Behavioral heterogeneity across multiple financial markets

Behavioral Heterogeneity Across

Multiple Financial Markets

Econometrics MSc. Thesis, Financial Track

Faculty of Economics and Business

August 14, 2018

Author Supervisor

Alexander C.B. Schmitt BSc.„ prof. dr. Peter H. Boswijk…

Second reader prof. dr. Cars H. Hommes§

Abstract

This paper proposes a bivariate extension of the heterogeneous agents model and provides empirical validation. The aim lies in finding evidence of the existence of fun-damentalists and chartists across multiple financial markets, and their behavior toward expectations over the assets. The results substantiate the presence of fundamentalists and chartists, and explain certain parts of historical asset pricing bubbles. Moreover, agents switch between strategies depending on the their past performance.

Keywords: Asset pricing; Behavioral finance; Bounded rationality; Evolutionary switch-ing; Heterogeneous expectations; Nonlinear economic dynamics

„_{Student number: 10589546, email: alexander@schmitt.nl}

…_{Professor of Financial Econometrics at the University of Amsterdam} §_{Director of CeNDEF at the University of Amsterdam}

Statement of Originality

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

Acknowledgements

I would like to take this opportunity to thank a number of people who have made it possible for me to successfully write my master’s thesis.

First, I want to show my gratitude to my supervisor prof. dr. H. P. Boswijk from the University of Amsterdam for his great guidance during the last couple of months. In between the econometric technicalities, he made sure that I stayed on the right track of finishing this master’s thesis. The door to his office was always open and he was always able to provide fundamental insight and clarifications to any problems.

Secondly, I would like to thank prof. dr. C. H. Hommes from the University of Amsterdam as the second reader of this thesis. In addition, I am very grateful for the knowledge I acquired from studying his book about Behavioral Rationality as well as his excellent academic work. It has proven to be a great source of information that helped me write this master’s thesis.

Lastly, I would like to thank my family and friends for their continuous support and encouragement throughout my years of study and through the process of re-searching and writing this thesis. Therefore I am very grateful.

I hereby present my master’s thesis to obtain the master’s degree in Financial Econo-metrics at the University of Amsterdam.

Contents

1 Introduction

1

2 Theory & Related Literature

3

3 The Model

7 3.1 Asset specification . . . 8 3.2 Wealth dynamics . . . 9 3.3 Conditional variance . . . 10 3.4 Heterogeneous beliefs . . . 11 3.5 Evolutionary dynamics . . . 12 3.6 Market maker . . . 14

3.7 Full dynamical model . . . 14

4 Data & Estimation Methodology

16 4.1 Reduced form model . . . 16

4.2 Data . . . 17 4.3 Fundamental value . . . 18 4.4 Estimation method . . . 19 4.5 Variance estimation . . . 20

5 Results

21 5.1 Estimation results . . . 21 5.2 Switching . . . 24

6 Conclusion

32

Bibliography

34

Appendix A Extended Model Form

36

Appendix B Additional Figures

37 B.1 S&P500 model . . . 37

1

Introduction

Expectations play a crucial role in economics, as they are one of the fundamental differences between economics and the natural sciences. Due to the working of ex-pectations, the dynamics of the economy exhibit non-linearities and feedback loops that have a great impact on financial markets and economic fundamentals. Recent financial crises have shown examples where positive expectation feedback loops can result in asset price bubbles that suddenly burst. Moreover, monetary policy makers also become more and more concerned about the impact of these phenomena. They observe complexities and fluctuations in asset price dynamics and financial markets that are highly unpredictable and do not seem to have clear explanations from a tra-ditional point of view. Therefore, it is of great importance to understand expectations and how they play a role in our economy.

Economic agents make decisions based on their expectations about future states of the economy. Economic models therefore depend heavily upon the underlying expectations hypothesis. The traditional paradigm of working with expectations in economics is the Rational Expectations (RE) hypothesis. Since the seminal papers of Muth(1961),Lucas(1972) andSargent (1973), the RE hypothesis has become the standard in characterizing the single representative agent in an economic model. This framework deals with expectations feedback systems by imposing that all agents make unbiased predictions based on all information. This results in a rather simplistic view of the economy where every agent acts in the same rational way. Nowadays, it has become evident that these standard economic models miss certain stylized facts in the behavior of economic agents, shown in financial markets for example. Especially in times of crises, sudden jumps in financial markets cannot be explained by standard representative agent models, and can certainly not be viewed as rational behavior.

In 1955, Simon proposed the idea of bounded rationality as an alternative to the standard representative agent models. Together with the lack of empirical evidence of traditional economic models, the idea of bounded rationality ultimately contributed to the rise of agent-based models (ABMs), a type of model that contains various types of agents and is primarily designed to explain complexities and fluctuations in asset price dynamics that before were labeled ‘anomalies’. ABMs incorporate complex expectation dynamics and frameworks (a discussion follows in the next sec-tion), and therefore they could become very complex and highly non-linear, due to which estimation would become quite tedious. Therefore, ABMs were mainly used for simulation-based studies. This was one of the reasons Heterogeneous Agent Models (HAMs) were developed, which are simpler in nature as they usually contain only two types of agents. More specifically, most HAMs consider a type of agent that is

Introduction

fundamentalist in nature, expecting mean reversion in prices, and a second type of agent that is chartist in nature, a type of agent that expects trend continuation1. Another advantage of HAMs is that they are much easier to test empirically. As Lux and Zwinkels (2018) rightfully point out, it might be interesting to turn our focus more toward estimating HAMs on individual stock level data instead of stock indices, which is the type of asset most HAMs focus on. Existing literature on mainstream approaches of explaining asset pricing in individual stock level data have observed anomalies that might have a behavioral explanation. Estimating HAMs on stock level data can provide explanations for those anomalies observed in practice.

This paper attempts to capture those behavioral effects in asset pricing of stocks using behavioral heterogeneity in a way that, to my best knowledge, has not been studied yet. A bivariate HAM is developed that aims to simultaneously explain asset pricing in two different assets, the S&P500 and IBM Stock. By allowing agents to choose between an index and a stock, a more representative environment is created for the estimation of the HAM. Furthermore, the choice of the assets is also made such, that by considering a combination of an asset and an index, agents have the opportunity to switch to an asset with the level of risk suitable for them. Agents are boundedly rational and belong to either the fundamentalist group or the chartist group. They use a performance measure to compare the two strategies and have the ability to switch to the most profitable one. The model is set under a market-maker scenario, which is assumed to be in control of the market of the S&P500 and IBM Stock2_{. The model, and specifically its extension, is aimed to find empirical evidence}

of behavioral heterogeneity of traders that are active on both markets. Moreover, the model is expected to capture the typical behavior shown in financial markets: Trends in asset prices that are created by fundamentals, upon which speculation takes over, until the trend diminishes. Furthermore, the model is expected to explain bubble formation in asset prices, a phenomenon characterized by excessive speculation, which can by no means be explained by rational expectations. Concluding, this paper aims to answer the question whether fundamentalists and chartists that are active traders in one market, can span across other financial markets, and whether asset pricing in those markets can be explained by behavioral heterogeneity.

The outline of the paper is as follows. Section2 discusses some related literature and in-depth theory. Whereas Section 3 describes the derivation of the bivariate heterogeneous agents model, Section 4 describes the data used and the estimation methodology and Section 5 contains the estimation results and discusses their impli-cations. Finally, Section 6 concludes our findings.

1_{In the literature of ABMs, chartists are also referred to as trend followers.}

2_{Since the S&P500 constitutes of stocks trading on the NYSE and NASDAQ, and IBM is traded}

on the NYSE which makes use of a single market maker, it is appropriate to assume a single market maker for both assets.

2

Theory & Related Literature

Taking agent-based models (ABMs) to the data is not very straightforward and in-volves taking many choices regarding the dynamics. Since it is of great importance to understand the underlying frameworks, this section first discusses some in-depth theories before introducing the model. Subsequently, since our model will be closely related to some existing papers, this section discusses similarities and differences with those papers and considers the choices that were made.

