Long-Run Purchasing Power Parity and General Relativity

Transcription

1 Long-Run Purchasing Power Parity and General Relativity Jerry Coakley a, Robert P. Flood b ; Ana M. Fuertes c and Mark P. Taylor dy a University of Essex b International Monetary Fund and NBER c Cass Business School, City University, London d University of Warwick and CEPR May 2004 Abstract This paper develops and implements novel tests of general relative purchasing power parity (PPP), which we de ne as a long-run unit elasticity of the nominal exchange rate with respect to relative national prices. We test this in a exible framework that accommodates both temporary and permanent shocks to the level of the real exchange rate. The tests build on panel estimators whose nite-sample properties are analysed using Monte Carlo simulations of a stylized PPP regression that interacts country heterogeneity, cross-sectional dependence and non-stationary disturbances. General relative PPP is tested for a unique sample of industrialized and developing countries over the period 1970:1-1998:12 using both CPI and PPI indices. We establish that in ation di erentials are re ected one-for-one in long-run nominal exchange rate depreciation: i.e that general relative PPP holds. We draw out the implications of our results for the research agenda in this area. Keywords: Real exchange rate; cross sectional dependence; real shocks; panel estimator; Monte Carlo. JEL Classi cation: C32; F31 This paper was partly written while Mark Taylor was a Visiting Scholar in the Research Department of the International Monetary Fund. Any views expressed in the paper are those of the authors alone and do not necessarily represent the views of the International Monetary Fund or any of its member countries. We are grateful to Ron Smith for comments on an earlier version of the paper, although the usual disclaimer applies. y Corresponding author: Professor Mark P. Taylor, Department of Economics, University of Warwick, Coventry CV4 7AL, UK. mark.taylor@warwick.ac.uk 1

2 1 Introduction The equilibrium condition known as purchasing power parity (PPP) involves a relationship between a country s foreign exchange rate and the level or movement of its national price level relative to that of a foreign country. Two versions of PPP are commonly referred to in the literature. Absolute PPP states that the purchasing power of a unit of domestic currency is exactly the same in the foreign economy, once it is converted into foreign currency at the absolute PPP exchange rate while relative PPP implies that relative changes in national price levels are o set by commensurate changes in the nominal exchange rate between the relevant currencies or in other words that the elasticity of the nominal exchange rate with respect to the price relative is unity. The voluminous empirical research literature on PPP over the course of recent decades has been driven largely by econometric problems relating to unit root tests of long-run PPP (see Taylor, 1995; Sarno and Taylor, 2002; and Taylor and Taylor, 2004 for overviews). These include issues such as the low power of unit root tests, possible structural breaks in kong spans of data, mixture of stationary and non-stationary error terms in the relevant regressions and neglected cross-sectional dependence when real exchange rate panel data is used. In this paper, we seek to make several contributions to the literature. The rst is that we generalise the concept of relative PPP to the case where the long-run elasticity of the nominal exchange rate with respect to relative national prices is unity: this is what we term general relative PPP. We then develop methods to test for general relative PPP, with no restriction on the innovation sequence. Accordingly, we deploy tests in a panel regression framework that is robust to country heterogeneity and cross-sectional dependence, as well as to most importantly permanent as well as transitory shocks to the real exchange rate. Thus, we allow for the long-run equilibrium real exchangre rate to shift while still testing for a long-run unit elasticity in the nominal exchange rate with respect to the price relative. In contrast, the extant empirical analysis of PPP generally precludes precludes permanent shocks to the long-run real exchange rate. It is widely accepted, however, that over long periods, real shocks may permanently impact on the long-run equilibrium real exchange rate level for example due to productivity di erentials as in the Harrod-Balassa-Samuelson e ect (Froot and Rogo, 1995; Sarno and Taylor, 2002; Lothian and Taylor, 2004; Bergin, Glick and Taylor, 2003). By exploiting recent developments in the 2

3 econometrics of non-stationary panel data, we can accommodate such shocks alongside transitory or monetary shocks. Moreover, Taylor (2001) has demonstrated how two problems trading costs and risk aversion on the one hand and temporal aggregation on the other can make the regression errors of nominal exchange rates on price relatives appear rather persistent or even non-stationary. The main theoretical econometric results that we build on in developing a methodology to address these issues are those of Pesaran and Smith (1995), Phillips and Moon (1999, 2000) and Kao (1999), which establish that long-run relationships are not exclusively associated with cointegrating equations in the Engle and Granger (1987) sense: in a static regression setup, these authors show that irrespective of whether the process generating the errors is stationary or contains a unit root, several panel estimators are consistent for the true long-run parameter. Hence, although a timeseries regression between two processes with random walk or unit root components produces a spurious correlation if the variables do not cointegrate, this is not the case in panel regressions for a large number of countries. The intuition is that pooling or averaging over individuals lessens the noise the individual covariance between the non-stationary error term and the non-stationary regressor that induces the spurious correlation problem and so leads to a stronger overall signal. A second contribution of our research is that we compare the nite sample properties of several panel estimators using Monte Carlo simulations of a stylized PPP regression that incorporates plausible features such as parameter heterogeneity, cross-section dependence and permanent disturbances. These simulations extend the aforementioned theoretical econometric literature where cross-sectional dependence and mixed stationary and non-stationary errors are ruled out by assumption (e.g. Phillips and Moon, 1999). While panel estimators of the slope coe cient may remain unbiased and consistent in the presence of cross-sectional dependence, they are generally ine cient. However, they will also be inconsistent if the cross-sectional dependence stems from unobserved common factors correlated with the regressors (Fuertes and Smith, 2004). The estimators we consider include a seemingly unrelated regression mean group (SUR-MG) estimator suggested by Coakley et al. (2004), the cross-sectionally augmented mean group estimator (CMG) proposed by Pesaran (2003), a mean group estimator based on cross-sectionally demeaned data (DMG) and pooled estimates with country and time e ects. These estimators allow us explicitly to address the issue of cross-sectional dependence between real exchange rates stemming from a common numéraire currency and foreign price level or from common macroeco- 3

