# How to fit ARMA-GARCH parameters for any distributions

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.
1. 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?
2. 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.
3. This is all repeated to optimize the distribution's parameters
4. Once this is all done, how is this used to construct a prediction for tomorrow?

Is my understanding correct? Thanks!

2. $$\epsilon_i$$ 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).
• Thank you. So when finding the parameters through MLE, what are we testing on? For example, in ARMA and GARCH we have $\epsilon_i$'s. Are these randomly taken from the distribution? Do you have any references I can look further into? