A. Abraham International Review of Economics and Finance 8 1999 213–222 217
trading induced by these expectations will now drive a wedge between the relative interest rates. As the parameters in the information set F are revised based on
randomly arriving information, the relative interest rates could drift apart. To see this more clearly combine now a version of the International Fisher Effect:
LnE
t
e
t
1
k
| F
t
5 Lne
t
1 kr
d t|t
1
k
2 r
f t|t
1
k
3 From Eqs. 2 and 3, it can be seen that the expected future exchange could
drift away from the spot exchange rate, inducing non-stationarity in the interest rate differential. To place this in context with the extant literature on cointegration in the
Spot-Forward exchange rates, note that the observed cointegration in the Spot-For- ward system in a floating rate regime is, as argued by Brenner and Kroner 1995,
driven by a stationary interest rate differential. In the case of fixed exchange rates, however, the arrow of causality is reversed running from expectations about the future
fixed rate rule to interest rates.
The dynamic properties of interest rates and, by symmetry, bond prices in general has received considerable attention in the literature, particularly in the area of contin-
gent claim pricing. A variety of stochastic characterizations have been proposed, including those by Vaicek 1977, Brennan and Schwartz 1979 and Cox, Ingersoll,
and Ross 1985. Spot rate processes need to take into account factors such as mean reversion and more importantly since zero coupon bonds have a fixed value at maturity,
time dependent volatilities. In a recent development Heath, Jarrow and Morton 1992, propose a new methodology that models the forward rate process rather than the
spot rate process itself, where a constant forward rate volatility is consistent with fixed par values for bonds. In keeping with this, tests conducted in this paper therefore use
implied forward rates computed from the observed term structure of interest rates, rather than the level of interest rates.
3
3. Statistical model
In the first stage of the analysis, I test for stationarity of the implied forward rate process obtained from the level of interest rates in each currency. Specifically I test
whether the forward process f
t
has a stationary I1 integrated of order 1 ARMA representation, implemented through a unit root test using the Augmented Dickey
Fuller methodology for each of the two currencies indexed by j. D
f
j t
5 a 1 a
1
f
j t
2
1
1 a
2
D f
j t
2
1
1 e
t
j 5 1,2
The presence of a unit root is implied by the coefficient on the lagged value of the series a
1
being zero. In the second part of the analysis, I test whether the two forward rates, which may
be independently non-stationary, can be viewed as a cointegrated system, equivalently, examining whether there exists a linear combination of the two series which is station-
ary. If the central bank policy is credible, one should expect to find a stable relationship between the two forward rate processes, however, if agents perceive that the current
exchange rate policy to be unsustainable, speculative activity should cause the forward
218 A. Abraham International Review of Economics and Finance 8 1999 213–222
Table 1 Unit root test in the forward rate process
a
Currency Coeff. Value a
1
ADF T-Value Eurodollar
20.0035 20.2689
Saudi Riyal 20.0089
20.5564
a
This table provides the results of unit root tests using the Augmented Dickey Fuller ADF test. The model tested is:
D f
t
5 a 1 a
1
f
t
2
1
1 a
2
D f
t
2
1
1 a
3
D f
t
2
2
1 e
t
Where D is a difference operator, f
t
the forward interest rate at time t, e
t
is a white noise process, and a
, a
1
, a
2
, a
3
are parameters. Mackinnon Critical values are 23.5213, 22.9012, and 22.5875 at the 1, 5, and 10 respectively.
rate processes to drift apart. The Johansen 1988 maximum likelihood estimator is used to test for the presence of cointegrating vectors in the interest rate processes.
The Johansen methodology is essentially a multivariate extension of the univariate Dickey-Fuller test. Specifically, consider the following simple VAR representation
for the two implied forward rates.
f
t
5 Qf
t
2
1
1 e
t
where, f
t
, f
t
2
1
, and e
t
are 2 3 1 vectors and Q a 2 3 2 coefficient matrix. Rewriting in terms of the differences, we have
Df
t
2 Q 2 If
t
2
1
1 e
t
.
The rank of Q 2 I, measured by the number of non-zero characteristic roots, determines whether the two series are cointegrated. A rank of zero implies that the
two series are unit root processes with no cointegrating vector. If we cannot reject the null hypothesis that the rank is zero, we are unable to
attribute a stable relationship between the two time series, suggesting therefore the existence of speculative activity, stemming from a lack of credibility in the central
bank’s posture.
