus on threshold cointegration tests, estimation strategy, and power consideration; and(ii) they study the behavior of domestic price differences measured across U.S. cities by employing a set of U.S. disaggregatedprice data. In our paper, on the other hand, we are primarily interested in the international price adjustment process, andtherefore focus on (i)estimation of TVECMs so as to investigate the dynamics of the nominal exchange rate and relative pricesusing data on exchange rates and prices, (ii) investigation e by way of testing restrictions that imply compatibility of theTVECM and the TAR model e of the consequences when the interactions of the nominal exchange rate and relative prices arenot taken into consideration as in the univariate TAR models, and (iii) comparison of our multivariate results with thoseobtained for the univariate threshold models of real exchange rates.

2 To gain statistical power, Lothian and Taylor (1996) use two-centuries of data on real exchange rates and examine the long-run properties of real exchange rates. Frankel and Rose (1996), among others, use wide panel data and apply panel unit roottests. Even with the increased test power, the speed of mean reversion of real exchange rates appears to be slow.

H. Nakagawa / Journal of International Money and Finance 29 (2010) 770e790772

exchange rate and relative prices suggests that the difficulty may be attributable to an unjustifiedpooling of nonstationary and stationary data, ignoring the nonlinear nature of the adjustment process.In our multivariate framework in which threshold nonlinearity is incorporated, the interactionsbetween, and different individual dynamic behavior of, the nominal exchange rate and relative pricesare explicitly taken into account. This systems approach allows us to detect threshold cointegration inthe system; that is, to estimate the critical threshold values jointly with the adjustment-speedparameters for nominal exchange rate and relative prices when parity reversion is indeed in force. Inour empirical application using data on aggregated and disaggregated price indices for 24 countriesand exchange rates for 23 currencies measured relative to the U.S. dollar, TVECMs are estimated, andconditions are tested to confirm the importance of considering threshold behavior in a multivariatesetting. Our estimation results elucidate the individual short-term dynamics of the nominal exchangerate and relative prices, enabling us to measure the degree to which reversion toward PPP occursthrough the nominal exchange rate versus prices. From our findings about the dominant role played bymovements in the nominal exchange rate and asymmetries in its adjustment, we will also draw somepolicy implications.

The rest of the paper proceeds as follows. The next section provides a brief review of the theoreticalbackground and empirical findings in the emerging literature on nonlinearities in real exchange rateadjustment. Section 3 introduces a TVECM involving the nominal exchange rate and relative prices, andalso compares it with a univariate model of the real exchange rate. Section 4 then discusses econo-metrics methodologies to deal with TVECMs. In section 5, an empirical application is conducted. Thelast section offers conclusions of the analysis and discusses some policy implications as well asdirections for future research.

2. An overview of nonlinear real exchange rate studies

Nonlinearities in real exchange rate adjustment can arise from the presence of internationaltransactions costs. The idea e whose origin can be traced to Heckscher’s (1916) writing e thattransaction costs should create “commodity points” has stimulated a renewed interest in the PPPdoctrine, theoretically and empirically. On the theoretical side, a number of researchers (for example:Williams and Wright, 1991; Dumas, 1992; Sercu et al., 1995) present theoretical models that incor-porate this transaction costs view.3 Overall, in these models, transport costs generate a band of inactionfor the real exchange rate: if the cost of transport exceeds the benefit of arbitrage, price differentialswill not be eliminated; only when arbitrage becomes profitable enough to outweigh the shipping cost,will the forces of arbitrage push the real exchange rate toward its equilibrium. As a result, the realexchange rate exhibits nonlinear dynamics: a nonstationary randomwalk process inside the band anda stationary mean-reverting process outside the band.

Looking at the empirical aspect, evidence uncovering this type of nonlinear real exchange rateadjustment is growing. In this strand of literature, a standard approach taken is to apply a variant ofunivariate nonlinear models (which fall in the class of single-equation analysis of nonlinear time seriesmodels) to the time series of real exchange rates. Obstfeld and Taylor (1997) fit threshold autore-gressive (TAR)models and find evidence of nonlinearly adjusting real exchange rates.4 The study pointsto the relevance of the market frictions to the nonlinear behavior. The nonlinearly mean-revertingbehavior of the real exchange rate is also confirmed by Michael et al. (1997) and Taylor et al. (2001),who employ exponential smooth transition autoregressive (ESTAR) models in which transitionbetween regimes is treated as smooth rather than discrete adjustment. For a parsimonious specifi-cation to model the nonlinear adjustment, Obstfeld and Taylor (1997) adopt and estimate a TAR modele a so-called band-TAR model that takes the form:

3 Taken more broadly, transaction costs may result from the presence of sunk costs of arbitrage under uncertainty. As thetheory of investment under uncertainty demonstrates, uncertainty affects agents’ wait-and-see attitude and the extent of priceinertia. See Dixit (1989), Dumas (1992), and Krugman (1989).

4 O’Connell (1998) also employs TAR models in examining PPP deviations. The author reports evidence of thresholdnonlinearity only for some intra-EC real exchange rates.


>< lðOUTÞðzt�1 � cÞ þ 3ðOUTÞt if zt�1 > c;

Dzt ¼

8>: 3ðINÞt if jzt�1j � c;

lðOUTÞðzt�1 þ cÞ þ 3ðOUTÞt if zt�1 < �c;

(1)

where 3ðOUTÞt wNð0; sðOUTÞ2 Þ and 3ðINÞt wNð0; sðINÞ2 Þ. zt is the equilibrium error in the price difference; thatis, the demeaned and detrended component of the real exchange rate. The band of inaction is definedby threshold c. l(OUT) is the speed of adjustment outside the band. Inside the band (jzt�1j � c), arbitrageis absent and thus the real exchange rate exhibits no central tendency, following a random walkprocess; outside the band, mean reversion to the edge of the band takes place as arbitrage operates. Intheir empirical application using monthly (1980:01e1995:12) data on disaggregated as well asaggregated goods prices for 32 city and country locations, the authors estimate TARmodels, presentingthe usefulness of such a relatively parsimonious specification in describing the nonlinear aspect of realexchange rate adjustment. In light of these findings of nonlinearity in real exchange rates that havecome out of the analysis based on univariate nonlinear models of real exchange rates, we now pursuea multivariate extension in conjunction with the idea of cointegration.

3. A threshold vector error correction model (TVECM) of the nominal exchange rate andrelative prices

In this section, a threshold vector error correctionmodel (TVECM) of the nominal exchange rate andrelative prices is introduced. As a motivation for our multivariate nonlinear modeling, we first verybriefly review a conventional linear vector error correction model (VECM). Then we set up a TVECM,and also compare it with a TAR model.

Let st be the logarithm of the nominal exchange rate, expressed in terms of domestic currency perunit of foreign currency, and pt and p�t the logarithms of the domestic and foreign price levels. The log ofthe real exchange rate is then

qthst þ�p�t � pt

�:

To simplify notation, we denote the relative prices by pthp�t � pt. Also note that when disaggregatedprices are used, rather than aggregated national price indices, qt may be interpreted as a measure of thedeviation from PPP. If long-run PPP holds, while short-run deviations from PPP are allowed, deviationsare not allowed to get larger over time (ruling out st and pt to diverge without bound). For thisrequirement to be met, the real exchange rate qt needs to be stationary. When st and pt (both nonsta-tionary and integrated of order one) are cointegrated with a cointegrating vector of (1,1), the realexchange rate qt is said to be stationary, implying that there is a long-run relationship consistent withlong-run PPP. Moreover, if indeed there is this cointegration relationship, an error-correction repre-sentation exists, as verified by Engle and Granger (1987). AVECMwith lag length p� 1 can bewritten as

Dst ¼ Pp�1i¼1 4

ssi Dst�i þ

Pp�1i¼1 4

spi Dpt�i þ lsqt�1 þ 3st

Dpt ¼ Pp�1i¼1 4

psi Dst�i þ

Pp�1i¼1 4

ppi Dpt�i þ lpqt�1 þ 3pt

By applying cointegration tests in univariate settings and by testing for cointegration in a VECM (amultivariate ECM), a number of studies have investigated whether or not the nominal exchange rateand relative prices are cointegrated (A partial list includes Taylor, 1988; Mark, 1990; Cheung and Lai,1993; Kugler and Lenz, 1993; MacDonald, 1993. For a survey, see Froot and Rogoff, 1995.). However,overall, judging from these studies, whether or not we can conclude that long-run PPP indeed holdsremains contentious. The volatile and persistent behavior of the exchange rate appears to havefrustrated the efforts of econometricians to obtain robust results. Although, in many studies that usethe Johansen procedure, cointegration is claimed to be found for trivariate systems ðst ; p�t ; ptÞ withan unconstrained cointegrating vector, interpreting their results appears difficult: estimates of coin-tegrating vector obtained are often inconsistent with the theoretically predicted values. In particular,


proportionality and symmetry conditions e i.e., cointegrating vector with coefficients (1,1,�1) e

implied by PPP are often rejected.5

Looking at the empirical difficulty described above at one end and the recent evidence onnonlinearity in real exchange rates at the other end, one would conjecture thatewhile the adjustmentprocess is treated as a linear process in the previous studies e if the actual adjustment process isnonlinear, then unjustified pooling of nonstationary data and stationary data results in the empiricaldifficulty that researchers have encountered in detecting cointegration between the nominal exchangerate and the relative prices.

