The Heuristics Switching Model in Learning-to-Optimize experiments

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The Heuristics Switching Model in

Learning-to-Optimize experiments

Faculty of Economics and Business, University of Amsterdam

Econometrics Bsc. Thesis

By: Olivier Go, 6236847

Supervisor: Tomasz Makarewicz, Msc.

June 27, 2014

Abstract

This paper tests the Heuristics Switching Model in a Learning-to-Optimize experiment. This experiment was carried out in a positive feedback market. It is shown that the Heuristics Switching Model, which consists of an adaptive learning heuristic and a trend-following heuristic, is a better model for predicting one-period ahead prices than any of the single heuristics. This shows that the subjects have hetero-geneous expectations. Also, in this experiment, the input parameters of the Heuristics Switching Model are tested and optimized.

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1 Introduction

In the past 15 years, the dot-com bubble and more recently the housing market bubble in the US have led people to doubt some of the underlying assumptions of classical economic models. It is clear that these bubbles, and mainly the crash that followed them, are of major economic importance. These bubbles only occur in markets where there is the possibility of speculation; where a higher expected price will lead to higher demand and therefore to a higher realized price, i.e. positive feedback markets.

Positive feedback markets have shown more bubble formation than negative feedback markets. (Heemeijer et al., 2009). This result is rather surprising since bubble are not supposed to form in an economy at all. Until recently, it was usually assumed that economic agents’ behavior followed the Rational Expectations Theory. (Muth, 1961)

But how do agents calculate their expectations? And are agents always rational? Questions like these have led to the beginning of behavioral eco-nomics. Through prospect theory (Kahneman, 1979) and more importantly, the finding of cognitive biases and heuristics (Kahneman, 1973) economists have started questioning the assumption of Rational Expectations.

Rational Expectations Theory is based upon the following two assump-tions. The first assumption is that agents have a good understanding of the structure of the market. The second assumption is that agents make an op-timal decision given their knowledge of the market. The first assumption is usually considered unrealistic, but the conditionally optimal behavior is nor-mally accepted as a valid concept. However, from Bao et al. (2014), it becomes clear that agents are also not very capable of performing conditionally optimal. It seems that an important boundary of rationality is the complexity of the task. It looks like Rational Expectations can be definitely rejected.

Bounded Rationality was introduced by Herbert Simon (1972) and has been accepted as the leading decision model by some of economic researchers instead of Rational Expectations Theory. The theory of Bounded Rationality

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states that people try to behave rational under given circumstances such as limited time, information and, maybe even more important, capacity. Bounded Rationality is consistent with most of the existing economic theory, whilst si-multaneously leaving room for some ’irrational’ decisions. Nowadays, bounded rationality is finally being included in some models about predicting and hu-man interaction.

The problem is that bubbles can only be observed after they crash. At a specific time, anyone will have different expectations of a certain asset and therefore overpricing is harder to identify. After a crash it is retrospectively visible that there was a bubble. It would be preferable if these bubbles could be predicted so a counter measure can be taken. Therefore, researchers started simulating markets to try and identify bubble formation. In order to do so, researchers use experiments. This way it is possible to test specific hypotheses regarding bubble formation. There are two types of experiments that are most widely used.

In Learning-to-Forecast (LtF) experiments, participants are asked to pre-dict the price in the next period (Marimon et al., 1993). It turns out that the type of market, positive or negative, is an important factor for the outcome of the experiment (Heemeijer et al., 2009). A striking result of the LtF research is that people tend to coordinate on the price by following the trend in positive feedback markets or by using an Adaptive Learning heuristic in negative feed-back markets, instead of behaving like Rational Expectations Theory would expect them to do (Bostian and Holt, 2009).

In most markets however, people do not predict prices, but they only imply prices by demanding or selling a certain quantity. The first researchers to observe bubbles in this more realistic setup, were Smith et al., (1988). In their experiment, subjects were asked to make trading decisions about an asset that paid random dividends. Therefore, the asset price was the present value of the future dividends, which decreased over time. The researchers found significant overpricing in the beginning and a crash near the end of the experiment. This was one of the first experiments in which subjects were asked to trade and

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optimize their trading behavior. Therefore, these types of experiments are called Learning-to-Optimize (LtO) experiments.

Various LtO experiments have taken place since then. One experiment was carried out by Kirchler et al. (2012). Their main point of criticism on some previous experiments was that participants in the experiment did not fully understand the concept of a declining fundamental price, which could have lead to irrational decisions. Therefore, they suggest that the fundamental price should not change over time.

Since the price forecasts in an LtF experiment with positive feedback are better with a Heuristics Switching Model (HSM) than any of the single heuris-tics (Anufriev et al., (2012)), it is worthwhile investigating how this model performs in LtO experiments. Therefore, this paper aims to find answers to the following questions: 1. Is the Heuristics Switching Model a more accurate model to predict individual forecasts than a single heuristics model? 2. What are the optimal parameter values of the HSM?

