The eect of trading coordination on price stability in
learning to optimize experiments
Isabel de Heus 10002993
Supervisor: Tomasz A. Makarewicz MSc Bachelor's thesis Econometrics
University of Amsterdam 27 June 2014
Abstract
The eects of trading coordination on price stability in learning to optimize experiments varies over time and per market. The realized prices in these markets are mostly very volatile and include bubbles and crashes. During the initial periods of an experimental asset market, trading coordination has a strong positive eect on the price stability. After those periods the prices do not converge but agents learn to ride the bubbles. Also the price stability across markets can be explained by the heterogeneity of trading behaviour when controlling for a reference point, the maximum realized price in the ve previous periods. By contrast the deviation between the realized price and the fundamental price does not work as a reference point in the trading behaviour.
Contents
1 Introduction 3
2 Learning to optimize experiments 5
2.1 Experimental economy . . . 5 2.2 Research method . . . 6 3 Results 7 3.1 Data overview . . . 7 3.2 Analysis . . . 9 3.2.1 Pooled estimation . . . 9
3.2.2 Group variant eects . . . 9
3.2.3 Time variant eects. . . 12
3.2.4 Reference points. . . 14
4 Comparison and relevance 16
5 Conclusion 18
A Estimations model (6) 21
1 Introduction
The dot-com crash and the 2007 nancial meltdown oer empirical evidence of bubble formation on asset markets. The existence of bubbles contradicts the theory of rational expectations, which has been the leading theory since its introduction by Muth (1961) and Lucas (1972). This theory states that 1) agents forecast rationally without systematic forecasting errors and therefore their expectations are model consistent and 2) they use these expectations to trade optimally. To form rational expectations agents need to know the true distribution of all endogeneous variables. In this situation the price will converge to the rational expectations equilibrium, this implies that over- or undervaluation of assets is not possible, hence the existence of such nancial bubbles is not explainable. These bubbles generate price instability since the value of an asset during a bubble is not based on the intrinsic value. This uncertainty in the price leads to an uncertain return of the asset. Kahneman and Tversky (1973) already disputed the theory of rational expectations by stating that agents have a certain loss aversion and exhibit biased behaviour.
Since uncertainty in the price leads to an uncertain asset return, the agent's utility will mostly decrease with the associated risk of an asset. At the same time the agents themselves have a huge in uence on the realized price. Namely their decisions to buy or sell the asset on the market sets the market clearing price. In order to understand the price dynamics of an asset it is necessary to look at the extent to which agents behave homogeneous. When agents make the same trading decisions, the agents are coordinating on their trading behaviour. The rational expectations theory implies that there exists no heterogeneity, which means that there is no dierence in individual quantity decisions. In the situation of rationally behaving agents the rational expectations theory predicts that the realized price will converge to the rational expectations equilibrium. This implies a positive relationship between the trading coordination and the price stability. In order to be able to anticipate the trading behaviour it is neccessary to obtain accurate knowledge in the price dynamics of assets.
The aforementioned dot-com crash and nancial meltdown are typical examples of overvaluation of assets. There are many factors which cause this overvaluation on the asset market, so it is di cult to distinct the causalities and in uencing variables. Experiments are a useful tool since there is full control on the changing variables. These experiments can be used to reliably mimic an asset market to compare and estimate the eects of the in uemcing factors. Smith et al. (1988) were one of the rst to reproduce bubbles with learning to optimize experiments. In these experiments the subjects have to decide on the quantity to buy or sell of a certain asset, and the price of the asset is determined by the excess supply or demand of the asset on the market. Thereupon more interest arised in the
formation of expectations which led to the introduction of learning to forecast experiments. The learning to forecast experiment was introduced by Marimon et al. (1993), in this experiment subjects are asked to make price predictions of an asset.
