Combining Economic Fundamentals to Predict Exchange Rates
1. Combining Economic Fundamentals to Predict Exchange
Rates
Author:
Brant Munro
University of Guelph
Supervisor:
Ilias Tsiakas
Second Reader:
Alex Maynard
Abstract
This research utilizes statistical and economic methods to evaluate exchange rate predictability
out-of-sample. The forecast performance of five widely used empirical models that predict nine one-
month ahead exchange rates quoted in U.S. dollars is evaluated using these statistical and economic
methods. The ex ante forecasts of the empirical models are also combined using model averaging
techniques so as to eliminate issues of model uncertainty. Designing a portfolio that uses a dynamic
asset allocation strategy that relies on a models one-step ahead forecasts assesses the economic
gains. The analysis finds that statistically and economically the empirical models used for rolling
out-of-sample forecasting do not outperform the benchmark random walk model in predicting the
different exchange rates, and the combination models are not anymore successful. When recursive
estimation is used, the Taylor Rule and most forecast combination models outperform the random
walk model in predicting most exchange rates using the statistical methods. PPP, the Taylor Rule,
and some of the combination forecasts outperform the random walk model economically.
2. 1 Introduction
The practical value of exchange rate predictability is that it provides useful information to
investors in asset allocation, business firms in risk hedging, and governments and central banks for
policy making. The extent to which exchange rates are predictable has significant implications for
efficient market hypothesis and for theoretical modeling in international finance. Macroeconomic
theory has proposed many different models that endogenously determine exchange rates based on
differences in nations’ exogenous economic fundamentals such as the interest rates, prices, money
supply, output, etc. However, much of the literature finds that exchange rates are not forecastable,
starting with the seminal contribution by Meese and Rogoff (1983) that finds that no model that
conditions on economic fundamentals provides more accurate forecasts out-of-sample than the naive
random walk (RW) model that does not condition on economic fundamentals, and has lead to the
prevailing view that exchange rates are not predictable especially at short horizons. Since this
contribution, the RW has become the standard benchmark model in which conditional models that
predict exchange rate returns are compared to. Since Meese and Rogoff (1983) some conditional
models that have been developed have successfully predicted better out-of-sample than the RW
benchmark depending on the forecast horizon and data frequency.
This analysis is following new trends in the literature that go beyond estimating empirical
exchange rate return models that condition on economic fundamentals, as it also uses the ex ante
forecasts of these empirical models and combines them in an attempt to eliminate model uncertainty.
The individual empirical exchange rate return models that will be estimated both in-sample (IS)
and out-of sample (OOS) are fairly general but also well renowned macroeconomic models that have
been estimated in previous literature which are uncovered interest parity (UIP), purchasing power
parity (PPP), monetary fundamentals (MF), and the Taylor Rule (TR). A benchmark random walk
(RW) with drift model will also be estimated. These models will be discussed further in section
2. Model averaging techniques will be utilized on the OOS forecasts of these five empirical models
where the weights are allocated based on simple model averaging like mean, median, and trimmed
mean techniques, as well as other model combinations that allocate weights to the forecasts based
on their previous statistical and economic performance. These forecast combination methods will
be discussed further in section 3. Section 4 will discuss the statistical evaluation of the exchange
rate models, that compares the predictive ability of the alternative models to that of the null RW
model based on the out-of-sample mean squared error (MSE) generated by the forecast models, and
tests whether the difference in MSE between an alternative and the null RW model is statistically
significant.
1
3. This analysis also goes beyond most of the traditional literature in that it not only evaluates
the statistical performance of the conditional exchange rate return models, but also evaluates the
economic value of exchange rate predictability. A statistical analysis of a forecast model’s predictive
performance does not well-inform an investor about the monetary gains that it can accumulate for
them. This is why assessing dynamic asset allocation strategies has become a new trend in the
existing literature. This analysis uses the same dynamic asset allocation strategy as Della Corte
and Tsiakas (2011) and Li, Tsiakas, and Wang (2014), where a U.S. investor is exposed only to
foreign exchange (FX) risk, and can allocate their wealth between domestic and foreign bonds, and
uses the exchange rate forecasts to predict the one-month ahead return on buying foreign bonds,
then uses that information to rebalance the portfolio. It is not the economic value of predicting
an individual exchange rate return that is assessed, but rather the portfolio excess returns that are
generated by each models’ predictions on all exchange rate returns. The economic gains generated
from the alternative and null RW model are compared by determining which model generates the
highest Sharpe Ratio. The FX strategy and method of evaluation is discussed further in section 5.
Section 6 discusses the in-sample (IS) results of our predictive regressions, and the results
from the out-of-sample statistical and economic evaluation of the empirical exchange rate return
and the combined forecast models. The OOS estimation predicts one month ahead exchange rate
returns, and employs monthly FX data ranging from January 1976 to June 2012 for the ten most liq-
uid (G10) and most widely traded currencies in the world which are: the Australian dollar (AUD),
Canadian dollar (CAD), Swiss franc (CHF), Deutsch mark/Euro (EUR), British pound (GBP),
Japanese yen (JPY), Norweigen krone (NOK), New Zealand dollar (NZD), Swedish krona (SEK),
and U.S. dollar (USD). These are the currencies that the analysis attempts to predict one-step
ahead, and are embedded in the portfolio that is used to assess the economic gains of exchange rate
predictability. Section 7 concludes the analysis.
2 Traditional Exchange Rate Predictability Models
In this section the five empirical models that will be employed for exchange rate predictability
will be reviewed. Four different exchange rate return models are estimated that each use a different
economic fundamental as the regressor, as well as a random walk benchmark model. All predictive
regressions have the same linear structure and are defined as follows:
∆st+1 = α + βxjt + t+1 (1)
where ∆st+1 = st+1 −st and st+h is the log of the exchange rate at time t+h. α and β are constants
that will be estimated, subscript j represents the predictive regressor for j = 1...4, that is used to
2
4. forecast exchange rate returns.
2.1 Random Walk (RW) Benchmark
The first model is a Random Walk (RW) with drift (i.e. α is not restricted to zero), β = 0 as
the RW does not condition on any economic fundamentals to predict exchange rate returns. Since
Meese and Rogoff (1983) determined that economic fundamentals could not consistently outperform
the RW model, it has become a standard benchmark model to compare the performance of the
conditional models to in exchange rate predictability literature. The RW captures the prevailing
view in international finance research that economic fundamentals cannot predict exchange rates,
especially at short horizons.
2.2 Uncovered Interest Parity
Fisher (1886) provides a general framework of how interest rates can be related to exchange
rate returns known as Uncovered Interest Parity (UIP). This theory takes the perspective of a
rational and risk neutral investor who knows at time t the domestic interest rate (it+1) and the
foreign interest rate (i∗
t+1), and can choose between investing in either one at time t. If the investor
chooses to invest 1 unit of domestic currency in foreign bonds, the foreign bond will pay the one
unit invested plus the foreign interest rate at time t + 1. After the return on the foreign bond is
realized it is then converted back into domestic currency at a price of (1/St), where (St) is the cost
of buying one unit of foreign currency with domestic currency. The return on holding the foreign
bond is therefore equal to
rt+1 = i∗
t + ∆st+1, (2)
where a return is not just realized on holding the foreign bond but also on the foreign currency
which may be positive or negative, as the investor is only exposed to foreign exchange (FX) risk.
