THE FORMATION OF EXPECTATIONS
IN
EXPERIMENTS
A thorough analysis of the dynamics underlying the formation of expectations.
The Netherlands, 2014
Edited by
LAURENS VOOGD
Supervised by
FLORIAN WAGENER
The University of Amsterdam
June 2014
Abstract
Economic agents are modeled as being rational and homogeneous. This for mainly styl-ization purposes that make it possible to model economic activity structurally. The ratio-nality and homogeneity hypothesizes, help model and analyze market dynamics, with the ultimate goal of understanding the price formation process. Prices are determined by a large multiple of factors such as expectations of the agents on the market. This research analyzes experimental data on positive and negative feedback markets. It concludes on the hypothe-sis of rationality being valid for the data used, while in contrast it rejects the homogeneity hypothesis for the used dataset. Finally it designs and evaluates a prediction rule.
Contents
1 INTRODUCTION 5
1.1 Why is it important? . . . 7
2 LITERATURE 8 2.1 An earlier research in review . . . 8
3 DATA 10 3.1 Overview of the data . . . 10
4 RESEARCH 14 4.1 Research Methods . . . 14
4.2 Building blocks of the research . . . 14
4.3 Rationality . . . 14
4.4 Homogeneous Expectation Formation . . . 16
4.5 Prediction Rule . . . 19
5 CONCLUSION 31 6 SUGGESTIONS & SHORTCOMINGS 32 7 REFERENCES 33 8 APPENDIX 34 8.1 Table 1: Basic statistics for the negative feedback group. . . 34
8.2 Table 1: Basic statistics for the positive feedback group. . . 35
8.3 Table 1: Basic statistics for the negative and positive feedback group.for the total mean prediction error T M P EI= 6T1 PTt=1P6i=1ei,t . . . 36
8.4 Homogenious model estimates P = XβI+ . . . 36
8.5 Individual negative feedback model estimates pe i = Xβi+ . . . 37
8.6 Individual positive feedback model estimates pei = Xβi+ . . . 38
8.7 Individual negative feedback model estimates pe i = α + β1pt−1+ β2pt−2+ β3pet−1+ β4pet−2+ β5ψ1t + β6ψt2+ . . . 39
8.8 Individual positive feedback model estimates pe i = α + β1pt−1+ β2pt−2 + β3pet−1+ β4pet−2+ β5ψ1t + β6ψt2+ . . . 40
8.9 Individual negative feedback model estimates pei = α + β1pt−1+ β2pt−2+ β3ψ1t + . . . 41
8.10 Individual positive feedback model estimates pei = α+β1pt−1+β2pt−2+β3ψt1+ 42 8.11 Individual negative feedback model estimates pe i,t= α + P k∈{1,2,3}β p kpt−k+ P k∈{1,2,3}β e kp e i,t−k+ P k∈{1,4}β pattern k ψ k t + P k∈{1,2}β change k φ k t . . . 43
8.12 Individual positive feedback model estimatespei,t= α +Pk∈{1,2,3}β p kpt−k+ P k∈{1,2,3}β e kpei,t−k+ P k∈{1,4}β pattern k ψ k t + P k∈{1,2}β change k φ k t . . . 44
8.13 Individual positive feedback model estimatespe i,t= α + P k∈{1,2,3}β p kpt−k+ P k∈{1,2,3}β e kpei,t−k+ P k∈{1,4}β pattern k ψ k t . . . 45
8.14 Individual positive feedback model estimatespe i,t= α + P k∈{1,2,3}β p kpt−k+ P k∈{1,2,3}β e kp e i,t−k+ P k∈{1,4}β pattern k ψ k t . . . 46
8.15 Individual positive feedback model estimatespe i,t= α + P k∈{1,2,3}β p kpt−k+ P k∈{1,2,3}β e kp e i,t−k+ β pattern k ψ k t . . . 47
8.16 Individual positive feedback model estimatespei,t= α + P k∈{1,2,3}β p kpt−k+ P k∈{1,2,3}β e kp e i,t−k+ β pattern k ψ k t . . . 48
8.17 Individual positive feedback model estimatespei,t= α +Pk∈{1,2,3}β p kpt−k+ βe 1pei,t−1 . . . 49
8.18 Individual positive feedback model estimatespe i,t= α + P k∈{1,2,3}β p kpt−k+ βe1pei,t−1 . . . 50
8.19 Individual positive feedback model estimatespei,t= α +
P k∈{1,2,3}β p kpt−k+ βe 1pei,t−1 . . . 51
8.20 Individual positive feedback model estimatespei,t= α +
P k∈{1,2,3}β p kpt−k+ βe 1pei,t−1 . . . 52
8.21 Individual negative feedback model estimates pei = α + β1pt−1+ β2pt−2+
β3ψ1t + . . . 53
8.22 Individual positive feedback model estimates pei = α+β1pt−1+β2pt−2+β3ψt1+ 54
8.23 One period prediction error of the first model negative and positive feedback groupP t∈(0,T )e 1 t = PT t=4[ ep 1 t − pt]2 . . . 55
8.24 One period prediction error of the second model negative and positive feed-back groupP t∈(0,T )e 1 t = PT t=4[ ep 1 t− pt]2 . . . 55
8.25 One period prediction error of the third model negative and positive feedback groupP t∈(0,T )e 1 t = PT t=4[ ep 1 t − pt]2 . . . 55
8.26 One period prediction error of the fourth model negative and positive feedback group . . . 55 8.27 One period prediction error of the fifth model negative and positive feedback
groupP t∈(0,T )e 1 t = PT t=4[ ep1t − pt]2 . . . 56
8.28 One period prediction error of the sixth model negative and positive feedback groupP t∈(0,T )e 1 t =PTt=4[ ep 1 t − pt]2 . . . 56
8.29 Three period prediction error of the first model negative and positive feedback groupP t∈(0,T )e 3 t =PTt=4[ ep 3 t − pt]2 . . . 56
8.30 Three period prediction error of the second model negative and positive feed-back groupP t∈(0,T )e 3 t = PT t=4[ ep 3 t− pt]2 . . . 57
8.31 Three period prediction error of the third model negative and positive feed-back groupP t∈(0,T )e 3 t = PT t=4[ ep 3 t− pt]2 . . . 57
8.32 Three period prediction error of the fourth model negative and positive feed-back groupP t∈(0,T )e 3 t = PT t=4[ ep 3 t− pt]2 . . . 57
8.33 Three period prediction error of the fifth model negative and positive feedback groupP t∈(0,T )e 3 t = PT t=4[ ep 3 t − pt]2 . . . 57
8.34 Three period prediction error of the sixth model negative and positive feed-back groupP t∈(0,T )e 3 t = PT t=4[ ep3t− pt]2 . . . 58
8.35 Price processes belonging to each of the seven sessions . . . 58 8.36 FIGURE: Estimated coefficients for the large and individual models pe
I,t =
1
INTRODUCTION
(Micro)-economic science, is interested in the interaction between individuals that are active on certain markets. It aims to explain and predict the behavior of buyers and sellers and as a result the market price formation process. The clearing price on markets is linked to the expectations of the individuals active on those markets. In order to analyze the process of price formation, it is necessary to take certain assumptions about the nature of the market and it’s agents. Let us first briefly discuss the economic agent active on the market. A broadly used assumption in economics is that agents act rational. While there are many different definitions of ”rationality”, this research uses the following: An economic agent is said to be rational, if he or she is unbiased with respect to his or her expectations about future realized prices, i.e. |T |1 P
t∈(0,T )[p e
i,t− pt] → 0 as | T |→ ∞ for i ∈ I. With i, the
i’th agent in the market I is denoted. The set I is an index set of a market. The definition of rationality is chosen since it is a rather easily testable statement. Since this research focuses on experimental data generated by the research of Heemeijer et al. Price Stability and Volatility in Markets with Positive and Negative Expectations Feedback: Experimental Investigation 2005 ([Heemeijer et al 2005])], this research also needs to take into account further assumptions and clarifications that were used in this research. All agents in this experiment performed by [Heemeijer et al 2005], were informed in a similar way. In other words, they all used the same information set, hence the above-mentioned definition of ”rationality” needs to be made more restrictive, as |T |1 P
t∈(0,T )(p e
i,t− pt | Ft−1) → 0 as
| T |→ ∞. Which in words can be stated as that given a certain filtration or information set an economic agent on average is unbiased in predicting the future realized prices.
