Statistical and Economic Methods
for Evaluating Exchange Rate Predictability
Pasquale Della Corte
Imperial College London
Ilias Tsiakas
University of Guelph
October 2011
Abstract
This chapter provides a comprehensive review of the statistical and economic methods used
for evaluating out-of-sample exchange rate predictability. We illustrate these methods by assessing the forecasting performance of a set of widely used empirical exchange rate models using
monthly returns on nine major US dollar exchange rates. We …nd that empirical models based
on uncovered interest parity, purchasing power parity and the asymmetric Taylor rule perform
better than the random walk in out-of-sample forecasting using both statistical and economic
criteria. We also con…rm that conditioning on monetary fundamentals does not generate out-ofsample economic value. Finally, combined forecasts formed using a variety of model averaging
methods perform better than individual empirical models. These results are robust to reasonably
high transaction costs, the choice of numeraire and the exclusion of any one currency from the
investment opportunity set.
Keywords: Exchange Rates; Out-of-Sample Predictability; Mean Squared Error; Economic
Value; Combined Forecasts.
JEL Classi…cation: F31; F37; G11; G17.
This paper is forthcoming as a chapter in the Handbook of Exchange Rates. The authors are grateful to Lucio Sarno and an anonymous referee for useful comments. Contact details: Pasquale Della Corte, Finance Group,
Imperial College Business School, Imperial College London, 53 Prince’s Gate, London SW7 2AZ, UK. Email:
[email protected]; Ilias Tsiakas, Department of Economics and Finance, University of Guelph, Guelph, Ontario N1G 2W1, Canada. Email: [email protected].
1
Introduction
Exchange rate ‡uctuations are regularly monitored with great interest by policy makers, practitioners
and academics. It is not surprising, therefore, that exchange rate predictability has long been at
the top of the research agenda in international …nance. Starting with the seminal contribution of
Meese and Rogo¤ (1983), a large body of empirical research …nds that models which condition on
economically meaningful variables do not provide reliable exchange rate forecasts. This has lead
to the prevailing view that exchange rates follow a random walk and hence are not predictable,
especially at short horizons.
Several well known puzzles in foreign exchange (FX) are responsible for this view. First, the
“exchange rate disconnect puzzle” concerns the empirical disconnect between exchange rate movements and economic fundamentals such as money supply and real output (e.g., Mark, 1995; Cheung,
Chinn and Pascual, 2005; Rogo¤ and Stavrakeva, 2008). Second, the “forward premium puzzle”
implies that on average the interest di¤erential is not o¤set by a commensurate depreciation of the
investment currency, which is an empirical violation of uncovered interest rate parity. As a result,
borrowing in low-interest rate currencies and investing in high-interest rate currencies forms the basis
of the widely used carry trade strategy in active currency management (e.g., Fama, 1984; Burnside,
Eichenbaum, Kleshchelski and Rebelo, 2011; Brunnermeier, Nagel and Pedersen, 2009; and Della
Corte, Sarno and Tsiakas, 2009). Third, there is extensive evidence that purchasing power parity
holds in the long run (e.g., Lothian and Taylor, 1996).
A recent contribution by Engel and West (2005) provides a possible resolution to the di¢ culty of
tying exchange rates to economic fundamentals. Speci…cally, Engel and West (2005) show analytically
that exchange rates can be consistent with present-value asset pricing models and still manifest nearrandom walk behaviour if two conditions are met: (i) fundamentals are integrated of order one, and
(ii) the discount factor for future fundamentals is near one.1
A model that is nested by the Engel and West (2005) present value relation is a variant of the
Taylor (1993) rule used for exchange rate determination. The Taylor rule postulates that the central
bank adjusts the short-run nominal interest rate in response to changes in in‡ation, the output gap
and the exchange rate. Using alternative speci…cations of Taylor rule fundamentals, Molodtsova and
Papell (2009) provide strong evidence of short-horizon exchange rate predictability, and hence o¤er
renewed hope for empirical success in this literature. In short, one way to summarize the state of
the literature is that it has come full circle: from the Meese and Rogo¤ (1983) “no predictability
at short horizons,” to the Mark (1995) “predictability at long but not at short horizons,” to the
Cheung, Chinn and Pascual (2005) “no predictability at any horizon,”to …nally the Molodtsova and
1
The assumption of integrated fundamentals of order one is widely accepted in the literature. The assumption that
the discount factor is close to one has been empirically validated by Sarno and Sojli (2009).
1
Papell (2009) “predictability at short horizons with Taylor rule fundamentals.”
This chapter aims at connecting these related literatures by providing a comprehensive review
of the statistical and economic methods used for evaluating exchange rate predictability, especially
out of sample. We assess the short-horizon forecasting performance of a set of widely used empirical
exchange rate models that include the random walk model, uncovered interest parity, purchasing
power parity, monetary fundamentals, and symmetric and asymmetric Taylor rules. Our analysis
employs monthly FX data ranging from January 1976 to June 2010 for the 10 most liquid (G10)
currencies in the world: the Australian dollar, Canadian dollar, Swiss franc, Deutsche markneuro,
British pound, Japanese yen, Norwegian kroner, New Zealand dollar, Swedish kronor and US dollar.2
The vast majority of the FX literature uses a well established statistical methodology for evaluating exchange rate predictability. This methodology typically involves statistical tests of the null
hypothesis of equal predictive ability between the random walk benchmark and an alternative empirical exchange rate model. The tests are based on the out-of-sample mean squared error (MSE) of
the forecasts generated by the models. In this chapter, we discuss the main recent contributions to
this methodology.
The most popular method for testing whether the alternative model has a lower MSE than the
benchmark is using the Diebold and Mariano (1995) and West (1996) statistic. By design, however, all
the models we estimate are nested and this statistic has a non-standard distribution when comparing
forecasts from nested models. Therefore, we focus on the recent inference procedure by Clark and
West (2006, 2007), which accounts for the fact that under the null the MSE from the alternative
model is expected to be greater than that of the RW benchmark because the alternative model
introduces noise into the forecasting process by estimating a parameter vector that is not helpful in
prediction. For a comprehensive statistical evaluation, we also implement the encompassing test of
Clark and McCracken (2001) and the F -statistic of McCracken (2007) using bootstrapped critical
values. Finally, following Campbell and Thompson (2008) and Welch and Goyal (2008) we also
2 measure and a root MSE di¤erence statistic.
report the out-of-sample Roos
In addition to the extensive literature on statistical evaluation, there is also an emerging line of
research proposing a methodology for assessing the economic value of exchange rate predictability.
A purely statistical analysis of predictability is not particularly informative to an investor as it falls
short of measuring whether there are tangible economic gains from using dynamic forecasts in active
portfolio management. We review this approach based on dynamic asset allocation that is used,
among others, by West, Edison and Cho (1993), Fleming, Kirby and Ostdiek (2001), Marquering
2
Note that we will not be discussing two recent approaches to predicting movements in exchange rates: the microstructure approach that conditions on order ‡ow as a measure of net buying pressure for a currency (e.g., Evans
and Lyons, 2002, and Rime, Sarno and Sojli, 2010); and (ii) the global imbalances approach (e.g., Gourinchas and Rey,
2007, and Della Corte, Sarno and Sestieri, 2011).
2
and Verbeek (2004), Abhyankar, Sarno and Valente (2005), Bandi and Russell (2006), Han (2006),
Bandi, Russell and Zhu (2008), Della Corte, Sarno and Thornton (2008) and Della Corte, Sarno and
Tsiakas (2009, 2011).
We …rst design an international asset allocation strategy that exposes a US investor purely to
FX risk. The investor builds a portfolio by allocating her wealth between a domestic and a set of
foreign bonds and then uses the exchange rate forecasts from each model to predict the US dollar
return of the foreign bonds. We evaluate the performance of the dynamically rebalanced portfolios
using mean-variance analysis, which allows us to measure how much a risk averse investor is willing
to pay for switching from a portfolio strategy based on the random walk benchmark to an empirical
exchange rate model that conditions on economic fundamentals. In contrast to statistical measures of
forecast accuracy that are computed separately for each exchange rate, the economic value is assessed
for the portfolio generated by a model’s forecasts on all exchange rate returns. This contributes to
our …nding that even modest statistical signi…cance in out-of-sample predictive regressions can lead
to large economic bene…ts for investors.
Our review also includes an assessment of the economic value of combined forecasts. We use a
variety of model averaging methods, some of which generate forecast combinations in a naive ad hoc
manner, some exploit statistical measures of past out-of-sample forecasting performance, and some
that use economic measures of past predictability. All forecast combinations we explore are formed
ex ante using the full universe of individual forecasts of each model for each exchange rate. It is
important to note that the combined forecasts do not require a view of which model is best at any
given time period and therefore provide a way for resolving model uncertainty.
To preview our key results, we …nd strong statistical and economic evidence against the random
walk benchmark. In particular, empirical exchange rate models based on uncovered interest parity,
purchasing power parity and the asymmetric Taylor rule perform better than the random walk in outof-sample prediction using both statistical and economic criteria. We also con…rm that conditioning
on monetary fundamentals does not generate out-of-sample economic gains. The worst performing
model is consistently the symmetric Taylor rule. Finally, combined forecasts formed using a variety
of model averaging methods perform even better than individual empirical models. These results
are robust to reasonably high transaction costs, the choice of numeraire and the exclusion of any one
currency from the investment opportunity set.
The remainder of the chapter is organized as follows. In the next section we brie‡y review
the empirical exchange rate models we estimate and their foundations in asset pricing. Section 3
describes the statistical methods we use for evaluating exchange rate predictability. In Section 4 we
present a general framework for assessing the economic value of forecasting exchange rates for a risk
averse investor with a dynamic mean-variance portfolio allocation strategy. Section 5 explains the
3
construction of combined forecasts using a variety of model averaging methods. Section 6 reports
our empirical results and, …nally, Section 7 concludes.
2
Models for Exchange Rate Predictability
In this section we review the empirical models we use for evaluating exchange rate predictability. We
begin by describing the Engel and West (2005) present value model that nests and motivates many
of the predictive regressions we estimate.
2.1
A Present Value Model for Exchange Rates
The Engel and West (2005) model relates the exchange rate to economic fundamentals and the
expected future exchange rate as follows:
st = (1
b) (f1;t + z1;t ) + b (f2;t + z2;t ) + bEt st+1 ;
(1)
where st is the log of the nominal exchange rate de…ned as the domestic price of foreign currency,
fi;t (i = 1; 2) are the observed economic fundamentals and zi;t are the unobserved fundamentals that
drive the exchange rate. Note that an increase in st implies a depreciation of the domestic currency.
This is a general asset pricing model that builds on earlier work on pricing stock returns by Campbell
and Shiller (1987, 1988) and West (1988).
Iterating forward and imposing the no-bubbles condition leads to the following present-value
relation:
st = (1
b)
1
X
bj Et (f1;t+j + z1;t+j ) + b
j=0
1
X
bj Et (f2;t+j + z2;t+j ) :
(2)
j=0
Engel and West (2005) show that the exchange rate will follow a random walk if the discount factor
b is close to one and either: (1) f1;t + z1;t
I (1) and f2;t + z2;t = 0; or (2) f2;t + z2;t
I (1). Some
other well-known exchange rate models take the general form of Equation (1), and in what follows,
we discuss two examples.
2.1.1
Monetary Fundamentals
Consider …rst the monetary exchange rate models of the 1970s and 1980s, which assume that the
