# ACF and PACF functions for AR(2) model

I am trying to practice calculating the autocovariance function and the partial autocorrelation functions of a time series $$X_t$$. I am familiar with how to calculate the autocovariance functions of an AR(2) model, but I am not sure how I to go about calculating the autocovariance and partial autocorrelation functions of an AR(2) model such as $$X_t=aX_{t-2} +e_t$$ where $$|a|<1$$. Not that here we are missing the $$X_{t-1}$$ term. Can anybody please help?

• if you can calculate for the generic AR(2) model, why don't you just equate its coefficient to $0$ in the equations you found? Mar 16, 2020 at 8:32

By Yule-Walker, $$Cov(X_t,X_{t-k}) = aCov(X_{t-2},X_{t-k})$$ $$γ(k) = aγ(k-2)$$ divide by $$γ(0)$$, we get autocorrelation function $$ρ(k) = aρ(k-2)$$ We know $$ρ(k) = ρ(-k)$$ and $$ρ(0) = 1$$, so that when $$k = 1$$, $$ρ(1) = aρ(1)$$ $$ρ(1) = 0$$ and when k = 2, $$ρ(2) = aρ(0) = a$$ For $$k \geq 3$$, we used repeated substitution in $$ρ(k) = aρ(k-2)$$