Nonlinear purchasing power parity under the Gold Standard.

1. Introduction

Recent theoretical analysis of purchasing power deviations (see, e.g., Dumas 1992; Sercu, Uppal, and Van Hull 1995; and O'Connell and Wei 1997) demonstrates how transactions costs or the sunk costs of international arbitrage induce nonlinear adjustment of the real exchange rate to purchasing power parity (PPP). Globally mean-reverting this nonlinear process has the important property of exhibiting near unit root behavior for small deviations from PPP since small deviations from PPP are left uncorrected if they are not large enough to cover the transactions costs or the sunk costs of international arbitrage.

A parametric nonlinear model, suggested by the theoretical literature, that captures the nonlinear adjustment process in aggregate data is the exponential smooth transition autoregression model (ESTAR) of Ozaki (1985). A smooth, rather than discrete, adjustment mechanism is motivated by the theoretical analysis of Dumas (1992). Also, as postulated by Terasvirta (1994) and demonstrated theoretically by Berka (2002), in aggregate data regime changes may he smooth, rather than discrete, given that heterogeneous agents do not act simultaneously even if they make dichotomous decisions. (1) Recent empirical work (e.g., Michael, Nobay, and Peel 1997; Taylor, Peel, and Sarno 2001, Peel and Venetis 2002) has reported empirical results that suggest that the ESTAR model provides a parsimonious fit into a variety of data sets, particular for monthly data for the interwar and postwar floating period as well as for annual data spanning 200 years, as reported in Lothian and Taylor (1996). In addition, nonlinear impulse response functions derived from the ESTAR models show that although the speed of adjustment for small shocks around equilibrium will be highly persistent, larger shocks mean-revert much faster than the glacial rates previously reported for linear models (Rogoff 1996). In this respect, the ESTAR models provide some solution to the PPP puzzle outlined in Rogoff (1996). (2)

The ESTAR model can also provide an explanation of why PPP deviations analyzed from a linear perspective appear to be described by either a nonstationary integrated I(1) process, or alternatively, described by fractional processes (see, e.g., Diebold, Husted, and Rush 1991). Taylor, Peel, and Sarno (2001), and Pippenger and Goering (1993) show that the Dickey-Fuller tests have low power against data simulated from an ESTAR model. Michael, Nobay, and Peel (1997) and Byers and Peel (2003) show that data that is generated from an ESTAR process can appear to exhibit the fractional property. That this would be the case was an early conjecture by Acosta and Granger (1995). Given that the ESTAR model has a theoretical rationale, whereas the fractional process is a relatively nonintuitive one, the fractional property might reasonably be interpreted as a misleading linear property of PPP deviations (Granger and Terasvirta 1999).

Whereas the empirical work employing ESTAR models provides some explanation of the glacially slow adjustment speeds obtained in linear models, there is one aspect of the empirical work that is worthy of further attention. A second way of explaining the Rogoff puzzle, raised by Rogoff himself, (3) is to relax the assumption that the equilibrium real exchange rate is a constant (see, e.g., Canzoneri, Cumby, and Diba 1996; and Chinn and Johnston 1996). Theoretical models, such as that of Balassa (1964) and Samuelson (1964), imply a nonconstant equilibrium in the real exchange rate if real productivity growth rates differ between countries. (4) Nonlinear models that incorporate proxies fur these effects are found to parsimoniously fit post-Bretton Woods data for the main real exchange rates (see Venetis, Paya, and Peel 2002; and Paya, Venetis, and Peel 2003). Naturally, models that ignore this effect may generate misleading speeds of PPP adjustment to shocks. In this regard, the empirical results of Hegwood and Papell (2002) for the Gold Standard period are particularly interesting. Balassa-Samuelson effects are one of the major arguments for the numerous equilibrium mean shifts found in Hegwood and Papell (2002) for the real exchange rates in the 16 real exchange rate series analyzed in Diebold, Husted, and Rush (1991) for the period 1792-1913 under the Gold Standard. Hegwood and Papell (2002) assume linear adjustment around an occasionally changing equilibrium determined on the basis of the Bai-Perron (1998) test for multiple structural breaks. They report that quick mean reversion around an occasionally changing mean provides a more reasonable representation of the data than does fractional integration, which was originally reported by Diebold, Husted, and Rush (1991) for their data set. They conclude that long-ran PPP (LRPPP) does not hold but instead it is quasi-PPP (QPPP) theory-the one supported by their analysis of the data. They also state that the slow convergence of LRPPP is due to the unaccounted mean shifts in the equilibrium rate and that a reduction of more than 50% is achieved in the half-lives of shocks when those shifts are included in the model.

