Macroeconomic Fundamentals
and the Forward Discount Anomaly
Adrian Ma
It has been widely documented that currencies with high interest rates tend to appreciate. This paper examines the macroeconomic conditions that accompany high interest rates and currency appreciations. In particular, when outputs exhibit high growth
rates, interest rates tend to be high and currencies are slightly more likely to appreciate. An econometric model demonstrates that macroeconomic comovements are able
to generate the widely documented negative correlations even though exchange-rate
movements are weakly correlated with macroeconomic fundamentals. Moreover, the
generalized-method-of-moments estimates of the uncovered interest parity condition
suggest that the forward discount anomaly can be reconciled with rational expectation
once the effects of macroeconomic comovements are taken into account.
JEL classification: C32, F31
Keywords: Exchange rate; Forward discount; Weak identification
Current draft: January 29, 2006
Department of Economics, Trent University, Peterborough, Ontario, Canada K9J 7B8. E-mail: adrianma@trentu.ca. I thank the Trent University Social Sciences and Humanities Research Committee for
financial support.
1
1 Introduction
The uncovered interest parity (UIP) is founded on a simple arbitrage argument. Under risk neutrality, a currency with high interest rate should be expected to depreciate
so that the expected returns on domestic and foreign short-term bonds are equalized.
Empirical studies, however, have produced strong evidence against the hypothesis. Numerous studies have found that the currency depreciation rates are negatively correlated
with the cross-country interest differentials. In other words, currencies with high interest rates tend to appreciate. This finding is often referred to as the forward discount
anomaly.
Because this widespread finding poses a serious challenge to our understanding of
the international financial markets, the nature of this anomaly has been the focus of an
enormous literature. Fama (1984) attributes the negative correlation to a time-varying
risk premium. Bacchetta and van Wincoop (2005) develop a model of rational inattention that is consistent with the negative correlation. Another strand of the literature
interprets the forward discount anomaly as a rejection of the rational expectation hypothesis. For instance, Froot and Frankel (1989) attribute the anomaly to expectation
errors. Mark and Wu (1998) examine the effect of noise trading. Gourinchas and Tornell (2004) attribute the negative correlation to a systematic distortion in the investors’
beliefs about the interest rate process.
This paper examines the macroeconomic conditions that accompany high interest
rates and currency appreciations. The model is based on a simple observation on the
cyclical behavior of the cross-country interest differentials and exchange-rate movements. Because interest rates are generally procyclical, countries with high interest
rates are likely to be experiencing positive output growths. If there is a slight tendency
for currencies to appreciate when output growths are high, the negative correlations
between the interest differentials and exchange-rate movements will simply reflect the
variables’ comovements with output growths.
The pro-cyclicality of the interest rates has been well documented. For instance,
Fama (1990) points out that the one-year U.S. interest rate is lower at the business
trough than at the preceding or following peak in every business cycle of the 19521988 period. King and Watson (1996) report that interest rate, price level and money
supply are generally procyclical. Because the forward discount is equal to the cross-
2
country interest differential, the forward discount is also procyclical in general.
The main challenge lies in establishing the relationship between exchange-rate
movements and output growths. Given the evidence of the disconnect between exchangerate movements and economic fundamentals, the literature has rarely examined the
role played by macroeconomic variables in explaining the forward discount anomaly.
Meese and Rogoff (1983) demonstrate that the random walk provides a better out-ofsample fit than the major exchange rate models. Subsequent studies generally support
their finding. To the extent that exchange-rate movements are correlated with fundamentals, the relationship is not stable enough for exchange rate models to provide
better out-of-sample fits than the random walk. For instance, Mark (1995) develops a
monetary model that outperforms the random walk at longer horizon. However, Faust,
Rogers and Wright (2003) find that Mark’s result can be replicated only within a twoyear window around his vintage. Cheung, Chinn and Pascual (2005) make a similar
observation on the robustness of the empirical exchange rate models in general.
Given the literature on exchange-rate disconnect, robustness is a main concern in
this paper. Two econometric exercises attempt to shed light on the role played by
macroeconomic fundamentals. The first exercise ascertains whether macroeconomic
comovements are able to generate the negative correlations even though exchangerate movements are very weakly correlated with macroeconomic fundamentals. An
econometric model demonstrates that the covariance between the forward discount and
exchange-rate movement can be decomposed into two components. While the first
component reflects the contribution of macroeconomic comovements, the second component contains factors that are orthogonal to macroeconomic fundamentals. The empirical significance of fundamentals can be gauged by comparing the signs and magnitudes of the two components. For the dollar exchange rates against the Deutsche
mark, the British pound and the Japanese yen over the post-1976 period, the macroeconomic contributions are negative and very close to the covariances between the forward discounts and exchange-rate movements. Thus, the covariance decompositions
demonstrate that macroeconomic comovements play an important role in explaining
the forward discount anomaly.
It is important to stress that exchange-rate predictability is not essential to the argument in this paper. Instead, the macroeconomic contribution stems from the covariance between the projections of the forward discount and exchange-rate movement on
3
macroeconomic fundamentals. Because the two projections tend to move in the opposition directions, the covariance between the forward discount and exchange-rate movement tends to be negative. In other words, the macroeconomic conditions that lead
to high interest rates also give rise to currency appreciations. The regression results
below show that the correlations between exchange-rate movements and macroeconomic fundamentals are weak and unstable over time. The weak correlations between
exchange-rate movements and fundamentals are consistent with the findings of Engel
and West (2005), who show that exchange-rate movements Granger-cause macroeconomic fundamentals.
Given the prevalence of structural instability, subsample analysis is essential. Especially at shorter horizons, some subsample covariances between the forward discounts
and exchange-rate movements are positive. In subsamples of various lengths, macroeconomic comovements are able to match both positive and negative covariances between the forward discounts and exchange-rate movements. In fact, the unstable relationship between exchange-rate movements and fundamentals helps to explain why the
covariances between the forward discounts and exchange-rate movements are unstable
and not always negative.
Because of the effects of macroeconomic comovements, the forward discount anomaly does not necessarily imply that UIP does not hold. In order to control for the effects of macroeconomic comovements, the second econometric exercise tests the UIP
hypothesis using the generalized method of moments (GMM). The GMM estimates
provide favorable evidence for UIP. In all fourteen specifications of the joint estimations, the GMM estimates of the slope coefficient in the UIP condition are positive and
very close to one. Thus, the forward discount anomaly can be reconciled with rational
expectation once the simultaneity bias has been taken into account.
The rest of this paper is organized into five sections. Section 2 discusses the nature of the forward discount anomaly. Section 3 derives the econometric model that
ascertains whether macroeconomic fundamentals are able to generate the widely documented negative correlations. Section 4 presents the empirical results of the covariance
decompositions. Section 5 discusses the GMM estimations of the uncovered interest
parity condition. Section 6 contains some concluding remarks.
4
2 Forward discount anomaly
The uncovered interest parity (UIP) is founded on a simple arbitrage argument. Under
risk neutrality, a currency with high interest rate should be expected to depreciate so
that the expected returns on domestic and foreign short-term bonds are equalized. Empirical studies, however, have produced strong evidence against the hypothesis. The
following regression equation represents a common way of testing UIP.
st+1 − st = α + β (ft − st ) + εt+1
(1)
The dependent variable is the currency depreciation rate, which is defined as the change
in the logarithm of the spot exchange rate. The independent variables include a constant
and the forward discount, which is the difference between the log one-period-ahead
forward rate and the log spot exchange rate. Because the forward discount is equal to
the cross-country interest differential, the uncovered interest parity implies that α is
equal to zero and β is equal to one. However, Froot and Thaler (1990) report that the
average of 75 published point estimates of β is -0.88. Only a few studies have reported
positive estimates, but none reports an estimate that is greater than one. Although
Engel (1996) points out that the point estimates of β are larger than one for the dollar
exchange rates of the French franc and the Italian lira for the period of February 1987
to May 1995, recent studies generally support the Froot-Thaler finding. For instance,
McCallum (1994) reports several estimates that are smaller than -3. Flood and Rose
(1996) report some estimates that are positive but less than one.
This paper focuses on the U.S. dollar exchange rates against the British pound,
the Deutsche mark and the Japanese yen. Given the concern about robustness, all
econometric exercises in this paper will be performed on monthly and quarterly data.
For the monthly data, the sample period is January 1976 to June 2004 for the Japanese
yen and the British pound. The sample period is January 1976 to December 1998 for
the Deutsche mark. The quarterly data cover the same sample periods. Except for
the interest rates, all variables are in logarithm and multiplied by 100. Thus, the first
differences of the variables are approximately equal to the percentage changes over a
month or a quarter. The exchange rates are expressed as the numbers of currency units
per U.S. dollar. The data were obtained from Datastream.