The answers to the above mentioned questions will be found by applying the Heuristics Switching Model to a LtO experiment in a positive feedback market (as in Bao et al., (2014)). To the best of my knowledge, this has never been done before. This paper is organised as follows. In section 2, different forecasting methods will be explained. The main focus of this section will be on the Heuristics Switching Model. In the third section, the experimental setup will be explained. Section 4 will consist of the results of applying the HSM in LtO experiments. Section 5 will be a robustness check of the HSM and the input parameters will be optimized. Section 6 will conclude.

2 Forecasting methods

In order to predict future bubbles and crashes, it is necessary to know how individual people forecast prices. A starting assumption is that people forecast homogeneously. This means that people only use one heuristic to predict the next-period forecast.

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2.1 Homogeneous Heuristics

Economists would often consider that the most basic assumption regarding forecasting behaviour is the assumption of naive expectations. In that case, the price in the next period is determined as

pnaive_t+1 = pt (1)

Under this assumption, the price forecast is equal to the price in the previ-ous period. This model however, does not fit experimental data very well (Heemeijer et al., 2009).

Another option is that subjects take the average of their previous predic-tions as the next price forecast.

pave_i,t = 1 t t−1 X j=1 pj (2)

This model is also inconsistent with laboratory experiments (Heemeijer et al., 2009).

And how would Rational Expectations Theory perform? Under the as-sumptions of this theory every subject trades optimal and maximizes their own profit, which is the payoff they receive after the experiment. The payoff

will be maximized at pt= pf, with pf = the fundamental price. This implies

that

pRET_t+1 = pf (3)

Details about the payoff can be found in Bao et al. (2014). This model too performs poorly in positive feedback markets (Anufriev et al., 2012). Another possible heuristic is Adaptive Learning.

pAda_t+1 = w ∗ pt+ (1 − w) ∗ pAdat (4)

The next period price depends on the price in the current period, on the subjects’ expectation for the price in this period and on w ∈ [0, 1], which is a

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weighting factor that determines whether people anchor mainly on the price or on their expectation for the current period. The last homogeneous heuristic considered in this part is a Trend Following heuristic.

pT rend_t+1 = pt+ γ(pt− pt−1) (5)

This heuristic implies that the next price forecast follows the trend in the past two periods. The γ is the degree of trend following.

Since people do not have rational expectations about the one period ahead

price, then how do they forecast? It turns out that simply following the

trend works sometimes. In many types of markets trend-following does not fit data very well. It is therefore not a reliable model. Also, naive expectations, which can be considered a trend-following model with trend-coefficient γ=0, fits most experimental data rather poorly as well. Also, adaptive learning tends to perform well in negative feedback markets, but poorly in positive feedback markets (Anufriev et al., 2012).

2.2 Heuristics Switching Model

To counter the homogeneity assumption, Brock and Hommes introduced a Heuristics Switching Model (HSM) in 1997. In this model it is assumed that people do not use a single forecasting rule, but tend to switch between differ-ent heuristics, depending on the performance of these heuristics in previous periods. If a heuristic was more accurate in the past, the prediction using this heuristic will be valued higher than if the heuristic performed poorly in the past. The two most commonly used heuristics in this model are Adaptive Learning and Trend Following.

In LtF experiments in a positive feedback market, the HSM performed slightly better than Trend Following, and also better than Adaptive Learning (Anufriev et al., 2012). So the HSM proved to be a better model for forecasting than any of the homogeneous heuristics. Therefore the HSM looks promising for predicting individual behavior. This could potentially lead to building a

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model for predicting bubbles and crashes on the stock markets, and any other positive (and maybe also negative) feedback market.

The Heuristics Switching Model used in this thesis is similar to the one in Anufriev et al., (2012), because the experiments are carried out in similar markets. At first, the price will be forecast using the single heuristics (formulas

4 and 5). This will give pAda_t+1 and pT rend_t+1 . To test the performance Uh,t, the

Mean Squared Error of each method will be calculated. In this calculation, there is also a one lag memory.

