The effects of endogenously updated impacts on the aggregate outcome in an asset pricing model : a quantitative analysis based on simulated and experimental data

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The Effects of Endogenously Updated Impacts on

the Aggregate Outcome in an Asset Pricing Model

~ Master Thesis Econometrics ~

A Quantitative Analysis based on Simulated and Experimental Data

June 2015 First Supervisor: Professor Dr. Jan Tuinstra Second Supervisor: Dr. Domenico Massaro Student: Sacha van Duren (5772303)

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S.M.C. van Duren UvA - Master Thesis Econometrics 2

These days, especially in light of the recent crisis, the formation of individual expectations regarding economic variables has gained a lot of attention since these expectations turn out to influence the actual realizations of economical variables. In this sense economics is opposed to the natural sciences where individual expectations cannot influence the actual realizations. Hommes (2011) gives an insightful example. He states that daily weather forecasts, either by the public or by experts, do not affect the real probability of rain. On the other hand, overly optimistic expectations about the economic prospects have encouraged the strong rise in worldwide financial markets in the nineties and the tremendous growth in housing prices in 2000-2008. In the same way overly pessimistic expectations about the economic prospects have encouraged the recent financial crisis and deepened the current economic crisis. Therefore analyzing individual expectations of economic agents in the right way is of great importance to understand how bubbles and crashes emerge on financial markets.

After the seminal work by Muth (1961) and Lucas (1972) it has become common practice in economics to assume that all individuals have rational expectations. In a rational environment individual expectations coincide, on average, with market realizations. On top of that markets are assumed to be efficient with prices fully reflecting economic fundamentals (Samuelson, 1965; Fama, 1970). Note that in this traditional economic view, market psychology and “irrational” herding behaviour are not taken into account. An important underpinning of the rational expectations approach comes from an early evolutionary argument made by Friedman (1953) that “irrational” traders will not survive competition and will be driven out of the market by rational traders, who will trade against them and earn higher profits. However research using so-called learning-to-forecast experiments (Hommes et al., 2000, 2005, 2007, 2008; Fehr and Tyran, 2008; Heemeijer et al., 2009; Sonnemans and Tuinstra, 2010) revealed that in positive feedback markets, like the asset market, it is profitable to imitate the behaviour of others, so in that sense it could be rational to behave irrational. In game theory this phenomenon is also known as strategic complementarity (Haltiwanger and Waldman, 1989). In fact the presence of strategic complementarity makes it more likely that agents imitate each other’s behaviour.

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S.M.C. van Duren UvA - Master Thesis Econometrics 4 Nagel (1995) provided an interesting feature of bounded rationality in so-called number guessing games and stated that individuals think accordingly a certain depth of reasoning. During the number guessing game a certain amount of players simultaneously state a number between [0,100] and the winner is the person whose number is closest to p times the mean of all chosen numbers. Note that p is a predetermined parameter and known to everyone. Nagel (1995) performed an experiment with four consecutive rounds of the number guessing game for different values of p (p=1/2, p=2/3 and p=4/3) and found that most participants in the first round, starting from initial reference point 50, state a number in the neighbourhood of 50p or 50p2_{, while the only Nash equilibrium is 0. This result was independent of the parameter}

p. In repeated number guessing games choices typically converge quickly to this Nash

equilibrium, but the participants in the experiment (Nagel, 1995) appeared to behave quite boundedly rational in the first round. The bounded rationality theory behind this game assumes that every individual thinks that he or she is the most sophisticated person of all individuals considered. Then level zero means that an individual makes a random decision, level 1 means that a certain individual thinks that everybody else makes a random decision and gives his/her best reply. This best reply in case of the number guessing game stated above would have been 50p, since the mean of all random decisions would have been 50. Level 2 means that a certain individual thinks that everybody else is level 1 and again gives his/her best reply (50p2_{) and so forth.} Sonnemans and Tuinstra (2010) linked this number guessing game to positive feedback experiments and showed that the feedback strength (i.e. the ‘p-value’ in standard number guessing games) is essential for the results. Furthermore it is likely that individuals are able to learn and therefore behave differently over time in terms of rationality (Fehr and Tyran, 2005).

In these aforementioned learning-to-forecast experiments (LtFEs) individual expectations, their interactions at the micro-level and the aggregate market behaviour they create at the macro-level are studied. In this way insight is obtained in the individual predicting behaviour on financial markets. Human subjects participating in the LtFE represent investors on a financial market. Within the laboratory experiment they typically only have qualitative information about the market available and their only task is to forecast the price of a certain asset for i.e. 50 consecutive periods. The

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S.M.C. van Duren UvA - Master Thesis Econometrics 5 realized market price in each period of the experiment is determined by a certain law of motion, depending on the average of all individual expectations. In this sense every participant or every investor within the experiment has the same impact on the realized market price. However looking at the real world it would make sense to presume that different investors have a different impact on the realized market price. To be more specific, successful investors earn more, attract more funds and therefore are likely to have a larger market impact, while the impact of unsuccessful investors is likely to diminish. So the impact of an investor on the realized market price will differ over time and will depend on the investor’s success in the previous period(s). The investors impact on the realized market price could then be computed endogenously in the model similar to the updating mechanism proposed by Brock and Hommes in the Heterogeneous Agents Model (1997). An extension of this model, the so-called Heuristic Switching Model (Anufriev and Hommes, 2012), appears to provide a good approximation of the observed data in learning-to-forecast experiments by incorporating switching between heuristics at the micro level. Individuals are able to assign a certain weight to every heuristic based on the performance of this heuristic in the previous period(s) and they are able to update these weights for every period. However in the model proposed in this thesis the evolutionary selection occurs on

macro level by endogenously updated impacts. The market assigns larger impacts to

successful investors, instead of individuals who assign larger weights to successful heuristics.

The main research question of this thesis is whether these endogenously updated impacts on the macro level in the asset pricing model have a stabilizing or destabilizing effect on the aggregate outcome. When prices are computed using the original exogenous impacts, the average price expectation for a certain period is used in the law of motion to compute the realized market price in that period. This realized market price appears to converge, diverge or shows persistent oscillations in the existing LtFEs. However if prices are computed using endogenously updated impacts, the price expectations of successful investors have a larger impact on the realized market price and these successful investors could either stabilize or destabilize the aggregate outcome due to their increasing market impact. Actually prices may converge quicker since the impact of unsuccessful investors on the realized price decreases and will

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S.M.C. van Duren UvA - Master Thesis Econometrics 6 eventually diminish in the endogenous impact market, while the impact of unsuccessful investors on the realized price in the exogenous impact market is equal to the impact of successful investors and does not decrease.