To begin with, we take a closer look at what motivated the idea of bounded rationality. Simon (1955) argued that the assumptions that underlie the RE hypoth-esis are not well suited to the representative agent for a number of reasons. First, rationality requires a tremendous amount of information processing: The rational expectation of a financial asset involves taking the mathematical expectation of all historical prices of the asset, which also requires agents to have perfect information about real markets, economic fundamentals and dynamic processes. This seems un-realistic since the implementation of this would require a great deal of knowledge and computational effort, a quality that is uncharacteristic for the representative agent. Secondly, the RE hypothesis assumes that the agents have perfect knowledge of the law of motion of an economy or financial market, which depends on the expectations of all other financial agents in the economy or market. Again, it is highly unrealistic to assume that agents know what the other agents will expect and how the underlying market equilibrium equations are formed. Simon(1955) therefore proposed modeling human behavior in economics with bounded rationality. The main idea of bounded rationality is that the ability of economic agents to form expectations is limited by factors such as non-transparency, time constraints and cognitive inabilities. At the time, Simon’s ideas did not win over his contemporaries, but in the last two decades ideas about modeling economies with boundedly rational agents acquired substantial interest. It was also in this area that he received the Nobel Prize in Economics in 1978.

In line with the concept of bounded rationality lies the heterogeneous expecta-tions hypothesis. It presumes that a dynamic system representing an economy is populated by several types of economic agents that interact with each other. Each type of economic agent has its own strategy of making decisions or forecasts, which are called heuristics. These heuristics are intended to represent the decision making of agents acting in an environment too complex for their understanding. The heuris-tics are therefore often focused on historical observations from a short time horizon, enabling agents to use them without great computing abilities. However, there are many different ways to specify heuristics. Most of them have a simple behavioral

Theory & Related Literature

interpretation, such as naive expectations, but more sophisticated rules that range from statistical analyses to trend extrapolating rules can also be used. The use of heuristics in practice has widely been studied through empirical experiments, where the behavior of human subjects is studied in an experimental setting. For an overview and discussion of these experiments see Hommes (2011).

The strategy switching framework ofBrock and Hommes (1997) further develops the heterogeneous expectational dynamic system. In this framework, the agents com-pete against each other through an evolutionary selection mechanism, which allows agents to switch to strategies that turned out to be performing well, while moving away from those that do not perform well. The performance of the strategies are calculated using a measure suitable to the environment that is considered, such as realized profits. This framework can further be extended by allowing the heuristics to adjust over time, following an adaptive learning process aimed to optimize the heuristics. These dynamics evolve over time and thus result in a complex dynamical system exhibiting many forms of non-linearities.

Current literature has proposed a number of other extensions to the original frame-work of Brock and Hommes(1997) and have estimated them. For example,Chiarella et al.(2012) estimate a simple HAM where they introduced Markov regime-switching state dependence. They argue that since financial markets exhibit periods of high and low volatility, incorporating a regime-dependent scheme for heterogeneous be-liefs could improve the empirical fit of the model significantly. Another example is

de Jong et al.(2009), who study shift-contagion in Asian markets through behavioral heterogeneity. They argue that by explicitly modeling co-movements between differ-ent financial markets, they can determine whether the source of price changes shifts across markets.

Since ABMs were originally built for simulation purposes, a bridge had to be made toward the empirical verification of HAMs, which is important for several reasons. First of all, HAMs are built on a framework that moves away from the traditional rational expectations, and since there are many shapes bounded rationality can take, empirical validation can serve as a test to see whether the model choices are jus-tified. Secondly, empirical validation provides a better understanding of the actual dynamics underlying market mechanisms than simulation based studies. Finally, em-pirical validation might make the ABM literature more appealing to the mainstream readership, which could serve as a bridge toward more practical applications of agent based models and eventually policy recommendations. Empirical evidence for ABMs and HAMs can be obtained from several different types of data, each with their own advantage and disadvantage. First of all, ABMs and HAMs can be studied from the micro level by using data such as experimental data, survey data and investment fund data. An early example is Schmalensee (1976), who used experimental data to

ex-Theory & Related Literature

plain expectation formation in wheat prices3. On the other hand, empirical evidence for ABMs and HAMs can be gathered from market data. Since market data on most asset prices is readily available, this type of data seems to have the upper hand. As a result, studies on HAMs where market data is used to explain asset pricing behavior are in abundance, although each study typically focuses on a different class of asset. Most of those studies have turned their focus on equities, and in particular on stock indices. However, a wide range of other asset classes have also been considered, such as options prices (Frijns et al.,2010), exchange rates (Manzan and Westerhoff,2007), volatility processes (Frijns et al.,2011), arbitrage opportunities (Frijns and Zwinkels,

2018) and oil prices (ter Ellen and Zwinkels,2010). Those that focus on stock indices range from assets such as the S&P500 (Boswijk et al.,2007), to Thai and Hong Kong stock markets (de Jong et al., 2009) and many more. Readers are referred to Lux and Zwinkels (2018) for an extensive survey of the current literature on empirical validation of asset-based models.

One of the first who empirically estimated a behavioral HAM wereBoswijk et al.

(2007), who successfully obtained evidence of the existence of heterogeneity in the S&P500. They formulated a simple model where agents can switch between the fundamentalist and the chartists type and formulate expectations over the price-to-earnings (PY) ratio of the S&P500. Though it is appealing to formulate HAMs in terms of absolute price level, this is rarely done in practice since most asset price time series are non-stationary. A solution is found by formulating the model in terms of deviation from a fundamental value, an approach applied byBoswijk et al.(2007) and

Chiarella et al. (2014) among others, including this paper. Though this fundamental value is assumed to be common knowledge among the agents, they will form their expectations in terms of deviation to this fundamental value. Another reason this paper studies the model in terms of deviation of its fundamental price is because we consider two assets: When one asset deviates from its fundamental and the other does not, agents should act upon this difference by switching their strategy in the one asset, while sticking with their strategy in the other.

As mentioned before, ABMs were originally built for simulation purposes due to their non-linear nature. As such, they exhibit more non-linearities, such as cost and learning parameters in the heuristics, and evolutionary learning with asynchronous updating. An example of a simulation-based study is Chiarella and He (2003a,

2003b), who extend the original framework of Brock and Hommes(1997) by allowing for heterogeneity in the agents’ risk appetite, and in the first and second moments of the price distributions. They investigate the effect of modeling an ABM with various different geometric decay processes (GDPs), which is the heuristic that the

trend-3_{See e.g. Assenza et al.(2014) for an extensive overview of experimental evidence on expectation}

Theory & Related Literature

followers in their model follow. This paper follows the specification of Boswijk et al.

(2007) for the trend followers. They focus on the simplest case of lag-length L = 1, since in their estimation the higher order lags were insignificant. This is in contrast to Chiarella et al. (2014), who use a limiting geometric decay process (LGDP) as heuristic for the trend followers, a special case of the general GDP where L → ∞. This implies that the agents’ memory is unbounded, but decays with a rate of ω.

Finally, we discuss the paper fromChiarella et al.(2014), which this paper closely follows. They estimate a HAM with fundamentalists and chartists on the deviation of the S&P500 from its fundamental value, and find empirical evidence of the presence of heterogeneous expectations and switching between groups. A two-step procedure is used to identify the structural parameters in their model, an approach used to solve the identification problem in their model. This is a very common problem when estimating HAMs, and this particular approach will also be used in this paper (more details about this procedure follow in Section 4). Furthermore, the setup and the aim of this paper is analogous to Chiarella et al. (2014), which is finding evidence of the existence of heterogeneous agents acting against each other through evolutionary switching. Therefore, the dynamics of the model in this paper will roughly follow those in Chiarella et al. (2014).

