Master Thesis
Prediction of Collective Expectations: Learning Models with Forgetting
Supervisor: Florian Wagener Student: Genxun Li
1. INTRODUCTION
Theorists are willing to model economic activities based on the assumption that individuals are equally informed and smart so that they can make the best use of information. Individuals make decisions using available information and their decisions consequently determine the economic variables, and therefore influence economic activities. One of the examples would be modeling price stability and volatility in markets over certain periods. In each period, individuals form expectations on the price for an asset in the next period, and the price in the next period is influenced by their expectations. Once the price is realized, individuals use newly released information to form expectations for the coming period. If all individuals had rational expectations, the market price would converge to the rational expectation price where every individual is reluctant to deviate from his expectation given the expectation prediction rules from others. However, recent evidence shows that under different characteristic feature of the market environment, individuals do not always make rational expectations. For example, under strategic complements that individual decisions mutually reinforce each other, they are likely to mimic the strategies of others; whereas under strategic substitutes those decisions mutually offset one another, individuals are inclined to make decisions opposite from what majority do (Haltiwanger & Waldman 1985). This phenomenon makes it attractive to investigate how individuals form expectation in a controlled environment. Recently, a number of experiments in a controlled environment were conducted, see, e.g., Hommes et al. (2005, 2007) , Heemeijer et al. (2009), Bao et al. (2012). In those experiments, two types of economic environment are defined, positive feedback and negative feedback markets. In positive feedback markets, the change in input corresponds to the change in the market variable in the same direction. Participants are likely to use momentum investing strategy under such environment. For example, in a speculative asset market, when individuals expect the price of an asset to rise, they are more likely to buy this asset, which in turn push up the price of this asset according to the law of supply and demand. If individuals’ expectations of an asset are low, they would be reluctant to buy this asset, and therefore, the asset price will go down. The opposite is true for negative feedback market, where the change in input corresponds to the change in the market variables in an opposite direction. For example, in a commodity market, the price of some commodity is expected to be high in the coming future. Producers are inclined to produce more in the next period. This makes the realized price going down in the future due to the large supply in the next period.
According to rational expectation hypothesis, individual participant’ expectation is equal to the true statistical expected value. It states that individual behaves as if they know the whole complicated economic structure, and every agent’s expectation is consistent with the theoretical value over time. Alternatively, Wagener (2014) interpret rational expectation as ‘collective’ version of the rational expectation hypothesis. This hypothesis tells that, instead every agent performs expectation rationally, on average, participants form rational expectation. And this ‘collective’ rational expectation invisibly controls market to converge to market clearing equilibrium. Under this scenario, individual agent has flexibility to form expectation, but the aggregate formation of a group is seen as rational. More importantly, in an economic structure, if time series of expectation and realized price are available, this collective rational expectation hypothesis can be tested (Wagener, 2014).
According to Te Bao et al. (2012), testing rational expectation hypothesis may be interpreted as 1): to test whether market price is a clearing equilibrium price when equilibrium is consistent; 2): to test whether market price can quickly converge to a new equilibrium price after a shock on equilibrium price.
In seventies and eighties, rational expectations hypothesis gets lots of popularity in modeling economic activities. When rational expectation hypothesis applies to experimental data investigating the formation of expectations, it does not perform that well. It can usually precisely explain the negative feedback market where quick convergences to the equilibrium price usually happen. It cannot consistently predict the positive feedback market where a slow oscillatory movement often occurs around the equilibrium price.
This thesis is motivated by the fact that ‘collective’ rational expectation hypothesis cannot simultaneously explain individual experimental expectation in positive feedback and negative feedback market. From the perspective of bounded rationality, we may get a different result that can explain individual expectation formation more efficiently and effectively.
Contemporary models forming individual expectations are likely to incorporate the concept of bounded rationality proposed by Herbert A. Simon. One advantage of bounded rationality is that it has more flexibility on individual’s expectation formation compared with rationality expectation hypothesis, since participants are not required to know the whole complicated economic structure. Individuals are assumed to act as an econometrician, knowing how to analyze data and estimate parameters. In some other models, expectations are modeled by
fixed-coefficients linear prediction rules, where the expected price fixedly linearly depends on past lag realized and expected price. This rule performs very well in Bao et al. experimental data (2012), with significant coefficients at the 5% level, high adjusted , low MSE and no serial correlation (Bao et al. 2012). One deficiency of this model sees this is a fixed coefficients model. Those fixed coefficients are calculated ex post, which means the best fitted variables can be only determined after all the experimental data are accessible. This disadvantage brings a new dynamic learning process, which allows individuals to update coefficients every time after price is realized, see, e.g., Wagener (2014). Hence, the coefficients are not fixed anymore; instead individuals updated each period with new information released.
Alternatively, adaptive expectations rule and trend rule are often applied. The former can explain experimental data pretty well in negative feedback market and the latter can describe positive feedback market very well. This dissimilar use of the best fit model indicates a heterogeneous expectation. One improvement is provided by Bao et al. (2012) that use evolutionary selection models to allow participants to choose among different models according to certain criteria. In general, although those models seem to be more practical, the rational expectation hypothesis does not necessarily perform poor. However, in those experiments, a single model of individual expectations is not able to simultaneously explain both markets precisely. Models that can explain one market precisely are usually not able to explain the alternative market precisely.