A possible nonlinear alternative to VECMs can be considered by incorporating threshold nonline-arity. Considering the stationary mean-reverting behavior of real exchange rates outside the trans-action cost-induced ‘band-of-inaction’ (region), we expect the cointegration effects between thenominal exchange rate and relative prices in the outer regions e the phenomenon referred to asthreshold cointegration. If the nominal exchange rate and relative prices are indeed threshold coin-tegrated, there is a bivariate threshold ECM involving these two key variables. As in Balke and Fomby(1997) and Tsay (1998), the model will be a VECM augmented with a threshold effect based on theerror-correction term. Among a large class of TVECMs, we employ the following model:

Dst ¼

8>>>>>>>><>>>>>>>>:

Pp�1i¼1 r

ðuÞssi Dst�i þ

Pp�1i¼1 r

ðuÞspi Dpt�i þ lðuÞsðqt�1 � cÞ þ 3ðuÞst if qt�1 > c;

Pp�1i¼1 r

ðmÞssi Dst�i þ

Pp�1i¼1 r

ðmÞspi Dpt�i þ 3

ðmÞst if jqt�1j � c;

Pp�1i¼1 r

ðlÞssi Dst�i þ

Pp�1i¼1 r

ðlÞspi Dpt�i þ lðlÞsðqt�1 þ cÞ þ 3ðlÞst if qt�1 < �c;

Dpt ¼

8>>>>>>>><>>>>>>>>:

Pp�1i¼1 r

ðuÞpsi Dst�i þ

Pp�1i¼1 r

ðuÞppi Dpt�i þ lðuÞpðqt�1 � cÞ þ 3ðuÞpt if qt�1 > c;

Pp�1i¼1 r

ðmÞpsi Dst�i þ

Pp�1i¼1 r

ðmÞppi Dpt�i þ 3

ðmÞpt if jqt�1j � c;

Pp�1i¼1 r

ðlÞpsi Dst�i þ

Pp�1i¼1 r

ðlÞppi Dpt�i þ lðlÞpðqt�1 þ cÞ þ 3ðlÞpt if qt�1 < �c:

(2)This is a (three-regime) band-TVECM.6e8

The error terms are two-dimensional white noise; 3ðkÞt wNð0;UðkÞÞ, where

3ðkÞt ¼

3ðkÞst

3ðkÞpt

!and UðkÞ ¼

�sðkÞss sðkÞspsðkÞps sðkÞpp

�for k ¼ u;m; l:

The threshold value c defines the band of inaction, or no-arbitrage region [�c, c] resulting from thepresence of transactions costs. Inside the band there is no arbitrage taking place, and thus the realexchange rate follows a nonstationary process, showing no central tendency; whereas outside the bandprice differentials outweighing the costs induce arbitrage and the real exchange rate tends toward theedge of the band. The bandwidth is symmetric about equilibrium, and we have symmetric upper and

5 Edison et al. (1997) find evidence somewhat more favorable to PPP.6 Although a more general formulation of TVECM can be considered, attention here is confined to the TVECM of the form (2); i.e.

the delay lag (in the transition variable) of unity and a three-regime, symmetric thresholds setting. This relatively parsimonious,representation fits naturally to the theory of arbitrage in the presence of transaction costs. Also, the present specification enablesus to contrast our multivariate framework of TVECM against the prevailing TAR application in the PPP literature.

7 In formulation (2), qt�1 measures the size of deviation of the real exchange rate (in period t � 1) from its equilibrium value.To be consistent with our empirical analysis in section 5, we have written the deviation to be centered around zero.

8 For expositional purpose, constant terms (i.e., intercepts) are not included in TVECM (2). This specification is plausible if weassume that there is no linear time trend in the variables, or if we deal with detrended series and focus on adjustment towardthe trend in real exchange rate equilibrium. More on this and other alternatives will be discussed in the first subsection ofsection 5 and in footnotes.15,16


lower band thresholds that define three regions e upper, middle, and lower. We allow differentcoefficient parameters for these three regions, indicated by superscripts u, m, l. The speed of adjust-ment parameters l(k)s and l(k)p respectively are the relative contribution of the nominal exchange rateand relative prices movements to correcting equilibrium errors in region k. The TVECM of system (2)posits that the movements of the nominal exchange rate and relative prices jointly bring in thenonlinear adjustment process of the real exchange rate.

In the single-equation dimension, in contrast to a multivariate setting just laid out, a band-TARmodel e often used to incorporate threshold behavior of real exchange rates that result from thepresence of transactions costs e is written as

Dqt ¼

8>>>>>>>><>>>>>>>>:

Pp�1i¼1 x

ðuÞi Dqt�i þ lðqt�1 � cÞ þ 3

ðuÞt if qt�1 > c;

Pp�1i¼1 x

ðmÞi Dqt�i þ 3ðmÞ

t if jqt�1j � c;

Pp�1i¼1 x

ðlÞi Dqt�i þ lðqt�1 þ cÞ þ 3

ðlÞt if qt�1 < �c:

(3)

If we take out lags of the dependent variable (Dqt�i, i ¼ 1, 2, . p � 1), this symmetric adjustment (i.e.,identical error-correcting coefficient l for outer regimes) band-TAR model reduces to the TAR model(TAR with autoregressive order of one) employed by Obstfeld and Taylor (1997) e the specificationwritten as Eq. (1) earlier. Comparing the univariate TAR model (3) with the multivariate TVECM (2), itcan be seen that the TVECM in general does not imply the TAR: the TVECM (2) can be simplified to theTAR representation (3) if certain requirements are met. In other words, the TAR model (3) correspondsto a reduced form of the TVECM (2) with certain restrictions imposed, the restrictions (conditions) tobe laid out below. From an empirical point of view, onlywhen the restrictions on parameters are indeedsupported in data, would the TAR estimates of its parameters be considered reliable and consistentwith those from the TVECM. The relevant conditions (restrictions) are

rðkÞssj þ rðkÞpsj ¼ rðkÞspj þ rðkÞppj for j ¼ 1;2; .; p� 1 and k ¼ u;m; l: (4)

lðuÞs þ lðuÞp ¼ lðlÞs þ lðlÞp: (5)

The first set of restrictions in (4) is needed if we are to derive a univariate model of TAR of the form (3)from the bivariate TVECM system (2) as a simplified representation. Thus if this restriction is notsatisfied, there existsno simpleunivariate representation: settingupaTARmodel apriori andestimatingit would not capture appropriately the threshold effect or the speed of mean reversion. The secondrestriction (5) is related to the fact that symmetric adjustment (i.e., identical speedof adjustment inouterregimes) in the TAR model (3) does not imply symmetric adjustment in the TVECM (2). On theoreticalgrounds, theremaybenoparticular reasonwhy the speedof adjustmentof the real exchange rate shouldvary depending on whether it is above or below the equilibrium level, in which case a symmetricspecification in TAR models is plausible. Looking into the components of the real exchange rate, on theother hand, such symmetric adjustment of the real exchange rate can be brought about by the jointdynamics of the nominal exchange rate and relative price. In that case, in the individual dynamics of thenominal exchange rate and relative prices, theremaywell be potential asymmetries. In the next section,methodological issues involved in estimating TVECMs and testing the restrictions are discussed.

4. Econometric methodologies

4.1. Model specification and tests for (threshold) cointegration

The thresholdvectorcorrectionmodel (TVECM) tobeestimated takes the form(2). It is a (three-regime,symmetric thresholds)band-TVECMof thenominalexchange rateandrelativeprices.Asaddressedearlier,this specification is employed for our empirical analysis because itfits comfortably to the transactions costtheory, andalsomakes itpossible toevaluate theempirical compatibilitybetweenthemultivariateTVECM


andtheTARmodel, and facilitates a systematic comparisonofourestimationresultsagainst those fromtheTAR application to PPP e a seminal work of which is Obstfeld and Taylor (1997).

In principle, specification tests (such tests as below) can be employed to formally test fora threshold, the number of regimes, and the presence of the band. For instance, in testing for thepresence of a threshold effect, a test of the specific two-regime TVECM alternative against the linearVECM null can be conducted by the use of likelihood ratio (LR) based statistic e a supLR statistic (i.e.,the maximum LR statistic over the possible threshold values). In the LR-type tests, there are nuisanceparameters under the null. Accordingly, for inference, empirical distribution and critical values must beobtained by simulation, or bootstrap method. A similar procedure may be followed to obtain specifi-cation tests, also using supLR statistics: a test of the two-regime TVECM null against the three-regimeTVECM alternative; and then a test of three-regime TVECM null against the (symmetric) band-TVECM.In these sequential steps, each test is based on nested models. These tests are carried out on a case-by-case basis; that is, it is not against the general threshold cointegrationmodel that the null can be tested.

As for the (threshold) cointegration tests, ideally it would be better if one could test for thresholdcointegration in the multivariate framework of the TVECM: while univariate unit root tests come shortof providing sufficient power, the system approach can serve as a useful platform for detecting thresholdcointegration.9 As a step in this direction, in the present paper we would like to focus mostly on theestimation of the TVECM to characterize the dynamics of the nominal exchange rate and relative pricesunderlying real exchange rate adjustment, and to test the restrictions that imply compatibility betweenthe TVECM and the TAR model, and less on (threshold) cointegration tests or specification tests.10

4.2. Estimation

In the TVECM (2), we estimate the threshold value and parameters jointly.11 This can be carried outby themaximum likelihood (ML) using a grid search method. For a given threshold value c, a likelihoodfunction can be defined for each of the partitions, and thus the likelihood function for the TVECM. (Fordetails on the likelihood function, see Appendix 2.) We search over all possible values of c and find thevalue that maximizes the likelihood function for the TVECM.12 Basically, we are employing ML esti-mation on the partitioned samples e upper, middle, and lower regimes. For practical matters, we formcandidate thresholds in a way that each regime contains a minimal number of observations, 10 percentof the total number of observations.