3

The Model

In this section a bivariate heterogeneous agents model (HAM) is constructed under a market-maker scenario that is assumed to adjust prices of both assets. Most of the dynamics that will be described here are closely related to those in Chiarella et al.

(2014). Our HAM focuses on the simple case where the market is populated by three types of participants: a group of fundamentalists, a group of chartists and a single market maker, a similar case as in Boswijk et al. (2007) and Chiarella et al. (2014). Fundamentalists focus solely on the deviation of asset prices from their fundamental values, whereas chartists are more refined agents who focus on past changes in the deviation of asset prices from their fundamental values. Agents from both groups form expectations over the assets prices, calculate their own profits, upon which they decide to update their strategies, according to the switching mechanism ofBrock and Hommes (1997).

First, let us discuss some preliminary definitions that will prove useful later. The agents in our model make decisions based on information they can easily observe and can gather themselves. Since the information of historical prices and dividends is commonly available, it is assumed that agents have access to all historical information about the asset. Furthermore, since this information is easily available, the model assumes the information set to be equal for all groups of traders. Therefore we will use the following definition regarding the beliefs of the agents.

Definition 1. _{Let E}t and Vt denote the conditional expectation and

condi-tional variance on a publicly available information set Ft containing past prices

pt−i and past dividends yt−i. Thus,

Et[·] = E[·|Ft],

Vt[·] = V[·|Ft],

where Ft= {pi, pi−1, ..., pi−L; yi, yi−1, ..., yi−L}, with lag length L → ∞.

While the information set Ft is available to each agent, a key feature of the model

is that each type of agent processes the information differently, resulting in different (subjective) expectations about future prices. In our case, this form of heterogeneity is modeled through two groups: the fundamentalists and the chartists. Furthermore,

The Model

let subscript f denote the fundamentalists and subscript c the chartists. We then arrive at the following definition.

Definition 2. _{Let E}h,t and Vh,t denote the (subjective) beliefs or forecasts of

trader type h ∈ {f, c} about the conditional expectation and conditional variance. Thus,

Eh,t[·] = Eh[·|Ft],

Vh,t[·] = Vh[·|Ft].

where subscript h denotes the subjective expectation of trader type h.

These definitions will be used in the specification of our HAM discussed in the next subsections.

3.1

Asset specification

The assets considered in this model are the S&P500 and IBM Stock. Agents are able to put together their portfolio with these two assets, including a riskless asset. The assets enter the model in terms of deviation from a fundamental value. The choice for the fundamental value will be the Gordon Growth model (Gordon, 1962), given as ¯ pt = 1 + gt ρt− gt yt, (1)

where pt denotes the price of the risky asset (ex dividend) at time t, yt denotes the

dividend process at time t, gt denotes the expected growth rate of dividends

condi-tional on the information available at time t, and ρt denotes the required expected

return on equity conditional on the information available at time t. All types of agents are assumed to hold the same expectation about the dividend process, and with that the same expectation about the fundamental price. This implies

Eh,t[¯pt+1] = Et[¯pt+1], (2)

since agents update their estimate of the true value of gt and ρt each period as more

data become available. More details about the calculation of the fundamental value follow in Subsection 4.3. The model is then written in terms of price deviations from their fundamentals given as

xi,t = pi,t− ¯pi,t. (3)

where xi,t denotes the price deviation of asset i to its fundamental price at time t,

The Model

given as

Ri,t = xi,t− Rxi,t−1+ δi,t, (4)

where R = 1 + r, with r the risk-free rate, and where δi,t denotes the error made in

forecasting next period’s fundamental price of asset i, that is

δi,t+1= ¯pi,t+1+ yi,t+1− Et[¯pi,t+1+ yi,t+1], (5)

where Et[δi,t+1] = 0 such that δi,t+1 is a martingale difference sequence.

3.2

Wealth dynamics

Agents construct their portfolio by choosing between two risky assets and one riskless asset. To determine their optimal portfolio, we specify the agents’ individual wealth equation as

Wt+1= (1 + r)Wt+ (p1,t+1+ y1,t+1− (1 + r)p1,t)z1,t

+ (p2,t+1+ y2,t+1− (1 + r)p2,t)z2,t,

(6)

which in vector notation becomes

Wt+1 = (1 + r)Wt+ pt+1+ yt+1− (1 + r)pt

0

zt, (7)

where the assets’ price vector is denoted by pt = (p1,t p2,t)0, the assets’ dividend

process by yt= (y1,t y2,t)0 and the demand vector of the traders by zt= (z1,t z2,t)0.

Agents are assumed to be myopic mean-variance maximizers, which allows us to specify their wealth maximization problem. Since we have two types of agents, the fundamentalist (f ) and the chartist (c), and since each type of agent maximizes their wealth over three assets4, we can describe the maximization problem of agent h as

max zh,t n Eh,t[Wt+1] − a 2Vh,t[Wt+1] o = max zh,t f (zh,t), (8)

where zh,t is the demand of trader h, which is solved by

∂f (zh,t) ∂zh,t = E h,t[pt+1+ yt+1] − (1 + r)pt− aVh,t[pt+1+ yt+1]zh,t = 0, zh,t = 1 a Vh,t[pt+1+ yt+1]
−1 Eh,t[pt+1+ yt+1] − (1 + r)pt
= 1 aΣ −1 t Eh,t[pt+1+ yt+1] − (1 + r)pt
,

4_{When maximizing over three assets, agents choose their demand for asset 1 and 2 and the rest}

they invest in the riskless asset. This implies that their maximization problem holds two degrees of freedom, z1,t and z2,t.

The Model

where a denotes the risk aversion factor, assumed equal for all agents5, and Σtdenotes

the 2 × 2 variance-covariance matrix of the two risky assets (more details about Σt

follow in the next subsection). We can now write this in terms of expected excess return as zh,t = 1 aΣ −1 t Eh,t[Rt+1], (9)

where Rt+1= (R1,t+1 R2,t+1)0. Since we want to create a framework where all agents

can construct their beliefs over both risky assets, we need to write this in a slightly different way. We introduce the matrix Zt, given as

Zt = z1,f,t z1,c,t z2,f,t z2,c,t ! = 1 aΣ −1 t Ef,t[R1,t+1] Ec,t[R1,t+1] Ef,t[R2,t+1] Ec,t[R2,t+1] ! = 1 aΣ −1 t Ht, (10)

which contains the optimal demand for the assets for each type of agent and for each asset. Here the matrix Ht contains the beliefs at time t of all types of agents over the

returns of the assets at time t + 1. In this new specification, agents have the ability to form different beliefs over asset 1 and asset 2.

3.3

Conditional variance

A specification of the conditional variance is proposed that follows Gaunersdorfer

(2000), who assumes that agents have homogeneous but time-varying expectations on conditional variances of returns. She argues this assumption is justified since there is generally more consensus among traders about the conditional variance than the conditional mean6. Moreover, the time-variance of the conditional variance expecta-tions are justified since agents observe prices and use their observaexpecta-tions to update their estimations of the variances of the returns. These two assumptions together hold Vh,t[Ri,t+1] = Vt[Ri,t+1] = σi,t2 . We then have

Vt[Ri,t+1] = Vt[xi,t+1− Rxi,t+ δi,t+1]

= Vt[xi,t+1− Rxi,t] + Vh,t[δi,t+1]

= Vt[ ˜Ri,t+1] + σδ2

= ˜σ2_i,t+ σ2_δ

5_{We would expect chartists to have more appetite for risk as they are tend to respond to trends}

they observe in asset prices. If desired, one can model this by specifying different parameters for both types of agents.

6_{This follows from the fact that variances are much easier to estimate, a result shown byNelson} and Foster (1995) among others.

