A comparison of different heuristic expectation
rules in a heterogeneous agent model
Jens Klooster
Bachelor’s Thesis to obtain the degree in Economics & Business
Specialization: Economics & Finance University of Amsterdam
Faculty of Economics and Business
Author: Jens Klooster Student nr: 10059229
Date: February 20, 2016 Supervisor: Dr. Marco van der Leij
Abstract
In this study we analyse the effect of changing the chartist rule in a heterogeneous agent model with a fundamentalists vs. chartists framework into a trend following type rule as introduced by Anufriev and Hommes (2012). To be able to make a good com-parison we use the heterogeneous agent model as introduced by Hommes and in ‘t Veld (2014), which is a heterogeneous agent model with a fundamentalists vs. chartists framework. We find that if we change the chartist rule by a weak trend following rule the performance of the model does not significantly change and could therefore be used instead of the chartist rule. The general trend following rule and the strong trend fol-lowing rule do not perform as good as the usual chartist rule and should therefore be avoided.
Keywords: stock prices, behavioural finance, heterogeneous expectations
Acknowledgements
First and foremost I would like to thank my thesis supervisor dr. Marco van der Leij for his guidance, valuable insights and support. Marco was always prepared to help me and really inspired me to write this thesis. I would also like to thank dr. Daan in ‘t Veld for letting me use his R code and data.
Contents
1 Introduction 1
2 Related literature 4
2.1 Important developments within the theory . . . 4
2.2 Heterogeneous Agent Models fitted to real data . . . 5
2.3 Different heuristics . . . 6
3 The Model 8 3.1 Heterogenous beliefs . . . 9
3.2 The different models . . . 10
3.2.1 The Hommes and in ‘t Veld Model . . . 10
3.2.2 The different Trend Following Models . . . 10
3.2.3 Strategy switching . . . 11
3.2.4 Econometric form . . . 12
4 Estimations and results 13 4.1 Data analysis . . . 13
4.2 Estimations . . . 15
4.3 Test statistics . . . 16
4.3.1 Computation of the AIC and BIC . . . 16
4.3.2 Out of sample forecasts . . . 18
4.3.3 Conclusion of tests . . . 20
4.4 Fractions . . . 21
5 Conclusion 25
1
Introduction
In his paper famous first bubbles Garber (1990) describes the first documented economic bubbles. The first documented bubble took place in the Netherlands and started in 1625, when the prices of special kinds of tulip bulbs were sold for 2000 guilders each (around $16, 000 in present time). By 1637 the rapid increase of tulip bulb prices attracted a lot of speculators and prices kept rising until January 1637. In February 1637 the bubble suddenly burst and bulbs could not be sold for more than 10% of the peak value leaving the Netherlands in economic despair.
Economic bubbles have caused controversy among some economists. On the one hand within the theory of conventional neoclassical economics there are, in theory, no financial bubbles due to the Rational Expectations Hypothesis (REH) and the Efficient Market Hypothesis (EMH). The REH states that individual expectations coincide, on average, with market realizations (Muth, 1961) and the EMH states that markets are efficient, meaning that prices fully reflect economic fundamentals and there is no way to earn excess earnings (Fama, 1970). From a neoclassical perspective, there is no room for market psychology and irrational herding behaviour, due to Friedman his argument that irrational traders will be driven out of the market by rational traders (Friedman, 1953).
On the other hand, there are economists, such as Simon (1957), who argue that the REH imposes unrealistic informational and computational requirements upon individual behaviour and therefore opt for a weaker version of the REH in which individuals are modelled as boundedly rational (Hommes, 2012, p. 36). In this alternative behavioural view economic agents do not know the true motion of the economy and therefore can not always make rational decisions, but rather make decisions based on expectations and available information. Within this new field of study called behavioural economics, which seeks to combine behavioural and cognitive psychology theory with conventional economics, there is room for bubbles and irrational behaviour. However, in contrast to the models in which agents are modelled as rational, there are a lot of ways to model an agent as boundedly rational.
One way of modelling boundedly rational agents is by using an evolutionary ap-proaching introduced by Nelson and Winter (1972), where agents choose from a set of simple, behavioural strategies according to their relative performance. This new ap-proach, together with developments in Mathematics and Computational Economics have motivated economists to use applications of nonlinear dynamics in economics and led
to the introduction of Heterogeneous Agent Models (HAMs) (Hommes, 2006).
HAMs disregard the Rational Agent Hypothesis and allow for heterogeneity by letting agents use different strategies according to their expectations of future prices. Within the research of HAMs agents or traders are usually being split up in two groups. On the one hand there are the fundamentalists who believe in mean-reversion of stock prices to their fundamental price and on the other hand there are the chartists who do not believe in fundamentals, but follow trends in the markets. Examples of these kind of models can by found in e.g. Boswijk et al. (2007), Slopek et al. (2010) and Zwinkels et al. (2010).
As mentioned before there exists a ’degrees of freedom’ problem, meaning that there are an infinite amount of ways to model boundedly rational agents. However, in most previous research concerning HAMs a fundamentalists vs. chartists framework is being used. The assumption that agents are either as fundamentalists or chartists already offers, in general, a more realistic view than modelling agents as perfectly rational, but is still only a simplifying assumption. A natural question to ask is what happens if we change the fundamentalist vs. chartists framework into something else? This ques-tion is the main motivaques-tion for our research and to analyse the effect of changing this framework we will use a recent paper written by Hommes and in ‘t Veld (2014) in which a HAM is estimated against S&P500 stock data. In their paper Booms, busts and behavioural heterogeneity in stock prices Hommes & in ‘t Veld use a HAM, with a fundamentalists and chartists framework, to investigate the value added of explain-ing asset pricexplain-ing movements usexplain-ing a heterogeneous agents model. Their model suggests that behavioural regime switching has significant and robust effects on asset prices and amplifies booms and busts. In this thesis we will continue to use the model of Hommes and in ‘t Veld (2014) and investigate the effect of changing their chartist rule by slightly different trend following rules as found by Anufriev and Hommes (2014). These rules have been found by Anufrief and Hommes (2014) by doing learning to forecast exper-iments within an experimental environment. For this research the same S&P500 stock market data will be used and the only difference can be found in the different rules. We will then use the Hommes & in ‘t Veld model as benchmark model and compare all models. For the different models we will compare the Akaike Information Criterium and Bayesian Information Criterium. To test the predictive power of the models we will use one step ahead out of sample forecasts and compare the root-mean-squared errors using a Diebold and Mariano test.
