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

Asymptotic behavior of unstable ARMA processes with application to least squares estimates of... 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 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

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

, Volume 5 (2) – Jul 14, 2005
21 pages

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Publisher
Springer Journals
Copyright © 1989 by Science Press, Beijing, China and Allerton Press, Inc., New York, U.S.A
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/BF02009747
Publisher site
See Article on Publisher Site

### 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 .

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jul 14, 2005

### References

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