Uh,t = −(pt− peh,t)

2_{+ η ∗ U}

h,t−1, (6)

where h∈ [AdaptiveLearning, TrendFollowing] The η ∈ [0, 1] is a fixed param-eter, which determines how much you value the previous performance of a method. Once the performance of every heuristic is calculated, the weighting factor n is calculated for both heuristics.

nh,t = δ ∗ nh,t−1+ (1 − δ) ∗

Exp(β ∗ Uh,t)

P2

h=1Exp(β ∗ Uh,t)

(7) The value for δ ∈ [0, 1] determines how eager people are to switch heuristic. A large δ means that people will tend to stick to their previous heuristic, even if there is another heuristic that performs better. Such behavior has been found by psychologists and behavioral economists such as in Kahneman (2003). The value of β > 0 is a measure of how strongly people will update their heuristic if there is a difference between the performance of both heuristics. The price forecast in the next period is a weighted average of the price forecasts using the two heuristics, given by

pHSM_t+1 =

H

X

h=1

nh,t+1peh,t+1 (8)

H is the number of heuristics used, which is 2 in this case. Since the Trend

Following requires 2 previous periods, pT rend

1 = pT rend2 = 50. Adaptive

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prices ∈ [0, 100]. For the first period, the weighting factors are equal between the two heuristics.

3 Market Dynamics

3.1 Explanation of the experiment

This thesis will use data from a paper by Bao et al. (2014). The authors

ran three different experiments. The first one was a Learning-to-Forecast

treatment, in which 4 groups of 6 subjects were asked to forecast prices of an asset in the next 50 periods. They were aware of the qualitative market dynamics. The second experiment was a Learning-to-Optimize experiment. In this LtO treatment, 6 groups of 6 subjects were asked to trade a certain amount of the asset. The third experiment was a mixed treatment consisting of 8 groups of 6 subjects. In this mixed treatment, people were first asked to give a price forecast, and after that, the amount they wished to trade. Most input variables are chosen in a similar way as in Anufriev et al. (2012). This research will focus mainly on the Learning-to-Optimize experiment, since the Heuristics Switching Model has already been applied in Learning-to-Forecast experiments by Anufriev et al. (2012).

The optimal quantity to trade is calculated as follows

z_i,t∗ = Ei,t(pt+1+ yt+1) − Rpt

aσ2 (9)

Where yt is the dividend paid on time t, R is 1+r, where r is the interest rate,

in this case 5%. This is a discount factor for the previous price. Furthermore,

aσ2 _{is equalized to 6, for mathematical convenience.}1 _{In this experiment, the}

expected dividend y is considered independent of time and set equal to 3.3 units. This way, the fundamental price does not change over time.

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The market is defined as in Beja and Goldman (1980).

pt+1 = pt+ λ(ZtD− Z

S

t ) + t (10)

For convenience, Rλ is equalized to 1, which makes λ = 20/21. ZD

t is the total

supply. ZS

t is the exogenous supply, which is equal to 0 in this experiment.

This makes the price adjustment formula

pt+1= pt+ 20 21 6 X i=1 zi,t+ t (11)

And an optimizing agent will have individual demand

z_i,t∗ = p

e

i,t+1+ 3.3 − 1.05pt

6 (12)

It is therefore equivalent to state that the price prediction of subjecti is equal

to

pe_i,t+1 = 6z_i,t∗ − 3.3 + 1.05pt (13)

Combining these formulas, it can be derived that

pt+1= 66 +

20

21(p

e

t+1− 66) + t (14)

Learning-to-Optimize behaviour can be explained by the underlying implied

price expectation pe

i,t+1. In this paper, the fundamental price equals 66 and

pe_t+1 is the average price prediction of the 6 people in a group. Under the

assumptions of rationality and profit maximizing, the Learning to Forecast experiment is equivalent to the Learning to Optimize treatment. Therefore, if the subjects behave rationally, it should not make a difference if they indicate a price forecast or submit a quantity decision. The two cases are mathematically equivalent. This can also be seen in formulas 12 and 13.

In this research, the Heuristics Switching Model will be applied in the Learning-to-Optimize experiments. The Mean-Squared-Error (MSE) of the Rational Expectation price forecast, Adaptive Learning forecast, Trend

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Fol-lowing forecast and the Heuristics Switching Model forecast will be calculated

for all the 6 Learning to Optimize groups in the experiment. After that,

these results will be scaled to compare the results to similar experiments in Learning-to-Forecast. When this is done, the input variables of the HSM will be optimized so that they minimize the MSE of the HSM.

3.2 Observations

This section will examine the results of applying the Heuristics Switching Model to the Learning-to-Optimize treatments. In Figure 1, the results of the Learning-to-Optimize experiment are reported. As stated before, this experi-ment consists of 6 groups with 6 people.

From Figure 1 can be concluded that none of the groups converges to the fundamental price. Group 2 and 5 seem stable but group 2 converges to a value

below pf = 66. Group 5 is steadily increasing and overshoots the fundamental

price. Groups 1, 3, 4 and 6 however show a more interesting result. In these markets, there is significant bubble formation. In group 1,3 and 4, there are multiple bubbles and crashes. Group 6 deviates from the rest of the groups. There is a major bubble at a price close to 100. After this, the price drops close to the fundamental price and remains very stable after that.

