Rosenberg-Young (1999) A Bayesian Approach to Understanding Time Series Data

Summary

Pedagogical introduction of Bayesian AutoRegressive (AR) models with simultaneous level and variance shifts to an actuarial audience, published in the North American Actuarial Journal 3(2): 130–143. Applies McCulloch-Tsay (1993, 1994) Gibbs sampling methodology to differenced log US unemployment rates (158 quarters, 1953Q2–1992Q4). Introduces a modified Akaike Information Criterion (AIC) based on posterior predictive paths and demonstrates prior sensitivity analysis. No new methodology; value is in the actuarial translation and the modified AIC contribution.

Key Claims

• The combined level-and-variance shift AR model provides more flexibility than AutoRegressive Conditional Heteroskedasticity (ARCH), whose variance structure is fully determined by the model specification; the Bayesian approach allows flexible uncertainty in when and how much shifts occur.
• Full conditionals for all six parameter blocks (AR coefficients, base error variance, level shift indicators/amounts, variance shift indicators/amounts, shift probabilities $e_1$ and $e_2$) are available in closed form: AR coefficients are Normal, $\sigma^2_{p+1}$ is Inverse-$\chi^2$, shift amounts are Normal, shift probabilities are Beta.
• Modified AIC — median over $30{,}000$ Gibbs paths of $\sum(y_t - \hat{y}_{t,i})^2/(2\sigma^2_{t,i}) / (n-p)$ + parameter penalty — provides a Bayesian-compatible model selection criterion; smaller is better and comparable only across models estimated via the same Gibbs scheme.
• For US unemployment: AR(1) chosen; level-shift probability $\hat{e}_1 \approx 0.015$ (negligible), variance-shift probability $\hat{e}_2 \approx 0.086$ (${\sim}9\%$); variance shifts concentrated around 1983–84; three alternative prior sets yield nearly identical posteriors.

Concepts Introduced or Extended

• Gibbs Sampler
• Markov-Switching VAR

Entities Mentioned

• Marjorie Rosenberg
• Virginia Young

Quotes

"Our Bayesian analysis, based on models developed by McCulloch and Tsay (1993, 1994), allows for shifts in the level and in the error variance of a process."

My Take

A competent bridge paper for actuaries encountering Markov Chain Monte Carlo (MCMC) for the first time. The modified AIC is a minor practical contribution; the main value is that it makes McCulloch-Tsay accessible without requiring readers to already know Bayesian econometrics. The sensitivity analysis in §5 is unusually thorough for this type of tutorial paper. Closely related to Albert-Chib (1993) which handles similar shifts in a Markov-switching context.