TESTING FOR PURCHASING POWER PARITY: ECONOMETRIC ISSUES AND AN APPLICATION TO DEVELOPING COUNTRIES* by DERICK BOYD University of East London and RON SMITH Birkbeck College, University of London and University of Colorado. 10 November 1998 ABSTRACT There is now a vast literature on testing Purchasing Power Parity (PPP). Any test is conditional on a particular econometric specification which embodies a set of auxiliary assumptions. This paper reviews the issues involved in econometric specification and estimation in the time-series and panel models used to test PPP. We start from a general model and then systematically examine the implicit restrictions that are imposed to obtain the standard procedures and discuss the implications of these procedures for estimation and inference. The issues are illustrated on data for a panel of 31 developing countries 1966-1990. Key Words: Purchasing Power Parity, Cointegration, Panel Data. JEL Classification: C23, C32, F31 * We should like to thank Philip Arestis, Jerry Coakley, Robert McNown, Bahram Pesaran, Hashem Pesaran, Martin Sola and two anonymous referees for comments, and Nora Zerrad for data assistance.
Introduction. There is now a vast literature on testing Purchasing Power Parity (PPP) using national data on exchange rates and prices. Rogoff (1996) and Taylor (1995) provide recent reviews. Any test is conditional on a particular econometric specification, which embodies a set of auxiliary assumptions. In this paper we provide a systematic exposition of the alternative specifications and illustrate the sensitivity of the results to specification on a sample of 31 developing countries over the period 1966-1990. Like real exchange rates themselves, views about PPP have shown persistent swings. The orthodoxy of the early 1970s in favour of PPP was largely abandoned in the 1980s as new time-series tests could not reject the hypothesis that there was a unit root in the real exchange rate. More recently failure to reject the unit root hypothesis has 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 historical data (e.g. Lothian and Taylor 1996); panel data (e.g. Frankel and Rose 1996) or long-horizon predictability (e.g. Mark 1995) have been more supportive of PPP. Although doubts remain about the size of the tests in long spans of data, e.g. Engel (1995), attention has tended to shift from testing PPP to measuring the speed of convergence to PPP and the determinants of real exchange rates. The consensus suggests a speed of adjustment of about 15% per annum equivalent to a half life of around 4 years. However, as the growth literature has made clear, measuring the speed of convergence from cross-country data faces major difficulties. Lee, Pesaran and Smith (1997) review those difficulties in the growth context. Most of the literature has assumed linear adjustment and we shall do the same. However, given transactions costs non-linear adjustment is more likely: slow adjustment close to equilibrium, fast adjustment further away, (see Michael et al, 1997; Bleaney and Mizen 1996). 2
Most of the recent cointegration analyses have used data from industrialised countries, but in terms of both policy and methodology the issue is more interesting for developing countries. In policy terms, exchange rate management is at the centre of many financial stabilisation plans in developing and transitional countries and PPP provides an important theoretical issue in external adjustment policy. For instance, in his influential article on the question of the contractionary impact of devaluation, Edwards (1986, p. 503) noted: An essential element in the traditional view of devaluations is the assumption that nominal devaluations generate an improvement in the domestic relative price of tradables to nontradables. That is, it is assumed that nominal devaluations result in real devaluations. PPP also provides an important basis for the financial stabilisation and structural adjustment policies proposed by the IMF and World Bank and plays a role in the choice between money, inflation or exchange rate targetting in the design of monetary policy. In methodological terms, developing countries show more cross-section variation, hyper inflations are rare in industrialised countries, and show more time-series noise. The noise arises both from severe measurement problems (official and market exchange rates often differ for instance) and from large policy shocks. If there is non-linear adjustment, return to PPP should be more obvious in developing countries, which are more often far from equilibrium. In this paper we examine the evidence for long run PPP using annual data for 31 developing countries over the 25 year period, 1966-1990 obtained from the International Financial Statistics Yearbook 1994. The nominal exchange rates used are the annual averages of the market rate and official exchange rate for countries where available, where unavailable the year end rates were used. The domestic prices are the various national consumer price indices. The foreign price is the USA consumer price index. It is well known that tests of PPP may be sensitive to measurement problems and the IFS data for 3
