Download An Empirical Examination of Heterogeneity and Switching in Foreign

An Empirical Examination of Heterogeneity and Switching
in Foreign Exchange Markets*
David Goldbaum†
Remco C.J. Zwinkels‡
March 2010
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. Switching,
however, does not occur between fundamentalism and chartism but
between a naïve and a sophisticated model that consists of both
fundamentalist and chartist elements. The sophisticated model is
increasingly used as the forecast horizon increases.
Keywords: Heterogeneity, Discrete Choice, Foreign Exchange
JEL Codes: F31, C35
*
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. The paper was written in part while the second author was
staying at the University of Technology Sydney, whose hospitality he gratefully acknowledges. The
authors wish to thank Adrian Pagan, Paul de Grauwe, as well as participants of seminars at the
University of Amsterdam, University of Technology Sydney, the second conference on heterogeneous
agents in financial markets in Rotterdam, the 2009 SNDE meeting in Atlanta, …...
†
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
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; for example Brock and Hommes (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
Goldbaum (2003) and 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, and their presence in the market is based on
their ability to extract information from the price.
Dynamics arise as traders switch between available information strategies or
levels of sophistication. The agents of the Brock and Hommes (1997) and Brock and
Hommes (1998) models consider the relative performance of different forecasting
strategies. The models employ the random element in the discrete choice model of
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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
divergent 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. Included among these is Boswijk,
Hommes, and Manzan (2007) that 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. De Jong et al. (2009) present empirical
evidence of behavioral heterogeneity in equity prices using a multi-market setting.
Frijns et al. (2010) illustrate the importance of allowing a multitude of strategies in
pricing options.
-3-
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.
A number of issues remain unresolved or are in need of empirical support.
The current project also seeks to examine markets for evidence of adaptive
heterogeneity and also to determine whether there is evidence in favor of the
fundamentalist – chartist dichotomy in foreign exchange markets. 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.
The evaluation offers the possibility of explaining market deviations as they
result from disequilibrium produced by competitive profit seeking in an environment
of imperfect information and adaptation.
The remainder of the paper is organized as follows. Section 2 presents the
underlying model. Section 3 introduces the survey data used in the empirical section
-4-
and Section 4 explains the empirical methodology applied. In Section 5 we present the
results, and Section 6 concludes.
2. Model
Each of N traders in a market maximize an expected negative exponential utility
function in next period’s wealth, Wt 1 , based on their information set, I ti . Formally,
max E (U (Wi ,t 1 ) | I ti )
at ,bt
subject to
Wi ,t  ai.t st  bi.t
Wi ,t 1  (1  rt* ) st 1 ai.t  (1  rt )bi.t
