Markov-Switching GARCH

Definition

A Markov-switching (regime-switching) GARCH model lets the parameters of a GARCH conditional-variance process change across a small number of discrete regimes, with the active regime governed by a hidden Markov chain. It combines volatility clustering (from GARCH) with the abrupt structural shifts between high- and low-volatility states (from Markov switching), and nests single-regime GARCH as the one-state case.

Key Ideas

Two sources of dynamics. Within a regime, the conditional variance follows an ordinary GARCH recursion; across time, a latent state $s_t\in\{1,\dots,K\}$ evolves as a Markov chain (transition probabilities possibly state- or information-dependent), selecting which GARCH parameters are active.
Path dependence — the core obstacle. Because $\sigma_t^2$ depends on $\sigma_{t-1}^2$ , which depended on $s_{t-1}$ , which fed back through the whole history, the time- $t$ conditional variance depends on the entire path $\{s_1,\dots,s_t\}$ . The likelihood therefore sums over $K^{t}$ paths, so exact maximum likelihood is computationally infeasible for realistic sample sizes.
Three responses to path dependence:
- Collapsing / approximation (Gray 1996; Klaassen 2002): integrate the lagged regime out of the variance recursion each period, restoring a path-independent recursion at the cost of an approximation.
- Path-independent respecification (Haas–Mittnik–Paolella 2004): run one GARCH recursion per regime in parallel, each depending only on its own lagged variance $\sigma_{k,t}^2 = \omega_k + \alpha_k y_{t-1}^2 + \beta_k \sigma_{k,t-1}^2$ , so the model is exactly estimable by ML and its stationarity/moment/autocorrelation properties are analytically tractable.
- Bayesian data augmentation (Bauwens–Preminger–Rombouts 2010; Das–Yoo 2004): keep the natural specification and augment the parameter space with the latent states; conditional on the full state path the recursion is deterministic, and a Gibbs sampler cycles over parameters and states.
Stationarity and moments. Sufficient conditions for stationarity and the existence of moments depend jointly on the regime-specific GARCH parameters and the transition matrix (an individually explosive regime can be consistent with a stationary switching process if it is visited rarely enough).

How It Works

Given returns $y_t = \sigma_t \varepsilon_t$ , in regime $s_t=k$ the variance follows $\sigma_t^2 = \omega_k + \alpha_k y_{t-1}^2 + \beta_k \sigma_{t-1}^2$ . The data-augmentation Gibbs sampler treats the state path $\{s_t\}$ as missing data:

Draw states $\{s_t\}$ given the parameters and data (a forward–backward / single-move or multimove step on the hidden Markov chain).
Draw GARCH parameters for each regime given the states — where, conditional on the path, the variance recursion is deterministic so the conditional posterior is a standard (if non-conjugate) GARCH posterior, sampled e.g. by griddy-Gibbs or Metropolis-within-Gibbs.
Draw the transition probabilities (conjugate Dirichlet given the state counts).

Why It Matters

It captures a stylized fact single-regime GARCH misses: volatility persistence is partly an artifact of unmodeled regime shifts (a high-volatility regime looks like near-integrated GARCH). See Markov-Switching Model.
It improves volatility forecasting and risk (VaR) measurement when the market alternates between calm and turbulent states.
It is a canonical example of using data augmentation to defeat a path-dependence / intractable-likelihood problem.

Open Questions

Which fix? Collapsing, path-independent respecification, and data augmentation give different models and different estimates; their relative accuracy and forecasting performance remain debated.
Label switching and identification of regimes, and choosing the number of regimes $K$ , are the usual mixture-model difficulties.

Markov-Switching GARCH

Definition

Key Ideas

How It Works

Why It Matters

Open Questions

Related