Detailed Notes on the
Santa Fe Artificial Stock Market (ASM) Model

Last Updated: 6 March 2012

Site Maintained By:

Leigh Tesfatsion

tesfatsi AT iastate.edu

Econ 308 Web Site:

http://www.econ.iastate.edu/classes/econ308/tesfatsion/

BACKGROUND REFERENCES:

• * Blake LeBaron, "Building the Santa Fe Artificial Stock Market" (pdf,126K), Working Paper, Brandeis University, June 2002.
  Abstract: This brief summary provides an insider's look at the construction of the Santa Fe Artificial Stock Market (ASM) model. The perspective considers the many design questions that went into building the model from the perspective of a decade of experience with agent-based financial markets. The model is assessed based on its overall strengths and weaknesses.

  W. Brian Arthur, John H. Holland, Blake LeBaron, Richard Palmer, and Paul Tayler, "Asset Pricing under Endogenous Expectations in an Artificial Stock Market" (pdf,120K), December 1996 preprint. Final version published as pages 15-44 in The Economy as an Evolving Complex System, II, edited by W. Brian Arthur, Steven Durlauf, and David Lane, Vol. XXVII, SFI Studies in the Sciences of Complexity, Addison-Wesley, 1997.

  Paul Johnson, "What I Learned from the Artificial Stock Market" (pdf,107K), Working Paper, Department of Political Science, University of Kansas, November 5, 2001.

  Abstract:This essay describes some of the changes that were incorporated in the ASM-2.2 revision of the code for the Santa Fe ASM. It also presents some important lessons for agent-based modelers that can be illustrated with the code.

  Blake LeBaron, "A Builder's Guide to Agent-Based Financial Markets" (pdf,207K), preprint of article published in Quantitative Finance 1 (2001), 254-261.

  Blake LeBaron, W. Brian Arthur, and Richard Palmer, "Time Series Properties of an Artificial Stock Market Model" (pdf,324K), Journal of Economic Dynamics and Control 23 (1999), 1487-1516.

  This paper is a rigorous technical discussion of the Santa Fe ASM model, including implementation details. Anyone interested in the actual implementation of this model should consult this paper in addition to the Arthur et al. (1997) paper cited above.

  Leigh Tesfatsion, "Overview of the Santa Fe Artificial Stock Market Model" (pdf,99K)

  Leigh Tesfatsion, resource site on Agent-Based Computational Finance

TABLE OF CONTENTS:

• Basic Concepts
• Basic Objectives of the Santa Fe ASM Model
• The Santa Fe ASM Model
  • Basic Model Features
  • Time-Line of Activities
  • GA Condition/Forecast Classifier Learning
  • Experimental Design and Findings

Basic Concepts

• Time Series Data (LeBaron 2001, p. 255):
  A successive recording over time of the values taken on by one or more specified variables. For example, a time-dated sequence (Q(1),Q(2),...), where the generic sequence element Q(t) gives the value of U.S. gross domestic product in period t.

  Trader Strategy (LeBaron 2001, p. 255):

  A contingency plan that identifies an intended action for each possible situation a trader can foresee.

  Trader Budget Constraint (LeBaron 2001, p. 255):

  A restriction on the actions of the trader that constrain the market value of his/her current market expenditures to be no greater than the market value of his/her current wealth.

  Zero-Intelligence Trader (LeBaron 2001, p. 255):

  Reference to a type of trader that randomly makes price bids (offers to buy) and/or price asks (offers to sell) subject only to a budget constraint preventing negative profits for sure. The term was introduced by Gode and Sunder (JPE, 1993) in a computational study of continuous double auctions.

  Exogenous vs. Endogenous (LeBaron 2001, p. 255)

  An aspect of a model is said to be exogenously determined if it has been determined outside of the model, and it is said to be endogenously determined if it is determined within the model as part of the model solution (outcome). For example, consider the model consisting of the two equations x + y = 1 and x - 2y = 0. The values 1 and 2 are determined outside the model, hence 1 and 2 are exogenous to the model. The values for x and y are determined within the model (by the model equations) to be x = 2/3 and y = 1/3, hence x and y are endogenous variables whose values 2/3 and 1/3 are endogenously determined.