where U (Wt )   exp(Wt ) , st is the spot exchange rate, rt is the domestic interest
rate, and rt* is the foreign exchange rate. Solving produces the optimal demand for
the foreign currency,
(1  rt* ) E ( st 1 | I ti )  (1  rt ) st
ai.t 
i2,t
(1)
where i2,t  (1  rt* ) 2 var( st 1 | I ti ) .
A market clearing Walrasian equilibrium requires supply equals demand,
N
a
i 1
Let xt  X t / N and at 
1
N
i ,t
 Xt .
(2)
N
a
i 1
i ,t
be the per capita supply and demand respectively
for the foreign currency so that (2) can be expressed as at  xt .
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.
-5-
2.1 Fundamental model
There is a fundamental exchange rate, st* . The realized market spot rate, st , 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,
Et f (st 1 )  ( st  st* ) .
(3)
Here,  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. Their traders
know the st 1 and st*1 as the most recently observed values of the spot and
fundamental rates
Capturing the forecast reports employed in the empirical section requires
modeling the k period ahead forecast of exchange rates. Let Z t f represent the vector
of time t fundamental information.
Further, let t st  k represents spot market
innovation st  k  st . For integer k  1 , the fundamental forecast is captured by the
following process:
Et f ( t st  k )   ' Z t f  1f Et 1 ( t 1 st  k 1 )   2f Et 1 ( t 1 st ) .
(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 t  1 and the current period t
forecast. The third term controls for information in Et 1 ( t 1 st  k 1 ) that is not useful
in forecasting Et f ( t st  k ) since st has already materialized such that potentially
useful information in it is incorporated into Z t f ; this reduces noise and increases the
usefulness in employing Et 1 ( t 1 st  k 1 ) 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
-6-
forecast period, Et ( t  k 1 st  k ) as well as any change in the forecast of t st  k 1 that
flows from the new time t information.
Individual trader fundamental forecasts are captured by the following:
Ei ,ft ( t st  k )   ' Z t f  1f Ei ,t 1 ( t 1 st  k 1 )   2f Ei ,t 1 ( t 1 st )  i ,t
(5)
There are thus two sources for heterogeneity among the fundamental traders. The
idiosyncratic term, i ,t , 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 ,t 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 Z tc represent the vector of time t chartist
information. The chartist forecast is captured by the following process:
Etc ( t st  k )   ' Z tc  1c Et 1 ( t 1 st  k 1 )   c2 Et 1 ( t 1 st )
(6)
Individual forecasts are captured by
Eic,t ( t st  k )   ' Z tc  1c Ei ,t 1 ( t 1 st  k 1 )   c2 Ei ,t 1 ( t 1 st )  i ,t ,
(7)
thereby capturing the same sources of heterogeneity that exists among the
fundamentalists.
2.3 Discrete choice
-7-
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.
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 i ,t  1 if the forecast by individual i is deemed to
originate from the fundamental strategy, with i ,t  0 otherwise. Further, let
Eˆ i ,ft ( t st  k )   ' Z t f  1f Ei ,t 1 ( t 1 st  k 1 )   2f Ei ,t 1 ( t 1 st )
(8)
Eˆ ic,t ( t st  k )   ' Z tc  1c Ei ,t 1 ( t 1 st  k 1 )   c2 Ei ,t 1 ( t 1 st )
(9)
and
represent the systematic components of each model. Thus,
Ei ,t ( t st  k )  i ,t ( ' Z t f  1f Ei ,t 1 ( t 1 st  k 1 )   2f Ei ,t 1 ( t 1 st ))
(1  i ,t )( ' Z tc  1c Ei ,t 1 ( t 1 st  k 1 )   c2 Ei ,t 1 ( t 1 st ))  i ,t
,
(10)
where


