Int. Fin. Markets, Inst. and Money 18 (2008) 413–424

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Journal of International FinancialMarkets, Institutions & Money

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Long-run PPP in a system context: No favorable evidenceafter all for the U.S., Germany, and Japan

David O. Cushmana,b,∗

a Department of Economics and Business, Westminster College, New Wilmington, PA 16172, USAb Department of Economics, University of Saskatchewan, Saskatoon, Canada S7N 5A5

a r t i c l e i n f o

Article history:Received 19 July 2006Accepted 2 May 2008Available online 7 May 2008

JEL classification:F31C32

Keywords:Purchasing power parityCointegration

a b s t r a c t

In a cointegration analysis of PPP in five-variable system for Ger-many, Japan, and the U.S., Sideris [Sideris, D., 2006. Testing forlong-run PPP in a system context: evidence for the U.S., Germanyand Japan. Journal of International Financial Markets, Institutionsand Money 16, 143–154] reports three cointegration vectors andconcludes that they are consistent with some form of PPP forall three exchange rates. The present paper reconsiders Sideris’sthree-country analysis with special attention to the specificationof deterministic terms in the cointegration testing. In addition, thepassage of time since the Sideris paper allows the data set to beextended. The present paper also applies the Johansen approachand longer data set to traditional two-country models for the sameexchange rates. In no case is any evidence in favor of PPP found.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

In a recent paper in this journal, Sideris (2006) applies Johansen (1991, 1995) cointegrationtests to prices and exchange rates simultaneously involving the U.S., Germany, and Japan in apurchasing-power-parity (PPP) analysis of the recent floating period. Sideris (2006) plausibly arguesthat applying the Johansen systems approach to a three-country setting could provide better testsby accounting for dynamic third-country relationships that are omitted in the standard two-countryapproach.

Sideris (2006) concludes that there are three cointegration vectors and that what he calls “weakPPP” exists for the German–U.S. and Japanese–U.S. exchange rates. He also reports what he calls a “PPP-

∗ Tel.: +1 724 946 7169; fax: +1 724 946 6158.E-mail address: cushmado@westminster.edu.

1042-4431/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.intfin.2008.05.001

414 D.O. Cushman / Int. Fin. Markets, Inst. and Money 18 (2008) 413–424

type relationship” for the German–Japanese exchange rate. Sideris (2006) concludes that his resultsprovide stronger evidence in favor of PPP than in many earlier papers, both those applying unit roottests to real exchange rates and those applying cointegration tests to prices and exchange rates usingthe standard two-country model. Papers in the latter vein that examine the same exchange rates as inSideris (2006) include Cheung and Lai (1993b), MacDonald (1993), Edison et al. (1997), and Xu (1999).1

The present paper reconsiders Sideris’s (2006) three-country analysis with special attention to thespecification of deterministic terms in the cointegration testing. In addition, the passage of time sincethe Sideris (2006) paper allows the data set to be extended. The present paper also applies the Johansenapproach and longer data set to traditional two-country models for the same exchange rates. In nocase is any evidence in favor of PPP found.

2. PPP among three countries

Let us begin by clarifying what cointegration vectors can tell us about PPP among three countries.Let pi be the log of prices in county i and ei,j be the country i price of country j’s currency in logs.There are three prices and two independent exchange rates. In accordance with well known empiricalresults, assume that all are nonstationary I(1) processes. A general equation for the exchange rate forcountries 1 and 2 is

e1,2,t = ˛1 + ˇ1p1,t − �1p2,t + ı1p3,t + ε1e1,3,t + �1t + u1,t . (1)

Suppose ˇ1 = �1 = 1 (the proportionality condition of Cheung and Lai, 1993b), ı1 = ε1 = �1 = 0, and u1is stationary around 0. Sideris (2006) calls this “strong” PPP.2 The real exchange rate, r1,2,t, defined ase1,2,t − p1,t + p2,t, is mean stationary. The relative competitiveness of the two countries remains constantin the long run, the usual textbook definition of relative PPP.3

Next, suppose ˇ1 = �1 but ˇ1 /= 1 (the symmetry condition of Cheung and Lai, 1993b). The exchangerate responds in the correct direction but not proportionately to any inflation differential. Sideris (2006)calls this “weak PPP.” However, the usual PPP definition is violated because the real exchange rate isunlikely to be mean stationary:

r1,2,t = (1 − ˇ1)(p2,t − p1,t) + ˛1 + u1,t . (2)

With non-unitary ˇ1, relative competitiveness as measured by r1,2,t will have no tendency to be main-tained; it will possess the deterministic and stochastic trends in the term (p2,t − p1,t).

