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Bayesian Model Averaging and Exchange Rate Forecasts
Jonathan H. Wright*
Abstract: Exchange rate forecasting is hard and the seminal
result of Meese and Rogoff (1983) that the exchange rate is well
approximated by a driftless random walk, at least for prediction
purposes, stands despite much effort at constructing other
forecasting models. However, in several other macro and financial
forecasting applications, researchers in recent years have
considered methods for forecasting that effectively combine the
information in a large number of time series. In this paper, I
apply one such method for pooling forecasts from several different
models, Bayesian Model Averaging, to the problem of pseudo
out-of-sample exchange rate prediction. Depending on the
currency-horizon pair, the Bayesian Model Averaging forecasts
sometimes do quite a bit better than the random walk benchmark (in
terms of mean square prediction error), while they never do much
worse. The forecasts generated by this model averaging methodology
are however very close to (but not identical to) those from the
random walk forecast. Keywords: Shrinkage, model uncertainty,
forecasting, exchange rates, bootstrap. JEL Classification: C32,
C53, F31.
* International Finance Division, Board of Governors of the
Federal Reserve System, Washington DC 20551. I am grateful to Ben
Bernanke, David Bowman, Jon Faust, Matt Pritsker and Pietro
Veronesi for helpful comments, and to Sergey Chernenko for
excellent research assistance. The views in this paper are solely
the responsibility of the author and should not be interpreted as
reflecting the views of the Board of Governors of the Federal
Reserve System or of any person associated with the Federal Reserve
System.
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1. Introduction.
Out-of-sample forecasting of exchange rates is hard. Meese and
Rogoff (1983) argued
that all exchange rate models do less well in out-of-sample
forecasting exercises than a
simple driftless random walk. Although this finding was heresy
to many at the time that
Meese and Rogoff wrote their seminal paper, it has now become
the conventional
wisdom. Mark (1995) claimed that a monetary fundamentals model
can generate better
out-of-sample forecasting performance at long horizons, but that
result has been found to
be very sensitive to the sample period (Groen (1999), Faust,
Rogers and Wright (2003)).
Claims that a particular variable has predictive power for
exchange rates crop up
frequently, but these results typically apply just to a
particular exchange rate and a
particular subsample. As such, they are by now met with
justifiable skepticism and are
thought of by many as the result of data-mining exercises.
However, in many contexts, researchers have recently made
substantial progress
in the econometrics of forecasting using large datasets (i.e. a
large number of predictive
variables). The trick is to combine the information in these
different variables is
combined in a judicious way that avoids the estimation of a
large number of unrestricted
parameters. Bayesian VARs have been found to be useful in
forecasting: these often use
many time series, but (loosely) impose a prior that many of the
coefficients in the VAR
are close to zero. Approaches in which the researcher estimates
a small number of
factors from a large dataset and forecasts using these estimated
factors have also been
shown to be capable of superior predictive performance (see for
example Stock and
Watson (2002a) and Bernanke and Boivin (2003)). Stock and Watson
(2001, 2002b)
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obtained good results in out-of-sample prediction of
international output growth and
inflation by taking forecasts from a large number of different
simple models and just
averaging them. They found the good performance of simple model
averaging to be
remarkably consistent across subperiods and across countries.
The basic idea that
forecast combination outperforms any individual forecast is part
of the folklore of
economic forecasting, going back to Bates and Granger (1969). It
is of course crucial to
the result that the researcher just average the forecasts (or
take a median or trimmed
mean). It is in particular tempting to run a forecast evaluation
regression in which the
weights on the different forecasts are estimated as free
parameters. While this leads to a
better in-sample fit, it gives less good out-of-sample
prediction.
Bayesian Model Averaging is another method for forecasting with
large datasets
that has received considerable recent attention in both
statistics and econometrics
literatures. The idea is to take forecasts from many different
models, and to assume that
one of them is the true model, but the researcher does not know
which this is. The
researcher starts from a prior about which model is true
(perhaps that all are equally
likely) and computes the posterior probabilities that each model
is true. The forecasts
from all the models are then weighted by these posterior
probabilities. It has been used in
a number of econometric applications, including output growth
forecasting (Min and
Zellner (1993), Koop and Potter (2003)), cross-country growth
regressions (Doppelhofer,
Miller and Sala-i-Martin (2000) and Fernandez, Ley and Steel
(2001)) and stock return
prediction (Avramov (2002) and Cremers (2002)). Avarmov and
Cremers both report
improved pseudo-out-of-sample predictive performance from
Bayesian model averaging.
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The contribution of this paper is to argue that Bayesian Model
Averaging is useful
for out-of-sample forecasting of exchange rates in the
1990s.
One does not have to be a subjectivist Bayesian to believe in
the usefulness of
Bayesian Model Averaging, or of Bayesian shrinkage techniques
more generally. A
frequentist econometrician can interpret these methods as
pragmatic devices that can be
useful for out-of-sample forecasting in the face of model and
parameter uncertainty.
The plan for the remainder of the paper is as follows. In
section 2, I shall describe
the idea of Bayesian Model Averaging. The out-of-sample exchange
rate prediction
exercise is described in section 3. Section 4 concludes.
2. Bayesian Model Averaging
The idea of Bayesian Model Averaging was set out by Leamer
(1978), and has recently
received a lot of attention in the statistics literature,
including in particular Raftery,
Madigan and Hoeting (1997), Hoeting, Madigan, Raftery and
Volinsky (1999) and
Chipman, George and McCulloch (2001).
Consider a set of n models 1,... nM M . The ith model is indexed
by a parameter
vector - this is a different parameter vector for each model,
but for compactness of notation I do not explicitly subscript by i.
The researcher knows that one of these models is the true model,
but does not know which one. The assumption that one of the
models is true is of course unrealistic, but it may still be a
useful fiction for getting good
forecasting results (for recent work considering Bayesian Model
Averaging when none of
the models is in fact true, see Bernardo and Smith (1994) and
Key, Perrichi and Smith
(1998)). The researcher has prior beliefs about the probability
that the ith model is the
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true model which we write as ( )iP M , observes data D, and
updates her beliefs to
compute the posterior probability that the ith model is the true
model:
1
( | ) ( )( | )( | ) ( )
i ii n
j j j
P D M P MP M DP D M P M=
= (1)
where ( | ) ( | , ) ( | )i i iP D M P D M P M d = is the
marginal likelihood of the ith model, ( | )iP M is the prior
density of the parameter vector in this model and ( | , )iP D M is
the likelihood. Each model implies a forecast. In the presence of
model uncertainty, our overall forecast weights each of these
forecasts
by the posterior for that model. This gives the minimum mean
square error forecast.
This is all there is to Bayesian Model Averaging. The researcher
needs only specify the
set of models, the model priors, ( )iP M , and the parameter
priors, ( | )iP M . The distinction between model uncertainty and
parameter uncertainty is a little
artificial in the sense that one could write all the models as
nested by a sufficiently
general model, and, within this nesting model, the only
uncertainty is then parameter
uncertainty. The Bayesian Model Averaging approach would however
imply very
particular and rather peculiar priors for these parameters.
