Can a heuristic switching model
explain expectations?
Bachelor thesis econometrics
Joël van Kesteren (10001962) University of Amsterdam Supervisors:
Prof. Jan Tuinstra
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Contents
1. Introduction ... 3
2. Theory ... 5
2.1 Overlapping generations model... 5
2.2 Heuristics ... 7
2.3 The heuristic switching model ... 9
3. Experiment ... 11
4. Results ... 12
4.1 Experimental data ... 12
4.2 Results model with individual heuristics ... 17
4.3 Results heuristic switching model ... 21
4.3.1 Heuristics Anufriev and Hommes (2012) ... 21
4.3.2 Optimal heuristics ... 22
5.Conclusion ... 27
6. References ... 30
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1. Introduction
Expectations play a prominent role in economic systems. Politicians and policy makers know that if the general public has negative expectations about economic performance, the chances of economic growth decline. Scared about the expected economic recession, the public becomes careful with its expenses. This causes less money to circulate in the economy and the public’s expectations turn out to be self-fulfilling: an economic downturn looms. As the expectations influence the realized value in the same direction, economists call this a positive expectations feedback system. Another typical example demonstrating the influential role of expectations in economics is the behavior of suppliers. Suppose suppliers expect the price of a good to go up. As they want to increase their profits, they will anticipate this expectation by producing the good as much as possible. However, this excess supply subsequently results in a
decrease of the price. Expectations now affect the realized value in an opposite direction, making this a negative expectations feedback system.
Both examples can be expressed in a more formal fashion; agents form
expectations about certain economic variables and adapt their behavior in accordance with these expectations. Consequently, this behavior influences the realized value of the economic variables. The expectations thus influence the realized value. Due to this important role of expectations in economic theory, it is evidently of key significance to know how these expectations are actually formed.
In classical economic theory rational expectations have been assumed for a long time (see seminal works of Muth, 1961, and Lucas, 1972), a presumption that is
nowadays increasingly rejected. The rational expectation hypothesis implies that every active economic agent has perfect information about the structure of the economy and the beliefs of all the other agents. In practice, agents do not have this knowledge and are unable to behave rational to this extent. Agents also typically show a significant amount
of irrational conduct.This makes the rational expectations hypothesis too demanding
and unrealistic.
Much academic research has since been done on more realistic expectation formation methods, thereby predominantly using the concept of boundedly rational agents (see e.g. Lucas, 1986, Marcet and Sargent, 1987). Boundedly rational agents are not perfectly rational as they are not familiar with the whole structure of an economy. They are more practical instead: they base their expectations on empirical data and in
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this process typically learn over time. Unfortunately, as Evans and Honkaphohja summarized (2001), research into bounded rationality so far resulted in a plethora of different outcomes and models, causing large academic discordance.
In order to theoretically describe how agents form expectations, economists have coined and developed a number of different expectation formation or learning rules. These learning rules all have in common that they base themselves on the previous realized economic variables and the previous expectations. The most basic learning rule is naïve expectations, using the last observed value of the realized economic variable for its current expectation. However, also many complex learning rules have been formulated. Which of these rules are most realistic and employed most often in practice, is still an issue of debate.
In an attempt to contribute to this debate on bounded rationality, this thesis examines whether a heuristic switching model can explain the expectation formation of agents. A heuristic switching model makes use of a predetermined number of learning rules or heuristics that are weighted for each period. These weights are determined based on the prior performance of the predictions of the heuristic. The model allows for heterogeneity between agents and is also able to explain how agents learn over time. The heuristic switching model is therefore theoretically a suitable tool for explaining expectation formation.
To test whether the heuristic switching model is useful in practice, this thesis applies it to a laboratory experiment conducted by the authors of Heemeijer et Al (2012). In this experiment participants are asked to forecast the inflation rate for fifty periods based on prior realized rates and predictions. By means of a standard
overlapping generations model this prediction determines how much a participant saves. The aggregates savings are consequently a key determinant of the actual rate of inflation. The model thus possesses an expectation feedback system, meaning that expectations influence realizations and vice versa.
This thesis first postulates eight learning rules based on the existent literature. These learning rules then serve as input for the heuristic switching model, which is then employed in an attempt to accurately describe the dynamics of the realized inflation rate in the experiment. This way it is tested whether the model can describe the realized inflation rate and thus the expectation formation of participants. Anufriev and Hommes (2012) found a heuristic switching model with four learning rules that was able to
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explain expectations in a different setting. This thesis tests whether these rules also perform well in a heuristic switching model that explains expectations in an overlapping generations model and, if not, it investigates a suitable alternative set of rules.
This thesis is structured as follows. Some relevant theory is presented in section 2, discussing first the overlapping generation model, then the learning rules and finally the heuristic switching model. Section 3 explains the experiment that is performed by the authors of Heemeijer et Al. (2012). The results are reviewed in section 4, and finally the main conclusions are reported in section 5.
2. Theory
As mentioned above, the relevant theory for this thesis will be clarified in three subsections: the overlapping generations model, the learning rules and heuristic switching model.
2.1 Overlapping generations model
The fundamental framework that is used is a standard overlapping generations model, similar to Bullard (1994) and Heemeijer et al. (2012). The overlapping
generations model is a dynamical economic model in which agents live for multiple periods. Successive generations of agents therefore exist together or overlap for a
certain number of periods. In the specific variant of the model that is used for this thesis, agents are assumed to live for two periods, a younger and an older period. In their
younger life they coexist with the previous generation and in their older life with the future generation.
Every agent in the model tries to maximize his utility function by means of consuming a single good 𝑐. The utility function of an agent or generation that is born at period t is expressed as 𝑈(𝑐𝑡,0, 𝑐𝑡,1), where 𝑐𝑡,0 and 𝑐𝑡,1 are the consumption in
respectively the agent’s younger and older life. The budget constraint for this agent then is 𝑝𝑡𝑐𝑡,0+ 𝑝𝑡+1𝑒 𝑐𝑡,1≤ 𝑝𝑡𝑤𝑡,0+ 𝑝𝑡+1𝑒 𝑤𝑡,1, where 𝑤𝑡,0 and 𝑤𝑡,1 denote income in younger
and later life, 𝑝𝑡 is the current price and 𝑝𝑡+1𝑒 is the price expectation for period t+1. If
the utility function satisfies some basic conditions, a unique solution (𝑐𝑡,0∗ , 𝑐𝑡,1∗ ) =
(𝑐0(𝜋𝑡𝑒), 𝑐1(𝜋𝑡𝑒)) exists, where 𝜋𝑡𝑒 designates the expected inflation rate 𝑝𝑡+1
𝑒
𝑝𝑡 . The amount of income that agents do not consume in their younger life is saved in a bank,
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Figure 1: Savings function S(πte). It can be seen that the graph has a steep slope between an
inflation factor of 1 and 1.1.
giving 𝑆(𝜋𝑡𝑒) = 𝑤𝑡,0− 𝑐0(𝜋𝑡𝑒) for the savings function.
The particular savings function 𝑆(𝜋𝑡𝑒) that is used for this thesis is as follows:
(1) 𝑆(𝜋𝑡𝑒) = 𝛿 + (1 − 𝛿) 𝑤0
1 + (𝑣𝜋𝑡𝑒) 𝜌 1−𝜌
𝛿 = 1.04, 𝜌 = 0.965, 𝑣 = 0.92, 𝑤0 = 0.9
Figure 1 shows a graph of this savings function. When the predicted inflation rate is smaller than minus 5%, agents save a maximum of around 0.95. If the forecasted inflation rate is above 15%, agents save a minimum of 0.4. The savings function further
has a steep decreasing slope between predicted inflation rates of 1% and 11%, ∆𝑆(𝜋𝑡𝑒)
∆𝜋𝑡𝑒
being -2.8 for these values.