We find that if we replace the chartist rule by a weak trend following rule the per-formance of the model does not significantly change and is able to help us describe economic phenomena particularly the impact of market psychology on economic mar-kets. The general trend following rule performs decent in terms of quality measures such as the AIC, BIC and RMSE, but is not able to help us describe economic reality as good as the weak trend following model due to strong switching behaviour of agents. The strong trend following rule does not perform well in terms of quality measures, but
the trading behaviour of agents in the strong trend following model does correspond with our expectations.
In section two we will start with a brief overview of the related literature and re-search about applications of HAMs. Subsequently, in section three we will introduce the model, as used by Hommes & in ‘t Veld and introduce the new rules. In section four we will present our results and the last section will contain our conclusion.
2
Related literature
This chapter will briefly summarize the results of seven papers related to our research. The first two papers are important because they introduced key elements used in todays HAMs. The last five papers fit a HAM to empirical data, as will be done in this thesis.
2.1
Important developments within the theory
As mentioned by Hommes (2006, p. 6) one of the first Heterogeneous Agent Models dates back to Zeeman (1974). In his paper Zeeman wrote about the unstable behaviour of stock exchanges based on catastrophe theory. Since catastrophe theory is a branch within mathematics the model is not based on any microeconomic foundations. Nevertheless, Zeeman used an important concept that is still being used in HAMs today, the idea is to split the market in two separate groups of investors, fundamentalists and chartists. The fundamentalists are investors who act on the basis economic aggregates such as the supply and demand, money supply, etc. Before a fundamentalist invests in a firm he inspects fundamental values of the firm, such as the growth potential and the market potential. An important aspect about fundamentalists is that they believe in mean-reversion of prices to some fundamental value. On the other hand chartists are investors who only invest in firms based on their technical analysis of the market. Technical or chartist analysis of financial markets involves providing forecasts or trading advice on the basis of largely visual inspection of past prices, without regard to any underlying economics or ’fundamental’ analysis (Taylor et al., (1992)). This implies that in theory chartists could e.g. buy stocks even if they are priced above their fundamental value pushing prices further up.
Brock & Hommes (1998) investigate a HAM with boundedly rational agents and introduce the concept of Adaptive Belief Systems. In their model agents can choose from a finite set of different beliefs about the future price of a risky asset. The selection process of most agents is then based upon a performance measure, such as the past realized profits, and is updated at each new date. They show how an increase in the ‘intensity of choice‘ to switch beliefs can cause market instability and complications in the dynamics of market asset prices. This gives a more realistic view of financial markets in which not only fundamental changes cause prices to fluctuate, but also market psychology.
2.2
Heterogeneous Agent Models fitted to real data
Boswijk et al. (2007) estimate a modified version of the model introduced by Brock & Hommes (1998) to annual US stock price data from 1871 until 2003. In their model they assume that the fundamental value of the risky asset is available to all agents, but beliefs about the persistence of deviations from the fundamental value differ. As introduced by Brock & Hommes an evolutionary selection mechanism is used. The performance measure is based on past performance and a strategy will attract more agents if it has performed relatively well compared to other strategies. Again, the model is set up in a fundamentalist and chartist framework. The model gives an explanation for the upward trend in stock prices from 2003 to 2007. Furthermore, the model shows that before the 90s the chartists was active only on occasion. However, in the late 90s at the time of the dot-com bubble the model shows an increase in chartists followers, which created an extraordinary deviation of stock prices from fundamentals.
Bolt et al. (2014) estimate a heterogeneous agent model to quarterly data of hous-ing prices for eight different countries, the US, UK, Netherlands, Japan, Switzerland, Spain, Sweden and Belgium. The paper shows that in all countries the data supports heterogeneity in expectations, with temporary switching between fundamentalist and chartist beliefs. For four countries, the US, UK, Netherlands and Spain they indentify strong housing bubbles in the period 2004-2007. For two other countries, Japan and Switzerland they indentify housing bubbles in the period 1980-1990. In all cases the bubbles are strongly amplified by chartist trader behaviour and when the bubbles burst the majority of agents switch to a fundamentalist trading strategy.
Slopek & Reitz (2008) estimate a HAM to US dollar market prices of WTI crude oil for the period 1986:1 - 2006:12. Their model shows that the more price deviates from its long-run equilibrium or fundamental value, the more the fundamentalists will become active. The increase in fundamentalists then drives price back to an equilibrium value. However, when the price of crude oil is close to the fundamental value the market of impact of chartists becomes relatively higher causing destabilizing random shocks which can cause an upward or downward swing in oil prices. The model suggests that heterogeneous traders and their nonlinear beliefs may be responsible for the swings in oil prices from 1986 - 2006.
Ter Ellen & Zwinkels (2010) also estimate a HAM to the oil market in which they divide the demand for oil into two types. The first types of demand is speculative demand and the second type is real demand. Their model is also based on a fundamentalists and chartists framework in which speculators base their strategy on past profitability. Their estimations on Brent and WTI oil show that both groups are active in the oil market, and that speculators often switch between groups. The HAM outperforms random walk and VAR models in out-of-sample forecasting.
Hommes & in ‘t Veld (2014) estimate a HAM using quarterly US stock price data from 1950:Q1 - 2012:Q4. The model relies on a fundamentalists and chartists framework
in which all agents know the fundamental value of the stock prices. The estimation of the model shows an improvement over representative agent models that is statistically and economically significant. It also shows that a behavioural switching regime has sig-nificant effects on asset prices which amplifies booms and busts in the market, particular in the dot-com bubble and financial crisis in 2008.
2.3
Different heuristics
As mentioned in the introduction there are an infinite amount of ways to model an agent as boundedly rational. The problem that arises is how to select realistic expectation rules that can be used in the heterogeneous agent model. In HAM literature we find two common ways to solve this problem. One way is to assume agents act as econometricians and use econometric techniques to make forecasts, an example can be found in Evans and Honkapohja (2009). Another way of solving this problem, which we will use, is to assume agents use simple rules of thumb to make forecasts. An example of a rule could be ”my expectation for tomorrow is that it will be the same as today”. If we combine this method with the assumption that agents behave as certain types, for example fundamentalists or chartists, we are able to introduce a simple rule that corresponds with a certain belief type.