As a result of arbitrage pressures and in the absence of transaction costs, the expected return
on holding the foreign bond should be the same as that of holding the domestic bond, that is
(1 + i∗
t )Et(∆st+1) = 1 + it, where Et(.) denotes the expectation operator at time t. The rational
investor would therefore be indifferent between holding domestic and foreign bonds. According to
this condition we can state our UIP regressor as follows:
x1,t = (it − i∗
t ). (3)
Assuming rational expectations, the UIP condition implies that α = 0 and β = 1 and
t+1 is serially uncorrelated. However, empirical studies have rejected the UIP condition, examples
include Engel (1996) and Sarno (2005). In previous literature, in-sample estimates of α have been
3
5. statistically significant from zero, and estimates of β tend to be closer to minus unity than plus unity.
These counter intuitive results are refered to as the ‘forward premium puzzle,’which imply that
high-interest rate currencies tend to appreciate rather than depreciate over time. Meese and Rogoff
(1983) find unfavorable evidence towards using interest rate differences for forecasting exchange
rate returns out of sample, and that the performance of the random walk is better. Clark and West
(2006) find slightly more encouraging results at short-horizons.
2.3 Purchasing Power Parity (PPP)
The second regressor is based on the PPP hypothesis, which states that the real price of
a comparable basket of commodities should be the same when expressed in a common currency.
That is, a consumer should be indifferent between purchasing goods and services domestically and
abroad as the domestic price level converted into the foreign currency by the nominal exchange
rate, should equal the same price level as that of the foreign country. This theory was developed
by Cassel (1918). The PPP regressor can be stated as follows:
x2,t = pt − p∗
t − st (4)
where pt is the logarithm of the domestic commodity price index (CPI) and p∗
t is that of the
foreign country’s. Rogoff (1996) states that the PPP condition is normally thought of as a long-run
condition rather than holding at each point in time. Out-of-sample evidence is not favorable to
PPP, for example Cheung, Chinn, and Pascual (2005) find that PPP does not out-perform the RW
at short-horizons, but does at long-horizons although it is never significantly better.
2.4 Monetary Fundamentals (MF)
The third regressor conditions on the monetary fundamentals (MF) of the two nations, sum-
marized as follows:
x3,t = (mt − m∗
t ) − (yt − y∗
t ) − st (5)
where mt is the logarithm of domestic money supply, m∗
t is the logarithm of foreign money supply,
yt is domestic real output, and y∗
t is the logarithm of the foreign country’s real output. It has been
well documented that equation (5) does not hold on a period-by-period basis and that deviations
from this relation can be persistent due to nominal rigidities. However, over time there might exist
a tendency for the exchange rate to gradually converge to a new long-run equilibrium in response
to either nominal or real shocks, therefore MF may contain useful information in forecasting future
exchange rates. Chinn and Meese (1995) find that monetary models do not predict well at one-month
to one-year horizons, and that short-run exchange rate variability appears to be disconnected from
the underlying fundamentals according to Mark (1995), although does find statistically significant
evidence in favor of the monetary model at very long horizons (three to five years). Groen (2000)
4
6. finds that fundamentals and nominal exchange rates do in fact move together in the long-run.
2.5 The Taylor Rule (TR)
The Taylor Rule (TR) relates movements of money supply to macroeconomic variables that
policymakers might target. This model is nested by Engel and West’s (2005) present value relation.
By letting πt = pt −pt−1, where πt is the inflation rate, and yg
t be the ”output gap.” We can assume
the domestic central bank follows a TR of the form
it = β1yg
t + β2πt + vt (6)
where β1 > 0, β2 > 1, and the vt term is a random shock. The TR states that the central bank
raises the short-term nominal interest rate when output is above its potential level, and/or inflation
rises above its desired level. The foreign country follows a similar TR except it explicitly targets
exchange rates, as follows:
i∗
t = −β0(st − ¯s∗
t ) + β1y∗
t + β2π∗
t + v∗
t (7)
where, 0 < β0 < 1 and ¯s∗
t is the targeted exchange rate.
Assuming that monetary authorities target the PPP level of the exchange rate so that the
following condition holds:
¯s∗
t = pt − p∗
t (8)
and since st is the amount of domestic currency required to purchase one unit of foreign currency,
the TR implies that the foreign country raises interest rates when the exchange rate depreciates
relative to the target (ceterus paribus). The exchange rate target is omitted from equation (6)
under the assumption that that domestic monetary policy has ignored exchange rates. Subtracting
the foreign TR from the domestic one, we obtain
it − i∗
t = β0(st − ¯s∗
t ) + β1(yg
t − y∗g
) + β2(πt − π∗
t ) + vt − v∗
t (9)
Recall that the UIP condition is given by (Etst+1 − st) = it − i∗
t + ρt, where ρt is the
expectation error term but can also be interpreted as a risk premium. Substituting the UIP condition
and equation (8) into equation (9), we can obtain an expression for st as follows:
β0
1 + β0
(pt − p∗
t ) −
1
1 + β0
[β1(yg
t − y∗
t ) + β2(πt − π∗
t ) + vt − v∗
t + ρt] +
1
1 + β0
Etst+1. (10)
5
7. The asymmetric Taylor Rule (TRa) can now be summarized as follows:
x4,t = 1.5(πt − π∗
t ) + 0.1(yg
t − y∗g
t ) + 0.1(st + pt − p∗
t ). (11)
The parameters on the inflation difference (1.5), the output gap (0.1), and the real exchange rate
(0.1) are fairly standard in the literature, for example Engel, Mark, and West (2007). The domestic
and foreign output gaps are computed via use of the Hodrick and Prescott (1997) (HP) filter 1
,
which is the percent deviation of real output from potential output.
Equation (10) can alternatively be expressed using the interest parity condition and the
equation for the target exchange rate (8), so that the interest rate differential is incorporated as a
fundamental, to account for smoothing, this version of the TR will not be estimated in this paper.
Molodtsova and Papell (2009) show that the Taylor Rule fundamentals forecast exchange rates
out-of-sample significantly better than the random walk for several countries, although it depends
on the exact specification of the model.
3 Forecast Combination Models
The performance of forecast combination models; will be evaluated in addition to the four
conditional forecast models discussed earlier. Considering a large set of conditional models that
capture different aspects of exchange rate behavior without knowing which model is ”true” or ”best,”
creates model uncertainty. Rapach, Strauss and Zhou (2010) argue that forecast combinations can
deliver statistically and economically significant out-of-sample performance because they reduce
forecast volatility relative to the individual forecasts, and they are linked to the real economy. The
combination forecast models that will be explored would attempt to resolve this uncertainty by
exploring the statistical significance of their performance relative to the RW, and whether portfolio
performance improves. Recall that N = 5 predictive equations are estimated that each provide
an individual forecast ∆ˆsi,t+1 for one-step ahead returns, i is the forecast model and i N. The
combined forecast models (∆ˆsc,t+1) are a weighted average of the N individual forecasts (∆ˆsi,t+1)
∆ˆsc,t+1 = ΣN
i=1wi,t∆ˆsi,t+1 (12)
where (wi,t)N
i=1 are the ex ante weights assigned to each of the N different model forecasts determined
at time t.
