THE DYNAMICS OF NOMINAL EXCHANGE RATES
BY
PETER CHIWAH LIU
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1989
U I.
This dissertation is dedicated to my wife,
Waihan Yu,
whose love and encouragement are my companions, always.
ACKNOWLEDGEMENTS
The completion of this dissertation would not have been
possible without the help of many people. First, I would
like to express my most sincere gratitude to my supervisor,
Dr. G. S. Maddala, for his great intuition in econometrics,
patient guidance, critical comments and long hours spent in
reading the many drafts all the way through to completion.
Second, I am deeply indebted to Dr. Anindya Banerjee for his
helpful suggestions and encouragement. Third, I would like
to thank Dr. Mark Rush for his kindness and continuous
support during my studies at the University of Florida.
TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS ........................................ 11iii
ABSTRACT ................................................ vi
CHAPTERS
I INTRODUCTION ...................................... 1
General Background.................................. 1
Purpose of the Study................................ 3
Data Description..................................... 7
The Survey Data................................. 7
The Spot and Forward Exchange Rates............ 8
The Data for Monte Carlo Experiments............ 8
Layout of the Dissertation ........................ 9
II TESTS FOR RATIONAL EXPECTATIONS: THEORY............ 11
Introduction ...................................... 11
Traditional Rational Expectations Tests............ 12
Weak and Strong Rational Expectations Tests... 13
Variance Bounds Tests for Rationality.......... 15
Weaknesses Associated with These Rationality
Tests ....................................... 16
Summary of the Theory of Cointegration............ 19
Tests for Cointegration .......................... 20
Relationship Between Cointegration and Rational
Expectations .................................... 24
III THE INFORMATION MATRIX TEST FOR RATIONAL
EXPECTATIONS .................................... 27
Introduction ...................................... 27
The Information Matrix Test........................ 30
The Information Matrix Test and Rational
Expectations .................................... 35
Monte Carlo Experiments on the Information
Matrix Test Statistic ............................ 38
Empirical Results of the Experiment............ 41
Interpretation of the Empirical Results........ 43
Summary ........................................... 47
iv
IV TESTS FOR RATIONAL EXPECTATIONS: EMPIRICAL
RESULTS ......................................... 49
Introduction ...................................... 49
Debate on the Dynamics of Exchange Rates........... 50
Debate on the Rational Expectations Hypothesis.... 54
Empirical Results................................... 57
Summary ........................................... 66
V TESTS FOR THE MARKET EFFICIENCY HYPOTHESIS: THEORY
AND EMPIRICAL RESULTS ........................... 68
Introduction ...................................... 68
The Market Efficiency Hypothesis .................. 72
Covered Interest Parity ........................... 75
Empirical Results................................... 76
Summary ........................................... 86
VI SUMMARY AND CONCLUSION ............................ 87
BIBLIOGRAPHY ............................................ 91
BIOGRAPHICAL SKETCH ..................................... 95
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THE DYNAMICS OF NOMINAL EXCHANGE RATES
By
Peter Chiwah Liu
December 1989
Chairman: Dr. G. S. Maddala
Major Department: Economics
This study describes the dynamic nature of nominal
exchange rates, with special emphasis on their random walk
structure. Although a lot of research favors the random
walk model, economists are quite reluctant to accept it. To
provide support for the random walk proposition, we apply
unit root tests to four different currencies. Results of
these tests confirm the existence of unit roots in the spot
exchange rate, its expected value as well as the forward
exchange rate. Our results show that we cannot reject the
nonstationarity property of all these variables in the
foreign exchange market.
Rational expectations and market efficiency hypotheses
are also tested for the exchange rate market. Since the
variables are shown to be nonstationary, the theory of co
integration is introduced. This theory provides a different
way to interpret the two hypotheses. In brief, it outlines
a long run relationship between the variables being studied.
By restricting the cointegrating factor to be unity, co
integration tests are applied to these hypotheses. The
results show that the rational expectation hypothesis is
accepted while the market efficiency hypothesis is rejected.
Failure of the latter hypothesis is induced by the existence
of risk premium.
An information matrix (IM) statistic is introduced to
test for rational expectations. The test is applied to a
subset of the parameter space to bypass problems raised by
nonstationarity. The procedure is theoretically valid,
though the Monte Carlo experiments are not encouraging.
However, the results lead to important conclusions.
The IM test is a very general specification test. It
is a portmanteau test for almost every sort of
misspecification. As such it will have less power than
tests designed for specific departures from the null
hypothesis. Our results indicate that in the case of
testing for unit roots, even specific tests like the Dickey
Fuller test have very low power. Thus it is not surprising
that the IM test has still less power. Our results suggest
that in testing for unit roots, general specification tests
are not useful.
CHAPTER I
INTRODUCTION
General Background
Modelling with the incorporation of expectations has a
long record in economic theory, both in the micro and macro
economic areas. Early models of expectations, in general,
postulated a 'model' outside the economic theory. The
common practice in such models, because it simplified the
analysis, was to use extrapolative or adaptive expectations.
The former proposed the use of the first lag of the variable
as the future prediction, while the latter was based on the
idea of 'error correction through learning'. Unfortunately,
both of them were built on ad hoc assumptions, casting
serious doubts on their validity in practice.
The concept of rational expectations, first introduced
by Muth (1961) suggested that theories of expectation
formation should be consistent with the economic model being
considered. Loosely speaking, an expectation is considered
to be rational if the agent, in forming it, makes use of all
the available information. This implies, in particular,
that the forecast error cannot be reduced any further.
Surprisingly, a number of papers have found that the
Muthian rational expectations hypothesis breaks down under
empirical testing. Indeed, more evidence has emerged
recently with the availability of survey data. Unlike the
forecasts from the expectations models, survey forecasts are
generated directly by the market participants. Evidence
that the survey data reject the Muthian rationality
hypothesis has been presented by Pesando (1975) and
Mullineaux (1978) for the Livingston price expectations
data, Frankel and Froot (1986, 1987) and Hsieh (1984) for
the foreign exchange expectations data, Friedman (1980) and
Froot (1989) for the interest rate expectations data, and so
on. They all share the conclusion that survey expectations
are irrational. In other words, these researchers find that
the survey data fail the rational expectations test designed
according to their interpretation of the Muthian hypothesis.
These rational expectations tests, classified under the
title 'weak', 'strong' and 'variance' test, will be
discussed in detail in the next Chapter.
However, some of these 'proofs of irrationality'
evidence is not strong enough to reject the hypothesis of
rationality. This is true, in particular, when the variable
under study is not stationary1. The idea that most economic
'in the present and future context, nonstationarity
and random walk process are considered equivalent. This is
somewhat restrictive. However, it will simplify our work
without affecting the main results.
variables are nonstationary came from the work of Nelson
and Plosser (1982) who pointed out that unit roots are
common in economic time series data. Dickey and Fuller
(1981) as well as Nelson and Kang (1981, 1984) showed that
conventional test statistics were inadequate in the presence
of unit roots. Mankiw and Shapiro (1986) suggested that in
the presence of unit roots, there was an overrejection of
the rational expectations hypothesis. All this work
demonstrates that traditional rational expectations tests
may not be applicable without checking in advance the
dynamics of the variables and so the conclusions drawn are
questionable. In fact, it can be shown that much of the
evidence has to be reinterpreted if the nonstationarity of
the variables is taken into account.
Purpose of the Study
Because of the inconclusive nature of the debate and
the failure to pay attention to nonstationarity, the
present dissertation examines the rational expectations
hypothesis in detail, with special emphasis on the dynamics
of the variables concerned. As it involves nonstationary
series, the theory of cointegration will also be included.
This concept is important because it describes the
relationship between two nonstationary series, providing
insight into the structure of the rational expectations
hypothesis. It will be shown that the rational expectations
hypothesis is equivalent to requiring that the variable and
its expectations be cointegrated with an integrating
parameter one. In addition, the residuals from these two
variables must be a white noise process. This means that
tests for the existence of cointegration between variables
need to be applied. Engle and Granger (1987) suggested
seven test statistics for this purpose. They are: the
Durbin and Watson (DW) test proposed by Sargan and Bhargava
(1983); the Dickey and Fuller (DF) test and Augmented Dickey
and Fuller (ADF) test suggested by Dickey and Fuller (1979,
1981); and the restricted and unrestricted vector
autoregression (RVAR and UVAR) tests and their augmented
counterparts. Most of these tests will be employed in this
dissertation to check the validity of the rational
expectations hypothesis. However, they will be applied only
after suitable modification, for it is not cointegration
but the hypothesis of rational expectations that we are
interested in testing. In later chapters, these modified
tests will be called restricted cointegrated tests. The
theory will be applied to the foreign exchange market. This
data set is chosen not only because of its popularity, so a
lot of research is available for comparison, but also
because of the easy availability of the data. Details about
the data used in the analysis will be discussed in the next
section.
In addition to all the tests mentioned above, a
different statistic based on the Fisher information matrix
will be introduced. The statistic was first developed by
White (1982) to test for model misspecification. By
applying the test to a subset of the parameter space, using
the Lagrangian Multiplier principle, we find that the
statistic can be used to test the rational expectations
hypothesis. Moreover, it also avoids the problem associated
with the nonstationarity of variables. However, the
information matrix statistic will only converge to a chi
square distribution asymptotically. This implies that the
statistic works only when there is a sufficiently large
number of observations. A series of Monte Carlo experiments
with one hundred observations each will be employed to check
the power of this statistic. Other test statistics, such as
the DW, DF and ADF, will also be included in the experiments
to serve as a reference. As it is believed that the nominal
exchange rate is not stationary, the experiments concentrate
on the ability to distinguish between a nonstationary
series and a stationary one. This means that a random walk
process will be taken as the null hypothesis of the
experiments, while the alternatives are first order
autoregressive series with parameters close to one.
Besides the rational expectations hypothesis, the above
tests can also be applied to test the market efficiency
hypothesis in the foreign exchange market. Most of the past
research concludes that the foreign exchange market is
inefficient and the forward exchange rate is consequently
not a good predictor of the future spot rate.
Unfortunately, the exact cause of failure for the hypothesis
has not yet been determined. Possible explanations include
the violation of rational expectations or risk aversion of
the agents. In most cases, the failure is attributed to the
existence of a risk premium, rather than the irrationality
of expectations. For example, both Hodrick and Srivastava
(1986) and Park (1984) reach this conclusion. Others,
however, take the opposite view: Frankel and Froot (1987)
and Froot and Frankel (1989) are examples. Since the issue
of rational expectations is addressed, it is natural to
extend the work to include the market efficiency hypothesis.
In fact, one major reason why there is no definite answer to
the existence of an inefficient market is that the debate on
the rational expectations hypothesis has never been settled.
Basically, the two hypotheses are very closely related and
it is difficult to separate them. For this reason, the
present dissertation will also study the efficiency of the
foreign exchange market. Once again, the restricted co
integration tests and the information matrix statistic will
be employed. If nominal exchange rates are nonstationary,
market efficiency requires the forward rate to be co
integrated with the spot rate with a factor one, and that
the residuals of the two variables be a white noise process.
Details of this will be elaborated in later chapters.
To summarize, the purpose of the present study is to
understand the dynamics of the nominal exchange rate, as
well as to study two important hypotheses in the foreign
exchange market. They are important for a proper
understanding of foreign exchange market behavior,
especially for the participants in the foreign exchange
market.
Data Description
Three different kinds of data will be employed in the
present study: (i) the survey data on future exchange rates,
which are a proxy for the market expectations (ii) the
actual exchange rates, both in spot and forward markets,
and, (iii) the data used in the Monte Carlo experiments.
The Survey Data
The data are supplied by the Money Market Services
(hereafter MMS) who collect survey data on four different
currencies: the British pound (), Deutsche mark (DM), Swiss
franc (SF), and Japanese yen (Y), all denominated in US
dollars per unit of the respective currency. The survey
provides on a weekly basis oneweek and onemonth (30day)
ahead expectations of the value in dollars of these
currencies. Participants of the survey include exchange
rate dealers, banking and corporate economists, as well as
market economists. The data period is October 24, 1984 to
May 19, 1989. We use the data to test the rationality of
expectations and the efficiency of the exchange market.
Because similar research has been conducted using monthly
data, our emphasis will focus on the weekly survey. This
provides us with the opportunity to examine the property of
exchange rates in the very short run.
The Spot and Forward Exchange Rates
Spot and forward rates for each currency are needed
when either the rational expectations or the efficient
market hypothesis is tested. Both these rates can be
obtained from the Wall Street Journal. Current spot and
oneweek from today exchange rates are used. These
correspond to the date of the survey within the time range
mentioned. However, the forward rate on a weekly basis is
not available in the market. To obtain an equivalent series
of the oneweek forward exchange rate, as suggested by Hsieh
(1984), the idea of covered interest parity will be used.
