Endogenous intensity of choice, depending on the variance of the prots, in heterogeneous agent models

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Faculty of Economics and Business, Amsterdam School of Economics University of Amsterdam

Bachelor's Thesis Econometrics

Endogenous intensity of choice, depending on the variance of

the profits, in heterogeneous agent models

Mandy van Oosterhout (10572821) Supervisor: prof. dr. Jan Tuinstra

26th June, 2018

Abstract

In this paper the discrete choice model is adjusted due to the insights of a recent experiment. The intensity of choice variable in this model is assumed to be endogenous, depending on the variance of the profits, instead of exogenous. The results of the adjusted model are compared to the original discrete choice model to find out if it is more robust.

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Statement of Originality

This document is written by Student Mandy van Oosterhout who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document are 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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1 Introduction

Heterogeneous agent models are nowadays popular models to describe financial markets. These models assume financial markets with different groups of agents having different expectations about future prices. De Long, Shleifer, Summers and Waldmann (1990), for example, distinguish between fundamentalists and chartists, where fundamentalists believe the price is determined by its efficient market hypothesis (EMH) fundamental value, while chartists believe future prices may be predicted by simple trading rules. Such heterogeneous agent models form the basis for this research.

A significant class of these models are heuristic switching models, as introduced by Brock and Hommes (1997). They assume a large population of agents, who have to choose between different heuristics every period. Agents make their choices based on the relative performance of the heuristics, which changes due to the changes in the distribution of the agents over the heuristics. Brock and Hommes (1997) use a demand-supply model in which agents choose between a naive expectations heuristic and a rational expectations heuristic, where a rational expectations heuristic uses more information and so comes at higher costs than a naive expectations heuristic.

A common way of modeling the choices made by agents is by using the discrete choice model, also described by Brock and Hommes (1997). An important parameter in that model is the intensity of choice, which measures how fast agents switch heuristics. Brock and Hommes (1997) prove that complicated dynamics occur in case of a high intensity of choice to switch between a costly rational expectations heuristic and a naive expectations heuristic. It is important to note that this proof is based on an exogenous intensity of choice.

Anufriev, Chernulich and Tuinstra (2018) follow up on this assumption and conduct an experiment to test whether the intensity of choice variable is exogenous. In the experiment they fit a simple discrete choice model to the participants' choices, where the naive heuristic is destabilizing the price when used by many participants and the rational heuristic is stabilizing the price. The goal of the participants was to maximize their profits.

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In the experiment, Anufriev et al. (2018) test whether the intensity of choice parameter could have different values in different market situations. Therefore two different empirical values of the intensity of choice parameter are computed for two different market situations. They find significant difference between these endogenous values, hence the intensity of choice parameter changes over time in response to the decision environment. This implies that the assumption of an exogenous intensity of choices parameter does not hold.

This paper describes adjustments to the original discrete choice model, by varying the intensity of choice parameter with the profits' variance. These adjustments are based on the results and insights of the experiment from Anufriev et al. (2018). This new model analyzes how the current results given in Brock and Hommes (1997) change due to endogeneity of the intensity of choice parameter.

The paper is organized as follows. Section 2 describes the heuristic switching model and discrete choice model with the intensity of choice as a key parameter. Furthermore it contains theoretical background on exogeneity versus endogeneity of this parameter. Section 3 focuses on the adjustments to the model. In addition, Section 4 describes the research method and in Section 5 the results of the research are given and analyzed. At last, Section 6 concludes.

2 Analysis

of

the

discrete

choice

model

and

possible

adjustments

The adaptive rational equilibrium dynamic (Brock & Hommes, 1997, pp. 1062-1065), also used in Grandmont and Laroque (1991), Brock and LeBaron (1996) and in several other papers, makes a connection between market equilibrium dynamics and the choices agents make to maximize their profit. To achieve the maximum profit, agents have to select a profit predictor out of a set of predictors every period. They base their decision on the predictor’s past performance.

