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Stability regions and spectra of discrete third-order autoregressive time-series

This paper presents an analysis of third order autoregressive time-series. The parameter regions, in the three dimensional parameter space, that produce the six separate types of power spectral density are analyzed. The study reveals that when a particular two dimensional cross section of the three dimensional parameter space is taken, the region of stability is always triangular. Within each triangular stability region in this two dimensional space, subregions which produce the six possible types of spectral shape are indicated. From these subregions it is possible to approximately choose the parameters necessary to model a process whose power spectral density contains at most two critical frequencies (maxima and minima).