Endogeneity in the Intensity of Choice : using the auto-correlation to model the Intensity of Choice and revise the results of a 1997 paper by Brock and Hommes

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Endogeneity in the Intensity of Choice

Using the auto-correlation to model the Intensity of Choice and revise the results of a 1997 paper by Brock and Hommes

Bachelor Thesis Econometrics Jorrim Prins

11038934 June 25th 2018

Supervisor: Jan Tuinstra Abstract

This research presents an updated version of an original heuristic switching model by Brock and Hommes from 1997. The Intensity of Choice in this model has always been regarded as exogeneous. This paper describes a model that contains an endogeneously computed Intensity of Choice. Computer simulations of the original and the new model are compared and the effect of an endogeneous specified Intensity of Choice is reviewed.

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Statement of Originality

This document is written by Jorrim Prins 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

The past couple of decades have marked the start of heterogeneous agent models becoming popular to describe and attempt to understand financial markets. These models explain economic behavior by not only expecting agents in the market to behave rationally, but also allowing groups of agents with different strategies to participate in and affect the market. Brock and Hommes (1997) were precursors in attractively describing a specific class of these models, the heuristic switching model. They examined a model in which agents are able to choose and switch between rational and na¨ıve expectations of future market prices, based on previous outcomes the heuristics generated.

Brock and Hommes (1997) use a cobweb model with simple supply and demand functions, in which the fraction of agents per heuristic and the market price depend on each other. An important parameter in this mutual dependence is the Intensity of Choice (IoC). This parameter measures the speed at which agents switch between heuristics. The higher the value of the parameter, the more likely agents are to adapt their behavior to the performance of the heuristics. In their research, Brock and Hommes (1997) assume the IoC parameter to be exogenous, meaning it is independent of previous prices and the distribution of agents. This model is competent in describing stylized facts of financial markets such as excess volatility, under the condition that the value of the IoC parameter is high enough.

Although the model by Brock and Hommes (1997) is still referred to in recent research, Anufriev, Chernulich and Tuinstra (2018) question the exogeneity of the IoC parameter. They conducted a lab experiment to test whether the IoC parameter could have varying values in different market situations. Using a similar cobweb and logit discrete choice model to the one Brock and Hommes (1997) used, Anufriev, Chernulich and Tuinstra (2018) gathered empirical data without assuming a constant IoC. University students were given the incentive to maximize their profits by choosing between an expensive stabilizing and a cheap destabilizing option. The distribution of students between the options and the profit they received affected each other, as is the case in the model by Brock and Hommes

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(1997). By varying the cost difference between cheap and expensive, Anufriev, Chernulich and Tuinstra (2018) calculated two different empirical values for the IoC parameter and discovered that there was a significant difference between them.

Building on the empirical data that suggests endogeneity of the IoC parameter, the next step is finding possible explanatory variables and implement these in the models used by Brock and Hommes (1997) and Anufriev, Chernulich and Tuinstra (2018). Earlier empirically based research by Anufriev, Bao and Tuinstra (2016) suggests that the IoC parameter might be correlated with the auto-correlation of realized profits. This paper describes a heuristic switching model which includes an endogeneous IoC parameter. To test and verify the model, data is generated by running a simulation and these data are compared to the original results by Brock and Hommes (1997). Similarities and differences between the results are discussed, to explore whether the conclusions by Brock and Hommes (1997) are still true under the new circumstances.

The paper is structured as follows; section 2 elaborates on the heuristic switching model, the logit discrete choice model and theoretical background on the exogeneity or endogeneity of the IoC parameter. Information about explanatory variables and simulation methods to test endogeneity of the IoC parameter can be found in section 3. Section 4 describes the results of the research and the final model it produced, it also analyzes the results. To round it up, concluding remarks are stated in section 5.

