Bauwens-Preminger-Rombouts (2010) Theory and Inference for a Markov Switching GARCH Model

Summary

Bauwens, Preminger and Rombouts develop univariate Markov-switching (regime-switching) GARCH models, in which the conditional variance follows one GARCH process in each regime and the active regime is governed by a hidden Markov chain (with possibly state-dependent transition probabilities). They provide sufficient conditions for stationarity and the existence of moments. Their central methodological point is that because the conditional variance is path dependent — it depends on the entire unobserved history of regimes — the likelihood requires summing over exponentially many state paths, so maximum-likelihood estimation is infeasible. By enlarging the parameter space to include the latent state variables (data augmentation), Bayesian estimation via a Gibbs sampler becomes feasible. The model is applied to NASDAQ daily returns. (Published in The Econometrics Journal 13(2): 218–244; circulated as CORE Discussion Paper 2006/11.)

Key Claims

• Regime-switching GARCH. The conditional variance switches over time from one GARCH process to another; the switching is driven by a Markov chain whose transition probabilities may depend on past information, nesting standard GARCH (single regime) and capturing structural change / high- vs. low-volatility regimes.
• Stationarity and moments. The paper gives sufficient conditions for (strict and second-order) stationarity and for the existence of moments of the regime-switching GARCH process.
• The path-dependence problem. In MS-GARCH, today's conditional variance depends on the previous conditional variance, which depended on the previous regime, and so on back to the start of the sample — so the variance at time $t$ depends on the whole path of regimes $\{s_1,\dots,s_t\}$. Evaluating the likelihood requires integrating over $K^{t}$ paths, making maximum-likelihood estimation computationally infeasible.
• Data-augmentation resolves it. Enlarging the parameter space to include the latent states $\{s_t\}$ removes the path dependence: conditional on the full regime path, each period's GARCH recursion is deterministic. A Gibbs sampler then cycles between drawing the GARCH/transition parameters and drawing the state path, delivering feasible Bayesian inference.
• Application. Estimated on NASDAQ daily returns, the model separates volatility regimes and improves the description of the return series relative to single-regime GARCH.

Concepts Introduced or Extended

• Markov-Switching GARCH
• GARCH and BEKK-GARCH
• Markov-Switching Model
• Data Augmentation

Entities Mentioned

• Luc Bauwens
• Arie Preminger
• Jeroen Rombouts

Quotes

"Because of path dependence, maximum likelihood estimation is infeasible. By enlarging the parameter space to include the state variables, Bayesian estimation using a Gibbs sampling algorithm is feasible."

My Take

This paper is the clean statement of why Markov-switching GARCH is hard and how to make it tractable. The path-dependence obstacle is genuinely fundamental: it is what forced the earlier literature into approximations that collapse the lagged regime out of the variance recursion (Gray 1996; Klaassen 2002) or that redefine the model so that each regime carries its own path-independent variance (Haas–Mittnik–Paolella 2004). Bauwens–Preminger–Rombouts instead keep the natural specification and pay for it with computation, using data augmentation to turn the intractable marginal likelihood into a tractable complete-data problem — exactly the same latent-state trick that powers Bayesian estimation of the Markov-switching model more generally (Albert–Chib; Kim–Nelson) and of stochastic-volatility models. It rounds out the GARCH toolkit's regime-switching corner and is the reference for the theory (stationarity, moments) as well as the Bayesian inference.