These results are potentially important and provide motivation for our study. Hegwood and Papell (2002) only consider the impacts of structural breaks in the context of linear adjustment. In this article, we further examine the real exchange rate adjustment mechanism in the 19th and early 20th centuries under the Gold Standard by employing an ESTAR framework that allows for both a constant and structural breaks in the equilibrium real rate. Because the gold standard era was a high point of international cooperation (Diebold, Husted, and Rush 1991, p. 1254) and it was a symmetric arrangement (both parts were committed to maintain parities), the symmetric nonlinear ESTAR model is an appropriate model of real exchange rate behavior at that time. We find that ESTAR models incorporating the structural breaks employed by Hegwood and Papell (2002) provide a parsimonious explanation of the data. We determine the significance of the structural breaks via bootstrap and Monte Carlo analysis. We then investigate the speeds of adjustment obtained from nonlinear impulse response functions in these models and compare them to the estimated models that exclude structural breaks. Our results provide further support, on a new data set, for the hypothesis that real exchange rates are stationary, symmetric, nonlinear processes that reverted in this time period to a changing equilibrium real rate. The half-life of shocks implied by the nonlinear impulse response functions were found to be dramatically faster than those obtained in models that do not include the breaks. Clearly, our results support those of Hegwood and Papell (2002).

The rest of the article is organized as follows. In section 2, we discuss the ESTAR model considered in our empirical applications and report empirical estimates of ESTAR models where the real exchange rate long run path is modeled both as a variable or a constant. Section 3 presents the Monte Carlo simulation exercise for the confidence interval of the statistics. Section 4 presents the results of the estimated impulse response functions for the nonlinear models. Finally, section 5 summarizes our main conclusions.

2. Nonlinear PPP

We analyze properties of a set of currencies (dollar, pound, Deutsche mark. French franc, Belgian franc, and Swedish krona) spanning the period 1792-1913. We use the same data set as in Diebold, Husted, and Rush (1991) and Hegwood and Papell (2002). (5) This data set includes 10 real exchange rates using wholesale price index (WPI) as the deflator of the nominal exchange rate, and six real exchange rates that use the consumer price index (CPI) as the deflator. We normalize all of the real rates so that the first observation is set equal to zero. Hegwood and Papell (2002) could not reject the null of a unit root on the basis of the augmented Dickey-Fuller (ADF) tests for 11 out of the 16 series. Six additional series reject the null of unit root when they apply the Perron-Vogelsang (1992) test for unit root while allowing a single mean shift in the data. The five remaining real exchange rates reject the unit root test in favor of a fractionally integrated alternative as reported in Diebold, Husted. and Rush (1991). These results are all consistent with the possibility that the real exchange rate followed an ESTAR process as noted in the previous section. (6)

We assume the true data-generating process for the PPP deviations ([y.sub.t]) modified for structural breaks has the simple form of the ESTAR model reported in Taylor, Peel, and Sarno (2001) and Paya, Venetis, and Peel (2003):

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [y.sub.t] is the real exchange rate ([y.sub.t] = [s.sub.t] - [p.sub.t] + [p.sup.*.sub.t]), [s.sub.t] is the logarithm of the spot exchange rate, [p.sub.t] is the logarithm of the domestic price level, and [p.sup.*.sub.t] the logarithm of the foreign price level; [alpha] is the constant equilibrium level of the real exchange rate, [gamma] is a positive constant (the speed of adjustment), [[beta].sub.i] are constants, [d.sub.1] ... [d.sub.n] are dummies for shifts in the equilibrium rate, and [u.sub.t] is a random disturbance term. (7)

The first model we estimate is the simple ESTAR model for the real exchange rate with a constant equilibrium ([d.sub.1] ... [d.sub.n] are set to zero). Tables 1 and 2 present the results of the estimations for Wholesale Price Index (WPI) and Consumer Price Index (CPI) real rates, respectively. The speed of adjustment parameter is significant in all cases except for the Belgium/U.S. and Belgium/Germany WPI rates and the Belgium/Germany CPI rate. The autoregressive structure of the ESTAR model (the [[beta].sub.i]) varies from an AR(1) to an AR(3). (8) Given the significance of [gamma] and that in all cases we cannot reject that the sum of the autoregressive terms adds up to 1, we impose this constraint in all estimations. (9)

Hegwood and Papen (2002), on the basis of the Bat and Perron (1998) test for multiple mean shifts, provide evidence that the real exchange rates do not exhibit a constant conditional mean for the whole sample, but instead they follow a mean reversion process to a changing mean. We include the same dummies that they found significant in the equilibrium process of the real exchange rates. (10) Tables 3 and 4 present the results for the estimation of the ESTAR model with changing equilibrium rates. Some of the initial dummies appeared to be insignificant when the real rates were allowed to follow a nonlinear mean-reverting process. We then removed the dummies that were insignificant in the new estimations. (11) The last column of Tables 3 and 4 display the F-test for the joint significance of all remaining dummies. In all cases, we can reject the null that all dummies were insignificant when taken all together, except in the case of the France/Sweden CPI real exchange rate. However, the residuals do exhibit significant nonnormality, and in this case the distribution of the statistics could follow nonclassical forms within a nonlinear framework. Consequently, we employ a bootstrap method in order to obtain appropriate test statistics.