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2 Adaptive Rational Equilibrium Dynamics and the Intensity

of Choice

The foundation for this paper is an original model by Brock and Hommes (1997), as it has been motivation for multiple other researches to be conducted. This section starts off by discussing the original heuristic switching model by Brock and Hommes (1997). After, the discrete choice model they used is elaborated on and some general info about discrete choice models is presented. An important parameter in discrete choice models is the Intensity of Choice (IoC) parameter, so its effects will be thoroughly discussed in section 2.2. Later research projects that have built on the original model will be shortly described in section 2.2.1 and 2.2.2. Empirical evidence for possible endogeneity of the IoC parameter is shown through research by Anufriev, Chernulich and Tuinstra (2018) and Anufriev, Bao and Tuinstra (2016).

2.1 Brock and Hommes

Brock and Hommes (1997) investigate a phenomenon called adaptive rational equilibrium dynamics (A.R.E.D.), which is a combination of market equilibrium dynamics and predictor selection. Original economic models were able to find and explain equilibria by expecting agents in the market to make a certain decision based on current market circumstances. As Brock and Hommes (1997) use a cobweb model for supply and demand, they allow fluctuation of market prices while giving agents the opportunity to base their expectations on previous prices. They add another dimension to their model by combining the cobweb structure with the existence of heterogeneous agents in the market. The agents are heterogeneous in the sense that they use different expectation techniques to base their decisions on. The combination of a cobweb model and agents with heterogeneous beliefs initiates mutual dependence of market equilibria and chosen expectation techniques, which is a key aspect of the so-called adaptive rational equilibrium dynamics.

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2.1.1 Heuristic Switching Model

The specific model that Brock and Hommes (1997) use, includes simple linear supply and demand functions that depend on two separate groups of agents. The two groups use different predictors for the price. One group of agents has rational expectations (denoted as H1) and

the other group has na¨ıve expectations (denoted as H2) for the price, the predictors are

formulated as follows:

H1( ~Pt) = pt+1 H2( ~Pt) = pt (2.1.1)

Denote the fractions of suppliers predicting rationally and na¨ıve as n1,t and n2,t respectively

and the following market equilibrium can be found:

D(pt+1) = n1,tS(pt+1) + n2,tS(pt) (2.1.2) with D(pt) = A − Bptand S H( ~Pt) = bH( ~Pt)

It is obvious that the rational predictor H1( ~Pt) is more sophisticated than H2( ~Pt),

the na¨ıve one, as it predicts the exact price in the following period. This gives agents with predictor H1( ~Pt) the opportunity to effectively maximize their profit. Consequently, the

availability of predictor H1( ~Pt) comes at a higher cost, with the price difference between the

rational and na¨ıve predictor being set at C.

Brock and Hommes (1997) make the assumption that agents base their predictor selection on the value of their realized profit. Combining this with earlier information about supply and demand functions and predictor costs, they find performance measures for rational and na¨ıve agents respectively:

π1(pt+1, pt+1) = b 2p 2 t+1− C π2(pt+1, pt) = b 2pt(2pt+1− pt) (2.1.3) Agents have a tendency to pick the predictor that generated the highest profit in one or more of the previous periods.

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2.1.2 Discrete Choice Model

An important modification that has to be made for heterogeneous agent models to work is regarding agents’ decisions to be discrete instead of continuous. A prominent model to implement this adjustment is the discrete choice model. Discrete choice models have different classifications, which describe the number of available alternatives, the executed regression method and the allowance of varying variables per alternative. Brock and Hommes (1997) decide to use a discrete choice model to describe the distribution of agents between the expensive rational predictor and the cheap na¨ıve predictor. They use a fairly simple version, with only two alternatives and a logit structure. This logit structure is helpful because it is limited to generating positive outcomes and contains a controlling parameter, the IoC parameter. The fractions of agents per predictor can be computed as follows, rational and na¨ıve predictor respectively:

n1,t+1= exp β b 2p 2 t+1− C ! /Zt+1 n2,t+1= exp β b 2pt(2pt+1− pt) ! /Zt+1 (2.1.4)

with parameter b coming from the supply function, Zt+1 being the sum of both

numerators and β being the IoC parameter which will be thoroughly discussed in the next section. Brock and Hommes (1997) introduce mt+1 as the difference between the two

fractions (n1,t+1− n2,t+1, with mt+1 = 1 meaning all agents choose the rational predictor

(H1)). Combining this with equilibrium equation 2.1.2, the market equilibrium price

becomes pt+1 = A − 1 + mt 2 bpt+1− 1 − mt 2 bpt ! /B (2.1.5)

Functions 2.1.4 and 2.1.5 can now be used to fully describe the model and the so-called adaptive rational equilibrium dynamics.

