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My master thesis on exchange rate modelling and imperfect knowledge. With a exchange rate model including risk, based on the Vector Auto Regressive (VAR) method.
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D E T S A M F U N D S V I D E N S K A B E L I G E F A K U L T E T K Ø B E N H A V N S U N I V E R S I T E T
Kandidatspeciale David Pedersen
Imperfect Knowledge Economics A solution to the exchange rate disconnect puzzle?
Vejleder: Michael Bergman
Afleveret den: 21/04/08
Summary Traditional exchange rate models based on the monetary approach have had a rather hard time at
explaining the fluctuations of exchange rates over the last thirty years. Mainstream exchange rate
models assume that macroeconomic fundamentals determine the value of the exchange rate, at
least in the long run. But this sharp prediction has been rejected in several empirical studies,
showing that there is a rather weak connection between the variables. A surprising result which
has been labeled “the exchange rate disconnect puzzle”.
The theory of Imperfect Knowledge economics (IKE), suggested by Michael D. Goldberg and
Roman Frydman (2007), puts forth a proposal for solving the apparent puzzle: Agents are
assumed to be heterogeneous. Furthermore, it is assumed that the agents acknowledge that their
understanding of the true model of the economy is limited. Thus, by definition, the overall result
of the IKE model has to be different from that of the rational expectations outcome given by the
monetary approach to exchange rates.
The structure of the thesis Overall, the thesis is divided into two parts: The theoretical part of IKE and the empirical test of
one of the assumptions of the theory, the importance of uncertainty in regards to exchange rate
determination. This is chapter 4 and 5 of the thesis, respectively. Before the discussion and test of
the IKE theory, chapter 2 and chapter 3 discuss the foreign exchange market and the economic
agents participating in the market (chapter 2), as well as the theory and empirical results of the
monetary approach (chapter 3).
Stylized facts – chapter 2 This chapter both discusses the time series properties of three of the most traded currencies, as
well as the agents and the market in which the currencies are traded. Several findings are
reported: Exchange rates are non-normal, has over-kurtosis; the market is decentralized and has a
rather low degree of transparency; and the agents are heterogeneous and use different methods
for the calculation of future exchange rates.
Exchange rates: models and puzzles – chapter 3 Chapter 4 discusses the mainstream approach to exchange rates, namely the monetary approach.
Two typical monetary models, the flexible price monetary model (FPMM) and the sticky price
2
monetary model, are presented and discussed. This is followed by a short study of the empirical
achievements of the monetary approach. The overall conclusion of the monetary approach’
empirical success is not supportive; hence the exchange rate disconnect puzzle, the purchasing
power parity puzzle and the forward premium puzzle. Thus, it seems that a new approach to the
modeling of exchange rates seems warranted. One such approach is discussed in this chapter as
well, the microstructure approach. The microstructure approach relaxes some of the more critical
assumptions of the monetary approach: That private information can have a significant effect,
that agents are heterogeneous, and that the institutions and structure of the exchange rate market
matter for the determination of exchange rates. Assumptions which are in line with the findings
of chapter 2. Testing a simple microstructure model with order flow included shows that private
information can have a significant effect; a result supported by several articles. Thus, the sharp
assumptions of the monetary approach seem to impede useful information on exchange rates.
The theoretical part – chapter 4 The main theme of the IKE theory is the importance of imperfect knowledge; and the
assumption that the agents recognize that their (individual) knowledge is limited. Hence, the
agents act like scientists, testing various strategies over time. Furthermore, the agents do not only
use strict macroeconomic models for forecasting the future exchange rate, but make use of
technical trading, insights from the microstructure approach and their experience. The IKE
theory adds the findings from prospect theory to the description of the agents utility: The aspect
of loss aversion (i.e. the disutility from a loss exceeds the utility of gains of the same size),
reference dependence (utility is defined relative to a reference point), and diminishing sensitivity
(i.e. the marginal utility of both gains and losses decreases with their size). Furthermore it is
assumed that loss aversion increases with the position size, preventing agents from taking
unlimited positions in the foreign exchange market. When these assumptions are coupled with
the “gap effect” – the difference between the historical benchmark value of the exchange rate
and the expected value of the exchange rate have an effect on the expected utility of agents – and
conservative revisions of strategy – i.e. agents stick to their strategies and are slow to revise them
– the exchange rate disconnect puzzle can be explained: The exchange rate diverge from PPP (i.e.
the historical benchmark) because agents are heterogeneous and thus have different expectations
(bulls or bears). But the gap effect, i.e. increases of loss aversion, pulls back the exchange rate if it
is “too” misaligned. Therefore the exchange rate does not entirely abandon the fundamental
value.
3
The empirical part – chapter 5 In the empirical part a VAR model is used when testing one of the main assumptions of the IKE
theory: The importance of uncertainty. Running a GARCH model on the macroeconomic
fundamentals provide a proxy for uncertainty, as the conditional variance changes according to
changes in the fundamentals. Adding the GARCH variables to a simple monetary model is tested
to be a stationary relation for Japan, pointing to the significance of uncertainty.
Main conclusion – chapter 6 The overall conclusion supports the IKE theory: From a theoretical point, the theory
incorporates several stylised facts, such as heterogeneous agents and the use of different methods
and models. From an empirical point, the test of the importance of private information as well as
the VAR model seems to support the IKE theory as well.
Thus, I cannot reject the hypothesis that the IKE theory can explain the exchange rate
disconnect puzzle.
4
Imperfect knowledge economics
- A solution to the exchange rate disconnect puzzle?
David Pedersen
April 2008
5
Preface For this thesis I have been inspired by the course in International Monetary Economics as well as
the seminar Empirical International Finance.
First of all a special thanks to my advisor, Michael Bergman, for guidance, critical questions and
useful discussions as well as a high degree of patience.
Other persons have given valuable remarks as well, which I appreciate: Thanks to Michael D.
Goldberg for answering several questions regarding imperfect knowledge economics (IKE) and
thanks to Katarina Juselius for introducing me to the interesting ideas of IKE in the early spring
of 2007. Thanks to Lars Christensen, Chief Analyst at Emerging Market Research at Danske
Bank, and Teis Knuthsen, Head of FX Research at Danske Bank, for interesting discussions of
exchange rates, seen from a practical point of view.
Last but not least, thanks to Gitte and my beautiful daughter, Freja, for putting up with me over
the months, and making the process much more fun.
6
Contents
CHAPTER 1: INTRODUCTION....................................................................................................9
1.1 INTRODUCTION ........................................................................................................................................9
1.2 DEFINITION: THE EXCHANGE RATE DISCONNECT PUZZLE..........................................................11
1.3 DELIMITATION........................................................................................................................................12
1.4 STRUCTURE OF THE THESIS ..................................................................................................................12
CHAPTER 2: STYLISED FACTS.................................................................................................. 15
2.1 INTRODUCTION ......................................................................................................................................15
2.2 THE FOREIGN EXCHANGE MARKET ...................................................................................................16
2.3 EXCHANGE RATE DATA.........................................................................................................................17
2.3.1 Descriptive statistics ......................................................................................................................................17
2.3.2 Stylised facts of exchange rate data ................................................................................................................21
2.4 THE STRUCTURE OF THE FOREIGN EXCHANGE MARKET...............................................................22
2.4.1 Features of the foreign exchange market ...................................................................................................22
2.4.2 Participants of the foreign exchange market..............................................................................................22
2.5 CONCLUSION ...........................................................................................................................................24
CHAPTER 3: EXCHANGE RATES: MODELS AND PUZZLES ...............................................26
3.1 INTRODUCING EXCHANGE RATE THEORY.............................................................................................26
3.2 THE MACROECONOMIC APPROACH TO EXCHANGE RATES............................................................27
3.2.1 Purchasing power parity...........................................................................................................................27
3.2.2 The interest rate – Uncovered interest rate parity .....................................................................................29
3.2.3 The money supply ....................................................................................................................................30
3.2.4 Summing up................................................................................................................................................31
3.3 THE MONETARY APPROACH ......................................................................................................................32
3.3.1 The flexible price monetary model..................................................................................................................33
3.3.2 The sticky price monetary model ....................................................................................................................34
3.3.3 The monetary approach – summing up..........................................................................................................36
3.4 EMPIRICAL STUDIES OF EXCHANGE RATES ............................................................................................36
3.4.1 Empirical results of the seventies ...................................................................................................................36
3.4.2 Modern empirical results ...............................................................................................................................37
3.4.3 Exchange rate puzzles..................................................................................................................................38
3.4.4 Puzzles after all? ..........................................................................................................................................40
3.4.5 Critique of the Rational Expectations Hypothesis .........................................................................................41
3.4.6 Different solutions to the exchange rate puzzles .............................................................................................42
7
3.5 THE MARKET MICROSTRUCTURE APPROACH .........................................................................................43
3.5.1 Introducing market microstructure .................................................................................................................43
3.5.2 Order flow as an important factor for exchange rates .....................................................................................45
3.5.3 A microstructure model of exchange rates ......................................................................................................47
3.6 CONCLUSION ...........................................................................................................................................50
CHAPTER 4: IMPERFECT KNOWLEDGE ECONOMICS.......................................................53
4.1 INTRODUCING IMPERFECT KNOWLEDGE ECONOMICS ......................................................................53
4.2 IMPERFECT KNOWLEDGE AND UNCERTAINTY.....................................................................................55
4.2.1 Knightian uncertainty....................................................................................................................................55
4.2.2 Imperfect knowledge ......................................................................................................................................55
4.3 MODELLING PREFERENCES AND FORECASTING STRATEGIES ..........................................................57
4.3.1 The expected utility hypothesis.......................................................................................................................57
4.3.2 Critique of the expected utility hypothesis.......................................................................................................58
4.3.3 Prospect theory ..............................................................................................................................................58
4.3.4 Prospect theory and the foreign exchange market – the IKE approach............................................................59
4.3.5 Equilibrium in the FX market under Prospect theory: UAIP ......................................................................63
4.3.6 Modelling forecasting strategies I: The Gap effect ...........................................................................................65
4.3.7 Modelling forecasting strategies II: Conservative revisions ...............................................................................67
4.3.8 Summing up .................................................................................................................................................68
4.4 IKE AND THE EXCHANGE RATE: A MONETARY MODEL ....................................................................69
4.4.1 A monetary model with IKE-expectations.....................................................................................................70
4.4.2 Money markets.............................................................................................................................................70
4.4.3 Goods markets .............................................................................................................................................71
4.4.4 Foreign exchange market ..............................................................................................................................71
4.4.5 The social context .......................................................................................................................................73
4.4.6 The solution to the model .............................................................................................................................74
4.4.7 The intuition of the result ..........................................................................................................................76
4.5 IKE AND THE EXCHANGE RATE DISCONNECT PUZZLE ......................................................................77
4.6 CRITIQUE OF IMPERFECT KNOWLEDGE ECONOMICS ........................................................................78
4.7 CONCLUSION ...........................................................................................................................................79
CHAPTER 5: EMPIRICAL TEST.................................................................................................82
5.1 INTRODUCING THE EMPIRICAL PART .......................................................................................................82
5.2 THE MODEL: MOTIVATION AND SET-UP............................................................................................83
5.3 THE MODEL: SPECIFICATION AND ESTIMATION..............................................................................85
5.3.1 Introducing the empirical test .....................................................................................................................85
5.3.2 The data ...................................................................................................................................................85
8
5.3.3 A brief discussion of multivariate cointegration...........................................................................................86
5.3.4 GARCH(p,q) estimation .........................................................................................................................87
5.3.5 Cointegration analysis .............................................................................................................................89
5.3.6 Lag length, residual analysis and dummy variables ..................................................................................93
5.3.7 Testing the models ...................................................................................................................................94
5.3.8 Conclusion ........................................................................................................................................... 103
5.4 THE RESULT FROM AN IKE PERSPECTIVE ...................................................................................... 103
5.5 CONCLUSION ........................................................................................................................................ 104
CHAPTER 6: CONCLUSION ..................................................................................................... 105
LITERATURE.............................................................................................................................. 107
APPENDIX A – FIGURES ...........................................................................................................114
APPENDIX B – MODELS............................................................................................................118
APPENDIX C – EMPIRICAL RESULTS ....................................................................................121
9
Chapter 1
Introduction “To repeat a central fact of life, there is remarkably little evidence that macroeconomic variables
have consistent strong effects on floating exchange rates, except during extraordinary
circumstances such as hyperinflations. Such negative findings have led the profession to a certain
degree of pessimism vis-à-vis exchange rate research”
Frankel and Rose (1995), p. 1709.
1.1 Introduction For most modern economies the exchange rate is, in the words of Obstfeld and Rogoff (2000),
the single most important relative price, essential for a number of economic activities. Exchange
rate fluctuations are therefore carefully followed: By investors, as it affect the value of
international portfolios; by governments, as it affects prices on exports, imports and the value of
international debt; and by Central Banks, as it affects inflation objectives and the value of
international reserves. For the financial market in general, the fluctuations of exchange rates are
important as well: Directly, as the market for foreign exchange is the largest financial market in
the world; and indirectly, as currency fluctuations influence a range of other asset prices.
It is a puzzle, then, that exchange rate theories have had such a hard time explaining the currency
fluctuations since the free float in the 1970s, and that the link between macroeconomic variables
and the exchange rate appear almost non-existent. The empirical result of the weak relation
between exchange rates and macroeconomic variables, as reported by for example Meese and
Rogoff (1983), has since been dubbed the “exchange rate disconnect puzzle”. Meese and Rogoff
(1983) concluded that a simple random walk model would predict major-country exchange rates
as well as a range of exchange rate models. The assumption of mainstream exchange rate theories
that the value of a given currency is determined by macroeconomic fundamentals such as output,
money supply and interest rates therefore seems to be too simple. At least in the short to medium
run of six to twelve months. Looking at figure 1 below, depicting the value of the US real
effective exchange rate, it is evident that institutions such as the Central Banks also matter for the
10
pricing of a currency. The Plaza Accord, for example, apparently had an impact on the value of
the dollar – which was not solely grounded in changes of macroeconomic fundamentals1. Figure 1 – The US real effective exchange rate since 1970
.
70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 00 02 04 06 08
perc
ent
75
80
85
90
95
100
105
110
115
120
perc
ent
75
80
85
90
95
100
105
110
115
120 Index
T he L ouvre A ccord
P aul V olcker app. chairm an o f the F ed
T he P laza A ccord
Source: EcoWin database
In the foreign exchange market two rather large players, Central Banks and international
companies, do not solely have profit maximisation as their main objective but in addition
objectives such as price stability (Central Banks) or hedging their investments (companies). This,
obviously, affects the determination of exchange rates. Furthermore, the structure of the foreign
exchange market, which is both highly decentralised and has a quite low degree of transparency,
matters for the price setting as well. But most important for the exchange rate determination are
the economic agents in the market. Not only the aforementioned institutions and organisations,
but also the traders and investors who are buying and selling currency on a daily basis. In
traditional exchange rate theories, these agents are assumed to be homogenous and endowed
with rational expectations; i.e. they know the true model of the economy and do not
systematically over– or underestimate the value of the exchange rate. Furthermore, private
information is not deemed important in the pricing of currencies, as all agents have access to the
same relevant information, according to rational expectations.
The question, then, is whether the hypothesis of fully rational individuals is where the many
problems and puzzles of modern day exchange rate theory come from? The answer to that
1 The Plaza Accord was an agreement between USA, Germany, France, UK and Japan to intervene in the foreign
exchange market with the objective of a depreciation of the dollar.
11
question is yes, according to the book “Imperfect knowledge economics: Exchange rates and
risk” by Roman Frydman and Michael D. Goldberg (2007). They consequently propose a new
theory for explaining the puzzles which are haunting exchange rates research: Imperfect
knowledge economics (IKE). The IKE theory instead suggests that agents are heterogeneous and
therefore have different expectations of the future price of a given currency. Furthermore, the
agents are fully aware that they do not know the true model of the economy. Hence, they use a
portfolio of models and methods, as well as experience and creativity, when forecasting the
future exchange rate. The result that exchange rates are not always perfectly correlated with
macroeconomic fundamentals, is therefore no surprise according to the IKE theory. Seemingly
the disconnect puzzle between fundamentals and the exchange rate is no longer a puzzle when
seen from the theoretical perspective of IKE.
From this introduction follows the purpose of the thesis: “Does Imperfect Knowledge Economics provide
a solution to the exchange rate disconnect puzzle?
1.2 Definition: The exchange rate disconnect puzzle The hypothesis of this thesis is whether the imperfect knowledge economics theory gives a
reliable solution to the exchange rate disconnect puzzle. It is therefore necessary first to define
the exchange rate disconnect puzzle.
Definition of the exchange rate disconnect puzzle: The finding that the fundamentals and
the (nominal) exchange rate are only weakly correlated has been defined as the “exchange rate
disconnect puzzle” by Obstfeld and Rogoff (2000).
In Obstfeld and Rogoff (2000: 373) the puzzle is defined as follows: “The weak relationship between
the exchange rate and virtually any macroeconomic aggregates”. The definition of Sarno (2005: 674):
“Fundamentals appear to be unable to explain both the actual level of exchange rates – not only on daily, but even
monthly, quarterly and annually – and their volatility”. Other authors (e.g. Lyons, 2001: 172) label it the
“exchange rate determination puzzle”, but this identical to the exchange rate disconnect puzzle.
The conclusion of a weak relationship between exchange rates and macroeconomic fundamentals
(e.g. output, money supply) dates back to the aforementioned influential article of Meese and
Rogoff (1983), which will be discussed further in the empirical survey in chapter 3.
Although often discussed apart from each other, the exchange rate disconnect puzzle and the
purchasing power parity (PPP) puzzle – mentioned in chapter 3 – are very much linked together.
The first puzzles’ subject is the disconnect between fundamentals and the exchange rate, whereas
12
the PPP puzzles’ subject is the long half-life when the exchange rates move towards the
fundamental value (given by PPP); half-lifes that reach as much as 3 or 4 years (Obstfeld and
Rogoff, 2000: 373; Rogoff, 1996). When looking at the disconnect puzzle, one can therefore not
go without discussing the PPP puzzle as well.
1.3 Delimitation First of all I have chosen to focus solely on the monetary approach and let the imperfect
knowledge economics theory be a critique thereof. I have not included discussions of, for
example, the new open economy macroeconomics (NOEM) or the portfolio balance models
approaches to exchange rates. Regarding the former, the NOEM approach has not produced
empirical exchange rate equations that alter the Meese and Rogoff (1983) result (cf. Lyons, 2001:
294). The monetary approach, on the other hand, has been the dominating theory of exchange
rate research since the 1970s, despite its empirical shortcomings discussed in chapter 3. Nor do I
discuss the results of behavioural economics, interesting as it may be. This could be a focus of
another thesis.
Secondly, this thesis try to answer whether the imperfect knowledge economics theory put forth
a constructive solution to the disconnect puzzle. Other solutions to the puzzle outside of the
realm of the imperfect knowledge theory (e.g. transport costs, bubbles, behavioural economics
result etc.) will therefore only be touched upon briefly, relevant as they might be for the
disconnect puzzle itself.
1.4 Structure of the thesis The thesis is structured as follows:
Chapter 2 – Stylised facts: This chapter examines the foreign exchange market from three
angles: The exchange rates and the time series property thereof; the structure of the foreign
exchange market; and the economic agents in the market with focus on the traders. This insight is
used in the following chapters, especially in regards to the IKE theory in chapter 4 which builds
on several of the stylised facts from chapter 2.
Chapter 3 – Exchange rates: Models and puzzles: The focus of this chapter is the monetary
approach and the empirical results of this theory. The main idea of the monetary approach is
13
presented by two often used models, the flexible price monetary model (FPMM) and the sticky
price monetary model. The empirical study following the models demonstrate the shortcomings
of the monetary approach, one of them the exchange rate disconnect puzzle. The rational
expectations hypothesis, the backbone of the monetary approach, is discussed and criticised.
Then follows a discussion of the microstructure approach to exchange rate which serves to: i)
introduce a micro foundation to exchange rate research missing from the monetary approach,
and ii) be an introduction to the ideas of imperfect knowledge in the following chapter, which
builds on several of the insights from microstructure theory.
Chapter 4 – Imperfect Knowledge Economics: This chapter discuss the imperfect knowledge
economics (IKE) theory with the main focus on the book “Imperfect knowledge economics:
Exchange rates and risk” by Frydman and Goldberg (2007). A theoretical model based on the
IKE theory is set up, and it is discussed why this provides a theoretical solution to the exchange
rate disconnect puzzle.
Chapter 5 – Empirical test: The empirical chapter tests a monetary model with the addition of
uncertainty. The IKE theory assumes that uncertainty plays an important part in the price setting
of exchange rates, and this hypothesis is tested using the Johansen method in a multivariate VAR
model on Norway and Japan against USA. Furthermore I test a simple monetary model for both
countries which supports the general result of the shortcoming of the monetary approach seen in
chapter 3. The tests of the models with uncertainty result in some support for the importance of
the uncertainty variables in the determination of exchange rates, at least for Japan. Furthermore,
the GARCH variables, proxying uncertainty, seem to be significant for the models.
One has to be aware, though, that some of the assumptions of the VAR model are violated.
Chapter 6 – Conclusion: This chapter presents the overall conclusion of the thesis, and answers
the question put forth in the introduction: Does the Imperfect Knowledge Economics theory
provide a solution to the exchange rate disconnect puzzle? The result of the thesis, and hence the
answer to this question, is based on two foundations: i) The theoretical foundation of IKE, which
appears quite strong as it is based on the stylised facts discussed in chapter 2 and chapter 3 as
well as the robust results of prospect theory and the microstructure approach; ii) The empirical
foundation from testing the IKE assumptions, which show that both private information (chapter
3 on microstructure) as well as uncertainty (the empirical test of chapter 5) play a role in regards
to exchange rate determination.
14
Based on this, I cannot reject the hypothesis that IKE could be a solution to the exchange rate
disconnect puzzle.
I have split the appendix in three parts: A) With figures; B) with models and calculations thereof;
and C) with results from various estimations and tests.
15
Chapter 2
Stylised facts “One of the most fascinating thing about the foreign exchange market is the huge sums of
money that are exchanged on a daily basis”
Keith Pilbeam, (2006), p. 4
2.1 Introduction A necessary condition for understanding the movements and the predictability of assets and asset
returns, in this case exchange rates, is understanding the financial data as well as the market in
which these prices are set. As a starting point for the analysis later in the thesis, it is therefore
relevant to look into the regularities and composition of the foreign exchange market. This
chapter does exactly that.
The results from the latest survey of the Bank of International Settlements (BIS, 2007) show the
size and trading structures of the foreign exchange markets. The statistical properties of time
series data of exchange rates, on the other hand, are important for an initial understanding of the
exchange rate movements. The statistical properties of exchange rates, and financial data as a
whole, are often referred to as “stylised facts”, i.e. a broad generalisation of empirical findings.
The descriptive statistics of exchange rates, as we shall see, very much follows that from other
financial time series data of bonds and equities; i.e. heavy tails, over-kurtosis, (left-) skewness and
rejection of the normality assumption (see for example Campbell et al, 1997: 19ff or Pagan,
1996).
This chapter is structured as follows: First a short look at the latest BIS (2007) survey of the
foreign exchange market. Then the descriptive statistics of three different exchange rates will be
discussed. Finally a discussion of the structure and agents of the exchange rate market, based on
the survey by Cheung and Chinn of American (2001) and English traders (Cheung et al, 2004).
The insights from this chapter is build upon in chapter 3, where the monetary and microstructure
approach to exchange rates are discussed, as well as in chapter 4 of the imperfect knowledge
economics.
16
2.2 The foreign exchange market The Bank of International Settlements (BIS, 2007a) triennial survey analyses the turnover of the
foreign exchange markets. According to Sarno and Taylor (2002: 271) it represents “the most
reliable source of information of foreign exchange market activity”. Table 1 below shows the result from the
survey. As can be seen, the daily turnover of the traditional (i.e. spot, forwards and swaps) foreign
exchange markets reached $3.2 trillion, an increase of almost 71 % from 2004. This increase is,
according to BIS (2007a: 1), driven by both increased activity of investor groups (e.g. hedge
funds) and technical trading. This is further supported by the second BIS report (2007b: 65),
which also points out that the foreign exchange market has been relatively attractive to leveraged
investors with short-term horizons, as well as investors with longer investment horizons trying to
diversify their portfolio. Table 1 – Foreign exchange turnover from the BIS survey, 2007
Source: BIS Triennial Survey (2007)
The dollar is the main currency although with a small downward trend since 2004 (BIS, 2007a: 7).
The Japanese Yen (JPY) and the Norwegian krone (NOK), used in the empirical test of chapter
5, are the third and tenth most traded currencies, respectively. The United Kingdom is the
geographical centre for foreign exchange trading followed by the United States and Japan (ibid.:
9). The most traded currency cross is the USD/EUR, amounting more than a quarter (27%) of
the total market turnover. Then follows USD/Other (19%), USD/JPY (13%) and USD/GBP
(12%). The largest part of the trades is between dealers (43%), followed by deals with other
financial institutions (40%) and non-financial customers (17%) (cf. BIS 2007a: 6).
17
Overall the volume of the foreign exchange market is enormous, and it dwarfs any other financial
instrument (Lyons, 2001: 41).
2.3 Exchange rate data
2.3.1 Descriptive statistics
This section serves to give a general description of the time series properties of exchange rates.
The data in the following section covers the euro/dollar (EUR/USD), dollar/Japanese Yen
(USD/JPY) and dollar/British pound (USD/GBP). The three crosses have been chosen as they
are the three most traded currencies (BIS, 2007a: 8).
The data is obtained from the EcoWin database. The frequency is on a daily, weekly and monthly
basis, and covers the period from the 1st of July 1974 to 31st October 2007. Figure 2, 3 and 4
below show the realisations of the exchange rates over the period. Figure 2 – Daily observations for EUR/USD from July 1974
0,6
0,7
0,8
0,9
1
1,1
1,2
1,3
1,4
1,5
1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006
Source: EcoWin
For the euro/dollar exchange rate, figure 2 above, the maximum value (i.e. weak dollar) over the
period is 1.45 (September 8th 1992) and minimum value 0.63 (February 26th 1985). For the
dollar/yen, figure 3 below, the maximum (here: strong dollar) is 306.8 (December 8th 1975) and
minimum 80.6 (April 18th 1995). For the dollar/pound in figure 4 below, the maximum is 0.95
(January 11th 1985) and minimum value 0.401 (October 8th 1980).
From figures 2 and 4 (EUR/USD and USD/GBP) it is evident that the US dollar reached a local
maximum (i.e. strong dollar) in the mid 1980s.
18
Figure 3 – Daily observations for USD/JPY from July 1974
75
105
135
165
195
225
255
285
1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006
Source: EcoWin
Figure 4 – Daily observations for USD/GBP from July 1974
0,4
0,5
0,6
0,7
0,8
0,9
1
1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006
Source: EcoWin
In the following the data for the three currencies is transformed into log-returns, given by
equation (2.2) below, which is defined as the natural logarithm of the gross return, equation (2.1).
(Campbell et al, 1997: 9-11).
(2.1) 1
1tt
t
PRP−
= −
(2.2) ( ) 11
log 1 log tt t t t
t
Pr R p pP −
−
≡ + = = −
19
Now the distribution of the log returns of the exchange rates can be computed. Figure 5 below
shows the plot of daily log returns. Simple “eyeball econometrics” shows a high degree of
volatility, some serial correlation and some volatility clustering (cf. Campbell et al, 1997: 482).
The graphical log return of weekly and monthly observations, respectively, are found in appendix
A, figure 1 and 2. Table 2 below shows the descriptive statistics of the log-returns for daily,
weekly and monthly observations.
Figure 5 – Daily log returns for USD/GBP, EUR/USD and USD/JPY
Table 2 – Descriptive statistics for daily, weekly and monthly observations of log returns
Daily Weekly Monthly Daily Weekly Monthly Daily Weekly MonthlyMean 0,00002 0,0001 -0,0005 -0,0001 -0,0005 -0,0024 0,0000 0,0000 -0,0003Maximum 0,0615 0,0707 0,1243 0,0415 0,0317 -0,1566 0,0382 0,1282 0,1282Minimum -0,0648 -0,0718 -0,0931 -0,0695 -0,0423 0,1153 -0,0343 0,2816 -0,1250Std. Dev 0,0063 0,0135 0,0292 0,0065 0,0077 0,0324 0,0049 0,0071 0,0299Skewness 0,0805 -0,0351 0,1464 -0,5952 -0,4731 -0,4869 0,7187 0,2816 -0,2079Kurtosis 5,0232 2,0099 1,0591 6,5260 2,1249 1,6655 9,2880 2,2127 1,5944Normality test 3528.7 ** 184.29 ** 16.899 ** 3965.5 ** 155.02 ** 28.259 ** 1253.2 ** 194.10 ** 31.495 **
EUR/USD USD/JPY USD/GBP
From table 2 the means are (slightly) different from zero, indicating that the euro has (on
average) depreciated against the dollar on a monthly basis, whereas the dollar has appreciated
against the yen and the British pound over the period. But note that none of the means are
20
significantly different from zero, given the standard deviations. The skewness and kurtosis in
table 2 are given by (Campbell et al, 1997: 16-17):
(2.3)
( )
( )
3
3
4
4
ˆ:
ˆ:
Skewness S E
Kurtosis K E
ε μσ
ε μσ
⎡ ⎤−≡ ⎢ ⎥
⎢ ⎥⎣ ⎦⎡ ⎤−
≡ ⎢ ⎥⎢ ⎥⎣ ⎦
Where ε is a random variable with mean of μ and variance of σ2. The skewness, S, measures the
asymmetry of the distribution, with the normal distribution having a skewness of 0. The
distribution of USD/JPY in table 2 thus has more negative than positive returns for all three
frequencies. For the other currency crosses, EUR/USD and USD/GBP, the skewness measure
changes sign over the frequencies. The kurtosis, K, in table 1 is the “excess” kurtosis, i.e. above
the normal distribution which has kurtosis of 3. According to the relatively large and positive
kurtosis of table 2, the returns of exchange rates have more mass in the tails than predicted by
the normal distribution. The excess kurtosis declines over all three currency crosses as the
interval increases. Both the kurtosis and the skewness figures for the daily frequencies in table 2
are highly statistically significant, as the standard error2 for the kurtosis is 0.052 ( 24T ) and for
the skewness 0.026 ( 6T ).The skewness turns insignificant for the EUR/USD at weekly and
monthly basis, and for the USD/GBP at the monthly frequency, whereas the kurtosis stays
significant over the frequencies. Finally, the normality (or Jarque-Bera) test jointly measures
whether the skewness and kurtosis equals that of the normal distribution (i.e., 0 and 3
respectively). This is soundly rejected for all three currencies.
