An Empirical Examination of Rational Expectations Hypothesis
in the Foreign Exchange Market
Fazlul Miah
Fayetteville State University
M. Kabir Hassan
University of New Orleans
Aminur Rahman
Independent University
Abstract
The Efficient Market (EMH) and Rational Expectation Hypothesis (REH) have
been examined for the Canadian Dollar / US Dollar and Swiss Frank / US Dollar
exchange rates using data from February 1988 to May 1999 published by the Financial
Times’ Currency Forecaster. Using unit root, and restricted co-integration techniques,
we find that one-month ahead forecast is rational for the Swiss Frank / US Dollar
exchange rate. However, one-month-ahead Canadian Dollar forecast is not rational.
The study shows that when forecasting errors are corrected for serial correlation
because of moving average process, three-month-ahead forecasts become rational for
both the currencies. The twelve-month-ahead forecasts are not rational for any of the
currencies.
Introduction
It has been documented in the empirical literature on foreign exchange that
forward discount is a biased predictor of actual exchange rate change. There are a large
number of studies that investigated the causes of this biasi. Fama (1984) argued that the
existence of a time-varying risk premium is the cause of the forward discount bias.
Another argument of expectational failure comes from Krasker (1980), who attributed
the failure to the “peso problem”ii. Lewis (1989) attributes it to the learning effect while
Bilson (1981) attributed it to simple irrationality.
The absence and presence of a risk premium has been of considerable interest to
economists and policymakers because of its far-reaching implications for the
substitutability of assets denominated in different currencies and, hence, for the efficacy
of sterilized foreign exchange market intervention. Dominguez and Frankel (1993)
showed that foreign exchange intervention had a significant impact on risk premium in
currency markets during the 1982-88 period. They also found that when rational
expectations are imposed, intervention became ineffective.
The use of survey data allows the direct measurement of a risk premium from
the observation of the forward discount (fdt)iii, which can be decomposed into the
expected currency depreciation and risk premium:
fdt+j = tf t+j -st
= (Et st+j -st) + rpt
where tft is the log of the forward exchange rate set in period t, Et is an expectations
operator based on the set of information available in period t, st+j and st are the log of the
spot exchange rate in period t+j and t respectively, and rpt is a risk premium.
Engel (1996) surveyed the literature that attempted to attribute the bias to a
foreign exchange risk premium. All these studies assume that agents’ forecasts are
rational. He concluded that the null hypothesis that the forward rate is a conditionally
unbiased predictor of the future spot rate is routinely rejected and that the models of the
risk premium have been unsuccessful at explaining the magnitude of this failure of
unbiasedness.
Using survey data on exchange rate expectations, a number of studies (for
example, Frankel and Froot (1987, 1989), Cavaglia, et al. (1993a, 1993b), Chinn and
Frankel (1994), tried to decompose the measure of the forward rate biasedness into a
component due to risk and a component due to irrationality, and a fair summary of the
literature suggests that both time-varying risk premia and irrationality are responsible for
the failureiv.
However, most of these studies do not investigate the time series properties of
the survey data and use traditional regression techniques, which are criticized for being
inappropriate. A good way to start an investigation of the causes of the forward discount
bias is to investigate whether agents’ forecasts satisfy the Rational Expectation
Hypothesis (REH). A number of studies such as, Dominguez (1986), MacDonald and
Torrance (1989), Liu and Maddala (1992a, 1992b), Miah, Hassan, Waheeduzzman and
Abual-Foul (2003) and Miah, Hassan and Alam (2003) investigated the REH using
survey data. The majority of these studies in this area found that survey data do not
satisfy REH, thus, implying that the failure is due to the existence of a risk premiumv.
