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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)),
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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),
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Siegel (1972)
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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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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.
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Ngama (1992)
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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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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
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Engel (1996).
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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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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.
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Copeland (1991) and
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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 Taylor and Sarno (1998), referred to as the Johansen
likelihood ratio (JLR) test.
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.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. Copeland (1991) and Lai and
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Lai (1991)
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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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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
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Johansen (1992)
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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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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 (Hansen and
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Hodrick (1980)).
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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,
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Moore (1992)
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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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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'
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Connell (1998), and
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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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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.
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Crowder (1994)
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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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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. These results are not reported here but are available upon request.
Therefore, the regression of forecasterror series on the forward premium series is balanced, and nonspurious.
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Hodrick (1992))
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find that the forecasterror series is a function of forward premium
values. The test results reported in Table 3 show that the forward premium enters
significantly in the CD, BP, SF, NG, and JY equations in the singlecurrency
framework.
15 In the multicurrency framework, there is evidence, at the 5 per cent
level, of crossmarket dependencies for
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