Finally, consider Eq. (1) with only the restrictions that ˇ1 and �1 are positive and ı1 = ε1 = �1 = 0.Sideris (2006) calls this a “PPP-type relationship,” but here I will call it “very weak PPP” to more clearlyrepresent Sideris’s (2006) implication that it is less PPP-like than his “weak PPP.” At least prices andthe exchange rate respond in the correct directions to each other.4 Still, the real exchange rate is atleast as likely to be nonstationary as in the weak PPP case:

r1,2,t = (1 − �1)p2,t − (1 − ˇ1)p1,t + ˛1 + u1,t . (3)

Taylor (1988) and Cheung and Lai (1993b), however, argue that this situation (non-unitary ˇ1 and �1)might not actually rule out strong PPP. They show how a certain form of price index measurementerror could lead to a measured very weak PPP relationship using the measured price indexes whenstrong PPP holds for the true price indexes.5 Meanwhile, some authors (Kargbo, 2006; Antonucci and

1 General summaries of past evidence are found in Froot and Rogoff (1995), Rogoff (1996), Taylor and Taylor (2004), andCushman(2008).

2 This is essentially the same terminology as used by MacDonald (1993), who calls this situation “strong-form PPP.”3 Theoretical textbook treatments of PPP generally define it as a proposition involving broad final-goods price indexes that

include both traded and nontraded goods (e.g., McCallum, 1996; Krugman and Obstefeld, 2003). Sideris’s (2006) empirical useof consumer price indexes is consistent with this. However, many papers have also used more narrowly defined indexes suchas the wholesale price index (e.g., Cheung and Lai, 1993b) or indexes for just traded goods (Xu, 2003).

4 MacDonald (1993) calls this situation “weak-form PPP.”5 The measurement error required for this effect is proportionately different at higher prices than at lower ones, implying

long-run non-neutrality of money in its effect on measurement error.

D.O. Cushman / Int. Fin. Markets, Inst. and Money 18 (2008) 413–424 415

Girardi, 2006) have considered that the above PPP relationships could still be defined and of interestwith a time trend, that is, even if �1 /= 0.

Now let us turn to the exchange rate between countries 1 and 3:

e1,3,t = ˛2 + ˇ2p1,t − �2p3,t + ı2p2,t + ε2 e1,2,t + �2t + u2,t . (4)

The same varieties of PPP could exist for e1,3,t as for e1,2,t if u2,t is stationary and the relevant parameterconstraints hold.

Finally, consider the exchange rate for countries 2 and 3. The cross-rate constraint gives

e3,2,t = e1,2,t − e1,3,t,e3,2,t = {(˛1 − ˛2)+(ˇ1−ˇ2)p1,t−(�1+ı2)p2,t+(�2+ı1)p3,t+(ε1−ε2)e1,2,t+(�1−�2)t+u1,t−u2,t}

/(1+ε1,t).(5)

Suppose strong PPP holds for e1,2,t and e1,3,t. Then it holds for e3,2,t. Next, suppose weak PPP holds fore1,2,t and e1,3,t. This does not imply weak PPP for e3,2,t unless additionally ˇ1 = ˇ2. Finally, suppose veryweak PPP holds for e1,2,t and e1,3,t. If ˇ1 = ˇ2, very weak PPP holds for e3,2,t.

Given I(1) variables, PPP testing involves determining whether stationary combinations such asEqs. (1), (4) and (5) exist and whether their parameters are also consistent with PPP. But, because Eq.(5) is simply a linear combination of Eqs. (1) and (4) with no new stationary process, PPP among allthree countries generates only two vectors. In the five-variable system, finding just one cointegratingvector implies some form of PPP for one exchange rate if the appropriate parameter constraints arealso met. Two cointegrating vectors can imply PPP for one to all three exchange rates depending onwhich parameter constraints are also met. A third vector must imply a third stationary relationshipthat is not related to PPP.6

3. The specification of deterministic factors

The Johansen procedure is based on a general VAR that includes an autoregressive lag order (cap-turing dynamics in ui) and possible deterministic variables (as in ˛i and t in Eqs. (1)–(5)). Regarding thedeterministic variables, Sideris (2006) concludes that he needs to allow for linear trends (drift in firstdifferences), seasonal factors, a permanent regime shift for German reunification, and three one-periodevents. The role and specification of these factors can be clarified by analyzing the error correctionform of the VAR along the lines of Hansen and Juselius (1995). For simplicity, let the autoregressive lagorder in levels be 2. The model is

�zt = �1�zt−1 + ˛ˇ′zt−1 + � + ıD + �t + εt (6)

where z and ε are vectors of variables and residuals, � 1 contains lag coefficients, ˛ and ˇ containthe error correction and cointegrating vector coefficients, � is a vector of constants, and D is a matrixcontaining the d deterministic factors consisting of various zero-one dummies.7 Let p be the numberof stochastic z variables and r the number of cointegrating vectors. Then ˛ and ˇ are p × r matrices offull rank. The �, ı, and � parameters can be decomposed into:

� = ˛�1 + ˛ ⊥ �2, (7)

ı = ˛ı1 + ˛ ⊥ ı2, (8)

� = ˛�1 + ˛ ⊥ �2, (9)

where �1 is an r-dimensional vector of constants in the cointegrating relations and �2 is a (p − r)-dimensional vector of linear trend slopes, or drift parameters, in the data. Similarly, ı1 is an

6 Accordingly, it is not correct to jointly consider three strong PPP relations among three vectors, one for each vector, as Sideris(2006) attempts in his H11 hypothesis. The third strong PPP relation is not independent from the first two. The correct jointstrong PPP test concerns only two sets of conditions.