Table I reports the well-known regression results that underlie the forward discount
5
anomaly. It can be seen that all six estimates of the slope coefficient β are negative, although the estimates for the monthly pound-dollar rate and the quarterly markdollar rate are statistically insignificant. For all three exchange rates, the magnitudes of
the correlation coefficients are higher at the quarterly frequency than the monthly frequency. The coefficients of correlation range from -0.1138 to -0.0710 for the monthly
data and -0.1418 to -0.3344 for the quarterly data. The small R2 ’s reflect the weak correlations between the forward discounts and currency depreciation rates. It can be seen
that the R2 ’s range from 0.0056 to 0.0130 for the monthly data and 0.0266 to 0.1169
for the quarterly data. Thus, while the forward discounts and currency depreciation
rates are negatively correlated, the correlations are not very high.
Engel (1996) points out that there are some dramatic changes in the estimates of the
slope coefficient β before and after the Louvre Accord. In particular, he reports positive
estimates for the dollar exchange rates of the French franc, Deutsche mark, Italian lira,
Dutch guilder and British pound for the period of February 1987 to May 1995. On the
other hand, the estimates are in the neighborhood of -3 for the period of September
1977 to June 1990, which is the sample period of the McCallum (1994) study. To
highlight the prevalence of structural instability, Table I also reports a fluctuation test
developed by Ploberger, Krämer and Kontrus (1989). It tests whether the regression
coefficients remain unchanged when the possible break dates are unknown a priori.
The fluctuation tests reject stability for the monthly yen exchange rate, the quarterly
pound rate, and the quarterly mark rate. The subsample analysis in Section 4 will
ascertain if macroeconomic comovements are able to account for the structural changes
in the covariances between the forward discounts and exchange-rate movements.
3 Role of macroeconomic fundamentals
The covered interest parity condition implies that the forward discount is equal to the
cross-country interest differential.
S
ft − st = it − iU
t
(2)
S
denotes the U.S. interest rate and it is a placeholder for the interest rates
where iU
t
in the other three countries. The negative estimates of β in regression (1) imply that
6
currencies with high interest rates tend to appreciate.
This paper focuses on the macroeconomic conditions that accompany high interest
rates and currency appreciations. It has been widely documented that interest rates
are procyclical. Being equal to the cross-country interest differentials, the forward
discounts are positively correlated with output growths. If there is a slight tendency for
currencies to appreciate when output growths are high, the forward discounts will be
negatively correlated with exchange-rate movements.
This section presents a simple way to gauge the role played by macroeconomic fundamentals. The key question is whether the macroeconomic conditions that give rise to
high interest rates also lead to currency appreciations. The set of macroeconomic variables consists of output growths, changes in money supplies and inflation rates in the
two countries for each country pair. This set of macroeconomic fundamentals has been
commonly used in the exchange rate literature. Let Xt be the set of macroeconomic
fundamentals for each currency pair.
¡
¢0
S
US
Xt = ∆ytUS , ∆yt , ∆mU
t , ∆mt , π t , π t
where ∆y denotes the change in output, ∆m denotes the change in money supply and
π denotes the inflation rate. Variables with the superscript U S are of the United States,
while variables without any superscript are placeholders for the variables of the other
three countries.
The model attributes the forward discount anomaly to three time-series properties. First, interest rates are positively correlated with output growths. Second, interest
rates are positively autocorrelated. Third, there is a slight tendency for currencies to
appreciate when output growths are high. Because interest rates are persistent, the forward discounts are also positively autocorrelated. Therefore, the covariance between
exchange-rate movement and lagged forward discount takes the same sign as the covariance between exchange-rate movement and contemporaneous forward discount;
and the covariance is negative because the macroeconomic conditions that lead to high
interest rates also give rise to currency appreciations.
While the pro-cyclicality and persistence of the interest rates have been well documented, the weak correlation between exchange-rate movements and output growths
seems to be at odds with the literature on the disconnect between exchange rates and
7
economic fundamentals. As a result, robustness is a main concern in the following
econometric exercise. In particular, the following model aims to ascertain whether
macroeconomic fundamentals are able to generate the widely documented negative
correlations even though exchange-rate movements are weakly correlated with macroeconomic fundamentals.
The model consists of three regression equations and a covariance decomposition.
Equation (3) captures the relationship between the forward discount and fundamentals.
Because interest rates are positively correlated with output growths, the forward discounts are negatively correlated with U.S. output growth and positively correlated with
the output growths of the other three countries.
S
= γ 0 + Xt γ 01 + ηt
it − iU
t
(3)
Equation (4) is an autoregression of the forward discount. Because interest rates
are positively autocorrelated, the coefficient on lagged forward discount ρ1 is generally
positive.
¡
¢
S
US
+
it+1 − iU
t+1 = ρ0 + ρ1 it − it
t+1
(4)
Exchange-rate equation (5) captures the weak correlation between exchange-rate
movements and macroeconomic fundamentals.
∆st+1 = θ0 + Xt+1 θ01 + ξ t+1
(5)
Given the literature on exchange-rate disconnect, this equation will receive special
attention in the following econometric exercise. In particular, the following section
shows that the coefficients on macroeconomic fundamentals θ1 are small and unstable. The weak correlations between exchange-rate movements and fundamentals are
consistent with the findings of Engel and West (2005), who show that exchange rates
Granger-cause macroeconomic fundamentals.
One important implication of the exchange-rate equation (5) is that the exchange
rates do not follow random walk because exchange-rate movements ∆st+1 are weakly
correlated with fundamentals Xt+1 , which are known to be persistent. If the exchange
rates follow random walk, exchange-rate movements will be independent of previously
8
available information. However, the forward discount anomaly is founded on the widespread finding that currencies tend to appreciate when lagged interest rates are high.
Hence, in order to account for the forward discount anomaly, one needs to reject either
the random walk hypothesis or the rational expectation hypothesis. This paper attempts
to reconcile the forward discount anomaly with rational expectation. In particular, the
generalized-method-of-moments estimates in Section 5 demonstrate that the negative
covariance between the forward discount and exchange-rate movement is consistent
with rational expectation.
Even though exchange rates do not follow random walk, it does not necessarily
imply that equation (5) will provide a better out-of-sample fit than the random walk.
Although exchange-rate movements are correlated with fundamentals, the correlations
are so weak and unstable that the random walk provides a better out-of-sample fit than
the exchange-rate equation (5). The following empirical section presents evidence for
these assertions.
To ascertain the role played by macroeconomic fundamentals, the covariance between the forward discount and exchange-rate movement is decomposed into two components. While the first component reflects the contribution of macroeconomic comovements, the second component contains factors that are orthogonal to macroeconomic fundamentals. In particular, equations (3), (4) and (5) are substituted into the
covariance between the forward discount and exchange-rate movement as follows.
=
=
=
=
S
cov(∆st+1 , it − iU
t )
1
S
cov(∆st+1 , it+1 − iU
t+1 − t+1 )
ρ1
1
cov(∆st+1 , Xt+1 γ 01 + η t+1 − t+1 )
ρ1
1
1
cov(∆st+1 , Xt+1 γ 01 ) + cov(∆st+1 , η t+1 − t+1 )
ρ1
ρ1
1
1
θ1 var (Xt+1 ) γ 01 + cov(∆st+1 , η t+1 − t+1 )
ρ1
ρ1
(6)
The following covariance decomposition (7) provides a useful gauge of the role
played by macroeconomic comovements.
9
S
cov(∆st+1 , it − iU
t )=
1
1
θ1 var (Xt+1 ) γ 01 + cov(∆st+1 , η t+1 −
ρ
ρ
|1
|1
{z
}
{z
contribution of
macroeconomic comovements
t+1 )
(7)
}
residual covariance that is unexplained
by macroeconomic fundamentals
If macroeconomic comovements are responsible for the widely documented negative covariance between the forward discount and exchange-rate movement, then the
first term in equation (7) has to be negative.
1
θ1 var (Xt+1 ) γ 01 < 0
ρ1
The expression
1
0
ρ1 θ 1 var(Xt+1 ) γ 1
represents the contribution of macroeconomic co-
movements to the negative covariance between exchange-rate movement and forward
discount. As a variance-covariance matrix, var(Xt+1 ) is semi-positive definite. Thus,
1
0
ρ1 θ 1 var(Xt+1 ) γ 1
is negative when the elements of θ1 and γ 1 generally take the oppo-
site signs because ρ1 is positive. That is, the macroeconomic conditions that give rise
to high interest rates also lead to currency appreciations.