The formula also uses the sevenday interest rates for the
currencies concerned, which are published in the Financial
Times. We will elaborate upon this procedure in a later
chapter.
The Data for Monte Carlo Experiments
These data are constructed by the random generator in
the Statistical Analysis System (SAS). The data are drawn
from a standard normal distribution. Without loss of
generality, the variance of the generated series is assumed
to be one. Based on these random series, several 'close to
random walk' autoregressive series with parameters equal to
0.99, 0.95 and 0.90 are created. In order to maintain the
same random property for the initial value, the data series
will be created as follows: First, by setting the initial
value equal to zero, a series of one hundred and fifty
observations is created. Second, after defining all the
lagged variables, the first fifty observations will be
deleted. This procedure has two advantages. First,
problems associated with the randomness of the initial value
are avoided. Second, there will be no loss of degrees of
freedom when the lagged variables are created. The Dickey
and Fuller (DF) test, Augmented Dickey and Fuller (ADF)
test, Durbin and Watson (DW) test, Box and Pierce (Q) test,
and the White Information Matrix (WIM) test will then be
applied to this final series. All these statistics will be
analyzed in full detail prior to the results of the
experiments.
Layout of the Dissertation
Several important issues will be discussed in the next
chapter. First, traditional tests for the rational
expectations hypothesis will be presented to provide a clear
picture of the development of this topic. Second, reasons
for their inapplicability if the variable follows a random
10
walk process will be explored. A description of the theory
of cointegration will follow immediately, acting as a
background material for the set of rational expectations
tests. These tests are based on the cointegration tests
suggested by Engle and Granger (1987). Finally, the concept
of cointegration, after a suitable modification, will be
applied to test for the rational expectations hypothesis.
The information matrix test will be discussed in Chapter
III. The chapter discusses the basic theory associated with
the White test, the procedure for calculating the statistic,
the Monte Carlo experiments and its results. A review of
the literature about the dynamics of exchange rates and the
rational expectations hypothesis will start Chapter IV. As
we claim that the exchange rates are nonstationary,
evidence to support such a proposition is necessary before
we proceed. For this reason, unit root tests will be
applied to the exchange rate series. The empirical results
of the tests will be listed. The market efficiency
hypothesis and the debate surrounding it will be presented
in Chapter V. In the same chapter we shall summarize
results of previous studies of the same issues and describe
the results of the restricted cointegration tests. The
final chapter concludes the thesis.
CHAPTER II
TESTS FOR RATIONAL EXPECTATIONS: THEORY
Introduction
The Muthian rational expectations hypothesis states
that the agent should make use of all available information
in forming expectations. Since there is no specific test
method associated with the hypothesis, researchers have
tried to test its validity from various aspects, according
to their interpretations of the hypothesis. In the next
section, the important tests of rationality will be
discussed. In case the variable under study is stationary,
these tests will have the correct statistical properties.
Unfortunately, most of the previous research did not take
account of the dynamics of variables, raising serious doubts
on their conclusion. It has been shown by Phillips (1986,
1987) that when the variable under study is nonstationary,
the estimators and statistics generated by OLS do not have
the distributions usually assumed. As most of the economic
time series data are believed to have a unit root, the
reliability of these traditional rationality tests becomes
questionable. Reasons why these tests fail if the variable
is nonstationary will also be discussed in the next
section. A set of tests of the rational expectations
hypothesis, applicable to nonstationary series, will also
be discussed.
In order to handle the dynamics of variables properly,
when testing the rational expectations hypothesis, the
theory of cointegration will be discussed. The theory is
used to investigate the long run relationship among non
stationary variables. In the present context, these refer
to the relationship between the spot and the forward rate or
the spot rate and its expected value. In order to
understand how cointegration theory fits into our context,
a special section will be devoted to explaining this
concept. This section will include the cointegration tests
suggested by Engle and Granger (1987). Finally we discuss
the modifications needed when the concept of cointegration
is applied to the test for the rational expectations
hypothesis. The full application will appear in Chapter IV.
Traditional Rational Expectations Tests
A general interpretation of 'exhausting all available
information in forming expectations' is that there should be
no systematic pattern in the forecast error. The forecast
error defined as the difference between the variable yt and
its expected values y7. More precisely, if the expectation
is rational, the forecast error should be a white noise
process. Based on this, several tests for rational
expectations were developed.
Weak and Strong Rational Expectations Tests
It is customary to start with a test of unbiasedness by
estimating the regression equation,
Yt= PO0 + P3yY + t
and testing the hypotheses p, = 0 and pi = 1, where yt and yt
is defined as above. Frequently, this unbiasedness test is
taken as a classic representation of the rational
expectations test. Because the rational expectations
hypothesis states that the variable should equal the sum of
its expected value and a random white noise error. The most
common test is to regress the forecast error on all the
variables in the information set It. Rationality ensures
that the parameters of this regression will be zero. The
only problem is that the hypothesis rarely specifies what
variables should be included in the information set. As
there is no universal rule governing the choice of the
variables, researchers must make their own choice, according
to their interpretation of the theory or the availability of
data. Clearly, y.1 is in the information set ItI. Hence,
the following equation is often estimated:
Yt Yt = oY + 1Yti + Et
and the hypotheses a0 = 0 and ai = 0 are tested. If the
null hypothesis is rejected, so is the rational expectations
hypothesis. Since y'i is also in It,, some tests are based
on the equation:
Yt Y't = o + alYt + 27Yt1 + et
or,
yt yt = 0 + p1(yt1 yt1) + eC
Rationality implies a0 = 0, a1 = 0 and a2 = 0 in the first
equation, or Pa = 0 and 0i = 0 in the second. The former is
an unrestricted test while the latter restricts the
coefficients of yet and yei to be the same. In addition to
these parameter tests, some tests focus on the behavior of
the estimated error terms. Rational expectations implies
that the error terms be serially uncorrelated. Thus, when
the forecast error exhibits a significant serial
correlation, indicating that the information contained in
previous forecast errors or related variables has not been
fully utilized in performing future predictions, the
rational expectations hypothesis is rejected.
Tests based on yt1 or (yt1 Y'i) or any combination of
these variables are called 'weak' tests of the rational
expectations hypothesis. The 'strong' tests, on the other
hand, require the forecast error to be uncorrelated with
any variable in the information set Iti. This includes any
available economic variables other than Yt or ye.
Variance Bounds Tests for Rationality
The main idea of Muthian rational expectations can be
embodied in the following equation
Yt = y + et
where et is the white noise error and is uncorrelated with
ye. This implies that the covariance between the variables,
Cov(y t) will be zero. For this reason,
Var(yt) = Var(yo) + Var(Et)
and hence,
Var(yt) > Var(ye).
This is the basic idea of the variance test for rationality.
If the reverse relationship is observed, the rational
expectations hypothesis is rejected.
Another way to test for the rational expectations, as
suggested by Lovell (1986), is to incorporate both the weak
rationality test and the variance condition together. This
16
joint test will reject the rational expectations hypothesis,
if either one of the two conditions fails.
Finally, a widely used procedure developed by Pesando
(1975), Carlson (1977), Mullineaux (1978) and Friedman
(1980) involves three steps:
1. Regress yt on the variables in the information set
It1.
2. Regress ye on the same set of variables in Iti.
3. Test the equality of the coefficients in the two
regressions using Chow tests.
As long as the variables in Iti are not uncorrelated with
Yt and y', these procedures can be used to test whether the
expectations are rational. Any significant difference in
the coefficients in the two equations can be interpreted as
evidence of irrationality.
Weaknesses Associated with These Rationality Tests
All of these tests mentioned are valid when the
variables under study are stationary. However, available
evidence shows that most of the economic time series data
are not stationary, making such tests possible invalid if
the usual critical values are used. Fortunately, in most of
the cases, it is the critical region rather than the test
procedure itself that makes the difference. This reduces
the difficulties involved in revising the test. It has been
17
shown that when variables are nonstationary, the estimators
of coefficients from the OLS regression do not follow the
usual t distribution. Thus the critical region is no
longer around 2.00 at 5% significance level. Instead,
according to Dickey and Fuller (1979, 1981), the critical
region should have an absolute value close to three. The
exact critical value depends on the degrees of freedom and
can be obtained from the Dickey and Fuller tables which are
designed for regressions involving nonstationary series.
To be more specific, the following example can be used
to clarify the problem. When the forecast error (y, y')
is regressed on the variable yti, the assumption of a
stationary white noise regression error becomes questionable
if yti is not stationary. Assume, for the moment, that the
rational expectations hypothesis holds so that the forecast
error is stationary. The regression error et in the
equation
Yt ye = ao + iYti + et
cannot be stationary if y t is not. Because it is not
meaningful to have a stationary series on the lefthand side
of an equality sign but a nonstationary series on the
right. This means that e, will violate the usual Gaussian
assumption and the Student t test for ai = 0 is no longer
applicable. As mentioned above, the hypothesis has to be
tested by using the Dickey and Fuller test.
Similar problems arise in the variance rationality
tests. When both y. and y' are random walk process, their
unconditional variances will tend to infinity as the number
of observations becomes sufficiently large. In that case,
it would be dangerous to draw any inference by comparing the
two variances. A rejection of the hypothesis based on the
fact that the variance of y, is smaller than that of yo is
not convincing. The problem does not disappear even if the
sample size is small. For example, assume that yt and yZ
have the following dynamic structures
Yt = Yt1 + Et and et N(0, ac)
Yt = yi + it and Tht N(0, o')
Then the variances of yt and y', given that both yo and y'
equal zero, are to, and to2 respectively. It is obvious
that even if the latter variable has a larger variance it
does not imply a rejection of the rational expectations
hypothesis.
As a whole, the usefulness of the rational expectations
tests mentioned in last section are doubtful when the
variable under study is nonstationary. In order to analyze
the problem suitably, the theory of cointegration, proposed
by Engle and Granger (1987), will be introduced. To have a
better understanding of this theory, the following section
will be devoted to explaining it.
Summary of the Theory of Cointegration
If xt is a vector of economic variables with no
deterministic component and has a stationary, invertible,
ARMA representation after differencing d times, then xt is
said to be integrated of order d, denoted by xt ~ I(d). An
important characteristic of such series is that if a and b
are any constants with b P 0, then a + bx, is also an I(d)
series.
This property forms the core of cointegration theory.
If wt and Yt are both I(d) processes, it is generally true
that the linear combination of the two, say zt, where
t = w cyt
will also be an I(d) process. However, it is also possible
that zt has a lower order, that is
zt I(d k)
with k > 0 and hence d k < d. This relationship reveals
that there exists a constraint operating on the longrun
components of the two series, wt and yt. A prominent
example which is very useful in our analysis is when
d = k = 1. This happens when both wt and Yt are 1(1)
series, in particular a random walk process, but a linear
combination of the two variables zt, is an 1(0) stationary
series.
In general, if both wt and Yt are components of the
vector xt, then they are said to be 'cointegrated of order'
(d, k), denoted by xt ~ CI(d, k), provided that both of
the following conditions are satisfied. These are
1. all components of xt are I(d) ;
2. there exists a vector a(? 0) so that zt = a'xt and
zt ~ I(d k). The vector a is called the co
integrating vector.
This is the fundamental idea of cointegration theory. By
using it, tests for the rational expectations hypothesis
when the variables are nonstationary can be formulated.
Before deriving these tests, we will discuss the tests for
cointegration proposed by Engle and Granger (1987).
Tests for Cointegration
Engle and Granger (1987) proposed seven different tests
for cointegration between two I(1) series, say, wt and yt.
The idea of these tests is to check, whether, after
regressing wt on yt, any unit root exists in the estimated
residual. Since wt and y, are 1(1) series, they are not co
21
integrated if the estimated residual series has a unit root.
On the other hand, if the estimated residual is stationary,
the null hypothesis of no cointegration would be rejected.
In that case, wt and yt are cointegrated and the estimated
coefficient will be the cointegrating vector.
Mathematically, a regression of the following form is
estimated by the OLS estimation method,
wt = ao + alYt + At
and the OLS residual At is obtained. All the tests for co
integration suggested by Engle and Granger (1987) are based
on this estimated residual. These are as follows:
1. Durbin Watson (DW) Test
The DW statistic is calculated based on the
following equations:
DW = 2
EDW = S2_ / S(^t 2 ,
where
A = t / T
The statistic was first proposed by Sargan and Bhargava
in 1983 to test for a unit root in the residual 4t If
there is any unit root in the residual, the DW
statistic should be close to zero, as the sum of
squares of the difference between residuals should be
small compared to the variance of the residual. Hence,
a large DW means a stationary residual and co
integration between wt and Yt.
2. Dickey and Fuller (DF) Test
The DF statistic is obtained from the regression
A = a + 3 +
and its value is equivalent to the Student statistic
for testing the hypothesis of P = 0. The critical
value for this statistic, however, is obtained from the
Enger and Granger's table instead of the Student t
table or the Dickey and Fuller table.
