Testing for Purchasing Power Parity: Econometric Issues and an Application to Developing Countries

TESTING FOR PURCHASING POWER PARITY:ECONOMETRIC ISSUES AND AN APPLICATION TO

DEVELOPING COUNTRIES*

byDERICK BOYD

University of East Londonand

RON SMITH{Birkbeck College, University of London, and University of Colorado

There is now a vast literature on testing purchasing power parity(PPP). Any test is conditional on a particular econometric speci¢cationwhich embodies a set of auxiliary assumptions. This paper reviewsthe issues involved in econometric speci¢cation and estimation in thetime series and panel models used to test PPP. We start from a generalmodel and then systematically examine the implicit restrictions thatare imposed to obtain the standard procedures and discuss theimplications of these procedures for estimation and inference. Theissues are illustrated on data for a panel of 31 developing countries,1966^90.

" Introduction

There is now a vast literature on testing purchasing power parity(PPP) using national data on exchange rates and prices. Rogo¡ (1996)and Taylor (1995) provide recent reviews. Any test is conditional on aparticular econometric speci¢cation, which embodies a set of auxiliaryassumptions. In this paper we provide a systematic exposition of thealternative speci¢cations and illustrate the sensitivity of the results tospeci¢cation on a sample of 31 developing countries over the period1966^90. As with real exchange rates themselves, views about PPP haveshown persistent swings. The orthodoxy of the early 1970s in favourof PPP was largely abandoned in the 1980s as new time series testscould not reject the hypothesis that there was a unit root in the realexchange rate. More recently failure to reject the unit root hypothesishas been interpreted as a product of the low power of the tests (e.g.Edison et al., 1997) and tests using either very long spans of historicaldata (e.g. Lothian and Taylor, 1996), panel data (e.g. Frankel and

ß Blackwell Publishers Ltd and The Victoria University of Manchester, 1999.Published by Blackwell Publishers Ltd, 108 Cowley Road, Oxford OX4 1JF, UK, and 350 Main Street, Malden, MA 02148, USA.

287

The Manchester School Vol 67 No. 3 June 19991463^6786 287^303

*Manuscript received 16.10.95; ¢nal version received 5.1.98.{We should like to thank Philip Arestis, Jerry Coakley, Robert McNown, Bahram Pesaran,

Hashem Pesaran, Martin Sola and two anonymous referees for comments, and NoraZerrad for data assistance.

Rose, 1996) or long-horizon predictability (e.g. Mark, 1995) have beenmore supportive of PPP. Although doubts remain about the size of thetests in long spans of data (e.g. Engel, 1996), attention has tended toshift from testing PPP to measuring the speed of convergence to PPPand the determinants of real exchange rates. The consensus suggests aspeed of adjustment of about 15 per cent per annum, equivalent to ahalf-life of around four years. However, as the growth literature hasmade clear, there are major di¤culties in measuring the speed ofconvergence from cross-country data. Lee et al. (1997) review thosedi¤culties in the growth context. Most of the literature has assumedlinear adjustment and we shall do the same. However, giventransactions costs non-linear adjustment is more likely: slow adjustmentclose to equilibrium, fast adjustment further away (see Michael et al.,1997).

Most of the recent cointegration analyses have used data fromindustrialized countries, but in terms of both policy and methodology theissue is more interesting for developing countries. In policy terms,exchange rate management is at the centre of many ¢nancial stabilizationplans in developing and transitional countries and PPP provides animportant theoretical issue in external adjustment policy. For instance, inhis in£uential paper on the question of the contractionary impact ofdevaluation, Edwards (1986, p. 503) noted: `An essential element in thetraditional view of devaluations is the assumption that nominaldevaluations generate an improvement in the domestic relative price oftradables to nontradables. That is, it is assumed that nominal devaluationsresult in real devaluations.' PPP also provides an important basis for the¢nancial stabilization and structural adjustment policies proposed by theIMF and the World Bank and plays a role in the choice between money,in£ation or exchange rate targeting in the design of monetary policy. Inmethodological terms, developing countries show more cross-sectionvariation (hyperin£ations are rare in industrialized countries) and showmore time series noise. The noise arises both from severe measurementproblems (o¤cial and market exchange rates often di¡er for instance) andfrom large policy shocks. If there is non-linear adjustment, return to PPPshould be more obvious in developing countries, which are more often farfrom equilibrium.

In this paper we examine the evidence for long-run PPP using annualdata for 31 developing countries over the 25 year period 1966^90 obtainedfrom the International Financial Statistics Yearbook (IMF, 1994). Thenominal exchange rates used are annual averages of the market rate ando¤cial exchange rate for countries where these are available; whereunavailable the year-end rates are used. The domestic prices are thevarious national consumer price indices (CPIs). The foreign price is the USCPI. It is well known that tests of PPP may be sensitive to measurement

288 The Manchester School

ß Blackwell Publishers Ltd and The Victoria University of Manchester, 1999.

problems and the IFS data for developing countries show many problems.1

The results are sensitive to choice of price index and base country. Inindustrial countries the wholesale price index tends to provide moresupport for PPP than the CPI. Nominal exchange rates measured againstthe deutschmark tend to provide more support than those measuredagainst the dollar, because of the long swings in the US real exchange rate;see for instance Coakley and Fuertes (1997) and Husted and MacDonald(1997).