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2.2 Intensity of Choice

The previous section mentioned the Intensity of Choice parameter briefly. However, as it is an important parameter in discrete choice models and especially in this paper, this section contains more information on the IoC. Simply put, the parameter describes the speed at which agents tend to switch between the rational and na¨ıve predictor.

2.2.1 Goldbaum and Mizrach

Goldbaum and Mizrach (2008) find an effective way to explain the effect of the IoC parameter more extensively. First of all, they use a similar technique to the one by Brock and Hommes (1997) to distinguish heterogeneous agents. A separate idiosyncratic component is added to a standard utility function to create two different groups. The utility functions for agents i = 1, 2 are modelled as follows:

Ui,k = Uk+ σi,k (2.2.1)

The idiosyncratic component is independent and identically distributed, with E(i,k) = 0 and V ar(i,k = 1). Given the assumed exponential distribution of , the

probability of agent i choosing heuristic 1 can be calculated as follows:

P (xi = 1) =

eρU1

eρU1 _{+ e}ρU0 (2.2.2)

Assuming a large number of agents, the Law of Large Numbers implies that the same formula holds for the fraction of agents choosing heuristic 1.

Clearly, Goldbaum and Mizrach (2008) use a logit discrete choice model like Brock and Hommes (1997) do. The ρ parameter in this formula is the IoC parameter and will be denoted as β from now on. It is an inverse function of the σ parameter, so the effect of the magnitude of σ can be easily seen. An increase in σ leads to a bigger difference in utilities for separate agents. It automatically results in a decrease in β and thus an increase in the probability that an agent chooses the inferior heuristic. An extreme example would be

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σ −→ ∞, which would cause β −→ 0. Use β = 0 in function 2.2.2 and the agents choose both heuristics with probability 1₂. This shows that an increase of σ results in a decrease of the predictability of an agent’s decisions, which is obvious as σ enlarges the effect of the random component in the utility function. Concludingly, the IoC parameter defines agents’ tendency to adapt their behavior to differences in utility levels. Translating this to the model by Brock and Hommes (1997); agents will adapt their predictor choice to previous profit levels faster as the IoC value increases.

2.2.2 Intensity of Choice in other research

However the research by Goldbaum and Mizrach (2008) describes a model that clarifies the effect of the IoC parameter adequately, lots of other researchers have tried to estimate models with different assumptions and interpretations of the IoC parameter.

Panchenko, Gerasymchuk, and Pavlov (2013) and Brock, Hommes and Wagener (2005) use a function similar to function 2.2.1 to distinguish different heuristics, as explained through Goldbaum and Mizrach (2008) in the previous section. Panchenko, Gerasymchuk, and Pavlov (2013) use it to study four network structures with different levels of information availability. The first network structure is fully connected, every agent in the structure has complete information about the market and is connected to every other agent. Their second, regular lattice, network gives all agents a connection to a fixed amount of other agents, with all agents having the same number of connections. The random graph network they use assigns random connections to other agents, and the small world network is a mix of the two previous ones. For all four networks, Panchenko, Gerasymchuk, and Pavlov (2013) explore price distributions for different levels of the IoC parameter.

Brock, Hommes and Wagener (2005) use their model to find that the possible stability of steady states depends on the value of the IoC parameter. They also conclude that an increase in the IoC value leads to more irregularity in expectation fluctuations, which could be helpful to explain excess volatility for example. Gaunersdorfer (2000) builds on original models by Brock and Hommes (1997, 1998) and has similar conclusions about the stability of

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steady states depending on the value of the IoC parameter.

A different approach on studying equilibrium dynamics is introduced by Chiarella, Dieci and Gardini (2001). They describe varying dynamics while keeping fractions of heterogeneous agents at the same level. Acknowledging the fact that their model is less realistic than models that were previously mentioned, they find results that are easier to manage from a mathematical point of view. Their research offers a different perspective on dynamics in markets with heterogeneous agents, as it is not mainly focused on the IoC parameter.

