an empirical examination of learning in foreign exchange ... · an empirical examination of...

An Empirical Examination of Learning

in Foreign Exchange Markets*

David Goldbaum†

Remco C.J. Zwinkels‡

September 2008

DRAFT

Abstract

Using a unique dataset of survey expectations, this paper examines the extent to which the classical fundamentalist – chartist dichotomy is valid for the foreign exchange market. By applying a recursive selection algorithm 1) respondents are classified into the two groups, and 2) the forecasting models are endogenously determined within the groups. We find that the largest part of the variation in expectations can be explained by the fundamentalist/chartist distinction. The majority of respondents use a simple chartist rule, while fundamentalists use a broad range of macro-economic information.

* Paper prepared for Investing Strategies and Financial Market Inefficiency Paul Woolley Centre for Capital Market Dysfunctionality University of Technology, Sydney. Financial support from the Paul Woolley Centre is gratefully acknowledged. † School of Finance and Economics, University of Technology Sydney; PO Box 123 Broadway; NSW 2007 Australia; email: [email protected]. ‡ Erasmus School of Economics, Erasmus University Rotterdam; PO Box 1738, 3000DR, Rotterdam, The Netherlands; email: [email protected].

1. Introduction

A substantial body of literature in economics and finance models investors as

heterogeneous and adaptive. The heterogeneity allows for interactions between

traders behaving differently to impact the market. The heterogeneity can exist in a

market at equilibrium or may keep the market out of equilibrium. The adaptation

allows traders to select behavior appropriate for the perceived, possibly changing,

market setting. The sensitivity of the market to the behavior of the traders can

produce market destabilizing feedback loops. Models based on adaptive

heterogeneous agents have offered insight explaining a variety of market phenomena

that are difficult to capture with representative agent models. In financial markets,

these include fat tails in returns, volatility clustering without auto-correlation in

returns, bubbles, excess volatility, and slow mean reversion; see e.g. Lux (1998), De

Grauwe and Grimaldi (2006).

Trader heterogeneity manifests in a variety of characteristics. A number of

papers emerged to explore models of dynamic heterogeneity. Some models consider

different levels of trader sophistication; famous example being Brock and Hommes

(1997, 1998). A sophisticated trader might employ rational expectations when

forecasting the behavior of market prices while other traders employ a more naïve

strategy. Alternatively, a model might explore the heterogeneity in information. A

fundamental approach might engage in research in order to gain a private signal about

future value while the market-based approach attempts to extract information from the

price, as in De Grauwe and Grimaldi (2005, 2006).

A theoretical foundation for sustainable market-based trading strategies

(technical trading, charting) is rooted in Grossman and Stiglitz’s seminal “On the

Impossibility of Informationally Efficient Markets” (1980). Their paper established

an equilibrium market condition in which market-based (uninformed) traders coexist

with and depend on fundamentalist (informed) traders. In Grossman and Stiglitz, the

uninformed traders are fully rational, but there presence in the market is based on their

ability to extract information from the price.

Dynamics arise as traders switch between available information or levels of

sophistication. The agents of the Brock and Hommes (1997) and Brock and Hommes

(1998) consider the relative performance of different forecasting strategies. The

models employ the random element in the discrete choice model of Manski and

McFadden (1981) to create heterogeneity in the individual level choice among the

available options. The environment highlights the inherent instability of markets.

The strategy that is in the minority performs better, but the superior performance

attracts members of the population.

Goldbaum (2005) introduces evolution in the strategies that are available to

traders. The evolution reflects the effort by traders to improve the performance of

inherently imperfect trading tools. A market populated by learning and adaptive

traders has the potential of transitioning the market from one of price stability noisily

reflecting the efficient market price, to a market in which the price is unstable and

able to move away from the fundamental value.

While heterogeneous adaptive agents models provide intuitively appealing

explanations for market phenomenon, do these explanations stand up empirically? If

heterogeneity exists, is it dynamic and can the evolution be captured by a model of

behavior? What dynamic model is most consistent with behavior? At present, there

are two major classes of models, offering different behavior in the population. The

parameters of these models, in particular the intensity of choice parameter, determine

the existence, uniqueness, and stability of the market equilibrium. Estimating the

parameters of these dynamic models can offer considerable insight into market

behavior. Finally, are the strategies being employed by traders static, even as the

proportion of the population employing them change, or are the strategies themselves

also evolving?

This paper contributes to the still emerging literature that empirically

examines markets based on heterogeneous adaptive agent models. Only a handful of

papers have sought to estimate these models and a number of issues remain

unresolved or in need of empirical support. Included among these is Boswijk,

Hommes, and Manzan (2007). The investigation finds evidence of switching by

traders between a trend following and mean reverting rule in the S&P500.

Goldbaum and Mizrach (2008) model the distribution of new funds between

active and passively managed mutual funds to estimate the intensity of choice model.

The success of the model in capturing the shift towards passively managed funds is

evidence in favor of adaptive heterogeneity.

Evidence in favor of switching has also been found in experimental settings.

Experiments involving market entry decisions often find a wide range of strategies

have been employed by the participants that still combined to bring the market to the

equilibrium number of entrants. Hommes et al (2007) have their participants forecast

an endogenously determined price that is influenced by their own forecast and the

forecast of the other participants. The participants are rewarded for accuracy the

accuracy of their forecasts. The authors identify four rule of thumb strategies

employed by participants. Hommes and Anufriev (2007) extend the analysis by

modeling the switching between strategies.

Branch (2004) empirically tests an adaptive heterogeneous agent model based

on survey respondents’ reported inflation forecasts. Branch models the population as

switching between three different models differentiated by there implicit level of

sophistication. Again, evidence is found in support of a switching model where

households respond to adopt the strategy that has performed well in the recent past.

MacDonald and Marsh (1996) document, also on the basis of survey data, that market

participants hold different beliefs on future price movements, and use different types

of models to form expectations.

The current project also seeks to examine markets for evidence of adaptive

heterogeneity and also to determine whether there is evidence in favor of learning in

the foreign exchange market. De Grauwe and Grimaldi (2006) and De Grauwe and

Markiewicz (2008) study heterogeneous agents and adaptation in foreign exchange

markets and show that they are well capable of explaining the stylized facts. Similar

to Branch, the current project seeks to model the reported forecast of survey

participants. In this case, the data being employed is the exchange rate forecasts

collected from participating international banks. Each period includes forecasts over

a number of horizons for a number in individual institutions. Using the same data,

Jongen et al. (2008) show that expectations are dispersed, and that panelists base

expectations on fundamentalist/chartist types of considerations.

