University of Amsterdam
Master Thesis in Mathematical Economics
Which evolutionary switching model can
explain expectations?
Monique de Haard Studentnumber 11120908 Supervisor: Dhr. Prof. Dr. J. Tuinstra Second reader: Dhr. Dr. T.A. Makarewicz University of Amsterdam January, 2017Abstract
This thesis examines whether a so-called Heuristic Switching Model is able to explain the expectation formation process in an overlapping generations model on the basis of two lab-oratory experiments with incentivized particpants. The first experiment is conducted for a single-agent economy whereas the second is conducted for a multiple-agent setting. In both experiments all participants had to predict the inflation rate for 50 consectutive periods based on prior predictions and prior realized inflation rates. After investigating this exper-imental data, first the fit of the homogeneous learning rules itself has been investigated. There was room for improvement and different forms of the Heuristic Switching Model were able to do so. Unfortunately the best model for both datasets was only able to explain the expectation formation of the participants for the first 15 a 20 periods leading to the conclusion that it is hard to explain the expectation formation process in an overlapping generations setting using the Heuristic Switching Model.
Key words: Heuristic Switching Model, Expectations, Expectation formation, Feedback sys-tem, Overlapping generations economy, Learning rules, Heuristics, Expectations rules, Ra-tional Expectations Hypothesis, Boundedly raRa-tional expectations.
This document is written by Student Monique de Haard who declares to take full responsi-bility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Contents
1 Introduction 3
2 Theory 6
2.1 The Overlapping Generations economy . . . 6
2.2 Different expectations rules . . . 8
2.3 Heuristic Switching Model . . . 13
3 Experimental Design 15 4 Methodology 18 4.1 Simulations of the learning rules . . . 18
4.2 Settings for the Heuristic Switching Model . . . 22
5 Results 24 5.1 Experimental Data . . . 24
5.1.1 A closer look at the experimental data . . . 24
5.1.2 Descriptive Statistics . . . 29
5.2 Homogeneous learning rules . . . 35
6 Results: Heuristic Switching Model 42 6.1 HSM Benchmark . . . 43
6.2 HSM with BM learning parameters . . . 44
6.3 HSM totally optimalized . . . 47
6.4 Own Benchmark Case . . . 50
7 Conclusion 53
1
Introduction
As it is well known, expectations play an important role in economic systems. Agents form ex-pectations about certain economic variables and adapt their behavior in accordance with these expectations. This affects the value of this economic variable. In other words, the expectations have an influence on the realized value of this economic variable. This process is known as a positive or negative feedback system.
An example of a positive feedback system is a speculative asset market, as is also mentioned by Heemeijer et al. (2009). If many agents expect that the price of an asset will increase they will all buy this asset. Hereby the aggregate demand increases and leads to a higher asset price by the law of supply and demand. The opposite holds for a negative feedback system. An ex-ample of this is the commodity market: if agents expect that the price of a good, for exex-ample hog, will increase for the next period, they will buy the good today instead of the next period. Consequently, this leads to no demand for for the good in the next period and by the law of demand and supply the price of this good will decrease now.
This thesis studies the negative expectations feedback system with the realized inflation rate as economic variable. Due to the important role of expectations in economic theory, it is essential to study how these expectations are actually formed.
For a long time the classical economy theory assumed that agents have rational expectations. This is also called the Rational Expectations Hypothesis. This means that the rational agent has perfect information about the structure of the economy and knows the beliefs of all other agents. This assumption is too restrictive and does not give a good description of the real ex-pectation behavior of agents in the markets, since in practice agents show sometimes irrational behavior. In recent years it has been shown that this hypothesis in most cases at best only holds for the long run behavior, since the agents are able to learn over time. Therefore the concept of boundedly rational expectations is introduced, which provide a better assumption for the formation of expectations. Here the agents do not act perfectly rational, since they do not know the structure of the economy, except for time series observations on certain economic variables. These time series observations are used for forming beliefs about the variable and based on these beliefs the agents make their decisions.
There are different ways of how the expectations for the next period can be modelled on basis of these time series observations. These different ways are called heuristics and learning- or expectations rules. Examples of important expectations rules are the naive, the adaptive and average expectations. Furthermore, in recent years also ’learning models’ are introduced where the learning process of the agents is examined. Different learning models are used, e.g. the model used by Bullard (1994) which leads to so-called learning-equilibra, or the model used by Tuinstra and Wagener (2007) where the dynamic system converges to the monetary steady state.
There has been a lot of research on expectation formation with experimental data. This is due to the fact that the expectations of the participants in the real markets are hard to observe and could also be affected by uncontrolled factors. Therefore it is more attractive to work with a clean and controlled dataset that is conducted in an experimental setting. One of the main papers that is used in this thesis is the paper of Anufriev and Hommes (2012). Their research to expectation formation is also done with experimental data. They tried to explain the expec-tation formation of participants that participated in an asset-pricing experiment by a so-called evolutionary selection model that Brock and Hommes (1997) develloped. This evolutionary se-lection model, also called the Heuristic Switching Model, makes use of a predetermined number of learning rules that are weighted each period. These weights are dependent on the prior per-formance of the predictions of the learning rules itself. This model is used to distinguish the heterogeneous expectations from evolutionary selection among the learning rules of the partic-ipants.
This thesis examines whether the Heuristic Switching Model that Anufriev and Hommes (2012) used is able to explain the expectation formation of participants that participated an over-lapping generations experiment. This dataset is conducted in the CREED laboratory by the authors of Heemeijer et al. (2012). In this experiment participants were asked to forecast the inflation rate for fifty periods, based on their own prior predictions and prior realized infla-tion rates. Two data sets were obtained from this experiment: the first data set is a so-called single-participant economy where each participant provides inflation estimates that determine the realized inflation rate that is not influenced by the actions of other participants. The sec-ond data set is a six participant, or also called a multiple-agent, economy where the realized inflation rate is determined by six individual inflation estimates. These two data sets differ in expectation formation and the goal of this thesis is to find out which evolutionary switching
model can explain these two data sets the best.
This thesis is structured as follows. Section 2 discusses relevant theory about the overlapping generations economy, the different forms of learning rules and how the Heuristic Switching Model works. In section 3 the experimental design of the used dataset is showed. Section 4 explains the methodology of this reasearch. Here the simulations of the learning rules and the evolution of the Heuristic Switching Model are discussed. The results of the experiment and Heuristic Switching Model are viewed in section 5 and 6 respectively. Finally the main conclusions are given in section 7.
2
Theory
In this section the background theory of the experiment is discussed. In the first subsection the concept of the overlapping generations model is introduced. This model can be closed after specifying how individuals form expectations. Expectation formation can take different forms and a number of rules are discussed in the next subsection. In the final subsection the evolutionary switching model will be introduced where the expectations rules of the previous subsection can be used.
2.1 The Overlapping Generations economy
This subsection studies the same standard overlapping generations exchange model, abbreviated to OG model, as studied by Bullard (1994) and Heemeijer et al. (2012). This standard model consists of generations that live for two periods, a single consumption good and a government that exists indefinitely.
The two generations in each time period t represent two households: a young household and an old household. The preferences of the households are represented by a utility function u(c0, c1)
where c0 and c1 denote the consumption of the good in the period when the household is young
(c0) and the period when the household is old (c1). Furthermore the households are endowed
with w0 and w1 consumption good units for the periods when the household is young and old,
respectively. The maximalization problem of each household that is born in period t is given by: max c0,c1 u (c0, c1) s.t. ptc0+ pet+1c1≤ ptw0+ pet+1w1, (1)
where ptdenotes the price of the consumption good in period t and pet+1 the price expectation
of the household for the consumption good’s price for period t + 1. This maximalization prob-lem has an unique solution when the following assumptions for u (c0, c1) are made: u (c0, c1)
is continuous, strictly increasing and stricly quasi-concave. Then the unique solution can be expressed as follows: (c∗0, c∗1) = c0 pe t+1 pt , c1 pe t+1 pt ≥ 0.