The models do not have to be linear regression models, but I
shall henceforth
assume that they are. The ith model then specifies that
y X = + where y is a time series that the researcher is trying
to forecast (such as exchange rate
returns), X is a matrix of predictors, is a px1 parameter
vector, 1( ,... ) 'T = is the disturbance vector and T is the
sample size. Motivated by the possibility of overlapping
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data in my subsequent application, I assume that the error term
is an MA(h-1) process
with variance 2 such that
2( , ) , 1t t jh jCov j h
h =
I shall define the models and the model priors in the context of
the empirical
application below. For the parameter priors, I shall take the
natural conjugate g-prior
specification for (Zellner (1986)), so that the prior for
conditional on 2 is 2 1(0, ( ' ) )N X X . For 2 , I assume the
improper prior that is proportional to 21/ .
This is a standard choice of the prior for the error variance,
that was made by Fernandez,
Ley and Steel (2001) and many others. Routine integration
(Zellner (1971)) then yields
the required likelihood of the model
/ 2 // 2( / 2)( | ) (1 ) p T hi TTP D M S
= +
where
1' ' ( ' ) '1
S Y Y Y X X X X Y 2 = +
The prior for is centered around zero and so within each model
the parameter is shrunken towards zero, which corresponds to no
predictability. The extent of this
shrinkage is governed by . A smaller value of means more
shrinkage, and makes the prior more informative, but this may help
in out-of-sample forecasting. Researchers
often try to make the prior as uninformative as possible
(corresponding to a high value of
), but at least in the exchange rate forecasting problem
considered in this paper, a more informative prior turns out to
give better predictive performance.
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One way of thinking about the role of is that it controls the
relative weight of the data and our prior beliefs in computing the
posterior probabilities of different models.
If =0, then ( | )iP D M is equal for all models and so the
posterior probability of each model being true is equal to the
prior probability. The larger is , the more we are willing to move
away from the model priors in response to what we observe in the
data.
3. Exchange Rate Forecasting
Each model for forecasting exchange rates that I consider is of
the form
't h t t te e X + = + (2) where te denotes the log exchange
rate, h is the forecasting horizon, tX is a vector of
regressors, and t is the error term, assumed to satisfy the
restrictions above (so that it is a moving average process of order
h-1). I shall consider prediction of the bilateral
exchange value of the Canadian dollar, pound, yen and mark/euro,
relative to the US
dollar. The models will consist of all possible permutations of
a set of potential predictor variables, including all of these
predictors and none of them, making a total of
2 candidate models. Each of these models will also include a
constant, except for the
model with no predictor variables at all which is simply the
driftless random walk
t h t te e + = Assigning equal prior probability to each model
means that models with a small number
of predictors may receive too little prior weight. A standard
approach is to specify that
the prior probability for a model with predictors is ( ) (1 )iP
M
=
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This is implemented by Cremers (2002) and Koop and Potter
(2003), among others. If
0.5 = , then all the models get equal weight. A smaller value of
this hyperparameter favors smaller models. The probability that the
true model has no predictors (i.e. the
driftless random walk) is (1 ) . The expected number of
predictors is . 3.1 Monthly Financial Dataset
I first consider pseudo-out-of-sample exchange rate prediction
using a dataset of financial
variables as the possible predictors. These data are available
at a monthly frequency, are
available in real-time and are never revised. Data vintage
issues can substantially affect
the results of exchange-rate forecasting exercises, as noted by
Faust, Rogers and Wright
(2003). The predictors are (i) the relative stock prices
(foreign-US) (logs), (ii) the
relative annual stock price growth rate, (iii) the relative
dividend yield, (iv) relative short
term interest rates, (v) the relative term spread (long minus
short term rates, motivated by
the finding of Clarida (2003) and others that term structure is
useful for exchange rate
prediction), (vi) exchange rate returns over the previous month,
and (vii) the sign of
exchange rate returns over the previous month. Data sources are
given in Appendix A.
The data cover the months 1973:01 to 2002:12. The models that I
consider use as
regressors all possible permutations of these 7 predictors
(including all 7 and none at all).
A constant is included in each model, except for the one with no
predictors which is
simply the driftless random walk. This gives a total of 27=128
models.
The pseudo-out-of-sample prediction exercise involves
forecasting the exchange
rate for 1993:01 to 2002:12 as of h months previously, for
3,6,9,12h = . For example, the first 3-month ahead forecast is the
prediction of the exchange rate in 1993:01 that was
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made in 1992:10. Of course, this forecast is constructed only
using data from 1992:10
and earlier.
3.2 Quarterly Macro and Financial Dataset
Although the monthly financial dataset has some advantages, it
is missing a great many
of the variables that researchers claim have predictive power
for exchange rates. To
include these, I switch to quarterly data, and give up on the
real-time feature of the
monthly asset price dataset.
This larger dataset contains all the same variables as the
monthly data, aggregated
to quarterly frequency. In addition it includes (i) relative
real GDP (foreign-US) (logs),
(ii) relative money supply (logs), (iii) the relative price
level (logs), (iv) relative annual
growth rates, (v) relative annual inflation rates, (vi) relative
annual money growth rates,
(vii) the relative ratio of current account to GDP and (viii)
relative labor productivity
(logs).
The data cover the quarters 1973:1 to 2002:4. The models I
consider have as
regressors all possible permutations of these 15 basic predictor
variables (including all 15
and none at all). A constant is included in each model, except
for the one with no
predictors which is simply the driftless random walk. This gives
a total of 215=32,768
models. Many standard models for forecasting exchange rates,
such as the monetary
fundamentals model of Mark (1995) and the sticky price monetary
fundamentals model
of Dornbusch (1976) are included in the set of models as these
are just particular
permutations of the 15 regressors (for example the Mark model
includes relative prices,
relative money supply and relative output). I cannot however
include the Mark
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fundamentals as one of the basic predictor variables because
this would induce a problem
of perfect multicollinearity.
3.3 Basic Results for Bayesian Model Averaging
I first considered the out-of-sample mean square prediction
error of the forecast obtained
by Bayesian Model Averaging using the monthly dataset, relative
to the out-of-sample
mean square prediction error for the forecast assuming that the
exchange rate is a driftless
random walk. Table 1 shows this relative out-of-sample root mean
square prediction
error (RMSPE) in the monthly dataset, for various values of the
hyperparameters and . A number greater than 1 means that Bayesian
Model Averaging is forecasting less well than a random walk. For
sterling, the out-of-sample RMSPE is nearly uniformly
slightly above 1 indicating that the random walk gives better
forecasts. But for the other
three currencies, the RMSPE is nearly uniformly below 1 in the
monthly dataset,
indicating that Bayesian Model Averaging gives better forecasts.
For small values of
and , the RMSPE is close to 1. As either of these
hyperparameters is raised, the relative
performance of Bayesian Model Averaging often improves, but for
larger values of
and , it can be erratic. Overall good results are obtained for
=1 and =0.1. In this case,
Bayesian Model Averaging can help quite a bit, and never hurts
much, relative to the
driftless random walk benchmark. Depending on the
currency-horizon pair, the mean
square prediction error may be 10% lower than that from the
random walk forcecast and
is never more than 1.4% higher.
