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2 Absence of riskneutral behavior, the nature of the policy regime, time deformation, and
misspecification of the functional form of the structural exchange rate model are some of the plausible
sources of nonlinearity in foreign exchange rates.
3 Longmemory methods have been applied extensively to financial asset price series: for stock prices
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(Lo (1991)),
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futures prices (Barkoulas, Labys, and Onochie (1999)), commodity prices (Barkoulas, Labys, and
Onochie (1997)), Eurocurrency deposit rates (Barkoulas and Baum (1997)), for example.
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of the plausible
sources of nonlinearity in foreign exchange rates.
3 Longmemory methods have been applied extensively to financial asset price series: for stock prices (Lo (1991)), futures prices (Barkoulas, Labys, and Onochie (1999)), commodity prices (Barkoulas, Labys, and
Onochie (1997)), Eurocurrency deposit rates (Barkoulas and Baum (1997)), for example. See
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Baillie (1996)
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for a survey of fractional integration methods and applications in economics and finance.
distant observations, its presence entails that past returns can help predict future returns,
and the possibility of consistent speculative profits arises.4
Applying rescaledrange (R/S) analysis to daily exchange rates for the British
pound, French franc, and Deutsche mark, Booth, Kaen, and Koveos
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As long memory represents a special form of nonlinear dynamics, it calls into question linear modeling
and invites the development of nonlinear pricing models at the theoretical level to account for longmemory behavior.
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Mandelbrot (1971)
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observes that in the presence of long memory, the arrival of new
market information cannot be fully arbitraged away and martingale models of asset prices cannot be
obtained from arbitrage.
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Statistical inferences
concerning asset pricing models based on standard testing procedures may not be appropriate in the
presence of longmemory series.
5 The classical rescaledrange (R/S) method has a number of drawbacks (see
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Lo (1991)).
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differencing) parameter for the foreign currency rates. We use the Gaussian
semiparametric method to estimate the longmemory parameter. Contrary to previous
findings of strong persistence, our evidence strongly favors the martingale model
against longmemory alternatives in the foreign currency markets, thus supporting the
market efficiency hypothesis in its weak form.
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The arbitrary restriction of d to integer values gives rise to the standard
autoregressive integrated moving average (ARIMA) model. The stochastic process ty is
both stationary and invertible if all roots of Φ(L) and Θ(L) lie outside the unit circle and
d<0. 5. Assuming that d∈0, 0.5() and d≠0,
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Hosking (1981)
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shows that the correlation
function, ρ(⋅), of an ARFIMA process is proportional to 2d−1k as k→∞. Consequently,
the autocorrelations of the ARFIMA process decay hyperbolically to zero as k→∞
which is contrary to the faster, geometric decay of a stationary ARMA process.
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For d∈0. 5,1[) the process is nonstationary (having an infinite
variance) but it is mean reverting, as there is no longrun impact of an innovation on
future values of the process.
We estimate the longmemory parameter using Robinson's Gaussian
semiparametric method.
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Robinson (1995)
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proposes a Gaussian semiparametric estimate,
GS hereafter, of the selfsimilarity parameter H, which is not defined in closed form. It is
assumed that the spectral density of the time series, denoted by f⋅(), behaves as
fξ()~G1−2Hξ as ξ→+0(3)
for G∈0,∞() and H∈0,1().
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For the
Italian lira, rejection of the unitroot null hypothesis is obtained but it is restricted to a
particular estimation window size, thus lacking robustness. This evidence for the major
currency rates does not confirm the longmemory evidence reported in Booth et al.
(1982) and
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Cheung (1993).
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For the remaining currencies, consistent evidence of long
memory is apparent for the Danish krone, Luxembourg franc, Portugese escudo, and
Spanish peseta. For the Belgian franc, Swedish krona, Finnish markka, and Greek
drachma, weak evidence of fractional dynamic behavior is found, but only for a specific
estimation window size.
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The GPH results are not reported here but they are
available upon request from the authors.
7 Monte Carlo simulations have shown that such bandwidth choices provide a good balance in the tradeoff between bias (which tends to increase with bandwidth) and sampling variability (which tends to
decrease with bandwidth).
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Henry and Robinson (1999)
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provide heuristic approximations of the minimum
mean squared error optimal bandwidth. Inferences drawn below remain unaltered for other plausible
bandwidth choices.
8 Alternative estimation methods of longmemory models include Sowell's (1992) exact maximum
likelihood method, Fox and Taqqu's (1986) frequency domain approximate maximum likelihood method,
and the condition
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Inferences drawn below remain unaltered for other plausible
bandwidth choices.
8 Alternative estimation methods of longmemory models include Sowell's (1992) exact maximum
likelihood method, Fox and Taqqu's (1986) frequency domain approximate maximum likelihood method,
and the conditional sum of squares (CSS) method (Chung and
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Baillie (1993)).
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The maximum likelihood
methods are computationally burdensome, especially in light of the repeated estimation in the sensitivity
analysis, and rely on the correct specification of the highfrequency (ARMA) structure to obtain consistent
parameter estimates.
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A sensitivity analysis suggests that, when evidence
of long memory is obtained, it is sporadic and generally temporally unstable. The
evidence of long memory in major currency rates reported in Booth et al. (1982) and
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Cheung (1993)
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would appear to be an artifact of the sample period and currencies
considered in those studies. For all but three of the broader set of currencies analyzed
here, we find that the unitroot hypothesis is robust to longmemory alternatives,
providing strong support for martingale behavior of currency rates and, therefore, for
foreign exchange market efficiency.
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These weekly rates represent Friday noontime bid prices from the New
York foreign exchange market and were obtained from the Federal Reserve Board of
Governors. When Friday prices are not available Thursday prices are used. (The
construction of the data set of weekly observations follows
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Cheung (1993)).
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For the U.S.
dollar nominal rates for the Austrian schilling, Danish krone, Luxembourg franc,
Norwegian krone, Finnish markka, Portugese escudo, and Spanish peseta, the frequency
of observation is monthly and the sample covers the period 08/1973 to 12/1995 for a
total of 268 returns observations.
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