Master’s Thesis
Stability analysis of the cobweb model
with SAC-learning versus naive
expectations
Explaining stylized facts from laboratory experiment with
human subjects
R.M.E. Laboyrie
Student number: 6133622 Date of final version: April 8, 2016
Master’s program: Econometrics
Specialization: Mathematical Economics Supervisor: Prof. dr. C. H. Hommes Second reader: Dr. A. G. Kopanyi-Peuker
Abstract
A stability analysis of the cobweb model with SAC-learning versus naive expectations is performed. When price dynamics are described under this cobweb model, the dynamics show a steady state, a 2-cycle, a 4-cycle and chaotic dynamics. For the second part of this thesis, the experiments conducted by Hommes, Sonnemans, Tuinstra and van de Velden (2007) are investigated. They observed three stylized facts of aggregate price fluctuations. The cobweb model in this thesis tried to explain these facts with a simulation and was able to explain the first fact, partly explain the second fact and unable to explain the third fact.
Keywords: cobweb model, SAC-learning, naive expectations, bifur-cations, stability, stylized facts
Contents
1 Introduction 2
2 Model 5
2.1 The cobweb model . . . 5
2.2 Sample autocorrelation (SAC)-learning . . . 6
2.3 Cobweb model with SAC-learning versus naive expectations . 8 3 Analysis 10 3.1 Dynamical System . . . 10
3.2 SAC-learning and naive expectations . . . 11
3.3 Price dynamics and fractions . . . 13
3.4 Bifurcations and phase plots . . . 15
3.5 Varying the intensity of choice parameter . . . 18
3.6 Stability analysis of cobweb model with naive expectations and autoregressive(1) rule . . . 20
4 Experiment 22 4.1 Setup and results experiment . . . 22
4.2 Simulation . . . 25 4.3 Parameter sensitivity . . . 28 5 Conclusion 31 5.1 Stability analysis . . . 31 5.2 Learning-to-forecast experiments . . . 32 Appendix 34 A Derivation Jacobian . . . 34 B Results experiment . . . 35
1
Introduction
Since Kaldor (1934) and Ezekiel (1938) introduced the cobweb model, it has been applied in many studies that investigate the dynamical behavior of expectation feedback systems. The cobweb model is a simple framework explaining price fluctuations in a market for a non-storable good that takes one time period to produce. These price dynamics depend on the demand and supply function and, most importantly, on the expectations hypothesis. For different expectations, the dynamics describing the price fluctuations will differ such that there can be continuous, divergent or convergent fluctuations. In economics, many behavioral models use the assumption that agents have homogenous beliefs, meaning all agents have the same preferences and expectations. Next to that it is often assumed that these agents are rational and therefore have perfect foresight. However, in a complex market this assumption is often seen as unrealistic, therefore boundedly rational agents with heterogeneous expectations were introduced by Conlisk (1996) as a more accurate and realistic description. The dynamical behavior of the cobweb model with heterogeneous beliefs is investigated by a number of studies. This heterogeneity can lead to periodic or chaotic price dynamics of the cobweb model.
Further development of the cobweb model is done by Brock and Hommes (1997). They introduce the concept of adaptively rational equilibrium dy-namics (A.R.E.D.) and focus on the cobweb model with costly rational ver-sus costless naive expectations and linear demand and supply. A.R.E.D. are dynamics where agents choose among a finite set of predictors, which are based on past performance. Agents choose between a perfect-foresight ra-tional predictor and a naive predictor, where the naive predictor equals the last observed price. They show that when the intensity of choice to switch predictors becomes higher, the model leads to a primary period doubling bifurcation and the existence of homoclinic bifurcations. These homoclinic bifurcations cause very complicated dynamics. For the cobweb model with nonlinear demand and supply Goeree and Hommes (2000) found bifurcation routes similar to the linear case.
The cobweb model of Brock and Hommes is further investigated by Laselle, Svizzero and Tisdell (2005) and Branch (2002) by considering a different set of forecasts. Laselle, Svizzero and Tisdell (2005) replace the costless naive expectations by costless adaptive expectations where they assume that adap-tive expectations are a weighted average of the last two prices. They state that next to the period doubling bifurcation, a new type of bifurcation exists, namely a Hopf bifurcation. Branch (2002) extends the Brock and Hommes model, by letting agents choose between rational, naive and adaptive beliefs.
When the adaptive expectations are costless, the steady state will be locally stable, while if the adaptive beliefs are more costly than the naive beliefs the steady state will be unstable.
Finding a more sophisticated rule than the rational expectations, Hommes (2013) applies the cobweb model to agents with contrarian beliefs versus naive beliefs. The contrarians expect negative first-order autocorrelation in realized prices and adapt their forecast by assuming that the next periods deviation from the fundamental price will be on the opposite side of the steady state. It is shown that the time series of prices show strong negative first-order autocorrelation, even when dynamics are chaotic. However, the realized prices have shown weaker first-order sample autocorrelation, so the contrarian belief is still inconsistent. In this thesis consider the situation where the contrarian belief is replaced by a more sophisticated linear fore-casting rule with time-varying parameters. Agents change their forefore-casting function within the class of autoregressive AR(1) beliefs, and update their belief parameters as additional observations become available. In each pe-riod the belief parameters are updated according to their sample mean and sample first-order autocorrelation. This way the dynamical system describes the price dynamics under sample autocorrelation (SAC)-learning.
All of the above applications of the cobweb model with different expecta-tion rules have different effects on the dynamics of the model. As menexpecta-tioned before, SAC-learning can possibly be more consistent than the contrarian belief. Furthermore, it can be found more realistic than the rational be-liefs. According to Hommes and Zhu (2014), another advantage of using the SAC-learning process is that the sample mean and first-order autocorrelation can be roughly guesstimated without any knowledge of statistical techniques. Therefore the first goal of this thesis is to analyze the dynamics of the cobweb model with SAC-learning versus naive expectations. With this analysis it can be investigated how the dynamical behavior of a nonlinear model depends on different parameters.
For the second goal of this thesis, the experiments conducted by Hommes, Sonnemans, Tuinstra and van de Velden (2007) will be investigated. They performed learning-to-forecast laboratory experiments in the classical cobweb model and examined whether agents can learn the unique rational expecta-tions (RE) steady state. They observed three stylized facts of aggregate price fluctuations with these experiments by considering three different treatments: a stable, an unstable and a strongly unstable treatment. During the stable treatment, subjects learned the RE steady state price and therefore the RE forecast rule is a good description of the subjects behavior. However, for the unstable treatments excess price volatility was present and the RE forecast rule is not a good description. They described the unstable cobweb economy
as a boundedly rational heterogeneous expectations equilibrium, since on av-erage subjects were able to learn the rational expectations equilibrium price, but the differences in the subjects beliefs caused the excess price volatility.
Even though these stylized facts are hard to explain by standard learning mechanisms, Hommes and Lux (2013) were able to explain all three styl-ized facts simultaneously with an agent based simulation where individual learning is modeled through genetic algorithms. Homogeneous expectations models are unable to explain the full set of stylized facts simultaneously, but Hommes (2009) suggests that heterogeneity of forecasting rules plays an im-portant role in explaining observed aggregate behavior. Therefore, next to the stability analysis, the model in this thesis will be investigated in order to find out if it can also explain all stylized facts simultaneously.
This thesis is organized as follows, Section 2 first introduces SAC-learning in the laboratory experiment after which the cobweb model with SAC-learning versus naive expectations will be explained. The analysis of this cobweb model will be performed in Section 3. In this analysis some parameters of the dynamical system will be varied in order to find out how this influences the dynamical behavior of the cobweb model. Thereafter, the experiment and its results will be discussed in Section 4. Furthermore, a simulation of the cobweb model with SAC-learning versus naive expectations will be per-formed to match the stylized facts of the experiment. Finally, Section 5 will discuss the findings of this thesis.
2
Model
The cobweb model is a simple framework used to illustrate the role and im-portance of expectation feedback systems. As mentioned before, the cobweb model describes fluctuations of equilibrium prices for a non-storable good that takes one time period to produce, such that the producers must form price expectations one period ahead. For a cobweb model with heterogeneous expectations, producers can choose between different forecasting rules, which are based on past information.