4 nomic or other (for instance, oil price) shocks. With the exception of a few contributions such as those of Abuaf and Jorion (1990), O Connell (1998) and Taylor and Sarno (1998), cross-sectional dependence has been largely ignored in the PPP literature. 1 A third contribution that we seek to make is that our empirical analysis is based on a unique large data set for the 1970:1-1998:12 period that comprises nineteen Organization for Economic Cooperation and Development (OECD) member countries and 26 developing countries and both consumer price index (CPI) and producer price index (PPI) data. Analysing panel time series of both developing and industrial countries using the same methodology is interesting. The former show larger heterogeneity, more cross-sectional variation (higher signal) but more time series noise also due to measurement error. Our simulation evidence suggests that the DMG, CMG, two-way xed e ects (2FE) estimator and a cross-section (CS) estimator based on time averaged data provide essentially unbiased measures of the long-run coe cient in the presence of permanent disturbances and cross-sectional dependence that may stem from unobserved factors correlated with the regressors. However, the standard errors of FE-type estimators appear severely downward biased. The latter problem can be circumvented by means of the sieve bootstrap approach suggested by Fuertes (2004). The standard errors of the CS estimates may also need to be corrected for heteroskedasticity stemming from coe cient heterogeneity. On the basis of our large sample, the robust conclusion from both an informal graphical analysis and formal empirical tests is that the common currency price of a basket of goods in the di erent countries will rise one-for-one over the long term. The remainder of the paper is organized as follows. Section 2 provides an overview of the literature and Section 3 discusses the notion of relative PPP. Section 4 presents the panel econometric framework and explores the nite sample properties of several estimators. In Section 5 we discuss the empirical results. A nal section concludes. 2 General Relativity Denote the price level of the domestic currency by P t and the corresponding foreign price level by P t. If S t is the nominal exchange rate (domestic price of foreign currency) and if absolute 1 Typically, panel unit root studies that accommodate cross sectional dependence support PPP less strongly or reject PPP as in the O Connell (1998) and Wu and Wu (2001) studies. 4

5 PPP holds, then expressing the domestic price in foreign currency terms must yield the foreign price, P t = P t =S t ; or taking logarithms (lower case letters) and rearranging we have the short-run relationship: s t = p t p t (1) which implies that the nominal exchange rate should be directly proportional to the relative price levels, even in the very short run. The short-run relative PPP conditon may be written: s t = p t p t (2) where denotes the rst-di erence operator, i.e. x t = x t x t 1, for x = s; p; p. While empirical tests of these short-run relationships were not uncommon in the early 1970s (e.g Frenkel, 1976 or the studies cited in O cer, 1982), the very high volatility of nominal exchange rates compared to relative national price levels or in ation rates during the recent oating rate period has adequately demonstrated that neither absolute nor relative PPP is a realistic description of shortrun exchange rate behaviour. Instead, over the past two decades or so, long-run absolute PPP has been extensively tested as a long-run equilibrium condition, largely through the application of unit root and cointegration techniques. Or, more precisely, a necessary condition for long-run PPP has been tested. E ectively, what these unit-root studies do is rstly to capture the short-run variation in real exchange rates by adding an error term to (1): s t = (p t p t ) + u t ; (3) and then to test the hypothesis that the process u t the deviation from PPP is non-stationary. 2 In particular, if u t contains a random walk or unit root component u t is I(1) then the implication is that deviations from PPP never settle down at an equilibrium level even in the long run. 3 Clearly, the existence of a long-run equilibrium level of the real exchange rate is a necessary condition for long-run absolute PPP to hold. This approach has spawned a vast literature which has, in general, found remarkable di culty in supporting the hypothesis of long-run PPP using data for the recent oating rate period alone (Sarno and Taylor, 2002; Taylor and Taylor, 2004). 2 The cointegration variant on this is to estimate slope coe cents in this relationship, rather than imposing that they be +1 and 1, and then to test for staionarity of u t (e.g. Taylor, 1988). 3 A series is said to be integrated of order d, I(d), if it must be di erenced d times to become stationary. 5

