Price forecasting- and trading-rules in the `Mixed Treatment' experiment

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Price forecasting- and trading-rules in the

‘Mixed Treatment’ experiment

by

David Wortman

A thesis submitted in partial fulfilment for the Bachelor’s degree Econometrics

Supervisor T. Makarewicz Examiners T. Makarewicz R. van Hemert in the

Faculty of Economics and Business Department of Economics and Econometrics Section of Mathematical Economics and Mathematics

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Abstract

Faculty of Economics and Business Department of Economics and Econometrics Section of Mathematical Economics and Mathematics

Bachelor’s of Econometrics by David Wortman

From the ‘Mixed-Treatment’ experiment in Bao et al. (2014) different price predictions and trading decisions were estimated between various subjects. This thesis identifies different rules to gather these estimations and examine the consequences of the price development in a positive feedback market (on a combination) of the different price estimation and quantity heuristics. One result found is that the price forecasting and trading rules used in an experiment have a strong influence on the market behaviour. Another results found is that the addition or subtraction of one subject using a trend reversing rule, has a large impact on the price development and the final price of the experiment.

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Although not new, the criticism on the neoclassical paradigm has grown since the burst of the ’dot-com bubble’ of the late 1990’s and the American housing price bubble that initiated the most recent world wide economic crisis. In retrospect, these bubbles seem to be created by an overenthusiastic expectation of the future returns on investments. One of the difficulties academics and observers have with this paradigm is the assumption of rational behaviour. In other words, this school of economics leans heavily on the assumption of rational behaviour and reasoning. The literature regarding this subject attributes (Muth, 1961) and (Lucas, 1972) as one of the firsts to lay down this paradigm. A part of its framework is the rational expectations theory, which in general states that expectations

of future economic indicators (i.e. stock price, income) are formed model consistent given the

information available to economic agents. Many academics were not convinced by the explanatory capabilities of this framework, in which all agents behave rationally and homogeneously, to explain the large over- and under-pricing observed in many economic markets over the last few decades. Shiller (1981) was one of the first to convince that the rationality assumption had to be modified. The author claimed that economic agents try to behave rationally, but they are limited by time, knowledge and capacity. This is referred to as Bounded Rationality, which will be discussed briefly in the next section.

Large over- and under-pricing of economic expectations and the possible result of bubbles and burst have an important impact on consumers, households, manufactures and businesses within an economy, as observed in the last decades. A better understanding of the forecasting and trading behaviours of agents should lead to economic policy that can address possible market inefficiencies and stabilise financial markets in the future.

1.2 Subject and Questions

The bubbles that have been observed in recent decades have been a much written about subject in scientific papers. One recent addition to this debate has been the experimental work by Bao, Hommes, and Makarewicz (2014). One of the findings of this paper is that when subjects are asked to predict a price in an asset market and are asked to trade the same asset on the same market afterwards, many subjects do not trade consistent with the rational expectations theory. In other words, test subjects not necessarily trade optimally given their price prediction in the experiments from Bao et al. (2014). Another result, which is consistent through out the literature on the same matter, is that different subjects use different rules to predict the price in the market for the next period in a similar pattern over the duration of the experiment. This thesis will use the results of the experiments from Bao et al. (2014) to discuss these heterogeneous predictions heuristics and their effects on the experimental asset market. Also, it is found that the quantity decisions subjects make in experiments can be described by simple heuristics (Anufriev and Hommes, 2009) and their ramification on price dynamics is of similar interest.

For those reasons, this thesis will ask the following questions:

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Introduction 2

Main question

How do economic agents make trade decisions conditional on a price forecast of the asset and what are the effects of these decisions on the market and the potential formation of bubbles? Auxiliary questions

• Are there distinct forecasting and trading strategies among agents?

• How do trading and forecasting heuristics effect the price development of a experimental market?

• Given the price forecasting heuristics, what would have been the optimal trade and its effect on the experimental outcome?

• How do heterogeneous forecasting and trading rules within a market effect the price dynamics?

1.3 Thesis Outline

This thesis is made up in the following way; after this section, it will lay out the theoretical framework and explain the experiment from Bao et al. (2014) that it will build upon. In third section the forecasting and trading heuristics from the experiment are identified. In section 4 the effects of the heuristics on price dynamics are modelled with subjects using homogeneous and heterogeneous forecasting and trading rules. This thesis will also look at different weighted rules and optimal trading and finally draw some conclusions from the research and give some recommendations for future experimental work.

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2. Theoretical Background

To gain more understanding on agents expectations many experimental research has been performed in the past. In this section this thesis will discuss these papers and together with other (non-experimental) work it will build a framework where this thesis will construct its research upon. Besides the discussion of the literature, this section will also consist of a summary of the experiment preformed by Bao et al. (2014) and an outline of the research performed in this thesis.

As mentioned before the expectations of agents have an important impact on economic indicators. Traditional work emphasises that in the equilibrium the predictions of agents have to be consistent with the model (Muth, 1961). The problem economist face when looking for an alternative for the Rational Expectations hypothesis is the so called ‘wilderness of bounded rationality’: there is an infinite amount of possible expectation mechanisms with varied limitations on human memory and computational capabilities (Anufriev et al., 2013). From this wide variety of possible forecasting heuristics Bao et al. (2014) build on prior experimental work (e.g. Heemeijer et al., 2009) for the estimation of individual behavioural rules, which will be discussed further on in this thesis.

