So I have the following AR(2) process:
$z_t = \delta + \phi_2z_{t-2} + \epsilon_t$
where $\epsilon_t$ is white noise $(0, \sigma^2)$
How can I transform this process to an MA($\infty$)?
My thought process is to recursively substitute as follows:
$z_t = \delta + \phi_2(\delta + \phi_2z_{t-4} + \epsilon_{t-2}) + \epsilon_t$
$z_t = \delta +\phi_2\delta + \phi_2^2z_{t-4} + \phi_2\epsilon_{t-2} + \epsilon_t$
Then substitute in for $z_{t-4}$ as follows:
$z_t = \delta +\phi_2\delta + \phi_2^2(\delta + \phi_2z_{t-6} + \epsilon_{t-4}) + \phi_2\epsilon_{t-2} + \epsilon_t$
$z_t = \delta + \phi_2\delta + \phi_2^2\delta + \phi_2^3z_{t-6} + \phi_2^2\epsilon_{t-4} + \phi_2\epsilon_{t-2} + \epsilon_t$
And then keep doing the same thing over and over again. Hopefully you see the pattern of what I am doing.
So this is kind of where I am stuck. If this is correct, how would I write the MA($\infty$) process in sigma notation?