3.1. Data, estimation and results Monthly data on one month and three month gulf currency interest rates and
corresponding rates on eurodollar deposits were collected from the Money and Bank- ing Statistics, published quarterly by the Saudi Arabian Monetary Agency. Data
collected covered the period January 1988 to March 1994,
4
from which implied forward rates were computed.
5
As a first step in the analysis, I test if the forward rate processes follow a random walk, that is, if the differenced system is fully stationary and thus is consistent with
a martingale representation. Table 1 reports the results using the Augmented Dickey Fuller methodology, which tests the coefficient on the lagged value of the forward
rate series. The results reported are robust to different choices of the number of lagged difference terms used to induce residual white noise in the regression. Details
A. Abraham International Review of Economics and Finance 8 1999 213–222 219
Table 2 Johansen cointegration test for forward interest rates
a
5 1
Likelihood Critical
Critical Sample period
ratio value
value Jan 88–Mar 94
NH1 10.16
19.96 24.60
NH2 0.93
9.24 12.97
Jan 88–Dec 92 NH1
10.38 19.96
24.60 NH2
1.01 9.24
12.97 Jan 88–Dec 91
NH1 8.18
19.96 24.60
NH2 0.81
9.24 12.97
a
The table gives the results for the number of non zero characteristic roots rank of the coefficient matrix p
2
in the following VAR model in differences. Results presented are for a lag of period two.
Df
t
5 p
1
Df
t
2
1
1 p
2
f
t
2
2
1 e
t
where D is a difference operator.
Two null hypotheses are tested: NH1: The p
2
matrix has rank zero i.e., both series are stationary and that the series are not cointe- grated.
NH2: The p
2
matrix has at most rank 1 i.e., that the matrix is not full rank, implying that the series are individually stationary.
are therefore provided only for an AR2 representation. Critical values are taken from Mackinnon’s tables.
It is apparent from the results presented that the forward rates, as expected, are non-stationary, and appear to follow a random walk, a reasonable finding from the
efficient markets perspective and consistent with the consensus opinion in the litera- ture. This follows, as the existence of a unit root cannot be rejected even at the mild
10 significance level for either series.
In the second step of the analysis, I investigate the structure of the relationship between the two forward interest rate processes which comprises the principal contri-
bution of the paper. The results of the cointegration tests are provided in Table 2. Two nulls are examined. The first, labeled NH1, is that there is no cointegrating
equation for the two series, and the second, labeled NH2, that each series is individually stationary. As in the earlier case, the conclusions are invariant to the choice of the
number of lagged first differences used, suggesting the robustness of the results pro- vided. Since our objective is to determine the credibility of the central bank’s policy
under differing economic conditions, test results are provided for the entire sample period and for sub-periods.
It is evident from the results provided in Table 2 that we cannot reject the null hypothesis NH1 that each series is individually non-stationary and that there is no
linear combination of the variables which is stationary for any of the periods examined.
220 A. Abraham International Review of Economics and Finance 8 1999 213–222
Cast in terms of the underlying forward interest rates, one cannot therefore attribute an equilibrium cointegrating association between the two time series. This is at the
same time both a surprising result and yet one which strikes a familiar chord of intuition. It is intuitive, since as shown by Diebold, Gardeazabal, and Yilmaz, nominal
exchange rates across economies tend to be not cointegrated, and thus attempts to force parity through a fixed exchange rule must be reflected in the absence of cointegration in
nominal interest rates. On the other hand, this is a surprising result, since as in the present case the central bank has demonstrated its resolve and ability to defend the
stated exchange policy over a sustained period of time nine years—with the singular advantage of hindsight, speculative attacks on the fixed rule could be executed, de-
facto, only at a loss. With respect to the second null NH2, as expected one cannot reject the null that the individual series are non-stationary, confirming the results
from Table 1. Not surprisingly, these findings generally support the notion of market efficiency, to the extent that information on the relative strengths and weaknesses in
the currency exchange rate are revealed through the medium of interest rates, even though the exchange rate is fixed by the central bank as in this case.
4. Conclusions