 
 


1 if E ( s )  Eˆ f ( s ) 2  E ( s )  Eˆ c ( s ) 2
i ,t t
t k
i ,t t
t k
i ,t t
t k
i ,t t
t k

.
i ,t  
2
2
0 if Ei ,t ( t st  k )  Eˆ ic,t ( t st  k )  Ei ,t ( t st  k )  Eˆ i ,ft ( t st  k )

(11)
3. Data
3.1 Survey Expectations
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
-8-
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 reestimated 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 transactions.
In this study we use the forecasts for the Japanese Yen and the Euro4 against the
U.S. dollar (i.e., in foreign currency per US Dollar) from 31 respondents for the
period of November 1995 through December 2007, which are 146 monthly
observations.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 the 1, 3,
and 12 months forecasting horizon in order to distinguish between the short-, middle-,
and long-run. The 1 month forecasts are also used as a control variable for the models
of the 3 and 12 months horizon (see Section 2).
4
Our database also contains U.K. Pound expectations. However, because due to to unknown reasons
the U.K. Pound expectations are only reported every other month, we do not use this currency because
of the limited number of observations.
5
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.
-9-
< Insert Table 1 Here >
Table 1 presents the descriptive statistics of the survey data. Respondents are
consistent across currencies and forecast horizons in answering the survey. The
median response rate per period is 70%, which results in an average total number of
observations per currency/horizon pair of approximately 2900.
The descriptive statistics of the expected exchange rate returns in panel b)
indicate there is a wide variety in answers, ranging from -30 to +45%. The kurtosis
indicates that, as is the case for market returns, the distribution of expectations is
heavy tailed. The expected returns are heavy auto correlated, which is inconsistent
with actual FOREX returns. Partly, the autocorrelation in expectations is due to
overlapping observations, i.e., the frequency of the data is higher than the forecast
horizon. The autocorrelations for the one-month forecast horizon, however, show that
this is not the complete story.
3.2 Forecasting Rules
All exogenous data is retrieved from Datastream. The chartist information Z tc
includes a constant and z1,c t , which is the most recent innovation in the spot rate,
t 1
st  st  st 1 .
The fundamentalist information Z t f includes a constant and a term representing
the partial mean reversion towards the fundamental value,  ( st  st* ) . Important
subsequent question is the choice of fundamental exchange rate st* . The issue here is
not necessarily to select a model that actually performs well in forecasting the
exchange rate, but that is perceived by the panellist to have a certain added value in
forecasting the exchange rate. In addition, the model should be such that the panellists
could have reasonably been able implement it practically. As such, we use a version
of the monetary model introduced by Mark (1995) given by
s t*  (mt  mt* )  ( y t  y t* ) ,
- 10 -
(12)
in which mt is the home money supply, mt* the foreign money supply, yt the home
income, and y t* the foreign income. The choice for this model is based on two
arguments. First, the study by Mark (1995) is one of the most well known and
persuasive studies illustrating the forecasting power of a fundamental model. Second,
this fundamental value is relatively simple to implement; that is, it does not require
any further estimation. This makes sure that we do not have to make any additional
choices or assumptions regarding the estimation process that could take us (further)
away from the fundamental value actually applied by panellists.
Data wise, we use M2 for money supply m and industrial production for
income y. Figure 1 displays the (log) exchange rates and fundamental exchange rates
for both the Yen and the Euro.
< Insert Figure 1 Here >
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 te1  t . 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.
Formally, the estimation procedure for the discrete choice model is as follows:
The model is estimated in a single equation 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
- 11 -
a certain path dependency conditional on the initial distribution of agents. We feel,
however, that this procedure yields the best results in that the final fit is maximized
and the initial distribution is credible as it is based on individual estimates.
Throughout, we perform a grid search to find the global optimum; we report the
solution with the best fit. Next, the two rules are estimated in a single equation 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. The autocorrelation in the residuals due to the overlapping data issue is
accounted for by calculating Newey-West standard errors, as in MacDonald (2000).
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.
< Insert Table 2 Here >
The estimation results in Table 2 generally indicate that both fundamentalist
and chartist information sources are being used significantly in forming survey
expectations. The model is able to capture a considerable amount of variation in the
expectations. The fit increases notably on the longer forecast horizons.
The fundamentalists act stabilizing vis-à-vis the fundamental exchange rate in
all but one cases; α<0. In three cases, this effect is significant. For the Euro, the mean
reversion clearly increases with the forecast horizon. For the Yen this is the case for
the 1 and 3 months horizon, but not for the 12 months horizon. The chartist
coefficient β is negative and highly significant for all currency-horizon combinations;
this implies that panellists expect a strong reversion of previous returns and therefore
act like contrarians. The reversion increases on the longer horizons. This is consistent
- 12 -
with the literature on survey expectations; see MacDonald (2000) for an overview6.
The lagged and the 1-month expectations, γ1 and γ2, 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 materialized previous 1-month expectation.
5.1 Discrete choice estimation
Table 3 presents the results of the model with static discrete weights. That is,
panellists are classified as either fundamentalist or chartist for the entire period
covered by the survey. The modified model being estimated is
Ei ,t ( t st  k )   i ( ' Z t f   1f Ei ,t 1 ( t 1 st  k 1 )   2f Ei ,t 1 ( t 1 st ))
(10’)
 (1   i )(  ' Z tc   1c Ei ,t 1 ( t 1 st  k 1 )   2c Ei ,t 1 ( t 1 s t ))   i ,t
where
T
T
2
2

f
ˆ
ˆ c ( s )
1
if
(
)
(
)
(
)
E
s
E
s
E
s
E








i
t
t
t
k
i
t
t
t
k
i
t
t
t
k
i
t
t
t
k
,
,
,
,






t 1
t 1
i  
(11’)
T
T
2
2
c
f
0 if
ˆ
ˆ
Ei ,t ( t st  k )  Ei ,t ( t st  k )   Ei ,t ( t st  k )  Ei ,t ( t st  k )


