Bauwens-Rombouts (2007) Bayesian Inference for the Mixed Conditional Heteroskedasticity Model

Summary

Bauwens and Rombouts (2007) develop a Bayesian estimation strategy for the Mixture-of-Normals GARCH (MN-GARCH) model of Haas, Mittnik, and Paolella (2004a), published in The Econometrics Journal 10(2): 408–425. Returns follow a $K$-component normal mixture where each component has its own GARCH(1,1) variance process. The Gibbs sampler uses data augmentation (latent state variables $S_t$), Dirichlet conjugate updating for mixing weights, jointly constrained Normal draws for the $K-1$ free means, and griddy-Gibbs for the non-conjugate GARCH parameters. A Laplace approximation to the marginal likelihood selects $K$. Applied to S&P 500 returns (1994–2005), $K=2$ is strongly preferred, and the near-integrated-GARCH (IGARCH) finding of standard GARCH(1,1) is reinterpreted as a misspecification artifact arising from ignoring mixture structure. An earlier version circulated as CORE Discussion Paper 2005/85 (December 2005).

Key Claims

• MN-GARCH model: $f(y_t|\mathcal{F}_{t-1}) = \sum_{k=1}^K \pi_k \phi((y_t - \mu_k)/\sqrt{h_{k,t}})/\sqrt{h_{k,t}}$, with each component having its own GARCH(1,1): $h_{k,t} = \omega_k + \alpha_k y_{t-1}^2 + \beta_k h_{k,t-1}$. Zero-mean constraint: $\mu_K = -\sum_{k=1}^{K-1}(\pi_k \mu_k)/\pi_K$.
• Weak stationarity: The mixture can be covariance-stationary even when individual components have $\alpha_k + \beta_k > 1$ (explosive), because the stationarity condition involves a weighted combination across components.
• Gibbs sampler — 4 blocks:
  1. $S_T$ (latent state sequence) — multinomial draw: $P(S_t = k) \propto \pi_k \phi(y_t; \mu_k, h_{k,t})$
  2. $\pi$ (mixing weights) — Dirichlet conjugate: $\text{Di}(a_{k0} + x_k)$ where $x_k = \#\{t : S_t = k\}$
  3. $\mu$ ($K-1$ free means jointly) — multivariate Normal enforcing the zero-mean constraint via a $(K-1)\times(K-1)$ linear system
  4. $(\omega_k, \alpha_k, \beta_k)$ per component — griddy-Gibbs over a 3-D grid (non-conjugate)
• Convergence diagnostic: Cumulative-sum statistic $CS_t = (\bar\theta_t - \bar\theta)/\hat\sigma_\theta$ (running mean minus overall mean, standardised by overall posterior std dev) is tracked for all parameters; convergence is declared when $CS_t \to 0$ across all chains. Applied to $B = 10\,000$ draws.
• Marginal likelihood for $K$ selection (Laplace): $\log m(y) \approx \log L(\hat\Psi|y) + \log \phi(\hat\Psi) + \frac{\dim(\Psi)}{2}\log(2\pi) + \frac{1}{2}\log|\hat\Sigma|$, where $\hat\Psi$ is the posterior mode and $\hat\Sigma$ is the posterior covariance at the mode.
• Simulation ($T=3000$, 2-component data-generating process, DGP): True $\pi_1 = 0.8$; Component 1: $(\omega_1, \alpha_1, \beta_1) = (0.003, 0.03, 0.94)$; Component 2: $(\omega_2, \alpha_2, \beta_2) = (0.03, 0.25, 0.85)$ — explosive ($\alpha_2 + \beta_2 = 1.10$) but mixture stationary (overall persistence condition $= 0.0024$). Bayesian estimates have smaller bias and variance than maximum likelihood estimation (MLE); $K=2$ correctly selected by marginal likelihood.
• S&P 500 application ($T=3047$, Jan 1994–Jun 2005): AR(3) (third-order autoregressive) mean filter applied first. $K=2$ strongly preferred (marginal log-likelihood $-4091$ vs. $-4139$ for $K=1$; Bayesian information criterion, BIC: 8206.1 vs. 8278.2 for $K=1$, difference 72 units). Component 1: $\hat\pi_1 \approx 0.79$, $\hat\alpha_1 + \hat\beta_1 \approx 0.98$; Component 2: $\hat\pi_2 \approx 0.21$, $\hat\alpha_2 + \hat\beta_2 \approx 1.14$ (explosive). Standard GARCH(1,1) estimate: $\hat\alpha + \hat\beta \approx 0.996$.
• Near-IGARCH as misspecification artifact: The GARCH(1,1) persistence estimate ($\approx 0.996$) is a weighted compromise between the two components' persistence levels, not evidence of a true unit root in variance. MN-GARCH explains this by assigning the high-persistence behavior to the minority explosive component.
• Predictive value-at-risk (VaR) (Table 8): The 5% one-day VaR under $K=2$ ranges from $-1.5505$ to $-1.6473$ across S&P 500 sub-periods, vs. $-1.2906$ to $-1.3648$ for GARCH(1,1) — a materially larger capital requirement driven by the mixture's fatter left tail (the explosive minority component puts substantially more mass in the extreme left than a single-component GARCH).

Concepts Introduced or Extended

• GARCH and BEKK-GARCH
• Mixture of Normals
• Gibbs Sampler
• Marginal Data Density

Entities Mentioned

• Luc Bauwens
• Jeroen Rombouts

Quotes

"The component with the larger weight (approximately 79%) has near-unit persistence while the minor component (approximately 21%) is explosive — yet the mixture as a whole is stationary."

"Near-IGARCH behaviour in the standard GARCH model is an artifact of imposing a single conditional distribution on return data that are better described by a mixture of two distributions with different persistence levels."

My Take

The most important contribution is conceptual: it explains a long-standing empirical puzzle (near-IGARCH in equity returns) as a model misspecification artifact rather than a genuine unit root in variance. The Bayesian machinery is careful — griddy-Gibbs for the non-conjugate GARCH parameters and a Laplace marginal likelihood for model selection — though the Laplace approximation is less principled than reversible-jump MCMC or full Bayes factors. The zero-mean constraint introduces a non-trivial $(K-1)\times(K-1)$ linear system at each Gibbs step for the means block. The main limitation is scalability: griddy-Gibbs over a 3-D GARCH parameter grid becomes expensive for $K>3$ or large $T$. The predictive VaR finding — MN-GARCH 5% VaR of $-1.55$ to $-1.65$ vs. GARCH $-1.29$ to $-1.36$ — is practically important for risk management and capital adequacy. A companion paper (Bauwens-Rombouts 2006, CORE DP forthcoming) conducts a Monte Carlo study comparing Bayesian and ML estimators for the MN-GARCH model, confirming smaller bias and variance for the Gibbs sampler across sample sizes.