The thesis is a continuation based on Wagener’s sOLS model. And the aim of this thesis is to use sOSL model and its extension with different evaluation selection criteria to investigate individual’s experimental expectation formation.
The thesis is organized as follows. Section 2 discusses a setup of an experimental design and Wagener’s (2014) sOLS model. Section 3 explains three different learning models and discusses their prediction performances. Second 4 presents a statistical and numerical comparison between different learning models. In Section 5, the best performing models are used to test the robustness. Finally, section 6 concludes.
2. PASEXPERIMENTS AND PAST RELATED WORK
A great number of experiments investigating positive and negative feedback markets have been conducted in the CREED laboratory at the University of Amsterdam. In those experiments, participants are asked to make one period ahead prediction and their earnings in each period are determined by the closeness between the expected price and the realized price. In general, numerical and graphic information about the past expected and realized price, as well as their earnings, is given during the experiments, but no qualitative information and other necessary information that can be used to derive the rational price (Wagener 2014). Moreover, participants get earnings each period according to the accuracy of their predictions.
2.1.Structural breaks experiment
In most experiments, fundamental price is set constantly in all periods, see, e.g., Hommes et al. (2005, 2007), Heemeijer et al. (2009). On the other hands, the fundamental price changes during the experiment. One example is the Te Bao et al.’s experiment in 2012. This experiment was conducted at CREED laboratory at the University of Amsterdam with 2 treatments (8 runs in positive feedback market; 8 runs in negative feedback market). In each run, 6 participants were asked to predict the price for each period up to 65 periods (except last period = 65). The realized price is a function of the average expectations:
( ⃐ ), where ̅̅̅ ∑ (3) Price generating function for positive feedback and negative feedback markets are given by:
( ⃐ ) positive feedback (4) ( ⃐ ) negative feedback (5)
where
In this experiment, is set as . The only difference between those two treatments is the sign of the slope. Absolute coefficients in both markets generating process are the same. It is easy to see that the fundamental price in both treatments is . The initial value of is set as 56, and during experiment, the value of changes twice, see follows:
{
Under ‘collective’ rational expectation hypothesis, participants were expected to forecast 56 in period 1-20, 41 in period 21-43 and 62 in period 44-65. Before the experiment, participants were not informed about the above price generating process. They were only informed about basic qualitative information about the function of the market. They did not know any information about the other participants, including their predictions, the number of participants. During the experiment, individual past prediction, past realized prices as well as some basic qualitative economic information about market were shown on a computer screen, numerically and graphically. Also, they are informed of the posibility of shocks during the experimental periods. Again, participants could not derive rational expectation price using information on the screen. After price was realized each period, participants got paid according to the closeness of their prediction to the realized price, which was expressed as quadratic error of the prediction in each period:
{
( ) } Hence, accuracy of expectation is the sole factor that affects the payoff.
2.2.Wagener’s related work the sOLS model.
In Wagener’s paper (2014), the switching ordinary least square (sOLS) model is applied for analyzing experimental data. Participants are assumed to be homogeneous and their expectation depends only on past realized price. This model consists of three prediction rules:
̂
̂ ̂ (6) ̂ ̂ ̂
The realized price, are defined in the Wagener’s paper (2014) as the price expectation for rule 0, 1, 2, at time respectively. The corresponding parameters ̂ are estimated based on past realized prices using ordinary least squares. Individuals are assumed to have homogeneous expectations, and their prediction is only based on past realized prices. Coefficients ̂ are non-fixed, which means in each period after price is realized; parameters are correspondingly estimated again with newly realized price information.
The prediction rule for next period is chosen based on lowest total squared prediction error ∑ ( )
. Hence, when computing switching criteria, participants are assumed to put
equal weights on each of prediction error, even a forecasting error made in the long past would be equally weighted with the recent prediction error.
This sOLS model has consistently explained Heemeijer et al.’s (2009) experimental data. However, when applying to Bao et al. (2012) experimental data with structural breaks, the sOLS model does not perform that well in explaining experimental behavior in negative feedback market. During that experiment, participants can immediately realize the change of fundamental price and make corresponding reactions in forming future expectations. One possible explanation is that participants get aware of the change of the environment and their prediction is based on the most updated environment. However, the sOLS model does not successfully capture this behavior.
2.3. The sOLS model with forgetting
One possible improvement from the sOLS model is the assumption that participants have a tendency to gradually forget past model performance when computing switching criteria. This idea can be implemented by incorporating forgetting rate such that information is lost over time. The implication of this forgetting hypothesis is that the recent prediction error is more important than more-distant past prediction results in determining which rule to choose. And this can be achieved by letting participants put less weights on long past prediction error but more weights on recent predictions error when choosing prediction rule. Let be the criteria and is defined as:
, and the is the forgetting rate of
rule at time . The evaluation at period t is expressed as a proportion of evaluation criteria from last period plus a fraction of current error square term. If (complete
forgetting), the criteria only depend on prediction error square at time . What happened before the last period would be neglected and will not be considered anymore. On the contrary, when
, participants do not update their criteria, the criteria are purely the same as last period
criteria.
