Kuschnig-Vashold (2021) BVAR: Bayesian Vector Autoregressions with Hierarchical Prior Selection in R

Summary

This Journal of Statistical Software article introduces BVAR, an R package for estimating Bayesian vector autoregressions with hierarchical prior selection following Giannone, Lenza, and Primiceri (2015). Rather than fixing the informativeness of the prior by hand, BVAR treats the prior hyperparameters as parameters to be estimated — assigning them hyperpriors and exploring the full posterior hyperparameter space via a Metropolis-Hastings step — using the Normal-inverse-Wishart conjugate family so the marginal likelihood is available in closed form. The package fills a gap: before it, there was no Bayesian counterpart to the frequentist vars package in R. It bundles the Minnesota (Litterman) prior plus sum-of-coefficients and single-unit-root dummy-observation priors (with a framework for custom ones), structural impulse-response analysis, forecasting, and FRED-MD data access behind an accessible, reproducible interface.

Key Claims

• The problem: over-parameterization. A VAR($p$) $y_t=a_0+A_1y_{t-1}+\cdots+A_py_{t-p}+\epsilon_t$, $\epsilon_t\sim N(0,\Sigma)$, has $M+M^2p$ coefficients, rising quadratically in the number of variables $M$ — the "curse of dimensionality" that degrades forecasts and structural inference. Bayesian informative priors shrink the model toward a parsimonious benchmark, reducing estimation error (a form of regularization, related to ridge/LASSO).
• Hierarchical prior selection (Giannone et al. 2015). Prior hyperparameters $\gamma$ are given hyperpriors, and inference proceeds from $p(\gamma\mid y)\propto p(y\mid\gamma)\,p(\gamma)$ where the marginal likelihood $p(y\mid\gamma)=\int p(y\mid\theta,\gamma)\,p(\theta\mid\gamma)\,d\theta$ integrates out the VAR parameters $\theta$. Maximizing $p(y\mid\gamma)$ is empirical Bayes; exploring its full posterior (as BVAR does) acknowledges hyperparameter uncertainty and yields robust inference. Conjugacy (NIW) makes $p(y\mid\gamma)$ closed-form and computation efficient.
• The Minnesota prior + its hyperparameters. The Minnesota (Litterman 1980) prior centers each variable on a random walk ($E[(A_1)_{ii}]=1$, all other coefficients $0$). Its key hyperparameters are $\lambda$ (overall tightness — $\lambda\to 0$ collapses to the prior, $\lambda\to\infty$ to OLS/data), $\alpha$ (rate of variance decay with lag order), and $\psi_j$ (own-vs-other-variable scale).
• Dummy-observation priors. Beyond Minnesota, BVAR ships the sum-of-coefficients prior (hyperparameter governing tightness toward no-cointegration/persistence) and the single-unit-root (dummy-initial-observation) prior (Sims 1993; Sims–Zha 1998), which allows cointegration by pushing variables toward their unconditional mean or toward at least one common unit root. Their hyperparameters get Gamma hyperpriors, and users can define custom dummy-observation priors via bv_dummy().
• Sampling. A Metropolis-Hastings algorithm explores the posterior over hyperparameters (with adjustable burn-in, saved draws, thinning, and manual or automatic proposal scaling to a target acceptance rate); conditional on the hyperparameters, the conjugate structure gives the parameter posterior directly.
• Applied toolkit. BVAR provides structural analysis of impulse responses (recursive/Cholesky and sign/zero restrictions), forecasts, forecast-error variance decompositions, convenient FRED-MD data access, and plotting — packaged transparently to improve reproducibility over ad-hoc scripts.
• Performance. The hierarchical approach of Giannone et al. (2015), which BVAR implements, delivers accurate impulse responses and forecasts, competitive with factor models, and has been adopted widely in applied macro research.

Concepts Introduced or Extended

• Bayesian VAR
• Minnesota Prior
• Sims-Zha Prior
• Marginal Data Density

Entities Mentioned

• Nikolas Kuschnig
• Lukas Vashold

Quotes

"The optimal choice of the degree of informativeness implied by these priors is subject of much debate and can be approached via hierarchical modeling."

"In the Bayesian hierarchical approach, the ML is used to explore the full posterior hyperparameter space, acknowledging uncertainty surrounding them."

My Take

BVAR is best understood as the reproducible, reference implementation of the Giannone–Lenza–Primiceri hierarchical-prior program — its intellectual content lives in that paper, but its contribution is making the method routine and inspectable in R (the Bayesian analogue of vars). For this wiki it is the concrete bridge between the Minnesota prior, the dummy-observation priors, and the marginal-likelihood-driven, empirical-Bayes-flavored selection of shrinkage that Bayesian VAR practice now takes for granted. The design choice worth noting is full posterior exploration of the hyperparameters via MH rather than mere ML-maximization — a deliberately more Bayesian stance that propagates hyperparameter uncertainty into impulse responses and forecasts.