McCulloch-Tsay (1993) Bayesian Inference for Mean and Variance Shifts in AR Series

Summary

McCulloch and Tsay (1993) develop a fully Bayesian framework for detecting and estimating mean shifts (random level-shift AR, RLAR) and variance shifts (random variance-shift AR, RVAR) in autoregressive time series, using the Gibbs sampler as the computational engine. Unlike classical methods that require fixing the number of break points in advance, the framework treats each observation's shift indicator as a latent Bernoulli variable, so inference automatically handles unknown numbers of shifts at unknown dates. A probit extension ties the shift probability to exogenous variables, enabling prediction of future structural changes.

Key Claims

• RLAR model: Mean level follows $\mu_t = \mu_{t-1} + \delta_t\beta_t$ with $\delta_t \overset{iid}{\sim}\mathrm{Bernoulli}(\varepsilon)$; the $\beta_t$ are normally distributed jump magnitudes. The AR process runs around this stochastically evolving mean. All parameters estimated jointly.
• RVAR model: Innovation variance shifts multiplicatively — $\sigma_t^2 = \sigma_0^2 \cdot \prod_{j \leq t} (\delta_j\beta_j^2 + (1-\delta_j))$ — with the same Bernoulli mechanism. Generalizes Inclan (1991) and Tsay (1988) deterministic variance-shift models to a stochastic framework that also handles multiple shifts and provides probability assessments.
• Gibbs sampler: All five full conditional posteriors are available in closed form under conjugate priors: (1) AR coefficients $\phi \sim \mathcal{N}$; (2) $\sigma^2 \sim \chi^{-2}$; (3) jump sizes $\beta_t \sim \mathcal{N}$ (RLAR) or $\beta_t^2 \sim \chi^{-2}$ (RVAR); (4) indicators $\delta_t$: binary, sampled from the closed-form posterior odds ratio; (5) shift probability $\varepsilon \sim \mathrm{Beta}(\gamma_1+k, \gamma_2+n-p-k)$. Drawing three indicators jointly rather than one at a time improves convergence.
• Probit extension: Replace fixed $\varepsilon$ with $\varepsilon_t = \Phi(w_t'\gamma)$. The probit coefficient $\gamma$ uses the Albert-Chib (1993) data augmentation. This enables prediction: $P(\text{shift at }t \mid w_t)$ is available for future time points.
• Prior specification discipline: A prior for $\varepsilon$ concentrated on small values ("shifts are rare") prevents overfitting without specifying the number of shifts. Given a tight prior on $\varepsilon$, the prior for $\beta_t$ can be diffuse.
• Empirical application: Monthly US unleaded gasoline price changes (AR(2), 1978–1991, n=158). Under RVAR: $\hat\varepsilon \approx 0.11$; detected variance shifts align with major oil-market events. Under RVAR-Probit with world-events indicator: $P(\text{shift} \mid \text{no event}) \approx 0.12$, $P(\text{shift} \mid \text{world event}) \approx 0.73$.

Concepts Introduced or Extended

• Gibbs Sampler — five-block Gibbs sampler for RLAR/RVAR; binary indicator conditionals; joint three-at-a-time drawing of shift indicators
• Structural Break Testing — fully Bayesian alternative to classical frequentist break-point tests; stochastic shift model with Bernoulli indicators replaces the fixed-number-of-breaks assumption

Entities Mentioned

• Robert E. McCulloch
• Ruey S. Tsay
• James H. Albert — Albert-Chib (1993) data augmentation for probit conditional
• Siddhartha Chib
• James D. Hamilton — Hamilton (1989) Markov switching noted as related but different (MS has fixed regime count vs. RLAR/RVAR iid Bernoulli per observation)

Quotes

"The Gibbs sampler makes analysis of the models presented in this article possible, because we are able to draw from the posterior distribution of parameters that describe the behavior of the series around a given time point, given all the other parameters."

"For instance, the exogenous variable is helpful in dating the time index of a variance shift."

My Take

The paper's key insight is that structural change need not be treated as a fixed-dimension parameter estimation problem. By placing a Bernoulli prior on each time point's shift indicator, the model sidesteps the model-selection problem of "how many breaks?" entirely — the shift probability $\varepsilon$ governs sparsity. This is conceptually analogous to spike-and-slab priors in variable selection. The conjugacy throughout the RLAR/RVAR models makes the Gibbs sampler clean and fast. The probit extension anticipates later work on regime prediction; in the gasoline application it correctly links variance episodes to known OPEC events. A limitation: the iid Bernoulli assumption means shifts at adjacent time points are independent — unlike Markov-switching models which allow persistent regimes. For data with genuine long-lived regime shifts, MS models are better suited; for data with discrete, isolated jumps, RLAR/RVAR is more natural.