Das-Yoo (2004) A Bayesian MCMC Algorithm for Markov Switching GARCH Models

Summary

Markov Switching GARCH (MS-GARCH) is intractable under maximum likelihood estimation (MLE) because the conditional variance $\sigma^2_t$ inherits path-dependence from the full history of latent states $S_1,\ldots,S_{T-1}$ , requiring $K^T$ likelihood evaluations. Das and Yoo resolve this by treating the state vector S as a latent variable in a Bayesian framework: conditioning on S makes the variance recursion tractable and the likelihood Gaussian. They develop a four-block Markov chain Monte Carlo (MCMC) sampler combining a single-move state scheme, conjugate Beta draws for transition probabilities, and two Metropolis–Hastings (MH) steps for regression and GARCH parameters (the latter using Nakatsuma's autoregressive-moving-average (ARMA) linearization). A simulation study with $T=500$ confirms posterior means close to true values, though the volatility regime indicator $\mu_1$ and the GARCH persistence parameter $\beta$ exhibit slow mixing.

Key Claims

Path-dependence makes MLE infeasible: In MS-GARCH, $\sigma^2_t$ depends on $S_1,\ldots,S_{t-1}$ , so the exact likelihood requires summing over all $K^T$ state paths; approximations either lose information or induce serial correlation in the state sequence.
Bayesian tractability: Treating the full state vector $S = (S_1,\ldots,S_T)$ as an additional parameter and conditioning on it at each MCMC step reduces the variance recursion to a deterministic function of past data and known states.
Model specification: $y_t = x_t'\gamma + \varepsilon_t$ , $\varepsilon_t = \sigma_t \omega_t$ , $\omega_t \sim \mathcal{N}(0,1)$ ; $\sigma^2_t = \mu_0 + \mu_1 S_t + \sum_j \alpha_j \varepsilon^2_{t-j} + \sum_j \beta_j \sigma^2_{t-j}$ ; $S_t \in \{0,1\}$ with Markov transitions $p_{00} = \Pr(S_t=0|S_{t-1}=0)$ , $p_{11} = \Pr(S_t=1|S_{t-1}=1)$ .
Prior specification: Normal priors on $\gamma, \mu_0, \mu_1, \alpha, \beta$ ; Beta priors on $p_{00}$ and $p_{11}$ .
Single-move state sampler (Block 1): For each $t$ , $\Pr(S_t=0 \mid Y, S_{\neq t}, \Theta)$ is computed by running the variance recursion forward from $t$ to $T$ — including all future terms $\prod_{s=t}^T (\sigma^2_s)^{-1/2} \exp(-\varepsilon^2_s/2\sigma^2_s)$ — then drawing from Bernoulli; handles path-dependence explicitly (eq. 4; following Carlin–Polson–Stoffer 1992 and Yoo 2004).
Transition probability draws (Block 2): $p_{00}$ and $p_{11}$ are drawn from conjugate Beta posteriors using transition count statistics $\eta_{ij}$ from the current state draw.
Regression parameter draws (Block 3): $\gamma$ is updated via Metropolis–Hastings with a tailored Normal proposal constructed from generalized least squares (GLS) on the current state and variance draw.
GARCH parameter draws (Block 4): $(\mu_0, \mu_1, \alpha, \beta)$ are updated via Metropolis–Hastings using Nakatsuma's (2000) ARMA linearization: $\varepsilon^2_t \approx \mu_0 + \mu_1 S_t + \sum_j(\alpha_j+\beta_j)\varepsilon^2_{t-j} + w_t - \sum_j \beta_j w_{t-j}$ , where $w_t = \varepsilon^2_t - \sigma^2_t \sim \mathcal{N}(0, 2\sigma^4_t)$ ; $(\mu_0, \alpha)$ proposed from state-0 observations, $\mu_1$ from state-1 increments, $\beta$ via first-order Taylor expansion around a nonlinear least squares (NLS) estimate $\beta^*$ .
Numerical study: MS-GARCH(1,1), $T=500$ , 6,000 MCMC draws (burn-in 1,000, thinning 3); true parameters $\gamma=0$ , $\mu_0=0.01$ , $\mu_1=0.05$ , $\alpha=0.2$ , $\beta=0.5$ , $p_{00}=0.98$ , $p_{11}=0.95$ ; posterior means are close to true values (Table 1); $\beta$ shows high first-order chain autocorrelation ( $\hat\rho \approx 0.98$ ); $\mu_1$ posterior mean 0.150 vs. true 0.050 (imprecise); MH acceptance rates 13–19%.
Contrast with Kaufmann–Frühwirth-Schnatter (KFS, 2002): KFS used switching ARCH (no lagged variance) precisely because multi-move state sampling breaks down under GARCH path-dependence; Das–Yoo show single-move sampling is workable for full MS-GARCH.
Companion paper: Yoo (2004) extends the approach to Markov switching models with ARMA-GARCH errors.
Limitation — slow state mixing: Single-move sampling for S_t (as opposed to multi-move or forward–backward) can yield high autocorrelation in the Markov chain, especially for persistent regimes.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"The difficulty of MLE for MS-GARCH is that the variance equation is dependent on the complete history of the Markov state process."

"By treating the state variable as another parameter in the model, the conditional variance is no longer path-dependent given the state, making the likelihood function tractable."

My Take

The key contribution is operational: they demonstrate that a single-move state sampler — which is slower than multi-move alternatives — is sufficient to make MS-GARCH Bayesian estimation feasible. The paper is honest about the limitations: slow mixing for $\beta$ and imprecision in $\mu_1$ . The working-paper status and the absence of an empirical application to real financial data are notable gaps. The approach is cleanly built on prior work (Nakatsuma 2000 for the GARCH block, Carlin–Polson–Stoffer 1992 for state sampling) and serves as a blueprint for subsequent MS-GARCH MCMC implementations.