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Long-Run Purchasing Power Parity with Asymmetric Adjustment.

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 test for symmetric adjustment (i.e., [[rho].sub.1] = [[rho].sub.2]), using a standard F-test. Since adjustment is symmetric if [[rho].sub.1] = [[rho].sub.2] the Engle-Granger test for cointegration is a special case of Equation 3.

Since the exact nature of the nonlinearity may not be known, it is also possible to allow the adjustment to depend on the change in [[micro].sub.t-1] (i.e., [delta][[micro].sub.t-1]) instead of the level of [[micro].sub.t-1]. In this case, the Heaviside indicator of Equation 4 becomes

[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (5)

Enders and Granger (1998) and Enders and Siklos (2001) show that this specification is especially relevant when the adjustment is such that the series exhibits more momentum in one direction than the other; the resulting model is called the momentum-threshold autoregressive (M-TAR) model. [2] Respectively, the F-statistics for the null hypothesis [[rho].sub.1] = [[rho].sub.2] = 0 using the TAR specification of Equation 4 and the M-TAR specification of Equation 5 are called [[phi].sub.[micro]] and [[[phi].sup.*].sub.[micro]]. Because there is generally no presumption regarding whether to use Equation 4 or Equation 5, the recommendation is to select the adjustment mechanism by a model selection criterion such as the Akaike Information Criterion (AIC).

If the errors in Equation 3 are serially correlated, it is possible to use an augmented threshold model for the residuals. In this circumstance, Equation 3 is replaced by

[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] + [[[sigma].sup.p].sub.i=1] [[beta].sub.i] [delta] [[micro].sub.t-i] [[epsilon].sub.t]. (6)

The distributions of [[phi].sub.[micro]] and [[[phi].sup.*].sub.[micro]] depend on the number of observations, the number of lags in Equation 6, and the number of variables in the cointegrating relationship. The empirical F-distribution for the null hypothesis [[rho].sub.1] = [[rho].sub.2] = 0 is tabulated by Enders and Siklos (2001). Table 1 reports the appropriate critical values for both [[phi].sub.[micro]] and [[[phi].sup.*].sub.[micro]] for the case of three variables in the cointegrating relationship. [3]

3. Empirical Results

We test for long-run PPP with asymmetric adjustment using data from France and Germany as the base countries. In particular, data from Italy, Austria, Belgium, Denmark, Netherlands, and Switzerland against the German mark and the French franc are considered. We found no additional evidence of PPP by including the United States, Japan, and the United Kingdom in the analysis. This is an interesting result in itself. Nevertheless, we focus on these seven countries to better examine the nonlinear aspects of the adjustment process (after all, there is no adjustment whatsoever if PPP fails). We measure [e.sub.t] by nominal, period average exchange rates and both prices (i.e, [p.sub.t] and [[p.sub.t].sup.*]) by the consumer price index. Monthly data from 1973.1 through 1997.1 are taken from the CD-ROM edition of the International Financial Statistics. [4]

Alternatively, using Germany and France as the base countries, we estimated Equation 1 using ordinary least squares (OLS) and saved the residuals in the sequence {[[micro].sub.t],}. For each type of asymmetry, we set the indicator function [I.sub.t] according to Equation 4 or Equation 5 and estimated an equation in the form of Equation 6. The AIC was used to select the most appropriate lag length p and adjustment mechanism (i.e., TAR versus M-TAR adjustment). [5] The sample value of the F-statistic for the null hypothesis [[rho].sub.1] = [[rho].sub.2] = 0 was compared with the appropriate critical value reported in Table 1. If the alternative hypothesis is accepted (i.e., the null of no cointegration is rejected), we then used Chan's (1993) methodology to find the consistent estimate of the threshold. After all, in the presence of measurement errors and/or adjustment costs, there is no reason to presume that the threshold is identically equal to zero. Once the threshold is properly estimated, we test for symm etric versus asymmetric adjustment (i.e., we test the null hypothesis [[rho].sub.1] = [[rho].sub.2]) using the usual F-statistic. Note that Hansen (1997) shows that small-sample properties of the OLS estimates of the individual [[rho].sub.1] and [[rho].sub.2] have inflated standard errors and the convergence properties of the OLS estimates can be poor. Hence, standard methods of inference concerning the individual values of [[rho].sub.1] and [[rho].sub.2] are problematic.