This unique solution can also be expressed in terms of the expected inflation rate πe t ≡
pet+1
pt
which gives the following expression:
The amount of income that the young households will use for consumption in the next period can be saved at the bank. This savings function is expressed in equation (2).
S p e t+1 pt = w0− c0 pet+1 pt . (2)
Furthermore, this function can also be expessed in terms of the expected inflation rate πet. For the savings function S (·) the next two assumptions will be made. The savings function must be positive S (·) > 0 and downward-sloping S0(·) < 0 for all values of p
e t+1
pt , i.e. the expected
inflation rate πte. The decreasing function implies that the consumption goods c0 and c1 are
gross substitutes.
In equilibrium it holds that the aggregate savings are equal to real money balances: Mt pt = S p e t+1 pt . (3)
Here Mtis the total money stock and is exogenously determined by the money creation policy
of the goverment that is given by the following formula:
Mt= θMt−1, θ > 1, (4)
where θ represents the rate of growth of the money stock.
When equation (3) is solved for Mt and is substituted in formula (4), the actual law of motion
is obtained. This actual law of motion is given by: pt+1 pt = θ S pe t+1 pt S pe t+2 pt+1 , (5)
or expressed in terms of gross inflation rate πt and expected inflation rates πte:
πt= θ
S (πte) S πe
t+1
. (6)
The monetary steady state of the dynamical system shown in formula (6) is given by πte= πt= θ
for all t. This means that the inflation rate in the steady state is constant and fully determined by the monetary policy of the government.
It is not difficult to see that in equation (6) the overlapping generations model, as used in this thesis, is an expectations feedback system. The key determinants of the realized inflation rate πtare the expectations of the inflation rate πteand πet+1. This model can be closed by
spec-ifying the expectation formation process. In the next subsection some well known expectation formation processes are discussed.
2.2 Different expectations rules
As already mentioned in the introduction a convenient approach is to assume that agents have rational expectations. Here the agents know the future inflation rates, so:
πte= πt, ∀ t. (7)
This results in the only feasible equilibrium trajectory:
πt= θ. (8)
The other trajectories diverge to infinity or become infeasible over time. What already is men-tioned in the introduction are the unsatisfactory characteristics of rational expectations, such as the too demanding assumptions that rational agents already know the next period inflation rate and the beliefs of the other agents. Therefore it is better to introduce other, less demanding, expectations rules, which are known as bounded rational expectations and can described with so-called learning rules. However, since agents are learning over time it is possible that they learn the rational expectations equilibrium. This implies that this rational expectation assump-tion could describe the behavior of the agents for the long run.
The simplest expectations rule is the static or naive expectations rule shown in the equation below. Here agents’ prediction for the next inflation rate is equal to the last observed inflation rate.
πt+1e = πt−1 (9)
Note that the realized inflation rate πt, shown in equation (6), depends on the two predicted
inflation rates πet and πt+1e . This implies that the agents cannot observe the realized inflation rate πtwhen they predict the inflation rate for period t + 1. Therefore in this model agents have
to predict the inflation rate two periods ahead. This explains why the the naive expecation rule shown in equation (9) has as last observed realized inflation rate the inflation rate of period t − 1. The naive expectations rule gives the following actual law of motion when the naive expectations rule is substituted in equation (6).
πt+1= θ
S (πt−1)
S (πt)
. (10)
The stability of the monetary steady state πt = θ depends on the sensitivity of the savings
inflation elasticity of savings:
a (π) = − π S (π)
∂S (π)
∂π . (11)
Since the savings function S (·) is positive and decreasing it holds that the inflation elasticity of savings is positive, a (π) ≥ 0 for all π ≥ 0. For the stability analysis of the monetary steady state πt = θ the necessary condition for local stability is that the absolute value of the derivative of
the actual law of motion must be smaller than one. The derivative of this function is the inflation elasticity of savings shown in equation (11) which induces the stability condition that a (π) < 1. When a (π) = 1 the dynamics of the actual law of motion undergoes a Neimark-Sacker bifur-cation, where an invariant circle is created. Along this invariant circle inflation rates constantly fluctuate.
Another example of a simple expectations rule is the Fundamentalist rule. Here the predicted inflation rate for the next period inflation is equal to the monetary growth level θ:
πt+1e = θ. (12)
The dynamics of this rule are as expected: it coincides with the steady state itself.
One other well-known expectations rule is the adaptive expectations rule. The predicted infla-tion rate for the next period depends on the last predicinfla-tion inflainfla-tion rate and is being adapted by the forecast error that was made in the previous period. The parameter α determines how fast the predictions are updated.
πet+1= πte+ α (πt−1− πte) , 0 ≤ α ≤ 1. (13)
This rule can also be expressed as a weighted average of the last realized and the last predicted inflation rate:
πet+1= απt−1+ (1 − α) πte, 0 ≤ α ≤ 1. (14)
Note that when α = 1 the adaptive expectations rule is equal to the naive rule. Furthermore, the adaptive expectations rule has similar dynamic behavior as the naive expectations rule. The monetary steady state θ is locally stable if and only if αa (θ) < 1.
The third expectations rule is the average expectations rule. The predicted inflation rate is determined by taking the average of the last k inflation rates. This expectations rule is shown
below: πt+1e = 1 k k X s=1 πt−s. (15)
Here k is the number of lags used. When k is equal to t − 1 the whole sample is used. This latter one is also known as the sample average expectations rule. The monetary steady state θ is locally stable if and only if a (θ) < a∗k, where a∗k is a threshold value that increases in k. Note that when k = 1 equation (15) is the naive expectations rule. This gives the conclu-sion that both the adaptive and averaging rule are generalisations of the naive expectations rule.
Now a couple of expectations rules will be introduced with a similar stucture and only based on past realized inflation rates. The first rule, of which the other rules are special cases, is the autoregressive expectations rules with two lags, abbreviated as AR(2) expectations rule:
πt+1= αAR+ β1,ARπt−1+ β2.ARπt−2. (16)
The dynamics created by this rule can be different depending on the parameter values and savings function. There could be for example monotonic convergence, converging or diverging oscillations and cyclic behavior. The AR(2) expectations rule can be rewritten as the following equation:
πet+1= αAR+ βπt−1+ δ (πt−1− πt−2) , (17)
where β ≡ β1,AR+ β2,AR and δ ≡ −β2,AR. Equation (17) is easier to interpret than the ’normal’
expectations rule shown in the equation above. The expected inflation rate for the next period depends on the last realized inflation rate and is adjusted to the trend in prices. When δ > 0 the agent believes that a trend in inflation rates will continue, while for δ < 0 the opposites holds.
An expectations rule with similar structure is the trend-following expectations rule. This ex-pectations rule can take different dynamics by only specifying the trend-following parameter γ differently. In equation (18) the trend-following expectations rule is shown.