á Measurement Issues

PPP is usually interpreted as implying that domestic and foreign pricesmeasured in a common currency should be equal. At any moment this willnot hold exactly, but deviations from PPP should only be transitory asthe system will adjust to reassert equality. Thus, using underlining toindicate true values, if sit is the logarithm of the true nominal spotexchange rate (domestic currency of country i per unit of some foreignbase currency), p

itand p�

tare the logarithms of the true values of domestic

prices and foreign prices, and eit is a mean-zero stationary error, thenPPP implies

sit ÿ pit� p�

t� eit t � 1; 2; . . . ; T ; i � 1; 2; . . . ;N �1�

Direct data on prices are rarely available; instead the data are for aprice index relative to a base year, say t � 0, i.e. we observe: p

itÿ p

i0.2

Thus in terms of the available data the relationship is

sit ÿ �pitÿ p

i0� � �p�

tÿ p�

0� � eit � p

i0ÿ p�

0

which is in the same units as the logarithms of base year nominal spot rate.If we remove this unit dependence by also measuring the spot relative tothe base year, the real exchange rate is

rit � �sit ÿ si0� ÿ �pitÿ p

i0� � �p�

tÿ p�

0� � eit ÿ �si0 ÿ p

i0� p�

0� � ri � eit �2�

where ri � ÿei0. Unless the base year deviation from PPP equals zero forall countries, even under PPP the measured real exchange rate for eachcountry will have a non-zero mean which will di¡er across countries. Intime series analysis or panel analysis with country-speci¢c intercepts, thisconstant will be picked up by country intercepts; but, as will be discussedbelow, it causes di¤culties for pooled or cross-section analysis.

1Heston and Summers (1996) discuss the measurement issues in more detail.2Given typical price index data, it is only possible to test relative PPP, which is what this

paper focuses on. Comparison of countries' actual price levels provides strong evidenceagainst absolute PPP (e.g. Balvers and Bergstrand, 1997).

Testing for Purchasing Power Parity 289


In practice the data are subject to substantial measurement errorsand we observe

sit � �sit ÿ si0� � v0it pit � �pitÿ p

i0� � v1it p�t � �p�t ÿ p�

0� � v2it

Normalizing on the measured nominal exchange rate, we can writeequation (2) as

sit � ri � pit ÿ p�t � �eit � v0it ÿ v1it � v2it� �3�which is then tested against an unrestricted equation of the form

sit � b0i � b1i pit � b2i p�t � uit �4�

with PPP implying the two restrictions b1i � ÿb2i � 1. If the jointrestriction is rejected, it may be useful to decompose the restrictions intoindividual tests of symmetry �b1i � ÿb2i � bi� and proportionality �bi � 1�to investigate which leads to rejection. This can be done by de¢ningdit � pit ÿ p�t , the price di¡erential in terms of observed price indices, andusing the intermediate equation

sit � b0i � bidit � uit �5�These tests are valid if eit � v0it is distributed normally and independentlyof the prices, with expected value zero and constant variance, andv1it � v2it � 0, for all i and t. These stochastic assumptions, and similarones for the dynamic case, are extremely strong.

It is straightforward to derive the bias of the estimates under theclassical errors in variables model. This suggests that the bias will dependon a matrix equivalent of the signal-to-noise ratio. The bias will bereduced when the signal (the variance of the true prices) is increasedrelative to the measurement error. Thus it is possible that if measurementerror variances are roughly constant and there is homogeneity of theparameters over time or across countries, tests which increase the signalby using long spans of data or cross-country variation will have smallerbias. However, the assumptions of the classical errors in variables modelare almost certainly too strong and much of the economic discussion oftests rejecting PPP has focused on a priori arguments about the likelynature of the stochastic processes. The fundamental disturbance eit willinclude policy shocks which are likely to be correlated with in£ation; thedi¡erence between the o¤cial and the free-market rate, v0it, is almostcertainly related to economic conditions; the measurement errors in theprices, v1it and v2it, are correlated with measured prices by de¢nition andmay be trended. For instance, it is argued that the US CPI overestimatestrue in£ation; this would imply a trend in the measurement error of theprice level.



Given the measurement problems, the equation is sometimes rewrittenas

sit � b0i � b1i pit � b2i p�t � b3it� uit �6�

The trend allows for the di¡erence between the true and the measuredprice indices and the non-unit coe¤cients for measurement error bias. PPPis then tested by the implication that uit should be stationary. Given thatthe variables are I(1), this implies that sit, pit and p�t should cointegrate,after allowing for a deterministic trend.

â Time Series Issues

The discussion of measurement issues suggests that PPP might be takenas implying that the three variables cointegrate, or that they cointegratewith unit coe¤cients, or that the measured real exchange rate is stationary.This is on the assumption that the variables are I(1), though it is possiblethat the price level may be I(2) in some countries: in£ation is stationary.For exposition, we will assume that the variables are I(1); we will imposethe symmetry restriction and express the models in terms of the logarithmsof the spot exchange rate st and the price di¡erential dt � pt ÿ p�t . Since itis not obvious whether st or dt adjusts to restore PPP, the natural timeseries model is the vector autoregression (VAR), which treats both asendogenous. In addition, PPP on its own is less interesting than howrapidly it is restored, and the VAR allows us to estimate the speed ofadjustment. Again for exposition, we shall assume a second-order VAR.We shall also assume that the parameters are constant. In practice it islikely that they will change in response to the policy regime. For instance,a change from a ¢xed to a £oating regime may result in adjustment beingmade by the exchange rate rather than the domestic price level.

