Asset pricing under rational learning about rare disasters : [Version 28 Juli 2011]

This paper proposes a new approach for modeling investor fear after rare disasters. The key element is to take into account that investors’ information about fundamentals driving rare downward jumps in the dividend process is not perfect. Bayesian learning implies that beliefs about the likelihood of rare disasters drop to a much more pessimistic level once a disaster has occurred. Such a shift in beliefs can trigger massive declines in price-dividend ratios. Pessimistic beliefs persist for some time. Thus, belief dynamics are a source of apparent excess volatility relative to a rational expectations benchmark. Due to the low frequency of disasters, even an infinitely-lived investor will remain uncertain about the exact probability. Our analysis is conducted in continuous time and offers closed-form solutions for asset prices. We distinguish between rational and adaptive Bayesian learning. Rational learners account for the possibility of future changes in beliefs in determining their demand for risky assets, while adaptive learners take beliefs as given. Thus, risky assets tend to be lower-valued and price-dividend ratios vary less under adaptive versus rational learning for identical priors. Keywords: beliefs, Bayesian learning, controlled diffusions and jump processes, learning about jumps, adaptive learning, rational learning. JEL classification: D83, G11, C11, D91, E21, D81, C6

Learning, adaptive expectations, and technology shocks

This study explores the macroeconomic implications of adaptive expectations in a standard real business cycle model. When rational expectations are replaced by adaptive expectations, we show that the self-confirming equilibrium is the same as the steady-state rational expectations equilibrium for all admissible parameters but that dynamics around the steady state are substantially different between the two equilibria. The differences are driven mainly by the dampened wealth effect and the strengthened intertemporal substitution effect, not by the escapes emphasized by Williams (2003). As a result, adaptive expectations can be an important source of frictions that amplify and propagate technology shocks and seem promising for generating plausible labor market dynamics.Equilibrium (Economics)

Heterogeneous Agents Models: two simple examples, forthcoming In: Lines, M. (ed.) Nonlinear Dynamical Systems in Economics, CISM Courses and Lectures, Springer, 2005, pp.131-164.

These notes review two simple heterogeneous agent models in economics and finance. The first is a cobweb model with rational versus naive agents introduced in Brock and Hommes (1997). The second is an asset pricing model with fundamentalists versus technical traders introduced in Brock and Hommes (1998). Agents are boundedly rational and switch between different trading strategies, based upon an evolutionary fitness measure given by realized past profits. Evolutionary switching creates a nonlinearity in the dynamics. Rational routes to randomness, that is, bifurcation routes to complicated dynamical behaviour occur when agents become more sensitive to differences in evolutionary fitness.

Learning, adaptive expectations, and technology shocks

This study explores the macroeconomic implications of adaptive expectations in a standard real business cycle model. When rational expectations are replaced by adaptive expectations, we show that the self-confirming equilibrium is the same as the steady state rational expectations equilibrium for all admissible parameters, but that dynamics around the steady state are substantially different between the two equilibria. The differences are driven mainly by the dampened wealth effect and the strengthened intertemporal substitution effect, not by the escapes emphasized by Williams (2003). As a result, adaptive expectations can be an important source of frictions that amplify and propagate technology shocks and seem promising for generating plausible labor market dynamics.Macroeconomics

Heterogeneous Agent Models in Economics and Finance, In: Handbook of Computational Economics II: Agent-Based Computational Economics, edited by Leigh Tesfatsion and Ken Judd , Elsevier, Amsterdam 2006, pp.1109-1186.

This chapter surveys work on dynamic heterogeneous agent models (HAMs) in economics and finance. Emphasis is given to simple models that, at least to some extent, are tractable by analytic methods in combination with computational tools. Most of these models are behavioral models with boundedly rational agents using different heuristics or rule of thumb strategies that may not be perfect, but perform reasonably well. Typically these models are highly nonlinear, e.g. due to evolutionary switching between strategies, and exhibit a wide range of dynamical behavior ranging from a unique stable steady state to complex, chaotic dynamics. Aggregation of simple interactions at the micro level may generate sophisticated structure at the macro level. Simple HAMs can explain important observed stylized facts in financial time series, such as excess volatility, high trading volume, temporary bubbles and trend following, sudden crashes and mean reversion, clustered volatility and fat tails in the returns distribution.

Learning the CAPM through Bubbles

Bubbles are generally considered the outcome of investor irrationality or informational asymmetry, both objectionable in efficient markets with rational investors. We introduce an Intertemporal-CAPM with market clearing between high- and low-risk-averse rational investors who learn the CAPM under incomplete, yet symmetric information. Periodic equilibrium prices make a lognormal price process that nests the classic CAPM with a potential for endogenous bubbles through learning. The absence of comparables through the introductory phase of new technologies results in unstable return dynamics that might burst to bubbles or decline to near-zero, “pink-sheet†valuations. When the technology shifts phase to generate real profits the return dynamics is convergent, revealing the classic CAPM. Once the real technology return is observable, over- and under-pricing can be assessed, resulting in prompt positive or negative price adjustments toward the CAPM valuation. Correspondence with the Abreu and Brunnermeier (2003) model of bubbles with rational arbitrageurs is presented as well.ICAPM; Bubbles; New Technologies; Rational Expectations

A nonlinear structural model for volatility clustering

A simple nonlinear structural model of endogenous belief heterogeneity is proposed. News about fundamentals is an IID random process, but nevertheless volatility clustering occurs as an endogenous phenomenon caused by the interaction between different types of traders, fundamentalists and technical analysts. The belief types are driven by adaptive, evolutionary dynamics according to the success of the prediction strategies as measured by accumulated realized profits, conditioned upon price deviations from the rational expectations fundamental price. Asset prices switch irregularly between two different regimes --periods of small price fluctuations and periods of large price changes triggered by random news and reinforced by technical trading -- thus, creating time varying volatility similar to that observed in real financial data.

Learning as a rational foundation for macroeconomics and finance

Expectations play a central role in modern macroeconomics. The econometric learning approach, in line with the cognitive consistency principle, models agents as forming expectations by estimating and updating subjective forecasting models in real time. This approach provides a stability test for RE equilibria and a selection criterion in models with multiple equilibria. Further features of learning – such as discounting of older data, use of misspecified models or heterogeneous choice by agents between competing models – generate novel learning dynamics. Empirical applications are reviewed and the roles of the planning horizon and structural knowledge are discussed. We develop several applications of learning with relevance to macroeconomic policy: the scope of Ricardian equivalence, appropriate specification of interest-rate rules, implementation of price-level targeting to achieve learning stability of the optimal RE equilibrium and whether, under learning, price-level targeting can rule out the deflation trap at the zero lower bound.cognitive consistency; E-stability; least-squares; persistent learning dynamics; business cycles; monetary policy; asset prices