6 Moreover, it has indicated that real exchange rates or deviations from long run PPP are extremely persistent a stylised fact which Rogo (1996) has described as the PPP puzzle. While it is clear that unit root analysis involves a test of long-run PPP, the unit root approach is rather restrictive since it implies that the level of the exchange rate is not subject to permanent shocks. The concept of general relative PPP that we are proposing here operationalises, int he context of testing ofr PPP, the traditonal other things equal assumption of traditonal economic analysis. Suppose that u t in (3) is observationally I(1), perhaps due to real shocks or Harrod- Balassa-Samuelson e ects. Now, further assume that the equation still holds in the sense that the coe cient on the relative price term the long-run elasticity of the nominal exchange rate with respect to relative prices is in fact unity, as in (3). This can be interpreted as a form of long-run relative PPP in the sense that a one percent increase in relative prices would still be o set one-toone by a long-run depreciation of the domestic currency of one percent, other things equal, even though long-run absolute PPP would be rejected in the conventional unit root framework and even though both short-run absolute and relative PPP would also be rejected. The other things equal clause here is crucial. The concept of long-run relative PPP that we are proposing is very general. It would allow, for example, for real shocks permanently to a ect the real exchange rate so long as any given percentage movement in relative prices would lead to a commensurate movement in the nominal exchange rate in the long-run, over and above any such permanent e ect on the real exchange rate. It is because of the generality of the concept that we term it general relative PPP. Note that general relative PPP does not imply any causality: although we are testing for a unit elasticity of the exchange rate with repsect to relative prices, the concept makes no assertions concerning which way or ways causality runs between the the variables concerned; it only aserts that, in the long run and other things equal, a given percentage change in relative prices will be o set by a commensurate percentage change in the nominal exchaneg rate. Another way of approaching this issue is to note that, irrespective of whether u t is I(0) or I(1), di erencing (3) yields: s t = (p t p t ) + u t ; (4) where u t must be I(0). Since u t represents deviations from relative PPP, (4) implies that relative PPP must hold in a long-run sense because deviations from relative PPP are stationary. Thus a test of general relative PPP is an economically meaningful test irrespective of the behavior 6

7 of u t. The drawback in estimating a regression in di erences, along the lines of equation (4), is that one discards the information in the levels and hence it is di cult to separate out short-run or high-frequency from long-run or low-frequency e ects. Equally, working with an error correction model that assumes cointegration between the variables imposes restrictions such as that u t is I(1) that appear invalid in nite samples. The object of our empirical analysis is on using the log-level equation s it = i + i (p it p it) + v it ; t = 1; :::; T (5) for i = 1; :::; N countries to estimate the mean e ect E( i ): In this context, the slope i ds it =d(p it p it ) is interpreted as the long-run relative price elasticity of the nominal exchange rate for country i. The innovation sequence can be I(0) or I(1). The null hypothesis that general relative PPP holds may then be tested as: H 0 : E( i ) = 1 (6) and the alternative is that 6= 1. The null hypothesis implies a unit relative price elasticity that can also be derived by taking the total di erential of equation (3). In this manner the paper revives the early focus of research on PPP on the slope coe cient (e.g. Frenkel, 1976) before the spurious regression problem came to light in the 1980s and the cointegration methodology took hold. We introduce below a robust panel framework to test for long-run relative PPP. Our approach can accommodate a number of factors germane to the PPP debate. One is the observed high persistence of real exchange rates that may re ect a combination of real and monetary (permanent and transitory) shocks. It may also relate to the noise stemming from measurement error, transaction costs and other market imperfections such as limits to arbitrage in foreign exchange markets (Taylor and Taylor, 2004) that may make the regression disturbances observationally equivalent to I(1) series. Another important factor is cross-sectional dependence, which, as we noted above, has been often neglected in empirical analysis. 7

8 3 The Econometric Framework 3.1 Non-stationary Panel Regression Suppose the data are generated according to the set of relationships y it = i + 0 ix it + v it ; i = 1; :::; N; t = 1; :::; T (7) where i = + ;i and i = + ;i : It is assumed that: (A1) the coe cients are constant over time but di er randomly across units, that is, ;i iid(0; 2 ), ;i iid(0; 2 ) and ( ;i; ;i ) 0 are distributed independently of the regressor x it and disturbances v it for all i and t; (A2) the disturbances have a zero mean, time-constant but possibly heterogeneous variance and are uncorrelated across di erent units, i.e. E(v it ) = 0; E(vit 2 ) = 2 i and E(v itv jt ) = 0 if i 6= j; (A3) the x is and v it are independently distributed for all t; s (strict exogeneity). This is the standard Random Coe cients Model (RCM) formulation. The focus of the estimators employed is a long-run relation between two non-stationary variables, namely, the parameter of interest is the mean e ect E( i ). 4 Consider the case of a single time series, N = 1, and let the vector time series (y t x t ) 0 evolve according to y t x t! = y t 1 x t 1! + u y;t u x;t! (8) where U t = (u y;t u x;t ) 0 is a stationary vector process with long-run covariance matrix = " yy xy yx xx # = lim T!1 E " 1 p T T X t=1 U t! 1 p T X T U 0 t t=1!# : (9) Analogously to the slope coe cient in the classical linear regression model, a long-run association between non-stationary y t and x t is de ned as: = lim E yt x p p T T!1 T T lim E T!1 xt p T x T p T 1 = yx 1 xx (10) where yx and xx are the long-run covariance and long-run variance, respectively. When has de cient rank, corresponds to a cointegrating coe cient. However, as Phillips and Moon (1999) emphasize, when has full rank and so y t and x t are not cointegrated, still represents a statistical long-run relationship. To provide some insight into what represents in the latter case, suppose 4 This section draws on Phillips and Moon (1999, 2000). 8

