Group Title: dynamics of nominal exchange rates
Title: The dynamics of nominal exchange rates
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Title: The dynamics of nominal exchange rates
Physical Description: vii, 95 leaves : ; 28 cm.
Language: English
Creator: Liu, Peter Chi-Wah, 1957-
Copyright Date: 1989
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Subject: Foreign exchange   ( lcsh )
Rational expectations (Economic theory)   ( lcsh )
Economics thesis Ph.D   ( lcsh )
Dissertations, Academic -- Economics -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Statement of Responsibility: by Peter Chi-Wah Liu.
Thesis: Thesis (Ph. D.)--University of Florida, 1989.
Bibliography: Includes bibliographical references (leaves 91-94)
General Note: Typescript.
General Note: Vita.
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Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: alephbibnum - 001572376
oclc - 22853901
notis - AHJ6199

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THE DYNAMICS OF NOMINAL EXCHANGE RATES


BY

PETER CHI-WAH 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,

Wai-han 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 Co-integration............ 19
Tests for Co-integration .......................... 20
Relationship Between Co-integration 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 Chi-wah 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

non-stationarity 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 non-stationary, 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 co-integrating 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

non-stationarity. 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, non-stationarity
and random walk process are considered equivalent. This is
somewhat restrictive. However, it will simplify our work
without affecting the main results.










variables are non-stationary 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 over-rejection 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 re-interpreted if the non-stationarity 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 non-stationarity, the

present dissertation examines the rational expectations

hypothesis in detail, with special emphasis on the dynamics

of the variables concerned. As it involves non-stationary

series, the theory of co-integration will also be included.

This concept is important because it describes the

relationship between two non-stationary 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 co-integrated 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 co-integration 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 co-integration

but the hypothesis of rational expectations that we are

interested in testing. In later chapters, these modified

tests will be called restricted co-integrated 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 non-stationarity 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 non-stationary

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 non-stationary,

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 one-week and one-month (30-day)

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

one-week 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 one-week forward exchange rate, as suggested by Hsieh

(1984), the idea of covered interest parity will be used.

The formula also uses the seven-day 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' auto-regressive 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 co-integration will follow immediately, acting as a

background material for the set of rational expectations

tests. These tests are based on the co-integration tests

suggested by Engle and Granger (1987). Finally, the concept

of co-integration, 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 non-stationary,

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 co-integration 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 non-stationary,

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 non-stationary will also be discussed in the next










section. A set of tests of the rational expectations

hypothesis, applicable to non-stationary series, will also

be discussed.

In order to handle the dynamics of variables properly,

when testing the rational expectations hypothesis, the

theory of co-integration 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 co-integration theory fits into our context,

a special section will be devoted to explaining this

concept. This section will include the co-integration tests

suggested by Engle and Granger (1987). Finally we discuss

the modifications needed when the concept of co-integration

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 It-I. Hence,

the following equation is often estimated:


Yt Yt = oY + 1Yt-i + 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- + 27Yt-1 + et



or,



yt yt = 0 + p1(yt-1 yt-1) + 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 un-correlated. 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 yt-1 or (yt-1 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 un-correlated with

any variable in the information set It-i. 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 un-correlated 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

It-1.

2. Regress ye on the same set of variables in It-i.

3. Test the equality of the coefficients in the two

regressions using Chow tests.



As long as the variables in It-i are not un-correlated 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 non-stationary, 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 non-stationary 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 yt-i, the assumption of a

stationary white noise regression error becomes questionable

if yt-i 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 + iYt-i + et



cannot be stationary if y t- is not. Because it is not

meaningful to have a stationary series on the left-hand side

of an equality sign but a non-stationary 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 = Yt-1 + Et and et N(0, ac)

Yt = y-i + 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 non-stationary. In order to analyze

the problem suitably, the theory of co-integration, 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 Co-integration

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 co-integration 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 long-run

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 'co-integrated 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 co-integration theory. By

using it, tests for the rational expectations hypothesis

when the variables are non-stationary can be formulated.

Before deriving these tests, we will discuss the tests for

co-integration proposed by Engle and Granger (1987).



Tests for Co-integration

Engle and Granger (1987) proposed seven different tests

for co-integration 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 co-integration would be rejected.

