Title: Fuzzy inductive reasoning, expectation formation and the behavior of security prices Nicholas S.P. Tay , Scott C. Linn
1Fuzzy inductive reasoning, expectation formation
and the behavior of security prices
Nicholas S.P. Tay , Scott C. Linn
 Introduction
 Expectation formation and market created
uncertainty  Models
 Experiments
 Results
 Conclusions

??? ???
2Introduction (1)
 This paper extends the Santa Fe Artificial
Stock Market Model (SFASM) in two important
directions  First, some might question whether it is
reasonable to assume that traders are capable of
handling a large number of rules.  We demonstrate that similar results can
be obtained even after severely limiting the
reasoning process.(by fuzzy reasoning)  Second, the kurtosis of SFASM simulated stock
returns is too small as compared to real data.  We demonstrate that with a minor
modification to how traders go about deciding
which of their prediction rules to rely on , we
can produce return kurtosis that is comparable to
that of actual returns data.
3Introduction (2)
 The actual market environment, not like
Neoclassical financial market models, is usually
illdefined, where the ability to exercise
deductive breaks down.  We conjecture that individual reasoning in an
illdefined setting can be described as an
inductive process in which individuals exhibit
limitations in their ability to process and
condense information.  The objectives of this paper are to first,
model an inductive reasoning process and second,
to investigate its implications for aggregate
market behavior and in particular for the
behavior of security prices.
4Expectation formation and market created
uncertainty
 Market created uncertainty and volatility
 Fuzzy reasoning and induction
5Market created uncertainty and volatility (1)
 No one knows for certain what essential
ingredients are necessary for explaining the
obtuse behavior seen in security prices.  A certain type of market created uncertainty
may be a potential explanation.  We mean the uncertainty created as a result of
the interactions among heterogeneous market
participants who must learn to form their
expectations in a market environment inherently
illdefined.
6Market created uncertainty and volatility (2)
 The environment is illdefined because for a
market participant to logically deduce his
expectations,he needs to know what others are
expecting. But since every other market
participants needs to do also, there is
circularity in the reasoning process. (beauty
contest problem)  The result is that no one will be able to
logically deduce his expectation.  Consequently, each market participant will by
necessity form his price expectations based on a
subjective forecast of the explanations of all
other market participants.  As a result, the market can develop a life of
its own and respond in ways not correlated with
movements in fundamental values.
7Market created uncertainty and volatility (3)
 In Arthurs words, if one believes that others
believe the price will increase, he will revise
his expectations to anticipate upwardmoving
pricesif he believes others believe a reversion
to lower values, he will revise his expectation
downward.  Keynes regarded security prices as the outcome
of the mass psychology of a large number of
ignorant individuals, with professional
speculators mostly trying to outguess the future
moods of irrational traders,and thereby
reinforcing asset price bubbles.  Dreman (1982) maintains that the thinking of the
groupheavily influences the forecast of future
events of individual investor,including
professionals.
8Market created uncertainty and volatility (4)
 These uncertainties create two types of risk for
potential arbitrageurs.  Identification riskIt is difficult for
arbitrageurs to exploit noise traders because
they can never be sure that any observed price
movement was driven by noise, which creates
profit opportunities, or by news that the market
knows but they have not yet learned.  Noise trader risk(future resale price risk)The
price may systematically move away from its
fundamental value because of noise trader
activity.So, an investor who knows, even with
certainty, that an asset is overvalued will still
take only a limited short position because noise
traders may push prices even further away from
their fundamental values.
9Market created uncertainty and volatility (5)
 Another type of risk, fundamental risk (inherent
in the market), although not due to the market
uncertainty,can also limit arbitrage, because
even if noise trader do not move away from
fundamental values, changes in the fundamentals
might move the price against the investor.  Altogether, these problems limit arbitrage
activity,and in turn impair the markets natural
tendency to return itself to a focus on
fundamentals.  Arbitrage plays an errorcorrection role in
the market, bringing asset prices into alignment
with their fundamental values.