This understanding or definition of ”rationality” is useful in modeling economic phe-nomena. It allows certain dynamic models to converge into stable fixed points, which makes performing analyses easier (and sometimes just possible).
In essence the rationality hypothesis states that rationality applies for all economic agents. So in the context of this research this means that |T |1 P
t∈(0,T )[p e
i,t− pt] → 0 : ∀i ∈ I. While
economic literature widely accepts the rationality hypothesis, the hypothesis is stated in different forms and in some cases is difficult to confirm or reject. Below a few definitions of ”rationality” are discussed.
The rational expectations hypothesis is one of the cornerstones of current economic theory. In the broadest scope, it defines rationality to be a scenario in which an agent, active in a dynamic system (market), is aware of the internal mechanics of the dynamic system. In context of a market, the trader, active on a market is aware of the markets clearing price process of the market and of all the active forces working in the market.
One alternative rationality definition was defined by Muth (Muth, 1961, p. 321). In summary this definition can be interpreted as follows: market participants do not have to be aware of the dynamics underlying the market, nor do they need to have similar beliefs, in the sense that they do not need to have to assign similar probabilities to future price outcomes. However, combining all the probability distribution as a whole of all the participants, will result in rational expectations about future prices.
A perhaps more intuitive definition was formulated by R. Friedman as follows: ”One might reply that the rationality of a person suggests his choice does not depend upon how much he knows, but only upon how well he reasons based on the information he has, how-ever this is incomplete. Our decision is perfectly rational provided that we face up to our circumstances and does the best we can”. [R. Friedman, 1979, p.23].
While several different forms of rationality are defined and widely discussed in economic literature, there is no structural, statistical evidence that confirms which ”rationality” as-sumption is the best one to use. The strong rationality asas-sumption, claims that all agents on a market, have an identical density function for the expected future prices and further more, that they are unbiased in the sense that they are in their expectations equal to the realized prices. This property on its own is difficult to test, moreover is it more difficult
since the question has been raised whether the nature of economic agents is rational, or the structure of the market makes agents rational. The unbiased rationality assumption, states that
i,t= pei,t− pt∼ F () : E() = 0, V AR() = σ2,t
This assumption can be relatively easily tested. However, this assumption is relatively weak and does not answer the question whether the rationality of the agents is caused by their nature or by the structure of the market they are in. The strong rationality assumption claims that
F (X1,t) = F (X2,t) = F (X3,t) · · · = F (Xi,t)∀i ∈ I, ∀t ∈ T
Where i ∈ I represents the fact that it must hold for every agent active in the market, which has index set I. Finally the rationality hypothesis states that the aggregate of expected prices is equal to the revealing price at every point in time.
X
i∈I
pei,t
| I | = pt, ∀t ∈ T
All these statements, whether they can be tested statistically in an experiment or not, do not answer the question which factors explain the rationality of the agents. Economic literature describes that a market has to have several types of structures. This research focuses on a structure of either positive feedback or negative feedback markets. This can be intuitively viewed as a situation in which an agent performed some form of action, for instance formulating an expectation of future prices, which can have either a positive or a negative reaction, depending on whether the market has a positive- or negative feedback structure.
The research aims to formulate and estimate a prediction rule that fits the behavior of agents in both positive- and negative feedback markets. In addition will the prediction rule be tested on its rationality in the sense that
ei,t= ˆpei,t− pt
will be analyzed on its density function and what the factors are that influence the den-sity function of γ. The question of rationality is often complemented by the question of homogeneity. The answer to whether economic agents are homogeneous in their expecta-tions formation, has a more complicated nature. The main reason for this complication, is the existence of information asymmetry. Factors such as skill and knowledge can influ-ence the preciseness with which an agent is able to estimate future prices and hinflu-ence form expectations. Asymmetry in information can easily occur in many different forms. A prin-cipal one is the knowledge about the nature of the market. Negative- and positive feedback markets are a basic example of a possible difference in information. The internal nature of information asymmetry is difficult to overcome. Cultural, educational and socio-economical characteristics of agents differ greatly and are difficult to measure in a mathematical suit-able way. Controlled experiments are assumed to be the best alternative to cover for the asymmetry problem.
The hypothesis of rationality and homogeneity are helpful tools in designing a prediction rule. In context of explaining behavior of economic agents, designing a prediction rule is perhaps the final process in the discussion on expectations formation. A prediction rule, aims to explain behavior of economic agents as precise as possible, in all possible market surroundings.
1.1
Why is it important?
Economic theory is characterized by modeling economic activities in a stylized manner. This simplification of the actual economic world is sometimes an oversimplification, how-ever useful in understanding the behavior driving discussion-making processes of economic agents. Rationality and homogeneity are two, such stylizations in the sense that they are assumption that make analyzing economic behavior possible. In financial economics for ex-ample, is risk neutral pricing a cornerstone of the field and is summarized by the statement s(t) = e−r(T −t)EQ[s(T )] [R.U. Sidle, Tools for computational finance 2009], which in words
means that the risk free rate ”contains all information about the density function of the future value of the asset” (rationality). From this starting point, a major part of the field of finance takes off into further theoretical conclusions. If the basis of these conclusions are wrong, than so is the conclusion itself. In economic theory, many of such assumptions are made: ”People have identical information”; ”People pay the same risk free rate”; ”People are confronted with identical tax rates” and so forth.
The following research is three-fold in the sense that it aims to tell something about the moments and density of the stochasts ei,t∀i ∈ I and cross compare all the errors ei,t, ej,t
i, j ∈ I ×I. To answer the question whether agents act homogeneously in the sense that they operate identical to each other if they have identical information sets. Finally the research will estimate a prediction rule and evaluate the proposed models.
2
LITERATURE
This section briefly discusses some earlier research. The discussion is kept brief since the current research differs in major parts of the earlier researches involving the process of expectation formation. We are particularly interested in the research by [Heemeijer et al, 2005] and the one performed by [Hommes et al, 2003]. Since this research is based on the data of [Heemeijer et al, 2005], during the review of literature the main attention is focused on the research of [Hommes et al, 2003].
2.1
An earlier research in review
The research of [Hommes et al, 2003], is rather similar to the one performed by [Heemeijer et al, 2005]. It also consisted of an experiment, which will be briefly discussed below.
The experiment consists of six subjects that are active in an experimental asset market. The subjects are only given the passed realization of the asset prices, and their own beliefs and expectations of the future prices. Unknowingly of how the equilibrium prices are formed, the subjects are asked to predict the asset price one time period ahead. The experiment had 51 time periods and was performed on 10 groups. Given the predicted prices, a computer calculates the actual realization of the next period asset price. Prior to the experiment, the only aspect unknown to the experiment was the formation process of the expectations of the subjects. The subjects were able to earn money according to the following formula.
eh,t= max 1300 −1300 49 (pt− p e h.t) 2 , 0
This reward structure is identical to the one used by [Heemeijer et al, 2005] as will be discussed in the chapter 3.
eh,trepresents the amount of points earned by participant h at time t. Dividend payments
are also included into the experiment, which are random variables whom are assumed to be i.i.d and to have mean ¯y and variance σ2. With this knowledge the fundamental price pf
can be calculated as follows.l
pf =y¯ r
Where r is the risk free rat. In the experiment, a fraction of the traders are fundamental traders, these traders force the market towards the fundamental price, in order to prevent the occurrence of bubbles in the market. The fraction of these fundamental traders at time t is denoted by η.