Table 2 reports the estimated values of the [[rho].sub.i] and the sample F-statistics for the null hypothesis [[rho].sub.1] = [[rho].sub.2] = 0 using the lag length and adjustment mechanism selected by the AIC. [6] The top portion of the table reports results for Germany and the lower portion of the table reports results for France. Comparing the estimated values of [[phi].sub.[micro]] (or [[[phi].sub.[micro]].sup.*]) with those reported in Table 1, it is clear that there is strong evidence of PPP for Germany with France, Italy, Austria, and the Netherlands. If we use the 10% significance level, there is also support for PPP between Germany and Switzerland.

In the Germany--France case, for example, the point estimates of the p, are both negative and the [[phi].sub.[micro]]-statistic for the null hypothesis [[rho].sub.1] = [[rho].sub.2] = 0 of 16.70 far exceeds the 1% critical value reported in Table 1. Given that we reject the null hypothesis of no cointegration, the point estimates [[rho].sub.1] and [[rho].sub.2] imply a reasonable amount of asymmetry--the real rate converges by 13.7% of a positive deviation from PPP and by 8.6% of a negative deviation.

The lower part of Table 2 reports the results using France as the base country. At the 1% significance level, the asymmetric model supports PPP for France with Germany (see the top portion of the table), Austria, and the Netherlands. If we use the 10% significance level, PPP also holds for France with Belgium, Denmark, and Switzerland.

One problem with the estimates presented in Table 2 is they constrain the threshold to equal zero. Consider modifying Equations 4 and 5 with the specifications

[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (7)

In either form of Equation 7, the threshold [tau] can be estimated data using Chan's (1993) method to find a consistent estimate of the threshold. For each of the 10 country pairs found to support PPP (see Table 2), we sorted the residual series in ascending order such that [[[micro].sup.[tau]].sub.1] [less than] [[[micro].sup.[tau]].sub.2] [less than]...[less than] [[[micro].sup.[tau]].sub.T], where T denotes the number of usable observations. The largest and smallest 15% of the {[[[micro].sup.[tau]].sub.i]} values were discarded and each of the remaining 70% of the values were considered as possible thresholds. For each of these possible thresholds, we estimated an equation in the form of Equations 6 and 7. The estimated threshold yielding the lowest residual sum of squares was deemed to be the appropriate estimate of the threshold.

As shown on the right-hand side of Table 3, the case for asymmetric adjustment is substantially strengthened when a consistent estimate of the threshold is used. For all 10 cases except Germany-Austria and France-Belgium, the null hypothesis of symmetric adjustment (i.e., [[rho].sub.1] = [[rho].sub.2]) is soundly rejected at the 5% significance level. For example, the point estimates of [[rho].sub.1] and [[rho].sub.2] for Germany-France are -0.161 and -0.081 and the F-statistic for the null hypothesis [[rho].sub.1] = [[rho].sub.2] has a p-value of 0.018.

For comparison purposes, we estimated the type of linear adjustment mechanism employed in the Engle-Granger procedure (see Eqn. 2). For each equation, the estimated value of [rho] and the associated AIC statistic is reported on the left-hand side of Table 3. Note that the null hypothesis [rho] = 0 cannot be rejected for Germany-Italy, Germany-Switzerland, France-Denmark, and France-Switzerland at the 10% significance level. Hence, the Engle-Granger linear methodology indicates that PPP fails in these four cases. Also note that the AIC selects the asymmetric model over the linear adjustment model for all cases except Germany-Austria.

Given that there is no presumption as to whether [[rho].sub.1] should be greater or smaller than [[rho].sub.2] it is also interesting to note that the speed of adjustment is always fastest when the real value of the mark is above its long-run equilibrium value. [7] No such pattern holds for France. Nevertheless, the entries in Table 3 do not indicate how the adjustment to long-run equilibrium occurs. In the next section, we examine the extent to which nominal exchange rates and national price levels respond to deviations from long-run PPP.

4. The Error-Correction Representation: Asymmetric Versus Symmetric Adjustment

Having found evidence supporting asymmetric adjustment, we present estimates of the error-correction models using consistent estimates of the thresholds. We used Chan's method to find the consistent estimate of the threshold for each of the three equations in the error-correction model. [8] For each model, we used the multivariate AIC to select the most appropriate lag length. Perhaps the most pronounced difference concerning the assumption of symmetric versus asymmetric adjustment occurs in the case of Switzerland-Germany. For these two countries, the estimated long-run PPP relationship is

[e.sub.t] = l.57[p.sub.t] - l.99[[p.sup.*].sub.t] + 1.75, (8)

where [e.sub.t] = log of the Swiss/German exchange rate, [p.sub.t] = log of the Swiss price level, and [[p.sup.*].sub.t] = log of the German price level.