The corresponding results for the quarterly dataset are shown in
Table 2. The
addition of the macro variables in the quarterly dataset in most
cases improves predictive
performance, though in some cases actually degrades it slightly.
In the quarterly dataset,
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the relative performance of Bayesian Model Averaging can be very
erratic for larger
values of and . But consistently good results are obtained for
=1 and =0.1, and
these results are better than their counterparts in the monthly
dataset except for sterling at
horizons of 2 quarters and longer and the one-year ahead
Canadian dollar forecast. In the
quarterly dataset, with small values of and , the Bayesian Model
Averaging procedure
gives better forecasts than the random walk for all
currency-horizon pairs, except for
sterling at horizons of 2 quarters and longer and even in this
case it does not
underperform the random walk by much.
Small values of and entail substantial shrinkage and give the
best results, or at
least the most consistent results. In this sense it does not pay
to make the prior as
uninformative as possible.
For any one currency-horizon pair and any one choice of and ,
the computation
in the quarterly dataset takes about 3 minutes on a 2.5 Ghz
computer. If the number of
models were much larger, it would be necessary to use simulation
based methods for
implementing Bayesian Model Averaging instead (see Madigan and
York (1995) and
Geweke (1996)).
Bootstrap p-values of the hypothesis that the out-of-sample
RMSPE is equal to
one in the monthly dataset are shown in Table 3. In each
bootstrap sample an artificial
dataset is generated in which the exchange rate is by
construction a driftless random
walk, using the bootstrap methodology described in Appendix B.
The p-values in Table
3 represent the proportion of bootstrap samples for which the
RMSPE is smaller than that
which was actually observed in the data. These are therefore
one-sided p-values, testing
the null of equal predictability against the alternative that
Bayesian Model Averaging
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gives a significant improvement over the driftless random walk.
The null is rejected for
several currency-horizon-hyperparameter combinations at
conventional significance
levels. In some other cases, the bootstrap p-values are between
0.1 and 0.2 which is at
least somewhat noteworthy, given the renowned difficulty of
exchange rate forecasting.
In the baseline case 1 = , 0.1 = , the p-values are 0.18 or
better at all horizons for all currencies except the pound.
Obtaining bootstrap p-values of the hypothesis that the
out-of-sample RMSPE is
one in the quarterly dataset was computationally slow because of
the large number of
models and I have therefore not simulated these p-values for all
hyperparameter
combinations. In Table 4, I report the bootstrap p-values in the
baseline case 1 = , 0.1 = for all currency-horizon pairs for the
quarterly dataset. Again, these p-values are
0.18 or better at all horizons for all currencies except the
pound, and are significant at
conventional significance levels in some cases.
Researchers are rightly suspicious of significant p-values in a
test of the
hypothesis that a particular model forecasts the exchange rate
better than a random walk.
The key reason is that these p-values ignore the data mining
that was implicit in choosing
the particular model to use. Researchers publish the results of
these tests only if they find
a model which forecasts the exchange rate significantly better
than a random walk, and
thus significant results can be expected to crop up from time to
time even if the
exchange rate is totally unpredictable. But to the extent that
the Bayesian Model
Averaging approach is starting out with a set of models that
spans the space of all models
researchers would ever want to consider, the results and
specifically the p-values in the
forecast comparison test are then immune to any such data-mining
critique. Clearly, I do
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not wish to claim that the set of models I am using does indeed
span the space of all
conceivable models, but still Bayesian Model Averaging does
mitigate the very
legitimate concern about data mining.
Bayesian Model Averaging forecasts are not necessarily very
different from
random walk forecasts. The driftless random walk forecast is of
course for no change in
the exchange rate. The root mean square forecast exchange rate
change in the Bayesian
Model Averaging gives a metric for how different this forecast
is from a random walk
forecast. I report this root mean square forecast exchange rate
change in Tables 5 and 6
for the monthly and quarterly datasets, respectively. In
general, the higher is or and the longer is the forecast horizon,
the larger is the magnitude of the forecast exchange
rate changes. But the key thing to note is that generally the
Bayesian Model Averaging
procedure is not forecasting large exchange rate fluctuations.
For 0.1 = and 1 = , where the Bayesian Model Averaging procedure
outperforms the random walk for most
currency-horizon pairs, the root mean square forecast exchange
rate change at a one-year
horizon is at most 1.95 percent.
Since most economists believe that the exchange rate is very
well approximated
by a random walk, this is a reassuring feature of the Bayesian
Model Averaging
procedure.
Bayesian Model Averaging predicts small exchange rate changes.
One question
of some interest is whether it predicts the sign of the exchange
rate change correctly or
not. Among other things, this metric is robust to the
possibility of outliers that
artificially enhance/inhibit predictive performance. Tables 7
and 8 report the proportion
of times that it does predict the correct sign of the exchange
rate change in the monthly
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and quarterly datasets, respectively. Bayesian Model Averaging
predicts the correct sign
more than half the time for the Canadian dollar and mark/euro,
at all horizons and for all
choices of . For the case 0.1 = and 1 = , the proportion of
times that Bayesian Model Averaging predicts the sign of the
Canadian dollar and mark/euro exchange rate
change correctly is at least one standard deviation above a coin
toss at all horizons.
Results for the yen and pound are mixed.
Cheung, Chinn and Pascual (2002) also considered forecasting
exchange rates
over the 1990s, but not using Bayesian methods. They found that
they were able to
predict the direction of exchange rate changes more than half
the time, but typically
underperformed the random walk in mean square error. Meanwhile,
Bayesian Model
Averaging methods are typically outperform the random walk both
in mean square error
and in the sign of the exchange rate change. These results are
all very much consistent
with the idea that model and parameter uncertainty are the
stumbling blocks to exchange
rate forecasting (given that the exchange rate is so close to
being a random walk), and
that the researcher who wants to get good out-of-sample
prediction, rather than in-sample
fit, should use shrinkage methods, such as Bayesian Model
Averaging.
3.4 How Stable Is the Predictive Power of Bayesian Model
Averaging?
In many forecasting applications, when a model does have
predictive power relative to
the naive time series forecast, this tends to be unstable. That
is, the model that has good
predictive power in one subperiod has no propensity to have good
predictive power in
another subperiod. Stock and Watson (2001, 2002b) found severe
instability of this sort
in prediction of inflation and output growth.
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To give a little evidence on whether Bayesian Model Averaging
prediction of
exchange rates suffers from this problem, I computed the
out-of-sample RMSPE in two
subsamples: 1993-1997 and 1998-2002 for all 16 currency-horizon
pairs in both datasets
for the case 0.1 = and 1 = . Figure 1, using a graphical device
adapted from Stock and Watson (2001, 2002b), plots this
out-of-sample RMSPE with the value in the 1998-
2002 subperiod on the vertical axis and the value in the
1993-1997 subperiod on the
horizontal axis. The figures are split into 4 quadrants: a
currency-horizon pair in the
lower left quadrant is forecast by Bayesian Model Averaging
better than a random walk
in both subperiods, a currency-horizon pair in the upper right
quadrant is forecast better
by the random walk in both subperiods, and currency-horizon
pairs in the other quadrants
are forecast better by the random walk in one subperiod but not
in the other.
