Heterogeneous expectation models in experimental asset markets

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Heterogeneous expectation models in experimental asset markets

Beschrijving van de Huizenmarkt met een

heuristiek switchmodel

Niels Heijenk

10379363

MSc Econometrics

Heterogeneous expectation models in

experimental asset markets

Faculty of Economics and Business

Amsterdam School of Economics

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Date of final version: 07-12-2017

Track: Free Track

Supervisor: C.H. Hommes

Second reader: A.G. Kopányi-Peuker

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Statement of originality

This document is written by Student Niels Heijenk who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

We fit two heterogeneous agent models, a heuristic switching model (HSM) (Anufriev & Hommes, 2012) and a genetic algorithms (GA) model (Anufriev, Hommes & Makarewicz, 2015), on experimental asset data with the same design as that of Smith, Suchanek & Williams (1988). The HSM uses multiple simple heuristics between which the model can switch based on past forecasting performance. The GA model does not have predefined heuristics but updates its heuristics periodically with four evolutionary operators: reproduction, mutation, crossover and election. This study provides a framework for the understanding of bubbles and crashes that occur in the experiment, and introduces two new heuristics based on the forecast beliefs submitted by the subjects. We find that the HSMs outperform the homogeneous agent models consisting each of one heuristic. More importantly, we find that the HSM also outperforms the behavioral models of Carle, Lahav, Neugebauer & Noussair (2015). Finally, we find that the GA does not outperform the HSMs considered in this paper because, among other things, the GA routine needs more periods to optimize its heuristics.

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TABLE OF CONTENTS

Chapter I.

Introduction & Theoretical framework

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Chapter II.

Lab Experiment

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Chapter III. Models

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3.2 The genetic algorithms model ...

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3.3 Empirical fit ...

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3.4 The models from Carle, Lahav, Neugebauer & Noussair (2015) ...

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Chapter IV. Empirical analysis

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4.2 The genetic algorithms model ...

32

Chapter V.

Conclusion

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Chapter VI. References

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Asset markets is based on expectations feedback systems. An expectations feedback system is a system where expectations influence the asset prices, and (historical) asset prices in turn affect the expectations of individuals. Therefore, individual expectations play an important role in predicting the asset prices. According to Muth (1961) and Lucas & Prescott (1971), all individuals have rational expectations which means that they act according to self-interest and choose the best action possible. Findings of Alchian (1950) and Friedman (1953) strengthen this statement because they argue that “irrational” traders trade against lower profits than rational traders which causes the “irrational” traders to be driven out of the market. However, the statement that every individual is rational also implies that every individual makes the best decision possible which is, according to Simon (1957), very unrealistic. He argued that rational expectations impose strict informational and computational assumptions for individuals. In later work, Simon (1972) introduced bounded rationality which implies that individuals act according to simple rules of thumb when making decisions. These rules of thumb are the underlying thought of the Heuristic switching model (HSM) in which agents can switch between simple heuristics. The simple heuristics of the HSM can be compared with the rules of thumb that Simon (1957) proposed.

Multiple laboratory experiments have shown that individuals do not act perfectly rational but instead use simple forecasting rules. This implies that experimental data can be better described by simple heuristics. (Tversky and Kahneman, 1974; Kahneman, 2003; Camerer and Fehr, 2006). Frankel and Froot (1987) find that expectations in asset markets are also heterogeneous and probably boundedly rational which supports the use of the HSM to describe experimental asset data. Multiple researchers (e.g., Sargent, 1993 and Evans & Honkapohja, 2001) suggest the use of adaptive learning for modeling the expectations of agents. Peseran (1987) also makes use of adaptive learning where agents form expectations based on the learning of unknown parameters and new techniques. This adaptive learning theory will be captured and applied by an adaptive expectations heuristic in the HSM.

The experimental asset data that is used for this study comes initially from an experiment of Haruvy, Lahav & Nouissair (2007). This experiment had a similar setup as that of Smith, Suchanek & Williams (1988), where also 4 markets of 15 periods were simulated. A

CHAPTER I. INTRODUCTION & THEORETICAL FRAMEWORK

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new feature to the experiment is that, at the beginning of every period, the participants were also asked to submit their expectations of the asset price in that period and all following periods. These beliefs are clearly heterogeneous, supporting the findings of Frankel and Froot (1987).

Heterogeneous agent models, such as the HSM, exist because of the heterogeneity in expectations and have shown to be less biased when describing asset prices than other homogeneous models (e.g., Brock and Hommes, 1997,1998; Anufriev & Hommes 2012). Baghestanian, Lugovskyy & Puzzello (2015) also used a heterogeneous agent model to describe an experiment similar to that of Smith et al. (1988). Their heterogeneous agent model used three types of agents: speculators, fundamental traders, and noise traders. The type is confirmed by specific exogenous characteristics like cognitive reflections and the accuracy of price forecasts which means a subject is not able to switch between different types during the experiment. Our study complements the existing literature by using a model, the heuristic switching model, where switching between different types of heuristics is possible. Baghestanian et al. (2015) find that if the speculators are selling their assets to the noise traders, a crash will follow. Carle, Lahav, Neugebauer & Noussair (2015) also use multiple heterogeneous agent models to describe the data of the experiment of Haruvy et al. (2007). They use the different beliefs submitted by the subjects which ensure heterogeneity among agents. The performance of the heterogeneous agent models in this study will be compared to the performance of the models of Carle et al. (2015).

The HSM has proven itself to be an efficient model to describe experimental data in ‘learning-to-forecast’ experiments, e.g., in Anufriev & Hommes (2012), where they use a 4-type HSM to describe the experimental data. Bolt, Demertzis, Diks, Hommes & van der Leij (2014) estimate a 2-type HSM with fundamentalists and trend-followers using house price data, where the fraction of trend-followers increases when a housing bubble occurs. Cornea, Hommes & Massaro (2017) estimate a 2-type HSM on the inflation dynamics with monopolistic competition, staggered price setting and heterogeneous firms. The estimation of the inflation dynamics by the HSM has a higher MSE than the estimation of Bolt et al. (2014) since inflation data is simply more volatile than the house market data which means it is harder to predict the movement and the magnitude of the movements. However, the MSE of the prediction of the inflation data is still lower than the MSE of the prediction of Hommes & in ‘t

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Veld (2017) where they use a 2-type HSM to describe S&P500 data. This difference is again caused by the volatility difference between both markets since the stock market is more volatile than the inflation dynamics. So, the higher the volatility of certain data, the bigger the estimation error by the HSM.

However, the HSM has a number of shortcomings and that is why a second model will be used in this research, a Genetic Algorithms (GA) model. One of the shortcomings of the HSM is that the set of heuristics is exogenously given with predefined parameters. These heuristics are usually based on experiments which does not necessarily mean that they will also be optimal for other experiments or data from, for instance, the stock market. The genetic algorithms model, however, does not have this problem since the heuristics used by this model are periodically updated by evolutionary operators which means they are not predefined but determined within the model.

The second shortcoming of the HSM is that the set of given heuristics in the HSM cannot adequately account for within treatment individual heterogeneity that is observed in the experiments (Anufriev, Hommes & Makarewicz, 2015). In the GA model, however, each agent has his own set of heuristics that he updates every period which ensures heterogeneity between agents. The third shortcoming of the HSM is that it only allows for agents to use particular heuristics that are not changing over time while in reality, you would expect agents to update their forecasting rules continually. The GA model is based on this updating and uses multiple evolutionary operators to update the set of heuristics periodically. This updating routine ensures that the ‘bad’ heuristics in the set will be replaced by heuristics that make better estimations and that a global maximum can be found in the parameter space.

