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Deviations from purchasing power parityunder different exchange rate regimes: Do they

revert and, if so, how?

Lucio Sarno

a,*

, Giorgio Valente

b

a Warwick Business School, University of Warwick and Centre for Economic Policy Research (CEPR),

United Kingdomb Department of Finance, Chinese University of Hong Kong, Hong Kong

Received 26 July 2005; accepted 9 December 2005Available online 12 July 2006

Abstract

We propose an empirical model for deviations from long-run purchasing power parity (PPP) thatsimultaneously accounts for three key features: (i) adjustment toward PPP may occur via nominalexchange rates and relative prices at different speeds; (ii) different exchange rate regimes may gener-ate regime shifts in the structural dynamics of PPP deviations; (iii) nonlinear reversion toward PPP inresponse to shocks. This empirical framework encompasses and synthesizes much previous empiricalresearch. Using over a century of data for the G5 countries, we provide evidence that long-run PPPholds, the relative importance of nominal exchange rates and prices in restoring PPP varies over timeand across different exchange rate regimes, and reversion to PPP occurs nonlinearly, at a speed thatis fairly consistent with the nominal rigidities suggested by conventional open economy models. 2006 Elsevier B.V. All rights reserved.

JEL classification: F31

Keywords: Real exchange rate; Purchasing power parity; Exchange rate regimes; Nonlinearity

0378-4266/$ - see front matter 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.jbankfin.2005.12.007

*

Corresponding author. Tel.: +44 2476 528219; fax: +44 2476 572871.E-mail address: [email protected] (L. Sarno).

Journal of Banking & Finance 30 (2006) 31473169

www.elsevier.com/locate/jbf
mailto:[email protected]:[email protected]


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1. Introduction

1.1. Overview

The purchasing power parity (PPP) hypothesis states that national price levels shouldbe equal when expressed in a common currency. A large literature in international financehas examined empirically the validity of PPP over the long run either by testing whethernominal exchange rates and relative prices move together in the long run or by testingwhether the real exchange rate has a tendency to revert to a stable equilibrium level overtime. The latter approach is motivated by the fact that the real exchange rate may bedefined as the nominal exchange rate adjusted for relative national price levels, and there-fore variations in the real exchange rate represent deviations from PPP, which must be sta-tionary if long-run PPP holds (see the surveys of Froot and Rogoff, 1995; Rogoff, 1996;Sarno and Taylor, 2002; Taylor and Taylor, 2004; Sarno, 2005).

Although long-run PPP is such a simple proposition about exchange rate behavior, ithas attracted the attention of researchers for decades because it has important economicimplications on several fronts. In particular, the degree of persistence in the real exchangerate can be used to infer what the principal impulses driving exchange rate movements are.For example, if the real exchange rate is highly persistent or close to a random walk, thenthe shocks are likely to be real-side, principally technology shocks, whereas if it is not verypersistent, then the shocks must be principally to aggregate demand, such as, for example,innovations to monetary policy (Rogoff, 1996). Further, from a theoretical perspective, ifPPP is not a valid long-run international parity condition, this casts doubts on the predic-

tions of much open economy macroeconomics that is based on the assumption of long-runPPP. Indeed, the implications of open economy dynamic models are very sensitive to thepresence or absence of a unit root in the real exchange rate (e.g. Lane, 2001; Sarno, 2001).Finally, estimates of PPP exchange rates are often used for practical purposes such asdetermining the degree of misalignment of the nominal exchange rate and the appropriatepolicy response, the setting of exchange rate parities, and the international comparison ofnational income levels. These practical uses of the PPP concept would obviously be of lim-ited use if PPP deviations contain a unit root.

Regardless of the great interest in this area of research, manifested by the large numberof papers on PPP published over the last few decades, and despite the increasing quality of

data sets utilized and the econometric techniques employed, the validity of long-run PPPand the properties of PPP deviations remain the subject of ongoing controversies. Specif-ically, earlier cointegration studies generally reported the absence of significant meanreversion of the real exchange rate for the recent floating experience (e.g. Mark, 1990),but were supportive of reversion toward PPP for the gold standard period (Dieboldet al., 1991), for the interwar float (Taylor and McMahon, 1988), for the 1950s USCana-dian float (McNown and Wallace, 1989), and for the exchange rates of high-inflationcountries (Choudhry et al., 1991). Some applied work on long-run PPP among the majorindustrialized economies has, however, been more favorable to the long-run PPP hypoth-esis for the recent float (e.g. Corbae and Ouliaris, 1988; Cheung and Lai, 1993, 1994, 1998;

Frankel and Rose, 1996; Coe and Serletis, 2002; Serletis and Gogas, 2004).One well-documented explanation for the inability to find evidence of long-run PPP is

the low power of conventional unit root and cointegration tests with a sample span cor-responding to the length of the recent float (Froot and Rogoff, 1995; Lothian and Taylor,

3148 L. Sarno, G. Valente / Journal of Banking & Finance 30 (2006) 31473169
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1997). Researchers have sought to overcome the power problem either through long-spanstudies (e.g. Kim, 1990; Lothian and Taylor, 1996; Taylor, 2002) or through panel unitroot studies (e.g. Abuaf and Jorion, 1990; Frankel and Rose, 1996; OConnell, 1998; Pap-ell, 1998; Sarno and Taylor, 1998; Taylor and Sarno, 1998) or through time-series models

that account for the possibility of nonlinear mean reversion toward PPP (e.g. Michaelet al., 1997; Obstfeld and Taylor, 1997; Baum et al., 2001; Coakley and Fuertes, 2001; Tay-lor, 2001; Taylor et al., 2001; Nakagawa, 2002; Imbs et al., 2003). However, whether ornot the long-span or panel-data studies do in fact answer the question whether PPP holdsin the long run remains contentious (e.g. Engel, 1999, 2000).1 With respect to studies usingnonlinear models of the real exchange rate, they unanimously support the validity of long-run PPP (see Michael et al., 1997; Obstfeld and Taylor, 1997; Baum et al., 2001; Coakleyand Fuertes, 2001; OConnell and Wei, 2001; Taylor et al., 2001; Imbs et al., 2003). How-ever, since they are based on univariate regressions for the real exchange rate and do notallow for the possibility that the dynamics of PPP deviations be affected by monetary andexchange rate regimes, to date these studies have not shed light on whether nominalexchange rates or prices drive the adjustment toward the PPP equilibrium and do notinvestigate the role of different nominal regimes on the behavior of PPP deviations.

1.2. Questions addressed and approach

In light of the evidence provided by this literature, there remain at least three importantissues in this area of research. First, it is still controversial whether long-run PPP is vali-dated by the data, even among the major G5 economies. Second, it is debatable whether,

when PPP is validated by the data, adjustment toward the long-run equilibrium leveldefined by PPP is driven primarily by the exchange rate, by relative prices, or by bothof them. Third, it is puzzling why the majority of studies find empirical estimates of thepersistence of PPP deviations that are too high to be explained by conventional nominalrigidities and cannot be reconciled with the high short-term volatility of real exchangerates.