πt+1e = πt−1+ γ (πt−1− πt−2) , γ > 0. (18)
This rule is equal to the AR(2) expectations rule, shown in equation (17), when αAR = 0,
β = 1 and γ = δ. As already mentioned the parameter γ represents the adjustment to the trend in the prediction for the inflation rate. When γ is equal to zero this rule is the same as the naive expectations rule. Anufriev and Hommes (2012) distinguished in their paper two
different forms of this expectations rule, namely a weak trend-following rule, where γ is a small number, and a strong trend-following rule where γ is larger than one. Another special case of the trend-following rule is when γ < 0. This is a so-called trend-reversing expecations rule. The inflation rate may either converge to or diverge from the monetary steady state θ under these different forms of learning rules. This depends on the trend parameter γ and the specification of the savings function.
A more sophisticated rule, with the same structure as equation (17), is the so-called Anchoring and Adjustment (AA) rule. This rule is introduced by Tversky and Kahneman (1974) and has the following form:
πt+1e = 0.5
πf + πt−1
+ (πt−1− πt−2) (19)
with πf the fundamental inflation rate, which is equal to the rate of money growth θ. The first
term is the reference point or anchor that describes the long run behavior. The second term is the last price change and extrapolates the expected inflation rate. This expectations rule is a special case of the AR(2) expectations rule, showed in equation (17), where αAR = 0.5 · πf,
β = 0.5 and δ = 1.
The disadvantage of this rule is that the agents do not know the fundamental inflation rate πf. Therefore there exists another more realistic learning rule that is similar to this, namely the learning anchoring and adjustment (LAA) rule. The fundamental inflation rate in equation (19) is replaced by the sample average inflation rate upon period t − 1 shown below.
πavt−1= 1 t t−1 X j=0 πj.
This value is known to the agents, leading to the following learning rule:
πt+1e = 0.5 πt−1av + πt−1 + (πt−1− πt−2) . (20)
The dynamics of this rule are similar to the dynamics of the anchoring and adjustment rule. Furthermore this expectations rule is not of the same form of the AR(2) expectations rule like AA does. This can be explained by the fact that the term αAR in equation (17) is now
time-dependent: αAR,t = 0.5 · πt−1av .
The first problem with the above expectations rules is that sometimes there could be auto-correlation in the timeseries of forecasting errors. Therefore more sophisticated expectations
rules have to be introduced where learning takes place. Antoher disadvantage of the introduced expectations rules is that most rules are short-term. Exceptions are the sample average ex-pectations rule and the learning anchoring and adjustment rule (shown in equations (15) and (20) respectively). These short-term rules take only into account the most recent predicted and realized inflation rates. Hereby a lot of useful information is lost. Therefore it is better to in-troduce learning rules which are long-term. Here agents adapt their beliefs to the structure in their forecasting errors each time. The last two learning rules that will be introduced in this subsection are of this form.
The first learning rule is introduced by Bullard in 1994. He already understood that agents adapt their forecasting rules to the information that they have. Using this timeseries informa-tion the agents estimate the perceived law of moinforma-tion for the inflainforma-tion rates. Bullard’s assumpinforma-tion for the beliefs of the agents is that the inflation factor should be constant πt= β. He expressed
the perceived law of motion for the agents in terms of prices:
pt= βpt−1. (21)
To obtain an estimate for this β the agents run a least squares regression on prices. The estimated βtis equal to their forecast for the inflation rate for period t: πte= βt. This expectations feedback
system introduced in Bullard (1994) can be described as the following recursive dynamical system: βt+1= βt+ gt θS (γt) S (βt) − βt γt+1= βt gt+1= " g−1t θS (γt) S (βt) −2 + 1 #−1 , (22) with gt = p2t−1 Pt s=1p2s−1 −1
representing the so-called gain. Bullard showed that when θ is high enough a Neimark-Sacker bifurcation occurs whereby an unstable or stable invariant closed curve vanishes or emerges respectively. Bullard defines this established equilibrium as a learning equilibrium, where the equilibrium paths do not converge to the rational expectations equilib-rium. This outcome is a contradiction with the intuition that in long run stationary equilibrium trajectories will convergence to the rational expectations steady state. The corresponding learn-ing rule that agents in this settlearn-ing use is deduced by Tuinstra and Wagener (2007):
Where Bullard (1994) showed that in the long run, and when learning takes place, inflation rates are not necessarily equal to the perfect foresight equilibrium, Tuinstra (2003) showed that Bullard’s equilibria are misspecified and that still learning over time may lead to this per-fect foresight equilibrium. Since Bullard’s perceived law of motion is based on non-stationary prices this caused the fluctuations in the inflation dynamics and beliefs that are summarized as the learning equilibria. The failure is easy to see in the dynamical system shown in (22). Least-squares learning is a decreasing gain algorithm, which gives the same weights to all time observations, but since the number of time observations is increasing the weights for each ob-servation will decrease over time. In the dynamical system this is shown by the gain gt which
should go to zero over time. When prices are bounded, or with other words stationary, this would be the case, but since prices are non-stationary or unbounded gt, shown in equation (22), will
not be equal to zero. The learning equilibrium proposed by Bullard is actually a constant gain algorithm. Instead of using the non-stationary prices in the perceived law of motion Tuinstra (2003) uses stationary constant inflation rates:
πt= β, (24)
where the expected inflation rate is equal to the least-squares estimate: πet = βt. This leads to
stable predictions over time. The corresponding learning rule is shown below: πet+1= 1 t − 1 t−1 X j=1 πj, (25)
which is conducted by the least squares regression on a constant and is the same as sample average learning, showed in equation (15), where k is equal to t − 1.
2.3 Heuristic Switching Model
The question for now is: do the participants use homogenous or heterogenous expectations during this experiment? When the participants use homogenous expectations their individual prediction strategies could maybe be described by one of the expectation or learning rules in-troduced in the previous subsection. If the participants use heterogenous expectations it means that an expectations or learning rule is chosen from a set of several expectations or learning rules for each period of the experiment. In the beginning of each period the participants choose the rule which performs the best in the past and base their inflation prediction for the next period on this rule. In the next period there could possibly be other rules which perform better that the participants would choose. This process can be summarized by the Heuristic Switching
Model proposed by Anufriev and Hommes (2012). This model is based on a similar model de-signed by Brock and Hommes (1997) where they describe the cobweb model with heterogenous expectations. The model is introduced as follows.
Let H represent the set of H individual expectations or learning rules where the agent can choose from to predict the inflation rate for each period. This set contains a subset of the rules introduced in the previous subsection. In the beginning of each period t every rule h ∈ H gives a prediction for the inflation for the next period t + 1 that is denoted by πe
h,t+1. This prediction
is based on a deterministic function fh that uses the information available for the agent:
πh,t+1e = fh πt−1, πt−2, ...; πh,te , πh,t−1e , ... . (26)
Each prediction is weighted with an impact nh,t. It must hold that the weights sum up to one:
PH
h=1nh,t = 1. Furthermore, the weights nh,t are time dependent which means that they are
evolving over time. This evolution depends on the past performance of the different learning rules. The performance measure for forecasting rule h ∈ H is based on the squared forecasting error which uses the data until t − 1:
Uh,t−1= − (πt−1− πh,t−1)2+ ηUh,t−2, (27)
with η ∈ [0, 1] representing the memory of the agents with respect to the past performance of forecasting rule h. Given this performance Uh,t−1 for each forecasting rule h the weights nh,t
can be updated by the discrete choice model with asynchronous updating which is shown in the equation below.
nh,t = δnh,t−1+ (1 − δ)
exp (βUh,t−1)
Zt−1
. (28)
Here Zt−1 is the normalization factor that is equal to Zt−1=PHh=1exp (βUh,t−1), δ ∈ [0, 1] the
inertia that agents have for updating the weight nh,t and β ≥ 0 the intensity of choice that
shows how sensitive the agents are with respect to differences in forecasting performance. When β = 0 this will lead to an equal distribution independent of past performance, while a high value for β makes agents switch faster to more successful rules.