If the predictive power of Bayesian Model Averaging were highly
unstable, one
would expect to see few observations in the bottom left
quadrant, but many observations
in the top left and bottom right quadrants. There is some such
forecast instability, but it
is not too severe. In both datasets, 9 out of the 16
currency-horizon pairs are in the
bottom left quadrant (consistently better predicted by Bayesian
Model Averaging). None
of the pairs are in the top right quadrant (consistently better
predicted by the random
walk) in either dataset.
3.5 Forecasting at Longer Horizons
In this paper, I have reported exchange rate forecasting at
short to medium horizons, up
to one year. These are the horizons considered by Meese and
Rogoff (1983). More
recent work has claimed that exchange rates are more
forecastable at longer horizons,
although some like Kilian (1999) have argued convincingly that
there is no evidence of
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higher predictability at longer horizons once the issues of
statistical inference in long-
horizon regressions are taken into account. I have also
experimented with using Bayesian
Model Averaging to forecast exchange rates at longer horizons,
but found that its superior
performance relative to the random walk actually becomes much
less consistent at
horizons of two years or more.
3.6 Comparison with Equal Weighted Model Averaging
The efficacy of Bayesian Model Averaging is conceptually related
to the idea that is very
much part of the folklore of the forecasting literature that
taking forecasts from several
different models and simply averaging them gives better
predictions than any one model
on its own (Bates and Granger (1969), Stock and Watson (2001,
2002b)).
I therefore experimented with taking all regression models of
the form of equation
(2) with a single right hand side predictor, plus a constant,
using all the predictors that I
have in the monthly and quarterly datasets, augmented with the
monthly long-term
interest rate and the quarterly monetary fundamentals of Mark
(1995). These last two
predictors could not be included in the Bayesian Model Averaging
exercise using all
possible permutations of regressors because this would have
generated a problem of
perfect multicollinearity. This gives a total of 8 models in the
monthly dataset and 17 in
the quarterly dataset. Needless to say, no one of these models
gives consistently good
forecasting performance. The out-of-sample relative mean square
prediction error
obtained from simply averaging all of these forecasts, relative
to the driftless random
walk benchmark is reported in Table 9. A number greater than 1
means that equal-
weighted model-averaging is forecasting less well than a random
walk. Except for the
Canadian dollar, most entries in these tables are greater than
1. Simple equal-weighted
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model averaging, that is such an effective strategy in many
forecasting contexts, does not
seem to buy us so much in exchange rate forecasting, at least
not with these models.
4. Conclusion
In this paper I have considered a specific approach to pooling
the forecasts from different
models, namely Bayesian Model Averaging, and argued that it
gives promising results for
out-of-sample exchange rate prediction. Bayesian Model Averaging
can help quite a bit
and cannot hurt much. That is, depending on the currency-horizon
pair, the Bayesian
Model Averaging forecasts sometimes do quite a bit better than
the random walk
benchmark in terms of mean square prediction error, while they
never do much worse.
In this paper, my sole focus has been on establishing whether
Bayesian Model
Averaging methods seem to work in the sense of beating random
walk forecasts out-of-
sample. One does not have to believe that the model and
parameter priors I have used in
this paper are genuine subjective prior beliefs to view Bayesian
Model Averaging as
practically useful. Moreover, viewing Bayesian Model Averaging
as practically useful
does not rule out the possibility that other related shrinkage
techniques are useful too. I
find that the results on using Bayesian Model Averaging for
out-of-sample exchange rate
prediction are at least promising, and stand in marked contrast
to many results obtained in
the exchange rate forecasting literature without using shrinkage
methods.
The forecasts generated by Bayesian model averaging are however
very close to
those from a random walk forecast.
In future work, it would be possible and interesting to include
nonlinear models,
notably Markov switching and threshold models in the model
averaging exercise. Such
models have been considered by Engel and Hamilton (1990), Meese
and Rose (1991) and
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Kilian and Taylor (2003) among others. These models may contain
information that
could make them useful as elements of a forecast pooling
exercise such as Bayesian
Model Averaging.
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Appendix A: Data Sources Exchange Rates: Monthly Average Rates,
OECD Main Economic Indicators (MEI).
Long Term Interest Rates: 10 year rates from MEI and IMF
International Financial Statistics (IFS).
Short Term Interest Rates: 3-month rates, except call money rate
for Japan, all from MEI.
Prices*: CPI from MEI. Not seasonally adjusted.
GDP*: MEI (real, sa).
Labor Productivity*: GDP (real, sa) divided by total civilian
employment (sa), both from MEI.
Money Stock*: M1 from MEI (sa).
Monetary fundamentals*: Computed from money stock, GDP and
prices.
Stock Prices: Morgan Stanley Capital International (MSCI) price
indices (local currency).
Dividend Yields: Computed from MSCI price and total return
indices.
Current Account*: MEI (sa).
Current Account/GDP ratio*: Current Account (sa) divided by GDP
(nominal, sa). both from MEI. *: included in quarterly dataset
only.
Appendix B: Construction of Bootstrap Samples To construct
bootstrap samples in the monthly dataset, I fitted a VAR(12) to the
exchange rate, log relative stock prices, the relative dividend
yield, relative short term interest rates, the relative term
spread. I estimated this 5-variable VAR but anchored the bootstrap
at the bias-adjusted estimates of Kilian (1998), not the OLS
estimates. I also imposed that all the coefficients in the exchange
rate equation were equal to zero, except for the coefficient on the
first lag of the exchange rate which I set to one. So the exchange
rate is a driftless random walk by construction (the null
hypothesis of no forecastability is imposed). I then generated 500
bootstrap samples of all of the variables in this VAR. All of the
predictors in the monthly dataset can be constructed from these 6
variables. So in this way I get a bootstrap sample of the exchange
rate and all of the predictors in which the exchange rate is a
random walk (but may affect future values of the predictors and so
is not strictly exogenous).
In the quarterly dataset, I fitted a VAR(4) to the exchange
rate, log relative stock prices, the relative dividend yield,
relative short term interest rates, the relative term spread,
relative log GDP, relative log money supply, relative log prices,
relative labor productivity and the relative ratio of current
account to GDP, and then proceeded in exactly the same way as for
the monthly dataset.
-
19
References
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20
Groen, J.J. (1999): Long Horizon Predictability of Exchange
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American Statistical Association, 92, pp.179-191.
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-
21
Stock, J.H. and M.W. Watson (2002a): Forecasting Using Principal
Components from a Large Number of Predictors, Journal of the
American Statistical Association, 97, pp.1167-1179.
Stock, J.H. and M.W. Watson (2002b): Combination Forecasts of
Output Growth in a Seven Country Dataset, mimeo.
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Econometrics, Wiley, New York.
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Bayesian Regression Analysis with g-Prior Distributions, in P.K.