The results above are in line with the results of Boothe and Glassman (1987: 303-304) for
exchange rate returns. They find clear signs of excess kurtosis, which declines as the interval
increases, a strong rejection of normality and some signs of skewness.
Another way to describe the distributions of the exchange rate returns is by using quantile-
quantile (QQ) plots. Then, the quintiles of a given sample are matched with the theoretical
quintiles. Figure 6 below shows the QQ-plots of the distributions against the normal distribution.
As is evident from the three plots, there are too many observations in the tails of the distribution
(red line) compared with the normal distribution (black line). Hence the returns are not normally
2 Following Campbell et al (1997: 17) the variances of the S and K estimators are 6/T and 24/T, respectively.
21
distributed, which is also the result of the normality test in table 2. The negative skewness from
table 2 of the USD/JPY, for example, is quite obvious from figure 6 (bottom chart).
Figure 6 – QQ plots for the three currency crosses
2.3.2 Stylised facts of exchange rate data
From the results above the following properties emerge:
• Exchange rates appear to be extremely volatile
• The distributions of exchange rate returns are non-normal
• Fat tails compared with the standard normal distribution. That is, large returns occur more often than
expected (kurtosis significantly larger than 3)
• The distributions are skewed, i.e. the distribution is not symmetric. The direction of the skewness is not
unequivocal, though, and turns insignificant – for some currency crosses - as frequencies decrease.
The stylised facts presented above are in line with the general result of exchange rates, see for
example Boothe and Glassman (1987), and for financial assets in general, see for example
Campbell et al (1997: 21, 67).
22
2.4 The structure of the foreign exchange market In the following section the foreign exchange market is, in the words of Sager and Taylor (2006),
put “under the microscope”. As discussed in the introduction, and further elaborated on in
chapter 3, the link between macroeconomic fundamentals and the exchange rate is mixed, to say
the least. To understand why this is so, looking at the market and the participants therein can give
useful information.
This section first looks at the features of the foreign exchange market, then the participants of
the market.
2.4.1 Features of the foreign exchange market
One important thing to notice regarding the foreign exchange market is the low level of
transparency (see for example Lyons, 2001: 41; Sager and Taylor, 2006; Sarno and Taylor, 2002:
266). Where equity and bond trades has to be disclosed within minutes in most markets, trades in
the foreign exchange market has no requirement of disclosure and hence the trades in the market
is generally not observable. Furthermore, as noted by Sager and Taylor (2006: 82), the foreign
exchange market is highly decentralized, which (further) implies some degree of lack of
transparency. As noted (ibid.: 82): “ It [the foreign exchange market] is opaque – or lacks transparency – in
the sense that the absence of a physical marketplace makes the process of price-information difficult to observe and
understand”. As mentioned by Sarno and Taylor (2002: 266), this decentralisation increases
inefficiency compared with more centralised markets such as the equity market. The high degree
of decentralisation furthermore implies that there is some degree of fragmentation; that is,
transactions may occur at the same time at different prices (ibid.: 267). The aspect of lacking
transparency, and its effect on exchange rate determination, will be discussed further in chapter 3,
and is an important part of understanding the foreign exchange market.
2.4.2 Participants of the foreign exchange market
Cheung and Chinn (2001) analyse the composition of foreign exchange traders in the US using
survey data. According to the result, traders in the US foreign exchange market can roughly be
divided into four groups (ibid.: 453)3: technical trading (29.5%), customer order (23.4%),
fundamental analysis (24.9%) and “jobbing” (21.1%). Jobbing refers to a trader continuously
3 The specific question in the survey is: The best way to describe your spot FX trading is: “Technical trading rules”,
“fundamental analysis”, “customer orders driven”, “jobbing approach”, “other”. Note that the sum of the categories
does not equal 1 as there, in some cases, are multiple responses or incomplete replies (Cheung and Chinn, 2001: 453)
23
buying and selling to take many (small) profits (cf. Cheung and Chinn, 2001: 454). Apparently,
only about a quarter of the respondents (state that they) use fundamental analysis as their
foremost strategy when forecasting exchange rate movements. This could, to some extent,
explain elements of the “exchange rate disconnect puzzle”, as the largest part of the traders seem
to base their strategy on other issues than the fundamental value. One should note, though, that
the number of respondents in the survey is only 142, and therefore not necessarily descriptive of
the US foreign exchange market as a whole. Furthermore the respondents in the survey, for a
large part, have rather small positions to manage.
But the overall conclusion of investor heterogeneity is supported by others. Frankel and Froot
(1990: 184), for example, concludes that the largest part of foreign exchange forecasting firms in
the years 1983-88 described themselves relying exclusively on technical trading. They state that
“shifts over time in the weight that is given to different forecasting techniques are a source of changes in the demand
for dollars, and that large exchange rate movements may take place with little basis in macroeconomic
fundamentals” (ibid.: 184). This apparent heterogeneity of traders is also supported by Frankel and
Rose (1995: 1712), De Bondt and Thaler (1994) and Sager and Taylor (2006: 91), and it seems to
be a robust finding. Menkhoff and Taylor (2007: 940) study the research on technical trading and
conclude: “Almost all foreign exchange professionals use technical analysis as a tool in decision making, at least
to some degree”.
The traders in the survey of Cheung and Chinn (2001: 459) assess that the fundamentals have
little to no effect on the shorter horizon, here intraday and medium run (up to six months). But a
large part (88.4%) of the respondents do believe that macroeconomic fundamentals influence the
exchange rate in the long run – here defined as longer than six months. This is, somewhat, in line
with the empirical results of exchange rate research, as mentioned in the introduction and
discussed further in chapter 3. As to why the exchange rate value differs from the fundamental
value, the respondent’s point to excess speculation (74%) and hedge fund/institutional
manipulation (68%). Around 40% of the traders in the survey believe that central bank
intervention cause the deviations from fundamental value, whereas 52% believe that this has no
effect on the exchange rates. As the most important macroeconomic fundamental the traders in
the survey point to unemployment (33.0%) and the interest rate (30.9%), whereas inflation
(18.3%) and money supply (1.6%) seem less important. A rather surprising result compared with
the mainstream view of macroeconomic variables and the exchange rate. Cheung and Chinn
(2001: 457) furthermore point out that the importance of different macroeconomic variables
shifts over time; but with interest rates always remaining important. Finally a large part (63%) of
the traders interprets the PPP model, discussed in chapter 3 below, as “merely academic jargon”
24
(Cheung and Chinn, 2001: 465). Furthermore, only 13% would sell dollars if the PPP model
indicated a dollar overvaluation. As with the macroeconomic fundamentals, the trader’s views of
the relevance of the PPP model change with the horizon: at the long horizon, 40% of the
respondents find that PPP in fact has some influence.
The main results of Cheung and Chinn (2001) are more or less reproduced in a survey by Cheung
et al (2004) of UK-based foreign exchange dealers. Cheung et al (2004) also find that the agents
are heterogeneous. Furthermore, the dealers in the survey think that over-reaction as well as
speculative and band-wagon effects are very important for exchange rate determination. The UK
dealers, on the other hand, find that fundamentals have significant effect at rather short time
horizons of around six months. But only 27% of the respondents would sell dollars if the PPP
model showed that it was overvalued (ibid.: 297), in line with the result from Cheung and Chinn
(2001).
2.4.3 Stylised fact of the foreign exchange market:
• The foreign exchange market is characterised by a rather low level of transparency • The foreign exchange market is highly decentralised • The agents in the market have heterogeneous expectations and employ different methods
• The agents assessment of the importance of different macroeconomic variables change over time
• The agents believe that macroeconomic fundamentals matter in the “long-run”, but have little to no effect
at shorter horizons.
• The agents believe that the PPP model is only valid in the long run.
2.5 Conclusion In this chapter the foreign exchange rate, the foreign exchange market, and the participants of the
market has been examined.
The foreign exchange market is, by far, the largest financial market in the world, and its size
(measured by turnover) has increased markedly (70%) over the last three years. The descriptive
statistics of exchange rate returns follow that of most other financial assets; i.e. the distribution of
returns is non-normal, skewed and fat-tailed. This change as the frequencies decreases, with less
extreme observations at monthly basis compared with the weekly and daily basis for all three
currencies.
25
The exchange rate market is characterised as being less transparent than other financial asset
markets, clouding the price information and further exacerbating the effect of the heterogeneous
agents as well as the tendency of asymmetric information.
Looking at the participants of the exchange rate market, the primary conclusion is that the traders
dealing with exchange rates are heterogeneous. Some use fundamental analysis as their primary
tool when forecasting exchange rates and deciding strategies; but a large part of the traders seem
to primarily use other methods which do not depend on fundamental values (e.g. technical
trading). Traders do believe that fundamentals have some importance, though. But which types
of fundamentals are important can differ over time, and it seems that the macroeconomic
fundamentals are only reckoned to be important in the long run.
26
Chapter 3
Exchange rates: models and puzzles “The clear conclusion is that exchange rates are moved largely by other factors than the obvious,
observable, macroeconomic fundamentals. Econometrically, most of the “action” is in the error
term.”
Rudiger Dornbusch and Jeffrey Frankel, (1987), p. 10
“Exchange rate economics is characterized by a number of anomalies, or puzzles, which we
struggle to explain on the basis of either sound economic theory or practical thinking… the
international finance profession has not yet been able to produce theories and, as a consequence,
empirical models that allow us to explain the behavior of exchange rates with a reasonable degree
of accuracy.”
Lucio Sarno, (2005), p. 674
3.1 Introducing exchange rate theory Before the free floating of exchange rates in the 1970s, and the following expansion of theoretical
exchange rate models, the approach to exchange rate determination was primarily based on the
goods market (Lyons, 2001: 2). That is, demand for foreign exchange was assumed to come
primarily from the sales (and purchases) of goods across borders; an increase in exports is
followed by an increase in the demand for domestic currency to pay for the goods. This
approach, at first, sounded plausible. But the trade balances turned out to be uncorrelated with
the exchange rate movements. Furthermore, trade in goods and services accounts for a very small
fraction of the daily foreign exchange trading, around 5 % (ibid.: 2). As a consequence of this the
asset market – or monetary – approach emerged in the 1970s, and this has since been the
dominant approach for exchange rate research (Sarno and Taylor, 2002: 46). In this chapter the
focus is on the monetary approach of exchange rate modelling and its empirical results. As an
alternative to the monetary models the market microstructure approach is discussed as well.
The chapter is structured as follows: First an introduction to the macroeconomic approach to
exchange rates. Then follows a discussion of the monetary approach to exchange rates, with
focus on the two most used models, the flexible price monetary model and the sticky price (or
27
overshooting) model. Then a survey of the empirical results of exchange rate studies. This leads
to a discussion of the exchange rate puzzles emerging from the empirical results. Then follows a
discussion of the rational expectations hypothesis (REH), a central part of exchange rate
modelling. Finally the microstructure approach is discussed, an alternative method of determining
exchange rate movements and a good starting point for understanding the IKE theory in chapter
4.
3.2 The macroeconomic approach to exchange rates As mentioned in the introduction to chapter 1 macroeconomic fundamentals are, from an
academic viewpoint, seen as important when evaluating the determinants of exchange rates. In
the following, two of the building blocks for the exchange rate models presented in section 3.3
will be discussed: the purchasing power parity and the uncovered interest rate parity.
3.2.1 Purchasing power parity
A first approximation of what determines the exchange rate is the purchasing power parity (PPP).
PPP states that arbitrage will, when goods are measured in the same currency, lead to equalisation
of goods prices internationally (Pilbeam, 2006: 126 or Sarno and Taylor, 2002: 51). That is, the
purchasing power of a US dollar, say, should be the same in both the Euro-zone and USA. PPP
is defined as follows by Sarno and Taylor (2002: 51): “.. The PPP exchange rate is the exchange rate
between two currencies which would equate the two relevant national price levels if expressed in a common currency
at that rate, so that the purchasing power of a unit of one currency would be the same in both countries”. If the
exchange rate is misaligned (i.e. either over- or undervalued according to PPP), arbitrage would
secure that the currency reaches the parity as investors seek to take profit. As with the uncovered
interest parity presented below, the PPP builds on the notion of market efficiency. The definition
of an informational efficient market is (Campbell et al, 1997: 20-21): “Price changes must be
unforecastable if they are properly anticipated, i.e. if they fully incorporate the expectations and information of all
market participants”. Or, in the words of Malkiel (1992): “A capital market is said to be efficient if it fully
and correctly reflects all relevant information in determining security prices. Efficiency with respect to an information
set … implies that it is impossible to make economic profits by trading on the basis of [that information set]”.
That is, a market in which the prices fully reflect the (available) information is efficient. For a
more thorough discussion of the efficient market hypothesis, see Campbell et al (1997) or Sarno
and Taylor (2002).
The absolute PPP condition states that:
(3.1) *t t ts p p= −
28
Where s is the (log) exchange rate, p is the (log) price level and an asterisk denotes a foreign
variable. In the following, s denotes domestic price of foreign currency, and hence an increase
(decrease) in s is seen as depreciation (appreciation). From the PPP condition, the real exchange
rate, q, can be obtained, which can be seen as a measure of deviation from PPP:
(3.2) *t t t tq s p p≡ − +
Figure 7 and 8 below plots the USD/JPY and EUR/USD against the (computed) exchange rate
as given by the PPP condition in equation (3.1). Figure 7 - USD/JPY and the PPP model, 1980 to 2007
.
80 82 84 86 88 90 92 94 96 98 00 02 04 06 08
perc
ent
7 5
1 0 0
1 2 5
1 5 0
1 7 5
2 0 0
2 2 5
2 5 0
2 7 5
3 0 0
perc
ent
7 5
1 0 0
1 2 5
1 5 0
1 7 5
2 0 0
2 2 5
2 5 0
2 7 5
3 0 0
P P P m o d e l
U S D / JP Y
Source: EcoWin
Figure 8 – EUR/USD and the PPP model, 1988 to 2007
.
88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07
perc
ent
0,8
0,9
1,0
1,1
1,2
1,3
1,4
1,5
1,6
perc
ent
0,8
0,9
1,0
1,1
1,2
1,3
1,4
1,5
1,6
PPP model
EUR/USD
Source: EcoWin
29
As can be seen from both figures 7 and 8 above there are large deviations between the factual
exchange rate (red dotted line) and the value given by the PPP model (blue line). Furthermore,
the swings away from parity appear to be relatively persistent, for example from 1999 to 2003 for
EUR/USD or 1991 to 1997 for USD/JPY.
For the USD/JPY cross, the fundamental value given by PPP somewhat trends the nominal
exchange rate over the period. For the EUR/USD, on the other hand, the PPP value appears
more stationary around a value of 1.1, and the spot exchange rate then “cycles” around this. The
primary conclusion based on visual inspection of the two charts follows the general conclusion
on the outcome of PPP against the spot exchange rate, see for example Pilbeam (2006: 131ff).
One important thing to notice from the two charts is that the exchange rate does not fully
abandon the relationship with the PPP value, although the swings away from it can take several
years. Instead, the PPP value seems to act like an anchor around which the exchange rate gyrates.
If the nominal exchange rate appears “too” misaligned, it is pulled back towards the fundamental
value of PPP.
The apparent disconnect in the figures 7 and 8 between the PPP value and the exchange rate is
part of the “disconnect puzzle”. That is, the exchange rate seems to be, at least in the short-to
medium term, disconnected from the fundamental value (here given by PPP). But the moves
away from PPP seem to be bounded to some extent. Furthermore the slow return towards the
PPP value (i.e. the high half-lifes of the return to the fundamental value) has been dubbed the
“PPP puzzle”, following Rogoff (1996). Both puzzles will be discussed further in section 3.4.3
below on exchange rate puzzles.
The imperfect knowledge economics theory, presented in chapter 4, tries to take into account the
empirical regularities from the charts above. Hence, two of the questions which the imperfect
knowledge economics theory seeks to answer are: i) Why is the exchange rate “disconnected”
from the fundamental value? And ii) why does the exchange rate return to parity?
3.2.2 The interest rate – Uncovered interest rate parity
Another macroeconomic fundamental looked at when discussing exchange rate movements is the
interest rate. Bacchetta and Wincoop (2007: 346) point out that “… FX changes are predictable by
interest rate differentials”. This leans on another cornerstone of foreign exchange rates: the
uncovered interest rate parity (UIP). As with the PPP condition above, the UIP is an arbitrage
condition, securing that no excess return can be earned in an efficient market (Sarno and Taylor,
2002: 5). The UIP is given in equation (3.3) below. It states that changes in the interest rate
differential are set off by equal changes in the (expected) exchange rate, securing equality between
foreign and domestic asset return.
30
(3.3) *1
et t ts i i+Δ = −
The domestic investor faces a choice of a secure domestic investment – with the payoff it in
period t+1 – or investing abroad with payoff it* plus the gain/loss from movements in the
currency. In an efficient market, defined as in section 3.2.1 above, the profit from the two
choices has to be equal.
The empirics on the UIP relationship are also mixed; a finding which has been dubbed the
“forward bias puzzle” or UIP puzzle (cf. Lewis, 1995) – another exchange rate puzzle that will be
discussed in section 3.4.3.
Figure 9 below shows the EUR/USD plotted against the 10 year bond differential between USA
and Euroland. There seems to be some connection between the two variables at some periods in
time, especially from 2002 until the beginning of 2005. But there is not a clear correlation over
the time span. Figure 9 – EUR/USD plotted against the US/EUR 10 year interest rate differential, 2000-2008
.
00 01 02 03 04 05 06 07 08
perc
ent
-1,00-0,75-0,50-0,250,000,250,500,751,001,251,50
perc
ent
0,8
0,9
1,0
1,1
1,2
1,3
1,4
1,5
1,6
<< EU R/U SD
USD 10 yrs. - EU R 10 yrs. >>
Source: EcoWin
3.2.3 The money supply
A third important fundamental when analysing exchange rates is the money supply. As the
money supply is, in the long run, assumed to correlate with the prices, this is just another side of
the argument from the PPP condition above.
Figure 10 below plots the relative money supplies against the exchange rate. There seems to be a
close relation between relative money supply and the exchange rate for a long period, around
1988 to the beginning of 1996. But this very close relationship clearly breaks down completely in
the middle of 2002 until 2007. Again, this does not coincide with the monetary models presented
31
in the next section. Why do macroeconomic fundamentals appear closely related to the exchange
rate for some period of time, but unrelated for other time periods? The imperfect knowledge
theory does have an explanation for this question as well. Figure 10 – USD/JPY plotted against the relative monetary base of USA and Japan, 1988-2007
.
88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07
perc
ent
70
75
80
85
90
95
100
105
110
115
perc
ent
80
90
100
110
120
130
140
150
160US monetary base/Japan monetary base (2000 = 100)>>
<< USD/JPY
Source: EcoWin
3.2.4 Summing up
Overall, the macroeconomic glance at the exchange rate seems less than clear cut at describing
the exchange rate movements. As the figures above suggest the conclusions regarding the
relationship between the macroeconomic fundamentals and the exchange rate are mixed. Sarno
and Taylor (2002: 264) conclude: “…there seem to be substantial and often persistent movements in exchange
rates which are largely unexplained by macroeconomic fundamentals”. Part of this could stem from the
survey of the traders in chapter 2: Only around a quarter of the traders (predominantly) use
fundamental analysis when assessing future exchange rate values. The traders also believe that the
relationship between fundamentals and the exchange rate is rather small at short to medium
horizons, but larger when looking beyond six months. Finally, the structure of the market itself –
e.g. the lack of transparency – could have an effect on the exchange rate determination.
In the next section, the two parity conditions from above – PPP and UIP – is the basis for the
monetary approach to the exchange rates. The empirical survey following the theoretical models
underlines the initial result from visual inspection of the charts above – the puzzling result that
32
fundamentals and exchange rates seem disconnected, at least for (longer) periods of time. A
result in sharp contrast with the conclusion of the two models, which we turn to now.
3.3 The monetary approach The monetary approach encompasses the flexible and the sticky price model, as well as several
other exchange rate models not discussed in this thesis.
The two models both starts from defining the exchange rate as the relative price of two countries
moneys (Sarno and Taylor, 2002: 108), and assume that the (relative) supply and demand for
money is the key determinant of exchange rates (Pilbeam, 2006: 152). Furthermore, the approach
assumes that there are no barriers (transaction costs) in the capital market (Frankel, 1995: 97).
Domestic and foreign assets are assumed to be perfect substitutes, i.e. the assets are equally risky.
An assumption that also covers the goods market. From this follows that the PPP condition
holds, and exchange rates are then given by the price difference between two countries. The price
level of a country is given by the demand and supply for money.
Another hypothesis shared by the two models is that of rational expectations on the part of the
agents, a key building block of most modern economic models (e.g. Frydman and Goldberg,
2007: 11; De Grauwe and Grimaldi, 2006a: 1). Rational expectation is given by the following
equation (Pilbeam, 2006: 226):
(3.4) 1| 1 1t t t tEs s u+ + += +
That is, the agents do not, on average, systematically over- or underestimate the future exchange
rate. The rational expectations hypothesis (REH) first took its form in the influential article by
Muth (1961: 316), who concluded that: “expectations… are necessarily the same as the predictions of the
relevant economic theory”. The agents in the market therefore utilise the same model as the
economist, and this is furthermore the true model of the economy. Thus, the agents know the
true model of exchange rate movements as well as the information which affects it. Hence,
private information does not matter for the determination of, in this case, exchange rates.
Furthermore information is assumed to be used effectively, cf. Muth (1961: 316). Markets are
therefore assumed to be efficient, following the discussion in section 3.2.1. Hence, the price of a
currency reflects the (relevant) information of the fundamental variables available to the agents. If
new information is revealed, this is (immediately) incorporated in the valuation of the asset. If the
agents did not use the new (relevant) information, they would pass up on profit opportunities –
which rational agents would not do.
The hypothesis of rational expectations has been criticised from several sides. The general
critique of the rational expectations hypothesis will be discussed further in section 3.3.4, as it is an
33
important component of understanding the imperfect knowledge economics theory presented in
chapter 4.
3.3.1 The flexible price monetary model
For the flexible price monetary model (FPMM), prices are assumed to be fully flexible. That is,
the prices react instantly to (for example) excess demand. Versions of the model were first
presented by Frenkel (1976) and Mussa (1976). The three equations below sum up the model
(following Pilbeam (2006: 152-153) and McNown and Wallace (1994: 397)):
(3.5) *t t ts p p= −
Equation (3.5) is the PPP condition discussed above. An asterisk denotes a foreign variable and
all variables are in logarithms.
(3.6) 1 2t t t tm p y iβ β= + −
(3.7) 1 2
* * * * * *t t t t
m p y iβ β= + −
Equation (3.6) and (3.7) is the domestic and foreign money demand, respectively. M is the level
of the money supply, y the income and i the interest rates. As the money market is assumed to be
in equilibrium, money supply equal money demand. Solving for st in equation (3.5), using (3.6)
and (3.7), yields the so-called generic monetary approach to the exchange rate (cf. Sarno and
Taylor, 2002: 109):
(3.8) ( ) ( ) ( )1 2
* * * * *1 2t tt t t t ts m m y y i iβ β β β= − − − + −
According to equation (3.8), an increase (decrease) in domestic money supply, relative to the
foreign money supply, should lead to depreciation (appreciation)4. A rise (fall) in domestic output,
on the other hand, induces an appreciation (depreciation) of the currency. And finally, an increase
(decrease) in the domestic interest rate induces depreciation (appreciation) of the domestic
currency.
Inserting the UIP condition ( ( ) *1t t tE s i i+Δ = − ) into equation (3.8) and assuming that *
1 1β β= and
*2 2β β= yields the following equation (cf. Sarno and Taylor, 2002: 109):
(3.9) ( ) ( ) ( )* *1 2 1tt t t t ts m m y y E sβ β += − − − + Δ
Rearranging equation (3.9) yields:
4 Note, once again, that st is defined as units of domestic currency per foreign currency. Hence, an increase in st
denotes a depreciation
34
(3.10) ( ) ( ) ( )* *1 21
2 2 2
11 1 1tt t t t ts m m y y E sβ β
β β β += − − − ++ + +
Forward iteration then yields the rational expectations solution to equation (3.10):
(3.11) ( ) ( )* *21
02 2
11 1 t i
i
t t t i t i t ii
s E m m y yβ ββ β +
∞
+ + +=
⎛ ⎞ ⎡ ⎤= − − −⎜ ⎟ ⎣ ⎦+ +⎝ ⎠∑
Since the rational expectations result of equation (3.11) yields a (potentially) infinite set of
solutions (cf. Sarno and Taylor, 2002: 110), the general exchange rate given by this equation –
denoted ts% – has numerous solutions:
(3.12) t t ts s B= +%
Where Bt is a bubble term given by:
(3.13) ( ) ( )1 22
1 1t t tE B Bββ+ = +
In the absence of rational bubbles, the exchange rate is thus given by equation (3.9) (and (3.11) as
well) above. Hence the flexible price monetary model delivers a sharp prediction of the
connection between the macroeconomic fundamentals (here: money supply and output) and the
exchange rate between two countries. In section 3.4 below, the empirical results of the model will
be discussed. But first the sticky price version of the monetary approach is put forth as a slightly
different approach to exchange rate determination.
3.3.2 The sticky price monetary model
The domestic country in the sticky price monetary model is assumed to be a small participant in
the capital market, and thus faces a given interest rate (Dornbusch, 1988: 62). Assets are still
assumed to be perfect substitutes, “given a proper premium to offset anticipated exchange rate changes”
(ibid.: 62), and perfect capital mobility is assumed. This is given by the uncovered interest rate
parity, discussed in section 3.2.2: If the domestic exchange rate is expected to depreciate, the
interest rate on domestic assets will rise to offset this depreciation. The UIP condition is
reproduced in equation (3.14) below.
(3.14) *i i Es= + &
The expectation of the exchange rate is formed as the difference between the long-run exchange
rate (given by PPP) and the current exchange rate; it is assumed that the current exchange rate
will converge towards the long-run value at a constant rate.
(3.15) ( )Es s sθ= −&
35
The current exchange rate value is denoted by s, the long-run value by s and θ is the coefficient
of adjustment (cf. Dornbusch, 1988: 63). The expected rate of depreciation is therefore
determined by the gap between the current exchange rate and the long run fundamental value
(which is assumed to be known by the agents), as well as the speed of adjustment given by the
parameter θ.
As in the flexible price model above, the demand for holding money in the domestic country is
given by:
(3.16) 1 2t t t tm p y iβ β− = −
Combining the three equations above yields the following relationship between the current spot
exchange rate, its long-run fundamental value and the price level:
(3.17) ( )*2 2 1t t t tp m i s s yβ β θ β− = + − −
As noted by Dornbusch (1986: 63), assuming a stationary money supply implies equality of the
interest rates in equation (3.14) as well as equality between the expected value of the exchange
rate and the current exchange rate in equation (3.15). This leads to the following equation for the
long-run equilibrium price level:
(3.18) ( )*2 1t t t tp m i yβ β= + −
Inserting equation (3.18) into (3.17) yields the following relationship between the exchange rate
and the price level:
(3.19) ( )2
1s s p pβ θ
= − −
Given the long-run values of the exchange rate and the price, the spot exchange rate is
determined by this equation. In the short run, an increase in the money supply m, with prices
fixed at p, is only held if interest rates drop (following equation (3.16)). Following the UIP
condition in equation (3.15) the lower domestic interest rate leads investors to require an
appreciation of the currency. This is achieved by an initial depreciation (i.e. overshooting) of the
currency, s larger than s , which is then followed by an appreciation to satisfy the UIP condition.
The increase in the money supply leads to a higher price level in the long run, from equation
(3.18) and the assumption of long-run neutrality of money, and a depreciated currency –
according to the PPP condition. But to uphold the UIP condition (expected appreciation of the
currency to offset the lower interest rate) the currency in the short-run overshoots the long-run
value, and then appreciates towards the new (albeit lower) long-run value. This is, in a short
version, the exchange rate “overshooting” model. In appendix B, a slightly more sophisticated
version of the model is shown (based on Sarno and Taylor, 2002: 104-7). From this can be seen
36
that equation (3.19) above is the saddle path of the model. For any given price level the exchange
rate adjusts accordingly (instantly) to clear the asset market, following the money market
equilibrium (equation (3.16)) and the UIP condition (equation (3.14)). As the exchange rate (in
the example above) is higher than its long run equilibrium (and thus cheap), domestic prices
slowly increase to restore equilibrium, pressured by the excess demand for domestic goods
(Dornbusch, 1988: 65).