The Model

and

Covt[Ri,t+1

,

Rj,t+1] = Covt[xi,t+1− Rxi,t+ δi,t+1

,

xj,t+1− Rxj,t+ δj,t+1]

= Covt[xi,t+1− Rxi,t

,

xj,t+1 − Rxj,t] + Vt[δi,t+1]

= Covt[ ˜Ri,t+1, ˜Rj,t+1] + σδ2

= ˜σ_ij,t2 + σ_δ2

where we used ˜Ri,t = xi,t−Rxi,t−1and where we assume Vt[δi,t+1] = σδ2 to be constant

and Covt[ ˜Ri,t+1, δj,t+1] = 0 ∀ i, j. In the remainder of this paper we will assume

σ2

δ = 0, i.e. agents assume they always forecast the next time period’s fundamental

price correctly (see (5)). The conditional variances are then used to determine the optimal demands for the agents, and enter the model in (10). Here the matrix Σt

contains the covariances of the returns of the assets, that is

Σt= Vt[R1,t+1] Covt[R1,t+1, R2,t+1] Covt[R2,t+1, R1,t+1] Vt[R2,t+1] ! = σ 2 1,t σ12,t σ21,t σ2,t2 ! . (11)

We assume that agents update their estimation of the variances with exponential moving averages, a procedure analogous to Gaunersdorfer (2000) but then in a bi-variate case. The matrix Σt will then be given as

Σt = wσΣt−1+ (1 − wσ)( ˜Rt−1− ˜µt−1)( ˜Rt−1− ˜µt−1) 0

, (12)

˜

µ_t = wµµ˜t−1+ (1 − wµ) ˜Rt−1, (13)

where ˜µ_t = (˜µ1,t µ˜2,t)0 contains the exponential moving averages of the returns

˜

Rt = ( ˜R1,t−1 R˜2,t−1)0, and where wσ, wµ∈ [0, 1] represent the weights in which past

information affects the moving averages.

3.4

Heterogeneous beliefs

The active agents trading in the assets are the fundamentalists and chartists, as specified before. Agents thus have the ability to choose between two forecasting strategies for each asset7_{. Fundamentalists are characterized by expecting the future}

market price of an asset to move back to its fundamental value. This is comparable to mean-reversion, only with the fundamental value as mean. Their expectations can be written as

Ef,t[xi,t+1] = xi,t− αxi,t = (1 − α)xi,t, (14)

7_{One could argue that agents use completely different forecasting rules for different assets.}

However, this will increase the amount of parameters and the complexity of the model significantly, which makes the estimation procedure more difficult.

The Model

where α ∈ [0, 1], and (1 − α) measures the fundamentalists’ observed speed of mean reversion. In terms of expected excess return this can be written as

Ef,t[Ri,t+1] = (1 − R − α)xi,t. (15)

Chartists, the second group, focus solely on past market prices and changes in them. We follow Boswijk et al. (2007) and assume that they predict the next period’s deviation by linearly extrapolation the past realizations with lag-length L = 18. This can be written as

Ec,t[xi,t+1] = xi,t+ d(xi,t− xi,t−1), (16)

where the parameter d measures the direction and degree to which chartists respond to the past realization of the price deviation. In terms of expected excess return, the chartists’ expectations can be written as

Ec,t[Ri,t+1] = (1 − R)xi,t+ d(xi,t− xi,t−1). (17)

We then collect all beliefs over all assets in Ht, which can be written as

Ht= Ef,t[R1,t+1] Ec,t[R1,t+1] Ef,t[R2,t+1] Ec,t[R2,t+1] ! = (1 − R − α)x1,t (1 − R)x1,t + d(x1,t− x1,t−1) (1 − R − α)x2,t (1 − R)x2,t + d(x2,t− x2,t−1) ! . (18)

This is the matrix that contains the heuristic coefficients α and d that are to be estimated. Since we assume that agents can choose from the same set of heuristics for both assets, these parameters are equal for asset 1 and 2.

3.5

Evolutionary dynamics

Agents update their beliefs according to the performance of their strategies. To de-termine this, the agents use realized profits as a performance measure9_{. This measure}

is obtained by multiplying the agents’ demand for each asset with the actual price change, denoted as

πi,h,t = (xi,t− Rxi,t−1+ δi,t)zi,h,t−1, (19)

8_{Boswijk et al.(2007) find that increasing the lag-length L in their model leads to insignificant}

estimations. The estimations in this paper on lag-lengths greater than 1 show similar results, so we focus on the simplest case with L = 1.

9_{Other performance measures exist, such as risk-adjusted realized profits. See Hommes(2013)}

The Model

where πi,h,t denotes the realized profit in asset i with strategy h at time t. We can

rewrite this in matrix notation as

Πt= π1,f,t π1,c,t π2,f,t π2,c,t ! = (x1,t− Rx1,t−1+ δ1,t)z1,f,t−1 (x1,t− Rx1,t−1+ δ1,t)z1,c,t−1 (x2,t− Rx2,t−1+ δ2,t)z2,f,t−1 (x2,t− Rx2,t−1+ δ2,t)z2,c,t−1 ! = (x1,t− Rx1,t−1+ δ1,t) 0 0 (x2,t− Rx2,t−1+ δ2,t) ! Zt−1, (20)

where we introduced the matrix Πtthat contains the profits of both assets and types,

and where δt = (δ1,t δ2,t)0. We then assume that the fraction of agents investing in

asset i that choose strategy h is given by a multinomial discrete choice model10, given as

ni,h,t =

exp βπi,h,t

exp βπi,f,t
+ exp βπi,c,t

, (21)

where β denotes the intensity of choice, measuring the traders’ speed of reaction to differences in profitability between the two strategies. By introducing mi,t =

ni,f,t− ni,c,t ∈ [−1, 1] we can rewrite the dynamics as

mi,t = tanh  β 2 πi,f,t− πi,c,t
 , (22)

where the hyperbolic tangent function11 _{transforms the profit difference into a value}

between −1 and 1. In the case of two strategies, agents thus make their decision to switch based upon the profit difference between those strategies. If we then consider the case where we have two assets, this becomes

mt = tanh β 2 π1,f,t− π1,c,t π2,f,t− π2,c,t ! , (23)

where mt = (m1,t m2,t)0, and where for convenience we used tanh(A)i = tanh(Ai),

i.e. an element-wise transformation. Note that we now have two variables, m1,t and

m2,t that respectively contain information of the updating of strategies of asset 1 and

asset 2. Note also that in this representation of the evolutionary dynamics we assume that agents have the same intensity of choice β when investing in each asset, which allows us to model the updating fraction with a single parameter.

10_{In the original framework ofBrock and Hommes(1997), cost parameters C}

h for each heuristic

are introduced. One can make complex heuristics, such as trend extrapolating rules, more costly than others. For simplification purposes, this paper assumes no costs, Ch= 0.

The Model

3.6

Market maker

The sum of the chartists’ and fundamentalists’ demand for each risky asset equals the total market demand traded via the market maker. It is important to note that, as a direct influence of mean-variance optimization, the demand of the assets zi,t+1

is determined by expectations over both assets. Expectations over asset 1 in period t + 1 thus influence the realization of asset 2 in period t + 1 and vice versa. The dynamical system describing the behavior of the market maker can be written as

xi,t+1= xi,t+ µ(ni,f,tzi,f,t+ ni,c,tzi,c,t) + ηi,t+1, (24)

where µ denotes the corresponding speed of price adjustment and where ηi,t+1 =

−(¯pi,t+1− ¯pi,t) + νt+1 where νt+1 represents an i.i.d. noise of any other type of traders

in the market as well as shocks to the supply of the assets. When using the notation mi,t = ni,f,t− ni,c,t, this becomes

xi,t+1= xi,t+

µ 2 

(1 + mi,t)zi,f,t+ (1 − mi,t)zi,c,t



+ ηi,t+1. (25)

Applying this in the bivariate case of two assets, the dynamical equation becomes

xt+1 = xt+ µ 2 (1 + m1,t)z1,f,t+ (1 − m1,t)z1,c,t (1 + m2,t)z2,f,t+ (1 − m2,t)z2,c,t ! + η_t+1. (26)

In this new representation of the market maker scheme we assume that the market maker applies the same speed of adjustment to both assets. This allows us to model the updating fraction with a single parameter µ. Note also that we keep ourselves from taking into consideration any microfoundation of the market maker, such as the dynamics of the market maker’s budget constraints and inventory.

