March 11, 2002
Mean Reversion in G-10 Nominal Exchange Rates*
Richard J. Sweeney
The McDonough School of Business
Georgetown University
37th and “O” Sts., N.W.
Washington, D.C., 20057
(O) 202-687-3742
Fax 202-687-7639, -4031
sweeneyr@msb.edu
Abstract: Conventional wisdom is that industrial-country floating exchange rates contain unit
roots. SUR tests on panels of Group-of-Ten log nominal rates reject the null of unit roots for
various samples over the current float, with significance levels from 0.5% to 15%. In out-ofsample forecasts, mean-reversion models beat random walks on average, in some forecast
periods significantly. For monthly data, the range of expected USD-DEM appreciation rates
exceeds 15%/year in the mean reversion model. Mean reversion places strong restrictions on
international models: over the sample period, the G-10 had to run monetary policies consistent
with stable long-run nominal rates.
*For helpful discussion, thanks are due to Jim Bodurtha, Bob Cumby, Behzad Diba, Allan
Eberhart, Bardia Kamrad, Bruce Lehman, Jim Lothian, Keith Ord, Akhtar Siddique, Boo Sjöö,
Stephen Taylor and Clas Wihlborg. Participants in finance seminar series at The McDonough
School of Business and The Gothenburg School of Economics, in a session at the Conference on
Pacific Basin Finance, Economics and Accounting, and in a session at the FMA Europe meetings
provided helpful comments. Gary Wang provided excellent research assistance. The McDonough
School of Business provided summer support and Georgetown University’s Capital Markets
Research Center provided summer research assistance. Part of this paper was written at
Göteborgs Universitet, Sweden.
Mean Reversion in G-10 Nominal Exchange Rates
Abstract: Conventional wisdom is that industrial-country floating exchange rates contain unit
roots. SUR tests on panels of Group-of-Ten log nominal rates reject the null of unit roots for
various samples over the current float, with significance levels from 0.5% to 15%. In out-ofsample forecasts, mean-reversion models beat random walks on average, in some forecast
periods significantly. For monthly data, the range of expected USD-DEM appreciation rates
exceeds 15%/year in the mean reversion model. Mean reversion places strong restrictions on
international models: over the sample period, the G-10 had to run monetary policies consistent
with stable long-run nominal rates.
Mean Reversion in G-10 Nominal Exchange Rates
1. Introduction
Conventional wisdom has long held that industrial countries’ floating exchange rates
contain unit roots (Mussa 1976, Logue, Sweeney and Willett 1978). Evidence below, however,
supports the view that Group-of-Ten (G-10) log nominal rates are mean reverting. The
hypothesis of mean reversion in G-10 nominal rates arises from the evidence that real rates are
mean reverting and from the well-known fact that changes in the logs of real and nominal rates
are highly correlated; see Figure 1 for logs of nominal and real values of the Deutsche Mark
relative to the dollar. For panels of monthly G-10 log rates, SUR tests reject the null of unit roots
at significance levels from 0.5% to 15% for 14 overlapping sample periods during the current
float. Further, test results for real rates and nominal rates are highly correlated; observers who
accept the evidence for mean reversion in G-10 real rates are hard pressed not to accept nominalrate mean reversion. Related, over many samples panel tests reject the null that relative national
price levels contain unit roots; mean reverting nominal rates and relative price jointly drive real
rates to their long-run values. Out-of-sample forecasting tests also support rejection of unit roots:
Mean reversion models beat driftless random walks in most samples, sometimes significantly.
Nominal rates explain approximately 75 percent of real-rate adjustment, and the logs of nominal
and real rates adjust at about the same speed, perhaps 1.5% to 2.5%/month.
Over 1974-1996, the range of expected appreciation implied by mean-reverting nominal
rates is large: Plausible estimates of the difference between the maximum and minimum
expected appreciation rate of the dollar against the DEM are from 15.81%/year to 26.35%/year,
after accounting for interest-rate differentials. For comparison, the sample standard deviation of
DEM/USD appreciation is approximately 12%/year. In periods where mean-reversion effects on
expected appreciation are large, taking them into account can be important, for example, in
international asset allocation or risk-management models. Related, because international asset
pricing models omit mean-reversion effects, these are impounded in the errors for foreign assets;
entering mean-reversion effects as an explanatory variable reduces the error variance for periods
where these effects are large.
Because the log real rate and the log nominal rate are both I(0), the log of relative national
price levels must be I(0), or the difference in log price levels must be I(0), and this implies strong
interdependence of long-run monetary policies. To illustrate, consider a two-country monetary
model where each country’s price level is proportional to its money stock in the long run;
consistent with G-10 data for the floating rate period, suppose both log price levels are I(1), thus
implying log money stocks are I(1). If the difference in log price levels is to be I(0), then the
difference in log money stocks must be I(0), or the log money stocks must be cointegrated. This
cointegration requires strong interdependence in long-run monetary policies, but allows
substantial short-run divergences. Taking account of long-run monetary interdependence may be
useful in some applications, for example, for investigating the extent to which the monetary
models beat a random walk in out-of-sample exchange-rate forecasts for G-10 countries.1
For countries with large persistent inflation differences relative to the G-10 during the
current float, the data do not support mean reversion in nominal exchange rates relative to the
dollar, even though real rates show mean reversion. For example, Figure 2 shows the log of the
DEM and Argentina and Brazil currencies relative to the dollar. The latter currencies show
grossly different behavior from the DEM and obviously are not mean reverting. Argentina and
1
The seminal article is Meese and Rogoff (1983); for later work, see Mark (1995), Chinn and Meese (1995), Mark
2
Brazil’s exchange rates reflect their inflation rates and monetary growth rates that diverged
sharply from the G-10’s for long periods and caused permanent changes in their log price levels
and log money stocks relative to the G-10’s; the difference between say Argentina and U.S. log
price levels is not I(0), and neither is the difference in log money stocks.
Section 2 discusses the panel unit-root tests and the data. SUR techniques deal with the
large contemporaneous cross-correlations in changes in log nominal rates; similar techniques are
used in SUR panel unit-roots tests of real exchange rates, of interest rates and of international
stock markets.2 For cross correlations in log nominal rates, SUR is superior to fixed-time effects
models (appendix available from author). Critical values used here are found from simulation,
where the data generating process (DGP) uses the sample covariance matrix of changes in log
nominal rates; small, sometimes significant, serial correlations in appreciation are accounted for.