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Brock and Hommes (1997) extend the the adaptive rational equilibrium dynamic by introducing the cobweb demand-supply model. In this model, agents choose between two predictors; a rational expectations heuristic and a naive expectations heuristic. A rational expectation uses more information to predict the profit and so comes at higher costs than a naive expectations heuristic. Therefore a rational expectation must be bought at positive information costs C, while a naive expectation is free, so obtainable at zero information costs. Agents want to achieve the maximum profit by paying minimal costs. Brock and Hommes (1997) suppose agents make the decision for the choice of the predictor based upon past performance.

2.1 The original discrete choice model

In the cobweb model (Brock & Hommes, 1997) with rational versus naive expectations, the market equilibrium is given by

D(pt+1) = n1,tS(H1(P#»t)) + n2,tS(H2(P#»t)), (2.1.1)

where H1 are rational expectations, H2 are naive expectations and n1,t, respectively n2,t

are the agents choosing H1 or H2. P#»t is a vector of past prices. The rational and naive

expectations are given by

H1(P#»t) = pt+1, H2(P#»t) = pt. (2.1.2)

Brock and Hommes (1997) choose the demand curve and supply curve to be linear:

D(pt) = A − Bpt, (2.1.3)

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Rational expectations use information and cognitive effort to predict the price for the maximum profit, which can be obtained at some price. This price to pay for the rational expectation predictor is called the information costs C ≥ 0. Therefore it is important to expand the equation with C. The naive expectations predictor has no information costs (C = 0). Using the equations above, the profits for choosing the rational, respectively the naive expectations, are given by

π1(pt+1, pt+1) = b 2p 2 t+1− C, (2.1.5) π2(pt+1, pt) = b 2pt(2pt+1− pt). (2.1.6) The fractions of agents using H1and H2 are updated in the next period, after the equilibrium

price pt+1 is known. These new fractions are

n1,t+1= e(β( b 2p 2 t+1−C))/Z t+1, (2.1.7) n2,t+1= e(β( b 2pt(2pt+1−pt)))/Z_t+1, (2.1.8)

where Zt+1 is the sum of the numerators and β is the intensity of choice, which measures how

fast agents switch heuristics. The difference between the fractions n1,t+1 and n2,t+1 is given

by mt+1: mt+1 = T anh β 2 b 2(pt+1− pt) 2_{− C} . (2.1.9)

Using n1,t = (mt+1)/2, n2,t= (1−mt)/2, the demand and supply (2.1.3) and the expectations

(2.1.2), the market equilibrium in 2.1.1 changes to

pt+1= A −1 + mt 2 bpt+1− 1 − mt 2 bpt /B. (2.1.10)

It is assumed that the rational agents have perfect knowledge about the beliefs of the other agents and solve equation 2.1.10 for pt+1. We normalize A to 0 such that pt represent the

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deviations from the steady state price. In addition, pt+1 and mt+1 change to: pt+1= −b(1 − m_t)pt 2B + b(1 + mt) , (2.1.11) mt+1= T anh β 2 b 2 b(1 − mt) 2B + b(1 + mt) + 1 2 p2_t − C !! . (2.1.12)

2.1.1 Results of the original discrete choice model

The stability of the dynamics are shown below for different situations.

Figure 1: Dynamics for different values of β, with A = 0, B = 0.5, b = 1.35 and C = 1

(a) β = 4 (b) β = 4.3

(c) β = 5 (d) β = 10

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4.3, the dynamics get unstable.

From these findings, Brock and Hommes (1997) conclude that complicated dynamics occur if the agent’s intensity of choice to switch predictors is high. They explain that this is because, in case of a price close to its steady state level, agents tend to choose the cheaper naive expectations heuristic because both expectation heuristics give comparable predictions. When the majority of the agents choose the naive expectation heuristic the price will fluctuate. Therefore the prediction error for the naive expectation increases. As a consequence, agents tend to switch to the rational expectations heuristic, which gives a smaller prediction error and more profit for the agents. In addition, when the majority of the agents use the rational expectations heuristic, prices will go back to their steady state value. Subsequently, the prediction errors of both the naive and the rational expectations heuristic will be the same. In that case, agents choose the naive expectations heuristic due to low costs and the process repeats.

