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the foreign exchange market, it must
hold true that
Ett+kS()=tf(1)
where tf is the log forward rate at time t for delivery k periods later, t+kS is the
corresponding log spot rate at time t+k, and tE⋅() is the mathematical expectations
operator conditioned on the information set available at time t.1 Assuming the
formation of rational expectations
 Exact

(Muth (1960)),
 Suffix

St+k=tEt+kS()+t+ku(2)
1 The justification for using logarithms as opposed to levels in (1) is connected to Siegel's paradox.
Assuming for a moment that the relationship in (1) is expressed in levels (without taking logs), Siegel
(1972) notes that such a relationship must hold true on both sides of the market, that is, i
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set available at time t.1 Assuming the
formation of rational expectations (Muth (1960)), St+k=tEt+kS()+t+ku(2)
1 The justification for using logarithms as opposed to levels in (1) is connected to Siegel's paradox.
Assuming for a moment that the relationship in (1) is expressed in levels (without taking logs),
 Exact

Siegel (1972)
 Suffix

notes that such a relationship must hold true on both sides of the market, that is, it must also
hold true that tE1
()St+k
=1
ft
. However, tEt+kS()=tf and tE1
()St+k
=1
ft
cannot simultaneously
hold when the variables are expressed in levels due to Jensen's inequality E1x()>1E(x)().
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In order for
the FRUH to be empirically supported, t+kS and tf should share one common
stochastic trend and the realized forecast error t+kS−tf should be a stationary (that
is, an I(0)) process.4
The empirical evidence on the existence of cointegration between t+kS and
ft is decidedly mixed.
 Exact

Baillie and Bollerslev (1989) and Hai et al. (1997)
 Suffix

find that
St+k and tf form a cointegrated system with a unitary cointegrating vector. Evans
and Lewis (1993) and Alexakis and Apergis (1996) fail to even find a longrun
relationship between forward and corresponding future spot rates.
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share one common
stochastic trend and the realized forecast error t+kS−tf should be a stationary (that
is, an I(0)) process.4
The empirical evidence on the existence of cointegration between t+kS and
ft is decidedly mixed. Baillie and Bollerslev (1989) and Hai et al. (1997) find that
St+k and tf form a cointegrated system with a unitary cointegrating vector.
 Exact

Evans and Lewis (1993) and Alexakis and Apergis (1996)
 Suffix

fail to even find a longrun
relationship between forward and corresponding future spot rates. Ngama (1992)
2 For alternative forms of testing the FRUH and a survey of the evidence and issues involved see
Baillie and McMahon (1989) and Engel (1996).
3 A series is integrated of order d, denoted by Id(), if it i
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Baillie and Bollerslev (1989) and Hai et al. (1997) find that
St+k and tf form a cointegrated system with a unitary cointegrating vector. Evans and Lewis (1993) and Alexakis and Apergis (1996) fail to even find a longrun
relationship between forward and corresponding future spot rates.
 Exact

Ngama (1992)
 Suffix

2 For alternative forms of testing the FRUH and a survey of the evidence and issues involved see
Baillie and McMahon (1989) and Engel (1996).
3 A series is integrated of order d, denoted by Id(), if it is rendered stationary after differencing it d
times.
4 Strictly speaking, the FRUH requires that
St+k−tf be a whitenoise process, a stronger condition
than
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Evans and Lewis (1993) and Alexakis and Apergis (1996) fail to even find a longrun
relationship between forward and corresponding future spot rates. Ngama (1992) 2 For alternative forms of testing the FRUH and a survey of the evidence and issues involved see
 Exact

Baillie and McMahon (1989) and Engel (1996).
 Suffix

3 A series is integrated of order d, denoted by Id(), if it is rendered stationary after differencing it d
times.
4 Strictly speaking, the FRUH requires that
St+k−tf be a whitenoise process, a stronger condition
than covariance stationarity.
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In that sense, the cointegration of t+kS and tf with a unitary
cointegrating vector is a necessary condition for the FRUH.
finds that the evidence of cointegratedness between t+kS and tf and the specific
cointegrating relationship varies across currencies and forward contract horizon.
 Exact

Evans and Lewis (1995)
 Suffix

find evidence of cointegration between t+kS and tf but they
reject the null of a unitary cointegrating vector.5 Through comprehensive testing
among alternative VAR specifications with respect to treatment of the constant
term and laglength structures, Luintel and Paudyal (1998) find robust evidence of
cointegration between t+kS and tf but they reje
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Evans and Lewis (1995) find evidence of cointegration between t+kS and tf but they
reject the null of a unitary cointegrating vector.5 Through comprehensive testing
among alternative VAR specifications with respect to treatment of the constant
term and laglength structures,
 Exact

Luintel and Paudyal (1998)
 Suffix

find robust evidence of
cointegration between t+kS and tf but they reject the unitary cointegrating vector
in the t+kS,tf() cointegrating relation. Copeland (1991) and Lai and Lai (1991) reject
the joint null of 0α=0 and 1α=1 in (4) in a cointegration analysis.
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unitary cointegrating vector.5 Through comprehensive testing
among alternative VAR specifications with respect to treatment of the constant
term and laglength structures, Luintel and Paudyal (1998) find robust evidence of
cointegration between t+kS and tf but they reject the unitary cointegrating vector
in the t+kS,tf() cointegrating relation.
 Exact