To observe how humans predict prices Hommes et. al (2005) did controlled exper-iments in which a game was played were human subject had to predict prices based on past realized prices and their own predictions. These ’learning to forecast’ exper-iments show that the realized prices differ significantly from the fundamental values and typically exhibit oscillations around, or slow convergence to the fundamental value. Anufriev and Hommes (2012) show that a model of individual learning can explain these different aggregate outcomes and use five different heuristics to describe the price prediction process. The five rules are adaptive heuristics (ADA), weak trend-following rule (WTR), strong trend-following rule (STR), anchoring and adjustment rule with a learning anchor (LAA) and anchoring and adjustment rule with a fixed anchor (AA) and are captured by the following equations (Anufriev & Hommes, 2012, p. 51):
ADA: E[p1,t+1] = 0.65pt−1+ 0.35E[pt], (2.1)
WTR: E[p2,t+1] = pt−1+ 0.4(pt−1− pt−2), (2.2)
STR: E[p3,t+1] = pt−1+ 1.3(pt−1− pt−2), (2.3)
LAA: E[p4,t+1] = 0.5(pavgt−1+ pt−1) + (pt−1− pt−2), (2.4)
AA: E[p5,t+1] = 0.5(pf + pt−1) + (pt−1− pt−2.) (2.5)
In this model the weak and strong trend following rules show similarities with the chartist rule, which is usually described as E[pt+1] = φ2(pt−1− p∗t−1), in which φ2 > 1
and p∗t−1 is the fundamental value of pt−1. This shows that chartists always believe that
the price of tomorrow will continue to depart from the fundamental value p∗t, either negative or positive. The trend following rules also show this type of behaviour and
could therefore act as interesting similar replacements for the chartist rule. However, there are two key difference if we compare the chartist rule to the trend following rules: firstly, the chartist rule is usually directly fitted to the data. Meaning that in order to find the value of φ2one usually fits it directly to the data, while the trend following rules
use fixed values which have been ’indirectly’ fitted, because they are not fitted to the data, but to the subjects in the experiments. Secondly, the trend following rules not also use information of ’yesterday’ (pt−1), but also of the day before yesterday (pt−2) to make
a forecast. This is a realistic assumption that uses more information to make a prediction about future prices. Due to these subtle differences we will use these rules as candidates for our new heterogeneous agent model in which we will use a fundamentalists rule and one of the trend following rules.
Lastly, to make a really fair comparison we will introduce a heterogeneous agent model with a general trend following rule of the form: E[pt+1] = pt−1+ φ3(pt−1− pt−2),
3
The Model
The model is a stylised asset pricing model with heterogeneous agents, firstly introduced by Brock and Hommes (1998) and Boswijk et al. (2007) and later generalized by Hommes and in ‘t Veld (2014). In this chapter we will briefly go over this model. In the model agents can invest in a risky asset that pays out an uncertain dividend Dt in period
t. The price of the risky asset is denoted by Pt in period t. The opportunity cost for
investing in the risky asset is given by the discount rate Rtin period t.
Firstly, it is assumed that investors have perfect knowledge of the fundamental value of the risky asset, but have different beliefs about what the price of the asset will be in the next period. The beliefs of an agent may change over time and these changes will result in fluctuating market sentiment. Secondly, it is assumed that agents can only disagree about the price of an asset in the next period and therefore we do assume that agents agree about the fundamental price of an asset.
The standard pricing equation is: Pt= ¯Et
Pt+1+ Dt+1
Rt+1
. (3.1)
The operator ¯Et[.] denotes the average expectation over all agents in period t. Since
we assume that the agents agree about the fundamental price the agents will also have identical beliefs about the dividend Dt+1= (1 + gt+1)Dt, where gt+1 denotes the growth
rate at time t + 1, and its discounted value: Et Dt+1 Rt+1 = Et 1 + gt+1 Rt+1 Dt. (3.2)
Now we can rewrite the pricing equation in terms of the price-dividend ratio (PD) δt:= Pt/Dtas δt= ¯Et Pt+1+ Dt+1 Dt· Rt+1 = ¯Et Dt+1(δt+1+ 1) Rt+1· Dt . (3.3)
The fundamental value Pt∗ is derived under rational expectations from the present value of all future cash flows in the same way as been done by Boswijk et al. (2007, p. 1965) and Hommes and in ‘t Veld (2014, p. 6). In general we will find
Pt∗= Et " ∞ X j=1 j Y k=1 Et+k−1 1 + gt+k Rt+k Dt # (3.4) for the fundamental price and thus for the fundamental PD ratio δ∗ we have
δ∗t = Et " ∞ X j=1 j Y k=1 Et+k−1 1 + gt+k Rt+k # . (3.5)
For the actual computation of these values Hommes and in ‘t Veld use some simplyfing restrictions to calculate the actual fundamental value, which we will use later as well. Lastly, we introduce the deviation from the fundamental ratio xt which is defined as
xt:= δt− δt∗.
3.1
Heterogenous beliefs
In this section we will model the heterogeneous beliefs of agents. It is assumed we have H ∈ N − {0} different types of investors with different expectations. In the standard pricing equation we find that Pt is dependent on the average expectation of all agents.
Since agents are allowed to have different beliefs we will take the weighted average of the beliefs as weights. Defining nh,t as the fraction of agents with beliefs Eh,t. We can
rewrite the standard pricing equation as
Pt= H X h=1 nh,tEh,t Pt+1+ Dt+1 Rt+1 (3.6)
and the PD ratio will become
δt= H X h=1 nh,tEh,t (1 + gt)(δt+1+ 1) Rt+1 . (3.7)
Due to the second assumption investors have homogenous beliefs about fundamental factors such as the growth rate gt and discount rate Rt. Because of independency we
can do the following
Eh,t (1 + gt)(δt+1+ 1) Rt+1 = Et 1 + gt Rt+1 Eh,t[δt+1+ 1]. (3.8)
We can use this to define the unconditional expectation 1/R∗ := E Et 1+gt+1 Rt+1 as the expected effective discount factor for pricing stocks in terms of the PD ratio. Now we can rewrite δt as δt= 1 R∗ H X h=1 nh,tEh,t[δt+1+ 1]. (3.9)
Also, because of the second assumption agents have homogeneous beliefs about funda-mental ratio δ∗ so that
Eh,t[δt+1∗ ] = Et[δt+1∗ ] = R∗δt∗− 1. (3.10)
Earlier we defined xt as the difference between the PD ratio and the fundamental PD
ratio: xt= δt− δt∗. We assume that the agent’s belief about the next period is separable
from his fundamental belief so that
Under these assumptions we can rewrite xtas: xt= δt− δ∗t = 1 R∗ H X h=1 nh,tEh,t[δt+1+ 1] − δt∗= 1 R∗ H X h=1 nh,tEh,t[xt+1]. (3.12)
We have now rewritten the standard pricing equation as a dynamical heterogeneous agent model.