The research set-up is as follows. To examine the behaviour of individual economic agents a general learning-to-forecast experiment will be conducted and their behaviour will be analysed based on OLS estimations generating linear prediction rules. On top of that a number guessing game with the same participants will be conducted to examine whether there is a correlation in behaviour between the number guessing game and the learning-to-forecast experiment. After the classroom experiment, the individual prediction rules will be estimated for every participant and used for artificial participants in the simulations to examine the effect of the endogenously updated impact on the aggregate outcome, see Figure 1.1 for a schematic overview of this research set-up.

Figure 1.1: Schematic overview of the research set-up

The results of the learning-to-forecast experiment indicate that the participants coordinate on a common prediction strategy as in Hommes et al. (2005). Furthermore the results indicate that if a robot trader with the strongest trend parameter obtains the largest weight at some point during the endogenous impact simulation, the realized price will be strongly affected by this trend parameter through the endogenously updated impacts. So the endogenously updated impact system seems to enhance the self-fulfilling prophecy property of positive feedback markets due to a strong sensitivity to extreme forecasting behaviour. In none of the simulations where prices for exogenous

Learning-to-Forecast Experiment Simulations OLS Estimation of Individual Prediction Rules Construction Artificial Participants based on Estimated Prediction Rules

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S.M.C. van Duren UvA - Master Thesis Econometrics 7 impacts diverge or show oscillations, the endogenously updated impacts have a stabilizing effect.

This thesis is structured as follows. First the theoretical framework considering the Heterogeneous Agents Model and existing learning-to-forecast experiments will be discussed in Chapter 2. Then the set-up and the results of the LtFE performed for this thesis will be explained in Chapter 3. Based on the estimated prediction strategies from the participants in the experiment several simulations will be conducted in Chapter 4 using the endogenously updated impacts. At last the discussion and some suggestions for future research will be stated in Chapter 5.

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S.M.C. van Duren UvA - Master Thesis Econometrics 8 2. Theoretical Framework

This chapter first describes the structure of the Heterogeneous Agents Model (HAM) and phenomena observed in financial markets. Then learning-to-forecast experiments are discussed and at last the extension of the theoretical framework using endogenously updated impacts is introduced.

2.1 Heterogeneous Agents Model

During the eighties and nineties models with bounded rationality were developed to capture the shortcomings of the Rational Expectation Hypothesis. These models used a more realistic approach based on agents with heterogeneous beliefs regarding their prediction strategies, so-called Heterogeneous Agents Models (HAM’s). An early version of this model can be found in Zeeman (1974), later examples include Haltiwanger and Waldmann (1989), Frankel and Froot (1988), DeLong et al. (1990), and Dacorogna et al. (1995). In these models, which turned out to be highly non-linear, different agents use different prediction strategies. These prediction rules are functions of information from the past and each rule has a certain performance level (i.e. profits or utility) which is known to all agents in the model. Some strategies are successful, while others are less successful. Agents in the HAM tend to switch to successful prediction strategies based on the past performance of that strategy. In fact, agents choose and switch between fixed prediction rules according to the discrete choice model (Manski and McFadden, 1981). A number of papers have found that this heterogeneity in beliefs may lead to market instability and complicated dynamics in financial markets (e.g. Chiarella, 1992; Day and Huang, 1990; DeGrauwe et al., 1993; Lux, 1995; Sethi, 1996; Brock and Hommes, 1997, 1998). Consider for example an asset market with fundamentalists and chartists. Fundamentalists are agents who believe that prices will return to their fundamental value according to the Efficient Market Hypothesis (EMH), where this fundamental value can be computed as the present discounted value of the stream of future dividends. On the other hand chartists believe that asset prices are not completely determined by their fundamental value and may be predicted more accurately by observing and extrapolating patterns in past prices. A large number of fundamentalists active in the market tends to stabilize prices, whereas a large number of chartists active in the

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S.M.C. van Duren UvA - Master Thesis Econometrics 9 market tends to destabilize prices. Fluctuations in asset prices are caused by the interaction between these stabilizing and destabilizing forces.

Note that asset markets are so-called positive feedback systems. This means that the higher the individual predictions, the higher the market price. To be more specific, higher market expectations will lead to an increase of speculative demand and therefore to an increase of the realized asset price. This phenomenon is related to the self-fulfilling prophecy property of positive feedback systems and can cause bubbles and crashes on financial markets. The Heterogeneous Agents Model (HAM) appeared to be quite successful in explaining these phenomena, however the HAM is a theoretical framework and to verify this theory with real data learning-to-forecast experiments are used. The underlying benchmark model used for these learning-to-forecast experiments is the asset pricing model (Brock and Hommes, 1998).

2.2 Learning-to-Forecast Experiments

The formation of individual expectations is an important part of modelling economic and financial markets. Laboratory experiments with human subjects are commonly used to study these individual expectations, because it is hard to acquire detailed information on individual expectations in real markets by survey data analysis. Exceptions are Shiller (1990) who analyzes surveys on expectations about stock market prices and real estate prices and Frankel and Froot (1987) who use a survey on exchange rate expectations. The underlying economic fundamentals and the exact information individuals can access are not fully under control in such studies which makes it harder to measure the individual expectations. Lucas (1986) already stressed the importance of laboratory experiments in studying adaptive learning and its stability. Hommes (2011) noted that early work in experimental economics focussed on market mechanisms, such as double auctions, the amount of information and the presence of future markets, ensuring that equilibrium would be reached (Smith, 1962; Plot and Smith, 1978; Plot and Sunder, 1982; Sunder, 1995). Thereafter unstable market environment designs also became available, where equilibrium may not be reached and instead bubbles and crashes could occur (Smith, 1988; Lei et al.,2001).

Hommes et al. (2005) pointed out two advantages of an experimental approach. An important advantage of conducting a learning-to-forecast experiment is that full control

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S.M.C. van Duren UvA - Master Thesis Econometrics 10 over the underlying fundamentals within the experiment can be obtained. The second advantage is that it is possible to obtain explicit information about individual expectations. In LtFEs subjects only have to predict the price of the asset and do not participate themselves directly in other market activities, like trading or producing. In this sense participants of the LtFE can be viewed as advisors to financial investors and by making this distinction between prediction choices and market-trading decisions, data are not disturbed by speculative trading behaviour or by changes in the underlying demand and/or supply functions of the participants.