3. Augmented Dickey and Fuller (ADF) Test
The process which obtains the ADF statistic is
similar to the one that provides the DF statistic. One
runs the regression,
ut = a + p t1 + E Si v^, + At
where i runs from 1 to any number p. In our context, p
equals four. Similar to the DF statistic, the value of
ADF statistics is equal to the Student t statistic for
testing P = 0. Once again the critical value for this
test is not obtained from a t table. Instead, it
comes from a table developed by Engle and Granger.
4. Restricted VAR (RVAR) Test
This and the following tests use a different
strategy. The test itself is similar to a two step
estimator. It is based on the following two
regressions:
Wt wt = p , + it ; and
Yt YtI = A t + 7(Wt wt1) + 2t
The RVAR test requires specification of the full system
dynamics. For simplicity, and to be consistent with
Engle and Granger (1987), a first order autoregression
dynamics is assumed. The test statistic is the sum of
squares of the Student t statistics from testing p = 0
and 7 = 0 in the two equations.
5. Augmented RVAR (ARVAR) Test
The ARVAR test is the same as RVAR except a higher
order system of lagged (wt wt_) and (yt Yt1) is
postulated in the equations where the statistics are
obtained. Again, the statistic is the sum of squares
of the two Student t statistics for testing p = 0 and
7 = 0 in the two equations.
6. Unrestricted VAR (UVAR) Test
This test is based on vector autoregressions in
the levels which are not restricted to satisfying the
cointegration constraints. Two regressions are
estimated:
wt wti = a1 + 1Pwt, + riYtI + Qit ; and
Yt yt1 = az + 32wtI + T2ytI
+ S(wt wt1) + 2zt
The statistic is equal to 2(Fi + F2), where FI is the F
statistic for testing both f1 and Tr equal to zero in
24
the first equation, while F2 is its counterpart in the
second equation. Again, this test assumes a first
order system.
7. Augmented UVAR (AUVAR) Test
This is an augmented or higher order version of
the UVAR test. That is, higher order lagged values of
(w, wt1) and (yt yt1) are included in the two
equations in the UVAR test. The statistic is obtained
by the same procedure as the UVAR test.
Relationship Between Cointegration and Rational
Expectations
Cointegration theory proposes a different way to
interpret the rational expectations hypothesis when the
variables under study are I(1) processes. If both yt and ya
are nonstationary, then the rational expectations
hypothesis, which postulates the following relationship,
Yt = y + Et
or,
Yt y = t
is equivalent to requiring yt and y' to be cointegrated
with a factor of unity, that is a = 1. In addition, the
25
error term, et, has to be a white noise process. This means
that rational expectations is a stronger requirement because
cointegration requires e, to be stationary, but rationality
goes one step further by requiring e, to be a white noise
error.
However, the tests of cointegration from Engle and
Granger (1987) cannot be applied directly to test for the
rational expectations hypothesis as the latter hypothesis
requires more than cointegration. Engle and Granger (1987)
require only a stationary estimated residual, but not any
restriction on the cointegrating factor or the randomness
of the estimated error term. It can happen that yt and y'
are cointegrated with a factor, say, 0.5 and that the error
term follows an ARMA stationary process. Obviously, the yt
in this situation is not a rational expectation. To be more
specific, cointegration is a necessary, but not a
sufficient, condition for rational expectations when yt and
ye follow random walks. Taken as a whole, an expectation is
said to be rational, if the three following conditions have
satisfied:
1. yt and ye must be cointegrated;
2. The cointegrating factor must be one;
3. The difference, yt yt, must be a white noise
process with no serial correlation.
In order to adjust the cointegration tests to fit the
above requirements, a slight modification can be used to
incorporate the first and second conditions above
automatically. Instead of using the estimated residual, jt,
a restricted residual Mt, defined as the difference Yt yZ,
could be used. This will automatically restrict the co
integration factor to one. We call these restricted co
integrated tests. Hence, if pt is stationary, Yt and yt are
cointegrated with a factor one, provided that both of them
are I(1) nonstationary series. The Box and Pierce (Q) test
will be applied to Mt to check for the existence of
systematic patterns in the residuals. Any serial
correlation that exists in pt will be captured by the Q
statistics. If all these tests provide positive results,
that is, yt and ye are cointegrated with a factor one, and
At is stationary and contains no systematic pattern, one can
conclude that the expectations are rational.
The ideas outlined in this section will be applied to
the foreign exchange market and these results will be
presented in Chapter IV. Before this, however, another
statistic for testing the rational expectations hypothesis
will be introduced. The test is built on the Fisher
information matrix. Its derivation, as well as the
computations, are discussed in the next chapter.
CHAPTER III
THE INFORMATION MATRIX TEST FOR RATIONAL EXPECTATIONS
Introduction
Since the method of maximum likelihood was developed in
the 1920s, it has become one of statistically most widely
used tools for estimation and inference. A fundamental
assumption of this method is that the stochastic law which
governs the true phenomenon is known to the researcher.
This means that the model proposed by the researcher must be
correctly specified.
However, there is no reason to assume that the
researcher has full knowledge about the data generating
process. What will happen if the model is not correctly
specified? Will there be any change in the properties of
the likelihood function, the estimators, and their
distributions? These questions were discussed by White
(1982), who suggested a method to test for the
misspecification of a model. The test is based on a well
known property of the likelihood function, namely, that if
the model is formulated correctly, the expected sum of the
second derivative of the natural logarithm of a likelihood
function (hereafter referred to as loglikelihood function)
and the square of its first derivative will be equal to
zero. That is,
Et[Fz{9)] + Et[Fi(E}Fi{e)'] = 0
where F2({) is the (pxp) matrix of the second derivatives of
the loglikelihood function, that is, a21ogft({)/e a89';
Fi{() is the (pxl) vector of the first derivatives of the
loglikelihood function, denoted by alogft{8)}/a; E[.] is
the expectation operator, generally defined as Jg[.]ft{O}de;
ft(e) is the likelihood function; and 9 is the (pxl) vector
containing all the p parameters in the system. The major
contribution in White (1982) is the derivation of the
asymptotic distribution of the components of this sum.
Thus, the specification of any model can be tested by means
of this distribution.
The next section derives this information matrix test.
Some of the principal assumptions that lead to the final
results are emphasized. Our major concern is to apply this
test to examine the validity of the rational expectations
and the market efficiency hypothesis in the exchange rate
market. In particular, we want to apply the test when the
variable under study is not a stationary series. As shown
by Phillips (1986, 1987), whenever there exists a unit root
in the series, the distribution of the test statistics can
be very complicated and will not converge to any standard
distribution like the normal or x2, asymptotically.
Phillips (1986) showed that if xt has a dynamic process of
the form
Xt = Xti + Et
then as the number of observations T tends to infinity, we
have the following results
T3/2Zxt a o;W(t)dt ;
T2EZX a. O1[W(t)dt ;
T Zxet a { (C [W()]2 n0) ;
T Z2 a.s. lim (TZEt() ) =
T
where a2 is the variance of E,, W(t) is a Wiener process on
C[0,1] and Z is the sign of summation summing from 1 to T.
Since the ordinary least squares estimators and the
statistics are functions of the above sums, the asymptotic
distribution of these random variables will not be simple
normal distributions. However, it may be shown that if the
information matrix test is applied to a subset of the
parameter space, such problems can be avoided. The choice
of parameters in the subset depends on the values of
parameters which are specified under the null hypothesis.
Details of this argument and its application will be
presented later in this chapter.
The final section gives the Monte Carlo experiment and
its results. The experiment is essential as it compares the
power of different statistics. Tests included in the
experiment are the DF, ADF, DW, three Box and Pierce (Q)
tests and the information matrix test. Since the test for
cointegration is similar to the test for a unit root in a
time series analysis, the experiment will concentrate on the
ability of the tests to distinguish a random walk from a
stationary AR(1) process that is close to being a non
stationary process.
The Information Matrix Test
The information matrix test is built on the assumption
that if the model is correctly specified, the Fisher
information matrix can be expressed either in the Hessian
form, that is, Et[F2(8)], or in the outer product form,
Et([F{( }F{86)']. In other words, it is built on the fact
that Et[F2(e}] + Et[Fi{9).*Fi{)'] = 0 when the model's
specification is correct. If this equality fails to hold,
it follows that the model is not specified correctly. To
derive the required statistics to test the hypothesis that
this equality holds, let's assume that the loglikelihood
function behaves properly, that is, the function is
continuous over the entire parameter space and
differentiable up to, say, the third order. In addition,
the existence of the first and second derivatives of the
loglikelihood function, the nonsingularity of Hessian and
outer product matrix are also needed so that the statistics
will converge, asymptotically, to a proper distribution. At
this moment, no dynamics for the variables are specified and
to simplify the derivation, all variables are assumed to be
stationary.
For a p parameter likelihood function, f,(8), there
are p2 elements in the Hessian and the outer product matrix.
Among them, only (p(p+l))/2 are different. To compare these
{p(p+l))/2 distinct elements in the two matrices, White
(1982) defined a statistics, dkt, which was equal to the sum
of the corresponding elements in the two matrices.
Mathematically,
dkt{e = alogfoe)/8i'8logft(8)/8Oj
+ 82logft{9)/a9ia8
where
k = 1, ......, p(p+l)/2 ;
i,j = 1, ..... p ;
t = 1, ..... T ;
To simplify the notation, let q = (p(p+l))/2. By means of
the matrix notation, dki(e) (k = l,...,q), for a particular
observation t, can be grouped into a (qxl) vector such that
d,{9) = (F2t(e} + Ft(e)FF E{)} ')c
where c stands for the stacking of a (pxp) matrix. Notice
that d,{6} is only a (qxl) vector because only the distinct
elements in the two matrices will be considered.
Assume the mean and partial derivatives of dt{e) exist
and define the mean as DT{{) such that
DT(E} = Et[dt({ }] ;
and the (qxp) Jacobian matrix of d,{8) as VDT{() which is
equal to
VDT({ ) = Et[(adt(e}/ak] ;
White (1982) argued that the random variable DT({) is
normally distributed with zero mean and a constant variance
V(e). That is,
DT{G1 N(O, V(Q))
where
V(E) = Et{d,(8) VDT(e}{Et[F2t()])EFit[Ft(8])
.{d{8e) VD{G) {Et[Ftz(e) ] EJ[Fit( ]) '
T is the total numbers of observations. White (1982)
proposed that the above test statistic can be calculated for
any sample by substituting for 6 its maximum likelihood
estimator, 8, and replacing the expectation operator Et[.]
by I/T*Z[.], where the summation sign represents a sum from
1 to T. Under the null hypothesis of a correctly specified
model, one can obtain the following sample statistics
1. JTDi(e) N(0, V{8))
2. V( )}  .. { ) ;
3. 'T = TDT{8)'V (@} DT(6)
where DT is the information matrix test statistic, which
follows a Xq distribution asymptotically.
The calculation of the above statistic is tedious,
especially when dealing with the variance V{6} since it
involves the third derivatives of the loglikelihood
function. To simplify the work, Chesher (1983) and
Lancaster (1984) defined a relatively easy formula to
calculate O9. Define a Tx(q+p) matrix Y{(} such that
Y{6} = [d,({ )': Fi,{)( ']
where t runs from 1 to T, dt(6)' is a (1xq) row vector
containing the sample values of the random variable d{(O)
for each observation t, and Flt{6)' is a (ixp) row vector
containing the sample values of the first derivative of the
loglikelihood function for the same observation. Hence
Y{(} is a combination of the row vector [d(6I)'I F1t(6}'
from 1 to T. Both Chesher (1983) and Lancaster (1984)
suggested that the information matrix statistic can be
simplified into the form
Tr = 'Y{( }) [Y{( ) Y{ }] AY{ )',
where L is a (Txl) vector such that I'l = T. The whole
expression is equivalent to T times the R2 statistic in the
regression of t on Y{[).
White (1982) emphasizes that, in many cases, it is
inappropriate to conduct the test on all p elements in the
system. There are several possible reasons for this: some
of the elements can be identically zero; some may be a
combination of others; and sometimes the degrees of freedom
are too large if all the parameters are included. Another
justification not mentioned by White (1982) is that in case
there exists any nonstationary variable in the analysis, it
35
is very likely then that the components in d,(9) containing
that variable are not stationary. It then turns out that
DT{O} will not converge to a normal distribution even when T
tends to infinity. If that happens, one has to avoid the
problem by considering a subset of the parameters that will
converge to an appropriate distribution.
