2005.RW.Beilare.Opong, Wydział Zarządzania WZ WNE UW SGH PW czyli studia Warszawa kierunki matematyczne, WNE UW

2005.RW.Beilare.Opong

2005.RW.Beilare.Opong, Wydział Zarządzania WZ WNE UW SGH PW czyli studia Warszawa kierunki matematyczne, WNE UW

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Journal of Banking & Finance 29 (2005) 1631–1643
www.elsevier.com/locate/jbf
Some evidence of random walk behavior of
Euro exchange rates using ranks and signs
Jorge Belaire-Franch
a,1
, Kwaku K. Opong
b,
*
Department of Economic Analysis, University of Valencia, Campus dels Tarongers,
Avgda dels Tarongers s/n, 46022 Valencia, Spain
b
Department of Accounting and Finance, University of Glasgow, 65-71 Southpark Avenue,
Glasgow, G12 8LE, UK
a
Received 25 November 2003; accepted 14 June 2004
Available online 22 September 2004
Abstract
This study utilises recently developed tests based on ranks and signs, in addition to the tra-
ditional variance ratio test, to examine the behavior of Euro exchange rates. We show that
adjustments for multiple tests must be employed in order to avoid size distortions. Overall,
such adjustments provide evidence consistent with random walk behavior of Euro exchange
rates.
2004 Elsevier B.V. All rights reserved.
JEL classiﬁcation: G12; G14; G15
Keywords: Euro exchange rates; Variance ratio; Ranks; Signs; Adjustment for multiplicity
1. Introduction
Past evidence suggests that nominal exchange rate series follow a random walk
process (see
Meese and Singleton, 1982; Baillie and Bollerslev, 1989; Giddy and
*
Corresponding author. Tel.: +44 141 330 5426; fax: +44 141 330 4442.
E-mail addresses:
J. Belaire-Franch),
(K.K. Opong).
1
Tel.: +34 96 3828246, fax: +34 96 3828249.
0378-4266/$ - see front matter 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.jbankﬁn.2004.06.031
1632
J. Belaire-Franch, K.K. Opong / Journal of Banking & Finance 29 (2005) 1631–1643
Dufey, 1975; Hsieh, 1988
; among others). This implies, therefore, that the behavior
of nominal exchange rates are weak form ecient, and hence not predictable (
Fama,
1970, 1991
). Nevertheless, a recent contribution by
Jamaleh (2002)
suggests that eco-
nomic fundamentals eﬀectively drive the dynamics of the Euro/Dollar exchange rate.
This suggestion implies that the notion of random behavior of Euro/Dollar exchange
rate return series is rejectable due to the impact of economic factors.
The examination of the behavior of asset returns, in the context of a weak form
ecient market, is of interest to not only academics, but also, practitioners and reg-
ulators. While academicians seek to understand the behavior of asset returns over
time, practitioners and investors are often interested in identifying market inecien-
cies that produce exploitable patterns in exchange rate returns. Regulators in con-
trast, are interested in improving the informational e
ciency of the securities
market in which exchange rates are traded. Knowledge of the behavior of exchange
rates, particularly in relation to eciency/randomness issues, are therefore of consid-
erable interest to a large number of interest groups.
The present work uses the alternative variance ratio tests based on ranks and signs
proposed by
Wright (2000)
, in addition to the tests suggested by
Lo and MacKinlay
(1989)
, to examine the behavior of some Euro exchange rates. On January 1, 1999,
the Euro became the currency of 11 European member states namely Austria, Bel-
gium, France, Finland, Germany, Ireland, Italy, Luxembourg, Portugal and Spain.
On this date, the national currencies of the member countries became non-decimal
subunits of the Euro and conversion rates between each of them and the Euro be-
came irrevocably ﬁxed. National banknotes and coins of old currencies were ﬁnally
withdrawn from use (end of dual circulation) and replaced by Euros from February
28, 2002.
The scope of this paper is limited to a preliminary test of aggregate market trans-
actions of the Euro. As
Lo (1997)
claims, a meaningful test for the ecient market
hypothesis should specify additional elements, such as information structure and
investors/traders preferences.