7 For consistency with the cointegration literature, I use standard notation from Hansen and Juselius (1995), and so the variousGreek letters do not have the same meaning as in Eqs. (1)–(5).

416 D.O. Cushman / Int. Fin. Markets, Inst. and Money 18 (2008) 413–424

(r × d)–dimensional matrix that allows changes in the constants in the cointegrating relations, andı2 is a (p − r) × d-dimensional vector that allows changes in the drift parameters. Thus, ı1 and ı2 mea-sure the seasonal, regime-change, and other factors that Sideris (2006) models as deterministic. Finally,�1 is an (r × d)-dimensional matrix that allows linear trends in the cointegrating relations, and �2 is a(p − r) × d-dimensional vector that allows linear drift. Eq. (6) can then be rewritten as

�zt = �1�zt−1 + ˛

⎛⎜⎝

ˇ�1ı1�1

⎞⎟⎠

′

z̃t−1 + ˛ ⊥ �2 + ˛ ⊥ ı2D + ˛ ⊥ �2t + εt, (10)

in which z̃′t−1 = (z′

t−1, 1, D, t).The Johansen procedures allow the following cases regarding �, ı, and � (see Hansen and Juselius,

1995). First consider � (and assume ı and � equal zero). We can have �1 and �2 equal to zero, �1unrestricted and �2 = 0, or both �1 and �2 unrestricted. These are models 1–3 in Hansen and Juselius(1995). In model 1, there are no constants or trends at all. In model 2, constants are present but restrictedto the cointegrating vectors. This means that no drift or trends are specified. In model 3, constants are inthe cointegrating vectors and there is constant drift. This does allow linear trends in the data. Analogouscases apply to ı, except it is the changes in the constants that are potentially restricted. Finally, considernonzero �. If �1 is unrestricted but �2 = 0, then there are linear trends in the cointegrating vectors butdrift is constant. Combined with unrestricted �1 and �2, this is model 4 in Hansen and Juselius (1995).And if both �1 and �2 are unrestricted, then linear trends are in the cointegrating vectors and there islinear drift (quadratic trends in levels). This is model 5.

The specification of the dummies involves an additional issue, which becomes relevant if we haverestricted �2 but not ı2. The issue is the centering of the dummies. Consider the case of the seasonaldummies. If the intention is to allow no drift or trend and thus we set �2 = 0 (and �i = 0), then theseasonal dummies must be centered. Otherwise, the seasonal drift will not average to zero over eachyear and, moreover, drift will be zero in one quarter of each year. Similarly, failure to center the otherdummies will allow drift for the periods when the dummy is set to one but not in the other periods,and thus there will be net drift allowed over the full sample period. On the other hand, if the constantis not restricted, centering makes no difference.

With respect to these issues, it turns out that Sideris (2006) does not accomplish in his specificationwhat he intends. First, he intends to allow for linear trends or drift, which seems correct given long-runprice inflation, yet he applies the constant restriction of �2 = 0 (and he sets the time coefficients �i = 0).Second, having applied the restriction, he nevertheless does not center any of the dummies.8 Thus,as far as the seasonal effects are concerned, his specification actually does allow drift except for thefourth quarter of each year (his seasonal dummies are for the first three quarters). As far as the regimechange is concerned, his specification allows drift in the latter part of the sample but not the first. Andeach of his impulse dummies allows drift in one time period only.

Another specification issue for the regime change dummy concerns its breakpoint date. To captureeffects from German reunification, Sideris (2006) chooses the first quarter of 1990 for the beginningof the new regime. But German reunification did not occur until early in the fourth quarter of 1990.Sideris’s (2006) breakpoint date could nevertheless be the better one if there were anticipatory effectsfrom, say, the election in the German Democratic Republic in March, 1990, of the Christian DemocraticUnion party, which was favorable to reunification. Below, I report on the difference the date makes tothe misspecification tests of the VAR.

