Haas-Mittnik-Paolella (2004) A New Approach to Markov-Switching GARCH Models

Summary

Proposes a Markov-switching GARCH specification that solves the path-dependence problem crippling earlier variants. Instead of a single variance recursion whose lagged term feeds back through the entire regime history, the model runs one parallel GARCH process per regime, each depending only on its own past conditional variance. This "disaggregated" variance structure makes the model exactly estimable by maximum likelihood and gives it analytically tractable dynamic properties (stationarity conditions, moments, autocorrelation of squares). Applied to several exchange-rate return series, the authors find a promising model is an independent-switching GARCH with a possibly skewed conditional mixture density.

Key Claims

• Parallel regime-specific variances. Each regime $k$ carries its own GARCH recursion $\sigma_{k,t}^2 = \omega_k + \alpha_k y_{t-1}^2 + \beta_k \sigma_{k,t-1}^2$; the observed variance is the one selected by the current state $s_t$. Because each $\sigma_{k,t}^2$ depends only on its own lag, the crippling dependence on the whole path $\{s_1,\dots,s_t\}$ is removed.
• Exact ML estimation. With path dependence gone, the likelihood no longer sums over $K^t$ regime paths, so exact maximum likelihood is feasible — unlike the Gray (1996) / Klaassen (2002) collapsing approximations that integrate the lagged regime out each period.
• Tractable dynamics. The parallel structure yields closed-form conditions for covariance stationarity and existence of moments, and expressions for the autocorrelation function of squared returns — properties that were "not well understood" in earlier MS-GARCH variants.
• Plausible variance disaggregation. Separating the variance into regime-specific components is argued to be a more interpretable decomposition of the volatility process than the collapsed alternatives.
• Empirical finding. On exchange-rate returns, an independent-switching GARCH with a skewed conditional mixture density performs well for density forecasting.

Concepts Introduced or Extended

• Markov-Switching GARCH

Entities Mentioned

• Markus Haas
• Stefan Mittnik
• Marc S. Paolella

Quotes

"Existing methods of combining the two approaches are unsatisfactory, as they either suffer from severe estimation difficulties or else their dynamic properties are not well understood. In this article we present a new Markov-switching GARCH model that overcomes both of these problems."

My Take

The paper that made Markov-switching GARCH estimable rather than merely conceptual. Path dependence had forced everyone into approximations (Gray, Klaassen) whose dynamic properties nobody could pin down; Haas-Mittnik-Paolella's move — refusing to let regimes share a variance recursion, running $K$ of them in parallel and selecting one per period — is the kind of respecification that trades a little modelling realism for a lot of analytical and computational tractability, and it won. It is one of the "three responses to path dependence" the MS-GARCH page catalogues (alongside collapsing and Bayesian data augmentation), and the one that keeps frequentist ML on the table. The skewed-mixture-density result also foreshadows the later emphasis on non-Gaussian conditional distributions for density forecasting and VaR.