The computer calculates the realized price at time accordingly based on the information above. pt= 1 1 + r " (1 − η) 6 X h=1 peh,t+ ηtpf+ t #
Where t ∼ N (0, 1). Finally the following dynamics for the fraction of fundamental
traders is defined to be as follows. ηt=
1 − e2001 |pt−pf|
The above is the basic framework in which the experiments were performed.
The main findings of the paper are that the realized prices differ significantly from the constant fundamental price. In some groups the realized asset price slowly converges to the fundamental price, while in others, the oscillation remains at a relatively constant amplitude or even increases over time. Analysis of the formation of expectation resulted in the finding that subjects in a group, use a similar rule to form their expectations. While
the subjects make forecast errors, they are similar in the way that they make these errors. Evidence is found that suggests that the subjects use autoregressive prediction methods, momentum trading and overreaction trading strategies. Eighty percent of the subjects are bounded rational in their predictions of the asset prices. Without fundamental traders, the markets become more volatile, where furthermore price bubbles of 1000 occurred, which is sixty times the fundamental price. In most groups the asset was undervalued and it’s price process excessive volatile. Participants tend to use simple linear forecasting strategies (autoregressive, of order 1). For a large group of the subjects, their forecasts are unbiased and their prediction errors are uncorrelated, which is evidence for rational behavior. The authors conclude that their research is evidence for rule of thumb expectation formation, specifically correlated imperfect trading momentum and overreaction trading.
3
DATA
The dataset is collected from a research performed by P. Heemeijer, C. Hommes, J. Son-nemans, J. Tuinstra, Price stability and volatility in markets with positive and negative expectations feedback: An experimental investigation, University of Amsterdam 2006. The research consisted of a number of 78 experiments in which a group of six individuals acted as economic agents. They all were equipped with an identical information set, meaning that they all were aware of each other and the fact that they all had identical information. None of the agents were informed about the working of the market in which they were trading. The experiments were done in two ways, one with a positive feedback dynamic and one with a negative feedback dynamic. The price formation associated with each type of experiment can be characterized by the following equation.
pt= 20 21 123 − 1 6 6 X i=1 pei,t ! + t
For the negative feedback dynamic market, and
pt= 20 21 1 6 6 X i=1 pei,t+ 3 ! + t
for the positive feedback dynamic market. The experiments had duration of 50 time periods and the agents were able to gain utility via the following equation.
Ui,t= max 1300 − 1300 49 (pt− p e i.t) 2 , 0
Hence, agents are assumed to maximizeP50
i=1Ui,ti ∈ I
3.1
Overview of the data
Prior to the actual analysis, we are briefly introduced to the data at hand. This is mainly for housekeeping purposes in the sense that a proper insight into the data helps better understand the results obtained during the research.
The table of figures below depicts the expectation formation process with the price pro-cess for each experiment group in the negative feedback treatment. The price propro-cess is
represented by a relatively thick line.
FIGURE 1: The price and expectation processes for the negative feedback markets. Comparing the table of figures above with the table of figures below, the difference between the natures of the markets becomes apparent. As was already briefly discussed during the introduction, a negative feedback market is characterized by an opposite driving force (i.e. if the agents expect prices to go up, which will result in supply going up which will finally lead to a drop in the clearing price due to an overproduction). In contrast a positive feedback market operates like a stock market (i.e. if agents expect the price of an asset to rise, the asset will become increasingly demanded and hence scarce which results in higher actual prices).
The hypothesis of rationality is one that this research aims to answer in the following chapter. However, for better understanding the data purposes, the figure below depict the average price process, that isphi(t) =¯ 1
t
Pt
k=1pi,k for i ∈ {1, 2 · · · 13}
FIGURE 3: Panel A: The average price processes belonging to the negative feedback experiments. Panel B: The average price processes belonging to the positive feedback
experiments.
The figure above confirms that for both the negative and the positive feedback groups, the average prices tend to the fundamental price of sixty. This suggests that the market as a whole acts rational in the long run. Typical however is the fact that the negative feedback groups tend to start at a price level above the fundamental price, whereas the positive feedback group tends to start from below the fundamental price.
FIGURE 4: Panel A: The S2, ¯µ) relationship for the negative feedback treatment agents.
Panel B: The S2, ¯µ) relationship for the positive feedback treatment agents. Looking at panel A of the figure above, it directly springs to our attention that the sample variances of the agents in the negative feedback group were in a small vicinity of each other -except for some outliers-. On the other hand, form panel B we obtain the positive feedback treatment agents, which almost seems like a reflection about the diagonal axis (¯µ = S2). A reflection of the pattern might be an overstatement however, it nearly aims to point out the switch from variation within sample mean (negative feedback group) to the variation within sample variance (positive feedback group). Further discussion on the data will take place in the chapter of the actual research.
4
RESEARCH
The research has been set up as broad as possible. The aim of this research is to asses the data as completely as possible in the sense that it aims to answer the question whether the agents in the experiment are rational, whether they act homogeneously and finally to predict a suitable prediction rule for the expectations process and hence for the price realizations.
4.1
Research Methods
The research is based on a variety of statistical and econometrical methods. Since the data was made available by an earlier research, this research only focuses on numerically analyzing the data and determining a model that precisely predicts the behavior of economic agents within, both the positive and negative feedback market dynamics.
4.2
Building blocks of the research
The research consists of three main parts, which in turn are further decided into smaller parts. The three main parts are as follows.
• Rationality
• Homogeneous expectation formation • Prediction rules
The Rationality paragraph assesses whether or not the agents who participated in the research, act as a rational agent. Earlier research using the Heemeijer et al. (2005) dataset, mainly focuses on establishing prediction rules, without rigorously testing the rationality of the agents involved. This part of the research aims to answer this question.
The Homogeneous expectation formation paragraph discusses whether the agents in-volved in the experiments, act in a similar way when forming their expectations about the future prices. The methods used to answer this question are Ordinary Least Squares (OLS). The structure of the experiments, allows for relatively simple data manipulation which en-ables for testing whether the elements, assumed to influence the formation behavior, have equal magnitude for the agents in a similar class (negative- or positive feedback markets).
Finally, the Prediction rule paragraph, uses the gathered insight into the behavior of the agents, to establish a prediction rule, that hopefully accurately predicts the formation of expectation in both the negative- as the positive feedback markets.
4.3
Rationality
As was discussed during the introduction of this research, are there a wide variety of ratio-nality definitions in use. To test whether or not the agents participating in the Heemeijer et al. (2005) experiments behave rational, the following method is used.
• Testing each individual on his or her rationality using the mean prediction error (MPE) statistic. P edi = 1 T T X t=1 ei,t= 1 T T X t=1 pei,t− pt
The sample mean converges to a normally distributed variable M P Ei ∼ N (µ, σ2), a
simple t-test can be used to test whether or not µ = 0. Formally stated, we are testing H0: µi= 0 ∀i ∈ I against Ha: µi6= 0 for at least onei ∈ I
This hypothesis is tested using the t-statistic1, defined to be t(M P Ei) = 1 T PT t=1ei,t S2(M P E i)
where S2(M P Ei) =T −11 PTt=1[ei,t− ¯ei]2 and e¯i= T1 PTt=1ei,t
Since it is also possible that the market2as a whole is rational instead of each individual
agent in the market. The rationality-test is also performed in the following way. • Testing whether individuals behave rational is one aspect of rationality as such, however
it may well be that the market as a whole acts rational. To test whether this is the case, we assess the following statistic.