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0 5 10 15 20 25 30 35 40 45 50 0 10 20 30 40 50 60 70 80 90 100 (a) Group 1 0 5 10 15 20 25 30 35 40 45 50 0 10 20 30 40 50 60 70 80 90 100 (b) Group 2 0 5 10 15 20 25 30 35 40 45 50 0 10 20 30 40 50 60 70 80 90 100 (c) Group 3 0 5 10 15 20 25 30 35 40 45 50 0 10 20 30 40 50 60 70 80 90 100 (d) Group 4 0 5 10 15 20 25 30 35 40 45 50 0 10 20 30 40 50 60 70 80 90 100 (e) Group 5 0 5 10 15 20 25 30 35 40 45 50 0 10 20 30 40 50 60 70 80 90 100 (f) Group 6

Figure 1: For every group, prices are displayed. Blue (circles) is the realized price, red (crosses) is the price according to the Adaptive Learning heuris-tic, black (diamond) is the price if a subject would follow the trend and pink (square) is the Heuristics Switching Model price.

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4 Results

4.1 Heuristics Performance

In this section, the Heuristics Switching Model is used to forecast prices in the next period. Figure 1 shows that under the same circumstances, groups behave in a different manner. In some of the groups there is a specific trend in the price. Some other groups however, show that adaptive learning could be better. This result implies that a switching model between these two heuristics would perform better overall. Therefore, this model uses two heuristics, Trend-Following and Adaptive Learning. Since this is a positive feedback market, it is expected that Trend-Following is a better heuristic than Adaptive Learning, which was proven by Anufriev et al., (2012). If we look at some plots of the weights of these heuristics, it becomes clear that especially in markets with bubble formation, the Trend-Following heuristic dominates Adaptive Learning. This can also be seen in Figure 2. In panel (a), the weighting factors of the HSM in group 1 are shown. Group 1 showed significant bubble formation. In this group, the Trend-Following heuristic is a much better approximation than the Adaptive Learning, and therefore, the pink line dominates the blue line in most periods. Group 2 shows a more stable price. Here, the heuristics are much more equal, with Adaptive Learning dominating Trend-Following slightly. For more details regarding these graphs, see Appendix.

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0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) Group 1 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) Group 2

Figure 2: In this figure, the weigths of every group are shown. The pink line (squares) is trend following and the blue line (circles) is adaptive learning

To test the performance of every heuristic, the Mean-Squared Error (MSE) of the 1-period ahead forecast is calculated. The idea is to test the ability of every model to predict the period t forecast conditional on the experimental data until period t-1. The Mean-Squared Error is calculated as follows

M SEh,t = (pt− peh,t)

2 ₍₁₅₎

Where h defines which model is used. After the MSE is calculated for every period, it is averaged over time, which yields an average MSE for a heuristic in a specific group. Five different heuristics are tested. naive expectations, rational expectations, adaptive learning, trend-following and the HSM. They are calculated as in section 2. The input parameter of adaptive learning is w = 0.75. This means that people anchor on the previous price for 75% of their next prediction, and the other 25% is determined by their last forecast of the price. The value of this parameter can be different for every individual, depending on their confidence in their own predictive quality. The trend-following coefficient is γ = 1. The value for the trend-coefficient is chosen as 1 because this is considered to be between weak trend following and strong trend following. Therefore, choosing this value as 1, gives us average trend-following behaviour. In the next section, there will be a distinction between these two.

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The input parameters of the HSM are β = 1.5, δ = 0.1 and η = 0.1. The value of sensitivity β tells how much people tend to update their weighting-factors of the Heuristics Switching Model. So a large value of β means that everybody will switch to the heuristic that performs best. A β equal to 1.5 is not extremely high, but definitely leaves room for switching between heuristics. The memory coefficients δ and η of 0.1 was found by Anufriev et al., (2012) by trial and error and will be used as starting value in this paper as well.

The next-period forecast by Adaptive Learning is calculated by formula (4). The Trend-Following forecast results from formula (5). The Heuristics Switching Model forecast is defined in formula (8). Naive Expectations are in formula (1) and Rational Expectations are defined in formula (3).

Heuristic Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 M SE Naive 18,2094 11,3897 24,4641 23,996 2,6596 14,7526 15,9119 Rational 201,4835 524,4509 540,6908 237,3497 146,0207 183,0629 305,50975 Adaptive 29,6793 10,6626 39,5363 37,7862 3,4861 22,0181 23,86143 Trend 2,7571 29,7876 23,4201 11,953 3,5657 6,9013 13,06413 HSM 2-rule 3,2257 15,2053 20,7835 9,4667 2,7477 7,108 9,75615 Table 1: Mean Squared Errors of every heuristic per group. The heuristic that

performs best is bold

It becomes clear from table 1 that the HSM is on average the best model for predicting the price in the next period in Learning-to-Optimize experiments. Although this was never shown before, it is an expected result, since this was already proven to be true in Learning-to-Forecast experiments. It is also striking to see how poorly the Rational expectations performs. This happens particularly in groups with either large bubbles, such as group 3, or groups that predict consistently too high or too low, such as group 2. It is therefore safe to say that Rational Expectations can be eliminated as a forecasting method in Learning-to-Optimize experiments.