Note that the macroeconomic contribution in equation (7) can also be written as
follows.
1
1
θ1 var (Xt+1 ) γ 01 = cov(Xt+1 θ01 , Xt+1 γ 01 )
ρ1
ρ1
While Xt+1 θ01 is the projection of the exchange-rate movement on fundamentals, Xt+1 γ 01
is the projection of the forward discount on fundamentals. The macroeconomic contriS
bution stems from the covariance between the projections of ∆st+1 and it+1 − iU
t+1 on
Xt+1 . Thus, the first term in the covariance decomposition (7) captures the effect of
macroeconomic comovements.
It is important to stress that exchange-rate predictability is not essential to the argument in this paper. Instead, the above derivation simply shows that the first term in
equation (7) contributes to the negative correlation when the elements of θ1 and γ 1 generally take the opposite signs. It has been well documented that the covariance between
the forward discount and exchange-rate movement is not always positive. For instance,
Engel (1996) reports positive covariances for the dollar exchange rates of the French
franc, Deutsche mark, Italian lira, Dutch guilder and British pound for the period of
10
February 1987 to May 1995. The subsample analysis below will highlight the prevalence of structural instability. In fact, the unstable relationship between exchange-rate
movements and fundamentals helps to explain why the covariances between the forward discounts and exchange-rate movements are unstable and not always negative.
Especially at short horizons, currencies do not always appreciate when output growths
are high. Thus, the macroeconomic contributions are positive in some subsamples because the elements of θ1 and γ 1 do not always take the opposite signs.
The above derivation makes use of the fact that the error term ξ t+1 in the exchangerate equation (5) is orthogonal to the macroeconomic fundamentals Xt+1 .
cov(Xt+1 , ξ t+1 ) = 0
This condition holds because the exchange-rate equation (5) will be estimated using the
ordinary least squares (OLS). Given the fact that exchange-rate movements are weakly
correlated with fundamentals, ξ t+1 will account for a substantial portion of the variations in ∆st+1 . The orthogonal component ξ t+1 could stem from expectation error,
risk premium or other factors that drive the exchange rates. Expanding the second term
in the covariance decomposition (7) demonstrates the effect of ξ t+1 on the unexplained
residual covariance that is unexplained by macroeconomic fundamentals.
¡
¢
1
cov ∆st+1 , η t+1 − t+1
ρ1
¡
¢
1
=
cov Xt+1 θ01 + ξ t+1 , η t+1 − t+1
ρ1
¢
¡
¡
1
1
= − cov Xt+1 θ01 , t+1 + cov ξ t+1 , η t+1 −
ρ1
ρ1
t+1
¢
Note that ξ t+1 and η t+1 are orthogonal to Xt+1 . The second term in decomposition
(7) represents the residual covariance that is unexplained by macroeconomic comovements. Hence, if the forward discount anomaly stems from sources that are independent
of the macroeconomic fundamentals, the residual covariance should be very close to
the covariance between the forward discount and exchange-rate movement. The role
played by macroeconomic comovements can therefore be ascertained by comparing
the signs and magnitudes of the two components in the covariance decomposition (7).
The following section reports the empirical results.
11
4 Empirical results
Given the concern about robustness, the above econometric model will be applied to
monthly and quarterly data. Subsample analysis will also be carried out. The International Financial Statistics is the main source of macroeconomic data. The sample
period is January 1976 to December 1998 for the German time series. The sample period is January 1976 to June 2004 for the time series of Japan, the United Kingdom and
the United States. The quarterly data cover the same time periods. For the quarterly
data, the change in output ∆y is defined as the first difference of the log seasonally
adjusted real GDP, which is obtained by dividing the seasonally adjusted nominal GDP
by the consumer price level. For the monthly data, the change in output ∆y is defined
as the first difference of the log seasonally adjusted industrial production. The change
in money supply ∆m is defined as the first difference of the log seasonally adjusted
money supply. Inflation rate π is defined as the first difference of the log consumer
price index.
This section will first discuss the estimation results of the individual regression
equations (3), (4) and (5). Section 4.3 reports the estimates of the covariance decomposition (7). Section 4.4 reports the results of the subsample analysis.
4.1 Forward-discount regressions
Table II reports the autoregressions of the forward discounts as in equation (4). As
expected, all six estimates of the coefficient on the lagged forward discounts ρ1 are
significantly positive. For the monthly data, the estimates of ρ1 range from 0.2565
to 0.4118. For the quarterly data, the estimates of ρ1 range from 0.7803 to 0.8396.
These estimates indicate positive autocorrelations. Since interest rates are known to
be persistent, this set of regressions simply shows that the forward discounts are also
persistent. However, the fluctuation tests reject stability for all six regression equations
at the 5% significance level. This is a strong indication of structural breaks in the
interest rate time series.
Table III reports the regressions of the forward discounts on macroeconomic fundamentals as in equation (3). The R2 ’s range from 0.1811 to 0.5188 for the monthly
data and from 0.4860 to 0.7798 for the quarterly data. The F -statistics reject the hypothesis that the regression coefficients are equal to zero at the 1% significance level
12
in five of the six cases. The fluctuation tests strongly reject stability for all six cases at
the 1% significance level. Hence, while the forward discounts are strongly correlated
with macroeconomic fundamentals, the relationships are unstable over time.
In Table III, all six coefficients on U.S. output growth are negative, although only
two of the six are statistically significant. As for the output growths of the other three
countries, four of six coefficients are positive, but none is significant. Because the forward discounts are very persistent, the heteroskedasticity-autocorrelation-robust standard errors tend to be large. Some preliminary regressions indicate that the correlations
between the forward discounts and output growths are much stronger at the businesscycle frequency. This is consistent with previous studies such as King and Watson
(1996), who report strong correlation between the U.S. interest rate and output at the
business-cycle frequency. As Baxter and King (1999) point out, the first-difference filter places heavy weight on high-frequency movements. As a result, most coefficients
on output growths in the forward-discount regressions are statistically insignificant.
Nevertheless, this paper chooses to use the first-difference filter because the focus of
the forward discount anomaly has been on the interest differentials and first-differenced
exchange rates. The business-cycle-frequency behavior will therefore be left for future
research.
The coefficients on inflation rates also show some interesting pattern. In all six
regressions, the coefficients on U.S. inflation are negative while the coefficients on
the inflation rates in the other three countries are positive. Moreover, five of twelve
coefficients on inflation rates are statistically significant. These coefficients reflect the
positive correlations between interest rates and inflation rates. On the other hand, the
coefficients on money growth rates do not show any consistent pattern.
This set of regressions suggests that when the forward discounts are high, U.S.
output growth and inflation rate are likely to be low while output growths and inflation
rates in the other three countries are likely to be high. The main question is whether
the currencies are more likely to appreciate against the U.S. dollar under these same
conditions. The following set of regressions attempts to answer this question.
13
4.2 Exchange-rate regressions on macroeconomic fundamentals
The estimates of the exchange-rate equation (5) are reported in Table IV. The R2 ’s
range from 0.0255 to 0.0790 for the monthly data and from 0.0556 to 0.2571 for the
quarterly data. The small R2 ’s reflect the weak correlations between exchange-rate
movements and fundamentals. Even though the R2 ’s of the exchange-rate equations are
small, it does not necessarily follow that macroeconomic comovements cannot account
for the forward discount anomaly. In five of six regressions, the R2 ’s of the exchangerate equations are larger than the R2 ’s of the forward discount anomaly regressions (1)
as reported in Table I. The only exception is the quarterly dollar-pound exchange rate.
The fluctuation tests reject stability for the monthly yen rate, the quarterly pound rate
and the quarterly mark rate. Coincidentally, the fluctuation tests also reject stability of
the forward discount anomaly regressions for these three exchange rates in Table I.
The estimates indicate a slight tendency for currencies to appreciate when output
growths are high. All six coefficients on U.S. output growth are positive, although only
two of the six are statistically significant. Five of the six coefficients on output growths
in the other three countries are negative, although only one of the six is statistically
significant. All six coefficients on U.S. inflation are positive. Four of the six coefficients on the inflation rates of the other three countries are negative. The coefficients
on money growth rates do not show any consistent pattern. Hence, Tables III and IV
suggest the coefficients on macroeconomic fundamentals generally take the opposite
signs in the forward-discount regressions (3) and the exchange-rate regressions (5). In
the notation of the above econometric model, the elements of θ1 and γ 1 generally take
the opposite signs. In other words, when U.S. output growth and inflation rate are low
and output growths and inflation rates in the other three countries high, the forward
discounts are likely to be high and the currencies are slightly more likely to appreciate
against the U.S. dollar.
