The 26 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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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.
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Baillie and Bollerslev (1989) and Hai et al. (1997)
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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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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.
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Evans and Lewis (1993) and Alexakis and Apergis (1996)
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fail to even find a long-run 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 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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Evans and Lewis (1993) and Alexakis and Apergis (1996) fail to even find a long-run 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
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Baillie and McMahon (1989) and 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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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.
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Evans and Lewis (1995)
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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 lag-length 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 lag-length structures,
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Luintel and Paudyal (1998)
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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 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 Lai and 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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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
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Taylor and Sarno (1998),
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referred to as the Johansen likelihood ratio (JLR) test. We motivate the relevance of employing a panel unitroot test on the basis of significant cross-sectional dependencies suggested by cross correlation and principal components analyses.
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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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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).
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Taylor and Sarno (1998)
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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 3-month (90-day) 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 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
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(Hansen and 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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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
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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 Dickey-Fuller and Phillips-Perron unit-root 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.,
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Meese and Singleton (1982), Baillie and Bollerslev (1989)).
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We establish similar evidence for the spot and forward rate series in our sample using Augmented Dickey-Fuller and Phillips-Perron unit-root 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 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),
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Im et al. (1995),
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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 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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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
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(Taylor and Sarno (1998)).
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The JLR test differs fundamentally from the standard panel-unit 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 unit-root 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 unit-root tests as it avoids the pitfalls of incorrect inference inherent in the standard tests. Through Monte Carlo simulations,
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Taylor and Sarno (1998)
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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 finite-sample bias, we also adjust the asymptotic critical value by the Reinsel-Ahn 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,
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Taylor and Sarno (1998)
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show that such an adjustment produces a reasonable approximation to the finite-sample 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 forecast-error series conditional on time t forward premium values in both single- and multi-currency contexts.13,14 A number of studies (e.g.
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Hodrick and Srivastava (1986),
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Bekaert and 12 Coefficient estimates, F-statistics, and adjusted 2R for the multi-market 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 forecast-error series on the forward premium series to be "balanced", or non-spurious, the forward premium
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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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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. Crowder (1994) finds ts−tf to be a unit-root process whereas
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Hai et al. (1997)
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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 unit-root process whereas Hai et al. (1997) reach the opposite conclusion.
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Baillie and Bollerslev (1994)
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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, time-varying currency risk premia based on equilibrium models reflect moments of relevant macroeconomic variables (see, for example,
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Hodrick and Srivastava (1984)).
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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 unit-root tests. Applied to a panel of forecast-error series for eight major currencies over the post-Bretton Woods era, the evidence overwhelmingly supports a slope coefficient of unity i n the coin
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