2. Model

Each of N traders in a market maximize an expected negative exponential utility

function in next period’s wealth, 1tW + , based on their information set, itI . Formally,

, 1,max ( ( ) | )

t t

ii t ta b

E U W I+

subject to

, . .i t i t t i tW a s b= +

*, 1 1 . .(1 ) (1 )i t t t i t t i tW r s a r b+ += + + +

where ( ) exp( )t tU W W= − −φ , ts is the spot exchange rate, tr is the domestic interest

rate, and *tr is the foreign exchange rate. Solving produced the optimal demand for

the foreign currency,

*

1. 2

,

(1 ) ( | ) (1 )it t t t t

i ti t

r E s I r sa ++ − +

=φσ

(1)

where 2 * 2, 1(1 ) var( | )i

i t t t tr s I+σ = + .

A market clearing Walrasian equilibrium requires supply equals demand,

,1

N

i t ti

a X=

=∑ . (2)

Let /t tx X N= and ,1

1 N

t i ti

a aN =

= ∑ be the per capita supply and demand respectively

for the foreign currency so that (2) can be expressed as t ta x= .

Consider a market in which the population of traders is informed by two

models of exchange rate determination. The fundamental approach presumes that the

market is driven by fundamentals. This may include notions of purchasing power

parity (PPP) or interest rate parity (IPP), among other fundamental determinants. A

trader relying on fundamentals trades in the currency market seeking to take

advantage of exchange rate deviations from the fundamentals. A chartist approach

employs past exchange rate innovations as a predictor for future innovations. The

chartist trades according to the predictions of the chartist approach.

2.1 Fundamental model

There is a fundamental exchange rate, *ts . The realized market spot rate, ts , can

deviate from the fundamental. The market has a tendency to revert to the fundamental

rate so that future innovations in the market spot rate are affected by the current

deviation. The fundamental traders form expectations about future innovations

accordingly,

*1( ) ( )f

t t t tE s s s+Δ = −ψ − . (3)

Here, ψ captures the rate at which the market reverts towards fundamentals.

In a similar environment, DeGrauwe and Grimaldi (2006) model the

fundamental rate as an exogenous process following a random walk. In addition,

traders know the fundamental value both 1ts − and *1ts − as the most recently observed

values of the spot and fundamental rates. In the present model, the actual

fundamental rate is never observed by the traders. Rather, the traders must attempt to

extract the fundamental value from available data.

Capturing the forecasts reports employed in the empirical section requires

modeling the k period ahead forecast of exchange rates. Let ftZ represent the vector

of time t fundamental information. Further, let t t ks +Δ represents spot market

innovation t k ts s+ − . For integer 1k > , the fundamental forecast is captured by the

following process:

1 1 1 1 2 1 1( ) ' ( ) ( )f f f ft t t k t t t t k t t tE s Z E s E s+ − − + − − −Δ = α + γ Δ + γ Δ . (4)

the second term on the right hand side is present to take advantage of the overlap in

the prediction period from the forecast made in period 1t − and the current period t

forecast. The third term controls for information in 1 1 1( )t t t kE s− − + −Δ that is not useful

in forecasting ( )ft t t kE s +Δ since tsΔ has already materialized such that potentially

useful information in it is incorporated into ftZ ; this reduces noise and increases the

usefulness in employing 1 1 1( )t t t kE s− − + −Δ as a control variable that is reported in the

survey of predictions. The first term is left to explain only the innovation in the

forecast from the previous period. It is thus capturing the new component of the

forecast period, 1( )t t k t kE s+ − +Δ as well as any change in the forecast of 1t t ks + −Δ that

flows from the new time t information.

Individual trader forecasts are captured by the following:

, 1 , 1 1 1 2 , 1 1 ,( ) ' ( ) ( )f f f fi t t t k t i t t t k i t t t i tE s Z E s E s+ − − + − − −Δ = α + γ Δ + γ Δ + ε (5)

There are thus two sources for heterogeneity among the fundamental traders. The

idiosyncratic term, ,i tε , captures trader specific differences between the forecasts of

individual traders. These can be seen as the result of private information not available

to the modeler, deviation in the objective function from the presumed utility function,

or simply the result of randomness in the traders forecasting method. The presence of

,i tε plus the fact that individuals can have different choice patterns cause the different

traders to have individual forecasts histories that appear in the second and third term

on the RHS of (5).

2.2 Chartist information

Chartist information is composed on past market information, namely previous

innovations in the exchange rate. Let ctZ represent the vector of time t chartist

information. The chartist forecast is captured by the following process:

1 1 1 1 2 1 1( ) ' ( ) ( )c c c ct t t k t t t t k t t tE s Z E s E s+ − − + − − −Δ = β + γ Δ + γ Δ (6)

Individual forecasts are captured by

, 1 , 1 1 1 2 , 1 1 ,( ) ' ( ) ( )c c c ci t t t k t i t t t k i t t t i tE s Z E s E s+ − − + − − −Δ = β + γ Δ + γ Δ + ε , (7)

thereby capturing the same sources of heterogeneity that exists among the

fundamentalists.

2.3 Discrete choice

Equations (5) and (7) capturing the forecasts of individuals will be examined

in a variety of settings. Included is an environment that allows each individual trader

to choose which strategy to employ for each given period. (The following is not

present in the current version of the paper, but will be examined: Each trader chooses

according to a fitness function as though solving a Manski and McFadden style

discrete choice problem. As introduced by Brock and Hommes (1997), the traders use

past performance as an indicator of future fitness.)

In the empirical examination, a forecast is labeled as either a fundamentally

derived forecast or a chartist forecast based on its relative proximity to systematic

component of (5) or (7). Let , 1i tθ = if the forecast by individual i is deemed to

originate from the fundamental strategy, with , 0i tθ = otherwise. Further, let

, 1 , 1 1 1 2 , 1 1ˆ ( ) ' ( ) ( )f f f f

i t t t k t i t t t k i t t tE s Z E s E s+ − − + − − −Δ = α + γ Δ + γ Δ (8)

and

, 1 , 1 1 1 2 , 1 1ˆ ( ) ' ( ) ( )c c c c

i t t t k t i t t t k i t t tE s Z E s E s+ − − + − − −Δ = β + γ Δ + γ Δ (9)

represent the systematic components of each model. Thus,

, , 1 , 1 1 1 2 , 1 1

, 1 , 1 1 1 2 , 1 1 ,

( ) ( ' ( ) ( ))

(1 )( ' ( ) ( ))

f f fi t t t k i t t i t t t k i t t t

c c ci t t i t t t k i t t t i t

E s Z E s E s

Z E s E s+ − − + − − −

− − + − − −

Δ = θ α + γ Δ + γ Δ

+ − θ β + γ Δ + γ Δ + ε (10)

where

( ) ( )( ) ( )