10The implication of this discussion
 In rational expectations, by fiat we eliminate
the market created uncertainty outlined above.  We allow the agents in our model the opportunity
to form their expectations based on individual
subjective evaluations, thus restoring the
potential for market created uncertainty to
emerge.
11Expectations formation
Fuzzy reasoning and induction
Fuzzy reasoning
 Market participants typically have very little
time to decipher the immense amount of
information from the stock market.  Some psychologists have argued that our ability
to process information efficiently is the outcome
of applying fuzzy logic as part of our thought
process.  This would entail the individual compressing
information into a few fuzzy notions that in turn
are more efficiently handled,and then reasoning
as if by the application of fuzzy logic.  Smithson (1987) also finds the evidence that some
categories of human thought are fuzzy,and the
mathematical operations of fuzzy sets as
prescribed by fuzzy set theory a realistic
description of how humans manipulate fuzzy
concepts.
12Induction (1)
 In real life, we often have to draw conclusions
based on incomplete information .In this
instances, logical deduction fails because the
information we have in hand leaves gaps in our
reasoning.  Induction is a mean for finding the best
available answers to questions that transcend the
information at hand.  Nonetheless, induction should not be taken as
mere guesswork, and we must find the answers both
sensible and defensible.
13Induction (2)
 Inductive reasoning follows a twostep process
 1.Possibility elaborationCreating a spectrum
of plausible alternatives based on our experience
and the information available.  2. Possibility reductionThese alternatives
are tested to see how well they connect the
existing incomplete premises to explain the data
observed.  The alternative offering the best fit connection
is then accepted as a viable explanation.
14How can induction be implemented in a security
market model ?
 Each individual in the market continually creates
a multitude of market hypotheses.(step.1)  These hypotheses, which represent the
individuals subjective expectational models of
what moves the market price and dividend, are
then simultaneously tested for their predictive
ability.  The hypotheses identified as reliable will be
retained and acted upon in buying and selling
decisionsunreliable ones dropped (step.2), and
ultimately replaced with new ones. ( Then
repeatedly learning and adapting)
15Modeling the expectations formation process
 The expectations formation process described
above can be modeled by letting each individual
form his expectations using his own personal
geneticfuzzy classifier system.  Each geneticfuzzy classifier system contains a
set of conditional forecast rules (subjective
market hypotheses) that guide decision making.  Inside the CS is a GA responsible for generating
new rules, testing all existing rules, and
weeding out bad rules.
16Models
 The market environment
 The sequence of events
 Modeling the formation of expectations
 Components of the reasoning model
 The specification of forecasting rules
 Coding market conditions
 Identifying expectation model parameters
 Fuzzy rule bases as market hypotheses
 An example
 Forecasting accuracy and fitness values
 Recapitulation
17The market environment ?
 The basic framework is similar to SFASM.
 Two tradable assets in the market
 A stock that pays an uncertain dividend dt
,and there are N units of the risky stock.  A riskfree bond that pays a constant r and is
in infinite supply.  The dividend is driven by an exogenous
stochastic process as following, which no agent
knows
AR (1)  et iid Gaussian (0,se2)
18The market environment ?
 N heterogeneous agents CARA utility function
U(W) exp(?W)  Agents are heterogeneous in terms of their
individual expectations.  Each agent is initially endowed with one share
per agent.  At each date, upon observing the information,
agents decide what their desired holdings of each
of the two assets should be by maximizing
subjective expected utility of next period
wealth.  ( Solving a sequence of single period
problem over an infinite horizon. Myopic)
19The market environment ?
 The maximizing problem would be
 Assuming that agent is prediction at time t
of the next periods price and dividend are
normally distributed with mean
and variance then agent is demand
for shares of the risky assets is given by  Since total demand must equal the total number of
shares issued,for the market to clear, we must
have
20The sequence of events
 The current dividend dt is announced at the start
of time period t.  Agents then form their expectations
based on all current information about
the state of the market ( including the
historical
)  Once their price expectations are established,
agents use Eq.(2) to calculate their desired
holdings of the two assets.  The market then clears at pt.( by market clearing
condition)the sequence of events is then
repeated and pt1determined.  At each step, agents learn about the
effectiveness of the geneticfuzzy classifier
they relied on, and the unreliable classifiers
are weeded out to make room for classifiers with
new and perhaps better rules.