4.3. Test of restrictions

As addressed earlier, we also examine empirically the restrictions (4) and (5): the following test isconducted to evaluate the empirical compatibility between the bivariate TVECM and the univariate TAR.Practically, the grid search procedure described above is applied, and a TVECM with the restrictions(overidentifying restrictions) imposed is estimated. (As for models that contain lags of the dependentvariables, restrictions (4) and/or (5) can be imposed; for models without lags, restriction (5) alone is

9 A direct test for cointegration and threshold jointly, that is, test of unit-root null hypothesis directly against the alternative ofgeneral threshold cointegration,wouldbe complicated, because of the nonstationarity of threshold variables (transitionvariables)under the null. Formoreon threshold cointegration tests, see Balke and Fomby (1997), Lo andZivot (2001), Hansenand Seo (2002).10 Our decision here to go ahead with the estimation of the TVECM is based on the following: (i) the power of these unit roottests is fairly poor in the present context; (ii) evidence on threshold stationary behavior of the real exchange rate has beenincreasingly reported; and (iii) testing the null of unit root against the general threshold cointegration alternative is compli-cated, given the large class of stationary threshold models. Concerning (i), tests results are available (on request) for a standardunit root test (the augmented DickeyeFuller test) and unit root tests that are designed to have power against the alternative ofstationary threshold adjustment, namely the Enders and Granger (1998) test.11 As for the number of lags of dependent variables to be included einstead of fitting an appropriate lag length model to thedata by using a information criteria or by executing the search on the estimates of the lag length optimally for each location-price pair in the panele our TVECM was estimated for each value of lag length in the range between 0 and 3. Given our modelestimates, we evaluate, for each permissible value of lag length, the TVECM against the corresponding TAR (TVECM with laglength p � 1 corresponds to the TAR model of autoregressive order p).12 For the grid search algorithm, we modified and used the Rats code from Obstfeld and Taylor (1997).


applicable.) By using the maximized log likelihood function for the restricted model together with themaximizedvalueof the log likelihood function for theunrestrictedmodel,weobtain a likelihood ratio (LR)statistic for testing thevalidityof theoveridentifying restrictions. Letting lnLrand lnLube the log likelihoodevaluated at the restricted and unrestricted estimates, respectively,�2(lnLr� lnLu) will be asymptoticallydistributed as chi-square (n), where n is the degree of freedom (the number of restrictions imposed).

5. Empirical application

5.1. Data

The data for our analysis are the same as those considered in the ObstfeldeTaylor study, and aredrawn from Engel and Rogers (1998). The exchange rate series are 23 currencies vis-à-vis the U.S.dollars. The price data come in the form of indices e we use both aggregated (CPI for all goods) anddisaggregated final goods price indices from 24 countries.13 Disaggregated indices are sub-categoriesof the CPI: price indices for clothing, food, and fuel. All series are monthly observations for the period1980:01e1995:12. We transform the data in the following way so as to standardize the mean ofequilibrium error to be zero around a possibly trending equilibrium. The data are demeaned, in light ofthe nature of the data involved (i.e., price series come in the form of indices; and there are differencesin units of measurement). In addition, the data may be initially detrended when a trend is present inthe data. Our intention is to focus on the adjustment toward the equilibrium real exchange rate, whichmay exhibit a long-run trend behavior driven, for instance, by the BalassaeSamuelson effect (orHarrodeBalassaeSamuelson effect).14 To account for this effect, a linear trend is often employed toproxy the effect. Alternatively, the effect can be measured explicitly by productivity differentials(Lothian and Taylor, 2008). In the present study, finding some evidence of a linear time trend in seriesto be analyzed, we adopted the preliminary detrending procedure. Practically, we regressed the dataon a constant and linear time trend, and used the residuals as the demeaned and detrended series forthe estimation of the TVECM and TAR as in formulations (2) and (3), respectively.15,16 The use of

13 In principle, the actual threshold value is likely to be different for each commodity. Accordingly, when a threshold-typemodel is employed to analyze aggregated price indices, the interpretation of the point estimate of threshold value c itselfbecomes somewhat problematic. In fact, for aggregated price indices, an alternative model such as a multivariate version ofsmooth-transition autoregressive model a la Rothman et al. (2001) would be more appropriate. Notwithstanding this limitationof a threshold-type model, estimated threshold values are still informative in the sense that they assess the magnitude oftransaction costs implied in the multivariate framework relative to that implied in the univariate counterpart. While nonlinearstudies on deviations from PPP are predominantly conducted in the univariate paradigm (for instance, the TAR technique isoften used), our threshold setting contrasts multivariate against univariate threshold models through testing restrictions e

failing to satisfy them implies distortions in the estimation based on the univariate setup.14 According to the Balassa-Samuelson model by Balassa (1964) and Samuelson (1964), induced by the differential rate ofproductivity growth in the tradable and nontradable sectors, rising prices of nontradable goods generate a long-run trend in thereal exchange rate. Moreover, this nontradable component may well be embedded in the tradable good prices under consid-eration; in reality tradables are not quite free from distribution and marketing costs.15 The demeaning and detrending procedure here is applicable as long as the deterministic component of the series remainsunaffected across regimes. In other words, we do not consider the threshold effect in the parameters which characterize thedeterministic component. With the demeaned and detrended data, the model to be estimated is written in terms of zero-meanand trend-free variables, and thus there would be no intercepts (constant terms) entering in the model, except for the implicitintercepts that are implied by a threshold value. In other words, a constant can enter only in the error-correction term asa threshold parameter; in this sense the model here is written in restrictive form.16 Note, however, that in addition to the use of the preliminary detrended series, we have investigated other alternatives.While we still do not consider threshold effect in the deterministic component of real exchange rate series, we can generalizethe dynamics of the nominal exchange rate and relative price by admitting the possibility of a regime-specific drift for each ofthese two variables. Under this scenario, the transformation of the nominal exchange rate and relative price series through theusual detrending method would no longer be appropriate; so these series are used without preliminary detrending, and in thiscase the model is written with intercepts in unrestrictive form (details in Appendix 1). Also, for the last remaining case, if theprocess does not contain deterministic time trends to begin with, then of course the data can be utilized without thepreliminary detrending, and the TVECM and TAR are estimated as in formulations (2) and (3), respectively. For the sensitivityanalysis with respect to the preliminary detrending, we conducted estimation for all three cases above. To conserve space andto be comparable with the results reported by Obstfeld and Taylor (1997), we report the results using the detrended data.


demeaned and detrended data in this way would permit us to analyze the dynamics around thetrending equilibrium.

5.2. Estimation results

We estimate the TVECM system (2) and, for comparison purpose, the TAR model (3). Tables 1and 2 present the TVECM estimates of the threshold value c (labeled “TVECM c”), the speed ofadjustment parameters (i.e., coefficients on error-correction terms), and the absolute t statistics ofthese parameters. Along with the TVECM results, we also show the TAR estimates of thethreshold value (labeled “TAR c”), the convergence speed parameter l, and the absolute t statisticof l. For both TVECM and TAR formulations, estimation results for the models with a lag length oftwo (i.e., first- and second-order lags of the dependent variables) and for those without lags arereported in Tables 1 and 2, respectively.17 Each table consists of 4 panels showing detailedestimation results based on price indices of (a) all goods, (b) clothing, (c) food, and (d) fuel. Alsoincluded in the tables are the results for the test of the overidentifying restrictions e therestrictions that need to be satisfied for empirical compatibility between the TVECM and the TARmodel.

From the estimates of threshold values, we find that threshold estimates based on the TVECMcan differ markedly (with no obvious patterns) from estimates based on the TAR model. Likewise,the difference is noticeable between convergence speed estimates for the TAR model and thoseimplied in the TVECM. These findings accord with the overall results indicated by the marginalsignificant levels of the LR statistics (columns labeled “LR1” and “LR2”) calculated to test theoveridentifying restrictions. In number of cases, the restrictions are rejected in the data, suggestingthat in estimating thresholds and convergence speeds, it is important to properly take into accountthe multivariate structure of relevant variables that brings about a threshold characteristic of realexchange rate behavior. In particular, the rejection of conditions (4) implies (see “LR1” in Table 1)empirical incompatibility between the TVECM and the TAR model. Here, an autoregression of thereal exchange rate on its own lags as in the univariate TAR would not be compatible with theestimation in the multivariate framework of the TVECM: the univariate approach that ignoresa multivariate structure can lead to distortions in threshold estimation and hence in otherparameters.

We may also mention, though, that in spite of the differences in parameter estimates obtained forthe TVECM and TAR model, and of the empirical incompatibility of the two models, there is a commonobservation that can be made from the TVECM and TAR results. Inspecting the range of the half-lifeestimates (included in Table 2) that can take either under the TAR model or TVECM, we observethat half-lives for PPP deviations tend to cluster in the range of 6 monthse2 years, much shorter thanconventionally estimated half-lives of 3e5 years.18 This suggests that, as demonstrated by Taylor(2001), the bias arising from the failure to recognize nonlinear adjustment can be an explanation forthe PPP puzzle.