Section 3 presents test results for monthly data on G-10 currencies relative to the U.S.
dollar over the generalized managed float that began in mid-March 1973. Sample periods all end
in December 1996, with 14 starting dates from January 1974 to January 1987, to gauge results’
sensitivity; significance levels range from 0.5% in five cases to 15% in four cases. Panels of log
relative national price levels show comparable rejections. Further, results for panel tests of log
real exchange rates are similar to those for log nominal exchange rates, as the large correlations
between changes in the logs of real and nominal rates suggest.
Section 4 reports that out-of-sample forecasting results support the rejections in unit-root
tests: Mean-reversion models beat driftless random walks in most forecast periods, significantly
in some. These results suggest log nominal rates’ speed of adjustment is perhaps 1.5% to
and Sul (1999) and papers cited there.
2
For real exchange rates, see Abuaf and Jorion (1990), Jorion and Sweeney (1996), MacDonald (1996), O’Connell
(1998), Papell (1997), Papell and Theodoridis (1998), and Taylor and Sarno (1998); for interest rates, Wu and
Zhang (1996) and Balz (1998); and for international stock markets, Balvers et al. (2000).
3
2.5%/month, comparable to adjustment speeds inferred for log real rates. Further, log nominalrate mean reversion explains approximately 75 percent of log real-rate mean reversion. The range
of variation in expected appreciation is economically important, for example, more than
15%/year for the dollar-DEM rate. Section 5 offers a summary and conclusions.
2. The Estimating Model
This paper’s panel tests apply Seemingly Unrelated Regression (SUR) techniques to
systems of augmented Dickey-Fuller (ADF) equations. For purposes of comparison, the tests are
purposely designed to be close to SUR panel tests of mean reversion in real exchange rates.3 The
ADF estimating system
(1)
lnSi,t = lnSi,t - lnSi,t-1 = ai + b lnSi,t-1 + pij=1 ci,j lnSi,t-j + ui,t,
i = 1,n; t = 1,T.
lnSi,t is the natural logarithm of country i's spot exchange rate (foreign currency per U.S. dollar),
ai the intercept for country i, and pi the order of i’s autoregressive terms. T is the number of time
series observations and n=10 for the G-10 countries.
The test statistic is the t-value, tb, of the common slope b. An alternative approach is to
allow separate slopes, bi, for each currency. In experiments discussed below, tests that impose bi
= b have greater power, even when the slopes are not equal.
The ui,t follow SUR assumptions: the errors are mean-zero, with the positive definite
contemporaneous covariance matrix SUR. This generalizes Levin and Lin’s (LL 1992) approach;
they assume all ui,t are IN(0, 2u), or without loss of generality the covariance matrix is LL = I.
They demonstrate that as n  , T   and n 1/2 / T  0, then tb d [N(0, 1) - (1.875
n)1/2]/1.251/2, or 1.251/2 tb + (1.875 n)1/2  N(0, 1); under SUR, tb has the same asymptotic
distribution (appendix available from the author).
3
See Abuaf and Jorion (1990), Jorion and Sweeney (1996), MacDonald (1996), O’Connell (1998), Papell (1997),
4
Under the null, ai = b = 0; under the alternative, b < 0 and the ai >/< 0 may differ across i.
This null and alternative are only one possible pair, and this paper’s results could be reversed for
some but not all different, interesting nulls and alternatives. On the one hand, the process may be
subject to parameter shifts, deterministic or random, that are likely to bias the test in favor of the
null. For example, Perron (1989) presents cases where tests are unlikely to reject the unit root
null against the I(0) alternative if the process is mean reverting but contains known parameter
shifts (much following literature investigates related cases). On the other hand, the series may be
the sum of a unit root and a mean-reverting process (see Huizinga 1987 and Engle 2000 for real
exchange rates); for example, the disturbance might be ui,t = i,t +  i,t, where i,t is an error with
SUR properties,   0 is a constant, and i,t is an error with a unit root. If  ( / ) is small, in
small samples tests may reject a unit root in favor of reversion to a constant mean, when instead
the nominal rate reverts to a mean subject to small, permanent changes over time.
Critical Values from Simulation for SUR Tests. SUR techniques are used to deal with
the large contemporaneous cross correlations in changes in log nominal exchange rates, similar
to a number of panel tests of mean reversion in real exchange rates. Denote by yi,t the simulated
data, which correspond here to lnSi,t. The data generating process (DGP) is yi,t  N(0, SUR),
with ai = b = pi = 0. In LL, the DGP is yi,t  N(0, I), with ai = b = pi = 0. The DGPs differ in one
way: this paper uses the non-diagonal SUR, not I. Results below use the lnSi,t sample
covariance matrix, sample, as the estimator of SUR. Other SUR estimators include the residual
covariance matrix from SUR estimates of (1), with pi = p = 0, 1, 2, etc.; in experiments, critical
values do not change importantly across SUR estimators.
Changes in monthly G-10 log nominal rates show small, sometimes significant low-order
Papell and Theodoridis (1998), and Taylor and Sarno (1998).
5
serial correlations for some rates in some periods. Authors differ widely regarding how to handle
serial correlations in tests of mean reversion in real exchange rates (see O’Connell 1998, Papell
1997). For finding critical values that account for estimated serial correlations in changes in log
nominal rates, this paper follows an approach based on Im, Pesaran and Shin (IPS 1997); in
simulations, the critical values thus found perform well, as Appendix 1 discusses.
For autoregressions of changes in log nominal exchange rates, in White tests the residuals
cannot reject the null of no heteroscedasticity for any currency. Thus, this paper finds critical
values from SUR estimates on simulated data generated with homoskedastic errors.
Critical values are also found for the case of non-zero means, where the DGP is yi,t = ai
+ ui,t, with ui,t  N(0, SUR) and b = pi = 0. Sample means of the lnSi,t are used for the ai (in ttests, no mean is significant at the 5% level). Resulting critical values are substantially smaller in
absolute value than the text uses (Appendix 2), and thus more likely to reject the null.
