for co-movement tested for the existence of a unit root in the forecast error. If the forecast error is nonstationary, then this would imply market inefficiency. The third test was a new
application of the common feature test, as proposed by Engle and Kozicki 1993, and tested for co-movement between the rate of depreciation and the forward premium. If the
markets are efficiently determined, than a common feature should exist between the rate of depreciation and the forward premium.
This study has several distinct features. First, we performed a variety of tests to investigate exchange rate efficiency across countries and within countries. Across coun-
tries, we investigated co-movement between the spot exchange rate of different countries and between forward exchange rates of different countries. Within countries, we per-
formed three sets of tests to ascertain the existence of efficiency between forward exchange rates and the corresponding future spot rate for a single country. Second, we
investigated both the daily spot and forward exchange rates for a large number of countries, and for an extended time period. Third, we allowed for nonstationarity of the
data in our analysis. Fourth, we utilized a new methodology to test for the stationary co-movement between variables which did not have nonstationary co-movement, as
indicated by the cointegration tests. This test, called common serial correlation feature test, allowed us to extend our test of inefficiency across countries where exchange rates
may not reject efficiency due to the lack of a cointegrating vector. This test extended our test of efficiency within countries where a lack of a cointegrating vector would imply
inefficiency but the data may actually contain stationary co-movement. Common feature testing also extended our test of efficiency across countries by investigating co-movement
between the forward premium and the rate of depreciation.
The paper is organized as follows. Section II presents the methodology utilized. Section III provides the empirical results and interpretation. Section IV concludes.
II. Methodology
The existence of long-term relationships among the spot and forward exchange rates was tested using the Johansen 1988 and Johansen and Juselius 1990 methodology for
cointegration. The existence of a cointegrating relation would imply causal ordering in at least one direction [see Granger 1986, p. 218]. The bivariate pairings which did not
demonstrate a cointegrating relation were subjected to a more stringent test for co- movement called “common serial correlation feature tests” developed by Engle and
Kozicki 1993. The finding of a common serial correlation between variables would imply at least one-way causality [see Engle and Kozicki 1993, p. 373].
The use of cointegration tests are relatively common in the literature, and the reader is referred to Johansen 1988 and Johansen and Juselius 1990 for a complete discussion.
However, an introduction to cointegration and its application to exchange rate efficiency testing is included here.
Given that two series, Y
1,t
and Y
2,t
, are nonstationary in levels and stationary in first differences, it is relevant to investigate the existence of a long-run relation through
cointegration. Following Engle and Granger 1987, a linear combination of I1 variables, Y
1,t
5 a
1 b
1
Y
2,t
1 e
t
, 1
will also generally be I1. However, if e
t
is covariance stationary, then Y
1,t
and Y
2,t
are defined to be cointegrated of order 1, 1, and e
t
is the transitory equilibrium error. If Y
1,t
Exchange Rate Market Efficiency
425
and Y
2,t
are cointegrated, then no standard finite-dimensional autoregressive approxima- tion is feasible. The error correction term, z
t2 1
, needs to be included to guarantee that the two variables, Y
1,t
and Y
2,t
, do not drift too far apart. The error correction model would be: DY
1,t
5 a
z
t2 1
1 g
i
O
DY
2,t21
1 b
i
O
DY
1,t21
1 e
t
. 2
Using Cointegration to Test Exchange Rate Efficiency Across Countries
Market efficiency implies that the prices from two efficient markets for different assets cannot be cointegrated [Granger 1986]. The first test of market efficiency across
countries is to test whether various spot exchange rates are cointegrated. The spot exchange rates of two countries which are under a flexible exchange rate regime are two
separate asset prices. In equation 1, let Y
1,t
be the spot exchange rate of country A, and Y
2,t
be the spot exchange rate of country B. If the two spot exchange rates are cointegrated, then an error correction model exists which implies that part of the change in the spot rate
of country A is predictable. That is, the two spot exchange rates are inefficiently determined in the market. The second test for market efficiency across countries is similar
to that of the spot exchange rates, except it involves the forward exchange rates. That is, in equation 1, let Y
1,t
be the forward exchange rate of country A, and Y
2,t
be the forward exchange rate of country B.
Using Cointegration to Test Exchange Rate Efficiency Within Countries
A test for efficiency within a country focuses on a single currency, and investigates the relation between the forward exchange rate and a future spot rate. The forward exchange
rate should be an unbiased and efficient predictor of the spot exchange rate. Therefore, there should exist a long-run relation between a forward rate and its corresponding future
spot rate. An unbiased predictor is one that is correct on average. That is, over the long run, the forward exchange rate is just as likely to overpredict the future spot rate as it is
to underpredict. If the forward rate is found to be an unbiased predictor, then this necessitates that the risk premium is zero. The risk premium, if one exists, serves as an
insurance premium to induce risk-averse traders to participate. However, if a risk premium exists, then the current forward rate would consistently overpredict the future spot rate. In
equation 1, Y
1,t
is the current forward rate for country A, and Y
2,t
is the corresponding future spot rate for country A. Within the context of daily exchange rates and 30-day
forward rates, it is natural to think that the relevant future spot rate is 22 days 22 working days in a month in the future, as this is the time period over which the expectation
typically takes place.