There aremore toour TVECMresults. The estimated speedsof adjustment fornominal exchange ratesand relative prices indicate that in majority of cases, it is the nominal exchange rate adjustment thateliminates deviations from PPP. Fig. 1 highlights this pointe the figure is produced based on the esti-mation results (ls and lp in Table 1) for TVECM with lag length of two. This finding on convergencespeeds implies that when the real exchange rate mean reverts outside the band defined by thresholds,the error correcting force induced by the nominal exchange rate adjustment is, in general, substantially

17 We first report our results for lag length 2 because most of serial correlation in the residuals from the TVECM esti-mation is eliminated. Changing the lag length to 1 or 3 yielded qualitatively similar results. Also, with regard to the modelspecification without lags of the dependent variables, despite the evidence of serial correlation found in the residuals, wereport its estimation results for comparison purpose (i.e., comparison with the TAR results reported in Obstfeld and Taylor(1997).18 When lags of dependent variable(s) are included in the model, calculating the half-lives from the coefficient in theerror-correction term alone would not give a precise measure of persistence of PPP deviations. Thus, in Table 1, wesimply report the error-correction parameter estimates as a measure of the degree of adjustment.

Table 1Estimation results for TAR and TVECM.

TAR TVECM

TAR c l TVECM c l(u)s l(l)s l(u)p l(l)p ls lp LR1

(a) Price index used: CPI AllCanada 0.015 �0.003 (0.18) 0.015 �0.001 (0.06) 0.003 (0.16) �0.013 (1.54) 0.005 (0.89) 0.001 �0.004 [0.38]Mexico 0.040 �0.026 (1.71) 0.042 �0.062 (4.24) 0.030 (2.27) 0.004 (1.54) �0.035 (4.60) �0.014 �0.016 [0.00]Austria 0.090 �0.044 (1.63) 0.144 �0.097 (2.08) �0.037 (0.94) �0.022 (2.88) �0.003 (0.27) �0.062 �0.011 [0.59]Belgium 0.170 �0.041 (1.20) 0.161 �0.077 (1.20) �0.053 (1.71) �0.006 (1.23) �0.004 (1.34) �0.061 �0.005 [0.42]Denmark 0.029 �0.031 (2.04) 0.049 �0.050 (1.90) �0.016 (0.91) �0.004 (0.98) �0.003 (0.57) �0.033 �0.003 [0.04]Finland 0.147 �0.055 (1.35) 0.039 �0.032 (1.56) �0.021 (0.96) �0.003 (0.88) 0.000 (0.17) �0.028 �0.001 [0.32]France 0.152 �0.042 (1.14) 0.153 �0.032 (0.38) �0.141 (4.99) 0.008 (1.30) �0.003 (1.17) �0.101 0.001 [0.00]Germany 0.070 �0.039 (1.70) 0.045 �0.113 (4.39) �0.018 (0.85) �0.002 (0.82) 0.000 (0.15) �0.070 �0.001 [0.02]Greece 0.060 �0.041 (1.85) 0.055 �0.071 (1.58) 0.000 (0.02) 0.000 (0.00) �0.018 (1.64) �0.039 �0.008 [0.11]Italy 0.221 �0.105 (0.97) 0.178 �0.168 (1.10) �0.081 (1.81) 0.032 (2.08) �0.006 (1.14) �0.121 0.011 [0.62]Netherlands 0.057 �0.036 (1.72) 0.056 �0.063 (1.81) �0.022 (0.90) �0.007 (2.38) �0.002 (0.81) �0.044 �0.005 [0.40]Norway 0.158 �0.068 (1.13) 0.044 �0.056 (2.15) �0.023 (1.26) �0.005 (0.69) �0.002 (0.43) �0.040 �0.004 [0.01]Portugal 0.142 �0.061 (1.68) 0.049 �0.044 (1.57) �0.018 (0.82) �0.011 (1.78) �0.018 (2.57) �0.030 �0.015 [0.55]Spain 0.023 �0.026 (2.02) 0.125 �0.086 (2.83) �0.015 (0.66) �0.002 (0.26) �0.008 (1.60) �0.048 �0.005 [0.38]Sweden 0.049 �0.026 (1.82) 0.027 �0.031 (1.95) �0.015 (0.74) �0.004 (0.90) �0.002 (0.56) �0.023 �0.003 [0.16]Switzerland 0.020 �0.041 (2.16) 0.023 �0.088 (2.51) �0.022 (1.20) �0.006 (2.07) �0.001 (0.27) �0.056 �0.003 [0.30]U.K. 0.169 �0.069 (1.30) 0.021 �0.038 (1.91) �0.015 (1.08) �0.001 (0.33) �0.002 (0.83) �0.026 �0.002 [0.64]Hong Kong 0.106 �0.054 (2.53) 0.098 �0.028 (0.99) �0.114 (7.77) �0.011 (1.04) 0.019 (0.78) �0.078 0.007 [0.01]Japan 0.131 �0.079 (1.51) 0.128 �0.116 (2.39) �0.074 (0.96) �0.035 (2.71) �0.008 (0.50) �0.096 �0.023 [0.01]Singapore 0.104 0.092 (0.56) 0.014 �0.005 (0.22) �0.011 (0.67) 0.003 (0.27) 0.007 (1.31) �0.008 0.005 [0.06]Taiwan 0.138 �0.088 (1.44) 0.084 �0.114 (3.89) �0.014 (0.84) 0.055 (1.05) 0.002 (0.13) �0.054 0.023 [0.00]New ZealandSouth Africa 0.238 �0.286 (2.53) 0.120 �0.041 (1.90) �0.017 (0.32) �0.031 (3.45) �0.033 (11.27) �0.028 �0.032 [0.93]

(b) Price index used: ClothingCanada 0.056 �0.203 (3.08) 0.059 0.019 (0.60) �0.025 (0.78) �0.158 (1.63) �0.247 (4.27) �0.005 �0.208 [0.76]Mexico 0.015 �0.058 (2.56) 0.015 �0.102 (8.02) 0.061 (3.51) �0.003 (0.14) �0.096 (6.54) �0.039 �0.039 [0.00]AustriaBelgiumDenmarkFinlandFranceGermanyGreeceItaly 0.152 �0.082 (1.49) 0.026 �0.050 (2.09) �0.011 (0.63) �0.012 (0.51) �0.022 (1.38) �0.033 �0.016 [0.00]Netherlands 0.021 �0.061 (2.20) 0.048 �0.053 (1.37) �0.018 (0.99) �0.017 (0.49) �0.040 (1.53) �0.038 �0.027 [0.00]Norway 0.109 �0.056 (1.00) 0.139 �0.649 (5.41) �0.016 (0.41) 0.317 (3.84) �0.044 (0.62) �0.284 0.108 [0.00]PortugalSpain

H.N

akagawa/Journal

ofInternational

Money

andFinance

29(2010)

770e790

779

Sweden 0.170 �0.084 (1.31) 0.154 �0.198 (1.71) �0.017 (0.66) �0.392 (1.99) �0.045 (1.18) �0.066 �0.138 [0.00]SwitzerlandU.K. 0.027 �0.038 (1.97) 0.128 �0.060 (1.42) �0.022 (0.86) �0.002 (0.10) �0.014 (0.82) �0.039 �0.009 [0.01]Hong Kong 0.015 �0.098 (2.62) 0.013 �0.053 (2.66) �0.013 (1.05) �0.022 (0.50) �0.091 (1.80) �0.031 �0.061 [0.18]Japan 0.047 �0.124 (2.81) 0.141 �0.127 (1.28) �0.046 (0.67) �0.259 (2.04) �0.097 (0.68) �0.094 �0.193 [0.06]Singapore 0.037 �0.065 (1.66) 0.093 0.022 (0.75) �0.099 (2.25) 0.040 (0.64) �0.028 (0.34) �0.050 �0.001 [0.01]Taiwan 0.095 �0.135 (0.89) 0.029 �0.043 (2.23) �0.012 (0.86) �0.109 (1.95) �0.093 (2.65) �0.028 �0.101 [0.17]New ZealandSouth Africa 0.112 �0.086 (1.32) 0.111 �0.083 (2.59) �0.068 (1.27) �0.024 (0.11) �0.036 (1.53) �0.077 �0.029 [0.00]

