Warne (2006) Bayesian Inference in Cointegrated VAR Models with Applications to the Demand for Euro Area M3

Summary

Warne (2006) develops a complete Bayesian framework for inference in cointegrated vector autoregression (VAR) / vector error correction model (VECM) systems extending Villani (2005b) in two directions: (1) adding a Minnesota-style block-diagonal shrinkage prior on the short-run dynamics $\Gamma|\Omega$ that enables joint rank-and-lag inference without Markov chain Monte Carlo (MCMC) at full rank; and (2) deriving an analytic posterior mode via a generalized eigenvalue problem that is algebraically identical to Johansen's reduced-rank regression on modified moment matrices. Applied to a six-variable euro area M3 demand system (1980Q4–2004Q4), the method selects lag order $k=2$ with near-certainty ($\approx 98\%$) but finds rank sensitive to the prior choice; income elasticity is robustly estimated near 1.37 while the opportunity cost semi-elasticity is data-uninformative.

Key Claims

• Flat-prior pathology on $\Psi$. A flat prior on the unconstrained cointegrating-space parametrization $\Psi$ does not correspond to a uniform distribution on the Grassmann manifold $\mathcal{G}(p,r)$; it overweights the region near normalization breakdown (Strachan and van Dijk 2003). Warne adopts Villani's (2005b) matrix-$t$ prior $\Psi \sim t_{(p-r)\times r}(0,c_\perp'c_\perp,c'c,0)$, which is algebraically equivalent to a uniform distribution over $\mathcal{G}(p,r)$.
• Informative prior on $\Gamma|\Omega$ (Warne's key extension). Block-diagonal shrinkage: $\Gamma_i|\Omega \sim \mathcal{N}(0, \Sigma_{\Gamma_i}\otimes\Omega)$ with $\Sigma_{\Gamma_i} = (\lambda_b^2/i^{2\lambda_l})I_p$. Hyperparameter $\lambda_b$ controls baseline shrinkage and $\lambda_l$ controls lag decay — a VECM analogue of the Minnesota prior.
• Full Gibbs sampler (Proposition 1). All six full-conditional posteriors are standard: $\Omega|\cdot \sim \mathrm{IW}$; $\Phi|\cdot, \Gamma|\cdot, \alpha|\cdot$ all $\sim \mathcal{N}$; $\Psi|\cdot$ is matrix-$t$. No Metropolis steps required.
• Posterior mode (Proposition 4). Integrating out $(\Phi,\Gamma,\alpha,\Omega)$ analytically, the posterior mode of $\Psi$ solves $|\lambda S_{11} - S_{10}S_{00}^{-1}S_{01}| = 0$ where $S_{ij}$ are Johansen-style concentrated moment matrices augmented by prior terms scaled by $(T+p+q+r+m+1)^{-1}$. Algebraically identical to Johansen maximum likelihood (ML) on modified moment matrices.
• Marginal Gibbs sampler (Proposition 5). Integrating $(\Phi,\Gamma)$ out analytically reduces the sampler to three blocks $(\alpha,\Psi,\Omega)$, decreasing the effective dimension and improving mixing.
• Chib (1995) marginal likelihood identity (MLI). $\log p(Y|r) = \log p(Y|\tilde\alpha,\tilde\Psi,r) + \log p(\tilde\alpha,\tilde\Psi|r) - \log p(\tilde\Psi|\tilde\alpha,Y,r) - \log \hat{p}(\tilde\alpha|Y,r)$. The first three terms are analytic; the fourth is estimated by Rao-Blackwellization $\hat{p}(\tilde\alpha|Y,r)=G^{-1}\sum_i p(\tilde\alpha|\Psi^{(i)},Y,r)$.
• Lag-order marginal likelihood (Corollary 1). Under the informative $\Gamma|\Omega$ prior, $p(Y|k)$ at rank $r=p$ (full rank, unrestricted VAR) is available in closed form — no MCMC required. This enables computation of the full joint posterior $p(r,k|Y)$ over all rank-lag combinations, with MCMC only needed for $r < p$.
• Bartlett's paradox. Improper flat priors on all VECM parameters render the posterior rank probabilities undefined. Villani's proper prior on $(\alpha,\Omega)$ is the minimal fix.
• Euro area M3 demand (6 variables, 1980Q4–2004Q4). Variables: real M3, real GDP, GDP deflator, short rate, long rate, own return on M3; broken linear trend from 2001Q4. Lag $k=2$ obtains $\approx 98\%$ posterior probability. Rank ambiguous: non-informative $\Gamma$ prior prefers $r=1$; informative prior spreads mass over $r=1,2$; fractional Bayes factors favor $r=4$. Income elasticity $\hat\beta_y \approx 1.37$–$1.39$ (robust across specifications); opportunity cost semi-elasticity $\hat\beta_o \approx -0.17$ to $-0.31$ with wide 95% credible intervals — data-uninformative.

Concepts Introduced or Extended

• Cointegration — Villani/Warne Grassmann-uniform prior; informative $\Gamma|\Omega$ shrinkage prior; posterior mode via generalized eigenvalue problem; joint rank-lag posterior $p(r,k|Y)$
• Gibbs Sampler — Proposition 1 full sampler (6 blocks); Proposition 5 marginal 3-block sampler; Rao-Blackwellization for MLI denominator
• Marginal Data Density — Chib (1995) identity applied to VECM; analytic numerator; matrix-$t$ second term; Rao-Blackwell third term
• Lag-Order Selection Criteria — Corollary 1 closed-form lag-order marginal likelihood; joint $p(r,k|Y)$ without MCMC at full rank
• Bayesian VAR — Minnesota-style shrinkage within a cointegrated VECM; informative prior on short-run dynamics

Entities Mentioned

• Anders Warne
• Mattias Villani
• Luc Bauwens
• K. Rao Kadiyala

Quotes

"A prior distribution that is flat on $\Psi$ does not correspond to a uniform distribution on the Grassmann manifold, but instead overweights the region near the normalization breakdown."

"Under the assumption that $r = p$, the marginal likelihood function $p(y|k)$ can be obtained analytically, making it possible to compute the joint posterior probability $p(r,k|y)$."

"The results for the income elasticity are robust across models, with a point estimate around 1.37, while the data are largely uninformative about the opportunity cost semi-elasticity."

My Take

Warne (2006) is an unusually complete Bayesian cointegration paper: it handles prior specification, posterior computation, marginal likelihood, and rank/lag determination within a single coherent framework. The key contribution relative to Villani (2005b) is the $\Gamma|\Omega$ shrinkage prior plus Corollary 1 (closed-form lag selection), which makes it practically possible to compare dozens of $(r,k)$ combinations without running a separate MCMC for each. The Grassmann-uniform prior is a genuine advance over the naive flat-on-$\Psi$ approach. The euro area application is honest about ambiguity: rank is sensitive to the prior, and the opportunity cost semi-elasticity remains wide regardless of sample size — the Bayesian framework quantifies this uncertainty rather than hiding it behind a point estimate.