2.2 Intensity of choice

The paper of Brock and Hommes (1997), describing the cobweb model with rational versus naive expectations, is the basis of many papers written thereafter. Several empirical studies have been done after the findings of Brock and Hommes (1997) to prove that the discrete choice model gives good insights in market behavior. For example, Branch (2004) estimates the discrete choice model as agents choose between three predictions, whereas Goldbaum and Mizrach (2008) use mutual fund allocations decisions to estimate the intensity of choice estimator. Boswijk, Hommes, and Manzan (2007) use both a fundamentalist heuristic and a chartist heuristic to estimate a discrete choice model.

However, in all the studies mentioned above, the intensity of choice needs to be jointly estimated with other parameters that describe behavior, in order to be able to give the conclusions made in the papers. Boswijk, Hommes, and Manzan (2007) find an insignificant intensity of choice parameter value. Other studies, although finding significant intensity of choice parameter values, give different values for the parameter. These different values are

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based on the specification of the heuristics. As an example, Goldbaum and Mizrach (2008) find three different values of the intensity of choice parameter for three different values of risk aversion.

Due to these insignificant and different findings for the value of the intensity of choice parameter, Anufriev et al. (2018) use an experiment on heuristic choice to estimate the intensity of choice parameter.

2.2.1 Experiment on endogenous intensity of choice

Anufriev et al. (2018) test the assumptions and implications of the discrete choice model given in Brock and Hommes (1997). They use a simplified version of the discrete choice model described in Section 2.1, the stylized discrete choice model, in the experiment.

In the experiment of Anufriev et al. (2018), participants have to choose between a costly complicated heuristic and a simple cheap heuristic, where the costly complicated heuristic stabilizes the price when used by many participants and the simple heuristic destabilizes the price. Two blocks are distinguished; a high block with high information costs and a low block with low information costs. Half of the participants have high-low treatment, starting with a block with high costs, followed by a block with low costs. The other half of the participants have low-high treatment. By distinguishing these reversed treatments and the different blocks, Anufriev et al. (2018) examine four situations. The participants are paid according to the performance of the chosen heuristic. This performance depends on the distribution of all participants’ choices.

First of all, Anufriev et al. (2018) test if there is a substantial difference in the volatility of both the fraction of agents choosing one specific predictor and the state variable between the high and low blocks. Anufriev et al. (2018, pp. 25) conclude that this hypothesis cannot be rejected. However, the difference between high and low blocks is smaller than expected.

Furthermore, Anufriev et al. (2018, pp. 26-27) find that the estimated intensity of choice coefficients in low blocks are about 50 times as high as those for high blocks. From

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this, they conclude that the participants are less sensitive to profit differences in the unstable high blocks than in the stable low blocks. These findings reject the hypothesis that there is no significant difference between discrete choice models estimated on data from high blocks, and the discrete choice models estimated on data from the low blocks.

The conclusions of the experiment from Anufriev et al. (2018) indicate that participants tend to adapt their choice behavior to their environment. Anufriev et al. (2018) question whether the results also hold for a bigger group of participants and for a longer time. The outcome of Anufriev et al. (2018, pp. 30) shows that this is true. They prove that participants become less sensitive to past performance over time in the high cost environment.

The experiment of Anufriev et al. (2018) suggests that a discrete choice model, including an endogenous intensity of choice parameter and negatively depending upon some measure of volatility of payoff differences, is likely to provide a better description of choice behavior (Anufriev et al., 2018, pp.32).

3 Adjustments to the original discrete choice model

This paper adjusts the original discrete choice model of Brock and Hommes (1997), by using the suggestions and findings of Anufriev et al. (2018). The first adjustment involves using an endogenous intensity of choice parameter in the model. An endogenous intensity of choice parameter, βendo, is derived from dividing β by a certain value that differs every period.

