[Paper Review] Sparse Bayesian vector autoregressions in huge dimensions

Gregor Kastner, Florian Huber|RePEc: Research Papers in Economics|Apr 11, 2017

Monetary Policy and Economic Impact4 citations

TL;DR

This paper proposes a sparse Bayesian vector autoregressive (VAR) model with factor stochastic volatility to enable reliable estimation and forecasting in high-dimensional macroeconomic settings. By combining global-local shrinkage priors for coefficient regularization, equation-wise estimation via factor stochastic volatility, and efficient MCMC sampling for high-dimensional Gaussians, the method overcomes the curse of dimensionality and enables fully Bayesian inference even with large p and moderate T.

ABSTRACT

We develop a Bayesian vector autoregressive (VAR) model with multivariate stochastic volatility that is capable of handling vast dimensional information sets. Three features are introduced to permit reliable estimation of the model. First, we assume that the reduced-form errors in the VAR feature a factor stochastic volatility structure, allowing for conditional equation-by-equation estimation. Second, we apply recently developed global-local shrinkage priors to the VAR coefficients to cure the curse of dimensionality. Third, we utilize recent innovations to efficiently sample from high-dimensional multivariate Gaussian distributions. This makes simulation-based fully Bayesian inference feasible when the dimensionality is large but the time series length is moderate. We demonstrate the merits of our approach in an extensive simulation study and apply the model to US macroeconomic data to evaluate its forecasting capabilities.

Motivation & Objective

• Address the challenge of estimating high-dimensional VAR models where the number of variables (p) is large relative to the sample size (T), a common issue in modern macroeconomic forecasting.
• Overcome the 'curse of dimensionality' in VAR estimation by applying shrinkage priors to reduce overfitting and improve forecast accuracy.
• Enable efficient posterior computation in high-dimensional settings by leveraging conditional independence structures and recent advances in sampling from high-dimensional multivariate Gaussians.
• Develop a fully Bayesian framework that supports uncertainty quantification and model comparison in large-scale macroeconomic systems.
• Demonstrate the model's forecasting performance on real US macroeconomic data, showing robustness and accuracy in high-dimensional contexts.

Proposed method

• Introduce a factor stochastic volatility structure for the reduced-form errors, enabling conditional equation-by-equation estimation and reducing the dimensionality of the covariance structure.
• Apply global-local shrinkage priors—specifically the Dirichlet-Laplace and normal-gamma priors—on VAR coefficients to induce sparsity and mitigate overfitting in high-dimensional settings.
• Use recent advances in MCMC sampling for high-dimensional multivariate normal distributions to efficiently simulate posterior distributions despite large p.
• Implement a fully Bayesian inference scheme that combines shrinkage priors with factor stochastic volatility to maintain computational feasibility and statistical efficiency.
• Leverage conditional conjugacy and block Gibbs sampling strategies to improve mixing and convergence in high-dimensional parameter spaces.
• Ensure scalability by avoiding full covariance matrix estimation through the factor structure, which reduces the number of volatility parameters from O(p²) to O(kp), where k ≪ p.

Experimental results

Research questions

• RQ1 Can a Bayesian VAR model with factor stochastic volatility and shrinkage priors achieve reliable estimation and forecasting in high-dimensional macroeconomic systems with large p and moderate T?
• RQ2 How does the proposed model compare to classical priors like the Minnesota prior in terms of forecast accuracy and variable selection performance?
• RQ3 To what extent does the use of global-local shrinkage priors improve estimation efficiency and reduce overfitting in high-dimensional VARs?
• RQ4 How effective is the factor stochastic volatility structure in simplifying the covariance estimation process while preserving predictive power?
• RQ5 What is the computational feasibility of fully Bayesian inference in large-dimensional VARs using the proposed method?

Key findings

• The proposed model achieves superior forecast accuracy compared to standard Minnesota and non-sparse VARs, especially in high-dimensional settings with moderate sample sizes.
• The use of global-local shrinkage priors—particularly the Dirichlet-Laplace—leads to better variable selection and improved out-of-sample performance by effectively shrinking irrelevant coefficients toward zero.
• The factor stochastic volatility structure significantly reduces the computational burden of estimating large covariance matrices, enabling scalable MCMC sampling.
• The model demonstrates robust performance across multiple macroeconomic variables, including GDP, inflation, and interest rates, in US data applications.
• Efficient MCMC sampling techniques allow for reliable posterior inference even when p exceeds 100, which is often infeasible with conventional Bayesian VARs.
• Empirical results show that the model maintains good predictive density scores and captures time-varying volatility patterns in macroeconomic data effectively.