Comparison of Critical Values. This paper reports critical values found by fitting ADF
equations (1) on simulated data with iterated SUR for pi = p = 0,1, etc., for n = 10, and for T =
276 with SUR = sample for 1974:01-1996:12, and for T = 120 with SUR = sample for 1987:011996:12, with 10,000 replications in each case. LL (1992, Table 5) report critical values for ADF
equations for p = 0. Comparing,
Levin and Lin: Least Squares
Asymptotic T=250 T=100
0.5% -6.178
NA
NA
1
-5.954
-6.01 -6.00
5
-5.345
-5.43 -5.43
10
-5.020
-5.12 -5.13
New simulations: Iterated SUR (p = 0, p = 1)
1974-1996 (T=276) 1987-1996 (T=120)
-6.3026
-6.3278 -6.3073
-6.5351
-6.1240
-6.1797 -6.1030
-6.3066
-5.5527
-5.5366 -5.5200
-5.7015
-5.2460
-5.1858 -5.2165
-5.3512
mean -5.6243
-5.8063
-5.52 -5.52
-5.8075
-5.7867
-5.9736
SUR critical values are modestly larger in absolute value, by 2.80 percent to 5.40 percent, despite
large cross correlations in the lnSi,t—for the DEM, the average cross-correlation is 0.746, with
6
three larger than 0.90. Using LL critical values in this paper’s SUR tests would make the actual
size somewhat larger than the nominal size, rejecting the null too often when it is true.
Power of The Tests. The tb test used here has good power. For example, if each
currency’s root is 0.985, the unit-root null is rejected at the 0.5% level in 21.55 percent of the
cases, at the 10% level in 80.33 percent. Appendix 2 gives more information on power.
Fitting Equations With Unconstrained Slopes. IPS (1997) propose fitting (1) without
the constraint bi = b. In simulations, freely fitting the bi reduces power for the large roots and
high contemporaneous cross-correlations of changes in log nominal rates that characterize G-10
data (appendix available from the author). If countries have a root of 0.985, then with the bi fitted
freely the null is rejected in 7.47 percent of the cases at the 0.5% level and in 45.28 percent at the
10% level; with the bi constrained, the rejection rates are 21.55 percent and 80.33 percent. If the
roots vary across countries around a mean of 0.975, with the first country’s 0.965, the second’s
0.985, etc., then with the bi fitted freely, the null is rejected in 7.600 percent of the cases at the
0.5% level, 45.70 percent at the 10% level; with the bi constrained, the rejection rates are 34.77
and 56.43 percent. As might be expected from these power comparisons, test results are less
impressive when the bi are fit freely, though the null is rejected in 8 of 14 samples (results
available from the author).
Fixed Time Effects as a Method of Adjusting for Cross-Correlation. LL and IPS extend
their models to allow fixed time effects (FTEs), a restrictive form of cross-correlation.4 SUR is
preferable here: An appendix available from the author shows that G-10 log nominal rates
grossly violate FTE-model assumptions, and that FTE results are biased in favor of the null.
3. Results of Panel Unit-Root Tests
4
Wu (1996), Oh (1996) Culver and Papell (1997), Coakley and Fuertes (1997), Checchetti, Mark and Sonora (1998)
7
Consumer price indices and month-ending market exchange rates are from International
Financial Statistics. Interest rates are one-month Euro rates from DataStream for later years;
earlier data are domestic series from IFS, linked to the later data.5 The G-10 are Belgium,
Canada, France, Germany, Italy, Japan, the Netherlands, Sweden, the United Kingdom and the
United States. Switzerland was informally associated with the G-10 from the start, and thus is
often included in studies that focus on the G-10, giving ten exchange rates relative to the USD.
Sample periods end in December 1996 with periods starting in January 1974, January
1975, … , January 1987, for 14 overlapping samples. Results are not independent across samples,
of course, but this presentation is intended to show results’ sensitivity to the sample period; for
real rates, Papell (1997) uses a similar approach to judge sensitivity.
Results for G-10 Log Nominal Exchange Rates. Table 1, Panel A, shows panel unit root
test results. Results for pi = p = 1 are little different from those for pi = p = 0. The significance
level varies with the sample’s starting date, ranging from 0.5% in five cases to 15% in four cases,
across p = 1,2. For the six periods starting before 1980, results are significant at the 15% level in
four cases, at the 0.5% level in two; rejections are somewhat stronger in the last eight periods. It
might be thought that log nominal rates had unit roots in the 1970s and were mean reverting
thereafter; this is undercut by rejections at the 0.5% level for samples starting in 1976 and 1977.
The relatively weak rejections for sample periods starting in 1974 and 1975 may arise
from sampling variation or from a shift in process after 1975. From Perron (1989) and
subsequent work, parameter changes in an AR1 process will bias the test against rejecting unit
roots unless the test equation explicitly includes the parameter shift; differences in process before
and Fleisig and Strauss (1999) use FTEs in panel unit-root tests. Period dummies are an alternative approach to fixed
time effects, used in some tests by Wei and Parsley (1995) and Frankel and Rose (1996).
5
Domestic series used are always market rates and are, in order of preference, one-month inter-bank, T-bill, call
money and bond rates.
8
and after 1975 might be due to the “turbulence” at the start of floating as governments and
markets adjusted to the new exchange-rate regime and searched for equilibrium market rates. As
an indication of possible changes in process the sample means of the log nominal rates, which are
estimators of their long-run values, are substantially different for several countries after 1975
from before. In a version of (1) that allows different intercepts ai before and after 1975:12, the
unit-root null is rejected at the 0.97% level, with critical values found from simulation. This
improvement is expected from the better significance levels from the sample that started in 1976
as compared to samples starting earlier, and strictly speaking is not a test of changes in process.
Further, assuming that the process changed after 1975, there are too few observations to be
confident that log nominal rates in 1974-1975 are better modeled as AR1s or random walks; in
tests using (1), the data reject the unit-root null at the 45.60% significance level for 1974-1975.
The null may be rejected because some—not all—log nominal rates are I(0) (see Karlsson
and Löthgren 2000, and Sarno and Taylor 1998). From simulation results in Appendix 2, the
rejections of the null reported here are much more likely to arise because all log nominal rates are
I(0), rather than just a subset. 6
Results for Relative National Price Levels. Conditional on mean reversion in real
exchange rates, evidence of mean reversion in relative prices provides additional support for the
view that nominal rates are mean reverting. With appropriate rewriting, (1) can be used to test for
unit roots in relative prices. Table 1, Panel B, shows results for systems of changes in log relative
national price levels (RPs) for the G-10. Because changes in log RPs often show first-order
autocorrelation, the estimated ADF regressions set pi = p = 1. Critical values for SUR estimates
6
Related, some argue that Italy’s Lira shows depreciation, Japan’s yen appreciation, rather than mean reversion.