  Endogenous Heterogeneity in ACE Models (LeBaron 2001, p. 255):

  Differences in agent behaviors and strategies that arise and evolve over time during the course of some ACE-modelled economic process, starting from exogenously specified initial conditions.

  Market Depth or Liquidity (LeBaron 2001, p. 256):

  The amount that the price of a financial asset moves in response to an imbalance between its demand and supply.

  Temporary Equilibrium (LeBaron 2001, p. 256):

  Essentially, a situation in which all spot markets are in balance (demand equals supply) even if forward markets are not. Spot markets are markets in which agreements for immediate delivery of goods/services are reached in exchange for immediate payments. Forward markets are markets in which agreements are reached today for payment and/or delivery of goods/services at a later time.

  Traditional Walrasian Economy (LeBaron 2001, p. 257):

  A basic model of an economy consisting of price-taking utility-maximizing consumers, price-taking profit-maximizing firms owned by the consumers, and an (implicit) "Walrasian Auctioneer" who adjusts prices until all markets clear. More precisely, the Auctioneer adjusts prices until supply is at least as great as demand in each market and the total value of excess supply is zero.

  Securities (LeBaron 2001, p. 257):

  Financial assets (e.g., stock shares, bonds) that are sold in financial markets subject to various types of rules and regulations.

  Noise Trading (LeBaron 2001, p. 258; Arthur et al. 1997, p. 17):

  Trading on the basis of incorrect information as if it were correct.

  Short Position (LeBaron 2001, p. 258):

  A trader in a short position has sold a financial asset that he does not currently own, with the intention of later buying back the financial asset at a lower price. Thus, the trader is trying to make money by selling high (now) and buying low (later). The trader then actually makes (loses) money if the price of the financial asset decreases (increases) subsequent to the short sale.

  (Price) Bubbles and Fundamental Values (LeBaron 2001, p. 258):

  A financial asset is said to exhibit a (price) bubble if its price differs from its fundamental value, where the latter is the "true" (objectively determined) expected present value of its associated payment stream.

  Stationary Environment (LeBaron 2001, p. 250):

  An environment whose basic structural aspects do not vary over time.

  Stock Share Return Rate as Defined in Empirical Work (LeBaron 2001):

  Letting p_t denote the price of a stock share at time t, empirical researchers commonly define the net return rate on this stock share from t to t+1 to be r(t,t+1) = ln(p_t+1/p_t) = [ln(p_t+1) - ln(p_t] approx= [p_t+1 - p_t]/p_t , where "ln" denotes the natural logarithm. Dividends and other payouts are ignored because they happen relatively infrequently.

  Persistent Volatility of Stock Share Net Return Rates (LeBaron 2001, p. 259):

  The empirical observation that the degree of variability (fluctuation) in the net return rate earned on a holding of stock shares tends to persist over time, so that a high (low) variability observed today tends to indicate that a high (low) variability will again be observed tomorrow. In consequence, periods of high variability are followed irregularly by periods of low variability. For this reason, persistent volatility is also commonly referred to as clustered volatility.

  Efficient Markets Hypothesis (Arthur et al. 1997, p. 16):

  Roughly, the contention that, at each time t, the current prices of financial assets reflect all available information relevant for forecasting the future net return rates on those assets.

  Rational Expectations (Arthur et al. 1997, p. 17):

  Agents in an economy are said to have (weak form) rational expectations if they optimally exploit all information available to them in forming their expectations. Formally, this requirement is typically represented by the assumption that the "average" or "representative" agent's subjective (personal) expectation for any random variable V conditional on his current information set I coincides with the "true" (objectively determined) expectation for V conditional on I.

  Stronger forms of the rational expectations hypothesis place restrictions on the agents' information sets. For example, as originally formulated by John Muth (1961), these information sets were presumed to contain complete information about the structure of the economy in which the agents resided.

  Technical Versus Fundamental Trading (Arthur et al. 1997, p. 17):

  Stock shares are ownership claims against real assets held by corporations. Technical trading is the practice of selecting stocks on the basis of price patterns or trading volume, whereas fundamental trading is the practice of selecting stocks on the basis of events directly affecting the earning power of the real assets underlying the stocks.

  Deduction versus Induction (Arthur et al. 1997, pp. 18-19):

  Deduction is reasoning from general properties to specific instances. Example: Even numbers are positive integers that are exactly divisible by 2. The number 4 is a positive integer exactly divisible by 2. Therefore the number 4 is an even number.

