Bayesian VAR

Definition

A Bayesian Vector Autoregression (BVAR) is a Vector Autoregression (VAR) model estimated under an explicit prior distribution over the coefficient matrices and error covariance. The posterior combines prior beliefs with the likelihood, providing regularization of the large parameter space that plagues unrestricted VARs.

Key Ideas

Parameter proliferation: A $p$ -variable VAR with $k$ lags has $p^2 k + p(p+1)/2$ free parameters — 486 for $p=6$ , $k=13$ . Most macroeconomic samples cannot support this without prior regularization.
Shrinkage toward a benchmark: Priors shrink coefficient estimates toward a low-dimensional belief (typically a random walk or white noise), reducing in-sample overfitting and improving out-of-sample forecasts.
Conjugate structure: Under a Normal-Wishart prior on $(\beta, \Sigma)$ and Gaussian likelihood, the posterior is also Normal-Wishart, enabling closed-form posterior moments. Non-conjugate or restricted priors require Markov Chain Monte Carlo (MCMC).
Steady state prior: The Steady State VAR reparametrization (Villani 2008) places priors directly on the unconditional mean — a more economically interpretable target than the lag matrices.

How It Works

Normal-Wishart Posterior

For $y_t = B x_{t-1} + u_t$ , $u_t \sim \mathcal{N}(0, \Sigma)$ , with a diffuse prior ( $N_0 = 0$ , $\nu_0 = 0$ ) the posterior is:

$\Sigma \mid Y \sim \text{IW}(\hat{S},\, T - k), \qquad \text{vec}(B) \mid \Sigma, Y \sim \mathcal{N}(\text{vec}(\hat{B}),\, \Sigma \otimes (X'X)^{-1})$

where IW denotes the Inverse-Wishart distribution and $\hat{B} = (X'X)^{-1}X'Y$ is the ordinary least squares (OLS) estimator. An informative Normal-Wishart prior shifts $\hat{B}$ toward the prior mean with weight determined by the prior precision.

Minnesota Prior

The Minnesota Prior (Litterman 1986) places a Normal-diagonal prior on VAR coefficients:

Own first lags centred at 1 (random walk belief); all other lags centred at 0.
Variance shrinks with lag order: $\text{Var}(\beta_{ij,\ell}) = (\lambda/\ell)^2 \cdot (\sigma_i/\sigma_j)^2$ .
Hyperparameter $\lambda$ controls overall tightness; $\lambda = 0.2$ is the Litterman benchmark.

Sims-Zha Prior

The Sims-Zha Prior generalises the Minnesota structure to allow cross-equation shrinkage, non-zero prior means, structural restrictions, and analytical marginal data density computation.

Restricted BVARs and Gibbs Sampling

When restrictions (zero exclusions, sign restrictions, steady-state priors) break the Normal-Wishart conjugate form, MCMC is needed. The Gibbs Sampler of Waggoner-Zha (2000) handles linear zero restrictions on structural VARs; Villani (2005, 2008) provides a three-block sampler for the steady-state VAR.

Marginal Data Density

The Marginal Data Density $p(Y \mid r, k)$ is available in closed form under conjugate priors. Non-conjugate and restricted BVARs use the Chib (1995) MCMC identity, enabling Bayesian rank and lag-order selection.

Why It Matters

BVARs consistently outperform unrestricted VARs and univariate models in multivariate macroeconomic forecasting. The fundamental argument — that prior information is unavoidable in econometrics and should be made explicit — is the central claim of Zellner (1985). Informative priors about steady states (Villani 2008) or long-run dynamics yield further forecast improvements at medium-to-long horizons.

Open Questions

Hyperparameter selection: Optimal tightness ( $\lambda$ ) is typically calibrated by marginal likelihood or held fixed; full hyperparameter posteriors are rarely computed.
Large BVARs: With $p = 20$ –100 variables, prior elicitation and MCMC become non-trivial; factor priors and hierarchical shrinkage are active research areas.
Non-linear extensions: Time-varying parameter BVARs and Markov-switching BVARs require more complex MCMC schemes and convergence monitoring.