Journal of International Money and Finance

19 (2000) 759764
www.elsevier.nl/locate/econbase

Purchasing power parity over two centuries:

strengthening the case for real exchange rate
stability
A reply to Cuddington and Liang
James R. Lothian a, Mark P. Taylor b, c,*

a
Graduate School of Business, Fordham University, 113 West 60th Street, Room 616, New York,
NY 10023, USA
b
Warwick Business School, University of Warwick, Coventry CV4 7AL, UK
c
Centre for Economic Policy Research, London, UK

Abstract

Cuddington and Liang (2000) [Purchasing power parity over two centuries? Journal of Inter-
national Money and Finance, 19, 751755] examine the long span of sterlingdollar real exchange
rate data of Lothian and Taylor (1996) [Real exchange rate behavior: the recent float from the
perspective of the past two centuries. Journal of Political Economy, 104, 488509] and claim to
reject long-run purchasing power parity by fitting time trends or by considering very high-order
autoregresssive representations. This reply demonstrates, however, that the central claims of Loth-
ian and Taylor are in fact strengthened by the implications of Cuddington and Liangs analysis
in that, while the economic importance of introducing trend terms is slight, this leads to a faster
estimated speed of mean reversion. 2000 Elsevier Science Ltd. All rights reserved.

JEL classification: F31; F41; C22

Keywords: Real exchange rates; Purchasing power parity

1. Introduction

We welcome the comment of Cuddington and Liang in this issue (hereafter CL)
on our 1996 paper on real exchange rates as contributing to the spirit of debate

* Corresponding author. Tel.: +44-24-7657-2832; fax: +44-24-7657-3013.

E-mail address: mark.taylor@wbs.warwick.ac.uk (M.P. Taylor).

0261-5606/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 2 6 1 - 5 6 0 6 ( 0 0 ) 0 0 0 3 0 - 9
760 J.R. Lothian, M.P. Taylor / Journal of International Money and Finance 19 (2000) 759764

which surrounds all worthwhile scientific endeavor. However, far from conceding
that Cuddington and Liang have dealt the knock-out blow to real exchange rate
stability over the past two centuries, it is clear to us that the central claims of Lothian
and Taylor (1996) (hereafter LT) are in fact strengthened by the implications of
CLs analysis.

2. Methodological issues

The central message of the CL analysis is that, in an analysis of our sterlingdollar

real exchange rate data over the period 17911990, if very long lags are considered in
the augmented DickeyFuller (ADF) auxiliary regressions, then the resulting ADF
statistics do not enable rejection of the unit root hypothesis at the 5% significance
level, while if very short lags are considered, then there are significant time trends.
CL conclude that this is evidence against long-run purchasing power parity (PPP).
In this section we shall briefly mention a number of methodological issues which
might be raised in this context.

2.1. Implausible lag lengths

Starting from a lag length of 15 years and sequentially testing down by looking
at the t-ratio of the coefficient of the longest lag, CL choose a lag length of 14 years
for the ADF statistic and find that the unit root hypothesis cannot then be rejected
at the 5% level. Since this involves 14 lags of the change in the real exchange rate,
this implies an AR(15) representation for the real exchange rate. This seems to us
both statistically and economically implausible.
It is statistically implausible because the sample autocorrelation and partial auto-
correlation functions for the sterlingdollar real exchange reported in LT (in Fig. 4
of that paper) reveal no evidence of serial correlation beyond AR(1).1
It also seems economically implausible why would any one entertain the idea
of adjustment lags in real exchange rates spreading over such a long period, some
15 years? CL argue that the longer lag lengths are perhaps not implausible given the
persistence in real exchange rates, citing Rogoff (1996). This does not seem to us,
however, a strong argument, since one should distinguish between the amount of
time taken for the full effects of a shock to be felt and the amount of time for which
a shock persists. To take an extreme example, with a random walk process the full
effects of a shock are felt after only one period but persist forever. With an AR(15)
unit root representation seriously, however, it would take 15 years for the full effects
of the real exchange rate shock to be felt, after which they would persist forever.

1
One way in which a high-order autoregressive representation might arise is through the presence of
moving average components, which are essentially inverted. This is not evident, however, either in the
sample partial autocorrelation function (LT, Fig. 4) or from direct estimation of moving average compo-
nents (LT, footnote 15, p. 495).
 J.R. Lothian, M.P. Taylor / Journal of International Money and Finance 19 (2000) 759764 761

Moreover, the whole thrust of Rogoffs (1996) purchasing power parity puzzle is
the implausibility of high real exchange rate persistence.
CL note that shorter lags in the autoregressive representation lead to the null
hypothesis of unit root behavior being rejected at the 5% level, although with the
presence of a statistically significant time trend.

