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Article Excerpt
1. Introduction
Empirical evidence on the distributional characteristics of common stock returns indicates: (1) A power-law tail index close to three describes the behavior of the positive tail of the survivor function of returns (pr(r > x) ~ [x.sup.-[alpha]]) (Gopikrishnan et al. 1999, Plerou et al. 1999), a reflection of fat tails; (2) general linear and nonlinear dependencies exist in the time series of returns (Scheinkman and LeBaron 1989, Hsieh 1991, Brock et al. 1991); (3) the time-series return process is characterized by short-run dependence (short memory) in both returns as well as their volatility, the latter usually characterized in the form of autoregressive conditional heteroskedasticity (Bollerslev et al. 1992, Glosten et al. 1993, Engle 2004); and (4) the time-series return process probably does not exhibit long memory (Lo 1991), but the squared returns process does exhibit long memory (Ding et al. 1993, Bollerslev and Wright 2000). Little is known, however, about what behaviors on the part of investors should give rise to jointly observing these phenomena. We propose a model of complex, self-referential learning and reasoning amongst economic agents that jointly produces security returns consistent with these general observed facts. The model features investors who reason inductively through experimentation with new hypotheses while compressing information into a few fuzzy notions that they can in turn process and analyze with fuzzy logic. Our approach is motivated first by the cogent argument set forth by Arthur (1991, 1992, 1994, 1995), Arthur et al. (1997), and LeBaron et al. (1999), who conclude that deductive reasoning must give way to inductive reasoning in complex, ill-defined settings and that real capital markets exhibit a high level of complexity. Second, we follow a stream of thought proposed by Smithson (1987) and Smithson and Oden (1999), amongst others, who conclude that human reasoning can be modeled as if the thought process is described by the application of fuzzy logic. Assuming mental behavior of this sort allows the agent to step outside the rigid confines of more traditional models. We embed this behavior in an artificial stock market model that is utilized as a vehicle for simulating the dynamics of a market from which market-clearing security prices emerge, allowing us to compute realized returns.
The structure of our model extends the important work done in developing the Santa Fe Artificial Stock Market Model studied by LeBaron et al. (1999). While there are important differences between the models, the two machine-learning methods produce similar results. We feel that this is important to understanding how agents learn and reason. One notable implication is that our framework requires, in principle, much less of the agent. We feel that the fact that our model produces results similar to the Santa Fe Institute (SFI) model is both an important statement about how individuals learn and reason, but is also an endorsement of the importance of the SFI model, because, turned around, we are saying that the SFI model produces results similar to a model in which agents learn in a much less structured fashion.
The learning system proposed by LeBaron et al. (1999) makes use of a model in which each agent is assumed to make choices predicated on a large number of rules for the mapping of market conditions into expectations, each with numerous conditions. (1) Agents in their model formulate new rules through the application of behavior emulated by a genetic algorithm. Agents in our model, in contrast, employ only a handful of hypotheses used to generate expectations. These hypotheses are composed of only four rules each. Each rule employs a selection of the information available, which will be used to construct a conditional assignment of values to the parameters of a model of predictions. An agent learns in two ways. First, new hypotheses are generated from existing hypotheses, and low-accuracy hypotheses are replaced, with high probability, with newly generated hypotheses. In this way, agents are selecting the variables to use in the construction of the parameters of their prediction models. Second, values for the parameters of the prediction model, conditional on the hypothesis, are formed by the agent applying fuzzy logic to the observed data. The latter allows us to reduce the number of hypotheses and rules to a quantity that is more like what one might expect versus the long list of rules utilized in the SFI model.
We show that with-dividend returns computed from simulated market-clearing prices for the environment we propose exhibit a tail index in their survivor functions characterized by a power law with exponent on the order of three, exhibit autoregressive conditional heteroskedasticity, and exhibit general nonlinear dependencies and long memory in the volatility process. We also document the presence of similar characteristics for a sample of 50 common stocks traded on the New York Stock Exchange, which act here as a benchmark. The appeal of our results is twofold: First, the behavioral model we propose generates return characteristics for risky securities that are similar to what are observed for actual stock returns; and second, it does so as a product of an environment in which economic agents are endowed with learning and reasoning processes that are close to what many disciplines believe is an accurate depiction of actual behavior.