3.7

Full dynamical model

When combining all the dynamics of the agents and the market maker, this results in the full dynamical model given as

Ht = (1 − R − α)x1,t (1 − R)x1,t+ d(x1,t − x1,t−1) (1 − R − α)x2,t (1 − R)x2,t+ d(x2,t − x2,t−1) ! , (27) Zt = 1 aΣ −1 t Ht, (28) Σt = wσΣt−1+ (1 − wσ)( ˜Rt−1− ˜µt−1)( ˜Rt−1− ˜µt−1) 0 , (29) ˜ µ_t = wµµ˜t−1+ (1 − wµ) ˜Rt−1, (30)

The Model xt+1 = xt+ µ 2 (1 + m1,t)z1,f,t+ (1 − m1,t)z1,c,t (1 + m2,t)z2,f,t+ (1 − m2,t)z2,c,t ! + η_t+1, (31) Πt+1 = (x1,t+1− Rx1,t+ δ1,t+1) 0 0 (x2,t+1− Rx2,t + δ2,t+1) ! Zt, (32) mt+1 = tanh β 2 π1,f,t+1− π1,c,t+1 π2,f,t+1− π2,c,t+1 ! , (33)

which is the full dynamical model that is applied to the data in the remainder of this paper. The parameters that are to be estimated are the coefficients of the agents’ behavior, α and d, the intensity of choice of the switching mechanism, β, and the speed of price adjustment of the market maker, µ. Finally, the importance of the timing the model is stressed: First, the agents form their initial demands. Then, when the next time period’s asset prices become known, they can calculate the profits they made, upon which they can update their beliefs accordingly and respecify their demands in the next period.

4

Data & Estimation Methodology

Taking ABMs to the data is not completely straightforward. Due to the number of unknown parameters and nonlinearities in the models, an irregular likelihood surface is created. Therefore, one needs to make choices regarding the model specification in order to be able to draw conclusions and perform estimations of the model. Our goal is to estimate the behavioral parameters α and d, which indicate the strength of the agents’ strategies, β, their intensity of choice, and µ, the speed of adjustment of the market maker. We are furthermore going to make some adjustments to improve the overall estimability of the model, which we will discuss in the following subsections.

4.1

Reduced form model

The dynamical model described by (27-33) is not fully identified empirically. To be specific, the heuristic parameters α and d cannot be estimated directly as they are scaled by the behavioral parameters β and µ. Therefore, empirically it is only possible to estimate their combined effect. Furthermore, another problem arises from the fact that the behavioral parameters µ and β can only be observed indirectly through the ratios β/2a and µ/2a. To address these identification issues, some simplifications are applied to the model, similar to those suggested by Chiarella et al. (2014). First, we decompose the term ηi,t+1= ci+ εt+1, where ci captures any fundamental drift from

asset i, and εi,t+1 is a white noise process with zero drift. Second, we set R equal to

one, which is a safe assumption given the monthly frequency of the data12. Third, δt+1

is omitted from the model since Et[δt+1] = 0. Fourth, we redefine α and d from being

the agents’ reaction to mis-pricing in the market to an aggregate measure of traders’ risk aversion and their reaction to the market. Mathematically, this translates to α0 = α/2a, d0 = d/2a. Finally, since α and d both (indirectly) enter (31) and (33), we can apply a two-step procedure in order to identify the structural parameters α, d, µ and β. We define combined parameters α∗ = µα, d∗ = µd, α∗∗= βα, d∗∗ = βd, which all can be observed empirically. This procedure ensures that all parameters can be retrieved indirectly from the combined parameters. These simplifications result in a reduced form model13 which, after rearranging some equations and plugging them

12_{Estimations in the next section show that using monthly historical 10-Year Government Bond}

yields from Shiller(2000) have no significant impact on the empirical results.

13_{Readers are referred to Lux and Zwinkels (2018) for a broad discussion about complications}

such as identification issues in the empirical validation of agent-based models, including reduced form models and estimation methods.

Data & Estimation Methodology

into each other, reads

xt+1− xt = c + (1 + m1,t)z1,f,t∗ + (1 − m1,t)z1,c,t∗ (1 + m2,t)z2,f,t∗ + (1 − m2,t)z2,c,t∗ ! + εt+1, (34) mt+1 = tanh β 2 π∗∗ 1,f,t+1− π ∗∗ 1,c,t+1 π_2,f,t+1∗∗ − π∗∗ 2,c,t+1 ! , (35) where                                      Π∗∗_t+1 = x1,t+1− x1,t 0 0 x2,t+1− x2,t ! Z∗∗_t , Z∗_t = Σ−1_t −α ∗_x 1,t d∗(x1,t− x1,t−1) −α∗_x 2,t d∗(x2,t− x2,t−1) ! , Z∗∗_t = Σ−1_t −α ∗∗_x 1,t d∗∗(x1,t− x1,t−1) −α∗∗_x 2,t d∗∗(x2,t− x2,t−1) ! , Σt = wσΣt−1 + (1 − wσ)(∆xt−1− µt−1)(∆xt−1− µt−1) 0 , µ_t = wµµt−1 + (1 − wµ)∆xt−1, (36)

in which (34) controls the price changes and (35) controls the switching. Note that the variables z_i,h,t∗∗ and π_i,h,t∗∗ in those equations are taken together in respectively the matrix Z∗∗_t and Π∗∗_t , which are given in (36). For more details of the model, see Appendix A, where the drawn-out equations of the model are written down.

4.2

Data

Data of the S&P500 index is obtained from the online dataset by Shiller (2000). Earnings instead of dividends are used as a measure of yt since earnings data is

less sensitive to management choices. Since earnings are not generated by the index itself, they are calculated via a weighted average of the price-earnings ratios of the constituent stocks within the index. Then, monthly earnings data from the S&P500 is computed by linearly interpolating its four-quarter totals. Data of IBM Stock is obtained from Datastream, where we will use figures from the 20th of each month14_.

For both assets, we collect data from January 1973 to December 2017, yielding 540 observations. After the initialization of the model, we are left with 533 observations. The resulting price series are depicted as the blue (solid) curves in Figure 1.

14_{Missing IBM earnings data is dealt with by linear interpolation, upon which the data is}

Data & Estimation Methodology

4.3

Fundamental value

In the specification of the fundamental value given in (1), the required return on assets ρt consists of two components: a capital gain component and a yield component.

Since the capital gain component equals the growth in earnings, the required return on assets is given by

ρt= gt− Et[yt+1/pt+1]. (37)

Therefore, the fundamental price of the risky asset according to the Gordon Growth model given by (1) equals

¯ pt=

1 + gt

Et[yt+1/pt+1]

yt. (38)

In this specification, the earnings growth gt is calculated as the average realized

growth rate up to period t and the expected earnings yield Et[yt+1/pt+1] is calculated

as the average realized earnings yield up to period t. As such, the stochastic discount factor is fully driven by the asset’s earnings. The resulting fundamental price time series for both assets are depicted as the red (dashed) curves in Figure 1. Figure 2

displays time series of the deviation from the fundamental price for both assets. Figure 1 and 2 show that for both assets the fundamental price closely follows the market price from the beginning of the sample up until 1997. After this period,

1975 1980 1985 1990 1995 2000 2005 2010 2015 0 1000 2000 3000 S&P500 1975 1980 1985 1990 1995 2000 2005 2010 2015 0 50 100 150 200 250 IBM Stock

Figure 1: Time series of prices and fundamental prices for the S&P500 and IBM Stock. The blue (solid) curves depict the price levels of the assets, whereas the red (dashed) curves depict the fundamental values of the assets.