  Induction is reasoning from specific instances to general properties. Example: A property is shown to hold for the smallest positive integer 1. Also, the property is shown to hold for integer n+1 if it holds for integer n, where n is an arbitrarily selected positive integer. Consequently, the property must hold for all positive integers n.

  Inductive Reasoning and Inductive Rationality (Arthur et al. 1997, p. 22):

  "(Formulating) hypothetical models to act upon, strengthening confidence in those that are validated, and discarding those that are not -- is called inductive reasoning. ... Agents who act by using inductive reasoning will be called inductively rational."

  Constant Absolute Risk Aversion (CARA) Utility Function (Arthur et al. 1997, p. 23):

  Let U(W) denote a trader's utility function in some time period, measuring the trader's happiness as a function of his wealth W in this time period. Let U'(W) and U''(W) denote the first and second derivatives of U(W), where U'(W) is assumed to be positive and U''(W) is assumed to be negative for each positive W level.

  The Arrow-Pratt measure of absolute risk aversion for U(W) is defined to be the value of [-U''(W)/U'(W)]. This measure is positively correlated with a variety of behavioral characteristics associated with risk aversion. For example, given utility functions UA(W) and UB(W) for traders A and B, it can be shown that trader A will accept participation in fewer small gambles (risky decision-making situations) than a trader B if and only if trader A is more risk averse than trader B in the sense that UA(W) has a larger Arrow-Pratt measure of absolute risk aversion than UB(W). [See, for example, Hal Varian, Microeconomic Analysis, W. W. Norton and Company, 1992, Chapter 11 on "Uncertainty."]

  Example: Consider the particular utility function U(W) = -exp(-lambda*W), where lambda denotes a positive constant. Calculating the Arrow-Pratt measure of absolute risk aversion for this utility function, one obtains -U''(W)/U'(W) = lambda. Thus, this utility function exhibits constant absolute risk aversion (CARA).

  Mean-Zero Gaussian (Normal) Distribution (Arthur et al. 1997, p. 23):

  A variable X has a mean-zero Gaussian (normal) distribution if the probability p(x) of the event X=x is given by

  p(x) = exp(-x²/2)/[2pi]^{1 / 2} .

  The graph of y=p(x) in the x-y plane is a symmetric bell-shaped curve centered around the line x=0 with thin infinitely elongated tails

  A Trader's If-Then Forecast Rule (Arthur et al. 1997, p. 24):

  An if-then rule used by a trader for forecasting, of the form IF CONDITION A HOLDS, THEN FORECAST B. Condition A relates to the current state of the market (as perceived by the trader), and B is an estimate for next period's stock price and dividend, together with a measure of the accuracy of the rule either based on past forecasting performance (if the rule has previously been used) or initialized to a parametrically specified constant.

  Genetic Algorithm (Arthur et al. 1997, pp. 25, 40):

  Applied to a population of strategies, a standard genetic algorithm (GA) evolves a new population of strategies from an existing population of strategies by applying the following five steps:

  1. evaluation, in which a fitness is assigned to each strategy in the existing population;
  2. recombination, in which offspring (new ideas) are constructed by combining the "genetic material" (structural characteristics) of pairs of parent strategies chosen from among the more fit strategies in the existing population;
  3. mutation, in which additional variations (new ideas) are introduced into the offspring strategies by mutating their structural characteristics with some small probability;
  4. elitism, in which a certain portion of the more fit strategies in the existing population (the elite) are selected for the new population without modification; and
  5. replacement, in which the remaining strategies for the new population are selected from among the mutated offspring strategies.

  Many variants of these five basic steps are possible and are under active exploration. See Melanie Mitchell, "Genetic Algorithms: An Overview," Complexity, Vol. 1, No. 1, 1995, for an introductory discussion of GAs.

  Skewness and Kurtosis (Arthur et al. 1997, p. 17):

  Skewness indicates the degree of asymmetry exhibited by a probability distribution (e.g., one tail longer than the other), whereas kurtosis indicates the relative flatness or peakedness of a probability distribution (e.g., strongly peaked with thin tails versus more flatly peaked with thicker tails). Skewness and kurtosis are measured by the third and fourth moments of a probability distribution. That is, letting V denote a random variable with mean vbar, and letting E(X) denote the expected value of X for any random variable X, the skewness and kurtosis of the probability distribution for V are given by:

  Skewness: E([V - vbar]³) .