4. The impact of dummies on the sampling distributions

Although seasonal and one-period impulse dummies make no difference, regime change dummiesaffect the asymptotic sampling distributions and critical values for the Johansen cointegration tests.The critical values generally become larger, with the precise amount depending on where in the sample

8 I confirmed this by replicating his results. I am grateful to Dimitrios Sideris for providing his data set.

D.O. Cushman / Int. Fin. Markets, Inst. and Money 18 (2008) 413–424 417

Table 1System misspecification tests on the four-lag unrestricted VAR

Vector error autocorrelation from lags 1 to 6 F(150, 341) = 1.092 (0.257)Vector normality test Chi square(10) = 14.103 (0.168)Vector heteroskedasticity using squares F(600, 727) = 0.726 (1.000)Chow breakpoint (1990:4) F(315, 180) = 1.039 (0.390)Chow breakpoint (1999:1) F(150, 341) = 0.856 (0.862)

Note: The tests are calculated using PcGive 10.40. p-Values are given in parentheses after the test statistics. The Chow breakpointscoincide with German reunification and the start of the Euro.

the regime change occurs (Johansen and Nielsen, 1993).9 Sideris (2006), however, uses the standardasymptotic distributions. Therefore, his p-values are too small, leading to the possibility of overesti-mating the number of cointegrating vectors. The solution proposed by Johansen and Nielsen (1993) isto simulate the distribution relevant to the case at hand. This is the approach I follow below.10

5. Empirical results

I now redo the empirical analysis with a corrected and more complete consideration of trends,centered rather than uncentered dummies, a regime change dummy breakpoint date that matches theGerman reunification date, and sampling distributions that account for the regime change dummy.The passage of time since Professor Sideris did his work also allows me to extend the quarterly dataset by three and one-half years so that it now runs from 1973:1 through 2006:2.11 To confirm that thenew conclusions from the present work are not dependent on this extra data, however, I also brieflyreport revised results for the original time period.

The first step in the testing is to decide on the specification of the general model. I start with anunrestricted eight-lag VAR (in levels) that includes linear trends and the regime change dummy. Lagsfive through eight are not significant but lags four through eight are. This suggests the use of a four-lag system, and such a system passes autocorrelation, heteroskedasticity, normality, and structuralstability tests at the 0.05 level after two impulse dummies are added to control for two large residuals(exceeding three standard errors). See Table 1 for the test results. The choice of four lags is the sameas in Sideris (2006).

Meanwhile, the revised regime change dummy date not only matches the historical date moreclosely than Sideris’s (2006) date, it also leads to better misspecification test outcomes. The four-lagVAR with the Sideris’s (2006) regime change date fails system autocorrelation and normality tests atthe 0.10 level (with p-values of 0.057 and 0.095) while the four-lag VAR with the revised date does not(with p-values of 0.257 and 0.168).12

With the lag order and dummy specification decided, the cointegration testing and further con-sideration of trends can proceed. I accomplish these tasks jointly using the procedure suggested byJohansen (1992) and Hansen and Juselius (1995) and recently revised by Hjelm and Johansson (2005).The procedure is an extension of the “Pantula Principle” first suggested by Pantula (1989) for univariateunit root tests. In general, one starts by moving sequentially from Hansen and Juselius’s (1995) model

9 The intuition is that an unrestricted regime change dummy (a sudden permanent change) is somewhat similar to an unre-stricted trend (a gradual permanent change), which also increases the critical values. (On the other hand, the lack of effect onthe asymptotic distributions from unrestricted seasonal and impulse dummies is noted in Johansen et al., 2000.)

10 The simulated random walks have 1000 steps and there are 20,000 replications of each. TSP 4.5 is used. Alternatively, onecould bootstrap the tests as in Cushman et al. (1996) and Cushman (2000).

11 The source is the IMF’s International Financial Statistics online, November, 2006. As in Sideris (2006), the prices are consumerprice indexes and, starting in 1999, the mark/dollar rate becomes the euro/dollar rate.

12 With nonzero values in 1974:1 and 1993:1, my two impulse dummies match two in Sideris (2006), but the third one heuses for 1991:1 is not needed here. This is most likely because of a change to the German consumer price index series made bythe IMF (or its source) since the data set used by Sideris (2006) was published. The series he uses has a presumably erroneousdownward jump in 1991:1, the first full quarter after German reunification. I suspect that the jump was a temporary error,because still older IMF data for the German CPI do not have it while otherwise matching the more recently published dataquite closely. Sideris’s (2006) 1991:1 and 1993:1 impulse dummies are not actually mentioned in his paper, but their use wasdescribed to me in e-mail from Professor Sideris dated May 31, 2006.

418 D.O. Cushman / Int. Fin. Markets, Inst. and Money 18 (2008) 413–424

Table 2Trace test results

Null Trend model

2 3 4 5

r = 0 96.84 (0.004) 87.94 (0.009) 103.79 (0.012) 92.05 (0.038)r ≤ 1 53.48 (0.162) 45.51 (0.278) 55.61 (0.369) 52.75 (0.323)r ≤ 2 29.98 22.26 32.35 29.63r ≤ 3 13.76 6.74 16.46 14.37r ≤ 4 4.60 0.38 6.33 4.30

Note: p-Values from the Monte Carlo simulations are given in parentheses.

