Stability analysis of individual rules in learning-to-forecast and -optimize experiments

University of Amsterdam

Bachelor Thesis

Econometrics and Operations Research

Stability Analysis of Individual Rules in

Learning-to-Forecast and -Optimize

Experiments

This thesis investigates whether the differences of individual rules in learning to forecast and -optimze experiments explain the differences in market behavior. In previous research it was found that more frequent and larger bubbles occured in learning to optimize experiments. I find that the individuals in the learning to optimize group have significantly higher trend coefficients, which may explain the more frequent bubbles. The price system of the estimated rules in the learning to optimize treatment also have complex eigenvalues more often. Since complex eigenvalues lead to price oscillations, this also supports the more frequent occurrence of bubbles.

Contents

4 Estimated Rules 8

4.1 Adaptive Expectations . . . 8 4.2 Trend Extrapolation Rule . . . 10 4.3 Rules in Learning to Optimize Treatment . . . 11

5 Comparison of the LtF and LtO rules 13

5.1 Comparison of the coefficients . . . 13 5.2 Stability comparison . . . 14

6 Comparison with Previous Research 17

7 Conclusion 21

A Steady State of the Process 23

B Proof of Proposition 3.1 24 C Proof of Proposition 3.2 25 D Proof of Proposition 3.3 26 E Proof of Proposition 4.3 27 F Proof of Proposition 4.6 28 G Proof of Proposition 5.1 29 H Proof of Proposition 5.2 29

1

Introduction

Individual expectations have a big effect on the market prices in financial markets. When people’s expectations are too optimistic, this can lead to prices that are significantly higher than the fundamental price. For instance, when stock are rising, traders may believe that the price will rise even more and buy more stocks. This results in even higher stock prices. There is feedback since the higher prices result in even higher prices. Over time the prices and expectations develop with mutual feedback. When the price finally comes down, pessimistic expectations may lead to prices that are significantly lower than the fundamental price. This effect can make a recession even worse than it already would have been. A recent example of this phenomenon is the recent housing prices bubble in the United States. For years overly optimistic expectations resulted in massive growth of the housing prices and when the price finally went down, overly pessimistic expectations may have caused a deeper crisis. Since people’s expectations have such a great influence on the economic market, it is important to study the way they form. In the last decades more attention has been devoted to studying these bubbles, e.g. De Bondt and Thaler (1987).

Historically, little attention has been devoted to bubbles. A possible reason for this is that bubble do not form according to standard economic theory (Muth, 1961; Fama, 1970; Lucas Jr, 1972). In early papers investigating financial market simple rules were used to predict individual behavior, e.g. Ezekiel (1938) amd Goodwin (1947). But in the 1960s the theory of Rational Expectations was introduced, which became widely used by economists. This theory is now considered part of standard economic theory. The Rational Expectations theory assumes that all individuals make optimal decisions and know the entire market structure including the demand and supply curves. Assuming this theory is correct, long-time over or under-evaluation of the market price would not occur, i.e. bubbles. Although some studies have shown bubbles can exist when people act rationally (Tirole, 1985), this only happens in very special circumstances (Santos and Woodford, 1997).

In order to explain the occurrence of bubbles in the market, a theory of Bounded Rationality was developed (Simon, 1957). Unlike with full rationality, bounded rationality does not assume that people know the entire workings of the market. Because of this, bounded rationality is more realistic than rational expectation theory, since it is unlikely that individuals know the entire market structure. Bounded rationality assumes that all individuals follow a certain underlying rule that can be estimated (Sargent, 1993). According to these rules, agents make decisions based on the previous prices and

expectations.

Laboratory experiments are used to study bubbles, since people usually fundamentally disagree about the fundamental price of an asset in empirical data. In laboratory experiments, the entire environment can be controlled, which makes it possible to quantify bubbles. The first experiments to successfully reproduce bubbles belong to a group called Learning-to-Optimize (henceforth LtO) experiments (e.g. Smith et al. (1988)). In LtO experiments people are asked to directly trade an asset. Marimon et al. (1993) introduce another type of experiments called Learning-to-Forecast (henceforth LtF) experiments. In these types of experiments the subjects are asked not to directly trade, but instead to forecast prices. These forecasts will be used to calculate the realized price in the next period.

A recent experiment using both a LtF and LtO treatment is done by Bao et al. (2014). The authors ran an experiment using an underlying market with positive feedback. This means that higher expected prices will result in higher realized prices. This type of feedback is characteristic of speculative asset markets. Bao et al. (2014) have three treatment groups: (1) a group that is asked to only make price forecasts, (2) a group that is only asked to make trading decisions and (3) a group where the subjects are asked to make both forecasts and trading decisions (a mixed design). They find that the prices in

the LtO and mixed treatment converged much slower or not at all. This is different than expected, since all three treatments should give the same results when subjects make forecasts and trading decision in the same way. A certain price forecast directly corresponds to a certain trading decision that is optimal, so this would result in the same market developments. Because of the big difference between the LtO and LtF treatments, the question arises why this is the case. What are the differences between the individual prediction rules estimated in the two groups? And can the difference in bubble forming be explained by these differences? I use the data from the Bao et al. (2014) experiment to compare the individual rules estimated for the LtF groups with those estimated for the LtO groups. I specifically analyze the stability of the rules, investigate whether the expected price development has oscillations, and if these things can explain the observed market patterns. Then I compare the estimated rules with those of similar experiments like the experiment done by Heemeijer et al. (2009).

This paper is organized as follows. Section 2 discusses experimental design of the experiment of which data is used. Section 3 describes and analyzes Benchmark rules. Section 4 describes and analyzes stability for the estimated rules in the experiments. Section 5 compares the rules of the different treatment groups. Section 6 compares the rules of the Bao et al. (2014) paper with those estimated in a similar experiment, and Section 7 concludes.

2

Data

I use data from an experiment done by Bao et al. (2014). The authors ran an experiment with three treatments: (1) a learning to forecast treatment, (2) a learning to optimize treatment and (3) a mixed treatment. The subjects in the LtO treatment were asked to make trading decisions for 50 periods. The LtO treatment consisted of 6 groups containing 6 individuals each. The individuals were all asked to submit a quantity between -5 and 5 each period to indicate how much they wanted to buy or sell. In the LtF group, there were only 4 groups containing 6 individuals each. The individuals were all asked to make one period ahead price forecasts for 50 periods. The subjects in the mixed treatment were asked to make both price forecasts and trading decisions. I focus on the first two groups.

In the experiment, the market price is determined using a price adjustment mechanism, given by pt+1= pt+ 20 21 6 X i=1 zi,t+ t, (1)

where t ∼ N ID(0, 1) is a small idiosyncratic shock, pt+1 is the price in period t+1 and zi,t is the

amount of the asset that subject i wants to trade in period t.