In the next section the cobweb model with SAC-learning and naive ex-pectations will be analyzed. However, to perform this stability analysis Sub-section 2.1 will first explain the cobweb model in further detail. The cobweb model used in this thesis is based on the framework introduced by Brock and Hommes (1997). Subsection 2.2 will give a description of the SAC-learning forecast rule. Next, Subsection 2.3 will introduce the dynamical system, based on the cobweb model with SAC-learning versus naive expectations, used for the stability analysis is the next section.
2.1
The cobweb model
Following the Brock and Hommes framework, consider the cobweb model with heterogeneous beliefs, where agents form their price expectations pe
j,t by
choosing from a set of m different expectations functions Hj for j = 1, ..., m.
These expectation functions depend on past prices and the fraction of how many agents choose for a specific forecasting rule depends on a publicly available performance measure. Since agents have heterogeneous beliefs, the supply function will be a weighted sum of each agent’s supply decision, where each agent’s optimal supply decision is found by maximizing their expected profits. Supply S is an increasing function of the expected price and demand D is a decreasing function and depends upon the current market price. The heterogeneous market equilibrium is described as follows
D(pt) = m
X
j=1
nj,tS(Hj(Pt)), (1)
where nj,t is the fraction of agents using predictor Hj at the beginning of
period t and Pt= (pt−1, ..., p0) is a vector of past prices.
Once the market equilibrium price ptis known, all predictors Hj are
eval-uated according to their publicly available performance measure Uj and the
new fractions nj,t+1 are determined. The updated fraction nj,t+1 of producers
with multinomial logit probabilities nj,t+1 = exp(γUj,t) Zt , (2) Zt= m X j=1 exp(γUj,t), (3)
where γ is the intensity of choice parameter, which measures how fast produc-ers switch between the prediction strategies and Zt is a normalization factor
so that the fractions nj,t+1 add up to 1. With these updated fractions the
equilibrium price in the next period is determined and so on. This dynami-cal system is dynami-called the adaptively rational equilibrium dynamics (A.R.E.D.). The performance measure Uj for predictor Hj is often equal to the realized
net profits by forecasting strategy h in period t Uj,t = πj(pt, H(Pt))
= ptS(Hj(Pt)) − c(S(Hj(Pt))) − Cj,
(4) where c(·) is the quadratic cost function and Cj are information costs for
obtaining predictor Hj. For naive expectations these information cost will be
zero. For a more sophisticated forecasting rule, like the SAC-learning, these costs may be positive, because of information gathering or computation costs. Another performance measure is given by the negative squared forecasting error. This measure will be used in the cobweb model analyzed in this thesis.
2.2
Sample autocorrelation (SAC)-learning
The introduction section introduced a number of expectation rules and their effect on the price dynamics. In the search for a more consistent and realistic belief in the cobweb model, this section introduces a simple adaptive learning scheme, namely the sample autocorrelation (SAC)-learning. Under SAC-learning agents follow an univariate linear autoregressive AR(1) model to forecast the economy with price forecast
pt+1e = α + β(pt− α), (5)
where α is the long-run mean and β ∈ [−1, 1] is the first-order autocorrela-tion of the AR(1) process. When price dynamics are described under SAC-learning, the belief parameters of the dynamical system are time-varying and updated according to their sample mean and sample first-order autocorrela-tion. Within the adaptive learning processes, agents often use statistic tools
for estimation. However, the SAC-learning is a more simple statistical learn-ing procedure where agents use time series observations to estimate the belief parameters. For any finite set of observations p0, p1, ..., pt the sample mean
is αt= 1 t + 1 t X i=0 pi, t ≥ 1, (6)
and the first-order sample autocorrelation coefficient is βt = Pt−1 i=0(pi− αt)(pi+1− αt) Pt i=0(pi− αt)2 , t ≥ 1. (7)
For β > 0, agents believe that the next price will be above average if the last observed price is above average and vice versa. If β < 0, they believe the next price will be below average if the last observed price is above average and vice versa.
Hence, in each period agents update their belief parameters as new in-formation becomes available using times series observations. The sample mean and first-order autocorrelation can be roughly guesstimated without any knowledge of statistical techniques. Define the price variance
Rt = 1 t + 1 t X i=0 (pi− αt)2, (8)
such that the following recursive dynamical system is obtained αt = αt−1+t+11 (pt− αt−1), βt = βt−1+ R−1t t+1 (pt− αt−1) pt−1+ t+1p0 − t 2+3t+1 (t+1)2 αt−1−(t+1)1 2pt − t t+1βt−1(pt− αt−1) 2 , Rt = Rt−1+ t+11 t t+1(pt− αt−1) 2− R t−1 . (9)
This derivation can be found in Appendix A from Hommes and Zhu (2014). To simplify the analysis, the dynamical system will be restricted and the decreasing gain t+11 will be replaced by a constant gain parameter κ.
αt = αt−1+ κ(pt− αt−1), βt = βt−1+ κR−1t [(pt− αt−1)(pt−1− αt−1) − βt−1(pt− αt−1)2], Rt = Rt−1+ κ[(pt− αt−1)2− Rt−1], (10)
where κ is a small and positive number. During the analysis in Section 3 κ will be fixed and equal to 0.05. To measure the parameter sensitivity in Subsection 4.3 the value of κ will be varied.
When the belief parameters αt and βt are updated according to formulas
in (10), the law of motion under SAC-learning becomes
pt+1= Fαt,βt(pt) := F (αt+ βt(pt− αt)). (11)
2.3
Cobweb model with SAC-learning versus naive
ex-pectations
In this thesis the cobweb model is analyzed where agents can choose between the two predictors H1 and H2. H1 are expectations based on SAC-learning
and H2 are expectations based on naive beliefs. Subsection 2.2 gave a
descrip-tion of the SAC-learning forecast rule in more detail. The naive expectadescrip-tions are a more simple forecasting rule and are equal to the last observed price. The demand and supply functions in this thesis will be the same as the func-tions used in the learning-to-forecast experiments performed by Hommes, Sonnemans, Tuinstra and van de Velden (2007). The demand curve D(·) is a linear decreasing function and the supply curve S(·) is a nonlinear, S-shaped and increasing function
D(pt) = a − bpt, a, b > 0, (12)
S(pej,t) = tanh(λ(pej,t− 6)) + 1, λ > 0. (13) The parameter λ measures the nonlinearity of the supply curve. With this nonlinear supply function price dynamics will be bounded. The market equi-librium can be written as
D(pt) = n1,tS(H1(Pt)) + n2,tS(H2(Pt)), (14)
where the predictor functions are defined as
H1(Pt) = αt−1+ βt−1(pt−1− αt−1), (15)
H2(Pt) = pt−1.
Solving the market equilibrium equation, the equilibrium price will be pt=
a − n1,tS(H1(Pt)) − n2,tS(H2(Pt))
b . (16)
The equilibrium price ptis determined by the fractions n1,t and n2,t. This
evaluate the predictors, such that the new fractions n1,t+1 and n2,t+1 can be
calculated afterwards. Hence, to determine the fractions n1,t+1 and n2,t+1,
the performance measure needs to be defined. The performance measure is given by the negative squared prediction error
U1,t = −[(pt− H1(Pt))2+ C], (17)
U2,t = −(pt− pt−1)2, (18)
where C ≥ 0 are the information costs for the SAC-learning. For the naive expectation forecast rule the observations for the calculation of the price forecast are freely available, hence there are no information costs. For the SAC-learning forecast rule the information costs C are often positive. How-ever, for the price forecast under SAC-learning the observations are freely available as well. The difference is that for the calculation of the price fore-cast under SAC-learning you will need more information and observations and it takes more effort, and therefore more costs, to calculate this forecast. During the analysis in the following section the value of costs C will be var-ied. Furthermore, in Section 4 the behavior of the dynamical system with C = 0 is investigated.
