Bayesian Forecasting with Large Vector Autoregressions

Provides fast and efficient procedures for Bayesian estimation and forecasting using state-of-the-art Vector Autoregressions. This package includes the model proposed by Chan (2020) <doi:10.1080/07350015.2018.1451336>, that is, a Bayesian Vector Autoregression with Minnesota priors and a flexible structure of the error term specification. The latter includes: conditional multivariate normal or Student’s t distributions, as well as homoskedastic or heteroskedastic specifications with a common volatility modelled by centred or non-centred Stochastic Volatility. Additionally, the package facilitates predictive analyses using density forecasting and forecast-error variance decompositions. All this is complemented by simple workflows, useful plots and summary functions, and comprehensive documentation. The 'bvars' package aligns with R packages 'bsvars' by Woźniak (2024) <doi:10.32614/CRAN.package.bsvars>, 'bsvarSIGNs' by Wang & Woźniak (2025) <doi:10.32614/CRAN.package.bsvarSIGNs>, and 'bpvars' by Woźniak (2025) <doi:10.32614/CRAN.package.bpvars> regarding objects, workflows, and code structure, and they constitute an integrated toolset.

Details

Models. All the BVAR models in this package are specified by two equations, including the reduced form equation: $$Y = AX + E$$ where $Y$ is an NxT matrix of dependent variables, $X$ is a KxT matrix of explanatory variables, $E$ is an NxT matrix of reduced form error terms, and $A$ is an NxK matrix of autoregressive slope coefficients and parameters on deterministic terms in $X$.

This package assumes that the error matrix follows a matrix normal distribution: $$E \mid X \sim \mathcal{MN}_{N \times T}(\mathbf{0}, \mathbf{\Sigma}, \mathbf{\Omega})$$ where $\Sigma$ is the NxN covariance matrix of the error term at time $t$, and $\Omega$ is a TxT diagonal matrix.

The diagonal elements of $\Omega$ determine the specification of the error term covariance structure. Specifically, the error term at time $t$ follows the multivariate normal distribution $$e_t \sim \mathcal{N}_N(\mathbf{0}, \sigma_t^2\lambda_t \mathbf{\Sigma})$$ where the scalar processes $\sigma_t^2$ and $\lambda_t$ determine the diagonal elements of $\Omega$. The process $\sigma_t^2$ specifies conditional variance and includes three options:

$\sigma_t^2 = 1$: homoskedastic error term
$\sigma_t^2$: estimated and following non-centred stochastic volatility
$\sigma_t^2$: estimated and following centred stochastic volatility

The process $\lambda_t$ specifies the conditional distribution of the error term and includes two options:

$\lambda_t = 1$: Gaussian error term specification
$\lambda_t$: estimated and following a priori an inverse gamma 2 distribution $\mathcal{IG}2(\nu - 2, \nu)$, where $\nu > 2$ is a degrees of freedom parameter

Prior distributions. The autoregressive matrix $A$ is assigned matrix-variate normal distribution: $$ \mathbf{A} \mid \underline{\mathbf{A}}, \mathbf{V}, \boldsymbol{\Sigma} \sim \mathcal{MN}_{N \times K}(\underline{\mathbf{A}}, \boldsymbol{\Sigma}, \mathbf{V}) $$ with the mean matrix $\underline{\mathbf{A}}$, and covariance matrices $\boldsymbol{\Sigma}_{N\times N}$ and $\mathbf{V}_{K\times K}$ defining the row- and column-covariance structures.

This is complemented by the inverse Wishart prior for the error term covariance $\boldsymbol{\Sigma}$: $$ \boldsymbol{\Sigma} \mid \underline{\mathbf{S}}, \underline{\nu} \sim \mathcal{IW}(\underline{\mathbf{S}}, \underline{\nu}) $$ with the scale matrix $\underline{\mathbf{S}}$ and degrees of freedom $\underline{\nu}$.

Note

This package is currently in active development. Your comments, suggestions and requests are warmly welcome!

References

Chan (2020) Large Bayesian VARs: A Flexible Kronecker Error Covariance Structure, Journal of Business and Economic Statistics, 38(1), 68–79, <doi:10.1080/07350015.2018.1451336>.

Author

Rui Liu rl3023@columbia.edu, Andres Ramirez Hassan aramir21@gmail.com, Tomasz Woźniak wozniak.tom@pm.me

Examples