(c) Price index used: FoodCanada 0.024 �0.009 (0.37) 0.006 0.001 (0.04) 0.003 (0.12) �0.021 (1.61) 0.009 (0.59) 0.002 �0.006 [0.66]Mexico 0.026 �0.036 (1.98) 0.026 �0.057 (4.00) 0.028 (1.73) 0.000 (0.01) �0.047 (5.27) �0.014 �0.024 [0.00]Austria 0.096 �0.053 (1.58) 0.141 �0.041 (0.70) �0.045 (1.16) �0.020 (1.32) �0.005 (0.39) �0.043 �0.011 [0.58]Belgium 0.027 �0.039 (2.28) 0.018 �0.045 (2.32) �0.019 (1.10) �0.012 (1.93) �0.007 (1.49) �0.033 �0.010 [0.17]Denmark 0.022 �0.041 (2.18) 0.015 �0.049 (1.89) �0.020 (1.03) �0.007 (1.00) �0.011 (1.39) �0.036 �0.009 [0.01]Finland 0.057 �0.031 (1.64) 0.035 �0.020 (0.90) �0.022 (0.90) �0.014 (2.22) �0.005 (0.64) �0.021 �0.010 [0.92]France 0.148 �0.053 (1.19) 0.039 �0.062 (1.94) �0.031 (1.71) �0.001 (0.08) �0.006 (1.28) �0.048 �0.003 [0.26]Germany 0.055 �0.044 (1.86) 0.092 �0.080 (2.13) �0.026 (0.80) �0.016 (2.80) 0.000 (0.01) �0.054 �0.008 [0.11]Greece 0.159 �0.123 (1.21) 0.031 �0.076 (2.34) �0.015 (0.77) �0.017 (0.86) �0.032 (1.40) �0.046 �0.024 [0.00]Italy 0.192 �0.118 (1.25) 0.105 �0.071 (1.66) �0.036 (1.17) �0.010 (1.53) �0.013 (1.77) �0.053 �0.012 [0.51]Netherlands 0.054 �0.042 (1.82) 0.081 �0.066 (1.59) �0.031 (1.12) �0.010 (1.78) �0.001 (0.26) �0.049 �0.006 [0.46]Norway 0.155 �0.133 (1.42) 0.018 �0.047 (1.97) �0.020 (1.06) �0.014 (1.88) �0.005 (0.54) �0.035 �0.010 [0.03]Portugal 0.128 �0.100 (2.17) 0.150 �0.093 (1.02) �0.059 (1.26) �0.110 (7.52) �0.029 (0.55) �0.075 �0.067 [0.16]Spain 0.047 �0.032 (2.12) 0.060 �0.043 (2.08) �0.013 (0.68) �0.015 (1.42) �0.016 (2.41) �0.029 �0.015 [0.38]Sweden 0.153 �0.092 (2.26) 0.188 �0.306 (3.91) �0.073 (1.62) 0.005 (0.05) �0.038 (1.07) �0.147 �0.024 [0.20]Switzerland 0.076 �0.052 (1.45) 0.059 �0.081 (1.51) �0.033 (1.37) 0.003 (0.31) �0.004 (0.59) �0.059 0.000 [0.19]U.K. 0.155 �0.070 (1.26) 0.159 �0.078 (1.21) �0.068 (1.11) �0.008 (0.84) �0.006 (0.49) �0.074 �0.007 [0.56]Hong Kong 0.113 �0.098 (2.35) 0.089 �0.048 (0.88) �0.023 (4.90) �0.030 (0.90) 0.014 (0.43) �0.036 �0.008 [0.03]Japan 0.137 �0.136 (1.74) 0.132 �0.134 (1.85) �0.111 (1.35) �0.029 (0.80) �0.017 (0.41) �0.123 �0.024 [0.05]Singapore 0.078 �0.096 (1.07) 0.017 �0.023 (1.15) �0.037 (1.74) 0.002 (0.07) 0.017 (1.27) �0.030 0.009 [0.04]Taiwan 0.094 �0.065 (1.17) 0.082 �0.143 (6.33) �0.013 (0.71) 0.170 (2.41) �0.022 (0.60) �0.074 0.068 [0.05]New Zealand 0.101 �0.081 (1.99) 0.100 �0.062 (1.30) �0.091 (1.31) �0.020 (0.97) �0.022 (0.98) �0.073 �0.021 [0.37]South Africa 0.291 �0.346 (2.56) 0.142 �0.059 (3.73) �0.035 (0.79) �0.018 (0.88) �0.016 (3.15) �0.048 �0.017 [0.47]

(d) Price index used: FuelCanada 0.069 �0.129 (2.06) 0.057 �0.032 (1.18) �0.067 (1.99) �0.037 (0.69) �0.080 (1.41) �0.053 �0.062 [0.05]MexicoAustria 0.101 �0.060 (1.58) 0.113 �0.094 (2.03) �0.015 (0.41) �0.020 (0.68) �0.017 (0.97) �0.056 �0.019 [0.26]Belgium 0.132 �0.087 (1.77) 0.092 �0.078 (2.22) �0.012 (0.33) �0.009 (0.27) �0.026 (1.35) �0.040 �0.019 [0.70]Denmark 0.036 �0.032 (1.91) 0.120 �0.082 (2.71) �0.012 (0.59) 0.010 (0.50) �0.006 (0.28) �0.052 0.003 [0.00]Finland 0.127 �0.037 (1.20) 0.058 �0.049 (2.14) �0.018 (1.12) 0.000 (0.01) 0.004 (0.28) �0.034 0.002 [0.27]France 0.145 �0.066 (1.50) 0.098 �0.060 (1.53) �0.041 (1.47) �0.008 (0.58) �0.013 (0.93) �0.049 �0.011 [0.39]Germany 0.110 �0.059 (1.65) 0.082 �0.070 (2.25) �0.020 (0.65) �0.013 (0.79) �0.010 (0.92) �0.044 �0.012 [0.50]

(continued on next page)

H.N

akagawa/Journal

ofInternational

Money

andFinance

29(2010)

770e790

780

Table 1 (continued)

TAR TVECM

TAR c l TVECM c l(u)s l(l)s l(u)p l(l)p ls lp LR1

GreeceItaly 0.180 �0.133 (1.60) 0.180 �0.089 (1.47) �0.122 (1.43) �0.017 (0.49) �0.120 (0.44) �0.104 �0.064 [0.44]Netherlands 0.104 �0.103 (1.71) 0.039 �0.086 (2.51) �0.024 (0.88) �0.012 (0.22) �0.032 (3.24) �0.057 �0.021 [0.05]Norway 0.168 �0.052 (1.28) 0.049 �0.037 (1.71) �0.016 (1.06) �0.019 (1.58) �0.006 (0.71) �0.027 �0.013 [0.95]PortugalSpainSwedenSwitzerland 0.252 �0.581 (2.02) 0.148 �0.066 (1.10) �0.041 (0.61) �0.112 (1.45) �0.055 (1.22) �0.053 �0.081 [0.59]U.K. 0.220 �0.114 (1.30) 0.018 �0.042 (2.18) �0.014 (1.12) 0.004 (0.72) �0.003 (0.26) �0.028 0.000 [0.64]Hong KongJapan 0.022 �0.044 (2.31) 0.022 �0.024 (0.98) �0.031 (1.69) �0.026 (1.92) �0.007 (0.94) �0.027 �0.016 [0.08]SingaporeTaiwanNew ZealandSouth Africa

TAR results are based on the TAR model in Eq. (3) for lag length of two.TVECM results are based on the TVECM in Eq. (2) for lag length of two.ls: weighted average of l(u)s and l(l)s (lp: weighted average of l(u)p and l(l)p), computed on the basis of the number of obs vations in the upper and lower regions.Figures in parentheses next to coefficient estimates denote the absolute t-statistics.LR1 is a likelihood ratio statistic for the null hypothesis that the restrictions given by Eq. (4) is true. The test statistics are di ributed as c2 (with the degree of freedom of three) under thenull. We report only the marginal significance level (in squared brackets).

H.N

akagawa/Journal

ofInternational

Money

andFinance

29(2010)

770e790

781

er

st

Table 2Estimation results for TAR and TVECM without lags of the dependent variable(s).

TAR TVECM

TAR c l TAR half-life TVECM c l(u)s l(l)s l(u)p l(l)p TVECM half-life LR2

(a) Price index used: CPI AllCanada 0.012 �0.001 (0.07) 769.8 0.017 �0.003 (0.16) 0.007 (0.31) �0.013 (1.50) 0.005 (0.86) 346.2 [0.21]Mexico 0.040 �0.036 (2.05) 19.1 0.042 0.064 (2.94) 0.069 (5.04) �0.098 (10.96) �0.107 (11.23) 18.9 [0.87]Austria 0.059 �0.050 (2.31) 13.6 0.064 �0.074 (2.07) �0.013 (0.55) �0.013 (3.19) �0.005 (1.09) 12.9 [0.02]Belgium 0.082 �0.048 (2.70) 14.0 0.158 �0.068 (1.47) �0.017 (0.50) �0.016 (3.08) �0.008 (2.39) 12.4 [0.13]Denmark 0.090 �0.049 (2.15) 13.8 0.064 �0.072 (2.31) �0.012 (0.61) �0.005 (1.11) �0.003 (0.46) 14.7 [0.01]Finland 0.158 �0.071 (1.74) 9.5 0.038 �0.030 (1.30) �0.018 (0.75) �0.003 (0.90) 0.000 (0.11) 26.8 [0.51]France 0.035 �0.039 (2.40) 17.4 0.018 �0.061 (2.85) �0.008 (0.47) �0.001 (0.64) �0.007 (2.11) 17.7 [0.02]Germany 0.115 �0.064 (2.31) 10.6 0.076 �0.079 (2.32) �0.017 (0.58) �0.012 (5.07) 0.000 (0.01) 12.5 [0.04]Greece 0.026 �0.041 (2.36) 16.5 0.055 �0.075 (1.68) 0.002 (0.10) �0.006 (0.51) �0.019 (1.53) 13.8 [0.01]Italy 0.175 �0.107 (1.89) 6.1 0.173 �0.193 (1.21) �0.033 (0.65) 0.012 (0.99) �0.012 (2.03) 5.8 [0.02]Netherlands 0.102 �0.057 (2.21) 11.9 0.065 �0.075 (2.30) �0.014 (0.54) �0.008 (3.16) �0.003 (0.98) 13.5 [0.02]Norway 0.083 �0.048 (1.68) 14.0 0.037 �0.053 (1.79) �0.017 (0.91) �0.005 (0.73) �0.002 (0.47) 17.7 [0.10]Portugal 0.142 �0.057 (1.66) 11.9 0.101 �0.038 (0.94) 0.009 (0.23) �0.025 (2.42) �0.034 (2.33) 15.4 [0.31]Spain 0.023 �0.031 (2.43) 22.0 0.025 �0.048 (2.33) �0.002 (0.11) �0.007 (2.34) �0.007 (1.99) 21.3 [0.01]Sweden 0.026 �0.027 (2.01) 25.1 0.027 �0.037 (1.93) �0.009 (0.44) �0.005 (1.23) �0.003 (0.89) 25.3 [0.10]Switzerland 0.057 �0.057 (2.35) 11.8 0.069 �0.096 (2.08) �0.022 (0.80) �0.018 (4.29) �0.001 (0.25) 9.8 [0.01]U.K. 0.169 �0.059 (1.14) 11.5 0.021 �0.038 (1.50) �0.015 (1.08) �0.002 (0.48) �0.003 (1.10) 23.6 [0.28]Hong Kong 0.099 �0.053 (2.59) 12.6 0.099 �0.041 (1.57) �0.118 (7.71) �0.006 (0.52) 0.010 (0.46) 8.6 [0.12]Japan 0.033 �0.041 (1.82) 16.4 0.131 �0.096 (1.10) �0.054 (0.61) �0.038 (2.94) �0.012 (0.56) 6.6 [0.36]Singapore 0.012 �0.008 (0.54) 86.3 0.019 �0.010 (0.40) �0.022 (1.13) 0.008 (0.73) 0.007 (1.11) 81.2 [0.51]Taiwan 0.143 �0.140 (1.79) 4.6 0.084 �0.104 (3.69) �0.015 (0.68) 0.038 (0.67) �0.002 (0.14) 16.4 [0.10]New ZealandSouth Africa 0.225 �0.187 (1.64) 3.4 0.021 0.002 (0.10) �0.021 (0.92) �0.023 (4.74) �0.013 (5.83) 24.9 [0.68]