The forgetting rate can be considered as constant over time or a function of error terms.
Constant forgetting rate assumes participants to forget each period. And the value of is not affected by other factors, such as the accuracy of past prediction. Alternatively, can be
considered as a function of error terms, and the value of can be calculated according to the
past prediction performance. That is, if one rule performs well in the last period, the criteria for that rule are like to be of little change. More explicitly, let be the weight coefficient that
participant wants to give to the last period prediction error at time t. After price is realized at , if the mismatch between prediction and realization is small, the rate of forgetting is supposed to become smaller. On the other hand, if the mismatch is high, is more likely to be a large number, more weight would be put on the current error term. Thus, the forgetting rate is assumed to be a continuous increasing function that satisfies:
| | | | (8)
In this thesis, the forgetting rate is assumed to be function of error term. While the specific mathematical expression of the function the forgetting rate is unascertained, any forms of can be considered as the potential candidates as long as they satisfy (8). For example, possible polynomial candidates are | |
| | . When is
small, | || | is almost equal to | | , and when Error is large, converging to infinity, | || | converges to 1. Alternatively, exponential candidates (| | ) satisfy above
conditions as well, if is a continuous increasing function with ( ) The choices of function is important in determining evaluation criteria, and hence
influence which rule to choose in the models. Thus, the key point of this thesis is to test different forms of and find a mathematical expression of the forgetting rate such that learning
3. The sOLS MODEL WITH FORGETTING AND THREE EXTENSIONS
In this section, first, the sOLS model with different forgetting rates are studied. In this paper, the forgetting rates are limited to six cases, | || |
and
| |
. The first three forgetting rate belongs to polynomial case and the last three is of exponential case. Furthermore, three extension models with above six forgetting rates are introduced. Also, all four different models with six different forgetting rates are applied to form one-period-ahead prediction and the dynamics movements of the expectation are evaluated afterwards. Finally, the explanatory powers of different models with different forgetting rates are compared with each other as well as the rational expectation hypothesis.
3.1. The sOLS model with forgetting
In general, pretty much the same as the sOLS model, the sOLS model with forgetting works as follows. Each time after the price is realized, the parameters of three rules are recalculated based on all past realized price using ordinary least square method. Meanwhile, evaluation criteria of three rules are updated according to a certain forgetting rate. And the rule with the smallest criteria value is selected for the next period prediction. Different forgetting rates, such as | |
| | | |
| | , and
| | | | , have been tested.
More specifically, firstly, evaluation criteria of three prediction rules are initially set as prediction error square at period 6:
( ) (9)
Forgetting rate at period 6 is also calculated using the corresponding forgetting rate formula, for example, | || | , , | | | | , | | and | | .
The rule with the smallest criterion value is selected as the prediction rule at period 7. For all the predictions after period 6, before the price is realized, the evaluation criteria of three prediction rules are compared. And the rule with the smallest evaluation criterion is selected as the prediction rule of the next period. Once the price is realized, the evaluation criteria for
next period rule selection are calculated as follows:
( ) (10)
Meanwhile, its corresponding forgetting rate is computed according to its form, for example,
| || | , , | | | | , | | , | | .
As the same procedure before, the evaluation criteria are progressively calculated using the evaluation criteria, the forgetting rate from last period and current error square. And the whole procedure is repeated until the end of the last period.
Figure 1 shows the one-step-ahead prediction of the sOLS model with selected forgetting rate for negative feedback market and positive feedback markets, and the function of corresponding forgetting rate over time. In the negative feedback market, one-step-prediction shows 2 inconsistent predictions in explaining data: the first inconsistency comes from the slow respond at first structural break when fundamental price drop to 41. It normally takes one or two periods for model to adjust its prediction rules. The possible reason causing this phenomenon is that model does not correctly select the best performing rules immediately after structural break. Those corresponding forgetting rates seem to perform well at first break. They suddenly jump to their limit at structural break point, and gradually move down as model adjusts its prediction. The second inconsistency is the incorrect explanation in participants’ behaviors. After second structural break, the graphs on left panel show a very inconsistent gap between model predictions and participants’ average expectations. This inconsistent prediction seems to slowly converge to fundamental level if periods are long enough. In terms of the forgetting rates, one can observe a slower down trend after second structural break, which means the predictions after second structural break result in long lasting high errors.
In positive feedback market, one-step-ahead sOLS model with different forgetting rate prediction all perform pretty well. No sudden jump at each structural break. Some forgetting rates have high frequency fluctuations over time. This means a good prediction is usually followed by a less good-performed prediction, but the prediction errors are in general small. Prediction smoothly follows the participants’ collective expectations.