Adaptive Learning is a method that works better in negative feedback mar-kets, because these markets tend to converge to the Rational equilibrium. In Positive feedback markets, Trend-Following is usually a better system because

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people expect that the price will move in the direction that it was moving already. In positive feedback markets, an increasing price will lead to an even higher price in the next period because a higher demand leads to a higher price. And an increase in the price will lead to higher price expectations in the next period forecast according to Trend-Following. Therefore, the trend-coefficient is positive.

It is therefore no surprise that Trend-Following is the second best perform-ing heuristic. The biggest difference between the HSM and Trend-Followperform-ing occurs in group 2. This is the only group with some price stability. The result from this group looks more like the result expected from a negative feedback market. It is therefore no big surprise that Adaptive Learning is the best performing heuristic. From this, it follows that Adaptive Learning is the dominating heuristic in this groups’ HSM. This can be seen in Figure 2.

Also, it makes sense that Naive Expectations performs relatively well in this group, because in the case of stable prices, the next price will be very close to the current price.

4.2 Two types of Trend-Following

It is important to know the intensity of the trend-following. So far, the subjects were considered neutrally trend-following, which was imposed in the model by setting the trend coefficient γ equal to 1. In this section, there will be two variables for the trend: the weak trend-following coefficient is 0.4, the strong trend-following coefficient is 1.3. These values are chosen as in Hommes (2011). This means that the weak trend following forecast becomes

pweak_t+1 = pt+ 0.4(pt− pt−1) (16)

and the strong trend following forecast is given by

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This changes the number of heuristics to three. Therefore, our new question is: Are people weak or strong trend-followers, and is the Heuristics Switching Model still a better model for predicting the future price?

A more detailed outline of the results can be found in Table 7 in Appendix B, but it turns out that the HSM is still on average the best. In fact, in the new scenario, the HSM is the best model in group 1, 3 and 4. In the very stable group 2, unsurprisingly adaptive learning is still the best heuristic. Also, weak trend-following is much better than strong trend-following, which could be expected, because there are relatively small changes in the price. A higher value for the trend-coefficient γ will amplify these small changes, and Strong Trend-Following is therefore not consistent with the realized expected prices. In group 5 and 6, Weak Trend-Following is the best forecasting model, just ahead of the HSM.

It is no surprise that the HSM is still the best system on average. It would seem that when more heuristics are added, more switching options arise. This should lead to more optimal forecasts. But this section also yields a strange result. The MSE of the Heuristics Switching Model is higher in every group, compared to the situation with only 1 trend heuristic. So even though the fact that the switching could lead to better predictions, something else is going on as well. The most plausible conclusion is that both these values (0.4 and 1.3) do not fit the data very well. This looks like a plausible explanation, because only in the relatively stable markets, Weak Trend-Following outperforms the single trend heuristic. In 4 out of 6 groups, a trend coefficient of 1 outperforms both weak and strong trend following. It appears that the optimal value of

this coefficient lies somewhere between 0.4 and 1.3.2

2_{It can be shown that this system is stable for any γ ∈ [0.4, 1.3]. This is however outside}

of the scope of this paper. For now, it is sufficient to say that this model meets the necessary stability requirements.

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5 Optimization of the HSM

5.1 Heuristic parameters optimization

So far, this paper has applied the Heuristics Switching Model (as in Anufriev et al., (2012)) to a Learning-to-Optimize experiment. The results were that the HSM had the smallest Mean-Squared-Error of 1-period ahead forecasts, averaged over all six groups. Trend-Following and Naive expectations turned out to be relatively good predictors as well. Also, it is shown that the subjects had no preference regarding weak or strong trend-following. And that neu-trally trend-following (γ = 1) fitted the data better than weak trend-following (γ = 0.4) or strong trend-following (γ = 1.3). It looks like the trend-coefficient γ should minimize the MSE at a value between 0.4 and 1.3. Also, the anchor-ing coefficient w ∈ [0, 1]. In this section, these values for trend followanchor-ing and anchoring are optimized to minimize the Mean-Squared Error of the HSM. The results are given in table 2.

w gamma HSM Adaptive Trend Naive Rational

Group 1 0,104 0,9099 2,6712 188,8985 2,5645 18,2094 201,4835 Group 2 0,1385 0,4 11,9549 17,305 17,7889 11,3897 524,4509 Group 3 1 1,1843 16,8394 24,4641 26,5838 24,4641 540,6908 Group 4 0,0307 0,8557 7,5771 325,3254 10,9271 23,996 237,3497 Group 5 0,4065 0,5063 2,2234 7,7179 2,5029 2,6598 146,0207 Group 6 0,2045 0,808 5,983 91,7615 5,9902 14,7526 183,0629 Average 0,31403 0,7774 7,87483 109,2454 11,05956 15,9119 305,50975

Table 2: The second and third column give optimal w and γ. The corresponding MSE is given in columns 4 to 8.

