Properties of ARMA models

Question 1

In the ARMA process model what do we assume about the associated polynomials to the AR and MA operators

Answer 1

They do not have common factors - avoids parameter redundancy

Question 2

How do we know if a unique solution of an ARMA process exists?

Answer 2

A stationary solution of (1) exists and (it is unique) if and only if the AR polynomial roots are not on the unit circle - must be inside or outside the circle

Question 3

How do we know if an ARMA(p,q) process is causal?

Answer 3

If Xt can be written as an infinite sum of white noise not dependent on the future or more simply if the roots of the AR polynomial lie outside the unit circle.

Question 4

How do we know if a ARMA process Xt is invertible?

Answer 4

ARMA(p,q) process is invertible if Wt can be written as an infinite sum of the present and past Xt values or more simply it is invertible whent he roots of the MA polynomial lie outside the unit circle