Goel and A. Zellner (eds.), Bayesian Inference and Decision
Techniques: Essays in Honour of Bruno de Finetti, North Holland,
Amsterdam.
-
Tab
le 1
: Out
-of-s
ampl
e R
MSP
E fo
r B
ayes
ian
Mod
el A
vera
ging
M
onth
ly F
inan
cial
Dat
a C
urre
ncy
Hor
izon
=
20
=5
=1
=0.
5
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
C
anad
ian
$ 3
mon
ths
0.98
4 0.
991
0.99
5 0.
973
0.98
0 0.
988
0.96
8 0.
978
0.98
6 0.
975
0.98
3 0.
990
6
mon
ths
0.96
6 0.
982
0.99
1 0.
935
0.96
1 0.
978
0.93
0 0.
955
0.97
3 0.
946
0.96
6 0.
980
9
mon
ths
0.95
3 0.
976
0.98
8 0.
912
0.94
7 0.
970
0.90
7 0.
939
0.96
4 0.
928
0.95
4 0.
973
12
mon
ths
0.94
2 0.
971
0.98
6 0.
893
0.93
7 0.
965
0.89
2 0.
930
0.95
9 0.
917
0.94
6 0.
969
Mar
k/E
uro
3 m
onth
s 1.
008
1.00
2 0.
993
0.97
2 0.
971
0.96
9 0.
934
0.94
4 0.
955
0.95
4 0.
967
0.97
9
6 m
onth
s 0.
873
0.87
5 0.
899
0.84
6 0.
853
0.87
5 0.
861
0.89
9 0.
935
0.91
9 0.
952
0.97
4
9 m
onth
s 0.
834
0.89
4 0.
942
0.80
2 0.
852
0.90
7 0.
861
0.91
6 0.
954
0.92
3 0.
960
0.98
0
12 m
onth
s 0.
840
0.92
3 0.
963
0.77
6 0.
870
0.93
2 0.
861
0.92
7 0.
964
0.92
3 0.
964
0.98
3 Y
en
3 m
onth
s 0.
991
0.99
4 0.
996
0.98
7 0.
990
0.99
3 0.
987
0.99
1 0.
994
0.99
0 0.
994
0.99
6
6 m
onth
s 0.
980
0.98
7 0.
992
0.97
6 0.
981
0.98
7 0.
983
0.98
8 0.
992
0.98
8 0.
993
0.99
6
9 m
onth
s 0.
978
0.98
7 0.
993
0.96
9 0.
978
0.98
7 0.
974
0.98
3 0.
990
0.98
2 0.
989
0.99
4
12 m
onth
s 0.
994
0.99
6 0.
998
0.98
8 0.
991
0.99
5 0.
983
0.98
9 0.
994
0.98
5 0.
992
0.99
6 P
ound
3
mon
ths
1.01
9 1.
004
1.00
1 1.
044
1.00
9 1.
003
1.00
8 1.
001
1.00
0 0.
999
0.99
9 0.
999
6
mon
ths
1.03
9 1.
010
1.00
4 1.
076
1.02
4 1.
008
1.03
5 1.
014
1.00
6 1.
017
1.00
7 1.
003
9
mon
ths
1.01
8 1.
004
1.00
1 1.
050
1.01
4 1.
004
1.03
4 1.
012
1.00
4 1.
019
1.00
7 1.
003
12
mon
ths
1.00
8 1.
001
1.00
0 1.
031
1.00
5 1.
001
1.02
3 1.
005
1.00
1 1.
012
1.00
3 1.
000
Not
es: T
his
Tabl
e re
ports
the
out-o
f-sa
mpl
e m
ean
squa
re p
redi
ctio
n er
ror
from
the
fore
cast
s ta
ken
by B
ayes
ian
Mod
el A
vera
ging
, re
lativ
e to
the
mea
n sq
uare
pre
dict
ion
erro
r fro
m a
drif
tless
rand
om w
alk
fore
cast
. A
num
ber l
ess
than
1 m
eans
that
Bay
esia
n M
odel
A
vera
ging
pre
dict
s be
tter
than
the
ran
dom
wal
k be
nchm
ark.
Th
e m
odel
s us
ed i
n th
e B
ayes
ian
Mod
el A
vera
ging
pro
cedu
re a
re
desc
ribed
in th
e te
xt.
-
Tab
le 2
: Out
-of-s
ampl
e R
MSP
E fo
r B
ayes
ian
Mod
el A
vera
ging
Q
uart
erly
Dat
a C
urre
ncy
Hor
izon
=
20
=5
=1
=0.
5
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
C
anad
ian
$ 1
quar
ter
1.23
2 1.
087
0.98
6 1.
145
1.03
2 0.
968
0.95
3 0.
948
0.95
2 0.
955
0.96
0 0.
967
2
quar
ters
1.
102
0.98
8 0.
983
1.08
3 0.
970
0.96
2 0.
950
0.94
4 0.
955
0.95
1 0.
955
0.96
6
3 qu
arte
rs
0.97
7 0.
956
0.96
9 0.
989
0.93
6 0.
945
0.93
3 0.
927
0.94
2 0.
937
0.94
2 0.
956
4
quar
ters
1.
012
0.98
5 0.
987
1.03
3 0.
976
0.97
1 0.
970
0.94
8 0.
955
0.96
0 0.
952
0.96
2 M
ark/
Eur
o 1
quar
ter
1.09
5 1.
106
1.10
6 1.
021
1.02
9 1.
032
0.93
2 0.
935
0.94
3 0.
940
0.95
0 0.
964
2
quar
ters
0.
873
0.87
8 0.
871
0.82
1 0.
828
0.83
5 0.
819
0.84
3 0.
883
0.88
6 0.
919
0.95
2
3 qu
arte
rs
0.73
9 0.
768
0.82
9 0.
710
0.74
6 0.
802
0.79
9 0.
857
0.91
5 0.
884
0.93
1 0.
965
4
quar
ters
0.
710
0.82
5 0.
918
0.69
0 0.
776
0.87
4 0.
821
0.89
3 0.
946
0.89
8 0.
947
0.97
6 Y
en
1 qu
arte
r 1.
054
1.04
7 1.
039
1.02
7 1.
019
1.01
5 0.
982
0.98
3 0.
987
0.98
0 0.
985
0.99
0
2 qu
arte
rs
1.00
3 0.
986
0.98
7 0.
986
0.97
2 0.
975
0.95
7 0.
967
0.97
9 0.
966
0.97
8 0.
987
3
quar
ters
0.
953
0.96
6 0.
982
0.93
7 0.
945
0.96
5 0.
927
0.95
2 0.
973
0.94
5 0.
968
0.98
3
4 qu
arte
rs
0.98
8 0.
992
0.99
7 0.
971
0.97
7 0.
988
0.94
5 0.
969
0.98
5 0.
954
0.97
6 0.
989
Pou
nd
1 qu
arte
r 0.
970
0.97
3 0.
987
0.96
3 0.
958
0.97
5 0.
940
0.96
3 0.
980
0.95
6 0.
975
0.98
7
2 qu
arte
rs
1.13
4 1.
042
1.01
3 1.