1
The output gap is estimated via use of the Hodrick and Prescott (HP) filter. The smoothing parameter is set
equal to 14,400, as is suggested for monthly frequencies. When estimating the (HP) trend out-of-sample, for any
given time t only data up until t−1 is used. The trend is then updated every time a new observation is added to the
sample, in order to closely capture the information available at the time a forecast is made and avoids look-ahead
bias
6
8. 3.1 Simple Model Averaging
These forecast combination models use a simple weighting scheme as they do not take into
account the historical performance of the N different forecast models. Three different simple model
averaging schemes are used: (i) the mean of all N forecasts so that wi,t = 1/N; (ii) the median
value of the (∆ˆsi,t+1)N
i=1 individual forecasts where wi,t = 1 on the median forecast and wi,t = 0 on
all other forecasts; (iii) the trimmed mean combination sets wi,t = 0 on the individual models that
forecast the highest and lowest values, and wi,t = 1/(N − 2) on all other forecast models.
3.2 Statistical Model Averaging
Statistical model averaging uses statistical information on the past OOS performance of each
individual model. These types of models are based on a Stock and Watson (2004) model, where
they compute a discounted MSE (DMSE) by setting weights as follows:
wi,t =
DMSE−1
i,t
ΣN
j=1DMSE−1
j,t
(13)
where:
DMSEi,t = ΣT−1
t=M+1θT−1−t
(∆st+1 − ∆ˆsi,t+1)2
(14)
where θ is the discount factor, θ 1, and when θ < 1 greater weight is attached to the most
recent forecast accuracy of the individual models. M represents the first in-sample observations
that are conditioned on to derive the first out-of-sample forecast. Only the case where θ = 1
will be considered as Li, Tsiakas, and Wang (2014) find the cases where θ = (0.9, 0.95) make
a negligible difference in forecast performance. This model will be called the cumulative MSE
forecast combination. A simpler case where ”the most recent best” will also be computed for
statistical model averaging where θ = 1 but the individual forecasts are weighted by their inverse
OOS MSE computed over the last k months, where k = (12, 36, 60).
3.3 Economic Model Averaging
Economic model averaging does not use statistical performance information to form weights
for the individual models, but instead exploits the economic information inherent in the individual
forecast models, it is used by Della Corte and Tsiakas (2011). The Sharpe Ratio (SR) of the excess
returns generated by an individual forecast model over a pre-specified period is used to extract its
economic information. The weights are computed as follows:
wi,t =
SRi,t
ΣN
j=1SRj,t
(15)
7
9. The Sharpe Ratio (SR) at time t is computed using the average excess portfolio returns and the
standard deviation of portfolio returns over the last k months, where k = (12, 36, 60). Hence, models
with the highest SR(k) will recieve a higher forecast weight for time t + 1. 2
The economic value
of the combined forecasts is assessed in the same manner as any of the other empirical models,
which will be described in Sections 4 and 5, the derivation of the Sharpe Ratios for each individual
forecast model i will be described further in Section 5.
4 Statistical Evaluation of Exchange Rate Predictability
The predictive ability of the conditional exchange rate models (UIP, PPP, MF, TR, and
the combined forecasts) will be evaluated via use of statistical procedures that are designed to
evaluate OOS predictability in comparison to that of the unconditional RW model. In essence
we are assessing whether exchange rate predictability is more adequately explained by a restricted
parsimonious model (the RW, where β = 0), versus unrestricted models (where β = 0).
4.1 Estimation Procedure
The empirical exchange rate models are estimated using ordinary least squares (OLS), both
in-sample and for the pseudo out-of-sample forecasting exercise. Given today’s realized observables
(∆st+1, xt)T−1
t=1 , the in-sample (IS) period is defined using observations (∆st+1, xt)M
t=1 and an out-
of-sample (OOS) period using (∆st+1, xt)t−1
t=M+1. This procedure produces P = (T − 1) − M OOS
forecasts, and our empirical analysis uses T −1 = 438 monthly observations (we lose one observation
after differencing), M=120, and P=317. The OOS monthly forecasts are obtained in two ways: (i)
using recursive regressions starting from the period of January 1986 to June 2012, where each new
parameter that is estimated conditions on one more observation so the estimation sample gets larger
and larger as more parameters are estimated, (ii) using rolling regressions that use a 10-year fixed
window of observations to condition on, that generates forecasts for the same time period that the
recursive estimation does. Both ways of estimating model parameters are utilized to assess whether
there is a trade off between having parameters that are more consistent as more information is taken
into account (recursive regressions), versus having parameters that are more ”stable” as they are
better equipped to take into account structural breakage, as pre-break observations are not used in
estimation (rolling regressions).
4.2 Statistical Criterion and Tests
The main statistical criterion that is employed for evaluating the OOS predictive ability of
the models is the Campbell and Thompson (2008) and Welch and Goyal (2008) OOS R2
statistic
2
Note that forecast models that have produced a negative Sharpe Ratio at time t receive a forecast weight of zero,
so as to penalize unsatisfactory economic performance.
8
10. (i.e. R2
oos). The R2
oos is defined as follows:
R2
oos = 1 −
MSE(∆ˆst+1|t)
MSE(∆¯st+1|t)
= 1 −
ΣT−1
t=M+1(∆st+1 − ∆ˆst+1|t)2
ΣT−1
t=M+1(∆st+1 − ∆¯st+1|t)2
(16)
where ∆¯st+1|t is the forecast of the unconditional RW model and ∆ˆst+1|t is the forecast of the con-
ditional model. A positive R2
oos implies that the alternative conditional model out performs the null
unconditional model, by having a lower mean squared error (MSE).
The significance of the R2
oos can be assessed by applying a testing procedure, which tests
for equal predictive ability between the alternative and null models that will be estimated. This
test is conducted to determine whether the model with the lower MSE just had ”good luck” on
these particular forecasts, or whether it actually is superior to the other model. This test is used
by Clark and West (2006 and 2007), it tests for equal MSE between the null RW and alternative
models. The test is defined as:
¯testt+1 = (∆st+1 − ∆¯st+1|t)2
− [(∆st+1 − ∆ˆst+1|t)2
− (∆¯st+1|t − ∆ˆst+1|t)]. (17)
¯testt+1 is then regressed on a constant, so as to test for the level of significance of the constant using
the t-statistic. The asymptotic distribution of this test is non-standard, but the standard normal
one-sided critical values provide a good approximation, and recommend that the null hypothesis
for equal predictive ability between the two models be rejected if the test statistic is greater than
+1.282 for a 10 percent level of significance, +1.645 for a 5 percent level of significance, and +2.326
for a 1 percent level of significance. 3
5 Economic Evaluation of Exchange Rate Predictability
This section discusses the procedure for economically evaluating exchange rate predictability
based on an international dynamic asset allocation methodology provided by Della Corte, and
Tsiakas (2011) and Li, Tsiakas, and Wang (2014).
