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I am interested in fitting an ARMA GARCH model by hand (that is without the use of a package such as rugarch), but am unclear on how the parameters are estimated. I have read that one should use MLE, but how exactly is this implemented?

I have asked a few questions on this and have also skimmed through previous answers but am still unclear.

Here are a few of the posts I have looked through:

Specifying an ARMA-GARCH model without rugarch

ARMA-GARCH model selection / fit evaluation

Fitting an ARCH/GARCH Model (Basics)

ARMA GARCH estimation process in practice

How to fit ARMA-GARCH parameters for any distributions

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  • $\begingroup$ This must be a duplicate of probably more than one question either here or on Quantitative Finance Stack Exchange. Could you indicate more precisely which part of the estimation is unclear to you? If you look at the other answers or textbooks that provide the likelihood of the model, which part do you not understand? Alternatively, if you look at the source code of packages that implement ARMA-GARCH models, which part is unclear? $\endgroup$
    – Richard Hardy
    Commented Apr 25, 2021 at 13:06
  • $\begingroup$ @RichardHardy I have skimmed through a few texts but most of the ones I have seen are handwavey when it comes to implementation. My biggest source of confusion is when implementing the AR or GARCH portions, how are we running a regression on the innovations if they are not observable? $\endgroup$
    – CBBAM
    Commented Apr 25, 2021 at 17:05
  • $\begingroup$ We are not. Estimation of ARMA and GARCH models is not regression-based. It is not by OLS, it is by maximum likelihood. Moreover, it is not by a naive optimization thereof (especially for ARMA models), but often via the state-space representation and Kalman filtering. But again, this really has been discussed on multiple occasions. I agree that finding a good account is hard and probably none of the answers on Cross Validated and Quantitative Finance SE is detailed enough and general enough, but the pieces are there to be put together. Have you tried Hamilton's "Time Series Analysis" textbook? $\endgroup$
    – Richard Hardy
    Commented Apr 25, 2021 at 18:16
  • $\begingroup$ Could you perhaps put up a list of threads you have looked at so that someone willing to help you could have a good starting point (the pieces of the puzzle)? You could edit the list into your question. $\endgroup$
    – Richard Hardy
    Commented Apr 25, 2021 at 18:20
  • $\begingroup$ @RichardHardy Thank you for your reply, I have edited my original post to include some of the threads I have gone through. I have also skimmed through the Hamilton book but did not find it of much use, perhaps I should give it another look. I found most of the sources I have gone through assume parameter fitting is being done with normally distributed innovations, however I am carrying this out of non-normal distributions. $\endgroup$
    – CBBAM
    Commented Apr 25, 2021 at 18:41

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