3. Robustness Analysis

Our null hypothesis is that the dummy variables for breaks have zero coefficients. Accordingly we simulate an "artificial" series for [y.sub.t] [??].sub.t] given the estimates of [alpha] and [gamma] obtained in Tables 3 and 4, with the coefficients on the dummy variables for structural breaks set to zero. The residuals [u.sup.b] are obtained from bootstrapping, with replacement, the estimated residuals obtained from the ESTAR models reported in those tables that include the dummies. (12) The resulting "artificial" series are given by

(2) [[??].sub.t] = [??] + [e.sup.[??]([[??].sub.t-1]-[[??]).sup.2] ([[??].sub.t-1] - [??]) + [u.sub.b].

We then estimate the nonlinear ESTAR model including the pertinent dummies in each case, and we repeat this experiment 10,000 times. The distribution of the t-statistics are computed as well as the distribution of the F-test for each real exchange rate. Tables 5 and 6 present the 90% and 95% confidence intervals for the t-statistics of each dummy and the F-test. On the basis of the F-statistics obtained in the nonlinear estimation and the critical values from the bootstrap, we can reject the null of joint nonsignificance of the dummies in all cases except for the France/Sweden and France/Germany CPI real rates. With regard to the particular dummy variables, some of them cannot be considered significant within this framework. (13) Hegwood and Papell (2002) provide some historical support for some of the dummies they found significant in their study. Our results support the significance of most of those dummies: the 1864 dummy in the U.S. real exchange rate coinciding with the American Civil War, the 1866 dummy in the German real exchange rates coinciding with the dissolution of the German Confederation, and the dummies of the 1940s when there was widespread protest, rebellion, and revolution in Europe (Cook and Stevenson 1998, p. 460).

4. Nonlinear Haft-Lives of Shocks

In this section we compare the speed of mean reversion of the nonlinear model of real exchange rates with constant equilibrium as well as comparing shifting equilibrium with the speed of mean reversion of the linear model as in Hegwood and Papell (2002). To calculate the half-lives of PPP deviations within the nonlinear framework, we must take into account that a number of properties of the impulse response functions of linear models do not carry over to the nonlinear models. (14) Koop, Pesaran, and Potter (1996) introduced the generalized impulse response function (GIRF) for nonlinear models. The GIRF is defined as the average difference between two realizations of the stochastic process {[y.sub.1]+h]}, which start with identical histories up to time t - 1 (initial conditions). The first realization is hit by a shock at time t, whereas the other one is not:

(3) GIR[F.sub.h] (h, [delta], [[omega].sub.t-1]) = E([y.sub.t+h] [u.sub.t] = [delta], [[omega].sub.t-1]) - E ([y.sub.t+h] / [u.sub.t] = 0, [[omega].sub.t-1]),

where h = 1, 2, ..., denotes horizon, [u.sub.t] = [sigma] is an arbitrary shock occurring at time t; and [[omega].sub.t-1] defines the history set of [y.sub.t].

Given that [delta] and [omega.sub.t-1] are single realizations of random variables, Equation 3 is considered to be a random variable. In order to obtain sample estimates of Equation 3, we average out the effect of all histories [[omega].sub.t-1]- that consist of every set ([y.sub.t-1] ..., [y.sub.t-p]) for t [greater than or equal to] p + 1, where p is the autoregressive lag length, and we also average out the effect of future shocks [u.sub.t+h]. (15) We set [sigma] = 5%, 10%, 20%, 30%, and 40%. The different values of [sigma]s would allow us to compare the persistence of very large and small shocks. As in Taylor, Peel, and Sarno (2001) and in Paya, Venetis, and Peel (2003), Tables 7 and 8 report the half-lives of shocks; that is, the time needed for [GIRF.sub.h] < 1/2 [sigma]. (16) The last columns of both Tables 7 and 8 correspond to the speeds of adjustment found in the linear models of Hegwood and Papell (2002). For the nonlinear models with constant equilibrium (Table 7), the half-life of the shocks decreases significantly for shocks of around 30%, or even 40% in some cases. However, compared with the linear case, even with the smallest shock of 5% the speed of mean reversion is faster in the ESTAR model. When the equilibrium is allowed to change over time (Table 8) the arbitrage band in the nonlinear model seems to lie around 20%, or even 10%. In this case, 10 out of the 16 real exchange rates need a shock of 20% to achieve faster adjustment than the linear case.