An advantage of an experimental approach, is that researchers have full control over the fundamental price of an experimental asset and the market dynamics. This allows scientist to focus their study on the forecasting heuristics of the subjects and gain understanding on how agents form their ex-pectations in the economy. One of the first to replicate an asset market successfully for the purpose of studying bubbles was Smith et al. (1988), after whom many other authors have recreated and replicated his method of research. This thesis refers to Sunder (1995) and Noussair and Tucker (2013) for an extensive survey.

The market dynamics used in Bao et al. (2014) is a so called positive feedback market. This means that high (low) price expectations lead to high (low) asset pricing, comparable with real world stock and housing markets. The opposite happens in a negative expectation feedback market; subjects expected the price to go up (down) leads to over- (under-)supply and gives a lower (higher) market price (e.g. commodities market). Bao et al. (2014), Hommes (2011) and Heemeijer et al. (2009) discuss these and other market structures in depth and give its micro fundamentals.

The experimental economy used in Bao et al. (2014) is based on the same asset market as in Heemeijer et al. (2009) with a fundamental price of 66 and a price adjustment mechanism in the positive feedback market

pt= pt−1+ 20 21 6 X i=1 qi,t+ εi, (2.1)

which is based on an economy with an interest rate of 5%, an asset dividend of 3.3 (paid per period)

and εi ∼ N ID(0, 1) being a small idiosyncratic shock simulation.

The maximisation problem over investments has a straightforward solution. The first order derivative conditional on the price expectation is given by

q_i,t∗ = p

e

i,t+1+ 3.3 − 1.05pt

6 . (2.2)

This is an important intuition, because when subjects give their price expectation first, the optimal quantity decision is easily computed from Equation (2.2).

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Theoretical Background 4

In the experiments conducted by Bao et al. (2014) subjects were divided into three different treatment groups with each assigned a distinct task within the market structure described above. In the ’Learning to Forecast’ treatment the subjects goal is to forecast the price of an asset for the next period and their payout will depend on their ability to predict the price accurately. The first to use this experimental framework was Marimon et al. (1993). Experiments where subjects are asked to make a trading decision by buying or selling a chosen amount of stocks belong to a class called ’Learning to Optimise’. Bao et al. (2014) combined these two types of experiments into a third they called the ’Mixed Treatment’. Here subjects were first asked to forecast the price of an asset, next they were asked to submit an explicit trading quantity decision in the same market. A more comprehensive discussion on the different experimental treatments is found in Hommes (2011), Duffy (2008) and Bao et al. (2014).

Hommes (2006) and Anufriev and Hommes (2012) give an extensive survey of the heterogeneous expectation models proposed in contemporary literature. For this thesis it will suffice to note that Bao et al. (2014) used two price expectation rules, the adaptive expectations and the trend ex-trapolation rule, to estimate the heterogeneous forecasting and trading heuristics of the subjects in each experiment. Both rules are often found in experimental work with a positive feedback market (Heemeijer et al., 2009). In the ‘Mixed Treatment’ the forecasting rule was estimated by

pe_i,t= β1+ β2pt−1+ β3pei,t−1+ β4(pt−1− pt−2). (2.3)

For the trading behaviour the model that was estimated is a mix between a AR process and a asset return extrapolation. Summarised by

qi,t = α1+ α2qi,t−1+ α3ρt−1+ α4ρei,t, (2.4)

with ρt−1 and ρet are respectively the asset return function and the expected asset return.1

ρt−1 = pt−1+ 3.3 − 1, 5pt−2 (2.5)

ρe_i,t = pe_i,t+ 3.3 − 1.05pt−1 (2.6)

Note that in Equation (2.6) the price expectation coefficient is estimated by Equation (2.3).

This thesis will interpret these estimation results using the heuristic labels found in Anufriev and Hommes (2009), Anufriev and Hommes (2012) and Heemeijer et al. (2009). They found that these estimates can be categorised into distinct groups, making the weights of the different elements of the heuristic more interpretable.

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The results of the estimations from the rules can be found in Table 9 of Appendix C of the paper of Bao et al. (2014).

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3. Forecasting and Trading Heuristics

To research the questions posited in the previous chapter this paper leans on the research and the experiments conducted by Bao, Hommes, and Makarewicz (2014), specifically the ‘Mixed Treatment’ experiment. In this chapter the estimated heuristics from this experiment are grouped, labelled and discussed.

3.1 Price Predictions Heuristics

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 −1 −0.5 0 0.5 1 1.5 AR(1)

Price prediction rule coefficients per rule

Past price Past trend TFR NR AD&TR TRR Outliers

Figure 3.1: Price Prediction Weights

The estimated weights of the price prediction model (Equation (2.3)) can be found in Figure 3.112

This gives a visual insight on the forecasting heuristics of subjects in the experiment. This paper identifies four different groups of prediction rules and specifies these in Table (3.1). Four observations do not fall within these four groups and are marked as outliers.

1

All figures and simulations in this paper are coded in Matlab and are made available by the author upon request.

