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22-06-2013 Group 6
T. Bao 2012/2013
BSc thesis Econometrics Semester 2
A simple heterogeneous agent model with rational and adaptive
expectations in positive and negative feedback markets.
Rens Dekker 10003132
2 Table of Contents
1 Introduction 3
2 Characteristics of positive and negative feedback markets 4
3 Heterogeneous agent models 5
4 Experimental design 6
5 Research method 7
6 Results and analysis 9
7 Conclusion and discussion 14
Appendix A: heuristic switching model 17
3 1. Introduction
In our society, markets and the way prices are formed on these markets are very important. These complex systems often consist of many different players: consumers, producers, investors and speculators among others. In earlier decades economists assumed that individuals have perfect knowledge of market conditions and use this information to make rational decisions and to form rational expectations. This rational expectations hypothesis was first proposed by Muth (1961) and later became influential through its usage by Lucas (1972). Recently more and more research appears on expectations formation rejecting the premise of rational expectations. Modeling expectations is important because under uncertainty, agents base their actions on their expectations of future economic conditions and the expectations of others. In the case of stock prices, the price depends on the expected future income of that stock. Similarly in wage negotiations, expectations on price inflation play an important role.
In the past two decades literature is growing on the subject of bounded rationality where agents use learning models to form their expectations. Many heterogeneous agent models have been introduced. In these models different groups of traders with various expectations and beliefs come together to form a market. Hommes (2005) distinguishes two typical trade types. Firstly there are rational traders (also called fundamentalists) that believe that prices are determined completely by economic fundamentals. Secondly there are chartists or technical analysts that do not believe prices are determined by fundamentals, but can instead be predicted by simple rules based on patterns in past prices, such as trends or cycles.
By acting upon expectations individuals can affect the aggregate market outcome. Economic agents form their expectations on the basis of market history and thus a market may be viewed as an expectations feedback system. A general distinction is made between positive and negative expectations feedback markets. In a positive feedback market a high average expectation leads to a higher realized market price. This can for instance be seen in a speculative asset market: if the price of one particular stock is expected to be high, more traders will want to buy the stock. Thus creating a bubble where the realized stock price is above the rational market price. Similarly in a negative expectations feedback market a high average expectation leads to a lower realized market price. If for example producers expect the price for their product to be high, they may decide to produce more. This in turn through the mechanism of supply and demand leads to a lower realized market price.
The central goal to this paper is to estimate a 2-type heterogeneous agent model (HAM) to explain the macro and micro behavior of experimental markets with positive and a negative expectations feedback. Can a simple heterogeneous model explain market prices
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as well as individual price expectations? The 2-type HAM is applied to the experimental data as collected by Heemeijer et al. (2009). Aggregate one-period-ahead price predictions are estimated using a combination of a rational expectations rule and an adaptive expectations rule. The fitted model as well as the fractions of agents using each particular rule are analysed across time and across the positive and negative feedback markets. Next the performance of the 2-type HAM is compared with a set of homogeneous models, a 2-type heuristic switching model (HSM) and a 4-type HSM that was developed by Anufriev and Hommes (2012).
The rest of this paper is organized as follows. First section 2 provides insight into positive and negative feedback markets. Then section 3 discusses heterogeneous agent models in general. Section 4 describes the experiments which are used for this paper. Section 5 goes into detail about the used research method. Results are presented and analysed in section 6. The last section concludes.
2. Characteristics of positive and negative feedback markets
The distinction between positive and negative feedback is related to the subject of strategic substitutes and strategic complements. If two actions are strategic substitutes (complements) then an increase in the action of one individual leads to a decrease (increase) in the action of another individual. Haltiwanger and Waldman (1985) argue that when actions are complimentary, agents have the incentive to imitate other agents. In other words it is profitable to predict a price close to what others have predicted. Because individuals coordinate their expectations the influence of irrational agents upon the realized price is enhanced. This in fact makes convergence towards the rational equilibrium unlikely. Contrastingly in the case where actions are substitutable, agents will have the incentive to deviate from other agents. This limits the impact of irrational agents, which makes convergence towards the fundamental price more likely. In this case agents coordinate their actions only after convergence has taken place.