Using Equation 8, the estimated error-correction equations using consistent estimates of the threshold are (with t-statistics in parentheses)

[delta][e.sub.t] = [A.sub.11](L)[delta][p.sub.t-1] + [A.sub.12](L)[delta][[p.sup.*].sub.t-1] + [A.sub.13](L)[delta][e.sub.t-1] - 0.062z_[plus.sub.t-1] - 0.151z_[minus.sub.t-1] (9)

(-2.80) (-4.34)

[delta][[p.sup.*].sub.t] = [A.sub.21](L)[delta][p.sub.t-1] + [A.sub.22](L)[delta][[p.sup.*].sub.t-1] + [A.sub.23](L)[delta][e.sub.t-1] - 0.001z_[plus.sub.t-1] - 0.012z_[minus.sub.t-1] (10)

(-0.129) (-2.24)

[delta][p.sub.t] = [A.sub.31](L)[delta][p.sub.t-1] + [A.sub.32](L)[delta][[p.sup.*].sub.t-1] + [A.sub.33](L)[delta][e.sub.t-1] + 0.024z_[plus.sub.t-1] - 0.007z_[minus.sub.t-1], (11)

(2.71) (-1.05)

where z_[plus.sub.t] = [I.sub.t]([e.sub.t] - 1.57[p.sub.t] + 1.99[[p.sup.*].sub.t] - 1.75), z_[minus.sub.t] = (1 - [I.sub.t])([e.sub.t] - 1.57[p.sub.t] + 1.99[[p.sup.*].sub.t] - 1.75), [I.sub.t] = the indicator function found by applying Chan's (1993) method to each equation, and [A.sub.ij](L) is a polynomial in the lag operator L.

The key feature in Equations 8-11 is the pattern of the estimated coefficients for z-plus and z-minus. In Equation 9, the Swiss franc-German mark rate falls (rises) whenever it lies above (below) its long-run PPP level. The point estimates imply that the exchange rate adjusts by 6.2% of a positive gap from long-run PPP and by 15.1% of a negative gap. The t-statistics imply that both coefficients are significant at conventional levels. Moreover, Equation 10 indicates that the German price level responds to negative, but not positive, deviations from PPP, whereas Equation 11 indicates that the Swiss price level responds to positive, but not negative, deviations. Thus, in response to a positive deviation from long-run PPP--say as a result of a positive German price level shock--the Swiss price level rises and the Swiss franc appreciates. Initially, less than 10% (0.062 + 0.024 [less than] 0.10) of the gap is eliminated. Instead, if there is a negative deviation from long-run PPP, the exchange rate eliminates abo ut 15% of the gap. As such, positive deviations from PPP have much longer half-lives than negative deviations. By way of contrast, the symmetric error-correction model for Switzerland-Germany is

[delta][e.sub.t] = [A.sub.11](L)[delta][p.sub.t-1] + [A.sub.12](L)[delta][[p.sup.*].sub.t-1] + [A.sub.13](L)[delta][e.sub.t-1] - 0.082[ec.sub.t-1] (-3.68) (12)

[delta][[p.sup.*].sub.t] = [A.sub.21](L)[delta][p.sub.t-1] + [A.sub.22](L)[delta][[p.sup.*].sub.t-1] + [A.sub.23](L)[delta][e.sub.t-1] + 0.003[ec.sub.t-1] (0.443) (13)

[delta][p.sub.1] = [A.sub.31](L)[delta][p.sub.t-1] + [A.sub.32](L)[delta][[p.sup.*].sub.t-1] + [A.sub.33](L)[dleta][e.sub.t-1] - 0.007[ec.sub.t-1], (-1.45) (14)

where [ec.sub.t] = ([e.sub.t] - l.57p, + l.99[[p.sup.*].sub.t] -- 1.75).

In the case of symmetric adjustment, only the error-correction term on the exchange rate is significant at conventional levels. The model implies that the exchange rate--but neither the German nor the Swiss price level--adjusts to eliminate deviations from PPP. In spite of the extra coefficient appearing in each equation of the threshold model, the multivariate AIC selects the asymmetric model over the symmetric model. The multivariate AIC is -8843.34 for the system given by Equations 9-11 and -8833.59 for the system given by Equations 12-14.

Table 4 reports the estimated error-correction terms for countries displaying PPP with Germany and Table 5 reports the error-correction terms for France. Examination of these tables reveals that the linear Swiss-German error-correction model reported above is quite representative of all the linear estimates. The linear estimates reported in the left-hand portions of the tables reveal that, in 9 out of the 10 cases (the exception being Italy-Germany), the nominal exchange rate adjusts to eliminate deviations from long-run equilibrium. Moreover, at the 5% significance level, the linear error-correction model indicates that there is little evidence of appropriate price level changes in order to correct short-term movements in real exchange rates. The sole exception is the case of Switzerland-France: Table 5 indicates that the Swiss price level changes by 1.4% of any deviation from PPP. (Although the error-correction term in the equation for [delta][p.sup.*] is significant in the case of Austria-Germany, the esti mated direction of change actually slows down the adjustment process.)