t 1
t 1








< Insert Table 3 Here >
Comparing the results from Table 3 with those of the benchmark model in
Table 2, there are a number of differences. First, with regards to the fundamentalists’
mean reversion, we observe that the effect size becomes notably stronger. The
significance level, though, does not gain much. As for the chartist coefficient β, the
effect size is also stronger compared to the benchmark model. The significance levels
are equally high. The estimates for the lagged expectations, finally, remain
comparable;  1f is typically somewhat smaller than  1c , while  2f is typically
somewhat larger than  2c . Strikingly, the fit of the model has hardly improved after
6
On shorter horizons, expectations are found to be stabilizing or exhibit bandwagon effects.
Apparently, the 1-month horizon is perceived to be the long run.
- 13 -
introducing static weights. The increase in R2 does not weigh up to the increased
number of estimated coefficients. The fit of the model for the 3 months Euro forecast
even decreases substantially. A final interesting observation is that the percentage of
panellists using the fundamentalist rule actually decreases as the forecast horizon
increases; from 35 to 13% for the Euro and from 21 to 5% for the Yen.
< Insert Table 4 Here >
Table 4 reports the results of estimating the original model captured by (10)
and (11) in which panellists are allowed to update their forecasting strategy each
period. Hence, instead of considering the average distance between the rule and the
expectation over the full sample period, as in (11’) and Table 3, the selection
procedure is applied per period.
The increased flexibility changes the estimation results substantially. The
fundamentalist mean reversion looses its significance for the Euro. For the Yen, on
the other hand, we observe an increase in significance and effect size for both the 3
and the 12 months horizon. The chartist contrarian behaviour becomes stronger for
both currencies and all horizons. The effect sizes of the auto-regressive terms for
fundamentalists,  1f and  2f , decrease substantially compared to the static setup in
Table 3. The autoregressive term for the 1 month Yen forecast also loses its
significance. For chartists, the opposite occurs and the effect sizes of the
autoregressive terms increase substantially.
The fit of the model is substantially higher in the dynamic case compared to
the static case. Whilst the model fit did not change when moving from the benchmark
model to the model with static classification, the added flexibility of having dynamic
classification does matter notably.
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.
- 14 -
One can draw a number of conclusions from the estimation results in Tables 2
through 4. First of all, both the fundamentalist and the chartist forecasting rule are
being used by the respondents in the survey. The fundamentalist-chartist dichotomy
often put forward in the literature therefore appears a 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,  1  0 . This makes sense as the period over which the
expectation is formed coincides for two and eleven periods with previous period’s
expectation for the 3 and 12 months forecast horizons. The same pattern, however, is
found for the 1-month forecast horizon where the overlapping data issue is not
present. Also, panellists’ expectations are consistently updated relative to previous
period’s expectation by subtracting the 1-month expectation of period t-1;  2  0 .
The flexibility of agents to change strategy is of great importance. 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). 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.
5.2 Robustness
Two results regarding the estimated behaviour of fundamentalists from the three
different models are puzzling. First, the estimated α is typically negative, but the size
and significance changes strikingly. This is especially the case for the Euro, where we
find no significant α. Second remarkable observation is the fact that the percentage of
fundamentalists decreases as the forecast horizon increases. This is inconsistent with
the existing literature.
In order to look deeper into the behaviour of the panellists, we proceed along
two lines. First, to ascertain ourselves of the validity of the estimation procedure we
simulate data and apply the estimation procedure on the generated numbers. Second,
to check the functional forms of the different groups, we estimate the different models
on the different groups separately.
- 15 -
The simulation model is calibrated to mirror the benchmark model of the 1month Yen forecasts. That is, we use the estimate coefficients from Table 2 and the
mean and variance characteristics of the Japanese fundamental exchange rate, the
market exchange rate, and the residuals from the benchmark model to simulate the
s ) given by
exchange rate expectations E ( ~
t t
t 1
Et ( t ~
st 1 )  c f   ( st  s t* )  s t  Et 1 ( t 1 ~
st )   t .
(13)
The model is simulated for 1,000 periods and 15 individuals.
Subsequently, we estimate the model given by (10) and (11) using the data
generated by (13). Because the data generating process is static, i.e., has no switching
between rules, the econometric model is misspecified. Comparing the estimation
results of the simulated data with those of the actual data gives a clue about the true
data generating process underlying the survey expectations.
Table 5 presents the estimation results of the models applied to simulated
data. We do not only show the benchmark and switching models, but also the
fundamental, chartist, and benchmark models estimated for the separate groups of
fundamentalists and chartists.
< Insert Table 5 Here >
The first column representing the benchmark model is highly comparable to
