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