21Components of the reasoning model
Modeling the formation of expectations
 We have argued that individuals apply fuzzy
logic and induction to the formation of
expectations, so we characterize individual
reasoning as the product of the application of a
geneticfuzzy CS, which based in part on the
design of the CS originally developed by Holland
(1986) .  Three essential components of CSA set of
conditional action rules, a credit allocation
system for assessing the predictive capability of
any rule and a GA by which rules evolve.  Our geneticfuzzy CS replace the conventional
rules in Hollands classifier with fuzzy rules.  These fuzzy rules also involve a
conditionaction format but they differ form
conventional rules in that fuzzy terms rather
than precise terms now describe the conditions
and actions.
22The specification of forecasting rules (1)
 The forecast equation hypothesis is
 (
Proof )  Where a b are the forecast parameters obtained
from the activated rule.  As SFASM, the linear forecasting model shown
above is optimal when,
 a) agents believe that prices are a linear
function of dividends.  b) a homogeneous rational expectations
equilibrium obtains.  While we place no such restrictions on the
system, a linear forecasting model serves as an
approximation for the structure likely to evolve
over time.
23A proof of this assertion (1)
 Recall that,dividend process is
 , and demand for the security
 Assuming that agents conjecture that price is a
linear fn. of the dividend ,that is
.  Then,

..(a)
24A proof of this assertion (2)
 Since all the agents are equally risk averse,
each agent must hold the same number of shares.We
substitute the the oneperiod ahead forecast into
demand fn.,then we obtain 
..(b)  The LHS is a constant, so the RHS cannot exhibit
any dependence on time.Therefore, dt must
vanish.This lead to  .Substituting into (b)
 Substituting f e into (a),we obtain
 Compare this to
25The specification of forecasting rules (2)
 We use five information bits to specify the
conditions in a rule, and two bits to represent
the forecast parameter a b.(Totally, seven
bits.)  The format of a rule when fuzzy rules prevail,
would therefore be If specific conditions are
satisfied then the values of the forecast
equation parameters are defined in a relative
sense.One example is If price/fundamental
value is low, then a is low and b is high.
26Coding market conditions (1)
 The five information bits used to specify the
conditions represent five market descriptors.  The five market descriptors are pr/d, p/MA(5),
p/MA(10), p/MA(100), and p/MA(500), where MA(n)
denotes an nperiod moving average of prices.  The first bit reflects the current price in
relation to the current dividend and it indicates
whether the stock is above or below the
fundamental value at the current timebits 25,
are technical bits which indicate whether the
price history exhibits a trend or similar
characteristic.
27Coding market conditions (2)
 Market descriptors are transformed into fuzzy
information sets by first defining a range of
possibilities for each information bit.  And second, by specifying the number and types of
fuzzy sets to use for each of these market
variables.
 First,we set the range for each of these
variables to 0,1.  Second, we assume that each descriptors has the
possibility of falling into four alternative
states low, moderately low, moderately
high and high.
28Coding market conditions (3)
 We let the possible states of each market
descriptor be represented by a set of four
membership functions associated with a specific
shape. 