This price adjustment mechanism is one of positive feedback, i.e. higher price expectations lead to higher prices. This is because higher demand (higher zi,t) will result in higher prices.

For the LtO treatment the realized prices are calculated using equation 1. For the LtF group, the price forecasts are first translated into trading decisions using the following function

zi,t =

pe_i,t+1+ 3.3 − 1.05pt

6 (2)

The realized prices are calculated by inserting equation 2 into equation 1 resulting in the following function pt+1= 66 + 20 21(¯p e t+1− 66) + t (3) where ¯pe t+1= 16 P6

For a more complete description of the experimental design, see the paper by Bao et al. (2014).

3

Benchmark Rules

Most of the early papers investigating financial markets used simple prediction rules to predict individual behavior (Ezekiel, 1938; Goodwin, 1947; Nerlove, 1958). An example of one of these rules is na¨ıve expectations. Here agents assume the price is exactly equal to the realized price of the previous period. When individuals use this rule they are always one step behind, resulting in systematic prediction errors. Similar errors occur with the other forecasting rules. Since the forecasting errors are systematic, some argued that it unlikely that they would actually be used. They believed that smart, rational agents would learn from their mistakes and would adjust their behavior. This led to Muth (1961) introducing the Rational Expectations theory, which assumes that all individuals know the underlying market and thus make perfect predictions. This theory became widely used and part of standard economic theory. Among supporters of it were Nobel Prize winners Thomas Sargent and Robert Lucas Jr (Lucas and Sargent, 1981). Critics argue that it is unrealistic that individuals know the entire underlying market including the demand and supply curves. Another thing Rational Expectations cannot explain is the formation of bubbles. Bubbles form when individuals over- or under-evaluate the price for multiple periods. Under rational expectations, this does not happen. In this section I investigate the prediction rules that were used before Rational Expectations theory became popular. These rules are mostly very simple and intuitive and despite the fact that they are so old, are still used in many recent studies investigating individual rules, e.g. Heemeijer et al. (2009), Hommes et al. (2005) and Assenza et al. (2013). I inspect the following three rules: (1) a rule of na¨ıve expectations, (2) a rule of average expectations and (3) an anchoring and adjustment rule. I compare the price development when all subjects use one of these rules with the realized price development in Bao et al. (2014)’s experiments to see how well they fit the observed results. I use the same market system as Bao

et al. (2014) analyze the rules, i.e. equations 1 and 3. When using this market system, the steady state of the price is equal to the fundamental price of 66 for all forecasting rules and most quantity rules (see Appendix A for a proof). Throughout this paper, I will investigate rules assuming homogeneous

expectations, i.e. all individuals use the same rule to determine their prices/quantities.

3.1 Na¨ıve Expectations

The first benchmark rule is the rule of na¨ıve expectations. All subjects using this rule expect the price to be the same in the next period as the one in this period:

pe_it= pt−1. (4)

The realized price only depends on the previous price:

pt=

66 21 +

20

21pt−1. (5)

Proposition 3.1. The price system of the na¨ıve expectations rule has an eigenvalue of 20₂₁. This means the steady state is asymptotically stable and the price will go to the steady state monotonically under homogeneous expectations.

The price development of a sample of this rule can be observed in Figure 1. Here the starting price forecast is chosen as 50 and in the remaining periods the 6 subjects use the na¨ıve expectations rule. The price goes slowly towards the fundamental price of 66, where it will stay. It can be seen that towards the end of the 50 periods the realized prices overshoot the fundamental price. The reason for this is that the same errors as in the Bao et al. (2014) paper are used, which are with the exception of one all positive in the last 8 periods. Since this overshooting of the price goes against the prediction, it might be possible that the overshooting of the fundamental value in the LtF groups may have been solely because of the shocks and not because of the price predictions.

The price development seen in the figure resembles the price development in the LtF groups of the Bao et al. (2014) experiment, but the expectations following the prices was not observed. Because of this it seems unlikely that most individuals were using na¨ıve expectations, but it is possible some of them were.

Figure 1: Realized market prices and expectations when all subjects use na¨ıve expectations.

3.2 Average Expectations

The second benchmark rule is the average expectations rule. When a subject uses this rule, it means he expects the price to be the average of all previous prices:

pe_it= 1 t − 1 t−1 X j=1 pj for t > 1. (6)

Proposition 3.2. The price system of the average expectations rule has an eigenvalue of 0 and an eigenvalue that converges from below to 1. This means that the steady state is asymptotically neutrally stable.

Proof. See Appendix C

Since one eigenvalue converges from below to 1, the price will go slowly towards the fundamental price. It will stay at the value it reaches after a while, even if this is not the fundamental price. The reason for this is that the subjects use all previous prices and give equal weight to all of them. When

the number of previous periods increases, the effect of the price prediction diminishes. The underlying market system gives realized prices closer to the fundamental price than the predicted prices, since the realized prices are a weighted average of the predicted price and the fundamental price. But when the number of previous periods is sufficiently large, this is not enough to get the price to go to the fundamental price. This can also be seen in Figure 2, where a sample is plotted of individuals using average expectations with an initial forecast of 50. In the beginning the price goes down because of negative shock and after that the price slowly goes towards the fundamental price. The price only rises very slowly and instead of the fundamental price, a steady state of around 51 is reached. This price development seen here does not look like the price developments observed in the experiment. From this it can be concluded that it is unlikely that the individuals in the experiment follow an average expectations rule.

Figure 2: Realized market prices and expectations when all subjects use average expectations.

3.3 An Anchoring and Adjustment Rule

The third and last benchmark rule I investigate is described by Hommes (2011) as an anchoring and adjustment rule. This rule was first theorized by Tversky and Kahneman (1974), and assumes a subject first makes an estimation and then uses this estimation as an anchor point from which he makes adjustments. Formally, the rule described by Hommes is as follows

pe_it= 66 + pt−1

2 + (pt−1− pt−2) (7)

Proposition 3.3. When all subjects use the anchor and adjustment rule, the price system has an eigenvalue of 0 and two complex eigenvalues with length

q

20 21.

Proof. See Appendix D

This rule is stable, since all eigenvalues lie inside the complex unit circle. Since two eigenvalues are complex, the price will not converge to the steady state monotonically, but instead the price will oscillate around the fundamental price. These oscillations dampen, since the rule is stable. This can also be observed in Figure 3, where a sample of individuals using this rule is plotted. Like before, the

is also taken as 50. It can be seen in the figure that the fundamental price is used as an anchor and the price oscillates around it. The size of the bubbles becomes smaller and the price will eventually become the fundamental price. However, this takes a long while and after 50 periods the steady state still is not reached.