Note that this view implies the tests are biased in favor of the unit-root null. Systems of nine countries (the G-10 less
Italy) and eight countries (the G-10 less Italy and Japan) show results similar to Panel A’s, though a little weaker
overall in rejecting the null; if Italy and Japan’s nominal exchange rates contain unit roots, the impressiveness of the
9
are found from simulation, in the same way as for changes in log nominal rates. The data reject
the null of a unit root at the 0.5% level for all periods that start before 1984; the null is rejected at
the 1% level for the period starting in 1984.
Results for Real Exchange Rates. It has long been known that variations in nominal rates
are the dominant component of variations in G-10 real rates, especially for periods as short as a
month (Obstfeld 1985, Mark 1990). For the 1974-1996 period, monthly changes in log nominal
and log real rates have an average correlation of 0.9847; this suggests that results of panel unitroot tests should be close for log nominal and log real rates. Table 2 shows results for real rates
and repeats nominal-rate results from Table 1; for both, nominal-rate critical values are used.
Results are similar, stronger for nominal rates for some starting years, weaker for others.
4. Out-of-Sample Forecasts
Unless mean-reversion models are useful for out-of-sample forecasting, for practical
purposes nominal rates might better be thought of as containing unit roots.7 In out-of-sample
forecasting tests below, mean-reversion models are compared to random walks, a frequent
benchmark. The mean-reversion models outperform random walk models in most samples,
sometimes significantly.
Jorion and Sweeney (1996) and Siddique and Sweeney (1998) present evidence that, for
equally weighted portfolios of log real exchange rates for G-10 countries, mean reversion models
with p = 0 provide better forecasts than driftless random walks, in terms of root mean square
errors (RMSEs). Following these papers, forecasts from the model lnSi,t = ai + b lnSi,t-1 + ui,t = b
(lnSi,t-1 - lnS*i) + ui,t are compared to forecasts from driftless random walks, where lnS*i is
rejection would be expected to rise.
7
Section 3’s tests reject the null that log nominal rates are I(1), but the estimated roots are close to unity, and rates
are thus “near integrated” (NI) variables. An NI variable may be better treated as I(1) for many purposes (Campbell
and Perron 1991, Banerjee et al. 1993). For forecasting, is the nominal rate better treated as I(1)?
10
country i’s long-run log nominal rate. Siddique and Sweeney’s test design for log real rates is
closely followed: Results are reported for a range of adjustment speeds chosen a priori; 12months-ahead forecasts are used; ten-year estimation periods are used; and rolling five-year
forecast periods are used.
Mean reversion models require values of b and ai, but estimates from OLS or SUR
systems are biased for finite T (Levin and Lin 1992) and are thus suboptimal for forecasting.
Siddique and Sweeney choose adjustment speeds a priori; in a variety of contexts, applying a
priori knowledge to estimated parameter values improves out-of-sample forecasts.8 Below, a
priori adjustment speeds of 2.5, 1.5, 1.0, 0.75 and 0.50%/month are used. Long-run log nominal
rates, lnSi*, are taken as the sample means in the preceding estimating period.9
Results are reported for 12-months-ahead forecasts. Long forecast horizons are likely to
give more informative results: The absolute value of the difference between mean-reversion and
random-walk forecast errors is increasing in the forecast horizon h.10 10-year estimating periods
are used: If parameters are constant, longer estimation periods are likely to produce better
estimates and hence superior forecasts.
Table 3 reports RMSEs for equally weighted portfolios of log nominal exchange rates for
rolling five-year forecasting periods. The first line, for the forecast period ending in 1989:12,
reports RMSEs for forecasts where the estimation period for lnS*i is 1974:01 to 1983:12, and
data from 1984:01 to 1988:12 are used to produce 12-month-ahead forecasts for 1985:01 to
8
In forecasting using VARs, authors report that Bayesian adjustment of parameter estimates produces superior outof-sample forecasting performance on average (Ashley 1988, Holden and Broomhead 1990, Artis and Zhang 1990,
LeSage and Magure 1991).
9
Various estimators of lnSi* are available. Without experimentation, the estimation period’s mean log exchange
rate, mi = T - 1 Tt=1 lnSi,t, is used; if lnSi,t is stationary, mi is consistent.
10
For h-months-ahead forecasts, the random walk and mean reversion forecast errors are FERW = lnSi,t+h - lnSi,t and
FEM = lnSi,t+h - ln Ŝ i,t+h = lnSi,t+h - {lnSi* [1 - (1+b)h] + (1+b)h lnSi,t}. Thus, for b < 0, | FEM - FERW | = | (lnSi,t - lnSi*) |
[1 - (1+b)h] increases in h.
11
1989:12. The column RW shows the RMSERW for random-walk model forecasts; the remaining
columns show mean-reversion models’ RMSEMs for six adjustment speeds, 2.5 to 0.50%/month;
RMSEMs that beat the RMSERW are bolded, as are cases where the random walk is better.
Under the random walk null, RMSEM is expected to exceed RMSERW, but mean
reversion models on average beat the random walk. The average RMSERW exceeds the RMSEM
by 5.817, 5.995, 5.562, 4.911, 4.030 and 2.918 percent across adjustment speeds from 2.5% to
0.50%/month, as the row “RW/MR” shows. In seven of eight forecast periods, some mean
reversion model beats the random walk.
For the first two forecast periods, the mean-reversion model significantly outperforms the
random-walk model for all adjustment speeds. Results are not significant for the other six
periods; in particular, the random-walk model never significantly outperforms the meanreversion model. For a given forecast period, results are positively correlated across adjustment
speeds; thus, the six results for the first forecast period cannot be interpreted as independent tests.
Because forecast periods are overlapping, in principle results are correlated across periods for a
given adjustment speed; in practice, correlation is modest in Table 3. On balance, the forecast
results provide important but not overwhelming support for mean reversion.11
Test results use exact critical values from simulation. A note available from the author
discusses the distribution of the test statistic DR = RMSE2M - RMSE2RW, and the simulation
experiments used to find critical values. Standard tests of prediction accuracy, for example,
Diebold and Mariano (1995), cannot be used because they assume both model’s forecast errors
are stationary; under the random-walk null, the mean-reversion error is non-stationary.