3.3.3 The monetary approach – summing up
As mentioned in the introduction, the monetary approach – either in the flexible or in the sticky
price version – has been the dominant method of modelling exchange rate movements since the
mid-seventies. Although the two models have differences – most notably whether prices are
assumed sticky or fully flexible in the short run – the models reach the same conclusion: The
value of the exchange rate – and its movements – are guided by macroeconomic fundamentals.
First and foremost the money supply (and demand) but also prices, output and interest rates.
Furthermore, both models employ the UIP and PPP condition.
In the following section the empirical results of the models are discussed, leading to a discussion
of the exchange rate disconnect puzzle. As shown in the beginning of this chapter, exchange
rates seem to be disconnected, at least in the short to medium run, from the fundamental value –
in contrast with the hypothesis of the two monetary models. Rudiger Dornbusch – the originator
of the sticky price model presented above – concluded in the late 1980s: “By now there are, I believe,
no more serious claims for the empirical relevance [of the simple monetary model]” (cited in Frankel, 1995:
139).
But not all empirical studies reach the same conclusion regarding the problems of the monetary
approach to exchange rate determination. Following the discussion of the exchange rate puzzles,
four different papers will be presented which show that the monetary approach actually has
important insights regarding exchange rate behaviour.
3.4 Empirical studies of exchange rates
3.4.1 Empirical results of the seventies
The macroeconomic overview in the beginning of this chapter hinted at potential problems when
explaining exchange rates from a purely macroeconomic standpoint. The PPP charts, for
example, show persistent discrepancies between the fundamental value and the nominal exchange
rate. The question, then, is how the monetary models of the exchange rate have performed
37
empirically since the free floating of the 1970s. The short answer to that question would be: Not
very good.
Initially, the result of Frenkel (1976) strongly supported the flexible price monetary model when
looking at the German exchange rate vis-à-vis the American dollar during the hyperinflation in
the 1920s. Frenkel found that the coefficient of the money stock estimate (in equation (3.8)) was
near unity, in line with theory. But, as pointed out by for example Sarno and Taylor (2002: 123),
he overlooked the fact that the time-series in his regression could have been non-stationary.
Following Frenkels supportive result of the flexible price monetary model, the model “ceases to
provide a good explanation of variations in the exchange rate data” (Sarno and Taylor, 2002: 124). The
seminal article by Meese and Rogoff (1983) concluded that a random walk model performs at
least as well as three different monetary models – including the flexible as well as the sticky price
monetary model – when forecasting 1-12 months ahead. A ground-breaking result, which has
been rather robust to different tests since then. And a result that has “had an enduring effect on the
profession … [leading] Frankel and Rose to advocate a move away from fundamentals based models”
(MacDonald, 1999: 675). Besides the poor out of sample forecast, the coefficient estimates of
models like equation (3.8) as well as the empirical fit thereof were only good in periods of
hyperinflations – as the result of Frenkel (1976) for example shows (Frankel and Rose, 1995:
1693). The article by Meese and Rogoff (1983) initiated the so-called “exchange rate disconnect
puzzle” (cf. Obstfeld and Rogoff, 2000), i.e. the finding that macroeconomic fundamentals and
the (nominal) exchange rate are only weakly correlated.
In the next section, the more recent empirical results of exchange rate research is presented and
discussed.
3.4.2 Modern empirical results
With the introduction of more advanced econometric methods for testing, the monetary models
have been examined numerous times over the last two decades.
McNown and Wallace (1994) test the flexible price model, using multivariate cointegration, on
three high-inflation countries and find strong support for long-run cointegration. But when
tested on industrialised countries, this relationship disappears. A result in line with the conclusion
from the previous section that the monetary models are most successful in periods with
hyperinflation. MacDonald and Taylor (1994) use multivariate cointegration tests as well, and find
some support for the monetary model. But the coefficients are of the wrong sign, compared with
equation (3.8) above. Groen (2000) uses panel-data set, thereby trying to reduce the small sample
bias. Using a panel of the G7 countries, he cannot reject the null-hypothesis of no-cointegration.
Cushman et al (1996) reach a less pessimistic result than Meese and Rogoff (1983), as they find
38
that the exchange rate seems to be related to fundamentals (Cushman et al, 1996: 358). But, on
the other hand, they conclude that the pure monetary model seems inadequate at explaining
exchange rates over the floating period. Furthermore, one should note that they chose the seven
countries which experienced the highest inflation rate during the current float. Mark (1995) finds
that fundamentals may be able to predict the exchange rate, but only at long horizons (three-to-
four years). Frankel (1995) tests a general monetary equation of exchange rate determination
(combining long-run monetary equilibrium path with short-run overshooting) on five currencies.
He finds wrong signs on most coefficients and low significance levels; only for France are all four
coefficients in line with the hypothesis. Finally, Cheung et al (2005) test a range of exchange rate
models, including the PPP and UIP. They reach the mixed conclusion that “the results do not point
to any given model/specification combination as being very successful. On the other hand, some models do well at
certain horizons, for certain criteria. And indeed, it may be that one model will do well for one exchange rate, and
not for another” (Cheung et al, 2005: 1171).
The overall conclusion drawn from the results mentioned above is summed up by Sarno and
Taylor (2002: 136): “Empirical work on exchange rates has still not produced models that are sufficiently
statistically satisfactory to be considered reliable and robust, either in-sample or in out-of-sample forecasting”. This
conclusion is further supported by the empirical test in chapter 5, on the Japanese Yen and the
Norwegian krone against the US dollar, where a generic version of the monetary approach is
tested and firmly rejected. Thus, the problems of the monetary models when describing the
exchange rate movements seem to be a rather robust finding.
3.4.3 Exchange rate puzzles
The conclusion from the brief survey of exchange rate studies in section 3.3.1 and 3.3.2 above
emphasizes the definition of the exchange rate disconnect puzzle from the introduction in
chapter 1: The finding that macroeconomic fundamentals and the (nominal) exchange rate are
only weakly correlated (Obstfeld and Rogoff, 2000). Furthermore that the correlation is almost
zero in the short to medium run but increases somewhat in the longer run. Thus at three to four
years the fundamentals can predict (some of) the trend in exchange rate movement.
The influential result of Meese and Rogoff (1983: 17) that “the [monetary models] do not perform
significantly better than the random walk model” underlines the disconnect puzzle: Models based on
fundamentals fare no better than a simple random walk at predicting the exchange rate. This is
also evident from the PPP charts above (figures 7 and 8), as the nominal exchange rate
overshoots (or undershoots) its fundamental value for longer periods of time – up to several
years. This could lend some support to the conclusion of the Dornbusch (1976) overshooting
39
model presented above – i.e. the exchange rate overshoots the long-run value – but studies rule
out this solution (see for example Eichenbaum and Evans, 1995).
As categorized by Obstfeld and Rogoff (2000: 380) the exchange rate disconnect puzzle is the
term for a broader class of puzzles regarding the weak link between the economy and the
exchange rate. Thus, the PPP puzzle (Rogoff, 1996) is a special case of the disconnect puzzle.
The PPP puzzle is best described by the question of Rogoff (1996: 647): “How can one reconcile the
enormous short-term volatility of real exchange rates with the extremely slow rate at which shocks appear to damp
out?”. The rather large half-lifes of the deviations from PPP is evident from figures 7 and 8 as
well. According to the result of Rogoff (1996), the PPP deviations die out at approximately 15
percent per year; implying half-lifes of roughly 3-4 years. Others (for example Murray and Papell,
2005) find even higher half-lifes than Rogoff. This finding further emphasizes the overall
disconnect puzzle of exchange rates.
Another puzzle of exchange rate research – though less related to the disconnect puzzle – is the
forward premium puzzle based on the result of Eugene Fama (1984). Fama estimated a
regression on the uncovered interest rate parity (equation (3.14) above) like the following:
(3.20) ( )1 0 1t t t ts f s uβ β+Δ = + − +
With ft being the forward rate, ut the disturbance (error) term and st the spot exchange rate. Given
that the agents are risk neutral and have rational expectations, the slope parameter should equal 1
and the disturbance term should be uncorrelated with information available at time t, following
the notion of efficient markets (Taylor, 1995: 15). But studies on regression equations resembling
equation (3.20) find that the β-parameter is closer to minus unity than 1 (see Lewis, 1995; Taylor,
1996; Frydman and Goldberg, 2007: 141ff.). Excess returns are apparently non-zero, i.e. they are
predictable given current information (the forward rate, ft). Furthermore, the variances of the
returns are relatively large given the expected exchange rate changes (Lewis, 1995: 1922). The
theoretical prediction of equalisation between the expected returns of two countries has thus
been rejected by the empirics. As concluded by Lewis (1995: 1914): “The behaviour of domestic relative
to foreign returns has decisively rejected this assumption [i.e. UIP] over the floating period”.
The two building blocks – PPP and UIP – of the monetary approach apparently have several
problems when tested empirically. This obviously feed into the overall empirical performance of
models based on these two parity conditions, such as the two monetary models.
In the next section, the view of the exchange rate puzzles and the empirical results of this and the
former section will be challenged.
40
3.4.4 Puzzles after all?
Not all economists subscribe to the mainstream view of exchange rate puzzles presented in
section 3.4.1 to 3.4.3. Among these, four different papers – Imbs et al (2005), Abhyankar et al
(2005), Engel et al (2007) and Alexius (2001) –question the normal view of exchange rate puzzles.
Imbs et al (2005) take on the PPP puzzle. They show that at a disaggregated level, prices revert to
parity rather quickly, with half-lifes of around eleven months (ibid.: 3). This is in stark contrast to
the mainstream belief of around 3-4 years (or even more) mentioned above. Supporting their case
of aggregation being central to the PPP puzzle, Imbs et al (2005: 3) reproduce the consensus view
when they do not correct for the heterogeneity in real exchange rates. As they point out in
regards to disagreggation being the solution to the PPP puzzle (ibid.: 2): “It is hard to think of
reasons why clothes and vegetables, say, should revert to parity at the same time”.
Alexius (2001) takes a shot at the uncovered interest rate parity (UIP). Using long interest rates
instead of short, as most articles tend to do, she finds that UIP appears to hold more often than
not. Again in contrast to the mainstream view presented above, for example put forth by Lewis
(1995).
Charles Engel et al (2007) find that monetary models actually forecast changes in exchange rates
better than the random walk using panel techniques. They conclude that “the array of evidence
presented in this paper … lends weight to the monetary models [i.e. flexible and sticky price] of exchange rates”
(ibid.: 45).
Finally, Abhyankar et al (2005) looks at the Meese and Rogoff (1983) result from a slightly
different perspective: Instead of using statistically measures of the forecasting accuracy, they test
whether using models based on fundamentals add economic value compared with the random
walk forecast. They conclude that the “gain from using the information in fundamentals in order to predict
the exchange rate out of sample (as opposed to assuming that the exchange rate follows a random walk) is often
substantial, although it varies somewhat across countries” (Abhyankar et al, 2005: 344). Furthermore, the
gains increase with the horizon and decreases with the level of risk aversion.
Based on the insight from these four papers, abandoning models based on fundamentals
altogether, as suggested by for example Frankel and Rose, does not seem to be the right solution
after all. Apparently, macroeconomic fundamentals do convey some information about exchange
rates. Simple as the monetary models may seem, they still carry insight regarding the movements
of exchange rates, albeit with some problems at the short to medium term. The IKE theory,
therefore, does not reject the monetary models, but see them as one method for forecasting
exchange rates.
41
3.4.5 Critique of the Rational Expectations Hypothesis
As mentioned in section 3.2, the monetary approach to exchange rates builds on the rational
expectations hypothesis (REH). In this section, the critique of the REH is discussed. REH is an
important part of the monetary models, and critique thereof seems relevant in regards to the
difficulty of explaining exchange rate behaviour from a strict rational viewpoint.
According to Frydman and Goldberg (1996: 869), only two possible explanations can account for
the observed shortcomings and problems of the traditional exchange rate models shown in
section 3.4.1 to 3.4.3: “Either … the assumptions concerning the underlying structure of the economy, or … the
assumptions used in modelling exchange rate expectations”. Frydman and Goldberg confront this problem
by exchanging rational expectations with the imperfect knowledge economics representation of
forecasting behaviour, which will be discussed in detail in chapter 4. Regarding the rational
expectations hypothesis and the validity thereof, Muth (1961: 330) adds that “the only real test is
whether theories involving rationality explain observed phenomena any better than alternative theories”. As
concluded by De Grauwe and Grimaldi (2006a: 2): “A scientific theory, unlike religious belief or work of
art, should not be judged by its elegance but by its capacity to withstand empirical testing”. Following these
arguments, the empirical results put forth in section 3.4.1 strongly question the validity of the
REH and call for alternative theories. Besides the empirical results discussed above, episodes of
bubbles and crashes in the foreign exchange market has put the REH under additional pressure.
As argued by De Grauwe and Grimaldi (2006a: 3-4) as well as Frankel and Froot (1986), the large
swings in the dollar in the 1980s were not due to positive nor negative macroeconomic news, as
“there was not enough negative news to explain the crashes afterward” (De Grauwe and Grimaldi, 2006a: 4).
The REH assumes that the agents fully understand the economy in which they act. As noted by
De Grauwe and Grimaldi (2006a: 45): “It [REH] has started from the proposition that there is a one-to-
one correspondence between the information set embedded in the world and the information set that can be stored
and processed in each individual’s brain. This is an extraordinary assumption, which is tantamount to assuming
that each individual agent is a godlike creature”. As will be further build upon in both section 3.5 below,
as well as in chapter 4 on the imperfect knowledge economics theory, the private information set
(and the individual expectations computed on this background) available to each agent seem to
matter for the formation of the exchange rate. In addition agents use heuristics (i.e. “rule of
thumb”), past experience as well as the expected action of other agents when forming their
beliefs of the future asset price. Regarding the latter, it can be “rational to behave irrational” for
an agent, i.e. when compared to the outcome of the REH, given that he expects other agents to
behave irrational (e.g. that they use the wrong model).
42
As noted by Lyons (2001: 75) the agents in the economy act non-strategic following the REH.
That is, the effect from individual trading on prices is not neglible, but still the traders in the
market act as price takers in the models; they completely disregard the effect from the other
agents’ trading. This paradox is referred to as the “schizophrenia problem” (Lyons, 2001).
Brock and Durlauf (2006) argue that if the assumption that the economic actors a priori know
the true parameters of a given dividend process was dropped, and instead replaced with an
assumption that the agents are learning about the parameters along the way, the puzzles (e.g.
equity premium puzzle) are solved. That is, the puzzles are solved by abandoning the strong
assumption of the rational expectations hypothesis. This could be the case for the range of
exchange rate puzzles as well. Brock and Durlauf (ibid.: 124) further claim that model uncertainty
increases the importance of how agents acquire information of the economy: “If an agent is
confident that he knows the true model of the economy, he is presumably going to treat the costs and benefits of
information acquisition differently than when the true model is unknown”. As they conclude, this is even
more relevant when the economy moves across regimes, “which in our context may be interpreted as
shifting models” (ibid.: 124). This, as we shall see, is in line with the conclusion of the imperfect
knowledge economics theory presented in the next chapter: Both in regards to the importance of
model uncertainty, the heterogeneity of agents and the shifting of the models they use.
Overall, the empirical anomalies of exchange rates – the exchange rate disconnect puzzle, the
PPP puzzle, the forward bias puzzle as well as the aforementioned bubbles and crash – seem to
invalidate the REH. Furthermore, the assumptions of the REH – e.g. that the agents are fully
informed – seem at odds with different surveys of currency trader behaviour, for example by
Cheung and Chinn (2001) and Cheung et al (2006) presented in chapter 2.
In the section on market microstructure below, some of the REH assumptions will be relaxed.
According to the microstructure approach, relaxing the assumptions of the monetary models
leads to a possible solution to the exchange rate disconnect puzzle. The imperfect knowledge
theory discussed in chapter 4 gives another solution to the exchange rate disconnect puzzle. This
theory, in parts, builds on some of the same assumptions and insights as the microstructure
approach; first and foremost a relaxation of the strict rational expectations hypothesis. Relaxing
the REH thus seems to be a possible solution to the exchange rate puzzles.
3.4.6 Different solutions to the exchange rate puzzles
Besides the critique of the REH discussed in the previous section, several economists have
attempted to explain the puzzling behaviour of exchange rates. Primarily two alternatives have
been put forth: to introduce either rational bubbles or non-rational behaviour into exchange rate
modelling (cf. Lyons, 2001: 172; Goldberg and Frydman, 2007: 105-6). Regarding the bubble-
43
explanation, the empirical evidence has, until now, been rather unconvincing (Lyons, 2001: 172).
Regarding the latter, this has been an unattractive alternative for economists.
Another explanation has been non-linearity in the demand for money functions (Sarno and
Taylor, 2002: 126), which would lead to misspecification of the log-linear forms of the monetary
models presented above. Along the same line, Taylor and Peel (2000) argue that the adjustment
towards the equilibrium value is non-linear. Others have argued (Obstfeld and Rogoff, 2000) that
(diminutive) international trading costs in the goods market combined with “pricing to markets”
(Sarno and Taylor, 2002: 57) account for a large part of the disconnect puzzle. The significant
result of Engel and Rose (1996) of border effects when looking at prices in Canada and USA lend
some support to this argument. These possible solutions to the puzzles are, more or less, based
on the monetary approach presented above. And as argued in the critique of the rational
expectations hypothesis, maybe this is where the problem (and the possible solution) lies?
In the next section the microstructure approach to exchange rate will be presented. This
approach relaxes the strong assumptions of perfect knowledge of the monetary models. The
microstructure approach thus adds a microeconomic foundation to the exchange rate research,
which is missing entirely from the monetary approach. Furthermore the finding that private
information matters for exchange rate determination is a first step towards understanding the
imperfect knowledge economics theory presented in chapter 4.
3.5 The market microstructure approach
3.5.1 Introducing market microstructure
The exchange rate models presented above only focuses on the macro-economic aspects of the
exchange rate. But as shown in chapter 2 there seems to be a case for looking into the micro-
economic aspects of exchange rate determination as well.
The microstructure approach to exchange rates is summed up as follows by Sarno and Taylor
(2002: 264): “The foreign exchange microstructure approach, unlike the conventional macroeconomic one, typically
does not assume that only public information is relevant to exchange rates, that foreign exchange market agents are
homogenous, or that the mechanisms used for trading is inconsequential”. As shown in chapter 2 this seems
plausible and in line with, for example, the result of the survey of American traders by Cheung
and Chinn (2001) and the apparent low transparency of the foreign exchange market. Following
Lyons (2001: 4), the microstructure approach relaxes the following three assumptions of the
monetary approach:
44
• Information: Microstructure models recognize that some information relevant to
exchange rates is not publicly available
• Players: Microstructure models recognize that market participants differ in ways that
affect prices
• Institutions: Microstructure models recognize that trading mechanisms differ in ways that
affect prices; i.e. trading matters for exchange rates
It is worth noticing, though, that the microstructure approach seeks to add a micro-foundation to
the macroeconomic monetary approach, not to disregard the monetary approach altogether
(Lyons, 2001).
Equation (3.21) below shows a generic version of the traditional asset-price models, as presented
in section 3.2 – e.g. equation (3.9):
(3.21) 1e
t t tS F Sβ α +′= +
Here, St denotes the spot exchange rate at time t, Ft a vector of fundamental variables with
explanatory power of the exchange rate (e.g. money supply, interest rate, output), β a vector of
factor loadings and St+1e the expected spot exchange rate at time t plus one (following Sager and
Taylor, 2006: 88). One notes that this class of models are strictly macroeconomic and, following
the bullet-points from above, omits the informational processing, agent heterogeneity, and
institutional structure in the market altogether. As noted by Sager and Taylor (ibid.: 88): “asset-
price models are essentially equilibrium models that marginalize the importance given to the means by which that
equilibrium is reached or the institutional setting in which currency prices are determined”. The market
microstructure approach thus focuses on the behaviour of the agents as well as the market
characteristics (cf. Taylor, 1995: 39); aspects which, following the quote by Sager and Taylor
above, the asset price models pay no attention to.
Regarding the former – information – it was shown in chapter 2 that agents in the market
process the information they receive differently (i.e. using methods bounded in the fundamental
value, in technical trading etc.). Below the concept of “order flow” will be discussed to shed light
at the importance of private information and its effect on the price setting of exchange rates.
Regarding the latter – institutions – the lack of transparency discussed in chapter 2 adds to the
importance of looking closely at the foreign exchange market structure. Apparently, the structure
of the exchange rate market (decentralised, non-disclosure etc) exacerbate the informational
asymmetries pointed out above.
45
3.5.2 Order flow as an important factor for exchange rates
First of all it is important to be aware of the distinction between order flow and transaction
volume: Order flow is transaction volume that is signed (Lyons, 2001: 6). Furthermore, order
flow takes on a negative sign5, and the initiator of the trade is on the sell side. Over time order
flow is measured as the sum of signed buyer- and seller initiated orders. In addition, as noted by
Lyons (2001: 7), order flow is a variant of excess demand; but it does not sum to zero since the
orders are initiated against a market maker who then soaks up the difference between buyers and
sellers.
The question is, then, why order flow should be important in regards to prices? First of all,
relaxing the assumption of public information leads to order flow conveying information relevant
for prices. Two assumptions of public information in typical macroeconomic models are (cf.
Lyons, 2001: 21):
• All information relevant for exchange rates is publicly known,
• The mapping from this information to prices is also publicly known.
The second assumption relates to the “right” model of exchange rates, which will be discussed
(and relaxed) further in the next chapter on imperfect knowledge economics. As seen in the
empirical study above, there is hardly consensus on which model is the true model of exchange
rate behaviour (cf. Cheung et al, 2005). Thus, as noted by Lyons (2001: 21), relaxing the second
assumptions is hardly controversial. From this, order flow suddenly becomes important. And this
has implication for the first assumption: not all relevant information is publicly known, as order
flow is not observable by the public or by all agents in the foreign exchange market. Thus, private
information matters for exchange rate determination. Private information is defined as (cf. Lyons,
2001: 26):
• Information not known by all agents; and
• Private information produces a better price forecast than public information alone
An example (following Lyons, 2001): Two FX traders (A and B) at different banks observe the
same macro news. If agent A does not know how agent B will interpret the effect on prices from
5 Following the example in Lyons (2001): If an agent approaches a dealer (marketmaker) and sells him 10 units (eg.
dollars), transaction volume is 10 but orderflow is -10.
46
the news (as the mapping of prices is not publicly known), observing the second agents trade
(order flow) gives some of this information. Hence the trade of B both influences the price as
well as agent A’s knowledge.
Since the foreign exchange market suffers from some lack of transparency, a dealer with
privileged access to order flow constitutes private information. Figures 11 and 12 below shows
the connection between order flow and exchange rate movements over a three months period
(May to August, 1996).
Figure 11 – Order flow (right hand scale) plotted against DM/USD
0,36
0,37
0,38
0,39
0,4
0,41
0,42
0,43
0,44
0,45
01/05 08/05 15/05 22/05 29/05 05/06 12/06 19/06 26/06 03/07 10/07 17/07 24/07 31/07 07/08 14/08 21/08 28/08-1200
-1000
-800
-600
-400
-200
0
200
400
600
DM/USD
Orderflow Source: Data from R.K. Lyons’ homepage, http://faculty.haas.berkeley.edu/lyons/fxdata.html
Looking at the two figures, there seems to be a rather strong correlation between the order flow
variable and the exchange rate movements over the period. This is supported by empirical tests.
Evans and Lyons (2002) find that order flow account for 40 to 60 per cent of the daily variation
in spot exchange rates in DEM/USD and YEN/USD. This result will be discussed further in the
following section with a test of a microstructure model. Furthermore, Evans and Lyons (2007)
find that the flows have significant forecasting power for macro fundamentals as well. That is,
order flow is central “to the process by which expectations of future macro variables are impounded into
exchange rates” (Evans and Lyons, 2007: 32). Lyons (2001), as well as Rime (2000), concludes that
private information plays a role, and that order flow conveys information on this. Gehrig and
Menkhoff (2004) finds, in a survey of foreign exchange market traders and fund managers, that
order flow analysis is almost as important as fundamental– and technical analysis, further
emphasizing the importance of the variable. Rime et al (2007) also find, using a year of data, that
orderflow have both explanatory and forecasting power for exchange rates. But as noted by
47
Sarno and Taylor (2002: 274), this does not necessarily imply that order flow is the main driver of
exchange rates. Instead, macroeconomic variables may still be the underlying force of exchange
rate movements. But conventional measures of macroeconomic variables could be imprecise at
the release date.
Figure 12 – Order flow (RHS) plotted against Yen/USD
4,61
4,62
4,63
4,64
4,65
4,66
4,67
4,68
4,69
4,7
4,71
4,72
01/05 08/05 15/05 22/05 29/05 05/06 12/06 19/06 26/06 03/07 10/07 17/07 24/07 31/07 07/08 14/08 21/08 28/08-500
0
500
1000
1500
2000
2500
3000
YEN/USD
Orderflow
Source: Data from R.K. Lyons’ homepage, http://faculty.haas.berkeley.edu/lyons/fxdata.html
As Ehrmann and Fratzscher (2004) argue, the data used for exchange rate research differs from
the real-time data available to market participants when making trading decisions – as revised
data with the “true” observation is first published months after the initial (real-time) data is
observed by market participants. Thus, order flow could be a precise proxy for the
macroeconomic fundamentals as well as the variation of the expectations of future fundamentals.
In the following section an exchange rate model incorporating order flow as an important
variable is presented and tested.
3.5.3 A microstructure model of exchange rates
In the model of Evans and Lyons (2002), each day of exchange rate trading is split into three
rounds. In the first round, dealers trade with customers. In the second round, dealers trade
among themselves (spreading the risk, cf. Sager and Taylor, 2006: 90). Then, the dealers observe
the order flow, xt., from the interdealer trade. In the third round, dealers again trade with the
public (based on the information obtained from the order flow variable), thereby sharing
overnight risk (Evans and Lyons, 2002: 174).
48
The model consists of two assets, one riskless and one risky (here: foreign exchange). There is T
trading periods (i.e. days), and the payoff on the risky asset is given by R:
(3.22) 1
1
T
tt
R r+
=
= Δ∑
The trΔ are i.i.d. (0, σr2) and observable before each trading day. There is N dealers on the foreign
exchange market, indexed by i, and a continuum of non-dealers (customers) indexed by
[ ]0,1z∈ . The round 1 price of dealer i is given by Pit1 and the customer order Cit
1. With Cit1 < 0
being defined as net customer selling.
The interdealer order flow at the end of round 2 is given by:
(3.23) 1
N
t iti
x T=
Δ =∑
In period 3, the dealers quote a scalar price Pit3 and total public demand is denoted by Ct
3. The
public demand can be written as:
(3.24) ( )3 3 31 3|t t tC E P Pγ +⎡ ⎤= Ω −⎣ ⎦
Here, γ is the public’s aggregate risk-bearing capacity and Ω the available public information
(Evans and Lyons, 2002: 175).
In period 3, the prices adjust such that Ct1+Ct
3 = 0. That is, 3 1t t
i
C C= −∑ . And since the order
flow indicates the customer order, Ct1, this can be written as: 1
t ti
C x= Δ∑ . Inserting into equation
(3.24) yields:
(3.25) ( )
3 31 3
1
1|
1,
t t t
t
i ii
P E P x
r x
γ
λ λγ
+
=
⎡ ⎤= Ω + Δ⎣ ⎦
= + Δ ≡∑
Equation (3.25) shows that the market clearing price in period 3 is the sum of the expected
payoff on the risky asset adjusted for a risk premium, defined by shift in order flow. This yields
the equation of price changes in the exchange rate:
(3.26) t t tP r xλΔ = Δ + Δ
Equation (3.26) shows the price change from period t-1 to period t. From this model, the
importance of order flow (the xt variable) is evident.
49
Evans and Lyons (2002) estimate an equation based on (3.26), with change in the payoff, trΔ ,
defined as the change in the interest rate differential: ( )*tt tr i iΔ = Δ − . Regressing on equation
(3.27) is then a test of the model given by equation (3.26) above:
(3.27) ( )*1 2t t t t tp i i Xβ β μΔ = Δ − + Δ +
The microstructure approach argues that the sign on the Xt variable should be positive, since this
signals a net purchase of (say) dollars, which should push up the price of the asset. For the β1
coefficient on the interest rate differential, the sign should be positive as well: following the
sticky-price model from section 3.2, an increase in the interest rate initially leads to a jump
depreciation of the currency (i.e. overshooting), followed by appreciation.
The data for the interest rates in the regression is the closing price for 3 months treasury bills
taken from the EcoWin database. In Evans and Lyons (2002) the interest rates are the policy
rates for the respective countries. I find that the treasury bills are a more precise variable, as they
represent the riskfree investment, in line with the interest rates in the monetary models above.
Secondly, the policy rates in the dataset used in Evans and Lyons (2002) is rather invariable; e.g.
the policy rate of Japan is unchanged over the period. In Appendix B figure 3, the interest rates
of the three countries (USA, Germany and Japan) are plotted over time. The order flow variable
is taken from the dataset of Richard K. Lyons.
Table 3 presents the results – along with the result from Evans and Lyons (2002). T-statistics are
shown in parentheses. The fit of the two models as well as the scaled residuals are shown in
appendix A figures 4 and 5.
