Finding covariance, $covar(y_t,y_{t-i})$ of AR(2)

Question

I am trying to derive a solution for the covariance, $covar(y_t,y_{t-i})$, of an AR(2) process of the following form: $$ Y_t={c}+{\phi_2}{Y_{t-2}}+{\varepsilon_t} $$ The variance of ${\varepsilon_t}$ is $\sigma^2$, $c$ and $\phi_2$ are constants, and $y_t$ is covariance stationary.

Any help is most appreciated.

Answer 1

$ cov(y_{t-k},y_{t•})= cov(y_{t-k},\phi_2 y_{t-2}+\varepsilon_{t}) $

$cov(y_{t-k},\phi_2 y_{t-2})$ + $ cov(y_{t-k},\varepsilon_{t}) $

$ \gamma_{k} = \phi_2\gamma_{k-2} $ for $k > 1 $

$ \gamma_{0} = \phi_2\gamma_{-2} +\sigma^2_{\varepsilon}$ for $k= 0 $

https://en.wikipedia.org/wiki/Autoregressive_model#Yule.E2.80.93Walker_equations