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
2.1. Fractional integration and cointegration The Autoregressive Fractionally-Integrated Moving Average ARFIMA model of
Granger and Joyeux 1980 and Hosking 1981 is an extension of the Autoregressive Integrated Moving Average ARIMA models as set forth in Eq. 4:
F L1 2 L
d
x
t
5 Q Le
t
4 where L is the lag operator, e
t
is a white noise, and d, the differencing operator, can take on integer and non-integer values. If d 5 0 the series x
t
is a standard mean- reverting Autoregressive Moving Average ARMA process, while it is a nonstation-
436 T. Choudhry International Review of Economics and Finance 8 1999 433–453
ary process non-mean-reverting if d 5 1. If 0 , d , 1, the series is a fractionally- integrated process and is known as a long memory process.
13
In the case of a fraction- ally-integrated process, an innovation has no permanent effect on the series, but its
mean-reversion properties are persistent. The study of a stationary long-run relationship between two or more nonstationary
variables by means of the cointegration test has been widely applied in empirical economics and finance. Consider two time series x
1
t and x
2
t and let X
t
be a 2x1 vector of x
it
. In all standard cointegration test methods, such as those of Johansen 1988
and Engle and Granger 1987, x
1
t and x
2
t are considered to be nonstationary in levels,
or d 5 1, and the cointegrating linear combination z
t
5 aX
t
with a as the cointegrating vector of the order Id2b with b 5 1. For a possible cointegration between x
1t
and x
2t
, it only requires the equilibrium error z
t
to be stationary, and therefore the integer values of 0 and 1 for d
z
of the equilibrium error term z
t
are too restrictive. If 0 , d
z
, 1, the equilibrium error z
t
will be fractionally integrated but mean-reverting. In this case the error term responds slowly to shocks so that deviations from equilibrium
are more persistent. Allowance for fractional integration provides flexibility in model- ling of mean-reverting dynamics Chou Shih, 1997. Baillie and Bollerslev 1994a
indicate that the traditional assumption that the spot and the forward rates are cointe- grated of the form C1, 1 may not be the proper presentation. A better representation
may be the fractional cointegration of the form C1, d with 0 , d , 1, so that all forward premia may have infinite variances but are nevertheless mean reverting.
14
The fractional cointegration approach applied in this paper integrates the notions of the Engle and Granger 1987 cointegration and of fractional differencing of Granger
and Joyeux 1980. In the first step the equilibrium error term from Eq. 1 2 and 3 is obtained by means of ordinary least square OLS and then in the second step
the stochastic structure of the error term is investigated. As shown by Yajima 1988 and Cheung and Lai 1993 the OLS estimate of a is consistent and converges at the
rate of T
1
2
d
if the elements of X
t
are fractionally cointegrated, i.e., the error term z
t
is fractionally integrated, 0 , d
z
, 1.
The values of d for the error term are estimated using the GPH semi-nonparametric estimator based on the following OLS estimating equation Eq. 5:
15
ln{Iw
j
} 5 b 1 b
1
ln{4Sin
2
w
j
2} 1 h
t
5 with b
1
5 2d , where Iw
j
is the periodogram of x
t
at frequency w
j
, w
j
5 2p
j
T j 5 1, . . . , T 2 1 and h are asymptotically i.i.d.
16
The number of low frequency ordinates n used in this test is equal to n 5 T
m
, where T is the number of observations. According to Cheung and Lai 1993 a large number of n will contaminate the estimate
of d and too few will produce imprecise estimates of d. Geweke and Porter-Hudak 1983 recommends the value of m to be 0.550 or above. The GPH test estimates the
differencing order dˆ in the first difference of the relevant series. The d of the level series is equal to 1 1 dˆ. The value of d can be used to test the null hypothesis of a
unit root. By testing the null d 5 1 dˆ 5 0 alternate null being d , 1 dˆ , 0 the presence of a unit root in the residuals may be investigated. Hypothesis testing regard-
ing d can be conducted by means of the t-statistics of the regression coefficients. The
T. Choudhry International Review of Economics and Finance 8 1999 433–453 437
variance of the estimate of b
1
is given by the usual OLS estimator while the theoretical asymptotic variance of e
t
is shown to provide a more reliable confidence interval. The GPH test is asymptotically normal, independently of the null hypothesis under
consideration Sephton, 1993. Thus, the GPH test can be applied to the residuals obtained from a cointegrating regression. As pointed out by Cheung and Lai 1993,
p. 108 while testing for cointegration, the power advantage of the GPH test over the Augmented Dickey-Fuller test ADF test is particularly relevant for a sample size
of 200 or less. They further indicate that the GPH test is more powerful than the ADF test in testing against ARFIMA alternatives. Critical values required for fractional
cointegration tests are provided by Sephton 1993.
17
2.2. Harris-Inder cointegration test The traditional cointegration tests of Engle and Granger 1987, Johansen 1988
and also the fractional cointegration tests, such as the GPH test, are based on the null hypothesis of no cointegration. Harris and Inder 1994 using the Engle-Granger
two-step method and the Kwiatkowski et al. 1992; KPSS unit root test provide a test of cointegration where the null hypothesis is the presence of cointegration between
the relevant variables.
18
The merit of the null hypothesis cointegration test is more visible in models where the variables are believed to be cointegrated, a priori Harris
Inder, 1994. They advocate the use of tests of both null hypotheses, if no a priori beliefs on the presence or absence of cointegration are imposed.
The test advocated by Harris and Inder 1994 is basically an extension of the test proposed by Engle and Granger 1987 mixed with the KPSS unit root test. The
Harris-Inder test is specified as y
t
5 x
t
9g 1 d
t
1 e
t
6 x
t
5 x
t
2
1
1 h
t
7 d
t
5 d
t
2
1
1 w
t
8 where y
t
is the dependent variable, x
t
is a vector of nonstationary independent variables and d
t
is a random walk in the residuals of the cointegration Eq. 6. If Eqs. 6, 7 and 8 are the true data-generating processes, then the presence of the random walk
component in the residuals will enable y
t
and x
t
to be cointegrated. However, if the variance of the random walk component v
2
is restricted to zero then the random walk component reduces to a constant for all t. In that case, Eq. 6 will represent a
cointegrating relationship between y
t
and x
t
with constant and stationary residuals. As indicated by Harris and Inder 1994, testing the null hypothesis of v
2
5 0 against
the alternative null v
2
. 0 will test the null hypothesis of cointegration against the
alternative of no cointegration. Also, in the case of the Harris-Inder test, the first step is to estimate Eqs. 1, 2, or 3 by OLS to obtain the error term, and then the
KPSS test is applied to check for unit roots in the residuals.
19
In the case of the Harris-Inder test, the value of d is restricted to 0 and 1. Lee and Schmidt 1996 show
438 T. Choudhry International Review of Economics and Finance 8 1999 433–453
that KPSS tests are also consistent against the stationary long-memory alternative. Harris and Inder 1994 provide the critical values required in this test.
3. The data and the results