5.1 The Dynamic Foreign Exchange (FX) Strategy
The international dynamic asset allocation strategy takes the perspective of a U.S. investor
who trades the U.S. dollar and nine other currencies: the Australian dollar (AUD), Canadian dollar
(CAD), Swiss franc (CHF), Euro (EUR), British pound (GBP), Japanese yen (JPY), Norwegian
krone (NOK), New Zealand dollar (NZD), Swedish krona (SEK). The U.S. investor builds a portfolio
3
The Clark and West (2006, 2007) statistic is testing the null hypothesis of equal predictive accuracy in population,
while the reported R2
oos reflects finite sample performance. Therefore, rejection of the null hypothesis may occasionally
be associated with a negative R2
oos
9
11. by allocating wealth between the U.S. bond and nine foreign bonds (Australia, Canada, Switzerland,
Germany, UK, Japan, Norway, New Zealand, and Sweden). The returns on these bonds are proxied
by the euro deposit rates. At time t + 1, the foreign bonds yield a riskless return in local currency
but a risky return (rt+1) in U.S. dollars. Recall equation 2, which can be re-stated as a conditional
expected return from investing in a foreign bond like so:
rt+1|t = i∗
t + ∆st+1|t. (18)
Since i∗
t+1 is known at time t, the investor is only exposed to FX risk between time t and t + 1. The
investor is assumed to borrow at the domestic risk free rate it to invest in foreign bonds i∗
t . The
investor’s expected excess return on each currency can therefore be defined as
re
t+1|t = ∆st+1|t − (it − i∗
t ) (19)
Every month the investor uses each forecast model to obtain one-month ahead foreign
exchange returns, then ranks the conditional expected excess returns from each country from highest
to lowest according to what the individual forecast model specifies. The investor goes long on the
top three expected excess returns and short on the bottom three expected excess returns with a
weight of one-third on all currencies that are chosen for the portfolio whether they are being bought
or shorted (this is known as a long-short portfolio). The purpose of this methodology is to evaluate
whether the predictive ability of the conditional models lead to a better performing asset allocation
strategy than what the unconditional RW benchmark does. Predictive accuracy is important in
implementing this strategy as predicting too high (low) an excess return may lead to going long
(short) on the currency when it is non-optimal to do so, although what is most important is that
the forecast model gets the rankings of the expected excess returns at t + 1 accurate.
The main performance measure that will be assessed is the Sharpe Ratio (SR) of the individual
forecast models. The SR is defined as the average excess return of the portfolio divided by the
standard deviation of the portfolio returns. The excess return on the portfolio can be defined as:
re
p,t+1 = (1/3)(st+1,1 + st+1,2 + st+1,3 + i∗
t,1 + i∗
t,2 + i∗
t,3 − st+1,7 − st+1,8 − st+1,9 − i∗
t,7 − i∗
t,8 − i∗
t,9),
(20)
where , l denotes the ranking from highest to lowest of the excess foreign exchange returns according
10
12. to the conditional expectation. The Sharpe Ratio is therefore defined as:
SR =
µe
p
σp
(21)
where µe
p is the average portfolio excess return, and σp is the standard deviation of the portfolio
excess returns.
6 Empirical Results
6.1 Data on Exchange Rates and Economic Fundamentals
The analysis uses spot exchange rates as well as macroeconomic variables for all of the coun-
tries whose currency we try to predict relative to the U,S. dollar. All variables are in monthly
frequency and range from January 1976 to June 2012 for a total of 439 observations. For exchange
rates end-of-month spot and one-month forward rates are used, they are obtained from the Down-
load Data Program of the Board of Governors of the Federal Reserve System. The exchange rate is
defined as amount of U.S. dollars to buy one unit of foreign currency, therefore an increase in the
exchange rate implies a depreciation in the U.S. dollar.
This study uses seasonally adjusted data on industrial production, consumer price indices
(CPI) and broad money from the OECD Main Economic Indicators. Real output is proxied by
industrial production (IPI) which is available in monthly frequency. The IPI of Australia, New
Zealand, and Switzerland is only available at quarterly frequencies so the monthly frequencies are
obtained via linear interpolation. The price level is measured by the (CPI), which is available at
monthly frequencies, except Australia and New Zealand where it is available quarterly. The annual
inflation rate is computed as the 12-month log difference of the CPI. Money supply is proxied by
the monetary aggregate M1 that is measured in the national currency, except for the United King-
dom where we use M0. End of month interest rates are proxied by Euro deposit rates which are
downloaded from datastream (This study uses the same data as Li, Tsiakas, Wang (2014)). With
the exception of interest rates all data is all in logs and multiplied by 100.
Table 1 reports the descriptive statistics for the monthly (%) FX returns ∆st+1, the inter-
est rate difference it − i∗
t , the difference in the (%) change in price levels ∆(pt − p∗
t ), the difference
in (%) change in real output ∆(yt − y∗
t ), the difference in (%) change in money supply ∆(mt − m∗
t ).
The Japanese yen has the highest mean return (0.31% monthly, 3.7% annually) over our sample
period and the Swedish krona has the lowest (-0.105% monthly, -1.26% annually). The standard
deviations for all currencies are close to 3% per month. Most FX returns are negatively skewed
(except the Japanese yen), and all returns show excess kurtosis (kurtosis greater than three). Not
11
13. only do these exchange rate returns not follow a normal distribution, but they experience negative
returns more than they do positive, and the probability that they will experience a negative return
is more likely than a positive return. The autocorrelation of the exchange rate returns are fairly
low and taper off as ρ increases, this is most likely due to the data being in monthly frequencies,
where the autocorrelation effects get washed up by the long time delays between observations.
The interest differentials (it −i∗
t ) show a very long persistent memory, as they exhibit very
high degrees of autocorrelation, and it tapers off very slowly. This is not surprising as short-term
interest rates tend to exhibit gradual rather than rapid changes throughout short periods of time,
in part due to being influenced by the country’s central bank, who intervene so as to not allow it to
greatly deviate from where it has recently been. ∆(pt −p∗
t ) is always negatively skewed, meaning the
percentage change in the price difference is negative more than it is positive, and always exhibits
excess kurtosis (although not always extreme) therefore there is a larger probability of extreme
negative % changes than positive % changes. ∆(yt − y∗
t ) has some very extreme excess kurtosis
most notably with Canada, Germany, Japan, Norway, and Sweden, meaning they have very widely
distributed tails. Lastly, ∆(mt − m∗
t ) exhibits some very extremely negative and positive skewness
relative to the other factors, meaning the distribution of observations has a much longer tail on one
side than the other. For instance Canada, Germany, the United Kingdom, Japan, Norway, New
Zealand and Sweden.
6.2 In-Sample (IS) Estimation Results
Table 2 shows the in-Sample (IS) regression information for uncovered interest parity (UIP),
purchasing power parity (PPP), monetary fundamentals (MF), Taylor Rule (TR), and the random
walk (RW). The regressions use all of the data from January 1976 to June 2012. The displayed
statistics from left to right are the coefficient estimate, standard error (SE), t-statistic, the p-value,
and the ordinary R2
for the constant (α) and the regressor coefficient (β).
The very low R2
s indicates that the empirical models explain the variation in exchange rate returns
very poorly, in fact no R2
is greater than 2.64%. However, it is of greater importance to assess the
slope (β) estimates of the empirical exchange rate models for each currency return, as this would
indicate that the RW model is under specified, except for UIP where the α should also be looked
at, as the theory assumes α = 0.
The IS estimates show strong consistencies with the literature on the forward premium
puzzle, as every one of the slopes that UIP estimates for each exchange rate return is negative. The
β is statistically significant from zero for four of the nine currencies at the 10% level. The (α) on
the other hand is only statistically significant from zero for two currencies, but is positive (the Swiss
12
14. franc and the Japanese yen), which would imply that the interest rate differential is not affecting
the exchange rate return one-for-one, as a risk premium may be necesary. The slope of the PPP
regressor is always positive which is expected as an increase in the difference between the price level
domestically and abroad at time t should lead to an depreciation of the U.S. dollar relative to the
foreign currency at time t + 1. The slope of the PPP regressor is statistically different from zero
for seven currencies at the 10% level of significance. MF only has one statistically significant slope
in the nine currency returns it attempts to explain (the Swiss Franc), and are always positive. MF
seems to do little to explain the variation of exchange rate returns. The TR’s β is negative for
all currency returns it predicts, and is statistically significant from zero for six out of nine models
at the 10% level of significance. In conclusion, PPP has the most statistically significant slopes of
all empirical exchange rate models, and in-sample (IS) economic fundamentals seem to explain the
Swiss franc, the British pound, and the Japanese yen as all of the regressor slopes are statistically
significant from zero at the 10% level for the Swiss, and three of four are for the British pound and
the Japanese yen (only MF’s slope is not statistically significant).