In that case, wt and yt are co-integrated and the estimated

coefficient will be the co-integrating 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 t-1 + 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 Yt-I = A t + 7(Wt wt-1) + 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 Yt-1) 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

co-integration constraints. Two regressions are

estimated:

wt wt-i = a1 + 1Pwt-, + riYt-I + Qit ; and

Yt yt-1 = az + 32wt-I + T2yt-I

+ S(wt wt-1) + 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, wt-1) and (yt yt-1) are included in the two

equations in the UVAR test. The statistic is obtained

by the same procedure as the UVAR test.



Relationship Between Co-integration and Rational
Expectations

Co-integration 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 non-stationary, 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 co-integrated

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

co-integration requires e, to be stationary, but rationality

goes one step further by requiring e, to be a white noise

error.

However, the tests of co-integration from Engle and

Granger (1987) cannot be applied directly to test for the

rational expectations hypothesis as the latter hypothesis

requires more than co-integration. Engle and Granger (1987)

require only a stationary estimated residual, but not any

restriction on the co-integrating factor or the randomness

of the estimated error term. It can happen that yt and y'

are co-integrated 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, co-integration 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 co-integrated;

2. The co-integrating factor must be one;

3. The difference, yt yt, must be a white noise

process with no serial correlation.










In order to adjust the co-integration 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

co-integrated with a factor one, provided that both of them

are I(1) non-stationary 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 co-integrated 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 log-likelihood 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 log-likelihood function, that is, a21ogft({)/e a89';

Fi{() is the (pxl) vector of the first derivatives of the

log-likelihood 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 = Xt-i + Et



then as the number of observations T tends to infinity, we

have the following results



T-3/2Zxt ---a o;W(t)dt ;


T-2EZX --a. O1[W(t)dt ;




T- Zxet --a {- (C [W()]2 n0) ;



T Z2 a.-s. lim (T-ZEt() ) =
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

co-integration 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 log-likelihood

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

log-likelihood function, the non-singularity 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. JT-Di(e) N(0, V{8))

2. V( )} -- -.. { ) ;

3. 'T = T-DT{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 log-likelihood

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

log-likelihood 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{ }] A-Y{ )',-



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 non-stationary 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 log-likelihood 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

Eyt-i.-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 non-stationary, their

difference, Et, has to be stationary if the null hypothesis

is true. Under H0, then, one can rewrite the log-likelihood

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 non-stationary. 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 non-stationary

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 Yt-1, 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 = PYict-n + 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 size-adjusted 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 size-adjusted 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 size-adjusted 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 size-adjusted 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 size-adjusted 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 non-stationarity 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 log-likelihood 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

co-integration, 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

co-integration, 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 co-integration 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 well-specified 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 non-stationary, 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 non-stationarity 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 out-of-sample

forecasting accuracy. The models they included were

flexible-price and sticky-price monetary models, as well as

a sticky-price model that incorporates the current account.

The out-of-sample 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 non-stationary. 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 non-random 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 out-of-sample and in-sample 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 non-stationarity 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 non-stationarity 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 non-interesting

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 cross-sectional

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. Non-stationarity

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 one-month and three-month 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 Carlson-Parkin 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 co-integration 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 + PXt-i + Pt



while the latter uses the restricted residuals defined as



9t = Xt Xt-1



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

log-normal 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 clear-cut evidence

that the nominal spot exchange rate follows a random walk

process.

As a whole, our results provide extremely strong

evidence on the non-stationarity 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 co-integration tests.

Results of the restricted co-integration 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 co-integrated with y', while the

Q statistics are used to test the randomness of the

residuals. Note that the null hypothesis for co-integration

tests is that the variables are not co-integrated. This is

equivalent to saying that a large value of the test



TABLE 4.5

RESTRICTED (UNRESTRICTED) CO-INTEGRATED 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) CO-INTEGRATED 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 co-integration between the variables. To

simplify the notation, the rejection of the co-integration

tests is not indicated in the tables because the tests

reject lack of co-integration. This means that nominal

exchange rates and their expectations are co-integrated. In










addition, they are co-integrated 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 co-integrating factor is

unity. This evidence strongly confirms the proposition that

nominal exchange rates are co-integrated 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 co-integration 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 non-stationarity of

the exchange rate expectations in logarithmic form. Hence

co-integration 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 non-stationary. The second

question is whether yt is co-integrated 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 co-integrated and the values of all the

statistics testing co-integration are very close to those

using unrestricted residuals. This implies that the

restricted residuals successfully reflect the true value and

using unity as a co-integrated 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. Co-integration 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 10-20% 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 co-integration tests will be used to test whether

the forward rate is co-integrated 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 co-integrated,

or at least not co-integrated in the way described.