3.2. The Structural weighted ordinary least square model (swOLS) with forgetting
In this section, the structural weighted ordinary least square (swOLS) with forgetting is applied to form predictions trying to solve the two inconsistent predictions at the first shock and after the second shock. The difference is that, at period in each rule, the swOLS model with forgetting uses weighted ordinary least square, instead of ordinary least square to calculate parameters.
Initially, the weights for observations at period are set as default value 0.5, individually. For each period after 5, the weights of the rest observations are defined according to the coefficient of the error square term in an expansion form of equation (7).
∏ ∑ ∏ ( ) (11)
That is, for any period , the weight of most recent observation is set as , and
, the weights of observation at (the coefficient of error square ) can be calculated by using the following formula:
∑ ∏ ( ) (12)
In this learning model, when we use past observations until , to make a prediction at period , the weight of observations at period is set as . And the weights of observation at period can be calculated by replacing respectively in (13), and so forth. For example, participants are likely to emphasize the most recent realized observation if the last period’s prediction error is high. This is expressed as a high
forgetting rate at period put on the most recent observation when making a prediction for period . Meanwhile lower weights put on distant observations and the value of weights drop sharply from the most recent observations to distant observations. On the other hand, when the forgetting rate at time period 1 is low, participants do not put a high weight on most recent realized observation. In this case, participants also put more weights on recent observations, less weights on distant observations. But the values of weights decrease less sharply from most recent observations to distant observations.
Figure 2 shows the one-step-ahead prediction using the swOLS model with selected forgetting rate. In negative feedback market, the forgetting rate suddenly reaches a high level at both structural breaks. The difference is that in the swOLS model, the value of the forgetting rates drop quickly after structural breaks, which means the swOLS model with forgetting can adjust its prediction much better than the sOLS model. Two inconsistencies from the sOLS model with forgetting are solved in the swOLS model with forgetting. Model responses to the first structural break quickly. And after the second structural break, predictions are smoothly overlapping collective expectation. However, at the first and second structural break point, prediction seems to over response to the shocks. The model often predicts a very extreme value at first and second structural break.
In positive feedback market, the swOLS model with forgetting predicts a comparatively lower absolute prediction error compared with the same forgetting rates used in sOLS model. And as expected, it does not show any sudden high at structural breaks
3.3. swOLS model with forgetting incorporating parameters penalty (
In terms of goodness-of-fit, a model with more parameters usually outperforms a model with fewer parameters. For example, among three prediction rules, the rule 3 with most parameters is like to fit data better than other two rules. Hence, when prediction models are compared, the Akaike information criterion (AIC) is normally applied to measure the relative quality of models, because it considers the balance between goodness-of-fit and the number of parameters in the models. A simple form of AIC using residual sums of squares can be expressed as follows:
( )
If ratio , AIC is normally biased. And the bias adjustment AICc is then used:
When the number of observations, , increase relative to the number of parameters, , the second part of , , gradually diminishes. In general, do not provide any tests of a model, but they can be used to calculate the relative probability that a rule is of the best quality among three prediction rules.
Starting from period 6 (because 6 is the smallest integer that ensure n-k-1 is positive), after previous period price is realized, of each rule is calculated. Then, the differences between the rules of the lowest are calculated separately as:
is the minimum value of all three rules at period .
The relative likelihood of each rule is defined as | , which is approximately equal to (Akaike ,1978, pp.218-220, and 1979, p.239). And we normalize this approximate relative likelihood for each rule as follows (Burnhamm & Anderson, 2002):
( )
where is interpreted as the probability that rule is the best approximating model at
period , given the data and other prediction rules. Based on evaluation criteria used in the sOLS model or the swOLS model, I propose new evaluation criteria that combine the forgetting rate and parameter penalty:
(( ) ) (13)
The new evaluation criteria at time is calculated by evaluation criteria used in the sOLS model or the swOLS model at time multiply by . The principle of rule selection tells that the smallest the criteria, the more accurate the previous prediction is. At time , the rule with the smallest evaluation criteria is selected as the prediction rule. Hence, one should not interpret as probability that rule is the not that best approximating rule at period . Instead, it can
also be interested as probability in the forgetting criteria that rule can be selected as the best approximating rule at period .
Figure 3 shows the one-step-ahead prediction of the model with selected forgetting rate. The results are pretty much the same as the sOLS model with corresponding forgetting rate. In negative feedback market, quick response to first structural break and consistent explanation after second structural break can be seen on graphs, solving the inconsistent predictions from the sOLS model with forgetting. The forgetting rate sudden reach high level at structural breaks, and sudden drop can be found afterwards. However, over responds can be still found at first and second structural break. Again, positive feedback market shows in no sudden jump at structural breaks.
