I am really trying to invest some time to estimate a GARCH(1,1) method, I know there is many statistical packages that will do this for me (Eviews, MATLAB, R), but I am trying to do this by hand, so that I can really understand the model.
Following the theory from Verbeek. 'A guide to modern econometrics',
I first estimate an AR(1) model,
$$Y_t= \alpha +\theta Y_{t-1} +\varepsilon_t $$
And then I save the residuals, $\varepsilon_t$,
I then want to model a GARCH(1,1) with these residuals.
$$\sigma_t^2=\omega+\alpha \varepsilon_{t-1}^2 +\beta\sigma_{t-1}^2$$
This is where I am struggling, I understand a few things, I understand how I can get the lagged $\varepsilon_{t-1}^2$, I already have it from the first equation, however my major question is how do I get a value for $\sigma_{t-1}^2$ this is what I am trying to estimate, so I am really struggling about how I derive this part. I suppose I need a series for $\sigma_{t-1}^2$, which in matrix terms is a long as the $\varepsilon_{t-1}^2$.
I am thinking I may need to use a for loop to get this $\sigma_{t-1}^2$, but any suggestions on how I might go about this would be helpful.
Second, when I do realise how to do the above step, I need to estimate the thing my maximum likelihood estimation. I think I read around this and try my best to solve it.
I would really like help on step one if anyone has got the time or direction to a good reference?