2
In any case, this is a timely addition to the extant
literature of a new ﬁnancial product, the Euro from its birth. Another contribution
of this paper is that, to the best knowledge of the authors, this study will be the ﬁrst
of its kind on the Euro. Also, the large volume of studies that have examined the
behavior of asset returns are based on parametric tests, whereas this paper reports
some ﬁndings based on a new non-parametric methodology. As pointed out by
Wright (2000)
, the non-parametric based tests will be robust to many forms of con-
ditional heteroskedasticity and ought to have power against a wide range of models
of serial correlation such as autoregressive moving average and fractionally inte-
grated alternatives with heavy tailed innovations. Finally, we apply p-value adjust-
ments for multiple tests in order to avoid size distortions due to sequential testing.
The remainder of the paper is organized as follows. Section 2 discusses some pre-
vious research on the issue. The research methodology, some Monte Carlo
2
We are grateful to an anonymous referee for pointing this out to us.
J. Belaire-Franch, K.K. Opong / Journal of Banking & Finance 29 (2005) 1631–1643
1633
simulations and the empirical results are discussed in Section 3. Section 4 provides a
summary of the main ﬁndings and some concluding remarks.
2. Previous studies
A number of studies have tested the hypothesis that exchange rate series follow a
random walk behavior. Among such studies are
Meese and Singleton (1982)
and
Baillie and Bollerslev (1989)
who report a unit root component in the exchange rate
series and
Giddy and Dufey (1975)
,
Cornell and Dietrich (1978)
,
Logue et al. (1988)
and
Hsieh (1988)
who suggest that exchange rate series contain uncorrelated
increments.
Lo and MacKinlay (1988)
criticized the traditional random walk tests of asset re-
turns and introduced a more robust volatility-based speciﬁcation test. Since most
asset returns often possess time-varying volatilities and deviations from normality,
the importance of developing a test which is robust to heteroskedasticity and non-
normality becomes important. Lo and MacKinlay developed tests based on assump-
tions of both homoskedasticity and heteroskedasticity which they applied to examine
the validity of the random walk hypothesis for weekly stock returns. Other studies
have applied the methodology to test the random walk hypothesis in world money
rates (e.g.
Liu and He, 1991; Chou et al., 1996; Pan et al., 1996; Fong et al., 1997
).
Liu and He (1991)
applied variance ratio tests based on
Lo and MacKinlay (1988)
and provided evidence that rejected the random walk hypothesis for ﬁve pairs of
nominal exchange rates. Their reported results suggest that autocorrelations are pre-
sent in weekly increments in nominal exchange rate series.
Ayadi and Pyun (1994)
applied the variance ratio test to the daily Korean stock market prices. Their ﬁndings
rejected the random walk hypothesis under the assumption of homoskedasticity.
However, when heteroskedasticity is assumed, their ﬁndings supported the random
walk hypothesis for daily data and longer horizons of data intervals.
Chou et al.
(1996)
use variance ratio to test the random walk hypothesis of interest rates for
eight world currencies. Their results indicate that most of the interest rates in the
countries examined do not follow a random walk in the short run and vice versa
for the long run.
3. Data and methodology
The behavior of the return series of the Euro-based exchange rates is examined by
ﬁrst applying the traditional variance ratio test, and then, by application of the non-
parametric tests suggested by
Campbell and Dufour (1997)
,
Wright (2000)
and
Luger
(2003)
.
The data for the study consists of the daily nominal exchange rates for the Aus-
tralian dollar, Canadian dollar, New Zealand dollar, Japanese yen, British pound,
Norwegian kroner, Singapore dollar, Swedish krona, Swiss franc and United States
dollar, all relative to the Euro from 5th January 1999 to 11 November 2002 (almost 4
1634
J. Belaire-Franch, K.K. Opong / Journal of Banking & Finance 29 (2005) 1631–1643
years of data). The number of daily observations totaled 1005. The exchange rate
data for the series examined were obtained from Datastream/Primark. The short his-
torical data available for the Euro is of course likely to have some impact on the
results.
The variance ratio test proposed by
Lo and MacKinlay (1988, 1989)
is based on
the fact that, for a random walk series, the variance of its kth diﬀerence is k times the
variance of its ﬁrst diﬀerence. The M
1
test is valid under the assumption of independ-
ent and identically distributed returns, whereas the M
2
is robust against
heteroskedasticity.
Table 1
shows the results of Lo and MacKinlays variance ratio tests on the log-
arithm of the daily Euro-based nominal exchange rates, for k = 2, 5, 10 and 30.