the estimation results of the 1 month Yen forecast in Table 2. This indicates that the
simulations are correct. The second column subsequently presents the estimation
results of the full switching model. As was the case for the actual 1-month Yen
forecasts, the fit of the model increases substantially. Clearly different compared to
the results for the actual data, is that α gains substantially in terms of significance
compared to the benchmark model. Also, the γ2’s of the two groups are equal and not
significantly different from the estimated value for the benchmark model. The latter
two results are clear indications that the actual data is generated by a non-static data
generating process.
Estimating the fundamental, chartist, and benchmark models for the separate
groups of fundamentalists and chartists, displayed in columns 3 to 8, yields very
- 16 -
similar results. That is, if we estimate the fundamental model for either the group of
fundamentalists or the group of chartists (columns 3 and 4), gives the same parameter
estimates. This is exactly as it should be, reflecting the simulated model. The same
conclusion holds for the chartist model and the benchmark model.
< Insert Table 6 Here >
Performing the same analysis on the actual 1-month Yen forecasts gives a
completely different picture. The first two columns of Table 6 give the estimation
results of the fundamental model applied to the group of fundamentalists and
chartists. First of all, the coefficients are not equal across groups, indicating that the
selection procedure does not result in spurious results.
Interestingly, though, it is not the fundamentalists for who we find a
significant α, but the chartists. Also striking is the large difference between the groups
in the estimated γ2 and the model fit. The same conclusions can be drawn after
estimating the chartist model for the groups of fundamentalists and chartists
separately, as presented in columns 3 and 4. The estimated β is significant for the
chartists, and insignificant for fundamentalists. Again, γ2 is much smaller for
fundamentalists. The final two columns, representing the benchmark model estimated
for the two groups, basically combine the aforementioned results: significance and a
high fit for chartists, not for fundamentalists.
The final conclusion to be drawn from these analyses is twofold. First of all, we have
ascertained ourselves of the fact that the selection and classification algorithm we
apply does not yield spurious results. Second, the panellists in the survey do apply
two different models, but they are not a fundamentalist and a chartist model. Instead,
they are a naïve or random walk model and a sophisticated model, consisting of both
fundamentalist and chartist characteristics. This directly explains the two puzzling
results from Table 4: i) The size and significance of α varies greatly because it is
estimated for the wrong group of panellists, and ii) the percentage of panellists using
the fundamentalist model decreases as the forecast horizon increases because the
naïve model is used predominantly for the shorter forecast horizons, and the
sophisticated model is used increasingly as the forecast horizon increases.
- 17 -
7. 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 switching, however, does
not occur between the fundamental model and the chartist model, but between a naïve
model and a sophisticated model consisting of both fundamental and chartist
elements.
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.
- 18 -
Bibliography
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Jongen, R., C.C.P. Wolff, W.F.C. Verschoor, and R.C.J. Zwinkels (2008). Dispersion
of
Beliefs
in
Foreign
Exchange,
- 20 -
CEPR
Discussion
Paper
6738.
Tables and figures
Table 1: Data
1 Month
Euro
3 Months
12 Months
1 Month
Yen
3 Months
12 Months
a) # Observations
Min. # panelists / period
Max. # panelists / period
Median # panelists / period
12
24
19
14
24
20
14
24
20
13
24
19
15
24
20
15
24
20
Min. # periods / panelist
Max. # periods / panelist
Median # periods / panelist
3
143
101
12
144
102
12
144
102
3
143
102
12
144
103
12
144
103
2825
2932
2930
2835
2941
2940
Total # observations
b) Descriptive statistics
Median
-0.0036
-0.0118
-0.0397
-0.0030
-0.0060
-0.0343
Maximum
0.1136
0.1366
0.2110
0.1985
0.3331
0.4512
Minimum
-0.1700
-0.2036
-0.2883
-0.1667
-0.1667
-0.3050
Standard deviation
0.0246
0.0370
0.0686
0.0301
0.0473
0.0857
Skewness
-0.3430
-0.2080
0.0565
0.3216
0.3332
0.5532
Kurtosis
5.7290
3.9952
2.8560
6.1241
4.7625
4.3544
Autocorrelation (1st lag)
0.3900
0.5090
0.7550
0.4170
0.5710
0.7410
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( E i ,t s t 1 )  ln(s t ) over
all panellists and periods (Panel b).
- 21 -
Table 2: Benchmark Model
c
α
β
1 Month
-0.0033***
(0.0004)
Euro
3 Months
-0.0052***
(0.0006)
12 Months
-0.0058***
(0.0007)
1 Month
-0.0020***
(0.0005)
Yen
3 Months
-0.0028***
(0.0006)
12 Months
-0.0042***
(0.0008)
0.0012
(0.0029)
-0.2468***
(0.0151)
-0.0028
(0.0038)
-0.3245***
(0.0195)
-0.0147***
(0.0041)
-0.5807***
(0.0204)
-0.0079*
(0.0042)
-0.3560***
(0.0141)
-0.0099*
(0.0057)
-0.4720***
(0.0189)
-0.0061
(0.0069)
-0.6862***
(0.0223)
0.7063***
(0.0249)
-0.1982***
(0.0372)
0.8835***
(0.0098)
-0.2055***
(0.0254)
0.7383***
(0.0244)
-0.1556***
(0.0378)
0.8785***
(0.0106)
-0.113***
(0.0283)
1
NA
2
0.4280***
(0.0175)
NA
0.4728***
(0.0158)
0.2747
0.4509
0.8198
0.3938
0.5459
0.8026
adj. R2
# Obs.
2494
2515
2512
2510
2528
2527
Notes: Table presents estimation results for the benchmark model. Newey-West standard errors are
given in parentheses. *, **, and *** represents significance at the 10, 5, and 1% level, respectively.
- 22 -
Table 3: Static Weights
1 Month
Fundamentalists
-0.0009
cf
(0.0007)
-0.0039
α
(0.0048)
 1f

