Explicit Stationarity Regions for AR(4), AR(5), and AR(6) via Unit-Circle Analysis

Determining the stationarity region of autoregressive (AR) processes becomes increasingly difficult to describe explicitly as the order increases: the classical Schur–Cohn criterion certifies stationarity through a recursive sequence of p reflectioncoefficient conditions, but does not directly yield a closed-form description of the stationarity region in terms of the original coefficients (ϕ1, . . . , ϕp). This paper derives such closed-form descriptions for AR(4), AR(5), and AR(6) by analyzing the inverse characteristic equation directly on the unit circle, and shows that the resulting regions can be described with strictly fewer explicit inequalities than the number of conditions in the corresponding Schur–Cohn recursion: three inequalities suffice for AR(4) (against four reflection-coefficient conditions), four for AR(5) (against five), and four for AR(6) (against six). This reduction simplifies both the theoretical characterization of the stationarity region and its practical verification in coefficient space. This reduction is not merely a relabeling of the Schur–Cohn conditions; several of the recursive conditions are shown to be algebraically redundant and therefore need not be imposed separately, and in the AR(6) case a single derived inequality is shown to subsume two successive steps of the reflection-coefficient recursion simultaneously. We also establish structural properties of the stationary parameter space, including zero-extension scalability, a sign-reversal symmetry, dimensional multiplicity, and a Gauss–Lucas-type reduction from AR(p) to AR(p − 1), and give a general, order-independent topological principle that replaces the repeated boundary-intersection arguments otherwise needed for each order individually. For AR(4), the derived region coincides exactly with the classical Schur–Cohn criterion; for AR(5) and AR(6) it is verified, numerically and via an independent Schur–Cohn implementation, to coincide with it as well. These results suggest that the same reduction in the number of independent conditions persists, and may grow, at higher orders; a general, order-independent inductive argument given here confirms this for the number of conditions itself, showing that at most ceil(p/2) + 1 inequalities suffice to describe the stationarity region at every order p ≥ 6, although writing these inequalities out explicitly beyond AR(6) remains open.