GA were first applied in economics by Arifovic (1994) where she used the GA to explain the results of an experiment by modeling both social and the individual learning. Nowadays, it is a leading tool to model the individual learning of agents (Sargent, 1993 and Dawid, 1996) and it has applications in different economic areas. GA are for instance used in the overlapping generation monetary economies (Arifovic, 1995), flowshop scheduling (Ishibuch, Murata & Tanaka, 1996) and exchange rate volatility (Arifovic 1996; Lux and Schornstein, 2005). In a recent application of Hommes & Lux (2013), the GA were used to update the heuristics instead of directly optimizing the prediction, an approach that is also used in Anufriev et al. (2015). This method will also be used for our research where every

CHAPTER I. INTRODUCTION & THEORETICAL FRAMEWORK

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agent gets a set of heuristics which will be periodically updated by GA learning. The exact routine of the GA model and functions of the operators will be discussed in section 3.

The first main question addressed in this thesis is: how does the Heuristic Switching

Model perform compared with the behavioral models from Carle et al. (2015)? The models will be compared based on two different criteria: the Spearman correlation and the mean squared error (MSE). The Spearman correlation coefficient is a measure of monotone association between the model and the real values, and the MSE will indicate the magnitude of the errors made by the model. A second question addressed in this thesis is:how does the

Heuristic Switching Model perform compared with homogeneous agent models existing each of one heuristic? The hypothesis is that the HSM performs better based on both the

Spearman correlation and the MSE. This expectation is stated since the HSM is a model in which agents can switch to better-performing heuristics every period. The result will be that only the ‘best’ heuristics will have a high fraction of agents while the worse performing heuristics will have a low fraction. Finally, this study will also aim to answer: how does the

GA model describe the experimental data of Haruvy et al. (2007)? A comparison with the

HSM will be made to determine how well the GA model performs. Usually, one would expect the GA model to perform better in the long run since it can optimize the set of heuristics to a global maximum by using the updating routine to make small adjustments (Anufriev et al., 2015). However, every market in this experiment only consists of 15 periods so it will be harder to find the best solutions for the GA model. Anufriev et al. (2015) also notice that it takes some time for the algorithms to optimize to an accurate value for the parameters.

This study contributes to describing and explaining experiments with the same design as Smith et al. (1988). To the best of our knowledge, a heterogeneous agent model where switching between heuristics is possible has never been applied to these experiments before. Our study also contributes to help understand how bubbles and crashes come to exist. Especially the HSM will contribute to the understanding of market crashes since one can expect a sudden change in fractions of the agents (e.g., Anufriev & Hommes, 2012 and Bolt et al., 2014). The introduction of two new heuristics that use the individual beliefs of the subjects will contribute to the literature of the heuristic switching model as well. If the HSM functions better with these heuristics, it might also be interesting to introduce these heuristics when estimating a HSM on real values of, for instance, the stock market. This seems feasible since

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there exist forecasting bureaus which can provide forecasts of individual stocks on a daily basis. An extended explanation of these heuristics can be found in Section 3.

This study also complements the literature on GA since it will provide more information on how the GA model will work with a small number of periods. In earlier work the GA model, where the heuristics will be updated every period, is only tested on data with 50 periods or more which ensured that it had enough time to update and find the global maximum. Here, however, every market only consists of 15 periods which may not be long enough for the GA operators to optimize and find the global maximum.

The thesis is organized as follows. Section 2 gives of a short explanation about the data and the experiment conducted by Haruvy et al. (2007). Section 3 explains both the HSM and the GA model that are used for this paper. Section 4 will report our findings and finally, Section 5 concludes.

II. LAB EXPERIMENT

The data used for this research initially comes from an experiment that was conducted by Haruvy et al. (2007) at the Emory University, located in Atlanta, Georgia, USA. Each of the 𝐺 = 6 sessions consisted of 𝑖 = 9 students1 _{who were inexperienced with asset market}

experiments. Except for the first session, every session consisted of 𝑀 = 4 consecutive markets with each 𝑇 = 15 periods2_{. The setup is based on the well-known experiment of Smith et al.}

(1988) where they used the same design for a single asset market as that is used in this experiment. Before the students could start with the experiment, they received a short explanation about the experiment i.e., the use of the market software and instructions about the course of the experiment.

The asset used for this experiment is a dividend paying asset meaning that, at the end of every period, the subjects receive dividend. The fundamental value of a unit of the asset is determined by multiplying the expected dividend and the number of periods left. The dividend that will be payed to the subjects is randomly chosen from the set 𝐷 = {0, 4, 14, 30}. The expected dividend value every period is thus equal to 12 francs so, multiplying that by 15

1 _{Session 4 had only 8 students.}

CHAPTER II. LAB EXPERIMENT

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gives the face value of the asset at the beginning of the market (180 francs). This also means that a unit of the asset is worthless after the final period when the last dividends are distributed since the value of the asset is equal to the expected return you could receive from owning an asset. Prior to the experiment, all the subjects received a table with an explanation about this calculation so that the face value of the asset in every period is known to all the subjects. The assets are distributed at the beginning of each market. Three random subjects receive an initial endowment consisting of 1 unit of the asset and 472 francs. Three random other subjects receive 2 units of the asset and 292 francs and the rest receives 3 units and 112 francs as initial endowment. Since the fundamental value of the asset is 180 at the beginning of the market, one can note that every initial endowment is equal in value and that the total value of the initial endowment is 652 francs.

The individual cash balances could change every period because dividends and sales increase them, and purchases of new assets decrease these balances. Afterwards, the cash balance will be carried over to the next period but disappears at the final period since they will receive a new initial endowment at the beginning of the next market. The total individual cash balances of all markets added together determines a part of the monetary reward the subjects receive for their participation in the experiment.

Every period, the subjects were asked to submit an ask price with a maximum number of shares, meaning the price for which they would sell a share and the maximum number of shares that they would like to sell for that particular price. The subjects also had to submit a bid which consisted of a bid price and a maximum number of shares the individual would like to buy for that price. After everyone submitted their bids and asks, the demand and supply curves were constructed where the intersection between the two curves led to the equilibrium price in that period. Subjects who submitted a bid higher than the market prices made purchases and subjects who submitted an ask lower than the market price were selling their shares. If there was a tie between the subjects over the last buy or sell, the buyer or seller was chosen randomly.

The subjects were also asked to submit their forecast beliefs about the price meaning that, at the beginning of every period, the subjects had to submit a forecast about the equilibrium price in that period (short-term belief) and about the equilibrium price in all future periods (long-term beliefs). This means that the subjects at the beginning of period 1 had to

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submit 1 short-term belief about 𝑡 = 1 and 14 long-term beliefs about period 2 to 15. At the end of every market, the subjects had made 120 forecasts of which were 15 short-term beliefs and 105 long-term beliefs. The more accurate these predictions, the higher the monetary reward for the subject which made sure the subjects had an incentive to submit reliable forecasts. The participants did not know the forecasts of the other participants, so they cannot use these beliefs to determine what the market price will be. The notation that will be used for the beliefs here will be 𝐵_{𝑚𝑔𝑖𝑡}𝑚𝑔,𝑡+𝑘, where 𝑚 and 𝑔 indicate the market and session respectively in which the belief is submitted, the 𝑖 stands for the number of the subject, the 𝑡 stands for the period the belief is submitted and 𝑡 + 𝑘 indicates the period where the forecast is about. About 51% of the short-term beliefs was above the actual realized price in markets 2-4 while in market 1, this was only 39%. The long-term beliefs, however, were three times as often above the realized prices which indicates that the subjects were optimistic in the long run.

The following information is known to every subject at the beginning of period 𝑡: the market prices of period 1 until period 𝑡 − 1, the number of units the subject bought and sold, the cash he or she received with sales and spent on assets, the dividend, the face value of the asset at every period, his or her cash balance that period, the income he or she earned with his or her predictions and finally the total income he or she has earned up until period 𝑡.