Without a surprise, the HSM is the best model because the optimization was aimed at minimizing the HSM. So the most interesting result is in column 2 and 3. It looks like w tends to be small. First of all, that means that our original thought of w = 0.75 as anchoring coefficient was not very accurate. A smaller w means that people rely more on their own prediction and less on

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the previous price.

Even more interesting is the trend-following coefficient, which is important because this is a positive feedback market and trend-following becomes the dominant heuristic. Section 4.2 suggested that the optimal value would lie somewhere between 0.4 and 1.3. Therefore, these values were used as the lower and upper bound. Only the MSE of the HSM in group 2 is minimized on this boundary. This is expected because group 2 is a very stable market, in which there is only a very small trend. It also matches the expectation from section 4.2, where we found that weak trend-following performs significantly better in group 2 than strong trend-following. So in this group, it was expected that this coefficient should be relatively low. All other groups are also consistent with the expectations from section 4.2. Also, the average trend coefficient γ is relatively close to 1, so the value chosen in section 4.2 is acceptable.

5.2 HSM parameters optimization per group

In this section, all the evolutionary parameters (δ for memory in the weights, sensitivity parameter β and performance memory coefficient η) of the HSM will be optimized. After that, sections 5.1 and 5.2 will be combined, and all 5 variables will be optimized simultaneously. This leads to the following table.

Group Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 (w,γ) (0,75;1) (0,75;1) (0,75;1) (0,75;1) (0,75;1) (0,75;1) (ˆδ,ˆη) (0;0) (0,6803;0,4668) (0,8689;0) (0;0,6303) (0;0) (0,7969;0,2381) ˆ β 0,6697 5,2237 3,2507 1,5078 0.0511 14,2599 MSE 3,0638 11,6038 16,7909 9,0141 2,4014 6,2145 ( ˆw,ˆγ) (0,0052;0,8968) (0,4105;1,3) (1;0,9755) (0,0036;0,8724) (0,5007;0,4955) (0,0790;1,0161) (ˆδ,ˆη) (0;0,3069) (0,7566;0,5081) (0,4699;0,1385) (0;0,4973) (0;0,42) (0,3191;0) ˆ β 54,4154 0,6045 5,3514 10,4973 4,5656 14,6611 MSE 2,5622 10,8337 14,5693 7,3975 2,1254 4,7654 Table 3: In the first column, the variables are defined. In the top part, w and

γ are fixed. Only the HSM parameters δ, η and β are optimized. In the bottom part, all 5 parameters are optimized.

It can be seen in this table that there are big differences between the groups. In groups 1, 4 and 6, a very small anchoring coefficient w is optimal, which

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means that these subjects tended to stick to their last forecast. In group 2 and 5, they predicted the next period by sticking to their own forecast for about half of their next prediction, and the other part of the next price depends on the last price. Subjects in group 3 would have performed best if they did not look at their own price prediction at all, and focused entirely on the last price. The trend coefficient γ is rather similar to the values estimated in section 5.1. Regarding the sensitivity parameters of the Heuristics Switching Model; In group 1, 4 and 5, δ is 0. So there should be no memory in the weights of a certain heuristic. This is the optimal value for our starting values of the anchoring coefficient w and trend coefficient γ and also after the optimization of these. This can be expected for groups 1 and 4, since there are multiple bubbles, so trend following and adaptive take turns in being the best heuristic. Therefore, switching should be quick as well. A high value for performance memory η implies that previous performance of a heuristic should be included in the next period performance. Also, high values of sensitivity β are observed in this optimization.

5.3 Optimization of all groups

Since there are big differences between the groups, it is interesting to see what the average values of the HSM parameters should be. So all groups are evaluated at the same time, yielding anchoring coefficient w = 0, 4511, trend coefficient γ = 0, 8338, memory coefficients δ = 0 and η = 0, 6052 and

sensitivity parameter β = 0, 6424.3_{. This leads to the following Mean-Squared}

Errors.

Model Naive Rational Adaptive Trend HSM

MSE 13,7572 269,8115 44,2535 7,3720 5,4040

Table 4: The first row denotes the different heuristics. The second row shows the Mean-Squared errors of every heuristic, averaged over all groups with the optimal parameter value.

3_{These values are found using fmincon and Global Search in Matlab with the following}

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So apparently people anchor almost half on their last prediction and half on the last price. The trend-following coefficient seems to be rather similar in every group, and consistent with the analysis in previous sections.