  Kurtosis: E([V - vbar]⁴) .

  Leptkurtosis (Lebaron et al., 1999):

  A probability distribution is said to exhibit leptokurtosis if it has high kurtosis (thick tails) relative to the normal distribution.

  Some Empirical Stylized Facts For Stock Markets (Arthur et al. 1997; LeBaron et al., 1999):

  Large trade volume; Net return rates exhibiting persistent volatility; Net return rates exhibiting approximately zero autocorrelation; Net return rate distributions exhibiting skewness and leptokurtosis; Autocorrelated trade volume; Crashes occurring in the absence of fundamental market news; Cross-correlation between trade volume and net return rate volatility.

Basic Objectives of the Santa Fe ASM Model

1. Understand impact of agent interactions and group learning dynamics

2. Provide a "first level" test of the rational expectations hypothesis by introducing agent interactions and group learning in a relatively traditional financial model.

3. Study dynamics around a well-studied equilibrium (fundamental pricing).

4. Examine whether the introduction of agent interactions and group learning helps to explain empirical observations.

5. In particular, does it help to explain well-documented "anomalies" (deviations from fundamental stock pricing)?

6. Stress on statistical characteristics of price and trading volume outcomes.

The Santa Fe Artificial Stock Market

Basic Model Features

• Time is broken up into discrete time periods t = 1, 2, ....

• The stock market consists of an Auctioneer together with N stock market traders, i = 1,...,N. Traders are identical except that each trader individually forms his expectations over time through an inductive learning process. In all experimental treatments, N=25.

• Each trader has the same positive wealth W₀ in the initial period 0. This initial wealth is not specified numerically by Arthur et al. (1997) or by LeBaron et al. (1999).

• There exists a risk-free financial asset F (e.g., Treasury bills), available in infinite supply. Asset F pays a constant risk-free net return rate, denoted by r. In all experimental treatments, r = 0.10.

• There exist N shares of a risky stock A available at the beginning of each period t. Stock shares are assumed to be divisible, i.e., traders can purchase share fractions. In all experimental treatments, N=25.

• Traders do not have precise knowledge of the fundamental value of a share of stock A in any time period t. This is because the dividend d_t paid by stock A in each period t is generated by an exogenous stochastic process unknown to the traders -- specifically, an AR(1) process (i.e., an "autoregressive process of order 1") having the form
  (1)....... d_t = 10 + 0.95*[d_t-1 - 10] + w_t

  where w_t is a mean-zero Gaussian process with finite variance, identically distributed for every t. In all experimental treatments, the variance is set at 0.0743. A numerical value for the exogenously given initial dividend d₀ is not specified.

• Traders have identical constant absolute risk aversion (CARA) utility-of-wealth functions of the form U(W) = -exp(-lambda*W), where W denotes a trader's current wealth. In all experimental treatments, lambda is set to 0.50.

• Each trader i at the beginning of each period t selects a financial asset portfolio (x_ti, W_ti - p_tx_ti), where the first term denotes the trader's holdings of the risky asset (shares of stock A) and the second term denotes the trader's holdings of the risk-free asset F. The trader selects this asset portfolio to maximize his expected utility of period-(t+1) wealth, E[U(W_t+1,i)], subject to the condition that his period-(t+1) wealth W_t+1 will equal the period t+1 value of the asset portfolio he purchases in period t:
  (2)...... W_t+1,i = x_ti*[p_t+1 + d_t+1] + (1+r)*[W_ti - p_tx_ti] .

  Additional quantity restrictions are also imposed in the implementation. Specifically, a trader is not permitted to trade more than 10 shares each period, and is not permitted to sell short more than 5 shares each period.

• At the start of the market process in period 1, each trader i has a set of K if-then forecasting rules (or predictors). Each forecasting rule generates an expectation of the sum of next period's stock price and dividend together with an updated estimate of the rule's "forecast variance." Each trader is an inductive learner, in the sense that he evolves his forecasting rule set over time in such as way that new rules are continually being introduced.

• A forecasting rule is said to be active (or activated) if its "if" condition statement matches the trader's current market state information, as will be more fully explained in the discussion of GA-classifier learning below.

