The Article considers the speciÞcation of models used to test Pur-
chasing Power Parity when applied to cross exchange rates. SpeciÞcally,
conventional dynamic models used to test stationarity of the real exchange
rate are likely to be misspeciÞed, except when the parameters of each ex-
change rate equation are the sam

Hunter, J

Simpson, M

English

The Article considers the specification of models used to test Purchasing Power Parity when applied to cross exchange rates. Specifically, conventional dynamic models used to test stationarity of the real exchange rate are likely to be misspecified, except when the parameters of each exchange rate equation are the same

Brunel University Research Archive

The speciÞcation of cross exchange rateequations used to test Purchasing Power Parity.John HunterDepartment of Economics and Finance, Brunel University,Uxbridge, Middlesex, UB8 3PH, EnglandMark SimpsonNational Audit Oﬃce, 157-197 Buckingham Palace Road Victoria,London SW1W 9SPNovember 18, 2004AbstractThe Article considers the speciÞcation of models used to test Pur-chasing Power Parity when applied to cross exchange rates. SpeciÞcally,conventional dynamic models used to test stationarity of the real exchangerate are likely to be misspeciÞed, except when the parameters of each ex-change rate equation are the same.Keywords : Arbitrage; misspeciÞcation; non-stationarity; Purchasing PowerParity; real exchange rateJEL Classifications : C32, F3111 IntroductionThe Article considers the problem of testing non-stationarity on cross exchangerates. Smith and Hunter (1985) addressed a theoretical proposition that onlyspeciÞc exchange rate models are coherent under cross arbitrage constraints.Here it is shown that cross arbitrage has similar implications for the speciÞcationof dynamic models used to test Purchasing Power Parity and the associatedproposition that the real exchange rate is stationary. To test the propositionthat the real exchange rate is stationary, either requires independent cross ratedata; that the coeﬃcients of every dynamic model on which a cross rate is basedare the same or that the cross rate equations are correctly speciÞed, when inaddition to the cross rate variables they include a set of dollar variables for oneof the two exchange rates used to derive the cross rate. Furthermore, arbitrageis also likely to bind, when the data collected is generated by transactions thatare calculated via an intermediary $ rate conversion; a common practice in manyoﬃces dealing in foreign exchange. The Þnal proposition implies that cross rateequations ought to have up to double the number of parameters of $ equations.By implication, the power and size of tests based on correctly speciÞed crossrate equations is likely to be aﬀected by the inclusion of the extra variables.2 Cross Equation Dynamics and tests of Sta-tionarity.Consider a small dynamic extension to the model of the exchange rate presentedin Smith and Hunter (1985):∆eijt = δoij + γijeijt−1 + βij(xi − xj)t−1 + β0ij∆(xi − xj)t + εijt (1)where eijt values the home currency in terms of country j, lowercase variablesare in logarithms, xi − xj is a j × 1 vector of parity conditions associated with,prices, interest rates and money, βij a 1× j vector of parameters and εijt is thedisturbance term for the ijth exchange rate equation. When international trans-actions occur through the intermediary of an international means of exchangeand country j deÞnes that denomination then application of the associated ar-bitrage condition means that eikt = eijt − ekjt and the cross rate equation is:∆eikt = δoik + γijeijt−1 − γikekjt−1 + βij(xi − xj)t−1 − βkj(xk − xj)t−1+β0ij∆(xi − xj)t − β0kj∆(xk − xj)t + εijt − εkjt (2)To simplify the exposition it will be assumed that εjkt = εijt − εkjt are whitenoise innovations, xm − xl are weakly exogenous for all m and l.1 and that the1 If the errors are not innovations and the regressors not weakly exogenous (see Ericssonand Irons(1994)), then the degree of inconsistency and the order of misspeciÞcation is likelyto be magniÞed.2the following reparameterisation can be applied:∆eikt = δoik + γijeikt−1 − (γkj − γij)eikt−1 + βij(xi − xk)t−1 −(βkj − βij)(xk − xj)t−1 + β0ij∆(xi − xk)t −(β0kj − β0ij)∆(xk − xj)t−1 + εikt.Hence:∆eikt = δoik + γijeikt−1 + βij(xi − xk)t−1 + β0ij∆(xi − xk)t−1 + εikt, (3)with εikt an innovation process, if and only if (γkj − γij) = 0, (βkj − βij) = 0and (β0kj + β0ij) = 0. Otherwise, (3) is misspeciÞed and estimates of γij , βijand β0ij are biased and inconsistent. To emphasize this point, models used totests long-run Purchasing Power Parity (PPP) and stationarity are considerednext.2Let xl = pl a price index for country l = i, j. In the context of (1), PPP issaid to hold in the long-run when βij = −γij and:∆eijt = δoij + γij(eij − (pi − pj))t−1 + β0ij∆(pi − pj)t + εijt (4)Tests of the proposition that ρijt = eijt−(pi−pj)t is integrated of order 1 or I(0)are based on the coeﬃcient on the error correction term γij < 0.The distributionof this test statistic under the null for the case considered here is asymptoticallynormal (see Kremers et al (1992)), but the power and performance of the testrelies on the proposition that the model is correctly speciÞed. In general, testsof PPP when the cross rate is considered need to be applied to:∆eikt = δoik + γij(eij − (pi − pj))t−1 − (γkj − γij)(eik − (pk − pj))t−1+β0ij∆(pi − pk)t − (β0kj − β0ij)∆(pk − pj)t−1 + εikt,which implies that the joint proposition γij < 0 and γkj < 0 be satisÞed, exceptfor the case where γkj − γij = 0 or γkj = γij . It follows that PPP can only betested coherently across all $ exchange rates and their cross rates using equationsof the form of (4), when:γkj = γij , βkj = βij and β0kj = β0ij .If PPP is tested using the test due to Dickey and Fuller (1979) via the propo-sition that the real exchange rate is I(0), then in addition to βij = −γij , it isrequired that β0ij = 1 and for the Dickey-Fuller test:∆eijt −∆(pi − pj)t = δoij + γij(eij − pi + pj)t−1 + εijtor∆ρijt = δoij + γijρijt−1 + εijt. (5)2 It should be noted that these results readily extend to the case where the variables usedare generated by a potentially cointegrating, vector moving average model, see the discussionof the Granger representation theorem in Engle and Granger(1987).3If j is the $, then the value of the currency for country k relative to the $ isgiven by the same type of dynamic equation:∆ρkjt = δokj + γkjρjkt−1 + εkjt (6)It follows from (5) and (6) that the cross rate equation is:∆ρijt −∆ρjkt = ∆ρikt = δoij + δokj + γijρijt−1 + γkjρjkt−1 + εijt + εkjtor∆ρikt = δoik + γikρikt−1 − (γik − γkj)ρjkt−1 + εikt.Hence the usual Dickey-Fuller test applied to the cross rate equation suﬀersfrom omitted variable bias as it assumes that (γik − γkj) = 0 or (γik = γkj).The following results associated with univariate tests of stationarity for real $exchange rates show γik is not usually equal to γkj .3Table 1 Summary of Augmented Dickey Fuller Tests and their RespectiveCoeﬃcientsCountry Parameter γik Test Statistic ResultItaly -0.0695 -1.96 n-sSpain -0.0504 -1.39 n-sBelgium -0.1188 -2.36 n-sDenmark -0.0612 -1.83 n-sFinland -0.0707 -1.36 n-sFrance -0.2207 -2.96 sGermany -0.0781 -1.76 n-sIreland -0.2208 -3.69 sLuxembourg -0.2023 -3.61 sHolland -0.1955 -2.70 n-sPortugal -0.0204 -0.84 n-sUK -0.1543 -2.83 n-s(n-s non-stationary t>-2.89, s stationary, t<-2.89)Based on table 1, Dickey-Fuller tests associated with cross rates will be similarwhen Italy, Spain and Denmark are analysed, but very diﬀerent when appliedto Portugal or France. The question of speciÞcation is further complicatedwhen one considers that the results in table 1 based on models with 5th orderlags are also likely to be aﬀected by the signiÞcance of diﬀerent lags and a keyassumption that prices are weakly exogenous.3 ConclusionAny diﬀerences in inference based on cross exchange rate data and their original$ equations is likely to be due to misspeciÞcation of the cross rate equations used3Quarterly observations on dollar real exchange rates were drawn from the DatastreamDatabase for the period (1980q1  1998q1) for twelve countries: Italy, Spain, Belgium, Den-mark, Finland, France, Germany, Ireland, Luxembourg, Holland, Portugal and UK.4to test either for PPP or stationarity of the real exchange rate. It is argued, onthe basis of the results in Smith and Hunter (1985), that the use of conventionaldynamic models (4) and (5) is inappropriate for cross rate equations except whenthe estimated parameters of the dollar rate equations are the same. Otherwise,the cross rate equations will be misspeciÞed and the parameter estimates ofγik used to test stationarity of the real exchange rate will be inconsistent andbiased. Where more appropriate equations with dynamic cross rate and dollarrate terms are used, then the power of tests based on cross rate equations wouldbe reduced in small and moderate samples due to the inclusion of an extra setof regressors. An appropriately specifoed augmented Dickey-Fuller test with klags on the dollar equation would require 2k lagged terms in the equivalent crossrate equation.The problem is not ameliorated by using an eﬀective exchange rate as anyrate generated from n dollar rates would require n−1 dynamic terms for compa-rable inference. Nor can it be fully resolved by considering a system or non-linearmodels. Estimating a system only obviates the requirement that the price, inter-est rate or money supply diﬀerentials are weakly exogenous. While nonlinearitymay eliminate exact cross restrictions, but not the inconsistency caused by theexclusion of linear or non-linear terms from the $ equations.4 ReferencesDickey, D.A. and Fuller, W.A., 1979. Distribution of the Estimators for Au-toregressive Time Series with a Unit Root. Journal of the American StatisticalAssociation, 74, 427-431.Engle, R.F. and Granger, C.W.J., 1987. Co-Integration and Error-Correction:Representation, Estimation and Testing. Econometrica, 55, 251-276.Ericsson, N.R. and Irons, J.S., 1994. Testing Exogeneity. Oxford UniversityPress, Oxford.Hunter, J and Simpson, M, 2001. A Panel Test of Purchasing Power Parityunder the Null of Stationarity. Paper presented at the Money Macro FinanceConference, Belfast, September 2001 and Brunel Discussion Paper, 01- 01.Kremers, J.J.M., Ericsson, N.R. and Dolado, J. 1992. The power of cointe-gration tests. Oxford Bulletin of Economics and Statistics. 54, 325-348.Smith, R.P and Hunter, J, Cross, 1985..Arbitrage and SpeciÞcation in Ex-change Rate Models. Economics Letters, 18, 375-376.5

The specification of cross exchange rate equations used to test Purchasing Power Parity

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