High-Dimensional Posterior Consistency in Bayesian Vector Autoregressive Models.

Area of Science:

• Statistics
• Econometrics
• Computational Neuroscience
• Functional Genomics

Background:

• Vector Autoregressive (VAR) models analyze temporal dependencies in multiple time series.
• High-dimensional VAR models are increasingly relevant in fields like genomics and neuroscience.
• Little is known about Bayesian VAR posterior distribution properties in high dimensions.

Purpose of the Study:

• Investigate the behavior of Bayesian VAR models in high-dimensional settings.
• Establish theoretical guarantees for posterior distributions with specific priors.
• Analyze the impact of temporal dependence on regularized parameter estimates.

Main Methods:

• Considered VAR models with matrix-normal and hierarchical (scale mixture of normals) priors.
• Established posterior consistency under standard regularity assumptions.
• Analyzed scenarios where model dimension (p) grows with sample size (n).

Main Results:

• Posterior consistency was established for both non-hierarchical and hierarchical priors.
• The findings hold when the model dimension grows with the sample size.
• A special case demonstrated a shrinkage prior inducing sparsity.

Conclusions:

• Provides theoretical foundations for Bayesian VAR models in high-dimensional regimes.
• The results are applicable to various fields utilizing time series analysis.
• Demonstrates the utility of specific priors for regularized estimation and sparsity.