Rational beliefs or distorted beliefs: the equity premium puzzle and micro survey data.

• Author: Park, Cheolbeom

1. Introduction

   Since Shiller (1982) and Mehra and Prescott (1985) questioned why the gap between the rates of returns from stocks and bonds is so large, the equity premium puzzle has attracted the attention of many economists. The numerous explanations for the puzzle that have been put forth can be categorized into three approaches. The first approach is to explain the puzzle under full rationality by introducing more complex utility functions. Epstein and Zin (1989) and Weil (1989) use a utility function that breaks the tight relation between the risk aversion coefficient and the elasticity of intertemporal substitution. Constantinides (1990) introduces the utility function with habit formation to the literature, and Campbell and Cochrane (1999) succeed in explaining the equity premium puzzle and stock return predictability by making use of a particular variant of habit formation.

   The second approach is to see the puzzle as being a result of distorted beliefs while maintaining the traditional power utility function. Rietz (1988) shows that the puzzle can be explained if consumers believe that a disastrous situation has a higher probability of arising than the historical data imply. Recently, Cecchetti, Lam, and Mark (2000) also succeeded in matching the first and second moments of the equity premium and the predictability of stock returns by assuming that the representative consumer, with the traditional power utility function, has distorted beliefs of a particular kind that lead him/her to expect that both expansions and contractions end more quickly than the data suggest.

   The third approach is to see the puzzle as being a result of market frictions such as borrowing constraints and short-sale constraints. Heaton and Lucas (1996) show the potential of this approach and Luttmer (1999) estimates the magnitude of such frictions using the first-order conditions from utility maximization.

   However, because of the success of Campbell and Cochrane (1999) and Cecchetti, Lam, and Mark (2000) in explaining various aspects of the asset pricing phenomena, including the equity premium puzzle, this paper mainly focuses on an investigation of the first two approaches with the use of survey forecasts. (1) In spite of the success of both approaches, the puzzle has not yet been resolved. Because both approaches use completely different mechanisms to explain the puzzle, we still do not know which one provides a better description of the cause of the puzzle. In other words, because the stylized facts in the financial market can be explained by both models, it is hard for economists to judge which is the correct mechanism by looking at financial and consumption data. (2) Hence, the main purpose of this paper is to derive different implications of consumers' expectations from both models and to examine which implication is supported more through an investigation of the survey forecasts.

   Because Campbell and Cochrane's model retains rational expectations, whereas Cecchetti, Lam, and Mark's model is based on distorted beliefs, the two models' implications for survey forecasts will differ. Whereas Campbell and Cochrane (1999) would predict that survey forecasts are rational, Cecchetti, Lain, and Mark (2000) would claim that forecasts are excessively pessimistic over expansions and excessively optimistic over contractions. Using the Livingston survey forecasts for the Standard and Poor's 500 index, which is managed by the Federal Reserve Bank of Philadelphia, this paper conducts a test of these implications. To check whether the expected excess returns from the survey forecasts are rational, a panel data model allowing a four-dimensional structure for forecast errors (individual bias, business cycle effect, aggregate shocks, and idiosyncratic errors) is analyzed. (3)

   The test results seem to support the model of Cecchetti, Lam, and Mark (2000). The expected equity premium implied by the survey forecasts is, on average, lower than the actual realized equity premium during the sample period. The average bias across individual economists is significant and appears to vary over the business cycle. As Cecchetti, Lam, and Mark (2000) assume, the fluctuations of the average bias imply that forecasts are excessively pessimistic over expansions and excessively optimistic over contractions.

   The structure of this paper is as follows. Section 2 briefly summarizes the two models of Campbell and Cochrane (1999) and Cecchetti, Lam, and Mark (2000) and derives implications from them. Section 3 discusses the data used and explains how the variables in the test are constructed. Section 4 presents the test methodology and the results from the analysis of survey forecasts. Section 5 contains concluding remarks.

