Covariance between $z_t$ and its first lag

Question

I am looking for $Cov[z_t, z_{t-1}]$

where $z_t = \delta + \psi_2 z_{t-2} + \epsilon_t$ and $\epsilon_t$ is a white noise with distribution $(0, \sigma^2)$

So I have started by doing the following:

$Cov[\delta + \psi_2 z_{t-2} + \epsilon_t, \delta + \psi_2 z_{t-3} + \epsilon_{t-1}]$

since $z_{t-1} = \delta + \psi_2 z_{t-3} + \epsilon_{t-1}$ I believe.

I believe this covariance comes out to $0$ since I think $z_t$ only depends on its even lags; that is $z_{t-2}, ..., z_{t-2n}$.

Is this line of reasoning correct or am I screwing up somewhere?

Answer 1

Well assuming $\delta$ and $\psi_2$ are constants, your $z_0, z_2, z_4, \dots$ are independent of $z_1, z_3, z_5, \dots$, so you've got that going for you. And in that case $\text{cov}(z_t, z_{t-1}) = 0$.