The Information Matrix Test and Rational Expectations
A basic formulation of a rational expectations model is
rather simple, that is
Yt = a + 3yt + E
with the hypothesis that a=0, P=1 and et is a white noise
process with mean zero and constant variance a2. Although
there is no special requirement on the distribution of the
error term, it is customary to assume that it follows a
normal distribution. This implies that the density function
of the variable yt, for a particular observation t, is
ft(a,l,a2) = 1/J(2raz) exp{(yt a /y) 2/a2)
and its loglikelihood function can be written as
logft(a,p,o2) = 1og(2r) log(U2)
(yt a /y:)2/a2
36
Since there are three parameters (a, P and a2) in the model,
there will be six distinct elements in the information
matrix. The first order condition for any observation t,
denoted by Fl,, is equal to
8logft(a,e,a2)/8 8 7
alogft(a,p ,a2)/8,a2
and the second order condition for the same observation,
denoted as F2t, is
82logft/aa8 8a 21ogf/8Ba* p a2logft/88a 2
a821ogft/8a f8a 21ogft/ap o3 a21ogft/8p a802
821ogf/aU2*a aa 82 lgf/82 20g/o2a 80a2
With this information in hand, one can work out the
vector dt{6), the D{(6) as well as the statistics tO, by
substituting the maximum likelihood estimators of a, p and
a2. However, when Yt, and most probably yZ as well, is non
stationary, elements in the Di(6} matrix involving them will
not converge to normal distributions as T tends to infinity.
That is, as shown in previous section if Yt is not
stationary, elements in Fi, or F2t that contain Syt, Sy2 or
Eyti.Et will be functions of the Wiener process after being
suitably scaled, preventing Fit and F2t from converging to
the normal or x2 distributions. This implies that the
statistic 0T will no longer have a x2 distribution.
Nevertheless, the problem can be avoided by considering
a subset of the parameter space. Under the null hypothesis
of rational expectations, a=0 and P=1. In this case, the
whole formulation can be rewritten in a much simpler form
6t = Yt Yt
Even though both yt and yt are nonstationary, their
difference, Et, has to be stationary if the null hypothesis
is true. Under H0, then, one can rewrite the loglikelihood
function in the form
logft(2) = 1log(27r) log(o2) /
Notice that only one parameter, a2, appears in the system.
Hence the vector dt({e) is reduced to a (1x1) scalar and the
whole analysis can be simplified to a large extent.
Two different approaches may be adopted for this
simplification. First, as White (1982) shows, it is
legitimate to use a subset of parameters to perform the
information matrix test. Based on the Lagrangian multiplier
principle, one can always use the value of the parameters
under the null hypothesis to work out the statistics. If
the null hypothesis describes the true values, this
restricted statistic will have the same distribution as the
unrestricted sample statistic. Thus, the parameters a and P
can be replaced by 0 and 1 respectively, reducing d{(6} to a
(1x1) scaler instead of a (6xl) vector for each observation
t. Also, the maximum likelihood estimator of az is replaced
by its restricted counterpart under the Lagrangian
multiplier principle, defined as
a2 = E/T
This, then, is the reformulation of the information matrix
statistic for testing rational expectations when the
variable under study is nonstationary. In brief, the idea
is to reduce the number of parameters in the system by
substituting parameter values under the null hypothesis. By
this method, the problem associated with any nonstationary
component in the system may be avoided. If the parameter
values under the null hypothesis are the true values, the
information matrix statistic will follow a x2 distribution.
Monte Carlo Experiments on the Information
Matrix Test Statistic
In order to check the performance of the information
matrix statistic, a Monte Carlo experiment was conducted to
compare its effectiveness with other unit root tests. For
the sake of simplicity, only three of the seven tests
suggested by Engle and Granger (1987) were included for
comparison. They are the Dickey and Fuller (DF) test,
Augmented Dickey and Fuller (ADF) test and the Durbin Watson
(DW) test. Besides them, Box and Pierce (Q) statistics will
be used to check the randomness of the restricted residual'.
These Q statistics are included because, they can be
calculated easily, and they show the properties of the
residual. The idea is that if a variable follows a random
walk process, the difference, yt Yt1, will be a white
noise process, and the Q statistic calculated will follow a
x2 distribution.
The experiments worked out the distributions of the
statistics under the null hypothesis and AR(1) alternatives
with p equal to 0.99, 0.95 and 0.90. These alternatives
were chosen because the purpose of the experiment was to
distinguish between the null of a random walk from a series
close to a random walk. The selection of alternatives is
somewhat arbitrary. But they are common in most recent
research. There are one thousand replications under the
null and each of the alternative.
IThe simplest form of the Q statistic is defined as
TZ J where i equals 1 to k and r2 is the ith sample
autocorrelation in the residuals. If the model is correctly
specified as a random walk process, then the Q statistic
will have a x2 distribution with k degrees of freedom.
The experiments proceeded as follows: Using the random
number generator in the SAS program, one thousand series of
random numbers, with 150 observations each, denoted by Eit
(i=l,...,1000, t=l,...,150), were generated from a standard
normal distribution. The initial condition yi0, for all
'i', equalled zero. Then a thousand series of yit was
generated based on the following equation
Yit = PYictn + eit i = 1,...,1000 and yio = 0
where p took the values of 1, 0.99, 0.95 and 0.90
respectively. When p equals one, yit is a random walk
process. By repeatedly running the tests on these series,
the size of the statistics can be obtained. In some cases,
the critical values obtained from the table may not have the
correct size and hence some minor adjustment will be needed.
For other values of p, yt is a simple first order
autoregressive AR(1) series. Since a fixed value of zero is
assigned to be the initial value, only the last hundred
observations were used for the experiment. The first fifty
observations will were deleted.
The following tests were conducted: The Dickey and
Fuller test (DF), augmented Dickey and Fuller test (ADF),
Durbin and Watson test (DW), Box and Pierce test (Q), and
the White information matrix test (WIM). When computing the
information matrix test statistic, Y{6), d,{6) and F,,(e}
were replaced by Y{(}, d,({) and Fnt{8) respectively. The
former were the maximum likelihood estimates of Y{(), d,{e)
and Fit(8}, while the latter were their counterparts using
the lagrangian multiplier estimators. The component of
dt{() and Fit(9) for a particular observation t, is defined
as
dt{() = a4 1e/a6 + K/a8 ; and
F1t{e) = a2 + /a4 ;
By substituting U2 for a2 in these expressions, the value of
dt(e} and Fit(O) can be obtained. Combining them together,
the value for Yt{e) as well as the information matrix
statistic 6. can be calculated easily.
Empirical Results of the Experiment
Results of the Monte Carlo experiment are summarized in
the following tables. Table 3.1 states the critical values
for various tests embodied in the experiment at the 10%, 5%
and 1% significance levels. Figures in the table are the
critical values obtained from tables associated with the
tests. For example, the critical values for the DF test are
obtained from the table in Fuller (1976), the critical
values for the Q test are obtained from a x2 table and so
on. These critical values mean that if a sample statistic
has a larger value, there will be 10%, 5% and 1% chance of
incorrectly rejecting a correct null hypothesis.
TABLE 3.1
CRITICAL VALUES UNDER NULL HYPOTHESIS
SIGNIFICANCE LEVEL
0.10 0.05 0.01
DF 2.58(2.514) 2.89(2.826) 3.51(3.338)
ADF 2.58(2.496) 2.89(2.775) 3.51(3.297)
DW N.A. (0.206) 0.259(0.248) 0.376(0.351)
Q4 7.779(8.963) 9.488(11.74) 13.277(16.37)
Q8 13.362(15.11) 15.507(18.50) 20.090(26.48)
Q12 18.549(20.57) 21.026(24.58) 26.217(32.12)
WIM 2.076(3.693) 3.841(4.822) 6.635(6.764)
Note: These critical values are obtained from the tables associated with each test statistics.
Values in parentheses are the corresponding values obtained from the nut( hypothesis. Very often,
they are the sizeadjusted critical values.
The critical values from published tables represent
asymptotic values. We have only 100 observations. The
figures in parentheses in table 3.1 represent the size
adjusted critical values obtained from the null hypothesis.
These figures will also be used as a reference when counting
the numbers of rejections under the different alternatives.
In some sense, these are better critical values because they
represent the true size of the tests in the experiment.
Tables 3.2 to 3.4 present the results of the experiment
when p takes the values of 0.99, 0.95 and 0.90 respectively.
The figures in the tables represent the number of rejections
in one thousand replications, based on the critical values
from the tables. For each alternative, the numbers of
rejections at 10%, 5% and 1% significance levels are
recorded. Figures in parentheses are the corresponding
numbers of rejections using the sizeadjusted critical
values.
Interpretation of the Empirical Results
The results from the experiment are not encouraging,
especially for the information matrix statistic. As
expected, the closer the value of p to one, the lower the
ability of the tests to differentiate the alternatives from
the null. When p equals 0.99, the power of all tests is no
greater than 10% at 0.05 significance level. The situation
gets worse when the 0.01 significance level is employed,
where no test can pick up the alternative against the null
hypothesis more than forty times out of one thousand. These
confirm what the finding of many researchers that all the
unit root tests have very low power in distinguishing an
AR(1) stationary series with a high value of p from a random
walk process.
Table 3.2
POWER OF THE TESTS (p=0.99)
SIGNIFICANCE LEVEL
p = 0.99
0.10 0.05 0.01
DF 0.107 (0.119) 0.060 (0.065) 0.012 (0.019)
ADF 0.091 (0.108) 0.052 (0.067) 0.008 (0.018)
DW N.A. (0.133) 0.073 (0.080) 0.014 (0.017)
Q4 0.151 (0.100) 0.087 (0.051) 0.034 (0.009)
Q8 0.152 (0.103) 0.100 (0.049) 0.037 (0.010)
Q12 0.149 (0.103) 0.092 (0.048) 0.035 (0.009)
WIM 0.170 (0.100) 0.094 (0.055) 0.010 (0.010)
Note: The critical values are obtained from various tables corresponding to each statistics.
Figures in parentheses are the corresponding numbers of rejections using the sizeadjusted critical
values obtained from Table 3.1.
Table 3.3
POWER OF TESTS (p=0.95)
SIGNIFICANCE LEVEL
p = 0.95
0.10 0.05 0.01
DF 0.237 (0.276) 0.129 (0.154) 0.029 (0.044)
ADF 0.182 (0.219) 0.097 (0.127) 0.023 (0.041)
DW N.A. (0.335) 0.186 (0.205) 0.049 (0.067)
Q4 0.176 (0.127) 0.112 (0.068) 0.045 (0.016)
Q8 0.181 (0.132) 0.125 (0.068) 0.051 (0.015)
Q12 0.191 (0.136) 0.121 (0.072) 0.054 (0.016)
WIM 0.163 (0.098) 0.090 (0.051) 0.010 (0.010)
Note: See note under Table 3.2.
Table 3.4
POWER OF THE TESTS (p=0.90)
SIGNIFICANCE LEVEL
p = 0.90
0.10 0.05 0.01
DF 0.528 (0.563) 0.353 (0.393) 0.099 (0.151)
ADF 0.380 (0.431) 0.224 (0.273) 0.054 (0.099)
DW N.A. (0.709) 0.504 (0.546) 0.170 (0.226)
Q4 0.299 (0.226) 0.203 (0.117) 0.085 (0.047)
Q8 0.316 (0.226) 0.215 (0.142) 0.112 (0.045)
Q12 0.309 (0.236) 0.223 (0.139) 0.113 (0.057)
WIM 0.154 (0.098) 0.091 (0.048) 0.070 (0.007)
Note: See note under Table 3.2.
Results improve significantly, when p decreases from
0.99 to 0.95 and even further to 0.90. The numbers of
rejections for some of the tests increase up to 200 and 500
out of one thousand, depending on the level of significance
used. Among them, the DW statistic has the best
performance. Its power is the highest at the 5%
significance level when compared to the other statistics.
The sizeadjusted figures provide a better picture.
The DW statistic dominates all the other tests in most of
the situations. When the alternative is an AR(1) process
with p equals to 0.90, the DW test rejects a random walk
more than 70% of the time at the 10% significance level.
This is an encouraging result, for it means that when the
alternative is not too 'close' to a random walk, the DW
statistic can be used to test against p = 1.0 with high
power. It would be too demanding, if we ask for a statistic
that can differentiate p = 0.99 and p = 1.00 correctly with
a high power.
The DF and ADF statistics improve gradually as p
decreases. At the 10% significance level, DF can pick up
more than half of the alternative when p equals to 0.90. In
addition, the sizeadjusted power of these two tests never
drops below 10% when the same significance level is adopted.
By comparison, the three Box and Pierce (Q) statistics
improve by a smaller extent when the value of p decreases.
They can only identify about 20% of the alternative if p
equals 0.90 after adjusting for the size.
The results for the information matrix statistics are
quite disappointing. Its power is never higher than 10%
and, unlike the other statistics, its power does not improve
as p decreases from 0.99 to 0.90. In fact, the number of
rejections seems to decrease as p moves further away from
one at all significance levels, both in the size unadjusted
and adjusted cases. Though the decrease is comparatively
small, it raises doubts about the usefulness of this
statistic.
