Impulse Response Inference for Matrix Autoregressions

Alain Hecq, Ivan Ricardo and Ines Wilms

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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.

Date: 2026-10
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