Impulse Response Inference for Matrix Autoregressions

2026-10-04

AI Summary

This study addresses the lack of effective inference tools for impulse response functions in matrix autoregressive (MAR) models by proposing ProBAB-MAR, a bias-corrected bootstrap method. The approach projects bias-corrected coefficients onto the Kronecker parameter space and integrates the Delta method with asymptotic distribution theory to construct an inference framework that achieves asymptotically correct coverage. Monte Carlo simulations demonstrate that ProBAB-MAR yields empirical coverage rates close to nominal levels while producing narrower confidence intervals in small samples. The proposed method is successfully applied to analyze the transmission of inflation shocks in the Eurozone. Overall, this work provides a reliable finite-sample inference solution for MAR models.

Abstract

Matrix autoregressive (MAR) models offer a parsimonious framework for modeling matrix-valued time series, yet tools for estimation and inference for their impulse response functions are lacking. We develop asymptotic and bootstrap-based inference for impulse responses of stable MAR($p$) models. We derive the joint asymptotic distribution of the coefficient and covariance estimators, which permits closed-form delta-method standard errors. To address finite-sample bias, we propose ProBAB-MAR, a bias-corrected bootstrap that projects the corrected coefficients back onto the Kronecker parameter space, and prove that it attains asymptotically correct coverage. Monte Carlo simulations show that delta-method intervals undercover in small samples, while ProBAB-MAR achieves near-nominal coverage with intervals considerably narrower than those from an unrestricted VAR. An application to euro area inflation illustrates how shocks transmit across countries and inflation categories.

Problem

• Matrix Autoregression
• Impulse Response
• Inference
• Time Series

Innovation

• Matrix Autoregression
• Impulse Response Inference
• Bias-Corrected Bootstrap
• Kronecker Parameter Space
• Delta Method