Q: Find the autocorrelation function (ACF) and the partial autocorrelation function (PACF) of the following AR(2) process up to and including lag 3:
I am trying to understand how to find the ACF and PACf of time series in general. Is there a general equation to solve them?
$$ y_t=\frac 13 y_{t-1} + \frac 29 y_{t-2}+\epsilon_t $$
Recall that in general for an AR(2) process $y_t = \phi_1 y_{t-1} + \phi_{2}y_{t-2} + \epsilon_{t}$, the ACF $\rho(h)$ satisfies $\color{blue}{\rho(0) = 1, \rho(1) = \dfrac{\phi_1}{1-\phi_2}}$, and
$$\color{blue}{\rho(h) = \phi_1 \rho(h-1) +\phi_2 \rho(h-2)}, \quad \text{for } h = 2,3,4,\ldots.$$
If you know how to solve linear recurrences, you will now be able to obtain a general formula for $\rho(h)$.
For the PACF of such an AR(2) process, recall that it is $\color{blue}{\phi_{11} = \rho(1) =\dfrac{\phi_1}{1-\phi_2}}$ and $\color{blue}{\phi_{22} = \phi_{2}}$, with $\color{blue}{\phi_{kk} = 0 \, \text{for all } k > 2}$ (in general the PACF of an AR(p) process cuts off after lag $p$).
See for example these notes (or your own notes if you have some) for some reference on AR(2) processes.