 We represent fuzzy information with the codes
1,2,3,4 for low, moderately low, moderately
high,and high.A 0 is used to record the absence
of a fuzzy set.(like before)  ExampleIf the condition part of the rule is
coded as 01302,this correspond to a state in
which the market price is less than MA(5) but
some what greater than MA(10) and is slightly
less than MA(500).
low
Moderately high
Moderately low
high
Fig1.Fuzzy sets of the market descriptors
29Identifying expectation model parameters (1)
 Now we turn to the modeling of the forecast part
of the rule.  We allow the possible states of each forecast
parameter to be represented by four fuzzy sets
which labeled respectively (with the shapes
indicated), low, moderately low, moderately
high, and high.  The universe of discourse for a b are set to
0.7,1.2 10,19, respectively.The shapes
and locations
30Identifying expectation model parameters (2)
 An example of the forecast part of the rule is
the string 2 4,which would indicate that
the forecast parameter a is moderately low and
b is high.  In general,we can write a rule asx1,x2,x3,x4,x5
y1,y2, where x1,x2,x3,x4,x5? 0,1,2,3,4and
y1,y2 ? 1,2,3,4. (for example,1 0 3 0 2 2 4
)  We would interpret the rule x1,x2,x3,x4,x5y1,y2
as  If pr/d is x1 ? p/MA(5) is x2 ? p/MA(10) is
x3 ? p/MA(100) is x4 ?and p/MA(500) is x5 ,  then a is y1 and b is y2 ...
31Fuzzy rule bases as market hypotheses (1)
 A geneticfuzzy classifier contains a set of
fuzzy rules that jointly determine what the price
expectation should be for a given market state.  Each rule base (a set of rules) represents a
tentative hypothesis about the market and
reflects a completebelief.  The rule If pr/d is high, then a is
low and b is high itself does not make much
sense as an hypothesis.Three addition rules
(specifying what ab should be for the case pr/d
is low,moderately low and moderately high)are
required to form a complete set of beliefs.  For this reason, each rule base contains four
fuzzy rules.
32Fuzzy rule bases as market hypotheses (2)
 Fig.4 shows an example of a rule base.
 In order to keep the model manageable yet
maintain the spirit of competing rule bases, we
allow each agent to work in parallel with five
rule bases.  Hence, each agent may derive several different
price expectations at any given time,and a agent
will acts on the one that has recently proven to
be the most accurate. (A modification will be
specified in the following page)
33Fuzzy rule bases as market hypotheses (3)
 In a separate experiment, we allow agents to
sometimes select the rule base to use in a
probabilistic manner. (Triggered by a random
event)  Specific event the polarization of
negative attitudes (group polarization phenomena)  We think of these negative attitudes as
doubts(with a very low probability to occur)
about whether the perceived best rule base is
actually best, and agent tie the probability of
selecting a rule to its relative forecast
accuracy.  This modification is a key extension of the
SFASM.It produces return kurtosis measures more
in line with observed data than that of
SFASM,while simultaneously generating return and
volume behavior that are otherwise similar to
actual data.
trigger
34An example (1)
 Consider a simple fuzzy rule base with the
following four rules.  If 0.5 p/MA(5) is low then a is moderately
high and b is moderately high.  If 0.5 p/MA(5) is moderately low then a is low
and b is high.  If 0.5 p/MA(5) is high then a is moderately low
and b is moderately low.  If 0.5 p/MA(5) is moderately high then a is
high and b is low.  Suppose that the current states is given by
p100,d10,and MA(5)100. 0.5 p/MA(5)0.5  Since 0.5 is outside 1st 3rd rules domain,
the forecast parameters will also have zero
membership value.
35An example (2)
 Only the 2nd and 4th rules contribute to the
resultant fuzzy sets for a b.  We employ the centroid method to translate the
resultant information into specific values for a
b.  We obtain 0.95 and 4.5 for a b.(Fig.6,8,9,10)
 Substituting these forecast parameter into the
linear forecasting model, give us the forecast
for the next period price and dividend  E(pd)0.95(10010)4.5109
36Fig.6
37Fig.8
38Fig.910
39Forecasting accuracy and fitness values (1)
 We measure that accuracy for a rule base by the
inverse of e2t,i,j.  Define
 The variable e2t,i,j is used for several purpose
 In each period the agents refer to e2t,i,j
when deciding which rule base to rely on.  It is used by agent i as a proxy for the
forecast variance s2t,i,j .  It is used to compute what we will label a
fitness measure.  (The fitness measure is used to guide the
selection of rule bases for crossover and
mutation in the GA.)