2 2

, , , ,

, 2 2

, , , ,

ˆ ˆ1 if ( ) ( ) ( ) ( )

ˆ ˆ0 if ( ) ( ) ( ) ( )

f ci t t t k i t t t k i t t t k i t t t k

i tc f

i t t t k i t t t k i t t t k i t t t k

E s E s E s E s

E s E s E s E s

+ + + +

+ + + +

⎧ Δ − Δ < Δ − Δ⎪θ = ⎨⎪ Δ − Δ ≤ Δ − Δ⎩

(11)

2.4 Continuous choice

Another environment examined is one in which the trades are allowed to

combine the two strategies in order to create a single estimate. In this case, the

estimate is a weighted average of the systematic components of (5) and (7). Let ,i tw

indicate the weight trader i places on the fundamental strategy. As a result,

, , 1 , 1 1 1 2 , 1 1

, 1 , 1 1 1 2 , 1 1 ,

( ) ( ' ( ) ( ))

(1 )( ' ( ) ( ))

f f fi t t t k i t t i t t t k i t t t

f c ci t t i t t t k i t t t i t

E s w Z E s E s

w Z E s E s+ − − + − − −

− − + − − −

Δ = α + γ Δ + γ Δ

+ − β + γ Δ + γ Δ + ε (12)

The weight is, again, is based on the relative distance of the individual’s

forecast from the fitted model.

1, ,(1 exp( ))i t i tw −= + −λ (13)

with

2 2

, , , ,, 2 2

, , , ,

ˆ ˆ( ( ) ( )) ( ( ) ( ))ˆ ˆ( ( ) ( )) ( ( ) ( ))

c fi t t t k i t t t k i t t t k i t t t k

i t c fi t t t k i t t t k i t t t k i t t t k

E s E s E s E sE s E s E s E s

+ + + +

+ + + +

Δ − Δ − Δ − Δλ =

Δ − Δ + Δ − Δ (14)

which results in , [0,1]i tw ∈ .

3. Data

To investigate the behavioral aspects of the forecasts of market participants, we use a

unique database of survey-based exchange rate forecasts. The individual forecasts are

obtained from a survey conducted by Consensus Economics of London on a monthly

basis among leading market participants in the foreign exchange market, investment

banks, and professional forecasting agencies. Examples of panelist companies are

Morgan Stanley, Oxford Economic Forecasting, Deutsche Bank Research, and BNP

Paribas. The panelists companies are located worldwide, although they are all from

developed economies. The forecasts are point forecasts for a large set of currencies

against the U.S. dollar and are available for horizons of 1, 3 and 12 months ahead.

The names of the panelist companies are revealed.

Although survey participants have a few days time to return their forecasts, we

learned that the vast majority send their responses by e-mail on the Friday before the

publication day, which is typically the second Monday of the month. We consider this

Friday to be the day on which the forecasts are formed and assume that the beliefs are

translated one-to-one in a point forecast. To verify that the information sets of market

participants are not too diverse, all of the analyses throughout this study were re-

estimated using spot data from various days surrounding this Friday, yet the overall

results remain virtually unchanged.

There may be reasons for panelists not to reveal their true beliefs, though. One

motive may be that agents do not want to expose their (private) information to other

market participants. This effect may be mitigated by the reputation effect that this

survey can have. When the names of the forecasters are given in the survey

publication (as is the case with our data), agents have an incentive to perform well in

order to attract customers.

All remaining data, i.e. spot rates and macro-economic data are obtained through

Datastream. Inflation is the percentage yearly change in CPI; interest rates are 3-

months interbank rates; income is the yearly change in industrial output; balance of

payments is the ratio of the net balance of payments to GDP. In this study we use the

forecasts for the U.K. pound, Japanese yen and Euro against the U.S. dollar from 31

respondents for the period of November 1995 through December 2004, which are 110

monthly observations.4,5 This period is of particular interest since it contains several

financial crises, the introduction of a single monetary currency unit, and several large

changes in the level of the exchange rates. The panel is unbalanced since the response

rate of the individual market participants is less than 100 percent and since market

participants left the panel and were replaced by others. Analyses are done on both the

3 and 12 months forecasting horizon in order to distinguish between the short- and

long-run; 1 month forecasts are used as a control variable (see Section 2).

< Insert Table 1 Here >

In addition to a constant, ftZ includes the following: 1,

ftz is the interest rate

differential, *t ti i− ; 2,

ftz is the inflation rate differential, *

t tπ − π ; 3,ftz is the growth rate

4 Prior to January 1999 we use forecasts on the Deutschemark versus the U.S. Dollar. We transform these forecasts into Euro / U.S. dollar forecasts using the official conversion rate. 5 UK Pound responses are bi-monthly.

differential, *t tg g− , measured as the yearly percentage increase in industrial

production; and 4,f

tz is the balance of payments differential, *t tx x− . The latter is net

balance of payments surplus as percentage of GDP.

The chartist information ctZ includes a constant and just 1,

ctz , which is the most

recent innovation in the spot rate, 1 1t t t ts s s− −Δ = − .

4. Methodology

Unique to the current examination (to our knowledge) is the fact that different models

under consideration are endogenous to the traders employing them. Branch (2004)

for example, considers three exogenous models of inflation. His naïve expectation

model has 1et t+π = π . The two more sophisticated models are a model of adaptive

expectations and a VAR. In both cases, the parameters of the model are chosen to fit

the data, so that the model is optimized to minimize the error of the forecast of

inflation, rather than to capture the model employed by the forecaster.

Our objective is to have those traders employing the model indicate the

parameters of the model. This is accomplished by choosing the parameters to

minimize the mean squared error of the forecast by those traders who employ the

forecast. This involves some degree of simultaneity as the estimation of the model

depends on how the individuals are sorted and the sorting depends on the model. Our

solution is to estimate, sort and then re-estimate over a number of iterations until the

sorting and the model parameters settle. Similarly, in the continuous choice model,

the weights and the model parameters are interdependent. As in the discrete choice

setup, weights and coefficients are determined through a number of iterations.

Experimentation with different starting points does suggest that there is some path

dependence in the estimation procedure, but not enough to change the implication for

the model.

Formally, the estimation procedure for the discrete choice model is as follows:

The model is estimated in a system of two equations, one equation per group, using

simple OLS. The initial distribution of agents over groups (or initial determination of

weights) is done by estimating the two expectation formation models individually per

respondent. Based on best fit, each respondent is subsequently classified as either

fundamentalist or chartist. There exists a certain path dependency conditional on the

initial distribution of agents. We feel, however, that this procedure yields the best

results in that the fit is maximized and the initial distribution is credible as it is based

on individual estimates. Next, the two rules are estimated in the system, in a pooled

setup, using the initial distribution of respondents. The distribution of respondents

across groups is subsequently updated based on the new estimation results, and the

system is again estimated. This procedure is repeated until convergence, i.e. until

respondents do not change groups anymore and coefficient estimates of the rules are

constant. Generally this occurs within ten iterations, conditional on the complexity of

the model. As such, the classification of agents and the actual expectation formation

rules are being learned endogenously in the iteration process.