As seen before, Anufriev et al. (2018) suggest a better description of choice behavior is provided by using an endogenous intensity of choice parameter, negatively depending upon some measure of volatility of payoff differences. In this paper β is divided by the variance of the difference between the profits of the naive and informative choices, over the last 10 periods (σ2_π). The profits of the last 10 periods are compared because it is supposed that the agents base their decision on the past performance of the heuristics up to 10 periods ago. The profits used, are the ones described by the original discrete choice model, given by equations

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2.1.5 and 2.1.6, but depending on the endogenous intensity of choice. The variance is taken to a power, α, which is the first new parameter included in the new model. The endogenous β in the adjusted model is:

βendo =

β (σ2

π)α

. (3.0.1)

Using βendo, the problem arises that σπ2 becomes 0 at some point and as a consequence

the prices also become 0. To get significant results for the prices, several adjustments can be made. For example, βendo can be derived from dividing β by 1 + (σπ2)α, or a noise term can be

added to the equation of the prices. The goal is to get the broadest range of values of βendo,

to be able to measure the most effect in the difference between the original and the adjusted model. For the first option, βendo can have a maximum value β, because if the variance of

the difference in the profits is very small, βendo is derived from dividing β by 1. However, by

including a noise term in the adjusted model, βendo can have values above β and thus there is

a broader range of values in that case. That is why a noise term is added to equation 2.1.11. The is a uniformly distributed random number in the interval (-0.5,0.5). The intensity of the noise term is variable by varying the second parameter θ. In the case of θ equal to 0, there is no error term included, and thus the prices will become 0 at some point. The price equation 2.1.11 changes to:

pt+1=

−b(1 − mt)pt

2B + b(1 + mt)

+ θ ∗ . (3.0.2)

To be able to compare the original with the adjusted model, the price pt+1 of the original

model (equation 2.1.11) must also be changed into equation 3.0.2.

By varying α, θ and β, the results of the adjusted model can be compared with the results of the original model in Section 2.1.

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4 Research method

Due to the three parameters in the adjusted model, α, θ and β, several situations can be studied. First, the stability of the dynamics of the adjusted model and the original model are compared. Two different values of the exogenous β in the original model are distinguished; the exogenous β can be the β that is the input for the adjusted model, or the exogenous β can be the median of the values of βendo, βmedian. The situations have different dynamics for

the original model.

Subsequently, the time series of βendo are plotted. In this way possible strange

behavior of βendo can be observed and studied.

After that, the models are analyzed by varying α for certain values of θ and β. In the case of α being equal to 0, the endogenous βendo is equivalent to the β in the original

model, because dividing β by the variance in the difference between the profits over the last 10 periods (σ_π2) to the power 0, is equal to dividing β by 1. If α is 1, βendo is derived from

dividing β by σ_π2. The goal is to analyze what happens if α is varied between 0 and 1. An other parameter that can be modified is θ. The bigger θ, the more the error term is included and the more similar the adjusted model and the original model are. The goal is to compare the outcomes of different values of θ, varying from 0.001 to 0.01.

The effects of the variations of α and θ are analyzed by the height of the variance of the prices for both models.

Later on, the time series of the prices are plotted for both models, to see how the prices change over time and to see the difference of the prices for both models.

In addition, β is compared with βmedian to study their relation.

5 Results of the adjusted model

This section describes the differences in the stability of the dynamics for both models. Possible explanations for these differences are tested and the effect of different values of α and θ are studied.

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5.1 Comparing the adjusted model with the original model by studying the stability of the dynamics

By comparing the original model with the adjusted model, the effect of including an endogenous intensity of choice parameter in the model is studied. Figures of the dynamics for different values of β are compared below. The original model uses β as exogenous β, whereas the adjusted model uses βendo.

Figure 2: Dynamics of the original and adjusted model for β = 1 and different values of θ and α, for 20000 periods

(a) Original model θ = 0.001, α = 0.25 (b) Adjusted model θ = 0.001, α = 0.25 (c) Original model θ = 0.001, α = 0.75 (d) Adjusted model θ = 0.001, α = 0.75

(e) Original model θ = 0.01, α = 0.25 (f) Adjusted model θ = 0.01, α = 0.25 (g) Original model θ = 0.01, α = 0.75 (h) Adjusted model θ = 0.01, α = 0.75

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Figure 3: Dynamics of the original and adjusted model for β = 3 and different values of θ and α, for 20000 periods

Figure 2 and Figure 3 show the stability of the dynamics of both models. For all values of β, α and θ, the dynamics of the adjusted model are more complicated than the dynamics of the original model. In these figures, the original model has stable dynamics for all cases, except for β = 3, θ = 0.01, α = 0.25 and β = 3, θ = 0.01, α = 0.75.