Consider the second-order VAR with deterministic trend written invector error correction model (VECM) form:

Dsit � a10i � a11isi;tÿ1 � a12idi;tÿ1 � a13iDsi;tÿ1 � a14iDdi;tÿ1 � a15it� ea1it �7a�Ddit � a20i � a21isi;tÿ1 � a22idi;tÿ1 � a23iDsi;tÿ1 � a24iDi;tÿ1 � a25it� ea2it �7b�

Given that sit and dit are I(1) we must ¢rst determine whether theycointegrate, in which case there is a variable zit such that zit � sit ÿ bidit isI(0). The Johansen trace and eigenvalue tests can be used to determinewhether they cointegrate. If they do the system can be written

Dsit � b10i � b11i�si;tÿ1 ÿ bidi;tÿ1� � b13iDsi;tÿ1 � b14iDdi;tÿ1 � b15it� eb1it �8a�Ddit � b20i � b21i�si;tÿ1 ÿ bidi;tÿ1� � b23iDsi;tÿ1 � b24iDdi;tÿ1 � b25it� eb2it �8b�The Johansen procedure provides estimates of the cointegrating



vector [1 bi] and the adjustment coe¤cients b11i and b21i, for each country.3

It can also be used to test the hypothesis that the cointegrating vector is[1 ÿ1]. If this hypothesis is accepted the system can be written

Dsit � c10i � c11i ri;tÿ1 � c13iDsi;tÿ1 � c14iDdi;tÿ1 � c15it� ec1it �9a�Ddit � c20i � c21i ri;tÿ1 � c23iDsi;tÿ1 � c24iDdi;tÿ1 � c25it� ec2it �9b�

The feedback coe¤cients c11i and c21i indicate the extent to whichdeviations of the real rate from equilibrium cause adjustments in thenominal rate or in prices.

Subtract these two equations and we obtain4

Drit � d0i � d1i ri;tÿ1 � d2iDri;tÿ1 � d3iDdi;tÿ1 � d4it� edit �10�where

d0i � c10i ÿ c20i d1i � c11i ÿ c21i d2i � c13i ÿ c23i

d3i � �c13i � c14i� ÿ �c23i � c24i� d4i � c15i ÿ c25i

If we add the further restriction of common dynamics, d3i � 0, we get

Drit � e0i � e1i ri;tÿ1 � e2iDri;tÿ1 � e4it� eeit �11�which is the equation used to calculate the augmented Dickey^Fuller(ADF(1)) with trend test for a unit root in the real exchange rate.

This structure suggests the following sequence of tests. (i) Testwhether sit and dit are I(1); (ii) estimate equation (7) and use the Johanseneigenvalue and trace tests for cointegration; (iii) estimate bi, thecointegrating vector in equation (8); (iv) test that bi � �1 ÿ1� conditionalon one cointegrating vector; (v) estimate c11i and c21i; (vi) test for commondynamics, d3i � 0 in equation (11); and ¢nally, (vii) conduct an ADF teston the real exchange rate. Conditional on symmetry, which has beenimposed, this structure allows us to identify exactly where each countryfails to provide evidence of PPP.

This sequence is easier to set out than to implement empirically. TheJohansen procedure and the ADF tests are both sensitive to choice of lagorder and treatment of intercepts and trends. Inference is sensitive to theuse of asymptotic or small-sample critical values. Trace and eigenvaluetests often give con£icting conclusions. If symmetry is not imposed there

3The trend is treated as unrestricted here, but in practice one would restrict it to enter thecointegrating relationship.

4Equation (9a) can be reparameterized

Dsit � c10i � c11i ri;tÿ1 � c13i�Dsi;tÿ1 ÿ Ddi;tÿ1� � �c13i � c14i�Ddi;tÿ1 � c15it� ec1it

� c10i � c11i ri;tÿ1 � c13iDri;tÿ1 � �c13i � c14i�Ddi;tÿ1 � c15it� ec1t

and similarly for equation (9b).



may be more than a single cointegrating vector. There are familiar pre-testing problems with such a sequence of tests. With a relatively parsi-monious model (e.g. ADF tests on the measured real exchange rate) thenull (unit root in rt) may not be rejected because of dynamic misspeci-¢cation and measurement error. On the other hand, with a £exibledynamic structure (an unrestricted VAR in st and dt) the null (nocointegration) may not be rejected because over-parameterization reducesthe power of the test.

ã Panel Econometric Issues

The tests of the preceding section did not use the cross-section dimensionof the data. There are a number of di¡erent ways that the panel structurecould be used. We shall discuss a variety of approaches, which will be usedin the empirical example. However, it should be noted that there are stilllarge gaps in the econometric theory for dynamic heterogeneous panels ofthis sort. Below we shall emphasize the possible biases in the variousestimators for the coe¤cients, but for inference, i.e. testing PPP,appropriate estimation of the standard errors is equally important.