Statistically significant evidence for the existence of heterogeneous agents and heuristic switching in empirical data is given by Boswijk, Hommes and Manzan (2007). They test their model, that is similar to the model by Brock and Hommes (1997), with the development of S&P500 stock prices from 1871-2003.

Branch (2004) also uses empirical data to describe the complexity of equilibrium dynamics that comes with finite values for the IoC parameter in discrete choice models. He uses a discrete choice model with three different heuristics and uses survey data to test the model. His utility function for j = 1, 2, 3 looks as follows:

Uj,t = −(M SEj,t+ Cj) (2.2.3)

with MSE representing the mean squared error of a predictor and C being the cost of that predictor. Apart from concluding that a higher MSE leads to a higher tendency to switch predictors, he concludes that agents seem to have a fundamental bias for one of the predictors.

2.2.3 Endogeneity of the Intensity of Choice

Heuristic switching models that are described in previous research are competent in describing market or survey data. However, these models often need to estimate the IoC parameter during the process. This is a problem as these estimates occasionally result in insignificant values for the IoC parameter. Even statistically significant values for the IoC can be very

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contrasting, as they vary over a large range of values.

Varying values for the IoC are caused by difference in modelling technique. For example, Goldbaum and Mizrach (2008) change the value of the risk aversion parameter and find different values for the IoC when they do. Panchenko, Gerasymchuk, and Pavlov (2013) find different equilibrium dynamics for four different networks with a range of IoC values. Their concluding remarks mention possible dependence of the value for the IoC on the number of connections in the network. Correlation between the IoC parameter and other parameters could be a sign of endogeneity of the parameter.

Anufriev, Chernulich and Tuinstra (2018) conduct a laboratory experiment and explain the outcome with a similar model to the one by Brock and Hommes (1997). They separate two different blocks for the experiment, one with a low cost difference and one with a high cost difference between the two predictors. The high cost environment is less stable and has a higher volatility, as the obtained profits are more irregular. Anufriev, Chernulich and Tuinstra (2018) calculate IoC values for both environments and find estimates that are about 50 times lower in the high cost environment. As participants in the experiment can see the low predictability of outcomes in the high cost environment, they become more cautious and have a lower tendency to switch between heuristics based on previous outcomes. This could indicate correlation between the IoC and the volatility in previous outcomes.

Another research that uses a model similar to the one by Brock and Hommes (1997) was conducted by Anufriev, Bao and Tuinstra (2016). They use time series with different data generating processes for an experiment similar to the previous one and use auto-correlations of profits to come to their conclusions. They specifically find that their estimated values for the IoC parameter depend on the auto-correlation of profits, the parameter will be larger when auto-correlation structure is more positive.

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3 Research Method

The original model by Brock and Hommes from 1997 is compared to the updated model. Comparisons are made with the help of 20000 computer iterations of the new model and the data resulting from those. The simulation gives insight on the stability of the steady states and the possible existence of periodic cycles.

3.1 Model specification

A considerably large part of the model is the same as the original one by Brock and Hommes (1997). Performance of heuristics is measured by the realized profits (equation 2.1.3) and fractions of agents per heuristic are identically computed (equation 2.1.4). Parameter values of the underlying cobweb model are also the same, with A = 0, B = 0.5, b = 1.35 and C = 1, so the two models can be effectively compared.

The first adjustment to the original model simply results from the nature of the model. Every time an iteration of the simulation has a price of exactly zero, the whole model converges to zero. An adjustment to the formula for the price is needed to solve this problem and make the model robust to these values. This can be easily done by adding a little bit of randomness to the price, created with an error term. Rewriting equation 2.1.5 and adding error term gives the following formula for the price

pt+1=

−b(1 − m_t)pt

2B + b(1 + mt)

+ (3.1.1)

with ∼ N(0,0.0001)

The second and most important adaptation is the endogeneous computation of the IoC parameter, which allows this parameter to vary over time. As earlier research by Anufriev, Bao and Tuinstra (2016) indicated a possible relation between the IoC parameter and the auto-correlation of the realized profits, the IoC parameter varies with the auto-auto-correlation of the previous 50 profits.