In the Bao et al. (2014) experiment similar price developments were observed, with the main difference that the bubbles were further apart. Since the individuals in the LtF treatment do not know the fundamental price, it seems unlikely that they would use this rule. In the LtO and mixed treatment however, the subjects do know the fundamental price (or can derive it) and it is possible they use a rule close to the anchor and adjustment rule. It seems unlikely most individuals use this rule, since in the experiment bubbles format most once every 10 periods, while in Figure 3 it can be seen that bubbles form on average more than once every 5 periods.

Figure 3: Realized market prices and expectations when all subjects use the anchoring and adjustment rule pe_it= 60+pt−1

2 + (pt−1− pt−2).

4

Estimated Rules

Since the benchmark rules cannot accurately describe the observed market developments in the experiments, individual rules are estimated. Heemeijer et al. (2009) ran a similar experiment as Bao et al. (2014) Instead of comparing forecasting individuals with trading individuals, they compared forecasting individuals subject to either positive or negative feedback. Their market system was almost the same as the one used by Bao et al. (2014) with the main difference that the fundamental price was slightly lower (60 compared to 66). Based on their results Bao et al. (2014) estimated two types of rules for all individuals in the Learning to Forecast treatment: (1) a rule of adaptive expectations and (2) a trend extrapolation rule. For the individuals in the Learning to Optimize treatment they estimated a quantity rule, consisting of an autoregressive and return extrapolation component.

4.1 Adaptive Expectations

A very frequently used expectations rule is the one of adaptive expectations, introduced by Nerlove (1958). When subjects use this rule, they predict the price to be a weighted average of the observed

and the expected price in the last period, or equivalently

pe_i,t = αpt−1+ (1 − α)pei,t−1 (8)

with α ∈ [0, 1].

The rules of na¨ıve and average expectations are examples of adaptive expectations rules. For the na¨ıve expectations rule α = 0 and for the average expectations rule α = t−2_t−1.

Proposition 4.1. When all agents use the adaptive expectations rule, the price system has eigenvalues of 0 and 1 − ₂₁1α. The steady state is asymptotically stable for 0 < α < 42 and unstable when α is smaller than 0 or bigger than 42.

Proof. The price system can be written as

pt= 66 + 20 21(p e t− 66) = 20 21αpt−1+ 20 21(1 − α)p e t−1+ 66 21 (9) pe_t = αpt−1+ (1 − α)pet−1 (10) =⇒ " pt pe_t # = " 20 21α 20 21(1 − α) α (1 − α) # " pt−1 pe_t−1 # + " 66 21 0 # = A " pt−1 pe_t−1 # + " 66 21 0 # where A = " 20 21α 20 21(1 − α) α (1 − α) #

This process is stable when the solutions λ of the equation det(A − λI) = 0 fall inside of the complex unit circle.

det(A − λI) =      20 21α − λ 20 21(1 − α) α (1 − α) − λ      = 20 21α − λ  ∗ (1 − α − λ) − α20 21(1 − α) = −λ 20 21α + (1 − α)  + λ2= 0 ⇒ λ₁ = 0 ∩ λ2 =  20 21α + (1 − α)  = 1 − 1 21α |1 − 1 21α| < 1 for 0 < α < 42

The adaptive expectations rule is neutrally stable when α is equal to 0 or 42 and is stable when α lies in between 0 and 42. Since the α is restricted to be between 0 and 1, this rule is stable for all subjects. This means that if individuals use this rule, the price would go towards the fundamental price.

4.2 Trend Extrapolation Rule

Another frequently used expectations rule is a trend extrapolation rule, introduced by Goodwin (1947). This rule is given by

pe_i,t = pt−1+ γ(pt−1− pt−2) (11)

Here subjects take the price in the previous period and extrapolate using the trend of the previous two periods. Na¨ıve expectations is a special case of this rule with γ = 0. The previously discussed anchor and adjustment rule also uses trend extrapolation, but uses the average of the fundamental price and the previous price instead of just the previous price. When subjects don’t know the fundamental price, they can only use the previous price as an anchor.

Proposition 4.2. When all individuals use the trend extrapolation rule the price system has eigenvalues of 0, 10₂₁(1 + γ) +1₂ q 20 21(1 + γ)
2 −80 21γ and 10 21(1 + γ) − 1 2 q 20 21(1 + γ)
2 −80 21γ.

Proof. The price system when the trend extrapolation rule is used is as follows pt= 66 + 20 21(p e t − 66) = 20 21pt−1+ 20 21γ(pt−1− pt−2) + 66 21 (12) = 20 21(1 + γ)pt−1− 20 21γpt−2+ 66 21 (13) =⇒ " pt pt−1 # = " 20 21(1 + γ) − 20 21γ 1 0 # " pt−1 pt−2 # + " 66 21 0 # = A " pt−1 pt−2 # + " 66 21 0 # where A = " 20 21(1 + γ) − 20 21γ 1 0 #

The eigenvalues are the solutions λ of the equation det(A − λI) = 0.

det(A − λI) =      20 21(1 + γ) − λ − 20 21γ 1 −λ      = 20 21(1 + γ) − λ  ∗ −λ + 20 21γ = λ2− λ20 21(1 + γ) + 20 21γ = 0 ⇒ λ_1,2= 10 21(1 + γ) ± 1 2 s  20 21(1 + γ) 2 −80 21γ

In this case the eigenvalues can be either real or complex.

Proposition 4.3. The eigenvalues of the market system, when all subjects use a trend extrapolation rule, are

• real and lie outside the complex unit circle for γ > 1.56 ∩ γ < −1.025 • complex and lie inside the complex unit circle for γ between 0.64 and 1.05 • complex and lie outside the complex unit circle for γ between 1.05 and 1.56 Proof. See Appendix E

The realized price development has dampening oscillations around the fundamental price for λ between 0.64 and 1.05 and monotonic convergence to the steady state for γ between -1.025 and 0.64. This means that a higher trend coefficient leads to a more unstable system. This makes sense, since this means individuals follow the trend more and the price is more likely to overshoot because of it. For all individuals positive trend coefficients are estimated. This makes sense, since trend following behavior is rewarded in a market with positive feedback. Only one individual has an estimated γ bigger than 0.64. All individuals for which the trend following rule is estimated have a stable rule.

4.3 Rules in Learning to Optimize Treatment

In the Learning to Optimize treatment subjects are not asked to make price predictions, but instead make trading decisions. Bao et al. (2014) estimated a rule composed of two different elements: (1) an AR(1) component and (2) an asset return component. The estimated rule is as follows

qi,t = constanti+ χiqi,t−1+ φi(pt−1+ 3.3 − 1.05pt−2) (14)

The AR(1) component of the rule is simply

qi,t= χiqi,t−1 (15)

Proposition 4.4. If all subjects use the quantity rule qi,t = χiqi,t−1, then the steady state is

asymptot-ically neutrally stable when χ ∈ [−1, 1] and unstable when χ > 1 or χ < −1.