Comparing the adjusted model with the original model, using the median of the values of βendo instead of β as exogenous β, better shows the relation between the models,

because more similar values are compared. This is due to the fact that the values of βendo are

bigger than the values of β, because βendo is derived from dividing β with a number between

0 and 1, namely (σ_π2)α, where α and σ_π2 are between 0 and 1. From now on, βmedian is used

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Figure 4: Dynamics of the original and adjusted model for β = 1 and different values of θ and α, for 20000 periods, with βmedian for original model

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Figure 4 and Figure 5 again show the stability of the dynamics. In the case of β = 1, θ = 0.001, α = 0.75 and β = 3, θ = 0.001, α = 0.25, there is a difference in the stability of the dynamics between the original model and the adjusted model. For the other values of θ and α for β = 1 and β = 3, both the original model and the adjusted model have unstable dynamics.

5.2 Time series of endogenous intensity of choice parameters

We are going to look at the time series of βendoto find out if possible outliers are an explanation

for the differences in stability of the dynamics, described in Section 5.1. By taking certain values of β, α and θ, the time series of βendo are plotted in Figure 6a, Figure 7a and Figure

8a. These time series are compared to the dynamics of both the original model, using the median of the endogenous β values as the exogenous β (Figure 6b, Figure 7b and Figure 8b)

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and the adjusted model, using an endogenous β for every period (Figure 6c, Figure 7c and Figure 8c).

Figure 6: Time series of βendo and dynamics of the original and adjusted model for β = 1,

α = 0.75, θ = 0.001, for 20000 periods

(a) Time series of βendo (b) Original model (c) Adjusted model

α = 0.75, θ = 0.001, for 20000 periods

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α = 0.75, θ = 0.001, for 20000 periods

(a) Time series of βendo (b) Original model (c) Adjusted model

These figures show the influence of the outliers of the time series of βendo on the

stability of the dynamics. Taking the median of the βendo values lead to values for the

exogenous β as input for the original model. Brock and Hommes (1997, pp. 1069) have proven that the original model has unstable dynamics as the values of the exogenous β exceed 4.3, as shown in Figure 1. Therefore, we assume that if the median of the βendo values,

βmedian, is below 4.3, the original model will have stable dynamics. In the figures above, this

is the case in Figure 6 (β = 1) and Figure 7 (β = 2). The values for βmedian in those cases

are 3.0873 and 3.9092. In Figure 8, representing β = 3, βmedian is equal to 5.1707, which is

above 4.3.

5.2.1 The effect of outliers of the endogenous intensity of choice on the stability of the dynamics

To test the assumption if the stability of the dynamics are different if βmedian is below 4.3,

Figure 4 and Figure 5 are simulated again in Appendix I, using ¯β = min{βendo, 4.3} as

endogenous intensity of choice parameter in the adjusted model. It was found that the outliers of βendo are not a good explanation for the difference between the stability of the dynamics.

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5.3 Comparing the variance of the prices for both models

To find a reason for the differences between the stability of the dynamics, the differences between the variances of the prices of both models are studied. This section shows the fluctuations of these variances.

5.3.1 Differing the error term

Figure 9 represents the effect of α and θ on the variances of the prices of both the original and the adjusted model. The values of θ differ from 0.001 to 0.01.

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Figure 9: The variances of the prices for the original model and the adjusted model, for different values of θ, for 20000 periods

(a) β = 1, α = 0.25 (b) β = 1, α = 0.75

(c) β = 3, α = 0.25 (d) β = 3, α = 0.75

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The figures show that the variances of the prices of the original model and the adjusted model are more equal as α is lower. This is due to the fact that for low values of α, the models are more similar.

Assuming that the variances of the prices serve well as a measurement for the stability of the dynamics, the model having lower variances of the prices should have more stable dynamics than the model having higher variances of the prices. In the results above, for β = 1 the variances of the prices of the adjusted model are lower than of the original model, for all values of θ. As β gets bigger, the variance of the prices of the adjusted model will become higher than of the original model. This is an explanation for the stable dynamics of the adjusted model for β = 1 and the unstable dynamics for bigger values of β.