5. Results

A benchmark version of the model is estimated without weights or division of

respondents. Both rules are estimated for the full sample of respondents and time. The

results are presented in Table 2.


The estimation results in Table 2 generally indicate that both fundamentalist and

chartist information sources are being used significantly in forming survey

expectations. For the 3-months horizon, we observe that different fundamental

information is used for different currencies; furthermore, information is also used

differently given the different signs. In general though, the interest rate and growth

differentials appear to be most influential. The sign of the growth differential in

Japan, positive, is counterintuitive; this might be due to the a-typical growth pattern

in Japan during the sample, i.e., negative growth. The chartist coefficient β is negative

and highly significant for all countries; this implies that panellists expect a strong

mean reversion. The lagged and the 1-month expectations, finally, are both highly

significant and carry the expected signs. This implies that panellists are depending

heavily on last period’s expectation due to the fact that the sampling frequency is

higher than the forecasting horizon. Also, expectations are being updated consistently

with regards to the 1-month expectation. For both the fundamentalists and the

chartists the model is able to capture a considerable amount of variation in the

expectations; especially so for the Yen and the Euro. The fit for the UK Pound is less

due to the lower sampling frequency.

For the 12-months horizon we observe that fundamental information is more

important. Both the effect sizes and the significance have increased compared to the

short horizon. The fit is also significantly better, but this might also be due to the fact

that the auto-correlation becomes stronger as a result of the 11 months overlap

between consecutive observations. Also the chartist rule is stronger. This increase in

effect sizes in both rules is a result of the fact that the forecasting horizon is longer,

and therefore that the expected variance in the exchange rate is larger.

5.1 Discrete choice estimation

Table 3 presents the results of the model with static discrete weights.

Panellists are classified as either fundamentalist or chartist for the entire period

covered by the survey. The modified model being estimated is

, , , , ,( ) ( ) (1 ) ( )f ci t t t k i t i t t t k i t i t t t kE s E s E s+ + +Δ = θ Δ + − θ Δ

in which each model is estimated separately according to

, 1 , 1 1 1 2 , 1 1 ,

, 1 , 1 1 1 2 , 1 1 ,

( ) ( ' ( ) ( ))

(1 ) ( ) (1 )( ' ( ) ( ))

f f f f fi i t t t k i t i t t t k i t t t i t

c c c c ci i t t t k i t i t t t k i t t t i t

E s Z E s E s

E s Z E s E s+ − − + − − −

+ − − + − − −

θ Δ = θ α + γ Δ + γ Δ + ε

− θ Δ = − θ β + γ Δ + γ Δ + ε (10’)

where

( ) ( )

( ) ( )

2 2

, , , ,1 1

2 2

, , , ,1 1

ˆ ˆ1 if ( ) ( ) ( ) ( )

ˆ ˆ0 if ( ) ( ) ( ) ( )

T Tf c

i t t t k i t t t k i t t t k i t t t kt t

i T Tc f


E s E s E s E s

E s E s E s E s

+ + + += =

+ + + += =

⎧Δ − Δ < Δ − Δ⎪⎪θ = ⎨

⎪ Δ − Δ ≤ Δ − Δ⎪⎩

∑ ∑

∑ ∑ (11’)


In terms of significant fundamental information, we observe a number of

changes relative to the benchmark model for the short horizon. The interest rate loses

its significance for the Euro and the Pound, while the growth rate and the balance of

payments gain significance. The 1-month expectation also loses significance for the

Yen. In general, though, the fit of the fundamentalist rule decreases. The results for

the chartist rule are opposite; both effect sizes and significance levels increase across

the board, and the fit improves as well.

For the long horizon, results are comparable. There are some shifts in the

significance levels of fundamental variables; the interest rate effect though, keeps its

significance and even improves for the Yen. Also for the balance of payments there

are large improvements for all currencies compared to the benchmark model. The fit

for the long horizon fundamental rule increases for all currencies. The changes for the

chartist rule are consistent; increase in effect sizes, significance levels, and model fit.

The percentage of fundamentalists in the survey ranges from 13 to 55 percent.

In other words, the majority of panellists use a chartist rule. Also, the fraction of

fundamentalists is lower at the long horizon compared to the short horizon.


Table 4 reports the results of estimating the original model captured by (10)

and (11) in which panellists update their forecasting strategy each period. Hence,

instead of considering the average distance between the rule and the expectation, as in

(11), the selection procedure is applied per period.

The flexibility substantially improves the fit of the model. For inflation,

growth, and the balance of payments, we find significant results for at least two

currencies, three for growth. The effect sizes of the auto-regressive terms, f1γ ,

decrease. The effect on the model fit differs per currency. For the chartist rule we

again observe an increase in effect sizes and significance levels; the model fit

increases dramatically for all three currencies.

The results for the long horizon are virtually identical compared to Table 3 in

terms of significance. Striking is the fact that the signs of the fundamental variables

change. Effect sizes and levels of significance again increase for the chartist rule. Fit

of the fundamental rule differs per currency while the fit of the chartist rule increases

considerably for all three currencies.

The percentage of panellist using the fundamental rule is generally larger than

in the static case, but still smaller than fifty percent. Fundamentalism is still less

common for the long than the short horizon. The autocorrelation in the chosen rule is

low. This means that strategies are chosen each period, independent of the choice in

the previous period.

5.2 Continuous choice estimation


Table 5 presents the results for the model in which panellists are allowed to

use a fixed weighted average of the two forecasting rules. The modified version of the

model depicted in (12) through (14) has individual predictions determined according

to

, , , , ,( ) ( ) (1 ) ( )f ci t t t k i t i t t t k i t i t t t kE s w E s w E s+ + +Δ = Δ + − Δ

As in the discrete choice version, the two models are estimated independently based

on

, 1 , 1 1 1 2 , 1 1 ,

, 1 , 1 1 1 2 , 1 1 ,

( ) ( ' ( ) ( ))

(1 ) ( ) (1 )( ' ( ) ( ))

f f f f fi i t t t k i t i t t t k i t t t i t

c c c c ci i t t t k i t i t t t k i t t t i t

w E s w Z E s E s

w E s w Z E s E s+ − − + − − −

+ − − + − − −

Δ = α + γ Δ + γ Δ + ε

− Δ = − β + γ Δ + γ Δ + ε (12’)

with

1(1 exp( ))i iw −= + −λ (13’)

2 2, , , ,

1 1

2 2, , , ,

1 1

ˆ ˆ( ( ) ( )) ( ( ) ( ))

ˆ ˆ( ( ) ( )) ( ( ) ( ))

T Tc f


i T Tc f


E s E s E s E s

E s E s E s E s

+ + + += =

+ + + += =

Δ − Δ − Δ − Δλ =

Δ − Δ + Δ − Δ

∑ ∑

∑ ∑ (14’)

The estimation results are highly comparable to those of the benchmark model in

Table 2. Only marginal changes in coefficients and standard errors can be observed.