Next, with these performance measures, the fractions can be written as n1,t+1=
exp(−γ[(pt− H1(Pt))2+ C])
exp(−γ[(pt− H1(Pt))2+ C]) + exp(−γ(pt− pt−1)2)
, (19)
n2,t+1= 1 − n1,t+1, (20)
where n1,t+1 is the fraction of agents choosing SAC-learning and n2,t+1 is the
fraction of agents using the naive expectations in period t + 1. As mentioned before, γ is the intensity of choice parameter and measures how fast agents switch between the predictions strategies. For γ = 0 the fractions will be fixed over time and equal to 12. For higher values of γ the fractions will not be fixed anymore and agents will choose more often their optimal forecast rule. For the extreme case γ = ∞, agents always choose their optimal forecast rule in terms of the performance measure.
3
Analysis
The cobweb model with SAC-learning versus naive expectations will be an-alyzed in this section using the software program E&F Chaos to simulate the model. The purpose of this analysis is to investigate the influence of the various parameters on the dynamics of the nonlinear cobweb model. First, Subsection 3.1 will determine the dynamical system used for the analysis. Subsection 3.2 shows how the dynamics behave if all agents either follow the SAC-learning rule or the naive expectations rule. Thereafter, the effect of the nonlinearity on the price dynamics and the influence of different costs on the fractions will be investigated in Subsection 3.3. A bifurcation diagram and further analysis of phase plots will be discussed in Subsection 3.4. In the last subsection the dynamical system will be simplified by taking a fixed value for α and β and a stability analysis will be performed.
3.1
Dynamical System
To analyze individual effects, some parameters in the demand and supply curve will be fixed. The stability parameter λ will take on different values, such that the effect of nonlinearity of the supply curve is measured. For λ ∈ (0.22, 0.5, 1, 2) and setting a = 2.3 and b = 0.25, the demand and supply curve are described by
D(pt) = 2.3 − 0.25pt. (21)
S(pej,t) = tanh(λ(pej,t− 6)) + 1, λ ∈ (0.22, 0.5, 1, 2). (22)
Figure 1 shows the demand and supply curve, where the supply curve is plotted for different values of λ. For larger values of λ the supply curve be-comes more steep. This steepness can be one of the features that cause more chaotic dynamics. Next to the nonlinearity parameter λ, other parameters may also effect the dynamics. In this section the effects of these parameters are investigated.
Combining the findings from Section 2, the following dynamical system is obtained pt = 2.3−n1,tS(H1(Pt))−n2,tS(H2(Pt)) 0.25 , n1,t+1 = exp(−γ[(pt −H1(Pt))2+C]) exp(−γ[(pt−H1(Pt))2+C])+exp(−γ(pt−pt−1)2), n2,t+1 = 1 − n1,t+1, αt = αt−1+ κ(pt− αt−1), βt = βt−1+ κR−1t [(pt− αt−1)(pt−1− αt−1) − βt−1(pt− αt−1)2], Rt = Rt−1+ κ[(pt− αt−1)2− Rt−1], (23)
where κ = 0.05 and the predictor functions are defined as H1(Pt) = αt−1+ βt−1(pt−1− αt−1),
H2(Pt) = pt−1.
During the analysis in this section, the system has initial states p0 = 2, α0 =
1, β0 = 0.5, R0 = 1, n1 = 0.5, n2 = 0.5 and parameter κ is set equal to 0.05.
Moreover, the parameters λ, C and γ will be adjusted to see the effect of these parameters on the dynamics.
3.2
SAC-learning and naive expectations
This subsection analyzes the cobweb model with homogeneous beliefs where all agents either follow the SAC-learning forecast rule or the naive expec-tations forecast rule. According to Hommes and Sorger (1998), the law of motion under SAC-learning in Formula (11) has three possible outcomes; a steady state, a 2-cycle or a chaotic attractor. For a steady state or 2-cycle, the belief parameters need to be predicted correctly. If the belief parameters αt and βt converge to constants and there is a strange attractor, the price
dy-namics become chaotic. In figure 2 the dydy-namics of prices pt, sample mean αt
and sample first-order autocorrelation βt are plotted for all agents following
the SAC-learning, meaning that n1 = 1. For these figures λ = 0.5, γ = 2 and
C = 1. Figure 2(a) shows that pt converges to the steady state with value
p∗ = 5.73, αt converges as well to the steady state value p∗ and therefore βt
2 3 4 5 6 7 8 9 0 50 100 150 200 pt t (a) pt 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0 50 100 150 200 αt t (b) αt 0.2 0.3 0.4 0.5 0.6 0.7 0 50 100 150 200 βt t (c) βt
Figure 2: Time series of prices pt, sample mean αt and sample first-order
autocorrelation βtfor n1 = 1, meaning that all agents follow the SAC-learning
forecasting rule.
As mentioned before, the price dynamics are bounded because of the nonlinear supply function. Because of this feature it is not possible to have unbounded price oscillations. Therefore, there are only two possible ways how the price dynamics can behave if all agents follow the naive expectation rule. Since the dynamical system reduces to a one-dimensional decreasing map pt= D−1(S(pt−1)), there is one unique steady state which is equal to the
intersection point of the demand and supply curve. For agents that make the correct prediction, the price will converge to this steady state p∗. Otherwise the dynamics will be converging to a stable 2-cycle. These dynamics will de-pend on the relative slope of the supply and demand curve at the steady state price p∗. For |S0(p∗)/D0(p∗)| < 1, the steady state is stable and the model will converge to the steady state. For |S0(p∗)/D0(p∗)| > 1, the steady state is unstable and prices converge to the stable 2-cycle. Therefore the stability depends on the nonlinearity parameter λ. Figure 3 shows the price dynamics
2 3 4 5 6 7 8 0 50 100 150 200 pt t (a) λ = 0.22 2 3 4 5 6 7 8 9 100 110 120 130 140 150 pt t (b) λ = 0.5
Figure 3: Time series of prices pt for λ = 0.22 and λ = 0.5. All agents follow
the naive expectation forecasting rule, meaning that n2 = 1.
naive expectation rule (i.e. n2 = 1). For the stable case when λ = 0.22 the
price converges to a stable steady state p∗ = 5.57, since |S0(p∗)/D0(p∗)| < 1. For higher values of λ, the steady state becomes unstable and the price con-verges to a 2-cycle with values (p∗1, p∗2) = (1.4, 9.2), as illustrated in Figure 3(b).
3.3
Price dynamics and fractions
Returning to the cobweb model with heterogeneous expectations, for different values of λ ∈ (0.22, 0.5, 1, 2), the price dynamics pt, sample mean αt and
sample first-order autocorrelation βtare plotted in Figures 4 to 7. Parameters
C and γ are fixed at C = 1 and γ = 2.
• For λ = 0.22, the price dynamics converge to a steady state with value p∗ = 5.57, meaning these are stable dynamics. Sample mean αt converges to the steady state p∗ = 5.57 as well. For price dynamics
converging to a constant, the first-order autocorrelation parameter βt
converges to an (arbitrary) constant, as visible in the figure.
• For λ = 0.5, the steady state p∗ becomes unstable and the price
dy-namics converge to a 2-cycle, with (p1∗, p2∗) = (5.4, 6.1). The average
of these two prices is equal to 5.75 and it is shown that αtconverges to
a value in the neighborhood of this average. βtconverges to a negative
value of -0.95, meaning an observed price above the steady state price is followed by a forecast below the steady state price. This makes sense since the price dynamics converge to a stable 2-cycle.
• For λ = 1 there are chaotic price dynamics. There is no clear conver-gence of αtand βtconverges to fluctuating values around −0.2, showing
weak negative autocorrelation.
• For λ = 2 the price dynamics converge to a 4-cycle (p1∗, p2∗, p3∗, p4∗) =
(1.1, 4.1, 8.0, 8.8) where the prices are going upwards before it returns to its original value. These prices have an average value of 5.5 and αt
converges to values around this average. βtconverges to values around
−0.1, meaning that there is almost no autocorrelation.