Several reasons may be offered to explain the poor
performance of the information matrix statistics. First, as
47
White (1982) suggested, the statistic will converge to a x2
distribution asymptoticallyy'. It is possible that a sample
size of 100 is not large enough for the statistic to
converge to its asymptotic distribution. Second, the choice
of alternatives may be a hindrance for the statistic. It is
likely that the test has a higher power in differentiating
between an ARMA, or MA from a random walk than the case used
in the present experiment. Finally, the most obvious answer
is the nonstationarity property of the variable. However,
the WIM depends only on e., and e, is drawn from a
stationary process irrespective of the null or alternative
hypothesis. Thus, there is no point in blaming non
stationarity for the poor performance.
Summary
This chapter derives another statistic to test the
rational expectations hypothesis. The intuitive idea is to
substitute the parameter value given by the null hypothesis
into the loglikelihood function, reducing it into a single
parameter function that depends only on a2. Then the
information matrix test is applied based on this function.
If the expectation is rational, or in other words, the null
hypothesis is true, the statistic should follow a x2
distribution. A Monte Carlo experiment is conducted to test
the ability of this statistic, compared to other tests of
cointegration, to differentiate an AR(1) process from a
random walk process. Although the information matrix
statistic performs poorly, it seems that the other tests of
cointegration, in particular the DW statistic, can serve
the purpose moderately well. This provides some confidence
in applying these tests to check the existence of a unit
root, the rationality of expectations, and the efficiency of
forward rate in the foreign exchange rate market in the next
two chapters.
The Monte Carlo experiments, though apparently
discouraging, lead to interesting conclusions. The
information matrix test is a very general test. It is
designed to test for all sorts of specification errors: non
normality, heteroskedasticity, lack of cointegration and so
on. As such, it is expected to have less power than the DW,
DF, ADF tests that are designed for specific alternatives,
that is wellspecified specification errors. Our Monte
Carlo results show that in the case of testing for unit
roots, even specific tests like the DW, DF, ADF tests have
very low power. Thus, it is not surprising that the
information matrix test has still less power. We are led to
conclude that in testing for unit root, very general
specification tests are not very useful. Our results cast
doubt about the general applicability of the information
matrix test.
CHAPTER IV
TESTS FOR RATIONAL EXPECTATIONS: EMPIRICAL RESULTS
Introduction
If exchange rates follow a random walk, the lagged
value of the exchange rate is the best predictor for the
future spot rate. Meese and Rogoff (1983) found that a
random walk model performs as well as any structural model
in the foreign exchange market. Others, like Somanath
(1986), claim that the evidence is not conclusive. As a
whole, there does not exist any definite answer to this
question.
As mentioned in the previous chapters, whether exchange
rates follow a random walk process greatly affects the
reliability of the tests of the rational expectations
hypothesis. If exchange rates are nonstationary, the
legitimacy of many of the tests that have been used becomes
questionable. It has been shown that if exchange rate is
not stationary, usual test statistics do not follow the
standard normal, Student t or X2 distributions. For this
reason, testing the existence of a unit root in exchange
rates becomes an important prerequisite for any further
analysis.
In order to have a better understanding of the
controversy over exchange rates, the next section will be
devoted to a review of some related research. Since so much
work has been conducted before, this review can hardly be
exhaustive. Emphasis will be on the random walk nature of
exchange rates and tests for the rational expectations
hypothesis in the exchange market. In particular, the
'evidence' in favor of rejecting the nonstationarity and
irrationality of exchange rates will be reviewed. The
purpose is to check whether there are strong enough reasons
to turn down these propositions. Finally, in the third
section, the empirical results of unit root and co
integration tests on weekly exchange rates are presented.
These results suggest possible answers to the questions of
whether nominal exchange rates follow a random walk and
whether expectations are rational.
Debate on the Dynamics of Exchange Rates
Perhaps the earliest and most influential study on the
dynamics of exchange rates is that by Meese and Rogoff
(1983), who compare the time series and structural models of
exchange rates on the basis of their outofsample
forecasting accuracy. The models they included were
flexibleprice and stickyprice monetary models, as well as
a stickyprice model that incorporates the current account.
The outofsample accuracy is measured by mean error, mean
51
absolute error and root mean square error. Their empirical
results show that all the structural models performed
poorly. In contrast, a simple random walk time series model
predicts exchange rates as well as any of the structural
models.
Though Meese and Rogoff (1983) never concluded that
nominal exchange rates follow a random walk process, they
did identify the unpredictable nature of nominal exchange
rates. Others who share a similar opinion are Mussa (1979)
who showed that the spot exchange rate follows approximately
a random walk, and Frenkel (1981) who showed that the spot
exchange rate is highly volatile. More recently, Huang
(1984) reported that, in general, random walk models perform
better than other models in characterizing exchange rate
behavior. Fratianni, Hur and Kang (1987) verified the
robustness of the random walk hypothesis using time series
of five major currencies; MacDonald and Torrance (1988) have
confirmed the existence of a unit root in monthly exchange
rate by direct testing. Hakkio and Rush (1989) also confirm
that nominal exchange rates are nonstationary. All these
show, either explicitly or implicitly, that nominal exchange
rates are highly volatile and that it is very likely that
they follow a random walk process.
The other side of the coin has also received some
attention. Examples claiming that the nominal exchange rate
is stationary can be easily found in any international
52
finance journal, particularly during the seventies. It was
not until the mid eighties that some economists started
challenging this view. There is still a substantial
literature in support of this view. Park (1984), for
example, rejected the random walk hypothesis in the foreign
exchange market because of a systematic nonrandom component
in the deviation of the current spot rate from the future
spot rate. Somanath (1986), responded to Meese and Rogoff
(1983) by considering a larger set of structural models.
Utilizing both the outofsample and insample evidence, his
results suggested that some structural models can dominate
the random walk model in various sample periods. More
importantly, he found that including lagged adjustment terms
can contribute towards better performance in any models.
Hakkio (1986) argued that the exchange rate is stationary
but 'close' to a random walk, and he maintained that the low
power of all unit root tests in distinguishing these two
cases is responsible for the controversy. Frankel and Froot
(1986) also rejected the nonstationarity hypothesis for
exchange rates, though they concede that the process of
exchange rate may be, once again, close to a random walk.
It is interesting to notice that even though many
researchers rejected the nonstationarity of nominal
exchange rate, they conceded that the rate is very 'close'
to a random walk. Moreover, upon examination, some of the
evidence appears weak. For example, Park (1984) rejected a
random walk because of the existence of a systematic
component in exchange rates, even though all the
coefficients of the lagged exchange rates are not
significantly different from one when tested using the
Dickey and Fuller test. In other words, it is also
legitimate to accept the random walk hypothesis if a direct
unit root test is used. Somanath (1986) found the random
walk model could not dominate the structural models.
However, his result also reveals that the ranking of the
random walk model is very close to the top. Hakkio (1986)
rejected the random walk hypothesis because it implied that
the exchange rate has an unbounded unconditional variance.
But he agreed that the evidence yields contradictory
conclusions. He explained this by pointing to the low power
of the unit root tests. Lastly, Frankel and Froot (1986)
claimed that the exchange rate will become a noninteresting
variable if it followed a random walk process. Like Park
(1984), they claimed to discover a systematic relationship
in expected depreciation. However, they did not conduct any
direct unit root tests. It is probably the case that since
unit root tests have low power, the power of an indirect
test will be even worse. Others, for example, Hodrick and
Srivastava (1984), simply ignored the existence of a unit
root and did not conduct any test on this particular issue.
We conclude that the evidence rejecting the random walk
hypothesis in nominal exchange rate analysis is usually not
54
strong enough to give a definite answer. We need to conduct
our own direct test.
Debate on the Rational Expectations Hypothesis
So far, the conclusions about the rational expectations
hypothesis in the foreign exchange market seem less
controversial. Most of the research, for instance,
Dominguez (1986) and Frankel and Froot (1986) reject the
rational expectations hypothesis. Hakkio and Rush (1989)
reject it for one market. Lacking an independent set of
expectations data, these papers have used the market forward
exchange rate as a proxy for expectations. However, it may
well be the case that the forward exchange rate contains
more than expectations, specifically, a risk premium may
also enter. Because this is a very important issue in
international finance, we will pay more attention to it in
the next chapter and so we postpone a discussion until then.
In this chapter we concentrate on the tests using market
survey data as a proxy for expectations.
The number of research papers using survey data has
increased tremendously in the eighties, and their
applications are spread over a wide range of areas in
economics; for instance, in consumer behavior, price level
forecasts, interest rate expectations and foreign exchange
rate expectations. Among these areas, we confine ourselves
to the foreign exchange market.
55
The earliest research that used survey data on foreign
exchange rates was Frankel and Froot (1986). The survey
data they used came from three different sources; the
American Express Banking Corporation; the Economist
Financial Report; and the Money Market Services, Inc. Using
ordinary least squares estimation, they rejected the
rational expectations hypothesis by finding an unconditional
bias in the survey errors. Dominguez (1986), who also used
forecast data in her work, regressed the actual spot
depreciation on the corresponding forecast depreciation.
The forecast data used in her paper were from the Money
Market Services. Based on these data, she rejected the
hypothesis of rationality in four foreign currency markets
by finding that the estimated coefficient was significantly
different from one. Ito (1988) also rejected the rational
expectations hypothesis by using a set of crosssectional
survey data conducted by the Japan Center for International
Finance. His results had two major conclusions. First,
market expectations are rather heterogeneous, and second,
many institutions are not expecting the future rationally.
These three papers constitute the leading research in
applying survey data to test the rational expectations
hypothesis. Their results are rather homogeneous in that
the hypothesis is rejected when applied to the foreign
exchange market. However, none of these papers takes
account of the statistical consequences of the non
56
stationarity of the exchange rate series. Nonstationarity
has crucial effects when inference is drawn from the
statistics involving that variable. For example, in Frankel
and Froot (1986), the number of rejections of rationality
decreases sharply if the critical value is obtained from a
Dickey and Fuller table instead of the usual Student t
table. In fact, the number of rejections is far less than
the number of acceptance if an absolute critical value of
3.00 is used (which is approximately the 95% level from the
Dickey and Fuller table). Similar results appear in Ito
(1988). None of the onemonth and threemonth coefficients
in his table 4 is significant if the critical value is
obtained from the Dickey and Fuller table. Dominguez (1986)
has a stronger evidence to support her results. She rejects
the rational expectations hypothesis in an overall sense.
However, her results show that she can only reject the
hypothesis using quarterly data, but not the monthly data.
This means that one should be very cautious when
interpreting the empirical results.
At the moment, the only paper that utilizes survey data
and accepts the rational expectations hypothesis is Taylor
(1989). Using individual rather than mean survey data, he
cannot reject the null hypothesis of rational expectations
because none of the coefficients in his regression is
significant. The survey he used is a qualitative data
survey, meaning that the data are recorded in a categorical
57
form. In fact, the participants respond to the survey only
by answering whether the exchange rates may go up, down or
stay the same twelve months hence. Taylor (1989) then
formulated his research by quantifying the qualitative data
using the CarlsonParkin method. He formulated a subjective
expectations distribution and used a scaling factor for each
individual to obtain a set of aggregate point expectations.
All his results are based on these aggregate estimates that
at least some of the survey results could pass the tests for
rationality. Although there may be some arguments about the
way Taylor constructed his data, he demonstrated the
robustness of his results by allowing the presence of random
measurement errors.
In the next section, we will present our empirical
results which use the restricted cointegration tests to
test the validity of the rational expectations hypothesis in
the foreign exchange market.
Empirical Results
Before we present the results of the restricted co
integration tests, it is necessary to know whether the
exchange rate follows a random walk. For this purpose,
three unit root tests, the DF, ADF and DW, and three Q
statistics, namely Q(4), Q(8) and Q(12), are used. The
first three are the standard unit root tests while the Q
statistics are included to test the randomness of the
residuals. Although the results in the Monte Carlo
experiments show the weakness of the information matrix
statistic, we will include it as well. In order to have
more confidence on the results of these tests, two different
versions of these tests will be used: the unrestricted and
restricted test statistics. The former uses the OLS
estimated residuals, At, from the equation
Xt = a + PXti + Pt
while the latter uses the restricted residuals defined as
9t = Xt Xt1
where Xt equals Yt, the nominal exchange rate, or y(, the
expected value of yt. The logarithm of nominal exchange
rates is used because it fits best most of the exchange rate
models. Hsieh (1984) argued that the only justification to
use the logarithmic form is if the exchange rate follows a
lognormal distribution. We present results using both the
level and the logarithmic form of exchange rate series.
