Jacquier-Polson-Rossi (1994) Bayesian Analysis of Stochastic Volatility Models

Summary

Introduces the first practical Markov chain Monte Carlo (MCMC) estimator for the discrete-time AR(1) log-variance stochastic volatility (SV) model. The latent variance path $\{h_t\}_{t=1}^T$ is treated as augmented data and sampled element-by-element via a cyclic independence Metropolis chain; parameters are drawn jointly from conjugate posteriors conditional on the full path. Monte Carlo experiments demonstrate that Bayesian root mean squared error (RMSE) for the persistence parameter is 3.7–4.1× smaller than method of moments (MM) and quasi-maximum likelihood (QML), and that the Bayesian smoother for latent variances dominates the approximate Kalman filter even when the Kalman filter is given the true parameters.

Key Claims

• The T-dimensional likelihood $p(y \mid \theta) = \int p(y \mid h, \theta) p(h \mid \theta)\, dh$ is intractable; data augmentation treats $h = \{h_t\}$ as a parameter, enabling exact posterior simulation via MCMC.
• The key conditional $p(h_t \mid h_{-t}, \theta, y_t) \propto h_t^{-1/2} \exp(-y_t^2/2h_t) \cdot h_t^{-1} \exp(-(\ln h_t - \mu_t)^2/2\sigma_t^2)$ has an inverse-gamma (IG)-shaped return piece and a lognormal-shaped state piece; the lognormal is approximated by moment-matching to an IG, giving a combined IG blanketing density for an independence Metropolis step with 70–80% acceptance.
• Parameters $(\alpha, \beta, \sigma_v^2)$ are drawn in one conjugate block (Normal-IG) conditional on the full variance path.
• Method of moments suffers corner solutions in >50% of simulated samples when persistence is high ($\beta \approx 0.98$) and vol-of-vol is low ($\mathrm{Var}(h)/E[h]^2 = 0.1$); QML is competitive only when $\sigma_v$ is large.
• Bayesian smoother $\hat{h}_t = E[h_t \mid y]$ achieves lower RMSE than the approximate Kalman smoother with true parameters (median relative RMSE 1.11; range 1.004–1.24) — the Gaussian approximation to $\ln z_t^2$ in the Kalman filter introduces irreducible error.
• Empirical application to 15 exchange-rate and equity datasets: estimated persistence $\hat{\beta} \approx 0.83$–$0.97$, half-life of volatility shocks 3–26 weeks.

Concepts Introduced or Extended

• Stochastic Volatility
• Bayesian Hierarchical Model

Entities Mentioned

• Eric Jacquier
• Nicholas Polson
• Peter E. Rossi

Quotes

"We propose a new Bayesian approach to the estimation of stochastic variance models... Our approach is exact in the sense that it does not require any approximation beyond the usual Monte Carlo error."

"It is remarkable that the Bayes smoothing solution dominates not only approximate Kalman filtering based on the MM and QML estimates but also the approximate Kalman-filtering solution using the true parameters."

My Take

JPR 1994 is the founding paper for MCMC estimation of SV models. The single-move Metropolis step is clever but slow to mix when volatility is persistent. Kim, Shephard, and Chib (1998) later improved mixing via multi-move sampling using a mixture-of-normals approximation to the log-chi-squared noise, but JPR's data-augmentation framing and the dominance result over approximate Kalman filtering remain the conceptual anchors for the field.