Because the correlations between exchange-rate movements and fundamentals are
weak and unstable, the exchange-rate equation (5) is not expected to provide a better
out-of-sample fit than the random walk. Table V reports the out-of-sample fit of the
exchange-rate equation. As in Meese and Rogoff (1983), the root-mean-squared error
(RMSE) of the exchange-rate equation is compared with that of the random walk in
order to assess the model’s out-of-sample fit. Table V also reports a test statistic that is
14
developed by Diebold and Mariano (1995). A positive value of the Diebold-Mariano
statistic indicates that the random walk has a smaller root-mean-squared error than the
exchange-rate equation (5).
For the monthly data, a moving window of 120 months is used to estimate the
exchange-rate equation and generate the out-of-sample fit. For the quarterly data, the
length of the moving window is 40 quarters. Not surprisingly, the out-of-sample fit of
the exchange-rate equation is worse than that of the random walk in sixteen of eighteen
cases. The Diebold-Mariano statistics indicate no statistically significant difference
between the exchange-rate equation and the random walk. Hence, although exchangerate movements are correlated with fundamentals, the correlations are so weak and
unstable that the random walk provides a better out-of-sample fit than the exchangerate equation (5).
4.3 Covariance decomposition
Table VI reports the results of the covariance decomposition according to equation (7).
The macroeconomic contributions are calculated from the estimated coefficients reported in Tables II, III and IV. The standard deviations of the simulated moments are
calculated from 2000 random draws from the empirical distributions. In all six cases,
the contributions of macroeconomic comovements are negative. Moreover, the magnitudes of the macroeconomic contributions are very close to the covariances between the
forward discounts and exchange-rate movements. Except for the quarterly pound rate,
the macroeconomic contributions are within one standard deviation of the covariances
between the forward discounts and exchange-rate movements.
Because many elements of θ1 and γ 1 are statistically insignificant, the standard deviations of the macroeconomic contributions are quite large. Fortunately, the residual
covariances that are unexplained by fundamentals also imply that the macroeconomic
variables play an important role. For the Deutsche mark, the residual covariances
are significantly positive at both monthly and quarterly frequencies. Thus, the negative covariances between the forward discounts and exchange-rate movements of the
Deutsche mark are generated entirely from macroeconomic comovements. As for the
other two exchange rates, the residual covariances are statistically different from the
covariances between the forward discounts and exchange-rate movements. Except for
15
the quarterly dollar-pound rate, the magnitudes of the residuals are much smaller than
the magnitudes of the macroeconomic contributions. Although the residual covariances
are small for the pound and the yen, they are negative and statistically significant. This
suggests that the macroeconomic variables do not fully account for the negative covariances between the forward discounts and exchange-rate movements of these two exchange rates. Overall, the covariance decomposition suggests that the macroeconomic
contributions help to explain the forward discount anomaly even though exchange-rate
movements are very weakly correlated with macroeconomic fundamentals.
4.4 Subsample analysis
Given the evidence of structural instability, this section reports the results of some
subsample analysis. For the monthly data, the above regression equations are estimated
with subsamples drawn from moving windows of 30, 60, 120 and 180 months. As for
the quarterly data, the above econometric model is estimated with subsamples drawn
from moving windows of 20, 40 and 60 quarters.
Figures 1 to 3 juxtapose the quarterly subsample values of the macroeconomic
contributions with the covariances between the forward discounts and exchange-rate
movements. The figures highlight two subsample features. First, there appear to be
substantial changes in the subsample values of the covariances between the forward
discounts and exchange-rate movements. Second, a substantial portion of the subsample covariances are positive, especially at shorter horizons. For the 20-quarter moving
window, the subsample covariances range from -2.7128 to 0.5865 for the yen, -3.0896
to 1.3268 for the pound, and -1.4345 to 1.1131 for the mark. Previous studies such
as Bekaert and Hodrick (1993) and Engel (1996) have also documented substantial
structural instability in the covariance between the forward discount and exchange-rate
movement.
It can be seen from the figures that the macroeconomic contributions are tracking
the subsample covariances quite closely. Thus, the macroeconomic contributions are
able to match the changing signs and magnitudes of the subsample covariances between the forward discounts and exchange-rate movements. To quantify the subsample
performance of the macroeconomic contributions, Tables VII and VIII report some
summary statistics of the rolling regressions.
16
Columns (iii) report the percentages of negative subsample covariances between the
S
forward discounts and exchange-rate movements cov(∆st+1 , it − iU
t ). In general, the
percentages of positive subsample covariances tend to be higher at shorter horizons.
For all three exchange rates, the percentages of negative subsample covariances are
larger at the 180-month moving window than at the 30-month moving window. The
percentages are also larger at the 60-quarter moving window than at the 20-quarter
moving window. In other words, currencies with high interest rates tend to appreciate
over the long term, but currencies with high interest rates often depreciate over the
short term.
Columns (v) report the percentages of subsamples in which the macroeconomic
contributions take the same signs as the covariances between the forward discounts
and exchange-rate movements. For seventeen of the twenty-one cases in Tables VII and
VIII, the percentages in columns (v) are larger than the percentages in columns (iii).
This indicates that the macroeconomic contributions are able to match both positive and
negative covariances between the forward discounts and exchange-rate movements. As
a result, Figures 1 to 3 show that the macroeconomic contributions are tracking the
subsample covariances quite closely.
Tables VII and VIII also present some summary statistics of the magnitudes of
the macroeconomic contributions. Columns (vi) report the numbers of subsamples in
which the following inequality holds.
¸2
·
1
US
0
cov(∆st+1 , it − it ) − θ1 var (Xt+1 ) γ 1
ρ1
·
1
S
<
cov(∆st+1 , it − iU
cov(∆st+1 , η t+1 −
t )−
ρ1
Inequality (8) holds when the macroeconomic contribution
S
to cov(∆st+1 , it − iU
t ) than the residual covariance
¸2
(8)
t+1 )
1
0
ρ1 θ 1 var(Xt+1 ) γ 1
1
ρ1 cov(∆st+1 , η t+1
is closer
−
t+1 ).
This provides an indication of the significance of the macroeconomic contributions.
Columns (vii) report the percentages of subsamples in which inequality (8) holds. For
the mark and the yen, the macroeconomic contributions are close to cov(∆st+1 , it −
S
iU
t ) in a majority of cases. The percentages range from 43% to 100% for the Deutsche
mark, and the percentages range from 31% to 100% for the Japanese yen. Moreover,
the percentages tend to be higher at longer horizons. The performance is less satis17
factory for the British pound, as the percentages range from 18% to 51%. In the full
sample, the quarterly dollar-pound exchange rate is the only case in which the macroeconomic contribution is not within one standard deviation of the covariance between
the forward discount and exchange-rate movement. However, although the macroeconomic contributions do not fully account for the negative covariances between the
forward discounts and exchange rate movements, they certainly play an important role.
The last two summary statistics indicate whether there is a slight tendency for currencies to appreciate when output growths are high. Columns (ix) report the percentUS
is positive in the exchange-rate
ages of subsamples in which the coefficient on ∆yt+1
equation (5). It can be seen that the percentages range from 61% to 100% for the 21
cases. Moreover, the percentages tend to be higher at longer horizons. This suggests
that the U.S. dollar tends to appreciate when U.S. output growth is high.
Columns (xi) report the percentage of subsamples in which the coefficient on ∆yt+1
is negative in the exchange-rate equation (5). The percentages are not as high as in
columns (ix). The percentages range from 0% to 100% for the mark, 45% to 100%
for the yen, and 19% to 70% for the pound. In particular, the percentage for the quarterly dollar-mark rate is equal to zero for 60-quarter moving window. This could be
the result of the dramatic structural change associated with German reunification in
the middle of the sample period. Figure 3 also indicates some dramatic changes in
the macroeconomic contributions for the dollar-mark exchange rate. Nevertheless, the
macroeconomic contributions are still able to generate the negative covariance between
the forward discount and exchange-rate movement of the mark. This is because the
coefficients on U.S. output growth take the opposite signs in the exchange-rate and
forward-discount equations. That is, when U.S. output growth is high, the forward discount tends to be low and the U.S. dollar tends to appreciate. Overall, this subsample
analysis suggests that macroeconomic comovement is an important factor behind the
forward discount anomaly.
5 Uncovered interest parity
Ordinary least squares (OLS) has been a common way of testing the UIP condition.