40Forecasting accuracy and fitness values (2)
 Agents revise their rule bases by GA on average
every k periods.  We specify the GA as following
 Selectionguided by fitness measure.
 Mutationby mutating the values in the rule
base array.  Crossovercombining part if one rule base
array with the complementary part of another.  In general, rule bases that fit the data well,
will more likely to produce whereas less fit
bases will have a higher probability of being
eliminated.
41Forecasting accuracy and fitness values (3)
 The fitness measure of a rule base is calculated
as follows  It imposes higher costs on rules leading
to larger squared forecast errors and employing
greater specificity.  ß is introduced to penalize specificity.The
purpose is to discourage agents from carrying
redundant bits because of limiting ability to
store and process information.  The net effect of this is to insure that a bit is
used only if agents genuinely find it useful in
predictions and in doing so introduces a weak
drift toward configurations containing only
zeros.
ßis a constant(a cost per unit) S is the
specificity of the rule base(eg.fig.4)
42Recapitulation (1)
 dt announced at start of time period t.
 Based on all current information
,  the five market descriptor pr/d, p/MA(5),
p/MA(10), p/MA(100), and p/MA(500)are computed
and the forecast models parameters identified.  Agents form Ei,tpt1, dt1by using ab from the
rule base most accurate.( Except pt
undeterminedEi,tpt1,dt1a(ptdt)b) )  On the other side,
 In their desired share holdings eq., only pt
is undetermined.  The market cleaning condition
determine pt.  ,and by substituting pt into e2t,i,j,we get
the accuracy of the rule base by the inverse of
e2t,i,j..
43Recapitulation (2)
 Learning in the model happens at two different
levels.  On the surface, learning happens rapidly as
agents experiment with different rule bases and
over time discover which rule bases are accurate
and worth acting upon and which should be
ignored.  At a deeper level, learning occurs at a slower
pace as the GA discards unreliable rule bases to
make room for new ones.The new, untested rule
base will not cause disruptions because they will
be acted upon only if they prove to be accurate.
44Experiments
 In these experiments, the primary control
parameter is the learning frequency constant k.  Recall that agents use inductive reasoning which
basically amounts to formulating tentative
hypotheses (rule base), and testing these
hypotheses repeatedly against observed data.  Under such a scheme, the learning frequency will
play a key role in determining the structure of
the rule bases and how well agents are able to
coordinate their price expectations.(The
reasonnext page)
45The reason why k plays a key role
 When learning frequency is high, agents revise
their belief frequently.Then,  They will not have adequate time to fully
explore whether their market hypotheses are
consistent with those belonging to other agents.  Their hypotheses are more likely to be
influenced by transient behavior of market
variables.  Difficult to converge on an equilibrium price
expectation.  In contrast, when learning frequency is low,
agents are more likely to converge on an
equilibrium price expectation.
46The models parameters
We conduct three sets of experiments
Table1common parameters
In the 1st and 2nd experiments, k is equal to
2001000, respectively. In both experiments,
agents form their forecasts using the most
accurate base. In the 3rd experiment, k is equal
to 200.We assume that there is a 0.1 probability
that an agent will decide to select the rule base
to act upon in a probabilistic fashion. ( Then,
all the remaining agents follow him.)
47Other features of the experiment
 In 3rd experiment,when a state of doubt arises,
the probability that agent i will select his rule
base j is then linked to the relative forecast
accuracy of the rule base by ranking the five
rule bases from 0 to 2, in increments of 0.5, and
compute the selection probability for each rule
base as 
Unique for each agent  We follow the LeBaron (1999) in refer to the two
cases (k2001000)as fast learning and slow
learning.(asynchronous)  We began with a random initial configuration of
rulesthen we simulate for 100,000 period to
allow any asymptotic behavior to
emerge.Subsequently, starting with the
configuration attained at t100,000 we simulated
an additional 10,000 periods to generate data for
the statistical analysis.We repeated the
simulation 10 times to facilitate the analysis of
regularities.