The threshold error-correction estimates, shown on the right-hand portions of Tables 4 and 5, tell a different story. Although the nominal exchange rate does most of the adjustment, the asymmetries found for nominal exchange rate adjustments are usually quite sharp. Even in the instances where both error-correction terms are statistically significant, the differences between [[rho].sub.1] and [[rho].sub.2] are large. Moreover, the asymmetric price adjustment estimations provide limited support for the notion that price levels react to close a gap between the real rate and its longrun value. In 5 of the 10 instances, the t-statistic for z_[plus.sub.t-1] in the equation for [delta][p.sub.t] is positive and exceeds 2.0. In no instance is the t-statistic on z[minus.sub.t-1] greater than 2.0. This is consistent with the notion that price levels rise more readily than they decline. However, no similar pattern emerges for the German or French price levels. In fact, when such a price adjustment equation contains a s ignificant error-correction term, it is often of a very small magnitude and the direction of change may act to perpetuate the deviation from PPP.

In addition to the Swiss-German case reported above, there are four other country pairs of interest where exchange rates and prices adjust consistently to correct deviations from PPP. The France--Germany and Italy--Germany cases are interesting because of the economic importance of these countries. The Belgium--France and Switzerland-France cases are of interest because of their close economic ties and because of the pattern of the error-correction coefficients.

France-Germany

Table 3 indicates that the real franc/mark exchange rate adjusts more rapidly to positive than negative deviations from long-run equilibrium. Recall that positive deviations from long-run equilibrium can be caused by decreases in the French price level (p) or by increases in the German price level ([p.sup.*]) and the nominal franc/mark rate (e). Regardless of the initial source of the discrepancy, Table 4 shows that the adjustment is done primarily by movements in the nominal exchange rate. If the real rate is above its long-run equilibrium value, the nominal franc appreciates by 14.8% of the discrepancy and the French price level rises by 1.8% of the discrepancy. However, the nominal franc appreciates only 6.8% of a negative discrepancy from long-run PPP and any price level adjustments are statistically insignificant.

Italy--Germany

Adjustment for the real lira/mark rate is similar to that for the real franc/mark rate. The nominal lira/mark rate falls by almost 21% of a positive discrepancy from long-run equilibrium but adjusts by 3.2% of a negative discrepancy. Price level adjustments are very small. When the real lira/mark rate is below its long-run equilibrium value, the Italian and German price levels adjust by about 1% of the discrepancy.

Belgium--France

The results in Table 3 suggest that there is symmetric adjustment in the Belgiuml/France case. However, the individual error-correction equations reveal a much more complicated dynamic process. Table 5 indicates that the nominal Belgian franc appreciates when the real Belgian franc is below (but not above) its long-run value. Instead, both price levels adjust when the Belgian franc is above (but not below) its long-run value. The point estimate is such that the nominal Belgian franc appreciates by 12.6% of any negative discrepancy from PPP. If there is a positive discrepancy, Belgium's price level rises by 3.4% of the gap and the French price level falls by 1.8% of the gap.

Switzerland-France

When the Swiss franc is undervalued relative to its purchasing power parity, the Swiss price level rises by 3.3% of the discrepancy with no significant adjustment in the Swiss franc or the French price level. When the Swiss franc is below its long-run value, the adjustment is carried Out through exchange rate changes: The Swiss franc closes 17.2% of the discrepancy.

5. Conclusions

In contrast with tests using panel data, bilateral tests of long-run PPP using a linear cointegration framework generally reject PPP for the industrialized nations during the post-Bretton Woods period. Implicit in such tests is the symmetry assumption that positive deviations from PPP are corrected in the same manner as negative deviations. However, the fact that central banks attempt to influence certain types of exchange rate movements and not others would seem to be a prima facie case against any type of symmetric adjustment. Similarly, there is evidence that national price levels increase more readily than they decrease. Using a bilateral cointegration test that has enhanced power in the presence of asymmetric adjustment, we find reasonable support for long-run PPP for Germany and France with other European nations. We also compare the estimates of linear and asymmetric adjustment error-correction models. It is shown prices and exchange rates have markedly different adjustment patterns for positive deviat ions from PPP than for negative deviations.

(*.) Department of Economics, Finance and Legal Studies, Culverhouse College of Business, University of Alabama, Tuscaloosa, AL 35487, USA; E-mail wenders@cba.ua.edu; corresponding author.

