Uniqueness of the Solution of the Yule-Walker Equations: A Vector Space Approach
Ana Paula Isais-Torres and Rolando Cavazos-Cadena
This work concerns the Yule-Walker system of linear equations arising in the study of autoregressive processes. Given a com- plex polynomial φ(z) satisfying φ(0) = 1, elementary vector space ideas are used to derive an explicit formula for the determinant of the matrix M(φ) of the Yule-Walker system of equations cor- responding to φ. The main conclusion renders the following non- singularity criterion: The matrix M(φ) is invertible if and only if the product of two roots of φ is always different form 1, a property that yields that the Yule-Walker system associated with a causal polynomial has a unique solution. The way in which this result is implicitly used in the time series literature is briefly discussed.
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