Hommes (2011) compares learning to forecast experiments of dierent bench-marks. The author found that people coordinate on trend-following behaviour which in turn leads to bubbles in most of these experiments. In addition, the author concluded that heterogeneity of individual price forecasts is the main rea-son of excess price volatility. But Bao et al. (2014) reject the hypothesis that nancial agents are trading consistently with their forecasts. The subjects seem to perform better in forecasting experiments than in optimizing experiments. Bao et al. (2013) explain this by the fact that the learning to optimize experiments have a higher cognitive load. Furthermore, in real life agents do not have to set their forecast explicitly but they only have to decide on their trading decision. Therefore the recent nancial bubbles will t the learning to optimize experiments better. Most large nancial institutions have separate departments for forecasting and optimizing (Bao et al., 2013). They explain the division of these tasks with `two heads are better than one'. Consequently I shall focus on learning to optimize experiments.
To gain an insight into the eects of trading coordination on price stability I shall look at the experimental asset pricing market of Bao et al. (2014). They designed three dierent treatments: learning to optimize, learning to forecast and a mixed treatment in which the subjects have to perform both tasks. In the learning to forecast experiment the supply and demand is determined as if the subjects are trading consistently with their forecasts. The price of the asset is determined by the price adjustment rule based on excess supply or demand as described by Beja and Goldman (1980). The learning to forecast treatment induced rather stable price bubbles, while in the learning to optimize and mixed treatment very volatile price patterns appeared. Furthermore they were the rst to nd repeated `super-bubbles' in the mixed treatments. They found that agents coordinate on the sign of the traded quantity, which could explain the existence of bubbles.
The eects of trading coordination on the price stability has not been investi-gated extensively yet, therefore I test whether there is a signi cant relationship. The heterogeneity is both signi cantly dierentiated in each group as well as per time period. To control for the dierences between each group panel data is used, which allows for market-speci c parameters.
For two groups it can be concluded that there is a signi cant positive rela-tionship between the trading coordination and the price stability. The trading behaviour of the participants can be explained by the maximum realized price of the ve previous periods, which works as a reference point. Also the participants
of all groups learn to ride the bubble after an average period of 25. The deviation between the fundamental price and the realized price does not have an eect on the price stability. The ignorance of the subjects of the fundamental price may therefore be a direct cause.
The second chapter starts with an introduction to the used learning to optimize experiment. Then the third section presents the results of the experiment with the corresponding analysis, after which I compare the results with the existing theories. The last section concludes.
2 Learning to optimize experiments
In learning to optimize experiments subjects have to decide on the amount to trade of a certain asset. After each subject's trading decision is set, the price of the asset is determined by a market clearing condition. The experimental asset pricing market of Bao et al. (2014) is based on the asset market of Heemeijer et al. (2009). In this learning to optimize experiment the subjects have to trade a certain asset for 50 successive periods from which they gain utility.
2.1 Experimental economy
The experimental economy of this asset market de nes the utility function based on a myopic mean-variance optimization model. In this model the agents want to maximize their expected wealth, and at the same time they want to minimize the variance of this wealth. So for an asset market each agent wants to maximize the return of the asset which is a function of dividend and return, and they want to minimize the risk associated with the asset. The reducing eect of the variance of wealth is depending on the risk aversion of the agent.
The price of the asset is determined by the same parametrization used in Heemeijer et al. (2009), which leads to the following price adjustment:
pt+1 = pt+ 20 21 6 X j=1 zj,t. (1)
The next-period's price is a function of the current price and the excess demand or supply on the market. Here zj,t is the amount agent j decides to trade at period t.
When an agent expects the price to increase and therefore buys the asset, the price will increase. The opposite holds for expecations of decreasing prices. This is the positive feedback mechanism of the price adjustment. The fundamental price is equal to pf = 66 for all 50 periods. The optimal demand of a rationally behaving
zj,t∗ = p
e
j,t+1+ 3.3 − 1.05pt
6 . (2)
The demand is derived from trading optimally conditional on the expected price pe
j,t+1.
In the learning to optimize experiment the subjects are assigned the role of a trader of a large rm. The subjects receive qualitative information about the market structure such that they know the law of motion of the price adjustment (1). This information is given in order to ensure that the subjects have the same information as actual traders of a large rm. The earnings of each subject are a function of the utitlity of the rm such that the optimization function of the subjects is equal to that of the rm.