5.3.2 Differing the power of the variance of the difference between the prices In Figure 10 α is varied between 0.1 and 1 for incremental values of β. Again, the differences of the variances of the prices of the adjusted and original model are showed.

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Figure 10: The variances of the prices of the original model and the adjusted model, for different values of α, for 20000 periods

(a) β = 1, θ = 0.001 (b) β = 1, θ = 0.01

(c) β = 3, θ = 0.001 (d) β = 3, θ = 0.01

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Figure 10 shows that the variances of the prices of both models are more equal as θ is bigger, so when the error term influence the prices more. This can be explained by the fact that for a bigger error term, the value of β as input for the prices becomes less important. So that is why, for higher values of θ, the variance of the prices of both models are more similar. Next, a particular part of the figure is investigated. Figure 10c suggests the variance of the prices of the adjusted model are higher than the variance of the prices of the original model for α below 0.4. Figure 11 shows the dynamics for both models using α = 0.3, θ = 0.001 and β = 3. These situation indeed leads to stable dynamics for the original model, whereas the adjusted model is not stable. This finding assumes that the variances of the prices measure the difference between the stability of the dynamics of the models well.

Figure 11: Dynamics for β = 3, α = 0.3, θ = 0.001, for 20000 periods (a) Original model (b) Adjusted model

5.3.3 The variances of the prices as stability measurement

In Section 5.3.1 and Section 5.3.2 the variances of the prices are used as measurement of the stability of the dynamics. It is assumed that lower variances would lead to higher stability of the dynamics. As shown in Figure 9 and Figure 10, the variances of the prices of the adjusted model are sometimes above and sometimes below the variances of the prices of the original model. To measure the effect of the variances of the prices, two situations are distinguished; the time series of the prices are plotted for a situation in which the original model has stable

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dynamics, while the adjusted model has not. We take β = 3, θ = 0.001, α = 0.25, referring to Figure 5a and Figure 5b. Also, the time series of the prices are plotted for a situation in which both models have unstable dynamics; β = 3, θ = 0.01, α = 0.75, referring to Figure 5g and Figure 5h. The results are shown below.

Figure 12: Time series of the prices for β = 3, α = 0.25, θ = 0.001, for period 5000-5100 (a) Original model (b) Adjusted model

Figure 13: Time series of the prices for β = 3, α = 0.75, θ = 0.01, for period 5000-5100 (a) Original model (b) Adjusted model

Comparing Figure 12a with Figure 12b, we see that the differences between the prices in the original model are smaller than the differences between the prices in the adjusted model. This leads to smaller variances for the original model. Since the original model has stable

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dynamics, the variances of the prices are a good explanation for the differences between the stability of the dynamics of both models.

Comparing Figure 13a with Figure 13b, we see that the time series of the prices barely differ for the models. This means that the variances of the prices of both models are very similar in case of instability of the dynamics for both models. Thus the variances of the prices are also a good explanation for the differences between the stability of the dynamics of both models in this case.

It can be concluded that the time series of both models are very similar when the difference between the stability of the models is small and so that the variances of the prices are also very similar in these cases. In the cases of differences between the stability of the dynamics, the time series of the prices differ. The variances of the prices of the stable dynamics are smaller than the variances of the prices of the unstable dynamics. Therefore, the variance of the prices are a good measure for the stability of the dynamics.

5.4 Comparing the variance of the prices of the original model with the adjusted model

In the original model, instability arises for increasing β. For example, for β = 2 the dynamics are stable, while for β = 5 the dynamics are unstable. In this model holds; the bigger β, the bigger the variance of the prices, as shown in Table 1.

β variance of the prices

1 0.0821

3 0.2660

8 0.2712

14 0.3092

Table 1: Variance of the prices for different values of β of the original model

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values of β, as shown in Figure 9 and Figure 10. This means that increasing values of β are corrected by similar values of the variance of the prices. In this way, the exogenous β, βmedian,

used as comparison to the original model, which is equal to the median of the values of βendo,

comes down to a specific value.