48Results
 Simulation results form our experiments show that
the model is able to generate behaviors similar
to many of the regularities observed in real
financial markets.  Asset price and returns
 Trading volume
 Market efficiency
49Asset price and returns (1)
 Fig 1113 present snapshots of observed price
behavior.
50Asset price and returns (2)

 In these three graphs, the market price is
consistently below the REE price.
51Discussion for the asset price and returns (1)
In our model, agents are not able to coordinate
their forecast perfectly.
This causes the market price more volatile than
the REE price.
REE price solution
where s2pd , e ,p
The market price is below the REE price.
The fact that difference between the market price
and REE price is larger for the two fast learning
cases is a result of high price
variability.(Table2)
52Table 2
of REE 5.4409
of REE2
Consistent with daily stock return (Table.3)
of REE0
53Discussion for the asset price and returns (2)
 The larger volatility in the fast learning
learning case can be attributed to the more
frequent revision of rules  The valueet1pt1 dt1ab(pt dt) N(0,4)
 ?s2pd 4 (for the parameter value listed in
Table.1)
Based on short horizon features of market
variable
Agents find it necessary to employ different rule
bases to form their expectation at different time
In turn, it gives rise to higher volatility
because they need time to adapt to the changes.
iid
54Table3
 Summary statistics for the daily withdividend
returns of three common stockDisney,Exxon, and
IBM.
Including 1987 market crash
24.7239
Excluding 1987 market crash
55Why we introduce a state of doubt to catch the
actual figure of kurtosis?
 Although during the first few hundred of time
steps, kurtosis is always rather large ( because
of initialized randomly and trying to figure out
how to coordinate), once agents have identified
rule bases that seem to work well, excess
kurtosis decrease rapidly.  From that point on, it is extremely difficult to
generate further excess kurtosis without
exogenous perturbation, because it is difficult
to break the coordination among agents.  We suspect the large kurtosis observed in actual
returns series may have originated from such
exogenous events as rumors or earnings surprises.  It is in this spirit that we introduced what we
earlier referred to as a state of doubt.
56Other characteristics of asset prices and returns
 The first two experiments show that there is
little autocorrelation in the residuals, which
corresponds to the low actual autocorrelations
shown in table3.  Because that actual security returns exhibit
conditional timevarying variability, we test for
ARCH dependence in the residual.( table2)  In Row 7,we present the first order
autocorrelation of the squared residuals.  In Row 8, we perform the ARCH LM test.
 Both reveal that there is ARCH dependence,
however, the effect is more pronounced for the
two fast learning cases.  The standard deviation, skewness and kurtosis
statistics for the simulated returns from
experiment3 are very similar to those shown for
the stocks in Table3.
57Trading volumeFig.1415
 Trading is active, consistent with real market.
58Trading volumeTable 4
 Experiment 3,can exceed 50 of the total number
of shares available in the market.And in both
experiments 1 2 the volume of trade can be as
high as 33.( No upper bound on traded shares, no
short sale restriction)
59Trading volumeFig.1619
One standard deviation away
Similar to
60Crosscorrelation between volume traded and
volatility Fig.2021
 We used the squared returns as a proxy for
volatility.  The volume is contemporaneously correlated with
volatility in the simulation similar to those for
actual security and those found by LeBaron (1999).
3
61Market efficiency (1)
 Fig.22 plots a snapshot of the difference between
the REE price and the simulated price.  If the simulated price tracks the REE price after
adjusting for volatility, we should expect a
constant difference between them.  But in Fig.22 the differences are not constant
across time, implying that the expectations held
by the agents are not always consistent with a
rational market equilibrium.
The simulated price is highly correlated with
the REE price. However,there are period of
sporadic wild fluctuations during which this
relation is broken.