Results of the unit root tests are summarized in tables
4.1 to 4.4. Tables 4.1 and 4.2 present results of testing
for unit roots in the spot exchange rate, Yt, while tables
4.3 and 4.4 present the corresponding results for the
expected spot exchange rate series, yt. In the tables the
top figures are the restricted test statistics while the
figures in parentheses under them are their counterparts
using the unrestricted residuals. Notice that none of the
three conventional unit root tests in any of the tables has
values large enough to reject the null hypothesis of non
stationarity. In fact, all of these statistics are so small
that they are far below the critical values. This lends
strong support to the proposition that spot exchange rates
and their expected values follow a random walk process, no
TABLE 4.1
RESTRICTED (UNRESTRICTED) UNIT ROOT TESTS
ON SPOT EXCHANGE RATES
CURRENCIES
Statistics DM SF
1. DW 0.0213 0.0072 0.0103 0.00612
(0.0209) (0.0071) (0.0102) (0.00596)
2. DF 1.5878 1.3136 1.2444 1.19573
(0.0000)+ (0.0000)+ (0.0000)+ (0.00000)+
3. ADF 1.6191 1.3566 1.3132 1.27130
(0.1706) (0.2046) (0.1812) (0.30353)
4. Q(4) 0.9270 1.7647 1.1378 3.80696
(0.9107) (1.5900) (1.0867) (3.50184)
5. Q(8) 1.4233 3.3393 2.0689 4.81714
(1.3806) (3.0365) (1.9470) (4.52426)
6. Q(12) 4.4297 5.0061 7.6403 7.26992
(4.4291) (4.7664) (7.5538) (6.96995)
7. WIM 4.9917" 3.6273 8.5263* 7.18138*
(4.7080)* (3.1616) (8.0708)* (7.18862)
Value is too small to report
Significant at 5% level
Note: Figures in parenthesis are values for each test using the unrestricted residuals.
TABLE 4.2
RESTRICTED (UNRESTRICTED) UNIT ROOT TESTS
ON THE LOGARITHM OF SPOT EXCHANGE RATES
CURRENCIES
Statistics DM SF
1. DW 0.0212 0.0066 0.0089 0.00490
(0.0209) (0.0064) (0.0088) (0.00474)
2. DF 1.6145 1.3669 1.2670 1.28157
(0.0000)+ (0.0000), (0.0000)+ (0.00000)
3. ADF 1.6271 1.4118 1.3433 1.35576
(0.1054) (0.1498) (0.1590) (0.31336)
4. Q(4) 0.6654 2.9799 1.6990 5.24761
(0.8134) (2.8956) (1.6647) (4.77394)
5. Q(8) 1.4652 5.0971 2.1408 6.43465
(1.5987) (4.8932) (2.0419) (5.97671)
6. Q(12) 3.9153 6.5092 6.4263 9.51398
(4.1114) (6.2864) (6.3407) (8.96821)
7. WIM 7.1709* 5.7239* 8.7964* 7.15721*
(6.4564)" (5.5730)* (8.4112)* (6.98218)*
Note: See note under table 4.1.
matter whether they are measured in the level or logarithmic
form. This may seem a bit odd since economists seldom
encounter cases where both the variable and its logarithm
are random walks. However, the Q statistics generally
confirm the white noise properties of the restricted
residuals. Almost all of the Q statistics are insignificant
as shown in both tables 4.1 and 4.2. This shows that the
increments of the spot exchange rate exhibit no serial
correlation. Furthermore, these results are not changed
when the restricted residuals are replaced by their
unrestricted counterparts. Values of all statistics in
parentheses are close to their restricted counterparts.
Combining these results together provides clearcut evidence
that the nominal spot exchange rate follows a random walk
process.
As a whole, our results provide extremely strong
evidence on the nonstationarity of spot exchange rates,
both in the level and logarithmic form. In addition, our
results show the survey expectations are also non
TABLE 4.3
RESTRICTED (UNRESTRICTED) UNIT ROOT TESTS
ON EXPECTED SPOT EXCHANGE RATES
CURRENCIES
Statistics DM SF
1. DW 0.0200 0.0075 0.0090 0.00588
(0.0199) (0.0074) (0.0090) (0.00580)
2. DF 1.2627 1.2318 0.5555 1.25603
(0.0000)+ (0.0000)+ (0.0000)' (0.00000)+
3. ADF 0.8557 0.4928 0.5309 0.59586
(0.1839) (0.5322) (0.1087) (0.33404)
4. Q(4) 6.7013 4.3505 3.8313 3.07689
(6.5578) (4.3528) (3.8438) (3.06462)
5. Q(8) 14.0664 18.8592* 14.3182 6.31395
(13.6559) (18.7621)* (14.2816) (6.28162)
6. Q(12) 16.0707 22.0778* 18.1758 6.96748
(15.6408) (21.9716)* (18.0243) (6.86930)
7. WIM 8.7954* 0.7460 0.0012 6.46704*
(9.1795)* (0.5861) (0.0187) (5.91558)*
Note: See note under table 4.1.
TABLE 4.4
RESTRICTED (UNRESTRICTED) UNIT ROOT TESTS
ON THE LOGARITHM OF EXPECTED SPOT EXCHANGE RATES
CURRENCIES
1. DW 0.0210
(0.0208)
2. DF 1.2849
(0.0000)
3. ADF 0.8695
(0.2551)
4. Q(4) 6.3661
(6.2494)
5. Q(8) 22.4215*
(21.8374)*
6. Q(12) 28.0525*
(27.2078)
7. WIM 10.7279*
(10.6235)'
Note: See note under table 4.1.
0.0070
(0.0070)
1.0843
(0.0000)+
0.3060
(0.6517)
1.5237
(1.4805)
23.6633*
(23.4625)
28.9086*
(28.5699)
4.9364*
(4.0890)
0.0085
(0.0085)
0.4965
(0.0000)+
0.4357
(0.0359)
1.6571
(1.6106)
22.0866*
(21.9455)"
29.9494*
(29.6674)*
3.8557*
(3.7502)
0.00464
(0.00456)
1.14455
(0.00000)+
0.33612
(0.54323)
2.56690
(2.53612)
6.66283
(6.60522)
8.56825
(8.44601)
7.15269
(5.70230)*
stationary, though the logarithm of the expectations may be
less likely to be. The robustness of these results provides
sufficient confidence for us to proceed further with the
restricted cointegration tests.
Results of the restricted cointegration tests are
presented in tables 4.5 and 4.6. Table 4.5 presents the
results using the level of exchange rates and their
expectations, while table 4.6 gives their counterparts in
the logarithmic form. As in the previous tables, figures in
63
parentheses are results based on the unrestricted residuals.
The first five statistics, DW, DF, ADF, RVAR and UVAR are
used to test whether Yt is cointegrated with y', while the
Q statistics are used to test the randomness of the
residuals. Note that the null hypothesis for cointegration
tests is that the variables are not cointegrated. This is
equivalent to saying that a large value of the test
TABLE 4.5
RESTRICTED (UNRESTRICTED) COINTEGRATED TESTS
IN LEVEL FORM
CURRENCIES
Statistics DM SF
1. DW 1.9397 1.9269 1.8420 1.7498
(1.9467) (1.9609) (1.8467) (1.7698)
2. DF 13.5792 13.4387 12.2783 12.3170
(13.6469) (13.5238) (12.2905) (12.3297)
3. ADF 5.4159 6.0112 5.8968 5.4539
(5.4982) (6.1214) (5.9014) (5.4587)
4. RVAR 3182.89 1714.22 873.19 1777.18
(3079.99) (2157.62) (994.49) (2021.59)
5. UVAR 3950.89 2707.09 1061.01 2564.35
(3950.89) (2707.09) (1061.01) (2564.35)
6. Q(4) 4.0587 7.0016 6.9495 4.8764
(2.7355) (6.2155) (6.4668) (4.3478)
7. Q(8) 9.6336 19.8609* 13.7977 7.2079
(8.0969) (18.5078)* (13.1403) (6.6025)
8. Q(12) 10.4481 23.2457* 18.8160 7.8899
(9.4921) (22.3792)* (18.3067) (6.9722)
9. WIM 8.5139* 0.1552 1.6293 9.6953*
(8.1194)' (0.3613) (2.0117) (10.0522)
Note: See note under table 4.1.
TABLE 4.6
RESTRICTED (UNRESTRICTED) COINTEGRATED TESTS
IN LOGARITHM FORM
CURRENCIES
Statistics DM SF V
1. DW 1.8927 1.9176 1.8137 1.7469
(1.9074) (1.9472) (1.8185) (1.7700)
2. DF 13.1739 13.2730 12.3619 12.3290
(13.3397) (13.3853) (12.3867) (12.3596)
3. ADF 5.4171 6.2789 6.0807 5.6250
(5.6019) (6.4363) (6.1025) (5.6424)
4. RVAR 2870.81 1727.20 958.67 1633.25
(2917.10) (1992.12) (1061.80) (1744.20)
5. UVAR 3753.25 2533.50 1172.87 2216.43
(3753.25) (2533.50) (1172.87) (2216.43)
6. Q(4) 4.6136 4.7108 4.6395 3.9556
(2.3079) (3.9299) (4.0208) (3.3544)
7. Q(8) 15.8527* 21.7234% 15.2921 7.6146
(13.0631) (20.4002)* (14.6575) (6.8609)
8. Q(12) 17.2410 26.3354* 23.1056* 9.3474
(14.8813) (24.8650)* (22.3648)* (7.9749)
9. WIM 10.6859* 1.9079 2.7424 5.3225*
(10.1548)* (1.5538) (3.0406) (5.5841)
Note: See note under table 4.1.
statistic means a rejection of the null hypothesis and hence
'acceptance' of cointegration between the variables. To
simplify the notation, the rejection of the cointegration
tests is not indicated in the tables because the tests
reject lack of cointegration. This means that nominal
exchange rates and their expectations are cointegrated. In
addition, they are cointegrated with a factor one as the
statistics are based on the restricted residuals. It should
be noticed that figures in parentheses are nearly the same
as their corresponding figures, showing no difference in
using the restricted or unrestricted residuals. In fact, a
careful examination of the auxiliary regressions, which give
the unrestricted residuals, reveals that the coefficients of
the regressors are very close to one, lending strong support
to the hypothesis that the value of cointegrating factor is
unity. This evidence strongly confirms the proposition that
nominal exchange rates are cointegrated with their
expectations with a factor of unity.
However, acceptance of the rational expectations
hypothesis requires more than this. The randomness of the
residuals is also an important factor. Absence of
randomness in residuals means that there exists some way of
predicting the future exchange rates, which violates the
basic requirement of rational expectations. In this
context, the Q statistic serves as an index to the
randomness of the residuals since it is designed to capture
any serial correlation between the residuals. In table 4.5,
other than the Q(8) and Q(12) statistics for the Deutsche
mark, which are significant at 5% but not at 1% level, all
the Q statistics show no serial correlation in the
restricted residuals. All the evidence in table 4.5 favors
the conclusion of cointegration with unit factor between
the variables Yt and yZ, and a random white noise process
for the residuals. In our terminology, these facts imply
that the variable yt is rationally expected by ye and the
rational expectations hypothesis in the foreign exchange
market cannot be rejected.
Results in table 4.6 are less convincing as more Q
statistics are significant at the 5% level. However, if 1%
significance level is used, the only significant statistics
will be Q(8) and Q(12) for Deutsche mark. As suggested
earlier, there is some doubt about the nonstationarity of
the exchange rate expectations in logarithmic form. Hence
cointegration tests in this case cannot be considered
conclusive. Nevertheless, most of the statistics in this
table still favor the proposition that expectations are
rational and so we have the same conclusion as in the
previous paragraph.
Summary
The purpose of this chapter is to examine three
important phenomena in the foreign exchange market. The
first question is whether the nominal exchange rates yt and
their expectations ye are random walk processes. The answer
to this question is in tables 4.1 to 4.4. Almost all the
evidence shows that these are nonstationary. The second
question is whether yt is cointegrated with y', with a co
integrating vector of unity. The answer to this question is
reported in tables 4.5 to 4.6. We see here that the two
variables are cointegrated and the values of all the
statistics testing cointegration are very close to those
using unrestricted residuals. This implies that the
restricted residuals successfully reflect the true value and
using unity as a cointegrated factor is a correct choice.
The last question is whether the residuals are stationary
and random. This can be answered by again inspecting tables
4.5 and 4.6. Cointegration between Yt and yZ with a co
integrating vector of unity means their difference is a
stationary process. Nearly all the Q statistics in the two
tables suggest that no serial correlation exists. This is
evidence that the residuals follow a white noise process.
Combining all three answers together confirms the
proposition that the rational expectations hypothesis is
accepted in the foreign exchange market.