In section 4.2 the initialization values and evolution of the Heuristic Switching Model are dis-cussed.
3
Experimental Design
The experiment was conducted on April 25, 2006, at the Centre for Research in Experimental Economics and political Decision making (CREED) at the University of Amsterdam. In two sessions of circa 90 minutes participants had to predict future inflation rates for 50 consecutive periods. The corresponding realized inflation rate is based on equation (6) but is slightly different because of an introduced disturbance to make the experiment more realistic. This actual law of motion is shown in the equation below.
πt= θεt
S (πet)
S πet+1 . (29)
Here εt is uniformly distributed on [0.975, 1.025]. This disturbance implies that the rational
expectation still would be πe
t = θ for each time period t, while the actual law of motion is equal
to πt = θεt. This makes the rational expectations equilibrium still constant for each period t
although the actual inflation rate fluctuates around the monetary steady state value θ.
The savings function is specified as follows:
S (π) = δ + (1 − δ) w0 1 + (νπ)
ρ 1−ρ
, (30)
with δ ∈ [0, 1] , ν > 0 and 0 6= ρ < 1. In the experiment the parameters in the savings equation are equal to δ = 0.4, ν = 0.92, ρ = 0.965 and w0 = 0.9. This specified savings function is shown
in Figure 1. Since ρ is relative high the savings decrease very quickly for inflation rates just above one.
The experiment has two treatments, namely a treatment with a high level of monetary growth where θ = 1.11, which is called the unstable treatment, and a treatment with a low level of monetary growth with θ = 1.01, also indicated as the stable treatment. These two different growth levels are indicated in Figure 1 with the vertical lines. For the case where θ = 1.01 the inflation elasticity of savings shown in equation (11), which is related to the slope, is smaller than one. This results in a stable treatment, since this ensures converging behavior for different heuristics. While for θ = 1.11 the opposites holds: the inflation elasticity of savings is bigger than 1 which indicates diverging away from this point. Therefore this treatment is referred as unstable.
Figure 1: Savings function showed in equation (30)
this experiment the multiple-agent economy consists of six participants). These two differ by the fact that in a single-agent economy each participant provides inflation estimates that deter-mine the realized inflation rate that is not influenced by the actions of other participants, while in a multiple-agent economy the realized inflation rate is determined by six individual inflation estimates. Equation (29) shows how the realized inflation rate is calculated for the single-agent economy, while equation (31) shows the same calculation for the realized inflation rate, but then for the multiple-agent economy:
πt= θεt P6 i=1S πei,t P6 i=1S πe i,t+1 . (31)
The different economies and different treatments will be indicated as follows: the single-agent economy with the stable treatment (θ = 1.01) is abbreviated to SS and the same economy with the unstable treatment (θ = 1.11) is equal to SU. For the multiple-agent economy this is MS and MU respectively. Participants were randomly assigned to each economy and each treatment with the following distribution. The single-agent economy has 32 participants and is distributed by 16 participants for SS and 16 participants for SU. The multiple-agent economy has a total of 78 participants and has seven groups consisting out of six partipants for the MS and six groups of six partipants for the MU. The distribution over the economies and treatments is summarized in Table 1.
Table 1: Distribution of the participants over the economies and treatments Treatment Economy Single-agent Multiple-agent Stable 16 7x6 Unstable 16 6x6
The information set for participant i at time period t consists of all previous predicted inflation rates of himself1 until t and the realized inflation rates until t − 1:
Ii,t = {{πti,1, πi,2e , ..., πei,t}, {π1, π2, ..., πt−1}}.
The participants had to predict the inflation rates for fifty consecutive periods. Since the real-ized inflation rate, shown in equation (29), is determined by πet and πt+1e the participants need to give a two periods ahead prediction. This explains why the realized inflation rates in the information set for participant i at time period t are until t − 1 and why the participants also need to predict the inflation rate for 51 periods.
To encourage the participants to submit the best predictions, they were rewarded for the ac-curacy of their inflation forecasts. This payment depends on the absolute difference between the prediction of participant i, πi,te , and the realized inflation rate πt according the following
formula:
Pi πei,t, πt = max{100 − 400|πei,t− πt|, 0}, (32)
with Pi(πi,te , πt) the number of points that participant i earned in period t. For each time
period the number of points were rounded, summed up and finally at the end of the experiment converted into euros with 200 points equal to 1 euro. If the participants perform optimally they could earn a maximum of 25 euros.
1
For generality in this thesis the assumption is made that the participants are males, while they also could be women.
Figure 2: Simulations of realized inflation rates by the learning rules in stable environment
4
Methodology
As already mentioned in the introduction and subsection 2.3 the goal of this thesis is to explain the data of the four different treatments (SS, SU, MS and MU) with evolutionary switching models and find out which of these different proposed models can explain these four datasets the best. To start with the Heuristic Switching Model first the heuristics or, with other words, the learning rules, that will be included in the set H have to be specified. This will be done in the first subsection. In the second subsection the initial conditions, the evolution and estimation techniques of the Heuristic Switching Model will be discussed.
4.1 Simulations of the learning rules
For the set H all learning rules of subsection 2.2 are included. It is important to realize that some learning rules depend on parameters, for example the adaptive and trend-following
ex-pectations. Together with the specified savings function (shown in equation (30)) these will determine the dynamics of the learning rules. Therefore simulations of the realized inflation rate of these learning rules in this experimental setting, including the shocks εtthat are used in
the experiment, are conducted for several parameter values and can be used as a good indication for the performance of these rules.
Figure 2 shows the dynamics of the learning rules in the ’stable’ environment when θ = 1.01. The dynamics of the learning rules for θ = 1.11, also known as the ’unstable’ environment, are shown in Figure 3. Now the dynamics of each learning rule h ∈ H will be discussed for both environments.
The naive expectations are shown in the top left figures of both figures. As can be seen, the dynamics are quite unstable for the first 50 periods in both treatments. Both dynamics of the realized inflation rate are similar to each other: both treatments show large amplitudes for the realized inflation rate, where the amplitudes for the realized inflation rate in the unstable envi-ronment are larger, and both treatments do not seem to converge to the monetary steady state. When the realized inflation rate in the experiment has similar behavior as the naive expecta-tions, this learning rule can be chosen for the Heuristic Switching Model.
The dynamics of the adaptive expectations rule are shown in the top right figure of Figure 2 and 3. Since this learning rule depends on parameter α two different functions are simulated with α = 0.25 and α = 0.65. As can be seen from Figure 2 the expectations rule with the lower value for α in the stable treatment is more stable and this holds already from the beginning of the simulations. The dynamics are also converging to the monetary state value θ = 1.01 with decreasing oscillations. When α = 0.65 the dynamics are more unstable compared with the dynamics of α = 0.25, but are converging with decreasing oscillations to the steady state.
In the unstable treatment, shown in Figure 3, the dynamics of α = 0.25 are very different when this is compared with the stable treatment. Here the dynamics are more unstable and the realized inflation rate does not seem to converge to the monetary steady state. When α is larger, this effect is strengthened. This result is logical, since when α gets closer to 1 the dynam-ics will look like the dynamdynam-ics under the naive expectations: unstable with large oscillations. This learning rule can be useful when the dynamics of the realized inflation rate in the stable
Figure 3: Simulations of realized inflation rate by the learning rules in unstable environment
treatment are quite stable and flat, while the dynamics in the unstable treatment have more fluctuations. This is indeed the case when the results are shown in subsection 5.1.