money market relation is described by:
mt = pt + yt
it + vm;t ;
where mt is the log of the domestic money supply, pt is the log of the domestic price level,
is the income elasticity of money demand, yt is the log of the domestic national income,
(3)
>0
> 0 is
the interest rate semi-elasticity of money demand, it is the domestic nominal interest rate and vm;t
4
is a shock to domestic money demand. A similar equation holds for the foreign economy, where the
corresponding variables are denoted by mt , pt , yt , it and vm;t . We assume that the parameters
f ; g of the foreign money demand are identical to the domestic parameters.
The nominal exchange rate is equal to its purchasing power parity (PPP) value plus the real
exchange rate qt :
s t = pt
p t + qt :
(4)
Finally, the interest parity condition is given by:
Et st+1
where
t
st = it
it +
t;
(5)
is the deviation from the uncovered interest parity (UIP) condition that is based on rational
expectations and risk neutrality. Hence
t
can be interpreted either as an expectational error or a
risk premium.
Using Equations (3) to (5) for the domestic and foreign economies and re-arranging, we get:
st =
1
1+
mt
mt
(yt
yt ) + qt
vm;t
vm;t
t
+
1+
Et st+1 :
(6)
This equation takes the form of the original model in Equation (1), where the discount factor is given
by b = =1 + , the observable fundamentals are f1;t = mt
fundamentals are z1;t = qt
2.1.2
vm;t
vm;t and z2;t =
mt
(yt
yt ), and the unobservable
t.
Taylor Rule
The second model to be nested by the Engel and West (2005) present value relation is the Taylor
(1993) rule, where the home country is assumed to set the short-term nominal interest rate according
to:
it = i +
g
1 yt
+
2( t
) + vt ;
(7)
where i is the target short-term interest rate, ytg is the output gap measured as the % deviation
of actual real GDP from an estimate of its potential level,
t
is the in‡ation rate,
is the target
in‡ation rate and vt is a shock. The Taylor rule postulates that the central bank raises the short-term
nominal interest rate when output is above potential output and/or in‡ation rises above its desired
level.
The foreign country is assumed to follow a Taylor rule that explicitly targets exchange rates (e.g.,
Clarida, Gali and Gertler, 1998):
it =
0 (st
st ) + i +
1 yt
5
g
+
2( t
) + vt ;
(8)
where 0 <
0
< 1 and st is the target exchange rate. For simplicity, we assume that the home and
foreign countries target the same interest rate, i, and the same in‡ation rate, . The rule indicates
that the foreign country raises interest rates when its currency depreciates relative to the target.3
We assume that the foreign central bank targets the PPP level of the exchange rate:
s t = pt
pt :
(9)
Taking the di¤erence between the home and foreign Taylor rules, using interest parity (5), substituting the target exchange rate and solving for st gives:
st =
0
1+
(pt
1
1+
pt )
0
1
0
ytg
yt g +
2( t
t)
+ vt
vt +
t
+
1
1+
Et st+1 : (10)
0
This equation also has the general form of the present value model in Equation (1), where the discount
factor is b = 1=1 +
2.2
0,
f1;t = pt
pt and z2;t =
1
ytg
yt g +
2( t
t)
+ vt
vt +
t
.
Predictive Regressions
Our empirical analysis is based on six predictive regressions for exchange rate returns, many of
which are nested and motivated by the Engel and West (2005) present value model. All predictive
regressions have the same linear structure:
st+1 =
where
st+1 = st+1
st ,
and
+ xt + "t+1 ;
(11)
are constants to be estimated and "t+1 is a normal error term.
The empirical models di¤er in the way they specify the economic fundamentals xt that are used to
forecast exchange rate returns.
2.2.1
Random Walk
The …rst regression is the random walk (RW) with drift model that sets
= 0. Since the seminal
work of Meese and Rogo¤ (1983), this model has become the benchmark in assessing exchange rate
predictability. The RW model captures the prevailing view in international …nance research that
exchange rates are not predictable when conditioning on economic fundamentals, especially at short
horizons.
3
The argument still follows if the home country also targets exchange rates. It is standard to omit the exchange
rate target from Equation (7) on the interpretation that US monetary policy has essentially ignored exchange rates
(see, Engel and West, 2005).
6
2.2.2
Uncovered Interest Parity
The second regression is based on the UIP condition:4
xt = it
it :
(12)
UIP is the cornerstone condition for FX market e¢ ciency. Assuming risk neutrality and rational
expectations, it implies that
= 0,
= 1, and the error term is serially uncorrelated. However,
numerous empirical studies consistently reject the UIP condition (e.g., Hodrick, 1987; Engel, 1996;
Sarno, 2005). As a result, it is a stylized fact that estimates of
tend to be closer to minus unity
than plus unity. This is commonly referred to as the “forward premium puzzle,” which implies that
high-interest currencies tend to appreciate rather than depreciate and forms the basis of the widely
used carry trade strategy in active currency management.5
2.2.3
Purchasing Power Parity
The third regression is based on the PPP hypothesis:
xt = p t
pt
st :
(13)
The PPP hypothesis states that national price levels should be equal when expressed in a common
currency and is typically thought of as a long-run condition rather than holding at each point in
time (e.g., Rogo¤, 1996; and Taylor and Taylor, 2004).
2.2.4
Monetary Fundamentals
The fourth regression conditions on monetary fundamentals (MF):
xt = (mt
mt )
(yt
yt )
st :
(14)
The relation between exchange rates and fundamentals de…ned in Equation (14) suggests that a
deviation of the nominal exchange rate st+1 from its long-run equilibrium level determined by the
fundamentals xt , requires the exchange rate to move in the future so as to converge towards its longrun equilibrium. The empirical evidence on the relation between exchange rates and fundamentals
is mixed. On the one hand, short-run exchange rate variability appears to be disconnected from
the underlying fundamentals (Mark, 1995) in what is commonly referred to as the “exchange rate
disconnect puzzle.” On the other hand, some recent empirical research …nds that fundamentals and
nominal exchange rates move together in the long run (Groen, 2000; and Mark and Sul, 2001).
4
An alternative way of testing UIP is to estimate the “Fama regression” (Fama, 1984), which conditions on the
forward premium. Note that if covered interest parity (CIP) holds, the interest rate di¤erential is equal to the forward
premium and testing UIP is equivalent to testing for forward unbiasedness in exchange rates (Bilson, 1981). For recent
evidence on CIP see Akram, Rime and Sarno (2008).
5
Clarida, Sarno, Taylor and Valente (2003, 2006) and Boudoukh, Richardson and Whitelaw (2006) also show that
the term structure of forward exchange (and interest) rates contains valuable information for forecasting spot exchange
rates.
7
2.2.5
Taylor Rule
The …nal two regressions are based on simple versions of the Taylor (1993) rule. We estimate a
symmetric Taylor rule (TRs ):
xt = 1:5 (
t
t)
+ 0:1 ytg
yt g ;
(15)
as well as an asymmetric Taylor rule (TRa ) that assumes that the foreign central bank also targets
the real exchange rate:
xt = 1:5 (
t
t)
+ 0:1 ytg
yt g + 0:1 (st + pt
pt ) :
(16)
The domestic and foreign output gaps are computed with a Hodrick and Prescott (1997) (HP)
…lter.6 The parameters on the in‡ation di¤erence (1:5), output gap di¤erence (0:1) and the real
exchange rate (0:1) are fairly standard in the literature (e.g., Engel, Mark and West, 2007; Mark,
2009). Alternative versions of the Taylor rule that we do not consider in this chapter may also account
for smoothing, where interest rate adjustments are not immediate but gradual, and heterogeneous
coe¢ cients for (i) the US versus foreign in‡ation, and (ii) the US versus foreign output gap (e.g.,
Molodtsova and Papell, 2009).
3
Statistical Evaluation of Exchange Rate Predictability
The success or failure of empirical exchange rate models is typically determined by statistical tests of
out-of-sample predictive ability. Our statistical analysis tests for equal predictive ability between one
of the empirical exchange rate models we estimate (UIP, PPP, MF, TRs or TRa ) and the benchmark
RW model. In e¤ect, we are comparing the performance of a parsimonious restricted null model
(the RW, where
model (where
= 0) to a set of larger alternative unrestricted models that nest the parsimonious
6= 0).7
We estimate all empirical exchange rate models using ordinary least squares (OLS), and then
run a pseudo out-of-sample forecasting exercise as follows (e.g., Stock and Watson, 2003). Given
today’s known observables f st+1 ; xt gTt=11 , we de…ne an in-sample (IS) period using observations
T 1
f st+1 ; xt gM
t=1 , and an out-of-sample (OOS) period using f st+1 ; xt gt=M +1 . This exercise produces
P = (T
1)
M OOS forecasts. Our empirical analysis uses T
1 = 413 monthly observations,
M = 120 and P = 293.8
6
Note that in estimating the HP trend in sample or out of sample, at any given period t, we only use data up to
period t 1. We then update the HP trend every time a new observation is added to the sample. This captures as
closely as possible the information available to central banks at the time decisions are made.
7
For a review of forecast evaluation see West (2006) and Clark and McCracken (2011).
8
The IS period for xt ranges from January 1976 to December 1985. The …rst OOS forecast is for the February 1986
value of st+1 that conditions on the January 1986 value of xt . The last forecast is for June 2010.
8
In what follows, we describe a comprehensive set of statistical criteria for evaluating the OOS
predictive ability of empirical exchange rate models. First, we compute the Campbell and Thompson
2 , that compares the unconditional forecasts of the benchmark RW
(2008) OOS R2 statistic, Roos
model to the conditional forecasts of an alternative model. Let
st+1jt denote the one-step ahead
sbt+1jt be the one-step ahead conditional forecast from the
unconditional forecast from the RW and
2 statistic is given by:
alternative model represented by one of Equations (12) to (16). Then, the Roos
2
Roos
PT
1
t=M +1
PT 1
t=M +1
=1
2
sbt+1jt
st+1
st+1
2:
st+1jt
2 statistic implies that the alternative model outperforms the benchmark RW by having
A positive Roos
a lower mean squared error (MSE).
Second, we compute the OOS root MSE di¤erence statistic,
RM SE, as in Welch and Goyal
(2008):
RM SE =
A positive
s
PT
1
t=M +1
st+1jt
st+1
P
s
2
PT
1
t=M +1
sbt+1jt
st+1
P
2
:
RM SE denotes that the alternative model outperforms the benchmark RW by having
a lower RMSE.
The most popular method for testing whether the alternative model has a lower MSE than the
benchmark is using the Diebold and Mariano (1995) and West (1996) statistic, which has an asymptotic standard normal distribution when comparing forecasts from non-nested models. However,
as shown by Clark and McCracken (2001) and McCracken (2007), this statistic has a non-standard
distribution when comparing forecasts from nested models and is severely undersized when using
standard normal critical values. Clark and McCracken (2001) and McCracken (2007) account for
this size distortion by deriving the non-standard asymptotic distributions for a number of statistical
tests as applied to nested models. We report the two tests with the best overall power and size properties: the EN C-F encompassing test statistic proposed by Clark and McCracken (2001) de…ned as
follows:
EN C-F =
PT
1
t=M +1
st+1
P
st+1jt
PT
1
2
st+1
1
t=M +1
st+1
st+1jt
st+1jt
sbt+1jt
st+1
2
;
and the M SE-F test of McCracken (2007):
M SE-F =
PT
1
t=M +1
P
st+1
PT
1
st+1jt
1
t=M +1
2
st+1
st+1
sbt+1jt
2
sbt+1jt
2
:
When the models are correctly speci…ed, the forecast errors are serially uncorrelated and exhibit
conditional homoskedasticity. In this case, Clark and McCracken (2001) and McCracken (2007)
numerically generate the asymptotic critical values for the EN C-F and M SE-F tests. When the
9
above conditions are not satis…ed, a bootstrap procedure must be used to compute valid critical
values, which we discuss later.
Finally, we also apply the recently developed inference procedure by Clark and West (2006, 2007)
for testing the null of equal predictive ability of two nested models. This procedure acknowledges
the fact that under the null the MSE from the alternative model is expected to be greater than that
of the RW benchmark because the alternative model introduces noise into the forecasting process
by estimating a parameter vector that is not helpful in prediction. Therefore, …nding that the RW
has smaller MSE is not clear evidence against the alternative model. Clark and West (2006, 2007)
suggest that the MSE should be adjusted as follows:
M SEadj =
1 XT 1
( st+1
t=M +1
P
sbt+1jt )2
1 XT 1
( st+1jt
t=M +1
P
Then, a computationally convenient way of testing for equal MSE is to de…ne
fbt+1jt = ( st+1
st+1jt )2
[( st+1
sbt+1jt )2
( st+1jt
sbt+1jt )2 :
sbt+1jt )2 ];
(17)
(18)
and to regress fbt+1jt on a constant, using the t-statistic for a zero coe¢ cient, which we denote by
MSE-t. Even though the asymptotic distribution of this test is non-standard (e.g., McCracken, 2007),
Clark and West (2006, 2007) show that standard normal critical values provide a good approximation,