9 that y t and x t are generated as follows: y t = z t + w t where z t = z t x t = z t 1 + " z;t and the innovation " z;t is I(0). If w t is I(1); then (10) is not a cointegrating coe cient. However, since the two series have a common driving non-stationary factor z t ; they will be correlated in the long run and this is precisely what captures. A di erent issue is how to estimate. As originally noted by Granger and Newbold (1974), a regression of y t onto x t by OLS when has full rank will yield the spurious (not consistent for ) estimator b = 1 T 2 TX t=1 y t x t 1 T 2! 1 TX x t x t =) t=1 Z B y B Z B x B x A 1 6= yx 1 xx (11) where (B y B x ) 0 is a vector Brownian motion with covariance matrix. Note, in particular, that b does not even converge to a constant but to a nondegenerate random variate which is the product of two functions of Brownian motion (Phillips, 1986). However, if we take expected values of these functions we have 0 Z B y B x A Z B x B x A5 1 = yx 1 xx : (12) Therefore, by obtaining statistical measures that converge to E R B y B x and E R Bx B x, a consistent estimator of can be constructed. This is straightforward with experimental data since a large number of replications of (y t x t ) 0 can be obtained and used to form averages which converge to the appropriate quantities. Panel data sets in some ways resemble experimental data. Accordingly, suppose that we have independent observations (y it x it ) 0 on a large number of countries i = 1; :::; N. Di erent procedures can be used to estimate the long run e ect E( i ): One is the xed e ects (FE) or within estimator, b F E, which involves combining the data by imposing common slopes and error variances but allowing for heterogenous intercepts. The latter is important since the intercepts capture, at a minimum, the non-zero real exchange rate level in the base year. An alternative approach is the mean group (MG) estimator, b MG ; suggested by Pesaran and Smith (1995) which involves averaging the individual OLS regression coe cients ^ i : 5 The MG estimator stands at the other extreme of pooled estimators, such as FE, in that it 5 The MG estimator of is de ned as ^ MG = N 1 P ^ i i with variance V (^ MG ) = N 1 f 1 P N 1 i (^ ^MG i ) 2 g: 9

10 allows for heterogeneous intercepts, slope coe cients and error variances. Another is the crosssection (CS) or between estimator, b CS, that involves averaging the data over time and estimating OLS regressions on individual means y i and x i : Pesaran and Smith (1995) note that the spurious correlation problem does not arise in the case of the CS estimator under the RCM assumptions. Phillips and Moon (1999) further show that both b F E and b CS are p N-consistent for when the error term v it in (7) is I(1). Coakley et al. (2001) use Monte Carlo simulations and response surface regressions to evaluate the nite-sample properties of the FE and MG estimator for panel dimensions N; T = f(15; 300); (30; 25)g typical of monthly and annual PPP studies, respectively. They show for a static regression such as (7) that both estimators appear unbiased with dispersion that falls at rate T p N when v it is I(0) and at rate p N when v it is I(1). 6 Moreover, standard t-tests based on the N(0; 1) quantiles have good size properties for the MG estimator irrespective of whether the regression disturbances are I(0) or I(1). In contrast, inference based on the FE estimator is likely to be misleading since the usual standard errors are severely underestimated when v it is I(1) and robust Newey-West type covariance matrices are not valid. The FE based t-tests are also unacceptably oversized when the true slope coe cients are heterogeneous even if v it is iid because the composite error term measures ( i )x it + v it which is I(1). Although for the present purposes the regression of interest (7) is static, neglecting slope heterogeneity is rather more serious in dynamic models even under stationarity since it introduces a bias in the pooled estimators which does not vanish for large N and T (Pesaran and Smith, 1995). These inference problems further underlie the potential superiority of MG type estimators. However, the asymptotic theory for panel estimators of non-cointegrating regressions in Phillips and Moon (1999) and the nite sample analysis in Coakley et al. (2001) rest on the assumption of cross-sectional independence. To gauge the plausibility of this assumption for our sample, Table 1 reports the average correlations across countries for relative in ation rates (d it ), deviations from relative PPP or real exchange rate changes (r it ) and individual OLS residuals ^u OLS it : [Table 1 around here] Some interesting features emerge. The largest average correlation among relative in ation rates 6 This Monte Carlo evidence is particularly relevant for the MG estimator since its asymptotic properties have not been established as yet for panel regressions with I(1) disturbances. 10

11 is in the industrialized, PPI panel (i.e. for industrialised countries and employing producer price index data the data are described in detail in Section 5) with pairwise correlations that are uniformly positive. The correlation coe cients among real exchange rate changes for industrialized economies are also invariably positive and some 15 times larger than for the developing countries, which exhibit both positive and negative dependencies. The average absolute cross-correlations in the equation-by-equation residuals of (5) range form 21% to 71%, with the industrialized panels exhibiting the highest levels. Thus, cross-sectional dependence has to addressed in testing for PPP using panel data. Before deploying panel estimators to test for general relative PPP, it is important to analyze their nite-sample performance in di erent settings that interact serial dependence (of I(0) or I(1) form) and cross-section correlation in the disturbances. This exercise is conducted below using Monte Carlo experiments. 3.2 A Monte Carlo Analysis of Panel Estimators Consider the stylized PPP equation y it = i + i x it + v it ; i = 1; :::; N; t = 1; :::; T (13) and underlying data generating processes (DGP1 hereafter) v it = v;i v i;t 1 + " v;it ; " v;it iidn(0; 2 v;i) (14) x it = x i;t 1 + " x;it ; " x;it iidn(0; 2 x;i) (15) where " v;it is independent of " x;is for all t; s. The iid assumption implies identically distributed and independent over time, that is, E(" v;it " v;is ) = E(" x;it " x;is ) = 0; for all s 6= t: However, rather than assuming that E(" v;t " 0 v;t) v and E(" x;t " 0 x;t) x we consider ! v! v 1! x! x! v 1! v v = ;!! x 1! x v < 1 and x = ! v! v 1! x! x ;! x < 1 respectively, where the diagonal terms in v are 2 v;i E("2 v;it ) = 1 and the o -diagonal terms are! v E(" v;it " v;jt ) 6= 0 and likewise for x. 11