5. Conclusions

Hegwood and Papell (2002) analyzed PPP adjustment under the Gold Standard. They allowed for novel structural breaks in the equilibrium real exchange rate and suggested that relatively quick linear adjustment to an occasionally changing mean provides a more reasonable representation of the data than does the fractional process, with its long memory property, for their data set--the latter was originally reported by Diebold, Husted, and Rush (1991).

In this article we have shown that the theoretically well-motivated nonlinear ESTAR process. embodying structural breaks in the equilibrium real rate, provides a parsimonious explanation of the data set. As conjectured by Rogoff (1996) and supported by our analysis and that of Hegwood and Papell (2002), allowance for a changing real equilibrium can have dramatic implications for the estimates of PPP adjustment speeds. On the basis of these results, empirical work exploring this possibility in postwar data may help solve the Rogoff puzzle.

Table 1. ESTAR Estimations of Wholesale Price Index (WPI) Real
Exchange Rates, 1792-1973

WPI               [[??].sub.0]  [[??].sub.1]    [[??].sub.2]

United States/       -0.076         1.28      1 - [[??].sub.1]
  United Kingdom     (0.019)       (0.13)
United States/       -0.033         1.09      1 - [[??].sub.1]
  Germany            (0.031)       (0.08)
Germany/             -0.21          1.25      -0.56
  United Kingdom     (0.039)       (0.09)     (0.16)
France/               0.008         1.14      1 - [[??].sub.1]
  United Kingdom     (0.014)       (0.10)
France/              -0.141         1.03      1 - [[??].sub.1]
  United States      (0.017)       (0.10)
France/Germany        0.186         1.23       0.48
                     (0.027)       (0.06)     (0.096)
Belgium/              0.262         1
  United Kingdom     (0.026)
Belgium/France        0.197         1
                     (0.019)

WPI                        [[??].sub.3]              [??]     s

United States/                                      -3.12   0.075
  United Kingdom                                    (0.90)
United States/                                      -2.52   0.095
  Germany                                           (0.71)
Germany/          1 - [[??].sub.1] - [[??].sub.2]   -1.52   0.076
  United Kingdom                                    (0.68)
France/                                            -11.86   0.057
  United Kingdom                                    (1.03)
France/                                             -7.42   0.060
  United States                                     (1.86)
France/Germany    1 - [[??].sub.1] - [[??].sub.2]   -2.55   0.051
                                                    (0.98)
Belgium/                                            -2.49   0.042
  United Kingdom                                    (0.95)
Belgium/France                                      -2.98   0.047
                                                    (0.94)

WPI               [R.sup.2]  q = 1   q = 4   q = 12

United States/      0.71      0.70    0.89    0.69
  United Kingdom
United States/      0.65      0.72    0.32    0.37
  Germany
Germany/            0.80      0.42    0.80    0.75
  United Kingdom
France/             0.57      0.95    0.40    0.75
  United Kingdom
France/             0.70      0.93    0.80    0.93
  United States
France/Germany      0.89      0.16    0.26    0.55

Belgium/            0.88      0.16    0.30    0.49
  United Kingdom
Belgium/France      0.87      0.31    0.31    0.22

Numbers in parentheses are the Newey-West (1987: NW) standard error
estimates. s denotes residuals standard error. The test for
autocorrelation for lags 1, 4, and 12 denotes the p-values of the
Eitrheim and Terasvirta (1996) Lagrange Multiplier test for
autocorrelation in nonlinear series.

Table 2. ESTAR Estimations of Consumer Price Index (CPI) Real
Exchange Rates, 1792-1913

CPI             [[??].sub.0]  [[??].sub.0]    [[??].sub.0]

Sweden/Germany    -0.123         1.26       1 - [[??].sub.1]
                  (0.039)       (0.085)
France/Sweden      0.001         1.51       1 - [[??].sub.1]
                  (0.010)       (0.11)
France/Germany    -0.004         1.23       1 - [[??].sub.1]
                  (0.037)       (0.12)
France/Belgium     0.189           1
                  (0.023)
Belgium/Sweden    -0.154         0.94       1 - [[??].sub.1]
                  (0.018)       (0.10)

CPI             [[??].sub.3]   [??]      s    [R.sup.2]