developing countries have many problems 1. The results are sensitive to choice of price index and base country. In industrial countries the Wholesale Price Index tends to provide more support for PPP than the CPI. Nominal exchange rates measured against the Deutschemark tend to provide more support than ones measured against 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). 2. Measurement issues PPP is usually interpreted as implying that domestic and foreign prices measured in a common currency should be equal. At any moment this will not hold exactly, but deviations from PPP should only be transitory as the system will adjust to reassert equality. Thus, using underlining to indicate true values, if s it is the logarithm of the true nominal spot exchange rate (domestic currency of country i per unit of some foreign base currency); p it and p* t are the logarithms of the true values of domestic prices and foreign prices; and ε it is a mean zero stationary error then PPP implies: (1) s it - p it + p* t = ε it. t=1,2,...,t, i=1,2,...,n. Direct data on prices are rarely available, instead the data is for a price index relative to a base year, say t=0, i.e. we observe: p it - p i0. 2 Thus in terms of the available data the relationship is: s it - (p it - p i0 )+ (p* t - p* 0 ) = ε it + p i0 - 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 to the base year, the real exchange rate is: 1 Heston and Summers (1996) discuss the measurement issues in more detail. 4
(2) r it = (s it -s i0 )- (p it - p i0 )+ (p* t - p* 0 ) = ε it -(s i0 -p i0 + p* 0 ) = r i + ε it where r i = - ε i0. Unless the base year deviation from PPP equals zero for all countries, even under PPP the measured real exchange rate for each country will have a non-zero mean which will differ across countries. In time-series analysis or panel analysis with country specific intercepts, this constant will be picked up by country intercepts; but, as will be discussed below, it causes difficulties for pooled or cross-section analysis. In practice the data are subject to substantial measurement errors and we observe: s it = (s it - s i0 )+ v 0it ; p t = (p it - p i0 )+ v 1it ; p * t = (p* t - p* 0 ) + v 2it. Normalising on the measured nominal exchange rate, we can write equation (2) as: (3) s it = r i + p it - p * t + (ε it + v 0it - v 1it + v 2it ) Which is then tested against an unrestricted equation of the form: (4) s it = β 0i + β 1i p it +β 2i p * t + u it with PPP implying the two restrictions β 1i = -β 2i = 1. If the joint restriction is rejected, it may be useful to decompose the restrictions into individual tests of symmetry: β 1i = -β 2i = β i ; and proportionality: β i =1, to investigate which leads to rejection. This can be done by defining d it = p it - p* t, the price differential in terms of observed price indices and using the intermediate equation: (5) s it = β 0i + β i d it + u it These tests are valid if ε it +v 0it is distributed normally and independently of the prices, with expected value zero and constant variance, and v 1it = v 2it = 0, all i and t. These stochastic assumptions, and similar ones for the dynamic case, are extremely strong. 2 Given 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 evidence against absolute PPP, e.g. Balvers and Bergstrand (1997). 5
It is straightforward, to derive the bias of the estimates under the classical errors in variables model. This suggests that the bias will depends on a matrix equivalent of the signalnoise ratio. The bias will be reduced, when the signal (the variance of the true prices) is increased relative to the measurement error. Thus it is possible that if measurement error variances are roughly constant and there is homogeneity of the parameters over time or across countries, tests which increase the signal by using long-spans of data or cross-country variation will have smaller bias. However, the assumptions of the classical errors in variables model are almost certainly too strong and much of the economic discussion of tests rejecting PPP has focussed on a priori arguments about the likely nature of the stochastic processes. The fundamental disturbance ε it will include policy shocks which are likely to be correlated with inflation; the difference between the official and the free-market rate, v 0it, are almost certainly related to economic conditions; the measurement errors in the prices, v 1it and v 2it are correlated with measured prices by definition and may be trended. For instance, it is argued that the US CPI overestimates true inflation, this would imply a trend in the measurement error of the price level. Given the measurement problems, the equation is sometimes rewritten as (6) s it = β 0i + β 1i p it +β 2i p * t +β 3i t + u it The trend allows for the difference between the true and the measured price indexes and the non-unit coefficients for measurement error bias. PPP is then tested by the implication that u it should be stationary. Given that the variables are I(1), this implies that (s it, p it and p * t) should cointegrate, after allowing for a deterministic trend. 3. Time series issues 6