In the learning-to-forecast experiments the only task of the participant is to predict the price of an asset for a number of periods. At the beginning of each period participants state their prediction for the next period, before the price of this period is announced. This type of predicting is called two periods ahead (Hommes et al.,2005); the realized price of today depends on the price predictions for tomorrow. So when traders have to make a prediction for the period they do not know the price in period yet, and they can only use information on prices up till time . The predictions are collected and used to compute the average expectation, which in turn is used for the actual realization of the market price. The price formation process is of the form

(2.1)

where is the average price forecast of all the participants. The average price expectation in (2.1) for a market with 6 participants is computed as

. (2.2)

The (unknown) price generating law of motion is given by

(2.3)

where is a random sequence, the equilibrium price of this system is 60 and the interest rate is 5%.

In LtFEs subjects typically have only qualitative information about the market. They are told that the price is an aggregation of individual predictions and derived from equilibrium between demand and supply. Furthermore the participants have access to

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S.M.C. van Duren UvA - Master Thesis Econometrics 11 past prices, their own past forecasts and performance. However the information available for participants in the LtFE is very limited. They do not receive information on forecasts of other participants, the exact market equilibrium equation or the exact demand and supply schedules. This shortage of information causes similarities to models of bounded rationality and adaptive learning, where agents try to learn a perceived law of motion based upon time series observations without knowing the underlying actual law of motion of the market.

Quite a number of studies using LtFEs have been conducted already. Hommes et al. (2005, 2008) performed different asset pricing experiments using (2.3) and showed that in most experiments prices deviate from the benchmark fundamental and bubbles emerge endogenously. These bubbles are inconsistent with rational expectations and seem to be driven by the trend-following behaviour of the participants in the positive feedback system. Furthermore they found that participants within a group tend to coordinate on a common prediction strategy. A number of experiments focussed on the type of feedback and the differences in individual expectations and aggregate outcomes (i.e. Fehr and Tyran, 2008; Sutan and Willinger, 2009). The study of Heemeijer et al. (2009), which uses one-period ahead predictions, showed by using LtFEs, that market behaviour depends to a large extent on whether realized market prices respond positively or negatively to average price expectations. In the case of negative expectations feedback, mostly in commodity markets, prices converge quickly to their equilibrium value, confirming the rational expectations hypothesis. In the case of positive expectations feedback, as is typical for speculative asset markets, large fluctuations in realized prices and persistent deviations from the fundamental price are seen. Next to observing different aggregate outcomes, Heemeijer et al. (2009) estimated individual prediction rules for the participants. Most of these estimated prediction rules can be described by a first order heuristic. This first order heuristic consists of a weighted average between the last observed price, the last prediction and a constant, and includes a trend following component, extrapolating the last price change. Trend following strategies appear not to survive in a market dominated by negative feedback, but play an important role in positive feedback environments. This is caused by the fact that in negative (positive) feedback markets rational agents have an incentive to choose an action that is very different from (similar to) the actions of the boundedly rational agents. This survival of trend following strategies contributes to the oscillatory price

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S.M.C. van Duren UvA - Master Thesis Econometrics 12 movements, large deviations of the equilibrium price and excess volatility in positive feedback markets (Heemeijer et al., 2009; Bao et al., 2011).

Anufriev et al. (2013) used the heuristic switching model (HSM), to reproduce the experimental data of the learning to forecast experiment of Heemeijer et al. (2009). Based on an adaptive heuristic and a trend-following heuristic in combination with the HSM they were able to explain the difference in experimental outcomes between positive and negative feedback systems. Within the HSM agents switch between heuristics learning to use those heuristics which performed better in the past. So active heuristics are chosen endogenously on the basis of their past performance. The impact of different heuristics on the aggregate price is therefore changing over time, explaining different aggregate outcomes observed in the LtFE. They showed that the out-of-sample predictive power of the HSM is higher compared to the rational or other homogeneous expectations benchmarks. The results furthermore show that heterogeneity in expectations is crucial to describe individual forecasting and aggregate price behaviour. The idea of heuristics was already mentioned by Tversky and Kahneman (1974) and indirectly incorporated in the adaptive belief framework of Brock and Hommes (1997). In fact the HSM can be viewed as an extension of this framework, since in both models evolutionary selection of forecasting rules, driven by their relative performance, is implemented. This evolutionary selection between heuristics at the micro level has shed new light on the existing concepts about individual expectations, their interaction and the aggregate outcome they create. The large differences in individual behaviour and aggregate outcomes between positive and negative feedback systems could finally be explained. Note again that the model examined in this thesis considers evolutionary selection at the macro level. The market assigns larger impacts to successful investors, instead of individuals who assign larger weights to successful heuristics.

2.3 Extension of the Theoretical Framework

This thesis aims to examine whether evolutionary selection at the macro level has a stabilizing or destabilizing effect on the aggregate market outcome in an asset pricing model. Evolutionary selection on the macro level is implemented by using endogenously updated market impacts. Generally the asset pricing model considers a law of motion based on exogenous impacts. These general exogenous impacts will be used in the

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S.M.C. van Duren UvA - Master Thesis Econometrics 13 learning-to-forecast experiment of Chapter 3. For the simulations in Chapter 4 endogenously updated impacts will be implemented, both continuously computed weights and fixed ranked weights. The difference between the technical computations for these two types of weights will be explained in Chapter 4. The simulations include

artificial participants, whose behaviour is based on the estimated prediction strategies

from the participants within the learning-to-forecast experiment, which will be described in the next chapter. See Figure 2.1 for a schematic overview of the extension of the theoretical framework.

Figure 2.1: Schematic overview of the extension of the theoretical framework. Asset Pricing

Model

Exogenous

Impact Endogenous Impact

Continuous

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S.M.C. van Duren UvA - Master Thesis Econometrics 14 3. Learning-to-Forecast Experiment and Number Guessing Game

This chapter first discusses the experimental design of the number guessing game and the learning-to-forecast experiment conducted for this thesis. Thereafter the results of the experiment will be discussed in paragraph 3.2 and the OLS estimations of the individual prediction strategies are conducted in paragraph 3.3.

