Calculate expected correlation for a AR(2) process

Question

Let $X_t = X_{t-1} + aX_{t-2} + e_t$ be a stationary AR(2) process. At $t = T$, what is the expected correlation between $X_{T+N}$ and $X_{T+N-1}$?

I know how to calculate $\gamma(1)$ but not conditional $\gamma(1)$.

Answer 1

Hint: for N=1 it is trivial that it is zero. N=2. This becomes $$X_{T+2}=X_{T+1}+aX_{T}+\varepsilon_{T}$$. So conditional on $X_{T}$ (subscripts are for $N$) covariance is

$$\gamma_{2}(1)=Var(X_{T+1}|X_{T})=Var(\varepsilon)$$. Repeat for $N=3$ $$ X_{T+3}=X_{T+2}+aX_{T+1}+\varepsilon_{T+3} $$ Compute the conditional covariance $$ \gamma_{3}(1)=Var(X_{T+2}|X_{T})+a\gamma_{2}(1) $$ Iterate forward and proceed similarly to compute the variance and arrive at an expression for the correlation coefficient