6.3 Statistical Evaluation of Out-of-Sample Forecasts
The statistical performance of the empirical exchange rate models and the combined forecast
models are evaluated by reporting OOS tests against the null RW model. First, the analysis focuses
on the R2
oos statistic of Campbell and Thomson (2008) and Welch and Goyal (2008). Recall that
it is a positive R2
oos that implies that the alternative model has a lower mean squared error (MSE)
and therefore higher predictive accuracy than the benchmark model. The significance of the R2
oos
is assessed using the Clark and West (2006, 2007) one-sided t-statistic (critical values are stated
in Section 4.2) to determine if one model actually is superior to the other, or if one model was
the ”winner” due to the data set that was used. The R2
oos for the rolling regressions are stated in
Table 3, and those of the recursive are stated in Table 4. The a,b, and c superscripts indicate the
level of significance of the Clark and West (2006, 2007) null hypothesis at the 10%, 5%, 1% level
respectively using a t-statistic.
For the rolling regressions in Table 3 we can see that almost all of the empirical exchange
rates models produce negative R2
oos for all currencies, although the Clark and West (2006, 2007)
t-statistic seems to indicate that the Purchasing Power Parity (PPP) model produces statistically
greater forecast accuracy for the British pound and the Swedish krona than does the RW model,
despite the negative R2
oos. This also applies to the Taylor Rule (TR)’s forecasts of the New Zealand
dollar. The TR also has a positive R2
oos for the Swiss franc, the euro, and the Swedish krona, and
their predictive accuracy is statistically superior to the RW benchmark for at least the 10% level of
significance. Other than these exchange rate forecasts, the empirical models do not seem to gener-
ate superior predictive ability to the null RW. The combination forecasts seem to more consistently
13
15. produce positive R2
oos than the empirical exchange rate models, some in fact produce very high
positive R2
oos, as the empirical exchange rate models never produce an R2
oos greater than + 1% for a
given currency. However, no combination model produces positive R2
oos for all nine currencies, and
only the mean and MSE (k = 36) combination models have statistically significant R2
oos which are
both for the Swiss franc at the 10% level of significance. Therefore, the out-of-sample statistical
predictive performance by combining the empirical exchange rate models that are estimated using
rolling OOS does not seem to beat that of the null RW model.
Table 4 shows exactly the same thing as table 3 except the statistical results are from the
recursive regressions. The empirical exchange rate models seem to be more successful when the
parameters are estimated recursively rather than using a rolling sample, as the empirical models
derive more positive R2
oos for each of the currencies that they forecast. For instance, the TR has
a lower MSE for forecasting seven of the nine currencies than the benchmark RW, and PPP has a
lower MSE for six of the nine currencies that it forecasts than the RW. Not only that, there is more
statistical evidence that the conditional models produce a lower MSE than the RW, from looking
at the Clark and West (2006, 2007) t-statistic. UIP forecasts for the Canadian dollar, the Japanese
yen, and the New Zealand dollar produce superior forecast accuracy to the RW that is statistically
significant at the 10% level. Once again PPP produces statistically significant R2
oos for the British
pound and Swedish krona. PPP also produces a positive R2
oos for the Swiss Franc, that is also
statistically significant, which was not the case in the rolling estimation. Monetary Fundamentals
(MF) also produces statistically significant R2
oos for the Australian dollar, the Swiss franc, the euro,
the Japanese yen, and the Swedish krona, despite all currencies showing a negative R2
oos other than
the Swedish krona, and the British pound. Much like with the rolling estimation with TR we reject
the null hypothesis for equal predictability at the 10% level of significance for equal MSE with that
of the RW, for the Swiss franc, the euro, the New Zealand dollar, and the Swedish krona, but we
also find statistically significant R2
oos for the Canadian dollar, and the Japanese yen as well at the
10% level of significance.
The combination forecasts also do much better than with the rolling regressions, and do at
least as good as the empirical models that are estimated recursively OOS, as the mean, median, and
trimmed mean, cumulative MSE, and Sharpe Ratio (k = 12) combinations all produce positive R2
oos
for eight of the nine different currency forecasts. The MSE (k = 12, 36), and Sharpe Ratio (k = 36)
combinations produce positive R2
oos for seven of the nine different forecasted currencies. The median,
trimmed-mean, cumulative MSE, MSE (k = 12, 36, 60), and the Sharpe Ratio (k = 60) combination
models have statistically significant R2
oos for all currencies they predict, at a 10% level of significance.
14
16. 6.3.1 Visual Inspection
Plots are shown in Figure 1 that are alike those from Welch and Goyal (2008), where the
difference in the cumulative MSE between the null RW model and the alternative model are shown
for all currencies on the same graph for the recursive regressions, the figures for the rolling regres-
sions are not shown due as the lack of statistically significant R2
oos making them less important
to discuss. For a given currency if the plot is above zero for a given time t the alternative model
has predicted more accurately up until that point than the null RW, conversely if at a given time
t the plot is below zero the null RW has predicted better than the alternative model. When vi-
sually inspecting the performance of the alternative models relative to the RW model, it is ideal
that the plot for a given currency is consistently above zero, as that is indication that the al-
ternative model is outperforming the null RW throughout the full OOS period, and the evidence
would be even more convincing if the plot was continually increasing or at least staying constant,
as reversion towards zero would indicate that the RW is improving relative to the alternative model.
The UIP forecasts of the Canadian dollar and the Japanese yen look to outperform the
null RW forecasts throughout the whole OOS period, especially in the first quarter of it, after that
the plots trend towards zero. PPP seems to predict the British pound throughout the whole OOS
period, and the Swiss franc and Swedish krona throughout most of the OOS period more accurately
than that of the RW. It is no surprise that the R2
oos for the Canadian dollar is not statistically sig-
nificant and is very negative as the plot is never above zero, and trends towards it very slowly.
MF does not predict any currency throughout the whole OOS period better than the null RW, but
does predict the British pound, the Swiss Franc and the Swedish krona more accurately throughout
most of the OOS period, although the plots all trend towards zero in the second half of the OOS
period. TR looks to outperform the RW in predicting the New Zealand dollar throughout the OOS,
especially in about the first quarter of the OOS period, but quickly reverts towards zero. The TR
seems to more accurately predict the Swiss franc throughout the OOS but the plot is never very
far from zero, so it is never greatly outperforming it.
The mean, trimmed-mean (t-Mean), median, cumulative MSE and MSE (k = 12) combi-
nations all predict the Canadian dollar, the Swiss franc, the euro, the British pound, the Swedish
krona more accurately than the RW throughout the whole OOS period. These combinations are
outperformed by the RW at predicting the other three currencies at first but seem to improve
relative to the RW, as the plots eventually get to or even above zero. The MSE (k = 36) and
the SR (k = 36) combination models predict only the British pound and the Japanese yen more
accurately throughout most of the OOS period, these model seem to be outperformed by the RW
when predicting any other currency for much of the OOS period, but finish about on par with the
RW by the end. The MSE (k = 60) and SR (k = 12, 60) combination models do not predict any of
the currencies more accurately than the RW throughout the whole OOS period, but most do seem
15
17. to end on par with it as the OOS period ends.