The fractions n1 and n2 depend on the performance measure that is given
by the negative squared prediction error. If the squared forecasting error (pt−
pt−1) from the naive expectations rule is smaller than the squared forecasting
error plus the costs (pt−H1)+C from the SAC-learning rule, most agents will
2 3 4 5 6 7 8 0 50 100 150 200 pt t (a) pt 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0 50 100 150 200 αt t (b) αt 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 50 100 150 200 250 300 βt t (c) βt
Figure 4: Time series of pt, αt and βt for λ = 0.22, C = 1 and γ = 2
2 3 4 5 6 7 8 9 0 50 100 150 200 pt t (a) pt 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0 50 100 150 200 αt t (b) αt -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0 50 100 150 200 250 300 βt t (c) βt
Figure 5: Time series of pt, αt and βt for λ = 0.5, C = 1 and γ = 2
2 3 4 5 6 7 8 9 0 50 100 150 200 pt t (a) pt 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0 50 100 150 200 αt t (b) αt -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 50 100 150 200 250 300 βt t (c) βt
Figure 6: Time series of pt, αt and βt for λ = 1, C = 1 and γ = 2
1 2 3 4 5 6 7 8 9 0 50 100 150 200 pt t (a) pt 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0 50 100 150 200 αt t (b) αt -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 50 100 150 200 250 300 βt t (c) βt
for the naive expectation rule becomes larger, agents will switch and prefer the SAC-learning rule. This means the costs C also has an influence on the switching of agents. These information costs are often positive, even though the observations needed for the forecast of the price under SAC-learning are freely available. Since more effort is needed to obtain this information and to compute the SAC-learning forecast, the information costs C can be positive. Therefore, in order to find out how agents switch between different expectations rules when costs are different, the time series of fraction n1,
where n1 is the fraction of agents choosing SAC-learning, are shown in Figure
8 for C ∈ [0, 1], λ = 0.5 and γ = 2. With C = 0, the convergence is rather
0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 n1,t t (a) C = 0 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 n1,t t (b) C = 1
Figure 8: Time series of fraction n1,t for λ = 0.5, γ = 2 and C = 0 or C = 1
quickly and the fraction n1 converges to 0.5, meaning that half of the agents
prefer the SAC-learning rule. When C = 1, the amount of agents preferring SAC-learning becomes smaller such that n1 converges to 0.25 and the fraction
of agents that follow naive expectations becomes larger. Moreover, for C = 1 the convergence is slower.
3.4
Bifurcations and phase plots
In the previous section time series of price dynamics with different period cycles were plotted. Another tool to find out how price dynamics behave when the value of a single parameter is changed, is the bifurcation diagram. In a bifurcation diagram the long term behavior of the dynamics are shown and the occurrence of different bifurcations is illustrated. This is shown in Figure 9(a) where C = 1, γ = 2 and the nonlinearity parameter λ is adjusted. As the value of λ varies, qualitative changes in the dynamics occur. This is called a bifurcation. The following bifurcations can be found in the diagram:
• For λ ∈ [0 − 0.32] prices converge to a stable steady state. • For λ = 0.32 a period-doubling bifurcation occurs.
1 2 3 4 5 6 7 8 9 0 0.5 1 1.5 2 pλ λ
(a) Bifurcation Diagram
0 0.1 0.2 0.3 0.4 0 0.5 1 1.5 2 LE λ
(b) Largest Lyapunov Exponent
Figure 9: Bifurcation diagram and Largest Lyapunov Exponent for λ ∈ [0, 2]
• For λ ∈ [0.56 − 1.30] there are chaotic price-dynamics. • For λ = 1.30 there is a period-doubling bifurcation.
In a period-doubling bifurcation, a stable steady state or 2-cycle becomes unstable and a stable 2-cycle or 4-cycle is created. The chaotic dynamics will be investigated further later in this section by creating phase plots to find out whether strange attractors occur.
To investigate whether these dynamics indeed behave chaotic for some values of λ, the Largest Lyapunov Exponent as a function of the nonlinear-ity parameter is plotted in Figure 9(b). The Largest Lyapunov Exponent measures the average exponential rate of divergence of nearby initial states. When there are positive values of the Largest Lyapunov Exponent, the dy-namics are chaotic. It is shown that in the Largest Lyapunov Exponent, there is a range of positive values for λ ∈ [0.56, 1.30], which provides numer-ical evidence that there are chaotic dynamics for these values of λ. This is the same result as what was found in the bifurcation diagram. Furthermore, when the Largest Lyapunov Exponent is equal to 0, bifurcations will occur. This happens for λ = 0.33, λ = 0.56 and λ = 1.3, meaning that period-doubling bifurcations will occur from a stable steady-state to a stable 2-cycle and from chaotic dynamics to a stable 4-cycle.
Additionally, in a parameter basin plot, two parameters are plotted and different colors are assigned to stable cycles of different periods. With this plot, the nonlinear dynamical system is analyzed in more detail. For this dynamical system the parameters λ and γ and λ and C are plotted in Figure 10. In the plots the light blue area corresponds to dynamics with a stable steady state. For the dark blue points there is a period 2-cycle and the yellow area belongs to a period 4-cycle. There is no convergence of the dynamics in
0 1 2 3 4 5 0 0.5 1 1.5 2 γ λ (a) λ and γ 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 C λ (b) λ and C
Figure 10: Parameter Basins for Periodic Cycles
the white area.
Last, phase plots plot attractors that appear as chaotic dynamics arise. A stable steady state is present when a single point attractor is plotted. When a finite number of points is plotted the attractor is called a stable periodic orbit. For a closed curved orbit, there is an invariant circle and a strange chaotic attractor occurs when there is a fractal structure in the plot.
For λ ∈ [0.22, 0.5, 2] the phase plots of p and n1 in Figure 11 show clear
points. Since there is only one point in Figure 11(a), it can be concluded that for λ = 0.22 there is a stable steady state. For λ = 0.5 there are 2 points in the figure, meaning that there is a stable 2-cycle. Last, for λ = 2 there is a 4-cycle because of the four plotted points.
0.05 0.1 0.15 0.2 5.5 5.55 5.6 5.65 nt p (a) λ = 0.22 0.2 0.25 0.3 0.35 5.4 5.5 5.6 5.7 5.8 5.9 6 nt p (b) λ = 0.5 0 0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8 nt p (c) λ = 2
Figure 11: Phase plots of p and n1 for λ ∈ [0.22, 0.5, 2], C = 1 and γ = 2
For λ ∈ [0.56 − 1.30] the chaotic dynamics are further investigated by computing phase plots of p and n1 for different values of λ in Figure 12.
In Figure 12(a) the orbit converges to an attractor consisting of two closed curves (around an unstable 2-cycle). As λ becomes higher, the closed curves break and converges to a strange attractor which is visible in Figures 12(b)
0 0.1 0.2 0.3 0.4 0.5 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 6.1 6.2 n1 p (a) λ = 0.576 0.1 0.2 0.3 0.4 0.5 0.6 0.7 5.2 5.4 5.6 5.8 6 6.2 6.4 n1 p (b) λ = 0.6116 0 0.2 0.4 0.6 0.8 1 3 4 5 6 7 8 n1 p (c) λ = 0.665 0 0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8 n1 p (d) λ = 0.7362 0 0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8 9 n1 p (e) λ = 0.9142 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 n1 p (f) λ = 1.1812
Figure 12: Phase plots of p and n1 for λ ∈ [0.56 − 1.30], C = 1 and γ = 2.
to 12(e). In 12(e) and 12(f) it is visible that agents switch quickly between the fractions. There is a horizontal line for n1 = 1 for low p and another
horizontal line for n1 = 0 for higher p. In between there are only a few
points, therefore agents make a quick switch between the naive expectation and SAC-learning forecasting rule.
3.5
Varying the intensity of choice parameter
In order to find out how agents switch between different expectations rules when the intensity of choice parameter varies, the time series of fraction n1
are shown in Figure 13 for γ ∈ [0, 0.15, 0.4, 0.8], λ = 2 and C = 1.
For γ = 0, the fraction n1,t has a fixed value and is always equal to 0.5,
meaning that half of the agents choose the naive expectations forecast rule and the other half chooses SAC-learning as forecasting rule. For γ = 0.15, the dynamics become messier and n1,t fluctuates around 0.5. For γ = 0.4, the
dynamics are chaotic and there is a lot of switching between the fractions. For γ = 0.8 the fraction converges to a 4-cycle. To investigate these dynamics further, phase plots for different values of γ are shown in Figure 14.
