Inner-Envelope Matrix Autoregression

We propose the Inner-Envelope Matrix Autoregression (IEMAR) for matrix-valued time series. IEMAR captures the necessary dynamics by restricting the MAR operators to the largest noise-reducing row/column subspaces contained in their coefficient spaces, yielding a strictly more economical parameterization while preserving the bilinear structure. We develop a Gaussian likelihood and a stable block algorithm with closed-form weighted least-squares updates for the dynamic cores, flip–flop updates for Kronecker-separable innovations, and mode-wise inner-SIMPLS envelope updates on the lattice of reducing subspaces. We prove existence and uniqueness of the inner envelopes (as subspaces), consistency of the estimated envelopes under mild mixing, asymptotic normality of the core parameters, and an efficiency ordering that favors IEMAR over EMAR (outer-envelope MAR) and unconstrained MAR whenever the necessary space is a proper subset of the sufficient one. Simulations show systematic parameter-efficiency gains with forecast risk comparable to EMAR at short/medium horizons, and a New York City taxi application illustrates how IEMAR isolates operative row/column dynamics and improves out-of-sample performance without sacrificing interpretability.