Nearly all previous work in this area uses one-month or

three-month 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 co-integrated 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 It-1 be Et_-[yt/It-i].

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/It-1].


y e = Et[y,/It-1]



and a testable form of the hypothesis derived from this

relationship is



Ut = Yt Yt



The theory states that E[yt/It-1] differs from Yt only by a

random error. Since the results from the last chapter

suggest that both yt and y' are non-stationary, 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/It-1]



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 non-stationary

series, the market efficiency hypothesis is equivalent to

saying that yt and y( are co-integrated 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 non-stationary, the concept of

co-integration 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 right-hand 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 non-stationary process. It would be rather surprising if

both the spot rate and its expectations are non-stationary

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 non-stationary. 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 7-day 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 non-stationary.

After determining the existence of a unit root in the

forward rate, the concept of co-integration can be applied.

The first set of variables to be tested for co-integration

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) CO-INTEGRATED 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) CO-INTEGRATED 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

co-integration 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 co-integration statistics in these two

tables are large enough to reject the null hypothesis of no

co-integration 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

co-integrated, because nearly all the statistics are

significant in rejecting the null hypothesis. But it is

very likely that they are not co-integrated 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 co-integration factor to unity introduces

serial correlation in the residuals. This suggests that the

forward and spot rate are not co-integrated 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 co-integration 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 co-integrated or that they are co-integrated 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 non-stationary, 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

non-stationary. 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 non-stationary. By using the theory

of co-integration, the rational expectations hypothesis is

reformulated. The hypothesis requires the nominal spot rate

and its expected value to be co-integrated with a unit

factor, and requires the residuals to follow a white noise

process. Based on this result, tests for co-integration 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

co-integration statistics reject the null hypothesis of no

co-integration, thus suggesting that the nominal spot rate

and its expected value are perhaps co-integrated. 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 co-integration 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 co-integrated, 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










co-integration 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 co-integration 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.















BIBLIOGRAPHY


Baillie, Richard T., Lippens, Robert E. and McMahon, Patrick
C. (1983), 'Testing Rational Expectations and
Efficiency in the Foreign Exchange Market,'
Econometrica, 51:553-563.

Carlson, John A. (1977), 'A Study of Price Forecasts,'
Annals of Economic and Social Measurement, 6:27-56.

Chesher, Andrew D. (1983), 'The Information Matrix Test:
Simplified Calculation Via a Score Test
Interpretation,' Economics Letters, 13:45-48.

Cornell, Bradford (1977), 'Spot Rates, Forward Rates and
Exchange Market Efficiency,' Journal of Financial
Economics, 5:55-65.

Dickey, David A. and Fuller, Wayne A. (1979), 'Distribution
of the Estimators for Autoregressive Time Series With a
Unit Root,' Journal of the American Statistical
Association, 74:427-431.

Dickey, David A. and Fuller, Wayne A. (1981), 'Likelihood
Ratio Statistics for Autoregressive Time Series With a
Unit Root,' Econometrica, 49:427-431.

Dominguez, Kathryn M. (1986), 'Are Foreign Exchange Forecast
Rational? New Evidence from Survey Data,' Economics
Letters, 21:277-281.

Dornbusch, Rudiger (1976), 'The Theory of Flexible Exchange
Rate Regimes and Macroeconomic Policy,' Scandinavian
Journal of Economics, 78:255-275.

Engle, Robert F. and Granger, C. W. J. (1987), 'Co-
integration and Error Correction: Representation,
Estimation, and Testing,' Econometrica, 55:251-300.

Fama, Eugene F. (1984), 'Forward and Spot Exchange Rates,'
Journal of Monetary Economics, 14:319-338.