For the average expectations rule the dynamics for k = 2 and k = 3 are shown in one figure. When k = 2, shown with the red graph in the left plots under the top, the dynamics of these expectations rule look similar to the naive expectations. For the stable treatment the dynamics of the realized inflation are more stable when this is compared with the dynamics in the un-stable environment. For k = 3, the blue graph, the dynamics in the un-stable treament are quite different when this is compared with the red graph in Figure 2. This graph has less oscillations, is closer to and converges to the monetary steady state value. For the unstable treament the average expectations rule with k = 3 has unstable dynamics. Therefore this expectations rule is useful in a Heuristic Switching Model where in the stable environment there are less oscillations
and a converging movement to the monetary steady state, while the opposite holds in unstable environment.
The dynamics of the three forms of the trend-following expectations are shown in the three consecutive figures. These dynamics depends on the trend parameter γ. Overall, the dynamics are similar in the two different treatments: there are many oscillations with high amplitudes and it does not seem to converge to the monetary steady state at all.
Now for different values for γ the three forms will be discussed. For the weak trend a ’high’ and ’low’ value for γ are chosen. When γ = 0.1, the amplitudes of the oscillations are smaller and are less frequent. Therefore this rule is more stable than when it is compared with the rule where γ = 0.6 is used. For both treatments this holds. The strong trend is simulated for γ = 1.1 and γ = 1.9. The higher value for γ has more oscillations and higher amplitudes compared with the dynamics of γ = 1.1. In the unstable economy the same holds, but then the amplitudes of the simulated inflation rate are higher. Compared with the other two forms of trend-following expectations the dynamics of the anti trend have the highest amplitudes and the most oscilla-tions. When γ is smaller in this case, here γ = −0.3, the dynamics are less explosive than when it is compared with γ = −1.3.
For the simulations of the anchoring and adjustment learning rule and the learning achorch-ing and adjustment, only the first learnachorch-ing rule is shown in Figure 2 and Figure 3, since the dynamics of these two learning rules are quite similiar. For the savings function used in this experiment the dynamics are quite unstable and have large and many oscillations that seem to be constant over time. Therefore the expectations of these learning rules are low for explaining the experimental data.
At last the dynamics of Bullard, Tuinstra, Fundamentalists and the AR(2) learning rules are simulated. Since these dynamics are also similar to each other, only Bullard’s dynamics are shown in the right bottom figure of both Figures. In the stable treatment the dynamics are already from the start stable and with small oscillations it moves around the steady state value.
Tuinstra’s and Fundamentalists rule have no parameters, but the other two learning rules do. For Bullard’s rule the beginning value of the vector g(·) must be chosen. The dynamics of
dif-ferent starting values in the stable environment does not differ that much, but in an unstable environment there is a substantial variation. This is shown in Figure 3 in the right bottom figure. The dynamics are smoother when g(1) = 0.1 compared to g(1) = 0.9. For the AR(2) rule the dynamics differ per chosen parameter values for αAR, β1,AR and β2,AR.
4.2 Settings for the Heuristic Switching Model
In this subsection the initial conditions of the Heuristic Switching Model will be given. Further the evolution and estimation procudere of the model will be discussed.
For the intitial conditions for each forecasting rule h ∈ H its inflation predictions, weights nh,0 and performance Uh,0 must be obtained. This thesis will work with four different
learn-ing rules when the Heuristic Switchlearn-ing Model is modelled. This number is chosen since the sufficient explanatory power that four learning rules together should have. Moreover previous papers (Anufriev and Hommes (2012)) also programmed the Heuristic Switching Model with four heuristics. Therefore for four heuristics the initial conditions must be given.
For the intitial weights nh,0 an equal weight of 0.25 is chosen. Furthermore for the first two
time periods the inflation predictions must be given, since the participants cannot base their predictions on realized inflation rates. For these predictions the average inflation rate of all participants for period 1 and 2 are calculated. This leads to πe1 = 1.03316 and πe2 = 1.03308 re-spectively. At last the performance Uh,0has to be initialized. This is set to zero for each heuristic.
Now the evolution of the Heuristic Switching Model for the successive periods will go as follows. First for each forecasting rule h ∈ H the performance measure Uh,t−1, shown in equation (27)
will be updated. Then the weight of each rule will be caculated following equation (28). After this, each learning rule will give a new inflation prediction for the next period πh,t+1 according
equation (26). Finally the realized model inflation rate can be calculated by using the actual law of motion (29). This law of motion will look as follows:
πm,t= θεt P4 h=1nh,tS πh,te P4 h=1nh,t+1S πh,t+1e . (33)
Since the savings function is decreasing a decline in average forecast between period t and t + 1
growth level θ. When the opposite holds, so πeh,t < πeh,t+1, the realized inflation rate would be larger than the monetary steady state. Only when the two predictions are equal a stable realized inflation rate could be established.
The three parameters β, η and δ, shown in equations (27) and (28) in subsection 3.3, are referred as the learning parameters and are constant over time. These parameters can be initialized be-fore setting up the heuristic switching model or can be estimated when there is experimental data available. In this thesis the latter is the case whereby the same estimation techniques are used as described by Anufriev and Hommes (2013). They estimated these learning parameters by minimizing the mean squared error between the realized model inflation πm,t and the realized
inflation rate πtfrom the experimental data. By assuming independent and normally distributed
errors for πm,t − πt this will lead to the maximum likelihood estimators ˆβ, ˆη and ˆδ when the
Figure 4: Example of obtained data for participant SU6. Left: predicted and realized inflation rate, right: corresponding forecast errors.
5
Results
In this section the results of the first part of this experiment will be discussed. In the first subsection a closer look will be taken at the experimental data where descriptive statistics and some figures will be shown. In the second subsection the homogeneous expectations rules, introduced in subsection 3.2, are investigated. Here it is examined whether some groups in the experiment can already be explained by one expectations rule, or whether more rules are needed. As expected the latter is the case. In section 6 the results of four different forms of the Heuristic Switching Model are presented.
5.1 Experimental Data
This subsection shows first how the experimental data looks like on basis of two examples. Further it categorizes the dynamics of the realized inflation rate into three different groups in subsubsection 5.1.1. In the second subsubsection the descriptive statistics are discussed.
5.1.1 A closer look at the experimental data
For the experiment a collection of data was obtained consisting out of predicted and realized inflation rates of both group and single agent economies. As already mentioned the data con-sists of seven multiple stable groups, indicated with MS, six multiple unstable groups, that are abbreviated to MU, and two groups each consisting of 16 participants for the stable and unstable environment, indicated with SS and SU respectivily. Each dataset consists out of 51 inflation predictions for each participant and 50 inflation realizations for each group or par-ticipant depending if the parpar-ticipant belongs to the multi- or single-agent economy. Due to a mistake in the computer software for four groups in the stable multiple-agent economy (MS4, MS5, MS6 and MS7) the last seven periods of the experiment are missing and only 43
data-Figure 5: Example of obtained data for group MS4 with realized and predicted inflation rate and correspoding forecasting errors.
points for these groups are available. Figures 4 and 5 give an example of what the data look like.
The big difference between these two figures is that in Figure 4 the realized inflation rate depends only on participant SU6, while the realized inflation in Figure 5 is a compostition of the six predictions for that period of the participants in that group.