Sweden/Germany                 -2.41   0.074    0.80
                               (1.23)
France/Sweden                 -13.77   0.033    0.82
                               (2.45)
France/Germany                 -7.17   0.081    0.70
                               (4.90)
France/Belgium                 -4.57   0.045    0.83
                               (1.54)
Belgium/Sweden                 -4.75   0.045    0.83
                               (1.40)

CPI                q = 1   q = 4   q = 12

Sweden/Germany      0.32    0.16    0.22

France/Sweden       0.12    0.39    0.89

France/Germany      0.34    0.22    0.52

France/Belgium      0.36    0.84    0.99

Belgium/Sweden      0.21    0.08    0.29

Numbers in parentheses are standard error estimates. s denotes the NW
residuals standard error. The test for autocorrelation for lags 1, 4,
and 12 denotes the p-values of the Eitrheim and Terasvirta (1996)
Lagrange Multiplier test for autocorrelation in nonlinear series.

Table 3. ESTAR Estimations of Wholesale Price Index (WPI) Real
Exchange Rates, 1792-1913 with Dummies

WPI               [[??].sub.0]  [[??].sub.1]  [[??].sub.2]

United States/       -0.124         0.228
  United Kingdom     (0.024)       (0.028)
United States/        0.027        -0.137         0.113
  Germany            (0.031)       (0.037)       (0.036)
Germany/             -0.068        -0.134        -0.237
  United Kingdom     (0.022)       (0.042)       (0.027)
France/              -0.028         0.098        -0.042
  United Kingdom     (0.001)       (0.021)       (0.017)
France/              -0.220         0.136         0.063
  United States      (0.014)       (0.037)       (0.022)
France/Germany        0.054         0.313         0.203
                     (0.020)       (0.025)       (0.030)
Belgium/              0.398        -0.320        -0.226
  United Kingdom     (0.017)       (0.020)       (0.018)
Belgium/             -0.002         0.295         0.251
  United States      (0.045)       (0.026)       (0.008)
Belgium/Germany       0.073         0.096
                     (0.023)       (0.030)
Belgium/France        0.292        -0.246
                     (0.012)       (0.017)

WPI               [[??].sub.3]  [[??].sub.1]    [[??].sub.2]

United States/                      1.28      1 - [[??].sub.1]
  United Kingdom                   (0.15)
United States/        0.157         1.07      1 - [[??].sub.1]
  Germany            (0.043)       (0.088)
Germany/             -0.091         1.27      1 - [[??].sub.1]
  United Kingdom     (0.029)       (0.15)
France/               0.032         1.32      1 - [[??].sub.1]
  United Kingdom     (0.013)       (0.10)
France/               0.230         1.02      1 - [[??].sub.1]
  United States      (0.026)       (0.08)
France/Germany        0.097         1.22      1 - [[??].sub.1]
                     (0.028)       (0.07)
Belgium/             -0.08          1.09      1 - [[??].sub.1]
  United Kingdom     (0.018)       (0.12)
Belgium/                              1
  United States
Belgium/Germany                     1.27      1 - [[??].sub.1]
                                   (0.12)
Belgium/France                        1

WPI                   [??]          s    [R.sup.2]

United States/      -4.28 (1.21)  0.071    0.75
  United Kingdom
United States/      -4.43 (0.78)  0.088    0.71
  Germany
Germany/            -8.50 (2.18)  0.075    0.81
  United Kingdom
France/            -30.9 (9.21)   0.039    0.65
  United Kingdom
France/            -34.1 (15.9)   0.050    0.81
  United States
France/Germany     -27.5 (1.75)   0.046    0.91

Belgium/          -186.7 (64.7)   0.036    0.91
  United Kingdom
Belgium/            -0.88 (0.32)  0.061    0.69
  United States
Belgium/Germany     -8.96 (3.32)  0.055    0.65

Belgium/France     -53.3 (17.73)  0.043    0.90

WPI                  q = 1   q = 4   q = 12     F

United States/        0.27    0.55    0.52
  United Kingdom
United States/        0.43    0.39    0.61     7.33
  Germany
Germany/              0.08    0.04    0.20     5.76
  United Kingdom
France/               0.40    0.95    0.74     8.84
  United Kingdom
France/               0.11    0.11    0.29    15.6
  United States
France/Germany        0.17    0.77    0.53    10.35

Belgium/              0.88    0.79    0.74    10.38
  United Kingdom
Belgium/              0.15    0.64    0.46     6.99
  United States
Belgium/Germany       0.79    0.62    0.74

Belgium/France        0.70    0.76    0.81

Numbers in parentheses are standard error estimates. s denotes
the residuals standard error.