2

Different angles of this scatter plot can be found in Appendix A

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Forecasting and Trading Heuristics 6 pt−1 pei,t−1 pt−1− pt−2 n symboll TFR 0.8 < β2 < 1.3 β3 = 0 β4 > 0 18 o NR β2≥ 0.8 β3 = 0 β4 = 0 11 + AD&TR 0.4 < β2 < 1 β3 > 0 β4 ∈ R 7 . TRR 0 ≤ β2< 0.8 β3 = 0 β4 ≤ 0 8 x

Outliers n.a. n.a. n.a. 4

Table 3.1: Price Prediction Weight Constrains

Trend Following Rule (TFR)

Most subjects in the ‘Mixed Treatment’ experiment use a trend following rule. In contrary with Anufriev and Hommes (2012) who identify a weak and a strong trend following heuristic

with trend weights β4 = 0.4 and β4 = 1.3 respectively, this thesis defines a general trend

extrapolation rule. The Trend Following Rule has a positive trend extrapolation weight and subjects seem to look at the past price to make a price expectation. Using the average weights of the subjects within the group and Equation (2.3) it is found that the Mean TFR is given by

pe_i,t = 1.489 + 1.011pt−1+ 0.89(pt−1− pt−2). (3.1)

Naive Expectation Rule (NR)

The Naive Expectation Rule is used by 23% of the subjects and the subjects that use this rule only look at the past price for their prediction. This thesis follows Heemeijer et al. (2009) and Anufriev and Hommes (2012) naming this rule. The Mean NR is given by

pe_i,t= 2.26 + 0.995pt−1. (3.2)

Adaptive and Trend following Rule (AD&TR)

In the ’Mixed Treatment’ experiment only five subjects have an Adaptive Heuristic as defined

by Anufriev and Hommes (2012).3 To account for the subjects that use an adaptive together

with a trend extrapolation weight, this thesis has chosen to include a positive trend following weight and naming it the Adaptive and Trend Following Rule. The Averaged AD&TR is

reported by4

pe_i,t = −3.687 + 0.678pt−1+ 0.376pei,t−1+ 0.22(pt−1− pt−2). (3.3)

Trend Reversing Rule (TRR)

The Trend Reversing Rule is similar to the TFR only that the sign of the trend weights β4 is

reversed. In some literature this rule is called the contrarian rule and is, just like the adaptive expectations rule, often observed in a Negative Feedback environment (Bao et al., 2012). Note

that this mean rule is the only rule with a double digit constant β1.

pe_i,t = 19.313 + 0.413pt−1− 0.117(pt−1− pt−2) (3.4)

Outliers

The four subject which this paper has marked as outliers, have very large coefficients in one or more elements of the estimated equation. This paper cannot identify any regularity in these (large) coefficients and therefore marks them as outliers. Bao et al. (2014) also identify outliers, mainly because of input error’s or misunderstanding of the experiment.

3_{Heemeijer et al. (2009) labelled this the First order rule} 4

This is the only averaged price prediction rule where there is an important difference between the mean and the median rule. The Median AD&TR is a Adaptive Heuristic (with no trend following weight) as defined by Anufriev and Hommes (2012). See Table B.1 for the median weights of the different heuristics.

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Forecasting and Trading Heuristics 7

3.2 Quantity Decision Heuristics

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 AR(1)

Quantity rule coefficients per rule

Exp.Return PastReturn NARR LAER AER PARR NSW NLPRR

Figure 3.2: Trading Heuristics Weights

The quantity decisions estimation model used in Bao et al. (2014) is given by Equation (2.4). Like the price forecasting estimations, this thesis reports a scatter plot visualising these estimated quantity weights (Figure 3.2) and identifies different groups with different coefficients that have similarities

between each other.5 In the case of quantity decisions six different groups are defined, with their

detailed information found in Table 3.2.6

qi,t−1 ρt−1 ρei,t n symboll

NARR α2 ≥ 0 α3< 0* α4 < 0* 4 o LAER α2 > 0 α3= 0 α4 ≥ 0 10 + AER α2 = 0 α3= 0 α4 ≥ 0 18 . PARR α2 = 0 α3> 0 α4 ≥ 0 6 x NSW α2 = 0 α3= 0 α4 = 0 8 NLPRR α2 < 0 α3> 0 α4 = 0 8 *

Table 3.2: Quantity Weight Constrains

Negative Asset Return Rule (NARR)

The Negative Asset Return Rule is the least rule used in the ’Mixed Treatment’ experiments. Subject that use the trading heuristics have a negative past asset return weight or a negative expected return coefficient. The Mean NARR is given by

qi,t = 0.164qi,t−1− 0.141ρt−1− 0.252ρei,t. (3.5)

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Different angles of Figure 3.2 can be found in Appendix A

6

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Lagged Asset Expectation Rule (LAER)

The Lagged Asset Expectation Rule is used by subjects that extrapolate their trading decision

from the previous period and the expected asset return for the next period.7 _{So that}

qi,t = 0.009 + 0.649qi,t−1+ 0.045ρei,t. (3.6)

Asset Expectation Rule (AER)

The Asset Expectation Rule is closely related to its lagged alternative. Subjects only have a significant asset expectation return weight and its mean rule is given by:

qi,t = 0.154ρei,t (3.7)

As noted by Bao et al. (2014), the optimal trade for all price expectations comes with α4 = 1/6.

This Quantity rule is mostly used in the ‘Mixed Treatment’ experiment and centres around

this optimal trade.8

Positive Asset Return Rule (PARR)

The Positive Asset Return Rule is used by six subjects in the ’Mixed Treatment’ experiment and has a positive past return weight and a non-negative asset return expectation. Its mean rule is

qi,t= 0.16 + 0.089ρt−1+ 0.011ρei,t. (3.8)

Notice that the constant α1 is positive, meaning that on average subjects that use this rule are

optimistic. This thesis refers to Bao et al. (2014) for more details on optimistic and pesimistic trading.