As can be seen from the two figures the data can show different behavior. Therefore these different behaviors will be categorized as follows. Some participants or groups are able to sta-bilize the inflation rates immediately after the experiment starts. These groups or individuals will be indicated with S from stable, while other groups or individuals are only able to do this after a while. This kind of dynamics will be indicated with US: from unstable to stable. The last group are the unstable ones, abbreviated with U, where these groups and individuals failed to stabilize the inflation rate at any time.
Sometimes it is hard to see which participant or group belongs to which category. Therefore boundaries are introduced to classify the groups and participants. These boundaries are based on the mean of the realized inflation rate for the first and last 25 periods. For the single-agent
Figure 6: Examples of realized inflation rates that belongs to category S
economies it holds that if the mean belongs to (0, 2) and (10, 12) for the stable and unstable treatment respectively it gets a S from stable. If this holds for the first and last 25 periods then this group or participant belongs to the S category. If the mean of realized inflation rate fell for the last 25 periods in this interval but not for the first 25 periods, this group or participant belongs to US category and otherwise it belongs to unstable. For the multiple-agent economies the same boundaries hold. In Figures 6, 7 and 8 some results of these three different categories are shown.
Note that in an unstable environment, so when θ = 1.11, it is harder to stabilize the infla-tion rate. This is due to the higher value of the monetary growth rate θ which causes higher volatility. When a participant deviates his prediction strategy each time this will increase or decrease the difference between predictions of two successive periods: πet − πe
t+1 causing more
volatile dynamics. Even this process is harder for multiple-agent economies, since the behavior of a participant does not influence the realized inflation rate by himself but with five other participants as well. Therefore if someone thinks to understand how to react or stabilize the inflation rate he is still dependent on what the other five agents do. If someone makes a really big mistake, as for example shown in Figure 5, the inflation rate will still be highly volatile.
The corresponding qualitative dynamics of the categories US and U are fluctuating but with decreasing amplitude and persistent oscillations, respectively. In Figure 5 an example of a group
Figure 7: Examples of realized inflation rates that belongs to category US
that belongs to the US-category is shown. Here the realized inflation rate is fluctuating but with decreasing amplitude (shown in the top left figure). In the first 10 to 15 periods of the experiment the amplitude of the realized inflation rate is high, while after this period, which is also reffered as learning phase, the amplitude seems to decrease. This can also be seen in the right top figure where the predictions of each participant are shown. In the beginning the participants need to figure out how this experiment works, but after period 15 some partici-pants know how to react. In the left bottom figure their forecast erros are shown. This plot confirms the participants’ behavior after period 15 when the learning phase ended by showing their decreasing forecast errors.
It looks like that the same dynamics are shown in Figure 4 for the single-agent economy. In the beginning of the experiment the amplitude of the realized inflation rate is fluctuating and slightly decreasing. After period 20 the participant figured out how the model works by giving the right predictions. This can also be seen in the right figure where his forecasting errors are declining over time. Still this qualitive dynamic behavior happens a lot faster in the single-agent economy commpared with the multiple-agent economy. The reason for this is that the realized inflation is only influenced by one participant here. Figure 7 shows more of this type of dynam-ics for the realized inflation rates. In Figure 8 the other dynamdynam-ics, persistent oscillations, are shown.
Figure 8: Examples of realized inflation rates that belongs to category U
Table 2 shows the distribution over the three categories for both economies. In the SS treatment there are significantly more participants that belong to the stable category than this number compared with the SU treatment, where none is able to stabilize the inflation rate. The same holds when the MS and MU treatments are compared with each other. When the multiple-agent treatment is compared with the single-agent treatment the relative number of participants and groups that belong to S are quite similar. Furthermore, the number of participants in the single-agent economy is constant for the US category comparing the two treatments. For the multi-agent setting this number is decreasing in treatment. This suggests that it is harder to stabilize the inflation rate in an unstable treatment with a multiple-agent setting. At last, the numbers of participants and groups that belong to the U-category are increasing in treatment. When the differences between the single- and multiple-agent settings are compared, the per-centage of participants that belong to U of the unstable treatment is 68.8% for the single-agent economy, while this is 83.3% for the multiple-agent economy. This confirms the fact that the realized inflation rate is more unstable when there are more participants available in the econ-omy. This suggest that the kind of treatment has a strong effect on stability.
Table 2: Distribution over the categories S, US and U
Economy Single Multi
SS SU MS MU
S 2, 8, 9, 11, 12, 14-16 - 2, 5, 7
-US 1, 3, 5, 7, 10 4-6, 10, 11 1, 3, 4, 6 5
U 4, 6, 13 1-3, 7-9, 12-16 - 1-4, 6
5.1.2 Descriptive Statistics
Tables 3 and 4 provide more useful information about the behavior of the participants in the single- and multiple-agent economies and the two treatments. In these tables the descriptive statistics are shown. However, since these two economies differ from each other it is not always possible to compare the same descriptive statistics. For example, it is interesting to examine if in the multiple-agent economy the participants coordinate on each other or just deviate. This effect, that cannot be calculated for the single-agent economy since the agents are all alone in the economies, can be calculated according the average individual quadratic forecast error, shown in equation (34): AvgIndQFE = 1 6 · (50 − LP) 6 X h=1 50 X t=1+LP πeh,t− πt2 . (34)
Here LP is the length of a possible present learning phase. For Tables 3 and 4 this is assumed to be zero. Hommes et al. (2005) showed that this error can be rewritten as the sum of the average dispersion error and average common error:
AvgIndQFE = 1 6 · (50 − LP) 6 X h=1 50 X t=1+LP πh,te − ¯πte2+ 1 (50 − LP) 50 X t=1+LP (¯πte− πt)2, (35) where ¯πte = 16P6
h=1πeh,t is the average prediction for period t in a group. The first term of
equation (35) represents the dispersion error and measures the dispersion between the individ-ual predictions. In other words, this number calculates the deviation from coordination on a common prediction strategy. When this number is small this indicates that the participants in the same economy are having the same prediction strategy. The last term of equation (35) is the average common error. What the name of this error already states, this error measures if the participants make the same forecast error. A large number shows that this is the case. This suggests that the participants coordinate on a common prediction strategy.
Table 3: Descriptive statistics of the single-agent economies
Part.
Stable treatment Unstable treatment
Cat. Inflation rate Forecast Error Cat. Inflation rate Forecast Error
Average Variance Average Variance Average Variance Average Variance
1 US 2.892 514.536 12.164 605.799 U 13.853 528.420 20.438 406.993 2 S 0.919 6.036 2.527 2.656 U 20.430 2522.093 45.462 2458.900 3 US 5.452 1058.152 27.299 1266.334 U 23.554 3381.872 60.004 2503.697 4 U 20.078 4980.793 103.932 5246.104 US 17.513 1757.814 34.118 2472.225 5 US 4.010 895.537 25.150 835.329 US 14.248 1006.926 21.959 1356.573 6 U 4.393 847.633 21.972 1523.165 US 13.803 574.846 16.803 672.315 7 US 5.023 1007.752 25.509 1907.286 U 21.419 2861.813 53.364 2115.376 8 S 1.084 5.153 2.169 2.859 U 25.826 3952.321 77.051 3245.300 9 S 0.945 4.853 2.051 2.531 U 27.131 4713.504 96.387 6091.462 10 US 7.742 173.717 37.732 2204.927 US 14.208 747.187 15.912 903.083 11 S 0.823 3.135 1.596 2.231 US 12.684 222.941 9.130 232.548 12 S 0.834 7.522 2.781 6.454 U 21.261 2909.183 56.690 2008.428 13 U 10.321 2319.684 51.207 1945.136 U 27.074 4257.389 87.697 4864.366 14 S 0.829 3.033 2.006 2.214 U 20.110 2102.931 63.365 2163.044 15 S 1.043 10.214 2.984 6.786 U 17.186 1548.868 32.505 1329.747 16 S 0.780 2.945 1.739 2.408 U 21.452 2936.789 62.534 2072.038
For the single-agent economy it is obvious that these two errors cannot be calculated. Therefore the average absolute forecast error, according to equation (36), and its variance are calculated.