(+.) Department of Economics, Southern Illinois University at Carbondale, Carbondale, IL 62901, USA; E-mail dibo@siu.edu.

We thank two anonymous referees for their helpful suggestions.

(1.) Similarly, the Johansen (1995) methodology uses the specification [delta][x.sub.t], = [pi][x.sub.t-1] + [v.sub.t], where [x.sub.t], is the (3 X 1) vector ([e.sub.t] [p.sub.t] [[p.sup.*].sub.t])', [pi] is a (3 X 3) matrix, and [v.sub.t], is a (3 X 1) vector of stationary disturbances that may he contemporaneously correlated. The crucial point to note is that the alternative hypothesis (i.e., rank[pi] [not equal to] 0) implicity assumes a symmetric adjustment process around [x.sub.t] = 0 in that, for any [x.sub.t], [not equal to] 0, [delta][x.sub.t+1] always equals [pi][x.sub.t].

(2.) Hansen (1997) presents a purely statistical argument for M-TAR adjustment. If {[[micro].sub.t-1]} is a near unit root process, setting the Heaviside indicator using [delta][[micro].sub.t-1] can perform better than the specification using pure TAR adjustment.

(3.) The critical values for the case of three variables in the cointegrating vector are not reported in the final version of Enders and Siklos (2001). Table 1 was constructed from their earlier working papers.

(4.) Given monetary union in Europe, we did not extend the estimation past 1997:1.

(5.) In using model selection criteria to select lag lengths, it is necessary to compare the alternative models over the same sample period. As such, we estimated Equation 3 for lags of 1-12 over the sample period, 1974:2-1997:1, and chose the lag length selected by the AIC. The results reported in the tables are the estimates using the longest possible sample period. In two instances, the use of the full sample period resulted in a situation such that the t-statistic for the last lag was not significant at the 10% significance level. In these two instances, the lag length was shortened by one.

(6.) Schwartz Bayesian criteria always chooses a lag length of unity. In an earlier version of the article, we reported results with a lag length of unity. These results are available from the authors on request. With one lag, we did not find evidence of PPP in the Germany-Austria and France-Belgium cases.

(7.) Even if price adjustment is asymmetric, there is no particular reason for positive discrepancies from PPP to be more persistent than negative discrepancies. After all, if the French franc/German mark real rate adjusts quickly when it is above its long-run equilibrium value, the German mark/French franc rate quickly adjusts when it is below its long-run equilibrium value.

(8.) We also estimated each error-correction model (1) setting the value of the threshold equal to zero and (2) setting the threshold equal to the value found for the {[delta][[micro].sub.t]} equation of Table 3. Results using the zero threshold value are quite similar to those reported here. However, method (2) often resulted in estimates that did not fit the data well. The implication is that the threshold for { [delta][[micro].sub.t]} need not be identical to those for {[delta][p.sub.t]}, {[[p.sup.*].sub.t] and {[delta][e.sub.t]}. All results are available from the authors.

References

Chan, K. S. 1993. Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. The Annals of Statistics 21:520-33.

Cheung, Y., and K. S. Lai. 1993. Long-run purchasing power parity during the recent float. Journal of International Economics 34:181-92.

Dornbusch, R. 1976. Expectations and exchange rate dynamics. Journal of Political Economy 84:1161-76.

Enders, Walter. 1988. ARIMA and cointegration tests of PPP under fixed and flexible exchange rate regimes. Review of Economics and Statistics 70:504-8.

Enders, Walter. and Barry Falk. 1998. Threshold-autoregressive, median unbiased, and cointegration tests of purchasing power parity. International Journal of Forecasting 14:171-86.

Enders, Walter, and C. W. J. Granger. 1998. Unit-root tests and asymmetric adjustment with an example using the term structure of interest rates. Journal of Business and Economic Statistics 16:304-1l.

Enders, Walter, and Pierre L. Siklos. 2001. Cointegration and threshold adjustment. Journal of Business and Economic Statistics 19:166-77.

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Froot, Kenneth A., and Kenneth Rogoff. 1995. Perspectives on PPP and long-run real exchange rates. In Handbook of international economics. Volume 3, edited by Gene M. Grossman and Kenneth Rogoff. Amsterdam: Elsevier, 1647-88.

Hansen, Bruce. 1997. Inference in TAR Models. Studies in Nonlinear Dynamics and Econometrics 1:119-31.

Johansen, Soren. 1995. Likelihood-based inference in cointegrated autoregressive models. Oxford: Oxford University Press.