1 to model 5, each time testing the null of r = 0, where r is the number of cointegration vectors. Onethen repeats the sequential consideration of models 1–5 under the null of r ≤ 1, then under the null ofr ≤ 2, and so on. The process stops when a null is accepted. The accepted r-value indicates the numberof cointegration vectors, and the model number that coincides with the acceptance indicates the con-stant or trend specification. The revision of Hjelm and Johansson (2005) is that if model 3 is indicatedat this point, the significance of the (restricted) linear trend in the cointegration space of model 4 istested. If significant, model 4 is chosen instead of 3. Hjelm and Johansson (2005) present Monte Carloevidence that shows that their modified Pantula Principle works quite well in simultaneously choosingthe correct trend model and number of cointegration vectors.

Hansen and Juselius (1995) and Hjelm and Johansson (2005) recommend that any of models 1–5that are unreasonable in a specific application be excluded from the sequential process. Certainly,model 1, with no constants or trends at all, is quite unreasonable in the present application, andmodel 2, which allows no trends, also seems unreasonable given average annual inflation rates overthe sample period of 2.8–4.6% for the three countries. However, I do consider model 2 since it wasinadvertently used by Sideris (2006).

The Johansen trace test values for the various models and hypotheses are given in Table 2. Thevalues incorporate the degree-of-freedom adjustment used by Sideris (2006).13 At the 0.05 or 0.10level, the conclusion from applying the Pantula principle is one cointegration vector and model 2. Asa response to low power, however, Juselius (1999) has suggested using higher significance levels ifthe result is a reasonable model, and Johansen and Juselius (1990) actually use a significance level of0.20 in one instance. Since the no-trend model 2 is not very reasonable, model 2’s p-value of 0.162 forthe null of r ≤ 1 could therefore be considered high enough to reject it. Johansen and Juselius’s (1990)test for restricting the constants as in model 2 compared to model 3 can also be applied. It generates2(4) = 7.96 with a p-value of 0.092.

These considerations lead us to model 3 and r ≤ 1, which can clearly be accepted. However, followingHjelm and Johansson (2005), I check the significance of the trend in the cointegration space of model4 (with one cointegration vector). The test generates 2(1) = 5.75 with a p-value of 0.016. The finalconclusion is therefore model 4 and one cointegration vector.14 The one-vector conclusion, however,follows regardless of which trend model is chosen; there is no support for three vectors as reported bySideris (2006). But one cointegration vector could still be consistent with a PPP relationship betweentwo of the three countries. This would require the vector coefficients to meet various constraints asdiscussed in Section 2. Because the evidence for determining the trend model is not decisive, let usconsider both models 2 and 4.

13 The degree-of-freedom adjustment is a correction for small-sample size distortion in the direction of an excessive tendencyto reject if the null is true. It has been suggested by Reimers (1992) and supported by simulations in Cheung and Lai (1993a),Gregory (1994), Cushman et al. (1996), and Cushman (2000). The simulations in the last two papers suggest that the correctionis also useful for various chi square tests in the Johansen cointegration procedures, and so I apply it here to all subsequent tests.

14 The restriction of the linear trend to the cointegration space in model 4 as compared to the unrestricted trend of model 5is easily accepted with a p-value of 0.58. Thus, suppose a general-to-specific testing approach (Hendry, 1995) were followedthat began with the most complicated trend specification, which is model 5. One would reject the no-cointegration null. Themodel would then pass the simplification of omitting the quadratic trend, but further eliminating the linear trend from thecointegration vector would not pass. The conclusion is again model 4 and one cointegration vector.

D.O. Cushman / Int. Fin. Markets, Inst. and Money 18 (2008) 413–424 419

Table 3Cointegration vector parameters and exclusion tests

Variable Trend model 2 Trend model 4

Coefficients 2(1) p-Value Coefficients 2(1) p-Value

eJ–U.S. −1.00 4.12 (0.042) −1.00 5.77 (0.016)pG 10.69 16.35 (0.000) 7.15 9.58 (0.002)eG–U.S. 0.93 3.59 (0.058) 0.99 5.32 (0.021)pU.S. −6.22 15.56 (0.000) −7.87 21.39 (0.000)pJ 2.78 5.87 (0.015) 6.10 10.92 (0.001)Constant −29.00 9.59 (0.002)Trend 0.026 5.75 (0.016)

Note: Ignoring any trend, very weak PPP for one country pair would otherwise follow if the signs and values for the vectorcoefficients of eJ–U.S. , pG, eG–U.S. , pU.S. , and pJ could meet the following constraints (with a, b, and c positive). For German–U.S. PPP:(0, +a, −b, −c, 0) or (0, −a, +b, +c, 0). For Japanese–U.S. PPP: (−1, 0, 0, −a, +b). For German–Japanese PPP: (−1, −a, +1, 0, +b).