179
1.08
0 1.
030
1.08
4 1.
044
1.02
1 1.
047
1.02
5 1.
013
3
quar
ters
1.
087
1.02
3 1.
008
1.16
4 1.
059
1.02
1 1.
093
1.04
5 1.
020
1.05
6 1.
029
1.01
4
4 qu
arte
rs
1.06
6 1.
017
1.00
6 1.
150
1.04
9 1.
017
1.10
0 1.
046
1.02
0 1.
062
1.03
1 1.
014
Not
es: T
his
Tabl
e re
ports
the
out-o
f-sa
mpl
e m
ean
squa
re p
redi
ctio
n er
ror
from
the
fore
cast
s ta
ken
by B
ayes
ian
Mod
el A
vera
ging
, re
lativ
e to
the
mea
n sq
uare
pre
dict
ion
erro
r fro
m a
drif
tless
rand
om w
alk
fore
cast
. A
num
ber l
ess
than
1 m
eans
that
Bay
esia
n M
odel
A
vera
ging
pre
dict
s be
tter
than
the
ran
dom
wal
k be
nchm
ark.
Th
e m
odel
s us
ed i
n th
e B
ayes
ian
Mod
el A
vera
ging
pro
cedu
re a
re
desc
ribed
in th
e te
xt.
-
Tab
le 3
: Tes
t Tha
t Bay
esia
n A
vera
ging
& R
ando
m W
alk
Hav
e O
ut-o
f-Sa
mpl
e R
MSP
E o
f 1
Mon
thly
Fin
anci
al D
ata
(p
-val
ues)
C
urre
ncy
Hor
izon
=
20
=5
=1
=0.
5
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
C
anad
ian
$ 3
mon
ths
0.06
0.
06
0.06
0.
06
0.06
0.
05
0.04
0.
03
0.04
0.
03
0.03
0.
03
6
mon
ths
0.05
0.
06
0.06
0.
04
0.05
0.
05
0.03
0.
03
0.04
0.
02
0.03
0.
03
9
mon
ths
0.05
0.
06
0.06
0.
04
0.04
0.
05
0.03
0.
03
0.04
0.
03
0.03
0.
03
12
mon
ths
0.05
0.
06
0.05
0.
04
0.05
0.
05
0.03
0.
04
0.04
0.
03
0.03
0.
04
Mar
k/E
uro
3 m
onth
s 0.
62
0.51
0.
04
0.04
0.
03
0.01
0.
00
0.00
0.
00
0.00
0.
00
0.00
6 m
onth
s 0.
00
0.00
0.
00
0.00
0.
00
0.00
0.
00
0.00
0.
00
0.00
0.
01
0.02
9 m
onth
s 0.
00
0.01
0.
01
0.00
0.
00
0.01
0.
01
0.02
0.
02
0.02
0.
04
0.06
12 m
onth
s 0.
01
0.02
0.
03
0.01
0.
02
0.02
0.
02
0.03
0.
05
0.04
0.
07
0.10
Y
en
3 m
onth
s 0.
05
0.04
0.
03
0.06
0.
05
0.05
0.
08
0.07
0.
07
0.10
0.
09
0.09
6 m
onth
s 0.
06
0.04
0.
04
0.08
0.
06
0.05
0.
13
0.11
0.
11
0.14
0.
13
0.13
9 m
onth
s 0.
08
0.08
0.
08
0.09
0.
08
0.08
0.
13
0.13
0.
12
0.14
0.
15
0.15
12 m
onth
s 0.
15
0.15
0.
16
0.16
0.
15
0.16
0.
18
0.18
0.
18
0.19
0.
19
0.20
P
ound
3
mon
ths
0.85
0.
72
0.66
0.
93
0.77
0.
62
0.56
0.
37
0.31
0.
33
0.30
0.
28
6
mon
ths
0.87
0.
79
0.73
0.
91
0.82
0.
74
0.85
0.
78
0.70
0.
79
0.71
0.
64
9
mon
ths
0.66
0.
54
0.47
0.
74
0.60
0.
51
0.73
0.
61
0.52
0.
68
0.59
0.
51
12
mon
ths
0.41
0.
32
0.29
0.
54
0.38
0.
32
0.53
0.
43
0.37
0.
50
0.43
0.
38
N
otes
: Thi
s Ta
ble
repo
rts th
e bo
otst
rap
p-va
lues
for
a o
ne-s
ided
test
of
the
hypo
thes
is th
at th
e dr
iftle
ss r
ando
m w
alk
and
Bay
esia
n M
odel
A
vera
ging
fore
cast
s ha
ve e
qual
out
-of-
sam
ple
mea
n sq
uare
pre
dict
ion
erro
r. S
peci
fical
ly th
e en
tries
are
the
frac
tion
of b
oots
trap
sam
ples
in
whi
ch th
e R
MSP
E is
bel
ow th
e sa
mpl
e va
lue
as re
porte
d in
Tab
le 1
. The
boo
tstra
p m
etho
dolo
gy is
des
crib
ed in
App
endi
x B
.
-
Table 4: Test That Bayesian Averaging & Random Walk Have
Out-of-Sample RMSPE of 1 Quarterly Data
(p-values, =0.1, =1) Horizon Canadian $ Mark/Euro Yen Pound 1
quarter 0.02 0.00 0.12 0.04 2 quarters 0.06 0.00 0.11 0.85 3
quarters 0.07 0.01 0.12 0.68 4 quarters 0.14 0.03 0.18 0.58
Notes: This Table reports the bootstrap p-values for a one-sided
test of the hypothesis that the driftless random walk and Bayesian
Model Averaging forecasts have equal out-of-sample mean square
prediction error. Specifically the entries are the fraction of
bootstrap samples in which the RMSPE is below the sample value as
reported in Table 2. The bootstrap methodology is described in
Appendix B. For computational reasons, results were simulated for
the case =0.1, =1 only.
-
T
able
5: R
oot M
ean
Squa
re F
orec
ast o
f Exc
hang
e R
ate
Cha
nge
with
Bay
esia
n M
odel
Ave
ragi
ng
Mon
thly
Fin
anci
al D
ata
Cur
renc
y H
oriz
on
=20
=
5 =
1 =
0.5
=0.
2 =
0.1
=0.
05
=0.
2 =
0.1
=0.
05
=0.
2 =
0.1
=0.
05
=0.
2 =
0.1
=0.
05
Can
adia
n $
3 m
onth
s 0.
23
0.12
0.
06
0.34
0.
20
0.12
0.
24
0.15
0.
09
0.16
0.
10
0.06
6 m
onth
s 0.
34
0.17
0.
09
0.55
0.
32
0.18
0.
41
0.26
0.
15
0.28
0.
17
0.10
9 m
onth
s 0.
45
0.22
0.
11
0.74
0.
43
0.24
0.
58
0.36
0.
21
0.40
0.
25
0.14
12 m
onth
s 0.
48
0.23
0.
11
0.87
0.
48
0.26
0.
72
0.44
0.
25
0.50
0.
31
0.18
M
ark/
Eur
o 3
mon
ths
2.59
2.
42
2.18
2.