Covariance decomposition (7) implies that macroeconomic comovements are partly responsible for the negative covariance between the forward discount and exchange-rate
18
movement. As a result, macroeconomic comovements generate a downward simultaneity bias in the OLS estimate of β in regression (1). Therefore, the widely documented
negative OLS estimates of β do not necessarily imply that UIP does not hold. McCallum (1994) makes a similar point, although the source of simultaneity is different
in his model. In particular, McCallum (1994) points out that simultaneity could result
from the fact that a central bank may raise its interest rate in order to stem a currency
depreciation.
Since UIP is a restriction on the expected exchange-rate-adjusted returns, instrumentalvariables regression is perhaps a more appropriate method of testing the hypothesis.
By controlling for macroeconomic comovements, the generalized method of moments
should be able to provide a more accurate assessment of UIP. Another advantage
of GMM is that it is robust to heteroskedasticity. Diebold and Nerlove (1989) and
Hsieh (1989) have provided evidence of autoregressive conditional heteroskedasticity
in the major exchange rates. The following GMM estimation therefore makes use of
a heteroskedasticity-robust weighting matrix. In particular, the GMM estimates minimize the robust continuous-updating GMM objective function:
#0
#
"
T
T
1 X
1 X
−1
√
φt (δ) V (δ)
φt (δ)
S(δ) = √
T t=1
T t=1
"
(9)
0
where δ = (α, β) is the vector of parameters to be estimated; φt (δ) = h(Yt , δ) ⊗
Zt , where h(Yt , δ) is the UIP moment condition and Zt is the vector of instruments;
¤£
¤0
PT £
and V (δ) = T1 t=1 φt (δ) − φ̄t (δ) φt (δ) − φ̄t (δ) is the robust covariance matrix.
This choice of the weighting matrix facilitates the use of weak-identification statistics
developed by Stock and Wright (2000).
Under rational expectation, the deviation from UIP should be uncorrelated with
previously available information. Thus, the moment condition of UIP can be written as
follows.
¢
¡
S
Et [ ∆st+1 | Zt ] = α + β it − iU
t
(10)
where Zt denotes the set of instruments. A constant is included as an instrument in
all specifications. Other instruments are drawn from the macroeconomic variables of
the four countries, ∆ytk , ∆mkt , π kt , k ∈ {G, J, U K, U S}. The superscripts G, J, U K
19
and U S refer to Germany, Japan, the United Kingdom and the United States respectively. The forward-discount equation (3) and the exchange-rate equation (5) imply
that macroeconomic fundamentals are relevant instruments. Using the macroeconomic
variables as instruments will therefore control for the effects of macroeconomic comovements.
Because conventional GMM test statistics often mistakenly assume that the objective functions are locally quadratic around the GMM estimates, this section will report
the concentrated S-sets for the parameters in addition to the J-statistics and robust
standard errors. The concentrated S-sets can be interpreted as confidence intervals that
are robust to weak identification. According to Theorem 3 in Stock and Wright (2000),
´
³
D
if a parameter θ is well-identified, then S δ 0 , θ̂ (δ 0 ) −→ χ2k−n , where S (δ 0 , α̂ (δ 0 ))
denotes the concentrated objective function evaluated at δ 0 , k is the dimension of the
weighting matrix, and n is the dimension of θ. In other words, a concentrated S-set
contains all parameter values such that the continuous-updating objective function is
smaller than the χ2k−n critical value.
In the first set of estimations, the parameters α and β are restricted to be the same
across the three exchange rates. The second set of estimations is carried out individually for each of the three exchange rates. Table IX reports the results of the joint
estimations for the monthly data. Table X reports the results of the joint estimations
for the quarterly data. It can be seen that all fourteen estimates of β in the two tables
are positive. They range from 0.1316 to 1.0334.
While the UIP condition implies that β is equal to one, β is equal to zero if exchange rates follow random walk. If the GMM estimate of β is significantly positive,
the hypothesis that the exchange rates follow random walk will be rejected. The fact
that many S-sets contain zero imply that the estimates are not precise enough to distinguish between the random walk hypothesis and the uncovered interest parity. That
is, the random walk hypothesis and the uncovered interest parity condition cannot be
rejected in most cases. In three of the fourteen specifications, the S-sets reject the hypothesis that β is equal to zero but cannot reject the hypothesis that β is equal to one.
The three specifications are Joint-M1, Joint-M4, and Joint Q4. Although these three
specifications provide strong support for UIP, the overall evidence is not sufficient for
a strong rejection of the random walk hypothesis.
The S-sets are empty for four of the fourteen specifications: Joint-M6, Joint-M7,
20
Joint-Q5 and Joint-Q7. Empty S-sets imply that there is no parameter value that satisfies the 27 or 39 orthogonality restrictions at the 5% significance level. As the number
of orthogonality restrictions decreases, a large set of parameters satisfies the orthogonality restrictions. When the GMM estimation procedure is applied to the individual
exchange rates, the numbers of orthogonality restrictions reduce to 3 and 7. As a result,
the S-sets are much wider for the separate estimations as reported in Tables X and XI.
In other words, the estimates of β tend to be much more imprecise for the individual
exchange rates. In all twelve specifications, the hypothesis that β is equal to one cannot be rejected. For specification GBP-M1, the random walk hypothesis is rejected
at the 5% significance level. However, the S-sets in the other eleven specifications
contain zero, suggesting that the random walk hypothesis cannot be rejected. Overall,
the GMM estimates suggest that the uncovered interest parity cannot be rejected even
though the OLS estimates of β are negative.
6 Conclusion
This paper provides a new perspective on the forward discount anomaly. The results
presented above suggest that the forward discount anomaly can be reconciled with
rational expectation. While this paper focuses on the empirical correlations between
the forward discount, exchange-rate movement and macroeconomic fundamentals, future research could therefore attempt to develop a structural model that captures the
correlations reported in this paper. In order to account for the forward discount anomaly, a general-equilibrium will need to generate procyclical interest rates and exchange
rates. Moreover, the above covariance decomposition indicates that macroeconomic
comovements do not fully account for the forward discount anomaly in some cases.
Future work should therefore ascertain whether a rational risk premium can fill in that
gap.
21
7 Bibliography
Andrews, Donald. 1991. “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation.” Econometrica, 59, 817-858.
Bacchetta, Philippe, and Eric van Wincoop. 2005. “Rational Inattention: A Solution to the Forward Discount Puzzle.” Manuscript.
Baxter, Marianne, and Robert King. 1999. “Measuring Business Cycles: Approximate Band-Pass Filters for Economic Time Series.” Review of Economics and
Statistics, 81, 575-593.
Bekaert, Geert, and Robert Hodrick. 1993. “On Biases in the Measurement of
Foreign Exchange Risk Premium.” Journal of International Money and Finance, 12,
115-138.
Cheung, Yin-Wong, Menzie Chinn and Antonio Pascual. 2005. “Empirical Exchange Rate Models of the Nineties: Are Any Fit to Survive?” Journal of International
Money and Finance, 24, 1150-1175.
Diebold, Francis, and Mark Nerlove. 1989. “The Dynamics of Exchange Rate
Volatility: A Multivariate Latent Factor ARCH Model.” Journal of Applied Econometrics, 4, 1-21.
Engel, Charles. 1996. “The Forward Discount Anomaly and the Risk Premium: A
Survey of Recent Evidence.” Journal of Empirical Finance, 3, 123-192.
Engel, Charles and Kenneth West. 2005. “Exchange Rates and Fundamentals.”
Journal of Political Economy, 113, 485-517.
Fama, Eugene. 1984. “Forward and Spot Exchange Rates.” Journal of Monetary
Economics, 14, 319-338.
Faust, Jon, John Rogers and Jonathan Wright. 2003. “Exchange Rate Forecasting:
The Errors We’ve Really Made.” Journal of International Economics, 60, 35-59.
Flood, Robert, and Andrew Rose. 1999. “Understanding Exchange Rate Volatility
without the Contrivance of Macroeconomics.” Economic Journal, 109, F660-F672.
Froot, Kenneth, and Richard Thaler. 1990. “Anomalies: Foreign Exchange.” Journal of Economic Perspectives, 4, 179-192.
Hansen, Lars, and Robert Hodrick. 1980. “Forward Exchange Rates as Optimal
Predictors of Future Spot Rates: An Econometric Analysis.” Journal of Political Economy, 88, 829-853.
22
Hsieh, David. 1989. “Modeling Heteroscedasticity in Daily Foreign-Exchange
Rates.” Journal of Business and Economic Statistics, 7, 307-317.