Experiment 2
Experiment 3
62Market efficiency (2)
 We conclude from these results that the market
moves into and out of various states of
efficiency.However, the simulated market prices
have some tendency to return to a constant
distance from the REE price after departures
occur.  In real financial markets, prices appear to be
set in an efficient fashion.However, there have
been occasions during which prices depart and
appear to exhibit a behavior unrelated to
fundamentals. (bubbles or crashesa example of
DJIA on 1997/10/27)  The movement into and out of efficient price
period entirely consistent with a world in which
the type of fuzzy reasoning and induction modeled
in this paper prevail.
63Conclusions (1)
 Our model can account simultaneously for several
regularities observed in real market, that have
been a struggle to rationalize within the
traditional RE paradigm.  1stOur model can give rise to active
trading.  2ndOur model supports the views of both
academicians and market traders.(rational
efficient psychological imperfectly
efficient.) Our model find that the market moves
in and out of various states of efficiency.  3rdWe find that when learning occurs
slowly,the market can approach the efficiency of
a REE. ( Fig.11)  4thDescriptive statistics for returns as
well as the resulting behavior of volume
autocorrelations and volume and volatility
crosscorrelations are shown to be consistent
with the results for actual data.And a
modification to allow for the intrusion of a
state of doubt is shown to produce return
kurtosis measures more in line with actual data.
64Conclusions (2)
 Our work is very much like the work of LeBaron
(1999) except on important difference with
respect to agents reasoning process.  While LeBaron also model reasoning as an
induction process, their model assigns to each
agent a total of 100 rules.  We have argued that the ability of agents to
process extensive amounts of data is limited, and
that existing evidence from other social sciences
suggests that individual reason as if by the
precepts of fuzzy logic.  We assume that agents work with only a handful of
rule bases,but these have special characteristics
that they are fuzzy rules.And we accomplish the
same objectives as LeBaron.  The fact that a model based upon a fuzzy
system generates market behavior similar to a
model based on crisp but numerous rules is
appealing because fuzzy decision making has
appeal as a reasonable attribute for individuals.
65Discussion
 Significance of Inductive Reasoning
 GeneticFuzzy Classifier Systems
 Population Size
 Role of K (Fast and Slow Learning)
 Doubt
 Time Complexity
 Connections to LeBaron et al. (1999)
 General Issues
66GeneticFuzzy Classifier Systems
 Uncertainty
 Psychological Foundation of Fuzzy Reasoning
 Why Linear Regression?
67Doubt
 It is so difficult to generate large kurtosis
(without relying on an exogenous perturbation) is
because it is difficult to break the coordination
among the agents in the model once they have
established some form of mutual understanding.
(p. 349)  Do you agree with the argument above?
68Role of Time in Learning Schemes
 Different learning schemes involves different
degree of time involvement. Some are very fast,
while some may be very time consuming.  Nonetheless, little has be said on how the time
pressure can influence the learning dynamics, in
particular, the evolution of learning schemes
itself.
69Connections to LeBaron et al. (1999)
 Points of Similarity
 Points of Difference
70Points of Similarity between LeBaron et al.
(1999) and Tay (2001)
 Consistent Deviation from the HREE Price (Figures
1113)  Excess Volatility (Figures 1113, Table 2)
 Excess Kurtosis (Table 2)
 Linear Dependence (Table 2)
 Nonlinear Dependence (Table 2)
 ARCH Effect
 Volume Persistence (Figures 1618)
 VolatilityVolume Relation (Figures 2021)
71Points of Difference between LeBaron et al.
(1999) and Tay (2001)
 No short sale restriction (footnote 28, p.350)
 Crashes and Bubbles (p.348)
72 Why is the market price consistently below the
REE price? (Figures 11, 12, and 13)  Do we observe the same phenomenon in LeBaron,
Arthur and Palmer (1999)?  Answer Increasing Price Variability (A higher
subjective perceived risk) (p.344)
73Fundamentals
 What may cause the market fail to return to the
fundamental? Is fundamental independent of the
socalled market created uncertainty?
74Matching Artificial Data with Real Data
 What is the technical issue when we try to
compare the statistical features of artificial
data with those of real data, say daily returns
of Disney, Exxon and IBM?