Bayesian Shrinkage in High-Dimensional VAR Models: A Comparative Study

TL;DR

This study tackles the problem of estimating high-dimensional VAR($p$) models with many coefficients ($d^2 p$). It systematically compares three Bayesian shrinkage priors (normal, horseshoe, Bayesian lasso) and two frequentist regularizers (ridge, nonparametric shrinkage) across three simulation settings and a Canadian macro data application. The results show that local-global priors, especially the horseshoe, provide strong parameter recovery and near-nominal coverage even under severe overfitting, while ridge excels in point forecasts but often understates uncertainty. The findings advocate using horseshoe-like shrinkage for reliable inference in high-dimensional VARs and indicate practical gains in macroeconomic forecasting when dimension or lag order is inflated. The Canadian data analysis confirms these patterns in practice, highlighting the horseshoe prior’s stability and accuracy across lag choices.

Abstract

High-dimensional vector autoregressive (VAR) models offer a versatile framework for multivariate time series analysis, yet face critical challenges from over-parameterization and uncertain lag order. In this paper, we systematically compare three Bayesian shrinkage priors (horseshoe, lasso, and normal) and two frequentist regularization approaches (ridge and nonparametric shrinkage) under three carefully crafted simulation scenarios. These scenarios encompass (i) overfitting in a low-dimensional setting, (ii) sparse high-dimensional processes, and (iii) a combined scenario where both large dimension and overfitting complicate inference. We evaluate each method in quality of parameter estimation (root mean squared error, coverage, and interval length) and out-of-sample forecasting (one-step-ahead forecast RMSE). Our findings show that local-global Bayesian methods, particularly the horseshoe, dominate in maintaining accurate coverage and minimizing parameter error, even when the model is heavily over-parameterized. Frequentist ridge often yields competitive point forecasts but underestimates uncertainty, leading to sub-nominal coverage. A real-data application using macroeconomic variables from Canada illustrates how these methods perform in practice, reinforcing the advantages of local-global priors in stabilizing inference when dimension or lag order is inflated.

Figures (9)

• Figure 1: Forecast RMSE by Method and Study (All Coefficients). Boxplots reflect the distribution of one-step-ahead RMSE across the 50 replications. Horseshoe achieves or ties for the lowest forecast error, especially in the high-dimension overfit scenario (Study 3).
• Figure 2: Parameter RMSE by Method and Study. Horseshoe is consistently lowest in overall parameter RMSE, while NonparamShrink (ns) occasionally performs well but can exhibit greater variance or undercoverage.
• Figure 3: Mean Interval Length by Method and Study. Shorter intervals may indicate overconfidence if coverage is below the nominal 95%; for instance, ns has narrower intervals but lower coverage in some scenarios.
• Figure 4: Coverage by Method and Study. A dotted line at $0.95$ indicates the nominal coverage target. Horseshoe, Lasso, and Normal usually achieve near 95%, while ns and Ridge can dip below this level for high-dimensional or overfit scenarios.
• Figure 5: The four Canadian macroeconomic variables in their original form (left) and once-differenced (right). Differencing helps remove trends and stabilize the series prior to VAR model estimation.