Heemeijer et al. (2009) created experimental markets with positive and negative expectations feedback. A far as the results of these experiments go, there are two interesting characteristics, which were already mentioned above: convergence of the aggregate price towards the ‘rational’ equilibrium price and coordination between individuals such that there is little dispersion between individual predictions. Heemeijer at al. (2009) find that with positive feedback individuals exhibit fast coordination, but slow convergence, whereas with negative feedback there is slow coordination, but quick convergence.
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In similar experiments Bao et al. (2012) introduced large unanticipated price shocks to the market. After a large shock in the positive feedback environment individuals quickly coordinate, but do so on a price different from the fundamental equilibrium. After a shock in the negative feedback market, predictions initially show a large variation, but rapidly reach a consensus on the new equilibrium price.
Differing convergence characteristics have been found by Fehr and Tyran (2008). Heemeijer et al. (2009) attribute this to differences in the local stability properties of the steady state. Whereas the slope near the steady state in Fehr and Tyran (2008) is zero, in both Heemeijer et al. (2009) and Bao et al. (2012) it is close to one (20/21). Sonnemans and Tuinstra (2008) have done research on positive expectations feedback with a lower value of 2/3 and found that it is indeed this value that causes a difference in convergence characteristics. Therefore the choice of the parameters requires careful attention.
3. Heterogeneous agent models
There has been much research into the validity of the assumption of heterogeneity. Numerous heterogeneous agent models using simple forecasting heuristics have been estimated for a variety of markets, such as hog and beef markets (Baak, 1999; Chavas, 2000), stock prices (Boswijk, 2007), stock option prices (Frijns et al., 2010), exchange rates (Westerhoff and Reitz, 2003) and commodities (Ellen and Zwinkels, 2010). This research shows that investors can be classified into different groups using different strategies. In addition it is found that there exists significant time-variation in the fraction of agents using these different strategies. Secondly there is literature using survey data (such as Allen and Taylor (1990) and Capistrán and Timmermann (2009)), which has the benefit that it can focus on the process of generating expectations. This provides evidence for the presence of heterogeneous forecasts among financial experts and professional forecasters. Furthermore laboratory experiments with human subjects can be used to validate learning models and expectations hypotheses (Hommes, 2011; Heemeijer et al., 2009).
As mentioned above it has been found that individuals make use of simple rules to make predictions. They may for instance use a naïve or adaptive expectation, a trend following or trend reversing rule, an average price expectation or rational expectations. Agents may switch between these different rules based on their past performances. In Bao et al. (2012) where shocks were introduced to an experimental market, individuals see the introduced shock and change their prediction rules accordingly. Thus similar subjects may use dissimilar rules in different situations and may in fact change between rules during the experiment.
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Brock and Hommes (1997, 1998) propose a model where heterogeneous individuals choose and switch between different trading strategies in an economic market. Anufriev and Hommes (2012), Anufriev et al. (2010) and Bao et al. (2012) apply such a heuristic switching model (HSM) to different experimental market data, such as that of Heemeijer et al. (2009). In this kind of model, individuals change between multiple different forecasting rules according to their relative performance. HSMs, like other heterogeneous models, are an attempt to explain expectations formation in various market settings.
4. Experimental design
This paper uses the experimental data as collected by Heemeijer et al. (2009). Computerized experiments were conducted at the CREED laboratory at the University of Amsterdam on February 18 and 19, 2003. Seven positive feedback markets of 50 periods were created, as well as six negative feedback markets. In each market six students were given the task to make a one-period-ahead forecast for the price in their market. Students would receive earnings based on the accuracy of their predictions. Subjects only had information on past realized prices and their own predictions. They were also given qualitative information on the way market prices were determined, but no quantitative info on the exact price generating mechanism. In both environments prices are determined via the simple form
( ) (1)
where ∑ the aggregate forecast. Supply and demand are assumed to be linear, so that the function is also linear. Symmetry in the parameters is chosen for the positive and negative feedback markets, so that they are comparable. The only difference between the two is the sign of the slope. For positive expectations feedback the price adjustment mechanism is:
( ) (2)
For negative expectations feedback it is:
( ) (3)
taking ( )⁄ a random term, representing small uncertainties in demand. The fundamental equilibrium price can be seen to be . Individual student’s earnings were determined by the quadratic error of their prediction:
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( ) (4)
1300 points correspond to 0.5 euro and on average earnings were approximately 22 euros in 90 minutes. For a more detailed description of the experiments, see the paper Heemeijer et al. (2009), particularly appendices A and B.