The simplest way to use the panel structure is to estimate equation(11) for all countries and conduct a panel unit root test of the jointhypothesis e1i � 0, for all i. Panel unit root tests have been suggested byLevin and Lin (1992) and Im, Pesaran and Shin (IPS) (1997). Monte Carloevidence suggests that the IPS test, which is based on the average ADFstatistic, has more power, and we will use it below. Notice that this is not avery strong null hypothesis. With a su¤ciently large T , it could be rejectedif one of the N real exchange rates was stationary. In principle, one couldconstruct panel cointegration tests; in practice there are a number ofdi¤culties in developing such a test and as yet no satisfactory tests areavailable.

The unrestricted VAR, equation (7), is a dynamic heterogeneouspanel, and Pesaran and Smith (1995) and Pesaran et al. (1996) discuss thevarious ways panels of this sort can be analysed. In this context we areinterested in testing the restrictions a11i � ÿa12i and a21i � ÿa22i and also intesting that at least one of the adjustment coe¤cients is non-zero in eachcountry. If the variables were I(0) and the disturbances were independentacross countries, this could be tested by a standard likelihood ratio test.But if the variables are I(1), testing these hypotheses is equivalent totesting for cointegration with a unit coe¤cient and the distributions of thetest statistics will be non-standard. Given the complexity and hetero-geneity of the data-generation process, designing Monte Carlo studies togenerate the appropriate critical values is not straightforward.

Ignoring the e¡ect of the unit root problem on the distribution ofthe test statistics, the simplest way to use the panel structure is to average



the coe¤cients across groups; Pesaran and Smith (1995) call this the meangroup estimator. Write an equation of the VAR (7) as

yi � Xibi � ui �12�where yi is a T � 1 vector, Xi is a T � 5 matrix, and all the variables aremeasured as deviations from group means to remove country-speci¢cintercepts. The least squares estimators for each group are bi � �X0iXi�ÿ1X0iyi,with variance^covariance matrices estimated by V �bi� � s2i �X0iXi�ÿ1, wheres2i is the usual unbiased estimate of the variance of each group. Themean of the bi is bM �Pi bi=N with variance^covariance matrixV �b� � �Nÿ 1�ÿ1ÿPi bib

0i ÿNbMbM0�. The mean group estimator can either

be the unweighted average bM or a weighted average such as the Swamy(1970) random coe¤cient estimator, which is bS �Pi Wibi, where

5

Wi ��X

i

�V �b� � V �bi��ÿ1�ÿ1�V �b� � V �bi��ÿ1 �13�

We could then test whether PPP held on average across the sampleby testing the restrictions on the average coe¤cients. For N large, theseaverages will have normal distributions. Hsiao et al. (1999) discuss theproperties of various mean group estimators. The attraction of thisprocedure is that since the individual estimates tend to show extremeheterogeneity, averaging may produce better estimates. This would be thecase when the heterogeneity is the product of country-speci¢c shockswhich happen to be correlated with the regressors but which cancel outwhen averaged across countries. The disadvantage of this procedure isthat, for small T , the coe¤cients will be biased, with the usual laggeddependent variable (LDV) bias, which biases a11i and a22i downwards.Since the bias is in the same direction in every country, averaging over N

will not reduce the problem. Bias corrections are available (e.g. thatsuggested by Kiviet and Phillips (1993)); unfortunately while these reducethe bias in the short-run coe¤cients, they may not reduce the bias in thelong-run coe¤cients since they are non-linear functions of the short-runcoe¤cients (see Pesaran and Zhao, 1999). In this case we are primarilyinterested in the long-run coe¤cient

bi � ÿa12i=a11i � ÿa21i=a22i

which should be equal to unity for all countries under PPP.If we impose the long-run unit coe¤cient we get equation (9) which

is a standard set of regression equations if the logarithm of the real

5It should be noted that unlike the individual estimates the Swamy estimator is not invariantto the treatment of the intercept. Because of the covariances it will give di¡erentestimates of the average slope coe¤cients if estimated with an intercept or on dataexpressed as deviations from the country mean.



exchange rate is stationary. Here the long-run coe¤cient is imposed;Pesaran et al. (1999) discuss free estimation of homogeneous long-runcoe¤cients in dynamic panels when short-run coe¤cients and variancesare allowed to di¡er.

Most panel studies impose cross-country homogeneity restrictions onslopes and variances and use ¢xed or random e¡ect models, which allowintercepts to di¡er. This gives the model

Dsit � a10i � a11si;tÿ1 � a12di;tÿ1 � a13Dsi;tÿ1 � a14Ddi;tÿ1 � a15t� e1it �14a�Ddit � a20i � a21si;tÿ1 � a22di;tÿ1 � a23Dsi;tÿ1 � a24Ddi;tÿ1 � a25t� e2it �14b�

The ¢xed e¡ect (FE) estimator, also called the least squares dummyvariable or within estimator, is a weighted average, like the Swamyestimator, of the individual time series estimates bi (if they exist). It alsohas the form

Pi Wibi, where Wi �

�Pi V �bi�ÿ1

�ÿ1V �bi�ÿ1. If a two-step

generalized least squares FE estimator is used, which weights theobservations for each country by the inverse of the standard error ofregression for that country, then V �bi� � s2i �X0iXi�ÿ1, the variance^covariance matrix of the individual estimates. If the usual FE estimator,which assumes a common variance for each country, is used, thenV �bi� � s2�X0iXi�, where s2 is an estimate of the variance for the wholepanel.