Lothian, J. R., and Taylor, Mark. 1996. Real exchange rate behavior: The recent float from the perspective of the past two centuries. Journal of Political Economy 104:488-510.

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Michael, Panos, A. R. Nobay, and David Peel. 1997. Transaction costs and non-linear adjustments in real exchange rates: An empirical investigation. Journal of Political Economy 105:862-79.

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Papell, David. 1997. Searching for stationarity: Purchasing power parity under the current float. Journal of International Economics 43:313-32.

Parsley, Davis, and Hellen Popper. 2001. Official exchange rate arrangements and real exchange rate behavior. Journal of Money, Credit and Banking. In press.

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Taylor, Mark. 1988. An empirical examination of long-run purchasing power parity using cointegration techniques. Applied Economics 20:1369-81.

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Wei. S. J., and D. Parsley. 1995. Purchasing power dis-parity during the floating rate period: Exchange rate volatility, trade barriers and other culprits. NBER Working Paper No. 5032.

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Table 1.

The TAR Model: [[phi].sub.[micro]]

Distribution for the F-Statistic for the Null Hypothesis, [[rho].sub.1]
= [[rho].sub.2] = 0

 No Lagged Changes One Lagged Change
Observed 90% 95% 99% 90% 95%

 50 6.53 7.80 10.90 6.24 7.49
100 6.35 7.53 9.94 6.34 7.47
250 6.29 7.44 9.61 6.27 7.30
500 6.24 7.32 9.70 6.23 7.32

 Four Lagged Changes
Observed 99% 90% 95% 99%

 50 10.15 5.71 7.79 9.17
100 10.00 6.24 7.36 9.79
250 9.73 5.22 7.32 9.69
500 9.64 6.20 7.30 9.64

The M-TAR Model:
 [[phi].sup.*].sub.[micro]]

 No Lagged Changes One Lagged Change
Observed 90% 95% 99% 90% 95%

 50 6.98 8.30 11.4 6.72 7.92
100 6.85 8.03 10.6 6.82 7.97
250 6.78 7.80 10.2 6.55 7.96
500 6.67 7.80 10.2 6.67 7.81

 Four Lagged Changes
Observed 99% 90% 95% 99%

 50 10.66 6.88 7.91 10.55
100 10.60 6.69 7.86 10.35
250 10.28 6.66 7.80 10.26
500 10.14 6.64 7.75 10.15
Table 2

The Estimated Adjustment Equations

Germany as the base country

Country [[rho].sub.1] [[rho].sub.2] [[phi].sub.[micro]] [a]

 France -0.137 -0.086 16.70 [***]
 Italy -0.104 -0.020 10.65 [***]
 Austria -0.084 -0.064 8.60 [**]
 Belgium -0.028 -0.017 2.43
 Denmark -0.040 -0.089 6.50
 Netherlands -0.163 -0.037 17.26 [***]
 Switzerland -0.054 -0.082 6.84 [*]

Country Lags [b] Flag

 France 10 TAR
 Italy 1 M-TAR
 Austria 12 TAR
 Belgium 1 TAR
 Denmark 1 M-TAR
 Netherlands 6 TAR
 Switzerland 1 M-TAR

France as the base country

Country [[rho].sub.1] [[rho].sub.2] [[phi].sub.[micro]] [a]

 Italy -0.051 -0.036 4.68
 Austria -0.109 -0.064 11.13 [***]
 Belgium -0.065 -0.081 7.72 [*]
 Denmark -0.031 -0.083 6.58 [*]
 Netherlands -0.121 -0.050 14.00 [***]
 Switzerland -0.097 -0.037 7.36

Country Lags [b] Flag

 Italy 3 TAR
 Austria 9 M-TAR
 Belgium 9 M-TAR
 Denmark 1 TAR
 Netherlands 10 M-TAR
 Switzerland 6 M-TAR

(a)Entries are the sample values of [[phi].sub.[micro]] or
[[[phi].sup.*].sub.[micro]] for the adjustment process shown in column
6.

(*), (**), and (***) indicate significance at the 10, 5, and 1% levels,
respectively.

(b)Entries in this column are the number of lags of
{[delta][[micro].sub.t]}.
Table 3.