As such, the fit is also similar, and thus smaller than that of the static discrete case in

Table 3.

The cross-sectional descriptive statistics of the weights indicate that on

average more panellists lean more towards chartism than fundamentalism. The

proportion lies roughly in the range between 0.40 and 0.55; there is not much

variation between individuals, given the relatively low standard deviation.


Table 6, finally, shows the estimation results of the model with flexible

continuous weights. Like in the comparison between Tables 3 and 4, we observe an

increase in significance of the fundamental variables. Especially information on

inflation and the balance of payments become more widely used. Because of the

decrease in both effect size and significance of theγ ’s, the R2’s do not increase. The

chartist rule again gains on effect, significance, and fit. The variance explained,

though, remains considerably lower compared to the dynamic discrete setup in Table

4.

The aggregation of the weights is consistent with what generated from the

fixed weight model. The average weight is somewhat below 50%, and the range is

limited to 0.27 to 0.73. The autocorrelation in weights is always negative, but low.

One can draw a number of conclusions from the estimation results in Tables 2

through 6. First of all, both the fundamentalist and the chartist forecasting rule are

being used by the respondents in the survey. For the fundamentalist rule, especially

relative economic growth 3α is influential in the short horizon, and both the interest

rate differential 1α and relative economic growth 3α for the long horizon. The fit of the

fundamental rules and the changing significance of variables indicates that the used

variables are relevant, but that there is no consensus amongst panellist on “the”

fundamental exchange rate. Also, the auto-correlation in expectations is not so large

that it drives a large part of the results, as it partly does for the chartists. Therefore,

fundamentalists apparently use a more sophisticated forecasting model.

The chartist rule is significant in models for all currencies. Also, it

consistently takes the form of a contrarian strategy; in other words, panellists expect a

reversion of the most recent change in the exchange rate. The chartist rule shows a

very large fit; in other words, chartists use a very simple forecasting rule based on

their past expectation combined with the most recent change in the exchange rate.

The fundamentalist-chartist dichotomy often put forward in the literature is therefore

a very relevant classification, consistent with the findings of, among others, Allen and

Taylor (1990, 1992), and Jongen et al. (2008).

Another interesting finding is that panellists lean heavily on their previous

period’s expectation: 01 >γ . This makes sense as the period over which the

expectation is formed coincides for two (eleven) periods with previous period’s

expectation for the short (long) horizon. Also, panellists’ expectations are consistently

updated relative to previous period’s expectation by subtracting the 1-month

expectation of period t-1; 02 <γ .

The flexibility of agents to change strategy is of great importance. For both

the discrete and the continuous case, there is a substantial improvement in the fit after

introducing switching. This is direct evidence in favour of the heterogeneous agents

models with switching, as introduced in Brock and Hommes (1997, 1998). Another

important finding in this respect is the fact that panellists appear to use one single

strategy instead of a combination of strategies. Panellists are either fundamentalist or

chartist. This shows from the better fit of the model with discrete weights (Table 4)

than the model with continuous weights (Table 6). A final important finding here is

that chartism is dominant. More than half of the panellists are classified as being

chartist. Again, this is consistent with Allen and Taylor (1992), who state that 90% of

market participants use some sort of technical analysis.

In order to gain somewhat more insights into the workings of the model, Table

7 presents the correlations between the different classifications and weights of the

estimated models.


The highest correlations can be found in the cells combining static with static,

and combining dynamic with dynamic. In other words, the models produce consistent

behaviour. Figure 1 illustrates the two forecasting rules together with the

expectations of one of the panellists; it concerns results from the dynamic discrete

model.

< Insert Figure 1 Here >

The figure illustrates a number of interesting issues. Firstly, the chartist rule

follows the actual expectations relatively close. The fundamentalist rule, on the other

hand, is more detached and does not follow the actual expectations. This is a

confirmation of the estimation results of theγ parameters. Clearly as well is the fact

that chartists are destabilizing, while fundamentalists are stabilizing. This shows from

the high volatility in the chartist rule compared to the low volatility in the

fundamentalist rule.

6. Conclusion

A model has been developed to examine the behaviour of banks when forming

forecasts of future exchange rate innovations over a variety of time horizons. The

model allows for market participants to switch between different strategies for

forming expectations. Two model based on two strategies is developed. The two

strategies examined are a fundamental strategy by which predictions concerning

future exchange rates are based on exchange rate fundamentals, and a chartist strategy

by which market based information serves as a predictor of future exchange rates.

The empirical analysis suggests that the switching model is useful for

explaining the heterogeneity in the forecasts of the different banking institutions that

took part in the survey. Allowing the banks to switch strategies during the sample

period improved the fit of the model. It also provides an attractive narrative of

market behaviour that is consistent with stylized facts. The forecasts produced by the

fundamental model are fairly stable, tending to produce predictions of only small

innovations in the exchange rate. Predictions of larger innovations are better captured

by the chartist model.

Allen and Taylor (1990) document the use of chartist techniques among

foreign exchange traders. Individual traders explain that it is not necessarily that they

believe that charting captures fundamentals, but that the market can be driven by

chartists since they are so plentiful in the foreign exchange markets. For this reason,

it is important to include chartist tools when considering trades. Presumably, the

same is true when forming predictions. The fact that bank forecasts appear to be

driven, at times, by a chartist models may be a reflection of the fact that bank believe

that the market based information is informative about market innovations away from

fundamentals. The results could also be considered supportive of the notion that

market based information is useful for predicting fundamental innovations supported

by private information not available to the modeller. The latter interpretation is

consistent with Grossman and Stiglitz (1980) and other papers that argue in favour of

the use of chartists techniques to extract information from the market.

The results raise a number of issues that remain to be examined. Preliminary

examination of the switching behaviour, for example, seems to suggest that there is

little predictability at the individual bank level. It does not, for example, appear to be

tied to past performance, as would be consistent with the body of literature developed

based on the work of Brock and Hommes (1997). Further investigation is clearly

warranted. The present model of behaviour by the exchange rate traders does not

include a model of switching, but one should be developed.