Table 3 – Estimates of the microstructure model (equation (3.27)) and the result of Evans and Lyons (2002)
No. obs: 80 β1 – interest rate β2 – order flow R2
DEM/USD 0.934 (1.6) 2.03 (10.3) 0.654
– Evans & Lyons 0.52 (1.5) 2.10 (10.5) 0.64
JPY/USD -0.96 (-0.671) 2.77 (7.25) 0.408
– Evans & Lyons 2.48 (2.7) 2.90 (6.3) 0.45 Source: Data taken from the EcoWin database and Richard K. Lyons homepage. T-statistics in brackets.
The first remarkable thing of the estimation is the relatively high fit of the model – an R2 between
40-65%. It is evident that using 3 months Treasury bill rates does not change much compared
with the result of Evans and Lyons (2002). In regards to the result above, Lyons (2001: 188)
concludes: “That order flow explains such a large percentage of price moves underscores the inadequacy of
50
…[the] public information framework”. Lyons (2001: 173) consequently suggests a hybrid version –
equation (3.28) below – of the generic monetary model given by equation (3.21) above.
(3.28) 1' et t t tS F X Sβ φ α +′= + +
Where Xt includes microstructure components. That could be order flow but also inventories or
other important variables from the microstructure research. Lyons (2001: 176) elaborates on the
hybrid model: “the model accommodates both possibilities: information that affects price directly and information
that affects price via order flow. Armed with models that allow for both, we can let the data determine their relative
importance”. The result of the microstructure model highlights the fact that traditional
macroeconomic models of exchange rate determination have omitted important variables; in this
case order flow. It further underscores the importance private information plays in the
determination of exchange rates. A conclusion also supported by Bacchetta and Wincoop (2006:
570), who conclude that “investor heterogeneity is key to understanding exchange rate dynamics”.
But one should note, though, that a problem with models incorporating microstructure variables
like order flow is that these variables are not publicly available and hence difficult to obtain for
researchers. This further amplifies the importance of order flow, of course, since only some of
the agents in the foreign exchange market have access to useful data on this variable.
The result above does not imply that macroeconomic variables should be abandoned altogether –
as Lyons (2001) also concludes. The microstructure approach rather highlights the shortcoming
of the strictly macro based models such as the FPMM or the overshooting model presented in
this chapter.
3.6 Conclusion This chapter commenced with a macroeconomic glance at exchange rate movements. Looking at
the charts, the exchange rate puzzle(s) emerged: there seems to be a disconnect between
macroeconomic fundamentals and the exchange rate, at least for some time periods.
Then two of the most influential exchange rate models – the flexible and sticky price monetary
models – were presented. Both models are part of the dominant monetary approach to exchange
rates, stating that exchange rates is the relative price of two moneys and determined by the
relative supply and demand hereof. Furthermore macroeconomic fundamentals, such as output,
are assumed to be key determinants of exchange rates.
The empirical study showed, however, that this approach may be too simple. Although several
researchers find some relation between the exchange rate and the fundamentals in the long run
(i.e. three to four years), the results are not conclusive nor do they point to one specific model for
51
exchange rate determination. And in the short to medium term, the relationship is almost non-
existent. On the other hand, some researchers actually find that the models give a relatively good
explanation of movements in the exchange rate. For some periods of time, for specific currency
crosses, specific models can give an explanation of the movements in the exchange rate, both in-
and out-of sample. But this is still a puzzling result and certainly not in line with the sharp
prediction of the two monetary models presented in this chapter.
This leads to several questions: Why can the currency be overvalued for periods of several years?
Why do different macro fundamentals seem to matter at different points in time? Why is the
exchange rate so slow to react on an overvaluation (undervaluation)? And when the exchange
rate reaches its fundamental value, why does it not stay there but continues its movement away
from parity?
One possible response to these questions could be that the underlying assumption of rational
expectations (REH) is flawed. And it is obvious that, at least to some extent, the REH seem a bit
unrealistic. Relaxing some of the assumptions, especially concerning the information set,
suddenly leads to increased importance of the microeconomic side of exchange rate
determination, as the microstructure approach argues. If private information plays a role,
variables like order flow convey important insights on exchange rate behaviour; insights that the
agents in the market utilise for setting their exchange rate expectations. As we have seen from the
test of the simple microstructure model, order flow does play an important role – at least in the
short run of up to a year. A result that has turned out to be rather robust.
In the following chapter some of the insights from the microstructure approach, as well as
conclusions from chapter 2, will be further elaborated on when presenting the imperfect
knowledge economics (IKE) theory: The criticism of rational expectations and the insights from
the microstructure approach of private information plays a central part in the IKE approach to
exchange rates, as well as the heterogeneity of agents. The IKE theory puts forth another
explanation of the exchange rate disconnect puzzle: The agents utilise a portfolio of models when
assessing the future value of the exchange rate, since they have only partial knowledge of the
market in which they participate. A portfolio which could include the monetary models discussed
above (but also technical analysis or other methods such as order flow analysis). Thus, the
outcome of the IKE model has to be different from that of the rational expectations outcome,
since the heterogeneous agents (by assumption) do not use the same models. But the IKE theory
does not abandon the macroeconomic fundamentals importance for exchange rates. Instead, the
importance of the macroeconomic variables (as seen in for example Cheung and Chinn, 2001)
changes over time. Nor does the IKE theory abandon rationality on part of the agents. Instead
52
the agents behave bounded rational, knowing that they do not perfectly know (or understand) the
market completely. Hence, the IKE theory combines the insight from both the macro and micro
founded models discussed in this chapter, as well as the insights of agent’s behaviour discussed in
chapter 2.
53
Chapter 4
Imperfect knowledge economics “… Macroeconomists face a choice: either they can continue to try to explain asset market
anomalies using the rational expectations hypothesis and live with the specter of gross
irrationality in assets markets, or they can reconcile that imperfect knowledge, which is an
inherent feature of the environment in which agents have to form expectations, may be the key
to reconciling the movement of asset prices with the postulate of individual rationality”
Roman Frydman and Michael D. Goldberg (2002), p. 2
”… I confess that I prefer true but imperfect knowledge, even if it leaves much indetermined and
unpredictable, to a pretence of exact knowledge that is likely to be false.”
Friedrich August von Hayek, Nobel Prize lecture (1974)
4.1 Introducing imperfect knowledge economics The central part of imperfect knowledge economics (IKE) is to place “imperfect knowledge on the
part of market participants and economists at the centre of our analysis” (Frydman and Goldberg, 2007: 6),
and thereby to replace the rational expectations hypothesis modelling of expectations.
In Frydman and Goldberg (2007: 8) knowledge is referred to as imperfect if: “... no one has access to
a fully predetermined model that adequately represents …the causal mechanism that underpins outcomes in all time
periods, past and future”. This is in accordance with Hayek (1945: 519): “The economic problem of society
is …a problem of the utilization of knowledge which is not given to anyone in its totality”.
Thus, the agents in the economy do not necessarily use the same (nor the “right”) model when
assessing the economy in which they live, and the agents therefore adopt different strategies in
forecasting the future; hence, the outcome (e.g. the expected value of the exchange rate) is bound
to be different from that of rational expectations.
Overall, two factors create the imperfect knowledge: a diffuse (in the sense: not fully intelligible)
as well as an unforeseeable world. Regarding the first, the agents in the society only have partial
information about the world they live in, “…market outcomes arise out of a division of knowledge whose
totality remains opaque to any one individual” (Goldberg and Frydman, 2007: 9). This is in line with the
54
microstructure approach from chapter 3 and the conclusion on the importance of private
information.
Regarding the second, the future states of the economy are uncertain and this affects the agents’
decisions as well. According to Frank Knight (1921: 198): “Hence it is our imperfect knowledge of the
future, a consequence of change, not change as such, which is crucial for the understanding of our problem”. The
aspect of Knightian uncertainty and its significance for the IKE theory will be discussed in the
next section.
Because of these two factors, imperfect knowledge and uncertainty, the agents use different
theories and models when forecasting and deciding which strategy to follow. As pointed out by
Frydman and Goldberg (2004: 1): “Financial markets are always in flux. Relationships that initially hold
are sooner or later replaced by new relationships. Strategies that initially generate profits lose their ability to produce
profits with the passage of time… In such a world of imperfect knowledge, profit seeking agents must engage in a
creative forecasting process.” The agents in the economy do not, therefore, behave irrationally, but
instead act (rationally) on the notion that their knowledge is not perfect: “Rational agents recognize
that their models and rules are not strictly correct and consequently behave like scientists... they test their models
and rules against available data and in the face of contradictory evidence revise their trading strategies” (Frydman
and Goldberg, 2002: 8). This is in line with, for example, the assumption of Frankel and Froot
(1986: 34) who, regarding the bubble of the American dollar in the 1980s, conclude that “… the
bubble is the outcome of portfolio managers’ attempt to learn the model”. This is also supported by Kurz et al
(2004: 2) who point out that: “agents do not hold rational expectations, since the environment is dynamically
changing, not stationary, and true probabilities are unknown to anyone. In such a complex environment agents use
subjective models. Some consider these agents irrational, but one cannot require them to know what they cannot
know”.
In the case of exchange rates, this means that agents have imperfect knowledge of the
relationship between the macroeconomic fundamentals and the exchange rate (Frydman and
Goldberg, 1996: 870). The agents consequently utilise a number of different theories – the set of
leading theories – to forecast the movements in the exchange rates. This set of theories could
include the flexible and sticky price monetary models discussed in chapter 3 as well as technical
trading methods or microstructure models. That is, the agents have a portfolio of models when
forecasting the exchange rate. This, as we shall see below, furthermore leads to different
forecasting strategies across agents, and hence heterogeneous expectations. Because of this, the
IKE theory does not fully prespecify the changes of preferences and forecasting behaviour from
the initial point and all other points in time (Frydman and Goldberg, 2007: 14). This means that
the IKE theory have only qualitative predictions regarding the possible outcomes (ibid.: 66). This
55
is one of the more critical points of the IKE theory, as argued by Papell (2003), and it will be
discussed further in section 4.6 below.
This chapter is structured as follows: First a discussion of imperfect knowledge as well as
uncertainty, including a discussion of the term Knightian uncertainty. Then follows a discussion
of preferences and forecasting strategies of the agents in an IKE setting, building on the results
from prospect theory. This is followed by the uncertainty-adjusted UIP, UAUIP, a new parity
condition on the foreign exchange market which implements the aforementioned prospect
theory. Then a version of a traditional monetary model which includes IKE specifications. This
model shows how the large swings away from the fundamental value (here PPP) can be sustained
in a model of exchange rates when the agents have imperfect knowledge. And why the price
eventually returns to parity. This leads to a discussion of the IKE theory as a possible solution to
the exchange rate disconnect puzzle.
4.2 Imperfect knowledge and uncertainty
4.2.1 Knightian uncertainty
The term “Knightian uncertainty” stems from the book of Frank Knight (1921), “Risk,
uncertainty and Profit”. Knight introduced a clear distinction between measurable uncertainty,
called risk, and true uncertainty, which is assumed to be immeasurable. As noted by Knight
(1921: 233): “The practical difference between the two categories, risk and uncertainty, is that in the former the
distribution of the outcome in a group of instances is known (either through calculation a priori or from statistics of
past experience), while in the case of uncertainty this is not true, the reason being in general that it is impossible to
form a group of instances, because the situation dealt with is in a high degree unique”.
As discussed by Richardson (1953: 136), some writers have argued that the distinction between
risk and uncertainty is of degree, not of kind (as Knight argues), i.e. that “no future event is ever
wholly unique” (ibid. 138). Richardson therefore makes another distinction, following the idea of
Knight, between objective and subjective estimates. The first being statistical estimates, which are
demonstrable and objective in the sense that “given the same information and same rules, everyone should
get the same result” (ibid. 140). The latter distinction can, on the other hand, not be fully
demonstrated or communicated, as the “estimate is private to the estimator and in this sense subjective”
(ibid. 140). This lies end to end with the notion of private information discussed in chapter 3.
4.2.2 Imperfect knowledge
The distinction of objective and subjective estimates leads to the hypothesis of imperfect
knowledge: Since some information is private, according to Richardson (1953), knowledge is
56
dispersed between the agents in the society. From this follows that the agents use different
models and methods when assessing the future value of financial assets, as argued by Frydman
and Goldberg (2007). Surveys of foreign exchange traders show that agents use different
methods when buying or selling currency (e.g. Cheung and Chinn, 2001). In addition, the agents
use rather simple methods – compared with the complexity of the market – when assessing the
future value of financial assets. As Kahneman and Tversky (1992: 317) concludes: “… when faced
with a complex problem, people employ a variety of heuristic procedures in order to simplify the representation and
the evaluation of prospects”. Secondly, a large part of the information available to the agents in the
market may be public, for example macroeconomic data, but the way the agents utilize this data
when making forecasts and decisions may be individual. As pointed out by Sarno and Taylor
(2006: 273) there is an increasing dispersion, measured by the standard deviation, in forecasts (i.e.
expectations) at longer forecast horizons.
The foreign exchange market specifically has a rather low transparency (as discussed in chapter
2), as trades are not generally observable. Hence, some agents have information which is not
available to all the participants in the market, for example order flow data. This private
information, as seen from the discussion of the microstructure approach in chapter 3, plays an
important part when the exchange rate is determined; at least in the shorter run. Private
information is an important part of the imperfect knowledge economics theory which, combined
with the uncertainty aspect discussed above, leads to a different result than given by the rational
expectations hypothesis.
It could be, though, that information of exchange rates was available to all agents but at a (small)
cost. In Grossman and Stiglitz (1980) agents are initially identical but can obtain information by
spending c. As Grossman and Stiglitz conclude (ibid.: 405): “We have argued that because information is
costly, prices cannot perfectly reflect the information which is available, since if it did, those who spent resources to
obtain it would receive no compensation”. Only when information is inexpensive, the market price will
reflect the informed agents (i.e., those who paid c) information. Even in the case of information
being available to all agents, the price of the asset (in this case the exchange rate) may not reflect
this, since it is costly (e.g. time consuming) to obtain the information. Another variant of this is
discussed in Brock and Durlauf (2006), mentioned in chapter 3, where benefits and costs of
information acquisition depend on the degree of model uncertainty – which is a central part of
the IKE theory.
Thus, in the theory of imperfect knowledge agents differ and the information available to the
agents differs, at least to some degree. This is coupled with the assumption that i) the future is
uncertain and ii) the “correct” model of the economy is unknown. Regarding the second
57
assumption, the empirical survey in chapter 3 showed that none of the different monetary models
are good at describing movements in various currency crosses (following e.g. Cheung et al, 2005).
Hence, this last assumption seems to be supported by empirical evidence.
This is the foundation of the IKE theory. In the following section, the IKE preferences and
strategies of the agents are modelled.
4.3 Modelling preferences and forecasting strategies In this section, the representation of preferences and forecasting strategies in the IKE framework
will be developed. First a discussion of the expected utility hypothesis, which is the conventional
representation of preferences, followed by a critique thereof put forth by Kahneman and Tversky
(1979, 1992). Then follows a presentation of prospect theory, which is the building block for the
modelling of preferences in the IKE theory.
4.3.1 The expected utility hypothesis
Normally, economic models are based on assumptions of how rational agents rank the different
uses of the resources (or actions) that are presented to them. One of these assumptions is that of
risk aversion: Agents tend to, when faced with comparable returns, to choose the less risky
alternative. Specifying the preferences, economists usually rely on the expected utility hypothesis
(Frydman and Goldberg, 2007: 155), stating that the utility of a gamble equals the expected value
of the utilities of the different outcomes compromising the gamble. According to Kahneman and
Tversky (1979: 263ff), expected utility theory is based on the following three tenets:
i) Expectation: ( ) ( ) ( )1 1 1 1, ;...; , ...n n n nU x p x p p u x p u x= + +
The overall utility of a given prospect, given by U, equals the weighted utility of the different
states (outcomes).
ii) Asset integration: ( )1 1, ;...; ,n nx p x p is acceptable at asset position w given that
( ) ( )1 1, ;...; ,n nU w x p w x p u w+ + > Thus, a prospect is acceptable given that the utility from
integrating the prospect with the assets exceeds the utility alone. As pointed out by Kahneman
and Tversky (1979: 264), the domain of the utility function is final states rather than gains or
losses.
iii) Risk aversion: u is concave ( )'' 0u < . An agent is risk averse if he prefers the certain prospect,
x, to any risky prospect (gamble) with expected value of x.
58
4.3.2 Critique of the expected utility hypothesis
The three assumptions above is, as mentioned, the mostly used normative and descriptive model
for decision making under uncertainty. But as noted by Kahneman and Tversky (1992: 297): “a
substantial body of evidence shows that decision makers systematically violate its basic tenets.”. Kahneman and
Tversky presents five phenomena of choice which violates the standard model presented above
(ibid.: 298):
Framing effects: Evidence of various experiments has shown that framing the options yield
different preferences, contrary to rational choice theory.
Non-linear preferences: Following the expectation principle, utility of a risky prospect is linear
in outcome probabilities. According to Kahneman and Tversky (1992), this has been challenged
by showing that the difference between probabilities of .99 and 1.00 has a larger impact on
preferences than the difference of 0.10 and 0.11.
Source dependence: People’s willingness to bet on an uncertain event does not only depend on
the degree of uncertainty, but on its source as well.
Risk seeking: People often prefer a small probability of winning a large prize over the expected
value of that prospect. Furthermore, risk seeking is common when people have to choose
between a sure loss and a substantial probability of a larger loss.
Loss aversion: That the disutility from a loss exceeds the utility of gains of the same size
(Frydman and Goldberg, 2007: 161). There is an observed asymmetry between gains and losses,
which is “far too extreme to be explained by income effects or by decreasing risk aversion” (Kahneman and
Tversky, 1992: 298)
The five “anomalies” presented above have led Kahneman and Tversky to present the alternative
theory for choices under uncertainty, prospect theory, which will be discussed in section 4.3.3.
The prospect theory is part of the basis for the imperfect knowledge economics theory.
4.3.3 Prospect theory
Prospect theory implies that the (risky) prospects are evaluated by a value function which has the
following three characteristics (Frydman and Goldberg, 2003: 17, and Kahneman and Tversky,
1991: 1039):
Reference dependence: “The carriers of [utility] are gains and losses defined relative to a reference point”.
Loss aversion: As shown above, the disutility from a loss exceeds the utility of gains of the same
size.
59
Diminishing sensitivity: “The marginal utility of both gains and losses decreases with their size”.
According to Kahneman and Tversky (1991: 309), this leads to the following two-part power
function:
(4.1) ( )( )
0
0
x if xv x
x if x
α
βλ
⎧ >⎪=⎨− − <⎪⎩
The v(.) is defined over gains and losses in wealth relative to a reference point. The parameter λ is
the degree of loss aversion and α and β the curvature of the utility function over gains and losses,
respectively (Frydman and Goldberg, 2003: 19). Loss aversion on the part of the agents is implied
for λ larger than 1. According to Frydman and Goldberg (ibid.: 19) empirical results have found
the λ to be above 2, which strongly indicates loss aversion.
According to the characteristic diminishing sensitivity, above, it can be verified that the utility
function in equation (4.1) is concave in gains and convex in losses for α and β < 1 (since, for x >
0, 0x′′ < ). In the following section the prospect theory is applied to the foreign exchange
market.
4.3.4 Prospect theory and the foreign exchange market – the IKE approach
In the following setup6 there are two countries, A and B, two types of non-monetary assets, A
and B bonds. Country A is referred to as the “domestic country”, and the bonds of type A and B
are denominated in A and B currencies, respectively. The nominal return on the bonds are
denoted by Ati and B
ti . The ex-post nominal return on a pure long position in foreign exchange,
Rt+1L is given by the following equation7:
(4.2) 1 1L B At t t t tR s s i i+ += − + −
Where “L” denotes the long position and st the log level of the spot rate at time t. The return on
the pure short position is given by:
6 This section is primarily based on Frydman and Goldberg, 2007: pp. 159ff. 7 At a pure long position, the agent borrows one unit of domestic currency (say DKK) at time t, paying the interest
rate itA. The agent then sell this currency (for EUR, say) at the foreign exchange market and receive 1ts
at interest
rate itB. In period t+1, the agent will sell 1ts
(1+ itB ) at the rate st+1. The total return at time t+1 would then be:
( ) ( )1 1 1B Att t
t
s i is+ + − + . Taking logs leads to equation (4.2) above. For a pure short position, the opposite holds
true, ie. the return is –Rt+1 (cf. Frydman and Goldberg, 2007: 140)
60
(4.3) 1 1S A Bt t t t tR s s i i+ += − + −
Only the problem faced from country A wealth holders is evaluated, as the decision faced by
both A and B holders are identical.
Following the notion of a reference level from prospect theory above the change in wealth
(denoted by W) relative to some reference level for individual i is given by equation (4.4)
(Frydman and Goldberg, 2007: 159):
(4.4) ( )1 1 1 1i i i i i A it t t t t t t t tW W W a R i p+ + +
⎡ ⎤Δ = −Γ = + + − −Γ⎣ ⎦
Here the portfolio share (i.e. the share of foreign assets, either short or long) is given by ati, pt is
the domestic rate of inflation and the individual reference level is given by Гti. Rt+1 is the return
on the position, as given by equation (4.2). Positive (negative) values of 1i
tW +Δ are equal to
individual i experiencing a gain (loss). The first problem, then, is to specify the reference level.
Following Frydman and Goldberg (2007: 160), one could define this reference level as the wealth
obtained from staying out of the market altogether, i.e. solely buying domestic bonds. This is
given by equation (4.5) below:
(4.5) ( )1i i At t t tW i pΓ = + − for all i and t
The potential gains (and losses) of agent i then comes from participating in foreign exchange
speculation, relative to this reference level. Inserting the value for the reference level, equation
(4.5), into the wealth equation (4.4) yields the following:
(4.6) 1 1i i i
t t t tW a R W+ +Δ =
For an agent holding a long position, ati > 0, a positive realisation of Rt+1 (denoted rt+1
+) leads to a
gain. On the other hand, an agent holding a short position, ati < 0, experiences a loss for rt+1
+. The
opposite is true for a negative realisation of Rt+1 (denoted by rt+1– ), which leads to a gain (loss) for
the short (long) position. An agent holding a long position is in the following defined as a “bull”,
whereas an agent holding a short position is defined as a “bear”. Hence, the IKE theory contains
heterogeneous agents. The notion of bulls and bears just serves as a simplification, though, as
there exists a continuum of different agents in the IKE theory.
First I will expand on equation (4.1) above, the utility function for the prospect theory. In the
foreign exchange market context, equation (4.6), the equation (in changes) will look like the
following:
(4.7) ( )( )
( )| | 0
| | 0
gi i
li i
W a r if WV W
W a r if W
α
βλ
⎧ Δ >⎪Δ = ⎨⎪− Δ <⎩
61
Where rg (rl) is the gain (loss) for both the bull and the bear, but equals either rt+1+ or rt+1
–
according to the type of the agent – see equation (4.8) below. That is, the bull (bear) gains (loses)
from an increase (decrease) in the exchange rate.
(4.8) 1 1
1 1
: 0, ,
: 0, ,
g li t t
g li t t
Bull a r r r r
Bear a r r r r
+ −+ +
− ++ +
> = =
< = =
Following Frydman and Goldberg (2003: 19ff.), the utility function in equation (4.7) can be
written in a simpler, linear, version:
(4.9) ( ) ( )( )| | , | |g l g i l i i ii i i i i i t i t Gt LtV W a r W a r r f r f V V
α βλ λ− = + = +
Here, the utility function is expressed as the sum of the expected gains (rig) and expected losses
(ril) from holding a position (fti) in foreign exchange times the investment, denoted by fti. Note that
loss aversion have an influence for λ strictly larger than 1. Then, i iGt LtV V+ is the expected utility
from a position of the size fti in the foreign exchange market (Frydman and Goldberg, 2003: 20).
Following the assumption of Frydman and Goldberg (ibid.: 21), it is furthermore assumed that
the “fear” of a potential loss, ril, increases with the position size, fti. This prevent the agent from
taking an unlimited open position, as the disutility of expected losses grows faster than the
expected gains when position size grows (Frydman and Goldberg, 2003: 20). This further
requires that α > β in equation (4.7), stating that the degree of concavity of the utility function is
larger than the convexity over its losses. Equation (4.10) below adds the position size to the
determination of the degree of loss aversion:
(4.10) 1 2
, 1i l i l i i i MAXi i tr r fλ λ λ λ λ= − < <
With MAXλ being a constant. From this, the degree of loss aversion is assumed to be a function of
the position size. Inserting equation (4.10) into (4.9) yields the following:
(4.11) ( ) ( ) ( )1 2 1 2
2g i i l i i i g i i l i ii i t i t t i t i tV r f r f f r f r fλ λ λ λ⋅ = + − = + −
Differentiating equation (4.11) with respect to the position size, setting equal to zero, and solving
yields the following:
(4.12) ( ) ( )12
102
i i g i lt i ii i
t
Vfor f r r
fλ
λ∂ ⋅
= = +∂
The important thing to notice is that the value of fti depends on whether 1
g i li ir rλ+ is larger or
smaller than zero8. Note that even if 0g li ir r+ > , and agent i thus expect to earn a profit on a
8 Note that lir is negative, as it is the expected loss and hence is a measure of disutility
62
given speculation, a large 1
iλ (i.e. a high degree of loss aversion) could shift the overall sign of the
equation causing the speculator to stay out of the market altogether. The expected return, in that
case, would not be large enough to compensate the agent for his large sensitivity to losses,
measured by the 1
iλ . As concluded by Frydman and Goldberg (2003: 22): “All … loss-averse
speculators require an expected return in excess of some individually determined positive value in order to take open
positions in foreign exchange”. This individual minimum premium, denoted by ˆtρ in the following,
can then be derived by rearranging equation (4.12):
(4.13) ( ) ( )1 12 2
1 1 ˆ12 2
i g l i l i it i i i t ti if r r r rλ ρ
λ λ +⎡ ⎤ ⎡ ⎤= + + − = −⎣ ⎦⎣ ⎦
Here, the i superscript on rt+1 can either indicate a bull (+) or a bear (-), following equation (4.8)
above. From equation (4.13), the risk premium can be written as:
(4.14) ( )1ˆ 1 0i i l
t irρ λ= − >
Thus, to take a position in foreign exchange, the agent (either bear or bull, in this case) has to be
compensated enough, i.e. ˆg l ii i tr r ρ+ >
The exchange rate forecast of the agents The agents’ forecast of the future exchange rate depends on both the current exchange rate, st, as
well as on a set of causal variables, Xt, which (can) differ among the agents in the economy9. The
causal variables (can) include macroeconomic variables (as well as for example order flow), but it
also includes the preferences and individual experience of agent i. The IKE forecast of the
exchange rate is then given by:
(4.15) ,| 1
ˆˆ , 0 1i IKE i it t t t t t ts X sβ δ δ+ ′= + < <
A change in the aggregate forecast of the exchange rate for period t+1 thus comes from either
changes in the underlying variables, Xti, (macroeconomic fundamentals, preferences etc.) or from
the level of the exchange rate itself, st. Furthermore, as argued by Frydman and Goldberg (2007:
265), changes in ˆ i it tXβ occurs because of either i) revisions of forecasting strategies at time t or ii)
changes that occur because of new realizations of the set of causal variables, Xti, leads the agents
to change their forecast of the exchange rate. In contrast to (typical) rational expectations models,
the ˆ itβ can change over time (Frydman and Goldberg, 2007: 186). Equation (4.15) is thus similar
9 Note that in Frydman and Goldberg (2007: 264), the causal variable, Xti, is also denoted ˆa
ts .
63
to the “hybrid” model (equation (3.28)) presented in the microstructure section in chapter 3, as
the causal variables can include macro- as well as micro fundamentals (such as order flow).
Which underlying variables included in Xt is not prespecified. Furthermore, the model allows for
changes in the variables determining the exchange rate: Innovations, structural changes or policy
changes may therefore change the composition of Xt. The IKE theory allows for this, as well as
for the creativity of the agents, who may change both their methods and the models used for
their individual forecasts of the exchange rate.
The total change10 in the forecast in equation (4.15) is given by (ibid.: 192):
(4.16) ( ),
| 1 1 1
,| 1,
| 1 1
ˆ ˆˆ
ˆˆˆ
i IKE i i i it t t t t t t t t t
i IKEt ti IKE i i
t t t tt
ds dX ds d X d s
sds dX
s
β δ β δ
β
+ − −
++ −
= + + +
∂= +
∂
Where the last part is the partial change in the forecast coming from changes in the forecasting
strategy and/or the exchange rate.
4.3.5 Equilibrium in the FX market under Prospect theory: UAIP
According to Keynes (1936: 170) “the market price will be fixed at the point at which the sales of the bears
and the purchases of the bulls are balanced”. That is, by definition, the market portfolio has to be hold
by the agents. Hence, aggregating the long (“purchases”) and short (“sales”) positions of agents
in equation (4.13) and equalling these leads to the (momentary) condition for equilibrium in the
foreign exchange market (Frydman and Goldberg, 2003: 23). The aggregate result for the bulls
(i.e. the long position) is:
(4.17) ( ),1
1
2
1ˆ , 112
l L S
L S
n n ni L L i
t t t n ni iii
f w r where w and wρ
λ
+++
+=
= − = =∑ ∑∑
%
Here, the “L” superscript denotes a long position, and “S” a short position. The analogue result
can be obtained for the bears (short position):
(4.18) ( ),1
1
ˆ ,Sn
i S St t t
if w r ρ−
+=
= − −∑ %
10 The result is achieved as follows (here shown for the part βtiXti , the same applies for δsti):
( ), ,| 1 1| 1 1 1 1
1 1 1 1 1 1
1 1 1
ˆ ˆˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ:
ˆ ˆ ˆ ˆ
i IKE i IKE i i i it t t t t t t t t t t t
i i i i i i i i i i i i i it t t t t t t t t t t t t t
i i i i i i i it t t t t t t t
s s X s X s
For X X X X X X X
X X X X
β δ β δ
β β β β β β β
β β β β
+ − − − − −
− − − − − −
− − −
− = + − −
− = − + − ⇔
− = Δ + Δ
64
The weights, wiL and wiS, are the part of the agents being either bulls or bears, respectively. These
weights are negative functions of the degree of loss aversion, given by λ2i. Thus, the distribution
of agents can change over time between bulls and bears. For example a falling dollar (as observed
in late 2007 and beginning of 2008) could shift some agents from being bulls to bears, as the
degree of loss aversion rises. As pointed out by Kurz et al (2004: 4), the descriptive statistics of
exchange rate returns (asymmetry and high density in the tails) shown in chapter 2 could stem
from an (asymmetric) distribution of the agents – a distribution given by the weight parameter w.