T M P EI= 1 6T T X t=1 6 X i=1 ei,t
Analyzing The Data A first step is to analyze whether or not a set of agents in a certain experiment is collectively rational 3. Figure 1 and 2, show a histogram and a scatterplot of the t-statistics observed during the experiments. The T M P EI= 6T1 PTt=1
P6
i=1ei,tvalues
all lie within a small range around zero. From the associated t-statistics we conclude that there is not sufficient evidence to reject H0: µi= 0 i ∈ I. This implies that the agents as a
group are from now on assumed to behave as collectively rational.
FIGURE 5: 1 Negative- and Positive feedback: panel A histogram of the mean grouped individual errors for negative feedback market. panel B A histogram of the mean grouped
individual errors for negative feedback market
1B.E. Hansen, Econometrics p.153
http://www.ssc.wisc.edu/ bhansen/econometrics/Econometrics.pdf
2The market in this context is the single experiment of six agents
3A set of agents is collectively rational, if they, as a group are rational in the sense that ¯e
t=|I|1 Pi∈Ipei,t− pt
T M P EI t-statistic P-value 1 -0.0052 0.9950 2 -0.0049 0.9758 3 -0.0054 0.7824 4 -0.0007 0.9031 5 -0.0034 0.9554 6 -0.0012 0.9554 T M P EI t-statistic P-value 7 -0.0070 0.9203 8 -0.0051 0.9859 9 -0.0025 0.8538 10 -8.8213e-05 0.9668 11 -8.8213e-05 0.9796 12 0.0013 0.8986 13 -0.0058 0.9744 TABLE 1: Negative- and Positive feedback: panel A Negative feedback t-statistics and P-values for grouped errors. Panel B: Positive feedback group t-statistics and P-values for
grouped errors.
The second step, is to investigate rationality for each individual agent. Since collective rationality has already been established, the next step is to test for rationality on an even smaller scale, that is, each individual on its own. In figure 3.A, a histogram of the mean prediction errors is plotted, for each individual, in the negative feedback treatment group. The mean prediction error (MPE) is defined as 1
T
PT t=1p
e
i,t− pt. The histogram shows a
slightly right-skewed pattern for the negative feedback group, while the positive feedback group is associated with a little left-skewed pattern. The overall conclusion is that there is not sufficient evidence to reject H0. For this reason, does the research embrace the rationality
hypothesis in the remainder of the text.
Figure 6 Negative feedback: panel A Histogram of the mean individual errors. panel B T-statistics for each individual
The Associated table with t-statistics and p-values, can be found in Appendix 1. As can be seen from this table, is there once more, insufficient evidence to reject the H0
-hypothesis. For this reason the remainder of the research will assume that both individuals behave rationally, collectively aswell as individually. In other words. on average both an individual as well as the market as a whole is unbiased in forming the price expectations in the sense that in the long run, the mean prediction error converges to zero.
4.4
Homogeneous Expectation Formation
The available information for the agents involved in the experiment is crucial in testing for homogeneous expectation formation. If asymmetry in information were to arise, it becomes hard to test for homogeneity, since the asymmetry needs to be neutralized, with some subjective measure. However, in the context of the experiments performed by Heemeijer et
al. (2005), this is only of interest in context of positive and negative feedback markets. The agents participating in the experiments, all have the same information set. All agents are aware about the presence of the other participants, the fact that they, also are not aware of the underlying price determination. The data is separately analyzed for the positive feedback group and the negative feedback group. Within these two groups, individual experiments are analyzed as well as the group as a whole. The method used to test for homogeneity is as follows. The analysis considers the two groups, denoted by Θpositive, and Θnegative.
Within each group, the experiment number is indicated by i, such that Θi
positive indicates
the i’th experiment in the positive feedback group. The procedure for testing homogeneity is as follows.
• Estimate the experiment equation. First the data is manipulated to obtain the following vectors and matrices.
Pe= pe1,3 pe 2,3 · · · pe 6,3 · · · pe 1,4 pe2,4 · · · pe i,t X = 1 p1 p0 1 p1 p0 · · · · 1 p1 p0 · · · · 1 p2 p1 1 p2 p1 · · · · 1 pt−1 pt−2
Secondly, the equation below is estimated P = XβI+
Where ∼ N (0, σ2Ω).
• The second step is to estimate the following equations pei = X
0
iβi ∀i ∈ {1, 2, · · · , 6}
Where ∼ N (0, σ2Ω), and X and pei are defined to be
Pie= pei,3 pe i,4 · · · pe i,t X = 1 p1 p0 1 p2 p1 · · · · 1 pt−1 pt−2
• Compare the estimated β0s. Within each experiment, seven β0s are estimated, each one of these has its own confidence interval, if all these β0s occur within a certain range, the homogeneity hypothesis can be accepted. The statement summarizes this test:
H0: βI= β1= β2β3= β4= β5= β6
against
Ha: βI6= βi for at least one i ∈ {1, 2, 3, 4, 5, 6}
The test is setup in such a way that it checks whether the group as a whole, given a certain information space acts similar to each individual in with identical information avail-able. In other words, it tests whether individuals, form their expectations, given a certain information set (X = (1, pt−1, pt−2)) in a similar fashion.
To test whether the estimated coefficients are significantly different from each other, so that H0 can be rejected, we define the following test statistic.
t(βI) = 1 6 X 0<i≤6 βI− βi q S2 1 n+ 1 n
As can be deduced from the test statistic, is equality of variances assumed. The variances are estimated using the following logic. Since ∼ N (0, σ2 the distribution of the coefficient vector is obtained to be β ∼ N (0, σ2(XTX)−1). As an estimate of σ2, the statistic S2is used.
i t(α) P − value(α) t(β1) P − value(β1) t(β2) P − value(β2)
1 83025.67 0.0 -1861.87 0.0 8.05 1.58e-10 2 83025.67 0.0 -1861.87 0.0 8.05 1.58e-10 3 74957.75 0.0 -1848.13 0.0 10.11 1.36e-13 4 30747.05 0.0 -1556.70 0.0 17.56 8.63e-23 5 44521.81 0.0 -1915.95 0.0 19.75 5.56e-25 6 28033.74 0.0 -1498.49 0.0 17.43 1.19e-22 TABLE 2: Coefficients t-statistics and P-values for the negative feedback experiments.
From the table above, it can be easily seen that H0 needs to be rejected. For α and β1
are the p-values not even specified other than zero. When looking for an explanation to this result, we take a look at the table depicted in appendix 4 and 5. From this table it can be seen that, for the first experiment (entry one in the table)- the α coefficients estimated -using the OLS-method-, vary from zero to approximately sixty-five. This results in a relatively large | αI− αi | for most of the individuals. The fact that the S2 in most cases is very
small results in very large t-statistics. The coefficients estimated for the large models in the negative feedback case are however, relatively constant. While constant are they in most of the cases above one hundred which is a lot higher than for the individual model estimates. If we now take a look at β1’s, it once again can be concluded that H0can be rejected in favor
of Ha. The large model estimates for β1 are all negative, while for the individual models
this is true for less than fifty percent of the agents. There also exists a large difference in the magnitude of the estimated coefficients. The large model estimates β1 to be a relatively
large, negative factor while for the individual agents this is closer to zero in almost all the cases. Finally β2 is considered. This coefficient estimate is estimated to be closest to each
other. Only one of the coefficients for the large negative feedback models is estimated to be negative, while for the individual model estimates, the individuals have seemed to explain this factor as an negative influence more often. The p-values that are associated with the β2’s are still significantly small and hence we can reject the H0.
Appendix 23 depicts scatterplots of the estimated coefficients for both the negative and positive feedback experiments. For the negative feedback experiments, it depicts a relatively clustered group of estimate triples, while it for the positive feedback group depicts a treading line of estimate triples.
