Geweke (1993) Bayesian Treatment of the Independent Student-t Linear Model

Summary

Geweke (1993) develops exact Bayesian inference for the linear model with i.i.d. Student-t disturbances by exploiting the Student-t = scale-mixture-of-normals equivalence: $y_i|X \sim t(x_i'\beta, \sigma^2; \nu)$ is identical to $y_i|X \sim N(x_i'\beta, \sigma^2\omega_i)$ with $\nu/\omega_i \sim \chi^2(\nu)$ and $n$ latent scale weights $\omega_i$. Augmenting with $\omega$ renders all full conditionals conjugate and enables a direct four-block Gibbs sampler. Applied to 14 Nelson-Plosser (1982) macroeconomic series, posterior odds strongly favor the Student-t over the normal for 13/14 series; lower degrees of freedom systematically reduce the posterior odds in favor of difference stationarity.

Key Claims

• Student-t = scale mixture of normals: $y_i|X \sim N(x_i'\beta, \sigma^2\omega_i)$ with $\nu/\omega_i \sim \chi^2(\nu)$ (Section 2.1, eq. 6) — augmenting with $\omega$ makes all full conditionals conjugate.
• Four-block Gibbs (Section 3): (1) $\omega_i|(\beta,\sigma,\nu)$ — Inverse-Gamma: $[(\sigma^{-2}u_i^2+\nu)/\omega_i]|(\beta,\sigma)\sim\chi^2(\nu+1)$ (eq. 15); (2) $\sigma^2|(\beta,\omega)$ — Inverse-Gamma: $[\sum_i u_i^2/\omega_i]/\sigma^2\sim\chi^2(n)$ (eq. 14); (3) $\beta|(\sigma,\omega)$ — generalized-least-squares (GLS)-form Normal $\hat\beta(\omega)=(X'\Omega^{-1}X)^{-1}X'\Omega^{-1}y$ (eq. 12); (4) optional $\nu|(\beta,\sigma,\omega)$ — non-standard kernel (eq. 18) $\propto(\nu/2)^{n\nu/2}\Gamma(\nu/2)\exp(-\eta\nu)$ with $\eta=(1/2)\sum_i[\log\omega_i+\omega_i^{-1}]+\lambda$, requiring specialized sampling.
• Posterior odds ratio (eq. 16): $\text{POR}(\nu^{(1)},\nu^{(2)})$ is an expectation under the $\nu^{(2)}$ posterior, computable from standard Gibbs output; $\lim_{\nu^{(1)}\to\infty}$ gives the odds ratio in favor of normality.
• Unknown $\nu$: flat $\pi(\nu)\propto\text{const}$ ($\nu>0$) forces normality as the limiting case; exponential prior $\pi(\nu)=\lambda\exp(-\lambda\nu)$ (eq. 17) is the recommended proper prior.
• Convergence: Theorem 4 (Gibbs chain converges in distribution) and Theorem 5 (a.s. ergodic convergence $m^{-1}\sum g(\theta^{(j)})\to E_p[g(\theta)]$) proved for fixed $\nu$ and for exponential prior.
• Posterior moments: $E[\beta]$ exists for $\nu>2$, $\text{Var}(\beta)$ for $\nu>4$ (Theorem 3, improper flat prior).
• Nelson-Plosser (1982) application: 13/14 macroeconomic time series favor Student-t over normal (Table I); posterior mean of $\nu$ typically 3–7 (Table II); lower $\nu$ systematically reduces posterior odds in favor of difference over trend stationarity (Table V) and shrinks posterior standard deviation (SD) of trend coefficient $\delta$.

Concepts Introduced or Extended

• Mixture of Normals — Student-t = scale-mixture-of-normals representation; $\omega_i$ latent scale weights as data augmentation device
• Gibbs Sampler — four conjugate blocks via scale-mixture augmentation; convergence Theorems 4 and 5
• Unit Root Inference — tail assumption sensitivity for Nelson-Plosser unit root conclusions
• Stochastic Volatility — $\omega_i$ latent weights are the static precursor to the dynamic $h_t$ in stochastic-volatility (SV) models

Entities Mentioned

• John Geweke

Quotes

"The main contribution is to provide a simple and stable computational method for full Bayesian inference in the independent Student-t linear model."

"For all but one of these series posterior odds ratios favour Student-t linear models with degrees of freedom in the range of 3 to 7, over normal linear models."

My Take

The canonical reference for the $\omega_i$ latent-weight data augmentation device in Bayesian regression with leptokurtic disturbances. The same structure reappears in Jacquier-Polson-Rossi (1994), where $\omega_t$ becomes time-varying and follows a log-AR(1) (first-order autoregressive) process — the SV model. Geweke-Keane (1999) extends the scale-mixture idea to the probit context. The Nelson-Plosser application is a useful reminder that unit root conclusions are not robust to tail assumptions, a point often neglected in applied work that assumes Gaussian disturbances.