I was given a model $r_{t} = ϕ_{0} + ϕ_{2}r_{t-2} + ϵ_{t}$ with $\epsilon_t \sim N(0,\sigma^2)$ and have to derive the likelihood of $(r_{3}, r_{4}, . . . , r_{T})$ conditional on $(r_{1}, r_{2})$ and find $ϕ_{0}$ and $ϕ_2$ that maximize the likelihood function given $σ^2$ is known
I think I managed to get the conditional likelihood function as followed: $$\begin{aligned} \ln L(r_{3}, r_{4}, . . . , r_{T}|r_{1},r_{2};\sigma^2) &= -\displaystyle\frac{T-2}{2}\ln(2\pi)-\displaystyle\frac{T-2}{2}\ln(\sigma^2)\\ \\ &-\displaystyle\sum_{t=3}^{T} \displaystyle\frac{(r_{t}-\phi_{0}-\phi_{2}r_{t-2})^2}{2\sigma^2}\\ \\ &=-\displaystyle\frac{T-2}{2}\ln(2\pi)-\displaystyle\frac{T-2}{2}\ln(\sigma^2)-\displaystyle\sum_{t=3}^{T} \displaystyle\frac{\epsilon_{t}^2}{2\sigma^2} \end{aligned}$$
However I am not sure how to derive MLE from this. Could someone let me know how to proceed further?
Okay. I'll take the derivatives and show you but you should practice with these things so you can get better with doing this sort of thing.
We have $L = \sum_{t=3}^{t=T} (r_t - \phi_2 r_{t-2} - \phi_{0})^2$
First take $\frac{\partial L}{\partial \phi_{0}}$ and set it to zero.
This gives $ -2 \sum_{t=3}^{t=T}(r_t - \phi_2 r_{t-2} - \phi_{0}) = 0$
$ (T-2) \phi_{0} = \sum_{t=3}^{t=T}(r_t - \phi_2 r_{t-2}) \longrightarrow $
$\phi_{0} = \frac{\sum_{t=3}^{t=T}(r_t - \phi_2 r_{t-2})}{T-2}$
So, that's the estimate for $\phi_0$ which we call $\hat{\phi}_{0}$. (Notice that it's a function of $\phi_2$ which we will do next)
So, now we take $\frac{\partial L}{\partial \phi_{2}}$ and set it to zero.
This gives $-r_{t-2} \sum_{t=3}^{t=T}(r_t - \phi_2 r_{t-2}) = 0 \longrightarrow \hat{\phi}_2 = \sum_{t=3}^{t=T} \frac{(r_t \times r_{t-2})}{r_{t-2} \times r_{t-2}}$
So, now that we have the MLE of $\phi_2$, we can use that in the MLE formula for $\hat{\phi}_{0}$ in order to obtain that.
I hope this is clear. If not, then let me know. Also, notice that all summations should be over the odd integers because the even integers don't come into play. Also, I just realized that you shouldn't divide by (T-2) but rather divide by the number of observations which is either ((T-2)/2) or ((T-2)/2 + 1) depending on the value of $T$.