2.2. Heteroskedasticity

In LT, we were careful to use heteroskedasticity-robust estimation methods, since

real exchange rate variability is known to vary widely across nominal exchange rate
regimes. This issue is ignored by CL, however, so that their ADF results are in fact
invalid: Whites (1980) test for heteroskedasticity applied to the residuals from their
ADF regression with zero lags of the dependent variable and a constant and time
trend, for example, yields a test statistic with a marginal significance level of virtually
zero.2 The PhillipsPerron statistic reported by CL is, however, valid in the presence
of heteroskedasticity and implies rejection of the null hypothesis of unit root behavior
of the sterlingdollar real exchange rate, albeit in the presence of a statistically sig-
nificant time trend.

3. Economic issues

Given the methodological issues raised in the previous section, the implication of
the CL analysis is that the sterlingdollar real exchange rate appears to be stationary
around a linear time trend over the period considered. The next question is whether
or not this trend factor is economically as well as statistically significant.
Re-estimating the AR(1) representation for the sterlingdollar real exchange rate
for the period 17921990 and including a linear time trend yields:
qt0.807qt13.11104tconstantresiduals, (1)
(0.045) 4
(1.4710 )

where the figures in parentheses are heteroskedasticity-consistent estimated standard

errors. This corresponds to the AR(1) specification estimated for sterlingdollar in
LT, augmented by a time trend, and to the ADF regression with zero lags reported
in Table 1 of CL, except that, in contrast to CL, we calculate heteroskedasticity-
robust standard errors.
We have to concede from Eq. (1) that there is, indeed, a statistically significant
time trend present. Moreover, re-estimating this equation with from two to 15 lags
led to quantitatively and qualitatively almost identical results in terms of the magni-
tude of the estimated time trend coefficient, its statistical significance, and the sum
of the autoregressive coefficients.

2
Whites (1980) statistic is asymptotically distributed as c2(3) under the null hypothesis of homosked-
asticity; the value of the statistic obtained was 51.95.
762 J.R. Lothian, M.P. Taylor / Journal of International Money and Finance 19 (2000) 759764

The economic significance of this factor is, however, another matter. The estimated
time trend coefficient is 0.000311. Since the real exchange rate is expressed in
logarithms, this implies a long-run trend depreciation of sterling against the dollar
of 0.0311/(10.807) or about 0.16% per annum. This does not seem to us to be of
great moment economically.
Over the sample period considered, Britain underwent the first industrial revolution
and became the worlds leading industrial and economic power before declining
relatively during the twentieth century. During the same period, the United States
transformed itself from an exclusively rural economy to take Great Britains mantle
as the leading international economic power. It therefore seems reasonable to suppose
that real effects such as the HarrodBalassaSamuelson (HBS) effect would have
made themselves felt over the sample. In LT we certainly did not deny the possibility
of permanent shifts in the real exchange rate; our overall objective was stated clearly:

It also seems likely that, over a period of 200 years, there will have been
important real shocks to the real exchange rate, some of which may have had
permanent components. Our aim is to examine whether the hypothesis of a station-
ary real exchange rate is a good first approximation that describes the salient
characteristics of real exchange rate behavior even over such a diverse period as
the last two centuries (LT, pp. 493494).

A particular concern, as the title of our article indeed suggested, was whether the
simple AR(1) model that we had estimated continued to perform adequately under
the current float. Based on both formal tests of stability and dynamic simulations,
we found that it did. The existence of a linear trend does not overturn that conclusion.
To have explained long-run real exchange rate behavior over the 200 year period as
a whole to within 0.16% per annum, moreover, seems to us to have attained our
overall objective quite well.

4. Strengthening the LothianTaylor results and resolving the PPP puzzle

On the other hand, including the time trend in the autoregressive representation
for the real exchange rate may be viewed as strengthening our claim for significant
mean reversion in the real exchange rate. Note that the estimated first-order autore-
gressive coefficient in Eq. (1) is 0.807, implying mean reversion in the real exchange
rate of some 20% per annum, as opposed to the speed of mean reversion of about
11% per annum according to the original LT estimate. Put another way, this implies
a half-life of shocks to the sterlingdollar real exchange rate of about 3 years, which
is almost exactly half the estimated half-life without allowing for a time trend (i.e.,
about 6 years see LT, p. 502). Hence, allowing for a time trend in the data, while
not strongly economically significant in itself, does appear to strengthen our claim
of significant mean reversion in the real exchange rate and goes some way to resolv-
ing Rogoffs PPP puzzle concerning the speed of mean reversion of real exchange
rates.
 J.R. Lothian, M.P. Taylor / Journal of International Money and Finance 19 (2000) 759764 763