3.1 Experimental Design

The participants are high school students (age 16-17) from the Barlaeus Gymnasium in Amsterdam and they all have basic economic knowledge. The experiment was conducted on Friday 16th_{of May 2014 at the computer lab of the Barlaeus Gymnasium}

and lasted for approximately one hour. The experiment consists of two parts, the learning-to-forecast experiment and the number guessing game. First the set-up for the learning-to-forecast experiment will be discussed, followed by the set up of the number guessing game.

3.1.1 Learning-to-Forecast Experiment

During the experiment 12 markets based on exogenous impacts are considered.1_The

experiment lasted for 50 periods and every market consists of five simulated robot players and one participant player, so in total the experiment considers 12 participants.2

The use of robot traders in addition to real participants is not uncommon in experimental asset markets (i.e. Bloomfield, 1996; Bloomfield and O’Hara, 1999 and Hommes et al., 2005). By including these computerized traders, it was possible to create a stable and an unstable treatment group based on the prediction strategies of the robots alone. The participants were divided equally over the two treatments, so every treatment considers six participants. Within a stable treatment prices are expected to converge after an initial period of oscillations, while within an unstable treatment prices are expected to keep on fluctuating. However the prediction strategy of the single participant in the experiment could either confirm the expected aggregate behaviour within the market or transform the stable (unstable) market into an unstable (stable) market.

1_{The LtFE is written and conducted in a Java environment, for details contact D. Massaro (}_{d.massaro@uva.nl}_). 2_{The original experimental design considers 24 participants and 24 markets in total. Within the omitted markets}

realized prices were computed according to the weights based on endogenous impacts described in formula (4.1). However due to a technical mistake the results of these markets were not valid. A description and the results of the omitted markets can be found appendix A.

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S.M.C. van Duren UvA - Master Thesis Econometrics 15 The predictions are made two periods ahead, so the realized price of today depends on the price predictions for tomorrow. When the participants have to make a prediction for the period they do not know the price in period yet, and they can only use information on prices up till time and information on expectations up till time . The robot traders are modelled based on the first order heuristic and every robot is endowed with a specific trend-following coefficient but the same values for and

3 Furthermore a lower bound of zero is included in the forecasting rules of the robot

traders, since prices and therefore price forecasts cannot be negative. There is no upper bound included in the forecasting rules of the robot traders. So the exact forecasting rules of the robot traders is as follows:

(3.1)

where is called the first order

heuristic. Note that this first order heuristic is linear in its parameters. A drawback of the first order heuristic is that the fundamental price 60 is included in the prediction strategy. In real life traders do not know the fundamental price of an asset, so it does not seem realistic to include this fundamental price in the prediction strategies of the robots. Hommes (2011) solved this problem by replacing the fundamental value by an observable proxy given by the sample average of the past prices. However to keep the model as simple as possible, for all robot traders within the LtFE holds and

, so the term cancels out.

Note that for , traders believe that an upward movement in prices will continue in the next period, whereas if they believe an upward movement in the prices will be (partially) offset by a downward movement in prices in the next period.Straightforward stability analysis reveals that the first order heuristic (3.1) with and produces converging prices in a positive feedback market for an average trend parameter .4_{The average trend parameter of the robots within the stable}

treatment is set to and the average trend parameter within the unstable treatment is set to . Some robots are strong trend followers, while others only have a small trend-following coefficient. This is consistent with estimated prediction

3_{Choosing the first order heuristic to simulate the robots is based on the fact that in positive feedback systems}

trend-following heuristics dominate the market (Heemeijer et al.,2009; Bao et al.,2011).

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S.M.C. van Duren UvA - Master Thesis Econometrics 16 rules from previous research (i.e. Heemeijer et al., 2009 and Hommes et al., 2005). Note that some of these previous experiments also included one period ahead learning-to-forecast experiments. The exact values of the trend parameter for the robots in the experiment are given in Table 3.1. For the stable treatment the dispersion is larger than for the unstable treatment, this will be relevant when analyzing the results. Typical aggregate behaviour and individual predictions for both treatments are shown in Figure 3.1. The left picture shows a stable market with an average trend parameter of 0.875, while the right picture shows an unstable market with an average trend parameter of 1.1. Note that these pictures are generated based on a market with 6 robots and no real participants.5_{During the experiment one robot will be replaced by a participant.}

Furthermore the initial price predictions of the robots are fixed for the first two periods and the same for both treatments.6

Stable Treatment Trend Parameter Unstable Treatment Trend Parameter

Robot 1 0.275 Robot 7 1 Robot 2 0.575 Robot 8 1.05 Robot 3 0.875 Robot 9 1.1 Robot 4* 0.875* Robot 10* 1.1* Robot 5 1.175 Robot 11 1.15 Robot 6 1.475 Robot 12 1.2 Average Average

Table 3.1: Overview of trend parameters for the robot traders (* robot 4 and 10 are replaced by the participant in the LtFE)

Figure 3.1: Typical aggregate behaviour for stable (left) and unstable (right) treatments.

5_{The simulations and pictures are all generated within MatLab version R2013a.}

6_{The average of the price predictions of the five robots in the first period in the LtFE is 41.6 and the average of the}

price predictions for the second period is 45.2. Aggregate results are robust to (small) changes in these initial values. The initial price predictions of every robot can be found in appendix B.

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S.M.C. van Duren UvA - Master Thesis Econometrics 17 The actual price in the experiment is computed based on the average price expectation and the law of motion of an asset pricing model with exogenous impacts, see (2.3). The winner of each treatment is the participant with the lowest total prediction error, where the total prediction error of a participant is computed as the sum of the participants prediction errors over all periods

. (3.2)

So at the end of the experiment each treatment group has one winner, giving two winners in total. Every winner receives two free tickets to the cinema, which seems a realistic incentive to motivate the students to perform the best they can and to avoid extreme and unrealistic behaviour.7

The experiment is explained to the students in the classroom in a simplified version of the general asset pricing experiment. After the classroom explanation of the experiment, the participants received a summary with the experimental instructions on paper and they could ask questions.8_{All participants are told that they are a financial advisor of an}

investor and that this investor has to make a decision concerning the assets he wants to sell and buy every day. In this sense it is important to accurately predict the prices of the assets. Usually the asset pricing experiment is explained more in detail to subjects of a LtFE, including a distinction between investing in a risk free asset and a risky asset. The risk free asset then has a risk free gross rate of return , where is the interest rate, while the risky asset pays uncertain dividends in period . The dividends are usually independently and identically distributed with mean , which is assumed to be common knowledge together with the interest rate . However since the participants in this LtFE are less experienced with economic variables, the choice to keep it simple was more important than explaining details to the participants regarding the interest rate. So the interest rate was included in the law of motion within the experiment without mentioning it to the students. The results will not be affected by the omission of this information since the focus of the LtFE is on the estimation of individual prediction rules in a limited information environment.