The discussion of measurement issues suggests that PPP might be taken as implying either that the three variables cointegrate, that they cointegrate with unit coefficients, or that the measured real exchange rate is stationary. This is on the assumption that the variables are I(1); though it is possible that the price level may be I(2) in some countries: inflation is stationary. For exposition, we will assume that the variables are I(1); we will impose the symmetry restriction and express the models in terms of the logarithms of the spot exchange rate, s t, and the price differential, d t = p t - p* t. Since it is not obvious whether s t or d t adjusts to restore PPP, the natural time-series model is the VAR, which treats both as endogenous. In addition, PPP on its own is less interesting than how rapidly it is restored, and the VAR allows us to estimate the speed of adjustment. Again for exposition, we shall assume a second order VAR. We shall also assume that the parameters are constant. In practice it is likely that they would change in response to the policy regime. For instance, a change from a fixed to a floating regime may result in adjustment being made by the exchange rate rather than the domestic price level. Consider the second order Vector Autoregression (VAR) with deterministic trend written in Vector Error Correction Model (VECM) form: (7a) (7b) s it = a 10i + a 11i s i,t-1 + a 12i d i,t-1 + a 13i s i,t-1 + a 14i d i,t-1 + a 15i t + ε a 1it d it = a 20i + a 21i s i,t-1 + a 22i d i,t-1 + a 23i s i,t-1 + a 24i d i,t-1 + a 25i t + ε a 2it Given that s it and d it are I(1) we must first determine whether they cointegrate, in which case there is a variable z it such that, z it = s it - β i d it is I(0). The Johansen trace and eigenvalue tests can be used to determine whether they cointegrate. If they do the system can be written, (8a) (8b) s it = b 10i + b 11i (s i,t-1 -β i d i,t-1 )+ b 13i s i,t-1 + b 14i d i,t-1 + b 15i t + ε b 1it d it = b 20i + b 21i (s i,t-1 -β i d i,t-1 )+ b 23i s i,t -1 + b 24i d i,t-1 + b 25i t + ε b 2it 7
The Johansen procedure provides estimates of the cointegrating vector [ 1 β i ] and the adjustment coefficients b 11i and b 21i, for each country 3. It can also be used to test the hypothesis that the cointegrating vector is [ 1-1 ]. If this hypothesis is accepted the system can be written, (9a) (9b) s it = c 10i + c 11i r i,t-1 + c 13i s i,t-1 + c 14i d i,t-1 + c 15i t + ε c 1it d it = c 20i + c 21i r i,t-1 + c 23i s i,t-1 + c 24i d i,t-1 + c 25i t + ε c 2it The feedback coefficients, c 11i and c 21i, i ndicate the extent to which deviations of the real rate from equilibrium cause adjustments in the nominal rate or in prices. Subtract these two equations and we obtain 4, (10) r it = d 0i + d 1i r i,t-1 + d 2i r i,t-1 + d 3i d i,t-1 + d 4i t + ε d it where, d 0i = c 10i - c 20i ; d 1i = c 11i - c 21i ; d 2i = c 13i - c 23i d 3i = (c 13i + c 14i ) - (c 23i + c 24i ); d 4i = c 15i - c 25i If we add the further restriction of common dynamics, d 3i = 0, we get, (11) r it = e 0i + e 1i r i,t-1 + e 2i r i,t-1 + e 4i t + ε e it which is the equation used to calculate the ADF(1) with trend test for a unit root in the real exchange rate. This structure suggests the following sequence of tests. (1) Test whether s it and d it are I(1); (2) estimate equation (7) and use the Johansen Eigenvalue and Trace tests for cointegration; (3) estimate β i, the cointegrating vector in equation (8); (4) test that β i = [1-1] 3 The trend is treated as unrestricted here, but in practice one would restrict it to enter the cointegrating relationship. 4 Equation (9a) can be reparameterised s it = c 10i + c 11i r i,t-1 + c 13i ( s i,t-1 - d i,t-1 ) + ( c 13i + c 14i ) d i,t-1 + c 15i t + ε c 1it = c 10i + c 11i r i,t-1 + c 13i r i,t-1 + ( c 13i + c 14i ) d i,t-1 + c 15i t + ε c 1t and similarly for equation (9b). 8