According to economic theory the aggregate savings should equal real money balances in equilibrium:
(2) � 𝑆�𝜋𝑡,𝑗𝑒 � =𝑀𝑝𝑡 𝑡 𝑛
𝑗=1
In equation (1) 𝜋𝑡,𝑗𝑒 is the inflation expectation of agent 𝑗 and 𝑀
𝑡 the total money supply
that is exogenously determined by a government. This 𝑀𝑡 is multiplied each period by a
factor 𝜃, giving:
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Combining (1) and (2) provides the fundamental law of motion that explains the relation between expected inflation and realized inflation.
(4) 𝜋𝑡 = 𝜃𝜀𝑡 ∑ 𝑆�𝜋𝑡,𝑗 𝑒 � 𝑛 𝑗=1 ∑𝑛 𝑆�𝜋𝑡+1,𝑗𝑒 � 𝑗=1 𝜀𝑡~𝑈𝑁𝐼𝐹𝑂𝑅𝑀(0.975; 1.025)
As in the real world economic laws are usually not deterministic due to the
unpredictable nature of humans, an error term 𝜀𝑡 with a uniform distribution is added.
The law of motion causes the overlapping generations model to be an expectations feedback system. The expectations of the inflation rate for period t and period t+1 are the key determinants of the actual rate of inflation.
However, this model does not suggest how expectations are formed. The overlapping generation model reacts and develops differently in its dynamics to
different expectations. Rational expectations would for instance imply that 𝜋𝑡,𝑗𝑒 = 𝜗 for
any 𝑡, resulting in the simple law of motion 𝜋𝑡 = 𝜃𝜀𝑡. Other learning strategies result in
different laws of motion. The next subsection discusses a general framework of some heuristics that are known in the literature and used for the purposes of this thesis.
2.2 Heuristics
This thesis initially defines a number of basic learning rules that can be used in the heuristic switching model (HSM). In the HSM these rules compete with each other in an evolutionary way. The exact definition of a learning rule h is that it is a function 𝑓ℎ
that provides a prediction for the next period based on past predictions and realized inflation rates. This is mathematically expressed as:
(5) 𝜋ℎ,𝑡+1𝑒 = 𝑓ℎ(𝜋ℎ,𝑡𝑒 , 𝜋ℎ,𝑡−1𝑒 , … ; 𝜋𝑡−1, 𝜋𝑡−2, … )
Notice that due to the two lags in the fundamental law of motion in the overlapping generations model, the last observed realized inflation rate for a prediction for period t+1 is the inflation rate of period t-1. The simplest heuristic rule is naïve expectations (NAI), which sets its prediction equal to this last realized rate of inflation:
(6) 𝜋ℎ,𝑡+1𝑒 = 𝜋𝑡−1
As mentioned in the introduction, this thesis tests the performance of a set of heuristics that Anufriev and Hommes (2012) find adequate in explaining expectation and learning. The first heuristic they use is adaptive expectations (ADH):
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(7) 𝜋ℎ,𝑡+1𝑒 = 𝛼𝜋𝑡−1+ (1 − 𝛼)𝜋𝑡𝑒, 0 ≤ 𝛼 ≤ 1
The interpretation of this heuristic is that a weighted average is taken from the last realized observed rate of inflation and the last own predicted inflation. For 𝛼 = 1 the adaptive expectations is equal to naïve expectations, and for 𝛼 = 0 the heuristic is not influenced by the values of the realized rates. A second heuristic that Anufriev and Hommes (2012) use is trend-following expectations:
(8) 𝜋𝑡+1𝑒 = 𝜋𝑡−1+ 𝛾(𝜋𝑡−1− 𝜋𝑡−2)
Like naïve expectations, this heuristic takes the last realized rate of inflation, but in addition adapts its prediction for a trend that is observable in the last two rates. The value of the parameter 𝛾 then determines how much this trend is taken into account in the formulated prediction. Anufriev and Hommes (2012) propose a weak
trend-following rule (WTR, 𝛾=0.4) and a strong trend-trend-following rule (STR, 𝛾=1.3). A special case of the trend-following heuristic occurs when 𝛾 < 0. In the literature this is called trend-reverse expectations or – and this is the designation this thesis follows – anti-trend behavior (ANT). A final heuristic that Anufriev and Hommes (2012) include in their model is the learning anchoring and adjustment (LAA) heuristic (Tversky and Kahneman, 1974):
(9) 𝜋𝑡+1𝑒 = 0.5 �� 𝜋𝑗 𝑡−1 𝑗=1
+ 𝜋𝑡−1� + (𝜋𝑡−1− 𝜋𝑡−2)
This heuristic takes an equally weighted average of the average realized inflation rates and the last observed realized inflation rate, which is called the reference point or anchor. To this anchor it adds a trend term, being simply the last change in inflation (so in contrast to the trend-following rules no parameter is used).
Despite LAA, the presented learning rules are short-term, signifying that they only take into account the most recent realized inflation rates and predictions. An alternative is a learning rule such as LAA that takes into account all past realized rates of inflation. Bullard (1994) for instance, assumes that agents believe that the inflation
rate 𝛽 is constant in a perceived law of motion 𝑝𝑡 = 𝛽𝑝𝑡−1. They then run a least squares
regression on the prices in order to estimate this rate of inflation 𝛽. Tuinstra and Wagener (2007) show that the estimate of 𝛽 can be rewritten into the following heuristic (which this thesis denotes Bullard’s rule, BUL):
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(10) 𝜋𝑡+1𝑒 = 𝑔𝑡−1𝜋𝑡−1+ (1 − 𝑔𝑡−1)𝜋𝑡𝑒
𝑔𝑡−1 = (𝑔𝑡−2−1(𝜋𝑡−2)−2+ 1)−1= 𝑝𝑡−12 / � 𝑝𝑠−12 𝑡 𝑠=1
BUL is an adaptive heuristic with a parameter 𝑔𝑡−1 that declines over time.
Tuinstra and Wagener (2007) however prove that due to the fact that 𝑝𝑡 increases as t
becomes larger, 𝑔𝑡−1 will not reach 0. This prevents BUL from stabilizing its predictions
for higher monetary growth rates 𝜃. An alternative that Tuinstra and Wagener (2007) propose as a more reasonable learning rule is average expectations (AVG):
(11) 𝜋𝑡+1𝑒 =𝑡 − 1 � 𝜋1 𝑗 𝑡−1 𝑗=1
AVG is also based on agents performing a least squared regression on the
inflation. However, they regress on the perceived law of motion 𝜋𝑡 = 𝛽, causing their
predictions to become stable over time.
