ARIMA models constraints on parameters and characteristic roots


fpp3(2021) Section 9.3 and Section 9.4, constraints on parameters:

restrictions for AR parameters
For p = 1, −1 < phi1 < 1.
For p = 2, the following three conditions must all be met: −1 < phi2 < 1, phi2 + phi1 < 1, phi2 − phi1 < 1. But what is the constraint on phi1?
What are the restrictions for p ≥ 3?

restrictions for MA parameters
For q = 1, −1 < theta1 < 1.
For q = 2, the following three conditions must all be met: −1 < theta2 < 1, theta2 + theta1 < 1, theta2 − theta1 < 1. What is the constraint on theta1?
Similarly, what are the conditions for q ≥ 3?

And are the restrictions the same if estimated jointly as an ARMA(p,q) model?

Section 9.7

I do not understand the following
"Any roots close to the unit circle may be numerically unstable, and the corresponding model will not be good for forecasting."

Also, what is the relationship and interpretation regarding the ARMA coefficients i.e. parameter values, (specifically being outside the +/- 1 bounds) and inverse roots being outside the complex unit circle? In otherwords, why is inside good and outside bad for the inverse roots, and how are those related/interpretable to the sign & magnitude of the coefficients in the ARIMA model?


If the inverse roots are outside the unit circle, the model is nonstationary.

This topic was automatically closed 21 days after the last reply. New replies are no longer allowed.

If you have a query related to it or one of the replies, start a new topic and refer back with a link.