conditional on one cointegrating vector; (5) estimate c 11i and c 12i ; (6) test for common dynamics, d 3i = 0 in equation (11); and finally, (7) conduct an ADF test on real exchange rate. Conditional on symmetry, which has been imposed, this structure allows us to identify exactly where each country fails to provide evidence of PPP. This sequence is easier to set out than to implement empirically. The Johansen procedure and the ADF tests are both sensitive to choice of lag order and treatment of intercepts and trends. Inference is sensitive to the use of asymptotic or small sample critical values. Trace and Eigenvalue tests often give conflicting conclusions. If symmetry is not imposed there may be more than a single cointegrating vector. There are familiar pre-testing problems with such a sequence of tests. With a relatively parsimonious model ( e.g. ADF tests on the measured real exchange rate ) the null (unit root in r t ) may not be rejected because of dynamic misspecification and measurement error. On the other hand, with a flexible dynamic structure (an unrestricted VAR in s t and d t ) the null (no cointegration) may not be rejected because over parameterisation reduces the power of the test. 4. Panel Econometric Issues The tests of the preceding section did not use the cross-section dimension of the data. There are a number of different ways that the panel structure could be used. We shall discuss a variety of approaches, which will be used in the empirical example. However, it should be noted that there are still large gaps in the econometric theory for dynamic heterogenous panels of this sort. Below we shall emphasise the possible biases in the various estimators for the coefficients, but for inference, i.e. testing PPP, appropriate estimation of the standard errors is equally important. 9
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 joint hypothesis d 1i =0, for all i. Panel unit root tests have been suggested by Levin and Lin (1992) and Im Pesaran and Shin (IPS, 1997). Monte Carlo evidence sugggests that the IPS test, which is based on the average ADF statistic, has more power, and we will use it below. Notice that this is not a very strong null hypothesis. With a sufficiently large T, it could be rejected if one of the N real exchange rates was stationary. In principle, one could construct panel cointegration tests; in practice there are a number of difficulties in developing such a test and as yet no satisfactory tests are available. The unrestricted VAR, equation (7) is a dynamic heterogenous panel and Pesaran and Smith (1995) and Pesaran, Smith and Im (1996) discuss the various ways panels of this sort can be analysed. In this context we are interested in testing the restrictions: a 11i = -a 12i and a 21i = -a 22i and also testing that at least one of the adjustment coefficients are non-zero in each country. If the variables were I(0) and the disturbances were independent across countries, this could be tested by a standard Likelihood ratio test. But if the variables are I(1), testing these hypotheses is equivalent to testing for cointegration with a unit coefficient and the distributions of the test statistics will be non-standard. Given the complexity and heterogeneity of the data generation process, designing Monte Carlo studies to generate the appropriate critical values is not straightforward. Ignoring the effect of the unit root problem on the distribution of the test statistics, the simplest way to use the panel structure is to average the coefficients across groups, Pesaran and Smith (1995) call this the mean group estimator. Write an equation of the VAR (7) as (12) y i = X i β i + u i 10
where y i is a Tx1 vector, X i is a Tx5 matrix, and all the variables are measured as deviations from group means to remove country specific intercepts. The Least Squares estimators for each group are b i =(X i X i ) -1 X i y i, with Variance Covariance Matrices estimated by V(b i ) = s 2 i (X i X i ) -1, where s 2 i is the usual unbiased estimate of the variance of each group. The mean of the b i is b M = i b i /N with Variance Covariance matrix V(b)= (N-1) -1 ( i b i b i - Nb M b M ). The Mean Group estimator can either be the unweighted average, b M, or a weighted average such as the Swamy (1971) Random Coefficient Estimator; which is b S = i W i b i, where (13) W i = [ i (V(b)+V(b i )) -1 ] -1 (V(b)+V(b i )) -1. 5 We could then test whether PPP held on average across the sample, by testing the restrictions on the average coefficients. For N large, these averages will have normal distributions. Hsiao, Pesaran and Tahmiscioglu (1997) discuss the properties of various mean group estimators. The attraction of this procedure is that since the individual estimates tend to show extreme heterogeneity, averaging may produce better estimates. This would be the case when the heterogeneity is the product of country specific shocks which happen to be correlated with the regressors but which cancel out when averaged across countries. The disadvantage of this procedure is that for small T, the coefficients will be biased, with the usual lagged dependent variable (LDV) bias, which biases a 11i and a 22i downwards. Since the bias is in the same direction in every country, averaging over N will not reduce the problem. There are bias corrections available (e.g. that suggested by Kiviet and Phillips 1993); unfortunately while these reduce the bias in the short run coefficients, they may not reduce the bias in the long-run coefficients, since they are non-linear functions of the short-run 5 It should be noted that unlike the individual estimates the Swamy estimator is not invariant to the treatment of the intercept. Because of the covariances it will give different estimates of the average slope coefficients if estimated with an intercept or on data expressed as deviations from the country mean. 11