¯ ei = 1 (50 − LP) 50 X t=1+LP |πei,t− πt| (36)
These descriptive statistics, the average realized inflation rate and its variances are shown in the two Tables 3 and 4 for both economies.
First the descriptive statistics of the single-agent economies are discussed. These are shown in Table 3. The most remarkable, but expected, result is that the participants in the stable treatment and who belong to the S-category have an average inflation rate close to the real value θ = 1.01 and very small variances with values under the 10. Participants who belong to the other two categories (US and U) have an average inflation rate around the 4 and above 7 respectively. Their variances are also increasing in the category.
Table 4: Descriptive statistics of multiple-agent economies
Group Category Realized inflation rate Forecast Error (average) Error (%)
Average Variance Individual Disp. Common Disp. Common
MS1 US 1.458 104.419 182.576 27.339 155.237 14.97 85.03 MS2 S 0.960 10.179 18.140 4.022 14.118 22.17 77.83 MS3 US 1.929 193.927 340.003 50.865 289.138 14.96 85.04 MS4 US 2.176 183.857 776.709 282.388 494.321 36.36 63.64 MS5 S 0.838 7.813 16.048 3.749 12.289 23.36 76.64 MS6 US 1.530 125.948 246.826 35.155 211.671 14.24 85.76 MS7 S 1.198 57.108 92.819 10.780 82.039 11.61 88.39 MU1 U 14.073 501.661 1062.312 311.289 751.022 29.30 70.70 MU2 U 13.333 407.973 886.428 249.255 637.173 28.12 71.88 MU3 U 13.286 409.009 750.174 108.859 641.315 14.51 85.49 MU4 U 12.957 257.307 455.959 95.278 360.681 20.90 79.10 MU5 US 14.197 640.691 1201.158 247.340 953.818 20.59 79.41 MU6 U 13.903 605.343 1319.098 241.185 1077.913 18.28 81.72
For the unstable treatment the same results hold. The average inflation rate needs to lie around 11 for belonging to the S category. As can be seen from Table 3, no one was able to do this. Participants 4, 5, 6, 10 and 11 correspond with the US-category and this can also be confirmed by the table, since they have average inflation rates more close to 11 than the rest of the par-ticipants. Furthermore, their variances are significantly lower than the rest. Still there is one remarkable result, namely participant SU1 that belongs to U has a low average and low vari-ance. Therefore it looks like this participant fits better in the previous category. But when this result is compared with the corresponding graph it looks like this partipant has learned, but in fact at the end of the experiment he did not. This is shown by heavily and fluctuating inflation rates at the end. For the average forecasting errors and variances the same story holds. When the participants belong to S, their forecast errors and variances are very small, since they are already predict the inflation rate quite well. While for the other categories this is increasing.
The results for the multiple-agent economies are now considered. These are shown in Table 4. For the groups in the stable treatment the realized average inflation rate lies close to the real value of the monetary growth rate θ = 1.01. The groups MS2, MS5 and MS7 are the most stable groups and have the smallest variances. This indicates that these three groups understood what
they were doing. The other four groups have higher variances. This is also confirmed by the higher amplitudes of the realized inflation rate. For the unstable treatment the mean realized inflation rate must be close to θ = 1.11. None of the six groups were able to do this, but their results are not that bad. Also, the variances of this treatment are a lot higher. This result was also as expected, since it is much harder to loose the fluctuations in an unstable environment for a multiple-agent setting.
Another remarkable result that Table 4 shows, is the relation between the mean and vari-ance of the realized inflation rate and the individual quadratic forecast error. When the mean is close to the real value of the monetary growth level and the variances are low, the individual forecast error is significantly smaller than when this is compared with the groups with higher mean and variances for the inflation rate. This could be explained by the fact that when the variance in a group is higher, it is harder for the participants to make the right predictions, whereby their individual forecast errors are higher as well. The individual quadratic forecast error can be split up in the dispersion and common error. When the dispersion error is small this gives an indication that the six participants in the group have a similar prediction strategy. Here no link can be found between the different categories and groups. For example, in group MS7 this number is around 14% and is close to S, while group MS3 is less stable but has a dispersion error around 15%. The same holds for the unstable environment. The other part that expains the individual error is the common error that participants make in the same group. This average percentage is in both treatments around the 80% and thus much higher than the average individual dispersion error around 20%. The high percentage of the common mistakes can also be seen in the bottom plot of Figure 5 where the forecast errors for the six participants are shown. As the plot shows, those forecast errors are correlated. From here the following con-clusion can be drawn: the participants coordinate on similar prediction rules, but these rules are not stabilizing the inflation rate. Therefore the participants make the same common errors.
It is interesting to ask what would happen with the descriptive statistics when the learning phase is deleted. After the plots of the single- and multiple-agent economies are studied the common learning phase for the single-agent economies is set at 20 periods, while the learning phase for the muliple-agent groups is only 15 periods. From here two new tables, Table 5 for the single-agent economy and Table 6 for the multiple-agent economy, can be constituted.
Table 5: Descriptive statistics single-agent economy with learning phase subtracted
Part.
Stable treatment Unstable treatment
Cat.
Inflation rate Forecast Error
Cat.
Inflation rate Forecast Error
Average Variance Average Variance Average Variance Average Variance
1 US 0.962 2.961 1.883 1.441 U 11.088 512.375 17.338 326.643 2 S 0.789 6.933 2.558 3.103 U 20.296 2359.333 43.667 2511.572 3 US 2.866 1121.854 23.519 1058.702 U 24.634 4409.189 68.255 3056.785 4 U 17.825 5200.905 112.612 5274.240 US 11.110 5.104 6.194 6.468 5 US 1.601 148.450 11.033 127.944 US 11.144 21.219 7.780 25.440 6 U 2.711 448.149 10.409 674.461 US 9.659 41.606 3.553 58.552 7 US 0.774 565.617 15.429 127.000 U 19.214 3062.805 55.300 2663.593 8 S 0.787 2.619 1.649 1.131 U 24.260 4241.735 84.151 3275.269 9 S 0.826 3.579 1.761 1.419 U 24.236 3956.714 91.169 6212.085 10 US 3.879 1359.125 35.584 2083.271 US 10.309 18.279 2.857 14.632 11 S 0.820 2.511 1.372 0.652 US 10.802 3.041 1.619 0.994 12 S 0.949 2.996 1.814 1.689 U 16.398 2271.698 46.323 1347.825 13 U 7.365 2206.083 47.991 1585.221 U 30.404 5131.503 97.999 5121.119 14 S 0.820 2.292 1.466 0.843 U 18.273 1811.770 61.051 2408.756 15 S 0.812 10.454 2.811 7.737 U 16.840 1356.825 26.969 1048.586 16 S 0.820 2.511 1.489 0.914 U 22.307 3386.461 69.889 2270.636
Table 6: Descriptive statistics of multiple-agent economies with learning phase subtracted
Group Category Realized inflation rate Error (average) Error (%)
Average Variance Individual Disp. Common Disp. Common
MS1 US 1.193 92.853 175.969 34.197 141.772 19.43 80.57 MS2 S 1.055 3.559 6.237 1.037 5.200 16.63 83.37 MS3 US 1.754 226.330 397.813 64.950 332.864 16.33 83.67 MS4 US 1.016 101.828 193.731 41.928 151.804 21.64 78.36 MS5 S 0.880 6.307 12.573 3.617 8.956 28.77 71.23 MS6 US 1.282 58.748 111.034 24.384 86.650 21.96 78.04 MS7 S 1.377 71.855 109.974 12.374 97.600 11.25 88.75 MU1 U 14.332 509.675 1152.723 406.942 745.780 35.30 64.70 MU2 U 11.785 322.135 605.565 165.692 439.874 27.36 72.64 MU3 U 11.858 202.686 345.562 85.188 260.374 24.65 75.35 MU4 U 12.585 206.694 356.891 88.541 268.351 24.81 75.19 MU5 US 12.774 259.362 646.151 251.744 394.407 38.96 61.04 MU6 U 12.546 339.880 684.275 192.710 491.566 28.16 71.84
As expected, in most cases the average inflation rate and variance of the realized inflation rate is much lower. Now 11 instead of 5 participants in the stable treatment can be categorized in S. The most notable results are the variances of partipant SS2, SS3, SS4, SS14 and SS16. Their average realized inflation rates and variances are increasing. The reason for this could by explained by the fact that by deleting the learning phase a period is deleted that showed less volatile dynames than the period after the learning phase, where more amplitudes and oscilla-tions were present. With other words, a ’more’ stable part is deleted than an unstable part.
