Sun-Ni (2005) Bayesian Analysis of Vector-Autoregressive Models with Noninformative Priors

Summary

Sun and Ni investigate which noninformative prior for the Vector AutoRegressive (VAR) coefficient-covariance pair $(\Phi, \Sigma)$ produces better-behaved Bayes estimators. They show that the popular constant-Jeffreys prior systematically over-estimates $\Sigma$ variances due to an over-dispersed Inverse Wishart marginal posterior, and that replacing it with Yang and Berger's (1994) reference prior for $\Sigma$ reduces frequentist average loss by ~60–70% across a range of simulations. Both posteriors are proved proper under mild sample-size conditions, and a hit-and-run Markov Chain Monte Carlo (MCMC) algorithm is developed for the reference prior case.

Key Claims

• Under the constant-Jeffreys prior, the marginal posterior of $\Sigma$ is Inverse Wishart with $T - Lp - 1$ degrees of freedom. The posterior mean of $\sigma_{ii}$ over-estimates the unbiased Maximum Likelihood Estimator (MLE) by $p/(T - (L+1)p - 1) \times 100\%$ — for $L=1, p=5, T=50$ this is ~13%.
• The Yang-Berger (1994) reference prior $\pi_R(\Sigma) \propto |\Sigma|^{-1}\prod_{i<j}(d_i - d_j)^{-1}$ (where $d_i$ are eigenvalues of $\Sigma$) yields much smaller bias under both entropy and quadratic loss.
• Propriety of the constant-Jeffreys posterior requires $T \geq (L+1)p + 1$; constant-reference posterior requires only $T > Lp + 1$.
• Under both priors, the Bayes estimator of $\Phi$ is the same (posterior mean = MLE); the priors differ only in their treatment of $\Sigma$.
• PRIAL (Percentage Reduction In Average Loss) of reference vs. Jeffreys: ~65% for $p=5, T=150$; ~66% for $p=5, T=50$; ~72% for $p=10, T=150$. The advantage of the reference prior increases with $p$.
• In a 10-variable consumption VAR (1967Q1–2000Q3), constant-Jeffreys posterior means of $\sigma_{ii}$ are ~20% above MLE; constant-reference estimates are close to MLE for large variances and slightly above for small variances.

Concepts Introduced or Extended

• Noninformative Prior for VAR
• Gibbs Sampler — hit-and-run algorithm for $\Sigma$ under reference prior

Entities Mentioned

• Dongchu Sun
• Shawn Ni

Quotes

"Our Bayesian computation results indicate that estimates using the constant prior on the VAR regression coefficients and the reference prior of Yang and Berger (1994) on the covariance matrix dominate the constant-Jeffreys prior estimates commonly used in applications of VAR models in macroeconomics."

My Take

A focused methodological paper that surgically isolates a concrete flaw in the constant-Jeffreys prior and proposes a tractable fix. The bias result is exact and generalizes: the over-estimation factor $p/(T-(L+1)p-1)$ grows with $p$ and shrinks with $T$, so the problem is most acute in exactly the large-VAR, small-sample regime that is standard in macroeconomics. The hit-and-run MCMC for $\Sigma$ is more complex than a standard Wishart draw, but the ~32% acceptance rate indicates it is practical. The paper does not address the VAR coefficient prior — finding a reference prior for $\Phi$ that exploits the VAR's AR structure remains open.