In the context of the foreign exchange market, REH says that if economic agents
use all available information in forming expectation about future exchange rates, then,
the expected exchange rate will be an unbiased predictor of the future spot rate. This is
known as the test of unbiasedness in the literature. Consider the following equation:
S t = α + β Set + εt
where St is the spot rate, Set is the expected future rate and εt is a random error. If the
economic agent is rational, then, a regression on the above equation will provide α = 0
and β=1 so that S t = Set + εt. In other words, the difference between the actual spot rate
and the expected rate for that period will be a random error.
Another test of the REH is based on the forecast error Set -St, which is known as
the test of orthogonality. If expectations are rational, then the forecast errors will be
uncorrelated with the variables in the information set, I t-1, containing all data available
as of time t t-1. In other words, forecast error, Set -St = f (I t-1). The orthogonality test
aims to assess whether economic agents use information that is available to them
efficiently in order to forecast future exchange rates. Variables that are used for the test
of orthogonality include past exchange rates, the forward discount, the inflation rate,
money supply and changes in the interest rate, and others.
In this paper, we test the REH using survey data published by the Financial
Times’ Currency Forecaster. Financial Times’ Currency Forecaster is one of the few
major sources of survey data on exchange rates. Takagi (1991) provided a detailed
survey on the studies that used different data sources to test the REH.vi There are only a
few studies that explored the rationality of survey data using data from the Financial
Times’ Currency Forecaster. However, these studies used the traditional regression
analysis and the time period considered for the test is very short.vii In this paper, we will
extend those previous studies using twelve years of monthly data and by applying the
more recent econometric methods recommended for time series data analysis. The
results from this research will provide a better perspective on the survey data and thus,
the forward discount bias.
2
We use the old as well the recent unit root tests and the restricted co-integration
test. These methods have not yet exploited by researchers using this type of data set.
This also shows an econometric example of whether the results provided by different
methods are similar, or different. The results obtained from this study can be utilized by
the practitioners to see the accuracy of exchange rate movements in the future.
This paper is divided into four sections. Section II discusses the estimation
techniques and Section III presents the analysis of the results. Section IV concludes the
paper.
Estimation Techniques
Two of the most widely used equations of the unbiasedness test are as follows:
S
where
s
t +k
e
t ,t + k
=α + β
S
e
t ,t + k
+ ε t +k
(1)
is expectation at time t of the exchange rate for time t+k,
realized exchange rate at time t+k and
S
t +k
− S t = α + β ⎛⎜
⎝
S
e
t ,t + k
ε
s
t +k
is the actual
is a random error.
t +k
− S t ⎞⎟ + ε t + k
⎠
Where st + k − st is actual depreciation and
(2)
st t + k − st is expected depreciation.
e
,
The
null hypothesis in this equation is that α = 0 and β = 1.
Recent studies on time series econometrics show that conventional regression
analysis may not be valid with non-stationary data. Nelson and Plosser (1982) showed
that most macroeconomic variables, including exchange rate data, are non-stationary.
This finding has an important implication on econometric analysis. The standard
asymptotic distribution theory often does not work with regressions when data is nonstationary, thus making inferences difficult to evaluate.
Besides being corrupted with the econometrics problems, the time horizon
considered by different studies is also very short which give rise to another problem,
known as the “peso problem”viii. The REH should be tested using a longer time period.
A longer time period provides ample opportunities to economic agents to learn about an
evolving environment, respond to changes in important policy variables, and correct for
their earlier mistakes and hence, eliminate the bias created by the peso problem. Thus,
Liu and Maddala (LM) (1992a, 1992b) and Osterberg (2000) argue that we should use
the co-integration technique to test for rationality in the exchange market.
Unit Root Tests
In this study, we will use two unit root tests. These are the Augmented DickyFuller (ADF) test, which is the most commonly used in the literature, and the other one
is the most recent one, the DF-GLS test.
The Augmented Dickey-Fuller (ADF) regression equation is as follows:
p
∆yt = α + δt + ρ yt −1 + ∑ β j ∆yt − j + ε t
(3)
j =1
Fuller (1976) and Dickey and Fuller (1981) tabulated the critical values for the
test statistics. There are two versions of this test: using a constant only or using a
constant and a trend. We will report results for both types of estimation.