• The fitness of a forecasting rule depends inversely both on the rule's "forecast variance" var and on its "specificity" s. (Var and s are more carefully defined in the discussion of GA classifier learning below.) The analytical expression for the period-t fitness of a forecasting rule j in the rule set of a trader i is given by Arthur et al. (1997) in Appendix A, equation (8), as follows:
  (3)....... f_tij = - var_tij - c*s_tij .

  In all experimental treatments, c=0.005.

• Each trader individually evolves his set of forecasting rules over time on the basis of fitness, retaining the most fit rules and replacing the least fit rules with variants of the more fit rules. This evolution is conservative, in the sense that each trader always has precisely K rules in his forecasting rule set. In all experimental treatments, K=100.

• THE PRECISE WAY IN WHICH THE FORECASTING RULE SETS ARE SPECIFIED FOR EACH TRADER IN PERIOD 1, AND ONE OR MORE RULES ARE ACTIVATED, IS NOT ENTIRELY CLEAR IN ARTHUR ET AL. (1997). The discussion in Appendix A of Arthur et al. (1997) suggests that K=100 forecasting rules are randomly specified for each trader's rule set in the initial period 1. This is taken up more carefully in the discussion, below, on GA-classifier learning.

• In each time period, each trader i decides how much of his wealth to invest in the risky stock A and how much to invest instead in the risk-free asset F. Trader i does this by generating a (net) demand for stock A as a function of: (a) his current expectations for next period's price and dividend; and (b) the yet-to-be-determined period-t price of stock A. This demand function is communicated to the Auctioneer. The Auctioneer in turn determines a market-clearing price for stock A and communicates this price back to each of the traders. Given this market-clearing price, each trader then goes ahead and purchases his corresponding demand.

Time-Line of Activities

Note: The following time-line is for an arbitrary period t with t greater than 1

• At the start of period t, the dividend d_t earned by each outstanding share of stock A is publicly posted.

• Each trader i then proceeds to use the newly posted dividend information, together with general information on the state of the market (including any past observed dividends and prices), to determine which of his forecasting rules is activated in the current period.

• Each activated forecasting rule j' generates an estimate of its current "forecast variance" var_tij' (explained below), together with an expectation regarding next period's price plus dividend of the form:
  (4)...... E_tij'[p_t+1 + d_t+1] = a_j'[p_t + d_t] + b_j'

  If trader i has multiple activated forecasting rules j in period t, a rule j' with the smallest actual current forecast variance is used to generate this expectation.

  NOTE: There is an ambiguity in Arthur et al. (1997) with regard to exactly how many forecasting rules are actually used to generate expectations when multiple forecasting rules are activated at the same time. On page 25 it is stated that a trader forms an expectation for next period's price and dividend by statistically combining the forecasts of the H most accurate forecasting rules that are currently activated. However, on page 39 it is stated: "If multiple (forecasting rules) are active, only the most accurate is used." According to an email clarification by Blake LeBaron (3/11/02), the latter procedure is the correct description: only the forecasting rule with the lowest actual forecast variance is activated.

  Given this expectation, trader i generates a demand function for stock A that gives his net quantity demand for stock A for each possible stock price p. This demand function is assumed to have the basic form

  (5)...... x_tij'(p) = (E_tij'[p_t+1 + d_t+1] - p_t[1+r])/(lambda*var_tij')

  The demand function (5) assumes that trader i's demand for stock A in period t is proportional to the deviation between stock A's period t price p_t and stock A's period t fundamental value (the period t present value of its period t+1 price p_t+1 plus period t+1 dividend payment d_t+1.) The proportionality factor is a function of both absolute risk aversion (measured by lambda) and confidence in the forecasted expectation (as measured by the forecast variance var_tij'). If stock prices have a Gaussian distribution, this demand function can be shown to maximize trader i's expected utility of period-(t+1) wealth subject to his budget constraint (2). Otherwise, (5) is an approximation for the optimal demand function.

  Each trader i communicates his demand function to the auctioneer. The Auctioneer uses the N demand functions from the N traders to determine a market-clearing price p_t for period t, i.e., a price that guarantees that the total demand for the risky stock is just equal to N, the total available supply. Consequently, the stock market is in temporary equilibrium.