A value for δ=0 is optimal, which means that there should be no memory in the weighting factors. Therefore, these weights should be updated every period. The performance memory coefficient η is approximately 0.6, which means that the performance of a certain heuristic is persistent. This, combined with a senstivity parameter β of 0,6 and a weights memory parameter δ close to 0, imply that if one heuristic performs well in a certain period, the weighting factor will change quickly. This can be seen in Appendix A, where many peaks can be observed, some for only 1 or 2 periods, after which a weight can drop to 0 again.

6 Individual Performance

To see how individuals perform, every participants implied price prediction has been compared to the actual price. The resulting graphs and table are in the appendix, but this section will only show a sample. As an example, the results for group 4 are shown here, which had significant bubble formation. There was however quite high coordination after period 25, so it can be expected that people have similar expectations (although possibly incorrect) about the next period.

Subject 1 Subject 2 Subject 3 Subject 4 Subject 5 Subject 6 23,4034 50,1558 25,7476 29,2264 47,0961 48,5635 Adaptive Trend-Following HSM Naive Rational

37,7862 11,953 9,4667 23,996 237,3497

Table 5: In the upper panel, the Mean Squared Errors of every participant of group 4 is displayed. The bottom panel shows the MSE of every heuristic for group 4.

For this group, the Mean-Squared Errors are much higher than HSM or Trend-Following would predict. However, Rational Expectations does not fit

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these numbers either. It looks like people switch between different heuristics, one of which could be naive expectations. But naive expectations is a model that is very easy to use, so if that would be the case, the MSE of 5 out of the 6 participants is still too high. The easiest explanation for these results is that people use more heuristics. That would imply also confirm the fact that the HSM is on average the best model. It is the only model that leaves room for switching.

Another possibility is that they are simply not good or consistent enough to follow the best heuristic. A similar result is found in all other groups. Maybe people are not capable of predicting the proper forecast following from a certain heuristic. This is a very promising research agenda, which is however outside of the scope of this thesis.

7 Discussion

This paper has attempted to fit the Heuristics Switching Model to Learning-to-Optimize experiments. The main aim of this paper was to find out if the HSM is a better model than any single heuristic for predicting in LtO experiments. Since this was never done before, the results are interesting. First of all, the HSM turns out to be a better predictor for next period quantities, which can be seen as implied price forecasts, than any of the single heuristics we have tested. However, looking at individual results, it can not be stated that the HSM fits individual forecasting well. So the HSM has to be extended to fit individual behavior better. The Trend-Following and Naive Expectations, which is a special case of Trend-Following (γ = 0), turned out to be better predictors of this positive feedback market than Adaptive Learning. This result can be viewed as a robustness check for previous papers on this matter such as (Anufriev et al., 2012).

This paper did not focus on the cause of the bubble formation, which could be subject for further investigation. The main aim was to look at the perfor-mance of the HSM. Also, input parameters of the HSM have been optimized

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seperate and together. It seems that an average amount of trend following is optimal. Not too strong, but also not too weak turned out to be optimal. Also, heuristics performance needs to be updated quickly. These values are higher than in negative feedback markets, which makes sense because there is less bubble formation in those markets. Therefore, it will not be as necessary to switch from a poorly performing heuristic. These parameter values certainly require a robustness check and verification from other experiments. That is a good place to start further research.

Fitting the HSM to individual predictions was outside the scope of this paper, but definitely interesting for future research as well. Also, a closer look at coordination and the effects of coordination on bubble formation are possible extensions of these findings.

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References

[1] M. Anufriev and C. Hommes. Evolution of market heuristics. The Knowl-edge Engineering Review, 27(02):255–271, 2012.

[2] M. Anufriev and C. Hommes. Evolutionary selection of individual expec-tations and aggregate outcomes in asset pricing experiments. American Economic Journal: Microeconomics, 4(4):35–64, 2012.

[3] T. Bao, C. Hommes, and T. Makarewicz. Bubble formation and (in) effi-cient markets in learning-to-forecast and-optimize experiments. Technical report, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance, 2014.

[4] A. Beja and M. B. Goldman. On the dynamic behavior of prices in

disequilibrium. The Journal of Finance, 35(2):235–248, 1980.

[5] A. A. Bostian and C. A. Holt. Price bubbles with discounting: A

web-based classroom experiment. The Journal of Economic Education, 40(1):27–37, 2009.

[6] W. A. Brock and C. H. Hommes. A rational route to randomness. Econo-metrica: Journal of the Econometric Society, pages 1059–1095, 1997. [7] W. A. Brock and C. H. Hommes. Heterogeneous beliefs and routes to

chaos in a simple asset pricing model. Journal of Economic dynamics and Control, 22(8-9):1235–1274, 1998.