2.2 Research method
In each of the six learning to optimize markets six subjects are trading an asset for 50 periods. To measure the price stability for market i the standard deviation of the price in ve subsequent periods takes the form of:
σpi,τ= v u u t 1 5 τ +4 X k=τ
(pi,k − ¯pi,τ)2 for τ={1, 6, ..., 46} (3)
The price stability measure at time τ is a function of the 5 following prices and
the mean ¯pi,τ = 15P5τ =1pi,τ of the same 5 prices. In order to compare this price
stability measure with the trading coordination the heterogeneity is de ned as following: hi,τ = 1 5 τ +4 X k=τ v u u t 1 6 6 X j=1
(qi,k,j − ¯qi,τ)2 for τ={1, 6, ..., 46} (4)
The heterogeneity of individual quantity decisions equals the averaged standard deviation of individual quantity decisions in ve successive periods. The mean of the quantity decisions of each subject in period isτde ned as ¯qi,τ = 16
P6
j=1qi,j,τ. If
all measurements from t=1 through t=46 would have been used, then all variables within four subsequent periods are highly correlated. This is because the same quantity decisions are used to compute the trading coordination measurement. In order to avoid this multicollinearity problem it is neccessary to use only the measurements within a 5 period interval. So with 50 trading periods, this results in 10 measurements per market, thus 60 usable observations remain.
In these experiments the only task of the participant is to decide on the quantity to trade for 50 periods after which the price is realized, thus there are practically
two variables available: realized prices and quantity decisions. To explain the price variability it is important to look at the deviation of the realized price from the fundamental price. Also the maximum or minimum price reached in 5 previous periods may explain the price (in)stability. Table 1 displays the additional variables along with their explicit formulas.
Variable Notation Formula
Deviation from the fundamental price DevPτ
(
Pf− Pt−5 f or τ > 1
0 f or τ=1
Maximum price of 5 previous periods M axPτ
(
max{pτ −5,pτ −4,pτ −3,pτ −2,pτ −1} f or τ > 1
0 f or τ=1
Minimum price of 5 previous periods M inPτ
(
min{pτ −5,pτ −4,pτ −3,pτ −2,pτ −1} f or τ > 1
0 f or τ=1
Table 1: Additional variables
3 Results
3.1 Data overview
As found in Bao et al. (2014) repeated bubbles appear in several groups, while in other groups prices remain stable. The subjects always need some initial periods before they are able to coordinate on their trading behaviour. In some groups the subjects never get coordinated, while in other groups the subjects are coordinating almost perfectly after about 20 periods. Figure 1 shows the realized prices for each group and the individual quantity decisions for all 50 periods.
The markets can be categorized into three groups. One of unstable prices and bubbles, one of stable prices and the last group is a combination of both. The unstable group consists of market 1, 3 and 4. In these markets the biggest bubbles appear, in all three groups even more than one bubble is realized. The prices uctuate around the fundamental price, especially in market 3 the price uctu-ates from 30% to 150% of the fundamental price. The heterogeneity of trading behaviour in these markets decreases over time and becomes even negligible after 25 periods. So in these markets the decreasing heterogeneity does not correspond with less volatile prices.
0 5 10 15 20 25 30 35 40 45 50 −2
0 2 4
Individual quantity decisions
Period (t) 0 10 20 30 40 50 0 50 100 Realized prices Period (t) (a) Group 1 0 5 10 15 20 25 30 35 40 45 50 −5 0 5
Individual quantity decisions
Period (t) 0 10 20 30 40 50 0 50 100 Realized prices Period (t) (b) Group 2 0 5 10 15 20 25 30 35 40 45 50 −5 0 5
Individual quantity decisions
Period (t) 0 10 20 30 40 50 0 50 100 Realized prices Period (t) (c) Group 3 0 5 10 15 20 25 30 35 40 45 50 −2 0 2 4
Individual quantity decisions
Period (t) 0 10 20 30 40 50 0 50 100 Realized prices Period (t) (d) Group 4 0 5 10 15 20 25 30 35 40 45 50 −2 0 2 4
Individual quantity decisions
Period (t) 0 10 20 30 40 50 0 50 100 Realized prices Period (t) (e) Group 5 0 5 10 15 20 25 30 35 40 45 50 −5 0 5
Individual quantity decisions
Period (t) 0 10 20 30 40 50 0 50 100 Realized prices Period (t) (f) Group 6
Figure 1: Overview of the data for groups 1-6. For each group the realized prices and quantity decisions are displayed. The left column shows the individual quantity decisions (blue dotted lines) with the average quantity decision (red line). The right column shows the realized prices (black line) with the fundamental price (blue dotted line).