2. Equity Premium Models and Their Empirical Implications

   To understand the equity premium puzzle, suppose that the representative consumer lives in an economy described by the Lucas (1978) tree model. He/she must decide whether to sacrifice some consumption in order to add another unit of the risky asset to his/her portfolio. The risky asset in this economy is a claim to the future stream of the nonstorable endowment called dividends ([D.sub.t]). Hence, the consumer's utility maximization problem can be written as

   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

   where [E.sub.o] denotes the conditional expectation given information at time O, [C.sub.t] denotes consumption at time t, U(.) denotes the utility function, [beta] [member of] (0,1) denotes the time discount rate, [P.sub.t] denotes the price of a share of the risky asset, and [A.sub.t] denotes the consumer's shareholdings at time t. The first-order conditions for the above problem can be reduced to

   (1) 1 = [E.sub.t][[M.sub.t+1].[R.sub.t+1]],

   where [M.sub.t+1] [equivalent to] [beta](U'([C.sub.t+1])) / (U'([C.sub.t])) and [R.sub.t+1] [equivalent to] (P.sub.t+1] + [D.sub.t+1]) / [P.sub.t]. In addition, the corresponding condition for the risk-free asset in this economy can be written as

   (2) 1 = [E.sub.t][[M.sub.t+1].[R.sup.f.sub.t+1]],

   where [R.sup.f.sub.t+1] is the return on the risk-free asset between time t and t + 1.

   The equity premium is the difference between [R.sub.t+1] and [R.sup.f.sub.t+1] in Equations 1 and 2. With the assumptions of the constant relative risk aversion (CRRA) utility function and the lognormal consumption growth, Equations 1 and 2 imply

   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

   where [sigma](.) is the standard deviation, [rho](,) is the correlation, and [gamma] is the coefficient of relative risk aversion. The ratio of mean excess returns to standard deviation is known as the Sharpe ratio. Equation 3 implies the largest possible bound of the Sharpe ratio in general, which is known as the Hansen-Jagannathan bound (Hansen and Jagannathan 1991).

   However, [[rho].sub.t]([M.sub.t+1], [R.sub.t+1]) = -1 (a perfect conditional correlation) is usually implied by most equilibrium-based consumption models, including the models in Lucas (1978), Campbell and Cochrane (1999), and Cecchetti, Lam, and Mark (2000) because consumption growth is the only source of uncertainty. But the perfect conditional correlation between [M.sub.t+1] and [R.sub.t-1] does not necessarily imply a perfect unconditional correlation between these variables. The reason is that the relation between [M.sub.t+1] and [R.sub.t+1] varies over time with the level of current consumption in the Lucas model. (4)

   The equity premium puzzle can be shown using Equation 3. In the United States, the postwar real aggregate stock return over real return on the Treasury bill rate is 8%, on average, while the standard deviation of the aggregate stock return is 16%. Because the standard deviation of the aggregate consumption growth in nondurables and services is about 1%, Equation 3 implies that the coefficient of relative risk aversion for the consumer should be at least 50. (5) Because traditional risk aversion values are 1 to 10, the Sharpe ratio is too large to be explained by the consumption theory reflected in Equation 3.

   Since Shiller (1982) and Mehra and Prescott (1985) pointed out the problem, numerous explanations have been put forward to resolve the puzzle. The models in Campbell and Cochrane (1999) and Cecchetti, Lain, and Mark (2000) can explain the equity premium puzzle, as well as other puzzles in the financial market. The former, which takes a more common approach, argues that the puzzle arises from complex utility functions under full rationality. The latter argues that the puzzle arises from distorted beliefs under the traditional power utility function. The following subsections briefly examine the mechanisms of each approach and derive empirically testable implications.

   Rational Expectation Approach with Habit Formation

   Campbell and Cochrane (1999) succeed in coherently resolving not only the equity premium puzzle, but also the predictability and volatility puzzles, by using the utility function with external habit formation. Their utility function is given by

   U([C.sub.t]) = ([C.sub.t] - [X.sub.t]).sup.1-[gamma]-1, / 1 - [gamma] (4)

   where [X.sub.t] is the level of habit. By defining the state variable, [S.sub.t] [equivalent to] ([C.sub.t] - [X.sub.t])/[C.sub.t], the stochastic discount factor under the utility function in Equation 4 can be written as

   [M.sup.c.sub.t+1] [equivalent to]...