Copeland (1991) and Lai and Lai (1991)
 Suffix

reject
the joint null of 0α=0 and 1α=1 in (4) in a cointegration analysis.
In this paper, we reexamine the FRUH using a new multivariate (panel) unitroot test recently proposed by Taylor and Sarno (1998), referred to as the Johansen
likelihood ratio (JLR) test.
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Copeland (1991) and Lai and Lai (1991) reject
the joint null of 0α=0 and 1α=1 in (4) in a cointegration analysis.
In this paper, we reexamine the FRUH using a new multivariate (panel) unitroot test recently proposed by
 Exact

Taylor and Sarno (1998),
 Suffix

referred to as the Johansen
likelihood ratio (JLR) test. We motivate the relevance of employing a panel unitroot test on the basis of significant crosssectional dependencies suggested by cross
correlation and principal components analyses.
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Section 2 outlines the
multivariate unitroot test employed. In section 3 we report the test findings.
Section 4 summarizes and concludes.
II. The Johansen Likelihood Ratio (JLR) Test
 Exact

Johansen (1992)
 Suffix

suggests a maximum likelihood method to determine the number
of common trends in a system of unitroot variables. Without any loss of generality,
a pdimensional vector autoregressive (VAR) process of kth order can be written
as follows:
∆tX = μ +1Θt−1∆X + ... + k−1Θt−k+1∆X + t−kΠX + tε (5)
where ∆ is the firstdifference lag operator, μ is a p×1() matrix of cons
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the
Johansen likelihood ratio (JLR) test statistic:
JLR=−Tln 1−pλ(),(8)
6 The rank of a matrix is equal to the number of its nonzero characteristic roots.
where pλ is the smallest eigenvalue of the generalized eigenvalue problem
λkkS−k0S00S−10kS = 0.(9)
The ijS matrices are residual moment matrices from the VECM in (5).
 Exact

Taylor and Sarno (1998)
 Suffix

show that the JLR test statistic in (8) is asymptotically distributed as
χ21() under the null hypothesis.
III. Data and Test Results
We analyze U.S. dollar spot and 3month (90day) forward rates for eight
major currencies: Canadian dollar (CD), Deutsche mark (DM), British pound (BP),
French franc (FF), Swiss franc (SF), Netherlands guilder (NG), Italian lira (
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If the forecasterror series
7 When the observation and forecast periods do not coincide, that is, when the time to maturity for t h e
forward contract exceeds the time interval between observations, the error term in the cointegrating
regression in (4) is a noninvertible moving average process
 Exact

(Hansen and Hodrick (1980)).
 Suffix

In that case,
Moore (1992) shows that the Johansen cointegration methodology is inapplicable, as the Granger
representation theorem breaks down in the presence of noninvertible moving average errors.
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error series
7 When the observation and forecast periods do not coincide, that is, when the time to maturity for t h e
forward contract exceeds the time interval between observations, the error term in the cointegrating
regression in (4) is a noninvertible moving average process (Hansen and Hodrick (1980)). In that case,
 Exact

Moore (1992)
 Suffix

shows that the Johansen cointegration methodology is inapplicable, as the Granger
representation theorem breaks down in the presence of noninvertible moving average errors. Therefore,
estimates and test results reported in previous empirical studies employing the Johansen technique are
invalid when the observation period is shorter than the duration of the forward contact
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The only exception is the Canadian dollar, which does
not appear to be closely linked to the behavior of the remaining currencies.
8 This methodology (imposition of the cointegrating vector implied by theory, and evaluation of the resulting residual series) is termed the "restricted cointegration test" in
 Exact

Liu and Maddala (1992).
 Suffix

9 The I1() behavior of
St+k and tf has been established in numerous papers (see, e.g., Meese and
Singleton (1982), Baillie and Bollerslev (1989)). We establish similar evidence for the spot and
forward rate series in our sample using Augmented DickeyFuller and PhillipsPerron unitroot tests.
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behavior of the remaining currencies.
8 This methodology (imposition of the cointegrating vector implied by theory, and evaluation of the resulting residual series) is termed the "restricted cointegration test" in Liu and Maddala (1992). 9 The I1() behavior of
St+k and tf has been established in numerous papers (see, e.g.,
 Exact

Meese and Singleton (1982), Baillie and Bollerslev (1989)).
 Suffix

We establish similar evidence for the spot and
forward rate series in our sample using Augmented DickeyFuller and PhillipsPerron unitroot tests.
Given that this evidence is a widely accepted stylized fact, it is not reported here but it is available
upon request.
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Given the above results, the application of a panel unitroot test to the system
of forecasterror series is bound to lead to substantial efficiency gains in estimation
by exploiting the crossequation dependencies. Standard panelunit root tests
suggested by Levin and Lin (1992, 1993),
 Exact