3
The ADF test is widely criticized for size distortion and poor power problem.
Elliot, et al. (1996) suggested a modification of the ADF test, which is known as DFGLS test. The DF-GLS equation is as follows:
∆ y =δ y
d
0
t
p
d
t −1
+ ∑ δ i ∆y
i =1
d
t −i
+ε t
(4)
Equation 4 looks very similar to equation 3, the original ADF model. The only
difference is that this model uses a de-trended series instead of the original series.
Restricted Co-integration technique
The co-integration technique that LM (1992a, 1993b) and Osterberg (2000)
applied in their studies is known as the Restricted Co-integration Test. This is rather a
direct approach. It is also possible to write a formal model as suggested by Granger
(1981). In this paper, we will follow their approach. The approach has been discussed
bellow.
The co-integrating regression equation takes the following form:
S t + k = α + β S t t + k + ε t + k , where st + k
e
,
s
e
t +k
is the actual realized spot rate at time t+k,
is the predicted rate for time t+k and ε t + k is a random error.
Note that if the actual rate s t + k follows a random walk, so should its rational
e
forecast s t + k , which means that the two time series should be co-integrated with a factor
of one and random residuals. In that case α=0 and β=1. Then the equation will look
like the following:
S t +k = S t t +k + ε t +k
e
,
Thus, we can simply take the difference between the actual and the expected
e
series, s t + k − s t + k and test for its stationary nature. If the residual series ε t + k is stationary,
then, the two series are co-integrated with factor 1. This is called the restricted cointegration test since we have restricted α=0 and β=1 in the above equation.
For the expected rates to be rational, we also require that the residual is a white
noise process. We will use Q-statistics to test for the serial correlation in the residual
series. If the residual series does not show any sign of serial correlation, we can say that
the expected exchange rate series satisfies the rational expectation hypothesis. Osterberg
(2000) argues that since the forecast periods are finer than the forecast horizon, the
residual series will follow a moving average process. So a Q-test will show serial
correlation even if the forecasts are rational. He applies a MA (3) estimation on the
residuals from the monthly series (the forecast periods were weekly) and analyzes the
residuals from the estimation and found that the residuals are white noise. Thus, he
concluded that the monthly forecasts are rational. Again, the actual residuals of the
three, six and the twelve-month-ahead forecast should be estimated as an MA (2), MA
(5) and MA (11) process respectively, as the forecasts were made each month. Thus, the
application of Q-test on the residuals from estimating these equations will be
meaningful. This technique has been used to the actual residual series, which are
stationary (co-integrated).
4
Analysis of Results
We analyze data collected from the Financial Times’ Currency Forecaster for
the Canadian Dollar/ US Dollar and Swiss Frank / US Dollar exchange rates from
February 1988 through May 1999. The survey horizons were one-month, three-month,
six-month and twelve-month.
Table 1 reports the results of the unit root test performed on the actual as well as
on the forecasted series at different horizons and Figure 1 provides the visual picture of
all the series: actual as well as expected rates. Looking at the series, it is clear that there
are trends in the data.
Using the DF-GLS test, we cannot reject the null of a unit root even at the 10%
level for all the series. However, using the ADF test, we cannot reject the null at the 5%
level for the actual spot rate, one-month and three-month-ahead forecasts and at the 1%
level for the six-month and twelve-month-ahead forecasts. So, in general, we can
conclude that the actual and the expected future rates are non-stationary.
For the Swiss Frank / US Dollar, both versions of the ADF and the DF-GLS
tests cannot reject the null at the 10% level for the actual and the one-month-ahead
expected exchange rates. Both versions of the DF-GLS test cannot reject the null at the
5% level for the three-month, six-month and twelve-month-ahead forecasts. The same is
true for the ADF test except for the three-month-ahead forecast. The ADFb cannot reject
at the 5% level, but the ADFa cannot reject at the 1% level. Thus, there is clear evidence
based on both the tests that the Swiss Frank / Dollar data are non-stationary.