The equality of the two aggregate positions in (4.17) and (4.18) yields the following:
(4.19) 1 1 1 ˆ ˆ ˆ, L St t t t t tr r r ρ ρ ρ+ −+ + += + = −% % %
The condition for equilibrium then yields:
(4.20) 1 1 1ˆ ˆ ˆL St t t t t tr r rρ ρ ρ+ −+ + ++ = − ⇔ =% % %
This is the uncertainty adjusted uncovered interest rate parity (UAIP), following Frydman and
Goldberg (2003: 24 and 2007: 210ff.). This can be rewritten as the following (Frydman and
Goldberg, 2007):
(4.21) | 1| 1 ˆˆ:
t tt tUAIP r ρ++ =
The UAIP represents equilibrium in the foreign exchange market and is the equality of the
aggregated uncertainty adjusted expected returns on foreign and domestic bonds (Frydman and
Goldberg, 2007: 211). That is, the expected return on long positions equals that of the short
positions. This follows the normal UIP condition, but here the expected returns have been
adjusted for by the larger sensitivity to potential losses (Frydman and Goldberg, 2003: 24). From
this it can be inferred that the sign of the uncertainty premium, | 1
ˆt t
ρ+
, depends on the sign of the
market expectation (i.e. the average of bulls’ and bears’ assessment of the expected exchange rate
movements), | 1t tr + . If the bears dominates the bulld, i.e. | 1t tr + < 0, then equilibrium in the foreign
exchange market necessitate that the bears either fear losses more than the bulls, or that they
picture a larger loss than the bulls, i.e. ˆ Stρ > ˆ L
tρ . From this concern of a higher expected loss
follows equilibrium between the long and short positions in the market. As Frydman and
Goldberg concludes (2003: 25): “The algebraic sign of the equilibrium uncertainty premium, | 1
ˆt t
ρ+
, will
change whenever the dominant weight behind the average opinion shifts between bulls and bears”.
From the context of the returns, equation (4.2), and the expected future exchange rate, (4.15)
follows:
(4.22) | 1
,| 1 ˆˆ
t t
i IKE B At t t t ts s i i ρ
++ − + = +
65
All else equal, an expected rise in the exchange rate will be followed by a equal rise in the
premium; this causes the demand to decline. In the following section I will add conservative
forecasting strategy as well as the “gap” effect which, together with the UAIP condition, implies
that a change in the causal variables lead to movements in the equilibrium exchange rate and the
premium (Frydman and Goldberg, 2007: 216). This further explains the long swings in the
exchange rate, as observed in chapter 3, and together it provides a possible solution to the
disconnect puzzle.
4.3.6 Modelling forecasting strategies I: The Gap effect
The expected gap is defined as the difference between the conditional forecast of the exchange
rate of agent i and the agents’ assessment of the historical benchmark value, which could be the
PPP value (Frydman and Goldberg, 2003: 27). But note that the benchmark value will be
determined each period by different agents, with (probably) different view on how to reach this
value, and hence the benchmark will in general be different over the agents. On the other hand,
the fundamental value given by PPP is available to everyone in the market, and is therefore a
fundamental alongside the output or interest rates.
The gap effect is then the effect from the gap (the difference between the historical benchmark
and the expected value of the exchange rate) on the agents’ expected loss. The expected gap is
given by the following (Frydman and Goldberg, 2003: 28 and 2007: 198)11:
(4.23) ( ) ( ) ( )| 1ˆ ˆHBt t t tgap z s z s z+= −
Following Frydman and Goldberg (2007: 197), the expected unit loss for bulls and bears,
respectively, is represented by:
(4.24) ,
, | 1 1
,, | 1 1
ˆ: 0 | 0
ˆ: 0 | 0
l L i ii t t t t t
l S i ii t t t t t
Bull r E R z
Bear r E R z
+ +
+ +
⎡ ⎤= < <⎣ ⎦⎡ ⎤= − < <⎣ ⎦
Here, zti includes both causal values used when forming forecasts of the unit loss by the agent as
well as current and past values of the exchange rate.
When combining the gap effect, equation (4.23), and the expected unit loss, equation (4.24), the
representation of an individuals’ forecast of potential unit loss will be:
(4.25) ( ) ( )( ) ( ) ( )( ), | 1 , | 1 , | 1 , | 1ˆ ˆ ˆ ˆl l i l l ii t t t i t t t t i t t t i t t t tr z r gap z r z r gap z+ + + +
⎡ ⎤= + −⎣ ⎦
11 Note that this is seen from the perspective of a domestic investor. Hence, an increase in st, ie. a depreciation of the
domestic currency, is a strengthening of the foreign currency – when having a long (short) position, a higher st
implies a gain (loss). See reference 6 as well.
66
Furthermore the gap effect will be qualitatively restricted by the following condition, which
depends on whether the agent is either a bull (L) or a bear (S) (Frydman and Goldberg, 2004: 26):
(4.26) , ,
, | 1 , | 1, ,
ˆ ˆ0, 0
l L l Si t t i t t
i L i St t
r rgap gap
+ +∂ ∂< >
∂ ∂
From equation (4.26) an increase in the gap – given by equation (4.23) – will have different
effects on the concern of a capital loss on the part of the bulls and the bears. That is, if the gap
increases (i.e. the currency is overvalued compared with PPP and is expected to move further
away from this benchmark value in the next period), the expected unit loss of a bull (bear)
increases (decreases)12. The bull, for example, expects that an overvalued currency continues its
movement away from the benchmark value, and therefore expects a higher return from a long
position in foreign exchange. But the increase in the gap effect from equation (4.23), on the other
hand, increases the concern of a movement back towards the benchmark value and a loss on the
position (Frydman and Goldberg, 2007: 200). This is further elaborated on in the example below.
Frydman and Goldberg (2004: 37) find that the gap effect and the uncertainty premium are
positively correlated. Empirical testing of the US dollar thus shows that the gap effect is highly
significant and positively related to the expected excess return on foreign exchange, in line with
the IKE assumption.
An example of exchange rate overvaluation In the following example it is assumed that the currency is overvalued compared with the
fundamental value (i.e. ( ) ( )| 1ˆ ˆHBt t ts z s z+ > ). For both the bulls and the bears it is uncertain
whether the (overvalued) exchange rate will continue its movement away from the fundamental
value or revert back to parity. On the one hand the bull assumes that the exchange rate will
continue its movement, and thus expects a greater return on a long position in the foreign
exchange market. But on the other hand the gap effect, equation (4.26), leads to an increased
concern of a loss for the bull – since the exchange rate could (suddenly) move back towards the
fundamental value (given by the benchmark) and the bull would incur a loss from the long
position. Hence, the bull’s expected unit loss increases as they become less confident that the
exchange rate will continue to be overvalued in the future (Frydman and Goldberg, 2007: 199ff.).
Furthermore, the insight from prospect theory (loss aversion as well as the fact that the degree of
loss aversion increases with the position) adds to the insecurity of the bulls. The increase in the
unit loss then feeds directly into the premium, equation (4.14), necessary for holding a long
12 Note that an increase in the unit loss is defined as a more negative value, whereas a decrease is a less negative value
67
position. The bears, on the other hand, expect a lower return on a short position. But the
increased gap leads to a smaller expected value of unit loss.
If the expected exchange rate thus moves further away from the PPP (the benchmark value), the
buyers (i.e. the bulls) raise their evaluation estimation of the losses, whereas the sellers (i.e. the
bears) lower their estimated losses. Then the uncertainty premium in equation (4.20) rises to
compensate the buyers for buying the foreign exchange. The exchange rate (st), therefore, does
not move one-for-one in the direction of the expected exchange rate | 1t ts + (Frydman and
Goldberg, 2001: 22). This happens because the buyers are unwilling to bid up the value of the
exchange rate, as their sensitivity to losses increases more than the expected return (due to loss
aversion), which then affects the risk premium. Furthermore, a continued movement away from
the benchmark value drives up the equilibrium uncertainty premium. The movement away from
PPP is thus self-limiting, as increases in the gap add to the risk assessment of the bulls, and hence
increases the risk premium. Eventually, this results in a revision of forecasting strategy which
drives the exchange rate back towards (and then away from) the fundamental value (i.e. the
benchmark). That is, the bulls’ concern of a capital loss grows so large that they no longer desire
to take on long positions (Frydman and Goldberg, 2007: 281). This explains the movements in
the exchange rate vis-à-vis the PPP value in figures 7 and 8 in chapter 3.
4.3.7 Modelling forecasting strategies II: Conservative revisions
A second qualitative description of the agents in the foreign exchange market is conservatism of
forecast revisions. As pointed out by Frydman and Goldberg (2007: 194): “Individuals can be slow to
change their beliefs in the face of new evidence”, which is supported by different psychological
experiments (e.g. Edwards, 1968). The definition of a conservative revision is a revision that is
“not too different” from the forecast before the change of strategy. As discussed in Frydman and
Goldberg (2007: 195) “not too different” depends on, among other, whether goods prices are
fully flexible or adjust slowly to their equilibrium value.
A way to set up the conservative restriction in the forecast of the individual is as follows (ibid.:
195):
(4.27) ,| 1 1
1 1
ˆ ˆˆ
ˆ ˆ
i IKE i i i it t t t t t
i i i i it t t t t
ds dX d X
d X E X X
β β
β β
+ −
− −
= + ⇒
⎡ ⎤< Δ⎣ ⎦
That is, the change in the individuals forecast revision at time t is less than the conditional
expectation of the change in the causal variables (in absolute terms). As pointed out by Frydman
and Goldberg (ibid.: 196), this does not necessarily imply that the change in the agents forecasts
are “small”, as this depends on the magnitude of the change in the causal variables. Furthermore,
68
the restriction that δ < 1 in equation (4.15) implies that changes in the exchange rate also lead to
a “conservative” revision, as the change in the forecast does not follow a change in the nominal
exchange rate one-for-one.
Furthermore, as briefly discussed in section 4.3.5, agents do not necessarily stick to their
respective strategy (in this case being either a bull or a bear, but it could also be whether to stick
to fundamental analysis or technical trading). As noted by Slovic (1973: 2): “It is hard to follow a
predetermined policy when your financial condition is riding the crest of good fortune or plummeting with a bear
market”. This is also assumed by Goldberg and Frydman (2007) – see for example equation (4.17)
above. Hence, the strategies of the agents may not be stable when environment changes; this is
also discussed in section 4.1 and section 4.3.4 above. This is in line with the discussion from
chapter 2 and 3 where the importance of different macroeconomic fundamentals seems to shift
over time (e.g. Cheung and Chinn, 2001). And for some periods of time, fundamentals appear
unimportant altogether and focus lies on, for example, technical trading (Frankel and Froot,
1986). Looking at the IKE forecast of the future exchange rate, equation (4.15), changes in the
causable variable (given by Xt) have an impact on the forecast of the future exchange rate. This
can have a direct effect on the loss aversion, equation (4.10), or an indirect effect via the gap in
equation (4.23). These changes will then alter the relative weight of the bulls and bears, as seen in
equation (4.17). The distribution of the two types of speculators does not, therefore, remain static
as time progresses. This follows for example Corrado et al (2007) who concludes that trading
rules generate (repeated) switches between bulls and bears. This switching process further
generates a significant misalignment of the exchange rate according to Corrado et al (2007: 264).
An alternative approach of agents switching types includes De Grauwe and Grimaldi (2006), with
chartists and fundamentalists being the two types of agents instead of bulls and bears.
4.3.8 Summing up
The main insights of the IKE theory from section 4.3 are the following:
• The agents are loss averse. That is, the disutility of a loss exceeds the utility of a gain of the same size.
• The utility of a loss/gain is dependent on a reference level. This reference level can be defined as the wealth
from refraining from speculating
• Fear of losses increase with the position size
• The agents forecast of future exchange rate movements are based on both causal factors (which (can)
include macroeconomic models, insight from the microstructure approach, technical trading as well as
individual experience) and the current level of the exchange rate
69
• Fundamentals matter for exchange rate determination. But the (relative) importance of different
macroeconomic variables can change over time
• The agents’ expectations are heterogeneous; in this context represented by bulls and bears. The weight of
either type can shift over time
• UAIP: The equilibrium in the foreign exchange market is the equality between the aggregate uncertainty
adjusted expected returns on foreign and domestic bonds
• The gap, i.e. the difference between the expected value of the exchange rate and the historical benchmark,
has an effect on the degree of loss aversion. The effect furthermore differs between bulls and bears
• The agents are conservative when revising their forecasts
The main conclusions pointed out above seem supported by the findings of chapter 2 and 3
above: Agents are heterogeneous (cf. Cheung and Chinn, 2001; Cheung et al, 2004) and hence
use different models when forecasting (including technical trading, experience etc.; private
information matters as well (cf. the microstructure approach). Fundamentals do matter, but the
relative importance changes over time (Cheung and Chinn, 2001), which was also evident from
figure 10 of the relative money supply. Other variables, such as order flow (discussed in chapter
3), matter as well. These results are magnified by the decentralised foreign exchange market and
the low level of transparency (discussed in chapter 2). Hence, both the agents as well as the
structure of the market play an important role for exchange rate determination in the IKE theory.
The IKE theory furthermore elaborates on the preferences of the agents by adding loss aversion
and conservative revisions. Finally, the agents have imperfect knowledge of the economy (i.e.
they do not know the correct model for exchange rate determination) as well as a degree of
uncertainty regarding future events.
In the following section the conclusions from above will be implemented in a IKE version of a
monetary model of the exchange rate. This leads to a discussion of the IKE theory as a possible
solution to the disconnect puzzle.
4.4 IKE and the exchange rate: A monetary model The anomalous behaviour of exchange rates, discussed in chapter 3, has forced economists to
introduce either rational bubbles or non-rational behaviour into exchange rate modelling. That is,
the conventional flexible price monetary model has to rely on large real shocks to explain the
swings away from the fundamental value (i.e., given by PPP) or the existence of the
aforementioned bubbles. The sticky price exchange rate model (i.e. the overshooting model), on
the other hand, can explain only one-time deviations (Frydman and Goldberg, 2007: 258).
70
Frydman and Goldberg therefore argue for introducing imperfect knowledge into the models to
account for the anomalies: “We find that as long as agents possess at least some degree of imperfect knowledge,
the monetary models of the exchange rate generate dynamics that are consistent with the anomalous behaviour
observed in the literature” (Frydman and Goldberg, 1996: 870). The solution of Frydman and
Goldberg (1996, 2007) to the exchange rate puzzles is not, therefore, to abandon the monetary
models of the exchange rates altogether, nor introducing irrational agents, but instead to try to
implement imperfect knowledge in the models.
4.4.1 A monetary model with IKE-expectations
In the following I present an exchange rate model13 building on a regular monetary model in the
vein of Dornbusch (1976) or Frenkel (1976) as presented in chapter 3. But with the insights from
the IKE theory discussed in the previous sections. The model consists of a two countries,
domestic and foreign, and three markets: a money market, a goods market and a foreign
exchange market. In the foreign exchange market, the usual assumption of UIP is changed with
the UAUIP, discussed in section 4.3.5 above.
In the following, the exchange rate, the interest rate and the price level are assumed to be
endogenous – the rest of the variables are determined exogenously.
4.4.2 Money markets
Equilibrium in the money markets is based on the same two equations as in section 3.3.1. This
leads to the following equilibrium for the money markets, here in terms of relative (i.e. domestic
minus foreign) magnitudes:
(4.28) t t t tm p y iφ λ= + −
As in section 3.1, m, p and y denote the log-levels of the (relative) money supply, output and
nominal interest rate, respectively. It is, as in traditional monetary models, assumed that supply
and demand for money is in equilibrium.
Rearranging such that the endogenous variables (here pt and it) are kept on the left hand side
yields:
(4.29) t t t tp i m yλ φ− = −
13 The following exchange rate model is primarily based on chapter 14, ”Imperfect knowledge and long swings in the
exchange rate”, in Frydman and Goldberg (2007: 258-291)
71
4.4.3 Goods markets
In the following it is assumed that domestic and foreign goods and assets are imperfect
substitutes, and hence (excess) demand for goods depends on international prices
competitiveness (given by the real exchange rate), as well as on the interest rate and income levels
(cf. Frydman and Goldberg, 2007: 261). Excess demand for domestic goods (relative to foreign
goods) is given by the following equation:
(4.30) ( ) ( )ˆpppt t t tEDG s p q iα η π= − − − −
Here, st is the log of the nominal exchange rate, qppp is the historical benchmark real exchange
rate (given by PPP), and π is the aggregation of agents expectations of relative levels of inflation
prevailing if the goods market cleared. Both qppp and π is assumed to be exogenous and constant.
Furthermore, α and η are the sensitivity of excess demand to movements in the exchange rate
and the interest rate, respectively (Frydman and Goldberg, 2007: 262).
The movement of goods prices is assumed to depend on both the underlying rate of inflation as
well as excess demand, which allows for a sluggish adjustment of the goods prices.
(4.31) ( ) ( ) ( )1 1ˆpppt t t t t t tp p s p q i p pδ α η π+ +
⎡ ⎤− = − − − − + −⎣ ⎦
The 1tp + is the value associated with goods-market clearing. As discussed in Frydman and
Goldberg (2007: 262-63), the assumption of flexible prices implies PPP if either of the two
following conditions hold: i) that domestic and foreign bonds are perfect substitutes (i.e. η =0) or
ii) that the real rates of interest across countries are assumed equal (i.e. ˆti π= ). If one of these
two conditions hold, and given flexible prices (i.e. ( )1 1t t t tp p p p+ +− = − ), PPP holds:
( ) 0pppt ts p q− − = .
In equilibrium (i.e. excess demand equals zero) the following holds:
(4.32) ( ) ( )ˆpppt t ts p q iη π
α− − = −
Rearranging such that the endogenous variables (pt, st and it) are on the left hand side yields:
(4.33) ˆ pppt t tp i s qη η π
α α+ − = −
4.4.4 Foreign exchange market
In the following, the UAUIP condition and the other assumptions from sections 4.3.4 and 4.3.5
is used. Equilibrium in the foreign exchange market is given by:
(4.34) ( )| 1ˆ ˆ,t t t t t t ts s x s i u+ − − =
72
Where | 1t ts + is the aggregate of the bulls’ and bears’ forecast of st+1 at time t, given the information
available. This information set is given by xt, which represents both current as well as past
realizations of factors that the agents utilize when making their forecasts (Frydman and
Goldberg, 2007: 263). Here, ˆtu is the uncertainty premium and is the aggregate uncertainty
premium at time t. The premium is discussed in section 4.3.4 above. This premium depends, as
seen in equation (4.35) below, on the degree of loss aversion, λ1, of both the bulls and the bears.
(4.35) ( ) ( )1 | 1ˆˆ 1 ,t t t t tu l s xλ += −
It is once again, see equation (4.15) above, assumed that the point forecast | 1t ts + depend linearly
on both st (the level of the exchange rate – the endogenous part) and xt, (causal variables, e.g.
macroeconomic variables and preferences – the autonomous part).
(4.36) | 1ˆ ˆt t t ts x sβ ρ+ = +
In this context it is, for simplicity, assumed that neither ρ nor β vary over time, in contrast with
equation (4.15) above. It is interpreted as if | 1t ts + is a weighted average of the parameters used by
the bulls and bears to forecast the nominal exchange rate. Normally, though, the IKE theory
assumes that the ˆtβ parameter could change (Frydman and Goldberg, 2007: 266). In such a case,
the change in tβ would stem from a change in the forecasting strategy for Rt+1, which lead to a
change in the forecasted mean of st+1 (ibid.: 264). Changes in the aggregate forecast, | 1t ts + , will
thus take place either because of i) new realizations of the causal variables (xt) (e.g. positive GDP
data for a given country) and/or ii) new realisations of the spot exchange rate (st) or because of
iii) changes in the autonomous part ( t txβ ) – for example a structural change in the variables in xt
(e.g. agents substituting the interest rate differential with the current account deficit when
forecasting).
The aggregate expected loss – i.e. the bulls’ minus the bears’ expected loss – is building on the
gap restrictions, as discussed in section 4.3.6. The aggregate gap is given by:
(4.37) | 1ˆ ˆHBt t t tgap s s+= −
It is assumed that the revisions of the expected return, Rt+1, is consistent with the gap restrictions,
discussed in section 4.3.6. This motivates the following specification of the uncertainty premium
(Frydman and Goldberg, 2007: 264):
(4.38) ( )| 1ˆ ˆ ˆHBt t t tu s sσ += −
73
Where σ is assumed to be strictly larger than zero, but smaller than one, and sHB represents the
bulls’ and bears’ aggregate historical benchmark level of the exchange rate. That is, an increase in
the gap, equation (4.37), increases the uncertainty premium through equation (4.38). This feeds
into the forecast of the expected exchange rate, given by equation (4.34) above, and thus the
expected return.
The agents’ assessments of the historical benchmark level of the exchange rate are collectively
represented by PPP:
(4.39) ˆ ˆHB PPPt ts q p= +
Solving for the difference between the weighted forecast and the exchange rate, ( )| 1t t ts s+ − using
equations (4.34), (4.38) and (4.39) yields the following:
(4.40) ( ) ( )| 1 | 1ˆ ˆ ˆPPPt t t t t t ts s i s p qσ+ +− = + − −
Now | 1t ts + in equation (4.40) is substituted by ˆt tx sδ+ , using equation (4.36) above. Remember
that β is invariant here (for simplicity), and is therefore unimportant for the result. Rearranging
yields:
(4.41) ( ) ( )ˆ ˆ ˆPPPt t t t t t tx s s i x s p qρ σ ρ+ − = + + − −
Again, rearranging equation (4.41) such that the endogenous variables (pt, st and it) are on the left
hand side yields:
(4.42) ( ) ( )ˆˆ1 , 1 1PPPt t t ti p hs x q hσ σ σ ρ σ− + = − + = − −
Using equation (4.28) and assuming that ˆti π= , the rational expectations steady state solution for
the spot exchange rate, *s , is given by the following:
(4.43) * * PPPt ts m y qφ λπ= − + +
This equation is used later on for the solution to the model.
4.4.5 The social context
As argued by Frydman and Goldberg (2007: 266) the institutional changes in the economy are
relatively infrequent when compared to the frequency of forecast revisions in the foreign
exchange market. Hence it is assumed that the parameters in both the goods and money market
demands, equation (4.31) and (4.28) respectively, are constant. Furthermore, the causal variables
(m, y and the different variables in x) are assumed to move as random walks with drift:
74
(4.44) 1
1
t t t t
t t t t
M mY g y v
μ ε−
−
= + += + +
With ε and v being white noise terms, μ and g drift terms. The t subscript on the drift terms
represents that the processes driving the system could change, and hence allow for shifts in the
drift components.
4.4.6 The solution to the model
The system of the money market, equation (4.29), the goods market, equation (4.33), and the
foreign exchange market, equation (4.42), can be written in matrix form as follows:
(4.45) ( )
1 0 0 0 01 01 1 0 0 0 1 0 ˆ
0 0 1 0 01
ˆ
t
tt
tt
t ppp
PPP
my
px
ish q
q
λ φη η
α πασ σσ
⎛ ⎞⎜ ⎟
−⎛ ⎞ ⎜ ⎟⎛ ⎞−⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟− = ⎜ − ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ −⎜ ⎟⎜ ⎟⎝ ⎠ ⎜ ⎟− ⎝ ⎠⎝ ⎠ ⎜ ⎟⎜ ⎟⎝ ⎠
The model consisting of equations (4.28) to (4.45) then implies the following steady state for the
exchange rate:
(4.46) ( ) ( )( )1 1ˆ ˆ ˆPPP PPPs m y x q qG G G G G
σ η αλα ση λασ α λασ α η αλϕ η π σα
− ++ − − += − + + − +
The calculations for the model can be found in Appendix B. As in the monetary approach, the
exchange rate is determined by macroeconomic fundamentals, here output and money supply, as
well as the causal variables, xt, included in the agents’ forecast. Furthermore, the expected
inflation rate and the PPP levels have an impact on the exchange rate as well. Using equation
(4.43) for the REH solution and rearranging yields:
(4.47)
( )( ) ( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )
*1
1 1ˆ ˆ ˆ ˆ
ˆ ˆ
ˆ0, 1 1 0
a REa REt tt t
RE PPP PPP
s s x sG G
q qG G
G h h h
η αλ σ η λσπ π
η λα σ η λα σπ
α η αλ σ ρ σ
+
+ − −= + − + −
+ +− + −
≡ + + − > ≡ − − >
Here, the RE superscript denotes the rational expectations solution. The solution for the real
exchange rate is given by equation (4.48):
75
(4.48) ( ) ( ) ( ) ( )
( )
1
1 1ˆ ˆ ˆ ˆ
ˆ ˆ
PPP a REa REt tt t
RE PPP PPP
hq q x s
G G
q qG
η σ η λπ π
ησ π ησ
+
− += + − − −
+ + −
Note that PPP PPPt ts p q= + and t t ts p q− = which implies that PPP PPP
t t ts s q q− = − . This is the
nominal exchange rate level relative to the PPP level (Frydman and Goldberg, 2007: 269). Using
this on equation (4.48), taking first differences to the equation and rearranging yields equation
(4.49) below. Note that the real exchange rate as well as the inflation rate are assumed to be
exogenous and do not change. Furthermore, for simplicity, I consider the case of flexible good
prices, i.e. q q= (Frydman and Goldberg, 2007: 269). This leaves us with:
(4.49) ( ) ( ) ( )1
1ˆ ˆPPP a REa
t t tt ts s x sG
η σ+
−Δ − = Δ −Δ
This is one of the key equations of the model, and describes how the difference between the IKE
forecast and the REH forecast results in divergence between the exchange rate and the PPP level.
As can be seen, a change in the causal variable, 1ˆat tx + , will push the nominal exchange rate
away/towards the PPP rate given by PPPts all else equal. In the words of Frydman and Goldberg
(2007: 269): “persistent trends in the causal variables will lead market participants to push the exchange rate
persistently toward or away from PPP”. Furthermore, if the policy environment includes deterministic
trends in m, y and the variables in xt, a persistent swing in the exchange rate away from (or
towards) the PPP level will arise in the model.
Note that if domestic and foreign assets are perfect substitutes (i.e. η =0), the nominal spot
exchange rate does not differ from the fundamental value. The effect from an increase in η on
the exchange rate is ambiguous as the sensitivity parameter affects both the nominator and the
denominator (through G). But overall, increases in the η parameter is assumed to amplify the
movement away from the fundamental value. Furthermore the σ parameter puts a bound on the
movement away from parity. A “high” σ value (i.e. close to unity) implies that the gap effect
weighs heavy on the agents, through equation (4.38), and thus increases the premium required by
the agents for holding a long position. This, in turn, pulls the exchange rate back towards the
fundamental value rather quickly as agents change their expected losses rather fast and hence
their forecasts. A “low” σ value (i.e. close to zero), on the other hand, indicates that the gap
effect has low influence on the uncertainty premium and hence does not act as a (strong)
boundary for movements in the nominal exchange rate away from the benchmark value.
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As the change in the agents expectations differ from the rational expectations result, by
definition, this leads to the disconnect between the change in the expected exchange rate and the
rational expectations outcome:
(4.50) ( ) 1 1 11ˆ ˆ , , 0a REat t t tt tE x s x y m− − −+
⎡ ⎤Δ − Δ ≠⎣ ⎦
Using equation (4.50) on equation (4.49) implies that persistent swings will arise in the model; i.e.
the nominal spot exchange rate will diverge from the fundamental exchange rate for long periods
of time. The influence from the gap effect, discussed in section 4.3.6, secures that the exchange
rate is bounded and does not diverge from the fundamental value indefinitely.
The result of the model is discussed further in the next two sections.
4.4.7 The intuition of the result
The agents in the market will, as seen from equation (4.49) above and given deterministic
trends14, persistently push the exchange rate either towards or away from the PPP level. As
concluded in Frydman and Goldberg (2007: 269): “…as long as the representation of forecasting
behaviour presumes imperfect knowledge, and this representation and the policy environment remain unchanged, the
monetary model with flexible goods prices implies a persistent movement of the exchange rate away from PPP”.
Hence, even if the participants’ base their forecast solely on macroeconomic fundamentals, the
imperfect knowledge (implying that the weights in | 1
ˆt t
s+
is different from the weights of | 1
ˆt t
REs+
, cf.
Frydman and Goldberg, 2007: 270) on part of the agents creates the persistent movements. The
intuition of the result is that with imperfect knowledge– i.e. the aggregated expected value of
change in the exchange rate is strictly different from the rational expectation hypothesis expected
value – the aggregate of the point estimates does not equal the PPP exchange rate (as this is given
by the REH). This implies that the exchange rate will be driven either away from or towards the
PPP rate. In the latter instance the exchange rate will, when reaching the fundamental value,
continue its movement away from the PPP value. Furthermore, the long swings occur even if
goods prices are assumed to be fully flexible, as in the model above.
