Villani (2005) Inference in Vector Autoregressive Models with an Informative Prior on the Steady State

Summary

Working paper precursor to Villani (2008). Develops the mean-adjusted Vector Autoregression (VAR) $\Pi(L)(x_t - \Psi d_t) = \varepsilon_t$ with a Normal prior on the steady state $\Psi$, a three-block Gibbs sampler with closed-form conditionals, and extensions to cointegrated VARs. Relative to the published version, the notable additions are a simulation study demonstrating Gibbs instability under flat priors and a 7-variable Euro area application showing that Maximum Likelihood (ML)/standard Bayesian VAR (BVAR) yield steady-state inflation estimates grossly inconsistent with institutional knowledge.

Key Claims

• The three full conditional posteriors ($\Sigma \mid \cdot$, $\Pi \mid \cdot$, $\Psi \mid \cdot$) are Inverse-Wishart, Normal, and Normal — enabling a clean three-block Gibbs sampler. The joint posterior is intractable because the model is non-linear in $(\Pi, \Psi)$.
• With a flat prior on $\Psi$, the precision of $\Psi$'s full conditional diverges as $\Pi$ approaches the non-stationary region: $Q_\Psi \to \infty$ as $\rho(I_p - \sum_i \Pi_i) \to 0$. This causes explosive Gibbs excursions. An informative prior clamps $Q_\Psi \to \Omega_\Psi^{-1}$ and stabilises the chain.
• Simulation study (Figures 1–3): bivariate AutoRegressive (AR)(1) with $\Pi = \mathrm{diag}(0.95, 0.95)$, $\Psi^* = (1,4)'$, $T=100$. Flat prior: chain diverges. Mildly informative ($\psi_i \sim \mathcal{N}(\cdot, 1.25\text{–}2.5^2)$): drastically better. Informative ($\psi_i \sim \mathcal{N}(\cdot,1)$): excellent mixing.
• Euro area application: 7 variables $(\pi, \Delta w, \Delta c, \Delta i, r, e, \Delta y)$, 1970Q1–2002Q4, monetary policy regime break at 1992Q4 (Bayesian Information Criterion (BIC)-selected), 4 lags. ML steady-state inflation post-1992: 6.97% (full sample) / $-0.62$% (1980-subsample). Mean-adjusted models: ~2.0% in both — consistent with European Central Bank (ECB) target and robust to sample-period changes.
• Cointegrated extension uses Clements-Hendry (1999) form: $\Gamma(L)(\Delta x_t - \gamma) = \alpha(\beta'x_{t-1} - \mu_0 - \mu_1 t) + \varepsilon_t$; unknown $\beta$ handled by augmenting the Gibbs sampler with a Metropolis step using a Normal proposal centred at the previous draw with Hessian-based covariance.

Concepts Introduced or Extended

• Steady State VAR

Entities Mentioned

• Mattias Villani

Quotes

"The decision maker will not be pleased to hear that while her prior information may easily be incorporated on the more obscure part of the model, such as the reduced form dynamic coefficients, her strong prior beliefs about the steady state cannot be used for 'technical reasons'."

My Take

The working paper and the published version (Villani 2008) cover the same methodology. The two things unique to the working paper are the three-panel Gibbs instability simulation (flat $\to$ explosive, mildly informative $\to$ adequate, informative $\to$ excellent) and the Euro area application with a regime dummy at 1992Q4. The Euro area results are arguably more striking than the Swedish results in Villani (2008): the ML estimate of Euro area post-break steady-state inflation is literally negative ($-0.62$%) when estimated on 1980Q1 data, making the case for the steady-state prior viscerally obvious.