For γ = 0.06 there is one point plotted, which corresponds to a stable steady state. Figure 14(b), 14(c), 14(d) and 14(e) show clear strange attrac-tors in the plot causing the dynamics to be chaotic. For γ higher than 0.8, four points are plotted, corresponding to the 4-cycle.
0.6 0.8 1 1.2 1.4 0 50 100 150 200 250 300 n1 t (a) γ = 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 300 n1 t (b) γ = 0.15 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 n1 t (c) γ = 0.4 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 n1 t (d) γ = 0.8
Figure 13: Time series of fraction n1 for different values of the intensity of
choice parameter γ. Other parameters are set at: λ = 2 and C = 1.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 5.85 5.9 5.95 6 nt p (a) γ = 0.06 0.2 0.3 0.4 0.5 0.6 0.7 0.8 5 5.5 6 6.5 7 n1 p (b) γ = 0.18 0 0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8 9 n1 p (c) γ = 0.24 0 0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8 9 n1 p (d) γ = 0.3 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 n1 p (e) γ = 0.48 0 0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8 nt p (f) γ = 0.84
Figure 14: Phase plots of p and n1 for different values of γ. Other parameters
3.6
Stability analysis of cobweb model with naive
ex-pectations and autoregressive(1) rule
In this subsection the general form of the dynamical system (23) will be simplified by taking fixed values for αt = α and βt = β. In this way, the
SAC-learning rule is replaced by the linear autoregressive AR(1) forecasting rule such that H1 = α + β(pt−1− α). This subsection describes a stability
analysis of the cobweb model with AR(1) rule versus naive expectations. When the demand curve is decreasing and the supply curve is increasing, there is a unique price where demand and supply intersect. This price p = p∗ is a steady state of the dynamical system and will be investigated. α is set equal to this intersection price, such that α = p∗. β and γ will be fixed parameters. For β fixed and α = p∗, the predictor functions are defined as
H1(pt−1) = p∗+ β(pt−1− p∗),
H2(pt−1) = pt−1,
and the following dynamical system is obtained (
pt = a−n1,tS(H1(pt−1))−(1−nb 1,t)S(pt−1),
n1,t+1 =
exp(−γ[(pt−[p∗+β(pt−1−p∗)])2+C])
exp(−γ[(pt−[p∗+β(pt−1−p∗)])2+C])+exp(−γ(pt−pt−1)2).
Setting pt= F (n1,t, pt−1), the dynamical system becomes
(
pt = a−n1,tS(H1(pt−1))−(1−nb 1,t)S(pt−1),
n1,t+1 =
exp(−γ[(F (n1,t,pt−1)−H1(pt−1))2+C])
exp(−γ[(F (n1,t,pt−1)−H1(pt−1))2+C])+exp(−γ(F (n1,t,pt−1)−pt−1)2).
To investigate the stability of the steady state p∗, the Jacobian in steady state p∗ needs to be determined. This derivation can be found in Appendix A. J (p∗, n1) = (n1(1 − β) − 1) S0(p∗) b 0 ∂n1,t+1 ∂pt−1 0 !
There is no need to derive ∂n1,t+1
∂pt−1 , since this term will be multiplied by 0 in the
derivation of the characteristic equation. The determinant of the Jacobian minus the identity matrix I2 is calculated as follows
Det(J (p∗, n1) − µI2) = Det
(n1(1 − β) − 1)S 0(p∗) b − µ 0 ∂n1,t+1 ∂pt−1 −µ ! = 0,
such that the characteristic equation that determines the eigenvalues at the steady state equals
((n1(1 − β) − 1)
S0(p∗)
b − µ)(−µ) − 0 = 0.
Solving this equation gives eigenvalues µ1 = 0 and µ2 = (n1(1 − β) − 1)S
0(p∗)
b .
If both eigenvalues are |µ| < 1, the steady state p∗ is stable. Since µ1 =
0 < 1, the stability of the model depends on the parameters that determine the second eigenvalue µ2. For the cobweb model with AR(1) rule versus
naive expectations when all agents are naive (i.e. n1 = 0), this means that
the second eigenvalue will be equal to µ2 = (0(1 − β) − 1) S0(p∗)
b = − S0(p∗)
b .
Therefore, the stability condition |µ2| = |−S
0(p∗)
b | < 1 depends on the relative
slope of the supply and demand curve at the steady state price p∗. When all agents follow the AR(1) rule (i.e. n1 = 1), the second eigenvalues becomes
µ2 = (1(1 − β) − 1)S
0(p∗)
b = − βS0(p∗)
b . Thus, next to the relative slope of the
demand and supply curve, the stability condition depends on the parameter β as well.
4
Experiment
The second goal of this thesis is to try to calibrate the cobweb model with SAC-learning versus naive expectations to the laboratory experiments con-ducted by Hommes, Sonnemans, Tuinstra and van de Velden (2007). They performed learning-to-forecast experiments in the classical cobweb model and examined whether agents learned the unique rational expectations (RE) steady state. Among economists, it is assumed that learning this rational expectations equilibrium requires more knowledge of the agents than what is ought to be realistic. Therefore, more theoretical studies with boundedly rational agents were investigated. Boundedly rational agents optimize their parameters by a statistical model as more information becomes available. A number of studies where agents follow ordinary least squares or genetic al-gorithm learning show convergence to the RE steady state. However, other studies in forecasting rules, such as the heterogeneous expectations cobweb model investigated by Brock and Hommes (1997), show no convergence to the RE steady state. This contradiction in theoretical studies was a reason for Hommes, Sonnemans, Tuinstra and van de Velden (2007) to perform these learning-to-forecast experiments. Moreover, there are not many empirical studies on the expectation formation hypothesis. Therefore, to investigate which forecasting rule is the most accurate in describing human forecast-ing behavior, they set up a controlled laboratory experiment with human subjects.
In Subsection 4.1, the setup of the experiments performed by Hommes, Sonnemans, Tuinstra and van de Velden (2007) will be explained. Further-more, the stylized facts observed from the results of the experiment will be illustrated. Subsection 4.2 will show the figures from the simulation of the cobweb model with SAC-learning versus naive expectation. Additionally, it will investigate whether this cobweb model is able to explain the stylized facts. In the last subsection the sensitivity of the parameters will be investi-gated by varying parameters and study the effect of these parameters on the dynamical behavior.
4.1
Setup and results experiment
The cobweb model has a unique rational expectations equilibrium, therefore the experiments are based on the cobweb framework. During the experi-ments, subjects are asked to predict the price of a product repeatedly for 50 periods. The subjects know the price is determined by market clearing and that their earnings are inversely related to their prediction error. This means that they receive higher earnings when their prediction is better. However,
they do not know the market equilibrium equations. The subjects earnings are defined as a quadratic function of squared forecasting errors
πi,t = max[1300 − 260(pt− pei,t) 2, 0],
where pei,t is subject i’s prediction of the market price in period t, for 1 ≤ t ≤ 50, and where pt is the realized market price in period t. The realized
market price pt is determined by solving the market equilibrium equation,
where demand and supply are set equal. The demand function is given by D(pt) = 2.3 − 0.25pt+ ηt,
where ηtis a small stochastic shock in period t drawn from a normal
distribu-tion. This small shock is added to make the demand function more realistic. Therefore, it becomes more difficult for the subjects to learn the RE steady state. The supply function is given by
S(pei,t) = tanh(λ(pei,t− 6)) + 1, λ > 0.
The parameter λ measures the nonlinearity of the supply curve and the sta-bility of the underlying cobweb model. Given the demand function, supply function and the individual forecasts of the market price, the realized equi-librium price is obtained as
pt = a −PK i=1S(p e i,t) b + t,
where t = ηbt ∼ N (0, 0.5) and K = 6 is the number of participants. Given
the parameter λ, the aggregate realized price pt depends on individual price
expectations of K participants as well as the realization of the stochastic shocks. Hommes, Sonnemans, Tuinstra and van de Velden (2007) conducted three different treatments, a stable, an unstable and a strongly unstable treatment. Setting λ = 0.22 for the stable treatment, λ = 0.5 for the unstable treatment and λ = 2 for the strongly unstable treatment.