Our empirical analysis is devoted to shed light on all of these three issues. We start fromnoting three features that we view as potentially important in designing a suitable modelfor the deviations from PPP. The first feature is that the model needs to allow for the factthat adjustment toward PPP is likely to occur at different speeds via nominal exchange

rates and prices. The vast majority ofempirical studies on PPP is based on univariate rep-resentations of the real exchange rate.2 This approach is only valid if certain common fac-tor restrictions in the unknown data generating process linking exchange rates and pricesare satisfied. We find that these common factor restrictions are generally rejected by thedata, implying a loss of power in testing the null hypothesis that the real exchange rate

1 As far as the long-span studies are concerned, the long samples required to generate a reasonable level ofstatistical power with standard tests may potentially be inappropriate because of differences in real exchange ratebehavior both across different historical periods and nominal exchange rate regimes (e.g. Baxter and Stockman,1989; Taylor, 2002). As for panel-data studies, these provide mixed evidence. While, for example, Abuaf and

Jorion (1990), Frankel and Rose (1996) and Taylor and Sarno (1998) find results favorable to long-run PPP, theempirical evidence reported by OConnell (1998) and Papell (1998) rejects PPP.2 Exceptions which have considered this issue include, inter alia, Edison (1987), Edison et al. (1997), Goldfajn

and Valdes (1999), Coakley and Fuertes (2000), Engel and Morley (2002) and Cheung et al. (2004). See alsoCoakley et al. (2005).

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is nonstationary (or that PPP is invalid) in conventional testing procedures. Employing amodel which does not impose these restrictions increases test power, whilst allowing us toshed light on the relative importance of nominal exchange rates and prices in restoring thePPP equilibrium. The second desirable feature is that the model allows explicitly for the

possibility that different monetary and exchange rate regimes generate regime shifts inthe structural dynamics of PPP deviations, especially when using long-spans of data.3

The third feature is that the model might be nonlinear, in accordance with the growingevidence that exchange rate dynamics displays statistically and economically importantnonlinearities. In particular, some recent literature provides evidence that real exchangerates display nonlinear mean reversion toward PPP for a discussion of nonlinear PPPmodels, see the relevant sections in the surveys of Taylor and Taylor (2004) and Sarno(2005). These nonlinear models generally imply an equilibrium level of the real exchangerate in the neighborhood of which the behavior of the log-level of the real exchange rate isclose to a random walk, becoming increasingly mean reverting with the absolute size of thedeviation from equilibrium. This is consistent with a number of recent theoretical contri-butions on the nature of real exchange rate dynamics in the presence of international arbi-trage costs (e.g. Dumas, 1992; Sercu et al., 1995). Also, the impulse response functionsimplied by these real exchange rate models generate, because of the nonlinearity, half-livesof shocks to real exchange rates that vary both with the size of the shock and with the ini-tial conditions. By taking account of statistically significant nonlinearities, the speed ofreal exchange rate adjustment implied by these models is found to be much faster than typ-ically recorded in the relevant literature based on data for the recent float (e.g. Tayloret al., 2001).

In this paper we extend the long-span data used by Obstfeld and Taylor (2004) andapply a general modelling methodology in which regime changes in the data generatingprocess are explicitly allowed for, focusing on the G5 countries. Taylor (2002) has exam-ined these time series employing single-equation and panel linear econometric methods.His empirical evidence is generally supportive of long-run PPP and shows that PPP devi-ations have similar half-lives across the different monetary regimes of the last century, butmuch larger shocks to the real exchange rate process have characterized floating exchangerate regimes relative to fixed exchange rate regimes. Our paper builds on and extends thework ofTaylor (2002) and other scholars who have contributed to the literature on long-span real exchange rate behavior including Edison (1987), Kim (1990), Lothian and Tay-

lor (1996), Michael et al. (1997) in several directions. We re-examine whether long-runPPP is valid and measure the relative importance of exchange rates and relative pricesin driving the adjustment toward the long-run PPP equilibrium across different exchangerate regimes, including the Gold Standard, the Bretton Woods period, and the recent float.With over a century of data and different exchange rate arrangements over our sample per-iod, we tentatively hypothesize that during floating exchange rate periods, the nominalexchange rate may be relatively more important in restoring departures from long-runequilibrium, while the relative price should restore long-run equilibrium during fixed

3 Indeed, a large literature has provided mounting evidence that the distribution of nominal and real exchangerate changes is well described by a mixture of normal distributions and that a regime-switching model may be agood characterization of exchange rate behavior (e.g. see Engel and Hamilton, 1990; LeBaron, 1992; Engel, 1994;Engel and Kim, 1999; Clarida et al., 2003; Dueker and Neely, 2005; Sarno and Valente, 2005).

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exchange rate periods.4 Over the sample period examined, the economic history of thecountries involved has seen a number of fundamental changes in monetary and exchangerate regimes, institutional structure and policy targets which, in addition to the continuousevolution of the financial system and various nominal and real shocks, represent serious

potential pitfalls to researchers attempting to find an empirical model of the deviationsfrom PPP that is stable over the full sample. The time-invariant, linear framework gener-ally adopted by the literature is not suitable for the purposes of this paper. We investigatewhether allowing for regime-switching in the underlying data-generating process is an ade-quate characterization that is capable of capturing the impact of the different monetaryregimes of the last century on the dynamics of exchange rates and relative prices. Thisis done through estimating a fairly general Markov-switching vector error correctionmodel (MS-VECM) which characterizes the dynamic relationship between exchange ratesand relative prices allowing for regime shifts in the parameters as well as for nonlinearreversion toward long-run PPP in each regime.

1.3. Main results

The results are supportive of long-run PPP for each of the four major exchange ratesexamined and of our basic conjecture: we find that during fixed exchange rate regimes, rel-ative prices adjust to restore deviations from long-run equilibrium, while nominalexchange rates bear most of the burden of adjustment during flexible exchange rateregimes. The estimated transition probabilities are consistent with the general result thatthe relative importance of exchange rates and relative prices in restoring the long-run equi-

librium level of the exchange rate varies over time and is affected by the nominal exchangerate arrangement in operation.Further, the estimated half-lives of the regime-dependent nonlinear exchange rate mod-

els are sensibly different for fixed and floating regimes. During fixed exchange rate regimes,shocks to the PPP equilibrium relationship may be very persistent, implying half livesonaverage across the exchange rates considered from over five years for large real exchangerate shocks of 20% to almost ten years for small shocks of 1%. However, the correspond-ing half-lives during floating exchange rate regimes are drastically shorter, since the nom-inal exchange rate is allowed to operate and contribute to restoring PPP. In fact, shockswill last for less than one year on average for 20% shocks, with a maximum of 1.78 years

for the smallest shocks of 1%.We conclude that the PPP puzzles recorded in the literature using data for the recent

float may have been generated to some extent by the fact that none of the studies in theliterature has explicitly allowed for all of the three features of PPP deviations describedabove, either imposing common factor restrictions to examine the stochastic properties

4 It is important to note that the latter statement is subject to the caveat that during fixed exchange rate regimesdevaluations may be used by policy makers to induce adjustments in the real exchange rate. Conversely, during afloating regime, policy makers may use heavily foreign exchange market intervention for the purpose of reducing

the variability of the exchange rate (e.g. Sarno and Taylor, 2001; Calvo and Reinhart, 2002). Given these caveats,it may be that, in fact, irregular shifts in nominal exchange rates may be responsible for inducing reversion to PPPin fixed exchange rate regimes or, vice versa, the role of nominal exchange rates in driving the adjustment to PPPduring floating regimes is not more important than the role of relative prices. Hence, the question of whether andhow PPP deviations dissipate remains an empirical issue.