3.2
The different models
3.2.1 The Hommes and in ‘t Veld Model
We start with a recap of the benchmark model which is the Hommes and in ‘t Veld (2014) model. For their empirical estimation the model uses a usual fundamentalists and chartists framework. For both model we will use two different strategies so that H = 2. On the one hand there are Fundamentalists. Fundamentalists believe in mean-reversion of stock prices to some fundamental value. On the other hand there are chartists who do not believe prices will move to a fundamental value but believe in a continuation of price trends.
In mathematical terms we can describe the expectation of the fundamentalists and chartists as follows:
Fundamentalists: E1,t[xt+1] = φ1xt−1, (3.13)
Chartists: E2,t[xt+1] = φ2xt−1, (3.14)
in which 0 < φ1 < 1 and φ2 > 1. The parameters φ1 and φ2 will give an impression
of how ’strong’ a belief type is. For a fundamentalists a φ1 close to zero shows that a
fundamentalists believes the value of tomorrow will be close to the actual fundamental value. A value of φ1 closer to one shows that a fundamentalists believes the price of
tomorrow will slowly converge to the actual fundamental price. For chartists we have the opposite, a relatively low value of φ2(close to one) means that the chartists believes
prices will slowly diverge from the fundamental value and a high value of φ2means that
a chartist believes the price value of tomorrow will diverge quickly from the fundamental value.
3.2.2 The different Trend Following Models
In this section we will introduce the three different trend following models. As mentioned in chapter two the trend following models use slighty different rules than the usual chartist rule. To be able to make a good comparison we wil only change the chartist rule by one of the trend following rules, but we will use the same fundamentalist rule as used by Hommes and in ‘t Veld. The first rule is the fundamentalist rule. For the fundamentalist rule nothing changes so fundamentalists still believe in mean-reversion of stock prices to their fundamental value. The other agents (called trend followers) believe that the price will move away from the fundamental value and expect a continuation
of the trend, this expectation will either be a weak belief or a strong belief which corresponds with lower and higher values of φ2 in the Hommes and in ’t Veld model.
In mathematical terms we can describe the three new models as following: 1. Weak trend following model:
Fundamentalists: E1,t[xt+1] = φ1xt−1, (3.15)
Weak trend followers: E2,t[xt+1] = xt−1+ 0.4(xt−1− xt−2). (3.16)
2. Strong trend following model:
Fundamentalists: E1,t[xt+1] = φ1xt−1, (3.17)
Strong trend followers: E2,t[xt+1] = xt−1+ 1.3(xt−1− xt−2). (3.18)
3. General trend following model:
Fundamentalists: E1,t[xt+1] = φ1xt−1, (3.19)
Trend followers: E2,t[xt+1] = xt−1+ φ3(xt−1− xt−2). (3.20)
In which for all models we have 0 < φ1 < 1 and for the general trend following model
we have φ3 > −1. As described in chapter two the different trend following rules have
been introduced by Anufriev and Hommes (2012) to describe forecasting behaviour of humans in a controlled environment.
3.2.3 Strategy switching
In both models the agents are allowed to switch between the two strategies and will do so if their predictions become too far-off from actual prices, meaning that agents are able to learn from their mistakes. The distribution of agents using each strategy is updated by a multinomial logit model as introduced by Brock and Hommes (1997):
nh,t+1=
eβUh,t
PH
j=1eβUj,t
, (3.21)
where β denotes the intensity of choice and Uh,t denotes the performance measure
of believe type h. Following Brock and Hommes (1998) the performance measure is dependent on the profits π of agent types. The profit is calculated as following:
πh,t+1= (Eh,t[xt+1] − R∗xt)(xt+1− R∗xt). (3.22)
Introducing memory parameter ω, we obtain performance measure:
Uh,t = (1 − ω)πh,t+ ωUh,t−1, (3.23)
in which the memory parameter ω tells us something about the memory strength. In a model with ω = 0 an agent bases its future strategy only on the most recent observed profit. In our model we will estimate the model to quarterly data in which we will have to account not only for the most recent observed profits but also for more recent profits so that ω ∈ (0, 1). We have now specified the model for the two belief types and we are able to write down the econometric form which allows us to estimate the model with data.
3.2.4 Econometric form
To be able to make an estimation of the model we need to write down the models in their econometric forms. The Hommes and in ‘t Veld Model can be written as an econometric first order autoregressive AR(1) model with time varying parameters and the trend following models can be written down as econometric second order autoregressive AR(2) models with time varying parameters as follows:
1. Hommes and in ’t Veld AR(1) model: xt=
1
R∗(n1,tφ1+ n2,tφ2)xt−1+ t. (3.24)
2. Weak trend following AR(2) model: xt=
1
R∗((n1,tφ1+ 1.4n2,t)xt−1− 0.4n2,txt−2) + t. (3.25)
3. Strong trend following AR(2) model: xt=
1
R∗((n1,tφ1+ 2.3n2,t)xt−1− 1.3n2,txt−2) + t. (3.26) 4. General trend following AR(2) model:
xt=
1
R∗((n1,tφ1+ (1 + φ3)n2,t)xt−1− φ3n2,txt−2) + t. (3.27)
In which for all models the term t represents an error term at time t. The error terms
are assumed to be i.i.d. and t∼ N (0, σ2).
An important element of these models is it allows agents to have different beliefs. The different beliefs cause our market to be divided by a fraction of fundamentalists and chartists (or trend followers). These different fractions will be important for our analysis and with equations 3.21, 3.22, 3.23 and one of the AR models these fractions depend nonlinearly on the four parameters and all previous values of xt. We find
n1,t = fn1(φ1, φ2, β, ω; xt−1, xt−2, . . . , x1), (3.28)
n2,t = 1 − n1,t, (3.29)
for the Hommes and in ‘t Veld model. For the general trend following model we replace φ2 by φ3 and for the weak and strong trend following model we omit the φ2 variable.