Slope and variance homogeneity are usually imposed without beingtested; when tested the homogeneity restrictions are almost invariablyrejected at conventional signi¢cance levels. For small T ;N going toin¢nity, the dynamic FE estimator is inconsistent because of the usualLDV bias. However, with a lagged dependent variable in the equation,slope heterogeneity also causes bias, and if the regressors are positivelyserially correlated this biases the coe¤cient of the lagged dependentvariable upwards, i.e. it is in the opposite direction to the LDV bias (seePesaran and Smith, 1995). Unlike the LDV bias, the heterogeneity biasdoes not decline with T .

Whereas the FE estimates are just a weighted average of theindividual time series estimates, the random e¡ect and pooled ordinaryleast squares OLS estimates also use the `between' variance from thecross-section regression using time averages for each country. As is wellknown, in dynamic models the `between' estimator gives inconsistentestimates and will be biased, even in the absence of the problems discussedabove. Thus the random e¡ect and pooled OLS estimators will also bebiased. To illustrate the issues, rewrite equation (11) as an autoregressionand set e2i � e4i � 0:

rit � a� li ri;tÿ1 � eit �15�where li � e1i � 1. De¢ne the means for each group over time as



ri �1T

XT

t�1rit

where ai � a� mi; li � l� Zi; and average (15) across time to get thebetween regression:

ri � a� lri;ÿ1 � �ei � mi � Zi ri;ÿ1� �16�The ¢rst two terms in the disturbance will certainly be correlated withri;ÿ1, the average disturbance ei because it includes T ÿ 1 terms ei;tÿ1, whichdetermine the ri;tÿ1, and the country-speci¢c e¡ect mi because it alsodetermines ri;tÿ1. If a country's speed of adjustment is a function of itsaverage deviation from PPP, because perhaps of non-linear adjustmentprocesses, the third term of the disturbance also will not be independent ofthe regressor. The composite disturbance is also heteroscedastic, but thiscan be dealt with by using robust standard errors.

A cross-section regression on a single year will avoid the problem ofaveraging the disturbance, but it will not avoid the problem ofheterogeneous coe¤cients correlated with the lagged dependent variable.The intercepts will be heterogeneous because of the deviation from PPP inthe base year, as discussed in Section 1. With the standard data it seemsvery unlikely that cross-section dynamic regressions will provide unbiasedestimates of the speed of adjustment to equilibrium. The problem has somesimilarity to those faced with cross-section regressions of growth on initialincome discussed by Lee et al. (1997).

The cross-section dynamic regression faces di¤culties because ofcorrelation of the intercepts with the regressor, which would be a potentialproblem in any levels regression, and because of dynamics. Both problemscan be avoided by taking ¢rst di¡erences of the long-run PPP relationship.This would suggest a cross-section regression of the form

Dsi � a� bDdi � mi �17�where Dsi and Ddi are the average changes in the logarithm of the spotand di¡erential over the sample for a particular country. Again PPPimplies b � 1, and also that a � 0. This equation, or a comparable panelequation, is less likely to su¡er econometric problems, but does not use theinformation in the levels or provide any information on the speed ofadjustment.

ä Empirical Analysis

The procedures described above were applied to the data for the 31developing countries for 1966^90. First, the order of integration of thevariables was investigated. Using an ADF(1) with trend test, the I(1) nullwas rejected once for the logarithm of the exchange rate and once for the



real exchange rate (in di¡erent countries) and was not rejected in any ofthe 31 countries for the logarithm of the price di¡erential. Using anADF(1) without trend, the I(2) null was rejected for 21 countries for theexchange rate, for 13 countries for the price di¡erential and for 24countries for the real exchange rate. In 10 countries the I(2) null wasrejected for all three variables and in only one country was it not rejectedfor all three variables. We will proceed on the basis that the logarithm ofthe exchange rate and price di¡erential are I(1), though this is far fromcertain.

The unrestricted second-order VAR, equation (7), was estimated forthe 31 countries and the Johansen procedure was applied. Table 1summarizes the cointegration results: the eigenvalue and trace teststatistics, the estimate of bi, conditional on there being one cointegratingvector, and the p value for the restriction that the cointegrating vector is[1 ÿ1]. The eigenvalue test rejects the non-cointegration null in 14 of the31 cases at the 10 per cent level, and the trace test in 18 cases. On the basisof these tests, therefore, there is weak evidence for PPP (allowing formeasurement error) for about half of the countries in the sample.