Bibliography

Allen, H. and M.P. Taylor (1990). Charts, Noise and Fundamentals in the London

Foreign Exchange Market, Economic Journal 100(400): 49-59.

Boswijk, H.P., C.H. Hommes and S. Manzan (2007). Behavioral Heterogeneity in

Stock Prices, Journal of Economic Dynamics and control 31: 1938-1970.

Branch, W.A (2004). The Theory of Rational Heterogeneous Expectations: Evidence

from Survey Data on Inflation and Expectations. The Economic Journal 114: 592–621

Brock, W. and C.H. Hommes (1997). A Rational Route to Randomness,

Econometrica 69: 1059-1095.

Brock, W. and C.H. Hommes (1998). Heterogeneous Beliefs and Routes to Chaos in a

Simple Asset Pricing Model, Journal of Economic Dynamics and Control 22: 123-

1274.

Taylor, M.P. and H. Allen (1992). The Use of Technical Analysis in the Foreign

Exchange Market, Journal of International Money and Finance 11(3): 304-314.

Hommes, C.H., Sonnemans, J., Tuinstra, J. and Velden, H. van de, (2007), Learning

in cobweb experiments, Macroeconomic Dynamics 11 (Supplement 1), 8-33.

MacDonald, R. and I.W. Marsh (1996). Currency Forecasters are Heterogeneous:

Confirmation and Consequences, Journal of International Money and Finance 15(5):

665-685.

Goldbaum, D. (2005). Market Efficiency and Learning in an Endogenously Unstable

Environment, Journal of Economic Dynamics and Control 29: 953-978.

De Grauwe, P. and M. Grimaldi (2005). Heterogeneity of Agents, Transaction Costs

and the Exchange Rate, Journal of Economic Dynamics and Control 29: 691-719.

De Grauwe, P. and M. Grimaldi (2006). Exchange Rate Puzzles: A Tale of Switching

Attractors, European Economic Review 50(1): 1-33.

De Grauwe, P. and A. Markiewicz (2008). Learning to Forecast the Exchange Rate:

Two Competing Approaches, CESifo Working Paper 1747.

Grossman, S.J., Stiglitz, J.E., 1980. On the impossibility of informationally efficient

markets. The American Economic Review 70(3), 393-408.

Lux, T. (1998). The Socio-Economic Dynamics of Speculative Markets: Interacting

Agents, Chaos and Fat Tails of Return Distributions, Journal of Economic Behavior

and Organization 33: 143-165.

Manski, C.F., McFadden, D., 1981. Structural Analysis of Discrete Data with

Econometric Applications (MIT Press, Cambridge, MA).

Jongen, R., C.C.P. Wolff, W.F.C. Verschoor, and R.C.J. Zwinkels (2008). Dispersion

of Beliefs in Foreign Exchange, CEPR Discussion Paper 6738.

Tables and figures

Table 1: Data 3 Months 12 Months U.K.

pound Japanese

Yen Euro U.K.

pound Japanese

yen Euro

a) # Observations Min. # panelists / period 14 15 14 14 15 14 Max. # panelists / period 23 23 24 23 24 24 Median # panelists / period 20 20 20 20 20 20 Min. # periods / panelist 6 12 12 6 12 12 Max. # periods / panelist 54 109 108 54 109 108 Median # periods / panelist 35 74 73 35 74 73 b) Descriptive statistics Median -0.0008 0.0003 0.0107 -0.0015 -0.0197 0.0442 Maximum 0.1333 0.3331 0.2036 0.1447 0.4512 0.2883 Minimum -0.1143 -0.1667 -0.1366 -0.2027 -0.2486 -0.2110 Standard deviation 0.0293 0.0464 0.0384 0.0481 0.0848 0.0695 Skewness -0.0062 0.4187 0.2967 -0.3023 0.6163 -0.0720 Kurtosis 4.2531 5.2638 4.0429 3.7750 4.6055 2.9608 Autocorrelation (1st lag) 0.3447 0.6144 0.6217 0.6771 0.8323 0.8578 Notes: Table presents the number of observations per period and per respondent (Panel a) as well as the descriptive statistics of the expected log-changes in the exchange rate, i.e. )ln()ln( 1, ttti ssE −+ over all panellists and periods (Panel b).

Table 2: Benchmark Model JPY/USD EURO/USD USD/UKP JPY/USD EURO/USD USD/UKP 3 months 12 months Fundamentalists

cf 5.13E-05 (0.0055)

0.0028 (0.0022)

0.0034 (0.0025) -0.0099

(0.0071) 0.0090*** (0.0026)

0.0050 (0.0031)

α1 -0.0654 (0.0629)

0.3292*** (0.0988)

0.0873*** (0.030) -0.0500

(0.0809) 0.3538*** (0.1172)

0.1273*** (0.0372)

α2 0.1597 (0.1990)

0.3434 (0.2730)

0.4684* (0.2731) 0.3654

(0.2575) -0.0882 (0.3222)

1.3299*** (0.3479)

α3 0.1186*** (0.0212)

-0.1711*** (0.0363)

0.0136 (0.0365) 0.1076***

(0.0271) -0.1519*** (0.0431)

0.0721* (0.0450)

α4 0.0267 (0.0613)

-0.04100 (0.0392)

0.1159 (0.0891) -0.0430

(0.0790) 0.0123 (0.0461)

0.1792* (0.1102)

f1γ 0.6319***

(0.0311) 0.6528*** (0.0301)

0.5059*** (0.0533) 0.8356***

(0.0147) 0.8517** (0.0137)

0.6785*** (0.0283)

f2γ -0.1189**

(0.0477) -0.1019** (0.0446)

-0.2764*** (0.0763) -0.1161***

(0.0414) -0.0882*** (0.0348)

-0.1709*** (0.0622)

R2 0.3816 0.4014 0.1462 0.6798 0.7384 0.4780 Chartists

cc 0.0001 (0.0007)

0.0048*** (0.0007)

-0.0006 (0.0009) -0.0027***

(0.0009) 0.0066*** (0.0009)

8.62E-05 (0.0011)

β -0.4525*** (0.0214)

-0.2952*** (0.0225)

-0.1654*** (0.0223) -0.6697***

(0.0263) -0.5420*** (0.0246)

-0.3358*** (0.0268)

c1γ 0.6931***

(0.0278) 0.6906*** (0.0282)

0.5085*** (0.0507) 0.8562***

(0.0126) 0.8747*** (0.0116)

0.7493*** (0.0251)

c2γ -0.1447***

(0.0430) -0.1638*** (0.0430)

-0.2478*** (0.0735) -0.1060***

(0.0346) -0.1698*** (0.0314)