Table 3 shows the vector coefficients and the corresponding chi square exclusion tests and p-values.The p-values imply that all five variables are required to generate the stationary relationship (regardlessof trend model) because each coefficient is statistically significant at (or very near) the 0.05 level, andsome are at the 0.01 level. This is not consistent with a PPP relationship involving any pair of the threecountries, because each PPP relationship (whether strong, weak, or very weak) requires one or twovariables to be excluded. The trend in the vector of preferred model 4 is also not consistent with PPPas usually defined. And even if we strictly apply the 0.05 significance level and thus accept model 2and omit eG–U.S. from the vector, the result still cannot reflect PPP between Japan and the U.S. becausethe German price remains.

6. Some extensions

I now discuss several extensions to the above results. The first is whether a modification to theJohansen procedure that potentially can achieve higher power is able to find evidence of PPP after all.It is not. But what is the economic meaning of the cointegration relationship that has been found?Considering this is the second extension. And the third extension is to see whether the conclusionthat PPP does not hold for the three exchange rates is an artifact of extending the data set beyond the2002:4 ending date in Sideris (2006).

6.1. A modification for more power

Greenslade et al. (2002) suggest that before testing for cointegration the researcher test for andimpose weak exogeneity on certain variables for which this seems reasonable a priori (or on what theycall theoretical grounds).15 This is meant to increase power in the cointegration tests. In the PPP case,one could argue that the three prices could be weakly exogenous under the assumptions that they aresticky in the short run, determined by central banks in the long run, and, thus, do not respond to PPPviolations. To test this in the Greenslade et al. (2002) fashion, one needs to assume some number ofvectors (before actually testing for them), and in the present case that could be two vectors as predictedby the PPP theory for three countries. When I do this, I find that only the U.S. price can be treated asweakly exogenous.16

The next step is to apply the standard Johansen cointegration test but with the U.S. price assumedweakly exogenous (see Hansen and Juselius, 1995; Pesaran et al., 2000). The resulting trace test resultsare in Table 4 and the conclusions are quite similar to those from Table 2. The modified Pantula principle

15 I am indebted to the referee for drawing this paper to my attention.16 For example, using the 2-test in Johansen and Juselius (1990), the weak-exogeneity p-values for the prices are: pG, 0.000,

pUS, 0.595, and pJ , 0.001 with trend model 4. The other p-values for weak exogeneity assuming two vectors are: eJ-US, 0.033 andeG-US, 0.152. Although eG-US is weakly exogenous at conventional significance levels, it is not a strong conclusion and there is notheoretical reason to impose it in the cointegration testing.

420 D.O. Cushman / Int. Fin. Markets, Inst. and Money 18 (2008) 413–424

Table 4Trace test results with weak exogeneity imposed on pU.S.

Null Trend model

2 3 4 5

r = 0 81.29 (0.005) 77.20 (0.004) 88.90 (0.009) 77.37 (0.032)r ≤ 1 39.72 (0.209) 35.80 (0.245) 40.80 (0.426) 38.08 (0.408)r ≤ 2 17.71 13.81 18.50 16.11r ≤ 3 8.05 5.40 6.73 4.33

Note: Pesaran et al. (2000) give critical values for various cases with weakly exogenous variables, but here there is also theregime-change dummy that will affect the sampling distributions. Thus, I have simulated the relevant distributions in order toobtain the p-values that are given in parentheses. The simulated random walks have 1000 steps and there are 20,000 replicationsof each.

Table 5Cointegration vector parameters and exclusion tests with weak exogeneity imposed on pU.S.

Variable Trend model 2 Trend model 4

Coefficients 2(1) p-Value Coefficients 2(1) p-Value

eJ–U.S. −1.00 4.59 (0.032) −1.00 5.89 (0.015)pG 10.49 15.91 (0.000) 7.05 9.50 (0.002)eG–U.S. 0.74 2.65 (0.103) 0.96 5.27 (0.022)pU.S. −6.10 15.58 (0.000) −7.91 21.13 (0.000)pJ 2.33 4.61 (0.032) 6.12 11.20 (0.001)Constant −26.35 8.34 (0.004)Trend 0.027 6.70 (0.010)

Note: See note to Table 3.

(with the 0.20 significance level stretched to 0.209) leads once again to one cointegration vector andtrend model 4. (The restricted trend in model 4 has a p-value of 0.010, and so model 3, which isotherwise indicated, is rejected.) And the cointegration vector estimates from this procedure (witheither trend model 2 or 4) are quite similar to those in Table 3. See Table 5. Thus, the approach ofGreenslade et al. (2002) leads to no more evidence of PPP than before. (Since it makes little differencehere, the approach is not used in the rest of the paper.)