coefficients, see Pesaran and Zhao (1997). In this case we are primarily interested in the longrun coefficient β i = -a 12i / a 11i =-a 21i /a 22i, which should be equal to unity for all countries under PPP. If we impose the long-run unit coefficient we get equation (9) which is a standard set of regression equations if the logarithm of the real exchange rate is stationary. Here the longrun coefficient is imposed, Pesaran, Shin and Smith (1997) discuss free estimation of homogenous long-run coefficients in dynamic panels, when short-run coefficients and variances are allowed to differ. Most panel studies impose cross-country homogeneity restrictions on slopes and variances, and use Fixed or Random Effect Models, which allow intercepts to differ. This gives the model: (14a) (14b) s it = a 10i + a 11 s i,t-1 + a 12 d i,t-1 + a 13 s i,t-1 + a 14 d i,t-1 + a 15 t + ε 1it d it = a 20i + a 21 s i,t-1 + a 22 d i,t-1 + a 23 s i,t-1 + a 24 d i,t-1 + a 25 t + ε 2it The Fixed Effect (FE) estimator, also called the Least Squares Dummy Variable or Within estimator is, like the Swamy estimator, a weighted average of the individual time series estimates, b i, (if they exist). It also has the form i W i b i, where W i = [ i V(b i ) -1 ] -1 V(b i ) -1. If a two step Generalised Least Squares FE estimator is used, which weights the observations for each country by the inverse of the standard error of regression for that country, then V(b i ) = s 2 i (X i X i ) -1, the Variance Covariance matrix of the individual estimates. If the usual FE estimator, which assumes a common variance for each country, is used then, V(b i ) = s 2 (X i X i ) -1, where s 2 is an estimate of the variance for the whole panel. 12
Slope and variance homogeneity are usually imposed without being tested; when tested the homogeneity restrictions are almost invariably rejected at conventional significance levels. For small T, N going to infinity, the Dynamic Fixed Effect is inconsistent because of the usual LDV bias. However, with a lagged dependent variable in the equation, slope heterogeneity also causes bias, and if the regressors are positively serially correlated, this biases the coefficient of the lagged dependent variable upwards, i.e. it is in the opposite direction to the LDV bias, see Pesaran and Smith (1995). Unlike the LDV bias, the heterogeneity bias does not decline with T. Whereas the FE estimates are just a weighted average of the individual time-series estimates the Random Effect and pooled OLS estimates also use the between variance, from the cross-section regression using time averages for each country. As is well known, in dynamic models the between estimator gives inconsistent estimates and will be biased, even in the absence of the problems discussed above. Thus the Random Effect and pooled OLS estimators will also be biased. To illustrate the issues, rewrite equation (11) as an autoregression and set e 2i =e 4i =0: (15) r it = α i + λ i r i,t-1 +ε it where λ i = e 1i +1. Define the means for each group over time as: r i = 1 T T t= 1 r it define α i =α+µ i, λ i =λ +η i ; and average (15) across time, to get the between regression: (16) r i = α + λr i,-1 +(ε i +µ i +η i r i,-1 ) The first two terms in the disturbance will certainly be correlated with r i,-1. The average disturbances, ε i because it includes T-1 terms ε it-1, which determine the r it-1 and the country specific effect, µ i, because it also determines r it-1. If a countries speed of adjustment is a 13
function of its average deviation from PPP, because perhaps of non-linear adjustment processes, the third term of the disturbance would also not be independent of the regressor. The composite disturbance is also heteroskedastic, but this can be dealt with by using robust standard errors. A cross-section regression on a single year, will avoid the problem of averaging the disturbance, but it will not avoid the problem of heterogeneous coefficients correlated with the lagged dependent variable. The intercepts will be heterogeneous because of the deviation from PPP in the base year as discussed in section 1. With the standard data it seems very unlikely that cross-section dynamic regressions will provide unbiased estimates of the speed of adjustment to equilibrium. The problem has some similarity to those faced by cross-section regressions of growth on initial income discussed in Lee Pesaran and Smith (1997). The cross-section dynamic regression faces difficulties because of correlation of the intercepts with the regressor, which would be a potential problem in any levels regression and because of dynamics. Both problems can be avoided by taking first differences of the longrun PPP relationship. This would suggest a cross-section regression of the form: (17) s i = α + β d i + u i where s i and d i are the average changes in the logarithm of the spot and differential over the sample for a particular country. Again PPP implies β=1, and also that α=0. This equation, or a comparable panel equation, is less likely to suffer econometric problems, but does not use the information in the levels or provide any information on the speed of adjustment. 5. Empirical analysis 14