The estimate of bi had the correct negative sign in 20 of the 31countries. Although there is very substantial dispersion in the estimatesfrom �6:0 to ÿ8:7, the mean value of ÿ0:8 is close to what would beexpected on theoretical grounds. In 11 of the 31 countries the unitcoe¤cient null could be rejected at the 5 per cent level. In two cases whereb had the incorrect sign the estimates were so imprecise that the PPP nullof [1 ÿ1] could not be rejected. Taking either cointegration or non-rejection of a unit coe¤cient as a positive result, there is evidence for PPPin 24 of the 31 cases. However, in only two countries is there evidence ofboth cointegration (at the 10 per cent level) and a unit coe¤cient (at the5 per cent level).

The unit root tests on the logarithm of the real exchange rate areconsistent with the negative results from the joint test. When no trend isincluded, the unit root null is not rejected in any of the countries, usingeither the DF or ADF(1) statistic. When a trend is included, the unit rootnull is rejected in only one country. This is the standard result in theliterature and contrasts with the greater support for PPP obtained fromthe unrestricted model. The di¡erence does not arise from the restrictionon the dynamics in the ADF test, since as Table 2 shows the hypothesisthat d3i � 0 in equation (10) is rejected at the 5 per cent level in only threecases. Table 2 also gives the estimates of the feedback coe¤cients inequation (9), c11i on the spot and c21i on prices. We would expect c11 < 0and c21 > 0, since a real over-valuation (r below its equilibrium) requires anominal devaluation (an increase in s) or a decline in domestic pricesrelative to foreign prices (a decline in d). The feedback on the spot ratewas negative in all but six cases (none of which was signi¢cant), and the



feedback on the price di¡erential positive in all but eight cases and in onlytwo cases were they signi¢cantly negative. This suggests that not only doesthe spot rate seem to adjust more than prices, but that in about a quarterof countries (8 out of 31) the feedback on prices is destabilizing. The meanfeedback on exchange rates is 23 per cent, on prices 6 per cent.

Im et al. (1997) tabulate means and variances for the average ADFin a panel under the null hypothesis of a unit root. In this case thesimulated average values are as follows: without trend DF ÿ0:89, ADF(1)

Table "Johansen Maximum Likelihood Estimates, Cointegration Likelihood Ratio Tests

and Parameter Estimates

Case Country

Eigenvalue* �14:0690�

** �12:0710�

Trace* �15:4100�

** �13:3250�b

estimate

Cointegratingrestriction�1 ÿ1�a

1 Barbados 25.2* 27.3* ÿ0.03 0.0002 Cyprus 13.6** 20.2* 6.0 0.0103 Dominican Republic 8.8 9.0 ÿ0.8 0.8624 Ecuador 9.2 9.7 ÿ1.5 0.0215 El Salvador 9.0 11.6 ÿ0.9 0.7656 Ethiopia 15.1* 17.6* 2.7 0.0127 Greece 11.9 19.2* ÿ2.2 0.0338 Guatemala 7.4 8.6 ÿ1.7 0.0609 Guyana 45.1* 49.9* 0.3 0.00010 India 9.5 11.0 ÿ1.6 0.10211 Jamaica 16.3* 21.1* ÿ4.8 0.00112 Korea 9.0 10.0 ÿ0.7 0.56313 Malawi 7.5 9.3 ÿ0.8 0.47914 Malta 20.8* 22.5* 3.1 0.00015 Mauritius 14.4* 23.5* 3.8 0.02516 Mexico 9.8 12.1 ÿ1.1 0.22017 Morocco 11.6 16.3* ÿ6.8 0.01018 Nepal 19.6* 19.7* ÿ1.2 0.06119 Pakistan 10.7 18.4* 0.4 0.20520 Paraguay 12.5** 13.5** ÿ2.0 0.00121 Peru 22.2* 28.7* ÿ1.1 0.00222 Philippines 12.6** 16.3* ÿ2.5 0.004.23 Singapore 7.5 13.2 ÿ2.9 0.83024 South Africa 10.5 12.6 1.5 0.00425 Sri Lanka 8.6 11.3 0.3 0.05426 Thailand 17.4* 22.3* 2.1 0.00127 Trinidad and Tobago 17.6* 22.0* ÿ5.8 0.00228 Turkey 3.1 3.3 ÿ0.6 0.66729 Uruguay 10.5 14.6** ÿ8.7 0.02630 Venezuela 26.2* 35.0* 0.6 0.00331 Zimbabwe 10.3 11.8 1.5 0.003

Mean (s.d.) 31 countries 13.9 (7.9) 17.4 (9.1) ÿ0.8 (3.0)

Notes: * Indicates rejection of the non-cointegrating null at the 5 per cent level.** Indicates rejection of the non-cointegrating null at the 10 per cent level.a Cointegrating restrictions on nominal exchange rate and relative domestic/foreign pricesobtained from zt � st � bdt where dt � pt ÿ p�t , the price di¡erential. Johansen likelihood ratio teston cointegrating vector, p value reported.