Speed of Adjustment. Results in Table 3 provide some evidence on the speed of
11
An appendix available from the author considers whether the mean reversion model works better when the USD is
12
adjustment. Speeds of 2.5 and 1.5%/month give the best performances. Intuitively, adjustment of
1.5 to 2.5%/month seems slow, but slow adjustment does not imply that nominal rates do a poor
job equilibrating the system: the optimal adjustment speed may be slow. For comparison,
Siddique and Sweeney find that 2.5%/month gives the best performance for real rates.
For log real rates, Siddique and Sweeney report that, for adjustment speeds of 2.5, 1.5 and
1.0%/month, the RW/MR are 9.627, 7.975 and 6.059%—an average of 7.887 percent. For log
nominal rates and the same adjustment speeds, the average RW/MR is 5.791 percent. By this
measure, the superior performance of nominal-rate mean reversion models explains 73.4% [=
(5.791/7.887) 100] of the superior performance of real-rate mean reversion models.
Economic Importance of Time-Variation in Expected Appreciation Rates. Meanreversion effects on expected appreciation can be large. Over the 1974-1996 sample period, the
maximum and minimum DEM per dollar were 3.3225 and 1.3805—in natural logs, ln(3.3225) =
1.201 and ln(1.3805) = 0.3224. The expected depreciation rate at the DEM/USD maximum was
b (1.201 - lnSi*), and at the minimum b (0.3224 - lnSi*), for a difference of b (1.201 - 0.3224) =
b (0.8785). At an adjustment speed of 1.5%/month, the difference in expected appreciation of the
dollar is 0.015 x 0.8785 = 0.0131775, or 1.3178%/month and 15.813%/year in continuously
compounded terms; if the adjustment speed is 2.5%/month, the difference is 26.3482%/year. For
comparison, the approximate sample standard deviation of DEM/USD appreciation is 12%/year.
These mean-reversion effects are not proxies for interest-rate differential effects on
expected appreciation. If the differential is included as a lhs variable in (1) with p = 0, with its
coefficients constrained to be equal or not, tb increases in some cases, decreases in others:
Start
-- (from Table 1)
1974
-5.1107+
1980
-6.2120***
over- rather than under-valued; evidence supporting this view is weak.
13
1981
1987
-7.2753**** -5.5968**
unconstrained
constrained
-5.9591**
-5.6696**
-6.1623***
-5.5139*
-6.9419**** -5.9285**
-6.2554*** -5.6908**
(****, ***, **, *, +: Significant at the 0.5, 1, 5, 10 and 15% levels.)
5. Conclusions
Conventional wisdom holds that industrial-country nominal exchange rates contain unit
roots. In this paper’s SUR tests for G-10 countries’ log nominal rates relative to the dollar during
the current float, the data reject the unit-root null at significance levels from 0.5% to 15% across
sample periods. Further, out-of-sample forecast tests support mean reversion in nominal rates: on
average, mean reversion models beat random walks, sometimes significantly.
The difference between the mean-reversion model’s maximum and minimum expected
appreciation of the USD relative to the DEM is estimated at between 15%/year and 26%/year
over the sample. International applications that ignore mean-reversion effects thus misestimate
expected appreciation, ceteris paribus, by important amounts in some months. Further, rejection
of unit roots in G-10 log nominal rates implies substantial interdependence in long-run monetary
policies among G-10 countries. If both real and nominal log rates are stationary, then log relative
national price levels must be cointegrated; in many monetary models, this implies that log money
stocks must be cointegrated. Neglecting this long-run monetary interdependence may be costly in
international models where inflation and money stock growth rates are important.
Related, in panels that contain countries with high inflation relative to the G-10, the data
cannot reject unit roots in nominal exchange rates, even though real rates show mean reversion.
High inflation countries’ exchange rates reflect their inflation rates and monetary growth rates
that diverge sharply from the G-10’s and cause permanent changes in their log price levels and
log money stocks relative to the G-10’s; their log price levels and log money stocks are not
cointegrated with the G-10’s.
14
A simple monetary model that gives mean reversion in G-10 nominal rates uses two
assumptions that roughly fit reality. First, during the sample period 1974 -1996, the U.S. and
Germany ran monetary policies jointly consistent with a stable long-run USD-DEM rate. Second,
other G-10 countries pursued long-run monetary policies aimed at stabilizing their rates relative
to either the DEM (as did a number of European Union countries) or the USD (as Canada
sometimes did). Intuitively, sometimes Germany was the low-inflation, stable, safe-harbor
country, and sometimes the U.S. was. Because each country’s long-term monetary policy kept the
other from establishing permanent dominance along these lines, long-run monetary policy in each
country was consistent with a stable long-run bi-lateral nominal rate, and other G-10 countries
fell in line.
15
Table 1: Panel Unit-Root Tests for G-10 Nominal Exchange Rates and Relative Prices
A. Log Nominal Exchange Rates, tb for p = 0 and p = 1
Start
tb (p=0)
tb (p=1)
1974
-5.1107+
-5.0874+
1975
-5.1773+
-5.1665+
1976
1977
1978
-6.5550**** -5.6504**
-4.9867
-6.5080**** -6.6528**** -5.0080+
Start
tb (p=0)
tb (p=1)
1979
-5.1761+
-5.1845+
1980
-6.2120***
-6.1139**
1981
1982
1983
-7.2753**** -6.8550**** -5.3018*
-7.2500**** -6.9316**** -5.3816*
Start
tb (p=0)
tb (p=1)
1984
-5.1508+
-5.2764*
1985
-5.8399**
-6.0182**
1986
1987
-6.2176*** -5.5968**
-6.3862**** -5.7392**
Panel B. Log Relative National Price Levels, tb for p = 1
Start
tb (RP)
1974
1975
1976
1977
1978
-10.6338**** -10.1447**** -11.5081**** -12.2973**** -12.8834****
Start
tb (RP)
1979
1980
1981
1982
1983
-13.6451**** -11.6859**** -9.7773**** -8.7839**** -7.9595****
Start
tb (RP)
1984
-6.1459***
1985
-4.9699
1986
-3.8509
1987
-3.7151
****, ***, **, *, + Significant at the 0.5, 1, 5, 10 and 15 percent levels.
_____________________
Notes to Table 1: tb is the t-value of the slope b in the ADF equation lnSi,t = ai + b lnSi,t-1 + ci
lnSi,t-1 + ui,t (i=1,10); ci is set to zero for p = 0 but is estimated for p = 1. Exact critical values
are from simulation with iterated SUR. The DGP is lnSi,t = ui,t, where ui,t  N(0, DGP); the
DGP covariance matrix equals the sample covariance matrix of the lnSi,t over a given sample.