From a theoretical view, this explains the large swings in the exchange rate relative to PPP
discussed in chapter 3.
14 As noted by Frydman and Goldberg (2007:269, note 16) stochastic trends in the variables x, y and m will still lead
to persistent deviations from the PPP value, but on average the divergence will not grow or sink
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4.5 IKE and the exchange rate disconnect puzzle As the model set up in section 4.4 shows, adding imperfect knowledge on part of the agents put
forth another possible solution to the exchange rate disconnect puzzle.
First of all, as the agents have heterogeneous expectations – and thus use different models and
methods when forecasting the exchange rate – the outcome has to, by definition, be different
from the rational expectations outcome. But this does not imply that the exchange rate moves
about randomly, entirely disconnected from the fundamentals. Since the benchmark value, given
by PPP, is available to all agents in the economy, an increase in the difference between the
nominal exchange rate and the PPP value (e.g. an overvaluation), raises the concern of the bulls’
(in this case) estimation of a possible return to parity and a subsequent capital loss. The agents in
the market therefore take the historical long swings into account when forming their forecast. As
the last thirty years have shown, evident by the charts in chapter 3, the nominal exchange rate can
diverge significantly from PPP for several years. Accordingly, it is rational for the bulls in the
foreign exchange market to utilize this knowledge and (initially) stay long when the exchange rate
moves above its benchmark value. As pointed out by Frydman and Goldberg (2007: 281): “[The
IKE monetary model]… assumes that market participants are aware of the long-swings nature of exchange rate
movements and take this behaviour into account when forming their forecast of the return and potential unit loss
from holding speculative positions in the market”. Frydman and Goldberg (2007b: 6) further exemplifies
this by the falling dollar in the late 2007: “A market participant may well decide that, because the
USD/EUR exchange rate is, say, 30 percent overvalued relative to PPP, as it currently is, she wants to be a net
seller of euros. However, in a world of imperfect knowledge, the gap between the actual and PPP exchange rates is
merely one of many fundamental factors that market participants might reasonably rely on in forming their
forecasts.” This secures that the exchange rate swings can be sustained for longer periods of time,
but also that these swings are bounded by the fundamental value via the gap effect. For the dollar
overvaluation in the 1980s, for example, sticking to a strict fundamental strategy when deciding
whether to invest in the dollar would have lead to an inferior result compared to a simple
technical trading rule. Hence, as a result thereof, the weighting of agents shifted towards more
use of technical trading in the mid 1980s (cf. Frankel and Froot, 1990).
According to the IKE theory the swing away from the benchmark value occurs because of i)
trends in the causal values and/or ii) revisions of forecast strategies (Frydman and Goldberg,
2007: 279). Regarding the former, a trend of positive macroeconomic data support a stronger
currency. Regarding the latter, the conservative behaviour of the agents further reinforces the
protracted movement away from PPP. Only when a certain threshold is breached, the forecasting
strategies are changed and the exchange rate (slowly) reverts to parity. Hence the important
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aspect here is not the conditional variance of returns of having a long position in exchange rates,
but instead the current gap between the nominal exchange rate and the benchmark value. As
concluded by Frydman and Goldberg (ibid.: 279), the fundamental PPP value acts as an anchor
for the nominal exchange rate and its movement. If the currency is “too overvalued”, increased
concern of capital loss induces a change in the forecast and thus a reversion of the exchange rate.
As already mentioned, abandoning PPP and other macroeconomic fundamentals when looking at
exchange rate determination is not what the IKE theory suggests. Instead, the macroeconomic
approach is part of the forecasting method used by the agents, and thus an important part of
understanding exchange rate behaviour. But the agents acknowledge that they have imperfect
knowledge and act accordingly; the nominal exchange rate as a result appear disconnected from
the fundamental value for longer periods of time.
On the theoretical level, the IKE theory seems to have a plausible solution for the exchange rate
disconnect puzzle. The question, then, is whether this is verified empirically as well. This is the
subject for next chapter. But first a critical discussion of the IKE theory.
4.6 Critique of Imperfect Knowledge Economics As mentioned in the introduction to this chapter, a central point in opposition to the IKE theory
is that in a world of imperfect knowledge (which implies indeterminacy) certain aspects cannot be
quantified (cf. Papell, 2003: 2). Thus IKE does not, per se, produce restrictions that can be tested
(and falsified) against the rational expectations hypothesis.
The IKE framework, on the theoretical part, synthesizes a theory of expectations consistent with
individual rationality, a part of rational expectations theory which has been criticised from many
sides (e.g. De Grauwe and Grimaldi, 2003: 1). But testing the hypotheses of the IKE theory
seem, at first, problematic. As noted by Papell (2003: 3): “With IKE, the scope of imperfect knowledge is
unlimited and nested tests cannot be constructed to compare IKE with either theories consistent expectations or RE.
In the absence of such tests, the concept of IKE is not empirically falsifiable”. Goldberg and Frydman
(2007:7) disagree: “IKE restricts its models sufficiently to enable an economist to distinguish empirically among
alternative explanations of economic phenomena”. Nonetheless, the IKE theory (as put forth in
Goldberg and Frydman, 2007) puts weight on a “realistic” description of the world, opposite to
the more instrumentalist view of, for example, Milton Friedman (1953: 15): “The relevant question to
ask about the assumptions of a theory is not whether they are descriptively realistic, for they never are, but whether
they are sufficiently good approximations for the purpose in hand. And this question can be answered only by
seeing whether the theory works… which means whether it yields sufficiently accurate predictions” And as argued
by Isard (1995: 182): “This situation [that few believe the behaviour of flexible exchange rates can
79
be accurately described by a model based on fully rational agents], however, does not imply that
economists should abandon efforts to model how flexible exchange rates would behave in a world of fully rational
and completely informed market participants”. Haavelmo (1944: 31), on the other hand, concludes that:
“To find such a basic system of highly autonomous relations in an actual case is not an analytical process, it is a
task of making fruitful hypothesis as to how reality actually is”. The question, then, is whether to put
weight on the “realistic part” or the “empirical/testable part” – the first being the realism view,
the latter being the instrumental view. The focus of the IKE theory, as mentioned, apparently
puts most weight to the first with emphasis on the qualitative description of the foreign exchange
market. It thus opens up for the aforementioned criticism of proponents of the instrumental
view.
The arguments of Frydman and Goldberg lie end to end with the thoughts of Hayek, as seen
from the introduction to this chapter, who in his Nobel laureate speech concluded (1974: 1):
“Unlike the position that exists in the physical sciences, in economics and other disciplines that deal with essentially
complex phenomena, the aspects of the events to be accounted for, about which we can get quantitative data, are
necessarily limited and may not include the important ones… in the study of such complex phenomena as the
market, which depend on the actions of many individuals, all the circumstances which will determine the outcome of
a process … will hardly ever be fully known or measurable”. This is further supported by Richardson
(1953: 156): “Instead of building models which, while complying with our rigorous canons of verifiability in
principle, fail lamently to pattern the real world, it would be better … to construct theories/models which do
satisfactorily explain the working of our economy”. All that aside, a necessary condition for a theory to
obtain validity is to be empirically testable and hence also falsifiable. A difficult task for theories
based on imperfect knowledge and uncertainty, such as IKE. But, as I will show in chapter 5, it is
possible to test the results and conclusions of the IKE theory by using models with IKE
“features” – in this case uncertainty.
Overall the problem of models with uncertainty, including the IKE theory, is the difficulty of
testing them. But that does not necessarily mean that we should abandon them altogether, as
these models can provide substantial knowledge in regards to exchange rate behaviour.
4.7 Conclusion The main point of the IKE theory is to put imperfect knowledge at the centre of the analysis.
That is, the agents in the economy do not have perfect information of neither the market they
speculate in (in this case the foreign exchange market), nor the different future possible states –
cf. “Knightian uncertainty”. Therefore the agents make use of different “rule of thumbs”
(heuristics) when deciding their actions; i.e. they utilize different models and stick (at least for
80
some time) to the “best” strategy. This strategy can be a portfolio of models and methods; for
example combining the result of the monetary approach with technical analysis or data on order
flow (if available, that is). Thus, the agents do not behave irrationally, but instead realize that they
cannot fully comprehend the entire economic system - or that it is too costly to obtain the
information, as argued by for example Grossman and Stiglitz (1980). Hence, the aggregate of the
agent’s forecasts differ from the rational expectations result. And this difference will lead to the
persistent movement away from or towards the fundamental value given by PPP. This explains
the first part of the exchange rate disconnect puzzle.
The utility function of the agents in the IKE theory is based on the insights from prospect
theory, which assumes that “losses loom larger than corresponding gains” (Frydman and Goldberg, 2007:
161). That is, the value function is steeper for losses than for (comparable) gains. Furthermore,
the utility of either gains or losses depends on a reference level, in our case given by the income
from staying out of the foreign exchange market. The agents are assumed to be heterogeneous,
which is in line with the conclusion of for example Cheung and Chinn (2001). In the text the
agents are assumed to be either bulls or bears, but they could also be “fundamentalists” and
“chartists”, as in Frankel and Froot (1986) or Vigfusson (1997). Equilibrium in the foreign
exchange market is then given by the equality of the bulls’ and bears’ aggregate expected returns
of investing in either foreign or domestic bonds. But in contrast to traditional analysis, the
expected return is adjusted for a greater sensitivity over losses than gains, following the
aforementioned finding from prospect theory. The equilibrium uncertainty premium is thus the
uncertainty premium of the bulls in excess of the bears (Frydman and Goldberg, 2003: 9). The
sensitivity to losses furthermore restrains the movement of the exchange not to follow the
expected exchange one-for-one. This is further magnified by the agent’s conservative revisions of
their forecasts as well as the gap effect. The gap effect is a central aspect, as this makes the
fundamental value (given by PPP) an anchor around which the exchange rate fluctuates. A large
“gap” – i.e. the absolute value of the expected exchange rate minus the benchmark value is large
– increases the loss aversion of the agents (either the bulls or the bears, depending on if it is a
over- or undervaluation) and hence the uncertainty premium, which tends to drive down the
exchange rate. This explains the movement of the exchange rate in the figures in chapter 3: The
exchange rate does move away from the fundamental value, but seem somewhat bounded to the
PPP benchmark – there seems to be an outer threshold15 which the nominal exchange rate does
not cross. Furthermore, a movement “too far” away from the benchmark value eventually leads
the exchange rate to revert back to the fundamental value as the agents change their expected 15 This treshold is normally assumed to be around ±20% of PPP, visible in the charts in chapter 3
81
gains/losses and forecasts. This is the second part of the theoretical explanation of the exchange
rate disconnect puzzle.
On the qualitative aspect the IKE theory seem to have a rather strong case, which is supported
by different papers (e.g. Kahnemann and Tversky (1992), Cheung and Chinn (2001)). But on the
quantitative side, the IKE theory seems to have some complications. Testing the conclusion of
the IKE theory in itself is a difficult task, given the assumption of both imperfect knowledge and
uncertainty. Furthermore the agents differ and the models the (heterogeneous) agents utilize
differ in ways not specified beforehand. Hence, testing the IKE model against the rational
expectations version is no easy task as Papell (2003) points out.
In the empirical chapter I have chosen to follow the idea of Cumperayotm (2005), who adds
uncertainty to traditional monetary models. The assumptions of this method – uncertainty is an
important aspect that help to describe exchange rate determination alongside macroeconomic
fundamentals – is in line with the IKE theory presented in this chapter.
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Chapter 5
Empirical test “Despite extensive efforts to capture and quantify what we perceive as the key macroeconomic
relationships, our knowledge about many of the important linkages is far from complete and, in
all likelihood, will always remain so. Every model, no matter how detailed or how well designed,
conceptually and empirically, is a vastly simplified representation of the world that we experience
with all its intricacies on a day-to-day basis… Moreover, we recognize that the simple linear
functions underlying most of our econometric structures may not hold outside the range in
which adequate economic observations exist”
Alan Greenspan (2004), pp. 5-6
5.1 Introducing the empirical part In this chapter I test a central part of the imperfect knowledge economics theory: The
importance of uncertainty with respect to determination of exchange rates.
The critique of IKE in the previous chapter showed that it can be a rather difficult task to test
the IKE model. First of all, the theory is based on imperfect knowledge of the “true model” of
exchange rate behaviour. Secondly, uncertainty of future states of the economy plays an
important part in the theory, further exacerbating the imperfect knowledge on part of the agents.
Thirdly, the agents in the model are assumed to be heterogeneous and utilise (a range of)
different models at different points in time when forecasting future exchange rate movement.
To circumvent these apparent problems, I have chosen to focus on one part of the theory only,
the aspect of uncertainty. In chapter 3 the microstructure approach showed that private
information plays a significant role in the determination of exchange rates. Combining the result
of the significance of private information with a test of the importance of uncertainty, two central
assumptions of the IKE theory is thus put to test.
The approach in this chapter is mainly based on the idea from the article of Phornchanok
Cumperayot (2003), “Dusting off the Perception of Risk and Returns in FOREX markets”.
Cumperayot (2003) shows that volatility of macroeconomic fundamentals – a proxy for
uncertainty – plays a central part in the determination of exchange rates for Canada, France, Italy,
83
Japan, the UK and the USA. The volatility – i.e. uncertainty – of macroeconomic fundamentals is
proxied by estimation in a GARCH model, which is discussed below. Testing a simple monetary
model – similar to the monetary model presented in chapter 3 – with the addition of uncertainty
variables points to significant cointegration between the variables in both the short and long run
in Cumperayot (2003). As a side note, the approach of Cumperayot is based on the idea of
Hodrick (1989) who tried to incorporate risk into exchange rate models, but without a useful
result.
In this chapter I have chosen to test the USD/NOK and the USD/JPY currency crosses in two
models: The model of Cumperayot (2003), presented in the next section, as well as an augmented
model which includes interest rates. Furthermore, I test a simple monetary model in line with the
models presented in chapter 3.
The chapter is structured as follows: First the motivation as well as set-up of the model presented
in Cumperayot (2003). Then follows the empirical test, which can roughly be split into three
parts: First a brief discussion of cointegration analysis, followed by a discussion and estimation of
GARCH and finally the cointegration analysis of the models. The test is compromised of both a
test of the flexible price monetary model as well as the model including uncertainty. Following
this is a discussion of the result seen from the perspective of the IKE theory.
5.2 The model: Motivation and set-up The model of Cumperayot (2003) is partly based on the monetary approach discussed in chapter
3, augmented with a gauge of macroeconomic uncertainty given by the volatility of the
macroeconomic data. Regarding the uncertainty of macroeconomic fundamentals, and its
importance for exchange rate determination, Cumperayot (2003: 2) states: “The uncertainty in
macroeconomic fundamentals may influence the perception of risk in the markets, and subsequently through the
risk premium it may price returns on the exchange rate”. Thus, uncertainty is assumed to play a part
alongside the current level of fundamentals and the expectation of future exchange rates – two
factors which the literature normally focuses on in relation to exchange rates (ibid.: 2). An
argument in favour of the importance of fundamental uncertainty is that “fundamental variances may
represent economic circumstances, namely whether the economy is in volatile or tranquil periods, in which the
expectations may be different.” (ibid.: 19). That is, in “turmoil” the variables, such as output growth, is
more volatile, and this feed into the expectations and hence the exchange rate. Following the
theory of imperfect knowledge, periods of high volatility (e.g. a recession) further feeds into the
loss aversion, thereby raising the risk premium and altering the relative weight of bulls and bears.
The expected future fundamentals, which affect the future exchange rate, therefore not only
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depend on the current level but on the expected variance of these fundamentals as well. From
this line of argument, the long run solution of the flexible price monetary model with uncertainty
is given by (Cumperayot, 2003: 5)16:
(5.1) 0 1 2 3 , 1 4 , 1t t t m t y ts m y h hβ β β β β− −= + + + +
The exchange rate, st, is thus determined by the current fundamentals, mt and yt, as well as the
conditional variances of the fundamentals given by hi,t.
For the empirical test below, I have chosen to augment the model of Cumperayot (2003) with the
interest rate, it, as this macroeconomic variable seems to have a significant impact on the
determination of exchange rates. The augmented model is given by:
(5.2) 0 1 2 3 4 , 1 5 , 1 6 , 1t t t t m t y t i ts m y i h h hβ β β β β β β− − −= + + + + + +
The reason for including the interest rate in the model is three-fold: First of all, increases in the
uncertainty have direct effect on the interest rate as the agents’ shifts towards (or away from)
more secure income streams, e.g. treasury bills; shifts which then affect the exchange rate.
Secondly, because interest rate changes feed into the expected exchange rate (through the UIP or
UAIP in the IKE theory), as the expected income from investing in dollars, say, changes with
interest rate variations. Thirdly, the interest rates proxy the alternative cost of holding money.
As the empirical survey in chapter 3 showed, macroeconomic fundamentals alone have a hard
time explaining the movements of the exchange rate. By adding macroeconomic risk, here given
by the time-varying conditional variances, the model in equation (5.1) catches the (possible)
significance of uncertainty in exchange rate determination. The time-varying conditional
variances, hi,t, of the macroeconomic fundamentals are based on a GARCH(p,q) (Generalised
Autoregressive Conditional Heteroskedasticity) model (see Bollerslev, 1987 or Tsay, 2002). The
GARCH(1,1) is given by the following equation:
(5.3) 20 1 1 1 1t t th u hα α β− −= + +
The ht variable is the conditional variance (often denoted by σ in the literature), ut-1 is the error
term, and ht-1 the lagged volatility (the GARCH part). Hence, the GARCH model assumes that
the conditional variance changes over time as a function of both the past errors, ut-1, and the past
conditional variance, ht-1. The GARCH(1,1) model therefore captures shifts in the uncertainty of
the fundamentals. Hence, a volatile (calm) ut-1 and/or ht-1 are expected to be followed by a volatile
(calm) ht (Cumperayot, 2003: note 14; Tsay, 2002: 94). This is also integrated in the IKE theory,
16 Note that I have chosen to use the lagged GARCH variables, since it is unreasonable to assume that increased
volatility affect the exchange rate immediately. The agents in the market, then, observe increased volatility of time t-1
at time t and react on this accordingly. But for the short run model Cumperayot uses the lagged GARCH variables.
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which assumes that the conservative behaviour of the agents prolongs the period of
disequilibrium. The concept of a persistent effect from the uncertainty of the fundamentals into
the exchange rate is thus included in the empirical model of equation (5.1) (via the conditional
variances of the macroeconomic fundamentals); i.e. a period of negative news (e.g. a falling GDP)
has negative effect on the exchange rate going forward, and vice versa. The empirical model
presented in this chapter thus assumes some inertia in the development of the exchange rate, in
line with the insight from the IKE theory. As pointed out by Frydman and Goldberg (2007b: 7):
“We find that the exchange rate will undergo a swing either toward or away from PPP in any period of time in
which individuals revise their forecasting strategies in conservative ways and trends in macroeconomic fundamentals
persist.” That is, negative (positive) news of macroeconomic data support a negative (positive)
trend in the exchange rate. Here it is assumed that increased volatility in both directions, i.e. both
falling or growing macrofundamentals, have an effect on the exchange rate.
In the next section, the cointegration method as well as the GARCH method will be discussed in
further detail.
5.3 The model: Specification and estimation
5.3.1 Introducing the empirical test
In this section, the empirical models of equation (5.1) and (5.2) above as well as the flexible price
monetary model will be tested. Before the actual testing commences, the data and the
econometric methods will be discussed.
5.3.2 The data
Two currency crosses are tested in this chapter, the USD/JPY and the USD/NOK. USA is in
both tests the foreign country, and the exchange rate thus measure domestic currency (i.e. JPY or
NOK) per unit of foreign currency (USD). For the money supply M2 is used and for the interest
rates I use 3 months treasury bills. As a proxy for GDP I use industrial production, since it is
difficult to obtain useful GDP numbers on a monthly basis. Furthermore, industrial production is
generally used as a GDP proxy, see for example McNown and Wallace (1994: 399) or
MacDonald and Taylor (1994: 280).
The data covers the period of 1978:2 until 2007:12 and is taken from the EcoWin database. The
estimation has been done in Ox Metrics and CATS 2.1 for RATS. All the data, except for the
interest rates, are in logs.
86
5.3.3 A brief discussion of multivariate cointegration
The monetary model in equation (5.1) assumes a long-run relationship between the exchange rate
and the macroeconomic fundamentals (here output and money supply). As pointed out by for
example Hendry and Juselius (1999: 1), most economic time series (such as the aforementioned
macroeconomic fundamentals) are not stationary, an assumption of regular econometric analysis.
Conventional regression techniques are therefore rendered useless, or with some risks of bias, as
the assumption of stationarity of the variables is a crucial assumption for the ordinary least
squares (OLS) properties. The properties of the OLS estimator, and the standard deviations, are
no longer valid under the circumstance of non-stationarity, and the inference on the significance
of the variables breaks down. As pointed out by Verbeek (2004: 313-14) as well as Hendry and
Juselius (1999), regressions based on non-stationary variables will lead to meaningless findings
(i.e. spurious regression). One way to avoid this problem is by using cointegration analysis based
on the two-step procedure of Engle and Granger (1987). This rather simple method (discussed in
for example Verbeek, 2004: 314ff.) has several drawbacks, though, one of them being that the
method can, at most, find one cointegrating relationship. This is further discussed in Verbeek
(2004: 328-29). Instead I use the so-called Johansen cointegration procedure (Johansen: 1988;
1991) in this chapter. As the models tested in this chapter are based on more than two variables
using the multivariate Vector AutoRegressive (VAR) method, as suggested by Johansen, is a
useful method. The VAR model tries to find whether there exist cointegrating vectors of the
variables that are stationary. That is, given p I(1) variables (i.e. variables integrated of order 1),
then there may be p–1 linear relationships that are I(0), cf. Verbeek (2004: 324-25). Stationary
cointegrating vectors can then be interpreted as if there exists a long-run relationship between the
variables; i.e. that they share a common stochastic trend. The cointegrating relations, if any, can
then be interpreted as long-run steady states, towards which the process moves over time
(Hansen and Juselius, 2003: 4). Furthermore, using the multivariate VAR model does not
condition on the exchange rate being endogenous. It is, for example, reasonable that changes in
the exchange rate (e.g. caused by changes in the money supply or interest rates) have an effect on
the industrial production, and not exclusively the other way around.
The p-dimensional VAR model in k-lags is given by the following (Hansen and Juselius, 2003;
Hendry and Juselius, 2001):
(5.4) 1 1 1 0...t t k t k t tY AY A Y Dμ ψ ε− − += + + + + +
Where Yt is a px1 vector of stochastic variables and Dt a vector of non-stochastic dummies (either
seasonal or intervention) or weakly exogenous stochastic variables excluded from the
cointegration space. The importance of dummies will be discussed further in the specification of
87
the models below, as including them can reduce problems with non-normality or autocorrelation.
The GARCH variables estimated below enters into the system as dummy variables, and are
therefore excluded from the cointegration space. The error terms, ε1,… εT, are niid.
Reformulating equation (5.4) to error-correction form allows for distinguishing between
stationarity by linear combinations and by differencing (Hansen and Juselius, 2003: 2):
(5.5) 0 1 1 1 1 1...t t k t k t t tY Y Y Y Dμ ψ ε− − − + −Δ = + Γ Δ + +Γ Δ +Π + +
The hypothesis of cointegration (for a given rank, r) in the system of equation (5.5) is tested by
maximum likelihood on the 1tY −Π part (since the 1 1 1 1...t k t kY Y− − − +Γ Δ + +Γ Δ parts are stationary):
(5.6) ( )1 :H r αβ′Π =
Here β is the matrix of cointegrating vectors, i.e. the long run coefficients, and α the weights by
which each of the vectors enter into the equations in (5.5), i.e. the short-run adjustment
coefficients (Nielsen, 2004: 122). The hypothesis, H1(r), implies that the process YΔ is stationary,
Yt non-stationary, and tYβ ′ stationary (Johansen, 1991). Finding the appropriate cointegration
rank, r, of the matrix Π, after checking for zero and non-zero eigenvalues, is the first step of the
cointegration analysis (Hendry and Juselius, 2001: 20). The rank determines the number of long-
run relations in the model, towards which the process is adjusting itself (cf. Juselius, 2005), and it
is an important part of the cointegration analysis.
For a more thorough discussion of cointegration analysis, please refer to the two introductory
papers of Hendry and Juselius (1999 and 2001), the comprehensive book of Juselius (2005) as
well as the aforementioned papers by Johansen (1989 and 1991).
Before the cointegration specification and estimation, the GARCH part of the empirical analysis
is discussed and estimated.
5.3.4 GARCH(p,q) estimation
As mentioned above, the GARCH(p,q) model suggests that the conditional variance is
determined by q lags of the error term (the innovation parameter) and p lags of the conditional
variance (the persistence parameter). The GARCH model is an augmented version of the
ARCH(q) model, which is given by:
(5.7) 20
1
p
t j t jj
h uα α −=
= +∑
The problem with the ARCH(q) model is that it requires a rather large number of lags to describe
the volatility process (Tsay, 2002: 93). The GARCH model, on the other hand, has been found to
be more flexible, and a GARCH(1,1) is often sufficient for describing the process. The GARCH
model has been found to be a good descriptor of the clustering of volatility as well as the
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relatively high kurtosis and fat tails (seen in chapter 2) of (financial) time series data. The
GARCH(p,q) is given by the following:
(5.8) 20
1 1
p q
t j t j j t jj j
h u hα α β− −= =
= + +∑ ∑
In the following, the macroeconomic fundamentals (output, money supply and interest rates) of
USA, Norway and Japan will, separately, be run through a GARCH(p,q) model. The conditional
variances of the estimation will be saved and used in the model for the cointegration analysis in
the next section. Including the GARCH variables endogenously in the models can give some
estimation problems, so they are instead added as dummies.
One has to be aware of potential problems with the GARCH estimation method used on
macroeconomic data on monthly frequency. First of all, several of the macroeconomic time-
series start in the late 1970s to early 1980s, resulting in approximately 360 observations. Thus, the
GARCH model can have problems converging to the true parameter values due to the relatively
short dataset. Secondly, the persistence of volatility is less when looking at monthly data,
compared with daily observations (as touched upon in chapter 2). This adds to the models
problem of finding persistence in the volatility. For this reason the GARCH model did not
converge for the UK macro fundamentals, and hence I chose to test the Norwegian krone
against the US dollar instead. Cumperayot (2003) does not post her results with respect to the
GARCH estimation in the paper, and does not report whether she had any problems with
convergence.
In figures 13–15 below, the differenced variables (money supply, output and interest rates) are
shown for USA, Norway and Japan. By visual inspection some problems appear, especially for
Japan where interest rates have been virtually unchanged (and close to zero) for the period 2002
to 2006. The results from the GARCH estimation is found in appendix C, result 1. All of the
GARCH parts are (highly) significant. As is evident, there have been some problems with the
estimation for some of the variables (e.g. industrial production for Norway and Japan). This has
been solved by adding either a constant or a trend to the model, as well as changing the order of
p and q. Overall, though, the result of the estimation is acceptable, and the models converge in all
9 instances. One problem with the conditional variances is the aspect of multicollinearity, i.e. high
(not perfect) correlation between two (or more) of the independent variables used when
estimating equation (5.2). To check for possible problems with correlation, the correlation
matrices for USD/JPY and USD/NOK are printed in appendix C result 2. As is evident, some
of the variables exhibit rather high correlations, especially the three months interest rates. To
avoid problems with biased estimators (Verbeek, 2004: 44), instrumental variable estimation has
89
been used on the most correlated variables (in bold). An instrumental variable, z, is uncorrelated
with the error of the model but correlated with the regressor, x (see for example Verbeek, 2004:
133). The error term from regressing two correlated variables against each other is thus a useful
instrument, as the error (by definition) is orthogonal to the regressor. The result of the estimation
is not posted, but the results are significant for all the regressions (JPYm2 against JPYtbill etc.).
The conditional variances (or the instrumental variables) from the estimation are saved and added
to the VAR models estimated in the next section.
5.3.5 Cointegration analysis
The two models below, equations (5.9) and (5.10) respectively, are estimated for USD/JPY and
USD/NOK. An asterisk denotes a foreign variable, in all cases USA. Adding the interest rates,
equation (5.10), is then a test of whether the interest rate has significant impact on the long-run
result.
But before I test the two models, the simple monetary model (FPMM) given by equation (5.11)
below is put to a test for Japan and Norway.