Table 4. ESTAR Estimations of Consumer Price Index (CPI) Real
Exchange Rates, 1792-1913 with Dummies

CPI              [[??].sub.0]  [[??].sub.1]  [[??].sub.2]

Sweden/Germany      -0.278         0.312         0.173
                    (0.017)       (0.028)       (0.028)
France/Sweden        0.000        -0.023         0.015
                    (0.007)       (0.012)       (0.008)
France/Germany      -0.030         0.241         0.048
                    (0.033)       (0.064)       (0.025)
France/Belgium       0.197        -0.015
                    (0.024)       (0.007)
Belgium/Sweden      -0.273         0.264         0.180
                    (0.009)       (0.011)       (0.011)
Belgium/Germany     -0.732         0.604         0.453
                    (0.015)       (0.017)       (0.033)

CPI              [[??].sub.3]  [[??].sub.4]  [[??].sub.5]

Sweden/Germany

France/Sweden

France/Germany

France/Belgium

Belgium/Sweden       0.078
                    (0.012)
Belgium/Germany      0.192         0.073         0.026
                    (0.040)       (0.029)       (0.022)

CPI              [[??].sub.1]    [[??].sub.2]      [??]     S

Sweden/Germany       1.43      1 - [[??].sub.1]  -17.41   0.060
                    (0.13)                        (4.69)
France/Sweden        1.50      1 - [[??].sub.1]  -21.18   0.033
                    (0.11)                        (4.35)
France/Germany       1.31           -0.62         -7.63   0.074
                    (0.09)          (0.13)        (4.25)
France/Belgium       1                            -5.20   0.046
                                                  (2.11)
Belgium/Sweden       1.16      1 - [[??].sub.1]   -2.55   0.037
                    (0.12)                       (71.8)
Belgium/Germany      1.39      1 - [[??].sub.1]  -19.88   0.062
                    (0.12)                        (5.67)

CPI              [R.sup.2]  q = 1  q = 4  q = 12    F

Sweden/Germany     0.88      0.28   0.15   0.00   22.80

France/Sweden      0.82      0.11   0.35   0.91    0.93

France/Germany     0.75      0.10   0.23   0.62    3.88

France/Belgium     0.85      0.27   0.71   0.99

Belgium/Sweden     0.89      0.30   0.67   0.59   13.16

Belgium/Germany    0.93      0.51   0.63   0.99    7.96

Numbers in parentheses are standard error estimates. s denotes the
residuals standard error. The test for autocorrelation denotes the
p-values of the Eitrheim and Terasvirta (1996) LM test. The
[[??].sub.3] coefficient in the France/Germany rates equals
1 - [[??].sub.1] - [[??].sub.2].

Table 5. Bootstrap Confidence Interval for t-Stat and F-Stat for
Dummies (a)

Wholesale Price Index               D1               D2

United States/United Kingdom
  90%                         (-3.2, 3.1)
  95%                         (-4.4, 4.0) *
United States/Germany
  90%                         (-2.8, 2.8)      (-2.5, 2.5)
  95%                         (-3.5, 3.4) *    (-3.2, 3.2) *
Germany/United Kingdom
  90%                         (-3.9, 4.0)      (-3.45, 3.40)
  95%                         (-4.9, 5.0)      (-4.35, 4.25) *
United States/France
  90%                         (-3.10, 3.35) *  (-3.30, 3.05)
  95%                         (-4.05, 4.80)    (-4.40, 4.15)
France/United Kingdom
  90%                         (-2.24, 2.36)    (-1.88, 2.10) *
  95%                         (-3.01, 3.08) *  (-2.34, 3.45)
France/Germany
  90%                         (-3.25, 2.36)    (-2.85, 2.65)
  95%                         (-4.25, 3.25) *  (-3.80, 3.45) *
Belgium/United Kingdom
  90%                         (-2.46, 2.36)    (-2.3, 2.1)
  95%                         (-3.02, 2.92) *  (-2.9, 2.7) *
Belgium/United States
  90%                         (-2.65, 2.55)    (-2.55, 2.40)
  95%                         (-3.32, 3.22) *  (-2.95, 3.00) *
Belgium/Germany
  90%                         (-2.48, 2.18)
  95%                         (-3.22, 3.02) *
Belgium/France
  90%                         (-1.60, 1.60)
  95%                         (-1.94, 1.92) *

