Note: this is not an assessed question, it is a practice question from a past exam.
If we are given the following time series: $X_t=\alpha^2X_{t-1}+Z_t-\alpha Z_{t-1}+2\alpha^2$
I am asked to find the values for $\alpha$ for which this is regular (i.e. the MA part is invertible and the AR part is causal, moreover the lack of a common root means that the process may not be simplified). Is what I have done correct?
I write the time series in the form $\phi(B)X_t=\theta(B)Z_t$ with $\phi(B)=(1-\alpha^2)$ and $\theta(B)=(1-\alpha B- \alpha B^2)$, where $B$ is the backshift operator.
We can then see that this is an $ARMA(1,2)$ model, and for an ARMA model we know that it is regular if the roots of both $\phi(B)=0$ and $\theta(B)=0$ lie outside the unit circle and they share no common roots.
Now the roots of $\phi(B)=0$ are $B=\frac{1}{\alpha^2}$ so this lies outside the unit circle if $|\alpha| < 1$
The roots of $\theta(B)$ are given by $\frac{-1}{\alpha},\frac{-1}{2\alpha}$ and these lie outside the unit circle is $|\alpha|< 1 $ and $|\alpha |<\frac{1}{2}|$
So in order for this process to be regular we require that $|\alpha| < \frac{1}{2}$, where we note that the roots will never be shared in this range.
Is the above correct/ the correct approach?