In conclusion, for the negative feedback experiments can be concluded that the agents do not form their expectations homogeneously. This effects the next section Prediction Rules in the sense that the results above contradict the existence of a uniform prediction rule. Still will the section Prediction Rules aim to find a single prediction rule that bees explain the process of expectations formation.
i t(α) P − value(α) t(β1) P − value(β1) t(β2) P − value(β2)
1 -1648.310 0.0 -11010.442 0.0 185.928 1.785e-71 2 -598.950 0.0 -10982.467 0.0 185.363 2.071e-71 3 586.163 0.0 -10238.664 0.0 172.720 6.568e-70 4 6466.881 0.0 -9599.395 0.0 160.409 2.445e-68 5 4015.826 0.0 -9492.212 0.0 159.528 3.201e-68 6 4542.949 0.0 -8845.035 0.0 148.706 9.934e-67 7 8766.147 0.0 -7692.297 0.0 128.219 1.392e-63 TABLE 3: Coefficients t-statistics and P-values for the positive feedback experiments.
Focusing our attention to the positive feedback experiments, makes the table above, us draw similar conclusions about the homogeneity of agents in the positive feedback group. For
most of the experiments, the large model estimate of β2 is close to the ones for individual
agents. However for the last experiment, the large model estimates a β2 close to one,
significantly larger than for most of the agents from this experiment.
Once again, the conclusion is that agents do not act homogeneously for both the negative asset as f for the positive feedback group.
As can be seen from the price processes during the experiments in the section DATA , is the price processes merited by the negative feedback experiments more stable than the once in the positive feedback group. The price process during the positive feedback experiments mostly depicted an slow but stable increase in the prices, with some bubble behavior in the sense that the stable increase of the prices was sometimes interrupted by a sharp decrease in the price for a relatively small period of time.
4.5
Prediction Rule
Now that rationality and homogeneity have been discussed, a logical next step is to inves-tigate is the performances of prediction rules. This research aims to estimate an prediction rule that explains the behavior of economic agents as accurate as possible, however a rel-atively simple model is preferable. The approach to designing a suitable prediction rule is rather structural in the sense that the optimal solution to the following problem will be calculated. minimize β θ(β) = X t∈(0,T ) X i∈I [pei,t− gpei,t(β))] 2 + Γ(λ, β) where Γ(λ, β) = λ | β |2
This optimization procedure is performed after first estimating the ordinary least squares estimates of the regression models.
The penalty factor λ is added to prevent over fitting. Price processes can be easily be characterized by patterns occurring in the process. Dummy variables can than be fitted to these occurrences, however the research is aiming to find a simple and short prediction rule that explains the behavior of economic agents as simple as possible.
Over fitting will be tested by considering the first and third prediction errorsP
t∈(0,T )e 1 t = P t∈(0,T )[pt(pgt−1)−pt]2andPt∈(0,T )e 3 t =Pt∈(0,T )[pt(pt−1, pgt−2, pt−3)−pt]2, of the models
merited by the optimization routine.
Given the results form the sections prior to the current, the rationality hypothesis is accepted where in contrast the homogeneity hypothesis is rejected. This may suggest it being more appropriate to estimate prediction rules for both the negative as well as the positive feedback experiments separately. This research however aims to determine a single prediction rule that best explains behavior of economic agents in both a negative feedback as well as a positive feedback market.
The analysis is two-fold in the sense that it first considers OLS-estimates for a one-period, three-period and recursive prediction model and they are evaluated on their performance.
Variables First off, let us consider the regression variables that are assessed. A reason-able assumption is that the prices and expectations of the past, give a proper indication of the price-expectation of today (and hence of the today’s prices in terms of the experiments). However, earlier research has shown that a simple autoregressive of order -n (AR(n)) ap-proach to model the expectation formation process, is little satisfactory. A problem with an autoregressive approach may be that the influence of past prices is averaged and typical pat-terns in price processes are not recognized by AR-models. With this in mind the following set of variables are considered in maximizing the above objective function.
• Autoregressive expectation variable pe
t−i, i = {1, 2, 3}
• Price pattern one. A dummy variable that takes on the value 1 or 0, according to the following condition.
ψ1i,t=
1 if pi,t−4< pi,t−3< pi,t−2< pi,t−1
0 otherwise
• Price pattern two. A dummy variable that takes on the value 1 or 0, according to the following condition.
ψ2i,t=
1 if pi,t−4> pi,t−3> pi,t−2> pi,t−1
0 otherwise
• Price pattern three. A dummy variable that takes on the value 1 or 0, according to the following condition.
ψ3i,t=
1 if pi,t−4> pi,t−3> pi,t−2< pi,t−1
0 otherwise
• Price pattern four. A dummy variable that takes on the value 1 or 0, according to the following condition.
ψ4i,t=
1 if pi,t−4< pi,t−3< pi,t−2> pi,t−1
0 otherwise
• Price pattern five. A dummy variable that takes on the value 1 or 0, according to the following condition.
ψ5i,t=
1 if pi,t−4> pi,t−3< pi,t−2> pi,t−1
0 otherwise
• Price pattern six. A dummy variable that takes on the value 1 or 0, according to the following condition.
ψ6i,t=
1 if pi,t−4< pi,t−3> pi,t−2< pi,t−1
0 otherwise
• Percent change pattern one. A dummy variable that takes on the value 1 or 0, according to the following condition.
φ1i,t=
1 if pi,t−2−pi,t−1
pi,t−1 > 0.05
0 otherwise
• Percent change pattern two. A dummy variable that takes on the value 1 or 0, according to the following condition.
φ2i,t=
1 if pi,t−2−pi,t−1
pi,t−1 < −0.05
0 otherwise
The above regress or variables (including a constant α), sum to a total of fifteen possible variables that can be included into the designed model.
The price pattern variables only have conditional statements that are influenced by four entry points maximum. This choice is mainly determinate by computational complexity constraints. A more fundamental argument is the fact that the experiment consists of fifty time periods, and it seems reasonable that agents only take into account medium term history to determine their expectations about future prices4.
4An extension of this research may be hyper-optimizing the parameter space of the currently found price
The influence of λ The factor λ in the punishment function Γ(λ, β) = λ | β |2, can be
viewed as an factor of aversion to including more variables into a model. The figure below depicts to plots. Panel A depicts the objective optimum as a function of λ, whereas panel B depicts a scatter plot, that associates a model, optimum and a λ. Surprisingly, over a range of ten for λ, do there only qualify a few different models for solving the objective function optimally.
FIGURE 6: Panel A: A graph of the minimized objective, as a function of λ. Panel B: A scatterplot that associates a value of λ, a value of the MSE and the minimizing model.
The following models solve the objective function optimally, given certain λ?s.
Model λ -Range pe i,t= α + P k∈{1,2,3}β p kpt−k+Pk∈{1,2,3}βkepei,t−k +P k∈{1,4}β pattern k ψ k t + P k∈{1,2}β change k φ k t 0 ≤ λ ≤ 0.25 pei,t= α +Pk∈{1,2,3}β p kpt−k+Pk∈{1,2,3}β e kp e i,t−k +P k∈{1,4}β pattern k ψ k t 0.25 ≤ λ ≤ 0.75 pei,t= α + P k∈{1,2,3}β p kpt−k+ P k∈{1,2,3}β e kpei,t−k +βpattern1 ψ 1 t 0.75 ≤ λ ≤ 1 pei,t= α +Pk∈{1,2,3}β p kpt−k+ βe1pei,t−1 1 ≤ λ ≤ 1.75 pe i,t= α + P k∈{1,3}β p kpt−k+ βe2pei,t−2 1.75 ≤ λ ≤ 2.75 pei,t= α + β1pt−1 2.75 ≤ λ ≤ 10
TABLE 4: The models that maize the objective function θ(β) over a range of 0 ≤ λ ≤ 10 From both the table swell as from the plot in panel B, can be seen that relatively quickly, the objective function finds a stable optimum at the naive model pe
i,t= α + β p
1pt−1. However
this only occurs for λ-values greater than 1.75. This is a high punishment factor and the research for this reason, does not neglect the solutions found in the λ-range below 1.75. In the following paragraph, all the models in the table will be assessed on their performance in both the positive swell as the negative feedback markets.