12 Albeit simple, this parameterization would be relevant if the cross-country dependencies stem from the variation in (or shocks to) the value of the dollar and the US price level. We set! x = 0:4 throughout, which roughly typi es the largest average correlation in x it across the four panels as Table 1 shows. We consider various levels of cross-section dependence in the regression disturbances! v = f0:0; 0:3; 0:5; 0:7; 0:9g: Second, another interesting setting (DGP2) in the context of PPP is v it = v;i v i;t 1 + i z t + " v;it ; " v;it iidn(0; 2 v;i) (16) x it = x;i x i;t 1 + i z t + " x;it ; " x;it iidn(0; 2 x;i) (17) with assumptions for " v;it, " x;it and v ; z matrices as above. The process z t represents unobserved common factors and is generated by z t = z z t 1 + " z;t ; " z;t iidn(0; 1) where " z;t is independent of (" v;t ; " x;t ) 0 : We consider z 2 f0:3; 1g: Note that for z = 1 the nonstationarity of x it is induced by z t and so x;i < 1 is speci ed, since otherwise x it would be I(2); the same applies to v it : DGP2 generalizes DGP1 in several directions: (a) The cross-sectional dependence is partially determined by the e ect of unobserved common factors z t global macroeconomic variables or shocks (e.g. oil price shocks) which may be correlated with the regressors; (b) The nonstationarity in disturbances and/or regressor may arise from unobserved global variables z t ; (c) The unobserved common factor has di erent e ects on the countries (random coe cients i and i ) and so heterogenous cross-sectional correlations are introduced. We adopt i iidu[a; b] and i iidu[a; b] such that E( i ) = E( i ) = 1 with (a; b) = f( 1; 3); (0:5; 1:5)g: As for the stationarity properties of the error term v it in (13) we allow for: (a) Weak dependence ( v;i = 0:3; 8i); (b) Unit root non-stationarity ( v;i = 1;for all i) and (c) A mix of highly persistent but stationary errors ( v;i = 0:9; i = 1; :::; N 1 ) and unit root errors ( v;i = 1, i = N 1 + 1; :::; N). We use N 1 = 7 or roughly N 1 =N ' 50%. Without loss of generality, we use i iidu[ 0:5; 5] and i iidu[0:5; 1:5] which roughly typify the range of variation in the individual OLS estimates for the industrialized, PPI panel. 7 7 The range is [ 0:43; 5:55] and [0:60; 1:70] for ^ i and ^ i, respectively. Using more widely dispersed parameters i U[ 1; 6] and i U[0:1; 1:9] we nd qualitatively similar results in the simulations. 12

13 The parameter of interest is the mean e ect of x it on y it with true value E( i ) = 1. The results are based on 5,000 replications of panel data sets with dimensions N = 15, T = T 0 that resemble our actual sample. The rst T 0 = 50 observations are discarded to reduce the initialization e ects, x i0 = v i0 = z 0 = 0. The estimators considered are the FE (1-way xed e ects), 2FE (2-way xed e ects), MG (mean group estimator based on individual OLS estimates), SUR-MG (mean group based on SUR-FGLS estimates), DMG (mean group based on individual OLS estimates for cross-sectionally demeaned data), CMG (mean group based on OLS estimates of regressions augmented by y i and x i ); and the CS (between) estimator. Note that the FE and MG estimators neglect cross-sectional dependence whereas the remaining approaches control for it in di erent ways. cross-sectional dependence is not an issue. The CS estimator averages the data over time and so By including time e ects, the 2FE estimator amounts to FE for cross-sectionally demeaned data, y it y t and x it x t ; and so it can be seen as the pooled counterpart of DMG. SUR-MG is suggested by Coakley, Fuertes and Spagnolo (2004) as a simple approach to increase the e ciency of MG in cross-sectional dependence settings. Pesaran (2003) proposes the CMG approach, namely, augmenting the regression of interest by the cross-section means of the variables to capture the unobserved factor(s) that induce cross-country dependence. He shows that the CMG estimator is consistent (as T! 1 and N! 1) in a general setup where the latent factors are I(0) or I(1) and may be correlated with the regressor. Tables 2 and 3 summarize the simulation results for DGP1 and DGP2, respectively. [Tables 2 and 3 around here] We report the sample mean of the estimates over replications, SM^; and its standard deviation, SSD^, to gauge the bias and relative e ciency. The SSD^ is compared with the average estimated standard error, SM s^ ; to assess the adequacy of conventional inference and the normality of ^ is examined using the Jarque-Bera test. For DGP1, all estimators are unbiased irrespective of whether the regression disturbances are I(0) or I(1), with or without cross-section dependence. However, the standard errors of FE (and 2FE) estimates are seriously underestimated for all designs. Correct standard errors can be obtained from Newey-West formulae in the autocorrelated I(0)-error case but these are not valid in the I(1)-error case nor do they account for cross-section dependence. The CS estimates are 13