The simple model averaging, cumulative MSE, and MSE (k = 12) combination models
seem to show the best visual predictive performance relative to the RW, as we see that they predict
most currencies more accurately throughout most of the OOS period. What is absent with all
of these alternative predictive models is the ability to sustain a constant or increasing gap in the
cumulative MSE against the RW, as even the models that predict better throughout the whole
sample for a given currency all seem to revert towards zero, meaning the RW is improving relative
to the alternative model somewhere mid-period and onward.
6.4 Economic Evaluation of Out-of-Sample Estimation
The economic evaluation of the predictive ability of the exchange rate return models is done
through analyzing the performance of the long-short dynamic portfolio discussed in section 5.1,
based on the one-month ahead forecasts generated by the recursive and rolling regressions. The
average annualized excess return (µe
p) summarized in equation 20, the annualized standard devia-
tion of the portfolio (σp), and the annualized Sharpe Ratio (SR), summarized in equation 21, are
displayed for each individual forecast model in Table 4. The Random Walk (RW) forecasts that
are estimated using rolling out-of-sample estimation appear to deliver the best monetary gains of
all forecast models as they produce both the highest annualized µe
p at about 6.83%, and the highest
annualized SR at 0.7930. This is consistent with most of the literature in international finance that
economic fundamentals do not yield higher monetary gains than the RW benchmark model. In
fact the UIP, PPP, MF, and TR that are estimated OOS using rolling regression are badly outper-
formed by that of the RW benchmark, as are the mean, MSE (k = 36), and Sharpe Ratio (k = 36)
combination models.
The RW forecast estimated using recursive regressions performs considerably worse than
its counterpart estimated using rolling regressions. It produces a µe
p at about 1.45%, and a SR at
about 0.1886. Nine of the fourteen alternative models that are estimated recursively outperform
this RW model in terms of both their µe
p and SR, seven of which are combination models. It might
be perplexing that the recursive median and Sharpe Ratio combination (k = 60) models are outper-
formed by the RW, because as shown in Table 4 these models had statistically significant R2
oos for
all currencies forecasted. What must be remembered with this long-short dynamic asset allocation
strategy (as mentioned in section 5.1) is that what is more important is how the forecast model
ranks the expected excess returns, and less so about its accuracy in comparison to other models.
16
18. 6.4.1 Visual Inspection
A visual illustration is displayed in Figure 2. $1 is invested in the initial period into the
long-short portfolio, which uses a dynamic asset allocation strategy and the $1 grows at a monthly
rate of return. Figure 2 shows the cumulative wealth of using each conditional model (red line)
to forecast returns for period t + 1 and dynamically rebalancing the portfolio according to what
the forecasts suggest, compared to using the benchmark random walk model (blue line) to follow
the very same strategy. These models are estimated out-of-sample using recursive regressions, and
ignore transaction costs. These plots are meant to show which time periods the conditional models
created more wealth relative to the benchmark RW. 4
We can see from Figure 2 that PPP, TR, the median, and the MSE (k = 12, 36, 60)
forecast combinations clearly accumulate more wealth than the RW benchmark model, in fact the
PPP, median combination, and TR models accumulate returns above that of the RW of 134.70%,
114.78%, and a 100.40% respectively by the end of the OOS period. The mean and trimmed mean
(T-Mean) combination models also look to accumulate more wealth than the benchmark RW, but
both alternative models seem to go through a stretch where many negative returns are realized
from about July 2002 onwards which allows the accumulated wealth of the RW model to close the
gap. The Sharpe Ratio (k = 12) is outperformed by the benchmark RW throughout the most of
the OOS period that the two models are compared to each other, but manages to forge ahead from
about october 2004 and onwards. UIP and MF have their moments where they are outperforming
the benchmark RW, but finish their respective time periods with less accumulated wealth than that
of the benchmark RW. It appears the Sharpe Ratio (k = 36, 60) combination models actually do
quite well in comparison to the benchmark RW throughout most of the OOS period but experience
mostly negative returns in 2008 which allows the RW model to forge ahead.
It is quite perplexing that the TR and median combination accumulate so much wealth
in comparison to the RW as they actually have lower Sharpe Ratios than the RW. Also, the MF
model actually has a much higher Sharpe Ratio than the RW but looks to accumulate less wealth
than the RW throughout most of the OOS sample. It appears that the TR and median combination
model have periods in about the first half of the OOS period that derive such substantial returns
relative to the RW that it is not able to catch up as the TR and median combination model do
look to stop growing wealth in about the last quarter of the sample. MF on the other hand derives
some very large negative returns in the last quarter of the OOS period, that it doesn’t recover from.
These plots do clear up some of the discrepancy between tables 4 and 5 with respect to the median
4
Note that in Figure 2 the time periods that the alternative model and the null RW are compared are shorter for
the cumulative MSE, MSE (k = 12, 13, 60), and Sharpe Ratio (k = 12, 36, 60) combination models as the forecast for
these models do not start until either 1, 12, 36, or 60 periods into the out-of-sample period, so it is ideal to compare
those models to the performance of the random walk over the same time period.
17
19. and Sharpe Ratio (k = 60) combination models, which show great statistical performance but less
than satisfactory economic performance, as we can actually see that these models accumulate more
wealth throughout most of the OOS period than the RW, but have periods of large negative returns
that negate their credibility economically.
7 Conclusion
Most previous literature in exchange rate predictability has used statistical criteria for out-
of-sample (OOS) tests of the null random walk representing the absence of predictability against
alternative models that condition on economic fundamentals. This research follows new trends that
have emerged in the literature, first by forming ex ante forecast combination models from a set of
empirical models that condition on one set of economic fundamentals, in order to resolve model
uncertainty, second by taking the view of an investor who builds a dynamic asset allocation strategy
that conditions on the forecasts from a set of empirical exchange rate models, by evaluating the
predictability from looking at the performance of a dynamically rebalanced portfolio.
The alternative models seem to do better when their forecasts are calculated using recur-
sive out-of-sample (OOS) estimation, than when rolling OOS estimation is used, in the statistical
evaluation. That is these models do have statistically greater predictive ability in comparison to the
random walk (RW) model they are being compared to. It is very possible that the RW’s predictive
accuracy is much worse when estimated recursively than rolling using OOS estimation, rather than
the alternative models predicting so much more accurately when recursive OOS estimation is used.
This problem can be avoided by estimating an RW model without drift (where α = 0) and the t+1
forecast is always zero, so that that the same benchmark is being compared to all models whether
they are estimated recursively or rolling OOS, however it seems unrealistic to assume the exchange
rate return is always going to be zero.