Table 2 provides the results of the co-integration tests, which are the unit root
tests on the forecast errors at different horizons for the Canadian Dollar / US Dollar
exchange rates. The null of no co-integration (i.e., null of a unit root) was rejected for
the one-month and three-month-ahead error series at the 1% level by both versions of
the ADF test and at the 1% level by the DF-GLSd test. However, both versions of the
ADF and the DF-GLS tests accept the null of no co-integration for the six-month and
twelve-month-ahead forecast errors.
Table 2 also presents the results of the co-integration test on the Swiss Frank /
US Dollar exchange rates forecast errors. The results for the one-month and threemonth-ahead forecast errors are highly significant. Both versions of the tests reject the
null at the 1% level. The results for the six and twelve-month-ahead forecast errors are
mixed. Both versions of the ADF test reject the null for the six-month-ahead forecast
error at the 1% level, but both versions of the DF-GLS test accept the null even at the
10% level. The ADFa rejects the null at the 10% level, but the ADFb and both versions
of the DF-GLS tests accept the null for the twelve-month-ahead forecast error.
The results of the Q-test are reported in Table 3. The lag lengths are chosen to
be 4, 8, 12, 24 and 36. The numbers in parentheses refer to the probabilities associated
with Q-statistics. We also estimated the residuals from the three-month-ahead forecasts
as an MA (2) process. Since the frequency of the data is monthly, a monthly series on
the three-month-ahead forecast error will display serial correlation even if the forecasts
are rational. The Q-test shows that there is serial correlation with the Canadian onemonth-ahead forecast error. However, one-month-ahead Swiss Frank forecast error does
not show serial correlation at the 1% level. MA estimation of the three-month-ahead
forecast errors for both the currencies cannot reject the null hypothesis of no serial
correlation. Osterberg (2000) estimated forecast errors from a monthly series, which was
forecasted weekly as an MA (3) process and found no evidence of serial correlation.
The three-month-ahead forecast error series would have a MA (2) moving average
process.
Table 3 does not report the MA estimation of the twelve-month-ahead forecast
errors since we found evidence that the twelve-month-ahead forecasts are not cointegrated. The table does show MA estimation of the six month forecast for comparison
purpose only, although the forecast error is not stationary.
5
Summary and Conclusion
The Rational Expectation Hypothesis (REH) is upheld for the three-monthsahead expectations for the Canadian Dollar/ US Dollar exchange rate and is rejected for
the one, six and twelve-month-ahead forecasts. The hypothesis is also upheld both at the
one and three-month-ahead expectations for the Swiss Frank / US Dollar exchange rate,
and the hypothesis is rejected for the six and twelve-moth-ahead forecasts. We have also
found that both the unit root tests provided fairly consistent results.
Our results support the earlier findings that the dispersion of expectations tends
to increase with the forecast horizon. However, this study does not necessarily confirm
other implications of the REH, such as the orthogonality of forecast errors with respect
to the information available at the time of the forecast. It seems that rationality is a
country specific phenomenon. At present the authors are investigating whether
application of different empirical methods and use of different time periods have any
implication on the current result.
Notes
I
See, for example Hodrick (1987), Engel (1996) for a review of literature on this issue.
The term was coined by a researcher who took as his classic case the behavior of the
Mexican Peso which, although notionally on a fixed exchange rate, traded consistently
at a forward discount to the US dollar in the mid 1970s, in anticipation of a devaluation
that duly materialized in 1976. So a test of REH for this time period will show that the
expectations are biased. In this situation, the standard assumption of normality of the
distribution of the test statistic will not be appropriate. This problem may arise in other
situations as well. For example, when there is a small probability of a large change in
the exchange rate each period, a bursting of speculative bubble, or a big change in
fundamentals and when the sample size is not large enough to invoke the central limit
theorem with confidence.