No Significant Weights (NSW)

Eight subjects within the ’Mixed Treatment’ experiment have no significant weight for any of

the estimations.9 This implies that subjects using this rule make no trade at all and therefore

its mean rule is

qi,t = 0. (3.9)

Negative Lag Positive Return Rule (NLPRR)

The Negative Lag Positive Return Rule is used by subjects who have a negative lagged and a

positive asset return extrapolation.10 The mean rule is given by

qi,t = −0.529qi,t−1+ 0.327ρt−1. (3.10)

7

Notice that the constant is larger then zero. See the PARR for more details.

8

The median of these weights is exactly α4= 1/6 9

Notice that for the NSW and NLPRR the price forecasting (rule) has no effect on the quantity traded

10

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3.3 Rule Distribution

The distribution of the different rule combinations used by the experimental agents in the ‘Mixed Treatment’ is given in Table 3.3.

Price rules

Quantity rules

NARR LAERR AER PARR NSW NLPRR n

TFR 0 5 8 3 0 2 18 NR 1 2 2 3 3 0 11 AD&TR 2 1 2 0 2 0 7 TRR 1 2 4 0 1 0 8 Outliers 0 0 2 0 2 0 4 n 4 10 18 6 8 2 48

Table 3.3: Rule distribution

This thesis also reports all the mean weights of the price prediction and trading rules in Table 3.4 and refers to Appendix B for the median weights.

Price prediction rule

cons. pt−1 peit−1 (pt−1− pt−2) β1 β2 β3 β4 TFR 1,489 1.011 0 0.89 NR 2.26 0.995 0 0 AD&TR -3.687 0.678 0.376 0.22 TRR 19.313 0.413 0 -0.117 Quantity rule

cons. qi,t−1 ρt−1 ρei,t

α1 α2 α3 α4 NAER 0 0.164 -0.141 -0.252 LAER 0.009 0.649 0 0.045 AER 0 0 0 0.154 PARR 0.16 0 0.089 0.011 NSW 0 0 0 0 NLPRR 0 -0.529 0.327 0

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4. Market Dynamics

In the previous chapter this thesis has identified different price expectations rules and quantity rules among agents. In this section the impact of these rules on the price development in the asset market are discussed by following the approach of Hommes et al. (2005).

4.1 Homogeneous Heuristics

The experimental market laid out before in this thesis is simulated on the different rule combinations

(4 · 6 = 24) with average coefficients of the individual subjects within the rules (see Table 3.4).1 In

this section all six subjects within the simulation follow the same rule combination, later different

subjects will be assigned different rule combinations. The initial price is set at p0= 43, as in the

laboratory experiment from Bao et al. (2014) and q₁= 1.1725 as the initial trade from the average

initial trade of all the 48 individuals.23

Figure Rules p_40:50 σp40:50

4.1a TFR and AER 130.78 4.9

4.1b TFR and LAER 151.11 0.73

4.1c TRR and AER 37.01 0.9

4.1d TFR and PRR 79.44 2.8

4.1e NR and PARR 77.64 3.0

4.1f NR and NSW 51.69 2.7

Table 4.1: Average Price and Standard Deviation of the Last Ten Periods.

In this thesis different graphs of the price developments are reported. Only the six most inter-esting/applicable graphs are displayed here (Figure 4.1), all the graphs for the 24 different rule combinations can be found in Appendix B.

The most used rule combination (see Table 3.3) is the Trend Following Rule (3.1) and the Asset Expectation Rule (3.7). In this rule combination (shown in Figure 4.1a) subjects use a trend ex-trapolation and trade nearly 1/6 times the expected return. Until halfway the experiment this trend extrapolation element gives subjects a higher expected price for all the periods, leading to positive trades (from the AER) and a realised market price of more than 2.5 times the fundamental price. Interestingly after period 30 the market price starts to fall but not as steep as in the ‘bull’ phase

of the experiment. After period 45 the price seems to stabilise around 130 (p_40:50 = 130.78) which

is still nearly double the fundamental price. A similar price development is found with the Trend Following Rule and the LAER (3.6) in Figure 4.1b. The difference is that there is no ‘bear’ phase

and the price stabilising at 150 (p_40:50 = 151.11). It seems that markets where subjects that use

the Trend Following Rule with a positive expected return coefficient (α4 > 0) lead to an

overshoot-ing of the fundamental price. Another result is that a significant lag coefficient in the quantity rule(LAER) makes sure the price does not ‘come down’. It is easily seen that subjects keep their traded quantity positive because of that. Another price dynamics using a rule combination with the

1_{The fifth rule are the outliers and for that reason will not me simulated.} 2

This thesis has found that for many of the rule combinations the average price of the last ten periods did not change for different initial prices and initial trading quantities

3

For modelling purposes this thesis has not imposed a restriction on the amount of digits behind the decimal. Bao et al. (2014) note that most subjects trade ’round’ levels.

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Market Dynamics 12 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

Plot Price rule 1 and Quantity rule 3

t

Price

(a) TFR(3.1) and AER(3.7)

0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (b) TFR(3.1) and LAER(3.6) 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (c) TRR(3.4) and AER(3.7) 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (d) TFR(3.1) and PARR(3.8) 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(e) NR(3.2) and PARR(3.8)

0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(f) NR(3.2) and NSW(3.9) Figure 4.1: Different Market Dynamics with Different Rule Combinations

Trend Following Rule is reported in Figure 4.1d. The market price is also increasing and passing the fundamental before alternating around 80 after period 30. It can be argued that the price does not overshoot as much as with other Trend Following rule combinations because the trading depends on small coefficients of the previous and expected return (PARR), leading to smaller trades by the test subjects.