2.3 The heuristic switching model
Having discussed the learning rules that are considered in this thesis, the
heuristic switching model itself can now be presented. The HSM is largely based on the one presented by Anufriev and Hommes (2012), though it is in this thesis applied to the overlapping generations model. Let 𝐻designate a set of individual learning rules. Any learning rule ℎ ∈ 𝐻 at each period t formulates a prediction 𝜋ℎ,𝑡+1𝑒 for the next period. In
addition, the heuristic is also assigned a relative weight 𝑛ℎ,𝑡, where ∑ℎ∈𝐻𝑛ℎ,𝑡= 1 must
apply. The weighted average of all heuristic predictions 𝜋𝑚,𝑡+1𝑒 = ∑ℎ∈𝐻𝑛ℎ,𝑡𝜋ℎ,𝑡+1𝑒 is then
inserted into the fundamental law of motion of the overlapping generations model: (12) 𝜋𝑚,𝑡 = 𝜃 𝑆(𝜋𝑚,𝑡
𝑒 )
𝑆(𝜋𝑚,𝑡+1𝑒 )
As the savings function is decreasing, this implies that if the weighted average of the heuristic predictions for t+1 is higher than the former weighted average
(𝜋𝑚,𝑡+1𝑒 ≥ 𝜋𝑚,𝑡𝑒 ), the predicted inflation of the model will be higher than the monetary
growth rate (𝜋𝑚,𝑡 ≥ 𝜃). Due to the relatively steep slope of the savings function, small
differences in consecutive weighted averages actually result in large deviations from the monetary growth rate. If, however, the two weighted averages of the heuristic
predictions are equal ( 𝜋𝑚,𝑡𝑒 = 𝜋𝑚,𝑡+1𝑒 ), the model inflation 𝜋𝑡,𝑚 stabilizes at the
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The relative weight 𝑛ℎ,𝑡of a certain learning rule h is determined by the prior
performance of this learning rule. This prior performance is denoted as 𝑈ℎ,𝑡−1. The
following equation applies for 𝑛ℎ,𝑡
(13) 𝑛ℎ,𝑡 = 𝛿𝑛ℎ,𝑡−1+ (1 − 𝛿)exp�𝛽𝑈𝑍 ℎ,𝑡−1� 𝑡−1
A weighted average is thus taken over the prior weight and a normalized current
performance measure. This normalized current performance measure is 𝑍𝑡−1 =
∑ℎ∈𝐻exp (𝛽𝑈ℎ,𝑡−1). In the special case that 𝛿 = 1 the impact of heuristic rule h (𝑛ℎ,𝑡) will
not change over time. If 𝛿 = 0, it will each period be fully adjusted to the most recent performance.
The performance measure of learning rule h is defined as 𝑈ℎ,𝑡−1= −(𝜋𝑡−
𝜋ℎ,𝑡−1𝑒 )2+ 𝜂𝑈ℎ,𝑡−2. It is therefore based on the sum of the negative quadratic forecasting
error for the current period and the previous performance multiplied by a factor 𝜂, with0 ≤ 𝜂 ≤ 1 . 𝜂 can be considered the memory; if 𝜂 = 0, only the most recent quadratic forecasting error is taken into account for the performance measure.
However, if 𝜂 gets closer to 1, previous quadratic forecasting errors are weighted higher. The described model is programmed in MATLAB. The predicted inflation rates for all heuristics are set at 3 for the first two periods, describing the first two
predictions of participants that cannot be based on any realized inflations. Also, all heuristics are in the first period assigned an equal weight (𝑛ℎ,𝑡) of 0.25. Given this initial
state, the model simulates the 50 consecutive periods that are used in the experiment. For any period t the model goes through the following steps:
- The performance 𝑈ℎ,𝑡−1until period t-1 is calculated for each heuristic.
- Relative weights 𝑛ℎ,𝑡 are assigned to these heuristics based on their performance.
- Forecasts 𝜋ℎ,𝑡+1𝑒 are calculated for each heuristic.
- The weighted average 𝜋𝑚,𝑡𝑒 is determined.
- The predicted model inflation 𝜋𝑚,𝑡 is calculated.
The only unknown parameters of the model are β, η and δ. This thesis estimates these parameters on the experimental data by means of minimizing the mean squared
error between the model prediction 𝜋𝑚,𝑡 and realized inflation rate of the
experiment 𝜋𝑡. Anufriev and Hommes (2013) show that, assuming independent and
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results in the maximum likelihood estimators 𝛽̂, η� and δ�. By considering the mean squared error, this thesis tests to what extent the HSM is able to describe the dynamics of the realized inflation rate and thus the expectation formation of participants in the experiment.
3. Experiment
The authors of Heemeijer et Al. (2012) designed an experiment that is based on the presented overlapping generations model. This thesis uses the data from this experiment. The experiment was conducted on April 25, 2006, in the laboratory of the Centre for Research in Experimental Economics and political Decision making (CREED) at the University of Amsterdam. A total of 78 students participated in the experiment, which lasted for approximately ninety minutes. Two treatments were used, one with a relatively high level of monetary growth (𝜃 = 1.11) and one with a low level (𝜃 = 1.01). Participants were randomly assigned to the treatment with a low level of monetary growth, the stable treatment, or to the treatment with a high level of monetary growth, the unstable treatment. In total seven groups of six participants were designated to the stable treatment and six groups of six participants to the unstable treatment. These treatments are for the purposes of this thesis particularly interesting as it makes it possible to observe which learning techniques are most successful under which economic circumstances.
Participants were provided with some qualitative information on the model. Most importantly, they were told that their individual expectations together with those of five other agents influenced how much they would save in the model, and therefore the value of the realized inflation rate. No quantitative substantiation of this statement was given.
For fifty consecutive periods participants had to forecast the rate of inflation. The program showed for each prediction for period t+1 the previous predictions until
period t and realized inflation rates until t-1. This lag of two periods between prediction and realized inflation is due to the law of motion of the overlapping generations model, which uses the prediction for t+1 to calculate the realized inflation for period t.
Participants therefore also needed to predict a 51th inflation rate.
To stimulate realistic performance, participants were incentivized and rewarded depending on the difference between their prediction and the actual rate of inflation.
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The following formula applied for the number of points that a subject i in a certain period t could earn:
(14) 𝑃𝑖�𝜋𝑖,𝑡𝑒 , 𝜋𝑡� = max�100 − 400�𝜋𝑖,𝑡𝑒 − 𝜋𝑡�, 0�
Points from each round were added and the total amount of points in the end was converted into euros. For this conversion 200 points were equivalent to one euro. If participants correctly predicted all inflation rates, they could earn a maximum of 25 euros.
4. Results
Having discussed the theory and experimental design, this thesis now turns to the question whether the experimental realized inflation rate can successfully be explained in a model. This section discusses the results of these simulations. Initially, the experimental data on the realized inflation and the predictions of participants are described in section 4.1. Eight heuristics, that are partly based on the literature (see 2.2) and partly derived from the data, are then individually used to simulate predictions. These predictions result in a modelled inflation that is compared to the realized inflation rate of the experiment in 4.2. In section 4.3 the heuristic switching model is discussed. Initially the model is optimized and compared to the realized inflations with a benchmark set of heuristics, those of Anufriev and Hommes (2012). Then, the optimal heuristics for the HSM are determined and the performance is discussed of this HSM when fitted to the data.