The procedures described above were applied to the data for the 31 developing countries 1966-1990. First, the order of integration of the variables was investigated. Using an ADF1 with trend test, the I(1) null was rejected once for the logarithm of the exchange rate and once for the real exchange rate (in different countries), and was not rejected in any of the 31 countries for the logarithm of the price differential. Using an ADF1 without trend, the I(2) null was rejected for 21 countries for the exchange rate, for 13 countries for the price differential and for 24 countries for the real exchange rate. In 10 countries the I(2) null was rejected for all 3 variables and in only one country was it not rejected for all 3 variables. We will proceed on the basis that the logarithm of the exchange rate and price differential are I(1), though this is far from certain. TABLE I ABOUT HERE The unrestricted second order VAR, equation (7), was estimated for the 31 countries and the Johansen procedure applied. Table 1 summarises the cointegration results: the eigenvalue and trace test statistics, the estimate of β i, conditional on there being one cointegrating vector, and the p value for the restriction that the cointegrating vector is [1-1]. The eigenvalue test rejects the noncointegration null in 14 of the 31 cases at the 10% level, and the trace test in 18 cases. On the basis of these tests, therefore, there is weak evidence for PPP (allowing for measurement error) for about half of the countries in the sample. The estimate of β i had the correct negative sign in 20 of the 31 countries. Although there is very substantial dispersion in the estimates from +6.0 to -8.7, the mean value of -0.8 is close to what would be expected on theoretical grounds. In 11 of the 31 countries the unit coefficient null could be rejected at the 5% level. In two cases where β had the incorrect sign the estimates were so imprecise that the PPP null of [ 1-1 ] could not be rejected. Taking either cointegration or non-rejection of a unit coefficient as a positive result, there is 15
evidence for PPP in 24 of the 31 cases. However, in only 2 countries is there evidence of both cointegration (at the 10% level) and a unit coefficient (at the 5% level). The unit root tests on the logarithm of the real exchange rate are consistent with the negative results from the joint test. When no trend is included, the unit root null is not rejected in any of the countries, using either the DF or ADF1 statistic. When a trend is included, the unit root null is rejected in only one country. This is the standard result in the literature and contrasts with the greater support for PPP obtained from the unrestricted model. The difference does not arise from the restriction on the dynamics in the ADF test, since as Table 2 shows the hypothesis that d 3i = 0 in equation (10) is rejected at the 5% level in only 3 cases. Table 2 also gives the estimates of the feedback coefficients in equation (9), c 11i on the spot and c 21i on prices. We would expect c 11 < 0, and c 21 > 0, since a real overvaluation ( r below its equilibrium) requires a nominal devaluation (an increase in s ) or a decline in domestic prices relative to foreign prices ( a decline in d). The feedback on the spot rate was negative in all but six cases (none of which were significant), and the feedback on the price differential positive in all but 8 cases and in only two cases were they significantly negative. This suggests that not only does the spot rate seem to adjust more than prices, but that in about a quarter of countries (8 out of 31) the feedback on prices is destabilising. The mean feedback on exchange rates is 23%, on prices 6%. TABLE II ABOUT HERE Im Pesaran & Shin (1997) tabulate means and variances for the average ADF in a panel under the null hypothesis of a unit root. In this case the simulated average values are: without trend DF -0.89, ADF1-1.08; with trend DF -1.55, ADF1-1.72. In all four cases, the actual averages are greater than the simulated values that would be expected if there were a unit root. Thus, like individual tests, panel tests would not reject a unit root in the real 16
