Abstract
In a macroeconomic framework, I quantitatively evaluate the theory of Loss Aversion/Narrow Framing (LANF) as a resolution to the Equity Premium Puzzle (EPP). The EPP is where the neoclassical asset pricing model cannot be reconciled with the empirical fact that stocks have much higher returns than risk-free assets. The prior predictive analysis employed follows a Bayesian approach that draws realizations for preferences that describe the degree of LANF characterizing consumer’s tastes. The analysis is also extended along two more dimensions: the variance of aggregate uncertainty and the elasticity of labor. The priors used are carefully defined from previous works in the literature. This Monte Carlo procedure finds that the theory is unable to jointly describe the equity premium and labor’s elasticity of supply. That is, only when the labor supply elasticity is unreasonably low can LANF preferences generate any equity premiums. Alternatively, when the elasticity is more realistically high, LANF preferences fail to generate significant premiums. My analysis therefore concludes that a resolution to the EPP via a theory of LANF must be modified along the description of labor’s choices. As ancillary result, the hybrid perturbation-projection method developed for this experiment is shown to be a robust technique.
- 1
Mehra and Prescott (1985) observe that, over the 90-year period 1889–1978, the average real annual yield on the Standard and Poor 500 Index is 7%. Whereas the average yield on short-term debt is <1%. The data they used for the risk-free rate includes asset returns on the 90-day Treasury bill, Treasury certificate, and 60-day to 90-day commercial paper.
- 2
See Fernández-Villaverde and Rubio-Ramirez (2006) for details
- 3
Data source: Boldrin, Christiano, and Fisher 2001, Danthine and Donaldson 2002, Guvenen 2009.
- 4
Boldrin, Christiano, and Fisher (2001).
- 5
Danthine and Donaldson (2002).
I am grateful to Stuart Fowler, Duane Graddy and Kevin Zhao for helpful comments and suggestions. I also appreciate contributive comments from Lars Peter Hansen at the American Economic Association meeting, and Edgar Ghossoub at the Southern Economic Association meeting.
Appendix
Details for hybrid perturbation-projection method
Perturbation with COV
The solution method here makes use of Taylor series expansion with changes of variables (Judd 1996, 2002; Fernández-Villaverde and Rubio-Ramirez 2006). Every policy function (i.e., lt, kt+1, etc.) is first approximated by a perturbation solution:
where λL=λH=λ=0 (i.e., no LANF). The solution in (10) is, presumably, accurate around {z, k, σ, λ}={0, kss, 0, 0}. Then, λ and σ are replaced by a change of variables (COV) defined by either a constant or a polynomial in the states. More specifically, let the COV be;
y3,t=τ0σ,
y4,t=α0λ+α1zt+α2(kt–kss).
To get a better understanding of how COV works, consider a simple example from Judd (2002). At first, the researcher has a basic second order Taylor series expansion of a function f(x):
where x has been expanded around a. The COV is then defined by y=Y(x) with an inverse function existing as x=X(y). The COV finds g(y)=f(X(y)) at y=b=Y(a). Note that g(y) can be approximated with Chain Rule at second order by:
More concretely, suppose a=1 and the COV is y=Y(x)=log(x). Then, we immediately see that b=0 and the inverse function is x=X(y)=exp(y). As a result, X′(b)=1 and X″(b)=1. The COV expansion is thus:
where {f(a), f′(a), f″(a)} are presumed to be known from equation (11).
In this study, the proposed transformation is:
where xt=[zt, kt, σ, λ]′. The inverse function is thus:
And, suppose that an initial second order perturbation gave equation (10) for kt+1 of:
where
Applying the COV gives:
where
My choice of COV gives Σ<1>=1/τ0, Σ<2>=0, Λ<1,0,0>=–α1/α0, Λ<0,1,0>=–α2/α0, and Λ<0,0,1>=1/α0 for example.
Projection methods
Given the COV transformations for the policy solution set:
In the next step, using (6), the optimality equations (1)–(3), (5), and (7) are evaluated using the COV perturbation solutions for any given set of states and unknown parameters: {kt, zt, σ, λ, α1, α2}. These equations are stacked into a vector that is denoted EER(kt, zt, σ, λ, α1, α2). By summing up EER by element and across sets of values for {k, z}, the minimization problem is:
Following Fernández-Villaverde and Rubio-Ramirez (2006), a grid for k is made by a grid of 70 points intended so that
Evaluation of solution method
If the Hybrid Perturbation-Projection method is an improvement, then the Euler equation errors (EERs) evaluated at the solutions should be in magnitude smaller than the EERs evaluated at the regular perturbation solutions represented in equation (10). Figure 5 shows the relationship between the relative EERs (the ratio of the EERs under the hybrid method to the EERs under the regular perturbation method) and realized variables σ, λH, and b0. We see that the relative EERs are all less than one implying that the Hybrid Perturbation-Projection method reduces the computational modeling error. Also, as expected, the relative EERs are related to σ, λH, and b0. Low realizations for σ, λH, and b0 reduce the importance of LANF preferences and technology shocks. This economy, with small distortions, is described well by the regular perturbation method. In total, the evidence suggests successful minimizations of the EERs using the proposed Hybrid Perturbation-Projection method.
Relationship between Euler errors and variable estimates (CD baseline model).
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