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of the real exchange rate rather than the full dynamic interaction of nominal exchangerates and relative prices, or ignoring the impact of different monetary and exchange rateregimes on PPP deviations, or else neglecting the possibility of significant nonlinearitiesin the adjustment to PPP. Allowing for all of these features simultaneously allows us to

build a generalization of existing empirical frameworks that sheds new light on the behav-ior of PPP deviations. The properties of PPP deviations during floating exchange regimesimplied by our model appear to be fairly consistent with standard models of open econ-omy macroeconomics and with their dynamic properties under conventional nominalrigidities.5 It is only during fixed exchange rate regimes, when the burden of adjustmenttoward PPP relies exclusively on relative prices, that we observe remarkably long half livesof PPP shocks.

1.4. Organization

The remainder of the paper is set as follows. Section 2 defines long-run PPP anddescribes the dynamic relationship between exchange rates and relative prices, outliningthe importance of common factor restrictions in this context. In Section 3 we set outthe econometrics of nonlinear Markov-switching multivariate models as applied to non-stationary processes and cointegrated systems. In Section 4 we describe our data set, whilein the following section we report and discuss our empirical results. A final sectionconcludes.

2. Long-run PPP and the dynamic interaction between exchange rates and prices

Defining st as the logarithm of the nominal exchange rate (expressed as domestic priceof foreign currency) and pt as the logarithm of the ratio of domestic to foreign prices, then

qt st pt 1

may be seen as the deviation from PPP or the logarithm of the real exchange rate. Abso-lute long-run PPP would allow qt5 0 in the short run, but it would require qt = 0 in thelong run. A less strict version of long-run PPP postulates that qt may have a non-zeromean but it has to be a realization of a stationary process. If both st and pt have a station-ary, invertible, non-deterministic ARMA representation after differencing once (i.e. st,

pt $ I(1)), this definition of long-run PPP implies that st and pt move together in the longrun and exhibit a common stochastic trend, cointegrating with one cointegrating vectorb 0 = [1 1].

The relationship between st and pt can be described by a cointegrating vector autore-gression (VAR) of the form:

gLyt et; 2

5 Notably Chari et al. (2002) find that theoretical autocorrelations for the real exchange rate at the quarterly

frequency are in the range between about 0.4 and 0.8, corresponding to half lives between less than one quarterand about three quarters. Indeed, the inability of this general equilibrium model to match the persistence of realexchange rates between 3 and 5 years recorded by much empirical literature constitutes one of the main criticismsagainst this important theoretical paper. However, our result that the half-life in a floating rate regime can beshorter than one year reduces the relevance of this criticism.



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where yt is a two-dimensional observed time series vector, yt = [st,pt]0; g(L) is a suitable

p-th order, 2 2 matrix polynomial in the lag operator L; et = [e1t,e2t]0 is a vector of white

noise processes with covariance matrix R, et $ NID(0,R).6 By the Granger Representation

Theorem (Engle and Granger, 1987), st and pt must possess a vector error correction

model (VECM) representation where the deviation from PPP, qt (the real exchange rate)plays the part of the equilibrium error:

Dyt CLDyt1 Pyt1 et; 3

where C(L) is a 2 2 matrix polynomial; and the long-run impact matrix P = ab 0, where aand b are 2 1 vectors, with b denoting the cointegrating vector (assumed to be [1, 1]under PPP) and a the vector of weights on the cointegrating vector in each of the twoequations of the VECM. Note that g(L) in Eq. (2) is unrestricted; hence, C(L) and a arealso unrestricted. The existence of one cointegrating relationship between st and pt impliesthat the rank ofP equals unity.

To understand the relationship between tests of PPP using the VECM(3) and unit roottests based on the augmented DickeyFuller (ADF) auxiliary regression or variants of itoften applied by researchers to the real exchange rate, let us start from noting that the lat-ter focus on the roots of qt = b

0yt rather than the properties of yt itself. This implicitlyimposes common factor restrictions, as it can be seen by pre-multiplying (3) by b0 to obtain

b0Dyt b0CLDyt1 b

0ab0yt1 b0et 4

or

1 GLLDqt qqt1 wt 5

where the coefficient q = b 0a; G(L) is a scalar polynomial in L, and

wt b0CL GLb0Dyt1 b

0et: 6

Eq. (5) is the conventional ADF regression used by a number of researchers for testing thenull hypothesis of a unit root in the real exchange rate. It is now apparent that the distur-bance term wt may contain valuable information for two reasons. First, unlessb 0C(L) = G(L)b0, lags ofD yt enter wt. Second, ifpt is not weakly exogenous and respondsto PPP disequilibria, then b 0et may be explained partly by the current value ofp. Both ofthese reasons imply a loss of information from testing for PPP by testing for a unit root in

the real exchange rate qt using an ADF regression of the form (5) rather than by analyzingthe full VECM linking the nominal exchange rate st and the relative price pt, namely Eq.(3). In turn, as demonstrated for several different data generating processes by Kremerset al. (1992), this loss of information leads to a substantial loss of power of the ADF testin rejecting the null hypothesis of a unit root, which, in this context, would bias the out-come of the test toward concluding that the real exchange rate is nonstationary and long-run PPP is invalid.

The discussion in this section suggests yet another reason, unexplored by the literatureto date, why unit root tests have low power in the context of testing for PPP. Several

6 For ease of exposition, in this section we do not allow for a constant term in the cointegrating VAR and donot allow for regime-switching in the VAR. However, it is important to note that none of the points made belowis dependent on this simplification. In fact, allowance for a constant term or generalization to a nonlinear VARwould simply increase the number of common factor restrictions on which we focus in this section.



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researchers have noted that conventional unit root tests have low power in rejecting thenull of real exchange rate nonstationarity for the recent float since 1973 or so, becausethe sample size is too short to yield sufficient test power (see Froot and Rogoff, 1995). Thishas led several researchers to develop panel unit root tests which exploit the cross-corre-

lation in exchange rate data in an attempt to increase test power (since at least Hakkio,1984; Abuaf and Jorion, 1990). However, these tests also impose the same common factorrestrictions as single-equation unit root tests. Indeed, the number of restrictions increaseswith the size of the panel. If the common factor restrictions are not satisfied, for any givensample size, the power of (single-equation and panel) unit root tests may be substantiallylower than the power of tests based on the full VECM because unit root tests applied tothe real exchange rate effectively ignore valuable information by assuming error ratherthan structural dynamics.

We investigate the empirical relevance of these issues by using the VECM representationof st and pt to test for PPP and to shed light on the relative importance of the nominalexchange rate and relative prices in restoring the PPP equilibrium level across different nom-inal regimes since the 19th century. Further, in order to take seriously into account the pos-sibility of regime shifts over long-spans of data, we use a generalization of the standardlinear VECM (3) which is capable of allowing all of the VECM parameters to change overtime and to identify the various regimes that characterize the long sample periods examined.

3. Nonlinear Markov-switching error correction

Consider the following M-regime pth order Markov-switching vector autoregression

(MS(M)-VAR(p)) which allows for regime shifts in the intercept term:gLyt mzt et; 7

where yt is a K-dimensional observed time series vector, yt = [y1t, y2t, . . . ,yKt]0; m(zt) is a K-

dimensional column vector of regime-dependent intercept terms, m(zt) = [m1(zt),m2(zt), . . . ,mK(zt)]

0; g(L) is a suitable pth order, K Kmatrix polynomial in the lag operatorL; et = [e1t, e2t, . . . , eKt]

0 is a K-dimensional vector of white noise processes with covariancematrix R, et $ NID(0,R). The regime-generating process is assumed to be an ergodic Mar-kov chain with a finite number of states zt 2 {1, . . . , M} governed by the transition prob-abilities pij= Pr(zt+1 = jjzt = i), and

PMj1pij 1 8i;j 2 f1; . . . ;Mg.