In Figure 1, the equilibrium prices averaged over all 6 sessions of the four consecutive markets are displayed.3 _{In the figure, market 1 exhibits a bubble and crash pattern where the}

price starts below the face value, rises until it reaches a peak and then plummets until it reaches the face value again. An explanation for this could be the lack of experience the participants have when it comes to asset markets. The magnitude of this bubble decreases in repeated markets. The growing experience of the participants when the market is repeated could be the explanation of this. The fact that there is almost no bubble and crash pattern anymore is in line with Lei & Vesely (2009) who conduct a comparable experiment but start with giving the subjects more information about the dividend which makes them more experienced when starting the experiment.

3 _{Note that the first session only consisted of 3 consecutive markets so the equilibrium prices in market 4 are} averaged over only 5 sessions.

CHAPTER II. LAB EXPERIMENT

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Figure 1 The equilibrium price averaged over all sessions and face value of the asset in every market

III. MODELS

3.1 The heuristic switching model (HSM)

The heuristic switching model that is going to be used in this research is a 4-type heuristic switching model which was first introduced by Anufriev and Hommes (2012). The model is based on the fact that every person uses different simple forecasting heuristics to predict future prices. These expectations are dependent on the observed market prices, and the realized market prices are dependent on what people forecast which makes this a market of mutual feedback. The HSM will estimate the one-period-ahead market price where a fitness measure of every heuristic is used as a weighting method to determine the fractions of every heuristic.

The forecast error that will be used to calculate the fitness measure is the mean squared error which will look like this:

(1) 𝑀𝑆𝐸ℎ,𝑡−1= (𝐸ℎ,𝑡−1(𝑃𝑡−1) − 𝑃𝑡−1) 2

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where 𝐸ℎ,𝑡−1(𝑃𝑡−1) represents the expectation of 𝑃𝑡−1 of heuristic ℎ at time 𝑡 − 1, and 𝑃𝑡−1 is

the equilibrium price at time 𝑡 − 1. Thus, only the prediction of period 𝑡 − 1 will have an effect on the fitness measure. Hereafter, the fitness measure can be calculated using the following formula:

(2) 𝑈ℎ,𝑡−1=

−

𝑀𝑆𝐸_ℎ,𝑡−1 ∑𝐻_𝑖=1𝑀𝑆𝐸_{𝑖,𝑡−1}

,

where 𝑈ℎ,𝑡−1 is the fitness measure for heuristic ℎ in period 𝑡 − 1. This formula ensures that a

smaller forecast error yield a higher value for the fitness measure.

When the fitness measure is determined for every heuristic, the fraction of agents following heuristic ℎ can be determined by a logistic switching model with a-synchronous updating:

(3) 𝑛ℎ,𝑡= 𝛿𝑛ℎ,𝑡−1+ (1 − 𝛿)

𝑒𝛽𝑈ℎ,𝑡−1 ∑4_𝑖=1𝑒𝛽𝑈𝑖,𝑡−1

,

where 𝑛ℎ,𝑡 is the fraction of people that follows heuristic ℎ at period 𝑡. It is a-synchronous since

only a part (1 − 𝛿) of the agents can switch from heuristic, where 𝛿 is called inertia. We will be using 𝛿 = 0.14_{, so at least 10% of the agents will have the same heuristic as last period. The} 𝛽 in the formula is the intensity of choice. The higher the 𝛽, the faster the agents will switch between heuristics if there is another heuristic that performs better. Since the 𝛽 is not estimated in the model, a grid search will be conducted to find its optimal value. Note that the sum of all fractions over all heuristics will be equal to 1. Also note that the fitness measure from 𝑡 − 1 is used which means the fraction can be determined without using values from period 𝑡.

When all the fractions are known, the estimated price of the model can be calculated by the following formula:

(4) 𝑃𝑡𝑒= ∑𝐻ℎ=1𝑛ℎ,𝑡𝐸ℎ,𝑡(𝑃𝑡)

Here 𝐸ℎ,𝑡(𝑃𝑡) is the expectation of heuristic ℎ in period 𝑡 which are determined by predefined

formulas described below. This model estimates the price one period ahead only using

CHAPTER IV. EMPIRICAL ANALYSIS

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information that is known up until period 𝑡 − 1. The heuristics that are going to be used are from Anufriev & Hommes (2012) which are the following:

Adaptive Expectations (ADA)

The adaptive expectations rule (e.g., Sargent, 1993 and Evans & Honkapohja, 2001) is based on the fact that people like to update their heuristic using the price from the last period and their expectation of that price. This means that the formula form of the heuristic looks like this:

(5) 𝐸1,𝑡(𝑃𝑡) = 𝑤𝑃𝑡−1+ (1 − 𝑤)𝐸1,𝑡−1(𝑃𝑡−1) 𝑤 𝜖 [0,1] ,

where 𝑤 is the weight of the equilibrium price of last period and 𝐸1,𝑡−1(𝑃𝑡−1) is the expectation

about the price of the adaptive heuristic from period 𝑡 − 1. Note that when 𝑤 = 1, we have naive expectations because the expectation of this heuristic will be equal to the price from period 𝑡 − 1. When 𝑤 = 0, the expectation of this heuristic will be equal to 𝐸1,1(𝑃𝑡−1) every

period, so the first expected price of this heuristic.

Weak trend-following rule (WTR)

Following Anufriev & Hommes (2012), two different trend-following rules are used. The first rule is the Weak trend-following rule which uses the last price and adjusts its direction using the last price change. The WTR heuristic looks like this:

(6) 𝐸2,𝑡(𝑃𝑡) = 𝑃𝑡−1+ 𝑔1(𝑃𝑡−1− 𝑃𝑡−2) 𝑔1 𝜖 (0,1)

Here 𝑔1, the extrapolation coefficient, is between 0 and 1 so that it will have a converging effect

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Strong trend-following rule (STR)

The third rule is the strong trend-following rule which has, instead of a coefficient between 0 and 1, an extrapolation coefficient higher than 1. This means that this heuristic has a diverging effect from the trend. Except for the fact that the coefficient borders are different, the rule looks the same as the weak trend-following rule:

(7) 𝐸3,𝑡(𝑃𝑡) = 𝑃𝑡−1+ 𝑔2(𝑃𝑡−1− 𝑃𝑡−2) 𝑔2 𝜖 (1, ∞)

Learning anchoring and Adjustment rule (LAA)

The next rule is the learning anchoring and adjustment rule (Tversky and Kahneman, 1974). This rule uses an ‘anchor’ that describes the long-run price level so, when the prices are deviating a lot from this level, the rule will estimate a value closer to this long-run price level than other rules. The expectation is that the fraction of people following this rule will increase when a bubble is about to crash. The rule looks like this:

(8) 𝐸4,𝑡(𝑃𝑡) = 0.5(𝑃𝑡−1+ 𝑃̅𝑡) + 𝑔3(𝑃𝑡−1− 𝑃𝑡−2) 𝑔3 𝜖 (0, ∞)

As can be seen, the second part of the rule looks identical to the trend-following rules but the first part contains 𝑃̅𝑡, the average equilibrium price up until 𝑡 − 1.5 This means that the value

of 𝑃̅𝑡 will differ every period so one can say that it ‘learns’ something every period about the

long-run price level.