King, Robert, and Mark Watson. 1996. “Money, Prices, Interest Rates and the
Business Cycle.” Review of Economics and Statistics, 78, 35-53.
McCallum, Bennett. 1994. “A Reconsideration of the Uncovered Interest Parity
Relationship.” Journal of Monetary Economics, 33, 105-132.
Meese, Richard, and Kenneth Rogoff. 1988. “Was it Real? The Exchange RateInterest Differential Relation Over the Modern Floating-Rate Period.” Journal of Finance, 43, 933-948.
Ploberger, Werner, Walter Krämer and Karl Kontrus. 1989. “A New Test for Structural Stability in the Linear Regression Model.” Journal of Econometrics, 40, 307-318.
23
Table I
Forward discount anomaly
st+1 − st = α + β (ft − st ) + εt+1
Spot exchange rates st and forward exchange rates ft .are in logarithm and multiplied
by 100. The exchange rates are expressed as the numbers of currency units per U.S.
dollar. In parentheses below the OLS estimates are the heteroskedastic-autocorrelationrobust standard errors that are computed with the quadratic spectral kernel as suggested
by Andrews (1991). In parentheses below the F -statistics and the fluctuation test statistics are the corresponding p-values. The F -statistics test whether all estimated coefficients in a regression are equal to zero. The fluctuation statistics test whether the
estimated coefficients are stable over time.
Exchange rate
Sample period
α̂
British pound
1976:01-2004:06
Deutsche mark
1976:01-1998:12
Japanese yen
1976:01-2004:06
0.1944
(0.2142)
-0.1289
(0.2375)
-0.4975
(0.2881)
British pound
1976:Q1-2004:Q2
Deutsche mark
1976:Q1-1998:Q4
Japanese yen
1976:Q1-2004:Q2
1.6655
(0.6886)
-0.8164
(0.9739)
-3.2187
(1.0961)
Coefficient
of correlation
corr(∆st+1 , ft − st )
F -statistic
Fluctuation
test
Monthly frequency
-0.6408 0.0056
(0.5445)
-0.6080 0.0130
(0.3627)
-1.0883 0.0119
(0.6609)
0.7687
(0.4645)
1.4069
(0.2471)
1.6845
(0.1874)
1.0646
(0.3233)
0.9544
(0.4749)
2.5707
(0.0000)
-0.0710
Quarterly frequency
-2.5649 0.1169
(0.8367)
-0.8269 0.0266
(0.9499)
-2.9011 0.0998
(0.9945)
4.9428
(0.0093)
0.5049
(0.6060)
4.7834
(0.0107)
1.7990
(0.0045)
3.9305
(0.0000)
0.9135
(0.5404)
-0.3344
R2
β̂
24
-0.1138
-0.0978
-0.1418
-0.2997
Table II
Autoregressions of the forward discounts
¡
¢
S
US
it+1 − iU
+
t+1 = ρ0 + ρ1 it − it
Exchange rate
ρ0
British pound
0.1289
(0.0208)
-0.1368
(0.0406)
-0.1749
(0.0211)
0.1038
(0.0527)
-0.0572
(0.0468)
-0.1566
(0.0704)
Deutsche mark
Japanese yen
British pound
Deutsche mark
Japanese yen
t+1
F -statistic
Fluctuation
test
Monthly frequency
0.3191
0.3094
(0.0529)
0.2565
0.1407
(0.0619)
0.4118
0.5704
(0.0484)
71.7913
(0.0000)
19.8722
(0.0000)
230.4132
(0.0000)
1.5458
(0.0254)
2.8644
(0.0000)
6.4689
(0.0000)
Quarterly frequency
0.7803
0.7464
(0.0641)
0.8396
0.7708
(0.0456)
0.8253
0.8790
(0.0639)
143.9273
(0.0000)
224.9447
(0.0000)
290.5198
(0.0000)
2.1863
(0.0001)
7.0870
(0.0000)
1.5660
(0.0220)
R2
ρ1
25
Table III
Forward-discount regressions on macroeconomic fundamentals
S
S
S
it − iU
= γ 0 + γ y1 ∆ytU S + γ y2 ∆yt + γ m1 ∆mU
+ γ m2 ∆mt + γ p1 π U
+ γ p2 πt + ξ t
t
t
t
Exchange rate
British pound
Deutsche mark
Japanese yen
British pound
Deutsche mark
Japanese yen
Estimated coefficients on the variables
S
S
∆yt
∆mU
∆mt
πU
t
t
R2
F -statistic
Fluctuation
test
7.4054
(0.0000)
6.0462
(0.0000)
16.4092
(0.0000)
2.5519
(0.0000)
3.4268
(0.0000)
3.7051
(0.0000)
1.7892
(0.1008)
3.2518
(0.0056)
9.8534
(0.0000)
2.6207
(0.0000)
3.7291
(0.0000)
3.1984
(0.0000)
constant
∆ytU S
0.2591
(0.0498)
0.1217
(0.0918)
-0.1547
(0.0609)
-0.0924
(0.0462)
-0.0825
(0.0659)
-0.0426
(0.0475)
-0.0004
(0.0233)
0.0071
(0.0163)
-0.0090
(0.0233)
Monthly frequency
-0.0421
-0.0083
-0.1514
(0.0516) (0.0115) (0.1057)
-0.4472
0.0710
-0.3671
(0.1220) (0.0493) (0.1412)
-0.1274
-0.0069
-0.2118
(0.0835) (0.0131) (0.0962)
0.0544
(0.0575)
0.0975
(0.1416)
0.0389
(0.0627)
0.2670
0.6924
(0.5428)
-0.6736
(0.7179)
-0.4033
(0.4338)
-0.3164
(0.2242)
-0.3013
(0.1940)
-0.3261
(0.1650)
0.0163
(0.0569)
0.0991
(0.0937)
0.1029
(0.1242)
Quarterly frequency
0.0363
0.0022
-0.3721
(0.0370) (0.0270) (0.3273)
0.3640
0.0368
-0.8225
(0.3229) (0.1015) (0.2464)
0.0628
-0.0393
-0.5580
(0.0276) (0.0349) (0.2211)
0.0772
(0.1920)
0.7074
(0.3421)
0.2263
(0.2367)
0.4860
26
πt
0.1811
0.5188
0.5554
0.7798
Table IV
Exchange-rate regressions on macroeconomic fundamentals
US
S
US
∆st+1 = θ0 + θy1 ∆yt+1
+ θy2 ∆yt+1 + θm1 ∆mU
t+1 + θ m2 ∆mt+1 + θ p1 π t+1 + θ p2 π t+1 + ξ t+1
Exchange rate
British pound
Deutsche mark
Japanese yen
British pound
Deutsche mark
Japanese yen
Estimated coefficients on the variables
S
S
∆yt+1
∆mU
∆mt+1
πU
t+1
t+1
R2
F -statistic
Fluctuation
test
1.4147
(0.1990)
2.7542
(0.0092)
1.0158
(0.4201)
1.5088
(0.1076)
1.4974
(0.1154)
1.9885
(0.0035)
0.5785
(0.7714)
2.5796
(0.0219)
2.6981
(0.0146)
2.2544
(0.0006)
1.7943
(0.0175)
1.0163
(0.8143)
constant
US
∆yt+1
0.0749
(0.3316)
-1.1913
(0.4609)
-0.9798
(0.4395)
0.4701
(0.3076)
0.9395
(0.3309)
0.3065
(0.3426)
-0.0317
(0.1554)
-0.1568
(0.0816)
-0.0361
(0.1681)
Monthly frequency
0.2021
-0.1332
0.3877
(0.3433) (0.0764) (0.7037)
0.9887
-0.0073
0.6462
(0.6127) (0.2478) (0.7093)
0.9020
0.1015
0.8599
(0.6020) (0.0945) (0.6936)
-0.6031
(0.3830)
1.2119
(0.7109)
-0.3614
(0.4522)
0.0361
2.3118
(1.9749)
0.2326
(2.9088)
-3.1180
(2.3347)
0.5655
(0.8159)
1.5536
(0.7861)
0.4885
(0.8882)
-0.0850
(0.2070)
0.5460
(0.3797)
-0.4719
(0.6682)
Quarterly frequency
-0.1118
-0.0976
0.5499
(0.1346) (0.0982) (1.1910)
-3.0704
0.2440
1.2800
(1.3084) (0.4111) (0.9985)
-0.3314
0.3434
2.9256
(0.1488) (0.1877) (1.1898)
-0.6239
(0.6987)
2.9147
(1.3861)
-1.8447
(1.2740)
0.0556
27
π t+1
0.0790
0.0255
0.2571
0.1760
Table V
Out-of-sample fit of the exchange-rate equation
∆st+1 = θ0 + Xt+1 θ0 + ξ t+1
For the monthly data, a moving window of 120 months is used to estimate the exchangerate equation and then generate the out-of-sample fit. For the quarterly data, the length
of the moving window is 40 quarters. A positive value of the Diebold-Mariano statistic
indicates that the random walk has a smaller root-mean-squared error (RMSE) than the
exchange-rate equation.