7_{The free tickets to the cinema were made available by cinema house The Movies in Amsterdam.} 8_{The instructions can be found in appendix E.}

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S.M.C. van Duren UvA - Master Thesis Econometrics 18 The task of the advisor (i.e., the participant) is to predict the price of an asset for 50 consecutive periods. Of course the participants do not know the exact underlying market equilibrium equation. Nevertheless they are told that the higher their prediction, the larger the fraction of money invested by the investor in the asset will be and the larger the demand for stocks will be. The participants are not explicitly informed about the fact that the actual price realization of the asset depends on their prediction. Furthermore all participants are told that the price is determined by the equilibrium between demand and supply of the asset. The participants have access to past prices, their own past forecasts and their forecast errors, typically shown in a table as well as in graphic form. Figure 3.2 illustrates the computer screen of the experiment as seen by the participants.

Figure 3.2: Screenshot computer screen as seen by the participants in the LtFE.

3.1.2 Number Guessing Game

The number guessing game is explained to the students as follows. Everybody simultaneously state a number between [0,100] on a small paper and the winner is the person whose number is closest to p=2/3 times the mean of all chosen numbers. The winner earns € 15 and in case of two or more winners this price is split evenly between them. This so called winner-takes-all pay off structure is commonly used for number guessing games and motivates participants to be as precise as possible since a small deviation can have large consequences. After the first round, the winning number and person is announced and a second (last) round of the number guessing game is played

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S.M.C. van Duren UvA - Master Thesis Econometrics 19 for the same value of p=2/3. Note that the participants now have more information available since they have a new reference point, the average of the numbers in round 1.

3.2 Experimental Results

First the aggregate behaviour in the learning-to-forecast experiment will be examined, followed by the results from the number guessing game and a comparison of the

participants behaviour in the number guessing game and during the learning-to-forecast experiment.

3.2.1. Aggregate Behaviour in the Learning-to-Forecast Experiment

Figure 3.3 and Figure 3.4 show the realized asset prices and the individual forecasts in the experiment for respectively the stable treatment and the unstable treatment. In all markets the fundamental price equals 60, which is shown in the pictures by the dotted line. As can be seen in Figure 3.3, market 2, 4, 5 and 6 generate aggregate behaviour which is in accordance with the stable treatment: the realized asset prices converge quite quickly to the fundamental price. However the first (upper left) and third (upper right) market show different aggregate behaviour. The cause for this unstable aggregate behaviour in market 1 is a typo of the participant in period 19.9_{This typo caused a huge}

increase in the realized asset price. Therefore this market is carefully taken into account regarding further analysis. The picture of the aggregate behaviour in market 3, Figure 3.3 upper right, suggests that participant 3 might have a large trend parameter since this originally stable market shows a large oscillation. This will be verified by the estimation of the individual prediction rules of the participants in the next paragraph.

9_{The participant stated 773.5 instead of 73.5. She asked me during the experiment whether she could undo her typo,}

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Figure 3.3: Results market 1-6 within the stable treatment.

Figure 3.4: Results markets 7-12 within the unstable treatment.

As can be seen in Figure 3.4, two participants were able to stabilize the market in comparison with a typical unstable market in Figure 3.1. Market 7 (upper left) and 8 (top centre) in Figure 3.4 both show only small oscillations around the equilibrium price. The aggregate behaviour in the other markets in Figure 3.4 is in accordance with the typical aggregate behaviour in unstable markets: prices show large oscillations and keep on fluctuating.

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S.M.C. van Duren UvA - Master Thesis Econometrics 21 The winner of the stable treatment was participant 2 (Figure 3.3, top centre) and the winner of the unstable treatment was participant 12 (Figure 3.4, bottom right). The table below shows the average forecasts and the total prediction error, which is used to determine the winner. As said before, the participant with the lowest total prediction error is the winner. One may expect that the winner of the unstable treatment was one of the participants who were able to stabilize the market, participant 7 or 8. However since in a positive feedback system it is most profitable to imitate the behaviour of other agents, within the unstable treatment it was most profitable to imitate the behaviour of the robots, who were simulated to generate an unstable market.10

Stable Markets

Average Forecast over

Periods Total Prediction Error Guessing Game 1 Guessing Game 2

Participant 1 _118.03* _334349.93* _0.10 _0.50 Participant 2 (Winner) _60.39 _318.77 _12.00 _2.50 Participant 3 _77.25 _6466.43 _0.10 _2.30 Participant 4 _62.04 _950.78 _0.25 _0.25 Participant 5 _59.34 _864.46 _5.00 _0.20 Participant 6 _61.05 _1157.64 _38.00 _4.30 Average _64.01 _1951.61

Table 3.2: Descriptive statistics learning-to-forecast experiment stable markets and number guessing game (* note that participant 1 is not taken into account in calculating the averages due to the aforementioned typo)

Unstable Markets

Average Forecast over

Periods Total Prediction Error Guessing Game 1 Guessing Game 2

Participant 7 57.65 2570.43 10.20 0.15 Participant 8 _56.31 _1764.52 _15.00 _2.66 Participant 9 _70.26 _8426.99 _32.66 _2.90 Participant 10 _89.38 _3205.81 _0.10 _0.60 Participant 11 _94.12 _4089.26 _6.00* _1.90 Participant 12 _69.67 _1069.73 _1.00 _0.19 Average _72.89 _3521.12

Table 3.3: Descriptive statistics learning-to-forecast experiment unstable markets and number guessing game (* participant 11 was the winner of the guessing game)

10_{The imitation of the actions of other agents is also known as strategic complementarity (Haltiwanger and Waldman,}

1989). Rational agents in a positive feedback systems have an incentive to choose an action that is very similar to the actions of boundedly rational agents. The impact of irrational individuals on the realized prices in this case is relatively large and convergence to the rational equilibrium price becomes unlikely (Heemeijer et al., 2009; Fehr and Tyran, 2008).