  Once the market-clearing price p_t is determined, the Auctioneer communicates this price back to each trader, who then goes ahead and purchases his corresponding stock share demand x_tij'(p_t).

  Each trader i then uses the posted period-t dividend d_t and the period-t stock price p_t determined by the Auctioneer to update the fitness of the forecasting rule(s) activated and used in the previous period t-1 to generate expectations for the sum of the period-t stock price and dividend.

  Subsequent to this fitness reassessment, with some fixed probability p_u each trader updates his entire forecasting rule set by means of a genetic algorithm involving elitism, recombination, and mutation operations. (See the discussion on GA-classifier learning, below).

  At the start of the next period t+1, a new exogenously determined dividend d_t+1 is posted, and the process repeats.

GA Condition/Forecast Classifier Learning

NOTE: See LeBaron et al. (1999) for a more detailed discussion of the design and implementation of GA condition/forecast classifier learning in Arthur et al. (1997).

• As detailed below, each trader i implements GA condition/forecast classifier learning to update his forecasting rule set. This learning is activated with a certain probability p_u at the end of each trading period, where p_u depends on the particular experimental treatment. Consequently, the exact updating times for each trader are independently determined, implying that this updating proceeds asynchronously (i.e., at different times) across traders. The updating probability p_u is an important parameter that determines the average speed of learning of the traders.

• The current market state is described by a J-bit array MarketState. Each of the J bit positions in this array corresponds to a distinct possible feature (descriptor) of the current market state. The bit in position k takes on the value 1 if the kth descriptor is a TRUE description of the current market state and the value 0 if the kth descriptor is a FALSE description of the current market state.

• In all experimental treatments, Arthur et al. (1997) set J=12. The first six descriptors correspond to "fundamental" measures of the current market state. Specifically, these six descriptors roughly indicate whether the risky stock is above or below its previous-period fundamental value at the current price p by setting six successively higher lower bounds (0.250, 0.500, 0.750, 0.875, 1.000, 1.125) for
  (6)........... pr/d ,

  where r denotes the risk-free net return rate and d denotes the latest dividend paid by the risky stock A. Note that pr/d is greater than 1.000 if and only if p is greater than the risky stock's previous-period fundamental value

  (7)........... [p + d]/(1+r).

  The next four descriptors correspond to technical trading measures of the current market state. They indicate whether a price trend is under way. Specifically, they indicate whether the current price p is greater than an n-period moving average of past prices, where n takes on the values (5,10,100,500).

  The last two descriptors contain no actual market information and hence serve only as experimental controls. They indicate the degree to which agents are learning to act in response to "useless" information. In all experimental treatments, the 11th descriptor is always set to 1 and the 12th descriptor is always set to 0.

• The intention of this descriptor specification is to permit testing for the possible emergence of fundamental trading (heavier reliance on the first six descriptors) versus technical trading (heavier reliance on the next four descriptors).

• Each forecasting rule j for each trader i in each period t is represented by a condition statement C and an associated array having the general form B=(var_tij,a,b).

  • The condition statement C is a J-bit array, where each bit takes on one of the following three values: 1, 0, or # ("don't care"). The J bit positions in the condition statement C correspond to the J different descriptors for MarketState. The specificity of C is defined to be the number of its bits that are set to 0 or 1. The condition statement C is said to match MarketState if the following two conditions hold: (a) C has a 1 or # symbol in every bit position that MarketState has a 1; and (b) C has a 0 or # symbol in every bit position for which MarketState has a 0.

  • Var_tij denotes the rule's forecast variance, which takes the form
    (8).... var_tij = (1-theta)*var_t-1,i,j + theta*([p_t+1 + d_t+1 - E_tij[p_t+1 + d_t+1])²

    Thus, var_tij is a "moving average" of the rule's squared forecast error (i.e., the difference between actual and forecasted price plus dividend). The parameter theta determines the weight (theta) placed on the most recent squared error versus the weight (1 - theta) placed on the sum of past squared errors. The value of theta is set differently in each experimental treatment (see the discussion on experimental design, below). The value of theta determines the horizon length that the traders consider relevant for forecasting purposes; the closer theta is to 1, the shorter this horizon becomes.