The comparison between the closely related Trend Following Rule and Naive Expectations Rule (the difference being the significant trend extrapolation element) is seen in Figure 4.1d and Figure 4.1e.

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Market Dynamics 13

Both price dynamics develop almost identical and the (large) bubbles that were found in previous

TFR figures is not observed here. The small past return and expected return weights, α3 and α4

respectively, associated with the Positive Asset Return Rule can be the cause of this relative ‘flat’ price dynamics and a trend extrapolation element does not seem to effect the development of the price much. In the case of subject assigning small weights to their price predication is even more applicative in Figure 4.1f where subject apply the No Significant Weights Rule and thus make no

trade at all.4 It is easy to see that in this case the price does not develop more than the small

idiosyncratic shock.

Figure 4.1c shows a market development that alternates around half the fundamental price. Instead of using the Trend Following Rule subjects in this dynamics use the Trend Reversing Rule. It seems that the negative trend weight from this rule forces the market price to stay down. The quantity heuristic does not seem to effect the outcome of a price dynamic when subjects use this price rule. From the above it is found that bubble formations and the price development in the simulations depend largely on the quantity and price expectation heuristics that test subjects use. This thesis suspects that quantity rules where the price expectation of a subject matters little or nothing for their trading decision have a price dynamic that is relatively straight. On the other hand if subjects use a quantity rule with larger weights it is found that for the often used rule combinations, the Trend Extrapolation Rule has the highest propensity to overshoot the fundamental (drastically) and for the Trend Reversing Rule to alternate underneath the fundamental price.

4.2 Weighted Rules

To account for all the estimated coefficients given by test subjects in the ‘Mixed Treatment’ exper-iment (including the outliers), this thesis has modelled two types of weighted price dynamics. The Conditional and Unconditional Weighted Rules. The unconditional weights are the average weights found by the different trading and forecasting rules relative to the amount of subjects that use those rules and calculated using

β_iw= 5 X l=1 nl 48βi,l for i = 1, . . . , 4 (4.1)

where nl is the amount of subjects using price rule l and

αw_i = 6 X k=1 nk 48αi,k for i = 1, . . . , 4 (4.2)

where nk is the amount of subjects using trading rule k.5

With these unconditional weights the price expectation heuristic becomes

pe_i,t= 4.2767 + 0.8414pt−1+ 0.0629pei,t−1+ 0.4023(pt−1− pt−2) (4.3)

and the quantity rule is given by

qi,t = 0.0035 + 0.1269qi,t−1+ 0.013ρt−1+ 0.0474ρei,t (4.4)

From the previous subsection it was found that small weights on the asset return elements α3 and

α4 create a ’straight’ price development. The unconditional trading rule also has small weights with

these coefficients and the price dynamics reported in Figure 4.2a seems confirm this view. 4_{Except the initial trade}

5

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Market Dynamics 14 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200 Weighted Rules t Price (a) Unconditional 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

Conditional Weighted Rules

t

Price

(b) Conditional Figure 4.2: Weighted Rule Price Dynamics

p_40:50 σp40:50

Unconditional 56.22 2.8

Conditional 57.18 2.9

Table 4.2: Average and standard deviation of the Weighted Heuristics

The conditional weighted dynamics looks very similar to the unconditional simulation, as seen in Figure 4.2, but they are not identical. Table 4.2 reports that the mean and standard deviation of the last ten periods of the price development of the two rules are not equivalent. The Conditional Weighted Rule are weighted quantity heuristics conditional on a price heuristic. This means that for every rule combination a weight can be assigned to a quantity rule based on the amount subjects

that use this rule after they made a forecast using a price prediction rule. So for every price rule pe_l

with l = 1, . . . , 5 the five different price rules, the conditional weights are given by

αw_i = 6 X k=1 nk,l 48 αi,k for i = 1, . . . , 4. (4.5)

The weights that are found using this equation (4.5) for all four price forecasting heuristics and the outliers is reported in Table 4.3.

For price Weights Quantity rules

forecasting cons. qi,t−1 ρt−1 ρe_i,t

l rule: α1 α2 α3 α4 1 TFR 0.0018 0.0456 0.0192 0.0310 3 NR 0.0013 0.0305 0.0026 0.0037 3 AD&TR 0.0002 0.0204 -0.0059 -0.0032 4 TRR 0.0003 0.0305 -0.0029 0.0094 5 Outliers -0.00004 0 0 0.0064

Table 4.3: Weights Conditional Heuristics

The summation of the different quantities that subjects chose to trade using the Conditional Weighted

Rule is used as the quantity trade (qi,t) for the price mechanism (see Equation (2.1)). From the

above this thesis suspects that using all the elements that subjects used in the ‘Mixed Treatment’ experiment to postulate a forecasting and trading decision and weigh them according to their fre-quency, gives a price dynamics that stays underneath the fundamental and moves ’straight’ with the price increasing limit over time.

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Market Dynamics 15

4.3 Optimal Trading Rules

One of the questions this thesis asks is what would have happend to the price development in the experimental market if subjects would have traded optimal. Using Equation (2.2) from Bao et al. (2014) the optimal trade decisions per price forecasting rule is easily computed and reported in Table 4.4.