Frankel, Jeffrey A. and Froot, Kenneth A. (1986), 'Using
Survey Data to Test Some Standard Propositions
Regarding Exchange Rate Expectations,' Research Papers
in Economics No. 86-11, Institute of Business and
Economic Research, University of California-Berkeley.

Frankel, Jeffrey A. and Froot, Kenneth A. (1987), 'Using
Survey Data to Test Standard Propositions Regarding
Exchange Rate Expectations,' American Economic Review,
77:133-153.

Fratianni, Michele, Hur, Hyung-Doh and Kang, Heejoon (1987),
'Random Walk and Monetary Causality in Five Exchange
Markets,' Journal of International Money and Finance,
6:505-514.
Frenkel, Jacob A. (1981), 'Flexible Exchange Rates, Prices,
and the Role of "News": Lessons from the 1970s,'
Journal of Political Economy, 89:665-705.

Friedman, Benjamin M. (1980), 'Survey Evidence on the
Rationality of Interest Rate Expectations,' Journal of
Monetary Economics, 6:453-465.

Froot, Kenneth A. (1989), 'New Hope for the Expectations
Hypothesis of the Term Structure of Interest Rates,'
The Journal of Finance, 44:283-305.

Froot, Kenneth A. and Frankel, Jeffrey A. (1989), 'Forward
Discount Bias: Is it an Exchange Risk Premium?' The
Quarterly Journal of Economics, 104:139-161.

Fuller, Wayne A. (1976), Introduction to Statistical Time
Series, New York: John Wiley & Sons.

Hakkio, Craig S. (1986), 'Does the Exchange Rate Follow a
Random Walk? A Monte Carlo Study of Four Tests for a
Random Walk,' Journal of International Money and
Finance, 5:221-229.

Hakkio, Craig S. and Rush, Mark (1989), 'Market Efficiency
and Cointegration: An Application to the Sterling and
Deutschemark Exchange Markets,' Journal of
International Money and Finance, 8:75-88.

Hansen, Peter Lars and Hodrick, Robert J (1983), 'Risk
Averse Speculation in the Forward Foreign Exchange
Market: An Econometric Analysis of Linear Models,' in
Jacob A. Frenkel, ed., Exchange Rates and International
Macroeconomics, Chicago: The University of Chicago
Press, 113-142.









Hsieh, David A. (1984), 'Tests of Rational Expectations and
No Risk Premium in Forward Exchange Markets,' Journal
of International Economics, 17:173-184.

Hodrick, Robert J. and Srivastava, Sanjay (1984), 'An
Investigation of Risk and Return in Forward Foreign
Exchange,' Journal of International Money and Finance,
3:5-29.

Hodrick, Robert J. and Srivastava, Sanjay (1986), 'The
Covariation of Risk Premiums and Expected Future Spot
Exchange Rates,' Journal of International Money and
Finance, 5:5-21.

Huang, Roger D. (1984), 'Some Alternative Tests of Forward
Exchange Rates as Predictors of Future Spot Rates,'
Journal of International Money and Finance, 3:153-167.

Ito, Takatoshi (1988), 'Foreign Exchange Rate Expectations:
Micro Survey Data,' NBER, Working Paper Series no.
2679.

Lancaster, Tony (1984), 'Notes and Comments: The Covariance
Matrix of the Information Matrix Test,' Econometrica,
52:1051-1053.

Lovell, Michael C. (1986), 'Tests of the Rational
Expectations Hypothesis,' American Economic Review,
76:110-124.

MacDonald, R. and Torrance, T. S. (1988), 'On Risk,
Rationality and Excessive Speculation in the
Deutschmark-US Dollar Exchange Market: Some Evidence
Using Survey Data,' Oxford Bulletin of Economics and
Statistics, 50:107-123.

Mankiw, N. Gregory and Shapiro, Matthew D. (1986), 'Do We
Reject Too Often?: Small Sample Properties of Tests of
Rational Expectations Models,' Economics Letters,
20:139-145.

Meese, Richard A. and Rogoff, Kenneth (1983), 'Empirical
Exchange Rate Models of the Seventies: Do They Fit Out
of Sample?' Journal of International Economics, 14:3-
24.

Mullineaux, Donald J. (1978), 'On Testing for Rationality:
Another Look at the Livingston Price Expectations
Data,' Journal of Political Economy, 86:329-336.




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