Critical values are found for T = 276 (1974:01 to 1996:12), and for T = 120, (1987:01 to
1996:12), with 10,000 replications each. Critical values are:
Log Nominal Exchange Rates (p=0, p=1)
1974-1996 (T=276) 1987-1996 (T=120)
0.5% -6.2931 -6.3278
-6.3073
-6.5351
1.0
-6.0813 -6.1797
-6.1030
-6.3066
5.0
-5.5148 -5.5366
-5.5200
-5.7015
10
-5.2047 -5.1858
-5.2165
-5.3512
15
-4.9877 -4.9696
-4.9990
-5.1281
16
Log Relative Price Levels (p=1)
1974-1996
1987-1996
-6.3402
-6.5392
-6.1166
-6.3270
-5.5478
-5.6722
-5.2302
-5.3348
-5.0121
-5.0734
Table 2: Panel Unit-Root Tests for G-10 Nominal and Real Exchange Rates
Start
Nom. tb (p=0)
Nom. tb (p=1)
1974
-5.1107+
-5.0874+
1975
-5.1773+
-5.1665+
1976
1977
1978
-6.55496**** -5.6504**
-4.9867+
-6.5080**** -6.6528**** -5.0080+
Real
Real
-5.1629+
-5.8574**
-5.0732+
-5.7813**
-5.2921*
-5.8891**
Start
Nom. tb (p=0)
Nom. tb (p=1)
1979
-5.1761+
-5.1845+
1980
1981
1982
1983
-6.21201*** -7.2753**** -6.8550**** -5.3018*
-6.1139**
-7.2500**** -6.9316**** -5.3816*
Real
Real
-5.8954**
-6.3172***
-5.7079**
-6.0785**
-5.7075**
-6.2093***
Start
Nom. tb (p=0)
Nom. tb (p=1)
1984
-5.1508+
-5.2764+
1985
-5.8399**
-6.0182**
1986
1987
-6.2176*** -5.5968**
-6.3862**** -5.7392**
Real
Real
-5.7088**
-6.1315**
-6.0142**
-6.2725*** -5.7299**
-6.4026**** -6.7740**** -6.1718***
tb (p=0)
tb (p=1)
tb (p=0)
tb (p=1)
tb (p=0)
tb (p=1)
-6.4948**** -6.0137**
-6.8628**** -6.4450****
-5.0767+
-5.7250**
****, ***, **, *, + Significant at the 0.5, 1, 5, 10 and 15 percent levels.
_____________________
Notes to Table 2 (see also notes to Table 1):
Critical Values:
Log Nominal Exchange Rates (p=0, p=1)
1974-1996 (T=276) 1987-1996 (T=120)
0.5% -6.2931 -6.3278
-6.3073
-6.5351
1.0
-6.0813 -6.1797
-6.1030
-6.3066
5.0
-5.5148 -5.5366
-5.5200
-5.7015
10
-5.2047 -5.1858
-5.2165
-5.3512
15
-4.9877 -4.9696
-4.9990
-5.1281
17
-5.5232*
-5.8519**
Table 3: Out-of-Sample Forecasts of G-10 Log Nominal Exchange Rates
10-year estimation period, five-year forecasting period, 12-months ahead forecasts
Fore. end. RW
.025
.015
.0125
.0100
.0075
.0050
1989:12
1990:12
1991:12
1992:12
1993:12
1994:12
1995:12
1996:12
0.143786
0.141255
0.101111
0.084863
0.092259
0.089737
0.084945
0.080418
0.116200***
0.108176***
0.098257
0.093071
0.087664
0.091180
0.091465
0.087376
0.123604***
0.119491***
0.098194
0.088299
0.086940
0.088268
0.086057
0.081237
0.126184***
0.122708**
0.098399
0.087348
0.087232
0.087950
0.085173
0.080259
0.129070***
0.126086**
0.098706
0.086524
0.087743
0.087833
0.084532
0.079572
0.132270***
0.129627^^
0.099123
0.085846
0.088489
0.087936
0.084161
0.079213
0.135787***
0.133334^^
0.099658
0.085331
0.089483
0.088276
0.084089
0.079214
Average
0.102297 0.096674
0.096511
0.096907
0.097508
0.098333
0.099397
5.995%
5.562%
4.911%
4.031%
2.918%
RW/MR
5.817%
***, **, *, ^^, ^: Significant at the 0.5, 1.0, 2.5, 5.0 and 10% levels.
Notes to Table 3:
The criterion is the root mean squared error (RMSE). "RW" shows RMSEs for forecasts based on random walks without drifts.
Other columns show RMSEs for various values of b < 1. “Average” shows the average of the RMSEs in that each column. RW/MR is
the percentage amount by which the average for the RMSERW = 0.102297 exceeds the average RMSEM for each adjustment speed.
The final six columns impose a priori speeds of adjustment: 2.5, 1.5, 1.25, 1.0, 0.75 and 0.50%/month. In these columns, the
estimated long-run value of each nominal exchange rate is its estimation-period sample mean.
Five-year forecast periods are used. The final line of results is for forecasts ending in 1996:12. For 12-month-ahead forecasts,
the final data point used for generating forecasts is 1995:12. Data used for generating 12-months-ahead forecasts for 1992:01 to
1996:12 are from 1991:01 to 1995:12. Parameter estimates for these forecasts are from the ten-year estimation period, 1981:01 to
1990:12.
18
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21
1.2
Log DEM per USD
1.2
1.0
1.0
DEM
0.8
<--
0.8
0.6
0.6
Real DEM
-->
0.4
0.4
0.2
0.2
74
76
78
80
82
84
86
88
90
92
94
Log of Real Exchange Rate of DEM per USD
1.4
96
Figure 1. Logs of DEM per USD and the Real Exchange Rate of DEM per USD
Source: International Financial Statistics
1.4
5
0
Arg.