and therefore recommend to reject the null if the statistic is greater than +1:282 (for a one sided
0:10 test) or +1:645 (for a one sided 0:05 test).9
The above statistical tests compare the null hypothesis of equal forecast accuracy against the
one-sided alternative that forecasts from the unrestricted model are more accurate than those from
the restricted benchmark model. Asymptotic critical values for these test statistics, whenever available, tend to be severely biased in small samples. In addition to the size distortion, there may
be spurious evidence of return predictability in small samples when the forecasting variable is su¢ ciently persistent (e.g., Nelson and Kim, 1993; Stambaugh, 1999). In order to address these concerns,
we obtain bootstrapped critical values for a one-sided test by estimating the model and generating
10; 000 bootstrapped time series under the null. The procedure preserves the autocorrelation structure of the predictive variable and maintains the cross-correlation structure of the residual. The
bootstrap algorithm is summarized in Appendix A.
4
Economic Evaluation of Exchange Rate Predictability
This section describes the framework for evaluating the performance of an asset allocation strategy
that exploits predictability in exchange rate returns.
9
This approximation tends to perform better when forecasts are obtained from rolling regressions than recursive
regressions.
10
4.1
The Dynamic FX Strategy
We design an international asset allocation strategy that involves trading the US dollar and nine
other currencies: the Australian dollar, Canadian dollar, Swiss franc, Deutsche markneuro, British
pound, Japanese yen, Norwegian kroner, New Zealand dollar and Swedish kronor. Consider a US
investor who builds a portfolio by allocating her wealth between ten bonds: one domestic (US), and
nine foreign bonds (Australia, Canada, Switzerland, Germany, UK, Japan, Norway, New Zealand
and Sweden). The yield of the bonds is proxied by eurodeposit rates. At the each period t + 1, the
foreign bonds yield a riskless return in local currency but a risky return rt+1 in US dollars, whose
expectation at time t is equal to Et [rt+1 ] = it +
st+1jt . Hence the only risk the US investor is
exposed to is FX risk.
Every period the investor takes two steps. First, she uses each predictive regression to forecast
the one-period ahead exchange rate returns. Second, conditional on the forecasts of each model, she
dynamically rebalances her portfolio by computing the new optimal weights. This setup is designed
to assess the economic value of exchange rate predictability by informing us which empirical exchange
rate model leads to a better performing allocation strategy.
4.2
Mean-Variance Dynamic Asset Allocation
Mean-variance analysis is a natural framework for assessing the economic value of strategies that
exploit predictability in the mean and variance. Consider an investor who has a one-period horizon
and constructs a dynamically rebalanced portfolio. Computing the time-varying weights of this
portfolio requires one-step ahead forecasts of the conditional mean and the conditional variancecovariance matrix. Let rt+1 denote the K
1 vector of risky asset returns;
the conditional expectation of rt+1 ; and
= Et
t+1jt
rt+1
rt+1
t+1jt
t+1jt
0
t+1jt
= Et [rt+1 ] is
is the K
K
conditional variance-covariance matrix of rt+1 .
Mean-variance analysis may involve three rules for optimal asset allocation: maximum expected
utility, maximum expected return and minimum volatility. Following Della Corte, Sarno and Tsiakas
(2009, 2011) our empirical analysis focuses on the maximum expected return strategy as this is the
strategy most often used in active currency management. For details on the maximum expected
utility rule and the minimum volatility rule see Han (2006).
The maximum expected return rule leads to a portfolio allocation on the e¢ cient frontier for a
given target conditional volatility. At each period t, the investor solves the following problem:
max
wt
n
p;t+1
s.t.
= wt0
2
p
t+1jt
= wt0
11
+ 1
wt0
t+1jt wt ;
rf
o
(19)
(20)
where
p
is the target conditional volatility of the portfolio returns. The solution to the maximum
expected return rule gives the following risky asset weights:
wt = p
where Ct =
t+1jt
rf
0
1
t+1jt
t+1jt
p
Ct
1
t+1jt
t+1jt
rf ;
(21)
rf + wt0 rt+1 :
(22)
rf .
Then, the gross return on the investor’s portfolio is:
Rp;t+1 = 1 + rp;t+1 = 1 + 1
Note that we assume that
t+1jt
= , where
wt0
is the unconditional covariance matrix of exchange
rate returns. In other words, we do not model the dynamics of FX return volatility and correlation.
Therefore, the optimal weights will vary across the empirical exchange rate models only to the extent
that the predictive regressions produce better forecasts of the exchange rate returns.10
4.3
Performance Measures
We assess the economic value of exchange rate predictability with a set of standard mean-variance
performance measures. We begin our discussion with the Fleming, Kirby and Ostdiek (2001) performance fee, which is based on the principle that at any point in time, one set of forecasts is better
than another if investment decisions based on the …rst set lead to higher average realized utility. The
performance fee is computed by equating the average utility of the RW optimal portfolio with the
average utility of the alternative (e.g., UIP) optimal portfolio, where the latter is subject to expenses
F. Since the investor is indi¤erent between these two strategies, we interpret F as the maximum
performance fee she will pay to switch from the RW to the alternative (e.g., UIP) strategy. In other
words, this utility-based criterion measures how much a mean-variance investor is willing to pay for
conditioning on better exchange rate forecasts. The performance fee will depend on , which is the
investor’s degree of relative risk aversion (RRA). To estimate the fee, we …nd the value of F that
satis…es:
T
X1
t=0
Rp;t+1
F
2 (1 + )
Rp;t+1
F
2
=
T
X1
Rp;t+1
t=0
2 (1 + )
2
Rp;t+1
;
(23)
where Rp;t+1 is the gross portfolio return constructed using the forecasts from the alternative (e.g.,
UIP) model, and Rp;t+1 is the gross portfolio return implied by the benchmark RW model.
We also evaluate performance using the premium return, which builds on the Goetzmann, Ingersoll, Spiegel and Welch (2007) manipulation-proof performance measure and is de…ned as:
" T 1
#
" T 1
#
1
1
1 X Rp;t+1
1
1 X Rp;t+1 1
;
P=
ln
ln
(1
)
T
Rf
(1
)
T
Rf
t=0
(24)
t=0
10
See Della Corte, Sarno and Tsiakas (2012) for an economic evaluation of volatility and correlation timing in foreign
exchange.
12
where Rf = 1 + rf . P is robust to the distribution of portfolio returns and does not require
the assumption of a particular utility function to rank portfolios. In contrast, the Fleming, Kirby
and Ostdiek (2001) performance fee assumes a quadratic utility function. P can be interpreted
as the certainty equivalent of the excess portfolio returns and hence can also be viewed as the
maximum performance fee an investor will pay to switch from the benchmark to another strategy.
In other words, this criterion measures the risk-adjusted excess return an investor enjoys for using
one particular exchange rate model rather than assuming a random walk. We report both F and P
in annualized basis points (bps).
In the context of mean-variance analysis, perhaps the most commonly used measure of economic
value is the Sharpe ratio (SR). The realized SR is equal to the average excess return of a portfolio
divided by the standard deviation of the portfolio returns. It is well known that because the SR
uses the sample standard deviation of the realized portfolio returns, it overestimates the conditional
risk an investor faces at each point in time and hence underestimates the performance of dynamic
strategies (e.g., Marquering and Verbeek, 2004; Han, 2006).
Finally, we also compute the Sortino ratio (SO), which measures the excess return to “bad”
volatility. Unlike the SR, the SO di¤erentiates between volatility due to “up” and “down” movements in portfolio returns. It is equal to the average excess return divided by the standard deviation
of only the negative returns. In other words, the SO does not take into account positive returns in
computing volatility because these are desirable. A large SO indicates a low risk of large losses.
4.4
Transaction Costs
The e¤ect of transaction costs is an essential consideration in assessing the pro…tability of dynamic
trading strategies. We account for this e¤ect in three ways. First, we calculate the performance
measures for the case when the bid-ask spread for spot exchange rates is equal to 8 bps. In foreign
exchange trading, this is a realistic range for the recent level of transaction costs.11 We follow the
simple approximation of Marquering and Verbeek (2004) by deducting the proportional transaction
cost from the portfolio return ex post. This ignores the fact that dynamic portfolios are no longer
optimal in the presence of transaction costs but maintains simplicity and tractability in our analysis.12
The second way of accounting for transaction costs acknowledges the fact that for long data
samples the transaction costs will likely change over time. Neely, Weller and Ulrich (2009) …nd that
11
In recent years, the typical transaction cost a large investor pays in the FX market is 1 pip, which is equal to 0:01
cent. For example, if the USD/GBP exchange rate is equal to 1:5000, 1 pip would raise it to 1:5001 and this would
roughly correspond to 1=2 basis point proportional cost.
12
Our empirical analysis uses the full bid-ask spread. Note, however, that the e¤ective spread is generally lower than
the quoted spread, since trading takes place at the best price quoted at any point in time, suggesting that the worse
quotes will not attract trades. For example, Goyal and Saretto (2009) and Della Corte, Sarno and Tsiakas (2011)
consider e¤ective transaction costs in the range of 50% to 100% of the quoted spread. Assuming that the e¤ective
spread is less than the quoted spread would make our economic evidence stronger.
13
the transaction cost for switching from a long to a short position in FX has on average declined from
about 10 bps in the 1970s to about 2 bps in recent years. If we were to keep transaction costs constant
over our sample period, we would spuriously introduce a decline in performance by penalizing more
recent returns too heavily relative to those early in the sample period. Therefore, we follow Neely,
Weller and Ulrich (2009) in estimating a simple time trend that assumes that the bid-ask spread was
20 bps at the beginning of our data sample and declined linearly to 4 bps by the end of the sample.
The actual one-way transaction cost is half of the bid-ask spread and hence declines from 10 bps
to 2 bps. Speci…cally, the net return from buying a currency at the spot exchange rate at time t
and selling at time t + 1 is equal to sbid
t+1
ask
ct+1 = 0:5 St+1
sask
= smid
t
t+1
smid
t
t+1 ,
where
t+1
ct+1 )
= ln (1(1+c
and
t)
bid =S mid is the one-way transaction cost (e.g., Neely, Weller and Ulrich, 2009).
St+1
t+1
Upper case St is the spot exchange rate and lower case st is st = ln St . In the …rst case we assume a
…xed bid-ask spread and hence
t
= , whereas in the second case
t
is time-varying.13
Third, we also calculate the break-even proportional transaction cost,
indi¤erent between two strategies (e.g., Han, 2006). We assume that
be ,
that renders investors
is a …xed fraction of the value
traded in all assets in the portfolio. Then, the cost of the dynamic strategy is jwt
each asset j
1 (1+rj;t )
j
1+rp;t
for
K. In comparing a dynamic strategy with the benchmark RW strategy, an investor
who pays transaction costs lower than
be
will prefer the dynamic strategy. Since
cost paid every time the portfolio is rebalanced, we report
5
wt
be
be
is a proportional
in monthly basis points.14
Combined Forecasts
Our analysis has so far focussed on evaluating the performance of individual empirical exchange
rate models relative to the random walk benchmark. Considering a large set of alternative models
that capture di¤erent aspects of exchange rate behaviour without knowing which model is “true”(or
best) inevitably generates model uncertainty. In this section, we resolve this uncertainty by exploring
whether portfolio performance improves when combining the forecasts arising from the full set of
predictive regressions. Even though the potentially superior performance of combined forecasts is
known since the seminal work of Bates and Granger (1969), applications in …nance are only recently
becoming increasingly popular (e.g., Timmermann, 2006). Rapach, Strauss and Zhou (2010) argue
that forecast combinations can deliver statistically and economically signi…cant out-of-sample gains
for two reasons: (i) they reduce forecast volatility relative to individual forecasts, and (ii) they are
13
The derivation is as follows:
bid
St+1
Stask
=
mid
St+1
0:5
Stmid +0:5
ask
St+1
Stask
(
(
bid
St+1
Stbid
)
=
)
Stmid
(1 ct+1 )
mid
ln sbid
ln sask
ln smid
ln (1+ct ) .
t+1
t+1 = ln st+1
t+1
14
For a slightly di¤erent calculation see Jondeau and Rockinger (2008).
14
bid
St+1
S mid
t+1
ask S bid
0:5 St
t
1+
mid
St
mid
1
St+1
ask
0:5 St+1
(
(
)
)
!
!
=
mid
St+1
(1 ct+1 )
Stmid (1+ct )
