Chib-Nardari-Shephard (2006) Analysis of High Dimensional Multivariate Stochastic Volatility Models

Summary

Chib, Nardari, and Shephard (2006) propose and estimate a flexible factor-based multivariate stochastic volatility (MSV) model for high-dimensional financial time series. Each of $p$ asset returns loads on $k$ latent factors, with both asset-specific and factor-specific log-volatilities following independent first-order autoregressive (AR(1)) processes. The model accommodates Student-t observation errors (via scale mixing) and asset-specific jumps. The key algorithmic innovation is a reduced blocking scheme that samples the factor loading matrix $B$ marginalized over the factors, which dramatically improves Markov chain Monte Carlo (MCMC) mixing. The log-likelihood is estimated by an auxiliary particle filter, enabling Bayes factor model comparison. The approach scales to $p=50$ series and 688 parameters — a scale previously infeasible for multivariate stochastic volatility (SV) estimation.

Key Claims

• Model (MSVJt): $y_t = Bf_t \cdot q_t + u_t$, where $f_t \sim \mathcal{N}_k(0, D_t)$ are latent factors, $q_t$ are Bernoulli jump indicators, $u_t$ has element-wise scale mixing $u_{jt} = \lambda_{jt}^{-1/2} e_{jt}$ with $\nu_j/\lambda_{jt} \sim \chi^2(\nu_j)$ (Student-t tails). Each log-volatility $h_{jt}$ (for asset $j=1,\ldots,p$ and factor $j=p+1,\ldots,p+k$) follows $h_{jt} = m_j + \phi_j(h_{j,t-1}-m_j) + \sigma_j Z_{jt}$.
• Identification: $B$ is lower-triangular with positive diagonal; this leaves $pk-(k^2+k)/2$ free elements $b$. At $p=50$, $k=8$: 688 total parameters.
• Four model variants: MSV (Gaussian, no jumps), MSVt (fat tails), MSVJ (Gaussian + jumps), MSVJt (fat tails + jumps).
• Key MCMC innovation — reduced blocking: Naive alternation of $b|f$ then $f|b$ produces inefficiency factors exceeding $1{,}000$. Instead, sample $b$ via a Newton-Raphson-tuned multivariate-$t$ Metropolis-Hastings (M-H) proposal marginalizing over $f$ (eq. 7), then draw $f|b, h, \lambda$ analytically (Gaussian: $\hat{f}_t = F_t B'(V_t^*)^{-1}\tilde{y}_t$). After this, the model separates into $p+k$ independent univariate SV state spaces.
• Univariate SV blocks (series-by-series): Given $B$ and $f$, the $p+k$ log-volatility paths are conditionally independent. Each is sampled via the Kim-Shephard-Chib (KSC, 1998) 7-component Gaussian mixture approximation to $\log\chi^2(1)$ plus the de Jong-Shephard (1995) simulation smoother — exactly the same device as in the 2002 paper but now applied independently to each series.
• Particle filter for likelihood: An auxiliary particle filter (Pitt-Shephard 1999a) with $M=20{,}000$ particles estimates the one-step-ahead predictive density $\hat{f}(y_t|F_{t-1}, c^*)$ at the posterior mode $c^*$. The simulation-consistent likelihood estimate feeds into the Chib (1995) marginal likelihood identity.
• Marginal likelihood: $\log m(y|M) = \log f(y|c^*) + \log p(c^*) - \log p(c^*|y)$. The posterior ordinate $\log p(c^*|y)$ is factored into blocks via sequential reduced MCMC runs, matching the Chib-Jeliazkov (2001) framework for M-H blocks.
• Simulation validation (Table 6): With $p=10$, $k=3$, true model MSVt3f — Bayes factors decisive against all alternatives in 84–100% of 50 replications; posterior ordinate estimates converge at $G\geq 10{,}000$ reduced draws (Table 5).
• Application — 10 international stock indices, weekly 1973–2003 ($T=1{,}605$): Bayesian model selection favors MSVt with 3 factors (fat tails, no jumps). Covariance forecasting: MSV outperforms all multivariate GARCH (MGARCH) alternatives (BEKK [Baba-Engle-Kraft-Kroner], MD-VECH [matrix-diagonal VECH], CCC [constant conditional correlation], DCC [dynamic conditional correlation], EWMA [exponentially weighted moving average], RollWin [rolling window]) on mean absolute deviation (MAD) at 1- and 2-week horizons (MSV MAD$_{1w}=3.45$ vs. BEKK 3.63, DCC 3.72). Value-at-risk (VaR) coverage: MSV best or tied across World/Europe/Pacific Rim/US portfolios at both 1% and 5%; CCC worst; RollWin fails conditional coverage.
• Scalability: $p=20$, $k=4$, $T=2{,}000$: full MCMC run $\approx 3$h; particle filter $\approx 5$h; reduced runs for posterior ordinate $\approx 4$h (C code, 3.4 GHz Linux).

Concepts Introduced or Extended

• Stochastic Volatility — multivariate factor-SV extension; reduced blocking scheme; particle filter for marginal data density (MDD)
• GARCH and BEKK-GARCH — empirical comparison (BEKK, MD-VECH, CCC, DCC, EWMA, RollWin) with MSV

Entities Mentioned

• Siddhartha Chib
• Federico Nardari
• Neil Shephard

Quotes

"Without use of our refinements, the inefficiency factors often exceed 1000, and with our refinement they range between 1 and 30."

"In terms of MAD, the performance of the MSV model is very satisfactory, especially at the shorter horizons, where MSV outperforms all other specifications."

My Take

The paper's lasting contribution is the reduced blocking scheme: the insight that sampling $b$ marginalized over $f$ (rather than conditional on $f$) breaks the near-collinearity that cripples naive alternating samplers. This is an instance of the general Chib-Carlin (1999) principle — integrate out the problematic block before updating the remaining parameters — here adapted to a non-conjugate multivariate setting requiring Newton-Raphson tuning. The particle filter/Chib marginal-log-likelihood (MLL) combination for model comparison is powerful but computationally demanding ($\approx12$h total for a 20-series model). The empirical finding that MSV beats multivariate GARCH on MAD but not always on root-mean-squared error (RMSE) reflects the MAD criterion's robustness to outliers noted by Ledoit-Santa-Clara-Wolf (2003).