The second group includes market 2 and 5. These markets realized rather stable price patterns. In both markets the price is increasing slowly with little uctuations. The only dierence between the markets is that the prices in market 2 are about 65% of the fundamental price, while the other market realized a price that ends above the fundamental price. The trading coordination is better in market 5 than in market 2, but it decreases over time in both groups. The subjects in these markets traded only little amounts, which could be the cause of the stable price pattern.
The last group is market 6. The price pattern is a combination of a bubble, crash and a stable price. It starts with large overvaluation after which the bubble crashes, but the price does not fall to a level below the fundamental price. After this crash the price stays stable. The subjects do not coordinate until period 25, but after that they are able to coordinate really wel. The decreasing heterogeneity in this group thus corresponds with a stable price pattern.
3.2 Analysis
3.2.1 Pooled estimation
To evaluate the relationship between the price stability and the trading coordina-tion the following model is estimated, henceforth called the standard model:
σpi,τ = α + βhi,τ + ετ. (5)
In the standard model the price stability is explained by the heterogeneity of trading decisions. Table 2 shows the estimations of the coe cients of the model (5), along with their standard errors and the probability value of the t-test.
Dependent variable: SD Price Coe cient Standard Error P-value
Constant 3.5369 0.6967 0.0000
hi,τ 3.3379 1.9216 0.0877
Table 2: Estimations of model (5)
The heterogeneity has a positive eect on the price deviation. But this esti-mation goes along with a probability of 8.8% that hi,τ has no eect on the price
stability. This indicates that there does exist a relationship but there are possibly omitted variables. The origins of this insigni cance could be in the dierences between groups and the time dependent coordination.
3.2.2 Group variant eects
Figure 2 shows the scatter plot of the standard deviation of the individual quantity decisions versus the standard deviation of the prices. Apparently, most groups have
its own dierentiated relationship with the price stability. Most observations are concentrated in the region of low heterogeneity where they correspond with both high and low price standard deviations.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 2 4 6 8 10 12 14 16 Heterogeneity
Price standard deviation
Group 1 Group 2 Group 3 Group 4 Group 5 Group 6
Figure 2: Trading heterogeneity versus price deviation per group
The standard model disregards the market structure of the observations because it treats all observations as if they belong to the same group. Panel data does account for the market structure. The dierences between the groups are modelled in terms of the market-speci c, xed parameters αi. In order to control for the
group eects, panel data is used. The following panel model is estimated:
σpi,τ = αi+ βhi,τ + εi,τ. (6)
The estimations of the parameters αi vary for each group from negative to positive
(see the appendix for all the market speci c intercepts). The xed eects are not redundant which means that the groups dier signi cantly. If allowed for the dierent intercepts per group, the eect of trading coordination remains positive (3.383) but still not signi cant with a p-value of 5.7%.
Besides the dierent xed group parameters, the eects of the trading behaviour in one group may dier from the eect in another group. Therefore model (7) includes dierent coe cients of heterogeneity per group, and still uses the panel model.
σp,i,τ = αi+ β1h1,τ+ β2h2,τ + β3h3,τ + β4h4,τ+ β5h5,τ + β6h6,τ+ εi,τ (7)
The error terms are assumed to be equal for all groups, in contrast with a seperate model for each group. Therefore this model (7) allows to test coe cients jointly.
Each group gets its own coe cient β1through β6 to estimate the individual eects.