III
The forward discount (premium) is the proportion by which a country’s forward
exchange rate falls below (exceeds) its spot rate. For example, assume that the 12month-ahead forward rate is 1US. Dollar = 0.60 Pound Sterling. If the spot rate at
present is 1 US. Dollar = 0.50 Pound Sterling, it follows that there is a forward premium
of 20% on dollars (and a forward discount on sterling), because dollars to be delivered in
twelve months cost 20% more than dollars to be delivered on the spot. In this paper, the
phrases forward discount or forward premium will be used interchangeably.
IV
See Frankel and Froot (1989) for a formal derivation of this decomposition of the
slope coefficient in a regression of the exchange rate change on the forward discount.
V
A recent survey of literature in this area is given in MacDonald (2000) and Osterberg
(2000).
VI
Takagi (1991) mentioned five major sources of survey data. They are American
Express Banking Corporation , London; Economist Financial Report, London; Money
Market Services, New York and London; Godwins, London and Japan Center for
International Finance, Tokyo.
VII
According to our knowledge, there are only two studies that test the rationality of
survey data using Financial Times’ Currency Forecaster. They are Chinn and Frankel
(1994) and Sobiechowski (1994) studies. The time period considered for the test of
REH in these studies do not exceed six years.
VIII
The peso problem arises when there is a small probability of a large change in the
exchange rate for each period, a bursting of a speculative bubble, or a big change in
fundamentals and when the sample size is not large enough to invoke the central limit
theorem with confidence. The peso problem invalidates the standard assumption of
normality of the distribution of the test statistic in a test.
II
6
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the Foreign Exchange Markets Empirical Economics 17, pp.303 – 314.
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7
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8
Figure 1: Future Spot and Expected Future Rates for Can. Dollar / US. Dollar and Swiss Frank / US
Dollar respectively
1.6
1.5
1.4
1.3
1.2
1.1
89
90
91
92
93
94
95
ST
S6
S3
96
97
98
99
97
98
99
S12
S1
1.8
1.6
1.4
1.2
1.0
89
90
91
92
93
94
S1
S12
S3
95
96
S6
ST
Note: St, S1, S3, S6 and S12 are actual, one-month, three-month, six-month and twelve-month-ahead
forecasts respectively.
9
Figure 2: The Residual Series for the Canadian Dollar / US Dollar and Swiss Frank / US Dollar
Exchange Rates respectively
0.1
0.0
-0.1
-0.2
-0.3
89
90
91
92
93
94
95
U1
U12
96
97
98
99
U3
U6
0.6
0.4
0.2
0.0
-0.2
-0.4
89
90
91
92
93
94
U1
U12
95
96
97
98
99
U3
U6
Note: U1, U3, U6 and U12 are one-month, three-month, six-month and twelve-month-ahead forecast errors
respectively.
10
Table 1: Unit Root Tests on Future (St) and Expected Future (Se t, t+i) Canadian Dollar / US Dollar
Exchange Rates
Statistics
CD$ / US$
ADFa
ADFb
DF-GLSc
DF-GLSd
SF / US$
ADFa
ADFb
DF-GLSc
DF-GLSd
Note:
Actual Spot
Rate
-0.1866
-3.0580
-0.0633
-1.3525
Survey Forecasts
One Month
Ahead
-0.3609
-3.1739
-0.2537
-1.5032
Three Month
Ahead
-0.5177
-3.3066
-0.4347
-1.6389
Six Month
Ahead
-0.5881
-3.5376
-0.5545
-1.7911
Twelve Month
Ahead
-0.7099
-3.7694
-0.7763
-1.6799
-2.2104
-2.3038
-2.2030
-2.2399
-2.3174
-2.4072
-2.2981
-2.3412
-3.0232
-3.1622
-2.7772
-2.8902
-2.9368
-3.1406
-2.6814
-2.8658
-2.8601
-3.3624
-2.5578
-2.7432
Lag lengths are chosen based on Baysian Information Criterion (BIC)
a. ADF test is performed with a constant only.
b. ADF test is performed with a constant and a trend.
c. DF-GLS test is performed on the series, which are de-trended using a constant only as described
in the paper.
d. DF-GLS test is performed on the series, which are de-trended using a constant and a trend as
described in the paper.