Performance of the models The models are analyzed in the above paragraph, in the sense that an OLS-estimation of these models minimizes the predetermined objective function θ(β). In this section we will analyze the performance of the models in the following way.
• We will look at each price process, as it occurred during the experiments. The analysis only focuses on the last 25 periods. This in the sense that we will assume all price real-izations prior to that point as given and from that point on, we will use our estimated
models, to predict the price realization. The models performance will be assessed in terms of squared prediction errors.Formally stated as
Example analysis of the model pei,t= α + β1pt−1 in a negative feedback market
X t∈(0,T ) ert = X t∈(0,T ) [pet− pt] 2 = X t∈(0,T ) " 20 21 123 − 1 6 6 X i=1 α + βpt−1 ! − pt #2
and for the positive feedback market X t∈(0,T ) ert = X t∈(0,T ) [pet− pt]2= X t∈(0,T ) " 20 21 1 6 6 X i=1 [α + βpt−1] + 3 ! − pt #2
• Next to the recursive procedure discussed in the item above, will the models be assessed on their one- and three period a head predictions. This will be once in the following way. X t∈(0,T ) e1t = T X t=4 [ ep1 t− pt]2
Finally the three-period ahead prediction is assessed via X t∈(0,T ) e3t = T X t=4 [ ep3 t− pt]2
The analysis starts -in this case- at period 9 to come to pi,10= α + βp9. This outcome,
will be recursively used as input for the next time period. The nest price processpetwill be compared to the actual price process pt.
Since the major part of the partitioned range of λ , the model pe
i,t = α + βpt−1 is
outputted as an optimal model. The analysis starts off with this model.
Starting point of the analysis We use the mean squared prediction error as a measure of fit. The positive feedback experiments perform dramatically worse than the negative feedback experiments in the sense that the MRE is relatively high in each experiment, compared to the negative feedback experiments. Furthermore should it be stated that the OLS-estimated expectation processes in the negative feedback experiments, move relatively erratic, compared to the positive feedback experiments. An even greater problem with this naive prediction rule, is that all estimated price processes converge at greatly differing prices. As can be seen below, other prediction rules converge at prices very close within each other?s vicinity. This result may indicate irrational behavior, however, this model is based on only a constant and a one period lagged price variable and hence contains too little information to make a reliable conclusion on rationality. In an earlier section it was discussed and shown that there is insufficient evidence to reject the hypothesis of rationality -based on the data used in this research-.
Experiment 1 |T | P t∈(0,T )e 2 t 1 0.3601 2 0.3533 3 0.4767 4 9.8041 5 0.3073 6 0.5042 - -Experiment 1 |T | P t∈(0,T )e 2 t 1 18.0224 2 9.6486 3 5.3996 4 66.0256 5 66.0256 6 37.5920 7 37.2818
TABLE 5: Panel A: Negative feedback experiments Panel B: Positive feedback experiments From the figure below, it can be seen that, while the model converges rapidly in the negative feedback experiments, its points of convergence are widely spread and while the points of convergence are in a small vicinity of each other in the positive experiments case, they converge relatively slowly.
For reason discussed above, this prediction rule is not considered as satisfactory and hence the investigation into prediction rules is extended.
FIGURE 7: Panel A: negative feedback experiments, convergence of the estimated price processes. Panel B: positive feedback experiments, convergence of the estimated price pro-cesses.
As can be seen from the figure above, is especially the three period prediction not accu-rate. It converges rather quickly to a wide variety of price levels. From append 21 and 22 it can be seen that for the negative feedback experiments, the constant α is often estimated to be relatively high, with a relatively low value for β1, where in contrast, for the positive
The second model The prediction rule pe i,t= α + P k∈{1,3}β p kpt−k+ βe2pei,t−2 will be
considered in the following section. From the table below, a similar pattern can be observed as showed in the first model of this section. In all the positive feedback experiments, the prediction rule has significantly more difficulty estimating the future prices in the sense that all the mean prediction errors are relatively high compared to the negative feedback experiments. In contrast to the first prediction rule discussed in this section, the current prediction rule converges to a set of price levels that are relatively close to each other. A surprising result of this prediction rule is its behavior from time period forty up to time period sixty. Starting from period forty the rule predicts and oscillating price process, which reaches a maximum amplitude at time period fifty. From period fifty to sixty, it reduces the amplitude of its oscillation to remain a stable pattern from time period sixty onwards.
Experiment |T |1 P t∈(0,T )e 2 t 1 0.3575 2 0.3771 3 4.4601 4 9.8096 5 0.3212 6 0.4779 - -Experiment |T |1 P t∈(0,T )e 2 t 1 26.4654 2 22.0301 3 8.5559 4 347.7277 5 347.7277 6 38.6045 7 36.4200
TABLE 6: Panel A: Negative feedback experiments Panel B: Positive feedback experiments It should be stated however that from the end of the experiment-in the graph time period forty, since we started the prediction after time period ten-, the lagged prediction variable pet−1, was substituted by pgt−1. This simplification is one that can be further investigated in future research. The prediction rule reacts servilely to the transition from the actual expectations obtained from the agents to the expectations prediction obtained from the prediction rule.
FIGURE 8: Panel A: negative feedback experiments, convergence of the estimated price processes. Panel B: positive feedback experiments, convergence of the estimated price
processes.
To further investigate the oscillating pattern of the price predictions and to see whether there is a recurring pattern of this oscillating behavior, the price process has been further expanded for thousand time periods. Still this is no solid mathematical evidence, based on the graph below, we assume that the oscillating pattern only occurred due to the transition from pe
t−1 to pt−1.
From the figure above, which depicts both the negative and positive recursive prediction and the three period prediction, it immediately spring to our attention that the three period prediction rather quickly converges to a stable price level and hence do not explain the actual price realization very well. Looking at table 19 and 20 in the appendix. The following is observed: compared to the previous model discussed, the negative feedback experiments estimate α being smaller where as the influence of the β’s increase. The positive feedback models however estimate a similar α, but the influence of the β’s decreases.
The conclusion of the quality of the prediction rule is as follows. While there has occurred some strange behavior in the sense that unexpected oscillation has occurred, does the model minimize the mean squared error with relatively little variables -which is preferred over a large number of regressor variables- and furthermore converge to a price level close to the fundamental price relatively quickly. The prediction rule converges in both the positive and the negative feedback markets. However in the positive feedback markets, the price of convergence varies over the experiments. This contradicts the inherent nature of the experiment, which brings forth a fundamental price of sixty. Especially when considering the positive feedback markets is there some room for improvement left and hence we now focus on the next prediction rule.
The third model The model pe i,t= α +
P
k∈{1,2,3}β p
kpt−k+ βe1pei,t−1can be considered
as a big improvement when viewing its performance in the positive feedback experiments. While it, similar to the previous to prediction rules, performs worse in context of the MSE than the negative feedback experiments, does it show a more stable range of convergence. Several of the predicted price processes follow the actual price processes of the experiments, rather accurate. A further improvement of this prediction rule is that it is less influenced by the transition from the pet−1to pt−1variable in context of the negative feedback experiments.
Experiment |T |1 P t∈(0,T )e 2 t 1 0.3376 2 0.3437 3 2.8627 4 9.7936 5 0.3278 6 0.4752 - -Experiment |T |1 P t∈(0,T )e 2 t 1 7.1782 2 6.1783 3 8.2267 4 274.3018 5 274.3018 6 42.8348 7 35.7413
TABLE 7: Panel A: Negative feedback experiments Panel B: Positive feedback experiments Overall this prediction rule is considered to be rather good. The model is a close match to the actual price processes, both the ones that occurred in the negative feedback experi-ments a swell as for the ones that occurred in the positive feedback experiexperi-ments. The few variables included in the model are also considered to be a quality of the prediction rule. The research aims to estimate a prediction rule that is as accurate as possible with as little variables as possible. The model currently discussed is one that satisfies major aspects of the characteristics that we find important.