3.4. The swOLS model with forgetting using constrained forgetting rates.
Both the swOLS model with forgetting and the model with forgetting can explain experimental data well, except the overshooting at first and second structural breaks. When there is an overshooting, the value of the corresponding forgetting rate is always 1. This gives a conjecture that the overshooting may come from the complete forgetting at shocks. The situation can happen that participants do not forget everything (complete forget) even when the prediction
error at last period is high. This idea can be implemented by introducing a new forgetting rate , which is defined as proportional to the original forgetting rate :
(14)
The new introduced model with forgetting is based on the swOLS model with forgetting and the difference is the forgetting rate in evaluation criteria. In the model with forgetting, the evaluation criteria are defined as:
( ) (15)
Where = and ( )
The forgetting rate is initially used to form a one-period ahead prediction. Overshooting at first and second structural break point disappears but undershooting occurs at those points. I try to optimize by eyeball (actually, predictions are pretty close if . And I found that when , the prediction matches the empirical experimental behavior best.
Figure 4 shows the one-step-ahead prediction of the model with a random chosen forgetting rate | |
| | (other forgetting rate perform almost the same). In this model, is
applied in evaluation criteria. In the negative feedback market, at two structural break points, overshooting inconsistency is solved and predictions almost perfectly match the participants’ expectations. At structural break points, forgetting rates show a sudden jump to its limit , and a very quick drop to a low level after a few periods and remain there for the rest of the periods. On the other hands, in the positive feedback market, it performs as good as in other models.
4. MODLE COMPARISION
Rational expectation and the sOLS model with forgetting are compared. In accordant with what used in Wagener’s paper (2014), the mean of the absolute prediction errors is applied to compare the quality of models in each run:
̅ ∑ | ̅ ̅ |
(16)
Different models (sOLS, swOLS, , swOL ) with forgetting are compared with rational expectations hypothesis and the results are shown in Appendix. Every single model and rational expectation hypothesis forms prediction for 16 runs, and the absolute prediction error are ordered collected into two sets according to the type of feedback markets. Paired difference t-test is used to analyze if the mean of any two sets differs in a significant way under the assumption that the paired differences are independent and identically and approximately normally distributed. Those sets of absolute prediction error in positive feedback market and negative feedback market are compared separately, since the price generating function differs. In order to investigate if any model with certain forgetting rate outperform rational expectation. In this test, the null hypothesis of the paired difference t-test is that rational expectation hypothesis outperforms a certain model. This is expressed as the mean of absolute prediction error from rational expectation significantly smaller than a certain model. The alternative is a certain model outperforms rational expectation. This test is a one-tailed t test and 5% critical level is used. The P value of t-test between a certain model and rational expectation hypothesis is shown 8th column of each table in Appendix.
4.1.The sOLS model with forgetting
As one can see from the Appendix table 1, all forgetting rates perform well. If six forgetting rates of the sOLS model are compared in terms of value of the absolute prediction error, one cannot easily tell which one performs best. In the negative feedback market (first 8 runs), no any sOLS model with a forgetting rate beats rational expectation:all forgetting rates predict a higher absolute prediction error than rational expectation and the paired difference t-test tells a p value
of 0 for all the six forgetting rate. Hence, we can conclude that rational expectation outperform sOLS model with all six forgetting rate in negative feedback market. The conclusion reversed in positive feedback market where each forgetting rate much significantly outperforms rational expectations.
4.2.The swOLS model with forgetting
After adding weights on observations, table 2 presents the comparison between the swOLS model with different forgetting rates and rational expectation. In negative feedback market, all swOLS predictions have a lower absolute prediction error than rational expectation except 6th run. Compared with rational expectation, the t-test on the swOLS model with forgetting rate | |
| | ,
| |, , | | individually gives a p-value higher than 5%. Hence, they do not
perform better than rational expectation, even they all have lower mean of absolute prediction error. Furthermore, compared with rational prediction, the forgetting rates | | | | and have a p-value less than 5% critical value. This indicates the swOLS model with those 2 forgetting rates individually outperforms rational expectation in negative feedback market
In positive feedback market, as one may predict, the swOLS model with different forgetting rate individually significantly outperform rational expectations with a p-value close to 0.
4.3. The model with forgetting
According to table 3, one can easily see, compared with rational expectation, no matter what forgetting rate is applied, model normally results in lower absolute mean error, but with very a very few exceptions. | | is the only forgetting rate in the model that has a
p-value less than 5%. Hence, it is the only forgetting rate that beats rational expectation.
In the positive feedback market, unquestionably, all forgetting rates in the model perform significantly better than rational expectations. And the p-value of each forgetting rate is close to 0. Hence, as the only forgetting rate in the model that beat rational expectation in positive and negative feedback market, | | is the best performing forgetting rate in model.
4.4. The model with forgetting
Table 4 shows that all forgetting rate in the model with performs very good. All 6 forgetting rates in the model predict a lower absolute prediction error than rational expectation in each run. And the p-value is extremely small, telling that the model with different forgetting rate all very significantly outperform rational expectation.
In positive feedback market, same conclusion as in other models, all forgetting rates in model outperform rational expectation.
5. ROBUSTNESS Forgetting rate | |
| | in the swOLS model, | | in the , and all forgetting rates in
the model with outperform rational expectations in positive feedback market and negative feedback market. And they can be seen as a proof of the assertion that learning models with forgetting can explain Te Bao et al’s experimental data (2014) better than the rational expectation hypothesis. For robustness test, all outperforming models are applied to Heemeijer et al. (2009) experimental data.
