Long-Run Purchasing Power Parity with Asymmetric Adjustment.

• Author: Enders, Walter
• Position: Statistical Data Included

Walter Enders [*]

Selahattin Dibooglu [+]

Tests of purchasing power parity (PPP) that use panel data are more supportive of the theory than are bilateral tests. The article uses threshold cointegration to explore long-run PPP. Using data from the post-Bretton Woods period, we show that cointegration with threshold adjustment holds for a number of European countries on a bilateral basis. Focusing on France and Germany as base countries, we show that the error-correction model has important nonlinear characteristics in that prices and the exchange rate have markedly different adjustment patterns for positive gaps from PPP than negative gaps.

1. Introduction

   Despite its intuitive appeal, there is inconclusive evidence supporting purchasing power parity (PPP) between countries with low inflation rates during the post-Bretton Woods period. For example, Enders (1988), Taylor (1988), and MacDonald (1993) use various forms of cointegration tests--called stage-three tests--and find that real exchange rates exhibit large fluctuations with slow rates of decay toward a long-run mean. Froot and Rogoff's (1995) literature review presents a consensus estimate that deviations from long-run PPP have a half-life of about three years. This is somewhat disappointing since stage-three tests seem to be well suited to the task--they require no assumptions concerning exogeneity and they imply a sensible dynamic relationship among price levels and the exchange rate. The vast literature on PPP is indicative of the importance of the issue and the ambiguity of the findings.

   It is well recognized that standard cointegration tests have low power to reject the null hypothesis of no cointegration. This observation is especially relevant for PPP since any mean reversion in real rates is very gradual and the length of the post-Bretton Woods period sample period is relatively short. Efforts to increase the power of unit-root and stage-three tests have had mixed success. For example, Lothian and Taylor (1996) and Mark (1990) have tried to circumvent the low power of stage-three tests by using long spans of time-series data. Unfortunately, the use of long time spans raises the possibility of structural changes occurring during the period being examined. Panel unit-root tests are generally more supportive of PPP than are bilateral tests of real exchange rate behavior. For example, Oh (1996), Wei and Parsley (1995), and Wu (1996) have used panel data in order to enhance the power of standard unit-root tests. Although these particular articles are supportive of PPP, panel studies must deal with the thorny issues of cross-sectional correlation and the choice of the nations to include in the panel. Papell (1997) finds that allowing for serial correlation substantially weakens the evidence in favor of long-run PPP. O'Connell (1998) applies generalized least squares to panel data to eliminate any contemporaneous correlation in the error structure and finds no evidence supporting PPP.

   The second problem with the standard unit-root and cointegration tests is that they implicitly assume symmetric adjustment. However, official intervention in the foreign exchange market means that nominal exchange rate adjustment may be asymmetric. Under a managed float, for example, one of the monetary authorities might be more willing to tolerate currency appreciation than depreciation. Similarly, a currency band mitigates exchange rate movements until the level of the band is altered. Furthermore, the slow adjustment of real exchange rates is often explained by the "stickiness" of national price levels. For example, in the well-known Dornbusch (1976) "overshooting" model, prices and the exchange rate move in the same proportion as the money supply in the very long run. However, in the short run, prices are sticky and monetary shocks cause PPP deviations since the exchange rate moves proportionately more than prices. Rhee and Rich (1995) and Madsen and Yang (1998) provide empirical evidence corroborating th e implications of the asymmetric price adjustment models. The key point to note is that, if prices are primarily sticky in the downward direction, there is no reason to presuppose that real exchange rate adjustment is symmetric.

   In spite of the evidence supportive of asymmetric exchange rate and price adjustments, there are only a few nonlinear tests of PPP. Although they do not explicitly test for PPP, Michael, Nobay, and Peel (1997) and Taylor and Sarno (1998) estimate real exchange rates as smooth-transition threshold adjustment processes. Enders and Falk (1998) formally apply various nonlinear unit-root tests to real exchange rates and find little evidence of PPP. Parsley and Popper (2001) use a large panel that includes a dummy variable representing one of five possible types of exchange rate arrangements. They show that real exchange rates exhibit the greatest degree of mean reversion under a dollar peg and that adjustments are asymmetric.