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3.2.2. Comparison of Behaviour in Number Guessing Game and LtFE

The number guessing game was conducted with 22 participants, see the two most right columns of Table 3.2 and 3.3 for the numbers stated by participant 1-12. The other players of the number guessing game were participant 13-17 and 20-24, see Appendix A for their chosen numbers. The average number of the first round was 9.78, so the winning number was , which corresponds to the number stated by participant 11, see table 3.3. For the second round the average number was 1.49, so the winning number was , which corresponds to the number stated by participant 16, see table A.2 in the appendix. Note that the average of all chosen numbers in round 1 was 9.78, while the average in round 2 was 1.49. This indicates that the participants indeed adapt their chosen number in the direction of the equilibrium. To compare the participants behaviour in the number guessing game with the behaviour in the LtFE, the correlation coefficient of the prediction errors is calculated. If the participants behave relatively similar in both games, the correlation coefficient of the prediction errors is expected to be high (>0.5). To compare the participants behaviour properly, a distinction between the first and second period should be made since the information available to the participants is different. Note that during the first round of the number guessing game, participants have no information available from previous rounds. This setting is similar to the predictions made by the participants for period 1 and 2 during the learning-to-forecast experiment. Furthermore the second round of the number guessing game is similar to the predictions made for period 3 in the LtFE, since the participants now have information available on the previous period. In the number guessing game after round 1 the average and winning number are announced, while in the LtFE the realized price is shown on the computer screen to the participant.

First the correlation coefficient of the prediction errors from the period without information on previous periods is calculated. Second the correlation coefficient of the prediction errors from the period with information on the single previous periods is calculated. The observations for these calculations include the original set of 22

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S.M.C. van Duren UvA - Master Thesis Econometrics 23 participants of the number guessing game since the results of the participants 13-24 within the LtFE were still valid for the first 2 periods. 11

Correlation Coefficients Correlation coefficient # observations Participants

e2_{LtFE # 2 – e}2_{Number Guessing # 1}

-0.19 22 1-17, 20-24

e2_{LtFE # 3 – e}2_{Number Guessing # 2}

0.55 22 1-17, 20-24

Table 3.4: Correlation coefficients of prediction errors for period #1 (no past information) and period #2 (information on previous period available).

As can be seen in Table 3.4 the correlation coefficient in the first row is slightly negative (-0.19) which indicates that the participants behave differently during the first period of the number guessing game and the LtFE. However the correlation coefficient in the second row (0.55) indicates that the participants behave rather similar in both games during the second period. So it appears that a participants basic intuition without information on previous periods is quite different between the two games, but that a participants intuitive reaction on a certain notification of the realized price/average is quite similar in both games. The behaviour in the first period is closer to random behaviour than in the second period since the participants have almost no information available in the first period. This could cause the slightly negative correlation coefficient of the prediction errors in the first period.

Furthermore notice that we analyzed the participants behaviour only according to the correlation of the prediction errors from the two games in the first and second period since we only have data available for two periods of the Number Guessing Game. To further analyze the participants behaviour one could increase the number of periods in the Number Guessing game. Another addition could be to compute the participants level of rationality for both games and calculate the so-called adjustment factor (Nagel, 1995). However this is beyond the scope of this research.

3.3 OLS Estimation of Individual Prediction Rules

The last step of analyzing the experimental results is the estimation of linear individual prediction rules, which will be used for the simulations of the positive feedback market based on endogenously updated impacts in the next chapter. The prediction strategies of the participants can be described by the following simple linear model

11_{The aforementioned technical error in the markets with exogenous impacts (participant 13-24) did not affect the}

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(3.3)

where is a constant and is an i.i.d. noise term. This linear model (3.3) could be estimated successfully (i.e. without autocorrelation in the residuals) for 11 participants, the residuals of participant 3 showed significant autocorrelations based on the Ljung-Box Q statistic (5%). The estimated coefficients of model (3.3) can be found in Appendix C. Note that the estimation sample takes a short learning phase of five periods into account. This is common in estimating such individual prediction strategies (i.e. Hommes et al., 2005 and Heemeijer et al., 2009). Furthermore a few special cases of (3.3) need to be mentioned explicitly:

(i) Naïve expectations ( = 1, all other coefficients are equal to 0)

(3.3a)

(ii) Adaptive expectations ( + = 1, all other coefficients are equal to 0)

(3.3b)

(iii) Obstinate expectations ( = 1, all other coefficients are equal to 0)

(3.3c)

(iv) First Order Heuristic ( all other coefficients are non zero).

(3.3d)

Note that the following parameter restrictions should hold to obtain the F.O.H. in (3.3d) from the AR(2) model in (3.3): . For most

participants indeed these restrictions apply. Only for participant 6 holds that the hypothesis is not rejected and for participants 1 and 6 holds that the hypothesis

is not rejected.12 Furthermore for four participants (1, 2, 4 and 9) holds that the

hypothesis is not rejected and for four other participants (3, 5, 6 and 11) holds

that the hypothesis is rejected. Nevertheless the First Order Heuristic is

estimated for every participant. Rewriting (3.3) with the aforementioned parameter restrictions and some parameter transformations gives the First Order Heuristic mentioned in (3.1). Note that the following parameter transformations have been applied to (3.3): , and .13 And the constant in (3.3)

is rewritten as . The estimation of the F.O.H. in (3.1) will be used for

12_{Based on the t-statistics in the AR(2) model.}

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S.M.C. van Duren UvA - Master Thesis Econometrics 25 the simulations in the next chapter. The OLS results are shown in the tables below.14_As

can be seen in Table 3.5 and 3.6, the F.O.H. provides a good estimation of the actual behaviour of the participants according to the high values of the adjusted R2_{. The}

estimated coefficients are in accordance with earlier findings in LtFEs (i.e. Heemeijer et al., 2009). For most participants holds that , which means that indeed the term cancels out.15_{As said before, it is not likely that participants use}

such anchoring based on the unknown equilibrium price. However the estimated F.O.H. of participant 4 and 6 includes a significant constant. This constant of the estimated F.O.H. of participant 4 equals , which indicates that participant 4 indeed seems to incorporate some sort of anchoring based on the unknown equilibrium price in his prediction strategy. The constant of the estimated F.O.H. of participant 6 equals which indicates that also participant 6 seems to incorporate some sort of anchoring based on the unknown equilibrium price. However note that the low value of the adjusted R2_{(0.21) of the estimated prediction rule for participant 6 indicates that}

this rule predicts the actual behaviour of participant 6 rather poor, so we should be careful with interpreting the estimated coefficients of this rule.

