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Article Excerpt
Summary. We show that in a Lucas endowment economy in which the process for dividends is described by a lattice tree subject to infrequent but observable structural breaks, in equilibrium recursive rational learning may inflate the equity risk premium and reduce the risk-free interest rate for low levels of risk aversion. The key condition for these results to obtain is the presence of sufficient initial pessimism. The relevance of these findings is magnified by the fact that under full information our artificial economy cannot generate asset returns matching the empirical evidence for any positive relative risk aversion.
Keywords and Phrases: Rational learning, Equity premium, Structural breaks.
JEL Classification Numbers: D83, E44, G12.
1 Introduction
Since Mehra and Prescott (1985, MP), we know that a Lucas-style economy with power, time-additive utility, complete markets, no frictions, and in which a representative agent forms rational expectations cannot reproduce the historical mean equity risk premium. This impasse is labeled the equity premium puzzle. In practice, unreasonably high levels of relative risk aversion must be used to get a sensible risk premium. Moreover, in MP's framework high risk aversion implies an implausibly low elasticity of intertemporal substitution that forces the real riskless rate to exceed historical averages, the risk-free rate puzzle. Following MP, a large effort has been made to solve the puzzles. Many papers have focused on the role of power utility constraining the elasticity of intertemporal substitution to be the inverse of relative risk-aversion. The completeness assumption has been removed, showing that the non-insurable uncertainty in consumption increases the equity premium. Another literature has experimented with borrowing constraints and transaction costs.
Less attention has been given to the assumption of full-information rational expectations as a way to close the model and impose some consistency on beliefs. Some Authors have approached the puzzles leveraging on events unique to the US history, particularly the Great Depression. Rietz (1988) shows that if agents are characterized by biases in beliefs reflecting catastrophic scenarios not present in the data, a high equity premium is generated under reasonable risk aversion. Cecchetti, Lam, and Mark (2000) study the effects of belief distortions on asset prices and show that some pessimism relative to the maximum-likelihood estimates generates plausible moments. However, the origin of such pessimistic fears is unclear. This literature relies on deviations between realized and subjectively perceived beliefs, often in arbitrary ways. On the opposite, in this paper we try to build a model in which persistent rational pessimism and crash fears may arise as a consequence of the application of optimal estimation methods.
When agents lack full information on some parameters, their subjective beliefs may rationally deviate from the (true) distribution of the state variables without postulating in an ad-hoc fashion that markets agree on a disaster state. A few papers have studied asset pricing under recursive learning (e.g. Barsky and De Long, 1993; Timmermann, 2001). However, the implications for the equity premium are not pursued. An exception is Brennan and Xia (2001): when an agent estimates the dividend drift using a risk aversion coefficient of 15, they derive an equity premium of 6 percent and a risk-free rate of 2.5 percent. However, they stress (p. 266) that learning has only second-order effects on the equity premium. (1) We show that first-order effects can be obtained using a plausible degree of curvature of the utility function.
Our paper offers two contributions. First, it removes the assumption lying at the core of the research on the equity premium puzzle that agents form full-information (FI) rational expectations. We focus on a restrictive rational learning mechanism that implies that prices reflect all possible, future perceived distributions of the parameters' estimates (future learning). Second, we prove that under the assumptions of MP, when dividends follow a binomial lattice process, in equilibrium recursive rational learning may inflate the equity risk premium and reduce the risk-free interest rate for low levels of risk aversion. The key condition for these results to obtain is the presence of sufficient initial pessimism.
One related literature has studied the ability of rational beliefs (RB) to explain the puzzles (see Kurz, 1994). Simulations in Kurz and Motolese (2001) have documented that under restrictions on the distribution of beliefs across agents, high premia may be generated along with low interest rates. There are obvious similarities between the learning-based and the RB approaches. In particular, we study a stable but non-stationary system characterized by infrequent structural breaks, i.e. we approximate non-stationarity with a sequence of time intervals over which the unknown DGP is invariant, see Kurz's (1994) "environments". Also the RB approach to the equity premium puzzle has generated results from restrictions on the structure of beliefs that can be read in terms of pessimism vs. optimism. The crucial difference consists in the fact that RB theory has focused on the effects of heterogeneity, in the sense that divergent beliefs are not only allowed, but they are essential for the approach. On the contrary, our paper adopts the fiction of a representative agent and thus concentrates on the effects of dynamics in beliefs. Moreover, RB theory lists future spot asset prices as part of the state space, that is therefore enlarged to imply endogenous uncertainty. In our model uncertainty remains strictly exogenous, although it is greatly amplified by learning.
2 The model
Two assets are traded: a one-period, risk-free, zero coupon bond in endogenous zero supply, yielding an interest rate of [r.sub.t.sup.f] (with price [B.sub.t] = 1/(1 + [r.sub.t.sup.f])), and a stock index with price [S.sub.t] in exogenous net unit supply. The stock pays an infinite stream of real dividends {[D.sub.t]}[.sub.t=1.sup.[infinity]]. These dividends are perishable; they cannot be reinvested and they must be consumed when received. The real growth rate of dividends [g.sub.t] [equivalent to] [[D.sub.t]/[D.sub.t-1]] - 1 follows a two-state Bernoulli process with parameter [pi]. At each date the real growth rate is either high ([g.sub.h]) or low ([g.sub.l] with - 1 < [g.sub.l] < < [g.sub.h]). When confronted with smooth processes such as US consumption growth, a zero first-order autocorrelation is realistic (see Barsky and De Long, 1993). [pi] may be subject to structural breaks. For simplicity, assume that breaks are observable. Events of the magnitude of the Great Depression and world energy crises are likely to be recognized....
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