4.1 Experimental data
For the thirteen groups of both treatments the data consists of six time series with predictions and a time series for the realized inflation, both for 50 periods1. The
predictions of each of the six participants and the realized inflation rate are illustrated in figure 2 for GS2 (second group op the stable treatment) and GU3 (third group of the unstable treatment). In the upper graph (GS2) the realized inflation rate in the first periods is relatively unstable, fluctuating between 10% and -10%. This instability is caused by participants that are changing their predictions within two periods, so that for agent i 𝜋𝑖,𝑡𝑒 − 𝜋
𝑖,𝑡+1𝑒 is large. From the fundamental law of motion in equation 3 (see
section 2.1) it can be seen that this causes the realized inflation rate to significantly
13 -15 -10 -5 0 5 10 15 20 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 In fla tio n Time (periods)
GS2: Predicted and realized inflation per period
Pred P1 Pred P2 Pred P3 Pred P4 Pred P5 Pred P6 Realized -40 -20 0 20 40 60 80 100 120 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 In fla tio n Time (periods)
GU3: Predicted and realized inflation per period
Pred P1 Pred P2 Pred P3 Pred P4 Pred P5 Pred P6 Realized
Figure 2: Predicted and realized inflation for GS2 and GU3. Participants of GS2 are able to stabilize the inflation rate, while those of GU3 are not. Note the scale differences for the inflation for both groups.
deviate from 𝜃 = 1.01. However, as the predictions itself vary less and appear to
coordinate on each other, GS2 manages to stabilize the realized inflation rate quite well, causing the average errors to become low. This is certainly not the case for GU3 in the lower graph. Throughout the experiment the realized inflation rate varies from -20% to 60% (notice the scale difference), again implying large differences in consecutive
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volatile easily. Although participants coordinate on each other for the first periods, after that the predictions deviate quite from each other. The average prediction errors of the participants are relatively large.
Table 1 provides further information on the average and variance of the realized inflation rate for each of the groups of both treatments. In addition, it also shows the average individual error, which as Hommes et Al. (2005) prove, can be written as the
sum of a dispersion error and a common error2. The dispersion error gives the average
distance between an individual prediction and the average prediction, and is therefore an expression of how much participants coordinate. The common error states the difference between the average prediction and the realized inflation rate.
From table 1 it can be seen that the average of all groups in both treatments approaches the monetary growth rate very closely. More remarkable is the variance, which is extremely high for almost all groups in the stable treatment (>100) and 2 𝑇ℎ𝑖𝑠 𝑖𝑠 𝑚𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑒𝑑 𝑎𝑠: 1 6 ∗ 50 � �(𝜋𝑖,𝑡𝑒 − 𝜋𝑡)2=6 ∗ 501 50 𝑡=1 6 ℎ=1 � �(𝜋𝑖,𝑡𝑒 − 𝜋�𝑡𝑒)2+ 50 𝑡=1 6 ℎ=1 1 50 �(𝜋�𝑡𝑒− 𝜋𝑡)2 50 𝑡=1 Descriptive statistics Group GS=Stable, GU=unstable Average realized inflation Variance realized inflation Avg individual error Avg common error Avg dispersion error Avg dispersion error (%) Avg common error (%) GS1 1.46 104.42 182.58 155.24 27.34 15% 85% GS2 0.96 10.18 18.14 14.12 4.02 22% 78% GS3 1.93 193.93 340.00 289.14 50.86 15% 85% GS4 1.87 183.86 776.71 494.32 282.39 36% 64% GS5 0.72 7.81 16.05 12.30 3.75 23% 77% GS6 1.32 125.95 246.83 211.67 35.16 14% 86% GS7 1.03 57.11 92.82 82.04 10.78 12% 88% GU1 14.07 501.66 1062.31 751.02 311.29 29% 71% GU2 13.33 407.97 886.43 637.17 249.26 28% 72% GU3 13.29 409.01 750.17 641.31 108.86 15% 85% GU4 12.96 257.31 455.96 360.68 95.28 21% 79% GU5 14.20 640.69 1201.16 953.82 247.34 21% 79% GU6 13.90 605.34 1319.10 1077.91 241.18 18% 82%
Table 1: Table shows descriptive statistics for the data from the experiment. The variance and average individual error are relatively high. Most of the individual error is due to the common error.
15 -100 -50 0 50 100 150 0 10 20 30 40 50 In fla tio n Time (periods) GU5-6 -60 -40 -20 0 20 40 0 10 20 30 40 50 In fla tio n Time (periods) GS6-1 Realized inflation unstable treatment (>400). Only the groups GS2 and GS5 seem to have been able to stabilize the realized rate of inflation at some period, as they have a variance that is much lower (around 10). The overall high variance shows that the realized inflation rate in general fluctuated heavily and often for all 50 periods. As the predictions of the participants did not stabilize the realized inflation rate, this proves that the
environment of the overlapping generations model is quite unstable.
The table further suggests a link between the variance and the average individual error. As the realized inflation for GS2 and GS5 has the lowest variance, participants of these groups also make the lowest individual error. GU5 and GU6 however both, have the highest variance and average individual error. The intuition behind this finding is that when the variance is higher, it is harder to predict the
inflation, making the average individual error higher as well. What table 1 also shows is that the average individual error is for around 80% due to a common error, and only for the remaining 20% due to a dispersion error. As individual errors are in general quite high, it can be derived that the predictions of the six participants are coordinated but mistaken: participants use prediction techniques that are quite similar, but collectively they are unable to predict the realized inflation rate.
In order to see what learning techniques participants used, figure 3 illustrates
Figure 3: For GS6-1 and GU5-6 figure shows the predicted inflation (blue) and a lag of two periods for the realized inflation (red). Learning strategies such as naïve and adaptive expectations can be observed.
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the predictions of two individual participants and the realized inflation rates. The realized inflation rate is shifted two periods, which is why the first realized inflation rate is in the third period. As the realized inflation from period t-2 was the last inflation rate that participants could see at period t, the shift makes it easy to see how the
prediction was based on the last observed realized inflation. The upper graph, depicting the expectations of participant one in group six of the stable treatment (GS6-1), shows that for the first 11 periods the expectation and realized inflation are equal, which
suggests that the participant uses naïve expectations (NAI). When he3 finds out that this
naïve behaviour probably does not result in large payoffs, he changes his strategy. He now appears to take a weighted average of his own last prediction and the last realized inflation rate (ADA). From period 20 he also occasionally shows some anti-trend following (ANT) behaviour, which can be explained by the intuitive reasoning that this might stabilize the system.
The second graph shows the sixth participant in the fifth unstable treatment (GU5-6), who also starts off with clear naïve behaviour. As this again does not perform satisfactorily, he decides to turn to an anti-trend following heuristic. He then changes again, and predicts one value for the remaining periods. This value might for instance be based on an estimation of the average realized inflation rates in the past.
In general participants start off with a heuristic such as naïve, adaptive or trend-following expectations. However, at a certain stage, almost all participants find out that the consistent use of a short-term heuristic does not perform satisfactorily. The
following period is characterized by much trial and error of different intuitions and strategies. Participants tend to switch much between different heuristics, which provides an argument for this thesis’ use of a heuristic switching model. However, to stabilize the system some constancy is required, implying that this switching usually causes to increase the volatility of the realized inflation even more. Nevertheless, some participants remain in this trial and error phase throughout the entire experiment. Other participants realize that short-term heuristics are not able to stabilize the system, and start taking into account longer periods. These participants then for instance revert to average expectations. In this process they let recent dynamics of the realized inflation only slightly influence their predictions. Note that the last paragraph is only based on
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observations of the predictions and is therefore not substantiated by any statistical evidence4.
4.2 Results model with individual heuristics
In its goal to explain the dynamics of the realized inflation rate, this thesis first examines whether individual heuristics are able to do so. For the model with a single heuristic it is assumed that all agents are homogeneous and consistent over time. The previous section showed that agents coordinate on each other, providing evidence for the homogeneity. However, the assumption of consistency is less convincing, as participants typically switched their prediction strategy over time. Nevertheless, the model with individual heuristics serves as a useful benchmark for the HSM. Moreover, having observed the performances of individual heuristics, it is also easier to grasp the dynamics of the HSM.
