On Steinʹs equation, Vandermonde matrices and Fisherʹs information matrix of time series processes. Part I: The autoregressive moving average process Original Research Article

This paper introduces several forms of relationships between Fisherʹs information matrix of an autoregressive-moving average or ARMA process and the solution of a corresponding Stein equation. Fisherʹs information matrix consists of blocks associated with the autoregressive and moving average parameters. An interconnection with a solution of Steinʹs equation is set forth for the block case as well as for Fisherʹs information matrix as a global matrix involving all parameter blocks. Both cases have their importance for the interpretation of the estimated parameters. The cases of distinct and multiple eigenvalues are addressed. The obtained links involve equations with left and right inverses, these can be expressed in terms of the inverse of appropriate Vandermonde matrices. A condition is set forth for establishing an equality between Fisherʹs information matrix and a solution to Steinʹs equation. Two examples are presented for illustrating some of the results obtained. The global and off-diagonal block case with distinct and multiple roots, respectively, are considered.