Proof. When all subjects use the quantity rule specified above, then the price system is as follows pt= pt−1+ 120 21 zt−1 (16) = pt−1+ 120 21 qt (17) = pt−1+ 120 21 χqt−1 (18) =⇒ " pt qt # = " 1 0 120 21χ χ # " pt−1 qt−1 # = A " pt−1 qt−1 # where A = " 1 0 120 21χ χ #

This process is stable when the solutions λ of the equation det(A − λI) = 0 fall inside of the complex unit circle.

det(A − λI) =      1 − λ 0 120 21χ χ − λ      = (1 − λ)(χ − λ) ⇒ λ1 = 1 ∩ λ2 = χ

The process is neutrally stable when all eigenvalues lies inside the complex unit circle and at least one lies on the complex unit circle. From this follows that the rule is neutrally stable when χ ∈ [−1, 1]. The rule is unstable when χ > 1 or χ < −1.

It seems unlikely that individuals use this rule, since the trading decisions will simply become smaller or larger with the same factor every period. Despite this, the rule was estimated for three individuals in the Learning to Optimize treatment (out of 27 estimated rules). Most estimated rules, however, have an asset return extrapolation element. Four individuals have a return extrapolation element in combination with an AR(1) element, but the majority (17 individuals) have solely a return component with possibly a constant.

Proposition 4.5. When all subjects use the quantity rule qi,t= constanti+ φi(pt−1+ 3.3 − 1.05pt−2),

the eigenvalues of the price system are equal to λ1,2 = 1₂ 1 +120₂₁φ
±1₂

q

1 +120₂₁φ
2

− 24φ. Proof. When all subjects use the return extrapolation rule, the price system is as follows

pt= pt−1+ 120 21 zt−1 (19) = pt−1+ 120 21 qt (20) = pt−1+ 120 21 (constanti+ φi(pt−1+ 3.3 − 1.05pt−2)) (21) =⇒ " pt pt−1 # = " 1 +120₂₁φ −6φ 1 0 # " pt−1 pt−2 # + " 120 21(3.3φ + constant) 0 # = A " pt−1 pt−2 # + " 120 21(3.3φ + constant) 0 # where A = " 1 +120₂₁φ −6φ 1 0 #

This eigenvalues are the solutions λ of the equation det(A − λI) = 0. det(A − λI) =      1 +120₂₁φ − λ −6φ 1 −λ      =  1 +120 21 φ − λ  ∗ −λ + 6φ = λ2− λ  1 +120 21 φ  + 6φ = 0 ⇒ λ1,2 = 1 2  1 +120 21 φ  ±1 2 s  1 +120 21 φ 2 − 24φ

Proposition 4.6. The eigenvalues of the price system when all subjects use the quantity rule qi,t =

constanti+ φi(pt−1+ 3.3 − 1.05pt−2) are

• real and lie inside the complex unit circle for φ ∈ [0, 0.113]

• real and lie outside the complex unit circle for φ ∈ (−∞, 0) ∪ [0.273, ∞) • complex and lie inside the complex unit circle for φ ∈ (0.113, 0.1667] • complex and lie outside the complex unit circle for φ ∈ [0.1667, 0.273) Proof. See Appendix F

For only one individual in the Learning to Optimize treatment a negative estimated trend coefficient φ was estimated. This means most individuals expect the asset return in the next period to have the same sign as the one in this period. The process is neutrally stable when φ = 0 or φ = 1₆, is stable when φ lies between 0 and 1₆ and unstable when φ lies outside these values. The price development will have oscillations for φ between 0.113 and 0.273. The estimated φ lies in between these values for most individuals (18 out of 27), so bubbles are expected based on the estimated rules.

5

Comparison of the LtF and LtO rules

5.1 Comparison of the coefficients

Bao et al. (2014) show that for an optimizing agent, the individual demand equals zi,t =

pe_i,t+1+ 3.3 − 1.05pt

6 (22)

The same equation is used to transform the forecasting decisions of those in the LtF into trading decisions to calculate the realized prices. Assuming that individuals try to make optimal decisions, the rules in the LtF and LtO groups can be directly compared because of this. To do this, one of the two rules needs to be rewritten.

Proposition 5.1. Assuming optimizing agents, the quantity rule

qi,t = constanti+ χiqi,t−1+ φi(pt−1+ 3.3 − 1.05pt−2)

can be rewritten as a price forecast rule. This forecasting rule equals

pe_it=χipeit−1+ 1.05(χi+ 6φi)∆pt−1+ (1.05 − 1.05χi− 0.3φi)pt−1

+ (6 ∗ constanti+ (χi− 1)3.3 + 19.8φi) (23)

Proof. See Appendix G.

Both the LtF and LtO rule can now be written as the general rule

pe_it= αpe_it−1+ γ∆pt−1+ βpt−1+ c (24)

For the quantity rule estimated for the LtO treatment α = χi

β = (1.05 − 1.05χi− 0.3φi)

γ = 1.05(χi+ 6φi)

c = (6 ∗ constanti+ (χi− 1)3.3 + 19.8φi)

For the LtF groups the rule is a combination of the two types of rules estimated for the LtF group. When α is taken as 0 (and β is taken as 1), the rule is one of trend extrapolation. When γ is taken as 0 and α as 1 − β, the rule is one of adaptive expectations.

In Figure 4 the α, β and γ for the LtF individuals are plotted against those for the LtO individuals. It can be seen that for the individuals in the Learning to Forecast treatment two types of rules are estimated. None of the individuals in the Learning to Optimize treatment follow the adaptive expectations rule, but a large number of them follow a rule very close to the trend extrapolation rule. The individuals that have a rule close to the trend extrapolation rule are those that have a forecast coefficient α of 0. This is because when the α is 0, the β will be close to 1 (1.05 − 0.3φ), which means the general rule reduces to the trend extrapolation rule. For these individuals the trend coefficient γ is on average 0.8670 and for all estimated rules in the LtO group the average γ is 0.8302. The individuals that follow a trend following rule in the LtF group have an average γ of 0.3833. This means the individuals that are asked to trade are a lot more trend following than those that are asked to make price forecasts. This explains the larger bubbles that form in the LtO groups, since larger trend coefficients lead to more unstable rules and more overshooting of the price. All estimated trend coefficients are positive. This makes sense, since the underlying market is one with positive feedback, which rewards trend following behavior.

The subjects in the LtO treatment that do not follow a rule close to the trend following rule are more scattered. They do not seem to follow one of the specific forecasting rules discussed in the previous sections. These individuals have an average α, β and γ of 0.2827, 0.7303 and 0.7325 respectively. This means they predict the price to be a weighted average of (or close to) the previous price and forecast plus the trend. For none of those in the LtF treatment was a constant estimated, while all subjects in the LtO treatment have a constant in their rule. The average of this constant is -0.8230, and is only slightly positive for four individuals. This means that on average those in the LtO treatment are slightly pessimistic about the price.