A standard case in economics and finance is that yt is nonstationary but first-differencestationary, i.e. yt $ I(1). Then, given yt $ I(1), there may be up to K 1 linearly indepen-dent cointegrating relationships, which represent the long-run equilibrium of the system(Granger, 1986; Engle and Granger, 1987). If indeed there is cointegration, the cointe-grated MS-VAR (7) implies a Markov-switching vector error correction model or MS-VECM of the form:

Dyt mzt CLDyt1 Pyt1 et; 8

where C(L) is a suitable K Kmatrix polynomial in the lag operator L of order p 1, suchthat Ci

Ppji1gj for i= 1, . . . ,p 1 are matrices of parameters, and P

Ppi1Pi I

is the long-run impact matrix whose rank r determines the number of cointegrating vectors(e.g. Johansen, 1988, 1991). An MS-VECM of the form (8) can be estimated using anexpectation-maximization (EM) algorithm for maximum likelihood estimation (Dempsteret al., 1977; Hamilton, 1993; Krolzig, 1997; Kim and Nelson, 1999).



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Although, for expositional purposes, we have outlined the MS-VECM framework forthe case of regime shifts in the intercept alone, shifts may be allowed for elsewhere. Thepresent application focuses on a multivariate model comprising, for each country ana-lyzed, the spot exchange rate and the relative price (hence yt = [st,pt]

0), for which, follow-

ing the reasoning of Section 2, one unique, independent cointegrating relationship,represented by the deviation from PPP qt, should exist. As discussed in Section 5 below,in our empirical work, after considerable experimentation, we selected a specification ofthe MS-VECM which allows for regime shifts in the intercept, the variance-covariancematrix and the whole set of parameters (including the autoregressive component, C(L)and cointegration matrix, P).

In each regime, moreover, the VECM allows for nonlinear reversion toward PPP of thesame exponential form proposed by the growing literature on nonlinear real exchange ratedynamics (e.g. Michael et al., 1997; Baum et al., 2001; Taylor et al., 2001). This model,the nonlinear Markov-Switching-Intercept-Autoregressive-Heteroskedastic-VECM orMSIAH-VECM, may be written as follows:

Dyt mzt CLztDyt1 Pztyt1U ut; 9

where P(zt) = a(zt)b0 , ut $ NIID(0,R(zt)), and zt 2 {1, . . . , M}. The nonlinear function

U 1 expq2td

r2qzt is an exponential function, where qtd= b

0ytd and the integer

d> 0 is a delay parameter. U() is bounded between zero and unity, U : R ! 0; 1, hasthe properties U[0] = 0 and limx!1U[x] = 1, and is symmetrically inverse-bell shapedaround zero. These properties are attractive in the present modelling context because theyallow symmetric adjustment toward PPP for deviations above and below the PPP equilib-

rium level (zero), as predicted by theories of real exchange rate determination under trans-actions costs (e.g. Dumas, 1992; Sercu et al., 1995). The size of the deviation from PPPqtd, standardized by dividing it by its regime-dependent variance, r

2qzt, then determines

the speed of mean reversion toward PPP, with lower PPP deviations being more persistentthan large deviations (of either sign), again as predicted by general equilibrium models ofreal exchange rate determination under transactions costs. In our empirical work, consis-tent with previous research in this context (e.g. Obstfeld and Taylor, 1997; Taylor, 2001;Taylor et al., 2001), we set d= 1, since economic intuition suggests a presumption in favorof smaller values of the delay parameter d rather than larger values, in that it is hard toimagine why there should be very long lags before the real exchange rate begins to adjust

in response to a shock.7

4. Data

The data set used in this study comprises annual observations for the nominal exchangerate (domestic price of foreign currency) and the price levels based on the consumer priceindex (CPI) or the gross domestic product (GDP) deflator, depending on availability rel-ative to the US for each of the G5 countries. Our data set is obtained from updating the

7 Although the transition probabilities are exogenous in our model, the speed of reversion to PPP is a functionof the absolute size of the deviation from PPP itself. An alternative formulation would involve endogeneizing theswitching probabilities, but this is not straightforward in a rich model of the type considered in this paper (e.g. seeKim et al., 2003).



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relevant time series in the data set constructed by Obstfeld and Taylor (2004) using theInternational Financial Statistics of the International Monetary Fund.8 From these datawe calculate the logarithm of the nominal exchange rate, st, the relative price, pt, andthe real exchange rate, qt = st pt, as defined in Eq. (1).

The sample spans from the late 19th or early 20th century to the late 20th century andthus covers a variety of international monetary arrangements, including the classical GoldStandard, the Bretton Woods era, the modern float and, for some countries, the ExchangeRate Mechanism of the European Monetary System. The exact sample period for eachcountry is as follows: 18702000 for the UK; 18801998 for France and Germany;18852000 for Japan. The start date was in each case dictated by data availability,whereas the end date is 2000 except for France and Germany, which joined the EuropeanMonetary Union and replaced their respective national currencies with the euro in January1999.9

5. Empirical results

5.1. Unit roots, cointegration, and common factor restrictions10

As a preliminary exercise, we test for unit root behavior of the nominal exchange rate(st) and the relative price (pt) time series, employing various unit root tests. These testsindicate that both st and pt are realizations from a stochastic process integrated of orderone. We then tested the common factor restrictions required for the VAR to reduce to aunivariate autoregression for the real exchange rate, using likelihood ratio (LR) tests. The

results indicate that, in each case, the restrictions are strongly rejected, providing the casefor using unrestricted VAR and VECM estimation for testing PPP rather than realexchange rate autoregressions. Thus, we employed the Johansen (1988, 1991) maximumlikelihood procedure in a VAR for yt = [st,pt]

0, which does not impose any common factorrestrictions. However, these cointegration tests do not detect a (proportional) cointegrat-ing relationship between exchange rates and relative prices, leading to the conclusion thatlong-run PPP does not hold for any of the countries examined over the full sample.

There are a variety of reasons for the failure to find a unique cointegrating relationshipbetween economic time series where one would normally be expected on the basis of eco-nomic theory (see Siklos and Granger, 1997; and the references therein). In particular, it is

8 We are thankful to Alan Taylor for graciously providing the data.9 We focus on the G5 countries rather than on the broader set of countries analyzed, for example, in Taylor

(2002), because the G5 countries are the subset of the world economy for which PPP is more likely to be areasonable approximation to the long-run behavior of the real exchange rate. In particular, given the relativelyfast technology diffusion that characterizes the G5 economies compared to other economies, HarrodBalassaSamuelson effects (shocks to productivity differentials) are unlikely to last forever. Even if some evidence existsthat the productivity differential is an important determinant of the real exchange rate, this set of countries is theone where long-run PPP is more likely to apply (see Rogoff, 1996; Sarno and Taylor, 2002). The only exceptionmight be Japan, occasionally referred to as a possible example of the HarrodBalassaSamuelson effect in

operation among major economies, since it has been on average the fastest-growing economy for much of the postWorld War II period (see Rogoff, 1996). This interpretation seems consistent with the findings in Taylor (2002),where a deterministic trend is often needed for a variety of countries outside the G5 to capture trendingdepartures from long-run PPP.10 The results discussed in this sub-section are not reported to conserve space but are available upon request.