Data & Estimation Methodology 1975 1980 1985 1990 1995 2000 2005 2010 2015 -200 0 200 400 600 800 1000 -100 -80 -60 -40 -20 0 20 40 60 80

Figure 2: Time series of the deviations from the fundamental price, xt, for both assets.

The blue (solid) curve depicts xt for the S&P500 (on the left y-axis), whereas the red

(dashed) curves depicts xtfor IBM Stock (on the right y-axis).

the deviation from the assets’ fundamental price becomes larger, a result that follows from the Dot-Com bubble15_{which caused prices to inflate. During its plunge in}

2000-2002, prices begin to revert back to the fundamental price. A noteworthy observation is the following: in 2009, when both assets’ prices have fallen drastically due to the 2007-2008 financial crisis, the fundamental price of the S&P500 also experiences this drop while the fundamental price of IBM Stock does not. As a result, x1 and x2 move

in opposite directions during the aftermath of the crisis.

4.4

Estimation method

Due to the non-linear nature of the model, the estimations will be sensitive to initial values. In order to ensure that the estimations are not a local optimum, a restricted version of the model is estimated where the market maker’s speed of adjustment is fixed to µ = 1, and where the switching between the agents is left out of the model, i.e. mt = 0. Since this model can be estimated with OLS, it is insensitive to initial

values. Its estimates will then be used as initial values for the unrestricted model. To be more specific, the restricted model produces initial values for c, α∗, and d∗. Since the unrestricted model requires two additional initial values, for α∗∗ and d∗∗, we perform a grid search in order to select the initial values that result in the best model fit. The model will be estimated using non-linear least squares, accompanied with heteroskedastic-consistent standard errors.

Subsequently, the estimations of the combined parameters α∗, d∗, α∗∗, d∗∗ allow for numerically solving the structural parameters α, d, µ and β, where we follow the

15_{The Dot-Com bubble was an economic bubble that occurred during 1997-2001. Characterized}

Data & Estimation Methodology

procedure of Chiarella et al. (2014). This can be accomplished by minimizing the loss function Z, given as

Z = |α∗_{− µα|} |α∗_{+ µα|} + |d∗_{− µd|} |d∗_{+ µd|} + |α∗∗_{− µα|} |α∗∗_{+ µα|} + |d∗∗_{− µd|} |d∗∗_{+ µd|}. (39)

In this loss function16_{, the numerators represent the difference between the combined}

parameters and their underlying structural coefficients, while the denominators rep-resent the order of magnitude of the coefficients, functioning as weights of the param-eters in order to prevent biased results. The initialization required for this procedure will follow the same initialization procedure as discussed in the previous paragraph.

4.5

Variance estimation

In the specification of the conditional variance (see Subsection 3.3), wµ and wσ

de-scribe the updating of the traders’ expectation of the variance. To lower the number of parameters that are to be estimated, both wµ and wσ are set equal to 0.8 in our

estimations. This implies that in estimating the variances of the returns, agents give more weight to their previous estimate relative to each new observation.

Another issue that arises when estimating the conditional variance is the well-known problem of error-maximization when applying mean-variance optimization in portfolio selection. This is a problem that is well documented and studied (see e.g. Jobson and Korkie, 1980). In our case, this problem arises when agents form estimations of the variances that are rather extreme. This causes the inverse variance-covariance matrix to assign extreme weights to assets, which are accompanied by great errors. A solution to this problem follows from shrinkage. This is a technique that aims to remove extreme values by shrinking the variance-covariance matrix to a more centralized one, called the shrinkage target. As shrinkage target the sample variance-covariance matrix (over all observations) is used. The shrinkage formula is given as

˜

Σt = δΣt+ (1 − δ)S (40)

where ˜Σt denotes the shrunk variance-covariance matrix, Σt denotes the agents’

estimations of the variance-covariance matrix as specified in (12), S denotes the shrinkage target, for which in our case the sample variance-covariance matrix is used, and δ denotes the shrinkage constant. In the estimations in the next section, it turns out that by shrinking the matrix by only 1 percent, i.e. δ = 0.99, the majority of the extreme values are removed and the estimation results are satisfactory.

16_{As a matter of fact, the model allows only for identification of the ratios α}∗_{, d}∗_{, α}∗∗_{, d}∗∗_.

This method merely attempts to infer the structural parameters without explicitly imposing the restriction α∗/d∗= α∗∗/d∗∗ which was used to obtain the reduced form model.

5

Results

This section presents the results from estimating the bivariate HAM discussed in Section 3 with the estimation procedure discussed in Section 4. In order to gain additional insight in the complexity of the model, the model is estimated in various ways. Besides estimating the model in the fashion previously discussed, we split our sample into two subsamples, 1973-1995 and 1995-2018, and estimate the bivariate HAM on these subsamples. Figure 2 shows that we can clearly distinguish these two periods, where one exhibits relatively stable returns (1973-1995), and one which exhibits relatively volatile returns (1995-2018). Second, we perform estimations of the model on each asset individually17.

5.1

Estimation results

The estimation results are shown in Table 1, where column (A) shows the restricted model with mt= 0 and µ = 1, and (B) the unrestricted model for the bivariate HAM

on the full sample. The results of the unrestricted model (B) are partially consistent with the hypothesis that fundamentalists and chartists are active on the markets of the S&P500 and IBM Stock. The results show a significant coefficient d∗ = 1.905 for the trend followers, in contrast to a coefficient α∗ for the fundamentalists that is not significantly different from zero. The coefficient d∗ = 1.905 implies that the trend followers extrapolate the previous time period’s deviation in xtby a factor 1.905

per month. These results thus constitute evidence of the presence of trend-following behavior in traders who are present on both markets of the S&P500 and IBM Stock. The insignificance of the coefficient α∗ for the fundamentalists means that there is no evidence of the presence of this type of trader.

In all models the coefficients α∗∗ and d∗∗ are not significantly different from zero. This can be explained by the properties of the tanh function, which is a transformation of R → [−1, 1]. Since this function transforms the profit difference πf,t− πc,t to mt,

variation in the profit difference when mt is near −1 or 1 has little effect, therefore

increasing standard errors of α∗∗ and d∗∗. These results imply that we cannot reject the null hypothesis of zero intensity of choice β = 0, i.e. the absence of switching. These are common results in estimating HAMs and are consistent with various other studies such asChiarella et al.(2014) andBoswijk et al.(2007). However,Ter¨asvirta

(1994) shows that the significance of β is not important as long as the groups are

17_{When estimating the model on a single asset, the variance-covariance matrix naturally becomes}

a single time-varying coefficient. This is in contrast to Chiarella et al. (2014), who perform a procedure where they estimated the combined effect of the variance and the relevant behavioral coefficient. Furthermore, shrinkage is not applied in the single asset models.

Results

significant and the model fit improves after introducing switching. We therefore assess the added value of introducing the switching mechanism by comparing the model fit of the restricted and unrestricted model. The unrestricted model (B) shows that introducing switching improves the model fit compared to the unrestricted model (A). Therefore we can conclude that switching is present in the unrestricted model. However, since the coefficient α∗ for the fundamentalists is not significantly different from zero, we can only say that the switching occurs between trend followers and ‘the rest’, i.e. the other types of agents that are active on the market.