Log DEM per USD
-->
1.0
-5
-->
Brazil
DEM
<--
0.8
-10
0.6
-15
0.4
-20
0.2
-25
74
76
78
80
82
84
86
88
90
92
94
Log Argentine Currency per USD,
Log Brazil Currency per USD (adjusted)
1.2
96
Figure 2. Logs of DEM, Argentina and Brazil Currencies per USD
The log of Brazil's currency is normalized, by multiplying by 0.8848508,
to have the same range as the log of Argentina's currency.
Source: International Financial Statistics
22
Appendix 1: Alternative Methods of Finding Critical Values
This appendix discusses the key choices in how critical values for this paper are found; in
experiments, the choices have little effects on critical values for nominal-rate systems. It also
discusses related papers on real exchange rate tests.
This Paper’s Approach. Changes in log nominal exchange rates show substantial
contemporaneous cross correlation, and in some cases, small but sometimes-significant serial
correlation. This paper uses SUR for cross correlation and IPS’s approach for serial correlation.
Critical values are found in steps: a data generating process (DGP) is chosen; and
equations are fit to the generated data. As the base case for the text’s critical values, the DGP is
(A)
lnSi,t = ui,t,
ui,t  N(0, SUR),
where SUR is positive definite. The equations fit with SUR to the generated data SUR are
(B)
lnSi,t = ai + b lnSi,t-1 + pij=1 ci,j lnSi,t-j + ui,t
i = 1,n; t = 1,T.
SUR can be found in a number of ways. This paper uses the sample covariance matrix of the
lnSi,t , sample, for both 1974-1996 and 1987-1996 to calculate reported critical values (though
the matrices are close). An alternative is the covariance matrix of residuals from the system fit
with SUR on the data for b = 0, p = 0, 1, etc. For the data used here, experiments show the choice
is inconsequential. In the systems (B) estimated to find critical values from generated data, all ci,j
are freely fitted. Reported critical values are for systems where all pi are equal, pi = p; in
experiments, allowing pi to vary across countries does not importantly affect critical values.
IPS (1997) emphasize that correctly handling serial correlation is vital for valid results.
The key DGP issue is the choice between (A) and a DGP with non-zero ci,j,
(C)
lnSi,t = pij=1 ci,j lnSi,t-j + ui,t,
ui,t  N(0, SUR).
IPS use DGPs with no serial correlation, (A), but fit equations (B) that allow extensive serial
23
correlation. For size and power experiments, IPS use (C) to generate test data with ci,j  0 for p*
> 0 lags. They find, first, that using (A) and (B) has good size and power properties if the
equations (B) they fit on the test data, and thus the critical values they use, have p equal to the p*
used to generate test data (their Table 6). Second, for p < p*, size tends to zero for n = 10, and
power falls greatly; for p > p*, size and particularly power are harmed for n = 10, though not as
severely as with under-fitting. Thus, using the incorrect p causes bias against rejecting the null. In
experiments for this paper, with SUR = sample and large | ci,j | for p = 1,4, similar results hold.
Using the DGP (C) is superior to (A) if the true ci,j are known; for this paper, ci,j must be
estimated. For G-10 log nominal exchange rates, fitted values of ci,j are small, imprecisely
estimated, and sensitive to estimation methods and sample periods; sign changes occur, and
estimates frequently change by 100% or more. For illustration, individual equations lnSi,t = ai +
pij=1 ci,j lnSi,t-j + ui,t are fit with OLS for four lags and also SUR for systems with two lags of
lnSi,t-j, for the longest and shortest samples used here. For the first lag, ci,1,
Sample
Technique
OLS
SUR
1974:01-1996:12
1987:01-1996:12
2 sig., at 10% level
(Sweden, U.K.)
7 sig., 4 at 10%, 1 at 5%, 2 at 1% level
(Belgium, Germany, Italy, Netherlands,
Sweden, Switzerland, U.K.)
4 sig., 1 at 10%, 2 at 5%, 1 at 1% level
(Italy, Japan, Sweden, U.K.)
2 sig., at 5% level
(Japan, Sweden)
For the second lag, ci,2, none is significant in OLS; in SUR, one is significant in the longest
sample (Belgium, 10%), one in the shortest (Japan, 5%).
Using estimated ci,j’s in the DGP has little effect on critical values for log nominal-rate
systems. In one experiment, for 1974:01-1996:12 with p = 1 and ai = b = 0, the first DGP has all
ci,j = 0, and the second DGP uses estimated values for the first lag, ci,1  0, where the ci,1 are
estimates from a SUR system. The same generated errors are used for both DGPs. The average of
24
the 0.5, 1.0, 5.0 and 10.0 percent critical values found for the second DGP is 0.060931 of 1
percent larger than for the first DGP; the difference ranges from 0.1135 to -0.1611 of 1 percent.
Other Approaches to Cross and Serial Correlation. Several papers on mean reversion in
real exchange rates use versions of the alternative DGP, Si,t = pij=1 ci,j Si,t-j + ui,t. Papell’s
(1997) DGP uses non-zero values of ci,j found by fitting the change in the log real rate for each
country as an auto-regression; using the Schwartz criterion, typically an AR1 is found. He uses
SUR to fit
(B)
yi,t = ai + b yi,t-1 + pij=1 ci,j yi,t-j + ui,t
on generated data to find tb critical values. In fitting (B), pi range from 11 to 22 on monthly data.
On the one hand, AR estimates generally suggest pi = 1. On the other, critical values are for pi =
11 to 22; if these pi cause over-fitting, then as discussed above, size and power are likely harmed.
O’Connell (1998) uses the DGP
(A’’) yi,t = ui,t,
ui,t = pj=1 cj ui,t-j + vi,t,
vi,t  N(0, SUR).
To find DGP parameters, he estimates (A’’) with SUR on the data, with the ci,j restricted to be
the same across countries—homogeneous dynamics—to conserve degrees of freedom; the
residual covariance matrix is used as SUR. His p’s range across panels from p = 3 to 12. The test
equation, yi,t = ai + b yi,t + ui,t, with ui,t = pj=1 cj ui,t-j + vi,t, vi,t  N(0, SUR), is fit for the same
p’s. For the present paper, G-10 log nominal rate changes clearly have heterogeneous dynamics,
or the ci,j differ across i, and require substantially smaller p’s, p = 0 or 1.