. Then,
linked to the real economy.15
Recall that we estimate N = 6 predictive regressions each of which provides an individual forecast
sbi;t+1 for the one-step ahead exchange rate return, where i
N . We de…ne the combined forecast
sbc;t+1 as the weighted average of the N individual forecasts
sbc;t+1 =
XN
i=1
! i;t sbi;t+1 ;
sbi;t+1 :
(25)
where f! i;t gN
i=1 are the ex ante combining weights determined at time t.
We form three types of combined forecasts. The …rst one uses simple model averaging that in
turn implements three rules: (i) the mean of the panel of forecasts so that ! i;t = 1=N ; the median of
the f sbi;t+1 gN
i=1 individual forecasts; and (iii) the trimmed mean that sets ! i;t = 0 for the individual
forecasts with the smallest and largest values and ! i;t = 1= (N
2) for the remaining individual
forecasts. These combined forecasts disregard the historical performance of the individual forecasts.
The second type of combined forecasts is based on Stock and Watson (2004) and uses statistical
information on the past OOS performance of each individual model. In particular, we compute the
discounted MSE (DM SE) forecast combination by setting the following weights:
where
DM SEi;t1
! i;t = PN
;
1
j=1 DM SEj;t
DM SEi;t =
XT
1
T 1 t
t=M +1
( st+1
sbi;t+1 )2 ;
(26)
is a discount factor and M are the …rst in-sample observations on which we condition to
form the …rst out-of-sample forecast. For
< 1, greater weight is attached to the most recent
forecast accuracy of the individual models. The DM SE forecasts are computed for three values
of
= f0:90; 0:95; 1:0g. The case of no discounting ( = 1) corresponds to the Bates and Granger
(1969) optimal forecast combination when the individual forecasts are uncorrelated. We also compute
simpler “most recently best” M SE ( ) forecast combinations that use no discounting ( = 1) and
weigh individual forecasts by the inverse of the OOS MSE computed over the last
months, where
= f12; 36; 60g.
The third type of combined forecasts does not use statistical information on the historical performance of individual forecasts. Instead it exploits the economic information contained in the Sharpe
ratio (SR) of the portfolio returns generated by an individual forecasting model over a prespeci…ed
recent period. We compute the discounted SR (DSR) combined forecast by setting the following
weights:
DSRi;t
;
! i;t = PN
j=1 DSRj;t
DSRi;t =
XT
1
t=M +1
T 1 t
SRt+1 ;
(27)
Finally, we also compute simpler “most recently best” SR ( ) forecast combinations that use no
discounting ( = 1) and weigh individual forecasts by the OOS SR computed over the last
15
months,
For a Bayesian approach to forecast combinations see Avramov (2002), Cremers (2002), Wright (2008), and Della
Corte, Sarno and Tsiakas (2009, 2012).
15
where
= f12; 36; 60g.
We assess the economic value of combined forecasts by treating them in the same way as any of
the individual empirical models. For instance, we compute the performance fee, F, for the DM SE
one-month ahead forecasts and compare it to the RW benchmark. Finally, note that where possible
we use these forecast combination methods not only for the OOS mean prediction but also for the
OOS variance covariance matrix that enters the weights in mean-variance asset allocation.
6
Empirical Results
6.1
Data on Exchange Rates and Economic Fundamentals
The data sample consists of 414 monthly observations ranging from January 1976 to June 2010, and
focuses on nine spot exchange rates relative to the US dollar (USD): the Australian dollar (AUD),
Canadian dollar (CAD), Swiss franc (CHF), Deutsche markneuro (EUR), British pound (GBP),
Japanese yen (JPY), Norwegian kroner (NOK), New Zealand dollar (NZD) and Swedish kronor
(SEK). The exchange rate is de…ned as the US dollar price of a unit of foreign currency so that an
increase in the exchange rate implies a depreciation of the US dollar. These data are obtained through
the download data program (DDP) of the Board of Governors of the Federal Reserve System.16
Table 1 provides a detailed description of all data sources we use. For interest rates, we use the
one-month euro deposit rate taken from Datastream with the following exceptions. For Japan, the
euro deposit rate is only available from January 1979 and hence before this date we use Covered
Interest Parity (CIP) relative to USD to construct the no-arbitrage riskless rate. The one-month
forward exchange rate required to implement CIP is taken from Hai, Mark, and Wu (1997). For
Australia, Norway, New Zealand and Sweden, euro deposit rates are only available from April 1997.
For Australia and New Zealand, we combine the money market rate from January 1976 to November
1984 taken from the IMF’s International Financial Statistics (IFS) and CIP relative to USD from
December 1984 to March 1997 using one-month forward exchange rates taken from Datastream. For
Norway and Sweden, we use CIP relative to GBP from January 1976 to March 1997, using spot and
one-month forward exchange rates from Datastream.
Turning to macroeconomic data, we use non-seasonally adjusted M1 data to measure money
supply. For the UK, we use M0 due to the unavailability of M1 data. To construct these times series,
we combine IFS and national central bank data from Ecowin.17 We deseasonalize the money supply
data by implementing the procedure of Gomez and Maravall (2000).
16
Before the introduction of the euro in January 1999, we use the US dollar-Deutsche mark exchange rate combined
with the o¢ cial conversion rate between the Deutsche mark and the euro.
17
For Germany regarding the period of January 1976 to December 1979, we construct the money supply using data
on currency outside banks and demand deposits from IFS.
16
The price level is measured by the monthly consumer price index (CPI) obtained from the OECD’s
Main Economic Indicators (MEI). For Australia and New Zealand, CPI data are published at quarterly frequency and hence monthly observations are constructed by linear interpolation. For the
in‡ation rate we use an annual measure computed as the 12-month log di¤erence of the CPI. We
de…ne the output gap as deviations from the HP …lter.
Since GDP data are generally available quarterly, we proxy real output by the seasonally adjusted monthly industrial production index (IPI) taken from IFS. For Australia, New Zealand, and
Switzerland, however, IPI data are only released at quarterly frequency and hence we obtain monthly
observations via linear interpolation.18 Orphanides (2001) has recently stressed the importance of
using real-time data to estimate Taylor rules for the United States, which are data available to central banks when the policy decisions are made. Since real-time data are not available for most of
the countries included in this study, we mimic as closely as possible the information set available
to the central banks using quasi-real time data: although data incorporate revisions, we update the
HP trend each period so that ex-post data is not used to construct the output gap. In other words,
at time t we only use data up to t
1 to construct the output gap. Using a number of detrending
methods, Orphanides and van Norden (2002) show that most of the di¤erence between fully revised
and real-time data comes from using ex post data to construct potential output and not from the
data revisions themselves.19
We convert all data but interest rates by taking logs and multiplying by 100. Throughout the
rest of the chapter, the symbols st , it , mt , pt ,
t,
yt and ytg refer to transformed spot exchange rate,
interest rate, money supply, price level, in‡ation rate„real output and output gap, respectively. We
use an asterisk to denote the transformed data (it , mt , pt ,
t,
yt and ytg ) for the foreign country.
Table 2 reports the descriptive statistics for the monthly % FX returns,
tween domestic an foreign interest rates, it
it ; the di¤erence in % change in price levels,
the di¤erence in % change in money supply,
output,
(yt
st ; the di¤erence be-
(mt
(pt
pt );
mt ); and the di¤erence in % change in real
yt ). For our sample period, the monthly sample means of the FX returns range from
0:138% for SEK to 0:296% for JPY. The return standard deviations are similar across all exchange
rates at about 3% per month. Most FX returns exhibit negative skewness and higher than normal
kurtosis. Finally, the exchange rate return sample autocorrelations are no higher than 0:10 and
decay rapidly. For the economic fundamentals the notable trends are as follows: (i) it
persistent with long memory; (ii)
and
(yt
(pt
pt ) are always negatively skewed; and (iii)
it are highly
(mt
mt )
yt ) have occasionally high kurtosis.
18
For New Zealand, IPI data are only available from June 1977. We …ll the gap using quarterly GDP data.
The output gap for the …rst period is computed using real output data from January 1970 to January 1976. In the
HP …lter, we use a smoothing parameter equal to 14,400 as in Molodtsova and Papell (2009).
19
17
6.2
Predictive Regressions
We test the empirical performance of the models by …rst estimating the six predictive regressions
for nine monthly exchange rates. The regressions include the random walk (RW) model, uncovered
interest parity (UIP), purchasing power parity (PPP), monetary fundamentals (MF), symmetric
Taylor rule (TRs ) and asymmetric Taylor rule (TRa ). Table 3 presents the OLS estimates with Newey
and West (1987) standard errors. We focus primarily on the signi…cance of the slope estimate
of
the predictive regressions since this would be an indication that the RW benchmark is misspeci…ed.
Consistent with the large literature on the forward premium puzzle, the UIP
is predominantly
negative. The PPP
is always positive and for TRa it is always negative. For these three cases (UIP,
PPP and TRa ), the
estimates are signi…cant for half of the exchange rates. The least signi…cant
slopes are for MF revolving around zero and for TRs for which they are always negative. Finally, the
2 of the predictive regressions is as high as 2:4% but in most cases it is below 1%. In conclusion,
Roos
the predictive regression results demonstrate that the empirical exchange rates models with the most
signi…cant slopes are the UIP, PPP and TRa .
6.3
Statistical Evaluation
We assess the statistical performance of the empirical exchange rate models (UIP, PPP, MF, TRs and
TRa ) by reporting out-of-sample tests of predictability against the null of the RW. We focus on the
2
following statistics: (i) the Roos
statistic of Campbell and Thompson (2008). Recall that a positive
2
Roos
value implies that the alternative model has lower MSE than the benchmark RW. However,
2
may be consistent with a better performing alternative because the
even a slightly negative Roos
2
calculation of the Roos
does take into account the adjustment in the MSE proposed by Clark and
West (2006, 2007) to account for the noise introduced in forecasting by estimating a parameter that
is not helpful in prediction; (ii) the
RM SE statistic, a positive value for which denotes superior
2 ; (iii)
OOS performance for the competing model but is subject to the same criticism as the Roos
the Clark and McCracken (2001) EN C-F statistic; (iv) the McCracken (2007) M SE-F statistic;
and (v) the Clark and West (2006, 2007) M SE-t statistic. The null hypothesis for the EN C-F ,
M SE-F and M SE-t statistics is that the MSEs for the random walk and the competing model are
equal against the alternative that the competing model has lower MSE. One-sided critical values
are obtained by generating 10,000 bootstrapped time series as in Mark (1995) and Kilian (1999).
The OOS monthly forecasts are obtained in two ways: (i) with rolling regressions that use a 10-year
window that generates forecasts for the period of January 1986 to June 2010; and (ii) with recursive
regressions for the same forecasting period that successively re-estimate the model parameters every
time a new observation is added to the sample.
18
Table 4 shows that most of the statistics tend to be negative and hence provide evidence against
the alternative model. In many cases, however, the results are not statistically signi…cant. If instead
we focus on the Clark and West (2006, 2007) M SE-t statistic, which makes the adjustment to the
MSE and is hence more reliable, a di¤erent picture emerges. For rolling regressions, the UIP and
PPP models have a positive M SE-t statistic for seven of the nine exchange rates, whereas the MF
and TRa models for six. The model that is most often signi…cantly di¤erent from the RW is the
TRa . The worse performing model is the TRs . The results are very similar for recursive regressions.
In short, therefore, a careful examination of the empirical evidence reveals that many of the models
perform well against the RW with the clear exception of the TRs .
It is important to note that in out-of-sample predictive regressions, lack of statistical signi…cance
does not imply lack of economic signi…cance. Campbell and Thompson (2008) show that a small
R2 can generate large economic bene…ts for investors. They use a mean-variance framework to
demonstrate that a good way to judge the magnitude of R2 is to compare it to the square of
the Sharpe ratio (SR2 ). Even a modest R2 can lead to a substantial proportional increase in the
