High-Dimensional Posterior Consistency in Bayesian Vector Autoregressive Models

ABSTRACT

Vector autoregressive (VAR) models aim to capture linear temporal interdependencies among multiple time series. They have been widely used in macroeconomics and financial econometrics and more recently have found novel applications in functional genomics and neuroscience. These applications have also accentuated the need to investigate the behavior of the VAR model in a high-dimensional regime, which provides novel insights into the role of temporal dependence for regularized estimates of the model’s parameters. However, hardly anything is known regarding properties of the posterior distribution for Bayesian VAR models in such regimes. In this work, we consider a VAR model with two prior choices for the autoregressive coefficient matrix: a nonhierarchical matrix-normal prior and a hierarchical prior, which corresponds to an arbitrary scale mixture of normals. We establish posterior consistency for both these priors under standard regularity assumptions, when the dimension p of the VAR model grows with the sample size n (but still remains smaller than n). A special case corresponds to a shrinkage prior that introduces (group) sparsity in the columns of the model coefficient matrices. The performance of the model estimates are illustrated on synthetic and real macroeconomic datasets. Supplementary materials for this article are available online.

Keywords:

• Bayesian lasso
• Posterior consistency
• Shrinkage prior
• Vector autoregressive models

Supplementary Materials

For the sake of brevity we move additional simulation and real data study, proofs of Lemmas 1–3, Theorems 1 and 2 and useful high-dimensional results of VAR models to the supplementary document.

Additional information

Funding

The authors gratefully acknowledge support from NSF grants DMS-1511945 (KK) and IIS-1632730 and CNS-1422078 (GM)and NIH grant R01 5R01GM11402902.