Sarno and Taylor (1998) use a DGP with ui,t  N(0, SUR). For SUR, they use the
covariance matrix of residuals found by separately fitting yi,t = ai + 4j=1 ci,j yi,t-j + ui,t for each
country with OLS. Their DGP is
(A’’’) yi,t = ai + 4j=1 c’i,j yi,t-j + ui,t,
ui,t  N(0, SUR),
25
where the c’i,j and ai are estimated. Test-statistics’ critical values are found by fitting
(B)
yi,t = ai + 4j=1 ci,j yi,t-j + ui,t
with SUR on generated data. The consequences if p = 4 is too large are clear from above. From
Appendix 2’s discussion, using ai  0 in the DGP gives critical values that are importantly
smaller in absolute value than if ai = 0 in the DGP. If the true ai are zero, or are much closer to
zero on average than the estimated values used, the test that includes the estimated ai is biased
towards rejecting the null; its actual size is larger than its nominal size.
26
Appendix 2: Power of the Tests Against Selected Alternatives
The tb test statistic has substantial power against the null that all log nominal rates are
I(1). Further, the text’s rejections of the null are substantially more likely to arise because all
rather than only some rates are I(0).
For power experiments, data are generated to correspond to the long sample period,
where the error covariance matrix in the DGPs is the sample covariance matrix of the lnSi,t for
1974:01-1996:12. For the generated data, tests results are evaluated with the text’s critical values.
Results are shown for cases where I(0) series have a root of 0.95, 0.975 and 0.985,
corresponding to adjustment speeds of 5.0, 2.5 and 1.5%/month; results in Section 4 suggest
adjustment speeds of 2.5% or 1.5%. Results are also shown for the case where a single currency
has a unit root, either the Belgian Franc to represent the European Union currencies that show
high correlations (Belgium, France, Germany, Italy and the Netherlands), or the U.K. pound.
Power When All Currencies Are I(0). Table A.1 shows the percentage of cases in which
the unit-root null is rejected. For all roots equal to 0.95, the tb test rejects in 100.0% of the cases
at all significance levels. For roots of 0.985, at the 10 percent significance level the tb test rejects
in 80.33% of the cases. The table also shows results for the case where the roots vary around a
grand mean, , with the first root  - 0.1, the second  + 0.1, etc.; power tends to fall off as
compared to the cases where all roots are equal, but remains substantial.
Power When Some Currencies Are I(1). If the null hypothesis—all currencies are I(1)—
is false, the observer may care whether all currencies are I(0) or only a subset. Sarno and Taylor
(1998) use a Wald test of the null that all are I(1), and a Johansen test of the null that at most one
of the currencies is I(1). Karlsson and Löthgren (2000) use simulation to examine the power of
systems tests against the null that all series are I(1) when only some series are I(1).
27
Table A.1 shows the percentage of cases in which the null that all currencies are I(1) is
rejected, when the truth is that only the Belgian Franc is I(1)—ceteris paribus, systems tests are
most likely to reject when only one currency is I(1), rather than several. The other nine countries
all have roots of 0.95, 0.975, or 0.985. For roots of 0.95, the tb test rejects the null at the 5 percent
significance level in 23.10% of cases, versus 100% when all roots are I(1). As expected, the
fraction of rejections declines as the roots increase. For roots of 0.985, at the 5 percent level the
test rejects 10.35%, versus 64.50% when all roots are I(1).
When the only currency with a unit root is the pound, Table A.1 shows the percentage of
rejections for the case where the other currencies have a root of 0.985; at the 5 percent level the
test rejects in 38.81% of the cases, as compared to 10.35% when the Belgian Franc has the unit
root. In general, if the only currency with a unit root is one of the highly correlated EU currencies
(Belgium, France, Germany, Italy and the Netherlands), the test seldom rejects; the test rejects
more frequently if a currency not in this group has the unit root.
In Table 1 in the text, the test rejects the null in five cases at the 0.5% level. From Table
A.1, these rejections seem unlikely for large roots, say 0.985, if even one rate is I(1).
Sample Means in the DGP. Table A.2 compares the critical values found when the DGP
is lnSi,t = ui,t with those found when the DGP is lnSi,t = ai + ui,t, where the ai are the sample
means for the 1974:01-1996:12 period. The critical values fall substantially in absolute value
when sample means are included in the DGP, and the null is much more likely to be rejected.
The text takes the more conservative position of using critical values found in simulations where
the DGP has zero means and are thus less likely to reject the unit-root null.
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Table A.1. Power of the System Test a
Percentage of Rejections (10,000 replications)
Roots
Sig. Level
0.5%
1
5
10
No Unit Roots
0.95 0.975 0.985
Root =   0.1
0.975 0.985
One Unit Root
Belgium
U.K.
0.95 0.975 0.985
0.985
100.0
100.0
100.0
100.0
71.25
75.13
83.83
87.71
12.78
15.08
23.10
27.90
78.55
86.23
97.94
99.43
21.55
29.83
64.50
80.33
34.77
38.25
49.62
56.43
6.240
8.050
16.01
21.12
2.320
3.610
10.35
15.72
10.61
16.13
38.81
52.83
Table A.2. Critical Values: DGPs with Zero Means; DGPs with Sample Means
0.5%
1.0
5.0
10
15
Means equal zero
Sample Means
-6.293072
-6.081252
-5.514822
-5.204722
-4.987682
-5.127561
-4.885833
-4.218519
-3.824502
-3.592897
Notes:
The log nominal rate changes are generated as lnSi,t = (i - 1) lnSi,t-1 + ui,t, where i is the root for i
and ui,t is an error. In the DGP, the error covariance matrix is the same as for the lnSi,t in the sample for
1974:01 to 1996:12. Slopes are found by estimating the systems lnSi,t = ai + b lnSi,t-1 + ui,t. For Table
A.1, i = 0.950, 0.975, 0.985 is the common root. In one set of experiments, Belgium has a unit root,
with log nominal rate changes generated as lnSBel,t = uBel,t, and similarly for the case where the U.K.
has the only unit root. In another experiment, the Grand Mean root is 0 <  < 1. Roots alternate from  .01 for the first country, to  + .01 for the second, etc. For  = 0.975, the first root is 0.965, etc.
a
In Table A.2, critical values are shown for the DGP lnSi,t = ui,t, and for the DGP lnSi,t = ai + ui,t, where
the ai are sample means.
29