Wholesale Price Index               D3           F

United States/United Kingdom
  90%
  95%
United States/Germany
  90%                         (-2.36, 2.1)     3.34
  95%                         (-3.0, 2.6) *    4.41 *
Germany/United Kingdom
  90%                         (-3.10, 3.15) *  2.84
  95%                         (-3.90, 4.0)     3.60 *
United States/France
  90%                         (-2.85, 2.45)    3.60
  95%                         (-4.20, 3.15)    6.00 *
France/United Kingdom
  90%                         (-2.01, 1.98) *  3.96
  95%                         (-2.75, 2.69)    5.02 *
France/Germany
  90%                         (-2.60, 2.25)    3.45
  95%                         (-3.35, 3.00) *  4.70 *
Belgium/United Kingdom
  90%                         (-2.2, 2.1)      4.55
  95%                         (-2.8, 2.6) *    5.55 *
Belgium/United States
  90%                                          3.71
  95%                                          4.75 *
Belgium/Germany
  90%
  95%
Belgium/France
  90%
  95%

(a) * denotes significant dummy variable reported in Table 3.

Table 6. Bootstrap Confidence Interval for t-Stat and F-Stat for
Dummies (a)

Consumer Price         D1               D2               D3
Index

Sweden/Germany
  90%            (-2.28, 2.91)    (-2.40, 2.40)
  95%            (-2.86, 3.72) *  (-3.00, 3.05) *
France/Sweden
  90%            (-2.60, 2.55)    (-2.20, 2.12)
  95%            (-3.30, 3.20)    (-2.66, 2.70)
France/Germany
  90%            (-2.40, 2.38)    (-2.35, 2.35)
  95%            (-3.10, 3.02) *  (-3.25, 3.05)
France/Belgium
  90%            (-3.15, 3.15)
  95%            (-4.15, 4.01)
Belgium/Sweden
  90%            (-2.10, 2.10)    (-2.05, 2.10)    (-2.04, 2.05)
  95%            (-2.60, 2.65) *  (-2.60, 2.70) *  (-2.60, 2.70) *
Belgium/Germany
  90%            (-2.80, 2.95)    (-3.00, 2.80)    (-2.95, 2.70)
  95%            (-3.46, 3.78) *  (-3.84, 3.60) *  (-3.60, 3.30) *

Consumer Price         D4               D5           F
Index

Sweden/Germany
  90%                                              3.25
  95%                                              4.25 *
France/Sweden
  90%                                              3.24
  95%                                              4.32
France/Germany
  90%                                              5.25
  95%                                              7.40
France/Belgium
  90%
  95%
Belgium/Sweden
  90%                                              3.60
  95%                                              4.55 *
Belgium/Germany
  90%            (-2.80, 2.55) *  (-2.55, 2.45)    2.50
  95%            (-3.45, 3.20)    (-3.20, 3.00)    3.12 *

(a) * denotes significant dummy variable reported in Table 4.

Table 7. Estimated Half-Lives Shocks in Months for Annual Model

                                                Shock

                                [??]   5%  10%  20%  30%  40%  Linear

Real rare WPI
  United States/United Kingdom  -3.12  5    4    3    2    2    4.04
  United States/Germany         -2.52  4    4    4    3    2    3.24
  Germany/United Kingdom        -1.52  6    6    5    2    2    5.72
  France/United Kingdom         -11.8  5    4    3    I    0    3.22
  France/United States          -7.42  3    3    2    1    0    7.85
  France/Germany                -2.89  5    4    4    2    1    5.77
  Belgium/United Kingdom        -2.49  7    7    7    6    4    9.24
  Belgium/France                -2.98  5    5    5    4    2    9.66

Real rate CPI
  Sweden/Germany                -2.41  5    5    5    4    3    5.99
  France/Sweden                 -13.7  4    4    2    1    0    5.24
  France/Germany                -7.17  3    3    2    2    2    4.08
  France/Belgium                -4.53  5    5    4    3    3    7.31
  Belgium/Sweden                -4.75  5    4    3    1    0    7.40

Table 8. Estimated Half-Lives Shocks in Months for Annual Model
with Dummies

                                                  Shock

                                 [??]    5%  10%  20%  30%  40%  Linear

Real rate WPI
  United States/United Kingdom    -4.28  5    4    3    2    1    2.51
  United States/Germany           -4.43  3    3    3    2    1    1.24
  Germany/United Kingdom          -8.5   3    3    2    1    0    1.25
  France/United Kingdom          -30.7   3    2    0    0    0    1.42
  France/United States           -34.1   2    2    0    0    0    2.54
  France/Germany                 -27.5   2    2    1    0    0    1.42
  Belgium/United Kingdom        -186     1    0    0    0    0    0.83
  Belgium/United States           -0.88  6    5    4    3    2    1.68
  Belgium/Germany                 -8.96  4    4    3    2    1    1.26
  Belgium/France                 -56.9   1    0    0    0    0    1.43

Real rate CPI
  Sweden/Germany                 -17.4   3    2    1    0    0    1.13
  France/Germany                  -7.63  2    1    1    1    0    1.13
  Belgium/Sweden                -255     1    0    0    0    0    0.61
  Belgium/Germany                -20     2    2    1    0    0    1.07

(1) Even in high-frequency asset markets, the idea of heteregeneous traders facing different capital constraints or percieved risk of arbitrage has been employed to rationalize employment of the ESTAR model. See, for example, Tse (2001) for arbitrage between stock and index futures.

