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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.
< 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. 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
i t t t k i t t t k i t t t k i t t t kt t
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’)
< Insert Table 3 Here >
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.
< Insert Table 4 Here >
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
< Insert Table 5 Here >
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 t t k i t t t k i t t t k i t t t kt t
i T Tc f
i t t t k i t t t k i t t t k i t t t kt t
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.
< Insert Table 6 Here >
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.
< Insert Table 7 Here >
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
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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