Kim-Shephard-Chib (1998) Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models

Summary

Kim, Shephard, and Chib (1998) develop a highly efficient multi-move Markov chain Monte Carlo (MCMC) algorithm for the standard discrete-time stochastic volatility (SV) model by transforming the observation equation via $y_t^* = \ln r_t^2 = h_t + \ln \varepsilon_t^2$ , approximating the non-Gaussian $\log\chi^2(1)$ measurement error with a fixed mixture of $K=7$ Gaussians. Conditional on discrete mixture-indicator variables, the model becomes a linear Gaussian state space, enabling exact Kalman filtering and Carter-Kohn (1994) multi-move simulation smoothing of the entire latent log-variance path in a single Gibbs block. They implement and compare three MCMC strategies — single-move (SM), mixture multi-move (MIX), and integration (INT) — finding that MIX achieves inefficiency factors of 1–3 versus 20–200 for SM; importance reweighting corrects the approximation error to recover exact Bayesian inference; and Bayes factors via the Chib (1995) identity decisively favor SV over GARCH and $t$ -GARCH for daily S&P 500 returns.

Key Claims

Model. The basic SV model is $y_t = \beta e^{h_t/2} \varepsilon_t$ , $\varepsilon_t \sim \mathcal{N}(0,1)$ , with state equation $h_{t+1} = \mu + \phi(h_t - \mu) + \sigma_\eta \eta_t$ , $\eta_t \sim \mathcal{N}(0,1)$ . Log-squaring gives $y_t^* = \ln y_t^2 = \ln\beta^2 + h_t + \ln\varepsilon_t^2$ , where the noise term $\xi_t = \ln\varepsilon_t^2$ follows a $\log\chi^2(1)$ distribution with mean $-1.2704$ and variance $\pi^2/2$ .
7-component Gaussian mixture (Table 4). The $\log\chi^2(1)$ distribution is approximated by $\sum_{i=1}^7 p_i \mathcal{N}(m_i, v_i^2)$ , with weights $\{p_i\}$ , means $\{m_i\}$ , and variances $\{v_i^2\}$ chosen by matching the first four moments plus the mode. The approximation reduces the log-likelihood error to below $10^{-4}$ across the entire support.
Three MCMC strategies. SM (single-move): draws $h_t$ one at a time; equivalent to a noisy version of the Jacquier-Polson-Rossi (JPR, 1994) Gibbs step in the log-squared space. MIX (mixture multi-move): given indicator draws $s_t \in \{1,\ldots,7\}$ , the model is linear Gaussian → Carter-Kohn (1994) simulation smoother draws the full path $\{h_t\}_{t=1}^T$ in one block. INT (integration): marginalizes over $\{h_t\}$ exactly using a particle filter; slowest but free of approximation error.
Inefficiency factors. INEF $= 1 + 2\sum_{k=1}^\infty \hat\rho_k$ measures sampling inefficiency relative to i.i.d. MIX achieves INEF $\approx 1$ –3 for $(\phi, \sigma_\eta^2, \mu)$ ; SM achieves INEF $\approx 20$ –200 — a 10–100× advantage for MIX.
Importance reweighting. Because the mixture approximates (not equals) the true $\log\chi^2(1)$ density, MIX draws are from a pseudo-posterior. The correction weight is $w_t \propto f_{\log\chi^2}(\xi_t) / \hat{f}_\text{mix}(\xi_t)$ ; after reweighting, all posterior summaries are exact. The effective sample size ratio exceeds 0.99 in simulations — the approximation is very accurate.
Parameter posterior block. Given the full path $\{h_t\}$ , the state equation $h_{t+1} = \mu + \phi(h_t - \mu) + \sigma_\eta \eta_t$ is a Gaussian AR(1) in $\{h_t - \mu\}$ . Parameters are drawn by Gibbs: $\sigma_\eta^2 \sim \mathrm{IG}$ (inverse-gamma), $(\mu, \phi)$ jointly Normal from the conjugate linear regression on $h_{t+1} - h_t$ given $h_t$ .
Marginal likelihood and Bayes factors. The Chib (1995) identity $\log p(y) = \log p(y|\theta^*) + \log p(\theta^*) - \log p(\theta^*|y)$ is evaluated at the posterior mode $\theta^*$ ; the log-likelihood $\log p(y|\theta^*)$ is computed via the Kalman filter under the 7-component mixture approximation. For daily S&P 500 returns (1981–1987), $\log_{10} BF(\text{SV vs. GARCH}) \approx +50$ ; SV also decisively beats $t$ -GARCH.
Empirical findings. Estimated $\hat\phi \approx 0.979$ (high persistence), $\hat\sigma_\eta \approx 0.148$ , $\hat\mu \approx -0.703$ for daily S&P 500. SV posterior implies slower mean-reversion and more time-variation than GARCH maximum likelihood (ML) estimates.

Concepts Introduced or Extended

Stochastic Volatility — multi-move MIX sampler as the practical solution for Bayesian SV inference
Mixture of Normals — 7-component fixed mixture approximating $\log\chi^2(1)$
Kalman Filter — simulation smoother for the entire latent log-variance path
Gibbs Sampler — multi-block Gibbs with indicator and path draws
State-Space Representation — log-squared transformation yields linear Gaussian state space
Marginal Data Density — Bayes factors via Chib (1995) identity comparing SV vs. GARCH

Entities Mentioned

Quotes

"The basic idea is that sampling the entire volatility path jointly, conditional on mixture indicators and parameters, is many times more efficient than sampling each $h_t$ from its own full conditional."

My Take

KSC (1998) is the paper that made Bayesian SV inference practical and is cited in virtually every subsequent SV paper. The 7-component mixture table is effectively a lookup table: once you fix those constants, the entire latent path can be drawn via an off-the-shelf Kalman smoother. The three-sampler comparison provides a controlled benchmark — MIX dominates SM on mixing speed while INT provides a correctness check on the approximation. The importance-reweighting device is elegant and shows the mixture error is negligible. The main limitation (noted by Yu (2005)) is that the log-squared transformation destroys the leverage effect: $\mathrm{Cor}(r_t, h_{t+1})$ is obscured when $r_t$ enters as $\ln r_t^2$ , so the multi-move step cannot be applied when leverage is a focus.