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for the first iterations. The rest of the 20000 periods calculate the price and the fractions of agents as discussed before, computing profits for rational and na¨ıve agents and the difference between them. The auto-correlation of 50 previous differences in profit with their first lag can then be used to model the IoC parameter endogeneously:

βendo,t= (1 + Rt−1)β0 (3.1.2)

with β0 the starting value for the IoC and Rt−1 the auto-correlation in the previous period,

calculated by the following formula:

Rt=

cov(πdif f,t, πdif f,t−1)

σtσt−1

(3.1.3)

3.2 Simulation

Apart from the information on the model that mainly comes from the research by Brock and Hommes (1997), other decisions have also been made on the simulation method. The first decision is on the number of iterations, which should simply be high enough to reach convergence and leave enough iterations for valuable results. Although this number is set to 20000, only the iterations starting from the point of convergence are important. The dynamics in the transient phase do not contain valuable information.

As mentioned, the auto-correlation is computed using the past 50 values for realized profit. It is also possible to do these computations using only 10 or for example 100 previous values. This decision has an effect on the steadiness of the auto-correlation. Using only 10 previous values, the auto-correlation will be fairly unstable and switch between different values fast. The more previous values that are used, the steadier the auto-correlation will become. Figure 1 and 2 show the auto-correlation calculated with 10 and 100 previous values respectively, to make the difference visible (β = 4).

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Figure 1: Auto-correlation, computed with endogeneous β

(a) 10 previous values (b) 100 previous values

These two examples are extreme, therefore we use a middle value for the computed simulation.

The last important step to execute correct iterations is the definition of the starting values. As mentioned before, some values for the price and the fractions of agents result in invalid results. Consequently, a starting value for the price and the fractions of agents needs to be determined carefully. The resulting attractors can also have different forms for different starting values, so this needs to be examined as well. Starting values for the price and the difference in fractions mtare −0.001 and −1 respectively. The starting value for the

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4 Results and Analysis

This section shows the comparisons between the old model by Brock and Hommes (1997) and the new model with an endogeneous IoC. A first check will be done on the distribution of the price and the fractions of agents. The auto-correlation and the IoC, the two most important parameters in this research, and their starting values will be thoroughly investigated in section 4.2. The final section elaborates on the endogeneous attractors for different starting values for the IoC. These attractors will be compared to the attractors that Brock and Hommes (1997) find.

4.1 Price and fraction distributions

Brock and Hommes (1997) take a look at the distributions for the price and the fractions of agents for their heuristic switching model with an IoC value of 5. The new model, with similar model parameters and an endogeneous IoC should generate similar values for these parameters. The price distributions for the model by Brock and Hommes (1997) and the new model respectively look as follows:

Figure 2: Price distribution with β = 5, A = 0, B = 0.5, b = 1.35

(a) with exogeneous IoC (b) with endogeneous IoC

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The price distribution of the new model reaches values close to zero more often, but the overall distribution is similar.

The distribution of variable mt is also worthy of inspection. It is the difference

between the fractions of agents, as explained in section 2.1.2. Here, a mtvalue of 1 corresponds

to all agents choosing the rational stabilizing predictor. The distribution of mtwith the Brock

and Hommes (1997) and the new model respectively:

Figure 3: Distribution of agents

(a) with exogeneous IoC (b) with endogeneous IoC

These distributions are even more similar than the price distributions that have been previously discussed. Both distributions have a tendency to attain values around -1, meaning all the agents pick the na¨ıve predictor. The opposite value of 1 is also attained several times, as well as values in between these two extremes.

4.2 Auto-correlation and Intensity of Choice

The auto-correlation of the realized profits is the underlying parameter for the endogeneous computation of the IoC. It is therefore important to understand the aspects of the heuristic switching model that have an impact on the auto-correlation.

An earlier discussed aspect is the amount of previous periods that is used for the computation of the auto-correlation. A big difference was shown between the use of 10 or 100

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periods. Both of these options are not optimal, as one of them varies over a range of values that is too large and the other one varies over a range that is small. Trial and error has led to an optimal amount of periods of 50. Auto-correlation developments for the model with endogeneous IoC are shown for starting values β0= 4, 4.3, 5, 10.