FIGURE 10: Panel A: negative feedback experiments, convergence of the estimated price processes. Panel B: positive feedback experiments, convergence of the estimated price
processes.
As can be seen from the figure above, does the model perform relatively well in all three assessments. For the positive feedback experiments, all three procedures result in a good prediction of the price realizations. In the negative feedback case only the three period prediction perforce relatively worse, however especially in comparison with the other models, this lack of perforce is minor compared to the good performance in all other areas of the research.
The fourth model The fourth model pei,t= α+
P k∈{1,2,3}β p kpt−k+Pk∈{1,2,3}β e kp e i,t−k+ β1patternψ 1
t is the first with a relatively high number of variables. As stated above, this
char-acteristic is considered to be a disadvantage. The research seeks an prediction rule that best predicts the price processes based on as little information as possible, as little as possible variables and predicts both negative as positive feedback markets.
While the prediction rule performs relatively well in the negative feedback experiments, it performs rather bad in the positive feedback experiments. The MPE is very large, especially for the later negative feedback experiments the prediction rule performs below a reasonable standard. For this reason we consider this prediction rule to be unfavorable.
Experiment 1 |T | P t∈(0,T )e 2 t 1 0.3904 2 0.3933 3 10.2822 4 9.8315 5 0.3308 6 0.5660 - -Experiment 1 |T | P t∈(0,T )e 2 t 1 6.9407 2 48.3602 3 6.3175 4 13.5548 5 13.5548 6 173.7826 7 37.6105
FIGURE 11: Panel A: negative feedback experiments, convergence of the estimated price processes. Panel B: positive feedback experiments, convergence of the estimated price
processes.
The figure above is typical in the sense that it summarizes the performance in the other two areas of the research. The over fitted model is an accurate OLS-estimate, however it dramatically fails to predict prices either recursively, one period a head or three periods a head. If we look at the table in appendix 15 and 16, it can be seen that the model has a relatively high estimate for α in both the negative swell as the positive feedback experiments, but all β’s are relatively small and alter from positive to negative.
We note that while this model minimizes the objective function θ(β), as was discussed in the beginning of this section, we nevertheless conclude that the prediction rule is unfavorable. This result is appropriate in the sense that the use of the objective function in combination with the mean prediction error - analysis, aims to find an accurate prediction rule, but prevent over fitting.
The fifth model The prior to last prediction rule is pe
i,t = α + P k∈{1,2,3}β p kpt−k+ P k∈{1,2,3}β e kp e i,t−k+ P k∈{1,4}β pattern k ψ k
t. Where the third prediction rule that we discussed
only has a relative small oscillating pattern at time period forty, is the oscillation in that occurs in the current prediction rule rather large. Moreover the prediction rule moves in an opposite direction for some of the positive feedback experiments. In summary, the model performs equally well for the negative feedback experiments as the third an fourth model. However it performs worse in predicting the price process for the positive feedback experiments compared to the third model especially.
Experiment |T |1 P t∈(0,T )e 2 t 1 0.3860 2 0.4645 3 10.0901 4 9.8209 5 0.3415 6 0.4896 - -Experiment |T |1 P t∈(0,T )e 2 t 1 14.3728 2 57.1878 3 12.8387 4 15.7492 5 15.7492 6 144.1385 7 34.3301
FIGURE 12: Panel A: negative feedback experiments, convergence of the estimated price processes. Panel B: positive feedback experiments, convergence of the estimated price
processes.
The sixth model The final model pei,t= α +
P k∈{1,2,3}β p kpt−k+ P k∈{1,2,3}β e kp e i,t−k+ P k∈{1,4}β pattern k ψ k t + P k∈{1,2}β change k φ k
t. From the figure below, it can be seen that the
prediction rule performs especially worse in the positive feedback experiments. While in the beginning it predicts the price rice?s suitable, it shortly afterwards starts to oscillate with extreme amplitude. This result alone makes the prediction rule incredible as an candidate for a accurate prediction rule. The rule performs relatively well for the negative feedback experiments, however the lack of accuracy for the positive feedback experiments overshadows its performance in the negative feedback experiments.
Experiment |T |1 P t∈(0,T )e 2 t 1 0.3385 2 0.3882 3 13.4667 4 12.1529 5 0.3122 6 0.5075 - -Experiment |T |1 P t∈(0,T )e 2 t 1 42.8918 2 29.6476 3 5.1832 4 33.4576 5 33.4576 6 68.5754 7 90.5765
TABLE 10: Panel A: Negative feedback experiments Panel B: Positive feedback experiments
FIGURE : Panel A: negative feedback experiments, convergence of the estimated price processes. Panel B: positive feedback experiments, convergence of the estimated price
processes.
Sub conclusion The six prediction rules suggested by the optimization of the objective function θ(β) all display different characteristics in the sense that each model has its own qualities. By this we mean that each model performs either very well in the negative feedback experiments or either in the positive feedback experiments. One model however performs relatively well in both experimental settings, namely the third prediction rule pe
i,t = α +
P
k∈{1,2,3}β p
kpt−k+β1epei,t−1. The prediction rule satisfies all of our requirements in the sense
that it has only a few regress or variables and perhaps more important it has relatively low mean prediction errors in both experimental settings. A final favorable characteristic is that in most of the cases the prediction rule converges to a price level, equal or close to the fundamental price of sixty.
5
CONCLUSION
Rationality is an characteristic that can be confirmed -for the dataset used in this research-, rather surely. Homogeneity in contrastresearch-, is rejected rather surely. The acceptance of the rationality hypothesis is useful in the sense that for designing a prediction rule, the prediction rule should converge to the fundamental price in the long run, this since in the long run, agents are assumed to converge their expectation formation to the fundamental price. The fact that homogeneity was rejected, suggest it being more appropriate to design separate prediction rules, however as was already stated, was the aim of this research to find, a single prediction rule, that best fits the expectations formation for both negative and positive feedback markets. Given these results obtained in chapter 4.3 and 4.4, a prediction rule was designed. The prediction ruled that was found to be optimal, is relatively simple : pe
i,t= α +
P
k∈{1,2,3}β p
kpt−k+ β1epei,t−1. Models with more regressor variables tend to over
fit the data. If the one - and three period prediction are considered, it can be seen that models with greater number of regressor variables than model three, behave rather chaotic in the sense that they well fit the data, however they perform dramatically in context of predicting the data. The third model discussed in chapter 4.5 perforce relatively well in all three prediction tests. Only in the negative, three period prediction evaluation is its accuracy relatively low, however it is still a great amount better in prediction the future prices than the other models discussed. In context of the rejection of homogeneity, the good performance of the third model can be explained by the fact that it only takes into account the first time lag of the individual?s expectations into account. In other words is there little direct influence of the individuals previous beliefs about future prices, instead it takes the collective beliefs of the agents in the experiment for three different lags into consideration. In summary are the main findings of this research that rationality were not to be rejected based on the data; homogeneity were to be rejected based on the data; and finally, that the relative simple third prediction rule, perforce rather well in both the negative and positive feedback experiments. This even given the fact that homogeneity was rejected and hence, the design of several prediction rules was more appropriate.
6
SUGGESTIONS & SHORTCOMINGS
The complete research took place in a relative short amount of time and hence some calcu-lations were simplified for computational purposes. This especially accounts for the test for homogeneity in chapter 4.4. Future research can improve the results by further investigat-ing the homogeneity test and calculatinvestigat-ing test outcomes fore precisely instead of aggregatinvestigat-ing over experimental groups. A final suggestion for future research is the possibility of applying deep machine learning [ T. Hastie et al. The Elements of Statistical Learning 2009]. The appliance of machine learning especially in finding optimal regression patterns (see chapter 4.5), can improve the prediction power of prediction rules.