Given, however, the economic history of the United States and Great Britain over
the sample period, if the time trend is proxying for HBS effects, it seems unnecess-
arily restrictive to assume that the effects are linear. Accordingly, we experimented
with adding higher-order trend terms into Eq. (1) and settled on the following cubic
trend specification (heteroskedasticity-robust standard errors given in parentheses,
sample period 17921990):
qt0.767qt19.36104t9.26108t3constantresiduals. (2)
(0.043) 4 8
(2.4210 ) (3.7410 )

The coefficients on the trend coefficients are again significantly different from zero
but small in magnitude, and it might again be argued that the composite trend term
is statistically significant but economically insignificant: it ranges between a
maximum of +2% and a minimum of 5% over the entire 200 year period.3
What is interesting, however, is the effect on the estimated autoregressive coef-
ficient. This has now shrunk again, implying mean reversion of the real exchange
rate of about 23% per annum, or a half-life of about 2.5 years. Thus, allowing for
HBS effects in this very simple fashion, while it does not affect LTs claim to have
provided a good first approximation, certainly appears to go some way towards
resolving Rogoffs (1996) PPP puzzle of apparently very slow adjustment in real
exchange rates.4

5. Conclusion

We are grateful to Cuddington and Liang for their interest in our work. Addressing
the issues they raise, as it turns out, actually has allowed us to strengthen the claims
that we made earlier in Lothian and Taylor (1996).
Methodological issues apart, there does seem to be evidence of statistically sig-
nificant trends in the real exchange rate over the past two centuries. Nevertheless,
the economic importance of these trends in one sense is slight, implying trend move-
ments in the equilibrium real exchange rate of only a fraction of a percentage point
per annum.
The inclusion of linear and, in particular, non-linear trends in the autoregressive
representation of the real exchange rate, however, also implies much smaller esti-
mates of the half-life of shocks to the real exchange rate as low as 2.5 years.
This, in fact, buttresses our claim of a stable real exchange rate and, at the same

3
A plot of the estimated trend component of the real sterlingdollar exchange rate from Eq. (2) (not
shown) reveals a non-monotonic shape of the trend which seems economically plausible and broadly to
fit the pattern that one might expect of a HarrodBalassaSamuelson effect of an appreciating currency
of a relatively fast-growing economy: a trend appreciation of sterling against the dollar until about the
mid-nineteenth century followed by a trend depreciation into the twentieth century.
4
Note that the estimated autoregressive coefficient in Eq. (2) is very close to that reported for the
francsterling real exchange rate in LT, suggesting that allowing for HBS effects in this simple fashion
between the US and Britain in fact leads to greater consistency in LTs results.
764 J.R. Lothian, M.P. Taylor / Journal of International Money and Finance 19 (2000) 759764

time, appears to go some way towards resolving Rogoffs (1996) PPP puzzle con-
cerning the slow speed of adjustment of real exchange rates.5
We are currently engaged in work in this area that seeks to identify the Harrod
BalassaSamuelson effect more directly, rather than relying on the proxy of linear
and non-linear trend terms. At the same time, we are exploring the possibility that
this adjustment is itself explicitly non-linear, due for example to the presence of
transactions costs in international goods arbitrage (Taylor et al., 1999). These
additional refinements are likely to shed further light on the real exchange rate adjust-
ment process.
In general, however, the present exchange only serves to strengthen our 1996
claim that long-run purchasing power parity does indeed provide a good first
approximation that describes the salient characteristics of real exchange rate behavior
even over such a diverse period as the last two centuries (LT, pp. 493494).

References

Cuddington, J.T., Liang, H., 2000. Purchasing power parity over two centuries? Journal of International
Money and Finance 19, 751755.
Hegwood, N.D., Papell, D.H., 1998. Quasi-purchasing power parity. International Journal of Finance and
Economics 3, 279289.
Lothian, J.R., Taylor, M.P., 1996. Real exchange rate behavior: the recent float from the perspective of
the past two centuries. Journal of Political Economy 104, 488509.
Rogoff, K., 1996. The purchasing power parity puzzle. Journal of Economic Literature 34, 647668.
Taylor, M.P., Peel, D.A., Sarno, L., 1999. Nonlinear mean reversion in real exchange rates: towards a
solution to the purchasing power parity puzzles. Mimeo, Warwick Business School. International
Economic Review, forthcoming.
White, H., 1980. A heteroskedasticity-consistent covariance matrix estimator and a direct test for heterosk-
edasticity. Econometrica 48, 817838.

5
An alternative statistical characterization that achieves much the same result is the use of dummy
variables as in Hegwood and Papell (1998). Using the LT sterlingdollar data, they identify breaks in
1863 and 1929. Including dummy variables to allow for these breaks reduces the half-life of adjustment
from 5.78 years to 2.32 years or by 60%. What these dummy variables are actually capturing HBS
effects, episodic phenomena as Hegwood and Papell (1998) suggest, or simple measurement errors in the
data is an open question.