Figure 4: Auto-correlation

(a) β0= 4 (b) β0= 4.3

(c) β0 = 5 (d) β0= 10

For every mentioned starting value, the auto-correlation of the realized profits fluctuates around a value close to 0.4. However, the graphs show that as β0 increases, the

auto-correlation loses the trend movement that it has for smaller values for the IoC. This indicates that the initial value β0 also has an effect on the development of the

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auto-correlation. To make sure the simulation will not overestimate the auto-correlation, the initial value for the auto-correlation is set at 0.3.

4.3 Attractors

The most important aspect of the model by Brock and Hommes (1997) is the form of their attractors. They find stable steady state cycles for IoC values up to approximately 4. Their model has a stable two-cycle for β = 2, two co-existing four-cycles for β = 4 and becomes chaotic for higher values. The new model is compared to the one by Brock and Hommes (1997) for β = 2, 4, 5, 10 in figure 5.

First of all, the thicker layers and the outliers are easy to explain and neglectable; as the endogeneous β varies, the prices and fractions will not return to the same values perfectly. Reviewing the attractors for the model with an endogeneous IoC, it becomes clear that the model only produces a simple attractor for low values for β0. The model produces

two co-existing stable five-cycles for β0= 2, but every other shown example produces chaotic

attractors. Even for β0 = 4 the attractor is chaotic, while the exogeneous model produces

two co-existing stable four-cycles for β = 4.

However, as the average auto-correlation of the realized profits moves around 0.4, the average value of the endogeneous IoC will be higher than β0. In the case of β0 = 4, the

average value of the endogeneous IoC will be around 5.4 and to make fair comparisons of the attractors this has to be compared to the attractor for the exogeneous Ioc of 5.4. Also, for an exogeneous IoC of 4, the model produces two co-existing stable four-cycles. A β0 that

produces an average endogeneous IoC of 4 could also have stable cycles, so this comparison is also made. Taking β0= 3.3 results in an average endogeneous IoC of 3.9 which has the same

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Figure 5: Attractors for model with exogeneous and endogeneous β

(a) β = 2, exogeneous (b) β0= 2, endogeneous

(c) β = 4, exogeneous (d) β0= 4, endogeneous

(e) β = 5, exogeneous (f) β0= 5, endogeneous

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Figure 6: Attractors for model with exogeneous and endogeneous β

(a) β = 5.43, exogeneous (b) β0= 4, endogeneous

(c) β = 3.9, exogeneous (d) β0= 3.35, endogeneous

Except for the small outliers for the endogeneous IoC, which has been explained earlier in this section, attractor a) and b) are very similar which suggests the model is specified correctly. The expectation for c) and d) would also be for them to be similar. However, the endogeneous IoC results in a chaotic attractor while the exogeneous IoC creates two co-existing stable four-cycles.

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5 Conclusion

The precursing heuristic switching model by Brock and Hommes (1997) has been motivation for lots of other researches in the following years. However, some of these researches have found flaws in the model. This research develops a new heuristic switching model based on the original one by Brock and Hommes (1997). The major difference between the two models is the specification of the IoC parameter, as the new model defines the IoC as an endogeneous parameter This was hinted upon by Anufriev, Bao and Tuinstra (2016). Outcomes of 20000 iterations of the new model are compared to a simulation of the original model. As the original model is unable to generate stable attractors for high values for the IoC, the ultimate goal is to find a model that is capable of doing so.

Observing the price distributions and the distributions of the fractions of agents, the development for both models looks similar. Both price distributions attain values between -2 and 2 and both fraction distributions attain the values -1, 1 as well as values in between. The auto-correlation of the realized profits, calculated over the previous 50 periods, is the parameter that influences the development of the IoC. Its progress is different for varying values of β0, the starting value for the endogeneous IoC parameter. The higher β0, the lower

the amount of variation in the auto-correlation.