How investors learn and interact in complex capital market environments is crucial to understanding the nature of financial security return distributions. Complexity demands an alternative approach to the analysis of markets and institutions. The approach we take has its roots in work begun and continuing at the Santa Fe Institute. Examples of such work focusing on the behavior of security prices include Arthur et al. (1997), Brock and Hommes (1998), LeBaron et al. (1999), and Tay and Linn (2001). Tesfatsion (2002) and LeBaron (2000, 2006) provide reviews of this literature. (2)
Our model is motivated by the discrepancy between the idealized well-defined environment that is commonly assumed in neoclassical financial market models and the complex ill-defined markets that are observed in practice. Neoclassical financial market models are generally designed within the context of a well-defined setting so that economic agents are able to logically deduce the expected prices of securities that they in turn employ when setting their demands for those securities. (3) Real stock markets do not, however, typically conform to the severe restrictions required to guarantee such behavior. The actual market environment is usually much more ill defined. The dilemma is that in an ill-defined environment the ability to exercise deductive reasoning breaks down, making it impossible for individuals to form precise and objective price expectations. This implies that participants would need to rely on some alternative form of reasoning to guide their decision making. We conjecture that individual reasoning in an ill-defined setting can be described as an inductive process. The application of inductive reasoning involves the formulation of tentative hypotheses to fill in the gaps left by incomplete information. These hypotheses are then continually tested in the market and revised as agents seek to improve their understanding of market behavior. Agents in our model generate predictions by the application of an inductive reasoning process in which they rely on fuzzy decision-making rules due to limits on their ability to process and condense information.
Our study is close in spirit to a contemporaneous study by Gabaix et al. (2006), which focuses on a model of large fluctuations in stock returns and is motivated by the presence of a power-law decay in the survivor function of returns as well as trading volume. Our study, however, differs in several important ways from theirs. First, we present a model that jointly produces a power-law decay in the survivor function for returns, autoregressive conditional heteroskedasticity, and general nonlinear dependencies. The central focus of Gabaix et al. (2006) is explaining what gives rise to the power-law decay. Second, our model focuses on the influence of learning and reasoning by agents and the influence of nontraditional aspects of these activities on the distributions of returns. Gabaix et al. (2006) present an insightful model built up from assumptions about the structure of trading and the search for trading partners. We, on the other hand, choose to minimize these influences to highlight how agents learn and reason in an ill-defined environment. In this way, both studies provide important and new insights into what factors potentially give rise to the features of stock returns already mentioned.
This paper is organized as follows. In [section][section]2-5, we begin by presenting empirical results on the existence of a power law in the behavior of the survivor function of common stock returns; on the presence of dependencies in the time series of returns, including autoregressive conditional heteroskedasticity as well as long memory in the volatility process; and on the general existence of nonlinear dependencies in stock returns. Our focus is on a sample of 50 common stocks traded on the New York Stock Exchange. Section 6 goes on to summarize how agents learn in the model and the market environment. Section 7 describes the dynamic simulation experiments of the model. Section 8 presents an analysis of the returns computed from the market-clearing prices generated in the artificial stock market, drawing comparisons with the results presented in [section][section]2-5 for the benchmark sample of stock returns. (4) We also investigate how the characteristics of the returns generated by the model vary with changes in values of key parameters of the model. Section 9 presents our conclusions.
2. The Data and Descriptive Statistics
The data examined herein are comprised of (a) daily with-dividend return series for 50 actively traded NYSE-listed common stocks, and (b) 540 return series computed from simulations of the artificial stock market model. The values of parameters of the model are varied to provide insight into their individual influence on the results of the simulations. Thirty simulations of the market are generated for each set of parameters examined. We defer our analysis of the data from the artificial stock market simulations until [section]8, following our discussion of the model's structure and the design of the decision-making algorithm ascribed to agents in the model. Instead, we begin by focusing on the characteristics of the stock returns for the 50 NYSE-listed stocks to establish a benchmark for comparison. The source of the stock return data is the Center for Research in Security Prices (CRSP) Daily Return file, and the data included are the 2,780 daily returns ending December 31, 1998. (5)
Table 1 presents summary statistics for the 50 actual stock return series along with results for the simulations, which will be discussed later in the paper. The results for the actual stock returns appear in the column headed "Actual." The sample statistic names are listed in the leftmost column. We present the average values for the statistics and, in parentheses, the standard errors of the point estimates of the statistics computed across the 50 cases. When a test statistic is reported in a table, we present the average value of the test statistic computed across the relevant cases and, in square brackets, the fraction of tests rejecting the null hypothesis.
Notable amongst the descriptive statistics for the actual returns is the high level of kurtosis and the positive skewness. Kurtosis for a normal distribution should equal three, while skewness should equal zero. Both measures for the sample return series deviate from these benchmarks. The Jarque-Bera test (not reported) rejects the null hypothesis of normality for each of the 50 sample series. (6) High kurtosis is consistent with the presence of heavy or fat tails in the distribution, but may also be a manifestation of a peaked distribution.
3. Power-Law Tail Behavior of the Survivor Function
Numerous investigators (see Mandelbrot 1997, Gopikrishnan et al. 1999, Plerou et al. 1999) have presented evidence indicating that the distributions of common stock returns exhibit tails with greater mass than would be predicted if the distributions were Gaussian normal, the conclusion being that these distributions exhibit fat or heavy tails. For a class of distributions characterized by what is referred to as regular variation in the tails, the far-right portion of the tail of the survivor function is a power-law function of the form (7)
pr([r.sub.i] > x) ~ 1/[x.sup.[[alpha].sup.i]], (1)
where [[alpha].sub.i] is the exponent characterizing the power-law tail index for security i and...
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