OLS Stable Treatment C Adjusted Type

Participant 116

0.88 1.01 0 1.26 0.99 Naive + Trend

Participant 2 (Winner I) _-1.36 _1.03 ₀ _1.00 _0.95 _{Naive + Trend}

Participant 317

3.64 0 0.94 0.68 0.97 Obstinate + Trend

Participant 4 _8.75* _0.86 ₀ _1.62 _0.96 _{Anchor, Naive + Trend}

Participant 5 _2.88 _0.53 _0.42 _1.35 _0.95 _F.O.H.

Participant 6 _34.68* ₀ _0.44 ₀ _0.21 _Obstinate

Average _8.245 _0.572 _0.300 _0.987

Dispersion (st. deviation) _13.383 _0.478 _0.378 _0.579

Table 3.5: Estimated coefficients F.O.H. of participants in the stable treatment.

14_{The estimated coefficients are conducted in Eviews 7 and estimated by eliminating the least significant variable}

until all p-values are below 5%. Furthermore again a short learning phase of five periods is taken into account, which means that these observations are excluded from the estimation sample.

15_{Results for the Wald statistic, testing} _{, can be found in Appendix I. Based on this test statistic}

does not hold for participant 3, 4, 6 and 10 (5% level).

16_{Since a typo occurred in period 19 in the LtFE of the first participant the following periods are excluded {19,..,28}}

from the estimation sample.

17_{Note that serial autocorrelation was found in the AR(2) estimation of participant 3, so the estimation of the F.O.H. is}

not reliable. However for completeness the estimated OLS coefficients are stated in the table. Furthermore note that this autocorrelation might be responsible for the oscillation in the upper-right market in Figure 3.3.

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OLS Unstable Treatment C Adjusted Type

Participant 7 _0.63 _0.28 _0.71 _0.98 _0.97 _F.O.H.

Participant 8 _0.09 _1.18 _-0.18 _0.95 _0.99 _F.O.H.

Participant 9 _6.72 _0.92 ₀ _0.90 _0.84 _{Naive + Trend}

Participant 10 _5.17 _0.47 _0.49 _1.23 _0.99 _F.O.H.

Participant 11 _3.17 _1.00 ₀ _1.42 _0.99 _{Naive + Trend}

Participant 12 (Winner II) _1.49 ₀ _0.98 _1.01 _0.99 _{Obstinate + Trend}

Average _2.878 _0.642 _0.333 _1.082

Dispersion (st. deviation) _2.640 _0.462 _0.463 _0.201

Table 3.6: Estimated coefficients F.O.H. of participants in the unstable treatment.

As discussed in the introduction, agents typically tend to imitate the behaviour of other agents in positive feedback systems due to the presence of strategic complementarity (Haltiwanger and Waldman, 1989). Note that the dispersion of the estimated trend parameters in the stable treatment (0.579) is much higher than in the unstable treatment (0.201), as can be seen in the last row of the fifth column of Table 3.5 and 3.6. The dispersion of the trend parameters of the robots in the stable treatment is also higher than of the robots in the unstable treatment, see Table 3.1. This suggests that the participants indeed imitate the behaviour of the robots in the LtFE. Furthermore the trend parameter of the average first order heuristic of the stable treatment (0.987) is smaller than the average trend parameter of the average first order heuristic of the unstable treatment (1.082). This also underpins the typical imitating behaviour in positive feedback markets. In fact the participants in the stable treatment approach the average trend parameter of the robots quite reasonable , however the participants in the unstable treatment approach the average trend parameter of the robots even better . This could indicate that coordination on a common prediction strategy is more likely in an unstable market. This could be caused by the fact that prediction strategies in an unstable market are more extreme in the sense of the trend parameter and therefore may be more visible to the participants through the realized prices. However further analysis is needed to verify this phenomenon, but that is beyond the scope of this research.

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S.M.C. van Duren UvA - Master Thesis Econometrics 27 4. Simulations Asset Pricing Model with Endogenously Updated Impacts Using the estimated forecasting rules from the previous chapter as predictions for artificial participants, asset prices are simulated based on endogenously updated impacts to see whether this mechanism might have a stabilizing or destabilizing effect on realized asset prices. Both the originally stable treatment and the unstable treatment are considered. The price mechanism based on endogenously updated impacts might be able to stabilize the unstable treatment, since successful investors (i.e. successful robot traders or artificial participants) will obtain a larger market impact based on their past performance, while the impact of less successful investors diminishes. First the simulations will be performed according to continuous endogenously updated weights, followed by simulations according to ranked endogenously updated weights. At last the results will be further examined with a special focus to extreme forecasting behaviour and the effect of different starting values.

4.1 Continuous Endogenously Updated Impacts

Every simulation considers six robots and 199 periods. For an experiment with human subjects the amount of periods is limited, but for simulations a longer time period is easily conducted and provides a better insight in the aggregate behaviour on the long term. The continuous endogenously updated impacts are described by a discrete choice model with asynchronous updating (Anufriev and Hommes, 2012)

(4.1)

where is a normalization factor. Note that this summation

adds six terms together, since within all simulations six robots are considered. In the special case , formula (4.1) reduces to the discrete choice model with synchronous updating used in Brock and Hommes (1997). In the more general case, , represents how fast the system responds to the successes of certain prediction strategies. Higher values of indicate a slower reaction of the system to successful prediction rules. Such inertia is widely reported in experiments on the individual level (Kahneman, 2003). The parameter represents the intensity of choice, measuring how sensitive the market is to differences in strategy performance. The higher the intensity of choice the larger impacts the system will assign to successful investors. If

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S.M.C. van Duren UvA - Master Thesis Econometrics 28 =0 the performances of the predictions are not taken into account at all, so the model will result in the classic model based on equal exogenous impacts. For a market considering 6 robot traders, we will then have for every robot trader and period

The performance in formula (4.1) is measured as follows

, (4.2)

where the parameter represents the intensity of memory. The combination of formula (4.1) and (4.2) is also used by Anufriev and Hommes (2012) and Anufriev et al. (2013) to match the data from different learning-to-forecast experiments (i.e. Hommes et al., 2005 and Heemeijer et al.,2009) with one-step ahead predictions of four different heuristics, prediction rules commonly used by individuals.18_{This so-called heuristic}

switching model provides a good approximation of the observed data by incorporating switching between heuristics at the micro level. Individuals are able to assign a certain weight to every heuristic based on the performance of this heuristic in the previous period(s) and they are able to update these weights for every period. The mechanism used for the simulations in this paragraph is a bit similar to the heuristic switching model, however as mentioned before one important difference is that in the model proposed here the evolutionary selection occurs on the macro level. The market assigns larger impacts to successful investors, instead of individuals who assign larger weights to successful heuristics.