ÿ1:08; with trend DF ÿ1:55, ADF(1) ÿ1:72. In all four cases, the actualaverages are greater than the simulated values that would be expected ifthere were a unit root. Thus, like individual tests, panel tests would notreject a unit root in the real exchange rate. However, the IPS test stronglyrejects the null that all the real exchange rates are I(2), whereas in someindividual cases this cannot be rejected. Inspecting the ADF equations,what is noticeable is how sensitive the estimate of the average adjustmentcoe¤cient is to the inclusion of a trend. When a trend is included in theADF(1) equation for the real exchange rate, the estimated speed ofadjustment more than doubles from 12 per cent to 28 per cent. While thedi¡erence is not statistically signi¢cant, since there is a very large degree of

Table áEstimates of Real Exchange Rate Feedback Coefficients and Common Dynamics

Case Country c11 t statistics c21 t statisticsd3 � 0

probability

1 Barbados 0.003 (0.06) 0.12 (1.39) (0.053)2 Cyprus ÿ0.30 (ÿ2.38) 0.20 (0.63) (0.891)3 Dominican Republic ÿ0.29 (ÿ1.16) 0.13 (1.76) (0.355)4 Ecuador 0.36 (0.17) 0.17 (1.55) (0.420)5 El Salvador ÿ0.86 (ÿ2.88) ÿ0.16 (1.32) (0.647)6 Ethiopia ÿ0.10 (ÿ1.53) 0.35 (2.03) (0.027)a

7 Greece ÿ0.31 (ÿ2.58) 0.01 (0.17) (0.234)8 Guatemala 0.11 (0.55) 0.17 (1.54) (0.026)a

9 Guyana 0.35 (0.99) ÿ0.29 (3.14) (0.871)10 India ÿ0.43 (ÿ1.04) 0.36 (2.12) (0.587)11 Jamaica ÿ0.41 (ÿ2.21) ÿ0.11 (1.74) (0.421)12 Korea ÿ0.47 (ÿ2.87) ÿ0.12 (1.21) (0.786)13 Malawi ÿ0.23 (ÿ1.14) 0.32 (1.97) (0.191)14 Malta ÿ0.31 (ÿ1.82) 0.03 (0.64) (0.783)15 Mauritius ÿ0.56 (ÿ2.75) ÿ0.14 (1.46) (0.664)16 Mexico ÿ0.30 (ÿ1.18) 0.28 (1.19) (0.318)17 Morocco ÿ0.20 (ÿ2.01) ÿ0.007 (ÿ0.20) (0.099)18 Nepal ÿ0.55 (ÿ2.90) 0.38 (2.25) (0.487)19 Pakistan ÿ0.22 (ÿ1.27) 0.35 (2.41) (0.050)20 Paraguay 0.09 (0.45) 0.06 (1.01) (0.729)21 Peru ÿ1.69 (ÿ3.43) ÿ1.31 (2.47) (0.237)22 Philippines ÿ0.28 (ÿ1.16) 0.09 (0.61) (0.817)23 Singapore ÿ0.13 (ÿ2.02) 0.16 (2.33) (0.190)24 South Africa ÿ0.09 (ÿ1.11) 0.01 (1.07) (0.472)25 Sri Lanka ÿ0.01 (ÿ0.06) 0.11 (2.06) (0.249)26 Thailand ÿ0.18 (ÿ2.07) 0.10 (1.05) (0.221)27 Trinidad and Tobago ÿ0.49 (ÿ2.71) 0.02 (0.34) (0.103)28 Turkey ÿ0.29 (ÿ1.51) ÿ0.14 (0.71) (0.041)a

29 Uruguay 0.12 (0.48) 0.26 (1.66) (0.116)30 Venezuela 0.38 (1.14) 0.45 (3.90) (0.761)31 Zimbabwe ÿ0.04 (ÿ0.99) 0.01 (0.88) (0.320)

Mean all countries ÿ0.23 0.06 0.393Standard deviation (0.38) (0.31) (0.288)

Notes: a Indicates d3 6� 0 at 5 per cent.



uncertainty, economically the di¡erence is large: quite di¡erent policyconclusions would be drawn from adjustment to equilibrium at 12 per centper annum and at 28 per cent per annum.

To investigate the pattern further the Swamy weighted averageestimates of the coe¤cients of equation (7) across all 31 countries werecalculated. These were estimated using data from which each countrymean had been subtracted. The estimates are as follows:

Dsit � ÿ0:24si;tÿ1 � 0:26di;tÿ1 � 0:17Dsi;tÿ1 ÿ 0:07Ddi;tÿ1 � 0:0001t(0.09) (0.11) (0.10) (0.15) (0.008)

MLL� 782.20

Ddit � 0:006si;tÿ1 ÿ 0:03di;tÿ1 � 0:03Dsi;tÿ1 � 0:21Ddi;tÿ1 � 0:0001t(0.08) (0.09) (0.09) (0.07) (0.0001)

MLL� 1207.44

Standard errors are given in parentheses; the maximized log-likelihood(MLL) is the sum over the individual equations. In the exchange rateequation, the lagged exchange rate and lagged price di¡erential would besigni¢cant if they had standard distributions and are roughly equal and ofopposite sign as PPP would suggest. In the in£ation equation, only thelagged change in price di¡erential would be signi¢cant.