7
REFERENCES
[1] R. Seydel, Tools for computational finance, Springer 2009
[2] Hastie. T, R Tibshirani, J. Friendman, The Elements of Statistical Learning, Springer 2009
[3] B.E. Hansen, Econometrics 2014
http://www.ssc.wisc.edu/~bhansen/econometrics/Econometrics.pdf
[4] Adam, K., 2007. Experimental evidence on the persistence of output and in?ation. Economic Journal 117, 603?636
[5] Arifovic, J., 1994. Genetic algorithm learning and the cobweb model. Journal of Economic Dynamics and Control 18, 3?28.
[6] Heemeijer. P, C Hommes, J. Sonnemans, J. Tuinstra Price stability and volatility in markets with positive and negative expectations feedback:
8
APPENDIX
8.1
Table 1: Basic statistics for the negative feedback group.
Agent # µ¯ S2 γˆ ˆk t-statistic P-value
1 0.051 66.594 -2.421 26.271 0.006 0.995 2 -0.157 26.664 0.669 23.020 -0.030 0.975 3 -1.541 30.822 -3.365 14.042 -0.277 0.782 4 -0.717 34.387 -0.351 16.476 -0.122 0.903 5 -0.548 95.306 -5.254 36.369 -0.056 0.955 6 -0.548 95.306 -5.357 37.053 -0.056 0.955 7 -1.369 185.823 -1.960 20.651 -0.100 0.920 8 -0.214 147.405 -1.674 21.957 -0.017 0.985 9 -2.151 134.926 -2.421 20.697 -0.185 0.853 10 -0.401 92.029 -0.199 19.203 -0.041 0.966 11 -0.247 93.706 2.694 22.177 -0.025 0.979 12 -2.119 274.293 -2.251 16.482 -0.127 0.898 13 -0.285 79.072 0.158 18.378 -0.032 0.974 14 -1.080 166.107 -2.098 18.135 -0.083 0.933 15 -2.154 144.648 -1.276 10.944 -0.179 0.858 16 -1.113 77.317 -3.614 21.662 -0.126 0.899 17 0.582 588.155 1.893 15.906 0.024 0.980 18 -0.763 98.972 -2.462 35.047 -0.076 0.939 19 -0.632 29.836 -5.993 45.781 -0.115 0.908 20 -0.289 12.057 -1.153 22.067 -0.083 0.934 21 0.105 22.376 -0.123 16.053 0.022 0.982 22 -0.194 18.971 -0.522 20.306 -0.044 0.964 23 0.737 25.991 0.403 15.189 0.144 0.885 24 0.737 25.991 0.417 15.277 0.144 0.885 25 -0.368 18.030 -3.827 24.838 -0.086 0.931 26 0.198 9.167 -1.493 14.296 0.065 0.947 27 0.547 33.471 4.040 26.396 0.094 0.924 28 -1.193 73.808 -5.613 37.213 -0.138 0.890 29 -0.336 6.984 -6.749 98.175 -0.127 0.899 30 -0.336 6.984 -3.027 17.015 -0.127 0.899 31 -0.645 37.097 -1.399 16.231 -0.105 0.916 32 -0.684 47.051 0.144 17.068 -0.099 0.920 33 0.416 102.355 -1.380 17.998 0.041 0.967 34 0.066 56.567 0.016 12.644 0.008 0.993 35 0.105 44.785 -0.485 13.699 0.015 0.987 36 0.105 44.785 -0.495 13.569 0.015 0.987
8.2
Table 1: Basic statistics for the positive feedback group.
Agent # µ¯ S2 γˆ ˆk t-statistic P-value
1 0.051 66.594 -2.421 26.271 0.006 0.995 2 -0.157 26.664 0.669 23.020 -0.030 0.975 3 -1.541 30.822 -3.365 14.042 -0.277 0.782 4 -0.717 34.387 -0.351 16.476 -0.122 0.903 5 -0.548 95.306 -5.254 36.369 -0.056 0.955 6 -0.548 95.306 -5.357 37.053 -0.056 0.955 7 -1.369 185.823 -1.960 20.651 -0.100 0.920 8 -0.214 147.405 -1.674 21.957 -0.017 0.985 9 -2.151 134.926 -2.421 20.697 -0.185 0.853 10 -0.401 92.029 -0.199 19.203 -0.041 0.966 11 -0.247 93.706 2.694 22.177 -0.025 0.979 12 -2.119 274.293 -2.251 16.482 -0.127 0.898 13 -0.285 79.072 0.158 18.378 -0.032 0.974 14 -1.080 166.107 -2.098 18.135 -0.083 0.933 15 -2.154 144.648 -1.276 10.944 -0.179 0.858 16 -1.113 77.317 -3.614 21.662 -0.126 0.899 17 0.582 588.155 1.893 15.906 0.024 0.980 18 -0.763 98.972 -2.462 35.047 -0.076 0.939 19 -0.632 29.836 -5.993 45.781 -0.115 0.908 20 -0.289 12.057 -1.153 22.067 -0.083 0.934 21 0.105 22.376 -0.123 16.053 0.022 0.982 22 -0.194 18.971 -0.522 20.306 -0.044 0.964 23 0.737 25.991 0.403 15.189 0.144 0.885 24 0.737 25.991 0.417 15.277 0.144 0.885 25 -0.368 18.030 -3.827 24.838 -0.086 0.931 26 0.198 9.167 -1.493 14.296 0.065 0.947 27 0.547 33.471 4.040 26.396 0.094 0.924 28 -1.193 73.808 -5.613 37.213 -0.138 0.890 29 -0.336 6.984 -6.749 98.175 -0.127 0.899 30 -0.336 6.984 -3.027 17.015 -0.127 0.899 31 -0.645 37.097 -1.399 16.231 -0.105 0.916 32 -0.684 47.051 0.144 17.068 -0.099 0.920 33 0.416 102.355 -1.380 17.998 0.041 0.967 34 0.066 56.567 0.016 12.644 0.008 0.993 35 0.105 44.785 -0.485 13.699 0.015 0.987 36 0.105 44.785 -0.495 13.569 0.015 0.987 37 -0.503 3.158 -5.465 86.154 -0.283 0.778 38 -0.061 0.856 -0.473 26.755 -0.066 0.947 39 -0.437 1.853 -2.898 16.407 -0.321 0.749 40 0.250 9.386 6.112 41.456 0.081 0.935 41 -0.560 13.210 -6.106 42.394 -0.154 0.878 42 -0.560 13.210 -6.302 42.823 -0.154 0.878
8.3
Table 1:
Basic statistics for the negative and positive
feedback group.for the total mean prediction error T M P E
I=
1 6T
P
T t=1P
6 i=1e
i,t Experiment i µ¯ S2 t-statistic 1 -0.581 12312.286 -0.005 2 -0.874 31782.367 -0.004 3 -0.925 29003.156 -0.005 4 -0.053 5281.687 -0.000 5 -0.227 4314.904 -0.003 6 -0.146 13742.773 -0.001 7 -0.260 1368.533 -0.007 8 -0.127 605.598 -0.005 9 -0.141 2989.024 -0.002 10 -0.004 2317.815 -8.821e-05 11 -0.004 2317.815 -8.821e-05 12 0.081 3742.213 0.001 13 -0.135 530.184 -0.0058.4
Homogenious model estimates P = Xβ
I+
PI β1 β2 α 1 -0.246 0.259 59.229 2 -1.008 0.022 119.191 3 -0.987 0.023 117.841 4 -0.631 0.013 97.135 5 -1.015 -0.032 122.934 6 -0.997 0.057 116.506 7 0.763 0.267 -1.801 8 0.394 0.609 -0.081 9 0.157 0.811 1.961 10 0.981 0.055 -2.229 11 0.981 0.055 -2.229 12 1.040 -0.002 -2.379 13 0.085 0.953 -2.381