The same results holds for the unstable treatments. Overall there is more improvement in both descriptive statistics. Since the participants in both treatments are much closer to the real monetary growth level, or there is been some improvement to the right direction, their average forecast erros are much smaller and its variance is also declined.
When the learning phase for the multiple-agent economies is deleted, the same results holds generally. For all groups in the stable treatment, except voor MS2 and MS7, the average infla-tion rate gets closer to the real value θ = 1.01. The other two groups were already relative stable and do not have a real learning phase. Here good and useful data is thrown away. The groups
of the unstable treatment are improved, only MU1 has stayed the same. What also is been suggested for the variance of the realized inflation rate in the single-agent economies applies also to the multi-participant setting. The groups where the variance increased have more heavy fluctuations after period 15 than before this period.
Still the same link between the mean and variance of the realized inflation and the individ-ual error holds. When the variance is still relatively high, the individindivid-ual error is also higher. After deleting the learning phase some percentages of the dispersion and common errors are increased, while others have been declined. Overall the average percentages of the dispersion and common errors are the same for the stable treatment (20% for the dispersion error and 80% for the common error). For the unstable treatment the average percentages are now around 30% for the dispersion error and 70% for the common error, while before deleting the learning phase this was 20% against 80%. This suggest that with learning phase deleted there is less coordination on the same strategies. A reason for this could be that the participants in those economies give predictions without reasoning. For them it is really hard to stabilize the inflation rate, because of the high monetary growth and the five other participants that were available in the economy.
5.2 Homogeneous learning rules
In this subsection the dynamics of the modelled inflation rate by the homogeneous learning rules, introduced in subsection 2.2 and 4.1, are investigated. In other words in this section the same will happen as in the Heuristic Switching Model, but only when the set of heuristics con-sists out of one heuristic and where the performance and weights (shown in equations (27) and (28)) are not needed any more.
In the model with the single heuristic it is assumed that all agents are homogeneous and con-stant over time. As can be seen by the high percentages for the common error in the tables of the previous subsection, the first assumption could be true. Here it is shown that the agents coordinate on each other, but only on the wrong rules. The second assumption that the agents are consistent over time does not seem to be the case, since the participants saw their earnings during the experiment and tried to maximize this. Therefore if some forecasting rules did not pay that much the participant had the incentive to find a better technique.
Although it could be the case that examining the homogeneous expectations rules should not give the best results, these outcomes could serve as a benchmark for the performance of the Heuristic Switching Model. Furthermore it provides information on how these expectations rules behave in the experimental setting.
The performance of the learning rules in group k is expressed in the Mean Squared Error, shown in the equation below:
MSE(k ) = 1
50 · (πm,t(k) − πt(k))
2
, (37)
where πm,t(k) is the model inflation of group k in period t calculated according to the learning
rule and πt(k) is the realized inflation rate of group k for the same period from the experiment.
After calculating the MSE for each group in MS or MU for each learning rule, this will be averaged over all groups k in either the stable or unstable treatment. The same happens for the single-agent economies, but due to large differences in realized inflation rates in these economies the MSEs are quite high compared to the multi-agent economies. Therefore the MSE is, besides the average MSE, also expressed in the median MSE over all groups k in both treatments.
It is important to notice that some expectations rules depend on a parameter. These are the adaptive, trend-following, Bullard and AR(2) learning rules, see subsection 4.1. To find the optimal parameters that explain the dynamics of the realized inflation rates of the single- and multi-agent economies the best, the average MSE for these learning rules are minimized. These results are shown in Table 7 for the two different treatments in the two different economies. Besides this, the sum of the MSE of the SS and SU treatment and the MS and MU treatment is minimized, since it is interesting to see for which optimal parameter values the learning rules are able to explain both treatments for the same kind of economy, while the first two columns of the single- and multi-agent economy show the minimum MSE with the corresponding optimal parameters to explain this dataset the best.
As can be seen from the table the MSEs for all groups and treatment are quite high. This indicates that the optimal homogeneous learning rules do not have accurate forecasts for the realized inflation rates. For all cases the MSE in the stable treatment (SS and MS) is lower than the unstable treatment.
Table 7: Minimum average MSE of the homogenous learning rules for the single- and multi-agent economy
Single-agent economy Multi-agent economy
SS SU Together MS MU Together
Naive 2195.473 6902.896 4549.185 1334.283 5730.655 3363.378
Adaptive 851.448 2294.143 1572.796 95.881 457.204 262.646
α 1.460e-09 1.043e-10 6.236e-11 2.055e-08 6.933e-11 3.249e-10
Average k=2 2211.118 5890.815 4050.967 464.727 3178.432 1717.206
Average k=3 1706.730 4985.679 3346.204 277.133 1820.491 989.452
Weak 1879.963 6902.896 4543.419 1334.283 5730.655 4543.419
γW 0.999 3.939e-10 0.020 6.944e-11 3.569e-11 2.034e-9
Strong 1871.948 6524.073 4350.894 2805.214 6754.811 4649.085 γS 1.021 1.999 1.486 1.000 1.098 1.000 Anti 2084.141 5417.336 3804.821 437.693 3156.671 1716.133 γA -0.936 -0.683 -0.639 -0.559 -0.472 -0.491 AA 1969.788 7357.679 4663.734 2476.948 6541.700 4352.988 LAA 2045.152 7653.211 4849.181 2518.070 6516.515 4363.506 Bullard 851.449 2294.414 1572.797 94.129 457.204 262.645
g(1) 4.042e-10 8.929e-14 1.812e-11 0.221 2.667e-11 4.541e-12
Tuinstra 1032.363 2369.966 1701.164 97.449 505.513 285.786 AR(2) 857.281 2278.218 1625.976 93.991 428.706 311.142 α 0.999 0.942 0.161 0.000 0.000 0.999 β1 0.000 0.849 0.102 0.447 0.935 0.014 β2 0.000 0.903 0.090 0.391 0.499 0.007 Fundamentalist 852.906 2280.006 1566.456 96.496 428.706 261.847