Instead of inserting the weighted average of all heuristics 𝜋𝑚,𝑡𝑒 in the law of
motion – as is done for the HSM - , the individual heuristics prediction 𝜋ℎ,𝑡+1𝑒 is now
inserted. To see how well the model performs, the mean squared error of the model prediction of heuristic h 𝜋𝑚,𝑡 is calculated for each group k:
(15) 𝑀𝑆𝐸(𝑘) = 501 (𝜋𝑚,𝑡(𝑘) − 𝜋𝑡(𝑘))2
The mean squared errors are then averaged over all groups k in either the stable or the unstable treatment. Table 2 shows the average MSEs for the eight heuristics that are discussed in section 2.3.1. The parameters of ADH, WTR and STR are copied from the specification of Anufriev and Hommes (2012). The parameter of ANT is set at -0.5, which is loosely based on observations and the results of the regressions.
The MSEs are in general high, indicating that the fit of the models is not very accurate. For all heuristics these MSEs are higher in the unstable treatment than in the stable treatment, which can be explained by the higher variance of the latter. A higher variance makes the model more prone to large errors and thus a higher MSE. Especially the trend-following rules (WTR, STR and LAA) have excessive MSEs. The other short-term heuristics ADH, NAI and ANT perform slightly better, but their predictions still deviate much from the realized inflation. The models with the longer term heuristics
4 This thesis endeavoured to perform regressions on the predictions of participants. It used the last three
observed realized inflation rates and last three predictions as regressors. This did not lead to significant results though, which can be explained by the amount of switching and inconsistency between prediction strategies of participants. See Appendix A for the results of the regressions.
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AVG and BUL explain the realized inflation best. BUL has the lowest MSE for the stable treatment, and AVG for the unstable treatment. However, note that in unstable
treatment the difference between AVG and BUL is quite high.
Figure 4 illustrates the realized inflation (red) of GS3 and the modeled inflation for the heuristics ADH, NAI, BUL, WTR, ANT and AVG. STR and LAA are not shown because they are largely similar to WTR. Typical of ADH is that it appears to lag behind one period to the realized inflation rate. This can be explained from the interaction of this heuristic with the law of motion from the overlapping generations model. In formulating its prediction 𝜋ℎ,𝑡+1𝑒 , ADH takes a weighted average of its previous
prediction 𝜋ℎ,𝑡𝑒 and the realized inflation of two periods ago 𝜋𝑡−1. Suppose that this
realized inflation rate 𝜋𝑡−1 is relatively high, so that it is larger than the previous
prediction 𝜋ℎ,𝑡𝑒 . This then implies that the current prediction will be higher than the
previous prediction 𝜋ℎ,𝑡+1𝑒 > 𝜋ℎ,𝑡𝑒 . The prediction of the model with ADH 𝜋𝑚,𝑡 is given the
law of motion then also relatively high (that is, higher than 𝜃). Although purely based on intuitive reasoning, the value of the realized inflation rate at period t-1 therefore in general corresponds to the value of the model prediction at period t, explaining that the dynamics of the model with ADH appear to lag behind one period.
The model with the naïve heuristic is typically more volatile than the dynamics of the realized inflation rate. Due to the specifications of the law of motion (see equation 7),
Heuristic Abbreviation Avg MSE
stable treatment
Avg MSE unstable treatment
Adaptive (0.65) ADH 661 3,963
Naïve NAI 1,344 5,753
Bullard’s rule BUL 95 1,063
Weak trend-following (0.4) WTR 2,003 6,386
Strong trend-following (1.3) STR 3,082 6,907
Anti trend-following (-0.5) ANT 470 3,178
Average AVG 98 509
Anchoring and adjustment LAA 2,519 6,554
Table 2: Average mean squared error (MSE) of models with individual heuristics. BUL and AVG have the lowest MSE and thus best performance in respectively the stable and unstable treatment.
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Figure 4: Realized inflation (red line) and modeled inflation for six individual heuristics for GS3. Left are shown from top to bottom ADH, NAI and BUL; right from top to bottom WTR, ANT and AVG. especially the steep savings function, relatively small changes in realized inflation rates lead to extrapolated predictions of the NAI model. Suppose for instance the realized inflation rate goes from 3% in period t-2 to 6% in period t-1, then the prediction of the model with naïve expectations for period t is 12%. If it conversely goes from 6% to 3%, its prediction is -9%. The NAI model extrapolates the large volatility in the realized inflation itself, and this leads to inaccurate predictions.
This effect is even stronger for the WTR model, as it not only takes into account the last observed realized inflation, but also a certain trend. Suppose that the realized inflation at period t-3 is for reasons of convenience equal to the realized inflation of period t-2, then the increase from 3% to 6% leads to a predicted inflation of the WTR model of 17% for period t. A converse trend leads to an inflation of -11%. This
demonstrates that small changes in the dynamics of the realized inflation cause the volatility of the WTR model to explode, which explains its large MSE.
ANT actually performs better than all other short-term heuristics as its specification causes the formulation of its expectation to be less volatile than the
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realized inflation. The difference between its prediction for t+1 and t ( 𝜋ℎ,𝑡+1𝑒 -𝜋ℎ,𝑡𝑒 ) is
lower, therefore reducing the tendency of the law of motion to explode. The models with the long-term heuristics AVG and BUL perform even better for GS3. For them, the difference between their predictions typically becomes smaller over time as the new observed realized inflation is weighted less. They are therefore not hindered by the tendency of the law of motion to forecast extreme model predictions out of relatively small differences in the heuristic predictions (𝜋ℎ,𝑡+1𝑒 -𝜋ℎ,𝑡𝑒 ). As these differences even
become zero over time, the model predictions 𝜋𝑚,𝑡𝑒 stabilize. Even though the realized
inflation rate is for some cases quite volatile, this stabilization leads to a lower MSE than the inaccurate extrapolations of the short-term heuristics.
What is remarkable from table 2 is the equal performance of AVG and BUL in the stable treatment, but the large difference in MSE for the unstable treatment. Tuinstra and Wagener (2007) show that as 𝜃 becomes larger, every new realized inflation has relatively more impact for the prediction of BUL. For a higher 𝜃 BUL becomes a more short-term rule, being largely similar to an adaptive heuristic. In contrast, AVG keeps weighing all realized inflations equally, so the weight of a new realized inflation declines. For the unstable treatment AVG therefore stabilizes its model prediction, while BUL does not (see figure 5 for the modeled predictions of AVG and BUL in GU3). Figure 5 also provides proof that BUL becomes similar to an adaptive heuristic, as the BUL model prediction clearly lags one period behind the realized inflation rate. In the unstable treatment the realized inflations itself are quite erratic, causing short-term heuristics including BUL to make predictions that are even more extreme than the discussed example of GS3. This causes the MSE to be much higher than in the case of a heuristic that is able to stabilize its prediction.
Figure 5: Realized inflation (red line) and modeled inflation for AVG (left) and BUL (right) for GU3. It can be seen that AVG stabilizes its prediction, while BUL does not.
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4.3 Results heuristic switching model
4.3.1 Heuristics Anufriev and Hommes (2012)
As section 3.1 showed that participants typically switched often between prediction rules and strategies, a heuristic switching model (HSM) might be a more adequate model to explain the realized inflation rate. First of all, it is interesting to see whether the heuristics that explained the expectation formation of participants in an asset pricing experiment, also do so in the overlapping generations model. Therefore the performance of a HSM with the heuristics ADH, WTR, STR and LAA is tested. Similar to Anufriev and Hommes (2012), a benchmark choice of learning parameters is used (η=0.7, δ=0.9, and β=0.4) and the parameters are fitted to minimize the mean squared error of the model. The mean squared error (MSE) is calculated as in equation 15. The average MSE of the benchmark and fitted HSM is shown in table 3, including the average optimized learning parameters. To see whether the HSM performs in comparison to the model based on individual heuristics, the ADH model is also included.