The ultimate test for the model is the comparison of attractors for different values of the IoC. The original model finds stable attractors for IoC values up to approximately 4. In an optimal situation, the new model also finds stable attractors for IoC values higher than 4. For β0 = 2 the attractor is a stable five-cycle, so it is stable but not comparable to the original

model. For higher values of β0, the attractors become chaotic. Even when lower values of

β0 are used, that result in an average endogenous IoC of approximately 4, the attractor is

chaotic. Concludingly, an endogeneous Intensity of Choice, when it is specified as clarified in this paper, does not achieve a model with stable steady state cycles as the original model does.

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the endogeneous IoC can be done in several ways. The model described in this paper uses a fairly simple specification with influence of the auto-correlation. It is possible that the correct specification is more complex than the one used in this paper, for example by adding more lags to the auto-correlation. Also, the current model allows the endogenous IoC to take values between 0 and 2β0 and every period is calculated from the starting value. It might

be better to let the endogeneous IoC evolve from its previous value, by adding a little bit of auto-correlation to it. This way, the IoC would be less limited to its starting value.

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Bibliography

Anufriev, M., Bao, T., & Tuinstra, J. (2016) Microfoundations for switching behavior in heterogeneous agent models: An experiment. Journal of Economic Behavior and Organization, vol. 129, pp. 74-99

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

Boswijk, H. P., Hommes, C. H., & Manzan, S. (2007) Behavioral heterogeneity in stock prices. Journal of Economic Dynamics and Control, vol. 31, pp. 1938-1970.

Branch, W. A. (2004) The Theory of Rationally Heterogeneous Expectations: Evidence from Survey Data on In ation Expectations. The Economic Journal, vol. 114, pp. 592-621 Brock, W. A. , Hommes, C. H. (1997) A Rational Route to Randomness. Econometrica, vol.

65, pp. 1059-1095

Brock, W. A. , Hommes, C. H. (1998) Heterogeneous beliefs and routes to chaos in a simple asset pricing model. Journal of Economic Dynamics and Control, vol. 22, pp. 1235-1274 Brock, W. A., Hommes, C. H., & Wagener, F. O. (2005) Evolutionary dynamics in markets

with many trader types. Journal of Mathematical Economics, vol. 41, pp. 7-42

Chiarella, C., Dieci, R., & Gardini, L. (2001) Asset price dynamics in a financial market with fundamentalists and chartists. Discrete Dynamics in Nature and Society, vol. 6, pp. 69-99

Gaunersdorfer, A. (2000) Endogenous fluctuations in a simple asset pricing model with heterogeneous agents. Journal of Economic Dynamics and Control, vol. 24, pp. 799-831 Goldbaum, D., & Mizrach, B. (2008) Estimating the Intensity of Choice in a Dynamic Mutual Fund Allocation Decision. Journal of Economic Dynamics and Control, vol. 32, pp. 3866-3876

Panchenko, V., Gerasymchuk, s., & Pavlov, O. V. (2013) Asset price dynamics with heterogeneous beliefs and local network interactions. Journal of Economic Dynamics and Control, vol. 37, pp. 2623-2642

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Appendix

Programs

clear all; clc;

global B b C beta betazero alpha B = 0.5; b = 1.35; C = 1; beta = 5; betazero = 5; alpha = 1; N = 20000; x = 1:N;

%% Exogeneous determination of prices, fractions and profits. p_exo = zeros(1, N); m_exo = zeros(1, N); profit_1_exo = zeros(1, N); profit_2_exo = zeros(1, N); auto_corr_exo = zeros(1,N); p_exo(1) = -0.001; m_exo(1) = -1; auto_corr_exo(1:50) = 0.3; for i = 2:N

p_exo(i) = p_tplus1(p_exo(i-1), m_exo(i-1)) + normrnd(0,0.0001); m_exo(i) = m_tplus1(p_exo(i-1), m_exo(i-1));