4
Estimations and results
In this chapter we will go over the estimations of all models and the results of all tests. The first section of this chapter will start with a brief analysis of the data. Subsequently, the next section will cover the estimations of both models. Lastly, the third section will contain some test statistics and quality measures in which we will start with a computation of the Akaike Information Criterium and Bayesian Information Criterium to measure the relative quality of all models. To be able to say something about the predictive power of both models we will compare the errors of one step ahead out of sample forecasts of both models and compute the root mean squared error and test if there is a significant difference between both models using the Hommes and in ‘t Veld model as benchmark model.
4.1
Data analysis
As mentioned in the introduction our goal is to estimate different heterogeneous agent models to stock market data. The stock market data that will be used for the estimations is quarterly S&P500 data from 1950:Q1 – 2012Q:4, which was used by Hommes and in ‘t Veld (2014). With this data and their model Hommes and in ‘t Veld computed the corresponding fundamental prices Pt∗, PD-ratios δt, and fundamental PD-ratios δ∗t,
which will also be used in our estimations. Figure 4.1 shows us the price movements of the S&P500 price index, which is the bold line, versus the estimated fundamental value, which is represented by the dotted line. Notice that in this plot up until around 1995 the estimations of the fundamental value of the index price were close to the index price, but in 2000 prices started to rise up to three times the estimated fundamental value with the start of the dot-com bubble. This bubble crashed in 2001 but was followed by another bubble reaching its peak in 2007. The bust of this bubble started the economic crisis of 2008.
Our estimation will not be made directly to the stock market data, but to the xt = δt− δt∗ value as described in section 3.1. Figure 4.2 gives an impression of the
movement of δtand δ∗t and figure 4.2 shows us the movement of the xtvalue. We notice
a few things in figure 4.3. First of all the value of xtoscillates roughly around zero from
1960 to 1990. The fact that xt ≈ 0 means that the actual prices were almost equal
to the estimated fundamental prices of stocks, which is something we would naturally expect. However, there are some exceptions, for example a large drop in the period 1973 - 1975 which corresponds with the 1970s recession. A few causes of this recession were
Figure 4.1
S&P500 Index Price vs. Fundamental Price
'50 '60 '70 '80 '90 '00 '10 0 500 1000 1500 2000 Years Pr ice in dollars Figure 4.2
S&P500 PD ratio vs. Fundamental PD ratio
'50 '60 '70 '80 '90 '00 '10 0 20 40 60 80 Years PD r atio
Figure 4.3 Xt ratio '50 '60 '70 '80 '90 '00 '10 0 20 40 60 Years Xt
the 1973 oil crises and the failure of the Bretton Woods system. Another large drop can be found in 1988, which corresponds with the early 1990 recession. The drops and rises during 1960 – 1990 pale in comparison to the rise and drop in the period of 1995 – 2001. Starting around 1997 the dot-com bubble started with the rise of stock values of new internet based companies such as ebay and amazon. In 2001 the bubble burst, which we can also in our graph by a drop in value from 60 to 20. The few years after the dot-com bubble the xtvalue oscillates around 20 and then drops again at the start of the global
economic crisis in which the xt value is back around its expected value 0. It seems that
whenever the actual value starts to deviate alot from its estimated fundamental value the market is experiencing a bubble and the value of xt is bound to be pulled back to
0 by some recession.
4.2
Estimations
In our models we will estimate the following parameters: φ1, φ2, φ3, β and ω. We will
assume that 0 < φ1 < 1, φ2 > 1, φ3 > −1, 1 ≤ β ≤ 10 and 0 < ω < 1. Due to
the non-linear nature of our model it is possible that there exists several local optimal points of the least square minimization problem based on the initial value we use for the estimations. This is especially the case for the intensity of choice parameter β and memory parameter ω, which will want to take into account. To solve this problem for β, Hommes and in ‘t Veld (2012) distinguish three different cases. The first case is setting the initial value equal to 5 and the other two cases are setting the initial value equal to either 1 or 10. As explained by Hommes and in ‘t Veld (2014, p. 15) it is justifiable to fix β = 1 for the Hommes and in ‘t Veld model. To be able to make a good comparison
we will fix β = 1 in the trend following models as well.
However, we will use different initial values to estimate ω. In the strong trend fol-lowers model we find that ω converges to two different values depending on the initial value. Setting the initial value of ω ∈ (0.55, 1], ω will converge to 0.88. If we choose a lower value of ω, it is possible that ω converges to either 0.88 or 0.99. It seems there is a bifurcation in the model at the initial value ω = 0.55. These different values of ω will have an impact of the switchin strategies of traders. In a model with a lower ω traders tend to switch more often with more people at the same time. The weak trend following model and the general trend following model seem to be more stable. The value of ω will only converge to an ’extreme’ value if the we set the initial value of ω equal to the boundery values, for example setting the initial value equal to 0.99 then it will also converge to this boundery. We find that within the strong trend following model the residual sum of squares is lower for ω = 0.99 and therefore we will use this case. Table 4.1 shows the results of the estimations of the parameters for both the Hommes and in ‘t Veld model and the trend following models.
Table 4.1: Estimations of the parameters for all models. Hommes-in ‘t Veld WTR STR GTR φ1 0.936∗∗∗ 0.935∗∗∗ 0.912∗∗∗ 0.913∗∗∗ φ2 1.026∗∗∗ - - -φ3 - - - 0.900∗∗∗ β 1 1 1 1 ω 0.824∗∗∗ 0.843∗∗∗ 0.997∗∗∗ 0.880∗∗∗
∗∗∗ denotes significance at the 1% level.
We notice that the values of φ1 and ω for the Hommes- in ‘t Veld model and the
weak trend following model are almost the same. In the strong trend following model we find that the value of φ1 is a little bit higher compared to the Hommes and in ‘t
Veld model, and together with the different value of ω this will have a big impact on switching behaviour, which we will see later. For the general trend following model we find a value of 0.9 for φ3, which is surprisingly almost in the middle of the weak and
strong trend following model. The value of φ1 therefore also converges to a value in the
middle of the φ1 values of the other trend following models. As mentioned before the
different values of ω, cause different trading behaviour. An ω value close to 1 causes agents to switch less frequent than lower values of ω.