   Given asymmetric price and/or exchange rate adjustment, the dynamic relationships implicit in testing PPP using the Engle and Granger (1987) and Johansen (1995) methodologies are misspecified. One aim of this article is to reexamine PPP using the Enders and Granger (1998) and Enders and Siklos (2001) threshold unit-root and cointegration tests. We show that allowing for threshold adjustment yields results that are more supportive of PPP than are other bilateral tests. The second aim is to examine the nature of the short-run adjustments toward PPP. If long-run PPP holds and if adjustment is shown to be nonlinear, the half-life of PPP deviations depend on the type of shock initially responsible for the deviation. We show that nominal exchange rate adjustment is quite asymmetric and that price level adjustment, when it occurs, can slow down the return to long-run equilibrium.

2. Threshold and Momentum Models of Cointegration

   The Engle and Granger (1987) methodology as applied to PPP begins by positing a long-run equilibrium relationship of the form

   [e.sub.t] = [[beta].sub.0] + [[beta].sub.1][p.sub.t] + [[beta].sub.2][[p.sup.*].sub.t] + [[micro].sub.t], (1)

   where [e.sub.t] is the logarithm of the nominal exchange rate expressed as units of domestic currency per unit of foreign currency, [p.sub.t] and [[p.sup.*].sub.t] are the logarithms of the domestic and foreign price levels; and [[micro].sub.t] is a stochastic disturbance term.

   The strong version of PPP implies that [[beta].sub.1] = -[[beta].sub.2] = 1, [[beta].sub.0] = 0, and [[micro].sub.t] is stationary. However, the homogeneity restriction [[beta].sub.0] = 0 is often relaxed due to the presence of transportation costs and other possible impediments to trade. Moreover, as shown by Cheung and Lai (1993), the proportionality ([[beta].sub.1] = -[[beta].sub.2] = 1) and symmetry ([[beta].sub.1] = -[[beta].sub.2]) restrictions can be relaxed due to measurement errors. In addition, national price levels and the nominal exchange rate are generally found to be nonstationary so that the estimated coefficients in Equation 1 are biased and do not have the usual t-distribution. For these reasons, cointegration tests of PPP do not usually impose restrictions on the values of the [[beta].sub.i] appearing in Equation 1.

   The next step in the Engle-Granger procedure focuses on the OLS estimate of [rho] in the regression equation,

   [delta][[micro].sub.t] = [rho][[micro].sub.t-1] + [[epsilon].sub.t], (2)

   where the estimated regression residuals from Equation 1 are used to estimate Equation 2.

   Rejecting the null hypothesis of no cointegration (i.e., accepting the alternative hypothesis -2 [less than] [rho] [less than] 0) implies that the residuals in Equation 2 are stationary with mean zero. As such, Equation 1 is an attractor such that its pull is strictly proportional to the absolute value of [[micro].sub.t-1] The change in [[micro].sub.t] equals p multiplied by [[micro].sub.t-1] regardless of whether [[micro].sub.t-1] is positive or negative. [1]

   The implicit assumption of symmetric adjustment is problematic if exchange rate adjustment is asymmetric or if prices are sticky in the downward, but not upward, direction. A formal way to introduce asymmetric adjustment is to let the deviations from the long-run equilibrium in Equation 1 behave as a threshold autoregressive (TAR) process. Thus, it is possible to replace Equation 2 with

   [delta][[micro].sub.t] = [I.sub.t][[rho].sub.1][[micro].sub.t-1] + (1 - [I.sub.t])[[rho].sub.2][[micro].sub.t-1] + [[epsilon].sub.t], (3)

   where [I.sub.t] is the Heaviside indicator such that

   [MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII]

   Asymmetric adjustment is implied by different values of [[rho].sub.1] and [[rho].sub.2]; when [[micro].sub.t-1] is positive, the adjustment is [[rho].sub.1][[micro].sub.t-1], and if [[micro].sub.t-1], is negative, the adjustment is [[rho].sub.2][[micro].sub.t-1]. A sufficient condition for stationarity of {[[micro].sub.t]} is - 2 [less than] ([[rho].sub.1], [[rho].sub.2]) [less than] 0. Moreover, if the {[[micro].sub.t]} sequence is stationary, the least squares estimates of [[rho].sub.1] and [[rho].sub.2] have an asymptotic multivariate normal distribution if the value of the threshold is known (or consistently estimated). Thus, if the null hypothesis [[rho].sub.1] = [[rho].sub.2] = 0 is rejected, it is possible to...