expected return by conditioning on the predictive variable xt . Indeed, regressions with large R2
statistics would be too pro…table to believe, which is equivalent to the saying: “if you are so smart,
why aren’t you rich?” In the limit, an R2 close to 1 should lead to perfect predictions and hence
in…nite pro…ts for investors. Furthermore, dynamic asset allocation is by design multivariate thus
exploiting predictability in all exchange rate series. In the following section, we discuss in detail
whether the predictive regressions can generate economic value.
6.4
Economic Evaluation
We assess the economic value of exchange rate predictability by analyzing the performance of dynamically rebalanced portfolios based on one-month ahead forecasts from the six empirical exchange
rate models we estimate. The economic evaluation is conducted both IS and OOS, but again the
main focus of our analysis is OOS. The OOS results we present in this section are based on forecasts
constructed according to a recursive procedure that conditions only upon information up to the
month that the forecast is made. The predictive regressions are then successively re-estimated every
month.
Our empirical analysis focuses on the Sharpe ratio (SR), the Sortino ratio (SO), the Fleming,
Kirby and Ostdiek (2001) performance fee (F), the Goetzmann, Ingersoll, Spiegel and Welch (2007)
premium return measure (P) and the break even transaction cost
be .
The F and P performance
measures are computed for three cases: (i) zero transaction costs; (ii) a bid-ask spread of 8 bps; and
(iii) a bid-ask spread of 20 bps at the beginning of the sample that linearly decays to 4 bps at the
end of the sample as suggested by Neely, Weller and Ulrich (2009). Following Della Corte, Sarno
19
and Tsiakas (2009, 2011) our empirical analysis focuses on the maximum expected return strategy
as this is the strategy most often used in active currency management. We set a volatility target of
p
= 10% and a degree of RRA
= 6. We have experimented with di¤erent
p
and
values and
found that qualitatively they have little e¤ect on the asset allocation results discussed below.
Table 5 reports the IS and OOS portfolio performance and shows that there is high economic
value associated with some of the empirical exchange rate models. We …rst discuss the IS results,
which demonstrate that all models outperform the RW, except for the symmetric Taylor rule (TRs ).
For example, SR = 1:30 for PPP, 1:28 for TRa , 1:18 for MF, 1:14 for UIP, 1:08 for RW and 0:96
for TRs . The SO have higher values ranging from 1:26 for TRs to 2:00 for PPP. Switching from
the benchmark RW to another model generates F = 285 annual bps for PPP, 202 bps for TRa ; 138
for MF and 83 bps for UIP. The P performance measure has similar value to F. Furthermore, both
measures are largely una¤ected by transaction costs. This can be exempli…ed by the very large value
of the monthly
be ,
which are 586 bps for UIP, 328 bps for MF and 138 bps for PPP.
The literature on exchange rate forecasting is primarily concerned with out-of-sample predictability and hence we turn our attention to the OOS results. The …rst thing to notice is that the value of
the OOS SR is smaller than IS. The RW has an OOS SR = 0:54 and is outperformed only by the
PPP (SR = 0:76), UIP (SR = 0:65) and TRa (SR = 0:65). Consistent with a very large literature
in FX, monetary fundamentals models do not outperform the RW and neither does the TRs . The
F values are 252 annual bps for PPP, 131 bps for UIP, 130 bps for TRa , 10 bps for MF and
384
bps for TRs . The P measure has slightly higher value than F. Transaction costs seem to be a bit
more important OOS than IS. For example, the
be
are 173 bps for UIP, 161 bps for TRa and 70
bps for PPP. However, it seems that whether we assume …xed transaction costs or linearly decaying
costs makes little di¤erence in the performance of the empirical exchange rate models. In short, our
…ndings demonstrate that it is worth using the UIP, PPP and TRa empirical exchange rate models
as their forecasts generate signi…cant economic value.
By design, the dynamic FX strategy invests in nine foreign bonds and thus exploits predictability
in nine exchange rates. Since we economically evaluate the performance of portfolios rather than
individual exchange rates, it would be interesting to assess whether the superior portfolio performance
of one versus another empirical model is driven by one particular currency. Table 6 reports the
economic value of exchange rate predictability when we remove one of the currencies (and hence
one of the bonds) from the investment opportunity set. For example, AUD in Table 6 denotes the
dynamic allocation strategy that invests in all currencies, except for AUD. The results for excluding
one currency at a time show that the best performing models are still the same as before. In sample,
all models but the TRs outperform the RW, whereas out of sample the UIP, PPP and TRa are still
the best models. Therefore, the empirical evidence suggests that our results are not driven by any
20
one particular currency.
A unique feature of the FX market is that investors trade currencies but all prices are quoted
relative to a numeraire. Consistent with the vast majority of the FX literature, we use data on
exchange rates relative to the US dollar. It is of interest, however, to check whether using a di¤erent
currency as numeraire meaningfully a¤ects the economic value of the empirical exchange rate models.
This is a crucially important robustness check since it is straightforward to show analytically that
the portfolio returns and their covariance matrix are not invariant to the numeraire. For example,
consider taking the point of view of a European investor and hence changing the numeraire currency
from the US dollar to the euro. Then, all previously bilateral exchange rates become cross rates and
nine of the previously cross rates become bilateral. Furthermore, converting dollar FX returns into
euro FX returns replaces the US bond as the domestic asset by the European bond. It also replaces all
US economic and monetary fundamentals by Europe’s fundamentals. The main question, however,
can only be answered empirically: if changing the numeraire also changes the portfolio returns, does
the economic value of the empirical exchange rate models also change?
Table 7 shows the IS and OOS economic value of exchange rate predictability from the perspective
of each of nine countries other than the US. For example, using the AUD as numeraire means that
all exchange rates are quoted relative to AUD, all predictive regressions are estimated using the
new exchange rates and the mean-variance economic evaluation is done from the perspective of an
Australian investor. The same holds when the numeraire changes to CAD, CHF, EUR, GBP, JPY,
NOK, NZD and SEK. We …nd that our main results remain robust across all numeraires: the best
IS and OOS models are consistently the UIP, the PPP and the TRa . In terms of Sharpe ratios and
performance fees, IS the PPP and TRa outperform the RW for all nine numeraires and UIP does
so six of nine times; OOS the PPP outperforms the RW seven of nine times, whereas the UIP and
TRa …ve of nine times. To conclude, the economic value of exchange rate predictability of the best
individual empirical exchange rate models remains robust regardless of the numeraire choice.
In addition to the results associated with individual models, even stronger economic evidence is
found for the combined forecasts reported in Table 8. In all cases, forecast combinations signi…cantly
outperform the RW model. In fact, the best performing model averaging strategies are those based
on the SR. For example, the SR( = 12) strategy generates: (i) SR = 0:76 compared to the RW
where SR = 0:54, and (ii) F = 254 annual bps with
be
= 128 monthly bps. It is noteworthy that the
simple model average strategy using the mean forecast also generates a high SR = 0:74 and F = 234
bps. Another trend worth mentioning is that the degree of discounting ( ) or the length of the most
recently best period ( ) have little or no e¤ect on the performance of combined forecasts. In short,
therefore, there is clear out-of-sample economic evidence on the superiority of combined forecasts
relative to the RW benchmark that tends to be robust to the way combined forecasts are formed.
21
Finally, Figure 1 illustrates that the OOS Sharpe ratios for the three best performing individual
models (UIP, PPP and TRa ) and the SR( = 60) forecast combination against the RW.
7
Conclusion
Thirty years of empirical research in international …nance has attempted to resolve whether exchange
rates are predictable. Most of this literature uses statistical criteria for out-of-sample tests of the
null of the random walk representing no predictability against the alternative of linear models that
condition on economic fundamentals. The results of these studies are speci…c to, among other things,
the empirical model and the exchange rate series. An emerging literature has moved in a di¤erent
direction by providing an economic evaluation of predictability. This second line of research takes 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. The results of these studies are also speci…c to the
empirical model, but instead of providing results for one exchange rate at a time, they evaluate
predictability by looking at the performance of dynamically rebalanced portfolios. Finally, there
is a third strand of empirical work that forms ex ante combined forecasts from a set of individual
empirical models. The results of these studies are not particular to an empirical model but rather
relate to forecast combinations that account for model uncertainty.
This chapter reviews and connects these three loosely related literatures. We illustrate the statistical and economic methodologies by estimating a set of widely used empirical exchange rate
models using monthly returns from nine major US dollar exchange rates. In line with Campbell and
Thompson (2008), we show that modest statistical signi…cance can generate large economic bene…ts for investors with a dynamic FX portfolio strategy. We …nd three main results: (i) empirical
models based on uncovered interest parity, purchasing power parity and the asymmetric Taylor rule
perform better than the random walk in out-of-sample forecasting using both statistical and economic criteria; (ii) conditioning on monetary fundamentals or using a symmetric Taylor rule does
not generate economic value out of sample; and (iii) combined forecasts formed using a variety of
model averaging methods perform better than individual empirical models. These results are robust
to reasonably high transaction costs, the choice of numeraire and the exclusion of any one currency
from the investment opportunity set.
A
Appendix: The Bootstrap Algorithm
This appendix summarizes the bootstrap algorithm we use for generating critical values for the OOS
test statistics under the null of no exchange rate predictability against a one-sided alternative of linear
predictability. Following Mark (1995) and Kilian (1999), the algorithm consists of the following steps:
22
T 1
1. De…ne the IS period for f st+1 ; xt gM
t=1 and the OOS period for f st+1 ; xt gt=M +1 . We gen-
erate P = (T
1)
regression:
M OOS forecasts f st+1jt ; sbt+1jt gTt=M1 +1 by estimating the predictive
st+1 =
+ xt + "t+1
and then computing the test statistic of interest, b.
2. De…ne the data generating process (DGP) as
st+1 =
+ xt + u1;t+1
xt =
+
1 xt 1
+ ::: +
and estimate this model subject to the constraint that
full sample of observations f
st+1 ; xt gTt=11 .
p xt p
+ u2;t ;
in the …rst equation is zero, using the
The lag order p in the second equation is determined
by a suitable lag order selection criterion such as the Bayesian information criterion (BIC).
3. Generate a sequence of pseudo-observations
st ; xt
st+1 = b + u1;t+1
xt
= b + b1 xt
1
T 1
1 t=1
+ : : : + bp xt
as follows:
p
+ u2;t :
The pseudo-innovation term ut = (u1;t ; u2;t )0 is randomly drawn with replacement from the set
of observed residuals u
bt = (b
u1;t ; u
b2;t )0 . The initial observations xt
1 ; : : : ; xt p
0
are randomly
drawn from the actual data. Repeat this step B = 10; 000 times.
4. For each of the B bootstrap replications, de…ne an IS period for
period for
st+1 ; xt
T 1
.
t=M +1
st+1 ; xt
M
,
t=1
and an OOS
Then, generate P OOS forecasts f st+1jt ; sbt+1jt gTt=M1 +1 by
estimating the predictive regression:
st+1 =
+
xt + u1;t+1
both under the null and the alternative for t = M + 1; : : : ; T
1, and construct the test statistic
of interest, b .
5. Compute the one-sided p-value of b as:
p-value =
B
1 X
I(b > b);
B
j=1
where I ( ) denotes an indicator function, which is equal to 1 when its argument is true and 0
otherwise.
23
Table 1: Data Sources
The table presents a detailed description of the sources of the raw data. The exchange rate data range from January 1976 to June 2010. The riskless rate and the money
supply data range from January 1976 to May 2005. Data on real output range from January 1970 to May 2010 and are used to construct the output gap. Data on the price
level range from January 1975 to May 2010 and are used to construct the inflation rate. The data are monthly but quarterly data are used to retrieve monthly observations