Most notably, the benchmark HSM performs for both treatments worse than the ADH model. When optimized, the HSM performs slightly better. Figure 6 illustrates the weights for GS3 (left) and GU4 (right) in each period for the fitted HSM, a graph that is called the transition path. It can be seen that after a short competition in the beginning, ADH comes out strongest and with a weight of one dominates and determines the entire
model5. Only in this beginning phase, the model performs a little better than the pure
ADH model. As explained in the previous section, the trend-following heuristics WTR,
5 A graph of the modeled inflation for the HSM with ADH, WTR, STR and LAA is not shown for this reason. It will
be largely similar to the graph of the ADH model depicted in figure 4. Average mean squared error and estimated
parameter value Stable treatment Unstable treatment
Benchmark HSM (η=0.7, δ=0.9, β=0.4) 1,143 4,539 Fitted HSM 620 3,701 Adaptive (0.65) 661 3,963 η 0.97 0.86 δ 0.27 0.27 β 9.33 4.65
Table 3: Average mean squared error of a benchmark and fitted HSM with the heuristics ADH, WTR, STR and LAA. The results are compared to the ADH-model of section 4.2. Also, the estimated learning parameters for the HSM are demonstrated.
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STR and LAA all have difficulties in keeping their predictions within reasonable
boundaries. This makes them susceptible to extreme prediction errors, decreasing their
performance 𝑈ℎ,𝑡−1 and weight 𝑛ℎ,𝑡. ADH predicts more stable, and is therefore best
equipped for the overlapping generations environment. This explains why it dominates the other rules and gains a weight of one.
4.3.2 Optimal heuristics
As this thesis aims to find a HSM that at least performs better than any model with an individual heuristic, it calculated the performance of all possible combinations of four out of six identified heuristics. It chose to not include STR and LAA in this initial set of candidates, as their performances in the model with individual heuristics were simply too weak to have any explanatory power in the HSM. Four heuristics are considered a perfect number, as this is relatively parsimonious but should still be enough to explain a wide range of dynamics. Table 4 shows the average MSE as calculated by equation 15 for both treatments for all possible combinations in a HSM where the learning parameters are optimized.
What should initially be remarked is the extremely weak performance of the HSM in the only case that neither AVG nor BUL are selected, indicating that these rules should be included. Besides this combination, all other combinations in the stable treatment have an average MSE of somewhat below the MSE of the optimal model with an individual heuristic (which is 95 for the BUL model). The optimal selection of
heuristics for the stable treatment is NAI, BUL, ANT and WTR with an average MSE of 78. Figure 6: Weights per period for the HSM with ADH, WTR, STR and LAA for GS3 and GU4. It can be seen that ADH in both cases gains a weight of one.
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Table 4: The average mean squared error is shown for all combinations of four out of six heuristics ADH, NAI, BUL, ANT, AVG and WTR. The optimal set of heuristics is NAI, BUL, ANT and WTR for the stable treatment and ADH, NAI, AVG and WTR for the unstable treatment.
The difference in performance between the model with individual heuristics and the optimal MSE is remarkably small though. The optimal set of heuristics for the unstable treatment (ADH, NAI, AVG and WTR) is not even able to outperform the lowest MSE of the individual heuristic models at all. All combinations of heuristics result in a higher MSE than the 509 for the AVG model. For both treatments, it should therefore be noticed that the performance of the MSE is disappointingly weak.
To explain this weak performance, this thesis further investigates the optimal set of heuristics for the stable and unstable treatment. For the optimal set of heuristics in the stable treatment (NAI, BUL, ANT and WTR), three graphs are depicted in figure 7. These graphs elucidate the performance of the model for the representative group GS3. The graph in the upper left corner shows the transition path for the four heuristics NAI, BUL, ANT and WTR in the model. It can be seen that the beginning phase is
characterized by a fierce competition between all four rules, most notably BUL and WTR. However, after approximately fifteen periods the competition has provided a clear winner in BUL, which stays in this position for the remainder of the experiment. The
ADH NAI BUL ANT AVG WTR Avg.
stable Avg. unstable x x x x 87 1,176 x x x X 88 689 x x x x 79 1,020 x x x x 88 709 x x x x 461 2,662 x x x x 82 518 x x x x 95 808 x x x x 82 1,111 x x x x 86 593 x x x x 85 645 x x x x 89 820 x x x x 78 1,204 x x x x 83 617 x x x x 81 648 x x x x 87 911
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Figure 7: Three graphs for the optimal HSM for the stable treatment (GS3). Heuristics that are included are NAI, BUL, ANT and WTR. Upper left graph shows the weights of these heuristics per period. Lower left graph shows the predictions of the heuristics. Graph on the right illustrates the realized inflation and model inflation.
predicted inflation of the HSM is from this time therefore completely determined by BUL, making it for the majority of the periods similar to the BUL model from section 4.2. This clarifies why the performance of the HSM is not significantly better than the
individual heuristic model.
The graph in the lower left corner shows the predictions of the four heuristics. It can be seen that in the beginning phase, all predictions coordinate reasonable well, explaining why their performances and weights remain relatively equal during this period. However, as soon as the short-term heuristics NAI, WTR and ANT are
confronted with the volatile dynamics of the realized inflation rate, their predictions become extreme. Between the periods of 20 and 30, which is typically the trial and error phase of participants, WTR and NAI predict inflation rates ranging respectively from -40% to 80% and -20% to 50%. Considering both, the volatile realized inflation due to the trial and error of participants and the extreme predictions of the short-term heuristics,
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the error of these predictions inevitably becomes large. This then causes the performance of these short-term heuristics to decrease and therefore explains why their weights become zero. In contrast, BUL is for this stable treatment able to stabilize its prediction already in an early phase, leading to relatively smaller prediction errors and a better performance. This better performance consequentially results in a weight of one.
What this graph further illustrates is that NAI, WTR and ANT actually coordinate quite well throughout the entire experiment, implying that their predictions do not deviate much from each other. The result of this is that the model, though formally consisting of four heuristics, only provides two significantly different alternatives. One of the usual benefits of the HSM is that it allows for heterogeneity but, with this
specification of heuristics, this advantage might not be fully utilized. In other words: with these coordinated heuristics, it is harder for the model to explain a large variety of dynamics in the realized inflation rate. The participants of the experiment might be more heterogeneous than what this model allows for, providing a possible reason why the HSM performs unsatisfactorily.
The third graph on the right depicts the realized inflation and the model inflation for each of the fifty periods. It shows that prior to the ultimate ‘victory’ of BUL the model actually performs quite well, as the blue line is able to simulate the dynamics of the realized inflation rate. Both go during the same periods up, down and then up again. This provides evidence that participants in GS3 actually used short-term heuristics such as WTR, BUL (or some sort of adaptive heuristic) and NAI. However, after 15 periods, when only BUL determines the model, the blue line (or model) is not able to explain the movements of the realized inflation rate anymore. BUL gaining a weight of one does therefore not mean that the participants unanimously use this rule; it rather implies that this heuristic is in some sense safest in explaining the behaviour of the realized inflation rate. Due to the stabilization effect it is after all not prone to make immense ‘lethal’ mistakes such as the other short-term heuristics. As the model only provides two relevant alternatives, BUL or the other short term heuristics, the fact that the short term heuristics make large errors automatically makes BUL the winner.