5.2 Stability comparison

In this section, I analyze the stability of the general rule in order to compare stability between the two treatment groups. To do this, I first calculate for which values of α, β and γ there is dynamic stability. Proposition 5.2. When all subjects use the general prediction rule

pe_it= αpe_it−1+ γ∆pt−1+ βpt−1+ c (25)

the steady state is asymptotically stable when the following inequalities hold • 20

21β + α − 1 < 0

• 40₂₁γ +20₂₁β + α + 1 > 0 • γ < 1.05

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 −1.0 −0.5 0.0 0.5 1.0

beta

alpha

gamma

Figure 4: Coefficients α, β and γ of the standard forecasting equation plotted for the estimated rules of the LtF individuals (filled black circles) and LtO individuals (open red circles).

The price development has oscillations when 20₂₁(γ + β) + α
2 < 80₂₁γ.

Proof. See Appendix H

The stability region can be observed in Figure 5.

All individuals in the LtF treatment have a stable rule. In the LtO treatment there are 7 individuals with an unstable rule, 3 with a neutrally stable one and the other 16 individuals have stable rules. For 6 out of the 7 subjects with an unstable rule, the instability is caused by a trend coefficient γ higher than 1.05. For the other subject 20₂₁β + α − 1 > 0.

The three stability restrictions only depend on two variables: (1) γ and (2) 20₂₁β + α
. This is not a coincidence, but is caused by the determinant of the matrix of the price system being equal to 0. The stability restrictions in a 3-dimensional system can normally be expressed in three variables including the determinant, so now two variables remain.

Figure 5: Stability region (shaded area) expressed in α, β and γ.

Proposition 5.3. The fundamental steady-state of the general forecasting rule pe_it= αpe_it−1+ γ∆pt−1+

βpt−1+ c is locally stable if

M < 1 (26)

−T − 1 < M (27)

T − 1 < M (28)

where M is the sum of the second order principal minors and T is the trace of the matrix A =    20 21(γ + β) 20 21α − 20 21γ γ + β α −γ 1 0 0   .

Proof. See Brooks (2004).

Proposition 5.4. The eigenvalues of the matrix A =    20 21(γ + β) 20 21α − 20 21γ γ + β α −γ 1 0 0  

 are all real if T

2 _≥

4M , where M is the sum of the second order principal minors and T is the trace of the matrix A. The eigenvalues are complex if the restriction does not hold.

Proof. According to Brooks (2004) the characteristic polynomial can be written as

λ3− T λ2_{+ M λ − D = 0 (29)}

where T is the trace of the matrix A, M is the sum of the second order principal minors of matrix A and D is the determinant of A. Since D = 0 it follows that

λ1 = 0 ∩ λ2,3 = 1 2T ± 1 2 p T2_{− 4M (30)}

From this it can be seen that λ2,3 are complex if T2 < 4M and real if T2 ≥ 4M .

Since the stability region can be expressed as restrictions only depending on two variables, the trace and the sum of second order principal minors, the stability region can be represented two dimensionally. This is done in Figures 6 and 7. It can be clearly seen that all rules estimated in the LtF treatment are stable and all except 1 have real eigenvalues. In the LtO treatment there are a few rules which are unstable and a lot of rules with complex eigenvalues. This means that in the LtO treatment larger and more bubbles are to be expected, which corresponds with the observed market price development.

6

Comparison with Previous Research

Heemeijer et al. (2009) ran a similar experiment as Bao et al. (2014). Instead of comparing forecasting individuals with trading individuals, they compared forecasting individuals subject to either positive or negative feedback. Their market system was almost the same as the one used by Bao et al. (2014) with the main difference that the fundamental price was slightly lower (60 compared to 66). In this section, I compare those in the positive feedback treatment of the Heemeijer et al. (2009) experiment with the LtF group in the Bao et al. (2014) experiment.

The market development of the prices was quite constant between the groups in the experiment by Bao et al. (2014), while in the Heemeijer et al. (2009) experiment some markets have significantly larger bubbles than others. In the Heemeijer et al. (2009) more bubbles were observed, since in the Bao et al. (2014) experiment the prices converged quite quickly.

Heemeijer et al. (2009) estimated the following rule

pe_i,t = α1pt−1+ α2pei,t−1+ (1 − α1− α2)60 + β∆pt−1 (31)

Like the Bao et al. (2014) rules, this rule falls under the general rule

Figure 6: Trace and sum of the second order principal minors of the dynamic system of the Learning to Forecast individual rules together with the stability region (triangular area between the three lines). The area below the parabola is where the eigenvalues are real. The area above the parabola is where the eigenvalues are complex.

where for the Heemeijer et al. (2009) rules α = α2

β = α1

γ = β

c = (1 − α1− α2)

The coefficients of α, β and γ for the Bao et al. (2014) and Heemeijer et al. (2009) LtF individuals are plotted in Figure 8. From this it can be clearly seen that in the Heemeijer et al. (2009) experiments the trend coefficients γ are significantly higher. The average γ’s are 0.6882 and 0.3833 respectively for the individuals with a positive γ from the Heemeijer et al. (2009) experiment and Bao et al. (2014) experiment. The trend coefficient is still a lot lower than the one in the LtO treatment of the Bao et al. (2014) experiment. This confirms that individuals that are asked to trade are more trend following.

Similar to the Bao et al. (2014) experiment, in the Heemeijer et al. (2009) experiment there is a group of individuals that follows something close to an adaptive expectations rule and a group that follows a rule close to a trend following rule. Different from the Bao et al. (2014) experiment is that there is also a significant amount of individuals that follow a rule that is a weighted combination of the

Figure 7: Trace and sum of the second order principal minors of the Learning to Optimize individual rules together with the stability region (triangular area between the three lines). The area below the parabola is where the eigenvalues are real. The area above the parabola is where the eigenvalues are complex.

forecasted price and realized price in the last period plus a trend. The same type of rule was observed in the LtO treatment of the Bao et al. (2014) experiment.

A possible explanation for the differences between the Heemeijer et al. (2009) and Bao et al. (2014) experiments is the different way the rules were estimated. The estimated rules were slightly different and could be the reason why different results are found. Quite a large group of individuals in the Heemeijer et al. (2009) experiment follow one of the rules estimated in the Bao et al. (2014) experiment. The ones that do not might not have been successfully estimated using the Bao et al. (2014) rules. This means that it is possible that individuals in the Bao et al. (2014) experiment do follow similar rules, but are just not succesfully estimated. Another explanation of the differences could be that fundamental price in the Heemeijer et al. (2009) experiment is slightly lower. This does not seem very likely, since the difference is small. Additional research needs to been done regarding the effect of a smaller or larger fundamental price. Apart from these two points, the differences can only be explained by the individuals not behaving in the same way in the two experiments. This questions the validity of the found result that the individuals in the Learning to Optimize group are more trend following.