The eects of group 1 and group 6 are both positive and signi cant. The eect of the trading coordination in group 1 (10.923) is not signi cantly dierent from the eect in group 6 (9.224). For group 2, 3, 4 and 5 the hypothesis of no eect cannot be rejected. As these groups do not have a signi cant relationship, it is reasonable to delete the regressors for these groups. The latter ndings are modelled as following:
σpi,τ = αi+ β1h1,τ + β6h6,τ+ εi,τ. (8)
Table 3 shows the estimations of the model (8). Both coe cients are signi cant and have a strong positive eect on the price deviation. So for these two markets the price stability can be explained solely by the trading coordination.
Dependent variable: SDprice Coe cient Standard Error P-value
Constant 3.7676 0.4052 0.0000
Heterogeneity Group 1 10.9227 4.2170 0.0124 Heterogeneity Group 6 9.2238 4.4122 0.0415
Group Fixed Eect
1 0.0641 2 -1.2291 3 2.9322 4 2.5200 5 -1.9164 6 -2.3707
Table 3: Estimations of model (8)
The xed eects support that market 6 realized a more stable price pattern than market 1. In market 1 the decreasing heterogeneity leads to a longer duration of the cycle of the bubble. This means that the prices are more stable, even though there is still bubble formation. For the other markets (3 and 4) within the same price and coordination dyamics the duration of the cycle does not change, and the maximum price reached during the bubble increases. Hence there is no eect of trading coordination on price stability in group 3 and 4. The eects of the trading coordination are more clear in market 6. In this market there is a big dierence in price dynamics since there appears one bubble as well as a stable price. The trading coordination after period 25 goes along with the stable price pattern.
The other markets (2 and 5) consisting of stable prices and increasing coor-dination do not have signi cant eects as well. The prices are stable throughout the whole session, which means that the decreasing heterogeneity does not lead to more stable prices or vice versa.
3.2.3 Time variant eects
The data of the experiment also shows that the trading coordination changes over time, this may have an eect on the relationship with the price stability. Figure 1 shows that the trading behaviour becomes less heterogeneous after a certain number of periods. This could indicate that the subjects learn to coordinate on their trading behaviour. In order to study the varying trading coordination, all observations are divided in three periods: 1-15, 16-31 and 32-461.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 2 4 6 8 10 12 14 16 Heterogeneity
Price standard deviation
Period 1−15 Period 16−31 Period 32−46
Figure 3: Trading heterogeneity versus price standard deviation
The division of the experiment into these three periods is displayed in the scatterplot in Figure 3 and gains more insight in the period eects. In the rst 15 periods the subjects are the most heterogeneous, but there does not seem to exist a strong relationship with the price stability. During the second part, from period 16 through 31, the participants are coordinating much better in contrast to the rst part. Also there seems to exist a strongly positive relationship with the price stability. At last during the third part of the session the subjects are able to coordinate really well on their trading decisions, but this corresponds with both high and low price stability.
Panel data can also be used to model the time structure of the observations. In this panel model each time period, 1-5 through 46-50, gets its own time-speci c parameter αt. The time xed eects are modelled as following:
1 The observations end at period 46 because the price stability at period 46 is
σpi,τ = αt+ βhi,τ + εi,τ. (9)
The estimation of β as well as the xed time eects are not signi cant. Thus it is not reasonable to model the time eects using panel data.
Instead of the panel model a structural break in the data can explain the time varying eects. Therefore the cumulative sum control chart (CUSUM) analysis is used, the chart is shown in Figure 4.
-30 -20 -10 0 10 20 30 5 10 15 20 25 30 35 40 45 50 55 60 CUSUM 5% Significance
Figure 4: CUSUM Test
The gure shows that the data never exceeds the thresholds, thus there is no signi cant structural break. There are two points in time that do signal a structural break, namely after period 5 and after period 25 (after observation 12 and 30).2 The Chow breaktest rejects the existence of a structural break after
period 15. Although not signi cant, the breaktest does signal a break after period 25. In order to allow for a structural break the following model is estimated
σpi,τ = α + βhi,τ + γ1D25i,τ + γ2D25i.τhi,τ + ετ. (10)
In this model a dummy is added for the observations after period 25. By adding the heterogeneity multiplied by the dummy the eect of the trading coordination on the price stability after period 25 is isolated.
2The data is ordered such that all 10 periods per group are pooled. Therefore
observation 1 through 6 corresponds with the observations in period 1-5 for all six groups.
