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Consider the stationary AR$(2)$ process of the form: $y_{t} = \alpha + \phi_{1} \ y_{t-1} + \phi_{2} \ y_{t-2} + u_{t}$ where $u_{t}$ is i.i.d. white noise.

Just as a head's up, we have not covered anything about Yule Walker equations, etc. in class.

Thus would the following be ok to calculate the autocorrelation?

$$ \begin{align} \text{Cov}(y_{t},y_{t-k}) &= \text{Cov}(\alpha + \phi_{1} \ y_{t-1} + \phi_{2} \ y_{t-2} + u_{t},y_{t-k})\\ &= \phi_{1}\ \text{Cov}(y_{t-1},y_{t-k}) + \phi_{2} \ \text{Cov}(y_{t-2},y_{t-k}) + \text{Cov}(u_{t},y_{t-k})\\ &= \phi_{1}\ \gamma_{k-1} + \phi_{2}\ \gamma_{k-2}\\ &= \gamma_{k} \end{align} $$

Then, we have

$$ \begin{align} \rho_{k} &= \frac{\gamma_{k}}{\gamma_{0}}\\ &= \frac{\phi_{1}\ \gamma_{k-1} + \phi_{2}\ \gamma_{k-2}}{\gamma_{0}}\\ &= \phi_{1}\ \rho_{k-1} + \phi_{2}\ \rho_{k-2}\\ \end{align} $$

for $k \ge 1$.

Now, $\rho_{k}$ is correct from what I have seen on the internet... but they all go through complex methods for calculating the variance, etc. Is the above ok, or am I making a huge algebra mistake somewhere, or are the restrictions on the above actually very strict and need to be specified? If so, what are they?

For reference, the above is the same method we used for the AR$(1)$ process autocorrelation function.

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That is correct as far as it goes. What is still missing are the initial values for your difference equation so as to be able to express correlations through the coefficients only.

We know that $\rho_{-k} = \rho_{k}$ by stationarity. We always have $\rho_0 = 1$. For $k=1$, this gives \begin{align*} \rho_1 & = \phi_1\rho_{0} + \phi_2\rho_{-1}\\ & = \phi_1 + \phi_2\rho_{1} \end{align*} so that $$ \rho_1 = \frac{\phi_1}{1-\phi_2}$$ Note: What you did is basically Yule-Walker.

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