Figure Forecasting Rule Optimal Trade

4.3a TFR q∗_i,t = 0.798 − 0.0065pt+ 0.1483(pt− pt−1)

4.3b NR q∗_i,t = 0.9267 − 0.009pt

4.3c AD&TR q∗_i,t = −0.0645 − 0.0062pt+ 0.063pei,t+ 0.37(pt− pt−1)

4.3d TRR q∗_i,t = 3.769 − 0.637pt− 0.0195(pt− pt−1)

Table 4.4: Optimal Trading Decisions per Price Prediction Rule

These equations give the optimal trade conditional on the price prediction that subjects make using the respective forecasting heuristic. The price development when all six subjects in the experimental market trade optimal for the different price predictions rules are found in Figure 4.3.

0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

Optimal Trading Rule With Price Rule 1

t

Price

(a) Optimal trade conditional on TFR

0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(b) Optimal trade conditional on NR

0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(c) Optimal trade conditional on AD&TR

0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(d) Optimal trade conditional on TRR Figure 4.3: Optimal Trading Dynamics

In this dynamics the subject only needs to postulate a price expectation and therefore this treatment is closely related to the ‘Learning to Forecast’ treatment. This thesis will compare this optimal treatment with the ‘Learning to Forecast’ experiment conducted by Bao et al. (2014).

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Market Dynamics 16

It seems that none of the four price developments in Figure 4.3 are similar to those in the LtF

from Bao et al. (2014). The four LtF dynamics are all ‘constantly’ increasing towards a price

slightly above the fundamental. Although a comparable ‘shape’, the Trend Following(4.3a) and Naive Expectation(4.3b) rules overshoot the fundamental much more than the LtF treatments from Bao et al. (2014). In three out of four experiments one or more subjects used a trend following rule. This thesis suspects that the small weights that subjects assign to the trend extrapolation element, alongside that not all subjects use a trend extrapolation rule, attributes to a smaller deviation of

the fundamental price in the LtF experiments from Bao et al. (2014).6

4.4 Heterogeneous Heuristics

In previous sections this paper has only looked at the price dynamics if the six subjects in the

experiment have the same price forecasting and quantity rule. In asset markets (real world or

experimental) the agents will not necessarily use the same predication and trading rules to postulated their expectations or their trade decisions. For that reason, this thesis investigates what the effect of these heterogeneous heuristics on the price development in the simulated experimental market. With four price predication heuristics and six quantity heuristics the amount of combinations that the six subjects in the price dynamics can use is large. In this subsection this thesis will report eight different price dynamics with often observed rule combinations (See Table 3.3). The heuristics used in the different price development, with the average price and standard deviation of the last ten periods, can be found in Table 4.5.

Mixed Rule 1 Mixed Rule 2 Mixed Rule 3 Mixed Rule 4

p_40:50 51.83 123.20 55.78 55.72

σp40:50 3.3 2.7 3.4 3.4

Price Quantity Price Quantity Price Quantity Price Quantity

Sub. 1 TFR PARR TFR PARR TFR PARR TFR PARR

Sub. 3 TFR AER TFR AER TFR AER TFR AER

Sub. 4 TRR PARR TFR PARR TRR PARR TRR PARR

Sub. 5 NR NSW NR NSW TFR NSW NR NSW

Sub. 6 NR NLPRR NR NLPRR TFR PARR TFR PARR

Mixed Rule 5 Mixed Rule 6 Mixed Rule 7 Mixed Rule 8

p_40:50 58,15 113,64 57,27 55,26

σp40:50 3,4 2,9 3,2 3,3

Price Quantity Price Quantity Price Quantity Price Quantity

Sub. 3 TFR PARR TFR PARR TFR AER TFR AER

Sub. 4 TFR AER NR AER NR AER NR NSW

Sub. 5 TFR AER NR PARR NR PARR NR PARR

Sub. 6 TRR PARR AD&TR NSW TRR PARR TRR PARR

Table 4.5: Mixed Rules

6_{The estimates of the Adaptive Rule and Trend Following rule from the experiment can be found in Appendix C}

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Market Dynamics 17 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200 Mixed Rules t Price

(a) Mixed Rules 1

0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200 Mixed Rules t Price (b) Mixed Rules 2 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200 Mixed Rules t Price (c) Mixed Rules 3 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200 Mixed Rules t Price (d) Mixed Rules 4 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200 Mixed Rules t Price

(e) Mixed Rules 5

0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200 Mixed Rules t Price (f) Mixed Rules 6 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200 Mixed Rules t Price (g) Mixed Rules 7 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200 Mixed Rules t Price (h) Mixed Rules 8 Figure 4.4: Different Mixed Rules with n=6 price expectations

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Market Dynamics 18

In homogeneous heuristics this thesis has claimed that positive feedback markets tend to overshoot if subjects use the Trend Following Rule and quantity rules with sufficient trading. In Figure 4.4e a price dynamic is reported with all the above features except that one subject uses the Trend Reversing Rule. The impact on the price dynamic is abundant. Instead of pricing the value of the experimental asset double its fundamental price, this heterogeneous dynamic has a average price of 58.15 over the last ten periods. An analogous result is found in the price dynamics from Figures 4.4a and 4.4b. The later having four subjects that use TFR and two NR. Both rules in the homogeneous case, cause bubble formation in the market and it seems that the combination of the two has a

equivalent result. Adding again a Trend Reversing Rule ensures the market to stay below the

fundamental price. It seems that in all reported dynamics where a trend reversing subject is present (4.4a, 4.4c, 4.4d, 4.4e, 4.4g and 4.4h), the price stay below the fundamental. This implies that a single test subject has the capability to ‘keep the price down’ by following the Trend Reversing Rule. In the price development of the second group in ‘Mixed Treatment’ experiment from Bao et al. (2014) two subjects have a negative weight for the trend extrapolation. The result found above seems to hold here. A bubble is not observed in the price dynamics with two subjects using a trend reversing rule, instead it alternates around the fundamental. In other experimental work (see Heemeijer et al., 2009; Hommes, 2011) a trend reversing rule is discussed in the Negative Feedback environment and others (Anufriev and Hommes, 2012) observe trend reversing rules in the Heuristic Switching Model. Both subjects that are not covered by this thesis.