    NOTE:As noted by LeBaron et al. (1999, p. 1496, footnote 12), the actual forecast variance (8) associated with a forecasting rule is calculated and recorded each time a rule is activated, but the specific forecast variance attached to a rule is kept constant between genetic algorithm evolutionary steps and only updated to its actual value during these evolutionary steps.

    The values a and b indicate that the rule generates a forecast for E[p_t+1 + d_t+1] having the form

    (9)....... a[p_t + d_t] + b.

• The discussion in Arthur et al. (1997, pp. 24-25 and Appendix A) and LeBaron et al. (1999, p. 1499) suggests that the traders' forecasting rule sets in the first period 1 are constructed as follows.

  • Ninety nine of the 100 forecasting rules in each trader's forecasting rule set are constructed using the same procedure. The condition statements are initialized as follows: a condition bit is set to # with probability 0.90, to 0 with probability 0.05, and to 1 with probability 0.05. The forecast variance var for each of these forecasting rule is set to 4.0, which is the variance estimate for the special case of a "rational expectations equilibrium" in which traders are assumed to have identical correct expectations regarding future prices and dividends. Finally, it appears that values for a and b are selected from uniform distributions over the intervals (0.7,1.2) and (-10,19.002), respectively. The expectation of the sum of next period's stock price plus dividend that is generated by (var,a,b) in any current period t is assumed to take the form
    (10)....... E_t[p_t+1 + d_t+1] = a*[p_t + d_t] + b.

  • Each trader's forecasting rule set always contains a specially constructed default rule. Each of the condition statement bits for this rule is set to #, implying it is always activated. The forecasting parameters (a,b) for this default rule are set to a weighted average of the values for each of the other rules, where the weight for each rule is one over it's current forecast variance estimate. The forecast variance var for this default rule in the initial period 1 is set at 4.0.

  NOTE: THE MANNER IN WHICH FORECASTING RULES ARE ACTIVATED AND SELECTED FOR USE IN THE INITIAL PERIOD 1 IS NOT EXPLAINED IN EITHER ARTHUR ET AL. (1997) OR LEBARON ET AL. (1999).

  More precisely, an initial market state in period 1 needs to be exogenously specified in order to determine which forecasting rules are activated by the "normal" matching process. A note from Blake LeBaron (3/12/02) clarifies how this was actually done: "There is an initialization period (500 steps) before trading begins. During this time, the dividend follows the given stochastic process [see (1)], and the price is set to d/r. There is no trade. Agents simply watch, and this history is loaded into all the bitstrings. When t=1 hits, there is a fully developed history which they can match their rules to. They then go from there, matching and forecasting."
  Presumably each trader in the initial period 1 selects for use a most fit forecasting rule from among all of his activated rules. Recall that all of the forecasting rules in each trader's forecasting rule set in period 1 have the same assigned forecast variance of 4.0, but the specificity of each rule is determined by a (generally) random assignment of 1, 0, and # values to its condition bits. Consequently, rules have different fitnesses (3). The default rule, by construction, has the lowest possible specificity of 0, and is always activated, hence it appears that this default rule will have the highest fitness in the initial period 1 by construction.

  Suppose the forecasting rule set of some trader i in some period t has been randomly selected for updating, where each rule in this rule set has a currently assigned fitness. The description of the updating is given by Arthur et al. (1997, Appendix A), but the description has several confusing aspects. The discussion in LeBaron et al. (1999) is clearer.

  • Elimination and Elitism: The default rule is maintained in each trader's forecasting rule set, with forecasting parameters always updated to be a weighted average of the forecasting parameters of all other rules currently in the set (see above). For all other rules, the worst-performing (least fit) rules are eliminated. In all experimental treatments, this replacement rate is set at 20 percent (20 rules out of the 99 rules eligible for elimination), hence the overall retention (elite) rate is 80 percent.

  • Replacement by Offspring: The eliminated rules are replaced with new "offspring" forecasting rules that are formed as variants of "parent" forecasting rules selected from among the retained (elite) rules. The construction of the condition statement, forecast parameters, and forecast variance for each offspring forecasting rule proceeds as follows.

    • With probability 1-P (where P depends on the experimental treatment), the offspring forecasting rule is generated purely by mutation operations. Specifically, one parent is chosen from among the elite rules by tournament selection. That is, two rules are randomly chosen from among the elite, and the more fit rule is retained as the parent. The condition bits of this parent are "flipped at random," and the parent's forecasting parameters have random values added to them. (See LeBaron et al., 1999, page 1498, footnotes 16 and 17 for the exact procedures.)

