Question

Suppose ?? is generated by an ARMA(2,2) process, such that ?? = ??−1 − 0.25??−2 + ?? − ??−1 + 0.25??−2; ??~??(0, ? 2 )

(a) Suppose we write the ARMA(2,2) process ϕ(L)?? = ?(?)?? how to define ϕ(L) and ?(?)?

(b) Is this ARMA(2,2) process invertible and stationary

Answer 1

Part (a)

this form of representing ARMA process is called lag operator notation

Part (b)

for ARMA (2,2) process to be stationary roots of the AR characteristic polynomial should exceed 1 in absolute value

for ARMA (2,2) process to be invertible roots of the MA characteristic polynomial should exceed 1 in absolute value

Since roots of AR characteristic polynomial and MA characteristic polynomial is greater than 1

Therefore, above ARMA (2,2) process is both stationary and invertible