14 highly dispersed and their standard errors are somewhat downward biased. The latter is because the disturbance term in the CS regression is e i = v i + ;i + ;i x i with variance V (e i jx) = V (v i ) x 2 i 2 and so heteroskedasticity e ects may be important. Robust standard errors for the CS estimates can be calculated using White s procedure. Note that, although ruled out here by assumption (A3), in practice there may be correlation between the country-speci c coe cients and the regressors which will render the CS estimator biased and inconsistent. For stationary disturbances (designs A1-A5) the standard errors of the mean group type estimators are correct. For our large T = 300, the MG, SUR-MG and CMG estimators are equally e cient whereas DMG incurs some e ciency loss because demeaning reduces the signal-noise ratio. For the non-stationary error designs (B1-B5) or mixed error designs (C1-C5), SUR-MG shows e ciency gains relative to MG but the standard errors (of both estimates) are somewhat underestimated and especially so the larger the level of cross-section dependence (! u ): The DMG and CMG approaches are the most e cient in accounting for the cross-section dependence in non-cointegration settings (B1-B5) and their standard errors are essentially correct. In the mixed error settings, the CMG estimator is the most accurate. Despite the relatively large T and the normality assumption for the innovations, the probability distribution of CS, FE and 2FE is generally not normal. By contrast, all mean group type estimators are essentially normal for the I(0) error design (A1-A5). However, when the error terms are all I(1) or a mixture, only DMG and CMG retain normality in the presence of cross-section dependence. For DGP2 where cross-section dependence is introduced by means of a latent common factor also, the CS, 2FE, DMG and CMG estimators are still unbiased when the latent factor is correlated with the regressor, in contrast with the remaining estimators (designs B1-B2, C3-C5). There is no bias for the cointegrating panels (designs A3-A4), despite the fact that the latent factor is present in disturbances v it and regressor x it ; because the correlation between the I(1) regressor and the I(0) disturbance goes to zero for large T. A comparison of design B2 and B1 (and C5 versus C4) suggests that the biases in FE, MG and SUR-MG are smaller when the factors a ect di erent countries in di erent directions (i.e. positive and negative e ects). 8 A further interesting result is that, due to the distinction between averaging and pooling, the bias in the MG estimator is larger than that of the FE estimator when the cross-section correlations 8 For the simple case x;i = v;i = 0 the theoretical bias for each unit is ux;i x;i = i i 2 z 2 i 2 z +2 x;i 14

15 are all positive (B2, C5) and vice versa. The opposite result emerges when the correlations are both positive and negative (B1, C3, C4). Similar considerations as for DGP1 apply regarding the e ciency ranking (and accuracy of standard errors) and the nite-sample probability distribution of the estimators. To sum up, these Monte Carlo experiments evaluate several panel estimators in the context of non-stationary data and cross section dependence. Pooled, cross-section and mean group type estimators are all unbiased in the context of I(1) regression errors when there is no cross-section dependence in line with the theory. When cross-section dependence is introduced, the unbiasedness remains as long as the latent common factors that induce the cross-section dependence are not correlated with the regressor. In the latter case, the CS, 2FE, DMG and CMG estimators are still unbiased and the latter is more e cient. One should be cautious about the standard errors of the pooled estimates which are severely downward biased. Robust standard errors may be necessary for the CS estimates to correct for heteroskedasticity induced by heterogeneity in the coe cients. The mean group standard errors are essentially correct. 4 Testing for General Relativity: Empirical Results Data were gathered from the International Monetary Fund s International Financial Statistics on the exchange rate of the national currency against the US dollar and two price measures, the consumer price index (CPI) and producer price index (PPI) with 1995 as base year. While the CPI series may contain a lower proportion of traded goods than the PPI series, it is generally more widely available than the latter series. The data are monthly over the period beginning January 1970 and ending December 1998 (since some of the important European currencies considered ceased to exist after the end of 1998 because of the introduction of the euro). This gives a total of T = 348 observations per variable. The nominal exchange rate data are middle rates, end-ofperiod quotes. To ensure a more satisfactory match between price and exchange rate data, the nominal exchange rates are scaled so that 1995=100. We use logged variables throughout. The cross-section dimension, N; of the panels is as follows. The industrialized CPI panel (Ind_CPI hereafter) comprises 19 economies. The industrialized PPI panel (Ind_PPI) includes 14 economies. The developing country CPI (Dev_CPI) and PPI panels (Dev_PPI) include 26 and 12 countries, respectively. The countries in each panel are detailed in Appendix A. 15