f
2
NA
0.4036***
(0.0310)
Euro
3 Months
12 Months
1 Month
-0.0013
(0.0009)
-0.0100*
(0.0063)
0.6263***
(0.0413)
-0.2215***
(0.0639)
0.0009
(0.0016)
-0.0205*
(0.0113)
0.8700***
(0.0270)
-0.3543***
(0.0676)
-0.0012
(0.0010)
-0.0030
(0.0096)
-0.0076***
(0.0008)
-0.4523***
(0.0236)
0.7250***
(0.0300)
-0.2019***
(0.0443)
-0.0063***
(0.0007)
-0.6515***
(0.0215)
0.8931***
(0.0099)
-0.1875
(0.0270)
NA
0.4197***
(0.0457)
Yen
3 Months
12 Months
-0.0002
(0.0017)
-0.0033
(0.0161)
0.7092***
(0.0666)
-0.2237*
(0.1171)
-0.0030
(0.0038)
-0.0875**
(0.0437)
0.8242***
(0.0476)
-0.4112***
(0.1126)
-0.0032***
(0.0007)
-0.5342***
(0.0201)
0.7373***
(0.0258)
-0.1420***
(0.0397)
-0.0041***
(0.0008)
-0.7030***
(0.0224)
0.8824***
(0.0105)
-0.0948***
(0.0292)
Chartists
cc
β
 1c

c
2
-0.0046***
(0.0005)
-0.3419***
(0.0183)
NA
0.4273***
(0.0208)
-0.0022***
(0.0005)
-0.4286***
(0.0155)
NA
0.4981***
(0.0163)
adj.R2
0.2991
0.3614
0.8241
0.4136
0.5553
0.8039
% fun
34.70
15.71
12.94
20.96
13.88
4.59
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.
23
Table 4: Dynamic Weights
1 Month
Fundamentalists
-0.0010**
cf
(0.0004)
0.0004
α
(0.0028)
 1f

f
2
NA
0.1135***
(0.0167)
Chartists
cc
-0.0045***
(0.0005)
β
-0.7692***
(0.0170)
 1c