(2) Namely, how to reconcile the enormous short-run volatility of real exchange rates with the extremely slow rate at which shocks appear to damp out (in linear models, around 3-5 years, which seems far too long to be explained by nominal rigidities).

(3) He suggests for instance that the sustained post-Bretton Woods war appreciation of Japan's real exchange rate against the dollar is consistent with the Balassa-Samuelson (BS) effects, in fact, he calls it the canonical example of BS effects.

(4) We know from the analysis of Taylor (2001) that if the true data generation process is nonlinear, then the use of the linear models can severely underestimate the speed of adjustment, particularly if the tow frequency data is temporally aggregated.

(5) We thank Hegwood and Papell for kindly providing as with the data.

(6) A key property of some ESTAR models (also shared by some threshold models) is that data simulated from them, although globally mean-reverting, can appear to exhibit a unit root. As a consequence, the test proposed in Froot and Rogoff (1995)--namely, that we impose unit coefficients and test directly, employing unit root tests, whether PPP deviations are mean-reverting--can have low power if the true data-generating process is nonlinear.

(7) We note in this article, as pointed out by a referee, that we test for nonlinear mean reversion while assuming that PPP holds. In this article, it is assumed that there is reversion to a changing mean and that what is being tested is the form of the reversion. As a consequence, the standard errors reported for the ESTAR have classical values. There is a common misconception that research on nonlinear adjustment to PPP tests a unit root null against a nonlinear mean-reverting alternative.

(8) This variation is not surprising if we assume that the true data generating process (DGP) of the real exchange rate is generated at a higher frequency, that is, monthly. In that case, Paya and Peel (2003) show that temporal aggregation of the true monthly DGP into, for instance, annual data induces additional autoregressive terms in the ESTAR model.

(9) We observe from Equation 1 that when ([n.summation over (i=1)] [[beta.sub.i] =1 if [??] = 0 PPP deviations follow a unit root.

(10) See Hegwood and Papell (2002) for an explanation of potential causes of the different breaks. We recognize that the breaks were obtained from estimates of a linear structure. Our Monte Carlo evidence suggests the breaks are in the majority significant.

(11) In particular, for the WPI rates we removed the third dummy of the Belgium/Germany, third dummy in the Belgium/U.S., fourth dummy in the France/U.K., and the fourth dummy in the U.S./Germany rate. For the CPI rates, we removed the first dummy of the France/Sweden rate and the third dummy of the Sweden/Germany rate.

(12) The bootstrapped residuals were centered on zero and scaled. The scaling factor is (n/n-k[)[??].sub.0.5].

(13) In particular, the first dummy of the Germany/U.K. WPI rate, second dummy of the France/U.S. WPI rate, second dummy of the France/U.K. WPI rate, second dummy of the France/Germany CPI rate, fifth dummy of the Belgium/Germany rate, and both dummies of the France/Sweden CPI rate.

(14) In particular, impulse responses produced by nonlinear models are (i) history dependent, so they depend on initial conditions, (ii) dependent on the size and sign of the current shock, and (iii) depend on the future shocks as well. That is, nonlinear impulse responses critically depend on the "past, present, and the future."

(15) For each available history, we use 300 repetitions (500 repetitions found similar result) to average out future shocks where future shocks are drawn with replacement from the models' residuals, and then we average the result across all histories. We set to max{h} = 48.

(16) The France/Sweden and France/Belgium CPI exchange rates have not been included in Table 8 because the dummy variables were not jointly significant according to the bootstrap confidence interval presented in Table 6. In the case of the France/ Germany CPI rate, we include the dummy that appears significant under the bootstrap confidence interval.

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Ivan Paya, Departamento Fundamentos Analisis Economico, University of Alicante, 03080 Alicante, Spain; E-mail ivanpaya@ merlin.fae.ua.es; corresponding author.

David A. Peel, Lancaster University Management School, Lancaster, LA1 4YX, UK; E-mail [e-mail].

The authors are grateful to participants of the Understanding Evolving Macroeconomy Annual Conference, University College Oxford. September 15-16 2003. The first author acknowledges financial support of Instituto Valenciano Investigaciones Economicas (VIE).

Received June 2003: accepted December 2003.