Villani (2008) Steady State Priors for Vector Autoregressions

Summary

Villani (2008) introduces the steady state Vector Autoregression (VAR) — a reparametrization of the standard VAR in which the unconditional mean $\mu_t = \Psi d_t$ is an explicit, separately identified parameter block, enabling informative Bayesian priors to be placed directly on it. The paper develops a three-block Gibbs sampler for the stationary case, extends the model to cointegrated VARs (steady state Vector Error Correction Model (VECM)), and generalizes the Waggoner-Zha (2003b) structural VAR sampler to non-zero prior means via the absolute normal distribution. Empirically, the steady state prior produces substantial Root Mean Square Error (RMSE) improvements over the Litterman Bayesian VAR (BVAR) for Swedish macro data (7 variables, 1980Q1–2005Q4), particularly for inflation and GDP growth at medium-to-long horizons.

Key Claims

• All existing Bayesian VAR priors (Litterman/Minnesota, Sims-Zha) shrink dynamic lag coefficients but are effectively non-informative about the steady state, which is a non-linear function of all lag matrices in standard form.
• Clements and Hendry (1998): badly estimated process mean is the dominant source of long-horizon forecast failure.
• The reparametrization $\Pi(L)(x_t - \Psi d_t) = \varepsilon_t$ makes the steady state $\mu_t = \Psi d_t$ directly accessible to prior elicitation (e.g., central bank inflation target, Dynamic Stochastic General Equilibrium (DSGE)-model implied calibration, historical pre-sample means).
• An informative prior on $\Psi$ also stabilizes the Gibbs sampler near unit roots: the data precision $T(I_p - \sum_i\Pi_i)'\Sigma^{-1}(I_p-\sum_i\Pi_i)$ vanishes at the unit root, so $\bar\Omega_\Psi \to \Omega_\Psi$ and the sampler remains well-behaved.
• The three full conditional posteriors $(\Sigma \mid \cdot)$, $(\Pi \mid \cdot)$, $(\Psi \mid \cdot)$ are Inverse-Wishart, Normal, and Normal respectively — enabling a clean three-block Gibbs sampler.
• For structural VARs, the Waggoner-Zha (2003b) sampling scheme generalizes to non-zero prior means $\mu_{\phi_i} \neq 0$ via the absolute normal distribution $f_{\mathrm{AN}}(x;\mu,\rho) \propto |x|\exp[-(x-\mu)^2/(2\rho)]$, approximated by a mixture of two Gaussians.

Concepts Introduced or Extended

• Steady State VAR
• Sims-Zha Prior (steady state as complementary prior dimension not addressed by existing priors)
• Marginal Data Density (Chib 1995 method applicable via a modified reduced Gibbs sampler in $(\Sigma,\Psi)$-space only)

Entities Mentioned

• Mattias Villani
• Christopher Sims
• Tao Zha
• Daniel Waggoner

Quotes

"The currently available Bayesian VAR methodology does not allow the user to specify prior beliefs about the unconditional mean, or steady state, of the system. This is unfortunate as the steady state is something that economists usually claim to know relatively well."

"Clements and Hendry (1998) show that a badly estimated mean of the process is the dominant source of forecast failure at longer forecast horizons."

"Decision makers will have a hard time accepting that their prior information may easily be incorporated on the more obscure parts of the model, such as the reduced form dynamic coefficients, but that their strong prior beliefs about the steady state cannot be used for 'technical reasons'."

My Take

The key move is a simple reparametrization that turns an implicit non-linear function into an explicit parameter — yet the payoff is large both in forecasting accuracy and in sampler stability. The unit root stability argument is especially elegant: the informative prior regularizes a singular information matrix, not as a numerical trick but as a reflection of genuine non-identification in the unit root region. The absolute normal extension in Appendix C is a clean generalization of Waggoner-Zha (2003b). One caveat: the Kronecker structure assumed for the prior covariance of $\Gamma$ in the VECM precludes the cross-equation shrinkage of Litterman (1986), a restriction that may matter in large systems.