5_𝑃̅ 𝑡 = 1 𝑡−1∑ 𝑝𝑖 𝑡−1

CHAPTER IV. EMPIRICAL ANALYSIS

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Trend-following rule with median short-term beliefs (TRM)

The next two heuristics in this research are based on the beliefs of the subjects that are available in this experiment. The first heuristic extrapolates the difference of the median of the short-term beliefs and then adds 𝑃𝑡−1, the equilibrium price of last period. For the differences in

short-term beliefs, the median of the beliefs at 𝑡 about 𝑡 deducted by the median of the beliefs at 𝑡 − 1 about 𝑡 − 1 is used. The medians are taken instead of the average since the median values are more robust for outliers. The heuristic looks like this:

(9) 𝐸5,𝑡(𝑃𝑡) = 𝑃𝑡−1+ 𝑔4(𝑀𝐵𝑚𝑔𝑡 𝑚𝑔𝑡

− 𝑀𝐵_{𝑚𝑔𝑡−1}𝑚𝑔𝑡−1) 𝑔4 𝜖 (1, ∞) ,

where 𝑀𝐵_𝑚𝑔𝑡𝑚𝑔𝑡 stands for the median of the short-term beliefs at period 𝑡. The short-term belief is submitted by the subject at the beginning of period 𝑡 about period 𝑡 and is based on his expectations. In real stock markets there are also certain forecast bureaus which give daily forecasts that can be compared with the median short-term beliefs in this experiment. Since this rule is based on the strong trend-following rule, it can be used instead of STR in the 4-type heuristic switching model.

Learning anchoring and adjustment rule with median short-term beliefs (LAM)

The following rule will also use the median short-term beliefs submitted by the subjects but now, the rule will be based on the learning anchoring and adjustment rule. This means that it will contain a learning anchor:

(10) 𝐸6,𝑡(𝑃𝑡) = 0.5(𝑃𝑡−1+ 𝑃̅𝑡) + 𝑔5(𝑀𝐵𝑚𝑔𝑡 𝑚𝑔𝑡

− 𝑀𝐵_{𝑚𝑔𝑡−1}𝑚𝑔𝑡−1)

This rule is a combination of LAA and TRM since this rule contains the first part of LAA and the second part of TRM. Since this rule also contains an anchor, it can be used instead of LAA in a 4-type HSM to compare the different correlations and MSEs of the different models.

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Fundamental price rule (FUN)

Looking at markets 3 & 4 in Figure 1, we notice that the price of the asset converges to the fundamental value of the asset which is why another heuristic will be added to the HSM. The heuristic will look as follows:

(11) 𝐸7,𝑡(𝑃𝑡) = 𝑃𝑡 𝑓𝑢𝑛𝑑

,

where 𝑃_𝑡𝑓𝑢𝑛𝑑 stands for the fundamental price in period 𝑡. This fundamental price is equal to the expected return of the asset in the remaining periods which corresponds to the expected dividend times the remaining periods. The fundamental price of the asset is known to all the subjects since it is explained in the beginning of the experiment. This heuristic will not replace one of the other heuristics but will be added to the HSM with ADA, WTR, STR & LAA so this HSM will transform into a 5-type HSM.

As can be seen in the heuristics, at least two lagged values of 𝑃𝑡 are necessary to

calculate the value of the WTR, STR and LAA heuristics. This means that the first two values for these heuristics need to be estimated in a different manner. The 𝑃1 and 𝑃2 can for instance

be replaced by the median of the short-term beliefs.6 _{Only the ADA heuristic is able to work}

in period 2 but it does use the prediction ADA made in period before. For this reason, we use the price of period 1 as the expectation of ADA in period 1.7_{There is a short recap of the main}

formulas used for the HSM:

𝑈ℎ,𝑡−1= − 𝑀𝑆𝐸ℎ,𝑡−1 ∑𝐻𝑖=1𝑀𝑆𝐸𝑖,𝑡−1 𝑛ℎ,𝑡= 𝛿𝑛ℎ,𝑡−1+ (1 − 𝛿) 𝑒𝛽𝑈ℎ,𝑡−1 ∑4 𝑒𝛽𝑈𝑖,𝑡−1 𝑖=1 𝑃𝑡𝑒= ∑ 𝑛ℎ,𝑡𝐸ℎ,𝑡(𝑃𝑡) 𝐻 ℎ=1

6 _{Taking the median of the short-term beliefs will not influence the difference in MSE if the HSM would} be compared with the MedBel or Medbel3Trad models of Carle et al. (2015). These models will be discussed in detail in Section 3.4.

7 _{It may occur that the heuristics predict a negative price. To avoid this problem, the maximum of zero} and the heuristics is taken.

CHAPTER IV. EMPIRICAL ANALYSIS

17

3.2 The genetic algorithms (GA) model

The genetic algorithms (GA) model consists of multiple evolutionary operators which are derived from the process of how DNA of biological organisms adapts to the environment. It is often used to solve difficult optimization problems since these operators are efficient in searching through the whole parameter space to find the optimal values. The GA model has proven itself to be a good forecasting model of positive & negative feedback markets (Anufriev et al., 2015). For this study, the GA model that will be used in this paper will be the same as the one that Anufriev et al. (2015) used.

The GA model consists of 6 agents who cannot observe the predictions of each other which means they all learn individually. Each agent has his own set of 20 heuristics which they update with the GA routine. At the start of the GA routine, 20 binary strings of 40 bits will be randomly chosen and will form the ‘chromosomes’ of the GA model. Each of the 𝐻 = 20 heuristics has its own values for the parameters in the following formula:

(12) 𝑝_{𝑖,ℎ,𝑡}𝑒 = 𝛼𝑖,ℎ𝑝𝑡−1+ (1 − 𝛼𝑖,ℎ)𝑝𝑖,𝑡−1𝑒 + 𝛾𝑖,ℎ(𝑝𝑡−1− 𝑝𝑡−2) 𝛼 𝜖 (0,1) & 𝛾 𝜖 (0,1.1) ,

where 𝑝𝑖,𝑡−1𝑒 stands for the price prediction of agent 𝑖 in period 𝑡 − 1 and 𝑝𝑖,ℎ,𝑡𝑒 is the price

expectation of agent 𝑖 of heuristic ℎ in period 𝑡. The values for 𝛼 and 𝛾 are determined by the binary chromosomes where the first 20 bits of each binary string will form the 𝛼 and the second 20 bits will be the 𝛾. Thus, the only difference between the 20 heuristics will be the value of 𝛼 (price weight) and 𝛾 (trend extrapolation coefficient). A version of this rule is also used in Heemeijer, Hommes, Sonnemans & Tuinstra (2009), where it also had an anchor at the fundamental price8_{. As can be derived from the extrapolation part of the heuristic, this formula}

can only be used if 𝑡 > 2. After the values for 𝛼 and 𝛾 are randomly chosen, the evolutionary operators will learn every period which will make the parameters changing over time. The HSM lacks this kind of learning in the heuristics since the parameters of the heuristics are set on a predefined value and thus are not changing over time. The evolutionary operators will be the following:

8 _{This anchor is set to zero as in Anufriev et al. (2015). The formula with the anchor at the fundamental} price is also tested for this study but it turns out that the value for Spearman correlation was not significantly different and the MSE was higher comparing it with their respective values of the formula without an anchor for the fundamental.

18

Mutation: There is a small possibility that every bit of the chromosome can mutate, which means that the value of the binary bit will change from zero to one or the other way around. When a mutation takes place on a chromosome, it will have an impact on the value of 𝛼 or 𝛾. This impact can be significant when the mutation takes place at the beginning of a chromosome but when it takes place at the end, the numerical change will be negligible.

Crossover: The next operator will be the crossover operator. Similar to DNA of biological organisms, crossover here means that a pair of chromosomes can exchange a predefined part of their binary string with each other. In this paper that will be the exchange of 𝛾, thus the last 20 bits of the chromosome. Not every chromosome will undergo a crossover but there will be a predefined probability with which a crossover will occur.

Election: After these two operators, an election (Arifovic 1991, 1994) will take place. During the election, heuristics from the new generation will be compared with heuristics from the old generation based on their MSE. Every heuristic from the new generation will be compared with a randomly chosen heuristic from the old generation. If the MSE of the heuristic from the new generation is lower, the new heuristic will take the old heuristic’s place. This operator makes sure that the routine will not diverge from the global solution due to unnecessary experimentation.