This experience is conducted at CREED laboratory in University of Amsterdam. The setup of this experiment is quite similar to Te Bao et al. experiment (2012). However, there are still some subtle differences. Each run of both experiments consists of 50 periods. And the price generating function are given by (4) and (5) with in both markets. Besides those changes, the main difference is that in Heemeijer et al. (2009) experiment, there is no structural break during the periods. Hence, the rational expectation, which is 60, is constant all over time. In this experience, the price quickly converges to the rational expectation price 60 under negative feedback market, and strong oscillations observed in most of the runs in positive feedback market.
The table on the next page presents the prediction performance from forgetting rate | | | | and
in the swOLS model,
| | in the model, and all forgetting rates in the
model with . The p-values tell none of them beat rational expectations hypothesis in negative feedback market.
One can notice their prediction quality is pretty much the same. It seems there is not much difference if one model incorporates a restriction on maximum forgetting ability or not. However, in the model, if the value of changes, the absolute prediction error can be diminished further. For example, using a random chosen forgetting rate | |
| | if , the
absolute prediction error from the model is about 0.26 (rounded number) in first run, almost the same as rational expectation; If , in 5th
run, absolute prediction error predicted by the model is 0.5944; pretty much close to 0.57; and if , the model can predict a result
with absolute prediction error of 0.43 in the last run. For the rest three runs, although I fail to find an appropriate value of that can make the result almost the same as the result from rational expectation hypothesis. It is still quite close. For example, the model can predict absolute prediction error of 0.72 in 2nd run, 1.47 in 3rd run and 0.28 in 4th run. However, an optimized choice of that works for a certain run sometimes can cause really bad results in other runs.
In positive feedback market, since rational expectation cannot catch the oscillatory expectations behaviors. It performs significantly worse than the other two models.
1 2 3 4 5 6 p-value | | | | 0.41 0.67 1.64 0.27 0.80 0.40 0.98 swOLS 0.42 0.70 1.82 0.30 0.84 0.42 0.98 | | 0.57 0.85 1.76 0.47 1.17 0.69 0.99 | | | | 0.38 0.75 1.65 0.28 0.70 0.43 0.98 0.37 0.69 1.70 0.29 1.02 0.43 0.98 | | | | 0.38 0.67 1.58 0.28 0.98 0.46 0.99 | | 0.40 0.72 1.91 0.31 1.28 0.49 0.98 0.41 0.72 1.92 0.34 1.16 0.51 0.99 | | 0.46 0.75 1.93 0.33 1.09 0.52 0.99 Rational expectation 0.26 0.46 1.11 0.14 0.57 0.40
6. CONCLUSION
This paper analyzed the experimental behavior in negative feedback and positive feedback markets in a structural break experiment and applies the best performing model to Heimeijin et al. (2009) experimental data to test robustness. The core idea in this thesis relies on the assumption that participants have the tendency to forget past prediction performance, only choose the strategy that performs good recently. And this idea is expressed in evaluation criteria where participants use to select the prediction rule for the next period.
In structural break experiment, the comparisons between several models tell a possible conjecture that possible learning models with forgetting can better explain experimental behavior in negative and positive feedback market than rational expectation. However, this is not always true when the same learning model applies to a non-structural break experiment; it failed to beat rational expectation in negative feedback market. Yet, it can still well explain the most important characteristics of the dynamics of the collective expectations. In both set ups, learning models shows an immediate convergence to fundamental price in negative feedback market, and a slow movement towards to equilibrium in positive feedback market. Those learning models have a high probability to beat rational expectation in structural break setup where it is necessary for participants quickly adjust their strategies after changes of fundamental prices
Appendix Table 1:
Below is the result of absolute prediction error in each run using the sOLS model with 6 different forgetting rates for negative feedback market (first 8 runs), and positive feedback market (last 8 runs). The first column shows the forgetting rate used in the sOLS model. And the last column is the P-value from a paired t-test on a corresponding forgetting rate of the sOLS model and rational hypothesis to test if two sets of absolute prediction error have the same mean. The highlighted number in each column represents the smallest absolute prediction error in each run (exclusive of rational expectation). Digits are rounded.
