I want to find the covariance for an AR(2) process I am not sure what the formula is for the covariance of an AR(2) process
$$
X_t = \phi_1 X_{t−1} + \phi_2 X_{t−2} + \epsilon_t
$$
where $\{\epsilon_t\}$ is white noise process (Gaussian) $N(0, \sigma^2)$.
What is the formula for $\text{Cov}(X_t, X_{t-j})$?
 A: Just use the bilinearity of the covariance function:
$$\operatorname{cov}\left(\sum_{i=1}^n a_iX_i, \sum_{j=1}^m b_j Y_j\right) = \sum_{i=1}^n\sum_{j=1}^m a_i b_j(\operatorname{cov}X_i, Y_j)$$
to get
\begin{align}
\operatorname{cov}(X_t,X_{t-j}) &= \operatorname{cov}(\phi_1X_{t-1}+\phi_2X_{t-2}+\epsilon_t, \phi_1X_{t-j-1}+\phi_2X_{t-j-2}+\epsilon_{t-j})\\
&= \phi_1^2 \operatorname{cov}(X_{t-1},X_{t-j-1})
+ \phi_1^2 \operatorname{cov}(X_{t-2},X_{t-j-2})\\
& \quad +\phi_1\phi_2 \operatorname{cov}(X_{t-1},X_{t-j-2})
+ \phi_1\phi_2 \operatorname{cov}(X_{t-2},X_{t-j-1})
\end{align}
which gives a linear recurrence relationship that you can 
try and solve for
yourself, or read up on how to go about doing so in your book. A very general relationship for time series defined by linear recurrences is that
the "output" autocovariance function is given by the convolution of the "input" autocovariance function and the "autocovariance" of the characteristic function of the recurrence:
$$C_o = C_i\, \star\,  \left(\phi(x)\,  \star\, \phi(x^{-1})\right).$$