6.2. What does the cointegration relationship mean?

In terms of statistical significance, it appears that the vector in either trend model might pri-marily involve a relationship among the three prices. Furthermore, at the 0.05 level and using thetest in Johansen and Juselius (1990), all variables are weakly exogenous except the German andJapanese prices regardless of the trend model.17 In fact, in trend model 4 the joint hypothesis thatthe exchange rates can be excluded from the cointegration vector and that all variables except theGerman and Japanese prices are weakly exogenous can be accepted: 2(5) = 8.059 with a p-valueof 0.153. The resulting cointegration vector for (eJ–U.S., pG, eG–U.S., pU.S., pJ, and t) is (0, −45.90, 0,39.57, −28.57, −0.107).18 These findings suggest that German and Japanese price or monetary pol-icy react to maintain some relationship to U.S. price or monetary policy. I pursue this possibility in theAppendix.

6.3. Results for the data set ending in 2002

If Sideris’s (2006) ending date of 2002:4 is used instead of 2006:2, the same four-lag/3-dummy VARspecification used to generate my primary results passes the misspecification tests, and the modified

17 E.g., the p-values under trend model 4 are: eJ-US, 0.912; pG, 0.000; eG-US, 0.058; pUS, 0.771; and pJ , 0.000.18 The situation is quite similar for trend model 2.

D.O. Cushman / Int. Fin. Markets, Inst. and Money 18 (2008) 413–424 421

Pantula principle (at either the 0.05 or 0.10 level) leads once again to one cointegration vector andtrend model 4.19 The vector coefficients are similar to those for the full data set results in Table 3. Butwhile the three price coefficients are all statistically significant at the 0.05 level as in Table 3, the twoexchange rate coefficients no longer are (each now has a p-value of about 0.11). Thus, the evidenceagainst PPP is at least as convincing as from the longer data set.

7. Comparison with previous cointegration findings on PPP among the U.S., Germany, andJapan

Although several papers that also apply Johansen cointegration procedures to two-country modelsfor the exchange rates in the present paper report less evidence than Sideris (2006) does in favor of PPPfor the recent floating period, they do report some. Thus, the present paper, which finds no favorablePPP evidence, not only contradicts Sideris (2006) but also these other papers.

Cheung and Lai (1993b) examine the German–U.S. case, Xu (1999) examines the Japanese–U.S. case,and MacDonald (1993) examines both. The three papers find very weak PPP for these two exchangerates. Edison et al. (1997) examine all three bilateral rates among the three countries. They also find veryweak PPP for the Japanese–U.S. case but not for the German–U.S. case. But they find strong PPP for theGerman–Japanese case.20 The divergence between these somewhat favorable PPP results and my ownunfavorable ones could reflect the fact that the earlier papers do not examine the three countries jointly,that they have much shorter samples, or that they only consider only one trend model (number 3).

To look further into the differences, I first investigate the result of using the joint three-countryfive-variable model for a shorter sample, using an ending date of 1990:3. This date represents a roughaverage of the ending dates in Cheung and Lai (1993b), Xu (1999), MacDonald (1993), and Edison et al.(1997) (whose ending dates range from 1989 to 1994) and it avoids the need for the regime-changedummy. In the five-lag model necessary to pass misspecification tests, the Johansen test cannot rejectthe null of zero vectors for any trend specification.21 Thus, the five-variable model with a similarsample period does not confirm the PPP findings of the earlier papers. This is not, however, a strongcontradiction of the results from the three-variable models. The sample period may be too short forthe larger five-variable model to have the power to detect the PPP relationships found in the otherpapers.

Now I investigate the result of applying bilateral three-variable models to the full data set of thepresent paper. According to my earlier results, these models are evidently misspecified (by each omit-ting two variables that are necessary in the cointegration vector), so the results indicate what isfound by ignoring this. For the German–U.S. case, a five-lag VAR passes all misspecification tests. TheJohansen-Pantula procedure at the 0.05 level then selects trend model 3 and r = 0. For the Japanese–U.S.case, a four-lag VAR is necessary and the Johansen-Pantula procedure at the 0.05 level chooses trendmodel 4 and r = 0. Finally, for the German–Japanese case, a six-lag VAR is necessary and the Johansen-Pantula procedure at the 0.05 level now chooses trend model 3 and r = 1. The resulting cointegrationvector, however, has the wrong signs for PPP. Finally, if the 0.10 significance level is used instead of the0.05 level for any of these analyses, the Johansen-Pantula procedure sometimes leads to different trendmodels or to one cointegration vector (where none was found before), but if one vector is suggested,it always has some insignificant or wrongly signed coefficients and is thus not supportive of PPP.22

For none of these exchange rates, therefore, do the bilateral models indicate that any form of PPPexisted for the longer estimation period. These findings do not appear to be very sensitive to trend

19 For the null of r = 0, the trace test p-values for models 2–5 are 0.001, 0.002, 0.005, and 0.025. For the null of r ≤ 1, the tracetest p-values for models 2–5 are 0.035, 0.050, 0.147, and 0.175. The restricted trend in model 4 has a p-value of 0.019.