    • With probability P, the offspring forecasting rule is instead generated purely by crossover operations, as follows:

      • An Offspring's Condition Statement: First, two parents are chosen from among the elite forecasting rules by tournament selection. That is, to choose each parent, two rules are randomly chosen from among the elite rules and only the most fit of these two rules is retained as the parent. The offspring's condition statement is then constructed from the resulting condition statements for the two parents via what LeBaron et al. (1999) refer to as uniform crossover of the two parents' condition statements. Specifically, the offspring's condition statement is constructed one bit at a time, choosing the corresponding bit from either parent with equal probability.

      • An Offspring's Forecasting Parameters: For the forecasting parameter vector (a,b), an offspring's vector is formed from its parents' vectors by one of the following three methods, selected with equal probability: (a) Component-wise crossover (take the value of a from one parent and the value of b from the other); (b) Take a weighted average of the two parent vectors (with each rule's weight equal to the inverse of its current forecast variance, i.e., 1/var); or (c) randomly select one or the other parent's vector to be the offspring's vector.

      • An Offspring's Forecast Variance: An offspring forecasting rule inherits the average forecast variance of its two parents unless both parent forecasting rules have never been matched (activated). In the latter case, the offspring is assigned the median forecast variance of all (ELITE?) forecasting rules.

  

  Experimental Design and Findings

  • The experimental treatments considered in Arthur et al. (1997) consist of different settings for the three parameters (p_u,P,theta), keeping all other parameters at fixed values. For each experimental treatment, 25 trials were run using 25 different seeds for the pseudo-random number generator in an attempt to ensure sufficient "genetic diversity" in the traders' initial forecasting rule sets. Each trial consisted of a total of 260,000 successive stock trading periods, divided into a "preliminary" stage consisting of 250,000 periods and a "data collection" stage consisting of 10,000 periods. The preliminary stage was conducted to permit sufficient time for learning and die-out of transients. Time series data were collected and reported only during the data collection stage.

  • Two different experimental treatments are reported, as follows:

    1. Slow-Learning Regime: The updating probability p_u was set to 1/1000 (so that GA learning was invoked every 1,000 trading periods on average for each trader), the probability of crossover was set at P=0.30, and theta (in the forecast variance equation) was set at theta = 1/150.

    2. Medium-Learning Regime: The updating probability p_u was set to 1/250 (so that GA learning was invoked every 250 trading periods on average for each trader), the probability of crossover was set at P=0.10, and theta was set at theta = 1/75 (increasing the relative weight on the most recent forecast error).

    NOTE: LeBaron et al. (1999) apparently re-ran all of the experiments in Arthur et al. (1997) with the crossover probability P always set equal to P = 0.10 and theta always set equal to theta = 1/75. Thus, in LeBaron et al. (1999), the only difference between the two experimental treatments was a single parameter k, the "forecast horizon" dictating the average interval length between invocations of the GA for each trader. The authors apparently found no significant differences with the earlier obtained results in Arthur et al. (1997).

  • Detailed experimental findings are presented in Arthur et al. (1997, Section 4.B) and LeBaron et al. (1999, Section 3). A brief summary of these findings is given below.

  • For the slow-learning regime, the simulated time series data look similar to the data generated for a rational expectations equilibrium benchmark case in which the efficient markets hypothesis holds by construction.

  • In contrast, in the medium-learning regime, the market does not appear to settle down to any recognizable equilibrium. The simulated time series data for the latter regime are in accordance with many of the empirical stylized facts for actual financial time series data regarding predictability, volatility, and volume relations. Traders are also more likely to be using "technical trading" forecasting rules conditioned on price trends rather than "fundamental trading" forecasting rules conditioned primarily on indicators of fundamental value.

  • The authors conclude that the average forecast horizon of stock market traders appears to have a substantial impact on the nature of the resulting stock market outcomes. In particular, with a sufficiently fast learning rate, "the market becomes driven by expectations that adapt endogenously to the ecology that these expectations co-create" (Arthur et al., 1997, p. 38).

  

  Copyright © 2012 Leigh Tesfatsion. All Rights Reserved.