CHAPTER V
TESTS FOR THE MARKET EFFICIENCY HYPOTHESIS:
THEORY AND EMPIRICAL RESULTS
Introduction
It is impossible to study the rational expectations
hypothesis in the foreign exchange market without referring
to the market efficiency hypothesis. The two are so related
that they are two sides of the same coin. Before the
availability of a reliable set of survey data, economists
took the forward rate as a proxy for the market expectation
of the future value of exchange rate. For example,
Dornbusch (1976) assumed that the forward rate is an
unbiased predictor of the future spot rate, while Cornell
(1977) claimed that the forward rate can be used as a proxy
for the market expectations. This is natural because, the
forward rate was the only available set of data relating to
the future spot rate and, it was generated from the market.
What economists had in mind was that if the foreign exchange
market was efficient, the forward bias, defined as the
difference between the forward rate and the corresponding
spot rate at its date of maturity, should be unpredictable.
This is because every profit opportunity in the market would
be closed by the invisible hand and hence the forward rate
would be the same as the market's expectation. This basic
logic contributed to the core of the market efficiency
hypothesis: if the market is efficient, the difference
between forward exchange rate and future spot rate is a
random error.
However, a lot of recent empirical evidence showed that
the forward exchange rate is a biased predictor of the
future spot rate. Baillie, Lippens and McMahon (1983)
rejected the hypothesis for six currencies they considered.
Hansen and Hodrick (1983) found evidence to reject this
hypothesis from the 1920s and the 1970s. Hsieh (1984)
claims that his results provided the strongest rejection of
the hypothesis ever seen. This evidence is so strong that
from the early eighties, there exists only a little argument
about the failure of the market efficiency hypothesis in the
foreign exchange market. The only disagreement revolves
around what causes the failure of the hypothesis.
Different researchers came up with different arguments
to explain this failure. After much contention, the debate
seems to have resolved down to two possible reasons: either
the failure of the rational expectations hypothesis or the
existence of a risk premium, or both. Although the number
of possible choices has been reduced tremendously, the
debate does not seem to have ended. For example, Hansen
and Hodrick (1983) found that risk premiums are important in
at least two of the five currencies they studied. Fama
70
(1984) concluded that most of the variation in forward rates
is variation in the risk premium and the premium is
negatively correlated with the expected future spot rate
components of the forward rates. Park (1984) also provided
evidence in favor of a risk premium and claimed that it
accounts for 1020% of the total variance in future spot
rates. However, Frankel and Froot (1987) claim that the
forward bias cannot be attributed to a risk premium. Froot
and Frankel (1989), once again, found no sign of risk
premium in the bias of forward exchange rate.
In the present chapter, the problem of efficiency in
the exchange rate market is discussed. Since the results in
chapter four support the rational expectations hypothesis,
it is likely that the failure of standard market efficiency
tests is due to the risk premium. Using survey expectations
data, any forward bias can be decomposed into portions
attributable to the risk premium and expectational errors.
This decomposition allows us to determine whether a risk
premium is the real cause of the failure of the market
efficiency hypothesis. The next section will be devoted to
explaining how the decomposition helps in testing market
efficiency.
Similar to previous chapters, stationarity of the
forward rate is emphasized because whether the forward rate
has a unit root will affect the reliability of the results
from the previous research. Once again, the tests mentioned
in chapter two will be applied to the forward rate. The
restricted cointegration tests will be used to test whether
the forward rate is cointegrated with the future spot rate
with a unit factor and white noise residuals. Previous
research documenting the failure of market efficiency
suggests that these two variables will not be cointegrated,
or at least not cointegrated in the way described.
Nearly all previous work in this area uses onemonth or
threemonth data since forward rates are available only on
this basis. The present research differs from this by using
data on a weekly basis. This is quite distinctive because
the weekly forward rate is not regularly reported. Even so,
it can be generated by means of the covered interest parity.
We assume that this will be a good proxy because of the
profit seeking behavior in the foreign exchange market,
where all possible revenue opportunities will be driven
away. Details about the covered interest parity and this
generated forward rate will be included in the third
section. In the final section empirical results of the
restricted cointegrated tests will be presented. This will
serve as evidence for the failure of the market efficiency
hypothesis and the existence of risk premium as a possible
answer to its failure.
The Market Efficiency
Let the market expectations of
yZ and the rationally expected spot
information set It1 be Et_[yt/Iti].
rational expectations hypothesis is
That is
Hypothesis
the future spot rate be
rate conditional on the
One way to interpret the
that yt equals Et[yt/It1].
y e = Et[y,/It1]
and a testable form of the hypothesis derived from this
relationship is
Ut = Yt Yt
The theory states that E[yt/It1] differs from Yt only by a
random error. Since the results from the last chapter
suggest that both yt and y' are nonstationary, the
hypothesis in fact requires the two variables to be co
integrated with a unit factor and vt to follow a white noise
process.
Another hypothesis called 'no risk premium in the
forward rate' states that the forward rate should equal the
market expectations, that is, y[ = y'. Combining these two
hypotheses together forms the central core of the market
efficiency hypothesis:
yt = Et[y/It1]
Again, a testable implication of this hypothesis can be
written in the following form
At = Yt Yt
Market efficiency implies that gt has zero mean and is
uncorrelated with any information in It. The analysis of
this is similar to testing the rational expectations
hypothesis in chapter four. If yf is also a nonstationary
series, the market efficiency hypothesis is equivalent to
saying that yt and y( are cointegrated with a unit factor
and the restricted residuals Mt form a white noise error.
Any violation of the above conditions is evidence rejecting
the efficiency hypothesis.
Before the availability of a reliable set of survey
data, there was no way to separate the rational expectations
and the 'no risk premium' hypothesis. The only testable
form was a joint hypothesis that involved both yt and y[,
which were the only available data set. This is one
possible reason why there is no consensus on the causes of
the failure of the hypothesis. In fact, there is no way to
identify the risk premium in the analysis and hence no way
to prove its existence. It is not until recently that the
availability of survey expectations made it possible to
decompose the two hypotheses. Since y' is accessible from
the survey, the rational expectations hypothesis can be
tested directly. This has already been conducted in chapter
four and the results showed no sign of irrationality. Then
the 'no risk premium in the forward rate' can also be tested
by considering the difference between y[ and y', that is
yf ye
By allowing the existence of a random error, the
relationship becomes
ft = yt Yt
The 'no risk premium in the forward rate' requires nt to
follow a white noise process. In particular, if both the
variables in the equation are nonstationary, the concept of
cointegration can be applied. Any violation of these
conditions will become evidence in favor of the existence of
a risk premium.
Before the presentation of empirical results, it is
necessary to explain how the weekly forward exchange rate is
generated. The theory of covered interest parity is
employed. Details of this will be discussed in the next
section.
Covered Interest Parity
The idea of covered interest parity is very simple. It
says that by means of the forward exchange market the return
from investing one dollar will be the same whether the
dollar is invested in the domestic or the foreign market.
Let Yt stand for the spot exchange rate in units of US
dollars per unit of foreign currency, and let yt and yf
represent a similar exchange rate obtained from the survey
and forward market respectively. If an investor deposits
one dollar in the US market, his returns on this investment
after one period will be (1 + ius) dollars, where ius is the
interest rate in the United States. However, if he deposits
the dollar in a foreign country and covers it through the
forward market, his returns will be yf(l + i*)/yt after one
period, where i* is the interest rate in the foreign market.
These returns must be equal, that is,
(1 + ius) = y (l + i*)/yt
or in terms of y[
y[ = yt(l + ius)/(l + iC)
This provides a formula to calculate the forward exchange
rate whenever it is not available. Values of the variables
on the righthand side are easily available in the Wall
76
Street Journal or the Financial Times. The only problem in
this parity condition would be a large transaction cost that
prevents the equality of the two investments. However,
Hsieh (1984), who used the same method to obtain a series of
weekly forward rates, pointed out that the cost of
transactions are not large enough to alter any conclusions.
Additionally, any existence of a risk premium causes no
trouble to the parity as the values of all variables are
known in the current period, so that the gain or loss can be
known exactly at this moment. This means that there are
sufficient reasons to believe the robustness of using the
generated yf as a forward rate in later analysis.
Empirical Results
The empirical results are presented in tables 5.1 to
5.6. The first two tables present the statistics of unit
root tests on the generated forward rate. Table 5.1 shows
the results from testing the hypothesis in the level form
while table 5.2 gives the counterparts in the logarithmic
form. Using the same terminology as in chapter four, the
numbers in the tables are the restricted test statistics and
numbers in parentheses are the corresponding unrestricted
values. The first three statistics, that is DW, DF and ADF,
test the existence of a unit root. None of the statistics
in tables 5.1 and 5.2 are significant at 5% level, whether
in the restricted or unrestricted form. These results lend
77
a strong support to the proposition that the forward rate is
a nonstationary process. It would be rather surprising if
both the spot rate and its expectations are nonstationary
but the forward rate is a stationary process. Together with
the unit root tests in chapter four, our results strongly
suggest that all variables in the foreign exchange market,
at least, on a weekly basis, are nonstationary. The Q
statistics in tables 5.1 and 5.2 are used to test for serial
correlation between the residuals of forward rate and its
TABLE 5.1
RESTRICTED (UNRESTRICTED) UNIT ROOT TESTS
ON THE GENERATED FORWARD EXCHANGE RATES
CURRENCIES
Statistics DM SF
1. DW 0.0185 0.0070 0.0404 0.00538
(0.0183) (0.0069) (0.0397) (0.00526)
2. DF 1.4873 1.5142 2.0160 1.26423
(0.0000)+ (0.0000)+ (0.0000)+ (0.00000)+
3. ADF 1.5872 1.5500 1.3495 1.36515
(0.3796) (0.2513) (2.4327) (0.29068)
4. Q(4) 6.7880 1.7665 57.1981* 1.57655
(6.7990) (1.5048) (55.2307) (1.35989)
5. Q(8) 7.5078 4.4429 57.5441* 1.77425
(7.5266) (4.0733) (56.1332)* (1.58596)
6. Q(12) 12.3287 8.4421 80.0100* 5.79349
(12.2915) (8.2983) (80.0250) (5.60798)
7. WIM 8.9325* 4.6552 3.7118* 8.18977*
(7.6494)* (4.6310) (5.5148)* (7.33910)*
+ Value is too small to report.
Significant at 5% level.
Note: Figures in parenthesis are values for each test using the unrestricted residuals.
TABLE 5.2
RESTRICTED (UNRESTRICTED) UNIT ROOT TESTS
ON THE LOGARITHM OF GENERATED FORWARD EXCHANGE RATES
CURRENCIES
Statistics DM SF
1. DW 0.0185 0.0063 0.0446 0.00430
(0.0183) (0.0062) (0.0437) (0.00417)
2. DF 1.5000 1.5670 2.1387 1.37455
(0.0000)+ (0.0000)+ (0.0000)+ (0.00000)+
3. ADF 1.5825 1.6091 1.3553 1.48566
(0.3519) (0.1928) (3.2265) (0.32229)
4. Q(4) 5.8413 2.4359 69.7092* 2.16717
(5.8548) (2.3241) (65.1210) (1.78748)
5. Q(8) 6.7272 6.3839 71.1988* 2.30786
(6.7769) (6.1910) (66.6002)* (1.94912)
6. Q(12) 10.7412 8.5794 88.2966* 7.89605
(10.7935) (8.4622) (87.4711) (7.39278)
7. WIM 9.6574* 2.4697 12.7123* 12.83453*
(9.7281)* (2.1208) (11.3970)* (11.16364)*
Note: See note under table 5.1.
lagged value. Except for the Swiss Franc, all the other
currencies show no sign of serial correlation in the
residuals at the 5% significance level. In particular,
figures for the restricted and unrestricted statistics are
very close to each other, showing that there is no
difference in using the restricted or unrestricted residuals
in performing these tests. The equivalence of these two
residuals gives further confidence about the results of the
unit root tests.
79
The highly significant Q statistics for the Swiss Franc
show that this currency is quite different from the others
as its residuals exhibit high serial correlation. After a
careful examination, these high serial correlations are
found to be the consequence of a highly volatile 7day Swiss
interest rate. The sample variances of the interest rate
for all five countries, namely, the United States, Great
Britain, West Germany, Switzerland and Japan, are 1.2037,
2.8569, 0.7936, 10.7336 and 1.5833 respectively. Notice
that the interest rate for Switzerland is at least three and
a half times to thirteen times more volatile than in the
other countries. Since the interest rate is an important
component in covered interest parity, its high volatility
has significant effects on the generated forward rate. For
this reason, it would be dangerous to draw any conclusion,
either positively or negatively, from the results of the
Swiss Franc. Nevertheless, the high significance of all the
unit root statistics, together with the strong evidence from
other currencies, is sufficient to support the proposition
that the forward exchange rate is nonstationary.
After determining the existence of a unit root in the
forward rate, the concept of cointegration can be applied.
The first set of variables to be tested for cointegration
is the forward rate and the corresponding spot rate. This
is equivalent to testing the market efficiency hypothesis.