16 For the nominal exchange rate and relative price series, we found little or no evidence against the unit root hypothesis whereas it is clearly rejected when applied to their rst di erences. Therefore, in keeping with the literature, it seems reasonable to consider s it and d it as I(1) series Graphical Analysis: Scatter Plots As a preliminary motivation to our formal empirical work, we construct scatter plots of depreciation of the dollar exchange rate against the relative (to the US) in ation rate over a number of time horizons. For each of the panels, we construct non-overlapping measures of the twelve-month percentage change in the nominal exchange rate and of the twelve-month percentage change in the relative price level. This is done for each country in the panel and a scatter plot of annual depreciation against relative annual in ation is drawn. Thus in the scatter plots of annual movements, 29 points are plotted for each of the countries in the panel. We then repeat this procedure using averages of the annual movements over non-overlapping ve, ten, fteen and 29 years. 10 In the latter cases we have one point per country in the scatter plots. 11 This graphical analysis is shown in Figures 1 and 2 for the CPI and PPI panels, respectively. [Figures 1 and 2 around here] Perhaps somewhat surprisingly, the scatter plots even at the one-year horizon appear to show a reasonably close degree of correlation between relative in ation and exchange rate depreciation for industrialized and developing countries and as a whole (world panels), although one needs to be careful about the di erent scales in these graphs when interpreting them. What is most striking, however, is the tendency of the scatter plots to collapse towards the 45-degree ray through the origin as the horizon over which the averages are taken increases. A question naturally arises as to the degree to which these plots are distorted by the e ects of outliers. In the industrial panels, for example, the outlier is Israel, due to that country s years of very high in ation from 1979 to annual Israeli CPI in ation was some 120 percent in The results for the augmented Dickey-Fuller and the Phillips-Perron unit root tests are available from the authors on request. The tests are not applied to the residual series for two reasons. On the one hand, they are unlikely to yield conclusive results over a span of just 29 years (Lothian and Taylor, 1997). Panel unit root tests are more powerful but have weaknesses (Taylor and Sarno, 1998; Karlsson and Löthgren, 2000). On the other hand, the novelty of our econometric framework is precisely that it sidesteps the cointegration debate. 10 Since our sample spans 29 years, the ve-year averages consist of ve true 5-year averages and one 4-year average for each country; the ten-year averages are two ten-year averages and one nine-year average, and so on. 11 This informal graphical approach is an extension of the analysis in Flood and Taylor (1996). 16

17 and just over 180 percent in 1985, and reached a peak of a little over 440 percent in While Israel did indeed appear to conform closely to relative PPP over the sample period, the e ect of including Israel in the scatter plots is vastly to increase the scale so that deviations from relative PPP for lower in ation countries are diminished in visual importance. Similarly, in the developing country panels, the very large exchange rate devaluations carried out by Argentina and Peru in the late period show up as outliers in the scatter plots. Hence, the 29-year averages are redrawn in Figure 3 excluding Israel, Argentina and Peru. [Figure 3 around here] These scatter plots tell the same story, namely that relative PPP appears to have held reasonably closely on average over the 29-year period under examination. In the CPI panels, the single exception appears to be Chile, which recorded an average 92 percent annual devaluation over the period (due largely to very high in ation and devaluation in the early 1970s), but an average CPI in ation rate relative to that of the US of only 67 percent. Interestingly, however, the Chilean relative PPI in ation rate over the period is a closely matching gure of some 96 percent. The two points furthest from the 45-degree ray in the developing PPI scatter plot in Figure 3 correspond to Sudan (average annual depreciation 61 percent, average relative annual in ation 41 percent) and Ghana (average annual depreciation 59 percent, average relative annual in ation 36 percent) Overall, therefore, the scatter plots provide strong visual con rmation of relative PPP holding quite closely for both developed and developing countries, at least on average over the 29-year period under consideration. Although this evidence is only informal, it does provide a strong motivation for pursuing our formal econometric exercise. 4.2 Panel Estimates and Inference The parameter of interest is the long-run elasticity of the exchange rate with respect to the relative price: The speci ed model is the log-levels equation (5) where the error sequence v it is allowed to be stationary or to contain a unit root. The long run relative PPP hypothesis is H 0 : = 1 and test statistics for H 0 : = 0 are also reported. Following the discussion in Section 4.2, in order to exploit the panel structure of the data in di erent ways we use three basic methods cross section, pooled and mean group and variants of them to account for cross-country dependence. 17

18 Tables 4 and 5 present the results for the Ind_CPI and Ind_PPI panels, respectively and Tables 6 and 7 those for the Dev_CPI and Dev_PPI panels, respectively. [Tables 4 to 7 around here] The overall picture is that the between and pooled estimates are quite close to the mean group estimates in each case. This suggests that the factors behind the coe cient heterogeneity are likely to be uncorrelated (or weakly so) with the price relative since otherwise biases would appear in the between and pooled estimates. Insigni cant diagnostic tests for heteroskedasticity in the between (or CS) regressions indicate that the OLS standard errors are appropriate. 12 The hypothesis that = 0 is strongly rejected for both industrialized and developing economies (using either CPI or PPI) and the coe cient is insigni cantly di erent from unity even at the 10% level. The standard error of the CS estimates is quite similar for the industrialized and developing countries. This may be because the larger cross-section variation in the developing country panels that increases the signal the variance in the regressor set, V ( d i ); increases from (Ind_CPI) to 8.88 (Dev_CPI) and from 1.21 (Ind_PPI) dramatically to (Dev_PPI) is counteracted by the higher volatility of policy and other shocks (noise) to developing countries. We report the conventional asymptotic standard errors for the pooled estimates allowing for unobserved country e ects (F E) and both country and time e ects (2F E) possibly correlated with the regressor. The arguments in Section 3 suggest that the standard errors are likely to be downward biased. This motivates calculating bootstrap standard standard errors using a procedure that preserves I(0) or I(1) autocorrelation and cross-section dependence in the disturbances. 13 In line with the simulation results, the OLS standard errors are dwarfed as compared to those for the mean group estimates (discussed below) and about 10 times smaller than the bootstrap standard errors. 14 The bootstrap t-statistics support long run relative PPP uniformly for all four panels. 12 For the industrialized countries, White s LM test statistic (p-value) is 1.07 (0.58) and 0.96 (0.62) using CPI and PPI data, respectively. For the developing countries, the counterpart statistics are 0.21 (0.90) and 1.66 (0.44). 13 We use a residual-based sieve bootstrap approach that is shown to provide a reasonably good approximation to the true variability of FE estimates in the presence of autocorrelated errors which may or may not be stationary (Fuertes, 2004). The resampling approach is based on pseudo-disturbances that follow AR(I)MA processes by construction. Cross-section dependence in the residuals is preserved by resampling rows with replacement. 14 Newey-West standard errors increase only slightly relative to OLS, e.g. at (FE) and (2FE) for the Ind _CPI panel. This is not surprising given that the Newey-West covariance matrix is inappropriate when the regression errors are I(1) and it does not capture cross-section dependence either. 18