26
2.14
1.
96
1.13
0.
95
0.73
0.
58
0.40
0.
26
6
mon
ths
3.99
3.
11
2.23
3.
68
3.07
2.
34
1.67
1.
17
0.74
0.
79
0.47
0.
26
9
mon
ths
3.53
2.
19
1.29
3.
75
2.60
1.
64
1.62
0.
98
0.55
0.
77
0.41
0.
21
12
mon
ths
2.56
1.
31
0.67
3.
24
1.87
1.
02
1.45
0.
77
0.40
0.
72
0.35
0.
17
Yen
3
mon
ths
0.78
0.
51
0.30
0.
85
0.64
0.
44
0.51
0.
38
0.26
0.
32
0.23
0.
15
6
mon
ths
1.22
0.
70
0.38
1.
52
1.03
0.
64
0.92
0.
64
0.40
0.
58
0.40
0.
24
9
mon
ths
1.45
0.
78
0.40
2.
01
1.28
0.
74
1.31
0.
88
0.53
0.
85
0.57
0.
34
12
mon
ths
1.53
0.
77
0.39
2.
37
1.42
0.
79
1.69
1.
10
0.65
1.
12
0.73
0.
43
Pou
nd
3 m
onth
s 0.
37
0.14
0.
06
0.64
0.
29
0.14
0.
39
0.21
0.
11
0.24
0.
13
0.07
6 m
onth
s 0.
44
0.16
0.
07
0.90
0.
37
0.17
0.
62
0.32
0.
16
0.39
0.
21
0.11
9 m
onth
s 0.
46
0.18
0.
08
0.98
0.
44
0.21
0.
80
0.42
0.
22
0.53
0.
29
0.16
12 m
onth
s 0.
48
0.20
0.
09
1.08
0.
49
0.23
0.
95
0.51
0.
26
0.65
0.
36
0.19
N
otes
: Thi
s Ta
ble
repo
rts th
e ro
ot m
ean
squa
re f
orec
ast o
f ex
chan
ge r
ate
chan
ges
from
Bay
esia
n M
odel
Ave
ragi
ng.
The
exch
ange
ra
te w
as t
rans
form
ed b
y ta
king
log
s an
d th
en m
ultip
lyin
g by
100
, so
the
elem
ents
in
this
tab
le c
an b
e in
terp
rete
d as
app
roxi
mat
e pe
rcen
tage
poi
nt fo
reca
st c
hang
es.
-
Tab
le 6
: Roo
t Mea
n Sq
uare
For
ecas
t of E
xcha
nge
Rat
e C
hang
e w
ith B
ayes
ian
Mod
el A
vera
ging
Q
uart
erly
Dat
a C
urre
ncy
Hor
izon
=
20
=5
=1
=0.
5
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
C
anad
ian
$ 1
quar
ter
1.34
0.
97
0.64
1.
17
0.86
0.
58
0.48
0.
32
0.24
0.
25
0.17
0.
13
2
quar
ters
1.
20
0.54
0.
29
1.34
0.
69
0.43
0.
61
0.39
0.
27
0.37
0.
25
0.17
3 qu
arte
rs
1.12
0.
63
0.35
1.
33
0.86
0.
56
0.79
0.
55
0.38
0.
51
0.36
0.
24
4
quar
ters
1.
07
0.55
0.
28
1.44
0.
90
0.54
0.
95
0.66
0.
43
0.63
0.
44
0.29
M
ark/
Eur
o 1
quar
ter
2.60
2.
65
2.61
2.
17
2.24
2.
22
1.13
1.
08
0.94
0.
63
0.51
0.
36
2
quar
ters
4.
73
4.72
4.
30
3.93
4.
02
3.76
1.
92
1.67
1.
24
0.98
0.
70
0.42
3 qu
arte
rs
5.54
4.
64
3.22
4.
60
4.23
3.
22
2.00
1.
49
0.93
1.
02
0.64
0.
35
4
quar
ters
4.
92
2.96
1.
54
4.53
3.
28
1.95
1.
89
1.19
0.
65
1.01
0.
57
0.29
Y
en
1 qu
arte
r 2.
05
1.84
1.
57
1.78
1.
58
1.37
0.
93
0.77
0.
62
0.58
0.
45
0.34
2 qu
arte
rs
3.14
2.
24
1.42
2.
91
2.30
1.
67
1.58
1.
20
0.86
0.
99
0.73
0.
51
3
quar
ters
3.
48
2.06
1.
15
3.68
2.
62
1.70
2.
14
1.57
1.
07
1.38
1.
00
0.67
4 qu
arte
rs
3.60
1.
93
1.01
4.
33
2.88
1.
75
2.70
1.
95
1.29
1.
78
1.28
0.
85
Pou
nd
1 qu
arte
r 0.
64
0.24
0.
10
0.77
0.
40
0.19
0.
38
0.22
0.
12
0.22
0.
13
0.07
2 qu
arte
rs
0.66
0.
22
0.08
1.
03
0.45
0.
19
0.58
0.
30
0.16
0.
35
0.19
0.
10
3
quar
ters
0.
61
0.21
0.
09
1.11
0.
47
0.21
0.
72
0.37
0.
20
0.45
0.
24
0.14
4 qu
arte
rs
0.61
0.
22
0.10
1.
22
0.51
0.
24
0.88
0.
45
0.24
0.
57
0.30
0.
17
Not
es: T
his
Tabl
e re
ports
the
root
mea
n sq
uare
for
ecas
t of
exch
ange
rat
e ch
ange
s fr
om B
ayes
ian
Mod
el A
vera
ging
. Th
e ex
chan
ge
rate
was
tra
nsfo
rmed
by
taki
ng l
ogs
and
then
mul
tiply
ing
by 1
00, s
o th
e el
emen
ts i
n th
is t
able
can
be
inte
rpre
ted
as a
ppro
xim
ate
perc
enta
ge p
oint
fore
cast
cha
nges
.
-
Tab
le 7
: Pro
port
ion
of T
imes
Bay
esia
n M
odel
Ave
ragi
ng P
redi
cts C
orre
ct S
ign
of E
xcha
nge
Rat
e C
hang
e M
onth
ly F
inan
cial
Dat
a C
urre
ncy
Hor
izon
=
20
=5
=1
=0.
5
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
C
anad
ian
$ 3
mon
ths
0.58
0.
58
0.58
0.
58
0.58
0.
58
0.58
0.
58
0.58
0.
58
0.58
0.
58
6
mon
ths
0.68
0.
68
0.68
0.
68
0.68
0.
68
0.68
0.
68
0.68
0.
68
0.68
0.
68
9
mon
ths
0.68
0.
68
0.68
0.
68
0.68
0.
68
0.68
0.
68
0.68
0.
68
0.68
0.
68
12
mon
ths
0.74
0.
74
0.74
0.
74
0.74
0.
74
0.74
0.
74
0.74
0.
74
0.74
0.
74
Mar
k/E
uro
3 m
onth
s 0.
60
0.59
0.
59
0.60
0.
60
0.59
0.
59
0.60
0.
60
0.60
0.