Estimated Adjustment Equations Using the Consistent Estimate of the
Threshold

Germany as the base country

 Linear Adjustment Threshold Adjustment
 [rho] [a] AIC [[rho].sub.1]

 France -0.106 [***] -1004.38 -0.161
 (-5.54) (-5.37)
 Italy -0.054 -688.92 -0.250
 (-3.62)
 Austria -0.075 [*] -1816.90 -0.087
 (-4.09) (-3.38)
 Netherlands -0.097 [***] -1533.39 -0.206
 (-4.58) (-6.74)
 Switzerland -0.068 -800.73 -0.262
 (-3.62) (-3.42)

 Threshold Adjustment
 [[rho].sub.2] [[phi].sub.[micro]] [b]

 France -0.081 18.44
 (-3.75)
 Italy -0.017 25.80
 (-1.14)
 Austria -0.062 8.72
 (-2.59)
 Netherlands -0.034 22.80
 (-1.38)
 Switzerland -0.055 10.10
 (-2.86)

 Threshold Adjustment
 [[rho].sub.1] = [[rho].sub.2] [c] AIC Flag

 France 0.018 -1008.28 TAR

 Italy 0.000 -721.95 M-TAR

 Austria 0.416 -1815.60 TAR

 Netherlands 0.000 -1554.00 TAR

 Switzerland 0.009 -805.52 M-TAR


France as the base country

 Linear Adjustment Threshold Adjustment
 [rho] [a] AIC [[rho].sub.1]

 Austria -0.089 [**] -988.45 -0.177
 (-4.53) (-5.99)
 Belgium -0.072 [*] -1016.92 -0.064
 (-3.90) (-3.32)
 Denmark -0.051 -1045.44 -0.025
 (-3.23) (-1.31)
 Netherlands -0.094 [***] -1019.92 -0.121
 (-4.80) (-5.36)
 Switzerland -0.070 -765.51 -0.054
 (-3.46) (-2.55)

 Threshold Adjustment
 [[rho].sub.2] [[phi].sub.[micro]] [b]

 Austria -0.043 18.46
 (-1.93)
 Belgium -0.140 8.68
 (-2.77)
 Denmark -0.104 8.20
 (-3.84)
 Netherlands -0.042 14.49
 (-1.42)
 Switzerland -0.170 8.22
 (-3.26)

 Threshold Adjustment
 [[rho].sub.1] = [[rho].sub.2] [c] AIC Flag

 Austria 0.000 -1001.90 M-TAR

 Belgium 0.152 -1017.06 M-TAR

 Denmark 0.017 -1049.22 TAR

 Netherlands 0.019 -102.64 M-TAR

 Switzerland 0.038 -767.94 M-TAR


(a)With three variables in the cointegrating relationship, Phillis and
Outliaris (1990) find the critical values of the t-statistic for the
null hypothesis [rho] = 0 to be -4.73, -4.11, and -3.83 at the 1, 5, and
10% significance levels, respectively. For this column, we let (*),
(**), and (***) indicate significance at the 10, 5, and 1% levels,
respectively.

(b)Entries in this column are the sample values of [[phi].sub.[micro]]
or [[[phi].sup.a].sub.[micro]], depending on the adjustment process
shown in the last column of the table. These values utilize the
consistent estimates of the threshold and, as such, should not be
compared to the values in Table 1.

(c)Entries in this column are the p-values for the sample F-statistics
for the null hypothesis that the adjustment coefficients are equal.
t-statistics are in parentheses.
Table 4.

The Estimated Error-Correction Models forGermany

France

 Linear model Threshold Adjustment
 [rho] [[rho].sup.1]

 [delta]e -0.103 -0.148
 (-4.69) (-4.55)
 [delta]p 0.004 0.018
 (0.683) (2.47)
 [delta][p.sup.*] 0.000 0.006
 (0.027) (0.743)

 Threshold Adjustment
 [[rho].sup.2] Threshold ([tau])

 [delta]e -0.068 0.019
 (-2.40)
 [delta]p -0.010 -0.033
 (-1.43)
 [delta][p.sup.*] -0.006 -0.033
 (-0.703)

Italy

 Linear model Threshold Adjustment
 [rho] [[rho].sup.1]

 [delta]e -0.060 -0.208
 (-0.364) (-5.66)
 [delta]p 0.004 -0.007
 (0.978) (-1.11)
 [delta][p.sup.*] -0.005 0.002
 (-1.87) (0.488)

 Threshold Adjustment
 [[rho].sup.2] Threshold ([tau])

 [delta]e -0.032 0.021
 (-1.92)
 [delta]p 0.008 0.066
 (1.88)
 [delta][p.sup.*] -0.008 0.000
 (-2.76)

Austria

 Linear model Threshold Adjustment
 [rho] [[rho].sup.1]

 [delta]e -0.072 -0.037
 (-3.79) (-1.38)
 [delta]p -0.022 0.040
 (-0.635) (0.824)
 [delta][p.sup.*] 0.064 0.096
 (2.27) (2.86)

 Threshold Adjustment
 [[rho].sup.2] Threshold ([tau])

