Let's say we have the following AR(2) model:
$y_t=\phi_0+\phi_1y_{t-1}+\phi_2y_{t-2}+e_t, \; e_t\sim N(0,\sigma^2_e)$
with T observations in total.
Working out the conditional log-likelihood is straight forward:
$LLF(\theta)=-\frac{T-2}{2}\ln(2\pi\sigma^2_e)-\sum_{i=3}^T\frac{(y_i-\phi_0-\phi_1y_{t-1}-\phi_2y_{t-2})^2}{2\sigma_e^2}$
where $\theta=(\phi_0, \phi_1, \phi_2)$.
Now, to find the exact likelihood we need to know the densities of $y_1$ and $y_2|y_1$. Assuming stationarity, the mean and variance of $y_1$ are simply the unconditional mean, $\mu$, and unconditional variance, $\sigma^2$, of the process which gives us
$f(y_1|\theta)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{(y_1-\mu)^2}{2\sigma^2}\right)$
where $\mu=\frac{\phi_0}{1-\phi_1-\phi_2}$ and $\sigma^2=\frac{\sigma^2_e}{1-\phi_1^2-\phi_2^2}$.
However, I don't know how to find the conditional density of $y_2|y_1$. Specifically, how do you find the conditional mean and variance? I suspect that $E[y_2|y_1]$ will be of the form $a+by_1$ but I don't know how to find a and b. Any help on this would be great!
