Autoregressive models of singular spectral matrices☆

This paper deals with autoregressive (AR) models of singular spectra, whose corresponding transfer function matrices can be expressed in a stable AR matrix fraction description [Formula: see text] with [Formula: see text] a tall constant matrix of full column rank and with the determinantal zeros of [Formula: see text] all stable, i.e. in [Formula: see text]. To obtain a parsimonious AR model, a canonical form is derived and a number of advantageous properties are demonstrated. First, the maximum lag of the canonical AR model is shown to be minimal in the equivalence class of AR models of the same transfer function matrix. Second, the canonical form model is shown to display a nesting property under natural conditions. Finally, an upper bound is provided for the total number of real parameters in the obtained canonical AR model, which demonstrates that the total number of real parameters grows linearly with the number of rows in [Formula: see text].