Jensen-Maheu (2008) Bayesian Semiparametric Stochastic Volatility Modeling

Summary

Jensen and Maheu (2008) propose a semiparametric Bayesian SV model (SV-DPM) that combines a parametric AR(1) log-volatility process with a nonparametric Dirichlet Process Mixture (DPM) for the return innovation distribution. The key finding is that misspecifying the return distribution as Gaussian causes severe upward bias in the volatility-of-volatility parameter ($\sigma_v^2$): on CRSP daily returns (1980–2006), the Gaussian SV model overestimates $\sigma_v^2$ by 2.7× relative to SV-DPM; on simulated data from a mixture DGP the inflation reaches ~20×. The DPM flexibly captures skewness and excess kurtosis without restricting them a priori, while correctly attributing these features to the innovation distribution rather than to volatility dynamics.

Key Claims

• Distributional misspecification inflates vol-of-vol. When return innovations are non-Gaussian (fat-tailed, skewed), a Gaussian SV model confounds excess kurtosis with volatility variation, leading it to overestimate $\sigma_v^2$. The DPM acts as a prior-robust filter, correctly separating distributional shape from volatility dynamics.
• SV-DPM model. Returns $y_t = \exp(h_t/2)\varepsilon_t$ where the innovation $\varepsilon_t \sim G$ with $G$ drawn from a Dirichlet Process Mixture: $G = \sum_{j=1}^\infty V_j \delta_{(\mu_j,\sigma_j^2)}$, so $y_t | h_t \sim \sum_{j=1}^\infty V_j \mathcal{N}(\mu_j,\, \sigma_j^{-2}\exp\{h_t\})$. The stick-breaking weights satisfy $V_j = U_j \prod_{l<j}(1-U_l)$, $U_j \sim \mathrm{Beta}(1,\alpha)$.
• Log-volatility is standard AR(1). $h_{t+1} = \mu + \phi(h_t-\mu) + \eta_t$, $\eta_t \sim \mathcal{N}(0,\sigma_v^2)$, independent of $\varepsilon_t$. No leverage effect is modelled.
• Base distribution. $G_0 = \mathcal{N}(\mu_0, B_0) \times \mathrm{Inv\text{-}Gamma}(a_0, b_0)$ for the cluster mean-precision pairs; conjugate Normal-IG form enables exact analytic updates for cluster parameters.
• MCMC — 4 blocks. (1) Truncated-Normal/IG for $(\phi, \sigma_v^2)$; (2) Fleming-Kirby (2003) random-length block sampler for $\{h_t\}$ — groups consecutive observations into randomly sized blocks and runs a Kalman smoother within each block; (3) Chinese Restaurant Process updates for cluster assignments $\{k_t\}$ and cluster parameters $\{(\mu_j,\sigma_j^2)\}$ via West-Müller-Escobar (1994) / MacEachern-Müller (1998); (4) Gamma posterior update for concentration parameter $\alpha$.
• CRSP results (6,815 daily returns, 1980–2006). SV-DPM: $\hat\phi=0.9877$, $\hat\sigma_v^2=0.0103$, avg. $k=7.16$ clusters used. SV-N: $\hat\phi=0.9795$, $\hat\sigma_v^2=0.0276$ — 2.7× larger. Persistence bias is also present but smaller in magnitude.
• Simulation study. Under a 3-component Gaussian mixture DGP, SV-N's $\hat\sigma_v^2$ is ~20× the true value; SV-DPM recovers the true parameters accurately and assigns most mass to 3 clusters.
• DPM recovers correct cluster count. With $k=7.16$ clusters on average in the CRSP data, the DPM identifies a multi-modal return innovation density consistent with documented fat tails and mild left skew in daily equity returns.

Concepts Introduced or Extended

• Dirichlet Process Mixture
• Stochastic Volatility
• Mixture of Normals

Entities Mentioned

• Mark J. Jensen
• John M. Maheu
• Sangjoon Kim
• Neil Shephard
• Siddhartha Chib

Quotes

"By modeling the error distribution nonparametrically with a Dirichlet process mixture, we allow the data to determine the shape of the innovation distribution without restricting it a priori."

"The misspecification of the return distribution as Gaussian causes the volatility of volatility parameter to be overestimated by a factor of about 2.7 in the CRSP data."

My Take

The paper makes a clean, important point: the vol-of-vol parameter ($\sigma_v^2$) in SV models is a nuisance absorber for any distributional misspecification. The DPM fix is elegant and computationally feasible. The main limitation is the absence of a leverage effect — the model is symmetric in returns, which is empirically problematic for daily equity data (see Omori-Chib-Shephard-Nakajima (2007) for the leverage extension). A natural extension would combine the bivariate mixture sampler of OCSN (2007) with the DPM return distribution, attacking both misspecification axes simultaneously. The Fleming-Kirby block sampler for $\{h_t\}$ is an interesting alternative to the Carter-Kohn smoother used in KSC-type algorithms and deserves attention in its own right.