One must be critical of the fact that whatever instructions the test subjects are given, will shape the outcome of the experiments. As such participants were given just the basic knowledge to fulfill their task. It is good to note that previous research by Fehr and Tyran (2008) obtained similar results in experiments where the subjects were given more information about their market, such as the number of other participants, other participants’ past forecasts as well as the best response to the average forecast of others.
5. Research method
Now a simple heterogeneous model is constructed to capture the macro and micro behavior of both a positive and a negative feedback market. The model is applied to positive feedback group 4 and negative feedback group 2 from Heemeijer et al. (2009). The aggregate one-period-ahead price prediction is modelled using a combination of two forecasting heuristics:
( ) (5)
A fraction of individuals uses the first heuristic to predict prices, whereas the remaining fraction of agents uses the second heuristic. In this paper a rational expectations rule (RE) is used for :
( ) (6)
This rule is different from the general price adjustment mechanism in that the ‘irrational’ noise term is excluded. For the adaptive expectations rule (ADA) is used:
( ) (7)
This consists of the previous ADA prediction plus an adjustment for the amount the previous ADA prediction failed to capture the realized price. For the first ADA prediction, i.e. when t=2, there is no previous prediction available, so here the aggregate forecast is used instead.
For the ADA rule a coefficient of 0,75 is used. Among all individuals from all groups a general model with three lags in prices and expected prices is estimated. Individuals with a
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significant coefficient for adaptive expectations are labeled as the ADA group. In the ADA group the median is determined from among the different coefficients. This median serves as the ADA rule coefficient, so as to make sure this really is a typical ADA rule.
The fractions of agents using each of the forecasting rules are estimated, by calculating alpha from equation 5 for every period in both markets:
{ } (8)
Using these alphas the fitted expectation ̂ is created. The fitted price ̂ is then given by
̂ ( ̂ ) (9)
Next the performance of this fitted 2-type heterogeneous agent model (HAM) is compared with that of a variety of other homogeneous, heterogeneous and heuristic switching models (HSM). For this the mean squared error (MSE) is calculated. At the aggregate level the model must provide a good fit for realized market prices, so that:
∑ ( ̂ ) (10)
However at the individual level the model must be able to explain the individual price forecasts made by each agent. For the homogeneous models this comes down to averaging over each individual:
∑ ∑ ( ̂ ) (11)
For the HAM and HSMs the MSE at the individual level becomes a combination of the accuracy of the different forecasting rules weighted by the fraction of agents using each rule. For example:
∑ ∑ ( ) ( ) ( ) (12)
The intuition behind this is that at the individual level each agent uses one heuristic exclusively. Thus each of the included heuristics must provide a good fit for what some individuals are forecasting in reality.
In the homogeneous models all agents use the same one forecasting rule. The different forecasting rules that are used are detailed in table 1. The two HSMs considered are a 4-type HSM that was developed by Anufriev and Hommes (2012) and a 2-type adaptation of the same model. The 4-type HSM uses the ADA, WTR, STR and A&A rules. The 2-type
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HSM uses the same rules that are used in the 2-type HAM, namely the rational and ADA rules. The workings of these heuristic switching models are further detailed in appendix A.
Table 1. Forecasting rules detailed
Fundamental:
Rational: ( )
Adaptive (ADA): ( ) Weak Trend Rule (WTR): ( )
Strong Trend Rule (STR): ( )
Anchoring and Adjustment (A&A): (
∑ ) ( ) Naïve: Contrarian: ( ) Average: ∑
6. Results and analysis
Figure 1 shows time series of the estimated alphas for the 2-type HAM in the positive (left) and negative (right) feedback markets. The corresponding fitted models are shown in the bottom panels. In the case where alpha is closer to 1, the rational rule is a better predictor than the adaptive expectations rule, vice versa for the case where alpha is closer to 0. Also if alpha lies between 1 and 0, i.e. unequal 1 or 0, the model has a perfect fit.