Reproduction: After the GA routine with the three operators above ran once, the reproduction operator will be the first operator of the new GA routine. This operator makes sure that heuristics with a higher fitness will have a better chance of reproducing itself comparing with the other heuristics. The fitness measure that will be used can be calculated by the following formula:

(13) Π𝑖,ℎ,𝑡=

exp(−𝑀𝑆𝐸_{𝑖,ℎ,𝑡−1)} ∑𝐻 exp(−𝑀𝑆𝐸_{𝑖,𝑘,𝑡−1)}

𝑘=1

CHAPTER IV. EMPIRICAL ANALYSIS

19

Every agent independently updates his heuristics every period following the routine above where after they pick, with the probabilities as in formula (13), one of the heuristics to predict the price of period 𝑡. The average of the 6 prices predicted by the 6 agents will form the price predicted by the GA model.

For the sake of reproducibility, this is a short recap what happens at time 𝑡 (note that this routine will start at 𝑡 = 3) 9_:

I. The market prices until 𝑝𝑡 are known and can be observed by agents.

II. Now agents are going to update their heuristic independently using the GA routine described above:

a. Reproduction: 20 ‘child’ heuristics are sampled from the 20 ‘parent’ heuristics (heuristics of last period) using the fitness measure Π𝑖,ℎ,𝑡 as

the corresponding probabilities. Note that the reproduction operator will work from period 4 since it needs the routine to have run at least once otherwise the fitness measure cannot be calculated.

b. Mutation: Every bit of the chromosomes of the child heuristic has a 𝛿𝑚 =

0.07 probability to change its value from 0 to 1 or vice versa

c. Crossover: child chromosomes are picked with probability 𝛿𝑐 = 0.7 and

randomly paired for a crossover of the last 20 bits, thus exchanging the value of 𝛾.

d. Election: the MSE of each child heuristic is compared with a randomly chosen parent heuristic’s MSE. If the child has a lower MSE, it takes the place of the parent. If this is not the case, the parent will join the set of heuristics of period 𝑡 + 1.

III. When the new set of heuristics is formed, agent 𝑖 will choose a heuristic with the probabilities equal to their fitness from formula (13).

9_{Since the GA routine will not work in period 1 and 2, the short-term median of the beliefs is taken as} the predicted price by the GA model. This will make sure that the difference in MSE will not be caused by these 2 periods but by the rest of the periods.

20

3.3 Empirical Fit

The variables that are going to be compared to each other between the different models are the Spearman correlation coefficient and the mean squared error (MSE). The Spearman correlation coefficient is a measure of a monotone association that is used when the distribution of data makes Pearson’s correlation coefficient undesirable or misleading. It assesses how well an arbitrary monotonic function can describe a relationship between two variables, without making any assumptions about the frequency distribution of the variables (Hauke & Kossowski, 2011). The coefficient is equal to 1 when there is a perfect positive monotone relationship and equal to -1 when there is a perfect negative monotone relationship. This correlation coefficient does not require the normality assumption like the Pearson correlation coefficient which makes it a non-parametric statistic.

Because the Spearman correlation will only say something about the similarity in direction between the compared models, the MSE will also be calculated to compare the models concerning the estimation error. The MSE will be calculated by the following formula:

𝑀𝑆𝐸 =1 𝑇∑(𝑃𝑖 𝑒_{− 𝑃} 𝑖)2 𝑇 𝑖=1

The MSE will give a measure of the magnitude of the errors made by the model.

3.4 The models from Carle, Lahav, Neugebauer & Noussair (2015)

Now, the models of Carle et al. (2015), henceforth CLNN15, will be shortly discussed since they will serve as comparing material for the HSM and GA model. The comparing variables, the Spearman correlation and MSE, are calculated and mentioned below.

MedBel is a model that uses the median of the short-term belief within the cohort as the prediction of the market price of that period. The median is used instead of the average since the median is more robust for outliers. The average Spearman correlation, averaged over all sessions, is 0.8735. The mean squared error of this model is 1522.3, also averaged over all sessions.

MedBel3Trad is the next model that is used by CLNN15 which is almost the same as MedBel, but now, the median of the best three traders is used to forecast the market price of

CHAPTER IV. EMPIRICAL ANALYSIS

21

that period. The average Spearman correlation coefficient of this model is 0.8776, and the MSE is 1436.6. Note that MedBel and MedBel3Trad require information about the current beliefs of all subjects and moreover the best traders can only be determined after the final period of the last market in a session.

The Next model is AdapPer. This model, unlike the previous ones, uses only lagged values of the individual beliefs to estimate the observed prices. Conjectured is that the stated belief changes proportionally to the forecasting error of the last period as follows:

𝐵_{𝑚𝑔𝑖𝑡}𝑚𝑔𝑡

𝐵_{𝑚𝑔𝑖𝑡−1}𝑚𝑔𝑡 = 𝛼 (

𝑃𝑚𝑔𝑡−1

𝐵_{𝑚𝑔𝑖𝑡−1}𝑚𝑔𝑡−1)

where 𝐵_{𝑚𝑔𝑖𝑡−1}𝑚𝑔𝑡 means the belief about time 𝑡 submitted at time 𝑡 − 1 by subject 𝑖 in market 𝑚 of session 𝑔. The left-hand side is a ratio which shows the adjustment of current short-term beliefs from prior beliefs about the current price. The right-hand side represents the observed price from last period relative to the short-term prediction introduced by subject 𝑖 in period 𝑡 − 1. Now a logarithmic transformation can by conducted which can be followed by a random-effects regression. The result of this regression will be fitted to the price trajectory as a one-period-ahead estimation model:

ln(𝐵_{𝑚𝑔𝑖𝑡}𝑚𝑔𝑡) ≡ 𝑏_{𝑚𝑔𝑖𝑡}𝑚𝑔𝑡 = −0.202 + 0.597𝑏_{𝑚𝑔𝑖𝑡−1}𝑚𝑔𝑡 + 1.033𝑝𝑚𝑔𝑖𝑡−1− 0.591𝑏𝑚𝑔𝑖𝑡−1 𝑚𝑔𝑡−1

where 𝑏 and 𝑝 denote the logarithm of 𝐵 and 𝑃. During the fitting process, the median of the beliefs is taken since it is known that this will create a better result. The average Spearman rank correlation coefficient of this model is 0.8456, and the MSE averaged over all sessions is 2085.7.

22

AdapIni is the last model viewed by CLNN15. This model is almost the same as AdapPer but now, only beliefs submitted in period 1 are used. The result is the following linear model:

ln(𝐵_{𝑚𝑔𝑖𝑡}𝑚𝑔𝑡) ≡ 𝑏_{𝑚𝑔𝑖𝑡}𝑚𝑔𝑡 = −0.168 + 0.213𝑏_{𝑚𝑔𝑖1}𝑚𝑔𝑡 + 1.064𝑝𝑚𝑔𝑖𝑡−1− 0.245𝑏𝑚𝑔𝑖1 𝑚𝑔𝑡−1

This model will also be used to estimate the price trajectory but now by using only the median of the initial beliefs. The result is an average Spearman correlation coefficient of 0.8415 and an MSE of 2307.1.