60
0.62
6 m
onth
s 0.
64
0.63
0.
63
0.64
0.
63
0.63
0.
65
0.64
0.
64
0.65
0.
69
0.71
9 m
onth
s 0.
67
0.67
0.
66
0.68
0.
67
0.67
0.
69
0.68
0.
69
0.71
0.
72
0.68
12 m
onth
s 0.
66
0.66
0.
66
0.66
0.
66
0.66
0.
73
0.71
0.
69
0.70
0.
71
0.71
Y
en
3 m
onth
s 0.
53
0.50
0.
48
0.55
0.
49
0.49
0.
56
0.50
0.
48
0.55
0.
50
0.47
6 m
onth
s 0.
54
0.53
0.
53
0.56
0.
53
0.52
0.
56
0.52
0.
50
0.55
0.
49
0.44
9 m
onth
s 0.
48
0.49
0.
50
0.47
0.
46
0.49
0.
47
0.44
0.
44
0.46
0.
43
0.44
12 m
onth
s 0.
44
0.48
0.
47
0.40
0.
44
0.47
0.
40
0.40
0.
44
0.38
0.
40
0.45
P
ound
3
mon
ths
0.53
0.
56
0.56
0.
54
0.55
0.
56
0.56
0.
55
0.56
0.
55
0.56
0.
55
6
mon
ths
0.48
0.
46
0.47
0.
49
0.48
0.
47
0.51
0.
49
0.47
0.
50
0.48
0.
47
9
mon
ths
0.48
0.
48
0.49
0.
49
0.48
0.
48
0.50
0.
48
0.48
0.
49
0.48
0.
48
12
mon
ths
0.47
0.
48
0.50
0.
46
0.48
0.
48
0.47
0.
48
0.49
0.
47
0.49
0.
49
Not
es:
Asy
mpt
otic
sta
ndar
d er
rors
can
be
obta
ined
fro
m t
he f
orm
ula
for
the
varia
nce
of a
bin
omia
l di
strib
utio
n, a
djus
ting
for
the
over
lapp
ing
fore
cast
s. T
he st
anda
rd e
rror
s so
obta
ined
var
y, b
ut a
re a
ppro
xim
atel
y 0.
08, 0
.11,
0.1
4 an
d 0.
16 a
t hor
izon
s of 1
, 2, 3
and
4
quar
ters
ahe
ad, r
espe
ctiv
ely.
-
Tab
le 8
: Pro
port
ion
of T
imes
Bay
esia
n M
odel
Ave
ragi
ng P
redi
cts C
orre
ct S
ign
of E
xcha
nge
Rat
e C
hang
e Q
uart
erly
Dat
a C
urre
ncy
Hor
izon
=
20
=5
=1
=0.
5
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
=
0.2
=0.
1 =
0.05
C
anad
ian
$ 1
quar
ter
0.55
0.
58
0.63
0.
53
0.53
0.
58
0.53
0.
53
0.60
0.
55
0.60
0.
63
2
quar
ters
0.
65
0.68
0.
70
0.65
0.
68
0.70
0.
65
0.68
0.
70
0.68
0.
70
0.70
3 qu
arte
rs
0.60
0.
63
0.63
0.
60
0.63
0.
63
0.63
0.
65
0.65
0.
63
0.65
0.
65
4
quar
ters
0.
58
0.58
0.
58
0.60
0.
60
0.63
0.
63
0.63
0.
63
0.63
0.
63
0.63
M
ark/
Eur
o 1
quar
ter
0.55
0.
55
0.55
0.
55
0.55
0.
55
0.55
0.
58
0.55
0.
58
0.55
0.
55
2
quar
ters
0.
70
0.68
0.
68
0.68
0.
68
0.68
0.
68
0.68
0.
68
0.68
0.
68
0.70
3 qu
arte
rs
0.70
0.
68
0.65
0.
70
0.68
0.
65
0.68
0.
68
0.68
0.
70
0.70
0.
70
4
quar
ters
0.
70
0.70
0.
68
0.70
0.
70
0.70
0.
70
0.70
0.
73
0.68
0.
73
0.68
Y
en
1 qu
arte
r 0.
60
0.63
0.
60
0.60
0.
60
0.60
0.
65
0.60
0.
65
0.60
0.
55
0.60
2 qu
arte
rs
0.63
0.
60
0.55
0.
63
0.60
0.
60
0.63
0.
60
0.50
0.
63
0.60
0.
43
3
quar
ters
0.
58
0.58
0.
50
0.58
0.
60
0.53
0.
58
0.58
0.
45
0.58
0.
58
0.43
4 qu
arte
rs
0.48
0.
45
0.50
0.
50
0.48
0.
43
0.50
0.
45
0.48
0.
50
0.38
0.
50
Pou
nd
1 qu
arte
r 0.
58
0.55
0.
55
0.60
0.
55
0.55
0.
58
0.55
0.
50
0.55
0.
53
0.50
2 qu
arte
rs
0.53
0.
53
0.53
0.
53
0.55
0.
53
0.53
0.
53
0.53
0.
50
0.53
0.
53
3
quar
ters
0.
50
0.50
0.
50
0.50
0.
50
0.50
0.
48
0.50
0.
50
0.48
0.
50
0.50
4 qu
arte
rs
0.45
0.
45
0.48
0.
48
0.45
0.
45
0.45
0.
45
0.48
0.
43
0.43
0.
48
Not
es:
Asy
mpt
otic
sta
ndar
d er
rors
can
be
obta
ined
fro
m t
he f
orm
ula
for
the
varia
nce
of a
bin
omia
l di
strib
utio
n, a
djus
ting
for
the
over
lapp
ing
fore
cast
s. T
he st
anda
rd e
rror
s so
obta
ined
var
y, b
ut a
re a
ppro
xim
atel
y 0.
08, 0
.11,
0.1
4 an
d 0.
16 a
t hor
izon
s of 1
, 2, 3
and
4
quar
ters
ahe
ad, r
espe
ctiv
ely.
-
Table 9: Out-of-Sample RMSPE for Equal-Weighted Averaged
Forecasts
Horizon Canadian $ Mark/Euro Yen Pound Monthly Financial
Data
3 months 0.949 1.001 1.006 1.022 6 months 0.883 1.011 1.019
1.036 9 months 0.834 1.016 1.031 1.044 12 months 0.799 1.026 1.055
1.030
Quarterly Data 1 quarter 0.916 0.990 0.990 0.990 2 quarter 0.876
1.010 1.011 1.048 3 quarter 0.840 1.018 1.019 1.055 4 quarter 0.834
1.037 1.055 1.056
Notes: This Table reports the out-of-sample mean square
prediction error from the forecasts taken by simple equal-weighted
averaging of different model forecasts, relative to the mean square
prediction error from a driftless random walk forecast. The models
each have a constant and just one other predictor, as described in
the text.
-
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1993-1997
1998
-200
2
Fig. 1: Out-of-sample Bayesian Model Averaging RMSPE for all
currency-horizon pairs in two subsamples ( =0.1, =1)Monthly
Data
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1993-1997
1998
-200
2
Quarterly Data