 [delta]e -0.133 -0.002
 (-3.45)
 [delta]p -0.132 0.004
 (-1.88)
 [delta][p.sup.*] 0.008 -0.004
 (0.170)

Netherlands

 Linear model Threshold Adjustment
 [rho] [[rho].sup.1]

 [delta]e -0.178 -0.043
 (-4.41) (-1.24)
 [delta]p 0.043 0.111
 (1.81) (2.95)
 [delta][p.sup.*] 0.025 0.063
 (1.13) (2.40)

 Threshold Adjustment
 [[rho].sup.2] Threshold ([tau])

 [delta]e -0.185 -0.028
 (-5.57)
 [delta]p -0.005 0.012
 (-0.151)
 [delta][p.sup.*] -0.017 0.024
 (-0.632)

Switzerland

 Linear model Threshold Adjustment
 [rho] [[rho].sup.1]

 [delta]e -0.082 -0.062
 (-3.68) (-2.80)
 [delta]p 0.003 0.024
 (0.444) (2.71)
 [delta][p.sup.*] -0.007 -0.001
 (-1.45) (-0.129)

 Threshold Adjustment
 [[rho].sup.2] Threshold ([tau])

 [delta]e -0.151 -0.034
 (-4.34)
 [delta]p -0.007 0.006
 (-1.05)
 [delta][p.sup.*] -0.012 0.009
 (-2.24)

(t)-Statistics are in parentheses, [p.sup.*] is the log of the German
price level, p is the log of the home country's price level, and e is
the log of the number of units of the home currency/German mark.
Table 5.

The Estimated Error-Correction Models for France

Austria

 Linear model Threshold Adjustment
 [rho] [[rho].sup.1] [[rho].sup.2]

[delta]e -0.093 -0.178 -0.050
 (-4.83) (-6.12) (-2.32)
[delta]p 0.006 0.015 -0.020
 (0.871) (1.75) (-1.31)
[delta][p.sup.*] 0.000 -0.003 0.008
 (0.021) (-0.586) (0.999)

 Threshold
 Adjustment
 Threshold([tau])

[delta]e 0.005

[delta]p -0.008

[delta][p.sup.*] -0.007


Belgium

 Linear model Threshold Adjustment
 [rho] [[rho].sup.1] [[rho].sup.2]

[delta]e -0.121 -0.034 -0.126
 (-5.45) (-0.524) (-5.66)
[delta]p 0.012 0.034 0.008
 (1.72) (3.10) (0.871)
[delta][p.sup.*] -0.005 -0.018 -0.001
 (-0.995) (-2.23) (-0.145)

 Threshold
 Adjustment
 Threshold([tau])

[delta]e 0.018

[delta]p 0.005

[delta][p.sup.*] 0.004


Denmark

 Linear model Threshold Adjustment
 [rho] [[rho].sup.1] [[rho].sup.2]

[delta]e -0.055 -0.001 -0.121
 (-3.30) (-0.043) (-4.01)
[delta]p -0.003 0.016 -0.030
 (-0.259) (1.12) (-1.66)
[delta][p.sup.*] 0.009 0.044 -0.037
 (1.80) (5.63) (-3.91)

 Threshold
 Adjustment
 Threshold([tau])

[delta]e 0.031

[delta]p -0.024

[delta][p.sup.*] 0.013


Netherlands

 Linear model Threshold Adjustment
 [rho] [[rho].sup.1] [[rho].sup.2]

[delta]e -0.098 -0.141 -0.019
 (-4.95) (-6.24) (-0.669)
[delta]p 0.003 0.014 -0.017
 (0.007) (1.89) (-1.80)
[delta][p.sup.*] -0.000 0.004 -0.012
 (-0.071) (0.701) (-1.50)

 Threshold
 Adjustment
 Threshold([tau])

[delta]e 0.000

[delta]p 0.000

[delta][p.sup.*] -0.002


Switzerland

 Linear model Threshold Adjustment
 [rho] [[rho].sup.1] [[rho].sup.2]

[delta]e -0.056 -0.039 -0.172
 (-2.67) (-1.76) (-3.43)
[delta]p 0.014 0.033 -0.000
 (2.67) (4.62) (-0.033)
[delta][p.sup.*] -0.002 0.001 -0.005
 (-0.641) (0.274) (-1.81)

 Threshold
 Adjustment
 Threshold([tau])

[delta]e -0.030

[delta]p 0.005

[delta][p.sup.*] 0.004


t-Statistics are in parentheses, [p.sup.*] is the log of the French
price level, p is the log of the home country's price level, and e is
the log of the number of units of the home currency/franc.
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