In practice, participants seem to use different techniques than the available heuristics in this HSM, proven by the fact that the model is not able to explain the dynamics of the realized inflation rate after 15 periods. As described in section 4.1,
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participants typically enter a trial and error phase during this phase. Only after this phase, when the participants are usually able to stabilize the realized inflation rate, the difference between the model prediction and realized inflation declines again. The dominance of BUL therefore does seem to make sense in the long run, indicating that participants eventually use a long-term heuristic. However, there is a large middle part (the ‘trial and error’ phase) that the current HSM is not able to explain. Due to the typical inconsistency and volatility of such a trial and error phase, it is however questionable whether any model can do so.
The argument that a stabilizing heuristic is the safest and therefore has the best performance applies even more to the unstable treatment. Remarkable for the unstable treatment is that any combination that includes the stabilizing AVG, is able to keep its MSE below 1000. The combinations that omit AVG however, all have a MSE higher than 1000. This serves as evidence for the indispensability of AVG as the only stabilizing heuristic in the HSM for the unstable treatment. As AVG is present in all optimal combinations of heuristics for the HSM, but these combinations do not perform better than a model with AVG individually, this suggests that AVG must be the clear winner in the HSM. Indeed this is the case; as the right graph in figure 8 illustrates for GU2 AVG gains an impact of one after approximately ten periods.
Most reasoning of the previous paragraph actually applies to the optimal HSM of the unstable treatment (containing the rules ADH, NAI, AVG and WTR) as well. Again, the rules coordinate on each other for the first periods, not affecting their weights, but
decently explaining the realized inflation in its dynamics6. This implies that participants
do employ the short-term heuristics ADH, NAI and WTR. However, after five to ten periods, the short-term heuristics do not satisfactorily explain the realized inflation anymore as they are inclined to make extreme predictions (for instance ranging from -60% to 80% for WTR) and therefore large errors due to the unstable environment. Similar to the optimal HSM for the stable treatment, the short-term heuristics
coordinate on each other during the entire experiment, indicating that the HSM only has two significantly deviant options. As the short-term heuristics inevitably make large errors, the result is that the alternative heuristic AVG, the most stable, consistent and therefore least ‘dangerous’ option, obtains an impact of one. The weights of the
6 In general, though, the HSM describes the dynamics in the beginning phase less accurately than the AVG
model. As the AVG performs better and the heuristic AVG reaches an impact of one for all unstable groups, it after all has to perform worse in this beginning phase.
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Figure 8: Three graphs for the optimal HSM for the unstable treatment (GU2). Heuristics that are included are ADH, NAI, AVG and WTR. Upper left graph shows the weights of these heuristics per period. Lower left graph shows the predictions of the heuristics. Graph on the right illustrates the realized inflation and model inflation.
term heuristics dwindle to zero.
For this particular group though, the domination of AVG at 50 periods still does not seem to make sense, as the realized inflation is not stabilized and still fluctuates. Arguably, if the experiment would have had more periods, AVG would become accurate. The winning of AVG within these 50 periods does therefore not imply that the
participants unanimously employ this heuristic, but solely indicates that this is the least erroneous heuristic in the model. How the participants actually predict during the middle and later stages of the experiment (the trial and error phase), is not
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5. Conclusion
This thesis has investigated whether a realized inflation based on expectations in an experimental setting can be simulated and explained by a heuristic switching model. In doing so, it aimed to find out how the participants of the experiment and agents in general form expectations, a question that is of great relevance to economic theory. It found that the optimal heuristic switching models were not able to simulate the realized inflation satisfactorily, being only accurate during the very first and – for the groups that were able to stabilize its inflation – last periods. The models were therefore not able to explain most of the dynamics of the realized inflation rate, causing their mean squared errors to be quite high. In summary, this thesis has provided three reasons for this failure of the heuristic switching model: the instability of the environment, the unpredictable behavior of the participants and the specification of the heuristics.
To start off the overlapping generations model proved to be an unstable
environment. This is mainly due to the specification of the savings function, which has a
steep slope, making the outcome of the fundamental law of motion 𝜋𝑡 very sensitive to
small differences between the expectation for t+1 𝜋𝑖,𝑡+1𝑒 and the expectation for t 𝜋 𝑖,𝑡𝑒 ,
and leading it to easily deviate from the monetary growth rate. This caused both, the realized inflation rates from the experiment and the simulated model inflations, to be very volatile. To fit a model that is sensitive to small changes to an erratic time series, proved to be hard and resulted in large mean squared errors.
Secondly, the way participants of the experiment reacted to this unstable environment, made a good fit of the HSM even more complicated. In the beginning period, participants typically used short-term strategies such as NAI, WTR or ADH. In the long-run, many of them reverted to longer term stabilizing heuristics such as AVG. As mentioned before, these periods are quite reasonably explained by the HSM. However, the HSM is generally not able to cope with the dynamics of the realized inflation throughout the entire middle part of the experiment. During this period participants – having found out that their short-term strategies do not perform
convincingly – generally enter a trial and error phase, in which they try many different forecasting strategies and techniques. This trial and error phase is due to its
inconsistency, illogicality and unpredictability hard to reproduce in a model, which is another explanation why the HSM performed relatively weak.
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adequate to explain all dynamics of the realized inflation rate. Most notably, it is shown that the coordination of the short-term heuristics is generally high, giving the model not enough alternatives to simulate the realized inflation rate. A more diverse set of
reasonable heuristics should therefore be able to give a better fit. Also, short-term heuristics such as NAI, WTR and ADH proved to be only accurate in the beginning phase, as the unstable environment over time induced them to explosive predictions. These predictions generally had extreme squared errors. The fact that they are only suitable for the beginning explains why the heuristics that were identified by Anufriev and Hommes (2012) performed weak in the overlapping generations environment.
Following this reasoning, it can be argued that the HSM should include more heuristics that are less unstable in their predictions, and take into account a longer term.
The first and second reasons that are given to explain the unsatisfactory performance of the model discuss factors that cannot be altered. However, the third reason provides suggestions for further research. It is after all not generally proven that the HSM is not able to explain the dynamics of the realized inflation in the overlapping generations model, it is only showed that this is hard. While considering the predictions of participants much switching and heterogeneity is observed, providing enough
evidence to retain the claim of the introduction that the HSM is a suitable tool to explain expectations. However, the specific heuristics that this thesis examines are not able to capture all of the dynamics observed in the realized inflation. This indicates that more research should be done on the identification and specification of a decent set of heuristics. In this research, this thesis can serve as a good starting or reference point.
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6. References
Anufriev, M. and C. Hommes (2012): “Evolutionary Selection of Individual Expectations and Aggregate Outcomes in Asset Pricing Experiments,” American Economic Journal: Microeconomics, 4(4), 35-64.
Anufriev, M. and C. Hommes (2013): “Evolutionary Selection of Expectations in Positive and Negative Feedback Markets”, Working paper.
Bullard, J. (1994): “Learning equilibria,” Economic Theory, 64, 468-485
Evans, G.W. and S. Honkaphja (2001): “Learning and Expectations in Macro-economics,” Princeton: Princeton University Press.
Heemeijer, P., C. Hommes, J. Sonnemans and J. Tuinstra (2012): “An Experimental Study on Expectations and Learning in Overlapping Generation Models,” Studies in Nonlinear Dynamics and Econometrics, 16(4), 1.