The only difference between the estimated rule in the Heemeijer et al. (2009) paper and the one described in the previous section, is a difference in constants. This has no effect on the stability conditions, so the same conditions as in the last section can be used. From this it can be derived that

like in the Bao et al. (2014) experiment all subjects have stable rules in the Heemeijer et al. (2009) experiment. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 −1.0 −0.5 0.0 0.5 1.0

beta

alpha

gamma

Figure 8: Coefficients α, β and γ of the standard forecasting equation plotted for the estimated rules of the LtF individuals of the Bao et al. (2014) experiment (filled black circles) and Heemeijer et al. (2009) experiment (open blue circles).

7

Conclusion

Bao et al. (2014) investigated the price development in markets between three treatment groups. The first treatment was a so-called Learning to Forecast treatment, where individuals were asked to make price forecasts. Based on these forecast the realized price was calculated. The second treatment was a so-called Learning to Optimize treatment, where individuals were asked to make trading decisions. There was also a third treatment, where subjects were asked to make both forecasts and trading decisions. I focused on the first two treatment groups. One would assume that the two treatments would result in similar market developments, since an expected price corresponds with a certain optimal trading decision. However, the authors found that bubbles were much larger and more frequent in the treatment where the subjects were asked to trade. This can also be explained by the differences in the coefficients of the individual rules. I found that those in the Learning to Optimize treatment have much higher trend coefficients on average, which might explain the difference in market behavior. The rules of those in the LtO treatment were also more unstable and a higher percentage had complex eigenvalues of the price system. Since complex eigenvalues result in price oscillations, these findings support the found differences between the two groups.

Heemeijer et al. (2009) ran a similar experiment, where they also had a Learning to Forecast treatment. In the Heemeijer et al. (2009) paper more bubbles formed and not always a quick convergence to the fundamental price. The coefficients of the estimated rules for the individuals in their experiment were different from the one in the Bao et al. (2014) experiment. The subjects in the Heemeijer et al. (2009) experiment were more trend following, resulting in less stable behavior. This complies with the

observed differences between the two experiments. Since the main difference in the underlying market between the two papers is the difference in fundamental price, future research should investigate what the effect of a difference in fundamental price has on the behavior of individuals.

A question that arises is why the rules used in the Learning to Optimize treatment are so different from the ones used in the Learning to Forecast treatment. A possible experiment to investigate this would be one with a LtF group and a group where individuals are asked to first make forecasts and then trading decisions. Bao et al. (2014) already had a treatment group where individuals were asked to make both price forecasts and trading decisions, but I suggest that the optimal trading decision is immediately calculated from the forecast. That way, the individuals would only have to fill in a trading decision when they want to change the calculated trading decision. This design would significantly reduce the mental tax of the trading task, which is usually presumed to be more difficult than forecasting. When subjects make trading decisions different from the automatically calculated ones, they can be asked why they made a different decision.

References

Assenza, T., Brock, W. A., and Hommes, C. H. (2013). Animal spirits, heterogeneous expectations and the emergence of booms and busts. Technical report, Tinbergen Institute Discussion Paper. Bao, T., Hommes, C., and Makarewicz, T. (2014). Bubble formation and (in)effcient markets in

learning-to-forecast and -optimize experiments. CeNDEF Working paper.

Brooks, B. P. (2004). Linear stability conditions for a first-order three-dimensional discrete dynamic. Applied Mathematics Letters, 17(4):463–466.

De Bondt, W. F. and Thaler, R. H. (1987). Further evidence on investor overreaction and stock market seasonality. Journal of finance, 42(3):557–581.

Ezekiel, M. (1938). The cobweb theorem. The Quarterly Journal of Economics, 52(2):255–280. Fama, E. F. (1970). Efficient capital markets: A review of theory and empirical work*. The journal of

Finance, 25(2):383–417.

Goodwin, R. M. (1947). Dynamical coupling with especial reference to markets having production lags. Econometrica, Journal of the Econometric Society, 15(3):181–204.

Heemeijer, P., Hommes, C., Sonnemans, J., and Tuinstra, J. (2009). Price stability and volatility in markets with positive and negative expectations feedback: An experimental investigation. Journal of Economic Dynamics and Control, 33(5):1052–1072.

Hommes, C. (2011). The heterogeneous expectations hypothesis: Some evidence from the lab. Journal of Economic Dynamics and Control, 35(1):1–24.

Hommes, C., Sonnemans, J., Tuinstra, J., and Van de Velden, H. (2005). Coordination of expectations in asset pricing experiments. Review of Financial Studies, 18(3):955–980.

Lucas, R. E. and Sargent, T. J. (1981). Rational expectations and econometric practice, volume 2. U of Minnesota Press.

Lucas Jr, R. E. (1972). Expectations and the neutrality of money. Journal of economic theory, 4(2):103–124.

Marimon, R., Spear, S. E., and Sunder, S. (1993). Expectationally driven market volatility: an experimental study. Journal of Economic Theory, 61(1):74–103.

Muth, J. F. (1961). Rational expectations and the theory of price movements. Econometrica: Journal of the Econometric Society, 29(3):315–335.

Nerlove, M. (1958). Adaptive expectations and cobweb phenomena. The Quarterly Journal of Economics, 72(2):227–240.

Santos, M. S. and Woodford, M. (1997). Rational asset pricing bubbles. Econometrica: Journal of the Econometric Society, 65(1):19–57.

Sargent, T. J. (1993). Bounded rationality in macroeconomics: The arne ryde memorial lectures. OUP Catalogue.

Simon, H. A. (1957). Models of man; social and rational. Wiley, New York.

Smith, V. L., Suchanek, G. L., and Williams, A. W. (1988). Bubbles, crashes, and endogenous expectations in experimental spot asset markets. Econometrica: Journal of the Econometric Society, 56(5):1119–1151.

Tirole, J. (1985). Asset bubbles and overlapping generations. Econometrica: Journal of the Econometric Society, 53(6):1499–1528.

Tversky, A. and Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. science, 185(4157):1124–1131.

A

Steady State of the Process

In the steady state of price development all prices all equal each period and all forecasted prices are equal each period. The underlying market is given by

pt+1= 20 21p¯ e t+1+ 66 21 (33) or equivalently pt+1= pt+ 20 21 6 X i=1 zi,t (34)

Proposition A.1. All discussed forecasting rules have a steady state equal to the fundamental price of 66 under homogeneous expectations.

Proof. Let the expected price be described by the following rule

pe_it= αpe_it−1+ γ∆pt−1+ βpt−1 (35)

Since in the steady state the prices and forecasts don’t depend on the time, the expected price equation in the steady state can be written as

pe = αpe+ βp (36)

=⇒ pe= β

(1 − α)p (37)

From this follows that the price in the steady state is p = 20 21 β (1 − α)p + 66 21 (38) =⇒ p = 66 211 −20_21(1−α)β  (39)

When α = 1 − β, the expression above gives p = 66. This is the case for the na¨ıve expectations rule, the average expectations rule, the trend extrapolation rule and the adaptive expectations rule.

