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Article Excerpt
Summary. A series of financial anomalies motivated the development of new theories that modify the rational expectations ideal. Two possibilities have been systematically explored. The literature on behavioral finance relaxes the assumption that agents form beliefs according to the laws of probability and assume, instead, that simpler heuristic rules are used. Another stream of the literature assumes that agents process information according to Bayes' rule, but do not posses sufficient information to know the true data generating process. In this paper, Bayesian and Behavioral agents coexist and trade in a standard dynamic asset pricing model. A long-standing conjecture is demonstrated. It is shown that, under suitable assumptions, Bayesian agents drive Behavioral, non-Bayesian agents out of the market. Hence, asset prices are eventually determined under the Bayesian paradigm.
Keywords and Phrases: Bayes' Rule, Wealth accumulation.
JEL Classification Numbers: D83.
1 Introduction
The models of the traditional asset pricing literature are based on the premise that agents maximize expected utility and hold rational expectations. However, several financial anomalies (i.e. price behavior that is inconsistent with rational expectations models) have been documented. They include excess volatility of asset prices relative to dividends fluctuations, positive short-term autocorrelation of stock returns, and negative autocorrelation of stock returns separated by long lags (see Daniel et al., 1998, for a literature review).
It is important to determine the causes of these financial anomalies. This identification will reveal the appropriate ways that the rational expectations assumption should be modified. Two possibilities stand out. It is possible that agents do not process information according to Bayes' rule and, therefore, suffer from cognitive biases when forming beliefs. Alternatively, it is possible that agents process the available information according to Bayes' rule, but do not have sufficient information about the structure of the economy to hold correct beliefs.
Financial anomalies have been attributed to both cognitive biases and incomplete information. Daniel et al. (1998, 2001), Barberis et al. (1998), Odean (1998), and Hong and Stein (1999), among others, developed models in which financial anomalies are attributed to agents' cognitive biases. Zeira (1999), Hansen et al. (2000), and Lewellen and Shanken (2002), among others, developed models in which financial anomalies are attributed to incomplete information. Brav and Heaton (2002) and Brandt et al. (2003) show that the predictive powers of theories based on cognitive bias and the ones based on incomplete information are similar. Hence, it is difficult to distinguish among these competing theories empirically.
Laboratory evidences seem to support the idea that agents systematically deviate from the laws of probability when forming beliefs. Tversky and Kahneman (1971, 1974) note that subjects often rely on simple heuristic rules to access probabilities. This leads to judgmental operations such as the law of small numbers, where agents make strong inferences from small samples; confirmatory bias, where agents hold dogmatic beliefs and ignore information that contradicts cherished viewpoints; and representativeness, where agents stereotype situations and neglect prior probabilities (see Camerer, 1995; Rabin, 1998, for surveys).
In the behavioral finance literature, agents' beliefs are assumed to be closer to the experimental evidence than to the standard laws of probability (see Thaler, 1993, for details. There are three main criticisms to this literature. First, there is the concern that this approach allows for any belief and essentially explains any behavior. This criticism can be countered because beliefs are not arbitrary in the behavioral finance literature. The experimental evidence restricts the set of possible beliefs and imposes some intellectual discipline on the existing models. The second criticism is that agents should use the existing data to form rational expectations. This criticism can also be countered because in a realistic environment the available information is not sufficient to determine the true structure of the economy and, therefore, the true data generating process cannot be learned from the existing data. The third (and crucial) criticism is based on the idea that each agent does not influence prices equally. In particular, asset prices may not reflect the actions of most individuals. Instead, asset prices are eventually determined by agents who accumulate wealth during the trading process. The traditional efficient markets hypothesis is supported by the long-standing idea in economics that agents who do not predict as accurately as others will be unable to accumulate sufficient wealth and, hence, will be driven out of the market (see Friedman, 1953; Alchian, 1950). However, it has not been demonstrated that agents who process information according to Bayes' rule accumulate wealth at the expense of behavioral agents who do not process information according to the standard laws of probability. This conjecture is the central theme in this paper. Thus, before addressing the contribution of this paper, it is useful to briefly review the existing literature on market selection.