via linear interpolation when monthly data are not available. The raw money supply is not seasonally adjusted but the raw real output is.
Country
Description
Source
Australia
Canada
Switzerland
Germany
Spot
Spot
Spot
Spot
Spot
Spot
Spot
Spot
Spot
Spot
Federal
Federal
Federal
Federal
Federal
Federal
Federal
Federal
Federal
Federal
UK
Japan
Norway
New Zealand
Sweden
Australia
Canada
Switzerland
Germany
UK
Japan
Norway
New Zealand
Sweden
US
USD/AUD
CAD/USD
CHF/USD
DEM/USD
USD/EUR
USD/GBP
JPY/USD
NOK/USD
USD/NZD
SEK/USD
Money Market Rate
Spot AUD/USD
1M Fwd AUD/USD
1M Euro Deposit Rate
1M Euro Deposit Rate
1M Euro Deposit Rate
1M Euro Deposit Rate
1M Euro Deposit Rate
Spot JPY/USD
1M Fwd JPY/USD
1M Euro Deposit Rate
Spot NOK/GBP
1M Fwd NOK/GBP
1M Euro Deposit Rate
Money Market Rate
Spot NZD/USD
1M Fwd NZD/USD
1M Euro deposit rate
Spot SEK/GBP
1M Fwd SEK/GBP
1M Euro deposit rate
1M Euro deposit rate
Range
Nominal Exchange Rate
Reserve Board
76:01-10:06
Reserve Board
76:01-10:06
Reserve Board
76:01-10:06
Reserve Board
76:01-98:12
Reserve Board
99:01-10:06
Reserve Board
76:01-10:06
Reserve Board
76:01-10:06
Reserve Board
76:01-10:06
Reserve Board
76:01-10:06
Reserve Board
76:01-10:06
Frequency
Series
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
DDP [RXI$US_N.B.AL]
DDP [RXI_N.B.CA]
DDP [RXI_N.B.SZ]
H.10 Historical Rates
DDP [RXI$US_N.B.EU]
DDP [RXI$US_N.B.UK]
DDP [RXI_N.B.JA]
DDP [RXI_N.B.NO]
DDP [RXI$US_N.B.NZ]
DDP [RXI_N.B.SD]
Riskless Rate
IMF IFS
76:01-84:11
Barclays Bank
84:12-97:03
Barclays Bank
84:12-97:03
Thomson Reuters
97:04-10:05
Thomson Reuters
76:01-10:05
Thomson Reuters
76:01-10:05
Thomson Reuters
76:01-10:05
Thomson Reuters
76:01-10:05
Hai, Mark and Wu (1997)
76:01-78:12
Hai, Mark and Wu (1997)
76:01-78:12
Thomson Reuters
79:01-10:05
Not Specified
76:01-97:03
Not Specified
76:01-97:03
Thomson Reuters
97:04-10:05
IMF IFS
76:01-84:11
Barclays Bank
84:12-97:03
Barclays Bank
84:12-97:03
Thomson Reuters
97:04-10:05
Not Specified
76:01-97:03
Not Specified
76:01-97:03
Thomson Reuters
97:04-10:05
Thomson Reuters
76:01-10:05
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Ecowin [ifs:s19360b00zfm]
Datastream [BBAUDSP]
Datastream [BBAUD1F]
Datastream [ECCAD1M]
Datastream [ECAUD1M]
Datastream [ECSWF1M]
Datastream [ECWGM1M]
Datastream [ECUKP1M]
Nelson Mark’s website
Nelson Mark’s website
Datastream [ECJAP1M]
Datastream [NORKRON]
Datastream [NORKN1F]
Datastream [ECNOR1M]
Ecowin [ifs:s19660000zfm]
Datastream [BBNZDSP]
Datastream [BBNZD1F]
Datastream [ECNZD1M]
Datastream [SWEKRON]
Datastream [SWEDK1F]
Datastream [ECSWE1M]
Datastream [ECUSD1M]
(continued)
24
Table 1: Data Sources (continued)
Country
Description
Australia
Canada
Switzerland
UK
Japan
Norway
M1
M1
M1
M1
Currency in Circulation
Demand Deposits
M1
M0
M1
M1
New Zealand
M1
Sweden
M1
US
M1
Range
Money Supply
Reserve Bank of Australia
76:01-10:05
Bank of Canada
76:01-10:05
IMF IFS
76:01-84:11
Swiss National Bank
84:12-10:05
IMF IFS
76:01-79:12
IMF IFS
76:01-79:12
Deutsche Bundesbank
80:01-10:05
Bank of England
76:01-10:05
Bank of Japan
76:01-10:05
IMF IFS
76:01-86:12
Norges Bank
87:01-10:05
IMF IFS
76:01-77:02
Reserve Bank of New Zealand
77:03-10:05
IMF IFS
76:01-98:02
Sveriges Riksbank
98:03-10:05
Federal Reserve United States
76:01-10:05
Australia
Canada
Switzerland
Germany
UK
Japan
Norway
New Zealand
Sweden
United States
Industrial Production Index
Industrial Production Index
Industrial Production Index
Industrial Production Index
Industrial Production Index
Industrial Production Index
Industrial Production Index
Gross Domestic Product
Industrial Production Index
Industrial Production Index
Industrial Production Index
IMF
IMF
IMF
IMF
IMF
IMF
IMF
IMF
IMF
IMF
IMF
Australia
Canada
Switzerland
Germany
UK
Japan
Norway
New Zealand
Sweden
United States
Consumer
Consumer
Consumer
Consumer
Consumer
Consumer
Consumer
Consumer
Consumer
Consumer
OECD
OECD
OECD
OECD
OECD
OECD
OECD
OECD
OECD
OECD
Germany
Source
Frequency
Series
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
[ew:aus12045]
[ew:can12042]
[ifs:s14634000zfm]
[ew:che12045]
[ifs:s13434a0nzfm]
[ifs:s13434b0nzfm]
[ew:deu12990]
[boe:lpmavaa]
[ew:jpn12066]
[ifs:s14234000zfm]
[ew:nor12045]
[ifs:s19634000zfm]
[ew:nzl12045]
[ifs:s14435l00zfm]
[ew:swe12010]
[ew:usa12010]
69:12-10:06
70:01-10:05
69:04-10:06
70:01-10:05
70:01-10:05
70:01-10:05
70:01-10:05
69:12-77:05
77:06-10:06
70:01-10:05
70:01-10:05
Quarterly
Monthly
Quarterly
Monthly
Monthly
Monthly
Monthly
Quarterly
Quarterly
Monthly
Monthly
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
[ifs:s1936600czfq]
[ifs:s1566600czfm]
[ifs:s1466600bzfq]
[ifs:s1346600czfm]
[ifs:s1126600czfm]
[ifs:s1586600czfm]
[ifs:s1426600czfm]
[ifs:s19699b0czfy]
[ifs:s19666eyczfq]
[ifs:s1446600czfm]
[ifs:s1116600czfm]
74:12-10:06
75:01-10:05
75:01-10:05
75:01-10:05
75:01-10:05
75:01-10:05
75:01-10:05
74:12-10:06
75:01-10:05
75:01-10:05
Quarterly
Monthly
Monthly
Monthly
Monthly
Monthly
Monthly
Quarterly
Monthly
Monthly
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
EcoWin
[oecd:aus_cpalcy01_ixobq]
[oecd:can_cpaltt01_ixobm]
[oecd:che_cpaltt01_ixobm]
[oecd:deu_cpaltt01_ixobm]
[oecd:gbr_cpaltt01_ixobm]
[oecd:jpn_cpaltt01_ixobm]
[oecd:nor_cpaltt01_ixobm]
[oecd:nzl_cpalcy01_ixobq]
[oecd:swe_cpaltt01_ixobm]
[oecd:usa_cpaltt01_ixobm]
Real Output
IFS
IFS
IFS
IFS
IFS
IFS
IFS
IFS
IFS
IFS
IFS
Price Level
Price
Price
Price
Price
Price
Price
Price
Price
Price
Price
Index
Index
Index
Index
Index
Index
Index
Index
Index
Index
MEI
MEI
MEI
MEI
MEI
MEI
MEI
MEI
MEI
MEI
25
Table 2. Descriptive Statistics
The table presents descriptive statistics for nine major exchange rates and a set of economic fundamentals.
s
is the % change in the US dollar exchange rate vis-à-vis the Australian dollar (AUD), Canadian dollar (CAD), Swiss
franc (CHF), Deutsche markneuro (EUR), British pound (GBP), Japanese yen (JPY), Norwegian kroner (NOK), New
Zealand dollar (NZD) and Swedish kronor (SEK); i is the one-month interest rate; p is the % change in the price
level; m is the % change in the money supply; y is the % change in real output; and the asterisk denotes a non-US
value. The exchange rate is de…ned as US dollars per unit of foreign currency. l is the autocorrelation coe¢ cient with
l lags. The data range from January 1976 to June 2010 for a sample size of 414 monthly observations.
M ean
AU D
s
i
CAD
i
(p p )
(m m )
(y y )
s
i
CHF
i
(p p )
(m m )
(y y )
s
i
EU R
i
(p p )
(m m )
(y y )
s
i
GBP
i
(p p )
(m m )
(y y )
s
i
JP Y
i
(p p )
(m m )
(y y )
s
i
N OK
i
(p p )
(m m )
(y y )
s
i
N ZD
i
(p p )
(m m )
(y y )
s
i
SEK
i
(p p )
(m m )
(y y )
s
i
i
(p p )
(m m )
(y y )
Std
Skew
Kurt
1
3
6
12
0:089
0:175
0:088
0:350
0:026
3:237
0:290
0:405
1:528
0:785
1:396
0:092
0:803
2:880
0:218
9:440
4:547
4:304
35:059
4:438
0:056
0:956
0:560
0:009
0:278
0:046
0:873
0:138
0:167
0:035
0:002
0:795
0:103
0:078
0:038
0:100
0:643
0:259
0:040
0:068
0:009
0:063
0:005
0:256
0:016
1:907
0:140
0:365
1:004
1:089
0:672
0:223
0:739
0:354
0:890
10:792
3:646
6:657
4:124
11:287
0:033
0:885
0:008
0:076
0:216
0:017
0:732
0:079
0:230
0:062
0:121
0:648
0:080
0:128
0:035
0:002
0:440
0:240
0:102
0:067
0:210
0:256
0:157
0:037
0:015
3:528
0:279
0:419
1:531
0:949
0:038
0:443
0:123
0:443
0:105
3:921
3:768
3:724
10:496
4:278
0:013
0:970
0:231
0:173
0:355
0:023
0:890
0:054
0:072
0:036
0:092
0:834
0:330
0:205
0:020
0:029
0:747
0:509
0:003
0:033
0:119
0:116
0:132
0:183
0:071
3:218
0:239
0:411
1:442
1:807
0:147
0:249
0:770
0:929
0:011
3:730
3:615
5:174
30:164
9:269
0:024
0:961
0:171
0:080
0:286
0:036
0:875
0:031
0:009
0:079
0:057
0:827
0:078
0:167
0:007
0:022
0:723
0:504
0:060
0:003
0:073
0:176
0:063
0:091
0:121
3:065
0:220
0:559
0:942
1:340
0:205
0:674
1:325
0:475
0:281
4:793
5:378
8:335
6:410
6:516
0:092
0:921
0:087
0:033
0:266
0:018
0:763
0:012
0:195
0:068
0:078
0:573
0:235
0:220
0:063
0:002
0:296
0:551
0:108
0:063
0:296
0:247
0:202
0:132
0:026
3:361
0:221
0:529
1:795
1:604
0:365
0:327
0:399
4:414
0:568
4:325
3:446
5:214
52:393
5:895
0:039
0:952
0:065
0:036
0:081
0:042
0:829
0:104
0:018
0:105
0:105
0:671
0:110
0:050
0:028
0:035
0:445
0:468
0:086
0:024
0:037
0:181
0:038
0:475
0:033
2:965
0:277
0:534
3:099
4:358
0:489
0:013
0:884
1:950
0:696
4:695
3:043
5:370
17:999
28:252
0:067
0:934
0:192
0:184
0:425
0:022
0:831
0:112
0:097
0:034
0:038
0:744
0:162
0:114
0:006
0:082
0:499
0:427
0:025
0:095
0:093
0:296
0:169
0:225
0:073
3:366
0:350
0:495
1:925
1:071
1:232
1:666
1:414
0:213
1:232
11:815
7:590
7:351
4:944
9:601
0:036
0:957
0:690
0:207
0:477
0:185
0:859
0:361
0:212
0:113
0:035
0:736
0:326
0:124
0:062
0:098
0:546
0:330
0:168
0:001
0:138
0:148
0:045
0:184
0:107
3:192
0:277
0:546
1:443
2:803
0:876
0:826
1:103
1:650
0:086
6:427
4:141
6:226
21:307
38:324
0:104
0:937
0:199
0:010
0:297
0:062
0:837
0:048
0:050
0:047
0:104
0:749
0:102
0:124
0:030
0:034
0:543
0:342
0:023
0:026
26
Table 3. Predictive Regressions
The table reports the least squares estimates of the predictive regression st+1 = + xt +"t+1 for nine major
exchange rates de…ned as US dollars per unit of foreign currency.
st is the monthly % exchange rate return.
The random walk (RW) model sets
= 0; the uncovered interest parity (UIP) model sets xt = it it , which
is the interest rate di¤erential between the home and foreign country; the purchasing power parity (PPP) model
sets xt = pt
pt st , where pt pt is the log price di¤erential; the monetary fundamentals (MF) model sets
xt = (mt mt ) (yt yt ) st , where mt mt is the the log money supply di¤erential and yt yt the log
g
real output di¤erential; the symmetric Taylor rule (TRs ) sets xt = 1:5 ( t
ytg , where t
t ) + 0:1 yt
t
g
is the in‡ation di¤erential and yt
xt = 1:5 (
t
t )+0:1
ytg
ytg
ytg the real output gap di¤erential; and the asymmetric Taylor rule (TRa ) sets
+0:1 (st + pt pt ), where st +pt pt is the log real exchange rate. Newey-
West (1987) standard errors are reported in parentheses. The superscripts a, b, and c indicate statistical signi…cance
at the 10%, 5%, and 1% level, respectively. The sample period comprises monthly observations from January 1976 to
June 2010.
RW
0:089
AU D
(0:170)
2
Roos
(%)
CAD
0:009
(0:097)
U IP
0:154
PPP
0:047
0:368
(0:160)
(%)
CHF
0:210
(0:182)
(%)
EU R
0:119
(0:171)
(%)
GBP
0:073
(0:168)
(%)
0:296 a
JP Y
(0:178)
N OK
0:037
(0:161)
(%)
N ZD
0:093
(0:184)
(%)
SEK
0:138
(0:182)
(0:092)
0:049
0:816
0:070
0:288
(0:116)
(%)
0:640
(0:185)
a
0:015
0:008
(0:929)
a
(0:097)
0:003
0:025
0:273
(0:172)
a
0:139
(0:586)
(0:008)
(0:004)
(0:440)
(0:074)
0:512
0:828
0:328
0:001
0:790
0:498 b
1:735
(0:250)
1:632
(1:149)
0:023
0:322
(1:075)
a
0:009
(0:236)
a
0:483
1:673
(1:067)
a
0:226
(0:720)
(0:014)
(0:005)
(0:686)
(0:129)
0:793
1:028
0:690
0:173
1:104
0:192
0:386
(0:190)
(0:374)
0:016
0:225
(0:304)
0:004
0:214
(0:194)
0:458
0:338
(0:335)
0:157
(0:789)
(0:011)
(0:004)
(0:606)
(0:103)
0:218
0:691
0:531
0:182
0:732
0:342 a
2:079
(0:199)
(1:387)
a
0:028
1:620
(1:504)
0:005
0:106
(0:165)
0:302
1:896
(1:218)
a
0:257
(0:887)
(0:018)
(0:005)
(0:511)
(0:155)
1:204
1:196
0:366
0:135
1:224
0:881 c
9:156 a
(0:211)
c
(5:379)
(0:655)
(0:010)
2:431
0:977
0:018 a
0:128
3:466
(0:173)
(2:295)
0:020
0:349 a
(0:193)
0:487 b
9:685 a
(0:194)
a
(5:145)
(0:002)
(0:399)
(0:096)
0:332
0:396
1:150
0:003
0:065
(1:240)
0:001
0:654
0:046
(0:162)
0:253
0:186 a
3:076
(2:022)
0:174
(0:551)
(0:013)
(0:003)
(0:503)
(0:115)
0:225
0:779
0:001
0:161
0:800
0:387 a
0:041
(0:218)
(0:274)
a
0:009
0:590
(2:797)
0:001
0:165
(0:191)
0:284
0:018
(0:239)
0:095
(0:516)
(0:011)
(0:006)
(0:318)
(0:097)
1:073
0:222
0:029
0:212
0:295
2:619 a
0:130
(0:183)
(1:579)
0:051
2
Roos
(0:174)
(0:486)
0:995
2
Roos
0:178 a
0:033
0:507
2
Roos
0:208
(0:154)
(0:005)
2:369
2
Roos
(%)
0:001
0:821
1:529
2
Roos
0:019 a
(0:010)
0:630
2
Roos
T Ra
0:076
0:108
1:126
2
Roos
(1:380)
T Rs
0:120
(0:433)
0:974
2
Roos
(0:174)
MF
0:251
0:015
1:862
(1:636)
0:008
0:145
(0:189)
0:125
2:668
(1:642)
0:150
(0:863)
(0:009)
(0:007)
(0:465)
(0:096)
0:002
0:807
0:493
0:024
0:849
27
Table 4. Statistical Evaluation of Exchange Rate Predictability
The table displays out-of-sample tests of the predictive ability of a set of empirical exchange rate models against the null of a random walk (RW). In addition to RW,
we form exchange rate forecasts using five alternative models: uncovered interest parity (UIP), purchasing power parity (PPP), monetary fundamentals (MF), symmetric
Taylor rule (TR ) and asymmetric Taylor rule (TR ). The out-of-sample monthly forecasts are obtained in two ways: (i) with rolling regressions that use a 10-year window
generating forecasts for the period of January 1986 to June 2010; and (ii) with recursive regressions for the same forecasting period that successively re-estimate the model
2 is the Campbell and Thompson (2008) statistic. ∆  is the root mean squared error difference
parameters every time a new observation is added to the sample. 
between the RW and the competing model.  - is the Clark and McCracken (2001) F -statistic,  - is the McCracken (2007) F -statistic and  - is the
Clark and West (2006, 2007) t-statistic, all of which test the null hypothesis of equal mean squared error (MSE) between the RW and the competing model; the alternative
hypothesis is that the competing model has lower MSE. One-sided critical values are obtained by generating 10,000 bootstrap samples as in Mark (1995) and Kilian (1999).
Significance levels at 90%, 95%, and 99% are denoted by a, b, and c, respectively.