19 The hypothesis that = 0 is also strongly rejected. As illustrated in the simulations, by including time e ects the 2FE regression allows for a exible trend to pick up any common shocks and so the FE and 2FE estimates can di er markedly when these common shocks are correlated with the regressors. A Hausman test can be deployed to explore this issue. Under the null, the common shocks are uncorrelated with the regressors and so both FE and 2FE are consistent but the former is e cient. Under the alternative, only the 2FE is consistent. 15 For all four panels, the ^ F E and ^ 2F E estimates are insigni cantly di erent suggesting that the latent common shocks (or macroeconomic variables) that induce the cross-section dependence are not correlated (or weakly so) with the regressors. The di erent mean group estimators point in the same direction: the long run elasticity is signi cantly di erent from zero and insigni cantly di erent from unity. An exception is the CMG approach for the Ind_CPI panel where the unit elasticity hypothesis is rejected at the 5% level. The simulation evidence in Section 4.2 suggested that, when cross-section dependence is present, the CMG estimator is more e cient. However, all four panel samples give relatively large standard errors for the CMG estimates which suggests that elements in the true DGP not captured in (14)- (17) make the CMG estimates less accurate in our context. Therefore, it seems reasonable to conclude that the deviation from unity of the long run elasticity suggested by the CMG estimate in the Ind_CPI panel is small in economic terms. As the averaging is sensitive to outliers, we checked for their potential e ects by removing the most extreme values, namely, countries for which Z(^ i ) j^ i ^CMG j ^ > 2 where ^ is the sample standard deviation of ^ i ; i = 1; :::; N. Interestingly, the outlier-robust CMG estimate for the Ind_CPI panel at moves closer to unity and the distribution of ^ i is now normal. For completeness, Tables 4-7 report (in italics) the outlier-robust mean group estimates for all MG, SUR-MG, DMG and CMG variants. 16 To sum up, the strong conclusion that emerges from these tests is that the long run relative price elasticity is insigni cantly di erent from unity. These results together with the informal empirical evidence presented in the scatterplots suggest that long run relative PPP holds. Nominal exchange rates and price di erentials tend to move one-for-one in the long run. 15 The test statistic is H = ^q 0 [V (^q)] 1 ^q: In this case ^q = ^ F E ^2F E, V (^q) = V (^F E ) V (^2F E ) and the squared bootstrap standard errors are used to compute the latter. The p-values from a 2 (1) are 0.79 (Ind _CPI), 0.83 (Ind _PPI), 0.74 (Dev _CPI) and 0.68 (Dev _PPI). 16 The (unreported) outlier-robust t-tests are qualitatively similar. The outlier countries, one or two at most, vary depending on the estimator and so the economic interpretation is not obvious. Details are available upon request. 19

20 5 Conclusion This paper proposes and implements the rst robust tests of general relative PPP, which we have de ned as a long-run unit elasticity of the nominal exchange rate with respect to the price di erential. Our work builds on novel asymptotic theory for cross section and pooled estimators (Phillips and Moon, 1999; Kao, 1999) and Monte Carlo evidence for the mean group estimator (Coakley et al., 2001). These estimators are shown to be consistent for the true parameter in regressions with non-stationary disturbances. The intuition is that by pooling or averaging over countries one can attenuate the e ect of the noise while retaining the strength of the signal. Several estimators are implemented in the empirical analysis to capture di erent aspects of the panel structure of the data such as country heterogeneity and cross-section dependence. The estimator nite sample properties in stylized PPP regressions are analysed using Monte Carlo simulations. In a log-levels equation relating the nominal exchange rate to the relative national price differential, general relative PPP implies a long-run unit slope coe cient with no restrictions on the error term. Measurement errors, transaction costs or limits to arbitrage in foreign exchange markets can make the latter appear observationally equivalent to a unit root sequence. In addition, real exchange rates can be subject to transitory (nominal) or permanent (real) shocks. For this reason we deploy panel estimators that are able to identify the true long run relationship between non-stationary variables even if they do not cointegrate. Our panel framework accommodates heterogeneity and controls from cross-sectional dependence. The latter may arise, by construction, from the variation in the value of the dollar and the US price index (numeraire e ect) and unobserved common macroeconomic variables or shocks. Our empirical analysis is based on a unique large dataset for 19 industrialized and 26 developing countries, 1970:1-1998:12. Both an informal graphical analysis and formal statistical tests unambiguously support the long-run relative PPP hypothesis. We conclude that in ation di erentials are re ected one-for-one in nominal exchange rate depreciation in the long run, in other words that general relative purchasing power parity holds. There are two further implications of our results that are worth drawing out. The rst is that the speed of adjustment towards long-run equilibrium may not be constant not only because 20