Dependent variable: Standard Deviation Price Coe cient Standard Error P-value
Constant 2.9256 1.0320 0.0064
Heterogeneity 4.8052 2.3129 0.0423
Dummy period 21-25 3.0247 1.7354 0.0868
Dummy*Heterogeneity -16.6934 8.3798 0.0515
Table 4: Regression output model (10)
Table 4 shows the estimations of the model (10). The estimations point out that the price stability is higher after period 25, namely 3.02. After period 25 the prices are more volatile but the eect of trading coordination on price stability is strongly negative (-12.693). Also the eect of heterogeneity is positive (4.805) in the periods before 25, this is due to the strong negative eect of heterogeneity after period 25. Although the estimation of the dummy at period 25 is not strongly signi cant, but deleting the dummy leads to even less signi cant estimations. This is because in that situation the isolated eect has to take account for the break as well as the negative eect. As seen in Figure 2 the eects are positive in the rst periods of the experiment, in the beginning the subjects' behaviour is very heterogeneous, causing unstable prices. After these intial learning periods the prices do not become less volatile. Hence it can be concluded that subjects learn after 25 periods to ride the bubble, instead of learning to converge to a stable price. In other words: subjects coordinate their trading behaviour on the sign of the trading quantity. In this situation the low heterogeneity in trading behaviour corresponds with unstable price patterns. This mechanism is enabled by the positive feedback price adjustment: if all subjects expect the price to increase and therefore buy the asset, the price becomes even higher.
3.2.4 Reference points
From the last two paragraphs it can be concluded that there does exist a rela-tionship between the trading coordination and the price stability, but there are dierences in eects between groups and dierences over time. The subjects are able to coordinate but it takes about 25 periods before this coordination is reached. At the same time the prices do not always stabilize when subjects are coordinat-ing. During the experiment a subject only has information concerning the current price and all previous prices. Therefore the subjects can use the previous prices as a reference point in their trading behaviour.
Figure 1 shows that in the groups with an unstable price pattern (1,3,4 and 6) the realized prices are oscillating around a certain price level. For group 1 this level is above the fundamental price, but for groups 3,4 and 6 the prices are oscillating around the fundamental price. The stable group 2 exhibits a price
pattern where the price is is about 60% to 80%of the fundamental price. The other stable market, group 5, realizes prices that start slightly below the fundamental and grows to about 120% of the fundamental. So most groups seem to focus on the fundamental price. When the realized price is close to the fundamental price this indicates that there is no under- or overvaluation of the asset. Hence it is expected that subjects will coordinate in this situation which in turn leads to a stable price. The deviation between the price in the beginning of the period t through t+5 can be seen as an anchoring point, from which the agents will set their behaviour and coordination. In order to estimate this eect the variable DevPF at period t is de ned by the deviation of the price from the fundamental price in period t-5. The variable DevPF is added to the model (9) and is estimated with market speci c intercepts:
σpi,τ = αi+ βhi,τ + δDevP Fi,τ + εi,τ. (11)
This model does not show any form of relationship with the deviation from the price stability since the estimations are not signi cant.
In addition the variable DefPV is added to model (8) where the eects of group 1 and 6 are seperately estimated, but this does not give signi cant estimations as well. Afterwards a model is estimated with the absolute value of the deviation between the realized price and the fundamental price, also in this model the devia-tion does not have a signi cant eect on the price stability. Another explanadevia-tion of the data can be given by editing the model (11) so that it becomes a second order polynomial with terms of the deviation from the fundamental price. Likewise this model does not show a signi cant relationship. So it can be concluded that over-or undervaluation by dierentiating signi cantly from the fundamental price does not have an eect on the price stability.
