Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise

TL;DR

This paper establishes necessary and sufficient conditions for the existence of strictly stationary solutions in multivariate ARMA equations with i.i.d. noise, using characteristic polynomials and matrix decompositions.

Contribution

It provides a comprehensive characterization of stationarity conditions for multivariate ARMA equations without prior assumptions on noise or coefficients.

Findings

Conditions expressed via characteristic polynomials and moments.

Additional characterization for p=1 using Jordan decomposition.

No assumptions on noise sequence or coefficient matrices.

Abstract

We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of multivariate ARMA equations with independent and identically distributed noise. For general ARMA$(p,q)$ equations these conditions are expressed in terms of the characteristic polynomials of the defining equations and moments of the driving noise sequence, while for $p=1$ an additional characterization is obtained in terms of the Jordan canonical decomposition of the autoregressive matrix, the moving average coefficient matrices and the noise sequence. No a priori assumptions are made on either the driving noise sequence or the coefficient matrices.