1 2 3 4 5 6 7 8 P_value | | | | 2.19 2.08 2.07 2.68 1.96 2.93 1.86 2.07 0.99 2.02 1.84 1.89 2.53 2.09 2.98 1.68 2.07 0.99 | | | | 2.12 1.70 1.82 2.45 2.23 3.05 1.57 2.07 0.99 | | 2.12 1.75 1.81 2.44 2.23 2.95 1.58 2.07 0.99 2.12 1.52 1.82 2.40 2.23 3.03 1.55 2.08 0.99 | | 2.12 1.52 1.82 2.40 2.23 3.05 1.55 2.07 0.99 Rational expectation 1.30 1.03 0.94 1.26 1.36 2.22 0.95 1.15 9 10 11 12 13 14 15 16 P_value | | | | 0.25 0.60 0.26 0.43 0.41 0.31 0.38 0.46 5.37E-09 0.25 0.59 0.35 0.44 0.44 0.30 0.40 0.46 7.59E-09 | | | | 0.25 0.59 0.36 0.44 0.45 0.30 0.40 0.48 7.84E-09 | | 0.25 0.60 0.27 0.42 0.34 0.31 0.37 0.46 5.04E-09 0.25 0.61 0.31 0.45 0.40 0.30 0.40 0.48 6.01E-09 | | 0.25 0.59 0.36 0.45 0.37 0.30 0.40 0.48 6.95E-09 Rational expectation 5.67 7.02 5.14 5.48 5.47 6.06 5.90 6.32
Table 2:
Below is the result of absolute prediction error in each run using the swOLS model with 6 different forgetting rates for negative feedback market (first 8 runs), and positive feedback market (last 8 runs). The first column shows the forgetting rate used in the swOLS model. And the last column is the P-value from a paired t-test on a corresponding forgetting rate of the sOLS model and rational hypothesis to test if two sets of absolute prediction error have the same mean. The highlighted number in each column represents the smallest absolute prediction error in each run (exclusive of rational expectation). Digits are rounded.
9 10 11 12 13 14 15 16 P value | | | | 0.27 0.47 0.33 0.40 0.51 0.42 0.54 0.46 1.10E-08 0.25 0.55 0.34 0.43 0.62 0.39 0.57 0.67 9.69E-09 | | | | 0.29 0.51 0.37 0.45 0.65 0.40 0.60 0.70 1.21E-08 | | 0.28 0.44 0.35 0.38 0.59 0.36 0.52 0.60 1.31E-08 0.24 0.44 0.48 0.40 0.61 0.40 0.56 0.62 1.83E-08 | | 0.27 0.44 0.58 0.42 0.61 0.38 0.60 0.64 2.35E-08 Rational expectation 5.67 7.02 5.14 5.48 5.47 6.06 5.90 6.32 1 2 3 4 5 6 7 8 P value | | | | 0.84 0.76 0.66 0.92 1.11 2.53 0.67 0.81 0.01 0.89 0.80 0.70 0.95 1.09 2.57 0.69 0.83 0.02 | | | | 0.91 0.82 0.72 0.99 1.10 3.03 0.72 0.85 0.18 | | 0.91 0.82 0.69 0.95 1.19 2.73 0.70 0.81 0.06 1.07 0.76 0.70 1.18 1.19 3.00 0.69 0.82 0.22 | | 1.07 0.77 0.83 1.21 1.19 3.10 0.68 0.83 0.30 Rational expectation 1.30 1.03 0.94 1.26 1.36 2.22 0.95 1.15
Table 3:
Below is the result of absolute prediction error in each run using the model with 6 different forgetting rates for negative feedback market (first 8 runs), and positive feedback market (last 8 runs). The first column shows the forgetting rate used in the model. The last column is the P-value from a paired t-test on a corresponding forgetting rate of the model and rational hypothesis to test if two sets of absolute prediction error have the same mean. The highlighted number in each column represents the smallest absolute prediction error in each run (exclusive of rational expectation). Digits are rounded.
1 2 3 4 5 6 7 8 P value | | | | 0.94 0.79 0.69 1.13 1.34 2.97 0.73 0.87 0.24 1.03 0.84 0.79 1.15 1.42 2.48 0.78 0.96 0.08 | | | | 1.08 0.85 0.76 1.18 1.47 2.55 0.78 0.96 0.16 | | 0.98 0.86 0.72 0.90 1.37 2.33 0.80 0.89 0.01 1.14 0.82 0.72 1.14 1.35 2.95 0.73 0.90 0.31 | | 1.12 0.83 0.92 1.14 1.36 2.62 0.72 0.92 0.17 Rational expectation 1.30 1.03 0.94 1.26 1.36 2.22 0.95 1.15 9 10 11 12 13 14 15 16 P value | | | | 0.32 0.47 0.34 0.37 0.47 0.41 0.57 0.51 1.02E-08 0.28 0.44 0.33 0.38 0.47 0.42 0.58 0.53 1.08E-08 | | | | 0.30 0.43 0.36 0.38 0.49 0.48 0.63 0.76 1.0E-08 | | 0.33 0.46 0.35 0.37 0.55 0.42 0.59 0.58 1.16E-08 0.30 0.45 0.35 0.38 0.61 0.48 0.51 0.72 1.17E-08 | | 0.31 0.45 0.41 0.42 0.50 0.48 0.65 0.63 1.27E-08 Rational expectation 5.67 7.02 5.14 5.48 5.47 6.06 5.90 6.32
Table 4:
Below is the result of absolute prediction error in each run using the model with 6 different forgetting rates for negative feedback market (first 8 runs), and positive feedback market (last 8 runs). The first column shows the forgetting rate used in the model. The last column is the P-value from a paired t-test on a corresponding forgetting rate of the model and rational hypothesis to test if two sets of absolute prediction error have the same mean. The highlighted number in each column represents the smallest absolute prediction error in each run (exclusive of rational expectation). Digits are rounded.