1
If the variables are cointegrated, then an error correction model is correctly specified. The model would be misspecified and yield inconsistent estimates of
the parameters if we simply regressed the first difference of the future spot exchange rate on the first difference of the forward exchange rate, as this would not include the
equilibrating error from the long-term model. The forward rate and future spot rate cannot drift too far apart if the markets are efficient. If the variables are cointegrated, then this is
a necessary but not sufficient condition for market efficiency. In the cointegrating vector,
1
The data is weekday data and the average month has 22 working days. The precedence of the use of the 22 days to represent the relevant spot rate associated with the 30-day forward rate is given in Baillie and
Bollerslev 1989 and Hodrick 1987.
426
T. A. Rapp and S. C. Sharma
represented by equation 1, we must test the hypothesis that the intercept, a , equals 0 and
the slope, b
1
, equals 1. If this hypothesis is true, then market efficiency is supported.
Use of Common Feature Testing to Investigate Efficiency
Cointegration tests investigate long-term relationships by analyzing forms of co- movement of variables which are nonstationary. In order to investigate the forms of
co-movement that are stationary, common features can be analyzed. Common serial correlation is tested by using the test statistic developed by Engle and Kozicki 1993. If
two variables are not found to be cointegrated, then it would be relevant to test for common features to ascertain the existence of a long-run relation. There was a possible
total of four applications of common feature testing in this study. First, for analysis of efficiency across countries, there exists two possible applications. If there was no cointe-
grating vector found to exist between the spot forward exchange rates of the various countries, then it would be relevant to investigate common features. In the case of testing
efficiency across countries, the first differences of the logs of the spot forward exchange rates should not share common features if the spot forward exchange rates are being
determined efficiently in the market. The common feature tested for was serial correlation. The finding of a common serial correlation feature between two spot forward exchange
rates would imply at least one way causality. Therefore, the past information of one exchange rate could be used to predict another, and this violates the EMH for exchange
rates across countries.
Second, for analysis of efficiency within countries, there exists two possible applica- tions. The first application within countries is that the forward exchange rate and the
corresponding future spot rate should share some type of co-movement. This co- movement could be detected through the use of cointegration tests or common feature
tests. The lack of a cointegrating relation between the forward rate and future spot rate would not support market efficiency, due to lack of co-movement. However, we could
extend the test for co-movement, if necessary, by testing for a common feature between the two variables. The second application within countries is that the rate of depreciation
S
t1 22
2 S
t
should share a co-movement with the forward premium F
t
2 S
t
. This co-movement could be established through the use of common feature testing.
2
Engle and Kozicki 1993 stated that if a feature is present in each individual series, and if there exists a non-zero linear combination of the series which does not have the feature,
then the feature is common between the series. They developed some regression-based tests for common features. Consider the following regression model:
y
t
5 x
t
b 1 z
t
g 1 e
t
. 3
We can test the null hypothesis H : g 5 0 against the alternative H
1
: g Þ 0, where rejection of the null indicates the presence of a feature. Testing for serial correlation, we
could specify { z} to include lags of y, and { x} could be a constant.
2
The rate of depreciation is generally accepted to be stationary, and we later provide unit root tests to confirm this. The forward premium does not have to be stationary by definition; however, we provide unit root
tests to indicate that it is stationary. As both of the variables in question are stationary, then the appropriate methodology to investigate co-movement is common feature testing.
Exchange Rate Market Efficiency
427
Assuming weak exogeneity of { x, z}, and joint stationarity of { y, x, z}, we can then compute a Lagrange multiplier LM statistic, under the simplifying assumption that there
are no { x} in the model, which is given by: s~ y 5 y9z~ z9z
21
z9y s
y 2
5 T~ y9z~ z9z
21
z9y y9y,
4 where s
y 2
is a consistent estimate of the residual variance, and the statistic equals TR
2
from the regression of y against z.
Next consider two series, y
1t
and y
2t
, each being tested for the presence of a feature within their individual series using the following regression model:
y
1t
5 x
t
b
1
1 z
t
g
1
1 e
1,t
; 5
y
2t
5 x
t
b
2
1 z
t
g
2
1 e
2,t
, where the set of regression { z, x} is the same for both series. To test for a common feature,
we tested whether there is a d such that u
t
5 y
1t
2 dy
2t
does not have the feature. The parameter d was chosen to minimize d as follows:
s~u 5 min s~ y
1
2 dy
2
5 u9M
x
z~ z9M
x
z
21
z9M
x
u s
u9 2
, 6
where M
x
is the projection matrix, M
x
5 I 2 x x9x
21
x9 . An alternative estimator, given
by the LIML estimate of d in the following regression, was shown to have the same asymptotic properties:
y
1t
5 dy
2t
1 x
t
b 1 e
t
, 7
where the instruments are { x, z}. This resulted in the following statistic: S~u
n
5 u9M
x
z~ z9M
x
z
21
z9M
x
u
n
s
u 2
. 8
The statistic was computed, in its LM form, as TR
2
from the regression of the LIML residuals on { x, z}, and the number of degrees of freedom equaled the number of
over-identifying restrictions. The feature can be said to be common if the null that the linear combination of the two series fails to have the feature, even though each of the
series individually has it, cannot be rejected. Intuitively, we were testing whether the dependence of one of the variables with the past is only through the channels that relate
other variables to the past.
III. Empirical Results