Results for individual k values suggest that the random walk assumption is vio-
lated in the Canada, Norway, Singapore and Switzerland cases, using M
1
test. How-
ever, using the M
2
test, the null is rejected just for Canada, Norway and Singapore
exchange rates.
Table 1
Lo and MacKinlays VR tests results
Australia
Canada
Japan
UK
US
M
1
k = 2
0.716
2.296*
0.824
1.049
0.415
k =5
0.487
2.318*
0.935
1.296
1.355
k =10
1.060
1.902
0.814
1.309
0.979
k =30
0.806
1.228
0.670
1.220
0.333
M
2
k = 2
0.623
1.877
0.753
0.863
0.390
k =5
0.454
2.028*
0.807
1.102
1.276
k =10
0.994
1.734
0.706
1.113
0.922
k =30
0.773
1.177
0.605
1.053
0.314
New Zealand Norway
Singapore
Sweden
Switzerland
M
1
k =2
0.404
0.725
2.558*
1.852
0.293
k =5
1.277
1.907
2.877**
0.017
1.223
k =10
1.178
1.960*
2.217*
0.803
1.682
k =30
1.520
2.059*
1.293
1.548
2.267*
M
2
k =2
0.373 0.649
2.484* 1.691 0.168
k =5
1.232
1.744
2.778**
0.015
0.735
k =10
1.126
1.799
2.117*
0.703
1.061
k =30
1.463
1.993*
1.233
1.396
1.633
The entries are Lo and MacKinlays VR tests at aggregation interval k. * indicates signiﬁcance at (indi-
vidual) 5% level, whereas ** indicates signiﬁcance at 5% level using the SMM(a;m;
1
) asymptotic dis-
tribution, a = 0.05 and m =4.
J. Belaire-Franch, K.K. Opong / Journal of Banking & Finance 29 (2005) 1631–1643
1635
However, the application of VR tests for multiple k values leads to overrejection
of the null hypothesis, above the nominal size. Therefore, as suggested by
Chow and
Denning (1993)
, we compare the maximum M
1
and M
2
statistics (in absolute value)
with the asymptotic a-point critical value of the studentized maximum modulus,
SMM(a;m;
1
), where m is the number of k values.
3
The results allow us to reject
the null just for the Euro/Singapore nominal exchange rate, using both M
1
and
M
2
tests.
3.1. Ranks and signs-based random walk tests
In a recent paper,
Wright (2000)
proposes the use of signs and ranks of diﬀerences
in place of the diﬀerences in the Lo and MacKinlay tests. Wright demonstrates that
his non-parametric variance ratio tests based on ranks (R
1
and R
2
) and signs (S
1
and
S
2
), can be more powerful than the tests suggested by Lo and MacKinlay. They have
high power against a wide range of models displaying serial correlation, including
fractionally integrated alternatives. The tests based on ranks are exact under the
independence and identical distribution assumption, whereas the tests based on signs
are exact even under conditional heteroskedasticity. Moreover,
Wright (2000)
shows
that ranks-based tests display low size distortion, under conditional heteros-
kedasticity.
Given T observations of asset returns {y
1
, ..., y
T
}, Wrights proposed R
1
and R
2
are deﬁned as:
!
Tk
P
t
¼
k
ð
r
1t
þþ
r
1t
k
þ
1
Þ
2
1
1
R
1
¼
T
P
t
¼
1
r
1t
1
/
ð
k
Þ
1
=
2
;
ð
1
Þ
Tk
P
t
¼
k
ð
r
2t
þþ
r
2t
k
þ
1
Þ
2
1
!
/
ð
k
Þ
1
=
2
;
R
2
¼
T
P
t
¼
1
r
2t
1
ð
2
Þ
where
ð
T
1
Þð
T
þ
1
Þ
12
r
r
1t
¼
r
ð
y
t
Þ
T
þ
1
2
;
r
2t
¼
U
1
ð
r
ð
y
t
Þ=ð
T
þ
1
ÞÞ;
/
ð
k
Þ¼
2
ð
2k
1
Þð
k
1
Þ
3kT
;
r(y
t
) is the rank of y
t
among y
1
, ..., y
T
,andU
1
is the inverse of the standard normal
cumulative distribution function. The tests based on the signs of returns are given by:
3
For a number of k values equal to four, the asymptotic critical value is 2.491, at the 5% signiﬁcance
level.
1
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