(5.9) Model I: * ** *
0 1 2 3 4 5 , 1 6 7 , 1 8, 1 , 1t t t t t m t y tm t y ts m m y y h h h hβ β β β β β β β β− −− −= + + + + + + + +
(5.10) Model II – with interest rates: **
0 5 6 11 , 1 12 , 1... ...t t t i t i t
s i i h hβ β β β β− −= + + + + + +
(5.11) Monetary approach – FPMM: * *0 1 2 3 4t t t t ts m m y yβ β β β β= + + + +
Before the actual testing I will briefly discuss the models in relation to the cointegration analysis
from the previous section. In equation (5.12) below I assume, for simplicity, that r = 1; i.e. there
is only one stationary relation between the variables in (5.10), and this relation is furthermore
assumed to be the monetary model with uncertainty (i.e. the GARCH variables). Then the last
part of the cointegrated VAR model in equation (5.5) can, for model II above, be written as the
following (for 1 lag) (cf. Juselius, 2005: 99 or Nielsen, 2004: 123):
90
(5.12) ( ) ( ) ( )1 1
1
2*
3* *
4 1 2 1 7 1*
5
6
* 7
... ...t
t t
t
t
t
t
t t t i t t t
t
sm
m
y s m i b GARCHm GARCHi D
y
i
i
αααα β β ψ μ εααα
− −− − −
⎛ ⎞Δ ⎛ ⎞⎜ ⎟ ⎜ ⎟Δ⎜ ⎟ ⎜ ⎟⎜ ⎟Δ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ′⎜ ⎟Δ = + + + + + + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟Δ ⎜ ⎟⎜ ⎟ ⎜ ⎟Δ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠Δ⎝ ⎠
Here the long-run relation has been normalized on the exchange rate, st. Note that the GARCH
variables enter exogenously into the system and is thus assumed to excluded from the
cointegrating space. Furthermore, I test for whether the GARCH variables lagged one period
have effect on the exchange rate at time t. The exchange rate equation, Δst, from (5.12) is then
given by (still for 1 lag of the model and rank equal 1):
(5.13) [ ] [ ]1 1 1 1 1 1t t t t ts s x b GARCH Dα β ψ μ ε− − −Δ = Γ Δ + + + + +
From the monetary approach discussed in chapter 3 we should expect that β2 = 1 (m), β3 = -1
(m*), β4 < 0 (y), β5 > 0 (y*), β6 > 0 (i), and β7 < 0 (i*) in equation (5.12). But since this is a test of
one of the assumptions of the IKE theory, and not the monetary approach per se, a strict
conditioning on the coefficients could potentially be a flawed method. According to the IKE
theory, the economic fundamentals do matter for the exchange rate but the structure of the
causal relations may change at different points in time and in different ways. The way in which
the fundamentals influence the exchange rate is therefore not prespecified in the IKE model. But
since this is a test of whether uncertainty, as assumed by the IKE theory, plays an important part
in relation to the determination of exchange rates, the sign on the endogenous variables is not
important. I therefore restrict on β2 = 1 (domestic money supply) and β3 = -1 (foreign money
supply) in the tests below, following the assumption of the flexible price monetary model. The
interesting part, in our context, is whether the GARCH variables in equations (5.12) and
(especially) (5.13) have significant effect on the exchange rate, suggesting that uncertainty is
important for the price setting of currencies alongside macroeconomic fundamentals. This is
tested in the last part of section 5.3.7 below. Note that the signs of the GARCH variables are not
prespecified; it is just tested whether they influence the exchange rate at all, compared with the
result from the tests of the simple FPMM.
First the graphs of the individual time series are inspected. Figure 13, 14 and 15 below show the
data in levels and in first differences. For the American data – figure 13 below – the money
supply (lower chart) and output (upper chart) looks trending in levels, but (slightly) mean
91
reverting in differences. The US 3 months treasury-bills (middle chart) display substantial changes
in the beginning of the 1980s, as the US Federal Reserve Bank tried to combat high
unemployment by lowering the interest rate drastically, then hiking the interest rate in response to
the ensuing inflationary pressure. This is also reflected in the Japanese interest rate in figure 14
below. For Japan, there is a clear shift in the money growth (lower charts) around the beginning
of 1990 – as well as an evident drop in interest rates. This seems to be a regime shift, and has to
be taken into account when estimating the model. For the Norwegian data, figure 15 below, the
interest rate (middle chart) exhibits large changes in December 1992 as the Norwegian central
bank gave up on defending the krone and allowed it to float. Dummy variables taking this into
account seems appropriate. For all three countries the series seem somewhat stationary in
differences, although with some problems. These problems will be sought dealt with below by
including dummies.
Figure 13 – US data in levels and first differences (RHS)
92
Figure 14 – JPY data in levels and first differences (RHS)
Figure 15 – NOK data in levels and first differences (RHS)
93
5.3.6 Lag length, residual analysis and dummy variables
Now I turn to the specification of the VAR model. First of all, the lag-length of the VAR model
is determined based on information criteria. As mentioned by Juselius (2005: 88) a VAR(2) is
most often used. The result of the lag-length is found in appendix C result 3. For the two models
for USD/NOK the result points to k = 1, whereas for USD/JPY the lag-length determination is
ambiguous, pointing at either k = 1 or k = 2. I have chosen to specify a VAR(2) model, though,
for both the USD/NOK models and the USD/JPY models, noting that the VAR(1) could be
sufficient for the models. Testing VAR(1) on the models does not change the results below,
though, and I therefore use the VAR(2) for all four models.
Secondly, the VAR model is based on the assumption of multivariate normality, from which it
follows that the model is: i) linear in parameters; ii) has constant parameters and iii) normally
distributed errors (Juselius, 2005: 56). But as Juselius (ibid.: 57-58) points out, this is seldom
satisfied in practice. This can, potentially, be a problem but can be solved by adding dummies to
the model. It is therefore important to perform residual analysis ensuring validity of the VAR
model (cf. Hendry and Juselius, 2001). The results of the residual analysis for the four models
(not shown) point to problems with normality and some ARCH problems, as well as significant
skewness and kurtosis. According to Hendry and Juselius (2001: 6), some degree of both excess
kurtosis and residual heteroskedasticity can be accepted, whereas non-normality is a problem. A
solution to these problems, as already mentioned, can be to introduce dummies of different kind.
As seen from the figures 13-15 above visual inspection points to several troubling periods. For
the Norwegian data, the floating of the krone in late 1992; for the Japanese data the structural
shift in the beginning of the 1990s, marking the start of the deep recession which affected money
growth, interest rates and output; and for the American data, the changes in the interest rate in
the beginning of the 1980s as the Fed tried to combat inflation as well as September 11th 2001.
To address the problems with the residuals, these have been inspected with focus on residuals
above 3.3 (Hendry and Juselius, 2001). For USD/NOK model I, I have chosen to include five
dummies. For Norway a dummy covering the banking crisis in late 1991 and a dummy covering
the free floating at October 1992. Furthermore, I have included a dummy for the May 1986
devaluation of the Norwegian krone. The two dummies covering the devaluation and the free
floating have been set to be impulse dummies, i.e. shifting the system, whereas the other
dummies (including the dummies for USA) have been set to be transitory (i.e. only temporary).
For USA, a May 1980 dummy covering a large change in industrial production, and a “September
11th” dummy have been included in the model. The dummies appear significant, and including
them improve the residual analysis somewhat. But the problems of non-normality and
94
autocorrelation are still present in the modified model. For USD/NOK model II, the free
floating dummy starts in August 1992 instead, and the US dummy for 1980 starts in March.
Including the dummy in model II once again improves the residuals somewhat, but with some
problems.
For the USD/JPY model I, a dummy marking the beginning of the recession in Japan, 1991:2, is
included. This is set to be an impulse dummy. The September 11th dummy is kept, and so is the
dummy for 1980. The dummies once again improve the model, but the hypothesis of normality is
still rejected. For the USD/JPY model II, a dummy for 1983 is included as well.
Finally, running a new lag-length determination on the four models including the dummies does
not change the lag length for any of the four models, and I keep the VAR(2) specification.
5.3.7 Testing the models
Long run structure – the flexible price monetary models As mentioned above, the determination of the cointegration rank is a crucial part of the empirical
analysis, as it affects the following inference (Juselius, 2005: 157). The rank (r) is equal to the
number of cointegrating vectors in the system (cf. Verbeek, 2003: 329), and the (correct)
determination thereof is thus very important. For finding the correct rank I use the
“top→bottom” procedure following Juselius (2005: 160), in which case we will accept the correct
rank 95% of the time. The rank is determined by the trace test, including a Bartlett correction.
First of all I test the monetary model, given by equation (5.11) above, for both USD/NOK and
USD/JPY. The same dummies, discussed in the previous section, are used and k set to 2. Since
dummies are included in the models, the critical values for the rank tests are simulated in CATS
for all the models in this section (cf. Harbo et al, 1998).
The rank test results for the simple monetary models are found in appendix C, result 4. For
USD/NOK, the rank is set to 1 with p-value of 37.8%. For the USD/JPY monetary model the
rank is set to 1 as well. Setting the rank to 1, the estimated exchange rate equation for
USD/NOK is given by the following:
(5.14) * *0.660 1.211 1.004 0.413t t t t ts m m y y= + − +
As is evident from equation (5.14) the coefficient to the US money supply (m*) is of the wrong
sign, whereas the coefficients on the other three variables are in accordance with what is expected
from theory. The signs are furthermore almost of the same size, and relatively close to one (for
the money supply). But as pointed out by Juselius (2005), the beta coefficients should only be
95
taken as indicative, not conclusive. For the alpha coefficients, the Norwegian industrial
production seems to have a significant equilibrium correction effect on the system.
For the monetary model for USD/JPY, equation (5.15), the coefficients on both money supply
variables are of the wrong sign. Furthermore, the signs are significantly larger than 1.
(5.15) * *16.706 47.791 1.785 24.566t t t t ts m m y y= − + − +
The result of a cointegration rank of 1 does not necessarily mean that a linear combination of the
variables in the monetary model results in a stationary long-run process. The cointegration of the
system could be any combination of the variables included, so a test with imposed restrictions on
the cointegrating vector is necessary. The H(0) of this test is then whether the restricted linear
combination of the process is stationary; i.e. whether the FPMM is a stationary relation. The
hypothesis on the cointegration vectors is given by the following (Juselius, 2005: 206; Hansen and
Juselius, 2003: 41):
(5.16) , 1, 2,...,i i iH i rβ ϕ= =
Where φi is a si x 1 vector of unknown parameters and Hi a known design matrix p x si. The
design matrices define the number of free parameters in each cointegrating vector. The test then
investigates whether a “known” cointegrating vector lies within the space of β. In the case of the
two monetary models above, the design matrix of the beta vector is given by [s, m, m*, y, y*] =
[1, 1, -1, *, *]. Here, a star denotes a “free” parameter.
The hypothesis of equation (5.16) is tested by the likelihood-ratio test (Juselius, 2005: 210), here
given by:
(5.17) ( )( )
( )2
max 22
max
~CT a
T
L H Rv
L H Rχ
−
−
⎛ ⎞⎜ ⎟Λ =⎜ ⎟⎝ ⎠
Where v = rm given by the m restrictions on the r cointegrating vectors; i.e. 3 for both monetary
models since I set the restrictions to 3 and the rank is set to 1 for both models. Rejecting the
hypothesis, according to Juselius (2005: 211), implies that (at least) one of the restricted relations
is no longer significantly mean-reverting; i.e. the relation is not stationary and the restricted
relation is therefore not cointegrating.
The result of the tests of the two monetary models is shown in table 4 below. For both monetary
models the null hypotheses are firmly rejected as the Chi-Square critical value with three degrees
of freedom is 7.81. Apparently the data include a cointegrating relation, but this relation is not
given by the simple monetary model for neither NOK or JPY. This is in line with the conclusion
on the empirical results discussed in chapter 3 and it is not a surprising result.
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Table 4 – Test of the simple flexible price monetary models in VECM form for Japan and Norway
USD/NOK FPMM: TEST OF RESTRICTED MODEL:
BARTLETT CORRECTION:
CHISQR(3) = 76.213 [0.000]
CHISQR(3) = 64.522 [0.000]
USD/JPY FPMM: TEST OF RESTRICTED MODEL:
BARTLETT CORRECTION:
CHISQR(3) = 45.802 [0.000]
CHISQR(3) = 36.121 [0.000]
Note: Tested with the restriction on the beta vector [s, m, m*, y, y*] that [1, 1,-1, *, * ]
The question, then, is whether introducing uncertainty proxied by the GARCH variables
estimated in the previous section improves the model result significantly. This is tested in the
next section.
Long run structure – models with uncertainty First of all, the rank is determined for the four models – the results hereof are found in appendix
C result 4. Once again, the critical values are simulated for all four models to correct for the
inclusion of dummy variables in the models. When the rank is determined, the lag-length
determination is run again. This, however, does not change the chosen lag for any of the four
models.
Note that the following results are, initially, based on models with the GARCH variables lagged
one period. With respect to the rank and restriction tests, adding 2, 3, or 4 lags of the GARCH
variables to the models change the result somewhat. For the USD/NOK model II, for example,
where the rank test for two GARCH lags turn borderline significant for r=2 with a p-value of
10.2% compared with a rank of 1 for just one lag of the GARCH. And for the USD/JPY model
II, where the rank is 2 instead of 3 when a GARCH lag length of 2 is included in the model. But
for the other two models the rank determination is not changed by adding more lags of the
GARCH variables. Hence in the following only the result from the simple model with GARCHt-1
is presented. Then I run a test for the USD/JPY model I where I include a second lag of the
GARCH variables, which changes the result. When testing for whether the GARCH variables
enter into the system with significance in the next section, I will test with more lags as well (i.e. 2,
3 and 4).
For the USD/NOK model I the H(1) is accepted with a (corrected) p-value of 97.5%. Looking at
the characteristic roots and the alpha coefficients does not change r = 1, and the cointegrating
relation (see appendix A, figure 6) does look stationary. Furthermore the two charts of the
cointegrating relation (Appendix A figure 6) look rather alike, suggesting that the statistical
97
analysis is valid (cf. Hansen and Juselius, 2003: 22-23). Thus, for the USD/NOK model I rank =
1 in the further analysis. For the USD/NOK model II, the rank is set to 2 with p-value of 39.4%.
The cointegrating relations, figure 7 in Appendix A, look stationary – but with some problems
around the floating of the Norwegian krone in 1992.
For the USD/JPY model I rank = 1 with a corrected p-value of 75.0%. This is the same result as
Cumperayot (2002: 9)17. Finally, for the USD/JPY model II the rank is set to 3.
The long-run estimated equation for the exchange rate for USD/NOK model I is given by:
(5.18) * *0.617 1.283 1.026 0.406 0.002t t t t ts m m y y t= + − + −
This is more or less in line with the result from the simple monetary model, equation (5.14)
above. The coefficient on the foreign money supply is still of the wrong sign, but rather close to
1 for all variables.
The long-run estimated equation for the exchange rate for USD/JPY model I is given by:
(5.19) * *6.839 29.884 0.302 14.677 0.094t t t t ts m m y y t= − − − +
For the USD/JPY the coefficients on the domestic output variable are of the wrong sign whereas
the money supply signs are correct, but significantly larger than 1, as for the simple FPMM for
USD/JPY in equation (5.15) with a small improvement. But as mentioned above, one has to be
aware of drawing strong conclusions on the beta coefficients.
The tests for whether the linear combination of the variables in the models is stationary changes
somewhat from the test of the monetary model in the previous section (table 4), see table 5 and
table 6 below. For the USD/NOK model I and model II the test soundly rejects H(0). The result
for the USD/NOK models are not changed by adding more lags of the GARCH variables. But it
changes the result for the USD/JPY models below (table 6).
Table 5 – Test of the flexible price monetary model in VECM form with GARCHt-1 for USD/NOK
USD/NOK model I: TEST OF RESTRICTED MODEL:
BARTLETT CORRECTION:
CHISQR(3) = 91.609 [0.000]
CHISQR(3) = 77.830 [0.000]
USD/NOK model II (with interest rates): TEST OF RESTRICTED MODEL:
CHISQR(2) = 23.289 [0.000]
Note: Tested with the restriction on the beta vector [s, m, m*, y, y*, i, i*] that [1, 1,-1, *, *, *, * ]
17 Note, however, that Cumperayot (2003) does not report whether she uses GARCHt or GARCHt-1 in the long-run
test, nor whether the GARCH variables appear endogenously or exogenously in the models.
98
Table 6 below posts the result for the two USD/JPY models. The null hypothesis is rejected for
model I for Japan, whereas the H(0) is accepted at 95% level for USD/JPY model II.
Thus, according to table 5 and 6, adding the GARCH variables in 1 lag to the model is somewhat
of an improvement over the results from the test of the simple monetary model above (table 4),
at least for Japan model II. Table 6 – Test of the flexible price monetary model in VECM form with GARCHt-1 for USD/JPY
USD/JPY model I: TEST OF RESTRICTED MODEL:
BARTLETT CORRECTION:
CHISQR(3) = 26.832 [0.000]
CHISQR(3) = 20.246 [0.000]
USD/JPY model II (with interest rates): TEST OF RESTRICTED MODEL:
CHISQR(1) = 3.100 [0.078]
Note: Tested with the restriction on the beta vector [s, m, m*, y, y*, i, i*] that [1, 1,-1, *, *, *, * ]
Adding a second lag of the GARCH variables change the result for the USD/JPY model I – see
table 7 below – such that it is accepted at 99% level. Adding a second lag of the GARCH
variables do not change the results for any of the other three models, however. Table 7 – Test of the flexible price monetary model in VECM form with both GARCHt-1 and GARCHt-2
USD/JPY model I: TEST OF RESTRICTED MODEL:
BARTLETT CORRECTION:
CHISQR(3) = 16.169 [0.001]
CHISQR(3) = 10.512 [0.015]
Note: Tested with the restriction on the beta vector [s, m, m*, y, y*, i, i*] that [1, 1,-1, *, *, *, * ]
In Appendix C result 5 the restricted estimated cointegrated equations for the models are posted.
The coefficients on output are larger than 1 for most of the estimated equations (except for
USD/NOK model I) and of the wrong sign (except for USD/JPY model I). But, compared with
the equation for the FPMM above, closer to the acceptable range between zero and 1. Especially
for Japan, where the coefficients of the FPMM were very large; in the model including the
GARCH variables, this is significantly improved.
Overall, the result of the tests seem to be an improvement over the simple flexible price model,
especially for Japan.
In the following I will test specifically whether the GARCH variables in different lags enter
significantly in the equations for the four models.
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Testing the GARCH variables – lagged one period Now the specific significance of uncertainty in regards to exchange rates is inspected. First I look
at the models with the GARCH variables lagged one period, then a test with the GARCH
variables lagged two periods included as well. I have also tested with 3 and 4 lags of the GARCH
variables, but this does not change the overall result.
For the model with the GARCH variables lagged one period, I look at the t-statistics from the
cointegration analysis. The coefficients and p-values on the GARCH variables from the four
models are found in table 8 below.
For both USD/NOK models none of the GARCH variables are significant at 95% level. In
model I, though, the Norwegian money supply is significant at 90% level (with a t-value of -1.78).
The overall significance of the GARCH variables seem to increase with the introduction of
uncertainty of the interest rate variables in the NOK/USD model II as the foreign output and
domestic interest rate turn borderline significant (t-values of 1.41 and 1.36). Based on the
coefficients it seem that increased volatility of domestic output and money supply appreciate the
Norwegian krone, with the opposite effect from increased volatility in American output and
money supply. The effect from the increased volatility of the interest rates, on the other hand,
seems to depreciate the NOK overall. Table 8 – Coefficients and p-values on the GARCHt-1 variables from the cointegration analysis
Model hy hy* hm hm* hi hi*
NOK I -0.012 (0.78) 0.001 (0.27) -0.001 (0.07) 0.004 (0.43) - -
NOK II -0.054 (0.27) 0.002 (0.15) -0.00 (0.26) 0.003 (0.50) 0.000 (0.17) 0.017 (0.49)
JPY I -0.00 (0.75) -0.002 (0.11) 0.005 (0.16) -0.005 (0.41) - -
JPY II 0.001 (0.68) -0.003 (0.01) -0.007 (0.60) -0.001 (0.88) -0.014 (0.63) -0.322 (0.02)
Note: An asterisk denotes a US variable for all four models. P-values in brackets. Bold indicates rejection of H0 at 95% level
For the USD/JPY model I none of the GARCH variables are significant in table 8, although the
p-values on the conditional variance of the foreign output (11%) and domestic money supply
(16%) are borderline significant. But for USD/JPY model II the conditional variances of
American output and American interest rates appear significant. Based on the t-statistics it is
rejected at 95% level that increased uncertainty (i.e. volatility) of these two US variables have no
effect on the Yen.
100
For all four models the results from table 8 show that especially hy*, i.e. the conditional variance
on US output, has an effect on the exchange rates. But the effect is depreciating for the NOK
and appreciating for the JPY. The negative coefficients on the domestic money supply in table 8
(except for USD/JPY model I) could stem from volatility caused by the Central Bank, i.e. due to
intervention to stabilize the economy which then have a positive effect on the future expectation
of the domestic currency (cf. Cumperayot, 2003: 12). It furthermore seems that adding one lag of
the conditional variance of the interest rates to the model increases the influence from the
conditional variance, at least for Japan.
One should be careful to conclude too much on the results from table 8, though. First of all,
most of the variables appear insignificant; and secondly, these are taken from models with the
GARCH variables lagged only one period.
Testing the GARCH variables – lagged more than one period For the models with GARCH lagged more than 1 period, I estimate the exchange rate equation
in error correction form given by the following equation (here for the rank set to 1 and the
GARCH variables lagged 2 periods):
(5.20) ( ) ( ) ( )1 1 1 1 1 ,1 1 ,2 2ˆ
t t t t i t i ts A x x D b GARCH b GARCHα β ψ μ ε− − − −Δ = Δ + + + + + +
Here the 1β is defined as the estimated β from the cointegration analysis and 1 1ˆ
txβ − is thus the
cointegrating relation(s)18. The test is then whether the bi,j on the GARCH variables in equation
(5.20) enter significantly into the model(s). The results from running OLS on equation (5.20) with
2 lags for the four models are found in Appendix C result 6.
Initially, looking at the t-statistics from the OLS the same GARCH variables seem to be
(borderline) significant in both models I and II for Norway as in table 8 above. But for
USD/JPY model II none of the variables appear significant when looking at the t-statistics from
the OLS regression. The same tests has been done for 3 and 4 lags of the GARCH variables (not
shown) as well, and the result thereof does not change much to the result obtained with only two
lags of GARCH. I therefore focus on two lags of the GARCH variables only.
The next step is then to test whether the GARCH variables can be excluded from the model.
This is done with a likelihood-ratio test, similar to equation (5.17) above, which is given by
(Verbeek, 2004: 173):
(5.21) ( ) ( )ˆ2 log logLR L θ θ⎡ ⎤= −⎣ ⎦%
18 Note that I have run the VAR models again with the extra lag of the GARCH variables.
101
Where Log(θ ) is the unrestricted estimator, i.e. including the GARCH variables in the model,
and Log(θ% ) is the restricted, with (some of) the GARCH variables excluded from the model.
The test is Chi-squared distributed with J degrees of freedom based on the number of restricted
variables. The null hypothesis is that restricting the model does not take change the goodness-of-
fit of the model significantly, i.e. that the extra variables in the unrestricted model do not include
extra information, and subsequently can be left out. The test is thus whether or not the more
complex model, the unrestricted model, fits the dataset significantly better than the simpler
mode, the restricted model, in which (some of) the GARCH variables are left out.
The LR-test has been run three times on the four models with 2 lags of the GARCH variables
against the alternatives of: i) no GARCH variables in the model (NO GARCH), i.e. both 1 and 2
lags GARCH excluded; ii) no first lag GARCH in the model but including second lag GARCH
(NO GARCH_1); and finally iii) no second lag GARCH variables in the model but first lag
GARCH (NO GARCH_2). The result from the LR-tests is found in table 9 below.
For the USD/NOK model I, leaving the GARCH variables out altogether does not take away
much from the model as H(0) cannot be rejected, although it is borderline. But leaving out the
GARCHt-1 variables seem to be an incorrect restriction as the null hypothesis is rejected at 95%
level. The same result is obtained when leaving out GARCHt-219. Apparently, there seems to be
some problems with correlation between the two GARCH variables which affects the overall
result of the LR-test for the model. I have chosen to model the GARCH variables univariately,
not multivariate, and therefore I do not assume that uncertainty on the part of output, for
example, affect the uncertainty of interest rates. This could be the case, of course, but wielding
“Occams’ Razor” I have chosen the simpler version of the model, in line with Cumperayot
(2003). Furthermore there is, as seen from appendix C result 2, some cross-correlation between
the GARCH variables, and this apparently affects the likelihood-ratio test in table 9. This
problem has been sought dealt with by using instrumental variables, but there could still be some
impact between the variables.
For USD/NOK model II a similar result is obtained. The null hypothesis of leaving out the
GARCH variables altogether is not rejected, although it is once again relatively close to the 90%
level. But leaving out GARCHt-1 is clearly too strong a restriction and the null hypothesis of
leaving out GARCHt-2 is rejected at 90% level.
19 Testing the USD/NOK model I with 1 and 2 lags of the GARCH does not change anything with respect to the
rank test or the restricted test from the previous section.
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Table 9 – LR-test results on the error correction model with two lags of the GARCH variables
LR-test NOK I NOK II JPY I JPY II
b1 = b2 = 0 (NO GARCH)
13.14 (0.107) 17.23 (0.14) 14.99 (0.06) 21.61 (0.042)
b1 = 0 (NO GARCH_1)
11.7 (0.02) 13.81 (0.032) 5.65 (0.227) 9.52 (0.146)
b2 = 0 (NO GARCH_2)
9.78 (0.04) 11.73 (0.068) 10.27 (0.037) 8.72 (0.19)
Note: Chi-square results from the LR-tests. P-values in brackets. Bold indicates rejection at 95% level
For the USD/JPY model I, the null hypothesis of leaving out the GARCH variables altogether is
rejected at 90% level (p-value of 6%). And GARCH lag 2 appears significant at 95% level,
whereas it seems that GARCH lag 1 can be left out. This is no surprise, since including the
second lag of GARCH variables in the model leads to a near acceptance of the H(0) in the
USD/JPY model I in table 7 (compared with table 6). For the USD/JPY model II, the null
hypothesis of leaving out the GARCH variables is rejected at 95% level. But the result at the
same time points to no inclusion of the individual GARCH variables, although both variables
have relatively low p-values (14.6% and 19%, respectively). Once again it could be that
correlation between the variables influence the tests.
As mentioned above I have run the test (not shown) with the GARCH variables lagged 3 and 4
times as well, and the tests clearly point to a maximum of 2 lags of the GARCH variables in the
models. For example, in the USD/NOK model I with 4 lags of the GARCH variables the LR-
test result of leaving out GARCHt-3 and GARCHt-4 altogether, but keeping GARCHt-1 and
GARCHt-2, gives a Chi-square result of 6.65 (critical value of 15.5) – a clear acceptance of the null
hypothesis. Apparently GARCHt-1 and GARCHt-2 includes most of the information regarding the
exchange rate.
Overall the tests point to some significance of the GARCH variables. For the models with one
lag of the GARCH variables only two variables appeared significant – both in the USD/JPY
model II. But some of the uncertainty variables appeared borderline significant in the other three
models. When adding a second lag of uncertainty, the LR-test pointed to some significance of the
GARCH variables. Leaving the GARCH variables out altogether was thus rejected for both
USD/JPY models but (borderline) accepted for the two USD/NOK models. Individually, five of
the eight tests of the GARCH variables pointed towards a significant improvement of the fit
103
when including the variables compared with the restricted models leaving the variables out of the
model.
5.3.8 Conclusion
The result from the empirical analysis is somewhat mixed. The restricted linear combination for
the simple flexible price monetary model for both Norway and Japan was tested to be (clearly)
not stationary, a result in line with the empirical survey in chapter 3. The enhanced models,
including the GARCH variables, improved this result to some extent. For Norway the two
models were still rejected, albeit with slightly lower Chi-square results for model II. But for the
USD/JPY model I (with two lags of the GARCH) as well as for the USD/JPY model II it could
not be rejected that the estimated models were stationary relations. Adding the uncertainty
variables, proxied by the conditional variance estimated by the GARCH model, thus seems to
enhance the result compared with the simple monetary models.
The result of the specific tests of the GARCH variables in 1 and 2 lags was somewhat supportive
for the models as well. With one lag of the GARCH variables, several of the parameters were
(borderline) significant in the four models tested. For two lags of the GARCH variables, leaving
the variables out altogether seemed (borderline) acceptable for two of the models. But on the
other hand, leaving out specifically either lag 1 or 2 of the GARCH variables seemed improper
for three of the models. A result which could stem from problems with correlation between the
GARCH variables, as the model assumed no effect between the uncertainty variables. This could,
perhaps, be too simple an assumption.
Overall, though, the idea and result of Cumperayot (2003) that uncertainty is important for
exchange rate determination cannot be rejected by the empirical test in this chapter. Furthermore,
augmenting the model with interest rates seems to be an improvement.
5.4 The result from an IKE perspective The question, then, is whether the empirical result from this chapter support the theory of
Imperfect Knowledge Economics presented in chapter 4.
To begin with, two issues are worth discussing. First of all, the test in the previous section is not
a test of the IKE theory per se, but rather a test of the monetary model of the exchange rate
augmented with a proxy for uncertainty. Thus, the rather mixed result of the test could stem
from the poor performance of the flexible price monetary model seen in chapter 3 and tested in
this chapter. And then, if the test is really a test of FPMM with extra variables, what about the
104
IKE theory? Here, it is worth noting that the flexible price model is a sub-set of the IKE theory,
and that the IKE theory does not reject the monetary approach. Furthermore, the FPMM
assumes that new information is instantly fed into the exchange rate – which is apparently not the
case when uncertainty variables lagged 2 months can affect the present exchange rate level, as was
the case for the USD/JPY model. Hence, it can be argued that the empirical test is a test of the
IKE assumptions regarding uncertainty; and this assumption seems to be supported by the result
– at least it cannot be rejected that uncertainty has significant effect on exchange rates.
Secondly, the question is whether the GARCH variable is a good proxy for uncertainty. The
problem, of course, is to quantify uncertainty, and the use of the GARCH variable can be
discussed. Furthermore, the model assumed that increased uncertainty of, for example, the
interest rate did not feed into the other uncertainty variables. Both aspects could be themes
worth exploring further.