Subjects were not given the exact fundamental price, but they were theoreti-cally able to compute it rationally by using the price formation rule. This ignorance can be a direct cause of the oscillations and weak coordination around this price level. In this situation where the subjects do not explicitly know the fundamen-tal value, it is di cult to coordinate on converging to an equilibrium. In groups that show unstable price patterns and bubbles, the minimum and maximum prices indicate the speed of the changing of prices. The minimum and maximum price of the foregoing periods does also imply a new standard price level around which it could be reasonable to uctuate. As was the case with the deviation from the fundamental price, the maximum or minimum price of the 5 previous periods may also work as an anchor. The estimations of a model including the minimum price show that the minimum price of the 5 previous periods does not have a signi cant eect on the price stability. In order to evaluate the eect of the maximum price of the 5 previous periods on the price stability this variable is added to the standard
panel data model (6) and becomes of the form:
σpi,τ = αi+ βhi,τ + ζM axPi,τ + εi,τ (12)
The estimations show that the maximum price does have a signi cant relationship with the price stability. Furthermore, the eect of heterogeneity becomes strongly signi cant. Table 4 shows the estimations of the model. The estimation of the eect of the reference point, the maximum price, is small but positive.
Dependent Variable: SD Price Coe cient Standard Error P-value
Constant -1.3114 2.0089 0.5168
Heterogeneity 4.6582 1.7355 0.0097
Maximum Price 0.0665 0.0264 0.0150
Table 5: Estimations of model (12)
To conclude there are three dierent paths to explain the price stability of an asset. The rst one explains the price variability by the group varying eects where only group 1 and 6 can be explained solely by the trading coordination. The second path is an explanation of the price stability by a structural break after period 25, afer this period the subjects learn to coordinate on riding the bubble. And the last path includes the reference point, the maximum price, which the participant use when making their trading decisions.
The combination of time varying and group varying eects leads to a model where there is allowed for structural breaks in groups 1 and 6. In this situation the eect in group 6 is not signi cant, neither are the structural break and isolated eects. Since the heterogeneity itself can explain the price stability for these two groups, the estimations may become insigni cant. Thus there does not exists a structural break in these groups.
Also when controlling for the maximum price in the model for groups 1 and 6, the reference point is signi cant. But the eect in group 6 of heterogeneity becomes insigni cant. So it can be concluded that in group 6 the trading coordination itself can explain the whole price stability. The other groups do use the reference point for their trading decisions.
4 Comparison and relevance
The repeated nding of bubbles on asset markets contrasts sharply with the ra-tional expectations theory. This boundedly rara-tional behaviour has impact on the trading coordination of agents. The rational expectations theory implies that all agents who use the same utility function exhibit similar trading behaviour. There-fore, in the situation of rational behaving agents, there is no heterogeneity to be seen in the trading decisions. Bao et al. (2013) concluded that the distribution of
individual quantity decisions is not rational. This nding is in line with the results of the learning to optimize experiments in my thesis. In my thesis subjects are not able to coordinate on rational trading behaviour because there are large deviations from the fundamental price. However the heterogeneity of trading decisions does mostly decrease over time. Shiller (2000) and Vissing-Jorgensen (2003) concluded from survey data that heterogeneity of forecasts exhibits signi cant variation over time. Furthermore Hommes (2013) found that the heterogeneity of forecasts does either decrease over time or will even diminish. The heterogeneity in the experi-ment of this thesis does never disappear, but it does diminish over time for most groups. In several groups signi cant heterogeneity remains throughout the whole experiment.
Learning to forecast experiments do not require the subjects to make a trading decision. So it is not clear to what extent the heterogeneity of forecasts will lead to heterogeneity of trading decisions. Hommes (2011) states that heterogeneity in individual price forecasts leads to excess price volatility. But it is concluded by Bao et al. (2013) that agents perform better on coordination in learning to forecast experiments than in learning to optimize experiments. Therefore it is not possible that agents used their expectations optimally to make trading decisions. This inconsistency makes it di cult to use the results of learning to forecast experiments to pursue these results to optimizing decisions. Nevertheless the results of my thesis do follow the ndings of Hommes (2011) of the positive relationship between coordination and price stability.
Bao et al (2014) conclude that agents are able to coordinate on the sign of the traded quantity, which is the reason of the price bubbles. This nding does follow the results of my thesis, but is only true for the periods after the initial periods of high heterogeneity. During the initial periods agents nd it harder to coordinate their behaviour.