(b) Price index used: ClothingCanada 0.086 �0.143 (1.61) 4.5 0.059 �0.016 (0.45) 0.023 (0.59) �0.200 (1.90) �0.189 (2.29) 3.3 [0.50]Mexico 0.016 �0.057 (2.54) 11.9 0.015 0.071 (2.71) 0.064 (3.69) �0.126 (6.08) �0.148 (8.32) 9.6 [0.41]AustriaBelgiumDenmarkFinlandFranceGermanyGreeceItaly 0.238 �0.394 (2.18) 1.4 0.026 �0.050 (1.88) �0.007 (0.38) �0.019 (0.80) �0.019 (1.11) 14.2 [0.12]Netherlands 0.242 �0.169 (1.47) 3.7 0.022 �0.058 (1.94) �0.011 (0.61) �0.017 (0.52) �0.045 (1.39) 10.2 [0.64]Norway 0.114 �0.112 (1.57) 5.8 0.142 �0.240 (1.61) �0.011 (0.24) �0.265 (0.52) �0.056 (0.77) 2.1 [0.01]

(continued on next page)

H.N

akagawa/Journal

ofInternational

Money

andFinance

29(2010)

770e790

782

Table 2 (continued)

TAR TVECM

TAR c l TAR half-life TVECM c l(u)s l(l)s l(u) l(l)p TVECM half-life LR2

PortugalSpainSweden 0.167 �0.344 (3.56) 1.6 0.152 �0.303 (2.14) �0.011 (0.35) �0. (1.46) �0.075 (1.87) 1.7 [0.00]SwitzerlandU.K. 0.169 �0.044 (1.06) 15.3 0.116 �0.042 (0.69) �0.017 (0.67) �0. (0.52) �0.012 (0.75) 16.2 [0.53]Hong Kong 0.025 �0.074 (2.67) 9.0 0.083 �0.131 (2.55) �0.022 (0.55) �0. (0.53) �0.246 (2.61) 2.8 [0.43]Japan 0.032 �0.094 (2.59) 7.0 0.082 �0.079 (1.28) �0.034 (0.79) �0. (1.50) �0.119 (1.64) 4.1 [0.92]Singapore 0.127 �0.111 (0.72) 5.9 0.058 �0.030 (0.97) �0.062 (3.01) �0. (0.72) �0.053 (0.72) 6.7 [0.58]Taiwan 0.080 �0.173 (2.08) 3.6 0.099 �0.068 (0.93) �0.016 (0.22) �0. (1.24) �0.557 (2.49) 1.1 [0.73]New ZealandSouth Africa 0.098 �0.085 (1.69) 7.8 0.111 �0.018 (0.40) �0.043 (0.79) �0. (0.26) �0.046 (0.51) 6.3 [0.70]

(c) Price index used: FoodCanada 0.041 �0.005 (0.20) 138.3 0.016 0.004 (0.14) 0.009 (0.41) �0. (1.64) 0.001 (0.09) 92.1 [0.24]Mexico 0.038 �0.041 (1.63) 16.8 0.026 0.064 (2.58) 0.060 (3.77) �0. (9.93) �0.106 (9.83) 16.6 [0.70]Austria 0.077 �0.054 (1.96) 12.4 0.037 �0.070 (2.19) �0.014 (0.69) �0. (1.91) �0.009 (1.08) 13.0 [0.03]Belgium 0.112 �0.058 (2.04) 11.6 0.040 �0.061 (2.52) �0.006 (0.31) �0. (3.07) �0.014 (2.31) 14.1 [0.02]Denmark 0.078 �0.053 (2.03) 12.8 0.016 �0.060 (2.14) �0.008 (0.41) �0. (1.03) �0.014 (1.73) 15.2 [0.06]Finland 0.190 �0.085 (1.54) 7.8 0.035 �0.016 (0.67) �0.014 (0.52) �0. (1.91) �0.005 (0.73) 28.5 [0.70]France 0.015 �0.034 (2.20) 19.9 0.015 �0.064 (2.61) �0.008 (0.48) �0. (0.35) �0.009 (2.46) 16.4 [0.03]Germany 0.071 �0.049 (2.04) 13.7 0.040 �0.075 (2.44) �0.020 (0.88) �0. (2.54) �0.001 (0.15) 12.7 [0.02]Greece 0.146 �0.111 (2.08) 5.9 0.042 �0.092 (2.25) �0.009 (0.42) 0. (0.06) �0.051 (1.56) 8.8 [0.47]Italy 0.190 �0.140 (1.80) 4.6 0.103 �0.105 (2.21) �0.019 (0.60) �0. (1.14) �0.013 (1.55) 9.3 [0.04]Netherlands 0.093 �0.052 (1.88) 12.9 0.081 �0.083 (2.57) �0.021 (0.66) �0. (1.83) �0.003 (0.43) 11.7 [0.07]Norway 0.183 �0.272 (1.78) 2.2 0.018 �0.029 (1.04) �0.013 (0.66) �0. (2.19) �0.006 (0.61) 21.3 [0.29]Portugal 0.056 �0.042 (1.80) 16.1 0.150 �0.026 (0.31) �0.025 (0.43) �0. (5.74) �0.065 (1.11) 6.8 [0.85]Spain 0.249 �0.170 (2.55) 3.7 0.045 �0.046 (1.81) 0.003 (0.14) �0. (2.30) �0.014 (2.23) 17.7 [0.01]Sweden 0.163 �0.091 (2.19) 7.3 0.077 �0.035 (1.07) �0.009 (0.38) �0. (2.76) �0.011 (1.16) 15.6 [0.04]Switzerland 0.024 �0.045 (1.94) 15.1 0.030 �0.081 (1.86) �0.028 (1.27) �0. (0.29) �0.004 (0.54) 11.6 [0.09]U.K. 0.156 �0.057 (1.13) 11.7 0.020 �0.042 (1.45) �0.019 (1.32) �0. (0.42) �0.002 (0.36) 21.0 [0.40]Hong Kong 0.144 �0.294 (3.92) 2.0 0.103 �0.074 (0.98) �0.039 (6.45) �0. (0.86) �0.002 (0.04) 8.8 [0.09]Japan 0.036 �0.053 (2.00) 12.8 0.081 �0.074 (1.28) �0.044 (0.84) �0. (1.41) �0.026 (0.92) 7.6 [0.53]Singapore 0.104 �0.389 (1.78) 1.4 0.019 �0.008 (0.33) �0.040 (1.45) �0. (0.17) 0.008 (0.51) 30.5 [0.51]Taiwan 0.026 �0.050 (1.80) 13.7 0.012 �0.051 (3.08) �0.018 (1.58) �0. (1.22) �0.004 (0.18) 10.9 [0.00]New Zealand 0.107 �0.068 (1.72) 9.8 0.100 �0.060 (1.32) �0.012 (0.15) �0. (1.27) �0.049 (2.05) 8.8 [0.66]South Africa 0.306 �0.289 (1.86) 2.0 0.149 �0.032 (0.95) �0.045 (0.93) �0. (0.28) �0.011 (1.86) 14.2 [0.75]

(d) Price index used: FuelCanada 0.071 �0.146 (2.40) 4.4 0.069 �0.070 (1.73) �0.075 (1.34) �0. (0.73) �0.078 (1.10) 4.6 [0.76]MexicoAustria 0.052 �0.054 (2.21) 12.6 0.105 �0.128 (2.75) 0.001 (0.02) 0. (0.23) �0.032 (1.89) 8.8 [0.04]

H.N

akagawa/Journal

ofInternational

Money

andFinance

29(2010)