Hommes, Sonnemans, Tuinstra and van de Velden (2007) compared the results from their experiment to the RE benchmark results to investigate whether subjects are able to learn the RE steady state or whether excess price volatility is present. When all agents follow the RE forecasting rule, it is assumed that agents have perfect knowledge about the market equilibrium equations. The price will then be equal to the steady state, which equals the intersection of the demand and supply curve, pe
t = p
∗. Given that all agents
Stability parameter RE Mean RE Variance Experiment Mean Experiment Variance Stable treatment 5.57 0.25 5.64 0.36 Unstable treatment 5.73 0.25 5.85 0.63
Strongly unstable treatment 5.91 0.25 5.93 2.62
Table 1: Mean and variance under RE and sample mean and sample variance of the realized market prices for the stable, the unstable and the strongly unstable treatment over the full sample of 50 periods.
The mean and variance of the realized market prices under RE are given in Table 1. For different values of λ, the mean of prices under RE is different. Furthermore, Table 1 shows the sample mean and sample variance of the realized market prices in the experiment for the three different treatments. By measuring the sample autocorrelation, they found a simple way to test for linear predictability of the realized market prices. See Appendix B for the time series of the realized market prices and autocorrelation plots of the six groups for the different treatments within the learning-forecast experiment. By comparing the sample mean, sample variance and sample autocorrelation of the realized market prices in the experiment with the RE characteristics, three stylized facts were observed.
• The sample mean of realized prices is very close to the RE benchmark in all three treatments.
• The sample variance of realized prices depends on the treatment. For the stable treatment it is close to the theoretical variance of the RE benchmark. The unstable and strongly unstable treatment exhibit ex-cess volatility in prices where the variance is significantly higher than the RE benchmark variance.
• For all treatments there is no significant linear autocorrelation between realized market prices.
It appears that for the stable treatment the RE forecasting rule is a good description of the observed aggregate price behavior of the subjects. However, for the unstable treatments, excess price volatility is present and therefore the RE forecasting rule is a less reasonable description.
Hommes (2013) stated that homogeneous expectations models are un-able to explain these facts and therefore suggests heterogeneous expectation models for the explanation of the three stylized facts. Hommes and Lux
(2013) were able to explain all stylized facts by using a model of heteroge-neous individual learning through genetic algorithms. In the next subsection, the three stylized facts are tried to be explained by the cobweb model with SAC-learning versus naive expectations.
4.2
Simulation
To explain the stylized facts, the simulation in this subsection has the same characteristics as in the experiment of the previous section. The dynamical system used for the analysis in Section 3 is adjusted by adding noise to the demand function, where the noise follows the Normal distribution N (0, 0.5), as in the experiments. During the simulation in this section, the system has initial states p0 = 5, α = 6, β = 0.5, R0 = 1, n1 = 0.5 and n2 = 0.5. For the
explanation of the first and second stylized fact, Figure 15 shows the time series of the prices for t = 50 for the different treatments. Setting λ = 0.22 for the stable treatment, λ = 0.5 for the unstable treatment and λ = 2 for the strongly unstable treatment. The other parameters are fixed at: κ = 0.05, γ = 2 and C = 1. 0 2 4 6 8 10 0 10 20 30 40 50 pt t
(a) Stable treatment
0 2 4 6 8 10 0 10 20 30 40 50 pt t (b) Unstable treatment 2 4 6 8 10 0 10 20 30 40 50 pt t (c) Strongly Unstable Treatment
Figure 15: Time series of realized prices pt for the stable, the unstable and
the strongly unstable treatment over the simulation of 50 periods with initial price p0 = 5. Other parameters are fixed at: κ = 0.05, γ = 2 and C = 1.
For all treatments the price dynamics move around the value of the RE mean. The amplitude of the price fluctuations for the strongly unstable treatment is larger than the amplitude of the stable and unstable treatment. Moreover, there is a high price volatility for the strongly unstable treatment, suggesting excess price volatility. These figures are similar to the time series of realized prices for all treatments in the experiment. However, the ampli-tude for the stable and unstable treatment is larger than the ampliampli-tude for these treatments in the experiment.
Stability parameter Sample Mean Sample Variance
Stable treatment 5.57 0.74
Unstable treatment 5.92 4.99
Strongly unstable treatment 6.13 12.70
Table 2: Sample mean and sample variance of the cobweb model with SAC-learning versus naive expectations for the stable, the unstable and the strongly unstable treatment over the simulation of 50 periods.
Table 2 shows the sample mean and sample variance of the simulation of the cobweb model with SAC-learning versus naive expectations over 50 peri-ods. The same as with the sample mean of the realized market prices in the experiment, the sample mean of the simulation is in the neighborhood of the RE mean for all treatments. Furthermore, the sample variance of the simu-lation is higher than the variance of the experiments in the stable treatment. Nevertheless, it gives a better description than the RE variance, since this value is still in the neighborhood of the variance of the experiment. Finally, for the unstable and strongly unstable treatment, excess price volatility is present because of the high variance. Compared to the sample variance of the experiment, the sample variance of the simulation is much higher.
For the explanation of the third stylized fact, no significant autocorre-lation for all treatments, the first-order autocorreautocorre-lation is plotted in Figure 16. For the stable treatment, the dynamics are reasonable stable.
There--0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 10 20 30 40 50 βt t
(a) Stable Treatment
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 10 20 30 40 50 βt t (b) Unstable Treatment -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 10 20 30 40 50 βt t (c) Strongly Unstable Treatment
Figure 16: Time series of the first-order autocorrelation βt for the stable,
the unstable and the strongly unstable treatment over the simulation of 50 periods with initial price p0 = 5. Other parameters are fixed at: κ = 0.05,
γ = 2 and C = 1.
fore, in Figure 16, the first-order autocorrelation converges to an (arbi-trary) constant. Moreover, Figure 16 shows dynamics that converge to −0.4 for the unstable treatment and values around −0.2 for the strongly
unstable treatment. For the higher-order autocorrelation, Figure 17 plots the autocorrelation function of order twenty with significance bandwidth of 5%. These bandwidths are an approximation determined by the formula ±2/p(N) = ±2/p(50) = ±0.2828.
(a) Stable Treatment (b) Unstable Treatment (c) Strongly Unstable
Treatment
Figure 17: Autocorrelation plots with significance bandwidth of 5% for the stable, the unstable and the strongly unstable treatment over the simulation of 50 periods with initial price p0 = 5. Other parameters are fixed at: κ =
0.05, γ = 2 and C = 1.
The autocorrelation function in the stable and unstable treatment shows significant autocorrelation for the first lags. The strongly unstable treatment has significant autocorrelation for a large number of lags. Furthermore, there seems to be a noisy period 3-cycle for the strongly unstable treatment, since there are repeatedly two negative lags followed by one positive lag.
In this subsection, a simulation of the cobweb model with SAC-learning versus naive expectations is performed to investigate the ability of this cob-web model to explain the stylized facts observed in the experiment. From Figure 15 and Table 2 it can be concluded that the cobweb model with SAC-learning versus naive expectations is able to explain the first stylized fact. The second stylized fact is partly explained by this cobweb model since the variance of the simulation is much higher than the variance of the results of the experiment for the unstable and strongly unstable treatment. How-ever, it gives a better description than the RE variance. The third stylized fact is more difficult to explain. The autocorrelation function of twenty lags in Figure 17 shows strong significant lineair autocorrelation in the first lags for the stable and unstable treatment. For the strongly unstable treatment, Figure 17 shows significant linear autocorrelation for a large number of lags. Even though for the stable and unstable treatment the majority of lags show no significant linear autocorrelation, the significant autocorrelation present in the first lags is strong. Therefore, the cobweb model with SAC-learning and naive expectations is unable to explain the third stylized fact for all treatments.
4.3
Parameter sensitivity
It appears that the cobweb model with SAC-learning and naive expectations shows a large number of significant autocorrelation for the strongly unstable treatment. By adjusting fixed parameters κ and C, the dynamical behavior of the cobweb model is investigated in this subsection. This will test if the number of significant lags in the strongly unstable treatment can be lowered and if the third stylized fact can be explained by the cobweb model with SAC-learning versus naive expectations.