In the figure below, the relation between β, the input of the adjusted model, is compared with βmedian, the input of the original model, for different values of α and θ.

Figure 14: The relation between β and βmedian for different values of α and θ, for 20000

periods

(a) α = 0.25 and θ = 0.001 (b) α = 0.25 and θ = 0.01

(c) α = 0.75 and θ = 0.001 (d) α = 0.75 and θ = 0.01

Figure 14 shows that the βmedian values are nearly linear with β. This insight is in

line with the theory described above. The figures show that it holds for all values of α between 0 and 1 and all values of θ.

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Thus, for the original model, increasing values of β lead to increased instability of the dynamics, because increasing values of β are corrected by increased values of the variance of the prices (Table 1). However, in the adjusted model, increasing values of β are corrected by a similar value of the variance of the prices, for every β. Due to this, the βmedian used to

compare the adjusted model with the original model, is an almost linear value.

To conclude, in the adjusted model it matters less for which β the model is estimated, in comparison with the original model. For this reason, the adjusted model always leads to almost the same instability of the dynamics, while the original model leads to stable dynamics for low values of β and as β increases, the dynamics get unstable.

6 Conclusions

This paper studies the stability of the dynamics for the discrete choice model. The original model of Brock and Hommes (1997) is adjusted and both models are compared. The adjusted model includes an endogenous intensity of choice variable for every period, depending on the variances of the profits of the last 10 periods, whereas the original model includes the median of the endogenous intensity of choice variables as exogenous intensity of choice variable. Also, an error term for the prices is added to both models, θ, and α is a parameter added to the adjusted model, measuring the influence of the variance of the difference between the profits on the endogenous intensity of choice.

It is found that the stability of the dynamics differ for the models. First, the time series for the endogenous intensity of choice variables are studies to find out if possible outliers could be the cause of the difference between the stability of the dynamics. It is concluded that the outliers of the endogenous intensity of choice are not a good explanation for the difference between the stability of the dynamics. Afterwards, the variances of the profits of both models are studied, for different values of α and θ. To find out if the variances of the prices could be a good way to measure the stability of the dynamics, the time series of the prices are plotted for two situations. In one situation the original model has stable dynamics, while the adjusted

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model does not. In the other situation both models have unstable dynamics. It is found that the variances of the prices are a good measure for the stability of the dynamics.

Furthermore it is concluded that the closer α is to 0, the more similar the adjusted model and the original model are, and thus also the more similar the variances of the prices are for both models.

The relation of θ on both models is studied as well. For higher values of θ, the error term is included more. In this case the value of an intensity of choice variable has less effect on the price, and so the prices are more similar for the models. This results in variances of the prices that are more similar as the value of θ becomes bigger.

In addition, it is found that the median of the endogenous intensity of choice variables is nearly linear with the exogenous intensity of choice. This insight is caused by the correction of the intensity of choice variable with nearly similar variances of the prices in the adjusted model, for different values of α and θ. So no matter what the intensity of choice variable is, the adjusted model always has unstable dynamics.

Finally, it can be concluded that using endogenous intensity of choice variables do not lead to stable dynamics and so the adjusted model is less robust than the original model of Brock and Hommes (1997).

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References

Anufriev, M, A. Chernulich, & J. Tuinstra (2018): A laboratory experiment on the heuristic switching model. Working Paper.

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Appendix

Appendix I

Below, Figure 4 and Figure 5 are plotted again, using ¯β = min{βendo, 4.3} as endogenous

intensity of choice variable in the adjusted model.

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Figure 15 and Figure 16 show that maximizing the value of βendo to 4.3 lead to stable

dynamics in for the adjusted model in the case of β = 3, θ = 0.001, α = 0.25 and β = 3, θ = 0.001, α = 0.75. However, it is not the case that the original model has stable dynamics, which was expected because by taking the median of βendo, as βendo was maximized to 4.3,

βmedian is maximum 4.3 in all cases. It can be concluded that maximizing βendo to 4.3 does

not lead to stable dynamics for the original model. Thus, the outliers in the time series of βendo are not a good explanation for the difference of the stability of the dynamics.

Endogenous intensity of choice, depending on the variance of the prots, in heterogeneous agent models