Albert-Chib (1993) Bayes Inference via Gibbs Sampling of Autoregressive Time Series Subject to Markov Mean and Variance Shifts

Summary

Albert and Chib (1993) develop a fully Bayesian estimator for autoregressive (AR)$(r)$ models with two-state Markov switching in both the intercept (mean shift $\gamma$) and the innovation variance (scaling factor $\omega$). The key insight is data augmentation: treating the latent state sequence $S_n = (s_1,\ldots,s_n)'$ as missing data converts an intractable $2^n$ mixture likelihood into a Gibbs sampler with closed-form full conditionals. Each iteration draws states by a backward pass, then cycles through conjugate blocks for regression coefficients, variances, and transition probabilities. Applied to quarterly U.S. Treasury bill (T-bill) rates and gross national product (GNP) growth, the model precisely recovers the Volcker high-variance episode (1979:4–1982:3) and favors an AR$(0)$ switching structure for GNP.

Key Claims

• Treating latent states as missing data (data augmentation) converts Hamilton's $2^n$ state-path integration into a simple backward draw; joint posterior $p(\theta, S_n \mid Y)$ has tractable conditionals.
• States $\{s_t\}$ are drawn by a single backward pass: $p(s_t \mid s_{t+1}, Y, \theta) \propto p(s_{t+1} \mid s_t)\, p(y_t \mid s_t, Y_{t-1}, \theta)$, a binary discrete draw.
• Transition probabilities have closed-form Beta posteriors: $a \mid S_n \sim \mathrm{Beta}(u_{01}+n_{01},\; u_{00}+n_{00})$; $b \mid S_n \sim \mathrm{Beta}(u_{10}+n_{10},\; u_{11}+n_{11})$.
• Regression block $(\beta, \gamma) \mid S_n, \sigma^2$ is truncated multivariate Normal (truncated to $\gamma > 0$); $\sigma^2 \mid S_n, \beta, \gamma$ is Inverse-Gamma; $\omega \mid S_n, \sigma^2$ is Inverse-Gamma truncated to $\omega > 0$.
• AR coefficients $\phi$ drawn conditional on a stationary region to guarantee identifiability of the two states.
• For 90-day T-bill data, state 1 maps exactly to 1979:4–1982:3 (Volcker Federal Reserve); posterior mean of $\omega \approx 2.8$, so state-1 variance is $\approx 3.8\times$ state-0 variance.
• Bayes Factors strongly prefer the switching model over constant-parameter AR for interest rates; GNP favors a simpler AR$(0)$ switching structure.

Concepts Introduced or Extended

• Markov-Switching Model
• Data Augmentation
• Markov-Switching VAR
• Gibbs Sampler

Entities Mentioned

• James H. Albert
• Siddhartha Chib
• James D. Hamilton

Quotes

"...we shall treat the state sequence $S_n = (s_1,\ldots,s_n)'$ as missing data and apply a data augmentation scheme..."

"The essential advantage of the data augmentation approach is that the complete-data likelihood is easy to work with, and the full conditional distributions required for the Gibbs sampler are of standard form."

My Take

The paper's main contribution is methodological: it shows that the entire inferential difficulty of Markov-switching models — Hamilton's filter must sum over $2^n$ state sequences — dissolves when the state path is augmented as a parameter block. Every conditional in the Gibbs cycle has a conjugate form, making implementation straightforward. The identification constraints ($\gamma > 0$, $\omega > 0$) are handled by truncation rather than reparametrization, which is simple but can slow mixing when the true parameters are near zero. The empirical findings on interest rates are sharp and the 1979–1982 identification is economically compelling; the GNP results are less decisive. This paper is the foundational univariate Bayesian reference for what later became the Markov-switching VAR literature.