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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?

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    $\begingroup$ 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? $\endgroup$
    – gunes
    Mar 16, 2020 at 8:32

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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)$

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