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well known that the presence of structural breaks or regime shifts may perversely affectcointegration tests, which appears to be especially likely when using long-spans of data.Such shifts may significantly alter the dynamic relationship that may exist between thevariables, and tests of the long term behavior of these variables should account for them.

Hence, as a check of adequacy of the models as well as an additional motivation foremploying a more general modelling framework, we test for the stability of the cointegrat-ing rank over time. In particular, we carry out two recently developed tests that are spe-cifically designed to test for stability within a cointegrating framework: the Hansen andJohansen (1999) test of the null hypothesis of stability of the eigenvalues of the stochasticmatrix; and the test statistic developed by Quintos (1997) for testing the null hypothesisthat the cointegrating rank is constant among breakpoints. The tests results suggest thatthe null hypothesis of temporal stability of the cointegrating rank is strongly rejected, fur-ther motivating the use of a regime-switching framework, to which we now turn.

5.2. Regime shifts and model selection

We investigate the presence of regime shifts by applying the bottomup proceduredesigned to detect Markovian shifts in order to select the most adequate characterizationof a nonlinear M-regime pth order MS-VECM for Dyt. Essentially, the bottomup proce-dure consists of starting with a simple but statistically reliable Markov-switching model byrestricting the effects of regime shifts on a limited number of parameters and checking themodel against alternatives. In such a procedure, most of the structure contained in thedata is not attributed to regime shifts, but explained by observable variables, consistent

with the general-to-specific approach to econometric modelling (Krolzig, 1997).

11

The VARMA representations of the time series suggests in each case that the number ofregimes is three. However, for any MS-VECM estimated the implicit assumption that theregime shifts affect alternatively only the variance-covariance matrix, the intercept term, orthe autoregressive component of the VECM was found to be inappropriate. In fact, wechecked the relevance of all these components by using three likelihood ratio (LR) testsof the type suggested by Krolzig (1997, pp. 135136). The results, reported in the first threerows ofTable 1 (LR1, LR2 and LR3, respectively), indicate strong rejections of the null ofno regime dependence, clearly suggesting that an MS-VECM that allows for shifts in theintercept, the variance-covariance matrix and the autoregressive component, namely an

11 In this paper we employ the testing procedure suggested by Krolzig (1997) to determine the number ofregimes, but we should point out several relevant caveats. It is well known that the determination of the numberof regimes in the data is a very cumbersome problem, which does not have a robust solution to date. Specifically,the presence of nuisance parameters gives the likelihood surface sufficient freedom so that the finding of somestatistically significant parameters could simply be due to sampling variation. The scores associated with theparameters of interest under the alternative hypothesis may be identically equal to zero under the null. Severaltesting procedures have been proposed, none of which is without problems. It is important to note here that theregularity conditions under which the tests suggested by Krolzig (1997) are valid may be violated here. Therefore,the distribution of the likelihood ratio tests may differ from the adjusted v2 distribution. We rely on the

asymptotic distribution in our empirical work and do not attempt to resolve the econometric problemssurrounding the above issues; this would be an ambitious aim beyond the scope of this paper, given the complexnature of our cointegrated Markov model and the presence of a nonlinear transition function in the errorcorrection term. For discussions of the problems related to hypothesis testing in this context, see Hansen (1992)and Garcia (1998).



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MSIAH-VECM(p), is the most appropriate model within its class in the presentapplication.

Further, in the same spirit of the tests previously computed, we carry out a further LRtest in order to select the most parsimonious MSIAH-VECM for characterizing thedynamic relationship between spot exchange rates and relative prices. In particular, we testthe null of MSIAH-VECM(1) against the alternative of a higher-order MSIAH-VECM(3),since three lags is the maximum lag length suggested by conventional information criteria.The fourth row ofTable 1 (LR4) indicates that we are not able to reject this null hypoth-esis at standard significance levels, hence concluding that one lag is appropriate for eachMSIAH-VECM estimated.

In order to discriminate between models allowing for two regimes against models gov-erned by a higher number of regimes we execute two further likelihood ratio tests, one

(LR5) for the null hypothesis that the MSIAH-VECM(1) with 3 regimes is equivalentto the MSIAH-VECM(1) with 2 regimes, and another (LR6) as a linearity test of the nullhypothesis that the selected MSIAH-VECM(1) is equivalent to a linear VECM(1). Theresults reported in Table 1 (fifth row) show very large test statistics and the corresponding

p-values suggest that three regimes may be appropriate in all cases to describe the dynam-ics of nominal exchange rates and relative prices. Finally, the linearity tests, reported in thelast row ofTable 1, indicate in each case the rejection of the linear VECM(1) in favor of itsnonlinear, Markov-switching counterpart.

Hence, the final result of this procedure identifies for all countries an MSIAH-VECMgoverned by three different regimes and one lag that can be written as follows:

Dyt mzt CLztDyt1 Pztyt1U ut; 10

where P(zt) = a(zt)b0, U 1 exp

q2t1

r2qzt, xt $ NIID(0,R(zt)) and zt = 1,2,3. We esti-

mate the MSIAH-VECM in Eq. (10) using an EM algorithm for maximum likelihood

Table 1Bottomup identification procedure

France Germany Japan UK

M 3 3 3 3

LR1 1.86 1068 3.73 1046 2.90 1022 1.19 1040LR2 9.90 105 3.28 108 7.76 107 9.85 104

LR3 1.68 1010 1.01 1014 2.14 103 1.01 103

LR4 2.58 101 1.25 101 2.74 101 3.44 101

LR5 1.82 1020 9.55 1051 1.33 1033 1.41 1012

LR6 5.38 1076 1.68 1074 3.67 1057 5.75 1053

Notes: LR1 is a test statistic of the null hypothesis of no regime dependent variance-covariance matrix (i.e.MSIA(M)-VECM(p) versus MSIAH(M)-VECM(p)). LR2 is a test statistic of the null hypothesis of no regimedependent intercept (i.e. MSAH(M)-VECM(p) versus MSIAH(M )-VECM(p)). LR3 is a test statistic of the nullhypothesis of no regime dependent autoregressive component (i.e. MSIH(M)-VECM(p) versus MSIAH(M)-VECM(p)). LR4 tests the null hypothesis that the model with one lag in the autoregressive component is

equivalent to another with a higher autoregressive order (i.e. MSIAH(M)-VECM(1) versus MSIAH(M)-VECM(3)). LR1,LR2,LR3,LR4 are constructed as 2(lnL* lnL), where L* and L represent the unconstrainedand the constrained maximum likelihood, respectively. These tests are distributed as v2(g) where g is the numberof restrictions. LR5 is the likelihood ratio test for the null hypothesis that the MSIAH-VECM(1) with 3 regimes isequivalent to the MSIAH-VECM(1) with 2 regimes. LR6 is a linearity test for the null hypothesis that the selectedMSIAH-VECM(1) is equivalent to a linear VECM(1). p-values relative to the LR5 and LR6 tests are calculated asin Ang and Bekaert (1998). For all test statistics only p-values are reported.



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(Dempster et al., 1977), for each of the countries under investigation. This algorithm hasbeen found to be fairly accurate in estimating MS-VECMs up to 4 states when the numberof observations available are larger than 100 (see Krolzig, 1997).12 Since unrestricted esti-mation of the MSIAH-VECM yielded parameter estimates for b 0 close to [1, 1], in thefinal estimation of the model we set the long-run cointegrating vector b0 = [1, 1], consis-tent with long-run PPP, to increase the accuracy of the remaining parameters estimated.