Exchange rate
Out-of-sample
horizon
1 month
6 months
12 months
RMSE of the
random walk
3.3906
8.4024
10.6584
RMSE of the
exchange-rate equation
3.3499
9.4194
12.3482
Diebold-Mariano
statistic
-0.3210
1.1013
1.1102
Japanese yen
1 month
6 months
12 months
3.5048
9.1754
11.9777
3.6888
10.6192
14.9191
1.4686
1.1977
1.0530
British pound
1 month
6 months
12 months
2.9651
7.1906
8.7570
3.0056
7.6619
9.6853
0.8812
1.8534
1.5836
Deutsche mark
1 quarter
2 quarters
4 quarters
6.0341
9.6568
11.9472
7.1038
11.7167
14.8636
0.7227
0.9006
1.0543
Japanese yen
1 quarter
2 quarters
4 quarters
6.1826
8.9440
11.4443
6.7141
9.6554
9.8631
1.1823
0.8937
-1.0205
British pound
1 quarter
2 quarters
4 quarters
5.2951
7.8656
9.7330
6.2447
10.2915
15.8539
1.5711
1.5825
1.3961
Deutsche mark
28
Table VI
Decomposition of the negative covariance between
the forward discount and exchange-rate movement
¡
¢
cov ∆st+1 , it − iUS
=
t
0
1
ρ̂1 θ̂ 1 var(Xt+1 ) γ̂ 1
+
¡
∆st+1 , η t+1 −
1
ρ̂1 cov
t+1
¢
The contributions of macroeconomic variables are calculated from the estimated coefficients reported in Tables II, III and IV. In parentheses are the standard deviations of
the simulated moments that are calculated from 2000 random draws from the empirical
distributions.
Exchange rate
¡
¢
S
cov ∆st+1 , it − iU
t
Deutsche mark
-0.2405
Japanese yen
-0.1075
British pound
-0.0766
Deutsche mark
-0.7062
Japanese yen
-1.3341
British pound
-1.1077
Contribution of
macroeconomic
comovements
0
1
ρ̂ θ̂ 1 var (Xt+1 ) γ̂ 1
1
Monthly frequency
-0.3945
(0.2387)
-0.0925
(0.0584)
-0.0580
(0.0725)
Quarterly frequency
-0.7741
(0.7302)
-1.2636
(0.5466)
-0.2681
(0.3757)
29
Residual covariance
that is unexplained by
macroeconomic variables
¡
¢
1
ρ̂ cov ∆st+1 , η t+1 − t+1
1
0.1541
(0.0481)
-0.0151
(0.0017)
-0.0186
(0.0027)
0.0678
(0.0038)
-0.0706
(0.0050)
-0.8397
(0.0623)
Table VII
Summary of rolling regressions at monthly frequency
The exchange-rate and forward-discount equations are estimated with subsamples drawn from moving windows of 30, 60, 120 and 180
months. Covariance decompositions are then generated for the subsamples.
(i)
(ii)
(iii)
Number of
months in
the moving
window
Negative subsample
¢
¡
S
cov ∆st+1 , it − iU
t
number
percentage
180
120
60
30
47
75
115
117
100%
70%
69%
59%
180
120
60
30
109
131
170
175
100%
78%
74%
68%
180
120
60
30
101
119
147
152
89%
69%
63%
58%
(iv)
(v)
(vi)
(vii)
Subsample estimates
Macroeconomic
of ρ̂1 θ̂1 var (Xt+1 ) γ̂ 01
contributions
1
take¡ the same sign ¢
¡ are close to U S ¢
S
cov
∆st+1 , it − it
as cov ∆st+1 , it − iU
t
number
percentage
number percentage
Deutsche mark
47
100%
47
100%
89
83%
74
69%
137
82%
111
66%
124
63%
84
43%
Japanese yen
109
100%
58
53%
134
79%
52
31%
186
81%
87
38%
185
71%
133
51%
British pound
101
89%
41
36%
116
67%
85
49%
101
43%
43
18%
139
53%
56
21%
30
(viii)
(ix)
(x)
(xi)
Exchange-rate equation
Positive subsample
Negative subsample
estimates of the
estimates of the
US
coefficient on ∆yt+1
coefficient on ∆yt+1
number percentage number percentage
47
107
158
183
100%
100%
95%
93%
47
107
147
152
100%
100%
88%
77%
104
151
157
159
95%
89%
69%
61%
69
82
106
124
63%
49%
46%
48%
96
151
147
172
85%
87%
63%
65%
21
39
106
143
19%
23%
45%
54%
Table VIII
Summary of rolling regressions at quarterly frequency
The exchange-rate and forward-discount equations are estimated with subsamples drawn from moving windows of 20, 40 and 60 quarters.
Covariance decompositions are then generated for the subsamples.
(i)
(ii)
(iii)
Negative subsample
¢
¡
S
cov ∆st+1 , it − iU
t
(iv)
(v)
Subsample estimates
of ρ̂1 θ̂1 var (Xt+1 ) γ̂ 01
1
take¡ the same sign ¢
S
as cov ∆st+1 , it − iU
t
number
percentage
(vi)
(vii)
Macroeconomic
contributions
¡ are close to S ¢
cov ∆st+1 , it − iU
t
number percentage
(viii)
(ix)
(x)
(xi)
Exchange-rate equation
Positive subsample
Negative subsample
estimates of the
estimates of the
US
coefficient on ∆yt+1
coefficient on ∆yt+1
number percentage number percentage
Number of
quarters in
the moving
window
number
percentage
60
40
20
25
31
49
100%
69%
75%
25
36
56
100%
80%
86%
Deutsche mark
22
27
41
88%
60%
63%
25
45
48
100%
100%
74%
0
5
19
0%
11%
29%
60
40
20
37
57
71
100%
100%
92%
37
57
73
100%
100%
95%
Japanese yen
37
55
51
100%
96%
66%
23
44
47
62%
77%
61%
37
41
35
100%
72%
45%
60
40
20
47
37
65
100%
55%
75%
45
46
66
96%
69%
76%
British pound
24
13
38
51%
19%
44%
46
56
62
98%
84%
71%
19
35
61
40%
52%
70%
31
Table IX
GMM estimates of the uncovered interest parity condition at monthly frequency
The uncovered interest parity condition is estimated using the continuous-updating generalized method of moments. The coefficients are
restricted to the same across the three exchange rates. A constant is included as an instrument in all specifications. The second column
indicates the other included instruments. In parentheses below the GMM estimates are the standard errors calculated from the robust
covariance matrix in the continuous-updating objective function. Open intervals beside the GMM estimates are the concentrated S-sets,
which contain parameter values such that the continuous-updating objective functions are smaller than the 95% χ2k−1 critical values. The
degree of freedom k is equal to the dimension of the weighting matrix, which is equal to the number of orthogonality restrictions. The
J-statistic is the value of the continuous-updating objective function at the GMM estimate. The corresponding p-value is reported in
parentheses below the J-statistic.
Instrument sets include
a constant and
the following variables
Specification
©
Joint-M1
©
Joint-M2
©
Joint-M6
Joint-M7
©
¯
ª
π kt ¯ k ∈ {G, J, U K, U S}
¯
ª
∆ytk , ∆mkt ¯ k ∈ {G, J, U K, U S}
©
Joint-M5
¯
ª
∆mkt ¯ k ∈ {G, J, U K, U S}
©
Joint-M3
Joint-M4
¯
ª
∆ytk ¯ k ∈ {G, J, U K, U S}
¯
ª
∆mkt , π kt ¯ k ∈ {G, J, U K, U S}
©
¯
ª
∆ytk , π kt ¯ k ∈ {G, J, U K, U S}
¯
ª
∆ytk , ∆mkt , π kt ¯ k ∈ {G, J, U K, U S}
GMM estimates, standard errors and 95% concentrated S-sets
α̂GMM
concentrated S-sets β̂ GMM
concentrated S-sets
-0.0368
(0.1355)
-0.0330
(0.1327)
-0.0035
(0.1345)
-0.0222
(0.1307)
-0.0341
(0.1271)
-0.0924
(0.1364)
-0.0409
(0.1315)
(-0.1349, 0.0614)
(-0.6549, 0.5965)
(-0.5855, 0.5851)
(-0.2329, 0.1892)
(-0.6227, 0.5585)
∅
∅
32
0.7721
(0.3638)
0.3933
(0.2791)
0.0851
(0.2925)
0.7608
(0.3503)
0.1316
(0.3137)
0.2282
(0.4607)
0.4569
(0.5215)
(0.4999, 1.0271)
(-1.0054, 1.7032)
(-1.1893, 1.3961)
(0.1421, 1.2820)
(-1.3799, 1.5421)
∅
∅
J-statistic
23.4228
(0.0756)
12.5619
(0.6361)
14.2318
(0.5080)
37.5827
(0.0847)
28.0993
(0.4059)
39.1353
(0.0616)
55.7938
(0.0396)
Number of
orthogonality
restrictions
15
15
15
27
27
27
39
Table X
GMM estimates of the uncovered interest parity condition at quarterly frequency
The coefficients are restricted to the same across the three exchange rates. Other notes on methodology can be found in Table IX.