The rational expectations theory implies that all agents are behaving ratio-nally. DeLong (1989) explains the irrational aggregate outcomes by pointing out the existence of the combination of rational and irrational agents. The irrational traders are called the noise traders because they disturb the convergence to a ra-tional equilibrium. Noise traders are traders with irrara-tional erroneous beliefs. The existence of such noise traders in modern economic environment, which is sup-ported by Abreu and Brunnermeier (2003), can explain the heterogeneity of the trading behaviour. The negative eect on price stability is also explained by noise traders.
In contrast with the results of no eect of price deviations from the fundamental price on the price stability are the ndings of Boswijk et al. (2007). The authors conclude that the trading behaviour can be divided into two groups. One group of trend followers and another group of fundamentalists, who believe that prices
will reverse to the mean. The prices will converge to an equilibrium in the case of fundamentalists. This is in contrast with the results of my thesis. Van Boening et al. (1993) state that the realized price will only converge to the fundamental price when highly experienced traders participate in three successive asset markets for 15 periods. In the experiments of Bao et al. (2014) the subjects were able to participate in one session of 50 periods, so there were no experienced participants. Another contribution to the eect of undervaluation on price stability was found already in 1994 by Caginalp et al. Their momentum model predicts that larger initial undervaluation of an asset leads to larger positive price movements. This is a glaring contrast with the results in my thesis, since the deviation from the fundamental price does not have a signi cant eect on the price stability. Although I found that maximum prices do aect the price stability, the eect of minimum prices on price stability is not signi cant.
5 Conclusion
Learning to optimize experiments show that agents are able to coordinate on their trading behaviour, but the time it takes to coordinate diers per market. The mean time in these experiments it takes to coordinate is about 25 periods, which is the half of the duration of the experiment. In the rst 25 initial periods the high heterogeneity leads to unstable price patterns, varying from bubbles and crashes to little uctuations in rather stable price patterns. After the initial periods prices are still oscillating and bubbles appear. But the dierence with the rst 25 periods is that agents learn to coordinate on riding bubbles, thus they coordinate on the sign of the trading quantity.
Next to the group varyting eects, the heterogeneity of trading behaviour dif-fers across markets too. Therefore panel data is used to model the market-speci c parameters. In the panel model where the price stability is explained by the hetero-geneity, the redundance of these xed eects is rejected. Most unstable markets' realized prices oscillate around a certain price level, close to the fundamental price. But no eect can be found of the deviation between the realized price and the fun-damental price, neither in absolute values nor in polynomial functions. The fact that the participants were not explicitly given the fundamental price could explain this result.
Apparently the price stability is not in uenced by largely over- or undervalua-tion of an asset. A possible explanaundervalua-tion is that the agents do not explicitly know the fundamental price. Though, the maximum price that has been realized in the 5 previous periods does aect the price stability. The dierence between the current price and the maximum price of the 5 previous periods indicates the speed of price movements. Thus when controlling for the maximum price the heterogeneity has
a positive signi cant eect on the price stability.
For future research it may be interesting to design learning to optimize exper-iments in which the subjects are given the fundamental price explicitly. In that case there will be controlled for participants' ignorance of fundamental prices. It would be interesting as well to conduct experiments that last longer, hereby it would be possible to nd the eects of learning and experienced players.
References
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Appendix
A Estimations model (6)
Dependent Variable: SD Price Coe cient Standard Error P-value
Constant 3.5237 0.6164 0.0000
Heterogeneity 3.3832 1.7415 0.0574
Group Fixed Eect
1 1.6843 2 -2.3012 3 1.8573 4 1.7307 5 -2.4122 6 -0.5589
B Estimations model (7)
Dependent Variable: SD Price Coe cient Standard Error P-value
Constant 3.5935 0.6198 0.0000 Heterogeneity 1 10.9227 4.3011 0.0144 Heterogeneity 2 4.9027 4.4628 0.2774 Heterogeneity 3 0.6337 2.9234 0.8293 Heterogeneity 4 -3.8949 4.5630 0.3976 Heterogeneity 5 0.3674 5.4652 0.9467 Heterogeneity 6 9.2238 4.5002 0.0459