In the positive feedback market the rational expectations rule is generally used by the larger fraction of subjects. Alpha seems to run in a cycle, where temporarily more agents use the adaptive rule and then gradually change again to mostly using the rational rule. This happens where prices go from a rising to a falling trend or from a falling to a rising trend and in the short timeframe after such an event. For negative feedback the adaptive rule is used by the largest fraction of agents for the first 30 periods or so, where the prices are most volatile. Afterwards when the market has converged around the fundamental price of 60, agents are about equally distributed amongst the two rules.
In the positive market the 2-type HAM has the least good fits around the time where the market switches between an upwards trend and a downwards trend. In the negative market it performs least well in the early stages where the market is most volatile and is yet to decide upon the price it will converge on. Overall the initial periods of the negative feedback market is where the model makes the largest errors.
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Figure 1. Estimated alphas for the positive feedback market (top left) and negative feedback market
(top right). Experimental and simulated prices using the 2-type heterogeneous agent model in the positive feedback market (bottom left) and negative feedback market (bottom right).
Four of the homogeneous models are plotted in figure 2. The best fit in the positive feedback market (left) is given by STR, but this model consistently overshoots in the negative feedback market (right). The best fit in the negative feedback market is given by ADA, but this model constantly lags behind the experimental values in the positive feedback market. Out of these four forecasting rules, the model that gives a reasonable, but not a great fit for both markets is the WTR.
Figure 3 shows plots of the 2-type HSM (second row) and 4-type HSM (fourth row) as well as the fractions of agents using each heuristic in the respective models (first and third row). In the 2-type HSM in contrast with the 2-type HAM, rational expectations are used by the majority of agents. It is good to note the subtle differences between these two models. In the HSM the forecasting rules are selected based on their performance in predicting the market price. In the HAM alpha, i.e. the fraction of agents using each heuristic is fitted directly. The HSM more or less tries to explain how agents form their expectations, whereas the HAM tries to explain which forecasting rules are used at what times.
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(a) Adaptive expectations rule (ADA)
(b) Weak trend rule (WTR)
(c) Strong trend rule (STR)
(d) Anchoring & adjustment (A&A)
Figure 2. Experimental and simulated prices for four of the homogeneous models in the positive
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Figure 3. Results for the 2-type HSM (top four panels) and the 4-type HSM (bottom four panels).
Shown are the fractions of users using the different heuristics (first and third row) and experimental and simulated prices (second and fourth row). Results for the positive feedback market (left) and negative feedback market (right)
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In the 4-type HSM most subjects use the ADA rule in the negative feedback market and both the WTR and STR rules in the positive feedback market. This illustrates how the more general 4-type model is able to have a good fit in both markets using different heuristics in different market situations. The intuitive mechanism of evolutionary selection and learning accomplishes this. The same 4-type HSM has previously been applied to these markets by Hommes (2011).
Table 2. MSEs of 12 models over 46 time periods of the experiment
MSEs for 9 homogeneous models, the 2-type heterogeneous agent model, a 2-type heuristic switching model and the 4-type HSM by Anufriev and Hommes (2012). The 2-type HSM uses the same prediction rules as the 2-type HAM. The 4-type HSM however uses different prediction rules. At the aggregate level MSE was calculated for prices, whereas at the individual level MSE was calculated and averaged for each agent’s individual price expectations. Performances are shown for the positive and negative feedback markets as well as the performance in both markets combined.
In table 2 the performance of the 2-type HAM is compared with the 9 homogeneous models, the 2-type HSM and the 4-type HSM. At the aggregate level the models need to be able to explain realized market prices. For this MSE is calculated for fitted prices. However at the individual level a model needs to be able to explain each agent’s individual price forecast. For this MSE is calculated for fitted price expectations. As a third requirement a good model must do well in the positive as well as the negative feedback market.