IV. EMPIRICAL ANALYSIS

4.1 The heuristic switching model

A 4-type and 5-type heuristic switching model are used to fit the experiment. Since we have seven different heuristics, multiple combinations can be made for the HSM. Here is a short recap of the heuristics that will be used:

(ADA) 𝐸1,𝑡(𝑃𝑡) = 𝑤𝑃𝑡−1+ (1 − 𝑤)𝐸1,𝑡−1(𝑃𝑡−1) 𝑤 𝜖 [0,1] (WTR) 𝐸2,𝑡(𝑃𝑡) = 𝑃𝑡−1+ 𝑔1(𝑃𝑡−1− 𝑃𝑡−2) 𝑔1 𝜖 (0,1) (STR) 𝐸3,𝑡(𝑃𝑡) = 𝑃𝑡−1+ 𝑔2(𝑃𝑡−1− 𝑃𝑡−2) 𝑔2 𝜖 (1, ∞) (LAA) 𝐸4,𝑡(𝑃𝑡) = 0.5(𝑃𝑡−1+ 𝑃̅𝑡) + 𝑔3(𝑃𝑡−1− 𝑃𝑡−2) 𝑔3 𝜖 (0, ∞) (TRM) 𝐸5,𝑡(𝑃𝑡) = 𝑃𝑡−1+ 𝑔4(𝑀𝐵𝑚𝑔𝑡 𝑚𝑔𝑡 − 𝑀𝐵_{𝑚𝑔𝑡−1}𝑚𝑔𝑡−1) 𝑔4 𝜖 (1, ∞) (LAM) 𝐸6,𝑡(𝑃𝑡) = 0.5(𝑃𝑡−1+ 𝑃̅𝑡) + 𝑔5(𝑀𝐵𝑚𝑔𝑡 𝑚𝑔𝑡 − 𝑀𝐵_{𝑚𝑔𝑡−1}𝑚𝑔𝑡−1) 𝑔5𝜖 (0, ∞) (FUN) 𝐸7,𝑡(𝑃𝑡) = 𝑃𝑡 𝑓𝑢𝑛𝑑

Here 𝑀𝐵_𝑚𝑔𝑡𝑚𝑔𝑡 stands for the median of all the beliefs about time 𝑡 submitted at the beginning of time 𝑡 in market 𝑚 during session 𝑔. In the heuristics TRM and LAM, the medians of the short-term beliefs are used which means these heuristics use exactly the same information about the beliefs as the MedBel model of CLNN15. Note however that the MedBel3Trad is more demanding with respect to the information used since it also uses the information about who are the best traders (which can only be determined after all 4 markets of the session are finished).

CHAPTER IV. EMPIRICAL ANALYSIS

23

First, the homogeneous agent models each consisting of one heuristic will be discussed. A grid search is conducted to find the best values for the parameters for each heuristic. The objective of the grid search was to find the best Spearman correlation. Table 1 will display these results:

Heuristic Spearman correlation Value Parameter

Adaptive expectations (ADA) 0.8276 𝑤 = 1.00

Weak trend-following rule (WTR) 0.8661 𝑔1 = 0.87 Strong trend-following rule (STR) 0.8657 𝑔2= 1.19 Learning Anchoring and adjustment (LAA) 0.8206 𝑔3= 1.46 Trend-following with median beliefs (TRM) 0.8717 𝑔4= 0.86 Anchoring with median beliefs (LAM) 0.8114 𝑔5= 1.10 Fundamental price rule (FUN) 0.3738 No parameter

Table 1 The Spearman correlation of the homogeneous agent models of every heuristic

The homogeneous agent model with the heuristic trend-following rule with median short-term beliefs (TRM) has the highest Spearman correlation. This heuristic also uses the beliefs submitted by the subjects which give a good indication about the direction of the prices.

As for the homogeneous agent models, a grid search will be conducted for the heuristic switching models to find the best values for all parameters in the heuristics and 𝛽 (intensity of choice). Note that these parameters will not necessarily also form the best model in terms of MSE. Because of the grid search, the parameters are all different when different combinations of heuristics are used. Below the results of the 4-type HSMs and 5-type HSM are shown:

In Table 2, a dash means that this parameter does not occur in the heuristics used in that HSM. From the four optimal combinations of the 4-type HSM that are found by the grid search, the

10 _{This HSM with ADA, WTR, STR and LAA has exactly the same values for the parameters as used in} Anufriev & Hommes (2012)

Model + Heuristics Correlation w 𝒈𝟏 𝒈𝟐 𝒈𝟑 𝒈𝟒 𝒈𝟓 𝜷 HSM with ADA, WTR, TRM & LAM 0.8802 1.00 0.82 - - 1.18 1.36 44

HSM with ADA, WTR, STR & LAM 0.8745 1.00 0.80 1.14 - - 1.36 44

HSM with ADA, WTR, STR & LAA 0.8720 0.80 0.99 1.23 1.24 - - 18

HSM with ADA, WTR, STR & LAA10 _{0.8483 0.65 0.40 1.30 1.00 - - 0.4}

HSM with ADA, WTR, TRM & LAA 0.8767 0.90 0.79 - 1.41 1.26 - 12

HSM with ADA, WTR, STR, LAA & FUN 0.8923 0.00 0.43 3.94 3.22 - - 4

24

model with ADA, WTR, TRM & LAM has the highest average Spearman correlation. This model has 𝑤 = 1 as parameter for ADA which means that the expectation of that heuristic is equal to the price of last period (naive expectations). The HSM with the heuristics ADA, WTR, STR and LAA with optimal parameters can also be compared with the HSM used by Anufriev & Hommes (2012) where different values for the parameters were used. As can be seen in Table 2, the correlation of the HSM with the optimal parameters found by the grid search has a higher Spearman correlation than the model used by Anufriev & Hommes (2012) which makes sense since the grid search looks for the optimal values for the highest possible correlation and the values for the parameters used by Anufriev & Hommes (2012) were also part of that parameter space.

While the fundamental price rule as homogeneous model has a very low Spearman correlation, the 5-type HSM outperforms all the 4-type HSMs. This indicates that adding the fundamental rule improves the HSM and that this rule is of added value for the HSM in periods when the HSM’s predictions could be improved.

The Spearman correlations of the HSMs also exceed the correlations of the homogeneous agent models. The least-performing HSM, the HSM with ADA, WTR, STR & LAA, found by the grid search already has a higher Spearman correlation (0.8720) than the best-performing homogeneous agent model (TRM heuristic with a Spearman correlation of 0.8717). This also makes sense since the HSM will switch to better-performing heuristics every period while that is not possible in a homogeneous agent model. Every period, the HSM will calculate the fitness measure which determines the fractions of every heuristic. The smaller the estimation error made by the heuristic, the higher the fraction will be. This will result in a higher Spearman correlation coefficient than the homogeneous models since the homogeneous agent models cannot switch from heuristics when a heuristic is underperforming.

The average Spearman correlation of 0.8802 of the best HSM here is also better than the correlation of MedBel (0.8735). So, with the same information used, it is possible to fit a model with a higher correlation on the experiment than MedBel. While MedBel3Trad uses more information than the best-performing HSM, the Spearman correlation of 0.8802 is still higher than the Spearman correlation of MedBel3Trad, which is 0.8776. However, this HSM cannot be compared with AdapPer or AdapIni since both of these models use less information about

CHAPTER IV. EMPIRICAL ANALYSIS

25

the beliefs than the heuristics TRM & LAM and predictions used for the first two periods.11 _For

this reason, a HSM with ADA, WTR, STR and LAA using only the initial beliefs for period 1 and 2 is fitted. The average Spearman correlation of that model was 0.8525 which is both higher than AdapPer and AdapIni while AdapPer even uses more information about the beliefs than this heuristic switching model. Thus again, the HSM gives a higher average Spearman correlation than the models used by CLNN15.

