A Bayesian Solution to the Equity Premium Puzzle

Summary

Jobert, Platania, and Rogers (2006) show that parameter uncertainty about dividend growth is sufficient to resolve the equity premium puzzle. When a CRRA agent treats the mean and precision of log-dividend growth as unknown — rather than imposing point estimates — the equity premium can be matched at plausible risk-aversion coefficients $R \in (1, 2)$ . The key insight is that the mean dividend-growth parameter $\mu$ requires roughly 1,550 years of data to estimate to $\pm 1\%$ at 95% confidence, so even lifetime investors face enormous posterior uncertainty. A doubly Bayesian particle-filter calibration on Mehra-Prescott (1889–1978) data recovers $R \approx 1.65$ (IQR $[1.62, 1.71]$ ) and $\beta \approx 0.95$ , dramatically reducing the residual sum of squares relative to the classical calibration.

Key Claims

The standard CRRA representative-agent model (Mehra-Prescott 1985) fails because it uses the classical point estimate of $\mu$ rather than averaging over posterior uncertainty.
The mean log-dividend-growth $\mu$ is so imprecisely estimated from historical data ( ${\sim}1{,}550$ years needed for $\pm 0.01$ precision at 95%) that parameter uncertainty alone suffices to resolve the puzzle.
The agent holds a Gamma-Gaussian conjugate prior over $(\mu, \tau)$ — mean and precision of log-dividend growth — whose posterior updates each period in closed form, yielding hyperparameters $K_t, m_t, \alpha_t, b_t$ .
Without a convergence-ensuring prefactor $\varphi(\mu, \tau) = e^{-c/2\tau^2}(1 - \beta e^{-\nu\mu+\nu^2/2\tau})^+$ ( $\nu = R - 1$ ), the stock price formula diverges for priors placing mass on $\beta e^{-\nu\mu+\nu^2/2\tau} \geq 1$ (Geweke 2001). The prefactor resolves this and yields $S_t = y_t E_t[1/(1 - \beta e^{-\nu\mu+\nu^2/2\tau})]$ .
Equilibrium price formulas are $S_t = y_t F_t(0,0)/(F_t(0,0) - \beta F_t(\nu,0))$ and $B_t = \beta(F_t(0,R) - \beta F_t(\nu,R))/(F_t(0,0) - \beta F_t(\nu,0))$ , where $F_t(\lambda,\rho)$ is a one-dimensional integral over the Gamma-Gaussian posterior.
A particle filter with $25{,}000$ particles calibrated to Mehra-Prescott (1889–1978) data recovers $R$ IQR $[1.62, 1.71]$ and $\beta \approx 0.9487$ , compared to $R \gg 10$ under classical calibration.

Concepts Introduced or Extended

Entities Mentioned

A. Jobert
A. Platania
L.C.G. Rogers
John Geweke — convergence limitation of CRRA utility (Geweke 2001)

Quotes

"The difficulty lies in the fact that the mean parameter μ is well-nigh impossible to estimate precisely: it would take of the order of 1550 years of data to estimate μ to within ±0.01 with 95% confidence."

"The particle filter recovers a risk aversion parameter R in the interquartile range [1.62, 1.71], values that are eminently plausible."

My Take

The central contribution is sharp: parameter uncertainty is not a second-order correction but the dominant effect, because $\mu$ has a standard error that shrinks so slowly as to be economically permanent on any realistic investment horizon. The doubly Bayesian structure — agent updating $(\mu, \tau)$ while an econometrician infers $(\beta, R)$ and hyperparameters via particle filter — is elegant and internally consistent. The convergence prefactor $\varphi$ feels like a technical patch rather than a behavioral primitive, though Geweke's (2001) criticism of CRRA divergence is well-founded and the fix is mathematically clean. The result is sensitive to the Gamma-Gaussian prior family choice; robustness to heavier-tailed priors or learning about higher-order moments is an open question.