Instrument sets include
a constant and
the following variables
Specification
©
Joint-Q1
©
Joint-Q2
©
Joint-Q6
Joint-Q7
©
¯
ª
π kt ¯ k ∈ {G, J, U K, U S}
¯
ª
∆ytk , ∆mkt ¯ k ∈ {G, J, U K, U S}
©
Joint-Q5
¯
ª
∆mkt ¯ k ∈ {G, J, U K, U S}
©
Joint-Q3
Joint-Q4
¯
ª
∆ytk ¯ k ∈ {G, J, U K, U S}
¯
ª
∆mkt , π kt ¯ k ∈ {G, J, U K, U S}
©
¯
ª
∆ytk , π kt ¯ k ∈ {G, J, U K, U S}
¯
ª
∆ytk , ∆mkt , π kt ¯ k ∈ {G, J, U K, U S}
GMM estimates, standard errors and 95% concentrated S-sets
α̂GMM
concentrated S-sets β̂ GMM
concentrated S-sets
0.2979
(0.3360)
-0.6882
(0.3540)
-0.2918
(0.3207)
-0.4201
(0.3460)
-0.8071
(0.3219)
-0.2802
(0.2594)
-0.6148
(0.2864)
(-1.5200, 2.0147)
(-1.6471, 0.2765)
(-1.9961, 1.2947)
(-1.0466, 0.2092)
∅
(-1.6696, 0.9888)
∅
33
0.8744
(0.2546)
0.5666
(0.2790)
0.1560
(0.2364)
1.0334
(0.2316)
0.3893
(0.2176)
0.7722
(0.2087)
0.9089
(0.1942)
(-0.4225, 2.3491)
(-0.1953, 1.3371)
(-1.0394, 1.5865)
(0.6104, 1.4575)
∅
(-0.2407, 1.9821)
∅
J-statistic
10.0542
(0.8163)
19.9838
(0.1726)
10.3994
(0.7939)
37.2292
(0.0909)
44.1910
(0.0198)
25.9548
(0.5211)
67.3659
(0.0032)
Number of
orthogonality
restrictions
15
15
15
27
27
27
39
Table XI
GMM estimates of the uncovered interest parity condition at monthly frequency
The estimations are carried out individually for each of three exchange rates. Other notes on methodology can be found in Table IX.
Specification
Instrument sets include
a constant and
the following variables
GBP-M1
∆ytU S , ∆ytU K
GBP-M2
S
UK
S
UK
∆ytU S , ∆ytU K , ∆mU
, πU
t , ∆mt
t , πt
DM-M1
∆ytU S , ∆ytG
DM-M2
S
G
US
G
∆ytU S , ∆ytG , ∆mU
t , ∆mt , π t , π t
JPY-M1
∆ytU S , ∆ytJ
JPY-M2
S
J
US
J
∆ytU S , ∆ytJ , ∆mU
t , ∆mt , π t , π t
GMM estimates, standard errors and 95% concentrated S-sets
α̂GMM
concentrated S-sets
β̂ GMM
concentrated S-sets
-1.2927
(0.6056)
-0.7315
(0.3970)
2.6646
(2.4752)
-0.2629
(0.1949)
1.2508
(2.2752)
-1.5101
(0.6077)
(-2.4091, 0.1477)
(-2.3752, 1.7030)
(-2.3868, 2.5955)
(-1.0949, 0.6197)
(-3.8013, 3.5660)
(-3.9762, 1.0632)
34
7.2521
(3.2792)
4.5294
(1.9041)
15.5747
(14.9684)
-0.9605
(0.7352)
4.6843
(7.4032)
-4.1701
(1.8425)
(0.0001, 71.8002)
(-11.0560, 28.2206)
(-85.1313, 89.6052)
(-4.5742, 2.5240)
(-35.6648, 38.2952)
(-19.2003, 3.7500)
J-statistic
0.0122
(0.9996)
5.2541
(0.6290)
0.2033
(0.9771)
3.8870
(0.7927)
0.0198
(0.9993)
2.8033
(0.9026)
Number of
orthogonality
restrictions
3
7
3
7
3
7
Table XII
GMM estimates of the uncovered interest parity condition at quarterly frequency
The estimations are carried out individually for each of three exchange rates. Other notes on methodology can be found in Table IX.
Specification
Instrument sets include
a constant and
the following variables
GBP-Q1
∆ytU S , ∆ytU K
GBP-Q2
S
UK
S
UK
∆ytU S , ∆ytU K , ∆mU
, πU
t , ∆mt
t , πt
DM-Q1
∆ytU S , ∆ytG
DM-Q2
S
G
US
G
∆ytU S , ∆ytG , ∆mU
t , ∆mt , π t , π t
JPY-Q1
∆ytU S , ∆ytJ
JPY-Q2
S
J
US
J
∆ytU S , ∆ytJ , ∆mU
t , ∆mt , π t , π t
GMM estimates, standard errors and 95% concentrated S-sets
α̂GMM
concentrated S-sets β̂ GMM
concentrated S-sets
0.8328
(1.3151)
0.9744
(0.6290)
-2.0229
(2.9791)
-0.4690
(0.4239)
0.6407
(6.9147)
-4.1427
(1.3608)
(-7.0524, 6.4170)
(-6.3053, 3.7913)
(-8.0770, 6.8565)
(-2.4020, 1.8771)
(-10.8705, 10.5056)
(-10.1367, 1.0748)
35
-1.4383
(2.3592)
-1.7693
(1.1553)
-2.8913
(5.7111)
-0.0797
(0.5778)
1.3510
(8.0696)
-4.0130
(1.4431)
(-25.0880, 25.8414)
(-6.7649, 23.0984)
(-29.7081, 34.5537)
(-3.2359, 3.2232)
(-13.1273, 16.0296)
(-11.0639, 1.1222)
J-statistic
0.2452
(0.9700)
3.4609
(0.8394)
0.2056
(0.9767)
3.1282
(0.8729)
0.1359
(0.9872)
4.8738
(0.6754)
Number of
orthogonality
restrictions
3
7
3
7
3
7
Figure 1
Summary of rolling regressions for the Japanese yen
Moving window of 60 quarters
-0.6
-0.8
-1
-1.2
Subsample values of covariances cov(∆st+1, it - iUS
)
t
-1.4
Contributions from macroeconomic comovements
-1.6
0
5
10
15
20
25
30
35
40
Moving window of 40 quarters
0
-0.5
-1
-1.5
-2
0
10
20
30
40
50
60
Moving window of 20 quarters
1
0
-1
-2
-3
0
10
20
30
40
50
60
70
80
Figure 2
Summary of rolling regressions for the British pound
Moving window of 60 quarters
0.5
0
-0.5
-1
Subsample values of covariances cov(∆st+1, it - iUS
)
t
-1.5
Contributions from macroeconomic comovements
-2
0
5
10
15
20
25
30
35
40
45
50
Moving window of 40 quarters
1
0
-1
-2
-3
0
10
20
30
40
50
60
70
Moving window of 20 quarters
2
1
0
-1
-2
-3
-4
0
10
20
30
40
50
60
70
80
90
Figure 3
Summary of rolling regressions for the Deutsche mark
Moving window of 60 quarters
0
-0.5
-1
Subsample values of covariances cov(∆st+1, it - iUS
)
t
Contributions from macroeconomic comovements
-1.5
0
5
10
15
20
25
Moving window of 40 quarters
2
0
-2
-4
-6
0
5
10
15
20
25
30
35
40
45
Moving window of 20 quarters
15
10
5
0
-5
0
10
20
30
40
50
60
70