How to fit ARMA-GARCH parameters for any distributions

Question

To better understand the ARMA-GARCH model I am working on implementing it while avoiding as many packages as I can. For data I am working on returns and for simplicity I am starting with ARMA (1,1) and GARCH (1). I have tried doing some reading but most sites refer to packages rather than explaining the implementation or theory.

From what I understand one begins with MA, then AR, then GARCH. If this is correct, I had a few questions on these processes:

1. For the moving average model, how do I go about parameter identification? Are the $\epsilon_i$ always taken to be from a normal distribution even if I want my innovations from ARMA-GARCH to not be Gaussian? Also, what is $\mu$ exactly? Some references claim it is the mean of the data, others say to take it as 0.

2. Similarly for MA, I am unsure how to obtain the parameters $c$ and $\varphi_i$ Here I assume we let the random term, $\epsilon_i$, be from the distribution we would like?
3. Once we have applied ARMA, are the innovations then fed into GARCH? My current understanding is the innovations from ARMA, $\epsilon_t$, are used in least squares to find the GARCH parameters.
4. This is all repeated to optimize the distribution's parameters
5. Once this is all done, how is this used to construct a prediction for tomorrow?

Is my understanding correct? Thanks!

Answer 1

Brief answers to some of your questions:

0. The ARMA model is estimated in one go (not split between MA and AR) e.g. using its state space representation and Kalman filtering.
1. $\epsilon_i$ are not always Gaussian. Their distribution depends on the assumption on standardized innovations in the GARCH conditional variance equation.
   $\mu$ is the constant and it can be either zero (a restricted version) or not zero (a nonrestricted version).
2. No, the distribution is as mentioned in 1.
3. It is advisable to estimate ARMA-GARCH simultaneously. There are multiple threads on Cross Validated that discuss this. However, you can estimate ARMA first and GARCH second. Least squares will however not help you. The standard way of estimating the GARCH model is by maximum likelihood.
4. Repetition is not needed if you estimate ARMA-GARCH simultaneously. Repetition is usually not done even if you estimate the two parts of the model separately.
5. This is really a different question. See some related threads on Cross Validated or open a time series textbook; you will definitely find a discussion there.