First, parameter κ is adjusted. During the analysis in Section 3 and the simulation in this Section, κ is set fixed at a value equal to 0.05. By letting κ be a fixed value instead of the time-varying parameter, the most recent prices are weighted more in the calculation of the price. Lowering this value of κ increases the weight of the most recent prices even more. Since human subjects probably take the most recent prices into account to forecast the next periods price, the outcome will probably be more realistic and comparable to the results of the experiment.
5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 6.1 6.2 0 20 40 60 80 100 120 140 αt t (a) κ = 0.05 5.5 5.6 5.7 5.8 5.9 6 6.1 0 20 40 60 80 100 120 140 αt t (b) κ = 0.03 5.5 5.6 5.7 5.8 5.9 6 6.1 0 20 40 60 80 100 120 140 αt t (c) κ = 0.02 5.5 5.6 5.7 5.8 5.9 6 0 20 40 60 80 100 120 140 αt t (d) κ = 0.01 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 20 40 60 80 100 120 140 βt t (e) κ = 0.05 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 20 40 60 80 100 120 140 βt t (f) κ = 0.03 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 20 40 60 80 100 120 140 βt t (g) κ = 0.02 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 20 40 60 80 100 120 140 βt t (h) κ = 0.01
Figure 18: Time series of sample mean αtand first-order sample
autocorrela-tion βt for different values of κ. Other parameters are fixed at: λ = 2, γ = 2
and C = 1
For κ ∈ [0.1, 0.2, 0.3, 0.5], sample mean α and the first-order autocorrela-tion β are plotted in Figure 18. For a lower value of κ, the time series of α and β are less volatile and β seems to converge to a value in the neighbor-hood of −0.1. The autocorrelation plots for the different values of κ in the strongly unstable treatment are shown in Figure 19.
(a) Strongly Unstable Treatment and κ = 0.03 (b) Strongly Unstable Treatment and κ = 0.02 (c) Strongly Unstable Treatment and κ = 0.01
Figure 19: Autocorrelation plots with significance bandwidths of 5% for the strongly unstable treatment over the full sample of 50 periods. Other pa-rameters are fixed at: λ = 2, γ = 2 and C = 1
For a lower value of κ, the number of lags with significant autocorrelation becomes less. When κ = 0.03, there is significant linear autocorrelation for 10 lags. When κ = 0.02, 7 lags show significant linear autocorrelation and for κ = 0.01, only 5 lags show significant linear autocorrelation. Therefore, when κ = 0.01, the number of significant linear autocorrelation lags is the lowest. However, the third stylized fact for the strongly unstable treatment can still not be fully explained by the cobweb model with SAC-learning and naive expectations.
Next, costs parameter C is investigated. As mentioned before, informa-tion costs C under SAC-learning are often positive, while the observainforma-tions are freely available. For the experiment, it can be more realistic to investi-gate the dynamical system with C = 0. Figure 20 plots the time series of the price pt, mean αt and first-order autocorrelation βt for C = 0, λ = 2, γ = 2
and κ = 0.05.
Figure 20 shows price dynamics ptfluctuating around the value of the RE
2 4 6 8 10 0 10 20 30 40 50 pt t (a) pt 5.4 5.5 5.6 5.7 5.8 5.9 6 6.1 0 10 20 30 40 50 αt t (b) αt -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 10 20 30 40 50 βt t (c) βt
Figure 20: Price dynamics pt with initial value p0 = 5, mean αt and
first-order autocorrelation βtfor the strongly unstable treatment for C = 0. Other
mean. Sample mean αt converges to the value of the RE mean and sample
first-order autocorrelation βt converges to a value around −0.1. Figure 21
plots the autocorrelation function for the strongly unstable treatment with C = 0. Only five lags show significant linear autocorrelation with C = 0,
Figure 21: Autocorrelation function for the strongly unstable treatment for C = 0. Other parameters are fixed at: λ = 2, γ = 2 and κ = 0.05.
which is a smaller number of lags with significant autocorrelation than with C = 1. Furthermore, there still seems to be a noisy period 3-cycle. Therefore, the third stylized fact is better explained if the costs parameter C equals zero. Additionally, for the costs parameter, a numerical analysis of the second eigenvalue calculated in the stability analysis in Subsection 3.6 is performed. From this analysis it follows that the model is stable if the second eigenvalue |µ2| < 1. For the strongly unstable treatment, the second eigenvalue will be
calculated in Appendix C with C = 0 and C = 1 to measure its effect on the stability of the model. The second eigenvalue µ2 = −1.93657 for C = 0
and µ2 = −6.3612 for C = 1. With both C = 0 and C = 1, the stability
condition of |µ2| < 1 is not satisfied, meaning that the steady state of this
simplified cobweb model is unstable. Moreover, µ2 has a lower negative value
for C = 1, such that the steady state is less unstable for C = 0. This was also visible in Figure 8 in Subsection 3.3, where the fraction n1 converges
quicker for C = 0, since C = 0 is more stable than C = 1, .
By adjusting the values of parameters κ and C, the number of significant lags with autocorrelation reduces. However, there is still strong negative autocorrelation present in the lags and therefore the cobweb model with SAC-learning versus naive expectations is unable to explain the third stylized fact. It appears that this cobweb model is too simple to explain the results from the experiment with the human subjects. However, to give the best fit to the results of the experiment, the cobweb model with SAC-learning versus naive expectations should fix the parameters at κ = 0.01 and C = 0, since for these values the number of significant lags with autocorrelation is the lowest.
5
Conclusion
The cobweb model is a simple framework used to illustrate the role and im-portance of the expectation feedback system. The cobweb model with het-erogeneous agents describes price fluctuations in a market for a non-storable good that takes one period to produce for agents with different beliefs. Brock and Hommes (1997) analyzed the cobweb model with rational versus naive expectations. The naive expectations are a free and simple forecasting rule, equal to the last observed price. The assumption of rational agents is that they have perfect knowledge about the market conditions and they believe all agents have this. However, in a complex nonlinear world this assumption does not seem realistic, since market equilibrium conditions are unknown.
In this thesis, the rational expectations forecast rule is replaced by the sample autocorrelation (SAC)-learning forecast rule. When price dynamics are described under SAC-learning, agents follow an univariate linear autore-gressive AR(1) process to forecast the economy. The belief parameters of the dynamical system are time-varying and updated according to their sam-ple average and samsam-ple first-order autocorrelation. The SAC-learning process has the advantage that the sample average and first-order autocorrelation can be roughly guesstimated without any knowledge of statistical techniques.
5.1
Stability analysis
The first goal of this thesis was to perform a stability analysis of the nonlin-ear cobweb model where agents can either follow the sophisticated forecast rule, the costly SAC-learning or freely obtain the simple nave expectations. This analysis is provided in order to find out whether this heterogeneous expectation model is a good description for modeling a complex nonlinear world.
For this stability analysis, the cobweb model with heterogeneous expec-tations developed by Brock and Hommes (1997) is used as a framework. The dynamics depend on both fixed and varying parameters. By varying parame-ters, its effect on the dynamical behavior of a nonlinear model is investigated. Time series of prices, bifurcations diagrams and a Lyapunov exponent are computed to analyze the dynamics. Furthermore, the occurrence of strange, chaotic attractors is investigated.
First, the parameter λ was analyzed. The parameter λ measures the nonlinearity of the supply curve and the stability of the underlying cobweb model. Time series of prices for different values of the nonlinearity parameter λ were computed. The price dynamics showed a steady state, a 2-cycle, a 4-cycle and chaotic dynamics. The bifurcation diagram confirmed these
dynamics. With phase plots, the chaotic dynamics were further investigated and strange attractors occurred. The first-order autocorrelation βt was also
plotted for the different values of λ. For price dynamics with a steady state, a 2-cycle and chaotic dynamics, βtconverges respectively to a random constant
value, a value in the neighborhood of −1 and to chaotic dynamics. However, for the price dynamics converging to a 4-cycle, the first-order autocorrelation converges to a value in the neighborhood of 0, meaning that there is almost no autocorrelation.
Thereafter, parameter C measuring the information costs for the SAC-learning is adjusted. These costs are often positive, even though the ob-servations needed for the forecast of the price under SAC-learning are freely available. Since more effort is needed to obtain this information, the informa-tion costs C can be positive. Therefore, C is varied and analyzed for C = 0 and C = 1. With C = 0, the fraction of agents using the SAC-learning quickly converges to 12, meaning that it is equal to the fraction of agents following the naive expectation rule. Setting C = 1, the convergence is more slowly and the fraction of agents using SAC-learning becomes smaller.