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end

for i = 1:N-1

profit_1_exo(i) = b/2*p_exo(i+1)^2 - C;

profit_2_exo(i) = b/2*p_exo(i)*(2*p_exo(i+1)-p_exo(i)); end

%% Determining the auto-correlation with exogeneous beta profit_exo_diff = profit_1_exo - profit_2_exo;

for i = 51:N-1

auto_corr_mat2 = corrcoef(profit_exo_diff(i-49:i),profit_exo_diff(i-50:i-1)); auto_corr_exo(i) = auto_corr_mat2(1,2);

end

%% Endogeneous determination of prices, fractions and profits. p_endo = zeros(1, N); m_endo = zeros(1, N); profit_1_endo = zeros(1, N); profit_2_endo = zeros(1, N); beta_endo = zeros(1, N); auto_corr_endo = zeros(1, N); p_endo(1) = -0.001; m_endo(1) = -1; beta_endo(1:50) = betazero; auto_corr_endo(1:50) = 0.3;

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%First 50 (exogeneous) values for p, m and profits

for j = 2:50

p_endo(j) = p_tplus1(p_endo(j-1), m_endo(j-1)) + normrnd(0,0.0001); m_endo(j) = tanh(betazero/2*(b/2*(b*(1-m_endo(j-1))/

(2*B+b*(1+m_endo(j-1)))+1)^2*p_endo(j-1)^2-C)); profit_1_endo(j-1) = b/2*p_endo(j)^2 - C;

profit_2_endo(j-1) = b/2*p_endo(j-1)*(2*p_endo(j)-p_endo(j-1)); profit_endo_diff = profit_1_endo - profit_2_endo;

end

% Rest of the computations for p, m, profits, auto-correlation and beta

for i = 51:N-1

p_endo(i) = p_tplus1(p_endo(i-1), m_endo(i-1)) + normrnd(0,0.0001); m_endo(i) = tanh(beta_endo(i-1)/2*(b/2*(b*(1-m_endo(i-1))/

(2*B+b*(1+m_endo(i-1)))+1)^2*p_endo(i-1)^2-C)); profit_1_endo(i-1) = b/2*p_endo(i)^2 - C;

profit_2_endo(i-1) = b/2*p_endo(i-1)*(2*p_endo(i)-p_endo(i-1)); profit_endo_diff = profit_1_endo - profit_2_endo;

auto_corr_mat = corrcoef(profit_endo_diff(i-49:i), profit_endo_diff(i-50:i-1)); auto_corr_endo(i) = auto_corr_mat(1,2); beta_endo(i) = (1+alpha*auto_corr_endo(i-1))*betazero; end %% Plots figure(1)

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plot(x, p_exo)

axis([10000 10200 -2 2]); ylabel(’p’);

xlabel(’t’);

title(’Time series for the price’); hold on

plot(x, p_endo);

legend(’Exogeneous IoC’, ’Endogeneous IoC’);

figure(2)

plot(x, m_exo);

axis([10000 10200 -1.1 1.1]); ylabel(’m’);

xlabel(’t’);

title(’Time series for the difference in fractions’); hold on

plot(x, m_endo);

legend(’Exogeneous IoC’, ’Endogeneous IoC’);

figure(3)

plot(p_exo(2000:end), m_exo(2000:end),’.’,’MarkerSize’, 12) axis([-2 2 -1.1 1.1]);

xlabel(’p’); ylabel(’m’);

title(’Attractors for exogeneous beta’);

figure(4)

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axis([-2 2 -1.1 1.1]); xlabel(’p’);

ylabel(’m’);

title(’Attractors for endogeneous beta’);

figure(5)

plot(x, auto_corr_exo);

axis([10000 12000 -0.3 1.1]);

title(’Auto-correlation for exogeneous beta’);

figure(6)

plot(x, auto_corr_endo); axis([10000 12000 -0.3 1.1]);

title(’Auto-correlation for endogeneous beta’);

%% Function for price in period t+1 function out = p_tplus1(pt, mt)

global B b

out = -b*(1-mt)*pt/(2*B+b*(1+mt)); end

%% Function for fractions in period t+1 function out = m_tplus1(pt, mt)

global B b C beta

out = tanh(beta/2*(b/2*(b*(1-mt)/(2*B+b*(1+mt))+1)^2*pt^2-C)); end

Endogeneity in the Intensity of Choice : using the auto-correlation to model the Intensity of Choice and revise the results of a 1997 paper by Brock and Hommes