1 2 3 4 5 6 7 8 P value | | | | 0.62 0.36 0.30 0.72 0.75 2.14 0.38 0.44 5.46E-05 0.62 0.42 0.33 0.70 0.76 2.15 0.41 0.46 5.36E-05 | | | | 0.63 0.46 0.35 0.69 0.79 2.09 0.43 0.46 2.55E-05 | | 0.60 0.39 0.31 0.70 0.74 2.21 0.38 0.44 1.14E-04 0.62 0.41 0.34 0.71 0.77 2.12 0.40 0.44 4.19E-05 | | 0.65 0.43 0.36 0.72 0.78 1.79 0.42 0.45 9.47E-08 Rational expectation 1.30 1.03 0.94 1.26 1.36 2.22 0.95 1.15 9 10 11 12 13 14 15 16 P value | | | | 0.29 0.57 0.32 0.39 0.51 0.40 0.56 0.55 7.09E-09 0.30 0.54 0.33 0.54 0.53 0.41 0.60 0.65 9.61E-09 | | | | 0.25 0.62 0.34 0.56 0.66 0.41 0.64 0.72 9.48E-09 | | 0.23 0.51 0.32 0.41 0.49 0.41 0.56 0.47 9.34E-09 0.28 0.50 0.32 0.49 0.51 0.39 0.58 0.66 9.69E-09 | | 0.30 0.58 0.36 0.50 0.5965 0.41 0.65 0.69 9.45E-09 Rational expectation 5.67 7.02 5.14 5.48 5.47 6.06 5.90 6.32
Figure 1:
Left panels show the experimental participants’ average expectation (blue line) and the expectation prediction (red line). First 6 pair graphs show the 1st to 6th run (negative feedback market) of the experiment and the prediction is calculated by using the sOLS model with forgetting rate | |
| | , , | |
| | ,
| |, and | |
respectively. Last 6 pair graphs show the the 9th to 14th run (negative feedback market) of the experiment and the prediction is calculated by using the sOLS model with forgetting rate | |
| | , , | |
| | ,
| |, and
| | respectively. The right panel displays the evolution of the corresponding forgetting rate. And the two red
vertical lines represent structural break points
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Figure 2:
Left panels show the experimental participants’ average expectation (blue line) and the expectation prediction (red line). First 6 pair graphs show the 1st to 6th run (negative feedback market) of the experiment and the prediction is calculated by using the swOLS model with forgetting rate | |
| | , , | |
| | ,
| |, and | |
respectively. Last 6 pair graphs show the the 9th to 14th run (negative feedback market) of the experiment and the prediction is calculated by using the swOLS model with forgetting rate | |
| | , , | |
| | ,
| |, and
| | respectively. The right panel displays the evolution of the corresponding forgetting rate. And the two red
vertical lines represent structural break points.
. Run 1
Run 2
Run 4
Run 5
Run 9
Run 10
Run 12
Run 13
Figure 3:
Left panels show the experimental participants’ average expectation (blue line) and the expectation prediction (red line). First 6 pair graphs show the 1st to 6th run (negative feedback market) of the experiment and the prediction is calculated by using the model with forgetting rate | | | | , , | || | , | |, and | | respectively. Last 6 pair graphs show the the 9th to 14th run (negative feedback market) of the experiment and the prediction is calculated by using the model with forgetting rate | | | | , , | || | , | |, and | | respectively. The right panel displays the evolution of the corresponding forgetting rate. And the two red
vertical lines represent structural break points.
Run 1
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Run 9
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Run 13
Figure 4:
Left panels show the experimental participants’ average expectation (blue line) and the expectation prediction (red line) calculated by the model with a random chosen forgetting rate | |
| | and the value of . The right
panel displays the evolution of the corresponding forgetting rate. And the two red vertical lines represent structural break points.
Run 1
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REFERENCES:
Bao T, Hommes CH, Sonnemans J.Tuinstra J. 2012. Individual expectations, limited rationality and aggregate outcomes. J. ECO.Dyn. Control 36:1101-20
Heemeijer P. Hommes CJ, Sonnemans J, Tuinstra J. 2009. Price stability and volatility in markets with positive and negative expectations feedback: an experimental investigation. J. Eco. Dyn. Control 33:1052-72
Aunfriev M, Hommes CH, Raoul, Philips H.S. 2013. Evolutionary Selection of Expectations in Positive and Negative Feedback markets. Journal of evolutionary economics, 23(3), 663-688
Wagener, F.O.O 2014. Expectations in Experiments. Annu. Rev. Econ. 6:18.1-18.23 Akaike, H. 1978. On the likelihood of a time series model. The statistician,27,217-235. Akaike, H. 1979. A Bayesian extension of the minimum AIC procedure of autoregressive
model fitting. Biometrika,66,237-242.
Burnham,K.P,& Anderson,D.R.(2002). Model selection and ultimodel inference: A partial information-theoreti approach. New YorkL springer-Verlag.