4.3
Test statistics
4.3.1 Computation of the AIC and BIC
In this section we compute the Akaike Information Criterium (AIC) and the Bayesian Information Criterium (BIC). The AIC was introduced by Akaike (1974) and measures
the relative quality of statistical models for a given data set. The AIC does not provide a test for a model but gives a relative measure of the quality of a model compared to another model. The lower the value of the AIC, the better. There are several formulas for the computation of the AIC, all of course different representations of the same formula. For the computation of the AIC we will use the formula as introduced by Verbeek (2014): AIC = ln1 N N X i=1 e2i+ 2K N ,
where e are the error terms, N denotes the amount of data points and K is the amount of parameters used to estimate the model. For the different models we find the following values for the AIC:
1. Hommes and in ’t Veld model: AIC = ln3391
252
+ 8
252 ≈ 2.63. (4.1) 2. Weak trend following model:
AIC = ln 3688 252 + 6 252 ≈ 2.82. (4.2)
3. Strong trend following model: AIC = ln5337
252
+ 6
252 ≈ 3.08. (4.3) 4. General trend following model:
AIC = ln 4579 252 + 8 252 ≈ 3.08. (4.4) To summarize we have the following table:
Table 4.2: Values of the Akaike Information Criterium. Hommes- in ‘t Veld WTR STR GTR AIC 2.63 2.82 3.08 3.08
Based on the value of the AIC the Hommes and in ‘t Veld model performs the best, followed by the weak trend following model. The strong trend following model and the general trend following model do not perform as good as the other models. It seems that a higher value of φ3 causes the sum of squares residuals to rise which has a negative
impact on the AIC.
Next we compare the Bayesian Information Criterion (BIC) of both models. The BIC was introduced by Schwarz (1978) and is the competitor of the frequentist AIC and is constructed from a bayesian point of view. The BIC is also not a test but a
quality measures which only measures relative performance. The lower, the better. The formula for the BIC is:
BIC = ln 1 N N X i=1 e2i +K ln(N ) N ,
where again e denote the error terms, N denotes the amount of data points and K is the amount of parameters used to estimate the model. For the different models we find:
1. Hommes and in ’t Veld model: BIC = ln 3391 252 +4 ln(252) 252 ≈ 2.69. (4.5) 2. Weak trend following model:
BIC = ln3688 252
+3 ln(252)
252 ≈ 2.75. (4.6) 3. Strong trend following model:
BIC = ln5337 252
+3 ln(252)
252 ≈ 3.12. (4.7) 4. General trend following model:
BIC = ln 4579 252 +4 ln(252) 252 ≈ 2.99. (4.8) To summarize the results we have the following table:
Table 4.3: Values of the Bayesian Information Criterium. Hommes-in ‘t Veld WTR STR GTR BIC 2.69 2.75 3.21 2.99
The results for the BIC are similar to the results we found for the AIC. The Hommes and in ‘t Veld model performs best and is followed by the weak trend following model. The general trend following model performance is better than the strong trend following model, but not as good as the weak trend following model. Based on the BIC the strong trend following model does not perform as good as the other models.
4.3.2 Out of sample forecasts
In this section we will compare the predictive power of all models by testing the difference in values of the root-mean-squared error (RMSE) using one step ahead out of sample forecasts. The RMSE is defined as:
RMSE = r Pn
t=1(ˆxt− xt)2
in which ˆxt is the estimated value of xtat time t. The RMSE measures the root of the
mean squared errors. If the RMSE would be equal to zero, our model would fit perfectly to the data and there are no forecast errors. If there are some forecasting errors the RMSE will become positive. Which shows us that RMSE ∈ [0, ∞) and we prefer a low RMSE. Another property of the RMSE is that it ’punishes’ a model for having large errors, because it squares the errors. Small errors (< 1) will have a relatively smaller effect than large errors (> 1) due to the squaring of the error term. For the computation of the RMSE we will use out of sample data for 33 steps back. For the different models we find:
Table 4.4: RMSE values for all models based on 33 forecasts. Hommes- in ‘t Veld WTR STR GTR RMSE 4.56 4.59 4.93 5.59
We notice that the Hommes-Veld model performs best and the weak trend following and general trend following models perform the same, which is a little bit worse than the Hommes-Veld model. Again the performance of the strong trend following model is of the poorest quality compared to the other models.
To evaluate if RMSE values significantly differ we will use a Diebold and Mariano (1995) test on a loss differential of forecasting the future output of an individual i, di,t.
This exact method has been used previously by many others, for example Ductor et al. (2014, p. 939). The loss differential di,t is defined as follows:
di,t := e2Ai,t− e2Bi,t
in which B is the benchmark model and A is the model which we will compare to the benchmark model. In our case model A will be the Hommes and in ‘t Veld model and B will be the one of the trend following models. To test if one model statistically outperforms the other model we test the following hypothesis:
1. H0: E[di,t] = 0,
2. H1: E[di,t] 6= 0.
In which under the null-hypothesis the Diebold-Mariano test statistic is: d q ˆ V (d)/n ∼ N (0, 1), where d = n−1P
i,tdi,t, is the average loss differential and ˆV (d) is a consistent estimate
of the asymptotic variance of √nd. Due to autocorrelation in the residuals we’re not allowed to use variance of the mean, but use a Newey-West estimator for the variance of the mean ˆV (d). Table 4.5 shows us the Diebold-Mariano test-statistics for all models.
If we assume an α equal to 5% we will come to the conclusion that we will not reject the H0 hypothesis for the weak trend following model, in fact we would not even reject
Table 4.5: Diebold-Mariano test statistics for all models using the Hommes-Veld model as a benchmark model. WTR STR GTR d 0.27 3.51 0.27 Newey-west variance of mean 7.05 26.66 2.98 DM test-statistic 0.58 3.33∗ 0.77
∗ denotes significance at the 1% level.
it at the 10% level. From which we can conclude that there is no significant difference in the RMSE of the Hommes-Veld model and the weak trend following model. We do reject the H0hypothesis for the STR model, which means the RMSE of the STR model
is significantly different than the RMSE of the Hommes-Veld model. Since the RMSE of the STR is higher than the Hommes-Veld model we conclude this model performce significantly worse at predicting future values of xt. The RMSE of the general trend
following model does not significantly differ from the RMSE of the Hommes-Veld model and would not even be rejected at an α equal to 10%.