Overall the result of the test pointed towards including uncertainty to the monetary model
compared with the monetary approach. This, therefore, seems to support the assumption from
the IKE theory that uncertainty plays an important part in the price setting of exchange rates.
5.5 Conclusion The result of this chapter showed that including uncertainty variables, proxied by the conditional
variance of the fundamentals, improved the simple monetary model somewhat. A result in line
with Cumperayot (2003). But the result was ambiguous, with no significant improvement for the
USD/NOK models, but support for the USD/JPY models. Furthermore, the tests did not
decisively support the effect of the GARCH variables on the models. But overall, the result of
the likelihood-ratio tests indicated that the GARCH variables had significant effect on the
exchange rate. Hence uncertainty seems important, to some extent at least, with respect to the
determination of exchange rates. A result in line with the conclusion from the IKE theory in the
previous chapter. Based on the empirical test, it cannot be rejected that including uncertainty
improves the monetary model of exchange rates.
105
Chapter 6
Conclusion
“It works in practice, but does it work in theory?”
French saying.
This thesis has concentrated on answering the question posed in the introduction to Chapter 1:
“Does Imperfect Knowledge Economics provide a solution to the exchange rate disconnect puzzle?”. Based on the
theoretical and empirical parts of the thesis, I cannot reject the hypothesis that IKE provides a
solution to the exchange rate disconnect puzzle.
Since the established result of Meese and Rogoff in 1983 the problems of explaining exchange
rate fluctuations have led to some gloom on the account of exchange rate research. Hence, the
imperfect knowledge economics theory, introduced by Roman Frydman and Michael D.
Goldberg, shows a new possible direction for economic research on the subject with less focus
on strictly rational and homogenous agents combined with a significant role for uncertainty.
Seen from a theoretical perspective, imperfect knowledge economics can explain why the
exchange rate is disconnected from the fundamental value. And secondly, why the currency
eventually revert to parity at least for some time. The IKE theory assumes that the economic
agents in the foreign exchange market are heterogeneous and hence use different models and
methods when deciding their individual strategy. This result in an outcome which is strictly
different from the rational expectations outcome given by the mainstream monetary models. The
IKE theory pairs the above assumptions with insight from both prospect theory, assumptions of
gap effects as well as conservative revisions of forecasting strategy. Combined, this explains both
the deviation from fundamental value of exchange rates, as well as the subsequent return.
Furthermore, it is assumed that people acknowledge that they do not have access to all important
information. Thus, the exchange rate disconnect puzzle is solved without neither removing
rationality from the model nor by including (irrational) bubble structures.
The assumption of agent heterogeneity have been thoroughly studied in several articles and it
seems to be a robust finding, as discussed in chapter 2. And the last two assumptions, the
importance of private information and the significance of uncertainty, have been tested and
discussed in chapter 3 and chapter 5: The hypothesis of the significance of private information,
using the order flow variable in a microstructure model, seem supported. Regarding uncertainty,
106
here proxied by GARCH estimation added to a flexible price monetary model, the hypothesis of
its significance could not be rejected in chapter 5. Thus, the IKE theory thus seems able to, both
theoretically and empirically, explain the exchange rate behaviour of the recent float.
That the model with uncertainty, apparently, can improve the result significantly when compared
with the simple flexible price monetary model is an interesting result. But, obviously, one has to
take caution when interpreting the results due to the result from the residual analysis of the VAR
model. Furthermore, investigating the behaviour of the GARCH variables and their correlation,
as well as allowing for the variables to influence each other, could be the next step. The aspect of
uncertainty and how to proxy it is, therefore, an interesting subject for future research.
But one thing is explaining the fluctuations in-sample. The next question would then be how to
utilise this information in out-of-sample forecasting of exchange rate movements at 1-12 months
horizons. That is, once again putting the Meese-Rogoff result to a test, but with a model
including uncertainty. This has not been the subject of the thesis, but could be an interesting
analysis as well.
Overall, it seems that the imperfect knowledge economics theory could be able to solve the
exchange rate disconnect puzzle. According to this result, future research on exchange rates
should therefore focus on trying to incorporate aspects such as uncertainty, imperfect knowledge
of the economy on part of the agents as well as agent heterogeneity. At least if the goal is to
explain as well as to understand the fluctuations of exchange rates.
107
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Appendix A – Figures Figure 1A – Log returns, weekly observations for USD/GBP, EUR/USD and USD/JPY
Figure 2A – Log returns, monthly observations USD/GBP, EUR/USD and USD/JPY
115
Figure 3A – Plot of 3 months Treasury rates: USA, Germany and Japan (RHS)
3,5
4
4,5
5
5,5
1.5.96 16.5.96 31.5.96 15.6.96 30.6.96 15.7.96 30.7.96 14.8.96 29.8.96
0,0000
0,0010
0,0020
0,0030
0,0040
0,0050
0,0060
0,0070
0,0080
US Germany Japan (RHS)
Figure 4A – Fitted returns and scaled residuals, DEM/USD microstructure model
116
Figure 5A – Fitted returns and scaled residuals, USD/JPY microstructure model
Cointegrating relations
Figure 6A: USD/NOK model I – cointegrating relation:
Beta1'*Z1(t)
1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006-0.20
-0.15
-0.10
-0.05
-0.00
0.05
0.10
0.15
Beta1'*R1(t)
1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Note: The upper panel plots tYβ ′ whereas the lower panel plots the concentrated likelihood function of the rank
regression 0t ktR Rαβ ε′= + (cf. Hansen and Juselius, 2003: 7).
117
Figure 7A: USD/NOK model II – 1st cointegrating relation:
Beta1'*Z1(t)
1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
Beta1'*R1(t)
1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
Figure 8A: USD/NOK model II – 2nd cointegrating relation:
Beta2'*Z1(t)
1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006-0.96-0.80-0.64-0.48-0.32-0.160.000.160.32
Beta2'*R1(t)
1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006-1.0
-0.8
-0.6
-0.4
-0.2
-0.0
0.2
0.4
118
Appendix B – Models
1B: Dornbusch Sticky price model (Sarno and Taylor, 2002: 104-107) The two first equations below is the same as in the text; the third is a Philips curve relation.
(6.1) *s i i= −&
(6.2) m p y iκ θ= + −
(6.3) ( )p s p yγ α μ= + − −⎡ ⎤⎣ ⎦&
The money supply is set to be exogenous and at its long-run equilibrium level. The long run
money market equilibrium is given by:
(6.4) *m p y iκ θ− = −
Where overbars denote long-run equilibrium values.Subtracting the long-run equilibrium value
from equation (6.2) yields the following:
(6.5) ( )*p p i iθ− = −
Inserting this into the UIP condition, equation (6.1), then yields:
(6.6) ( )1s p pθ⎛ ⎞= −⎜ ⎟⎝ ⎠
&
Looking at the goods market equilibrium, given by the Philips-curve in equation (6.3), the long-
run value is obtained for 0p =& , that is:
(6.7) ( ) 0s p yγ α μ+ − − =⎡ ⎤⎣ ⎦
Now, subtracting this long-run equilibrium from the Philips-curve, equation (6.3), yields:
(6.8) ( ) ( )p s s p pγμ γμ= − − −&
The two differential equations, (6.6) and (6.8), can then be written as:
(6.9) 10s s s
p p pθ
γμ γμ
⎛ ⎞ −⎡ ⎤ ⎛ ⎞⎜ ⎟= ⎜ ⎟⎢ ⎥ ⎜ ⎟ −⎣ ⎦ ⎝ ⎠−⎝ ⎠
&
&
As the determinant in the matrix is negative, the system has a unique (and convergent) saddlepath
(cf. Sarno and Taylor, 2002: 106). Given that λ is the (negative) stable root to the system, the law
of motion equation for the exchange rate s must obey:
(6.10) ( )s s sλ= − −&
Inserting this into the differential equation yields the saddlepath equation, which is the same as
equation (3.19) in section 3.2.2 (with β2 = λ).
(6.11) ( )1s s p pλθ
⎛ ⎞= − −⎜ ⎟⎝ ⎠
119
2B: IKE monetary model – solution
( ) ( )
( ) ( )
( ) ( )( )
( )
1
1 1 0 0 0 0
1 1 , 0 0 0 1 0 ,
1 0 0 1 0 0
1
1,
1 1
0A B
h
hh h hG G G G G G
hh h hA BG G G G G G
G G G G G G
G h
λ ϕη ηα α
σ σ
ϕ α η σ αλα η λη αλ λασ
ϕ α ασ σ αασ α η α ασ
ϕ α ησ σ η αλα ση λασ α λασ α η αλη σα
α η
σ
−
⎛ ⎞− −⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟= − = ⎜ − ⎟⎜ ⎟ ⎜ ⎟
⎜ ⎟⎜ ⎟ −⎝ ⎠⎝ ⎠
+ −⎛ ⎞+− −⎜ ⎟
⎜ ⎟⎜ ⎟− + −−
= − −⎜ ⎟⎜ ⎟
+ − +⎜ ⎟+ − − +− −⎜ ⎟
⎝ ⎠
= + +
−
( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )
( ) *
*
ˆ 0
1 1ˆ ˆ ˆ
1ˆ ˆ ˆ
1ˆ ˆ ˆ
:
ˆˆ 1
PPP PPP
PPP PPP
PPP PPP
REa RE RE
h
s m y x q qG G G G G
h h hp m y x q qG G G G G
hh hq s p m y x q q
G G G G G
Define
s s
s
αλ σ
σ η αλα ση λασ α λασ α η αλϕ η π σα
σ αλα η λη αλ λασϕ π
λη σ ησ η σ η α λα σ ηϕ π σ
ρ π
− >
− ++ − − += − + + − +
−+= − + + − +
− +⎡ ⎤− − + −⎣ ⎦≡ − = − − + − + +
= − +
=
( )
( )( )
* * ˆˆ 1 , 1
ˆˆ 1
RE PPP
RE REt
REa RE RE PPP RE
m y q
s s s
s m y q
ϕ λπ
θ θ π θ ρ
ρ ϕ λπ π
− + +
= + − + − =
⇒
= − − + + +
120
( )( ) ( ) ( )( )
( )( ) ( )( ) ( )( )
( )( )( ) ( )( ) ( )( ) ( )
( )( ) ( )( ) ( )
ˆ ˆ part in :
1 1ˆ ˆ ˆ
1 1 ˆˆ 1
ˆ1 1 1 ˆ1 .....
ˆ ˆ1 1 1 1...,
REa
REa REa
REa RE RE RE
RERE RE
RE RE
Add and subtract s to the x s
x s sG G
s m yG G
s m yG G
sG
σ η αλ σ η αλ
σ η αλ σ η αλρ ϕ λπ π
σ η αλ ρ ση α σ η αλϕ λ ρ π
σ ρ αλ η ρ σ α
− + − +− +
− + − + ⎡ ⎤⇒ = − − + +⎣ ⎦
⇒
− + − + + − += − + + − +
⇒
− − + − − += ⋅ +
( ) ( )( )
( )( ) ( ) ( ) ( )( )
( )
( )( ) ( ) ( )( ) ( ) ( )( )
ˆ1 1 *1 ...,
Rearrange the min # :
ˆ ˆ ˆ1 1 1 * 1
#
RE
RE RE PPP
RE RE RE
add and subtract G h h
h h hs
G
Now insert m y s q
no ator of
h h h
α η λα σ
σ ρ αλ η α η α λα σ
ϕ λπ
αλ σ ρ λα σ ρ ρ σ αλ η ρ ρ σ
= + + −
⇒
⎡ ⎤⎢ ⎥− − + + − + − −
= + ⋅ +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
− = − −
− − − − ⇔ − − + − ⇔ − −
14444444444244444444443
( )
( )( )( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )
( )( )
* *
*
.
:ˆ1 1 1
ˆ ˆ ˆ
ˆ
ˆ ˆˆ ˆRe 1 , . .
4.47
RE PPP
REREa RE
RE PPP PPP
REa REa RE RE RE
Now rearrange for and q as well
This yields
s s s x sG G G
q qG G
defining s such that s s i e
This yields equation
αλ η
λπ
σ η αλ ρ ρ σ η αλ η λσπ π
η αλ σ η αλ σπ
ρ π ρ ρ
+
− + − − + −= + + − + −
+ +− + −
= − + =
121
Appendix C – Empirical results
Result 1C: GARCH estimation for m, y and i for USD, JPY and NOK
GARCH(1,1) – JPY M Coefficient Std.Error t-value t-prob
ARCH(Alpha1) 0.0236471 0.01080 2.61 0.009
GARCH(Beta1) 0.972534 0.009243 133.0 0.000
GARCH(2,1) – JPY Industrial prod. (Y) Coefficient Std.Error t-value t-prob
ARCH(Alpha1) 0.015116 0.0059518 2.540 0.0115
GARCH(Beta1) 1.880461 0.050508 37.23 0.0000
GARCH(Beta2) -0.943878 0.054455 -17.33 0.0000
GARCH(1,1) – JPY 3 mths treasury rate (i) Coefficient Std.Error t-value t-prob
ARCH(Alpha1) 0.876853 0.1379 1.85 0.065
GARCH(Beta1) 0.599847 0.03463 6.59 0.000
GARCH(1,1) – US Industrial prod. (y)
Coefficient Std.Error t-value t-prob
ARCH(Alpha1) 0.202951 0.07812 2.27 0.024
GARCH(Beta1) 0.479973 0.1782 2.11 0.035
GARCH(1,1) – US M
Coefficient Std.Error t-value t-prob
ARCH(Alpha1) -0.041537 0.026141 -1.589 0.1129
GARCH(Beta1) 0.825147 0.058983 13.99 0.0000
122
GARCH(1,1) – US 3 mths treasury rate (i) Coefficient Std.Error t-value t-prob
ARCH(Alpha1) 0.652518 0.2011 2.61 0.009
GARCH(Beta1) 0.508761 0.07696 4.56 0.000
GARCH(1,1) – NOK M Coefficient Std.Error t-value t-prob
ARCH(Alpha1) 0.023194 0.0086671 2.676 0.0077
GARCH(Beta1) 0.971850 0.011404 85.22 0.0000
GARCH(2,2) – NOK industrial production (y) Coefficient Std.Error t-value t-prob
ARCH(Alpha1) -0.007616 0.027475 -0.2772 0.7818
ARCH(Alpha2) -0.050367 0.010895 -4.623 0.0000
GARCH(Beta1) 0.113356 0.055379 2.047 0.0414
GARCH(Beta2) 0.863513 0.054156 15.94 0.0000
GARCH(1,1) – NOK treasury bill (i)
Coefficient Std.Error t-value t-prob
ARCH(Alpha1) 0.528283 0.28068 1.882 0.0605
GARCH(Beta1) 0.691429 0.081104 8.525 0.0000
123
Result 2C: Correlation matrices for the GARCH variables: USD/NOK and USD/JPY
USD/NOK: nokm2 nokgdp noktbill usdm2 usdtbill usdind
nokm2 1.0000 -0.11071 -0.025966 0.10580 -0.33058 -0.33403
nokgdp -0.11071 1.0000 -0.13003 0.16897 0.056048 0.087314
noktbill -0.025966 -0.13003 1.0000 -0.0062130 0.14487 0.12666
usdm2 0.10580 0.16897 -0.0062130 1.0000 0.043690 0.041909
usdtbill -0.33058 0.056048 0.14487 0.043690 1.0000 0.54793
usdind -0.33403 0.087314 0.12666 0.041909 0.54793 1.0000
USD/JPY: jpym2 jpytbill jpyindu usdm2 usdtbill usdind
jpym2 1.0000 0.34024 0.012578 0.18035 0.21071 0.16157
jpytbill 0.34024 1.0000 0.11500 0.15243 0.57640 0.33298
jpyindu 0.012578 0.11500 1.0000 -0.096151 0.096950 0.023640
usdm2 0.18035 0.15243 -0.096151 1.0000 0.043829 0.041923
usdtbill 0.21071 0.57640 0.096950 0.043829 1.0000 0.54793
usdind 0.16157 0.33298 0.023640 0.041923 0.54793 1.0000
Result 3C: Lag-length determination for the four models Model I – USD/NOK: VAR(6) 6 352 70 8429.403 -42.064 -44.377 0.087 0.351
VAR(5) 5 352 61 8399.733 -42.645 -44.661 0.350 0.484
VAR(4) 4 352 52 8363.212 -43.187 -44.905 0.265 0.475
VAR(3) 3 352 43 8306.496 -43.615 -45.035 0.000 0.000
VAR(2) 2 352 34 8244.053 -44.009 -45.133 0.012 0.000
VAR(1) 1 352 25 8169.819 -44.337 -45.163 0.000 0.000
124
Model II – USD/NOK: Model k T Regr Log-Lik SC H-Q LM(1) LM(k)
VAR(6) 6 352 97 8913.158 -39.332 -43.819 0.015 0.000
VAR(5) 5 352 84 8840.611 -40.436 -44.321 0.221 0.202
VAR(4) 4 352 71 8762.112 -41.506 -44.790 0.095 0.001
VAR(3) 3 352 58 8662.432 -42.455 -45.138 0.001 0.000
VAR(2) 2 352 45 8544.920 -43.303 -45.385 0.000 0.000
VAR(1) 1 352 32 8414.903 -44.081 -45.561 0.000 0.000
Model I – USD/JPY: Model k T Regr Log-Lik SC H-Q LM(1) LM(k)
VAR(6) 6 352 70 9625.154 -48.858 -51.171 0.075 0.272
VAR(5) 5 352 61 9598.407 -49.456 -51.471 0.758 0.642
VAR(4) 4 352 52 9554.930 -49.958 -51.676 0.003 0.129
VAR(3) 3 352 43 9481.519 -50.291 -51.712 0.000 0.000
VAR(2) 2 352 34 9411.326 -50.642 -51.765 0.000 0.000
VAR(1) 1 352 25 9315.651 -50.848 -51.674 0.000 0.000
Model II – USD/JPY: Model k T Regr Log-Lik SC H-Q LM(1) LM(k)
VAR(6) 6 352 96 10780.499 -50.059 -54.499 0.000 0.604
VAR(5) 5 352 83 10703.276 -51.136 -54.975 0.388 0.042
VAR(4) 4 352 70 10615.729 -52.154 -55.392 0.016 0.177
VAR(3) 3 352 57 10512.806 -53.085 -55.722 0.000 0.000
VAR(2) 2 352 44 10421.458 -54.082 -56.117 0.008 0.001
VAR(1) 1 352 31 10195.009 -54.311 -55.745 0.000 0.000
125
Result 4C: Rank determination – with simulated critical values Model FPMM – USD/NOK: p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value*
5 0 0.253 155.916 151.832 88.554 0.000 0.000
4 1 0.089 52.249 50.899 63.659 0.322 0.378
3 2 0.027 19.164 11.207 42.770 0.969 1.000
2 3 0.017 9.422 5.613 25.731 0.940 0.998
1 4 0.009 3.297 2.613 12.448 0.831 0.905
Model FPMM – USD/JPY: p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value*
5 0 0.276 181.128 176.248 88.554 0.000 0.000
4 1 0.105 66.250 32.308 63.659 0.029 0.983
3 2 0.034 26.924 14.855 42.770 0.688 0.997
2 3 0.029 14.535 8.936 25.731 0.619 0.955
1 4 0.012 4.144 2.693 12.448 0.721 0.898
Model I – USD/NOK: p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value*
5 0 0.249 138.950 135.572 96.345 0.000 0.000
4 1 0.057 36.853 36.008 72.451 0.969 0.975
3 2 0.026 15.842 14.588 50.172 0.999 0.999
2 3 0.014 6.372 5.601 30.391 0.999 0.999
1 4 0.004 1.529 1.166 15.914 0.991 0.996
Model II – USD/NOK: p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value*
7 0 0.325 309.841 306.804 163.556 0.000 0.000
6 1 0.218 170.043 168.634 130.647 0.000 0.000
5 2 0.093 82.711 82.151 100.205 0.377 0.394
4 3 0.078 48.150 47.896 73.732 0.755 0.764
3 4 0.034 19.135 19.063 51.123 0.991 0.991
2 5 0.016 6.790 6.775 31.877 0.998 0.998
1 6 0.003 1.001 1.001 16.654 0.997 0.997
126
Model I – USD/JPY: p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value*
5 0 0.206 156.153 152.257 93.668 0.000 0.000
4 1 0.090 74.213 44.732 67.396 0.014 0.750
3 2 0.071 40.655 35.374 47.100 0.173 0.373
2 3 0.035 14.586 11.883 29.568 0.716 0.870
1 4 0.005 1.845 1.622 14.554 0.964 0.974
Model II – USD/JPY: p-r r Eig.Value Trace Trace* Frac95 P-Value P-Value*
7 0 0.266 311.063 298.795 160.271 0.000 0.000
6 1 0.202 201.032 188.037 127.096 0.000 0.000
5 2 0.146 120.890 111.827 96.840 0.001 0.004
4 3 0.107 64.551 36.545 72.140 0.156 0.971
3 4 0.032 24.286 17.440 49.719 0.920 0.995
2 5 0.025 12.642 9.279 30.765 0.864 0.970
1 6 0.010 3.633 3.233 15.733 0.827 0.864
Result 5C: The estimated equations from the restricted test of the models
USD/NOK model I: BETA(transposed)
S USM2 USY NOKY NOKM2
Beta(1) 1.000 -1.000 -3.156 0.660 1.000
(.NA) (.NA) (-4.351) (1.855) (.NA)
USD/NOK model II (with interest rates): S USM2 USY USTBILL NOKY NOKM2 NOKTBILL
Beta(1) 1.000 -1.000 -5.379 -0.064 -0.829 1.000 -0.384
(.NA) (.NA) (-54.934) (-1.143) (-1.983) (.NA) (-8.177)
127
USD/JPY model I: S USM2 USY JPY JPM2
Beta(1) 1.000 -1.000 1.933 -5.679 1.000
(.NA) (.NA) (3.657) (-4.410) (.NA)
USD/JPY model I with 2 lags of the GARCH: S USM2 USY JPY JPM2
Beta(1) 1.000 -1.000 2.938 -8.518 1.000
(.NA) (.NA) (4.365) (-5.259) (.NA)
USD/JPY model II (with interest rates): S USM2 USY USTBILL JPTBILL JPY JPM2
Beta(1) 1.000 -1.000 -3.285 -0.077 -0.098 9.582 1.000
(.NA) (.NA) (-5.986) (-4.436) (-3.584) (9.254) (.NA)
128
Result 6C: Result from OLS on the ECM, equation (5.20) (with GARCH lagged 1 and 2 periods) Note: The season- and dummy variables have been removed from the results below to obtain a higher
degree of readability.
USD/NOK model I – with GARCH lag 1 and 2 included (rank = 1)
Coefficient Std.Error t-value t-prob Part.R^2 deltas_1 -0.0182966 0.06472 -0.283 0.7776 0.0002
deltausm_1 0.698618 0.9933 0.703 0.4823 0.0015
deltausy_1 -0.377790 0.2533 -1.49 0.1368 0.0068
deltnokm_1 -0.107744 0.1169 -0.922 0.3573 0.0026
deltanoky_1 -0.0153218 0.05325 -0.288 0.7737 0.0003
usdnok_1 -0.0151984 0.01133 -1.34 0.1806 0.0055
betaxt_1 0.0267313 0.04203 0.636 0.5253 0.0012
garchnokm -0.00373487 0.001152 -3.24 0.0013 0.0314 garchnokgdp 0.0248382 0.1188 0.209 0.8345 0.0001
garchusdm -0.00107685 0.006366 -0.169 0.8658 0.0001
garchusdy 0.00293097 0.001298 2.26 0.0246 0.0155 garchusy_2 -0.00185606 0.001276 -1.45 0.1466 0.0065
garchusm_2 0.00456801 0.006264 0.729 0.4664 0.0016
garchnokm_2 0.00328394 0.001148 2.86 0.0045 0.0246 garchnoky_2 -0.00580293 0.1171 -0.0496 0.9605 0.0000
R^2 0.160249 F(31,324) = 1.994 [0.002]**
log-likelihood 779.987 DW 2.02
no. of observations 356 no. of parameters 32
129
USD/NOK model II – with GARCH lag 1 and 2 included (rank = 2)
Coefficient Std.Error t-value t-prob Part.R^2 deltas_1 -0.0210265 0.06484 -0.324 0.7459 0.0003
Constant -0.00424407 0.1001 -0.0424 0.9662 0.0000
deltausm_1 0.610285 1.019 0.599 0.5495 0.0011
deltausy_1 -0.306898 0.2649 -1.16 0.2475 0.0042
deltnokm_1 -0.119859 0.1176 -1.02 0.3087 0.0033
deltanoky_1 -0.0126254 0.05466 -0.231 0.8175 0.0002
usdnok_1 -0.0117838 0.01341 -0.879 0.3802 0.0024
betaxt1_1 0.0158675 0.03668 0.433 0.6656 0.0006
betaxt2_1 0.00273289 0.005307 0.515 0.6070 0.0008
garchnokm -0.00458189 0.001343 -3.41 0.0007 0.0353
garchnokgdp 0.0193226 0.1208 0.160 0.8730 0.0001
garchnoktbill -8.15115e-005 0.0005413 -0.151 0.8804 0.0001
garchustbill 0.0500393 0.03607 1.39 0.1664 0.0060
garchusdy 0.00482626 0.001892 2.55 0.0112 0.0200
garchusm -0.00127418 0.006479 -0.197 0.8442 0.0001
garchnoktbill_2 0.000451454 0.0005285 0.854 0.3936 0.0023
garchusy_2 -0.00302475 0.001824 -1.66 0.0983 0.0086
garchusm_2 0.00509965 0.006354 0.803 0.4228 0.0020
garchustbill_2 -0.0305858 0.03474 -0.880 0.3793 0.0024
garchnokm_2 0.00409883 0.001366 3.00 0.0029 0.0275
garchnoky_2 0.0113336 0.1205 0.0941 0.9251 0.0000
sigma 0.0284855 RSS 0.258032906
R^2 0.168353 F(36,318) = 1.788 [0.005]**
log-likelihood 779.031 DW 2.01
no. of observations 355 no. of parameters 37
130
USD/JPY model I – with GARCH lag 1 and 2 included (rank = 1)
Coefficient Std.Error t-value t-prob Part.R^2 deltas_1 0.0209497 0.05524 0.379 0.7047 0.0004
Constant -0.254139 0.2337 -1.09 0.2776 0.0036
deltajpy_1 0.0964622 0.3224 0.299 0.7650 0.0003
deltajpm_1 -1.50755 0.9182 -1.64 0.1016 0.0082
deltausy_1 -0.0488726 0.3489 -0.140 0.8887 0.0001
deltausm_1 2.41638 1.272 1.90 0.0583 0.0110
usdjpy_1 -0.0241005 0.01159 -2.08 0.0384 0.0131
garchusm 0.00133149 0.01025 0.130 0.8967 0.0001
garchusy 0.000879951 0.001561 0.564 0.5733 0.0010
garchjpy -0.00198297 0.006255 -0.317 0.7514 0.0003
garchjpym 0.0213338 0.009852 2.17 0.0311 0.0142
garchusm_2 -0.00812774 0.01007 -0.807 0.4202 0.0020
garchusy_2 -0.00351405 0.001398 -2.51 0.0125 0.0191
garchjpy_2 0.00170759 0.006249 0.273 0.7848 0.0002
garchjpym_2 -0.0176534 0.009840 -1.79 0.0737 0.0098
betaxt_1 0.00132937 0.0009028 1.47 0.1419 0.0066
sigma 0.0343583 RSS 0.3836598
R^2 0.0894027 F(30,325) = 1.064 [0.380]
log-likelihood 711.119 DW 2
no. of observations 356 no. of parameters 31
131
USD/JPY model II – with GARCH lag 1 and 2 included (rank = 2)
Coefficient Std.Error t-value t-prob Part.R^2 deltas_1 0.00684266 0.05595 0.122 0.9027 0.0000
Constant -0.208858 1.706 -0.122 0.9027 0.0000
deltajpy_1 0.0916255 0.3244 0.282 0.7778 0.0002
deltajpm_1 -1.47575 1.179 -1.25 0.2115 0.0049
deltausy_1 -0.0563921 0.3610 -0.156 0.8760 0.0001
deltausm_1 2.17195 1.292 1.68 0.0937 0.0088
usdjpy_1 -0.0287478 0.01585 -1.81 0.0707 0.0102
betaxt_1 0.00169142 0.001376 1.23 0.2200 0.0047
betaxt2_1 0.000291642 0.004134 0.0706 0.9438 0.0000
garchusm 0.00372217 0.01053 0.354 0.7239 0.0004
garchustbill 0.0578529 0.09256 0.625 0.5324 0.0012
garchusy 0.000936347 0.002512 0.373 0.7096 0.0004
garchjpy -0.00293045 0.006353 -0.461 0.6449 0.0007
garchjpym 0.0186926 0.03766 0.496 0.6200 0.0008
garchjpytbill -0.0859020 0.4101 -0.209 0.8342 0.0001
garchusml2 -0.00851043 0.01028 -0.828 0.4084 0.0021
garchustbill2 -0.0266271 0.08691 -0.306 0.7595 0.0003
garchusy2 -0.00441672 0.002491 -1.77 0.0772 0.0097
garchjpy2 0.00289187 0.006325 0.457 0.6478 0.0007
garchjpyml2 -0.0136210 0.03737 -0.364 0.7157 0.0004
garchjpytbill2 -0.116313 0.4260 -0.273 0.7850 0.0002
sigma 0.0342891 RSS 0.376237437
R^2 0.107019 F(35,320) = 1.096 [0.332]
log-likelihood 714.597 DW 1.97
no. of observations 356 no. of parameters 36
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