Only the last 46 periods are used to calculate the MSEs. This was chosen so as to give each model enough time to be able to provide a reasonable forecast. In the process some of the characteristic volatile initial periods of the negative feedback market are left out, nevertheless a balance must be struck.
The homogeneous model using the rational rule is one of the best performing in the positive feedback market at both the aggregate and individual level. It however does not work that well for the negative feedback market. Contrastingly the ADA rule is one of the best
Model Positive Negative Both Positive Negative Both
Fundamental 100,856 0,931 50,893 112,924 3,112 58,018 Rational 0,229 3,259 1,744 1,772 5,763 3,768 ADA 8,856 0,899 4,878 11,284 3,161 7,223 WTR 1,876 1,564 1,720 3,588 3,197 3,393 STR 1,367 7,305 4,336 3,026 10,223 6,624 A&A 37,330 3,795 20,562 42,679 6,353 24,516 Naive 5,000 0,931 2,966 7,032 3,197 5,114 Contrarian 8,512 1,232 4,872 10,905 3,529 7,217 Average 180,288 3,736 92,012 200,295 6,289 103,292 2-HAM 0,095 0,578 0,337 2,202 3,259 2,730 2-HSM 0,325 1,997 1,161 3,081 4,999 4,040 4-HSM 0,729 0,849 0,789 6,819 4,225 5,522
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performing in the negative feedback market, but fails to capture the positive feedback market. No homogeneous model fulfils the requirement to have a good fit across the feedback regimes and across the aggregate and individual level. Out of the models considered the best is the WTR, which does well in explaining individual prices and reasonable at explaining market prices, with a good balance across the feedback regimes. Results for the Fundamental rule suggest that in the negative feedback market, individuals are able to learn the equilibrium.
The 2-type heterogeneous agent model does an excellent job explaining market prices at the aggregate level for the positive as well as the negative feedback market. It is outperforming every single homogeneous model as well as the more complex heuristic switching models. In explaining individual expectations it is outperformed in the positive market by the rational expectations rule and in the negative market by the fundamental rule. However when looking at both markets combined, the 2-type HAM outperforms every other model.
Naturally the 2-type HAM also does better than the 2-type heuristic switching model making use of the same two heuristics. In the HAM, alpha was freely and optimally chosen for each period to provide the best fit. The HSM is restricted in its choice of the fractions in that it has some inertia and is reluctant to changing parameters too much between subsequent periods. In a sense the HAM here is a special case of the HSM. Overall the 2-type HSM performs only slightly worse than the 2-2-type HAM and performs third best at the aggregate level as well as fourth best at the individual level. Because of the large fraction of agents using the rational heuristic in the negative feedback market, its fit here could be better.
The more complex 4-type HSM by Anufriev and Hommes (2012) does better than the 2-type HSM for negative feedback, but worse for positive feedback. The 4-type HSM also does worse at the individual level, but better at the aggregate level. The relatively bad fit at the individual level for positive feedback is largely caused by the weak performance of the A&A rule even if only a 5% fraction is using it. The two HSMs only have the ADA heuristic in common. Thus the chosen heuristics for a HSM are a big factor in determining whether the model may be more successful in one area and less successful in another. At the individual level it might be interesting to break down the MSE per heuristic
7. Conclusion and discussion
This paper reviewed a simple 2-type heterogeneous agent model (HAM) using rational and adaptive expectations. The goal with this model was to explain the macro and micro behavior
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of experimental markets with positive and a negative expectations feedback. The type of expectations feedback determines the characteristics of the different markets and thus the performance of different forecasting heuristics in those markets. Research shows that economic agents may use simple rules to make predictions. Furthermore different agents may use different heuristics and may change their heuristics over time. HAMs and HSMs attempt to capture these traits across various market settings. A good model can explain aggregate prices at the macro level as well as individual price expectations at the micro level. Controlled experiments with human subjects are used to validate learning models and expectations hypotheses. In this limited investigation the 2-type HAM was applied to two experimental markets by Heemeijer et al. (2009). Its performance was compared with a set of homogeneous models, a 2-type HSM and a 4-type HSM that was developed by Anufriev and Hommes (2012). No single homogeneous model was able to have a good fit in both the positive and negative feedback markets. Results find that the simple heterogeneous model provides a better fit than any of the homogeneous models at both the aggregate and individual level for positive and negative expectations feedback. This provides a case for the use of heterogeneous models instead of homogeneous models to describe market behavior. The 2-type HAM also outperformed a 2-type HSM that uses the same heuristics as well as a benchmark 4-type HSM. In the 2-type HAM, however, the fraction of agents using each rule is estimated for each period. This alpha is chosen optimally and without restrictions, so that the fitted model is the best one possible for this model. Alpha is not determined one period ahead; neither does it depend on other historic values of alpha. It is not realistic that the participants in the experiments or real market agents for that matter have the perfect foresight that allows them to choose which heuristic will work well in the next period.
