Expected value of $z_t$ given all of its history including itself

Question

So I have the following time series process AR(2):

$z_t = \delta + \psi_2 z_{t-2} + \epsilon_t$

where $\epsilon_t$ is a white noise with distribution $(0, \sigma^2)$

I am looking for the $E[z_t|z_1, ..., z_{t-1}, z_t]$.

Hence, I am looking for the conditional expectation of $z_t$ given its past history and itself.

So does this just turn into $E[z_t]$; that is, an unconditional expectation and this would just be $0$ assuming this is a mean reverting process with that mean being $0$?

Or does this just equal $z_t$ itself; that is $\delta + \psi_2 z_{t-2} + \epsilon_t$

I think it is the former but not entirely sure so any help is greatly appreciated!

Answer 1

Whenever the value of $z_t$ is given, then that is what it is expected to be.

$$\mathsf E(z_t\mid z_t,\textit{yadda yadda yadda}) = z_t$$

(Assuming $z_t$ is measurable over the sigma algebra generated by $z_t$ and the rest.)

Note, this is not $\mathsf E(z_t)$.   The conditional expectation is the random variable itself, and that is not the expectation of the random variable (which is a constant).