If we reparameterize the VAR as an equation for the real exchangerate, by subtracting the two equations, the Swamy weighted averageestimates are

Drit � ÿ0:29ri;tÿ1 � 0:06di;tÿ1 ÿ 0:26Dri;tÿ1 ÿ 0:15Ddtÿ1 ÿ 0:0001t(0.08) (0.11) (0.13) (0.16) (0.0002)

MLL� 763.36

If the variables with insigni¢cant average e¡ects are deleted, the result is

Drit � ÿ0:12ri;tÿ1 � 0:15Dri;tÿ1(0.04) (0.06)

MLL� 620.48

The lagged real exchange rate now suggests a much slower speed ofadjustment, 12 per cent rather than 29 per cent, and has a t ratio of ÿ3:29.This matches the result with the unweighted averages. The unrestrictedequation involves estimating 186 coe¤cients �6� 31�, the restricted equationinvolves estimating 93. The hypothesis that the restrictions hold in eachcountry would be rejected by a likelihood ratio test.

We now estimate the three equations by the FE estimator using robuststandard errors. The estimates of the VAR are as follows:



Dsit � a1i ÿ 0:05si;tÿ1 � 0:11di;tÿ1 � 0:11Dsi;tÿ1 � 0:70Ddi;tÿ1 ÿ 0:0006t(0.03) (0.04) (0.11) (0.16) (0.001)

R2 � 0:63, SER � 0:17, MLL � 262:72

Ddit � a2i ÿ 0:02si;tÿ1 ÿ 0:07di;tÿ1 � 0:10Dsi;tÿ1 � 0:73Ddi;tÿ1 ÿ 0:001t(0.02) (0.03) (0.09) (0.14) (0.001)

R2 � 0:74, SER � 0:14, MLL � 430:37

Robust standard errors are in parentheses and SER is the standard errorof the regression. The speed of adjustment in the exchange rate equation isnow much slower, as one would expect from the heterogeneity bias of theFE estimator. The proportionality restriction would be rejected in bothequations (t � 2:01 and t � 2:38). The real exchange rate equation broadlyshows the features that one would expect from the equations for its twocomponents:

Drit � a3i ÿ 0:04ri;tÿ1 � 0:002di;tÿ1 � 0:01Dri;tÿ1 ÿ 0:06Ddtÿ1 � 0:0008t(0.02) (0.009) (0.18) (0.12) (0.002)

R2 � 0:07, SER � 0:13, MLL � 436:80

If time e¡ects as well as group e¡ects are included, allowing a £exible trendto pick up any common shocks, the estimates are very similar, and the yeardummies are not signi¢cant. Likelihood ratio tests would reject thehypothesis that the coe¤cients were equal across countries. However,conditional on coe¤cient equality, equality of intercepts would not berejected for the exchange rate and real exchange rate equations, though itwould be for the price equation.

The panel estimates of the static equation (4) are supportive of PPP,

sit � ai � 0:99684pit ÿ 0:95375p�t(0.0171) (0.0426)

R2 � 0:9871, SER � 0:31, MLL � ÿ171:96as is the cross-section regression of the average change in the log of thenominal exchange rate on the average rate of in£ation:

Dsi � ÿ0:05 � 0:95Ddi R2 � 0:94, SER � 0:034(0.008) (0.02)

The intercept is signi¢cantly less than zero �t � ÿ5:85� and the slope issigni¢cantly less than unity �t � ÿ2:73�, but the di¡erences are not large ineconomic terms. Although we would expect a downward bias frommeasurement error, the estimate from the reverse regression (R2 dividedby the estimate from the direct regression) is 0.99, also below unity.



å Discussion and Conclusions

This paper provides a systematic exposition of the econometric issues intesting PPP using time series and panels, interpreting the alternative testsin terms of the restrictions and reparameterizations they implied for anunderlying heterogeneous VAR. The models were estimated on a panel of31 developing countries. The empirical results were broadly in accord withthose reported in the literature. Using time series data one cannot rejectthe null of a unit root in the real exchange rate, though one can reject thenull of no cointegration between nominal exchange rates and pricedi¡erentials for over half the countries. Although the systematic testingprocedure helped identify which aspects of speci¢cation the tests weremost sensitive to, it did not solve the basic problem, the low power of thetests. Using panel estimates, either averages of coe¤cients or constrainingcoe¤cients to be the same across countries provided much more evidencefor PPP. In the cross-section dimension, PPP holds almost exactly; thedi¡erences from unity may be statistically signi¢cant, depending on how thestandard error is calculated, but are small in economic terms.

The estimates in this paper conform to the trend in the literaturewhich concludes that relative PPP holds almost exactly in the long run. Ofcourse, this conclusion is not very interesting for most policy purposesunless convergence to equilibrium is reasonably fast. However, ourtheoretical discussion and estimates cast doubt on the trend in theliterature to conclude that the speed of convergence is about 15 per cent ayear. There are major di¤culties in measuring the speed of convergencein cross-country panels unless T is large and the structural parameters areconstant across countries and time. Otherwise, estimates of the coe¤cientsof the lagged dependent variables are subject to a variety of biases andare sensitive to speci¢cation and estimation method. Small changes inspeci¢cation can cause the estimated speed of adjustment to vary between10 and 30 per cent. Not only may we not be able to estimate the speed ofconvergence from panels, but there may be nothing to estimate if the speedof convergence is not a constant but itself a function of the deviation fromequilibrium or other variables.

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Testing for Purchasing Power Parity: Econometric Issues and an Application to Developing Countries

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