exchange rate. However, the IPS test strongly rejects the null that all the real exchange rates are I(2), whereas in some individual cases this cannot be rejected. Inspecting the ADF equations, what is noticeable is how sensitive the estimate of the average adjustment coefficient is to the inclusion of a trend. When a trend is included in the ADF1 equation for the real exchange rate, the estimated speed of adjustment more than doubles from 12% to 28%. While the difference is not statistically significant, since there is a very large degree of uncertainty, economically the difference is large: quite different policy conclusions would be drawn from adjustment to equilibrium at 12% p.a. and 28% p.a. To investigate the pattern further the Swamy weighted average estimates of the coefficients of equation (7) across all 31 countries, were calculated. These were estimated using data from which each country mean had been subtracted. The estimates are: s it = -0.24 s i,t-1 + 0.26 d i,t-1 + 0.17 s i,t-1-0.07 d i,t-1 + 0.0001 t (0.09) (0.11) (0.10) (0.15) (0.008) MLL=782.20 d it = 0.006 s i,t-1-0.03 d i,t-1 + 0.03 s i,t-1 + 0.21 d i,t-1 + 0.0001 t (0.08) (0.09) (0.09) (0.07) (0.0001) MLL=1207.44 Standard errors are given in parentheses, the maximised log-likelihood (MLL) is the sum over the individual equations. In the exchange rate equation, the lagged exchange rate and lagged price differential would be significant if they had standard distributions and are roughly equal and of opposite sign as PPP would suggest. In the inflation equation, only the lagged change in price differential would be significant. If we reparameterise the VAR as an equation for the real exchange rate, by subtracting the two equations, the Swamy weighted average estimates are: 17
r it = -0.29r i,t-1 +0.06 e i,t-1-0.26 r i,t-1-0.15 e t-1-0.0001 t (0.08) (0.11) (0.13) (0.16) (0.0002) MLL = 763.36 If the variables with insignificant average effects are deleted, the result is: r it = -0.12r i,t-1 + 0.15 r it-1 (0.04) (0.06) MLL=620.48 The lagged real exchange rate now suggests a much slower speed of adjustment, 12% rather than 29% and has a t ratio of -3.29. This matches the result with the unweighted averages. The unrestricted equation involves estimating 186 coefficients (6x31), the restricted equation involves estimating 93. The hypothesis that the restrictions hold in each country would be rejected by a Likelihood Ratio test. We now estimate the three equations, by the Fixed Effect estimator, using robust standard errors. The estimates of the VAR are: s it =α 1i -0.05 s i,t-1 + 0.11 d i,t-1 + 0.11 s i,t-1 + 0.70 d i,t-1-0.0006 t (0.03) (0.04) (0.11) (0.16) (0.001) R 2 =0.63, SER=0.17, MLL= 262.72 d it =α 2i -0.02 s i,t-1-0.07 d i,t-1 + 0.10 s i,t-1 + 0.73 d i,t-1-0.001 t (0.02) (0.03) (0.09) (0.14) (0.001) R 2 =0.74, SER=0.14, MLL=430.37 Robust standard errors are in parentheses and SER is the standard error of the regression. The speed of adjustment in the exchange rate equation is now much slower as one would expect from the heterogeneity bias of the fixed effect estimator. The proportionality restriction would 18
be rejected in both equations (t=2.01 and 2.38). The real exchange rate equation broadly shows the features that one would expect from the equations for its two components: r it = α 3i -0.04r i,t-1 + 0.002 d i,t-1 + 0.01 r i,t-1-0.06 d t-1 + 0.0008 t (0.02) (0.009) (0.18) (0.12) (0.002) R 2 =0.07, SER=0.13, MLL= 436.80 If time effects as well as group effects are included, allowing a flexible trend to pick up any common shocks, the estimates are very similar, and the year dummies are not significant. Likelihood Ratio tests would reject the hypothesis that the coefficients were equal across countries. However, conditional on coefficient equality, equality of intercepts would not be rejected for the exchange rate and real exchange rate equations, though it would for the price equation. The panel estimates of the static equation (4), are supportive of PPP: s it = α i + 0.99684 p it - 0.95375 p * t (0.0171) (0.0426) R 2 = 0.9871, SER = 0.31, MLL = -171.96. as are the cross-section regression of the average change in the log of the nominal exchange rate on the average rate of inflation: s i = -0.05 + 0.95 d i R 2 = 0.94 (0.008) (0.02) SER 0.034 The intercept is significantly less than zero (t=-5.85) and the slope is significantly less than unity (t=-2.73); but the differences are not large in economic terms. Although we would expect a downward bias from measurement error, the estimate from the reverse regression (R 2 divided by the estimate from the direct regression) is 0.99, also below unity. 19