In all simulations in the remainder of this chapter, parameters are fixed at the benchmark values and the initial impact of every robot trader is . 19_{These impacts are used to compute the realized prices for period 1, 2 and 3}

according to (2.3). In the third period the performance is measured for the first time using formula (4.2). Then the weights based on this performance measured are computed endogenously according to (4.1) and used to compute realized price in period 4.

18_{Note that Anufriev and Hommes (2012) and Anufriev et al. (2013) used different letters for the parameters:}

intensity of choice , intensity of memory .

19_{These benchmark values for and are consistent with the values in Anufriev and Hommes (2012) and Anufriev}

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4.1.1. Six Artificial Participants

The first two simulations consider only artificial participants. Their prediction strategies correspond to the estimated first order heuristic (3.1) of the participants in the LtFE, see Table 3.5 and 3.6 for the estimated coefficients. For example, the prediction rule of artificial participant 1 is based on the estimated coefficients of the first order heuristic for participant 1 in the LtFE, see (4.3).

(4.3) The prediction rule of artificial participant 2 is based on the estimated first order heuristic of participant 2 etc. Note that also the prediction rules of the artificial participants include a lower bound of zero and that there is no upper bound included. Furthermore the initial price predictions of the artificial participants correspond to the initial price predictions stated by the participants in the LtFE. Figure 4.1 shows the simulation for artificial participant 1-6. Their forecasting rules are based on participants 1-6 from the original stable treatment in the LtFE, see Table 3.5 for the estimated F.O.H. coefficients. Figure 4.2 shows the simulation for artificial participant 7-12 from the original unstable treatment in the LtFE, see Table 3.6 for the estimated F.O.H. coefficients.

Figure 4.1: Simulations artificial participant 1-6. Top left: endogenously updated impacts, top right: weights of

endogenous impacts. Bottom left: exogenous impacts, bottom right: constant weight of 1/6. 20 40 60 80 100 120 140 160 180 0 50 100 150 200

Time Series Exogenous Impacts

Period P ri c e a n d P re d ic ti o n s 20 40 60 80 100 120 140 160 180 0 50 100 150 200

Time Series Endogenous Impacts

Period P ri c e a n d P re d ic ti o n s 20 40 60 80 100 120 140 160 180 0 0.2 0.4 0.6 0.8 1

Time Series Weights for Endogenous Impacts

Period W e ig h ts 20 40 60 80 100 120 140 160 180 0 0.2 0.4 0.6 0.8 1 Period W e ig h ts

Time Series Weights Exogenous Impacts participant 1(beta=1.26) participant 2(beta=1.00) participant 3(beta=0.68) participant 4(beta=1.62) participant 5(beta=1.35) participant 6(beta=0)

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S.M.C. van Duren UvA - Master Thesis Econometrics 30 The top left picture of Figure 4.1 shows the price predictions and the realized prices based on continuous endogenously updated weights, while the top right picture shows the associated weights. To compare the results with the original mechanism based on exogenously constant weights (i.e. for every and ) the picture at the bottom left shows the aggregate results for the original mechanism with the associated exogenous weights in the bottom right picture (which are logically constant over time). The average trend parameter of the six artificial participants considered in Figure 4.1 is , which should yield converging prices according to the stability analysis for the average estimated parameters and in Table 3.5.20_{Indeed the bottom left picture of}

Figure 4.1 shows converging prices to the fundamental value 60 for the original mechanism based on exogenous impacts, but the top left picture in Figure 4.1 shows oscillating prices for the updating mechanism based on endogenous impacts. The associated weights in the top right picture indicate that the weight of one trader, artificial participant 4, dominates all weights. This dominating weight seems to have a destabilizing effect on the market, as can be seen in the top left picture. Note that the trend parameter of this artificial participant 4 is the strongest trend parameter in the simulation ( =1.62 see Table 3.5), which suggests that the endogenous

impact system is more sensitive to extreme prediction rules considering a strong trend parameter than the original exogenous impact system. In fact the strongest trend parameter might determine the aggregate market behaviour within endogenous impact systems instead of the average trend parameter, which is the case in the classical mechanism based on exogenous impacts. This phenomenon will be examined further in the next simulations.

The results of the second simulation considering artificial participants 7 – 12 are shown in Figure 4.2

20_{Stability analysis for} _{(Table 3.5) reveals that convergence occurs to the fundamental value in}

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Figure 4.2: Simulations artificial participants 7-12. Top left: endogenously updated impacts, top right: weights of

endogenous impacts. Bottom left: exogenous impacts, bottom right: constant weight of 1/6.

The average trend parameter of the six artificial participants considered in Figure 4.2 is 1.082, which is outside the convergence interval.21_{Indeed the market price in the}

bottom left picture, the original exogenous impact system, does not converge and seems to diverge rather fast after one small oscillation. The top left picture also shows divergence after three increasing oscillations for the endogenous impact system. So the aggregate behaviour in both systems is approximately the same, however the weights develop quite different. As can be seen in the top right picture the weight of artificial participant 9 is rather large in the beginning due to initial values close to the realized price in the starting periods within the endogenous impact system. After period 30, the weight of artificial participant 11 starts to increase with systematic fluctuations and dominates the other weights until period 90. This dominating and fluctuating weight of artificial participant 11 coincides with the price oscillations as can be seen in the top left picture. Note that the trend parameter of artificial participant 11 is 1.42, which is again the strongest trend parameter considered in the simulation. However around period 90 the weight of artificial participant 12 starts to dominate the other weights and this coincides with divergence in the top left picture. Note that the trend parameter of artificial participant 12 (=1.01) is not the strongest trend parameter. This suggests that it is not the strongest trend parameter alone which determines the aggregate behaviour

21_{Stability analysis for} _{(Table 3.6) reveals that convergence occurs to the fundamental value}

in a positive feedback market for . The analytical derivation of the stability analysis can be found in Appendix D. 20 40 60 80 100 120 140 160 180 0 50 100 150 200