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Fuzzy inductive reasoning, expectation formation and the behavior of security prices Nicholas S.P. Tay , Scott C. Linn


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Title: Fuzzy inductive reasoning, expectation formation and the behavior of security prices Nicholas S.P. Tay , Scott C. Linn

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

  • ??? ???

Introduction (1)
  • This paper extends the Santa Fe Artificial
    Stock Market Model (SFASM) in two important
  • 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.

Introduction (2)
  • The actual market environment, not like
    Neoclassical financial market models, is usually
    ill-defined, where the ability to exercise
    deductive breaks down.
  • We conjecture that individual reasoning in an
    ill-defined 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.

Expectation formation and market created
  • Market created uncertainty and volatility
  • Fuzzy reasoning and induction

Market 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

Market created uncertainty and volatility (2)
  • The environment is ill-defined 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.

Market 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 upward-moving
    pricesif he believes others believe a reversion
    to lower values, he will revise his expectation
  • 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

Market created uncertainty and volatility (4)
  • These uncertainties create two types of risk for
    potential arbitrageurs.
  • Identification risk-It 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.

Market 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
  • Arbitrage plays an error-correction role in
    the market, bringing asset prices into alignment
    with their fundamental values.

The 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

Expectations 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
  • 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

Induction (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
  • 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.

Induction (2)
  • Inductive reasoning follows a two-step process
  • 1.Possibility elaboration-Creating a spectrum
    of plausible alternatives based on our experience
    and the information available.
  • 2. Possibility reduction-These alternatives
    are tested to see how well they connect the
    existing incomplete premises to explain the data
  • The alternative offering the best fit connection
    is then accepted as a viable explanation.

How 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
  • 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)

Modeling the expectations formation process
  • The expectations formation process described
    above can be modeled by letting each individual
    form his expectations using his own personal
    genetic-fuzzy classifier system.
  • Each genetic-fuzzy 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.

  • 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

The 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 risk-free 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
    AR (1)
  • et iid Gaussian (0,se2)

The 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
  • ( Solving a sequence of single period
    problem over an infinite horizon. Myopic)

The 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

The 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
  • 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 genetic-fuzzy classifier
    they relied on, and the unreliable classifiers
    are weeded out to make room for classifiers with
    new and perhaps better rules.

Components 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
    genetic-fuzzy 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 genetic-fuzzy CS replace the conventional
    rules in Hollands classifier with fuzzy rules.
  • These fuzzy rules also involve a
    condition-action format but they differ form
    conventional rules in that fuzzy terms rather
    than precise terms now describe the conditions
    and actions.

The 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.

A 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)

A 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 one-period 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

The 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
  • 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.

Coding 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 n-period 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 2-5,
    are technical bits which indicate whether the
    price history exhibits a trend or similar

Coding 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
  • 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.

Coding 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
  • 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).

Moderately high
Moderately low
Fig1.Fuzzy sets of the market descriptors
Identifying 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

Identifying 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
  • 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 ...

Fuzzy rule bases as market hypotheses (1)
  • A genetic-fuzzy 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.

Fuzzy 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)

Fuzzy 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
  • 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
  • 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.

An 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.

An 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
  • 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

Forecasting 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.)

Forecasting 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

Forecasting 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

ßis a constant(a cost per unit) S is the
specificity of the rule base(eg.fig.4)
Recapitulation (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

Recapitulation (2)
  • Learning in the model happens at two different
  • -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
  • -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.

  • 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)

The 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
  • Difficult to converge on an equilibrium price
  • In contrast, when learning frequency is low,
    agents are more likely to converge on an
    equilibrium price expectation.

The 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.)
Other 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
  • 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

  • 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

Asset price and returns (1)
  • Fig 11-13 present snapshots of observed price

Asset price and returns (2)
  • In these three graphs, the market price is
    consistently below the REE price.

Discussion 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
Table 2
of REE 5.4409
of REE2
Consistent with daily stock return (Table.3)
of REE0
Discussion 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 dt1-ab(pt dt) N(0,4)
  • ?s2pd 4 (for the parameter value listed in

Based on short horizon features of market
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.
  • Summary statistics for the daily with-dividend
    returns of three common stockDisney,Exxon, and

Including 1987 market crash
Excluding 1987 market crash
Why 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.

Other 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 time-varying 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.

Trading volumeFig.14-15
  • Trading is active, consistent with real market.

Trading 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)

Trading volumeFig.16-19

One standard deviation away
Similar to
Cross-correlation between volume traded and
volatility Fig.20-21
  • We used the squared returns as a proxy for
  • The volume is contemporaneously correlated with
    volatility in the simulation similar to those for
    actual security and those found by LeBaron (1999).

Market 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
Market 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
  • 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.

Conclusions (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.
  • 1st-Our model can give rise to active
  • 2nd-Our 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.
  • 3rd-We find that when learning occurs
    slowly,the market can approach the efficiency of
    a REE. ( Fig.11)
  • 4th-Descriptive statistics for returns as
    well as the resulting behavior of volume
    autocorrelations and volume and volatility
    cross-correlations 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.

Conclusions (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.

  • Significance of Inductive Reasoning
  • Genetic-Fuzzy Classifier Systems
  • Population Size
  • Role of K (Fast and Slow Learning)
  • Doubt
  • Time Complexity
  • Connections to LeBaron et al. (1999)
  • General Issues

Genetic-Fuzzy Classifier Systems
  • Uncertainty
  • Psychological Foundation of Fuzzy Reasoning
  • Why Linear Regression?

  • 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?

Role 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

Connections to LeBaron et al. (1999)
  • Points of Similarity
  • Points of Difference

Points of Similarity between LeBaron et al.
(1999) and Tay (2001)
  • Consistent Deviation from the HREE Price (Figures
  • Excess Volatility (Figures 11-13, Table 2)
  • Excess Kurtosis (Table 2)
  • Linear Dependence (Table 2)
  • Nonlinear Dependence (Table 2)
  • ARCH Effect
  • Volume Persistence (Figures 16-18)
  • Volatility-Volume Relation (Figures 20-21)

Points of Difference between LeBaron et al.
(1999) and Tay (2001)
  • No short sale restriction (footnote 28, p.350)
  • Crashes and Bubbles (p.348)

  • 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)

  • What may cause the market fail to return to the
    fundamental? Is fundamental independent of the
    so-called market created uncertainty?

Matching 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?
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