This only leaves the anchor and adjustment rule. This rule is as follows pe_it= 66 + pt−1

2 + (pt−1− pt−2) (40)

The following must hold in the steady state

pe= 33 + 1

2p (41)

and the price in the steady state is

p = 20 2133 + 20 21 1 2p + 66 21 (42) =⇒ 11 21p = 242 21 (43) =⇒ p = 66 (44)

Proposition A.2. The quantity rules have a steady state equal to the fundamental price if the asset return coefficient φ is not equal to 0 and the constant is 0.

Proof. For the quantity rules it can be seen from function 34 that under homogeneous expectations in the steady state

q = 0 (45)

The general quantity rule is

qi,t = constanti+ χiqi,t−1+ φi(pt−1+ 3.3 − 1.05pt−2) (46)

filling in q = 0 and assuming homogeneous expectations this becomes

constant + φ(3.3 − 0.05p) = 0 (47)

=⇒ p = 66 − 20 ∗ constant

φ = 66 if the constant is 0 (48)

B

Proof of Proposition 3.1

Proof. Assuming na¨ıve expectations, all subjects use the following rule:

pe_it= pt−1. (49)

From this follows that the following market system exists

pt= 66 21 + 20 21pt−1 =⇒ h pt i = h 20 21 i h pt−1 i + h 66 21 i = Ah pt−1 i +h 66₂₁ i where A = h 20 21 i

This process is stable when the solutions λ of the equation det(A − λI) = 0 fall inside of the complex unit circle.

det(A − λI) =    20 21− λ    ⇒ λ = 20 21

C

Proof of Proposition 3.2

For the average expectations rule the following prediction rule is used

pe_it= 1 t − 1 t−1 X j=1 pj for t > 1 (50)

and from this follows

pe_it−1= 1 t − 2 t−2 X j=1 pj for t > 2 (51) Combining 50 and 51 pe_it= (t − 2)pe_it−1+ pt−1
1 t − 1 for t > 2 (52)

From this follows that the following market system exists

pt= 66 21 + 20 21p e t−1 (53) = 20 21  (t − 2)pe_t−1+ pt−1
1 t − 1  +66 21 (54) = 20(t − 2) 21(t − 1)p e t−1+ 20 21(t − 1)pt−1+ 66 21 (55) pe_t = (t − 2) (t − 1)p e t−1+ 1 (t − 1)pt−1 (56) =⇒ " pt pe t # = " ₂₀ 21(t−1) 20(t−2) 21(t−1) 1 (t−1) (t−2) (t−1) # " pt−1 pe t−1 # + " 66 21 0 # = A " pt−1 pe_t−1 # + " 66 21 0 # where A = " ₂₀ 21(t−1) 20(t−2) 21(t−1) 1 (t−1) (t−2) (t−1) #

This process is stable when the solutions λ of the equation det(A − λI) = 0 fall inside of the complex unit circle.

det(A − λI) =      20 21(t−1) − λ 20(t−2) 21(t−1) 1 (t−1) (t−2) (t−1)− λ      =  20 21(t − 1) − λ  ∗ (t − 2) (t − 1) − λ  − 1 (t − 1) 20(t − 2) 21(t − 1) = −λ  20 21(t − 1) + t − 2 t − 1  + λ2= 0 ⇒ λ1= 0 ∩ λ2=  20 21(t − 1) + t − 2 t − 1  = t − 22 21 t − 1 < 1 and λ2→ 1 when t → ∞

D

Proof of Proposition 3.3

Proof. The anchor and adjustment rule is as follows pe_it= 66 + pt−1

2 + (pt−1− pt−2) (57)

The following market system exists

pt= 66 21+ 20 21p e t−1 (58) = 20 21  66 + pt−1 2 + (pt−1− pt−2)  + 66 21 (59) = 10 7 pt−1− 20 21pt−2+ 242 7 (60) pe_t = 33 +3 2pt−1− pt−2 (61) pt−1 = pt−1 (62) =⇒    pt pe_t pt−1   =    10 21 0 − 20 21 3 2 0 −1 1 0 0       pt−1 pe_t−1 pt−2   +    242 7 33 0    = A    pt−1 pe t−1 pt−2   +    242 7 33 0    where A =    10 21 0 − 20 21 3 2 0 −1 1 0 0   

This process is stable when the solutions λ of the equation det(A − λI) = 0 fall inside of the complex unit circle.

det(A − λI) =        10 21 − λ 0 − 20 21 3 2 −λ −1 1 0 −λ        = 10 7 − λ  ∗ λ2− λ20 21 ⇒ λ₁ = 0 ∨ λ2−10 7 λ + 20 21 = 0 ⇒ λ1 = 0 ∩ λ2,3 = 10 7 ± q 10 7
2 − 80 21 2 = 5 7± i r 65 147 ⇒ |λ_2,3| = s  5 7 2 + 65 147 = r 20 21 < 1 The two eigenvalues are complex and have a length of

q

20 21.

E

Proof of Proposition 4.3

Proof. When all subjects use the trend extrapolation rule, the eigenvalues are equal to 0, 10₂₁(1 + γ) +

1 2 q 20 21(1 + γ)
2 −80₂₁γ and 10₂₁(1 + γ) −1₂ q 20 21(1 + γ)
2 −80₂₁γ (see Proposition 4.2).

These eigenvalues can be either real or complex and are stable when they lie inside the complex unit circle.

The λ’s are complex when

 20 21(1 + γ) 2 −80 21γ < 0 ⇒ 20 21(1 + γ) 2 < 80 21γ ⇒20 21+ 20 21γ < r 80 21 √ γ ⇒20 21γ − r 80 21 √ γ +20 21 < 0 This last inequality is an equality when

√ γ1,2 = q 80 21± q 80 21 − 80 21 20 21 40 21 = 1.0247 ± 0.2236 ⇒γ₁ = 1.56 ∩ γ2 = 0.64

For γ ∈ (0.64, 1.56) the inequality holds, so for these γ the eigenvalues are complex. For the other values of γ, the eigenvalues are real.

When the eigenvalue λ is real, then λ = 1 or λ = −1 would result in |λ| = 1. When they are filled into the equation

λ2− λ20 21(1 + γ) + 20 21γ = 0 (63) it follows that λ = 1: 1 −20 21(1 + γ) + 20 21γ = 0 ⇒ 1

21 = 0 this is not possible, so λ = 1 is not possible λ = −1: 1 +20 21(1 + γ) + 20 21γ = 0 ⇒ 41 21 + 40 21γ = 0 ⇒ γ = − 41 40

The eigenvalues are real for γ < 0.64 and γ > 1.56, so the eigenvalues are real and lie inside the complex unit circle for γ ∈ [−1.025, 0.64]. The eigenvalues are real and lie outside the complex unit circle for γ ∈ (∞, −1.025) ∪ [1.56, ∞).

