The 12 reference contexts in paper Natalya Delcoure, John T. Barkoulas, Christopher F. Baum, Atreya Chakraborty (2000) “The Forward Rate Unbiasedness Hypothesis Revisited: Evidence from a New Test” / RePEc:boc:bocoec:464

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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
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    (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 long-run 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 white-noise 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 white-noise 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 lag-length 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 lag-length 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 unit-root 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 unit-root variables. Without any loss of generality, a p-dimensional vector autoregressive (VAR) process of k-th order can be written as follows: ∆tX = μ +1Θt−1∆X + ... + k−1Θt−k+1∆X + t−kΠX + tε (5) where ∆ is the first-difference lag operator, μ is a p×1() matrix of cons
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    If the forecast-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
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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 unit-root test to the system of forecast-error series is bound to lead to substantial efficiency gains in estimation by exploiting the cross-equation dependencies. Standard panel-unit 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 non-spurious, the forward premium series must be of the same integration order as the forecast-error series, that is, it must be a stationary process (given the obtained evidence of stationarity for t h e forecast-error 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 unit-root 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 forecast-error series on the forward premium series is balanced, and nonspurious.
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    Hodrick (1992))
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    find that the forecast-error 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 single-currency framework. 15 In the multi-currency framework, there is evidence, at the 5 per cent level, of cross-market dependencies for
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