In the homogeneous section the suggestion is made that both the Trend Following and the Naive Expectation heuristics have a penchant to form bubbles in the experimental market if subjects make sufficient trades. In the heterogeneous price dynamics this result appears to be confirmed. Excluding the cases that a subject uses a Trend Reversing Rule (see above), a rule combination with a mixture of TFR and NR subjects results in bubbles in the price dynamics (see Figures 4.4b and 4.4f). The quantity rules seem to not have such a profound impact on the price development in this treatment. The homogeneous section reported that in a price dynamic with subjects that have small weights, the price stayed underneath the fundamental. In the heterogeneous approach one subject using the No Significant Weight heuristic can not effect the outcome as drastic. In this subsection price dynamics are reported with a subject using the NSW heuristic and still form a bubble.

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5. Conclusion

This thesis has tried to lay out a framework of standard economical expectations theory and has looked closely at the bounded rationality theory, which states that agents in an economy develop their expectation rationally being limited by time, knowledge and capacity. As mentioned earlier, the expectations of economic agents are important for policy makers to understand because their be-haviour influence the economy as a whole (i.e. stock prices, (business) investments and consumption levels).

From the literature discussed in this thesis, Hommes (2011), Anufriev and Hommes (2012) and Bao et al. (2014) are the most insightful for the subject discussed. The first for its introduction of the expectation models, the discussion of the different experimental feedback markets, and the survey of the literature. The second paper for the insight on heuristics models and the final paper for the (scientific) justification of the experimental market setup and the results of the estimating of the price expectation (2.3) and quantity decision (2.4) rules.

First this thesis has grouped distinct forecasting and trading heuristics, labelling and discussing them using the prior work from Heemeijer et al. (2009) and Anufriev and Hommes (2012).

This thesis has found that the development of the price in the experimental market postulated by Bao et al. (2014) is largely a product of the price prediction rules that subjects use in the experiment. This means that in a positive feedback market where agents have a strong trend extrapolation element in their forecasting behaviour markets tend to overshoot the fundamental price, sometimes even leading to a realised market price twice as high as the fundamental market price. Although the effect of the price prediction is great, the influence of the quantity decisions made by the subjects can not be overstated. If subjects have no or small weights assigned to the coefficients of Equation (2.4), the price development is effected substantially.

Another result this thesis has found is the capability of one subject to alter the price development in an experiment. More specific, one subject using the Trend Reversing Rule can offset the formation of a bubble that would have acquired if another forecasting heuristic was chosen. This could have some implication for future experimental work on bubbles in a positive feedback environment. Because bubbles are a robust finding within this treatment, a case of low price development could be caused by one subject using a forecasting heuristic that has a negative trend extrapolation element. The results from this paper show that the way economic agents postulate their price expectations and trading decisions have a convincing effect on the market price, whether they are in a market with agents with the same or with different expectations and trading behaviour and their impact on the potential formation of bubbles are thorough.

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6. Bibliography

M. Anufriev and C. Hommes. Evolutionary selection of individual expectations and aggregate out-comes in asset pricing experiments. American Economic Journal: Microeconomics, 4(4):35–64, 2012.

M. Anufriev and C. H. Hommes. Evolution of market heuristics. Knowledge Engineering Review, 27 (2):255–271, 2009.

M. Anufriev, C. Hommes, and T. Makarewicz. Learning to forecast with genetic algorithm. Technical report, Tech. rep.(February 2013), 2013.

T. Bao, C. Hommes, J. Sonnemans, and J. Tuinstra. Individual expectations, limited rationality and aggregate outcomes. Journal of Economic Dynamics and Control, 36(8):1101–1120, 2012. T. Bao, C. Hommes, and T. Makarewicz. Bubble formation and (in)efficient markets in

learning-to-forecast and -optimize experiments. CeNDEF Working paper, 14-01, March 2014.

J. Duffy. Macroeconomics: a survey of laboratory research. Handbook of experimental economics, 2, 2008.

P. Heemeijer, C. Hommes, and J. Tuinstra. Price stability and volatility in markets with positive and negative expectations feedback: An experimental investigation. Journal of Economic Dynamics and Control, 33(05):1052–1072, 2009.

C. Hommes. Heterogeneous agent models in economics and finance. Handbook of computational economics, 2:1109–1186, 2006.

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

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

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

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

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

C. N. Noussair and S. Tucker. Experimental research on asset pricing. Journal of Economic Surveys, 27(3):554–569, 2013.

R. J. Shiller. The use of volatility measures in assessing market efficiency*. The Journal of Finance, 36(2):291–304, 1981.