770e790

783

p

291

013046082051314

101

029100011015007013002010001006008016078020032003002036031005050030007

056

007

Belgium 0.066 �0.061 (2.26) 11.1 0.132 �0.114 (2.24) �0.005 (0.10) �0.005 (0.11) �0.045 (1.74) 7.9 [0.20]Denmark 0.033 �0.029 (1.76) 23.4 0.033 �0.060 (2.45) �0.010 (0.73) 0.006 (0.49) �0.005 (0.28) 19.7 [0.08]Finland 0.030 �0.025 (1.62) 27.0 0.058 �0.047 (1.81) �0.021 (1.12) 0. 4 (0.30) 0.002 (0.14) 22.0 [0.28]France 0.053 �0.050 (2.37) 13.5 0.023 �0.061 (2.66) �0.006 (0.33) �0. 4 (0.33) �0.015 (1.32) 15.8 [0.09]Germany 0.043 �0.057 (2.73) 11.7 0.041 �0.081 (3.47) �0.010 (0.45) �0. 2 (0.98) �0.011 (0.82) 11.8 [0.01]GreeceItaly 0.178 �0.131 (1.72) 4.9 0.180 �0.117 (1.52) �0.064 (0.71) �0. 3 (0.24) �0.093 (0.22) 4.5 [0.89]Netherlands 0.198 �0.516 (3.09) 1.0 0.040 �0.114 (3.34) �0.009 (0.32) �0. 6 (0.31) �0.035 (3.27) 7.6 [0.02]Norway 0.122 �0.049 (1.76) 13.7 0.049 �0.042 (1.54) �0.013 (0.79) �0. 5 (1.26) �0.005 (0.53) 18.1 [0.09]PortugalSpainSwedenSwitzerland 0.250 �0.627 (3.46) 0.7 0.178 �0.118 (2.03) �0.100 (0.72) �0. 9 (1.68) �0.045 (0.41) 2.8 [0.27]U.K. 0.133 �0.047 (1.20) 14.4 0.114 �0.065 (1.03) �0.022 (0.96) 0. 6 (0.42) �0.006 (0.16) 15.6 [0.43]Hong KongJapan 0.022 �0.035 (1.76) 19.6 0.022 �0.015 (0.56) �0.024 (1.15) �0. 6 (1.87) �0.006 (0.71) 19.2 [0.69]SingaporeTaiwanNew ZealandSouth Africa

TAR results are based on the TAR model in Eq. (3) without lags of the dependent variable (i.e., TAR model with autoregr sive order of one).TVECM results are based on the TVECM in Eq. (2) without lags of the dependent variables.TAR half-life is calculated as ln(0.5)/ln(1 þ l).TVECM half-life is calculated as ln(0.5)/ln(1 þ lTVECM), where lTVECM is the implied convergence speed of the real exchan rate, which is computed from the average of upper and lowerregion convergence speeds (i.e., average of l(u)s þ l(u)p and l(l)s þ l(l)p).Figures in parentheses next to coefficient estimates denote the absolute t-statistics.LR2 is a likelihood ratio statistic for the null hypothesis that the restriction given by Eq. (5) is true. The test statistics are dis ibuted as c2 (with the degree of freedom of one) under the null.We report only the marginal significance level (in squared brackets).

H.N

akagawa/Journal

ofInternational

Money

andFinance

29(2010)

770e790

784

000001

010101

1600

02

es

ge

tr

Fig. 1. Estimated speeds of adjustment for nominal exchange rates and relative prices.


stronger than that induced by changes in relative prices.19 This is a noteworthy property of the nominalexchange rate, and we will draw some policy implications from this later in the concluding section.

It is also interesting to observe the apparent asymmetry in adjustment of exchange rates: thenominal exchange rate (expressed as units of U.S. dollars per unit of foreign currency) exhibitssubstantially stronger error-correcting force when the real exchange rate is above its equilibrium thanwhen it is below the equilibrium level. This point is made starkly in Fig. 2 (we are comparing l(u)s and

19 Caution should be exercised in interpreting these results on adjustment speed estimates, however. If, as in the sticky-pricemodel of exchange rate determination, in response to a nominal shock the exchange rate overshoots and deviates from itsequilibrium more than prices do, then it is not surprising that the exchange adjusts more. On this issue, Engel and Morley(2001) stress the distinction between: (i) the response of exchange rate and relative prices to a deviation from PPP; and (ii)the response of exchange rate and relative prices respectively to exchange rate and prices gaps. In our study, the focus is on theformer aspect of adjustment.

Fig. 2. Asymmetry in nominal exchange rate adjustment.


l(l)s from Table 1). For the relative prices, no such pattern is found. Possibly, such an asymmetricadjustment process of exchange rates may be reflecting official foreign exchange intervention andaccompanyingmarket expectations. For instance, monetary authorities of countries, whose exports areof great concern, maywell be sensitive to an appreciation of their currencies against the U.S. dollar, andthus may have a tendency to intervene in the foreign exchange market to prevent their home currencyfrom appreciating against the dollar. In order to obtain a precise picture, further study would benecessary, of course. Given the space limitations, we defer the issue to future research.

6. Conclusions and discussions

Inspired by the increasing evidence of nonlinear mean reversion in real exchange rates or nonlinearPPP reversion, this paper has investigated the driving forces underlying nonlinear real exchange rateadjustment. The dynamics of the nominal exchange rate and relative prices are analyzed in the contextof threshold error-correction models (TVECMs) e bivariate nonlinear systems in which one can detectthreshold cointegration between the two key variables, and thus capture properly the thresholdbehavior of real exchange rates. In our TVECM application, we have estimated threshold values jointly


with adjustment-speed parameters. Also, equally important, our investigation of the multivariatestructure of TVECM as compared with the univariate approach of TAR model e by way of testing theconditions that imply the empirical compatibility of these two models e has provided confirmationthat it is important to take into consideration the interactions between, and the individual dynamics of,the nominal exchange rate and relative prices.

Estimates of speeds of adjustment for the nominal exchange rate and relative prices in TVECMsunravel their relative contribution to real exchange rate adjustment when parity reversion is indeedpresent. Our estimated adjustment-speed parameters imply that most of the reversion toward PPPis ascribed to the exchange rate movements. (There is, of course, a caveat in interpreting theestimates on adjustment speeds. Although the nominal exchange rate may adjust more than pricesto correct deviations from PPP, the speed of adjustment of the nominal exchange rate to itsequilibrium may not be as fast. This point is stressed by Engel and Morley, 2001, as mentioned infootnote.19)

From the above finding, some policy implications can be drawn as to the choice of exchange rateregime. As advocates of a flexible exchange rate regime contend, in the world of price rigidity,exchange rate changes under the floating regime can insulate the economy from effects of an exog-enous price shocks originated abroad. Exchange rate flexibility buffers shocks, and if inflation abroadis not transmitted to affect domestic prices (i.e., exchange rate movements help restore PPP), this willleave the real side of the economy unaffected. More generally, given the unresponsiveness of prices, inresponse to real shocks to the economy (e.g., export demand shock, productivity shocks), theexchange rate movement e despite its short-run fluctuations e can play a role in promotingadjustment over the longer term. In that case, the view that exchange rate movements are erraticfluctuations (and send the real exchange rate off the long-run equilibrium path) does not give anaccurate picture of the international adjustment mechanism. Although the one-sided story cannot bepushed too far, in light of this potentially stabilizing role of the nominal exchange rate and unre-sponsiveness of prices, a policy regime aimed at limiting exchange rate flexibility can lead to exchangerate misalignments, forcing infrequent but significant parity changes that bring about a kind of painfuleconomy-wide adjustment we have seen in exchange rate realignments under the fixed or adjustablepeg system. Another interesting observation from our empirical results is the apparent asymmetry inexchange rate adjustment. The pronounced asymmetry may be reflecting the lack of internationalcoordination of macroeconomic policies. This issue e probably in relation with official foreignexchange intervention, for example e may be of interest to future work.

Ourempirical frameworkhas studied thedynamicsof nominal exchange ratesand relativeprices thatdrive the nonlinearly mean-reverting process of real exchange rates. The theoretical side e the mech-anism that addresses the observedmovements of exchange rates and pricese is yet to be fully explored,however. To better understand the nature and behavior of the variables (e.g., nonlinear reversion in realexchange rates; exchange rate variations as a considerable source of adjustment; price stickiness),a theoretical model is needed. To generate the observed price stickiness, amodel based, for example, onpricing-to-market framework with sunk costs may serve. The real challenge is how to reconcile sucha framework with the nominal exchange rate movement that displays day-to-day volatility butcontributes to international price convergence, as judged by the adjustment-speed measure.

On the empirical side, a possible avenue of future research would be to consider out-of-sampleforecasting ability of TVECMs, or a more general class of nonlinear VECM. Nonlinearity can becharacterized in other forms besides threshold, and a multivariate framework can include variablese exchange rates, prices, or others e that bear a long-run relationship through cointegration. Somestudies have shown that standard empirical models with a multivariate framework (e.g., a linearVECM involving relevant variables) outperform a random walk model in out-of-sample forecastingof future exchange rates at certain horizons. Accordingly, empirical exploration of multivariatemodels that incorporate nonlinearities e such as TVECM here e may prove fruitful in exchange rateforecasting.20

20 Recent work by Clarida et al. (2003), for example, conducts an empirical investigation in this spirit. They apply a multi-variate Markov-switching VECM to exchange rate forecasting and report encouraging results.


Acknowledgements

This paper is based on chapter 3 of my Ph.D. dissertation at Columbia University. I am heavilyindebted to Richard Clarida for his helpful guidance along this line of research. The main idea that hesuggested has grown to be this paper. I am grateful to Benjamin Broadbent, Richard Clarida, AnnHarrison, Robert Hodrick, and Hideshi Itoh for offering helpful suggestions. For detailed and thoughtfulcomments, I would like to thank the editor of this journal, James R. Lothian, and an anonymous referee.My thanks also go to Obstfeld and Taylor for providing the Rats code, and to Engel and Rogers formaking available the data. Any remaining errors are my own.