Hommes, C., J. Sonnemans, J. Tuinstra and H. Van de Velden (2005): “Coordination of Expectations in Asset Pricing Experiments”, Review of Financial Studies, 18, 955-980.
Lucas, R.E. (1972): “Expectations and the Neutrality of Money,” Journal of Economic Theory, 4(2), 103-124
Lucas, R.E. (1986): “Adaptive Behavior and Economic Theory,” Journal of business, 59, S401-S426.
Marcet, A., and T.J. Sargent (1987): “Convergence of Least Squares Learning Mechanisms in Self-referential Linear Stochastic Models,” Journal of Economic Theory, 48, 337-368.
Muth, J. F. (1961): “Rational Expectations and the Theory of Price Movements,” Econometrica, 29(3), 315-335
Tversky, A., and D. Kahneman (1974): “Judgment under Uncertainty: Heuristics and Biases,” Science, 185, 1124-1130.
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Appendix A
The following regression was performed on the predictions of participants: 𝜋𝑖,𝑡𝑒 = 𝛼 + 𝛽
1𝜋𝑡−2+ 𝛽2𝜋𝑡−3+ 𝛽3𝜋𝑡−4+ 𝛾1𝜋𝑖,𝑡−1𝑒 + 𝛾2𝜋𝑖,𝑡−2𝑒 + 𝛾3𝜋𝑖,𝑡−3𝑒
For all estimated models there is no autocorrelation in the residuals. The most economical model with significant coefficients is chosen. The following table shows the regression results for all participants (P1 until P6) in all groups of the stable (GS1-GS7) and unstable treatment (GU1-GU6).
Partic. 𝛼 𝛽 1 𝛽2 𝛽3 𝛾1 𝛾2 𝛾3 𝑅2 GS1P1 3,69 (1,04) -0,52 (0,15) -0,34 (0,15) 0,25 GS1P2 2,10 (0,79) -0,39 (0,15) 0,15 GS1P3 GS1P4 4,17 (1,46) -0,43 (0,15) 0,19 GS1P5 0,81 (0,63) -0,14 (0,06) 0,14 GS1P6 3,94 (0,97) 0,17 (0,09) 0,08 GS2P1 0,73 (0,17) -0,27 (0,07) 0,30 GS2P2 0,40 (0,30) -0,36 (0,12) 0,18 GS2P3 GS2P4 GS2P5 GS2P6 0,32 (0,25) 0,28 (0,10) 0,16 GS3P1 2,76 (0,85) 0,14 (0,06) 0,14 GS3P2 9,09 (2,5) -0,40 (0,15) 0,16 GS3P3 GS3P4 4,07 (0,7) 0,08 (0,05) 0,08 GS3P5 3,2 (0,9) -0,13 (0,06) 0,13 GS3P6 0,53 (0,34) -0,05 (0,02) -0,04 (0,02) 0,13 GS4P1 3,42 (1,45) -0,22 (0,11) 0,11 GS4P2 4,01 (1,70) 0,57 (0,14) 0,45 (0,15) 0,37 GS4P3 3,15 (1,08) 0,19 (0,09) 0,43 (0,09) -0,49 (0,12) 0,31 (0,11) 0,64 GS4P4 0,48 (1,43) 0,4 (0,15) 0,48 (0,14) 0,34 GS4P5 6,3 (0,8) 0,31 (0,07) 0,39 GS4P6 2,87 (0,30) -0,23 (0,05) -0,34 (0,05) 0,68 GS5P1 1,11 (0,42) 0,61 (0,09) -0,32 (0,116) 0,31 (0,17) 0,12 (0,12) 0,71 GS5P2 0,50 (0,25) 0,30 (0,08) -0,27 (0,089) 0,45 (0,15) 0,47 GS5P3 0,74 (0,21) 0,17 (0,03) 0,03 (0,033) 0,57 (0,10) 0,72 GS5P4 1,60 (0,39) 0,16 (0,15) -0,30 (0,14) 0,12 GS5P5 2,12 (0,64) 0,82 (0,15) 0,17 (0,20) -0,45(0,15) 0,64 GS5P6 1,02 (0,24) -0,14 (0,05) 0,40 (0,14) 0,26 GS6P1 3,94 (1,13) 0,44 (0,01) 0,33 GS6P2 1,28 (0,58) 0,460 (0,05) 0,19 (0,049) 0,13 (0,052) 0,75 GS6P3 6,47 (0,80) 0,00 GS6P4 3,94 (0,75) 0,40 (0,06) 0,26 (0,077) -0,45 (0,17) 0,63 GS6P5 2,43 (1,91) 0,43 (0,17) 0,15 GS6P6 1,76 (0,91) 0,32 (0,08) 0,31 GS7P1 2,22 (0,88) 0,23 (0,12) 0,09 GS7P2 2,49 (0,59) 0,00 GS7P3 1,83 (0,76) 0,07 (0,15) -0,31 (0,15) 0,10 GS7P4 4,76 (0,36) 0,09 (0,02) 0,03 (0,018) -0,15 (0,08) 0,44
32 Partic. 𝛼 𝛽 1 𝛽2 𝛽3 𝛾1 𝛾2 𝛾3 𝑅2 GS7P5 0,87 (0,42) 0,14 (0,05) 0,27 (0,14) 0,28 GS7P6 4,26 (0,43) 0,07 (0,06) -0,13 (0,057) 0,19 GU1P1 8,83 (2,38) 0,00 GU1P2 14,95 (5,51) 0,00 GU1P3 GU1P4 13,43 (1,71) -0,19 (0,07) 0,18 GU1P5 3,37 (2,9) 0,37 (0,11) 0,23 GU1P6 GU2P1 15,82 (2,05) -0,36 (0,15) 0,13 GU2P2 15,65 (2,36) -0,42 (0,15) 0,18 GU2P3 21,13 (3,01) -0,42 (0,13) 0,22 GU2P4 11,23 (3,46) 0,00 GU2P5 17,88 (5,06) 0,00 GU2P6 10,25 (2,11) 0,00 GU3P1 GU3P2 GU3P3 18,64 (2,88) -0,31 (0,11) -0,3 (0,10) 0,22 GU3P4 GU3P5 15,57 (1,87) -0,45 (0,15) 0,20 GU3P6 GU4P1 GU4P2 GU4P3 -0,12 (0,48) 0,06 (0,02) 0,77 (0,11) 0,58 GU4P4 4,65 (1,76) 0,22 (0,07) 0,19 (0,07) 0,24 GU4P5 16,6 (2,73) 0,2 (0,09) -0,31 (0,15) 0,19 GU4P6 8,33 (3,01) 0,21 (0,09) 0,39 (0,14) 0,29 GU5P1 6,80 (3,898) 0,035 (0,12) 0,26 (0,12) 0,12 GU5P2 18,16 (2,814) 0,00 GU5P3 10,51 (1,844) -0,427 (0,06) 0,51 GU5P4 3,58 (3,901) 0,280 (0,12) 0,26 (0,12) -0,22 (0,15) 0,15 GU5P5 10,90 (4,667) 0,021 (0,14) -0,30 (0,14) 0,13 GU5P6 11,25 (4,036) 0,195 (0,11) -0,28 (0,17) 0,23 (0,180) 0,10 GU6P1 15,65(4,341) 0,4667 (0,15) 0,17 GU6P2 8,45 (2,629) 0,00 GU6P3 3,69(2,467) 0,373 (0,08) 0,32 (0,11) 0,47 GU6P4 7,31 (1,558) 0,523 (0,06) 0,66 GU6P5 8,83 (0,131) 0,36 (0,14) 0,13 GU6P6