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

S. Sunder. Experimental asset markets: a survey handbook of experimental economics ed j kagel and a roth, 1995.

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A. Scatter Plots

A.1 Price Prediction Coefficients

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 −1 −0.5 0 0.5 1 1.5 AR(1)

Past price Past trend TFR NR AD&TR TRR Outliers

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Scatter Plots −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −1 −0.5 0 0.5 1 1.5 AR(1)

Past price

Past trend

Figure A.2: PPRC: Angle 2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 1 2 −1 −0.5 0 0.5 1

1.5 Price prediction rule coefficients per rule

AR(1) Past price

Past trend

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Scatter Plots −1 −0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −1 −0.5 0 0.5 1 1.5 AR(1) Past price

Past trend

Figure A.4: PPRC: Angle 4

Rule pt−1 pe_i,t−1 pt−1− pt−2 n symboll

TFR 0.8 < β2 < 1.3 β3 ∈ R β4 > 0 18 o

NR β2≥ 0.8 β3 = 0 β4 = 0 11 +

AD&TR 0.4 < β2< 1 β3 > 0 β4 ∈ R 7 .

TRR 0 ≤ β2< 0.8 β3 = 0 β4 ≤ 0 8 x

Outliers n.a. n.a. n.a. 4

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Scatter Plots

A.2 Quantity Coefficients

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 AR(1)

Exp.Return PastReturn NARR LAER AER PARR NSW NLPRR

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Scatter Plots −1 −0.5 0 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 AR(1) Exp.Return

PastReturn

Figure A.6: Angle 2

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Exp.Return

AR(1)

PastReturn

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Appendix B: Price dynamics −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Exp.Return

AR(1)

PastReturn

Figure A.8: Angle 4

Rule qi,t−1 ρt−1 ρe_i,t n symboll

NARR α2 ≥ 0 α3< 0* α4 < 0* 4 o LAER α2 > 0 α3≥ 0 α4 ≥ 0 10 + AER α2 = 0 α3= 0 α4 ≥ 0 18 . PARR α2 = 0 α3> 0 α4 ≥ 0 6 x NSW α2 = 0 α3= 0 α4 = 0 8 NLPRR α2 < 0 α3> 0 α4 = 0 8 *

Table A.2: Quantity rules.

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B. Price Dynamics

B.1 Median Coefficients

Price prediction rule

constant pt−1 peit−1 pt−1− pt−2 β1 β2 β3 β4 TFR 0 1 0 0.891 NR 0 1.004 0 0 AD&TR 0 0.661 0.347 0 TRR 19.8695 0.485 0 0 Quantity rule

constant qi,t−1 ρt−1 ρei,t

α1 α2 α3 α4 NARR 0 0 -0.16 -0.101 LAER 0 0.675 0 0 AER 0 0 0 0.167 PARR 0 0 0.091 0 NSW 0 0 0 0 NLPRR 0 -0.529 0.37 0

Table B.1: Median from price prediction rules and quantity rules coefficients

B.2 Average and Standard Deviation

TFR NR AD&TR TRR

p_40:50 σp40:50 p40:50 σp40:50 p40:50 σp40:50 p40:50 σp40:50

NARR 0 0 0 0 61.13 2.8 n.a. n.a.

LAER 151.11 0.73 102.73 3.5 52.37 1.7 37.57 1.1

AER 130.78 4.9 99.28 3.5 9.7 3.8 37.01 0.9

PARR 79.44 2.8 77.64 3.0 73.22 3.2 50.88 3.3

NSW 51.69 2.7 51.69 2.7 51.69 2.7 51.69 2.7

NLPRR n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a.

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Price Dynamics

B.3 Trend Following Rule

0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(a) _p TFR and NARR

40:50 = 0 σp40:50 = 0 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (b) _p TFR and LAER 40:50= 151.11 σp40:50 = 0.73 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (c) _p TFR and AER 40:50 = 130.78 σp40:50 = 4.9 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (d) _p TFR and PARR 40:50= 79.44 σp40:50 = 2.8 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (e) _p TFR and NSW 40:50= 51.69 σp40:50 = 2.7 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(f) TFR and NLPRR Figure B.1: Trend Following Rule

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Price Dynamics

B.4 Naive Expectation Rule

0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(a) _p NR and NARR

40:50 = 0 σp40:50 = 0 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (b) _p NR and LAER 40:50= 102.73 σp40:50 = 3.5 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (c) _p NR and AER 40:50= 99.28 σp40:50 = 3.5 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (d) _p NR and PARR 40:50= 77.64 σp40:50 = 3.0 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (e) _p NR and NSW 40:50= 51.69 σp40:50 = 2.7 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(f) NR and NLPRR Figure B.2: Naive Expectation Rule

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Price Dynamics

B.5 Adaptive and Trend Following Rule

0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(a) _p AD&TR and NARR

40:50= 61.13 σp40:50 = 2.8 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(b) _p AD&TR and LAER

40:50= 52.37 σp40:50 = 1.7 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(c) _p AD&TR and AER

40:50= 9.7 σp40:50 = 3.8 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(d) _p AD&TR and PARR

40:50= 73.22 σp40:50 = 3.2 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(e) _p AD&TR and NSW

40:50= 51.69 σp40:50 = 2.7 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(f) AD&TR and NLPRR Figure B.3: Adaptive and Trend following Rule

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Price Dynamics

B.6 Trend Reversing Rule

0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(a) TRR and NARR

0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (b) _p TRR and LAER 40:50= 37.57 σp40:50 = 1.1 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (c) _p TRR and AER 40:50= 37.01 σp40:50 = 0.9 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (d) _p TRR and PARR 40:50= 50.88 σp40:50 = 3.3 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t Price (e) _p TRR and NSW 40:50= 51.69 σp40:50 = 2.7 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 120 140 160 180 200

t

Price

(f) TRR and NLPRR Figure B.4: Trend Reversing Rule

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Price forecasting- and trading-rules in the `Mixed Treatment' experiment