20 This last finding is not, however, confirmed by their application of the Horvath and Watson (1995) cointegration procedure.21 The model requires no regime or impulse dummies. The trace test statistics for the null of r = 0 for trend models 2 through

5 are 55.88, 52.65, 68.50, and 62.20. The p-values are all in the neighborhood of 0.50.22 Full details are available from the author upon request. The dummy variables needed in the three-variable models are not

identical to those in the full model. The German-U.S. case only requires the regime change dummy and the impulse dummyfor 1993:1. The Japanese-U.S. case only requires the impulse dummy for 1974:1. Finally, of the dummies in the full model, theGerman-Japanese case only requires the impulse dummy for 1993:1. But, to avoid a very large residual, it also requires a newimpulse dummy for 1997:2.

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specifications. They are consistent with the conclusions from the three-country five-variable modelfor the full sample and for the shorter sample ending in 1990:3. If PPP nevertheless existed for theseexchange rates during shorter time periods as reported in the earlier papers, it has apparently notsurvived the passage of time.23

8. Concluding summary

Sideris (2006) has usefully suggested cointegration analysis to jointly examine PPP among theprices and exchange rates of Germany, Japan, and the U.S. He reports three cointegration vectors andconcludes that they are consistent with some form of PPP for all three exchange rates. But after employ-ing more plausible trend and dummy specifications and using sampling distributions that account forinclusion of the German reunification regime change dummy, I find only one cointegration vector andit is not consistent with any form of PPP. The lack of any favorable evidence for PPP is robust to sev-eral trend specifications and estimation time periods. It also fails to support somewhat favorable PPPfindings for the same exchange rates in several earlier papers that apply the Johansen cointegrationprocedure to simpler two-country models with shorter data sets. The failure is not, however, depen-dent on use of the three-country model. The two-country models do not support PPP either, when thelonger data set is used.

Acknowledgment

I am indebted to an anonymous referee for useful comments.

Appendix A. The relationship among the three prices

The cointegration vector derived in section 6.2 shows a relationship among the three countries’prices. Since monetary policy drives prices in the long run, the result is consistent with a relationshipamong monetary policies. In fact, Clarida et al. (1998) find (for an estimation period of 1979 throughthe early 1990s) very similar monetary policy rules for the U.S., Germany, and Japan. Furthermore,those for Germany and Japan include quantitatively small but statistically significant responses to U.S.monetary policy. Consequently, it is not surprising that the three price levels, which the central bankscontrol in the long run, would be involved in cointegration relationships. For example, the Germanand Japanese central banks might wish to maintain some level of competitiveness with the U.S. and, ifthey believe that the exchange rate is relatively erratic or uncontrollable, they might focus on the U.S.price level.

The results in the present paper can be further analyzed to shed more light on the relationshipamong the prices. The restricted cointegration vector in Section 6.2 of the main text, when normalizedon the U.S. price coefficient, gives a long-run relationship of:

pU.S. = 1.160pG + 0.727pJ + 0.0027 t. (11)

Then, because one cointegration vector implies two unit root processes (or common stochastic trends)among the three prices, each price can be modeled as a function of the two (possibly unobserved) unitroot processes, u1 and u2. With the deterministic trend omitted for simplicity:

pU.S. = a1u1 + a2u2, (12)

pG = b1u1 + b2u2, (13)

pJ = c1u1 + c2u2. (14)

23 MacDonald and Marsh (2004) actually precede Sideris (2006) in pursuing the cointegration analysis of PPP jointly amongthe U.S., Germany, and Japan for the recent floating period. They find PPP relationships if, but only if, interest rates are alsoincluded in the model. Thus, their results are essentially in agreement with ones in the present paper because they find noevidence of PPP involving only the standard price and exchange rate variables.

D.O. Cushman / Int. Fin. Markets, Inst. and Money 18 (2008) 413–424 423

The hypothesis that Germany and Japan follow long-run price policy similar to that of the U.S. wouldbe reflected in some similarity between b1, b2, c1, and c2 on the one hand and a1 and a2 on the other.Combining Eqs. (11)–(14) leads to:

1.160 = a1c2 − a2c1

b1c2 − b2c1, (15)

0.727 = a2b1 − a1b2

b1c2 − b2c1. (16)

Then, with the U.S. as the numeraire country, the unit root processes can be normalized suchthat a1 = a2 = 1. But there is still not enough information to solve for the four remaining parametersindividually. However, it does turn out from Eqs. (15) and (16) that b1 + c1 > 0 and b2 + c2 > 0. In words,the two non-U.S. countries’ prices have combined responses to each of the two unit root processes thatare in the same direction as the U.S. price responses. Alternatively, assume that the U.S. price followsjust one unit root process such that a1 = 1 and a2 = 0. Then pU.S. = u1 and b1 + c1 > 0. In this case, the othertwo countries’ combined response to the U.S. price is positive, but they also respond to another unitroot process.24

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