Table 2: Restricted Co-integration Tests on Future (St) and Expected Future (Se t, t+i) Canadian Dollar
/ Dollar Exchange Rates: Unit Root Tests on Forecast Errors (Se t, t+i - St)
Statistics
CD$ / US$
ADFa
ADFb
DF-GLSc
DF-GLSd
ADFa
ADFb
DF-GLSc
DF-GLSd
One Month
Ahead
-12.4255
-12.7575
-1.2343
-10.8283
-9.6519
-9.6506
-9.5991
-9.6342
Survey Forecast Errors
Three Month
Six Month
Ahead
Ahead
-5.4993
-1.9097
-6.1614
-2.0403
-3.582
-0.7483
-5.69
-1.9465
-6.8333
-4.1978
-6.8129
-4.0988
-5.4918
-2.0971
-6.4967
-2.7093
Note: Legends are the same as in Table 1.
11
Twelve Month
Ahead
-2.4663
-3.1223
-1.0151
-2.8964
-2.8515
-2.8882
-2.0006
-2.3853
Table 3: Test of Serial Correlation on the Canadian Dollar / Dollar Exchange Rate Using Q-Statistics
CD$ / US$
Series
1 month ahead forecast error
3 month ahead forecast error
MA(2) estimation
6 month ahead forecast error
MA(5) estimation
12 month ahead forecast error
SF / US$
1 month ahead forecast error
3 month ahead forecast error
MA(2) estimation
6 month ahead forecast error
MA(5) estimation
12 month ahead forecast error
Note:
Q(4)
2.7667
(0.598)
87.866
(0.000)
2.4237
(0.298)
224.11
(0.000)
24.843
348.86
(0.000)
9.9306
(0.042)
92.267
(0.000)
8.3309
(0.016)
155.98
(0.000)
10.880
243.95
(0.000)
Q(8)
17.842
(0.022)
110.11
(0.000)
12.305
(0.055)
298.66
(0.000)
39.673
(0.000)
557.49
(0.000)
Q(12)
23.027
(0.028)
134.26
(0.000)
17.647
(0.061)
357.11
(0.000)
54.56
(0.000)
648.54
(0.000)
Q(24)
66.254
(0.000)
159.59
(0.000)
27.160
(0.205)
375.03
(0.000)
66.254
(0.000)
691.86
(0.000)
Q(36)
72.693
(0.000)
176.78
(0.000)
39.049
(0.253)
383.39
(0.000)
72.693
(0.000)
711.77
(0.000)
12.679
(0.123)
99.978
(0.000)
12.549
(0.051)
157.25
(0.000)
13.248
(0.004)
282.11
(0.000)
15.84
(0.199)
107.29
(0.000)
15.157
(0.126)
164.85
(0.000)
15.963
(0.25)
301.39
(0.000)
34.476
(0.077)
133.93
(0.000)
27.414
(0.196)
226.8
(0.000)
43.119
(0.001)
364.47
(0.000)
45.039
(0.144)
151.81
(0.000)
37.704
(0.304)
237.24
(0.000)
54.325
(0.006)
390.2
(0.000)
All Q(k) statistics are distributed as chi-square (k)
The Numbers in parentheses are the associated probabilities
The null of the Q-test is no serial correlation
MA(5) estimation for the six-month-ahead forecast is provided for comparison purpose only
although the series is not stationary.
12
13