The RW estimated using rolling OOS estimation seems to perform the best of any model
economically, as it has the highest Sharpe Ratio, so would therefore be the wisest forecast model
for a rational investor to use for deciding how to rebalance their portfolio for period t + 1. What
must be dully noted is that strong statistical predictive accuracy does not necessarily imply strong
economic performance, as some of the alternative models that were estimated using recursive OOS
estimation and had tremendous statistical results were badly outperformed by their counterparts
that used rolling OOS estimation (ex. the RW, and the trimmed mean, median, SR (k = 12, 60)
combination models). In fact the median and Sharpe Ratio (k = 60) combination models showed
very strong statistical performance in the recursive OOS estimation, but had lower Sharpe Ratios
than that of the RW. This is partly due to the dynamic asset allocation strategy we have imple-
18
20. mented that cares about how the forecast models predict the rankings of the excess exchange rate
returns, rather the predictive accuracy of the exchange rate returns.
It is surprising how PPP seems to have statistical significance in explaining the exchange
rate return of the Canadian dollar in-sample (IS), but predicts it very poorly using rolling and
recursive OOS estimation. Economic fundamentals seem to be the most successful (statistically)
in explaining and predicting the Swiss franc, the British pound, and the Japanese yen, at least IS
and in recursive OOS estimation. Economic fundamentals also seem to do a satisfactory job at
explaining the Swedish krona at least out-of-sample. For future research a mean-variance dynamic
asset allocation strategy should be used (whether it is a target return or volatility strategy) for
evaluating the economic value of predictability of exchange rate returns, as the portfolio returns
that are generated using this type of strategy are more sensitive to the predictive accuracy than
the long-short strategy implemented in this analysis. The mean-variance dynamic asset allocation
strategy is not implemented in this analysis due to the difficulty in computing the portfolio weights
of nine risky assets.
19
21. Table 1: Descriptive Statistics
This table presents the descriptive statistics for monthly log exchange rate returns as well as the
economic fundamentals. The exchange rate is defined as the price in U.S. dollars to purchase a
unit of foreign currency. ∆s is the percentage change in the U.S. dollar against the currency in
question. i is the one-month interest rate, ∆p, ∆m, and ∆y are the percentage changes in price,
money supply, and real output respectively. The (∗) denotes the non-U.S. economic fundamental,
and ρl is the autocorrelation coefficient for l lags. Data ranges from January 1976 to June 2012.
Mean St. dev Skew. Kurt. ρ1 ρ3 ρ6 ρ12
AUD ∆s -0.0468 3.315 -1.225 8.647 0.0204 0.062 0.0303 -0.0695
i − i∗
-0.18590 0.2846 0.188 4.621 0.956 0.874 0.797 0.644
∆y − ∆y∗
0.03 0.780 -0.246 3.445 0.302 0.0612 0.069 0.0085
∆p − ∆p∗
-0.08 0.392 -0.803 4.519 0.559 0.104 0.151 0.234
∆m − ∆m∗
-0.08 0.861 3.74 2.112 0.127 0.046 -0.089 0.44
CAD ∆s 0.005 1.941 -0.648 9.930 -0.060 0.034 -0.104 0.0367
i − i∗
-0.064 0.136 -0.223 3.86 0.88 0.72 0.63 0.415
∆y − ∆y∗
-0.012 1.069 -0.921 11.384 -0.213 0.0667 -0.029 -0.075
∆p − ∆p∗
0.006 0.35 -0.718 6.768 0.016 0.084 0.0751 0.236
∆m − ∆m∗
-0.0621 0.651 19.46 2.106 -0.196 0.302 0.145 0.339
CHF ∆s 0.207 3.602 -0.115 4.151 0.0196 0.035 -0.0637 -0.042
i − i∗
0.242 0.277 0.542 3.795 0.971 0.892 0.836 0.749
∆y − ∆y∗
0.0105 0.926 -0.0198 4.307 0.323 -0.062 -0.0117 -0.017
∆p − ∆p∗
0.162 0.413 -0.0867 3.805 0.23 -0.0613 0.339 0.506
∆m − ∆m∗
-0.00411 1.31 9.78 6.442 -0.166 0.283 0.356 0.341
EUR ∆s 0.118 3.249 -0.16 3.629 0.0044 0.0462 -0.0297 -0.0206
i − i∗
0.107 0.235 -0.146 3.602 0.961 0.877 0.829 0.723
∆y − ∆y∗
0.048 1.806 0.036 8.864 -0.277 0.087 -0.0115 -0.0067
∆p − ∆p∗
0.127 0.409 -0.747 5.0115 0.145 0.0176 0.0614 0.513
∆m − ∆m∗
-0.042 1.262 19.093 1.739 -0.344 0.315 0.282 0.508
GBP ∆s -0.0589 3.0434 -0.2053 4.7371 0.0771 0.0400 -0.0485 0.0103
i − i∗
-0.1676 0.2161 -0.7692 5.5503 0.9202 0.7650 0.5717 0.2772
∆y − ∆y∗
0.1317 1.3147 0.2820 6.5645 -0.2453 0.0585 -0.0794 -0.0605
∆p − ∆p∗
-0.0636 0.5452 -1.3491 8.8120 0.0891 -0.0199 0.2381 0.5331
∆m − ∆m∗
-0.0154 0.6615 -10.798 1.4859 -0.0810 0.1823 0.0066 0.4403
JPY ∆s 0.3058 3.3131 0.3417 4.3830 0.0379 0.0380 -0.1055 0.0291
i − i∗
0.2334 0.2219 0.4202 3.3686 0.9521 0.8318 0.6788 0.4629
∆y − ∆y∗
0.0505 1.8571 2.4582 25.1160 -0.1069 0.0274 0.0461 -0.0668
∆p − ∆p∗
0.2033 0.5121 -0.4540 5.4056 0.0788 -0.1122 0.1070 0.4141
∆m − ∆m∗
-0.0218 1.5527 -22.4256 2.0180 -0.3436 0.3455 0.3381 0.4283
NOK ∆s -0.0161 3.0493 -0.4439 4.4276 0.0225 -0.0034 -0.0047 -0.0596
i − i∗
-0.1800 0.2692 0.0039 3.2226 0.9338 0.8296 0.7412 0.4884
∆y − ∆y∗
-0.0219 4.3029 0.6631 28.1271 -0.4160 -0.0377 -0.0062 -0.0975
∆p − ∆p∗
-0.0283 0.5332 -0.8838 5.4419 0.1608 0.1216 0.1355 0.4317
∆m − ∆m∗
-0.0981 1.7305 -71.6757 9.0807 -0.5191 -0.2205 0.0819 0.0749
NZD ∆s -0.0597 3.4364 -1.1644 10.7674 0.0132 0.1954 0.0648 -0.0797
i − i∗
-0.2913 0.3407 -1.7522 8.0920 0.9571 0.8588 0.7355 0.5459
∆y − ∆y∗
0.0345 1.1167 -1.2014 7.6385 0.5268 0.1507 0.0431 0.0272
∆p − ∆p∗
-0.1577 0.4913 -1.5806 8.2116 0.6841 0.3384 0.2968 0.2791
∆m − ∆m∗
-0.0644 1.8561 18.5179 2.9365 -0.3774 0.0186 0.3301 0.5590
SEK ∆s -0.1045 3.2609 -0.7629 6.0117 0.0729 0.0734 -0.0732 -0.0242
i − i∗
-0.1468 0.2697 -0.8613 4.3832 0.9365 0.8372 0.7478 0.5274
∆y − ∆y∗
0.0935 2.7884 0.0847 37.1153 -0.2974 0.0387 0.0394 -0.0216
∆p − ∆p∗
-0.0392 0.5418 -1.0812 6.3185 0.1873 0.0550 0.0953 0.3483
∆m − ∆m∗
-0.0907 1.5555 -534.7871 74.8818 -0.0236 0.0479 -0.0364 0.1833
20
22. Table 2: In Sample Estimates
The following tables show the In-Sample (IS) regression information for uncovered interest parity
(UIP), purchasing power parity (PPP), monetary fundamentals (MF), the taylor rule (TR), and
the random walk (RW) for all currencies. The data ranges from January 1976 to June 2012. α
is the constant, and β is the regressor coefficient, The statistics shown from left to right are the
estimated value of the coeficient, the standard error (SE), t-statistic, p-value, and ordinary R2
.
