International Review of Economics and Finance 8 1999 433–453
Re-examining forward market efficiency Evidence from fractional and Harris-Inder
cointegration tests
Taufiq Choudhry
Southampton Management School, University of Southampton, Southampton SO17 1BJ, UK Received 3 February 1998; accepted 19 August 1998
Abstract
This article investigates the forward market efficiency by testing the unbiased forward exchange rate hypothesis using nine currencies vis-a`-vis the U.S. dollar. The empirical tests
are conducted using monthly data during the period between January 1985 and December 1996 and two different methods of cointegration tests, a fractional GPH test and the Harris-
Inder test. The two cointegration tests are based on two different null hypotheses. Results provide ample evidence of cointegration between the spot and the forward rate, but little
evidence of the unbiased rate hypothesis, which may be due to the nonstationary risk premium.
1999 Elsevier Science Inc. All rights reserved.
JEL classification: F30, F31
Keywords: Cointegration; ARFIMA; Mean-reversion; Forward premium
1. Introduction
Many studies investigate the foreign exchange market efficiency by looking at the relationship between the forward rate and the spot rate, especially the unbiased
forward rate hypothesis. The unbiased forward rate hypothesis states that the forward exchange rate is an unbiased predictor of the corresponding future spot exchange
rate.
1
The unbiasedness of the forward rate is important for the construction of macroeconomics models and for testing monetarist theories concerning the asset
market approach to the determination of the foreign exchange rate Bailey et al.,
Corresponding author. Tel.: 144-1703-593-966; fax: 144-1703-593-844. E-mail address
: T.ChoudhrySwansea.ac.uk T. Choudhry 1059-056099 – see front matter
1999 Elsevier Science Inc. All rights reserved.
PII: S1059-05609900023-4
434 T. Choudhry International Review of Economics and Finance 8 1999 433–453
1984. Current works usually study the unbiased forward rate hypothesis applying data in levels and the latest estimation methods such as cointegration that take nonsta-
tionarity of the data into consideration Phillips et al., 1996. This article provides a study of the unbiased rate hypothesis using nine currencies vis-a`-vis the U.S. dollar.
This hypothesis is often investigated using the following regression:
s
t
1
k
5 a 1 a
1
f
t
1 e
t
1
k
1 where s
t
1
k
is the log of the future spot rate k periods ahead, and f
t
is the log of the k
-period forward rate. Acceptance of the null hypothesis H : a
, a
1
5 0, 1 provides evidence that the forward exchange rate is an unbiased predictor of the future spot
rate.
2
This null hypothesis is based on the assumption that economic agents are risk neutral and have rational expectations Baillie McMahon, 1989. The forward
market is weak-form efficient and economic agents are risk neutral if the null is accepted and the residual e
t
1
k
is a white noise.
3
Deviation from the stated hypothesis is often attributed to a time-varying risk premium Fama, 1984; Hakkio Sibert,
1995.
4
However, as stated by Sibert 1989, market efficiency may be consistent with risk averse investors requiring a premium on forward contracts. Sibert shows using a
choice-theoretic, general equilibrium model of the risk premium that stochastic money supplies and endowments are enough to generate time-varying risk premium. In this
framework, forward contracts are used as a method of sharing risk. Premia are related to the efficacy of such contracts as hedges against risk. Other explanations, such as
expectation errors, institutional consideration, etc., have also been offered for the lack of solid empirical support for the unbiased hypothesis.
Furthermore, according to Frenkel 1977, semi-strong efficiency of the forward market may be examined by investigating whether the forward exchange rate summa-
rizes all the relevant information. In an efficient market the forward rate f
t
summa- rizes all the information concerning the expected value of s
t
1
k
that is available at t. One of the items of information available at t is the stock of information available at
t 2 1. If the forward market is semi-strong efficient then the forward rate lagged one
period f
t
2
1
added as an explanatory variable in Eq. 1 should have insignificant effect; in other words f
t
should summarize all the information including that contained in f
t
2
1
.
5
The following regression may be used to test the semi-strong efficiency Frenkel, 1977:
s
t
1
k
5 a 1 a
1
f
t
1 a
2
f
t
2
1
1 e
t
1
k
2 Acceptance of the null hypothesis H
: a , a
1
, a
2
5 0, 1, 0 provides further evidence that the forward exchange rate is an unbiased predictor of the future spot rate implying
a semi-strong efficient forward market.
6
Longworth 1981 provides a different form of a semi-strong efficiency test in the forward exchange market; his test involves
adding a lagged spot rate as an explanatory variable in Eq. 1. Thus, the Longworth 1981, semi-strong efficiency test may be presented
s
t
1
k
5 a 1 a
1
f
t
1 a
3
s
t
1 e
t
1
k
3 If the market is semi-strong efficient then a
and a
3
should be equal to zero and a
1
T. Choudhry International Review of Economics and Finance 8 1999 433–453 435
be equal to unity. In other words, acceptance of the null hypothesis a , a
1
, a
3
5 0, 1, 0 provides evidence of the unbiased hypothesis and the semi-strong efficiency.
Thus, a semi-strong efficient forward market implies that the coefficient on the lagged spot rate a
3
should have no significant effect on the future spot rate. Longworth 1981 using Eq. 3 and the U.S.-Canadian exchange rates, was able to reject the
forward market hypothesis. According to McFarland et al. 1994 results from previous research on whether the forward rate is an unbiased predictor of the future spot rate
are inconclusive.
7
As stated earlier, this article studies the unbiased rate hypothesis using Eqs. 1, 2, and 3 and nine currencies vis-a`-vis the U.S. dollar by means of a fractional
cointegration Geweke and Porter-Hudak or GPH test and the Harris-Inder cointe- gration test. The forward market efficiency has also been investigated by applying
the first difference of Eq. 1 see Fama, 1984 and Barnhart Szakmary, 1991. Using the spot and the forward rate in levels [as in Eqs. 1, 2 and 3] rather than first
difference has several advantages Corbae et al., 1992. The first advantage is that a differenced model yields estimates that converge to the true parameter estimates at
the rate T
12
where T is the sample size rather than rate T for levels. Due to the slower rate of convergence there is potential for spurious inference Phillips McFar-
land, 1997. Second, while a stationary stochastic risk premium may exhibit stochastic correlation with difference regressors, the levels approach is not affected by such
correlation since the order of nonstationary regressors dominates the order of the stationary risk premium. Furthermore, Phillips et al. 1996 found that the direct
regression [Eqs. 1, 2, and 3] has advantages when dealing with overlapping data.
8
During the earlier period most of the studies analyzing Eqs. 1, 2, and 3 have applied the standard linear regression method.
9
As indicated by Granger and Newbold 1986 the application of nonstationary variables in standard regression results in
spurious estimation.
10
In such a case, the application of the cointegration method is more appropriate.
11
Cointegration mimics the existence of a long-run equilibrium to which an economic system converges over time.
12
To our knowledge no one has studied the forward market efficiency using fractional cointegration or the Harris-
Inder cointegration test.
2. The estimation procedures