Columns (C-D) and (E-F) respectively show estimations of the model on the subsamples 1973-1995 and 1995-2018 in order to gain more insight in the difference on the model outcome between these periods. The results show that there is evidence of the presence of both fundamentalists and trend followers in the subsample 1973-1995, whereas we find no evidence for either in the subsample 1995-2018. This implies

Models

Both Assets Single Asset

1973-2018 1973-1995 1995-2018 1973-2018 (A) (B) (C) (D) (E) (F) (G) (H) c 0.861 0.846 0.129 0.090 1.516 1.546 1.912 −0.142 (0.751) (0.752) (0.300) (0.310) (1.436) (1.473) (1.465) (0.315) α∗ 0.185 0.178 0.199∗∗ 0.258∗∗ 0.467 0.656 −0.075 0.054∗ (0.152) (0.208) (0.096) (0.105) (1.511) (1.526) (0.242) (0.033) d∗ 1.141 1.905∗∗ 1.484∗ 1.298∗∗ 0.052 0.363 7.190∗∗∗ 0.079 (1.334) (0.956) (0.885) (0.619) (4.459) (5.127) (1.913) (0.186) α∗∗ −0.049 0.057 −0.016 −0.013 0.017 (0.100) (0.082) (0.225) (0.025) (0.059) d∗∗ 0.667 1.269 0.447 −0.023 0.348 (1.265) (1.392) (2.893) (0.091) (0.456) SSR 623753 623096 20033 19715 602394 602067 601764 21109 R2 _{0.9850 0.9850 0.9900 0.9902 0.9675 0.9675 0.9852 0.9235} Obs. 533 533 258 258 269 269 533 533

∗ _{Significant at the 10% level.} ∗∗ _{Significant at the 5% level.} ∗∗∗ _{Significant at the 1% level.}

Table 1: Estimations of the restricted model on the full sample (A), the unrestricted model on the full sample (B), the unrestricted model on subsample 1 (C), the unrestricted model on subsample 1 (D), the unrestricted model on subsample 2 (E), the unrestricted model on subsample 2 (F), the model on only the S&P500 on the full sample (G) and the model on only IBM Stock on the full sample (H). Estimations are shown without brackets, standard errors in round brackets. Significance levels for the null hypothesis H0 : ci6= 0 are denoted

Results

that the traders’ behavior in the first period does not continues in the second one. Hence, we can conclude that the markets of the S&P500 and IBM Stock are populated with fundamentalists and trend followers during a period ending in roughly 1995, after which the behavior of the fundamentalists and the chartists changes to a type currently unidentified.

Columns (G) and (H) respectively show estimations of the model on the S&P500 and IBM Stock individually. These results show that on the market with solely the S&P500, only trend followers are active, whereas on the market with solely IBM Stock, there is only evidence for the presence of fundamentalists. This suggest that when we provide the agents with the choice of investing in an extra asset, their behavior changes from predominantly sticking to one strategy to exploiting other strategies. This suggests that the multi-asset HAM is better able to describe the composition of different types of traders in these markets.

Table2shows the implied structural coefficients resulting from minimizing the loss function given in (39). The coefficients are largely consistent with the estimations from Table 5. The first column confirms our findings for the full sample, showing the absence of fundamentalists and the presence of trend followers. Column two and three show the results from the subsamples, where we must take into account that the values for α and d in column three result from optimizing values accompanied by large standard error, which therefore are unreliable. On the other hand, the values for α and d in column two are more reliable since the coefficients α∗ and d∗ were significant for that subsample. It shows that in the first period both fundamentalists and trend followers are present. As emphasized before, since the parameters α∗∗ and d∗∗ are not significantly different from zero, the implied coefficient β will not be significant, though this is not worrisome. This explains the negative value for β in column three. Furthermore, the impact of the market maker µ is estimated positive in all samples and is considerably smaller in the first subsample than in the second.

1973-2018 1973-1995 1995-2018

α −0.00668 0.29717 0.57948

d 1.38286 1.49819 0.32096

µ 1.37764 0.86655 1.13152

β 0.48235 0.84721 −0.02781

Table 2: Implied structural coefficients for all three samples. Values follow from minimizing (39). Column 1 shows the implied structural coefficients of model (B) using the full sample, column 2 for model (D) using the sample 1973-1995 and column 3 for model (F) using the sample 1995-2018.

Results

5.2

Switching

This subsection investigates the resulting switching in the models. First, the model on the full sample is studied, upon which we study the model on the two separate subsamples.

Figures 3, 4 and 5 show details of the resulting switching in the unrestricted model on the full sample. In Figure 3, subfigures (a-b) show mt, whereas subfigures

(c-d) show the 12-month moving average of mt plotted over xt. A value of mt equal

to 1 translates to all agents following the fundamentalist strategy, whereas a value of −1 translates to all agents following the chartists strategy. The figure shows a

1980 1990 2000 2010 -1 -0.5 0 0.5 1 S&P500 1980 1990 2000 2010 -1 -0.5 0 0.5 1 IBM Stock 1975 1980 1985 1990 1995 2000 2005 2010 2015 -500 0 500 1000 -1 -0.5 0 0.5 S&P500 1975 1980 1985 1990 1995 2000 2005 2010 2015 -100 -50 0 50 100 -0.6 -0.4 -0.2 0 0.2 0.4 IBM Stock

Figure 3: Time series of the resulting switching of the agents in Model (B) (using the full sample). Subfigures (a) and (b) depict the switching in strategies per asset, whereas subfigures (c) and (d) depict xt (left y-axis) against the 12-month moving average of mt

Results

great amount of switching taking place throughout the whole sample. Note that this implies that the composition of the market as a whole changes, instead of individuals continually switching strategies. Chiarella et al.(2014) find a strong relation between xtand mt: whenever xtmoves away from its fundamental value, mtmoves downwards,

increasing the chartists’ fraction (and vice versa). In Figure 3 (c-d), this relation is also present, though in a more moderate version. For example, in 1995-2000, the S&P500 moves away from its fundamental, upon which the fraction of the chartists starts to increase, but only after 1997. This partly explains the Dot-Com bubble, which occurred during that time: The slow but steady increase in the chartists’ fraction has a destabilizing effect, fueling the bubble, which continues to drive prices up. However, this increase in chartists during the Dot-Com bubble cannot be observed in IBM Stock, a finding that is unexpected. Despite this finding, IBM Stock does exhibit the same relation between mt and xt: Around 2016, when it exhibits the

greatest deviation from its fundamental value, agents switch to the chartists strategy to a point where mt almost reaches −0.6, which means that chartists form a little

less than 80% of the market’s population. Although parts of the moving averages of mt can be explained by this relation, they also show parts that seem to have no

clear explanation, such as the severe spikes around 1995 observed in both assets. Figure 4 shows the distribution of mt and effect of the profit difference on mt. It

confirms that agents switch to the fundamentalist strategy when there is a positive

-1 -0.5 0 0.5 1 0 50 100 S&P500 -1 -0.5 0 0.5 1 0 50 100 IBM Stock -10 0 10 -1 -0.5 0 0.5 1 -10 0 10 -1 -0.5 0 0.5 1 Mean = -0.06 S.D. = 0.51 Mean = -0.00 S.D. = 0.47

Figure 4: Detailed statistics for the switching in Model (B) for S&P500 (left figures) and IBM Stock (right figures). Subfigures (a) and (b) show histograms of the distribution of m1 and m2, whereas Subfigures (c) and (d) show the distribution of the profit difference

between the fundamentalist and chartists strategy of an each asset, and its effect on the switching within that asset.

Results

difference between the fundamentalists’ and chartists’ profit (and vice versa), which is an expected result. Furthermore, the histograms in Figure3shows that the majority of the time mt lies around 0, which implies that the composition of the market is

evenly populated by the fundamentalists and the chartists, while it also shows two spikes at −1 and 1. Next, we study the combined distribution of the traders across both markets. Table 3 shows how we define each group, while Figure 5 shows the 12-month moving averages of the fractions of all the groups. The groups show a constant amount of switching throughout the sample, except for the period around 1995. Again, there seems to be a disturbance in switching around 1995 that has no clear explanation.

S&P500 IBM Stock

Group 1 Fundamentalist Fundamentalist

Group 2 Chartist Fundamentalist

Group 3 Fundamentalist Chartist

Group 4 Chartist Chartist

Table 3: Definition of the different groups of traders that are active on the combined market of the S&P500 and IBM Stock.