-0.1727*** (0.0580)

R2 0.4908 0.4433 0.1815 0.7603 0.7895 0.5362 Notes: Table presents estimation results for the benchmark model. R2 is adjusted R2; standard errors in parenthesis

Table 3: Static Discrete Weights JPY/USD EURO/USD USD/UKP JPY/USD EURO/USD USD/UKP 3 months 12 months Fundamentalists

cf -4.18E-03 (0.0049)

0.0048** (0.0022)

0.0035 (0.0023) 0.0324***

(0.0112) 0.0094*** (0.0028)

0.0115*** (0.0030)

α1

0.0260 (0.0564)

0.0356 (0.1005)

0.0287 (0.0272) -0.5460***

(0.1016) 0.4299*** (0.1253)

0.3217*** (0.0400)

α2

0.2623 (0.1711)

0.6268** (0.2687)

0.2395 (0.2394) 1.1244***

(0.4084) 0.2713 (0.3291)

-0.2164 (0.3511)

α3

0.0689*** (0.0187)

-0.111*** (0.0352)

-0.1177*** (0.0346) 0.2523***

(0.0406) -0.2566*** (0.0460)

0.0206 (0.0464)

α4

0.0077 (0.0553)

0.0755* (0.0407)

-0.0640 (0.0824) 0.5455***

(0.1358) 0.1527*** (0.0505)

0.3836*** (0.1050)

f1γ 0.5505***

(0.0287) 0.6104*** (0.0302)

0.4315*** (0.0524) 0.8168***

(0.0179) 0.8157*** (0.0163)

0.6113*** (0.0274)

f2γ 0.0328

(0.0490) -0.1679*** (0.0463)

-0.3183*** (0.0839) -0.5615***

(0.0416) -0.2968*** (0.0368)

-0.2088*** (0.0706)

R2 0.3640 0.3232 0.1470 0.6534 0.7518 0.4943

Chartists cc -0.0006

(0.0007) 0.0069*** (0.0007)

-0.0002 (0.0009) -0.0024***

(0.0009) 0.0079*** (0.0008)

0.0015 (0.0010)

β -0.5498*** (0.0207)

-0.4611*** (0.0207)

-0.4435*** (0.0269) -0.6905***

(0.0247) -0.6252*** (0.0225)

-0.5358*** (0.0298)

c1γ 0.7293***

(0.0269) 0.7320*** (0.0259)

0.6458*** (0.0482) 0.8599***

(0.0119) 0.8761*** (0.0107)

0.7658*** (0.0238)

c2γ -0.1882***

(0.0406) -0.1876*** (0.0387)

-0.3639*** (0.0638) -0.0772**

(0.0332) -0.1453*** (0.0285)

-0.1893*** (0.0534)

R2 0.5542 0.5965 0.3547 0.7849 0.8358 0.6061 % fun 0.2258 0.4839 0.5517 0.1290 0.2258 0.2414 Notes: Table presents estimation results for the model with static discrete weights. R2 is adjusted R2; standard errors in parenthesis; % fun is the percentage of panellists using the fundamentalist rule.

Table 4: Dynamic Discrete Weights JPY/USD EURO/USD USD/UKP JPY/USD EURO/USD USD/UKP 3 months 12 months Fundamentalists

cf (0.0158)*** (0.0039)

-0.0045*** (0.0017)

0.0034* (0.0019) 0.0807***

(0.0066) 0.0008 (0.0022)

0.0174*** (0.0023)

α1

0.0074 (0.0433)

-0.0690 (0.0826)

0.2349*** (0.0226) -1.0340***

(0.0696) -0.2200** (0.1004)

0.4054*** (0.0310)

α2

-0.1985 (0.1310)

1.7549*** (0.2256)

0.9652*** (0.2028) 4.4184***

(0.2054) -0.3975 (0.2822)

0.2712 (0.2573)

α3

0.1444*** (0.0147)

-0.0532* (0.0287)

-0.0722*** (0.0276) 0.5074***

(0.0245) 0.1123*** (0.0370)

-0.0025 (0.0345)

α4

0.2291*** (0.0432)

-0.0870*** (0.0327)

-0.0320 (0.0667) 1.6339***

(0.0775) -0.0640* (0.0391)

0.1825** (0.0818)

f1γ 0.2840***

(0.0217) 0.4004*** (0.0232)

-0.2128*** (0.0412) 0.1509***

(0.0138) 0.7427*** (0.0111)

0.4603*** (0.0201)

f2γ -0.2101***

(0.0341) -0.2249*** (0.0342)

0.0156 (0.0569) -0.2220***

(0.0327) -0.5415*** (0.0289)

-0.3253*** (0.0469)

R2 0.2793 0.3125 0.2720 0.4272 0.7497 0.5175

Chartists cc -0.0010**

(0.0004) 0.0032*** (0.0004)

0.0047*** (0.0005) -0.0012**

(0.0005) 0.0052*** (0.0005)

-0.0019*** (0.0006)

β -0.8893*** (0.0127)

-0.8768*** (0.0137)

-0.5455*** (0.0168) -0.8634***

(0.0152) -0.9541*** (0.0143)

-0.8750*** (0.0183)

c1γ 1.0430***

(0.0161) 1.0274*** (0.0165)

1.0182*** (0.0310) 0.9730***

(0.0071) 0.9369*** (0.0067)

1.0104*** (0.0154)

c2γ -0.1387***

(0.0243) -0.1548*** (0.0252)

-0.42217*** (0.0460) -0.0885***

(0.0201) 0.0327* (0.0178)

-0.1791*** (0.0345)

R2 0.8790 0.8828 0.7607 0.9324 0.9493 0.8781 % fun 0.3981 0.4697 0.4614 0.2053 0.3346 0.4585

AC -0.0170 -0.0290 -0.0140 0.0320 0.040 -0.0060 Notes: Table presents estimation results for the model with dynamic discrete weights. R2 is adjusted R2; standard errors in parenthesis; % fun is the percentage of fundamentalist panellists; AC is the autocorrelation in the fundamentalist/chartist classification.

