Let ${X_t}$ be an AR(2) process where
$$
X_t = \phi_1X_{t−1} + \phi_2X_{t−2} + \epsilon_t
$$
where $\{\epsilon_t\}$ is a white noise process with $E(\epsilon_t) = 0$ and $\text{Var}(\epsilon_t) = \sigma^2$.
Define $$ Y_t = X_t + \phi_1^{-1}X_{t−1} - \phi_2^{-2}X_{t−2}. $$ Find $\text{Var}(Y_t)$.
I tried doing this: $$ \text{Var}(Y_t) = \text{Var}(X_t) + \phi_1^{-2}Var(X_{t−1}) + \phi_2^{-4}Var(X_{t−2}) + \text{Cov}(X_t,X_{t-1}) - \text{Cov}(X_t,X_{t-2}) - \text{Cov}(X_{t-1},X_{t-2}). $$ However, I am not sure how to simply this expression. Can someone tell me what how to simplify this?