c
2
NA
0.8329***
(0.0182)
Euro
3 Months
12 Months
1 Month
-0.0075***
(0.0006)
-0.0021
(0.0040)
0.3566***
(0.0252)
-0.2457***
(0.0378)
-0.0063***
(0.0008)
-0.0029
(0.0056)
0.6956***
(0.0130)
-0.5556***
(0.0323)
0.0012***
(0.0005)
0.0019
(0.0043)
-0.0028***
(0.0005)
-0.8411***
(0.0196)
1.0693***
(0.0241)
-0.2124***
(0.0363)
-0.0023***
(0.0006)
-0.9683***
(0.0183)
0.9701***
(0.0080)
-0.0093
(0.0225)
-0.0050***
(0.0005)
-0.7900***
(0.0147)
NA
0.1924
(0.0152)
NA
0.8390***
(0.0163)
Yen
3 Months
12 Months
0.0002
(0.0007)
-0.0237***
(0.0061)
0.3802***
(0.0257)
-0.2041***
(0.0411)
-0.0076***
(0.0012)
0.0404***
(0.0107)
0.3045***
(0.0171)
-0.0169
(0.0414)
-0.0032***
(0.0006)
-0.9193***
(0.0171)
1.0520***
(0.0218)
-0.1302***
(0.0329)
-0.0014**
(0.0006)
-0.9124***
(0.0180)
1.0008***
(0.0081)
-0.1155***
(0.0231)
R2
0.6395
0.7281
0.9047
0.6961
0.7905
0.8999
% fun
52.25
43.42
29.66
51.91
41.18
22.52
AC
-0.0300
-0.0320
0.0000
0.0010
0.0070
0.0450
Notes: Table presents estimation results for the model with dynamic 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.
24
Table 5: Estimation Results Simulated Data
Model:
Group:
Benchmark
All
Switching
All
cf
-0.0005
(0.0008)
-0.0063
(0.0065)
0.4560***
(0.0249)
-0.0233***
(0.0007)
-0.0259***
(0.0059)
0.4076***
(0.0235)
α

f
2
cc
β
 2c
R2
% fun
AC
NA
-0.3756***
(0.0220)
NA
0.3833
NA
NA
Fundamental model
Fundamentalists
Chartists
-0.0233***
(0.0007)
-0.0259***
(0.0061)
0.4076***
(0.0243)
0.0191***
(0.0007)
-0.0226***
(0.0065)
0.3901***
(0.0246)
0.0178***
(0.0007)
-0.2331***
(0.0207)
0.4022***
(0.0229)
NA
NA
NA
NA
NA
NA
0.7338
48.25
0.0590
0.3739
NA
NA
0.3371
NA
NA
Chartist Model
Fundamentalists
Chartists
NA
NA
NA
NA
NA
NA
-0.0203
(0.0007)
-0.2214***
(0.0199)
0.4112***
(0.0220)
0.0178***
(0.0007)
-0.2330***
(0.0200)
0.4022***
(0.0222)
0.4853
NA
NA
0.4629
NA
NA
25
Benchmark model
Fundamentalists
Chartists
-0.0208***
(0.0007)
-0.0177***
(0.0055)
0.4171***
(0.0219)
0.0176***
(0.0007)
-0.0107*
(0.0059)
0.4017***
(0.0221)
NA
NA
-0.2125***
(0.0199)
-0.2264
(0.0203)
NA
NA
0.4962
NA
NA
0.4653
NA
NA
Table 6: Estimation Results Comparison
Model:
Fundamental model
Group: Fundamentalists
Chartists
0.0012***
(0.0005)
0.0019
(0.0042)
0.1924***
(0.0150)
-0.0050***
(0.0009)
-0.0376***
(0.0076)
0.7906***
(0.0301)
cc
NA
NA
β
NA
NA
 2c
NA
NA
R2
0.1120
0.3888
cf
α
 2f
Chartist Model
Fundamentalists
Chartists
NA
NA
NA
NA
NA
NA
0.0012***
(0.0005)
-0.0185
(0.0133)
0.1909***
(0.0148)
-0.0050***
(0.0005)
-0.7900***
(0.0149)
0.8390***
(0.0165)
0.1132
0.8132
26
Benchmark model
Fundamentalists Chartists
0.0013***
(0.0005)
0.0027
(0.0043)
0.1924***
(0.0150)
-0.0051***
(0.0005)
-0.0107***
(0.0042)
0.8325***
(0.0166)
NA
NA
-0.0196
(0.0134)
-0.7854***
(0.0150)
NA
NA
0.1128
0.8140
Figure 1: Fundamental Exchange Rates
Euro
Japanese Yen
5.0
.2
.1
4.9
.0
4.8
-.1
4.7
-.2
4.6
-.3
4.5
-.4
96
97
98
99
00
01
s
02 03
04
05 06
96
07
97 98
99 00
01
s
s*
27
02
03
s*
04
05 06
07