4.3.3 Conclusion of tests
In the preceding sections we computed several quality measures and test statistics. In this section we will link these results together to see which model performs best based on these measures. Firstly, we found that based on the values of the AIC and BIC the performance of the Hommes and in ‘t Veld model is the best and is followed by the weak trend following model. Both the general trend following model and strong trend following model did not perform as good as the other models. The second performance measure was the RMSE. We found that the RMSE of the weak and general trend following model were not significantly different than the RMSE of the Hommes and in ‘t Veld model, meaning that the models perform equally good in out of sample forecasting. The RMSE of the strong trend following model is significantly worse than the Hommes and in ‘t Veld model, meaning that the strong trend following model does not perform as good as the Hommes and in ‘t Veld model in out of sample forecasting. Based on these results we conclude that in general the performance of the Hommes and in ‘t Veld model is the best, followed by the weak trend following model. The third best model is the general trend following model and lastly we have the strong trend following model.
However, we should not only base our conclusion on these results. We do not only want a model that performs as good as the Hommes and in ‘t Veld model based on quality measures. We also want our model to be able to help us describe the real world. To test if a model is able to do this we will analyse the switching behaviour of agents in different models and compare it to the Hommes and in ‘t Veld model.
Figure 4.4
Percentage of fundamentalists in Hommes and in ‘t Veld Model
'50 '60 '70 '80 '90 '00 '10 0% 20% 40% 60% 80% 100% Years P ercentage
4.4
Fractions
To analyse and compare the trading behaviour within different models we will start with a brief recap of the trading behaviour in the Hommes and in ‘t Veld model. Figure 4.4 gives an impression of the agents in the market with a fundamentalists belief in the Hommes and in ‘t Veld model. We notice that the amount of fundamentalists keeps oscillating around 50% until the start of the dot-com bubble when all agents switch to a chartist belief. For a few years all agents have a chartist belief until the bubble burst and all agents almost instantly change to a fundamentalist belief strategy. During the time that all agents used a chartist rule the price level of the S&P500 started to rise significantly and deviates a lot from the fundamental price as can be seen in figure 4.1. This motivates the fact that switching behaviour of agents can have a significant effect on economic markets and that market psychology can play a role in the formation of bubbles. This result shows us that the Hommes and in ‘t Veld model is not only a good model because it is able to perform well in terms of test statistics, but also helps us describe economic phenomena.
Next we compare the trading behaviour of agents within the trend following models, which can be found in figures 4.5, 4.6 and 4.7. The first thing we notice is that the trading behaviour of the weak and general trend following models are quite similar. It seems that the trading behaviour in the general trend following model is an amplification of the trading behaviour in the weak trend following model, which can be explained by
Figure 4.5
Percentage of fundamentalists in WTR Model
'50 '60 '70 '80 '90 '00 '10 0% 20% 40% 60% 80% 100% Years P ercentage Figure 4.6
Percentage of fundamentalists in STR Model
'50 '60 '70 '80 '90 '00 '10 32% 40% 48% 56% 64% 72% Years P ercentage
Figure 4.7
Percentage of fundamentalists in GTR Model
'50 '60 '70 '80 '90 '00 '10 0% 20% 40% 60% 80% 100% Years P ercentage
the higher value of φ3. The general trend following rule is a stronger rule than the weak
trend following rule and therefore triggers more drastic trading behaviour. We find that in the weak and general trend following model agents tend to switch to a fundamental belief whenever there is a recession or bust. For example in 1973 during the oil crisis agents switched to a more fundamental belief. This is also the case in 1988 at the start of the early 1990s recession. When we analyse the period of 1995 – 2002 we see that the amount of fundamentalists starts to decrease until it reaches zero percent. Due to trend following behaviour the actual prices diverge from their fundamental prices, which causes the actual price and the value of xt to rise. After the bust in 2001 all agents almost
instantly switched to a fundamental price belief strategy. The following years agents gradually switched to trend following behaviour until the start of the economic crisis in which agents these agents switched to a fundamental belief strategy.
If we compare the weak trend following model to the general trend following model we conclude that the weak trend following performs better. Firstly, this is based on the AIC and BIC values, but more importantly it is based on the trading behaviour. The trading behaviour in the general trend following model seems to be to extreme, even with some relatively weak market fluctuations all agents tend to switch strategies, which seems highly unlikely. The weak trend following model gives a more balanced picture with less heavier fluctuations.
to the significantly different value of memory parameter ω, which was close to one in this model, the trading behaviour is quite different than in the other trend following models. Firstly, we notice that the fluctuations are less heavy than the other model and until the dot-com bubble the amount of fundamentalists oscillates roughly around 56%. During the start of the dot-com bubble the percentage of fundamentalists dropped till around 40% and shoots back up to around 50% when the dot-com bubble went bust. We also notice that at the start of the global economic crisis the amount of fundamentalists did not change much, but during the crisis the amount of fundamentalists went up to around 65%. It seems that the general pattern of the switching behaviour in the strong trend following model works quite well.
If we compare the performance of the weak trend following model and the strong trend following model we notice that the trading behaviour within both models is quite good. However, due to the fact that the strong trend following model does not perform as good as the other models in terms of quality measures we prefer the weak trend following model. We finally conclude that within the trend following models the weak trend following model performs best. This is due to its ability to also help us describe economic reality and based on the test statistics we conclude that this new model actually works quite well and even helps us describe the effect of market psychology on the economy, for example with the dot-com bubble in which trend followers pushed up prices.
The next question to answer is if the weak trend following rule could replace the chartist rule. Based on our test statistics, AIC and BIC we favour the Hommes and in ‘t Veld model. Next if we compare the trading behaviour within the different models we notice that the trading behaviour within weak trend following model fluctuates a lot more than it does in the Hommes and in ‘t Veld model. To some extend this is useful to describe the behaviour of agents during economic recessions but it also seems to extreme. For example, just before 1960 the amount of fundamentalists went from 100% to 20% to 80% in just a few periods. This seems highly unlikely due to the fact that there we no real significant events during that time and only a small fluctuation in the xt value. The Hommes-Veld model is more robust in this sense and shows that
the amount of fundamentalists oscillates around 50% up until the dot-com bubble, in which chartist took over, which is very likely. We finally conclude that it is possible to change the chartist to a weak trend following model in a fundamentalist vs. chartists framework, which does not make the model significantly worse, but also not significantly better.