7. Discussion and Conclusions This paper provided a systematic exposition of the econometric issues in testing PPP using time series and panels, interpreting the alternative tests in terms of the restrictions and reparameterisations they implied for an underlying heterogeneous VAR. The models were estimated on a panel of 31 developing countries. The empirical results were broadly in accord with those reported in the literature. Using time-series data one cannot reject the null of a unit root in the real exchange rate, though one can reject the null of no cointegration between nominal exchange rates and price differentials for over half the countries. Although the systematic testing procedure helped identify which aspects of specification the tests were most sensitive to, it did not solve the basic problem, the low power of the tests. Using panel estimates, either averages of coefficients or constraining coefficients to be the same across countries, provided much more evidence for PPP. In the cross-section dimension, PPP holds almost exactly; the differences from unity may be statistically significant, depending on how the standard error is calculated, but it is small in economic terms. The estimates in this paper conform to the trend in the literature that concludes that relative PPP holds almost exactly in the long run. Of course, this conclusion is not very interesting for most policy purposes unless convergence to equilibrium is reasonably fast. However, our theoretical discussion and estimates cast doubt on the trend in the literature to conclude that the speed of convergence is about 15% a year. Measuring the speed of convergence in cross-country panels faces major difficulties unless T is large and the structural parameters are constant across countries and time. Otherwise, estimates of the coefficients of the lagged dependent variables are subject to a variety of biases and are sensitive to specification and estimation method. Small changes in specification can cause the estimated speed of adjustment to vary between 10% and 30%. Not only may we not be able to 20
estimate the speed of convergence from panels, but there may be nothing to estimate if the speed of convergence is not a constant, but itself a function of the deviation from equilibrium or other variables. 21
Table 1 Johansen MLE Cointegration LR Tests and parameter estimates Eigenvalue * (14.0690) **(12.0710) Trace * (15.4100) **(13.3250) β estimate Cointegrating Restriction ( 1-1 ) 1 Case COUNTRY # 1 Barbados 25.2 * 27.3 * -0.03.000 2 Cyprus 13.6 ** 20.2 * 6.0.010 3 Dominican Republic 8.8 9.0-0.8.862 4 Ecuador 9.2 9.7-1.5.021 5 El Salvador 9.0 11.6-0.9.765 6 Ethiopia 15.1 * 17.6 * 2.7.012 7 Greece 11.9 19.2 * -2.2.033 8 Guatemala 7.4 8.6-1.7.060 9 Guyana 45.1 * 49.9 * 0.3.000 10 India 9.5 11.0-1.6.102 11 Jamaica 16.3 * 21.1 * -4.8.001 12 Korea 9.0 10.0-0.7.563 13 Malawi 7.5 9.3-0.8.479 14 Malta 20.8 * 22.5 * 3.1.000 15 Mauritius 14.4 * 23.5 * 3.8.025 16 Mexico 9.8 12.1-1.1.220 17 Morocco 11.6 16.3 * -6.8.010 18 Nepal 19.6 * 19.7 * -1.2.061 19 Pakistan 10.7 18.4 * 0.4.205 20 Paraguay 12.5 ** 13.5 ** -2.0.001 21 Peru 22.2 * 28.7 * -1.1.002 22 Philippines 12.6 ** 16.3 * -2.5.004. 23 Singapore 7.5 13.2-2.9.830 24 South Africa 10.5 12.6 1.5.004 25 Sri Lanka 8.6 11.3 0.3.054 26 Thailand 17.4 * 22.3 * 2.1.001 27 Trinidad & Tobago 17.6 * 22.0 * -5.8.002 28 Turkey 3.1 3.3-0.6.667 29 Uruguay 10.5 14.6 ** -8.7.026 30 Venezuela 26.2 * 35.0 * 0.6.003 31 Zimbabwe 10.3 11.8 1.5.003 Mean (s.d.) 31 countries 13.9 (7.9) 17.4 (9.1) -0.8 (3.0) * Indicates rejection of the non-cointegrating null at the 5% level. ** Indicates rejection of the non-cointegrating null at the 10 % level. 1 Cointegrating restrictions on nominal exchange rate and relative domestic/foreign prices obtained from: z t = s t + βd t where d t = p t - p* t, the price differential. Johansen LR test on cointegrating vector, p value reported. 22
Table 2 Estimates of Real Exchange Rate Feedback Coefficients and Common Dynamics Case # COUNTRY c 11 (t stats) c 21 (t stats) d 3 =0(prob) 1 Barbados 0.003 (0.06) 0.12 (1.39) (.053) 2 Cyprus -0.30 (-2.38) 0.20 (0.63) (.891) 3 Dominican Republic -0.29 (-1.16) 0.13 (1.76) (.355) 4 Ecuador 0.36 (0.17) 0.17 (1.55) (.420) 5 El Salvador -0.86 (-2.88) -0.16 (1.32) (.647) 6 Ethiopia -0.10 (-1.53) 0.35 (2.03) (.027) * 7 Greece -0.31 (-2.58) 0.01 (0.17) (.234) 8 Guatemala 0.11 (0.55) 0.17 (1.54) (.026) * 9 Guyana 0.35 (0.99) -0.29 (3.14) (.871) 10 India -0.43 (-1.04) 0.36 (2.12) (.587) 11 Jamaica -0.41 (-2.21) -0.11 (1.74) (.421) 12 Korea -0.47 (-2.87) -0.12 (1.21) (.786) 13 Malawi -0.23 (-1.14) 0.32 (1.97) (.191) 14 Malta -0.31 (-1.82) 0.03 (0.64) (.783) 15 Mauritius -0.56 (-2.75) -0.14 (1.46) (.664) 16 Mexico -0.30 (-1.18) 0.28 (1.19) (.318) 17 Morocco -0.20 (-2.01) -0.007 (-0.20) (.099) 18 Nepal -0.55 (-2.90) 0.38 (2.25) (.487) 19 Pakistan -0.22 (-1.27) 0.35 (2.41) (.050) 20 Paraguay 0.09 (0.45) 0.06 (1.01) (.729) 21 Peru -1.69 (-3.43) -1.31 (2.47) (.237) 22 Philippines -0.28 (-1.16) 0.09 (0.61) (.817) 23 Singapore -0.13 (-2.02) 0.16 (2.33) (.190) 24 South Africa -0.09 (-1.11) 0.01 (1.07) (.472) 25 Sri Lanka -0.01 (-0.06) 0.11 (2.06) (.249) 26 Thailand -0.18 (-2.07) 0.10 (1.05) (.221) 27 Trinidad & Tobago -0.49 (-2.71) 0.02 (0.34) (.103) 28 Turkey -0.29 (-1.51) -0.14 (0.71) (.041) * 29 Uruguay 0.12 (0.48) 0.26 (1.66) (.116) 30 Venezuela 0.38 (1.14) 0.45 (3.90) (.761) 31 Zimbabwe -0.04 (-0.99) 0.01 (0.88) (.320) Mean All Countries -0.23 0.06 0.393 Standard Deviation (0.38) (0.31) (0.288) * Indicates d 3 0 at 5%. 23
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