The results are listed in tables 5.3 and 5.4, with the usual
80
format that the first table gives the results from using the
level of the variables while the second table gives similar
results for the logarithmic form. The figures in the tables
are statistics obtained from the restricted tests and the
figures in parenthesis under them are the unrestricted
statistics. As in the tables in chapter four, the first
TABLE 5.3
RESTRICTED (UNRESTRICTED) COINTEGRATED TESTS
IN LEVEL FORM
CURRENCIES
Statistics DM SF
1. DW 1.3134 1.2015 1.8148 0.7992
(1.3396) (1.4326) (1.9185) (1.1353)
2. DF 10.8832 9.9509 14.0278 7.4727
(11.2301) (11.4912) (14.9760) (9.6047)
3. ADF 4.1219 3.5910 3.8791 2.0349#
(4.4106) (4.6170) (4.6160) (3.2875)4
4. RVAR 103.84 58.67 74.63 52.09
(484.58) (842.55) (309.91) (343.48)
5. UVAR 756.96 1270.84 352.20 524.56
(756.96) (1270.84) (352.20) (524.56)
6. Q(4) 18.8437* 16.6717* 37.9994* 22.7927*
(14.1628)* (12.2973)* (31.8220)* (20.3231)
7. Q(8) 20.1405" 20.2191* 38.9374* 26.4670*
(15.3443) (14.9319) (32.8547)* (23.1337)*
8. Q(12) 23.9973* 23.3382* 56.8005* 31.7049*
(19.7549) (17.6883) (48.5617)* (26.7408)*
9. WIM 7.5910* 23.0573* 32.8941" 0.0108
(1.8002) (2.1769) (4.4429)* (2.3056)
Note: See note under table 5.1.
Not significant at 5% Level.
TABLE 5.4
RESTRICTED (UNRESTRICTED) COINTEGRATED TESTS
IN LOGARITHM FORM
CURRENCIES
Statistics DM SF V
1. DW 1.2184 1.4664 2.0539 0.9756
(1.3225) (1.4869) (2.0179) (1.1182)
2. DF 10.2712 11.6631 16.1632 8.5103
(11.2469) (11.8975) (15.9829) (9.5327)
3. ADF 3.8574 4.7883 4.8632 2.6105
(4.5481) (5.0074) (4.8124) (3.3489)
4. RVAR 108.50 64.45 87.57 60.80
(525.83) (961.18) (316.32) (345.63)
5. UVAR 807.76 1456.77 356.41 538.17
(807.76) (1456.77) (356.41) (538.17)
6. Q(4) 21.0980" 11.5887" 30.8954" 21.3194*
(10.5440)* (10.3323)" (30.3248)* (19.2380)*
7. Q(8) 22.7571* 15.1395 32.9769* 24.9134*
(11.8103) (13.5730) (32.2937)* (22.1948)*
8. Q(12) 25.7405* 17.6372 48.7884* 30.2574*
(16.1365) (15.9874) (48.1922)* (26.3840)*
9. WIM 0.2539 70.3592" 69.8353* 13.4684*
(7.5707)* (3.0687) (5.4836)* (2.1176)
Note: See note under table 5.1.
five statistics, DW, DF, ADF, RVAR and UVAR, are tests for
cointegration under the null hypothesis that no co
integration exists between the two variables. Hence a large
value of the statistics will represent a rejection of the
null hypothesis or, in other words, an acceptance of co
integration.
Almost all the cointegration statistics in these two
tables are large enough to reject the null hypothesis of no
cointegration at the 5% significance level. The only
exception is the restricted ADF statistics for the Japanese
Yen, both in the level and the logarithmic form. However,
both of them become significant if the unrestricted
statistics are considered instead. In fact, the results in
the tables outside and inside the parentheses are rather
different. This difference becomes extreme in the case of
the RVAR statistics, where the values in parentheses are at
least four times higher. The Q statistics also have high
significant values, showing serial correlations as being
important in the restricted residuals.
These results contain two important points. The
forward exchange rate and the corresponding spot rate may be
cointegrated, because nearly all the statistics are
significant in rejecting the null hypothesis. But it is
very likely that they are not cointegrated with a factor
one because of the great difference between the restricted
and unrestricted statistics. Additional evidence comes from
the restricted Q statistics, which have a higher rejection
rate than the unrestricted case. This means that
constraining the cointegration factor to unity introduces
serial correlation in the residuals. This suggests that the
forward and spot rate are not cointegrated with a factor
one, and their difference is not a random white noise error.
This is equivalent to saying that, as most of the recent
research did, the market efficiency hypothesis is rejected.
We can now decompose the forward bias into an
expectation error and a risk premium. Using the following
formula, the forward bias can be rewritten as
Yt yt = (Yt ye) (y: yt)
Since the rational expectations hypothesis is accepted based
on the results from chapter four, the difference between yt
and ye would be a white noise error, say et. In other
words, the equation becomes
Yt Yt = 6t (Y: yt)
Hence, any bias that exists on the left hand side must come
from the difference (yZ yf). A test of this hypothesis,
denoted as the 'no risk premium' hypothesis, can be written
as
yt yt = Ut
If there is no risk premium in the forward exchange market,
vu must be a white noise error. Any violation of this
formulation, either in the coefficient of the forward rate
or in the randomness of the error, is evidence of risk
premium and is support for the presumption that risk premium
is the major cause for the rejection of the market
efficiency hypothesis.
Results of this 'no risk premium' tests are presented
in tables 5.5 and 5.6, testing the level and the
logarithmic form of the variables. In general, we cannot
TABLE 5.5
RESTRICTED (UNRESTRICTED) 'NO RISK PREMIUM' TESTS
IN LEVEL FORM
CURRENCIES
Statistics DM SF
1. DW 0.4104 0.3816 1.7573 0.2545#
(0.4172) (0.5486) (1.8407) (0.4748)
2. DF 5.1249 4.7463 14.1153 3.6879
(5.1623) (5.9825) (14.7760) (5.2680)
3. ADF 3.2658 2.1398 3.8458 0.8313
(3.2931)# (2.8676)# (4.6563) (1.7349)#
4. RVAR 6.3667# 3.3106" 57.3573 4.4415#
(42.1648) (54.7821) (249.9081) (48.4715)
5. UVAR 43.4438 56.1800 262.9344 50.5202
(43.4438) (56.1800) (262.9344) (50.5202)
6. Q(4) 48.2803* 45.8696*' 44.9390* 29.1464*
(46.2234) (40.2844)* (44.6178) (32.9468)*
7. Q(8) 59.9273* 53.7556* 45.9077* 43.0062*
(58.1069) (49.7513)* (45.7182)* (47.6340)*
8. Q(12) 62.6342* 60.4681* 78.3148* 45.5293*
(60.5989) (56.4360)* (77.7783)* (50.3223)*
9. WIM 200.8925* 196.7702* 4.9741* 17.3664*
(2.3042) (2.3100) (11.9574)* (0.0044)
Note: See note under table 5.1.
Not significant at 5% Level.
TABLE 5.6
RESTRICTED (UNRESTRICTED) 'NO RISK PREMIUM' TESTS
IN LOGARITHM FORM
CURRENCIES
Statistics DM SF
1. DW 0.3866 0.6309 1.9752 0.4177
(0.4478) (0.6308) (1.9461) (0.5015)
2. DF 4.9713 6.3374 15.5185 4.9326
(5.4873) (6.3369) (15.2613) (5.4782)
3. ADF 3.1339# 2.8355# 4.8044 1.5026#
(3.4987)# (2.8341)# (4.7135) (2.0082)#
4. RVAR 6.7686# 3.4698# 65.5005 6.6391#
(45.9360) (60.6551) (265.6032) (60.5095)
5. UVAR 47.1672 61.6078 279.2266 62.8484
(47.1672) (61.6078) (279.2266) (62.8484)
6. Q(4) 55.0510* 37.5930* 48.0486* 26.9205*
(42.0222) (37.4967)* (48.5537) (28.1553)*
7. Q(8) 71.2517* 49.6387* 55.0649* 37.9044*
(59.5246)* (49.6676)* (55.4194)* (38.8121)
8. Q(12) 75.5385* 55.7035* 83.8934* 39.7021*
(63.1332)* (55.7357)* (85.0640)* (40.5636)*
9. WIM 190.2147* 209.3365* 55.8230* 87.2403*
(8.6399)* (1.0970) (17.9048)* (0.3741)
Note: Sre note under table 5.1.
Not significant at 5% Level.
reject the null hypothesis of no cointegration based on the
ADF and RVAR statistics. In particular, the values of the
RVAR statistics change greatly under the restricted and
unrestricted case, showing either that the two variables are
not cointegrated or that they are cointegrated with a
factor other than one. In any case, the 'no risk premium'
hypothesis is definitely rejected. This idea is further
confirmed by the Q statistics in both tables. None of them
has a value less than 25, showing that a high serial
correlation exists between the residuals. This conclusion
is not affected by using the unrestricted instead of the
restricted residuals, or by taking the variable in the
logarithmic form. This provides strong evidence that it is
the existence of a risk premium that causes the failure of
the market efficiency hypothesis.
Summary
This chapter tested the stationarity of the forward
exchange rate and the validity of the market efficiency
hypothesis in the foreign exchange market. From chapter
four, since we found that both the spot rate and its
expectations are nonstationary, it is very likely that the
forward rate follows a similar process. Tables 5.1 and 5.2
provide support for this. Another result obtained from
chapter four is the acceptance of the rational expectations
hypothesis. Because of this, any inefficiency in the
exchange market would be the consequence of a risk premium.
Tables 5.3 and 5.4 provide evidence for the failure of the
market efficiency hypothesis. The last two tables show
indirectly the existence of risk premium.
CHAPTER VI
SUMMARY AND CONCLUSION
Several important issues relating to the dynamics of
nominal exchange rates are studied in the previous chapters.
These include the stationarity of the nominal rates, the
validity of the rational expectations hypothesis and the
market efficiency hypothesis. Results in chapter four and
five confirm clearly that the nominal spot rate, its
expected future value and the nominal forward rate are all
nonstationary. This is at variance with the results of
previous research. This could be partly attributed to the
fact that this study uses weekly data, whereas the others
use monthly or quarterly data. It has been shown that the
distribution of the test statistics will be different from
the traditional normal distributions when the variables
under consideration are nonstationary. By using the theory
of cointegration, the rational expectations hypothesis is
reformulated. The hypothesis requires the nominal spot rate
and its expected value to be cointegrated with a unit
factor, and requires the residuals to follow a white noise
process. Based on this result, tests for cointegration are
applied to the nominal exchange rate and its expected value.
88
Four currencies, including the British pound, Deutsche mark,
Swiss Franc and Japanese Yen, against US dollar are employed
to test rationality and the results are presented in chapter
four. All the test results are conclusive. First, all the
cointegration statistics reject the null hypothesis of no
cointegration, thus suggesting that the nominal spot rate
and its expected value are perhaps cointegrated. Second,
all the three Q statistics show no sign of serial
correlation in the residuals, confirming that the residual
series is white noise. Finally, values of all the
restricted test statistics are extremely close to their
unrestricted counterparts, suggesting that there is no
difference in using either one of them. In other words,
restricting the cointegration factor to be unity is
legitimate. Combining all this evidence helps us to
conclude that the rational expectations hypothesis is
accepted in the foreign exchange market.
By contrast, the market efficiency hypothesis is
rejected when the same procedure is applied to the forward
rate and its corresponding future spot rates. Though the
two series may still be cointegrated, the residual is not a
white noise process. This conclusion is based on the high Q
statistics values in tables 5.3 and 5.4. Moreover, the
large difference between the restricted and unrestricted
test statistics, at least for some of the tests, can be
considered as supportive of the fact that restricting the
cointegration factor to be one is invalid. Putting this
evidence together suggests that even though there may be a
long run relationship between the forward rate and the spot
rate, they are not related in a way described by the market
efficiency hypothesis.
Earlier studies could not determine satisfactorily the
causes of the failure of the market efficiency hypothesis
because the forward bias could not be decomposed. With the
availability of survey data on expectations, the forward
bias can be decomposed into two components, expectations
error and risk premium. As the acceptance of the rational
expectations hypothesis is evidence in favor of a random
expectations error, the only cause of the failure of the
market efficiency hypothesis would appear to be the
existence of risk premium. This proposition is tested in
chapter five and the results are listed under the 'no risk
premium' tables. High Q statistics as well as large
difference in values between the restricted and the
unrestricted tests reject the null hypothesis of no risk
premium in the market.
Our empirical results are based entirely on the weekly
data which has seldom been used in previous research. This
is especially true with the studies of the market efficiency
hypothesis as the forward rate is not available on a weekly
basis. The difference in the time span used may be one
major reason why we get results contradicting earlier work.
90
In fact, we can observe higher absolute values of the DF and
ADF test statistics when the restricted cointegration tests
are applied to monthly survey data. It seems fair to say
that economic agents can predict accurately within a short
time span, but not so accurately when the time span is
extended, say, to a month.
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