The next grid search will be conducted with the objective to find the lowest MSE in the parameter space. The heuristics use two lagged values so, for comparing purposes, we use the median of the short-term belief in the first two periods since that will not affect the difference in MSE between the HSMs and the models of CLNN15.12 _{The results of the homogeneous agent}

models are shown in Table 3:

Heuristic MSE Value Parameter

Adaptive expectations (ADA) 2133.3 𝑤 = 1.00

Weak trend-following rule (WTR) 1696.6 𝑔1= 0.61 Strong trend-following rule (STR) 1790.6 𝑔2= 1.01 Anchoring and adjustment (LAA) 1901,9 𝑔3= 1.26 Trend-following with median beliefs (TRM) 1496.7 𝑔4= 0.66 Anchoring with median beliefs (LAM) 1946.2 𝑔5= 1.05 Fundamental price rule (FUN) 8111.1 No parameter

Table 3 The MSE of the homogeneous agent models of every heuristic

Just as before, the TRM heuristic performs best as homogeneous model. However, the value of 𝑔4 is different this time which implies that the optimal TRM model in terms of MSE is not

necessarily the best-performing model in terms of Spearman correlation.

11 _{AdapPer only uses beliefs submitted by the subjects in period 𝑡 − 1 and AdapIni uses only the beliefs} submitted in period 1. TRM and LAM however, use beliefs submitted at the beginning of period 𝑡 so the HSM with one of these heuristics cannot be compared with both AdapPer and AdapIni. The short-term median belief of period 2 used for the prediction of period 2 of the HSM is also not in the available information of AdapPer and AdapIni which is why the median of the beliefs about period 2 submitted in period 1 will be used instead.

12 _{However as said before, AdapPer and AdapIni do not use the median short-term belief of period 2 so} again a model with only the initial beliefs will also be considered.

26 The results of the grid search of the HSM are:

As before, the HSM with ADA, WTR, TRM & LAM performs best with the lowest MSE which does make sense since it simply uses more information than the other models considered. The model with the ‘old’ heuristics (ADA, WTR, STR & LAA) is again the least-performing model with an MSE of 1357.5. This MSE, however, is still lower than the MSE of the HSM of Anufriev & Hommes (2012) which is 1622.1. While the model with the ‘old’ heuristics is the least performing model of all four 4-type HSMs, the MSE of 1357.5 is still lower than the best-performing model of CLNN15, the MedBel3Trad, which has a MSE of 1436.6. Notice also that the MSE of all the 4-type HSMs is lower than the MSE’s of the homogeneous agent models. The explanation for this is again the same as by the Spearman correlation since the HSM can simply switch to better-performing heuristics every period while the homogeneous models do not have this attribute.

Again, the 5-type HSM is outperforming the 4-type HSMs so it can be concluded that the fundamental price rule is of added value to the HSMs used for this study. As can be seen in Table 3, FUN has a MSE that is much higher than the rest of the heuristics which implies that this heuristic makes significant errors in some periods. However, because there is an improvement when this heuristic is added to the HSM, it also implies that the fundamental price rule is a well-performing heuristic in some other periods which was also expected looking at Figure 1.

Now comparing the different values for the parameters with the grid search before, we see some differences. First, the value of 𝑤 is now equal to 0.00 in every HSM while it was a lot higher in the grid search before, and even equal to 1 in two of the four HSMs. In the homogeneous model of ADA, 𝑤 = 0.00 is not optimal but apparently, when this heuristic is used in combination with other heuristics in the HSM, 𝑤 = 0.00 is the optimal value to find

Model + Heuristics MSE w 𝒈𝟏 𝒈𝟐 𝒈𝟑 𝒈𝟒 𝒈𝟓 𝜷

HSM with ADA, WTR, TRM & LAM 1244.1 0.00 0.27 - - 2.07 1.01 5

HSM with ADA, WTR, STR & LAM 1267.1 0.00 0.01 1.50 - - 2.06 2

HSM with ADA, WTR, STR & LAA 1357.5 0.00 0.01 2.86 1.30 - - 2

HSM with ADA, WTR, STR & LAA7 _{1622.1 0.65 0.40 1.30 1.00 - - 0.4}

HSM with ADA, WTR, TRM & LAA 1288.8 0.00 0.01 - 1.31 2.25 - 3

HSM with ADA, WTR, STR, LAA & FUN 1142.1 0.00 0.01 1.64 3.46 - - 3

CHAPTER IV. EMPIRICAL ANALYSIS

27

the lowest possible MSE. The 𝑔1, however, is almost equal to 0 in three of the four HSMs which

makes the second part of the WTR heuristic almost negligible. The first part of the WTR heuristic with 𝑔1= 0.01 is almost equal to the ADA heuristic with 𝑤 = 1, naive expectations,

so a heuristic with naive expectations can be found in almost all HSMs.

Second, the value for 𝛽 is a lot lower in the last grid search implying that a higher intensity of choice is better when looking for a high Spearman correlation but also results in a bigger MSE. A reason behind this could be that, when searching for the best monotone association, the HSM fractions have to change faster if a heuristic performs better than another one than when the grid search objective is a low MSE. The third remarkable fact is that the coefficients of STR, TRM & LAM of the second grid search are overall higher than the coefficients in the grid search before. Apparently, this makes sure that the heuristics together will have a lower MSE.

To compare the HSM with AdapIni and AdapPer of CLNN15, the HSM with ADA, WTR, STR and LAA is again run with the initial beliefs. The MSE of that model is 1566.8 which is lower than both the MSE of AdapIni (2085.7) and AdapPer (2307.1). This means that with the same information used, the HSM performs better than the models of CLNN15, based on both correlation and MSE.

28

In Figure 2, the HSM is compared with the equilibrium price of each market and underneath the fractions of the heuristics in the same market. In each market, all the fractions up to period 3 are equal to 0.25 since the HSM cannot predict the first two periods and the fraction formula of period 3 also uses data from period 2. In market 1, the WTR and TRM heuristics are dominant when the equilibrium price is increasing which makes sense since both heuristics are trend-following. However, when the bubble is about to collapse, the ADA heuristic increases immediately while the WTR and TRM heuristics decrease. The same happens in market 2, except there, the equilibrium price is decreasing somewhat earlier. The last two markets barely have a bubble anymore and the equilibrium price from period 6 in market 3 (period 9 in market 4) to the end is almost equal to the face value of the stocks. From these periods until period 13 there is some kind of order in the heuristics since ADA has the highest fraction, then the fractions of WTR & TRM and finally the fraction of LAM is almost equal to zero. This implies that the LAM heuristic is not estimating the price very well when the price is almost equal to the face value.

Figure 3 HSM with ADA, WTR, TRM & LAM (grid search MSE)

Figure 3 contains the graphs of the HSM with ADA, WTR, TRM & LAM where the parameters of the heuristics have the optimal values to get the lowest possible MSE. Comparing this figure with Figure 2, we notice that the fractions are more stable, i.e., less volatile in this figure. The reason for this is the lower value for 𝛽 that is optimal for the lowest MSE. In market 1, one can

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also notice that only the ADA heuristic is low during the bubble while both ADA & LAM were low in market 1 in Figure 2. This is peculiar since expected was that LAM was a good heuristic to describe crashes but that it would underperform when there is a certain trend. However here, the fraction of LAM is approximately the same as the fractions of WTR and TRM. A cause of this could be that the values of the parameters of TRM and WTR, 2.07 & 0.27 respectively, are more dispersed than in these parameters in Table 2. For instance, in Table 2 we see that these parameters for TRM & WTR are 1.18 and 0.82 respectively which are both closer to 1.

When the price is about to crash in markets 1 & 2, notice that the fraction of the ADA heuristic suddenly increases. This indicates that the ADA heuristic is useful in predicting sudden crashes. However, when the market is converging to face value, we see in market 3 and 4 that the fraction of ADA is one of the lowest and at the end of market 4 almost equal to zero.

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In Figure 4, the heuristic TRM is replaced by STR and one can notice that it takes on the same role as that TRM had before in the fraction plot. In market 1, the STR heuristic is also dominant when the bubble is formed but decreases one period when the bubble is about to burst. Also in the last two markets, the fractions of the trend-following heuristics are in between the fractions of ADA and LAM when the equilibrium price is almost equal to the face value. The figures of the other 4-type HSMs where the objective of the grid search was Spearman correlation are also very similar to Figures 2 & 4 which is why they will not be discussed any further. These figures are however added to the appendix.