2

(%)
∆ (%)
-
-
-
2

(%)
∆ (%)
-
-
-
2

(%)
∆ (%)
-
-
-
−098
−002
187
−285
060
−334
−005
044
−946
011
−274
−004
−231
−780
−139


−188
−003
−186
−541
−143
 
−192
−003
−037
−551
−019
−138
−002
138
−399
060
 
−274
−004
253
−782
076
−133
−002
007
−385
005

−263
−004
−121
−751
−073
 
−214
−004
−117
−614
−044
−145
−002
−098
−420
−062
−309
−005
029
−878
009
 
 
−152
−003
176
−439
070
−319
−005
−090
−907
−037

053
001
289
156
173
−336
−005
254
−951
047

002
001
328
007
114
−081
−001
204
−235
070


 
Rolling Regressions

−135
−169
−263
−001
−002
−003
−104
−040
−292
−391
−487
−751
−079
−024
−130

057
001
375
167
174
−162
−003
−142
−468
−126

−087
−001
262
−252
142

−180
−003
022
−518
009
−205
−003
−178
−589
−105
−084
−001
130
−243
069
 
−111
−001
−007
−321
−004

129
002
498
384
225
−206
−003
125
−593
068

−180
−003
435
−518
100
−039
−001
387
−114
135
−638
−010
−410
−176
−131


−103
−002
214
−298
092

−060
−001
533
−174
162
−017
001
306
−050
095
 
−148
−002
396
−426
085
−044
−001
092
−128
064

−051
−001
200
−149
087

 
 
−182
−003
−123
−524
−057
−375
−006
010
−106
003
−126
−002
077
−364
044
−378
−006
−069
−107
−020
−156
−003
−063
−451
−030
039
001
260
113
125
(continued)
28
Table 4. Statistical Evaluation of Exchange Rate Predictability (continued)


2

(%)
∆  (%)
-
-
-
−026
000
−025
−075
−049
034
001
136
100
096
2
(%)

∆  (%)
-
-
-
−155
−002
030
−447
010
074
001
144
219
166
2
(%)

∆  (%)
-
-
-
−050
−001
−005
−146
−004
068
001
134
202
146

 
 
−121
−002
−071
−350
−053
 
−073
−001
324
−212
110

−142
−002
−143
−411
−129
 



 
Recursive Regressions

030 −132 −041
001 −001
000

126
−050 −056
089 −381 −120
116 −022 −113
−034
−001
−039
−098
−104
091
002
229
270
176
006
000
140
018
121
−106
−002
−074
−306
−056
−171
−003
−013
−493
−005
−122
−002
091
−354
029
122
002
265
362
193

−033
000
160
−097
088
−153
−002
−148
−442
−088
101
002
220
300
200
011
000
167
033
085
−037
−001
−039
−107
−061

−065
−001
−054
−190
−062
29
 



009
001
096
027
068
−168
−003
243
−485
064
111
002
241
329
188

−035
−001
573
−102
178
−069
−001
−079
−200
−114
164
002
386
489
287
071
001
427
209
178
030
001
098
087
091
 
−167
−003
410
−482
107
−006
000
058
−018
061
115
002
252
341
210
−100
−002
−122
−291
−085
072
001
140
213
159

−028
000
150
−082
071
 
 
−081
−001
−030
−236
−017
−237
−004
112
−679
036
−085
−001
−071
−246
−088
−164
−003
−074
−473
−039
−060
−001
−078
−174
−131
077
001
168
227
169
Table 5. The Economic Value of Exchange Rate Predictability
The table shows the in-sample and out-of-sample economic value of a set of empirical exchange rate models for
nine nominal spot exchange rates relative to the US dollar. We form exchange rate forecasts using six models: the
random walk (RW), uncovered interest parity (UIP), purchasing power parity (PPP), monetary fundamentals (MF),
symmetric Taylor rule (TRs ) and asymmetric Taylor rule (TRa ). Using the exchange rate forecasts from each model,
we build a maximum expected return strategy subject to a target volatility p = 10% for a US investor who every
month dynamically rebalances her portfolio investing in a domestic US bond and nine foreign bonds. For each portfolio,
we report the annualized % mean ( p ), % volatility ( p ), Sharpe ratio (SR) and Sortino ratio (SO ). F denotes the
performance fee a risk-averse investor is willing to pay for switching from the benchmark RW strategy to a competing
strategy. P is the premium return performance measure. F and P are computed for a degree of relative risk aversion
equal to 6 and are expressed in annual basis points. be is the break-even proportional transaction cost that cancels
out the utility advantage of a given strategy relative to the RW. be is only reported for positive performance measures
and is expressed in monthly basis points. F (P ) denote the performance fee (premium return) reported net of the
bid-ask spread, which is assumed to linearly decay from 20 bps in 1976 to 4 bps in 2010. F8 and P8 are computed
for a …xed bid-ask spread of 8 bps. The in-sample analysis covers monthly data from January 1976 to June 2010. The
out-of-sample analysis runs from January 1986 to June 2010.
Strategy
p
p
SR
SO
F
be
P
In-Sample
F
P
F8
P8
RW
18:0
10:8
1:08
1:46
U IP
19:1
11:1
1:14
1:68
83
92
586
83
91
83
91
PPP
21:5
11:6
1:30
2:00
285
297
138
285
297
285
297
MF
19:9
11:5
1:18
1:64
138
139
328
138
139
138
139
T Rs
16:9
11:1
0:96
1:26
127
128
127
128
127
128
T Ra
19:8
10:5
1:28
1:86
202
202
202
202
202
202
Out-of-Sample
RW
10:8
11:4
0:54
0:73
U IP
11:9
11:1
0:65
0:99
131
143
173
138
153
141
163
PPP
13:3
11:3
0:76
0:97
252
247
70
230
227
231
208
MF
11:1
11:7
0:55
0:73
10
4
3
3
11
5
T Rs
7:1
11:6
0:21
0:23
384
433
395
445
392
451
T Ra
12:1
11:4
0:65
0:83
130
121
162
154
156
174
30
161
Table 6. The Economic Value of Exchange Rate Predictability when Removing one Currency
The table shows the in-sample and out-of-sample economic value of a set of empirical exchange rate models when one of the nine foreign currencies is removed from
the investment opportunity set. The nine exchange rates include the Australian dollar (AUD), Canadian dollar (CAD), Swiss franc (CHF), Deutsche mark\euro (EUR),
British pound (GBP), Japanese yen (JPY), Norwegian kroner (NOK), New Zealand dollar (NZD), and Swedish kronor (SEK) relative to the US dollar (USD). For example,
AUD denotes an investment strategy that invests in all currencies except for AUD. We form exchange rate forecasts using six models: the random walk (RW), uncovered
interest parity (UIP), purchasing power parity (PPP), monetary fundamentals (MF), symmetric Taylor rule (TR ) and asymmetric Taylor rule (TR ). Using the exchange
rate forecasts from each model, we build a maximum expected return strategy subject to a target volatility  ∗ = 10% for a US investor who every month dynamically
rebalances her portfolio investing in a domestic US bond and nine foreign bonds. For each portfolio, we report the Sharpe ratio (SR) and Sortino ratio (SO). F denotes
the performance fee a risk-averse investor is willing to pay for switching from the benchmark RW strategy to a competing strategy. P is the premium return performance
measure. F and P are computed for a degree of relative risk aversion equal to 6 and are expressed in annual basis points.   is the break-even proportional transaction
cost that cancels out the utility advantage of a given strategy relative to the RW.   is only reported for positive performance measures and is expressed in monthly basis
points. The in-sample analysis covers monthly data from January 1976 to June 2010. The out-of-sample analysis runs from January 1986 to June 2010.
SR
SO

