Closed-form expression for finite predictor coefficients of multivariate ARMA processes

TL;DR

This paper presents a new closed-form expression for finite predictor coefficients of multivariate ARMA processes, enabling efficient linear-time computation and advancing understanding even for univariate cases.

Contribution

It introduces a novel explicit matrix-based formula for predictor coefficients, facilitating fast computation and analysis of ARMA processes.

Findings

Provides a linear-time algorithm for predictor coefficient computation

Determines asymptotic behavior in AR model fitting and bootstrap

Results are new for univariate ARMA processes

Abstract

We derive a closed-form expression for the finite predictor coefficients of multivariate ARMA (autoregressive moving-average) processes. The expression is given in terms of several explicit matrices that are of fixed sizes independent of the number of observations. The significance of the expression is that it provides us with a linear-time algorithm to compute the finite predictor coefficients. In the proof of the expression, a correspondence result between two relevant matrix-valued outer functions plays a key role. We apply the expression to determine the asymptotic behavior of a sum that appears in the autoregressive model fitting and the autoregressive sieve bootstrap. The results are new even for univariate ARMA processes.