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

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

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

[11] D. Kahneman. A perspective on judgment and choice: mapping bounded rationality. American psychologist, 58(9):697, 2003.

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[14] M. Kirchler, J. Huber, and T. Stockl. Thar she bursts: Reducing confusion reduces bubbles. The American Economic Review, 102(2):865–883, 2012. [15] R. Marimon, S. E. Spear, and S. Sunder. Expectationally driven market volatility: an experimental study. Journal of Economic Theory, 61(1):74– 103, 1993.

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1:161–176, 1972.

[18] V. L. Smith, G. L. Suchanek, and A. W. Williams. Bubbles, crashes, and endogenous expectations in experimental spot asset markets. Economet-rica: Journal of the Econometric Society, pages 1119–1151, 1988.

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A

HSM Results

In this section, the results of the experiment and the Heuristics Switching Model are shown. Figure 3 shows the weights of individual heuristics. The pink line (squares) shows trend-following. The blue line (circles) shows adaptive learning. It becomes clear that especially in the groups with bubble formation, trend-following performs much better. This is reflected in the results from section 4. In figure 4, individual quantity decisions are displayed. Quite decent coordination can be observed in some of the groups. Subjects are considered to be coordinating if they make similar trading decisions. This can be observed in group 1, group 3 after period 30, group 4 after period 20 and some coordination in group 5 and 6. This implies that people have different expectations at first, but they somehow converge to a similar trading strategy.

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Figure 3: Weights HSM 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) Group 1 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) Group 2 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (c) Group 3 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (d) Group 4 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (e) Group 5 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (f) Group 6

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Figure 4: Individual trade decisions 0 5 10 15 20 25 30 35 40 45 50 −5 −4 −3 −2 −1 0 1 2 3 4 5 (a) Group 1 0 5 10 15 20 25 30 35 40 45 50 −5 −4 −3 −2 −1 0 1 2 3 4 5 (b) Group 2 0 5 10 15 20 25 30 35 40 45 50 −5 −4 −3 −2 −1 0 1 2 3 4 5 (c) Group 3 0 5 10 15 20 25 30 35 40 45 50 −5 −4 −3 −2 −1 0 1 2 3 4 5 (d) Group 4 0 5 10 15 20 25 30 35 40 45 50 −5 −4 −3 −2 −1 0 1 2 3 4 5 (e) Group 5 0 5 10 15 20 25 30 35 40 45 50 −5 −4 −3 −2 −1 0 1 2 3 4 5 (f) Group 6

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4.2 shortly mentions these results. Aside from that, these results are posted mainly for individual interest and as a starting point for further research on this topic.

Subject # Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Subject 1 7,02 53,4206 60,4127 23,4034 32,5648 27,5294 Subject 2 39,0629 72,765 89,1874 50,1558 11,9271 24,2477 Subject 3 12,4337 108,8207 34,4377 25,7476 6,832 25,1681 Subject 4 11,0801 52,4088 130,5519 29,2264 33,6026 24,7678 Subject 5 31,5963 18,0263 24,2114 47,0961 6,5677 45,4594 Subject 6 10,7932 75,7286 34,2584 48,5635 14,2127 34,7134 Group Average 18,66436 63,5283 62,17658 37,36546 17,61782 30,3143 Table 6: Mean Squared Errors of every single participant. Every column represents a different group. The last line in the table is the group average.

B

Weak vs. Strong Trend-Following

In this section, the Mean Squared Errors with 2 trend heuristics are displayed. Weak trend following performs better than strong trend following in 5 out of 6 groups. It is therefore most likely that a γ of 1.3 is too high. Trend-Following however dominates the HSM, and the results of the HSM are better than weak-trend following in most groups. Therefore, it can be assumed that a value of 0.4 as a coefficient is too small. The optimal value will be somewhere in between.

Heuristic Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 M SE Naive 18,2094 11,3897 24,4641 23,996 2,6596 14,7526 15,9119 Rational 201,4835 524,4509 540,6908 237,3497 146,0207 183,0629 305,50975 Adaptive 29,6793 10,6626 39,5363 37,7862 3,4861 22,0181 23,86143 Weak Trend 7,0444 17,7889 24,2803 14,2462 2,4407 7,659 12,24325 Strong Trend 5,5291 38,8609 29,3931 17,2833 4,8333 10,4985 17,73303 HSM 3-rule 3,9298 16,8445 23,3423 11,2731 3,1112 7,9118 11,06878 HSM 2-rule 3,2257 15,2053 20,7835 9,4667 2,7477 7,108 9,75615 Table 7: Mean Squared Errors of every heuristic per group. The heuristic

that performs best is bold. The 2-rule HSM is also added to this table for comparison.

The Heuristics Switching Model in Learning-to-Optimize experiments