Omori-Chib-Shephard-Nakajima (2007) Stochastic Volatility with Leverage: Fast and Efficient Likelihood Inference

Summary

Omori, Chib, Shephard, and Nakajima (2007) extend the Kim-Shephard-Chib (1998) multi-move Markov chain Monte Carlo (MCMC) sampler from stochastic volatility (SV) models without leverage to the more realistic class with correlated return and volatility shocks. The core innovation is to approximate the joint density of the log-squared return innovation and volatility innovation by a 10-component mixture of bivariate normals (rather than the 7-component univariate mixture of KSC), using an analytically derived linear approximation of $\exp(\varepsilon_t^*/2)$ within each component. The resulting sampler inherits all the computational advantages of KSC — simplicity, use of the simulation smoother, weak serial dependence in draws — while being exact (up to a negligible approximation corrected by importance reweighting) for $\rho \neq 0$ . Applied to TOPIX (Tokyo Stock Price Index) daily returns, the leverage model is decisively preferred over the symmetric SV.

Key Claims

ASV model. The leverage SV model (ASV) is $y_t = \varepsilon_t\exp(h_t/2)$ , $h_{t+1} = \mu + \phi(h_t - \mu) + \eta_t$ , where $(\varepsilon_t, \eta_t) \sim N_2(0, R)$ with $R = \begin{pmatrix} 1 & \rho\sigma \\ \rho\sigma & \sigma^2 \end{pmatrix}$ . The parameter $\rho < 0$ captures the leverage effect; prior work (Jacquier-Polson-Rossi [JPR] 2004, Yu 2005) could not combine multi-move efficiency with $\rho \neq 0$ .
10-component bivariate mixture. The joint density $f(\varepsilon_t^*, \eta_t \mid d_t, \rho, \sigma)$ — where $\varepsilon_t^* = \log\varepsilon_t^2$ and $d_t = \text{sign}(y_t)$ — is approximated as $\sum_{j=1}^{10} p_j N(\varepsilon_t^* \mid m_j, v_j^2) N(\eta_t \mid d_t\rho\sigma\exp(m_j/2)\{a_j + b_j(\varepsilon_t^* - m_j)\},\, \sigma^2(1-\rho^2))$ . The linear approximation of $\exp(\varepsilon_t^*/2)$ has closed-form coefficients $a_j = \exp(v_j^2/8)$ , $b_j = \tfrac{1}{2}\exp(v_j^2/8)$ (mean-square-optimal). Crucially, the approximation quality is independent of $\rho$ .
MCMC algorithm (2 steps). Step 1: draw mixture indicators $s_t \in \{1,\ldots,10\}$ from a discrete full conditional. Step 2: (a) draw $\theta = (\phi,\rho,\sigma)$ via Metropolis-Hastings with a Kalman-filter-based proposal; (b) draw $(\{h_t\}, \mu)$ jointly via the Gaussian simulation smoother (Carter-Kohn 1994; de Jong-Shephard 1995) conditional on $\theta$ and $\{s_t\}$ .
Importance reweighting. Draws from the approximate posterior are reweighted using $w_k \propto \prod_{t=1}^n f(\varepsilon_t^{*k}, \eta_t^k \mid d_t, \rho^k, \sigma^k) / g(\varepsilon_t^{*k}, \eta_t^k \mid d_t, \rho^k, \sigma^k)$ to recover exact posteriors. For TOPIX, log-weight std dev $\approx 0.34$ (vs. $\approx 1.0$ for KSC without leverage) — the approximation error is negligible.
Efficiency. Inefficiency factors for TOPIX: $\phi$ = 9.3, $\sigma$ = 13.0, $\rho$ = 6.8, $b = \exp(\mu/2) = 2.7$ . All small, indicating near-i.i.d. draws; favorable comparison with JPR (2004) single-move MH.
Empirical (TOPIX 1998–2002, n=1232). Posterior means: $\hat\phi = 0.951$ , $\hat\sigma = 0.134$ , $\hat\rho = -0.362$ [95% credible interval (CI): $-0.593$ , $-0.107$ ]. Log Bayes factor (Chib 1995) for ASV vs. SV $\approx 2.24$ ; leverage is decisively present. Model ranking (log marginal likelihoods): ASV-t $\approx$ ASV $\approx$ SP (superposition) $\gg$ SV-t $\gg$ SV.
Extensions. (1) Superposition model: $h_t = \sum_i a_{i,t}$ where each $a_{i,t}$ is an independent AR(1) with its own $\phi_i$ and leverage $\rho_i$ . (2) Heavy-tailed return errors: Student- $t$ or Gaussian scale mixture. (3) General framework: $y_t = \varepsilon_t\exp(h_t/2)$ , $h_t = z_t'a_t$ , $a_{t+1} = b_t + T_t a_t + \gamma_t$ with correlated $(\varepsilon_t, \gamma_t)$ .

Concepts Introduced or Extended

Entities Mentioned

Quotes

"The approach relies on the novel idea of approximating the joint distribution of the outcome and volatility innovations by a suitably constructed ten-component mixture of bivariate normal distributions." (Abstract)

My Take

The paper is the natural completion of the KSC program. KSC (1998) established the mixture-sampler approach for $\rho=0$ ; the long-standing belief was that correlated innovations fundamentally prevented this approach from working. Omori et al. show the obstacle is purely technical and elegantly resolve it with a bivariate mixture whose quality does not depend on $\rho$ . The analytical derivation of $a_j$ and $b_j$ is clean and the negligible log-weight variance confirms the approximation is tight even at $\rho = -0.9$ . The main limitation is that it maintains the KSC log-squaring transformation, which discards sign information about $y_t$ (partially recovered via $d_t$ ) and implies a degree of precision loss relative to single-move exact samplers. For most practical purposes, however, the efficiency gains make it the preferred approach for discrete-time ASV estimation.