Im et al. (1995),
 Suffix

O'Connell (1998), and others,
have as their null hypothesis that all variables in the panel are realizations of unitroot processes. This null will be rejected if even one of the series in the panel is
stationary.
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Given the above results, the application of a panel unitroot test to the system
of forecasterror series is bound to lead to substantial efficiency gains in estimation
by exploiting the crossequation dependencies. Standard panelunit root tests
suggested by Levin and Lin (1992, 1993), Im et al. (1995), O'
 Exact

Connell (1998), and
 Suffix

others,
have as their null hypothesis that all variables in the panel are realizations of unitroot processes. This null will be rejected if even one of the series in the panel is
stationary.
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Under these conditions, rejection of the null leads to the misleading
inference that all series in the panel are realizations of stationary processes. T h e
rejection frequencies for such tests are sizable even when there is a single stationary
process (with a root near unity) in the panel
 Exact

(Taylor and Sarno (1998)).
 Suffix

The JLR test
differs fundamentally from the standard panelunit root tests in how it establishes
its null and alternative hypotheses. In the JLR test the null is that at least o n e of the
series in the panel is a unitroot process with the alternative being that all series i n
the panel are stationary processes.
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Therefore, the JLR
test provides a useful alternative to standard panel unitroot tests as it avoids the
pitfalls of incorrect inference inherent in the standard tests. Through Monte Carlo
simulations,
 Exact

Taylor and Sarno (1998)
 Suffix

show that the JLR test has good size and power
properties.
Table 2 reports the JLR test results. As the JLR test statistic is asymptotically
distributed as 2χ1(), its 5 per cent critical value is 3.84.
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To account for finitesample
bias, we also adjust the asymptotic critical value by the ReinselAhn scale factor
Τ
Τ−pk
, where T is the number of observations, p is the number of series in the
panel (dimension of the system), and k is the lag order in the VECM in (5). Using
response surface analysis,
 Exact

Taylor and Sarno (1998)
 Suffix

show that such an adjustment
produces a reasonable approximation to the finitesample critical values, thus
avoiding significant size distortions. The adjusted critical value at the 5 per cent
level is 4.6702.
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We next examine the forecastibility of the realized forecasterror series
conditional on time t forward premium values in both single and multicurrency
contexts.13,14 A number of studies (e.g.
 Exact

Hodrick and Srivastava (1986),
 Suffix

Bekaert and
12 Coefficient estimates, Fstatistics, and adjusted 2R for the multimarket framework are not reported
here but are available upon request.
13 The forward premium is defined as
st−tf.
14 In order for the regression of the forecasterror series on the forward premium series to be "balanced",
or nonspurious, the forward premium
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forward premium series to be "balanced",
or nonspurious, the forward premium series must be of the same integration order as the forecasterror
series, that is, it must be a stationary process (given the obtained evidence of stationarity for t h e
forecasterror series). There is a debate in the literature regarding the integration order of the forward
premium series.
 Exact

Crowder (1994)
 Suffix

finds ts−tf to be a unitroot process whereas Hai et al. (1997) reach
the opposite conclusion. Baillie and Bollerslev (1994) find ts−tf to be a fractionally integrated
process. We apply the JLR test to our panel of forward premium series for the eight currencies and
document strong evidence supporting that the forward premium series under consideration are
real
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premium series must be of the same integration order as the forecasterror
series, that is, it must be a stationary process (given the obtained evidence of stationarity for t h e
forecasterror series). There is a debate in the literature regarding the integration order of the forward
premium series. Crowder (1994) finds ts−tf to be a unitroot process whereas
 Exact

Hai et al. (1997)
 Suffix

reach
the opposite conclusion. Baillie and Bollerslev (1994) find ts−tf to be a fractionally integrated
process. We apply the JLR test to our panel of forward premium series for the eight currencies and
document strong evidence supporting that the forward premium series under consideration are
realizations of stationary processes.
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There is a debate in the literature regarding the integration order of the forward
premium series. Crowder (1994) finds ts−tf to be a unitroot process whereas Hai et al. (1997) reach
the opposite conclusion.
 Exact

Baillie and Bollerslev (1994)
 Suffix

find ts−tf to be a fractionally integrated
process. We apply the JLR test to our panel of forward premium series for the eight currencies and
document strong evidence supporting that the forward premium series under consideration are
realizations of stationary processes.
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In that case, the forward rate would be a conditionally biased forecast of the future spot rate i f
the risk premium is forecastable on the basis of time t information available to market participants.
For example, timevarying currency risk premia based on equilibrium models reflect moments of
relevant macroeconomic variables (see, for example,
 Exact

Hodrick and Srivastava (1984)).
 Suffix

It must be noted
that the JLR test is asymptotically immune to the omission of a stationary risk premium.
methodological advantages over alternative standard panel unitroot tests. Applied
to a panel of forecasterror series for eight major currencies over the postBretton
Woods era, the evidence overwhelmingly supports a slope coefficient of unity i n
the coin
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