 


 
 
111
115
125
119
096
121
153
174
194
171
130
178

 


 
 
101
107
119
110
097
127
141
162
185
152
131
183

 


 
 
098
105
115
112
087
115
124
150
176
153
108
171
F
P
 
SR
SO
163
99
410
−
454
094
099
120
106
081
120
125
140
193
152
105
161
146
109
153
−
62
102
111
122
110
089
118
133
154
174
145
115
159
218
130
171
−
314
110
117
130
119
096
123
152
183
208
174
134
192
AUD
56
219
128
−148
119
64
234
130
−147
120
F
P
In-Sample
CAD
EUR
67
235
121
−43
240
69
319
153
−151
253
SR
SO
443
169
271
−
176
094
102
118
102
089
119
128
149
188
125
118
175
526
157
216
−
103
096
098
118
107
084
116
130
146
184
148
112
172
76
249
118
−46
238
107
256
117
−134
155
4
149
393
−
88
104
106
124
118
093
128
145
154
187
176
127
192
31
85
278
141
−133
141
P
 
91
281
97
−58
245
100
296
83
−62
244
127
146
175
−
99
41
284
136
−129
208
41
103
259
−
162
JPY
113
265
117
−134
155
NZD
104
231
187
−120
190
F
CHF
GBP
NOK
89
215
181
−123
182
60
304
150
−150
258
 
32
271
137
−129
207
SEK
95
291
143
−131
146
35
249
161
−119
213
41
37
259
200
168
404
−118
−
214
206
(continued)
Table 6. The Economic Value of Exchange Rate Predictability when Removing one Currency (continued)
SR
SO

 


 
 
056
067
078
058
020
066
078
102
100
075
023
083

 


 
 
048
058
070
054
026
064
067
091
091
068
030
081

 


 
 
042
058
058
045
021
053
061
091
072
056
024
064
F
P
 
SR
158
70
−
−
104
046
052
068
056
011
064
SO
F
P
Out-of-Sample
CAD
060
069
71
79
088
251
253
077
110
117
012 −397 −439
070
206
176
260
55
149
−
132
050
061
066
048
017
057
067
092
086
065
019
072
135
50
2
−
112
052
065
077
052
016
060
069
092
095
069
017
076
AUD
118
249
10
−416
102
129
242
−1
−465
89
EUR
114
238
50
−271
172
SR
SO
169
89
357
−
136
046
057
071
047
023
068
065
087
093
059
026
085
178
59
−
−
82
052
056
070
054
017
065
068
079
092
076
019
084
123
228
37
−306
161
127
176
−39
−376
75
162
78
−
−
164
049
063
076
051
020
065
070
100
110
066
026
092
32
139
277
−12
−412
83
P
 
131
287
−1
−262
252
144
275
−21
−312
230
390
64
−
−
120
64
213
37
−434
147
57
63
19
−
185
186
312
8
−335
190
2693
114
32
−
157
JPY
139
173
−41
−420
68
NZD
188
166
−14
−261
86
F
CHF
GBP
NOK
178
176
10
−225
108
 
52
212
33
−392
151
SEK
150
270
−13
−471
77
179
314
27
−328
196
Table 7. The Economic Value of Exchange Rate Predictability for Alternative Numeraires
The table presents the in-sample and out-of-sample economic value of a set of empirical exchange rate models for alternative numeraires other than the US dollar. The
set of currencies includes the Australian dollar (AUD), Canadian dollar (CAD), Swiss franc (CHF), Deutsche mark\euro (EUR), British pound (GBP), Japanese yen (JPY),
Norwegian kroner (NOK), New Zealand dollar (NZD), Swedish kronor (SEK) and the US dollar (USD). For example, AUD denotes an investment strategy using AUD as the
domestic currency and expressing all exchange rates relative to AUD. We form exchange rate forecasts using six models: the random walk (RW), uncovered interest parity
(UIP), purchasing power parity (PPP), monetary fundamentals (MF), symmetric Taylor rule (TR ) and asymmetric Taylor rule (TR ). Using the exchange rate forecasts
from each model, we build a maximum expected return strategy subject to a target volatility  ∗ = 10% for a US investor who every month dynamically rebalances her
portfolio investing in a domestic US bond and nine foreign bonds. For each portfolio, we report the Sharpe ratio (SR) and Sortino ratio (SO). F denotes the performance
fee a risk-averse investor is willing to pay for switching from the benchmark RW strategy to a competing strategy. P is the premium return performance measure. F and P
are computed for a degree of relative risk aversion equal to 6 and are expressed in annual basis points.   is the break-even proportional transaction cost that cancels out
the utility advantage of a given strategy relative to the RW.   is only reported for positive performance measures and is expressed in monthly basis points. The in-sample
analysis covers monthly data from January 1976 to June 2010. The out-of-sample analysis runs from January 1986 to June 2010.
SR
SO

 


 
 
108
095
124
082
106
149
149
144
194
129
139
240

 


 
 
109
125
135
110
107
128
144
174
198
167
130
182

 


 
 
108
126
124
131
114
133
150
219
184
167
157
208
F
P
 
SR
−
54
−
−
212
108
110
132
107
093
117
F
P
In-Sample
CAD
147
146
55
55
203
306
317
186
−11
6
137 −163 −155
164
100
103
170
245
8
1
290
107
118
123
104
103
122
147
173
185
158
147
181
827
127
139
8
166
107
105
115
098
106
155
153
146
185
162
147
261
AUD
−141
242
−285
−13
370
−127
268
−269
−14
373
SO
EUR
159
329
13
5
204
SR
SO
114
116
−
−
316
109
128
141
104
102
126
144
182
227
153
126
182
61
124
−
−
147
109
116
138
108
102
124
147
201
201
151
142
168
162
342
24
1
208
133
222
−37
−50
157
−
28
−
−
401
108
100
133
101
111
139
150
151
179
158
136
206
33
−19
129
−114
−11
452
P
 
200
394
−38
−72
190
207
412
−26
−78
194
137
378
−
−
186
140
375
17
−80
183
92
251
23
−
277
JPY
145
236
−23
−49
157
NZD
150
248
239
74
265
F
CHF
GBP
NOK
146
235
243
73
258
 
115
367
11
−86
180
SEK
−14
152
−107
−7
452
−100
340
−62
45
292
−98
−
344 128
−46
−
37
5
294
21
(continued)
Table 7. The Economic Value of Exchange Rate Predictability for Alternative Numeraires (continued)
SR
SO

 


 
 
054
025
054
029
042
046
075
036
083
038
058
063

 


 
 
054
064
068
042
036
033
073
094
097
051
043
039

 


 
 
053
050
065
060
052
067
074
080
092
070
077
094
F
P
 
SR
054
066
066
054
024
060
SO
F
P
Out-of-Sample
CAD
074
088
143
141
088
130
131
070
20
5
029 −313 −329
077
76
57
053
064
051
031
049
074
074
099
071
043
072
102
053
045
077
036
029
071
075
067
103
042
041
093
AUD
−333
−15
−278
−111
−81
−326
−2
−280
−113
−71
−
−
−
−
−
EUR
113
155
−145
−235
−254
SR
SO
169
36
8
−
121
054
066
081
028
033
026
073
091
118
040
045
033
47
−
−
−
102
054
059
086
036
033
034
073
094
120
053
041
042
121
157
−183
−259
−284
89
56
−
−
−
122
−52
−250
−38
239
−
45
−
−
109
054
052
061
017
047
073
074
077
085
023
054
089
−
37
39
−
278
34
−123
270
−173
−223
205
P
 
134
318
−320
−227
−333
136
323
−310
−220
−349
92
103
−
−
−
72
381
−184
−225
−252
23
104
−
−
−
−14
73
−476
−150
198
−
22
−
−
903
JPY
137
−47
−244
−34
238
NZD
−20
126
49
2
163
F
CHF
GBP
NOK
−28
121
73
−3
164
 
49
379
−191
−222
−237
SEK
−111
268
−207
−225
197
−21
70
−471
−110
217
Table 8. The Economic Value of Combined Forecasts
The table presents the out-of-sample economic value of combined forecasts across a set of empirical exchange rate
models. We form forecasts for the nine exchange rates using combinations of six models: the random walk (RW),
uncovered interest parity (UIP), purchasing power parity (PPP), monetary fundamentals (MF), symmetric Taylor rule
(TRs ) and asymmetric Taylor rule (TRa ). Simple Model Averaging denotes ex-ante combining methods that disregard
the past out-of-sample performance of the individual models and use the mean, median and trimmed mean of the
individual forecasts. Statistical Model Averaging denotes ex-ante combining methods based on the past out-of-sample
mean-squared error (MSE) of the individual models. DM SE( ) use the inverse of the discounted MSE with as a
discount factor. M SE( ) use the inverse of the MSE over the most recent
months. Economic Model Averaging
denote ex-ante combining methods which use the past out-of-sample Sharpe ratio (SR) of the individual models.
DSR( ) use the discounted SR with as a discount factor. SR( ) uses the average SR over the most recent
months. Using the forecast combinations, we build a maximum expected return strategy subject to a target volatility
p = 10% for a US investor who every month dynamically rebalances her portfolio investing in a domestic US bond
and nine foreign bonds. For each portfolio, we report the Sharpe ratio (SR) and Sortino ratio (SO ). F denotes the
performance fee a risk-averse investor is willing to pay for switching from the benchmark RW strategy to a competing
strategy. P is the premium return performance measure. F and P are computed for a degree of relative risk aversion
equal to 6 and are expressed in annual basis points. be is the break-even proportional transaction cost that cancels
out the utility advantage of a given strategy relative to the RW. be is only reported for positive performance measures
and is expressed in monthly basis points. The in-sample analysis covers monthly data from January 1976 to June 2010.
The out-of-sample analysis runs from January 1986 to June 2010.
F
P
be
Averaging
0:74 0:89
0:61 0:74
0:60 0:72
234
76
65
206
52
37
81
21
23
Statistical Model Averaging
DM SE ( = 0:90)
13:2 11:7 0:72 0:87
DM SE ( = 0:95)
13:2 11:7 0:73 0:88
DM SE ( = 1:00)
13:3 11:6 0:74 0:89
213
218
232
186
191
204
71
74
81
M SE( = 60)
M SE( = 36)
M SE( = 12)
0:88
0:88
0:87
222
218
207
195
191
179
76
74
70
Economic Model Averaging
13:4 11:5 0:76 0:96
13:4 11:5 0:76 0:95
12:9 11:4 0:72 0:89
254
255
207
235
236
187
128
127
103
261
254
254
241
234
235
129
126
128
RW
M ean
M edian
T rimmed M ean
DSR ( = 0:90)
DSR ( = 0:95)
DSR ( = 1:00)
SR( = 60)
SR( = 36)
SR( = 12)
10:8
11:4
Simple
13:3
12:0
12:0
Model
11:6
12:0
12:1
13:2
13:2
13:1
13:5
13:4
13:4
11:7
11:7
11:7
11:5
11:5
11:5
35
SR
SO
0:54
0:73
0:73
0:73
0:72
0:76
0:76
0:76
0:96
0:95
0:96
Figure 1. Out-of-Sample Sharpe Ratios
The figure displays the out-of-sample annualized Sharpe ratio (SR) for selected empirical exchange rate models. We
show the results from forming exchange rate forecasts using uncovered interest parity (UIP), purchasing power parity
(PPP), asymmetric Taylor rule (T Ra ), and the ex-ante forecast combination method that uses the out-of-sample SR
over the past 60 months (SR(κ = 60)). All models (solid line) are displayed versus the random walk (RW) benchmark
(dashed line). Using the exchange rate forecasts from each model, we build a maximum expected return strategy subject
to a target volatility σp∗ = 10% for a US investor who every month dynamically rebalances her portfolio investing in a
domestic US bond and nine foreign bonds. The SR is computed using the out-of-sample portfolio returns for one year.
The out-of-sample period runs from January 1986 to June 2010.
36
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