Here a general heuristic switching model provides a solution. A HSM gives an intuitive explanation for how agents observe the market, learn from this, and change their forecasting rules accordingly. The HSM is restricted in its choice of the fractions in that it has some inertia and is reluctant to changing parameters too much between subsequent periods. In a sense the 2-type HAM is a special case with perfect foresight of the 2-type HSM. Thus the ‘unoptimized’ HSM cannot possibly provide a better fit than the HAM.
Both HSMs do better than all the homogeneous models in explaining the market price and do reasonable well, but not great at explaining the individual price expectations. If at the individual level a model gives a bad fit this may mean that the participants in the experiment, of which there were only 6, are either using different heuristics or that the fractions of agents using each rule do not reflect reality.
A fair comparison between the HSM and other models depends on the specification of this HSM. One issue for a good HSM is the choice of the parameters and starting values.
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In this paper the benchmark parameters were used, which are unoptimized for the fitted markets.As such the calculated forecasting errors may be a little higher than they should be. For example the mean squared errors of the models were calculated leaving only the first four periods out as a learning period for the models. For a HSM this period may be too short if the heuristics all start at equal weights and if some heuristics do not give a good fit. No learning period has preceded the market that determined that equal shares of participants prefer specific heuristics at the start of the experiment. These fractions were artificially chosen and may result in forecasting errors that are too high relatively to models that lack this evolutionary learning.
Another point is the choice of the heuristics used in the HSMs and HAM. The chosen heuristics are a big factor in determining whether the model may be more successful in one area and less successful in another. Results illustrate how the 4-type model is able to have a good fit in both markets using different heuristics in different market situations. This may suggest that the evolutionary mechanism in the HSM is very reasonable. In the two markets of this paper the A&A heuristic is hardly used and might be left out or be replaced by another heuristic at little expense or great benefit.
The inclusion of the rational expectations rule in the 2-type HAM may be a little odd. When looking at the markets as learning to forecast experiments, rational expectations are not realistic. No one of the participants in the experiment has the perfect foresight required for this rational expectation. If no individuals are using the rule then there is a good chance that the model could give a bad fit at the individual level. It must be said that the homogeneous rational model has a good fit for positive (but not negative) feedback for individual expectations as well as market prices. Alternatively this rule could be looked on as being used by really good forecasters.
17 Appendix A: heuristic switching model
The following heuristic switching model (HSM) was developed by Anufriev and Hommes (2012). Agents choose and switch between H different heuristics, so that the aggregate price expectation is:
∑ (A.1)
Where is the fraction of agents using heuristic h at time t. Agents choose between the different forecasting rules based upon their relative performance. The performance of the different rules is determined by:
( ) (A.2)
Here the parameter is the weight that subjects attribute to past forecasting errors. The evolutionary updating rule for the fraction of agents using heuristic h is given by:
( )∑ ( ( ) )
(A.3)
The parameter represents the fact that agents are reluctant to changing parameters too much between subsequent periods. Lastly parameter reflects the speed with which participants change to a different heuristic.
Specifically in their paper Anufriev and Hommes (2012) apply a 4-type HSM to asset pricing experiments. The four heuristics selected are the ADA, WTR, STR and A&A rules. Their parameter settings are with initial weights . The 2-type HSM is an adaptation of the model above, using the same parameter settings. In this the rational and ADA rules are used.
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