The estimation yielded fairly plausible estimates of the coefficients, including theregime-dependent adjustment coefficients in a(zt), which are generally found to bestatistically significantly different from zero in at least one regime. Before turning to theempirical results from estimating the MSIAH-VECMs, however, we carry out tests ofregime-dependent tests of the common factor restrictions which would make theMSIAH-VECM(10) reduce to a univariate regime-switching real exchange rate autore-

gression. The likelihood ratio tests results, reported in Table 2, indicate rejections of thecommon factor restrictions in at least two regimes for each MSIAH-VECM with fairlylow p-values, strengthening the case for testing for PPP in the context of a full VECMrather than in the context of univariate models of the real exchange rate.

5.3. MSIAH-VECM estimation results

Table 3 reports results relating to the classification of regimes implied by our estimatedMSIAH-VECM. In particular we calculated the average smoothed probabilities over theGold Standard (18701914), interwar (19151944), Bretton Woods (19451972) and

recent float (19732000) periods in order to see whether our preferred MSIAH-VECMswere able to correctly identify different nominal exchange rate regimes. These results indi-cate several interesting patterns. First, the estimated MSIAH-VECMs are able to disen-tangle quite precisely the different nominal regimes on the basis of the calculation ofthe regime classification measure (RCM) proposed by Ang and Bekaert (1998, 2002).The RCM, which is defined between 0 and 100, is easily interpretable as follows:when the RCM is 0, identification is perfectly accurate, whereas when the RCM is 100

12 This EM algorithm carries out iteration steps that are repeated until convergence is ensured on the basis of

several criteria based on the relative change in the log-likelihood and the parameters; see Krolzig (1997), Section6.4, pp. 109110. An alternative way to carry out estimation for this class of regime-switching models involvesusing Markov Chain Monte Carlo (MCMC) methods (e.g. see Chib, 2001; and the references therein). We choseto estimate the MS-VECM by maximum likelihood methods given their simplicity and faster convergenceproperties.

Table 2Tests for common factor restrictions: nonlinear MSIAH-VECM

LRcf(z = 1) LRcf(z = 2) LRcf(z = 3)

France 1.51 102 3.58 102 9.21 103

Germany 4.86 101 4.91 102 9.90 103Japan 1.42 101 3.14 102 4.56 102

UK 2.42 104 2.26 102 5.75 102

Notes: LRcf(z = i) is the likelihood ratio test for the null hypothesis that b0C(Ljz = i) = G(Ljz = i)b 0 conditional

on regime i= 1,2,3. Under the null hypothesis that the common factor restrictions are valid, the test statistic isdistributed as a v2(d), where d is the number of restrictions. Figures reported are p-values.



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no information about the regimes is revealed.13 It is comforting that our results in Table 3indicate that in all cases the RCM is very small and in some cases close to zero. Second,the MSIAH-VECMs are able to identify regimes with fixed exchange rates from regimeswith flexible exchange rates. In fact, in all cases we can see that the regime with the highestprobability under the Gold Standard period is the same exhibiting the highest probabilityunder the Bretton Woods period.14 However, we note that the results for the interwarperiod are less clear-cut in that the average smoothed probabilities display more variationacross countries, unlike other periods. This is understandable since the interwar period issomewhat mixed because it includes both floating and fixed (the return to the Gold

Standard) regimes.It is worth noting that our regime classification is implied by the estimated MSIAH-VECMs and is, therefore, data and model driven. This differs from analyses of real

Table 3Regime classification measure (RCM)

18701914 19151944 19451972 19732000

France p1 0.002 0.734 0.215 0.002

p2 0.994 0.199 0.587 0.002p3 0.004 0.067 0.197 0.996RMC(3) 8.15 102 2.74 106 1.44 102 4.13 107

Germany p1 0.012 0.333 0.072 0.001p2 0.988 0.587 0.807 0.004p3 0.000 0.080 0.122 0.995RMC(3) 6.07 103 3.49 103 9.90 103 6.00 108

Japan p1 0.037 0.242 0.215 0.001p2 0.689 0.467 0.709 0.000p3 0.273 0.291 0.076 0.999RMC(3) 3.19 106 8.78 101 5.48 105 1.59 106

UK p1 0.038 0.207 0.216 0.346p2 0.928 0.262 0.656 0.060p3 0.034 0.531 0.128 0.595RMC(3) 9.58 101 5.98 101 3.14 101 9.96 101

Notes: pj is the average smoothed probability relative to regime j= 1,..,3 calculated as pj PTj

1 pj;t=Tj, where Tjis the number of observations in sub-period j, and pj,t is the smoothed (ex-post) regime probability relative toregime j= 1,..,M at time t. RCM(M) is the regime classification measure proposed by Ang and Bekaert (2002).The statistic is calculated as RCMM 100M2 1Tj

PTji1QM

j1pj;t.

13 Since the true regime is a Bernoulli random variable, the RCM is essentially a sample estimate of its variance(Ang and Bekaert, 2002).14 While two of the regimes clearly identify fixed and floating exchange rate regimes, the third regime does not

lend itself to any easy economic interpretation. However, inspection of the transition probabilities suggests thatthis regime is largely characterized by outliers in the data, which do not fit easily in any of the first two regimes.Indeed, estimation of a more parsimonious two-regime VECM yielded similar results and economic implications

as the three-regime VECM, but in a less clear-cut fashion. This is because the allowance for more than tworegimes, which was suggested by the bottom-up procedure, is able to remove the outliers in the data and toidentify more precisely the fixed and floating exchange rate regimes, ultimately yielding a statistically superiormodel. We could not discriminate whether the outliers are extreme observations or inaccurate observations, butdecided to estimated the model using all of the available data rather than arbitrarily take out the outliers.



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exchange rate behavior across exchange rate regimes which are typically carried out usingregime classifications chosen a priori on the basis of the historical experience of the coun-tries examined (e.g. Taylor, 2002).

The interpretation of regimes also benefits from examining jointly the results in Table 3with the ones in Table 4, where we report the regime-dependent error correction coeffi-

cients for fixed and floating regimes. The results in Table 4 suggest that during fixedexchange rate regimes the error correction coefficients in the exchange rate equationsare not significantly different from zero at conventional significance levels, while the errorcorrection coefficients in the relative price equations are statistically significant and cor-rectly signed. This implies that during fixed exchange rate regimes the relative priceresponds to deviations from PPP in a way to restore the long-run exchange rate equilib-rium. Vice versa during flexible rate regimes we find that the error correction coefficientsin the exchange rate equations are strongly statistically significant, while the error correc-tion coefficients in the relative price equations, albeit correctly signed, are not statisticallydifferent from zero except for Japan.15 Even if one ignores the lack of statistical signifi-

cance of the error correction coefficients associated with relative prices, moreover, theseerror correction coefficients are quite small in size relative to the corresponding error cor-rection coefficients associated with nominal exchange rates, again except for Japan. Thisfinding is consistent with the view that during flexible rate regimes the exchange rate is pri-marily responsible for restoring deviations from long-run PPP equilibrium.