When the λ’s are complex |λ| = v u u t 1 4  20 21(1 + γ) 2 +1 4 80 21γ −  20 21(1 + γ) 2! = r 20 21γ = 1 for γ = 1.05 For γ < 1.05, |λ| < 1 and for γ > 1.05, |λ| > 1.

F

Proof of Proposition 4.6

Proof. The eigenvalues are λ1,2 = 1₂ 1 +120₂₁φ
±1₂

q

1 +120₂₁φ
2− 24φ.

The eigenvalues can either be real or complex and stable or unstable (or neutrally stable). The eigenvalues are stable when they lie inside the complex unit circle. The eigenvalues are complex when

 1 +120 21 φ 2 − 24φ < 0 (64) ⇒  1 +120 21 φ 2 < 24φ (65) ⇒1 + 120 21 φ < √ 24pφ (66) ⇒120 21 φ − √ 24pφ + 1 < 0 (67)

Normally an inequality like x2 < y has two solutions x <√y and −x >√y, but in this case there is only one solution, since φ has to be positive. Otherwise, the first inequality (64) cannot be met, since φ is always real. When the last inequality is taken as an equality, the border values can be calculated.

pφ1,2 = √ 24 ± q 24 −480₂₁ 240 21 = 0.429 ± 0.0935 ⇒φ₁ = 0.273 ∩ γ2 = 0.113

Since the inequality holds for all values in between 0.113 and 0.273, it means that the eigenvalues are real for φ ∈ (−∞, 0.113] ∪ [0.273, ∞) and complex for φ ∈ (0.113, 0.273).

If both λ’s are real, then λ = 1 or λ = −1 would result in |λ| = 1. When filled into the equation λ2− λ  1 +120 21 φ  + 6φ = 0 (68) it follows that λ = 1: −120 21 φ + 6φ = 0 ⇒ φ = 0 λ = −1: 2 +120 21 φ + 6φ = 0 ⇒ φ = − 7 41

The rule is not neutrally stable for φ = −₄₁7 , since the eigenvalue λ1= 1₂ 1 +120₂₁φ
+1₂

q

1 +120₂₁φ
2− 24φ lies outside the complex unit circle for all φ < 0. The eigenvalues both lie inside the complex unit circle and are both real for φ ∈ [0, 0.113]. They are both real and lie outside the complex unit circle for φ ∈ (∞, 0) ∪ [0.273, ∞).

When both λ’s are complex

|λ| = v u u t 1 4  1 +120 21 φ 2 +1 4 24φ −  1 +120 21 φ 2! =p6φ = 1 for φ = 1 6 For φ < 1₆, |λ| < 1 and for φ > 1₆, |λ| > 1.

G

Proof of Proposition 5.1

Proof. The general LtO quantity rule is as follows

qit= constanti+ χiqit−1+ φiρt−1 (69) and substituting qit= _pe it+3.3−1.05pt−1 6   pe it+ 3.3 − 1.05pt−1 6  = constanti+ χi  pe it−1+ 3.3 − 1.05pt−2 6  + φi(pt−1+ 3.3 − 1.05pt−2) =⇒ pe_it= 6 ∗ constanti+ χipeit−1+ (χ − 1)3.3 + 1.05pt−1− 1.05χipt−2+ 6φipt−1+ φi19.8 − φi6.3pt−2 = χipeit−1+ 1.05(χi+ 6φi)∆pt−1+ (1 − χi− 0.3φi)pt−1 + (6 ∗ constanti+ (χi− 1)3.3 + 19.8φi)

H

Proof of Proposition 5.2

Proof. The general forecasting rule is as follows

pe_it= αpe_it−1+ γ∆pt−1+ βpt−1+ c (70)

When all subjects use this rule, it results in the following price system pt= 66 + 20 21(p e t− 66) = 66 +20 21(αp e t−1+ γ∆pt−1+ βpt−1− 66) = 66 +20 21αp e t−1+ 20 21γ∆pt−1+ 20 21βpt−1− 20 2166 = 66 21 + 20 21αp e t−1− 20 21γpt−2+ 20 21(β + γ)pt−1 pe_t = αpe_t−1− γx_t−1+ (β + γ)pt−1 =⇒    pt pe_t pt−1   =    20 21(γ + β) 20 21α − 20 21γ γ1+ β1 α1 −γ1 1 0 0       pt−1 pe_t−1 pt−2   +    66 21 0 0    = A    pt−1 pe_t−1 xt−1   +    66 21 0 0    where A =    20 21(γ + β) 20 21α − 20 21γ γ + β α −γ 1 0 0   

This process is stable when the solutions λ of the equation det(A − λI) = 0 fall inside of the complex unit circle.

det(A − λI) =        20 21(γ + β) − λ 20 21α − 20 21γ γ + β α − λ −γ 1 0 −λ        = 20 21(γ + β) − λ  (α − λ)(−λ) − 20 21αγ + 20 21(α − λ)γ + λ 20 21α(γ + β) = λ220 21(γ + β) + λ 2_{α − λ}3₋20 21λγ = 0 ⇒ λ = 0 ∨ λ2− 20 21(γ + β) + α  λ +20 21γ = 0 (71) ⇒ λ1= 0 ∩ λ2,3= 1 2  20 21(γ + β) + α  ±1 2 s  20 21(γ + β) + α 2 − 80 21γ (72) It can be directly seen from the equation (72) that the eigenvalues are complex when

 20

21(γ + β) + α 2

< 80

21γ (73)

If all λ’s are real, then λ = 1 or λ = −1 would result in |λ| = 1. When filled into equation (71) λ = 1: 1 − 20 21(γ + β) + α  +20 21γ = 0 ⇒ 20 21β + α = 1 λ = −1: 1 + 20 21(γ + β) + α  +20 21γ = 0 ⇒ 40 21γ + 20 21β + α = −1

The steady state is asymptotically neutrally stable when one of the above equalities holds and all eigenvalues that do not lie on the complex unit circle lie inside it.

If λ2,3 are complex then

|λ_2,3| = v u u t 1 4  20 21(γ + β) + α 2 +1 4 80 21γ −  20 21(γ + β) + α 2! = r 20 21γ = 1 for γ = 1.05

So the system is stable when the following inequalities hold 20 21β + α − 1 < 0 (74) 40 21γ + 20 21β + α + 1 > 0 (75) γ < 1.05 (76)

I

Estimated individual rules

Figure 9: Estimated individual rules for the LtF treatment in the Bao et al. (2014) experiment

Figure 11: Estimated individual rules for the LtF treatment with positive feedback in the Heemeijer et al. (2009) experiment