Figure 5 HSM with ADA, WTR, STR and LAM (grid search MSE)

The fractions in Figure 5 look very similar to the fractions in Figure 3 and also here, the STR takes on the same role as that TRM had before, so the difference in coefficients apparently does not have much impact. Also notice that, while WTR with 𝑔1 = 0.01 is almost similar to ADA

with 𝑤 = 1, both heuristics have totally different fractions each period while you would expect them to substitute each other. The reason behind this is the big difference in parameter values of the other heuristics since only that could explain the difference in fitness measures of the heuristics. Since the figures of the other 4-type HSMs with the parameters found by the grid search to find the lowest MSE are nearly the same as Figure 3 & 5, these will not be discussed any further and can be found in the appendix.

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Figure 6 HSM with ADA, WTR, STR, LAA & FUN (grid search correlation)

Figure 6 shows the graphs of the 5-type heuristic switching model. All fractions now begin at 0.20 and again start to change in period 4. In market 1, the fraction of the fundamental price rule starts low but increases steadily when the equilibrium price increases. When the equilibrium price crossed the fundamental price, the fraction of FUN decreases again since the prediction error that FUN makes increases. In market 3 & 4, the fundamental price rule is one of the best performing heuristics since the equilibrium price is almost equal to the fundamental price. The figure with the 5-type HSM where a grid search is performed to get the lowest possible MSE is similar to Figure 6 which is why this figure will not be discussed any further and can be found in the appendix.

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4.2 The genetic algorithm model

Since the Genetic Algorithms model begins with randomly picking 20 values for 𝛼 and 𝛾 for all 6 agents, a Monte Carlo simulation with 1000 simulations is conducted. A Monte Carlo simulation is used here to get a significant average of the price predictions and the values for 𝛼 and 𝛾. Just as with the HSM, the GA model is not executable in the first two periods since the price of 𝑡 − 2 is needed (see formula (12)). For this reason, the short-term median of the beliefs in these periods is taken as the first two predictions of the model which resulted in an average Spearman correlation of 0.8635. This correlation is lower than the HSMs, MedBel and MedBel3Trad but higher than the rest of the models in CLNN15. An explanation for this can be the short length of the market. Anufriev et al. (2015) already noted that, in a positive feedback market, the GA take some time to optimize the heuristics. Since the market treated in this study is also a positive feedback market, it will take about 15 to 20 periods (numbers are from Anufriev et al. (2015)) for the parameters in the heuristics to converge to a certain value. This experiment only exist of 15 periods so, as can be seen in Figure 7, the parameters 𝛼 and 𝛾 will not converge to a certain value yet. Especially since the prices of the assets in this experiment take on a broader space of values i.e., higher volatility than the prices of the positive feedback market in Anufriev et al. (2015), the parameters need more periods to converge.

Another reason that the Spearman correlation of the GA model is lower than the Spearman correlation of the HSM could be that the values of the parameters of the heuristics in the HSMs are determined while already knowing all the prices. The grid search used for the parameters searches for the optimal values for every parameter by already using all the prices. This process ensures that the chosen parameters are optimal for these prices while in reality, these parameters are not known yet since the prices of the following periods are still unknown. So, by using the grid search in the HSM, the model uses more information than that is available while the GA model optimizes its heuristics after every period only using information about past prices.

Another difference between the models that could form an explanation for the differences in correlation is that underperforming heuristics will be replaced by better-performing heuristics. We saw by the HSM’s results that a heuristic while underbetter-performing for some periods, can make better estimations than other heuristics later in the market. For

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instance, the fraction graph of market 1 in Figure 2 shows that ADA is underperforming until period 10 but then increases steadily until period 15 where it is one of the best performing heuristics. The ADA is a heuristic that performs well when the market crashes in market 1 while it underperforms when the prices follow a trend. In the GA model, this heuristic would not survive until period 10 but would be driven out of the set of heuristics by better-performing heuristics in the first 10 periods. A consequence of this is that the group of heuristics will only consist of heuristics that perform well when the prices follow a trend but underperform when the bubble crashes. For this reason, the GA have some trouble predicting sudden changes in the prices after a few periods since there is no heuristic available anymore that can predict this.

While the Spearman correlation of the GA model is lower than all the HSMs, the MSE is 1272.5 which is higher than the models of CLNN15, the homogeneous models and most of the HSMs. Only the HSMs with the heuristic LAM have a MSE that is lower than the MSE found by the GA model. In Figure 7, the graphs of market 1 and 2 are shown:

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The graphs are split up by market where each market exist of 3 graphs. One graph with the estimation of the GA model, the equilibrium price and the face value, one graph with 𝛼, the price weight, and the last one with the 𝛾, the trend extrapolation coefficient, of the following GA equation:

𝑝_{𝑖,ℎ,𝑡}𝑒 = 𝛼𝑖,ℎ𝑝𝑡−1+ (1 − 𝛼𝑖,ℎ)𝑝𝑖,𝑡−1𝑒 + 𝛾𝑖,ℎ(𝑝𝑡−1− 𝑝𝑡−2) 𝛼 𝜖 (0,1) & 𝛾 𝜖 (0,1.1)

In Figure 7, the values for 𝛼 and 𝛾 each period are determined by the average of the coefficients of the chosen heuristic of every agent. In contrast to Anufriev et al. (2015), the values of 𝛼 and 𝛾 do not converge to a certain value but keep changing over time. An explanation for this can be that the prices also change frequently over time while the prices are more stable in Anufriev et al. (2015). The consequence is that other values for 𝛼 and 𝛾 will be optimal every period so these values will not converge. What is also noticeable is that the value of 𝛼, the price weight coefficient, suddenly increases in market 1 when the bubble collapses in period 13. This implies that the actual price of period 𝑡 − 1 becomes more important than the prediction in 𝑡 − 1 of agent 𝑖 submitted in period 𝑡 − 1.

There are also a few differences between the two markets that can be noted: for instance, market 2 has a lower bubble which collapses earlier than in market 1, and it also looks like the GA model has more problems predicting market 2 than market 1. The consequence of this is that the 𝛾 (the trend extrapolation coefficient) takes on values between 0.85 and about 1 in market 1 and values between approximately 0.5 and 0.95 in market 2. What this indicates is that the trend extrapolation coefficient becomes more volatile when the GA model has a more trouble predicting the equilibrium prices.

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Figure 8 The prices, 𝛼 and 𝛾 of market 3 and 4 of the Genetic Algorithms model

In both markets 3 and 4, it looks like the GA model acts one period too late. For instance, if the equilibrium price decreases, the GA model will decrease in the next period as well. A cause can be that the MSE used for the election operator is the MSE of the period before since the MSE of time t is not yet known when the election operator comes into play. It is also clear that the values for α and γ again do not converge to a particular value. However, the expectation was that they would converge in both markets since the equilibrium price also converges to the face value. α and γ seem to be fluctuating even more than in the markets before, especially for the 𝛾 in market 3 & 4 where the values fluctuate between 0.35 and 0.80. These fluctuations can be caused by the errors that the model still makes but, as said before, the length of 15 periods may also be too short for the parameters to converge to a particular optimal value so we cannot conclude much from these fluctuations.

Since in market 3 and 4, the price seems to converge to the fundamental price of the asset, the anchor of formula (12) is extended by the fundamental price. The Spearman correlation coefficient of this extension is 0.8700 which is not significantly13 _{different from the}

0.8634 found before. The MSE found by the new formula is equal to 1307.2 which is higher

13 _{Fisher’s z-transformation is used to test if the values are significantly different from each other (Myers} and Sirois, 2014)