5
Conclusion
The goal of this study is to analyse the effect of changing the usual fundamentalists vs. chartists framework into a different framework within a heterogeneous agent model. To analyse this effect we started with a heterogeneous agent model for the stock mar-ket introduced by Hommes and in ‘t Veld (2014) and analysed the effect of changing their chartist rule by different trend following rules. These trend following rules were introduced by Anufriev and Hommes (2012) by analysing some learning to forecast experiments done by Hommes et al. (2005). Anufriev and Hommes (2012) introduced an individual learning model that was able to explain different outcomes in these ex-periments. Their individual learning model uses five different heuristics of which two are trend following rules. These different trend following rules are quite similar to the chartist rule, because they also expect prices will deviate from their fundamental value and were therefore used as possible replacements for the chartist rule.
In section 3.2.2 we introduced three new trend following models in which agents are either fundamentalists or trend followers. These three new models are the weak trend following model, the strong trend following model and the general trend following model. The weak and the strong trend following models use trend following rules as introduced by Anufriev and Hommes (2012). These rules were not directly fitted to our data, but to the learning to forecast experiments done by Hommes et al. (2005). To make a really fair comparison we decided to introduce the general trend following rule. The general trend following model uses a trend following rule that is directly fitted to the data.
We decided to compare the new models to the Hommes and in ‘t Veld model based on two different features. On the one hand the model has to perform well in terms of quality measures and test statistics such as the AIC, BIC and RMSE. On the other hand the model has to be able to give a realistic view of reality.
Based on the test statistics introduced in section 4.3 we find that the Hommes and in ‘t Veld model performs best in terms of quality measures such as the AIC and BIC. We find that the weak trend following model is also able to perform quite well in these measures compared to the Hommes and in ‘t Veld model and the other trend following models do not perform as good. In section 4.3.2 we found that the RMSE of the Hommes and in ‘t Veld model, the weak trend following model and the general trend following models do not significantly differ. The RMSE of the strong trend following model is significantly worse than the RMSE of the Hommes and in ‘t Veld model. In general we find that the performance of the Hommes and in ‘t Veld model is the best and
is followed by the weak trend following model. The performance of the general trend following model is not as good as the weak trend following model, but better than the performance of the strong trend following model. The strong trend following model performs the worst based on all quality measures and test statistics.
The ’usefulness’ of the model was analysed in section 4.4.4 by investigating the trading behaviour within different models. We find that the trading behaviour in the general trend following model is an amplification of the trading behaviour in the weak trend following. The trading behaviour in the weak trend following model seems quite realistic, but the trading behaviour in the general trend following model is to extreme. The trading behaviour in the strong trend following model also seems realistic, but due to the weak performance in terms of quality measures and test statistics we prefer the weak trend following model over the strong trend following model.
The main conclusion of this study is that changing a chartist rule by a weak trend following rule is possible and this will not significantly change the heterogeneous agent model. The other rules do not perform as well as the weak trend following model and should therefore not be used.
For future research one could analyse the effect of using more than two rules in a heterogeneous agent model by for example estimating the full individual learning model introduced by Anufriev and Hommes (2012) to stock market data or other economic data such as housing prices or oil prices.
References
Akaike, H. (1974). A new look at the statistical model indentification. IEEE Transac-tions on Automatic Control, 19(6), 716 - 723.
Anufriev, M & Hommes, C.H. (2012). Evolutionary selection of individual expectations and aggregate outcomes in asset pricing experiments. American Economic Journal: Microeconomics, 4(4), 35 - 64.
Bolt, W., Demertzis, M. Diks, C., Hommes, C. & van der Leij, M. (2014). Indentifying booms and busts in house prices under heterogeneous expectations (Working Paper No. 450). Retrieved from De Nederlansche Bank.
Boswijk, H.P., Hommes, C.H., & Manzan, S. (2007). Behavioral heterogeneity in stock prices. Journal of Economic Dynamics and Control, 31, 1938 - 1970.
Brock, W.A., & Hommes, C.H. (1997). A rational route to randomness. Econometrica, 65(5), 1059 - 1095.
Brock, W.A., & Hommes, C.H. (1998). Heteregeneous beliefs and routes to chaos in a simple asset pricing model. Journal of Economic Dynamics & Control, 22, 1235 -1274.
Diebold, F,X. & Mariano, R,S. (2002). Comparing predictive accuracy. Journal of Busi-ness & Economic Statistics, 20(1), 134 - 144.
Ductor, L., Fafchamps M., Goyal S., & van der Leij, M.J. (2014). Social networks and research output. Review of Economics and Statistics, 96 (5), 936 - 948.
Evans, G., Honkapohja, S. (2009). Learning and Macroeconomics. Annual Review of Economics, 1, 421 - 449.
Fama, E. F. (1970). Efficient capital markets: a review of theory and empirical work. The journal of Finance, 25(2), 383 - 417.
Friedman, M. (1953). Essays in Positive Economics. University of Chicago Press. Garber, P.M. (1990). Famous First Bubbles. The Journal of Economic Perspectives,
4(2), 35 - 54.
Hommes, C.H., Sonnemans, J., Tuinstra, J. van de Velden, H. (2005). Coordination of Expectations in Asset Pricing Experiments. The Review of Financial Studies. 18(3), 955 - 980.
Hommes, C.H. (2006). Heterogeneous agent models in economics in finance. Handbook of computational economics, 2(23), 1109 - 1186.
Hommes, C.H., & in ‘t Veld, D. (2014). Booms, busts and behavioral heterogeneity in stock prices. Amsterdam: University of Amsterdam. CeNDEF working paper. Schwarz, G. (1978). Estimating the Dimension of a Model. The Annals of Statistics,
6(2), 461 - 464.
Simon, H. A. (1957). Models of man, Wiley, New York, NY.
speculators? German Economic Review, 10(3), 270 - 283.
Taylor, M.P. & Allen, H. (1992). The use of technical analysis in the foreign exchange market. Journal of International Money and Finance, 11(3), 304 - 314.
Muth, J.E. (1961). Rational expectations and the theory of price movements. Econo-metrica, 29, 315 - 335.
Nelson, R.R. & Winter, S. (1973). Toward an evolutionary theory of economic change, Harvard University Press, Cambridge, Mass.
Verbeek, M. (2014). A guide to modern econometrics. John Wiley & Sons, Ltd.
Zeeman, E. C. (1974). On the unstable behaviour of stock exchanges. Journal of Math-ematical Economics, 1(1), 39 - 49.
Zwinkels, R.C.J. & Ter Ellen, S. (2010). Oil price dynamics: A behavioral finance ap-proach with heterogeneous agents. Energy Economics, 32(6), 1427 - 1434.
