Asymptotic behavior of unstable ARMA processes with application to least squares estimates of their parameters

Abstract

A time seriesx(t),t⩾1, is said to be an unstable ARMA process ifx(t) satisfies an unstable ARMA model such as

$$x(t) = a_1 x(t - 1) + a_2 x(t - 2) + \cdots + a_s x(t - s) + w(t)$$

wherew(t) is a stationary ARMA process; and the characteristic polynomialA(z) = 1 −a ₁ z −a ₂ z ² − ... −a _s z ^s has all roots on the unit circle. Asymptotic behavior of$\sum\limits_1^n {x^2 (t)}$ will be studied by showing some rates of divergence of$\sum\limits_1^n {x^2 (t)}$. This kind of properties will be used for getting the rates of convergence of least squares estimates of parametersa ₁,a ₂, ...a _s .