1980 1990 2000 2010 0 0.1 0.2 0.3 0.4 0.5 0.6 Group 1 1980 1990 2000 2010 0 0.1 0.2 0.3 0.4 0.5 0.6 Group 2 1980 1990 2000 2010 0 0.1 0.2 0.3 0.4 0.5 0.6 Group 3 1980 1990 2000 2010 0 0.1 0.2 0.3 0.4 0.5 0.6 Group 4 Mean = 0.23 Mean = 0.27 Mean = 0.24 Mean = 0.26

Figure 5: Time figures for the fractions of all the different groups of traders in Model (B). For the definition of the groups of traders, see Table 3. Subfigures (a-d) respectively show the evolution of the fractions of groups 1-4.

Figures 6, 7 and 8 show details of the resulting switching in the unrestricted model on the subsample 1973-1995. In Figure 6, the relation between mt and xt

Results

described earlier can be observed more clearly: Whenever the assets move away (move towards) their fundamental value, mt decreases (increases), or the fraction of

the chartists (fundamentalists) increases. For example, in 1983, the S&P500 moves away from its fundamental value, upon which the traders react immediately and start to switch to the chartist strategy. The same relation can be observed in the time series of IBM Stock. For example, around 1992 IBM Stock moves away from its fundamental value, causing an increase in the chartists’ fraction. In comparison to the model estimated on the full sample, the switching in this sample is much more reactive to deviations from its fundamental, a result supported by our estimations of α∗ and d∗ (see Table1) since in this subsample these coefficients are more significant.

1975 1980 1985 1990 -1 -0.5 0 0.5 1 S&P500 1975 1980 1985 1990 -1 -0.5 0 0.5 1 IBM Stock 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 -100 0 100 200 300 -1 -0.5 0 0.5 S&P500 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 -10 0 10 20 30 -1 -0.5 0 0.5 1 IBM Stock

Figure 6: Time series of the resulting switching of the agents in Model (D) (using the sample 1973-1995). Subfigures (a) and (b) depict the switching in strategies per asset, whereas subfigures (c) and (d) depict xt (left y-axis) against the 12-month moving average

Results -1 -0.5 0 0.5 1 0 20 40 S&P500 -1 -0.5 0 0.5 1 0 20 40 IBM Stock -20 -10 0 10 20 -1 -0.5 0 0.5 1 -20 -10 0 10 20 -1 -0.5 0 0.5 1 Mean = -0.15 S.D. = 0.58 Mean = 0.01 S.D. = 0.58

Figure 7: Detailed statistics for the switching in Model (D) for S&P500 (left figures) and IBM Stock (right figures). Subfigures (a) and (b) show histograms of the distribution of m1 and m2, whereas Subfigures (c) and (d) show the distribution of the profit difference

between the fundamentalist and chartists strategy of an each asset, and its effect on the switching within that asset. Note the difference in scale of the x-axes of Subfigures (c) and (d). 1975 1980 1985 1990 0 0.1 0.2 0.3 0.4 0.5 0.6 Group 1 1975 1980 1985 1990 0 0.1 0.2 0.3 0.4 0.5 0.6 Group 2 1975 1980 1985 1990 0 0.1 0.2 0.3 0.4 0.5 0.6 Group 3 1975 1980 1985 1990 0 0.1 0.2 0.3 0.4 0.5 0.6 Group 4 Mean = 0.22 Mean = 0.29 Mean = 0.21 Mean = 0.29

Figure 8: Time figures for the fractions of all the different groups of traders in Model (D). For the definition of the groups of traders, see Table 3. Subfigures (a-d) respectively show the evolution of the fractions of groups 1-4.

Results

Figure7shows that the distribution of mtis different on this subsample in comparison

to the full sample. First of all, the allocation of mt at −1 and 1 is more extreme.

This result is supported by the larger implied coefficient β in this subsample than in the full sample (see Table 2), since β measures the traders’ speed of reaction to profit differences between the two strategies. Second, traders in the S&P500 are more chartist from nature, as the mean from mt is equal to −0.15. This is also shown in

the histogram of mt which shows a substantial spike at −1, whereas it shows none at

1. This is a result supported by the estimations of the model on solely the S&P500 (see column (G) in Table5), where there is only evidence for the presence of chartists. Figure 8also shows that the chartists’ strategy is most favorable with the agents, as

2000 2005 2010 2015 -1 -0.5 0 0.5 1 S&P500 2000 2005 2010 2015 -1 -0.5 0 0.5 1 IBM Stock 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 -500 0 500 1000 -0.4 -0.2 0 0.2 0.4 S&P500 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 -100 -50 0 50 100 -0.5 0 0.5 IBM Stock

Figure 9: Time series of the resulting switching of the agents in Model (F) (using the sample 1995-2018). Subfigures (a) and (b) depict the switching in strategies per asset, whereas subfigures (c) and (d) depict xt (left y-axis) against the 12-month moving average

Results -1 -0.5 0 0.5 1 0 20 40 60 S&P500 -1 -0.5 0 0.5 1 0 20 40 60 IBM Stock -10 0 10 -1 -0.5 0 0.5 1 -10 0 10 -1 -0.5 0 0.5 1 Mean = -0.04 S.D. = 0.43 Mean = 0.02 S.D. = 0.40

Figure 10: Detailed statistics for the switching in Model (F) for S&P500 (left figures) and IBM Stock (right figures). Subfigures (a) and (b) show histograms of the distribution of m1 and m2, whereas Subfigures (c) and (d) show the distribution of the profit difference

between the fundamentalist and chartists strategy of an each asset, and its effect on the switching within that asset. Note the difference in scale of the x-axes of Subfigures (c) and (d). 2000 2005 2010 2015 0 0.1 0.2 0.3 0.4 0.5 0.6 Group 1 2000 2005 2010 2015 0 0.1 0.2 0.3 0.4 0.5 0.6 Group 2 2000 2005 2010 2015 0 0.1 0.2 0.3 0.4 0.5 0.6 Group 3 2000 2005 2010 2015 0 0.1 0.2 0.3 0.4 0.5 0.6 Group 4 Mean = 0.25 Mean = 0.26 Mean = 0.23 Mean = 0.26

Figure 11: Time figures for the fractions of all the different groups of traders in Model (F). For the definition of the groups of traders, see Table 3. Subfigures (a-d) respectively show the evolution of the fractions of groups 1-4.

Results

the means of the fractions of groups following the chartist strategy in the S&P500, group 2 and 4, are bigger than the means the other groups (a difference of 15%).

Figures 9, 10 and 11 show details of the resulting switching in the unrestricted model on the subsample 1995-2018. Figure 10 shows confirmation of what we sus-pected earlier. It shows that the majority of the time mt lies around 0, which implies

that agents are less stimulated to switch to either the fundamentalist strategy of the chartist strategy. This suggests that in this subset, agents use other strategies, which they view as superior to the strategies in this model. This is also supported by the fact that the coefficients α∗ and d∗ are not significantly different from zero in this subset, implying that the chartist and fundamentalist strategies are not being used. There are several possible reasons for the insignificance of these coefficients: First, since we formulated our model in terms of deviation to a fundamental value, it could be that in this subsample the fundamental value becomes less accurate due to disturbances in earnings data during crises. Second, the fact that agents believe that fundamentals drive prices might no longer hold in this period. Third, agents can have strategies different than the fundamentalist and chartist strategy that they use in times where markets exhibit great volatility. The results of the model on this subset explain the contrast between the model on the full sample and on the sample 1973-1995 since during 1995-2018, the model is no longer fit to describe the agents’ behavior.

Additional figures depicting the switching from the single asset models (G) and (H) are provided in AppendixB. The figures substantiate the same results as discussed in the previous subsection: Since in each single asset model evidence is found for only one type of agent, switching does not happen frequently. Furthermore, the distribution of mt appears to be normally distributed as it is highly centered around