Table 5: Static Continuous Weights JPY/USD EURO/USD USD/UKP JPY/USD EURO/USD USD/UKP 3 months 12 months Fundamentalists

cf -0.0017 (0.0055)

0.0037* (0.0022)

0.0036 0.0025 -0.0098

(0.0072) 0.0091*** (0.0025)

0.0051 (0.0031)

α1

-0.0430 (0.0625)

0.2731*** (0.0994)

0.0839*** (0.0301) -0.0510

(0.0808) 0.3196*** (0.1168)

0.1329*** (0.0375)

α2

0.0464 (0.1995)

0.4161 (0.2735)

0.4195 (0.2751) 0.3511

(0.2606) -0.0642 (0.3200)

1.2238*** (0.3515)

α3

0.1147*** (0.0213)

-0.1579*** (0.0364)

0.0030 (0.0367) 0.1116***

(0.0274) -0.1436*** (0.0429)

0.0648 (0.0455)

α4

-0.0053 (0.0620)

-0.0170 (0.0397)

0.1163 (0.0902) -0.0430

(0.0808) 0.0329 (0.0460)

0.1858* (0.1116)

f1γ 0.6472***

(0.0307) 0.6570*** (0.0300)

0.5125*** (0.0535) 0.8391***

(0.0145) 0.8542*** (0.0135)

0.6823*** (0.0284)

f2γ -0.1377***

(0.0472) -0.1124** (0.0446)

-0.2824*** (0.0772) -0.1271***

(0.0414) -0.1008*** (0.0346)

-0.1800*** (0.0625)

R2 0.3919 0.3992 0.1496 0.6865 0.7414 0.4816 Chartists

cc -0.0002 (0.0007)

0.0051*** (0.0007)

6.31E-05 (0.0009) -0.0028***

(0.0009) 0.0069*** (0.0008)

0.0014 (0.0010)

β -0.4703*** (0.0208)

-0.3071*** (0.0224)

-0.2682*** (0.0264) -0.6839***

(0.0251) -0.5593*** (0.0238)

-0.5051*** (0.0305)

c1γ 0.7094***

(0.0276) 0.6946*** (0.0282)

0.5211*** (0.0506) 0.8593***

(0.0123) 0.8790*** (0.0113)

0.7536*** (0.0242)

c2γ -0.1601***

(0.0425) -0.1688*** (0.0428)

-0.2741*** (0.0726) -0.1070***

(0.0335) -0.1732*** (0.0303)

-0.1939*** (0.0556)

R2 0.5137 0.4538 0.2214 0.7738 0.8043 0.5832 % fun 0.4717 0.4923 0.4939 0.4596 0.4729 0.4846 max 0.5422 0.5448 0.5424 0.5328 0.5359 0.5535 min 0.3440 0.4304 0.4279 0.3245 0.4099 0.3972

st.dev. 0.0390 0.0246 0.0264 0.0388 0.0344 0.0329 Notes: Table presents estimation results for the model with static continuous weights. R2 is adjusted R2; standard errors in parenthesis; % fun is the percentage of fundamentalist panellists; max, min, st.dev. is the maximum, minimum, standard deviation of the individual weights, respectively.

Table 6: Dynamic Continuous Weights JPY/USD EURO/USD USD/UKP JPY/USD EURO/USD USD/UKP 3 months 12 months Fundamentalists

cf 0.0058 (0.0047)

-0.0017 (0.0020)

0.0037* (0.0022) -0.0065

(0.0068) 0.0069*** (0.0024)

0.0055** (0.0027)

α1

-0.0450 (0.0536)

0.2715*** (0.0913)

0.1638*** (0.0263) -0.1210

(0.0760) 0.2384** (0.1080)

0.2207*** (0.0335)

α2

0.1742 (0.1691)

0.9848*** (0.2515)

0.5365** (0.2429) 1.1666***

(0.2393) -0.0553 (0.3011)

1.4131*** (0.3012)

α3

0.1653*** (0.0183)

-0.1853*** (0.0325)

-0.0418 (0.0329) 0.1869***

(0.0258) -0.1111*** (0.0405)

0.0619 (0.0396)

α4

0.1398*** (0.0529)

-0.1050*** (0.0366)

0.0198 (0.0783) 0.1680**

(0.076) -0.0340 (0.0420)

0.0803 (0.0954)

f1γ 0.5053***

(0.0269) 0.5641*** (0.0264)

0.1641*** (0.0476) 0.7016***

(0.0140) 0.8002*** (0.0124)

0.5613*** (0.0238)

f2γ -0.2284***

(0.0420) -0.2590*** (0.0389)

-0.0940 (0.0677) -0.1670***

(0.0384) -0.2595*** (0.0308)

-0.2217*** (0.0548)

R2 0.3603 0.3412 0.1099 0.6400 0.7404 0.4919 Chartists

cc -0.0006 (0.0006)

0.0045*** (0.0006)

0.0017** (0.000) -0.0015**

(0.0007) 0.0051*** (0.0007)

0.0018** (0.0009)

β -0.7420*** (0.0170)

-0.5740*** (0.0196)

-0.4443*** (0.0224) -0.9116***

(0.0194) -0.8154*** (0.0192)

-0.7296*** (0.0249)

c1γ 0.8543***

(0.0223) 0.7861*** (0.0251)

0.7962*** (0.0432) 0.9377***

(0.0094) 0.9278*** (0.0093)

0.8705*** (0.0206)

c2γ -0.1164***

(0.0337) -0.0617* (0.0382)

-0.4001*** (0.0630) -0.0785***

(0.0262) -0.0835*** (0.0252)

-0.1639*** (0.0464)

R2 0.7318 0.6642 0.5033 0.8810 0.8888 0.7455 % fun. 0.4646 0.4780 0.4846 0.4205 0.4368 0.4729 max 0.7311 0.7311 0.7311 0.7310 0.7311 0.7311 min 0.2689 0.2689 0.2689 0.2689 0.2689 0.2689

st.dev. 0.1657 0.1592 0.1553 0.1601 0.1621 0.1599 AC -0.0120 -0.0130 -0.0590 -0.0030 -0.0030 -0.0450

Notes: Table presents estimation results for the model with dynamic continuous weights. R2 is adjusted R2; standard errors in parenthesis; % fun is the percentage of fundamentalist panellists; max, min, st.dev. is the maximum, minimum, standard deviation of the individual weights, respectively. AC is the auto-correlation in the weights.

-.2

-.1

.0

.1

.2

.30

1

10 20 30 40 50 60 70 80 90 100 110Expectation ChartistWeight (1=fun) Fundamentalist

Table 7: Correlations Weights static discrete static continuous dynamic discrete Yen Euro Pound Yen Euro Pound Yen Euro Pound

a) 3 months static continuous 0.701 0.790 0.661 dynamic discrete 0.074 0.131 0.079 0.103 0.117 0.133

dynamic continuous 0.113 0.138 0.091 0.150 0.125 0.126 0.779 0.769 0.778

b) 12 months static continuous 0.422 0.765 0.665 dynamic discrete 0.024 0.100 0.083 0.095 0.129 0.107

dynamic continuous 0.052 0.108 0.075 0.099 0.126 0.106 0.525 0.742 0.730 Notes: Table presents correlations between the classifications and weights of the different models. Figure 1: Panellist #1

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