Last, in order to find out how agents switch between different expectations rules, the intensity of choice parameter γ is varied. With γ = 0, the fraction n1 = 12, meaning that the fraction of agents following SAC-learning is equal
to the fraction of agents following the naive expectations rule. Agents switch more between the forecasting rules as the value of γ increases and a strange attractor occurs. As from γ = 0.8 the switching of the agents shows a clear pattern as the fraction n1 converges to a 4-cycle.
From the stability analysis it followed that the SAC-learning is a consis-tent forecast rule, since the first-order autocorrelation converges to a value in the neighborhood of −0.1. Furthermore, the dynamical behavior of the cobweb model with SAC-learning versus naive expectations depends on the nonlinearity parameter λ, intensity of choice parameter γ and costs parame-ter C in such a way that there can be both periodic as chaotic dynamics.
5.2
Learning-to-forecast experiments
For the second goal of this thesis, the learning-to-forecast experiments per-formed by Hommes, Sonnemans, Tuinstra and van de Velden (2007) were investigated. They performed experiments on human subjects to test the expectation hypothesis by considering three different treatments: a stable, an unstable and a strongly unstable treatment. Comparing the results of the experiment with the rational expectations (RE) characteristics they observed three stylized facts of aggregate price fluctuations.
sample mean of realized prices was very close the RE benchmark. For the stable treatment, the sample variance was close to the theoretical variance of the RE benchmark. However, the unstable and strongly unstable treat-ment exhibit excess volatility. For all treattreat-ments there was no significant linear autocorrelation between realized market prices. It appears that the RE equilibrium is only a good description of the experiments results for the stable treatment. The results from the unstable and the strongly unstable treatment are not explained by the RE characteristics.
Since the RE forecasting rule is unable to completely explain the findings of the experiments results, this thesis tried to explain the stylized facts with the heterogeneous expectations cobweb model. A simulation of the cobweb model with SAC-learning and naive expectations was performed and the sample mean, variance and autocorrelation function were calculated. These findings were compared to the experimental results of Hommes, Sonnemans, Tuinstra and van de Velden (2007) to investigate whether the model could explain the stylized facts.
The mean of the simulation prices is in the neighborhood of the mean of the experiment for all treatments. The variance of the simulation prices is higher, but still a better description than the variance of RE. For the stable and unstable treatment the first lags show strong significant autocorrela-tion. For the strongly unstable treatment there is significant autocorrelation present for a large number of lags. Therefore, for the strongly unstable treat-ment fixed parameters κ and C were varied to measure their effect on the dynamics. After lowering parameters κ and C, the number of lags with signif-icant linear autocorrelation became lower for the strongly unstable treatment. However there are still lags with significant autocorrelation. Therefore, the cobweb model is still unable to explain the third stylized fact after adjusting the parameters.
Concluding, the cobweb model with SAC-learning versus naive expecta-tions is a simple and intuitive model able to explain the first stylized fact and partly explain the second stylized fact. After lowering parameters κ and C, the number of significant lags reduces. However, this adjusted cobweb model is still unable to explain the third stylized fact. The model of Hommes and Lux (2013), where individual learning is modeled through genetic algorithms, was able to explain the three stylized facts. Compared to this model it ap-pears that the cobweb model with SAC-learning versus naive expectations is too simple to explain the results from the experiment with human subjects. However, to give the best fit to the results of the experiment, the cobweb model with SAC-learning versus naive expectations should fix the parame-ters at κ = 0.01 and C = 0, since for these values the number of significant lags with autocorrelation is the lowest.
Appendix
A
Derivation Jacobian
The dynamical system from Subsection 3.6 becomes ( pt = a−n1,tS(H1(pt−1))−(1−n1,t)S(pt−1) b , n1,t+1 = exp(−γ[(F (n1,t,pt−1)−H1(pt−1)) 2+C]) exp(−γ[(F (n1,t,pt−1)−H1(pt−1))2+C])+exp(−γ(F (n1,t,pt−1)−pt−1)2).
To investigate the stability of the steady state p∗, the Jacobian is determined.
J (pt, n1,t) = ∂pt ∂pt−1 ∂pt ∂n1,t ∂n1,t+1 ∂pt−1 ∂n1,t−1 ∂n1,t ! , = −n1,t∂S(H1(pt−1)) ∂H1(pt−1) ∂H1(pt−1) ∂pt−1 −(1−n1,t) ∂S(pt−1) ∂pt−1 b S(pt−1)−S(H1(pt−1)) b ∂n1,t+1 ∂pt−1 A , where A= 2 exp(γ(F (n1,t,pt−1)−H1(pt−1)))F0(n1,t) exp(−γ[(F (n1,t,pt−1)−H1(pt−1))2+C])+exp(−γ(F (n1,t,pt−1)−pt−1)2)
+(exp(γ[(F (n1,t,pt−1)−H1(pt−1))2+C]))(2 exp(−γ(F (n1,t,pt−1)−H1(pt−1)))F0(n1,t)+2 exp(−γ(F (n1,t,pt−1)−pt−1))F0(n1,t))
(exp(−γ[(F (n1,t,pt−1)−H1(pt−1))2+C])+exp(−γ(F (n1,t,pt−1)−pt−1)2))2 .
There is no need to calculate ∂n1,t+1
∂pt−1 , since this term will be multiplied
with 0 in the calculation of the characteristic equation. Next, fill in steady state p∗ J (p∗, n1) = −n1β S0(p∗) b − (1 − n1) S0(p∗) b S(p∗)−S(p∗+β(p∗−p∗)) b ∂n1,t+1 ∂pt−1 A ∗ ! , = (n1(1 − β) − 1) S0(p∗) b 0 ∂n1,t+1 ∂pt−1 0 ! . Since F0(n1,t) = ∂F (n1,t,pt−1) ∂n1,t = ∂pt ∂n1,t = 0 when p ∗ is filled in, A∗ = 0.
B
Results experiment
Figure 22: Time series of realized prices of the six groups in the strongly unstable treatment, the unstable treatment and the stable treatment from the learning-to-forecast experiments performed by Hommes, Sonnemans, Tu-instra and van de Velden (2007)
Figure 23: Autocorrelation plots over the full subsample of 50 periods for the six groups in the strongly unstable treatment, the unstable treatment and the stable treatment from the learning-to-forecast experiments performed by Hommes, Sonnemans, Tuinstra and van de Velden (2007)
C
Derivation second eigenvalue
For the calculation of µ2, first the fraction n1 and S0(p∗) needs to be
cal-culated. For the unstable treatment, λ = 2, the intersection point of the demand and supply curve equals p∗ = 5.91. Moreover,a, b, γ and β are set fixed at a = 2.3, b = 0.25, γ = 2 and β = −0.5. First, the part S0(pb∗) will be calculated: S(p) = tanh(2(p − 6)) + 1 S0(p) = 2(1 − tanh(2(p − 6))2) S0(p∗) = S0(5.91) = 2(1 − tanh(2(5.91 − 6))2) = 1.93657, S0(p∗) b = 1.93657 0.25 = 7.74628. Next, the fraction n1 is calculated.
n1 = exp(−γ[(p∗− H1(p∗))2+ C]) exp(−γ[(p∗− H 1(p∗)2+ C]) + exp(−γ(p∗− p∗)2) , = exp(−γ[(p ∗− p∗))2+ C]) exp(−γ[(p∗− p∗)2+ C]) + exp(−γ(p∗− p∗)2), = exp(−γC) exp(−γC) + 1.
For C = 0, n1 = 12 and the eigenvalue is:
µ2 = (n1(1 − β) − 1) S0(p∗) b = ( 1 2(1 − −0.5) − 1) ∗ 7.74628 = −1.93657. For C = 1, n1 = exp(−γ)
exp(−γ)+1 and the eigenvalue is:
µ2 = (n1(1 − β) − 1) S0(p∗) b = ( exp(−2) exp(−2) + 1(1 − −0.5) − 1) ∗ 7.74628 = −6.3612.