AUD
Model Coefficient Estimate SE t-Statistic p-Value R2
UIP α -0.0013 0.0019 -0.6922 0.4892 0.0015
β -0.4538 0.5574 -0.8141 0.4161
PPP α -0.0001 0.0016 -0.0452 0.9640 0.0043
β 0.0127 0.0093 1.3688 0.1718
MF α 0.0005 0.0018 0.2804 0.7793 0.0026
β 0.0077 0.0072 1.0736 0.2836
TR α -0.0007 0.0016 -0.4414 0.6592 0.0005
β -0.0509 0.0346 -1.4713 0.1419
RW α -0.0005 0.0016 -0.2956 0.7677 0.0000
CAD
Model Coefficient Estimate SE t-Statistic p-Value R2
UIP α -0.0006 0.0010 -0.5600 0.5758 0.0047
β -0.9732 0.6819 -1.4272 0.1542
PPP α -0.0023 0.0017 -1.3767 0.1693 0.0064
β 0.0129 0.0076 1.6892 0.0919
MF α 0.0002 0.0009 0.1912 0.8485 0.0015
β 0.0043 0.0055 0.7954 0.4268
TR α -0.0011 0.0011 -0.9643 0.3355 0.0080
β -0.0680 0.0368 -1.8485 0.0652
RW α 0.0000 0.0009 0.0491 0.9609 0.0000
CHF
Model Coefficient Estimate SE t-Statistic p-Value R2
UIP α 0.0047 0.0023 2.0463 0.0413 0.0069
β -1.0773 0.6219 -1.7323 0.0839
PPP α -0.0163 0.0090 -1.8122 0.0706 0.0099
β 0.0217 0.0104 2.0811 0.0380
MF α -0.0019 0.0028 -0.6845 0.4940 0.0077
β 0.0101 0.0055 1.8325 0.0676
TR α -0.0023 0.0031 -0.7512 0.4530 0.0074
β -0.0785 0.0455 -1.7262 0.0850
RW α 0.0021 0.0017 1.1998 0.2309 0.0000
EUR
Model Coefficient Estimate SE t-Statistic p-Value R2
UIP α 0.0018 0.0017 1.0587 0.2903 0.0018
β -0.5851 0.6617 -0.8842 0.3771
PPP α -0.0042 0.0033 -1.2586 0.2089 0.0076
β 0.0175 0.0096 1.8204 0.0694
MF α 0.0035 0.0021 1.6377 0.1022 0.0026
β 0.0067 0.0042 1.5710 0.1169
TR α 0.0005 0.0016 0.2773 0.7817 0.0124
β -0.0769 0.0432 -1.7792 0.0759
RW α 0.0012 0.0016 0.7610 0.4470 0.0000
21
26. Table 5: Economic Evaluation of Exchange Rate Predicability
The following table displays statistics that represent the economic value of the predictability of our empirical
exchange rate return and combination models. The statistics using rolling estimation are shown on the left,
which use a fixed window of observations to estimate the model parameters (10 years), the statistics using
recursive estimation are shown on the right where a new observation is added to the sample every time a
new parameter is estimated. The combined forecasts use information from all five empirical exchange rate
models. These values are derived using a long-short portfolio strategy, from the perspective of an American
investor. The investor uses one of the predictive models to forecast the excess return at time t + 1 (eq. 19),
then ranks the expected excess returns from highest to lowest, and invests long in the top three currencies
that are expected to perform the best, and invests short in the bottom three expected to perform the poorest.
The average excess portfolio return (µe
p), the standard deviation of the portfolio excess returns (σp), and the
Sharpe Ratio SR (eq. 21) for using each predictive model are reported in this table. All statistics are annualized.
Rolling Regressions Recursive Regressions
Model µe
p (%) σp (%) SR µe
p (%) σp (%) SR
Random Walk 6.83 8.61 0.7930 1.45 7.68 0.1886
Uncovered Interest Parity 1.49 7.95 0.1873 1.06 7.75 0.1364
Purchasing Power Parity 2.99 7.85 0.3813 3.29 7.07 0.4656
Monetary Fundamentals 2.78 8.23 0.3375 4.11 8.28 0.4969
Taylor Rule 0.79 7.45 0.1065 0.79 7.45 0.1065
Mean Combination 0.79 7.45 0.1065 3.51 7.22 0.4866
Trimmed Mean Combination 5.41 8.46 0.6397 1.69 7.56 0.2235
Median Combination 5.64 8.09 0.6969 0.85 7.41 0.1140
Cumulative MSE Combination 4.77 7.85 0.6084 3.75 7.16 0.5229
MSE Combination (k=12) 3.25 7.42 0.4380 3.20 7.44 0.4297
MSE Combination (k=36) 0.82 7.17 0.1146 2.11 7.18 0.2941
MSE Combination (k=60) 1.75 7.13 0.2453 1.75 7.13 0.2453
Sharpe Ratio Combination (k=12) 4.13 7.94 0.5199 2.27 7.98 0.2843
Sharpe Ratio Combination (k=36) -1.44 7.31 -0.1976 0.65 7.97 0.0811
Sharpe Ratio Combination (k=60) 3.78 7.76 0.4868 0.86 8.11 0.1061
25
27. Figure 1: Difference in Cumulative Mean Squared Error (MSE) between Conditional
Models and the Random Walk (RW) (Recursive Regressions) These figures plot the
out-of-sample performance of the monthly predictive exchange rate returns for all currencies.
These plots are similar to Welch and Goyal (2008) where the cumulative squared prediction error
of the alternative models is subtracted from that of the null random walk (RW) model. If a plot
is above the zero at time t, the alternative model has predicted more accurately the exchange rate
return for that currency than the null model. Alternatively, if a plot is below zero the null model
has predicted more accurately than the alternative for that currency’s exchange rate return.
26
31. Figure 2: Cumulative Wealth of Different Forecasting Strategies (Recursive Regres-
sions) These figures display the cumulative wealth of the out-of-sample dynamic investment
strategies conditioning on a set monthly economic fundamentals and combined forecasts
(red line) compared to that of the benchmark random walk (RW) (blue line). The ini-
tial investment is set at $ 1, which grows at the monthly return of the portfolio strategy
without any transaction costs. The out-of-sample monthly forecasts are obtained using re-
cursive regressions that generate forecasts for the period of January 1986 to June 2012.
30
32. Figure 2: Cumulative Wealth of Different Forecasting Strategies (Rolling Regres-
sions) These figures display the cumulative wealth of the out-of-sample dynamic investment
strategies conditioning on a set monthly economic fundamentals and combined forecasts
(red line) compared to that of the benchmark random walk (RW) (blue line). The ini-
tial investment is set at $ 1, which grows at the monthly return of the portfolio strategy
without any transaction costs. The out-of-sample monthly forecasts are obtained using re-
cursive regressions that generate forecasts for the period of January 1986 to June 2012.
31