Table 4Regime-dependent adjustment coefficients

Fixed regime Flexible regime

aDst aDpt aDst aDpt

France 0.002 0.227 0.302 0.041(0.004) (0.091) (0.129) (0.033)

Germany 0.004 0.028 0.340 0.024(0.069) (0.012) (0.079) (0.132)

Japan 0.012 0.073 0.137 0.161(0.009) (0.028) (0.038) (0.055)

UK 0.002 0.058 0.659 0.067(0.004) (0.017) (0.144) (0.072)

Notes: aDst denotes the estimated regime-conditional error-correction coefficient relative to the spot exchange rate

equation. aDpt

denotes the estimated regime-conditional error-correction coefficient relative to the relative priceequation, obtained from estimating the nonlinear MSIAH-VECM(10). Values in parentheses are asymptoticstandard errors.

15 The greater role of relative prices in bearing the adjustment toward PPP in Japan does not have an obviousinterpretation. However, we conjecture that one possible explanation may be that Japan is, within the G5, the

country which has engaged most in foreign exchange market intervention in order to manage exchange ratesduring the post-Bretton Woods period (e.g. Sarno and Taylor, 2001; McKinnon and Schnabl, 2003). This meansthat the yen might not have been allowed to bear as much of the adjustment toward the PPP equilibrium as itwould have under a pure floating regime, imposing that more of the adjustment ought to be borne by relativeprices.



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The above results are quite interesting. It is rare in the use of Markov-switching modelsthat one finds such strong evidence of being in one particular regime or the other. Theresults of the MSIAH-VECM in terms of classification of nominal regimes provide clearevidence that adjustment to long-run equilibrium takes different forms in fixed versus

floating exchange rate regimes.This finding appears to be in contrast, prima facie, with the findings of Engel and Mor-

ley (2002), who use a state-space representation of the VECM linking exchange rates andrelative prices for the recent float alone and find that nominal exchange rates respond lessthan relative prices to PPP disequilibria. Note, however, that Engel and Morley focus onthe speed of adjustment of exchange rate and prices toward their equilibrium levels i.e. inresponse to what they term the exchange rate gap and the price gap, respectively, neitherof which is directly related to PPP deviations rather than toward the PPP equilibriumlevel. In this sense, therefore, their estimates of speeds of adjustment are not comparableto the ones we report in this paper, where we focus on the response of exchange rates andprices to PPP deviations. Our results are also in line with the evidence provided by Gold-fajn and Valdes (1999) and Cheung et al. (2004), who argue that the nominal exchange rateis responsible for most of the adjustment toward PPP, rather than prices. However, ourempirical framework, by generalizing this line of research to a nonlinear regime-switchingVECM system, allows us to pin down the speed of adjustment toward PPP for a long-spanof data and across regimes.

5.4. PPP deviations are not as puzzling as we thought

We examined the dynamic adjustment in response to shocks through impulse responsefunctions which record the expected effect of a shock at time t on the system at time t + j.Estimating the impulse response function for a nonlinear model raises special problemsboth of interpretation and of computation since the shape of the impulse response func-tion is not independent with respect to either the history of the system at the time theshock occurs, the size of the shock considered, or the distribution of future exogenousinnovations. Exact estimates can be produced by multiple integration of the nonlinearmodel with respect to the distribution function of the j future innovations, which is com-putationally impracticable for the long forecast horizons required in impulse responseanalysis. We calculate impulse response functions to system-wide shocks conditional on

average initial history using the Monte Carlo integration method proposed by Koopet al. (1996). We estimated regime-dependent impulse response functions and half livesfor four different sizes of shock k2 {1,5,10,20}, conditional on average initial history.This range of shocks seems plausible in this context since it is compatible with theobserved standard deviation of the real exchange rates examined.16 The shocks considered

16 It is well documented that real exchange rates are more volatile during floating than fixed exchange rateregimes (e.g. Baxter and Stockman, 1989). For example, on our data, the demeaned real exchange rates have thefollowing standard deviations during the Bretton Woods period: 0.038 for France, 0.051 for Germany, 0.053 forJapan, and 0.039 for the UK. The corresponding standard deviations during the recent float are: 0.065 for France,

0.072 for Germany, 0.068 for Japan, and 0.057 for the UK. Therefore, the range of shocks from 1% to 20% allowsus to study deviations from PPP up 3 or 4 times the standard deviation of observed real exchange rates,depending on the regime. While other authors (e.g. Taylor et al., 2001), consider larger shocks up to 40%, werefrain from analyzing these shocks which would imply faster half lives because they are likely to be very rareoccurrences.

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are shocks to the log-level of the real exchange rate, i.e. shocks to the PPP equilibrium,essentially focusing on the analysis of the impulse response function ofqt to an innovationon qt1 = b

0yt1, by taking into account all of the dynamic interactions implied by the full

VECM. This is a particularly appealing way of measuring the response to PPP shocks inour context.

The estimated half lives of shocks, reported in Table 5, illustrate the nonlinear nature ofour models, with larger shocks dissipating much faster than smaller shocks. The regime-dependent half-lives are sensibly different between the fixed and floating regimes and fordifferent shock sizes. During the fixed exchange rate regimes, when the relative price is pri-marily responsible for bearing the burden of adjustment toward PPP, the effects of shockson the PPP equilibrium relationship will last for more than five years on average acrosscountries for a 20% shock, and for almost ten years on average for a 1% shock. These halflives are indeed longer than the half lives typically recorded by the literature using linear

methods on data for the recent float (Rogoff, 1996).However, the half-lives of system-wide shocks on the equilibrium relationships during

flexible exchange rate regimes are drastically shorter. On average across countries, 10%and 20% shocks will last for less than one year, as one might expect given that under

Table 5Regime-conditional half lives

Fixed regime Flexible regime

Shock = 1%

France 4.05 [1.12,6.04] 2.00 [1.00,2.12]Germany 15.01 [6.03,17.01] 2.01 [0.99,3.00]Japan 7.02 [5.03,9.01] 2.09 [1.04,4.01]UK 13.01 [5.02,16.00] 1.02 [0.29,2.13]Average 9.77 [4.30,12.02] 1.78 [0.83,2.82]

Shock = 5%France 3.03 [1.00,5.02] 1.08 [1.01,2.01]Germany 13.00 [6.03,16.00] 1.11 [0.99,2.14]Japan 6.04 [5.04,8.00] 2.03 [1.05,3.04]UK 8.02 [5.03,13.02] 0.70 [0.38,1.08]Average 7.53 [4.28,10.51] 1.23 [0.86,2.07]

Shock = 10%France 2.07 [1.01,3.04] 1.06 [1.04,1.08]Germany 10.02 [6.03,12.02] 1.04 [1.00,2.04]Japan 6.02 [5.04,7.01] 1.12 [1.06,2.06]UK 7.00 [5.05,9.02] 0.64 [0.53,0.85]Average 6.28 [4.28,7.77] 0.97 [0.91,1.51]

Shock = 20%France 2.05 [2.03,2.08] 1.06 [1.04,1.08]Germany 8.02 [7.00,9.01] 1.02 [1.01,1.09]Japan 6.02 [6.01,7.01] 1.10 [1.08,1.13]UK 6.04 [6.01,7.02] 0.61 [0.56,0.81]

Average 5.53 [5.26,6.28] 0.95 [0.92,1.03]

Notes: Fixed and Flexible exchange rate regimes denote the regimes identified by the nonlinear MSIAH(3)-VECM(1) corresponding to Fixed and Flexible. The figures reported are the half-lives of the real exchange rate,whereas figures in brackets are the 0.025 and 0.975 quantiles of the empirical distribution of the half lives,calculated by Monte Carlo integration methods.