For decades, the efficient market hypothesis has been investigated empirically with controversial results. Recently, several authors argued that this hypothesis lacks theoretical backing. The work of De Long et al. (1996), Shleifer and Vishny (1997), Kyle and Wang (1997), and Hirshleifer and Luo (2001), among others, demonstrated that in some settings agents with incorrect beliefs may drive agents with correct beliefs out of the market. Different intuitions have been provided for this claim, but the central idea is that agents maximize expected utility which is not the same as maximizing the rate of wealth accumulation. Thus, it is possible to distort correct beliefs, making them incorrect while simultaneously inducing a faster rate of wealth accumulation.
The aforementioned intuition holds in simple models, but it does not hold in a fully developed general equilibrium model. Sandroni (2000), Blume and Easley (2003) and Kogan et al. (2003) have shown that, in standard dynamic general equilibrium models, agents with correct beliefs drive agents with incorrect beliefs out of the market. The intuition behind this result is that agents allocate more wealth to events they believe are more likely to occur. Thus, only the agents with correct beliefs will allocate their wealth to events that are, in fact, likely to occur.
The market selection literature produced various results in different settings. The difficulty is centered at the apparently obvious claim that "agents allocate more wealth to events they believe are more likely to occur". If agents' behavior are derived from expected utility maximization, then a proof of this claim requires analytical details that can be found in a fully developed dynamic general equilibrium model. However, these analytical details may not be found in a simpler model such as a partial equilibrium model.
The market selection literature has given some qualified support to the claim that agents with correct beliefs drive agents with incorrect beliefs out of the market. These results focus on a comparison between correct and incorrect beliefs and, hence, they do not shed light on how the rational expectations assumption should be modified. This may be achieved by focusing on a comparison between agents who process information according to Bayes' rule and agents who use behavioral rules to update beliefs.
A significant inquiry on the role of different learning processes must be conducted in a fully developed dynamic general equilibrium, where the probabilities of the payoff-relevant variables are not known from the beginning of the economy. I consider an environment where agents must try to learn the fundamental structure of the assets and none of the agents is assured to have correct beliefs at all times. Instead, agents rely on empirical models that use the available data as inputs to form predictions for the future. These empirical models may differ across agents. Some agents rely on the Bayesian paradigm while others rely on Behavioral, non-Bayesian models. Some empirical models may lead to economic prosperity while other models may lead to economic ruin. Prices reflect all empirical models in the economy, but greater weight is given to the models held by wealthier agents. Hence, prices ultimately reflect the empirical models with predictive abilities that lead to greater wealth. The main question is which predictive properties are conducive to wealth accumulation and which empirical models posses these abilities.
It is useful to classify the fundamental structure of the assets into two groups. The asset structure is learnable when the available data eventually reveals the true probability of dividends with near certainty. In this case, the historical records become a perfect guide to the future. If the available data cannot reveal the true probability of dividends then the asset structure is unlearnable. In this case, historical records never become a perfect guide to the future.
Whether or not the asset structure is learnable depends on the availability of relevant information. If the relevant historical records eventually become sufficient large then the data may reveal the true probabilities of dividends. In this case, the asset structure is learnable and some agents may eventually have rational expectations. On the other hand, a sufficiently strong technological shocks may lead to a regime switch, making past data obsolete. If regime switches occur frequently then the relevant available information is always limited. In such a case, the asset structure is unlearnable and processing the available data does not lead to rational expectations.
Unlearnable asset structures seem, no matter how the information is processed, more akin to the complex real world. They are particularly interesting to study because they permit us to determine which types of empirical models are conducive to wealth accumulation when agents' beliefs cannot be correct. The main results in this paper are as follows: Assume that the fundamental structure of the assets is learnable. Then, Bayesian agents will learn the asset structure and survive. Some non-Bayesian agents may learn the asset structure and some may fail do so. Those agents who fail to learn the asset structure and those agents who do not learn as quickly as do Bayesian agents are driven out of the market. However, some non-Bayesian agents may learn the asset structure as quickly as do Bayesian agents. These non-Bayesian agents eventually forecast arbitrarily close to the Bayesian paradigm (hence close to rational expectations) and survive. Now assume instead that the fundamental structure of the assets is unlearnable. In this case, Bayesian agents will...
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