Appendix 1In below, we lay out a TVECMwith intercepts; i.e., it is written in unrestrictive form. Considering the

possibility of a long-run trend in real exchange rates, we decompose the real exchange rate qt into thedeterministic component and the detrended component ~qt as follows:

qt ¼ aþ bt þ ~qt ;

where a¼ asþ ap and b¼ b(k)sþ b(k)p for k¼ u,m,l; asþ b(k)st and apþ b(k)pt represent the deterministiccomponents of st and pt, respectively. While threshold effect is not allowed in the deterministiccomponent of qt, we may admit regime-specific drift in st and pt. In that case, the band-TVECM wouldbe written in unrestrictive form:

Dst ¼

8>>>>>>>><>>>>>>>>:

aðuÞs þPp�1i¼1 r

ðuÞssi Dst�i þ

Pp�1i¼1 r

ðuÞspi Dpt�i þ lðuÞs

�~qt�1 � c

�þ 3ðuÞst if ~qt�1 > c;

aðmÞs þPp�1i¼1 r

ðmÞssi Dst�i þ

Pp�1i¼1 r

ðmÞspi Dpt�i þ 3ðmÞs

t if��~qt�1

�� c;

aðlÞs þPp�1i¼1 r

ðlÞssi Dst�i þ

Pp�1i¼1 r

ðlÞspi Dpt�i þ lðlÞs

�~qt�1 þ c

�þ 3

ðlÞst if ~qt�1 < �c;

Dpt ¼

8>>>>>>>><>>>>>>>>:

aðuÞp þPp�1i¼1 r

ðuÞpsi Dst�i þ

Pp�1i¼1 r

ðuÞppi Dpt�i þ lðuÞp

�~qt�1 � c

�þ 3ðuÞpt if ~qt�1 > c;

aðmÞp þPp�1i¼1 r

ðmÞpsi Dst�i þ

Pp�1i¼1 r

ðmÞppi Dpt�i þ 3ðmÞp

t if��~qt�1

�� c;

aðlÞp þPp�1i¼1 r

ðlÞpsi Dst�i þ

Pp�1i¼1 r

ðlÞppi Dpt�i þ lðlÞp

�~qt�1 þ c

�þ 3

ðlÞpt if ~qt�1 < �c;

where aðkÞs ¼ bðkÞs �Pp�1i¼1 r

ðkÞssi bðkÞs þPp�1

i¼1 rðkÞspi bðkÞs and aðkÞp ¼ bðkÞp �Pp�1

i¼1 rðkÞpsi bðkÞpþPp�1

i¼1 rðuÞppi bðkÞp for k ¼ u,m,l. From the empirical point of view, the TVECM with the intercepts allows

us to work with exchange rate and price data without preliminary detrending.

Appendix 2The log likelihood function for the TVECM is written as

lnLðq; cÞ ¼ �X

IðuÞðqt�1Þ

12

�2lnð2pÞ þ ln

��UðuÞ��þ 3ðuÞ0t UðuÞ�1

3ðuÞt

��

XIðmÞðqt�1Þ

12

�2lnð2pÞ þ ln

��UðmÞ��

þ 3ðmÞ0t UðmÞ�1

3ðmÞt

��

XIðlÞðqt�1Þ

12

�2lnð2pÞ þ ln

��UðlÞ��þ 3ðlÞ0t UðlÞ�1

3ðlÞt

�

The errors are assumed to be i.i.d. Gaussian; 3ðkÞt wNð0;UðkÞÞ, where


3ðkÞt ¼ 3ðkÞst

ðkÞp and UðkÞ ¼�

sðkÞss sðkÞspðkÞps ðkÞpp

�for k ¼ u;m; l:

3t

!s s

Included in q are the coefficient parameters and the variance-covariance matrix of residuals; i.e.,

q ¼�lðkÞs; lðkÞp; rðkÞss; rðkÞsp; rðkÞps; rðkÞpp;UðkÞ� for k ¼ u;m; l:

The indicator functions are defined as

IðuÞðqt�1Þ ¼1 if qt�1 > c

0 otherwise

IðmÞðqt�1Þ ¼1 if jqt�1j � c

0 otherwise

IðlÞðqt�1Þ ¼1 if qt�1 < �c

0 otherwise

References

Balke, N.S., Fomby, T.B., 1997. Threshold cointegration. International Economic Review 38, 627e645.Balassa, B., 1964. The purchasing power parity doctrine: a reappraisal. Journal of Political Economy 72, 584e596.Cheung, Y.W., Lai, K.S., 1993. Long-run purchasing power parity during the recent float. Journal of International Economics 34,

181e192.Clarida, R.H., Sarno, L., Taylor, M.P., Valente, G., 2003. The out-of-sample success of term structure models as exchange rate

predictors: a step beyond. Journal of International Economics 60, 61e85.Dixit, A.K., 1989. Hysteresis, import penetration, and exchange rate pass-through. Quarterly Journal of Economics 104, 205e228.Dumas, B., 1992. Dynamic equilibrium and the real exchange rate in a spatially separated world. Review of Financial Studies 5,

153e180.Edison, H.J., Gagnon, J.E., Melick, W.R., 1997. Understanding the empirical literature on purchasing power parity: the post-

Bretton Woods era. Journal of International Money and Finance 16, 1e17.Enders, W., Granger, C.W.J., 1998. Unit-root tests and asymmetric adjustment with an example using the term structure of

interest rates. Journal of Business & Economic Statistics 16, 304e311.Engel, C., Morley, J.C., 2001. The Adjustment of Prices and the Adjustment of the Exchange Rate. NBER. Working Paper No. 8550.Engel, C., Rogers, J.H., 1998. Regional patterns in the law of one price: the roles of geography vs. currencies. In: Frankel, J.A. (Ed.),

The Regionalization of the World Economy. University of Chicago Press, Chicago IL and London, pp. 153e183.Engle, R.F., Granger, C.W.J., 1987. Co-integration and error correction: representation, estimation, and testing. Econometrica 55,

251e276.Frankel, J.A., Rose, A.K., 1996. A panel project on purchasing power parity: mean reversion within and between countries.

Journal of International Economics 40, 209e224.Froot, K.A., Rogoff, K., 1995. Perspectives on PPP and long-run real exchange rates. In: Grossman, G., Rogoff, K. (Eds.), Handbook

of International Economics, vol. 3. North-Holland, Amsterdam.Hansen, B.E., Seo, B., 2002. Testing for two-regime threshold cointegration in vector error-correction models. Journal of

Econometrics 110, 293e318.Heckscher, E.F., 1916. Vaxelkursens grundval vid pappersmyntfot. Ekonomisk Tidskrift 18, 309e312.Krugman, P.R., 1989. Exchange Rate Instability. MIT Press, Cambridge, MA.Kugler, P., Lenz, C., 1993. Multivariate cointegration analysis and the long-run validity of PPP. Review of Economics and Statistics

75, 180e184.Lo, M.C., Zivot, E., 2001. Threshold cointegration and nonlinear adjustment to the law of one price. Macroeconomic Dynamics 5,

533e576.Lothian, J.R., Taylor, M.P., 1996. Real exchange rate behavior: the recent float from the perspective of the past two centuries.

Journal of Political Economy 104, 488e509.Lothian, J.R., Taylor, M.P., 2008. Real exchange rates over the past centuries: how important is the HarrodeBalassaeSamuelson

effect? Economic Journal 118, 1742e1763.Mark, N., 1990. Real and nominal exchange rates in the long run: an empirical investigation. Journal of International Economics

28, 115e136.MacDonald, R., 1993. Long-run purchasing power parity: is it for real? Review of Economics and Statistics 75, 690e695.Michael, P., Nobay, A.R., Peel, D.A., 1997. Transaction costs and nonlinear adjustment in real exchange rates: an empirical

investigation. Journal of Political Economy 105, 862e879.Obstfeld, M., Taylor, A.M., 1997. Nonlinear aspects of goods-market arbitrage and adjustment: Heckscher’s commodity points

revisited. Journal of the Japanese and International Economics 11, 441e479.O’Connell, P.G.J., 1998. Market frictions and real exchange rates. Journal of International Money and Finance 17, 71e95.Rogoff, K., 1996. The purchasing power parity puzzle. Journal of Economic Literature 34, 647e668.


Rothman, P., van Dijk, D., Franses, P.H., 2001. Multivariate STAR analysis of money-output relationship. MacroeconomicDynamics 5, 506e532.

Samuelson, P., 1964. Theoretical notes on trade problems. Review of Economics and Statistics 46, 145e154.Sercu, P., Uppal, R., Van Hulle, C., 1995. The exchange rate in the presence of transaction costs: implications for tests of

purchasing power parity. Journal of Finance 50, 1309e1319.Taylor, A.M., 2001. Potential pitfalls for the purchasing-power-parity puzzle? Sampling and specification biases in mean-

reversion tests of the law of one price. Econometrica 69, 473e498.Taylor, M.P., 1988. An empirical examination of long-run purchasing power parity using cointegration techniques. Applied

Economics 20, 1369e1381.Taylor, M.P., Peel, D.A., Sarno, L., 2001. Nonlinear mean-reversion in real exchange rates: toward a solution to the purchasing

power parity puzzles. International Economic Review 42, 1015e1042.Tsay, R.S., 1998. Testing and modeling multivariate threshold models. Journal of the American Statistical Association 93,

1188e1202.Williams, J.C., Wright, B.D., 1991. Storage and Commodity Markets. Cambridge University Press, Cambridge, UK.

investigating nonlinearities in real exchange rate adjustment: threshold cointegration and the...

Documents