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flexible exchange rate regimes the nominal exchange rate can also contribute to restoringPPP in response to shocks. Even for very small shocks of 1 percent, on average across theG5 dollar exchange rates, the half life is well below two years. In comparison to the halflives reported by the literature on nonlinear mean reversion during the recent float using

univariate nonlinear models (e.g. Taylor et al., 2001), our nonlinear MSIAH-VECMproduces shorter half lives, presumably because of the lack of any common factor restric-tions in our multivariate nonlinear model.

The properties of PPP deviations implied by our model also appear to be fairly consis-tent with standard models of open economy macroeconomics and with their dynamicproperties under conventional nominal rigidities. For example, Chari et al. (2002) calibratea general equilibrium model which generates theoretical autocorrelations for the realexchange rate at quarterly frequency between about 0.4 and 0.8. These autocorrelationscorrespond to half lives between less than one quarter and about three quarters. However,the inability of this general equilibrium model to match the persistence of real exchangerates between 3 and 5 years recorded by much empirical literature constitutes one of themain criticisms against this important theoretical paper. Our result that the half life in afloating rate regime is below one year for shocks of ten percent or higher somewhatreduces the relevance of this criticism.

Our results are related to the work ofTaylor (2002), who studied twenty countries overa very similar sample and concluded that half-lives of PPP deviations are roughly the sameacross exchange rate regimes. In Taylor (2002), countries and regimes are analyzed eitherone by one or using pooled samples across both countries and time periods. In our paperwe find that half-lives are much shorter for flexible exchange rate periods than for fixed

ones. Specifically, we report longer half-lives than Taylor for the fixed exchange rateregimes but much lower ones for the flexible regimes. Clearly, this difference in results isa by-product of the different framework used in our paper where, unlike in Taylor(2002), we do not impose any common factor restrictions, select regimes on the basis ofthe regime classification implied by the estimates of our Markov switching models, andallow for nonlinear reversion to PPP. Further, this difference in results is also a by-productof the calculation of the half-life, obtained in our paper using Koop-Pesaran-Potterapproach because of the intrinsic nonlinearity of our model. Taylors (2002) finding thatshocks have been much larger during floating regimes than during fixed regimes then pro-vides further insight in understanding our results: clearly, if this is the case, ceteris paribus,

the larger shocks of the floating regime would imply faster mean reversion during theseperiods in our nonlinear model.

Overall, the fact that consensus estimates for the rate at which PPP deviations dampsgenerally range between three and five years may be due to several reasons. First, severalprevious studies using long-span data acknowledged the fact that the results were obtainedby blending fixed and floating rate data, but to the best of our knowledge this is thefirst explicit attempt to allow for the presence of regime shifts as well as nonlinear adjust-ment in investigating PPP with long-span data across countries. Second, much previousresearch has employed (single-equation or panel, linear or nonlinear) autoregressionsfor the real exchange rate rather than focusing on the full VECM linking nominal

exchange rates and relative prices, hence biasing the test outcome against long-run PPP.Our finding of very short half-lives for the recent float is presumably the product of takinginto consideration simultaneously three issues that are relevant in determining the powerof tests for PPP: examining the full VECM for exchange rates and prices rather than using

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real exchange rate autoregressions; using a long-span of data rather than only data for therecent float, while allowing for regime shifts; modelling the dynamics of PPP deviations ina nonlinear fashion.17

6. Conclusions

In this paper we have re-examined the behavior of deviations from PPP using a centuryof data for the four dollar exchange rates obtaining among the G5 countries. We begun bynoting the empirical success in establishing the validity of long-run PPP recorded by bothlong-span studies and studies based on nonlinear real exchange rate models, which con-trasts with the unfavorable evidence provided by studies based on standard unit root testsand most cointegration tests as well as with the mixed evidence provided by studiesemploying panel unit root tests. However, long-span studies have often been accused ofignoring the possible regime shifts that may characterize PPP deviations across differentmonetary and exchange rate regimes. We also illustrated how linear and nonlinear realexchange rate autoregressions impose potentially invalid restrictions on the unknown datagenerating process driving the relationship between nominal exchange rates and prices,and how these restrictions may prevent us from detecting reversion toward PPP.

This process of interpretation of the time series evidence of the relevant literature led usto employ a nonlinear empirical framework for PPP deviations which facilitates a study oflong-span data while taking into account possible regime shifts and at the same time inves-tigating whether, if PPP is validated by the data, adjustment toward the long-run equilib-rium level defined by PPP is driven primarily by the exchange rate, by relative prices, or by

both. Within this framework, we are able to shed light on three hotly debated questions:whether PPP holds; whether nominal exchange rates or relative prices or both of them areresponsible for responding to PPP disequilibria; and whether the half-lives of PPP devia-tions during floating regimes are consistent with standard dynamic general equilibriumopen economy models with nominal rigidities.

Our results are encouraging on a number of fronts. First, the nonlinear MSIAH-VECMs clearly indicate that long-run PPP holds for each of the four exchange rates exam-ined over the last century and under each of the monetary regimes that characterize it. Sec-ond, we find that during fixed exchange rate regimes, relative prices adjust to restoredeviations from long-run equilibrium, while exchange rates bear most of the burden of

adjustment during flexible exchange rate regimes. The estimated transition probabilitiesare consistent with the view that the relative importance of exchange rates and relativeprices in restoring the long-run equilibrium level of the exchange rate varies over timeand is affected by the nominal exchange rate arrangement in operation. Third, the esti-mated half-lives implied by the models are rather different for fixed and floating rateregimes. During fixed exchange rate regimes the effects of system-wide shocks on thePPP equilibrium relationship may be very persistent indeed, with half lives ranging fromtwo to almost ten years, depending on the shock size. However, the corresponding half-lives during flexible exchange rate regimes are drastically shorter, presumably because

17 A comparable result obtained from a different viewpoint is worth quoting. Imbs et al. (2005) show that failingto control for the heterogeneity of the speed of mean reversion of prices across different sectors may lead tooverestimation of the degree of persistence of the real exchange rate. When they correct for such aggregation bias,their estimates of the half life of the real exchange rate fall drastically.



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the nominal exchange rate is allowed to contribute to restoring PPP. In fact, large shocksto PPP will last for less than one year on average across countries.

Overall, we leave this study with some degree of optimism, which may be summarizedin the following three points: long-run PPP may hold after all; PPP deviations are reversed

more quickly under floating rate regimes, when nominal exchange rates are allowed torespond to shocks; the speed at which PPP deviations dissipate is fairly consistent withthe predictions of standard open economy macro theory.

Acknowledgements

This paper was partly written while Lucio Sarno was a Visiting Scholar at the Interna-tional Monetary Fund and at Norges Bank. The research was funded by the Economicand Social Research Council (ESRC, Grant No. RES-000-22-0404). The authors are

grateful to Giorgio Szego (editor), two anonymous referees, Yin-Wong Cheung, JerryCoakley, Stefan Gerlach, Andrew Rose, Shang-Jin Wei and participants to presentationsat the Australasian Macroeconomic Workshop in Hong Kong, the European CentralBank, the Bank of Italy, and the 2005 Annual Conference of the Royal Economic Societyfor helpful comments on previous drafts, in addition to Alan Taylor for kindly providingthe data set. The authors alone are responsible for any errors that may remain.

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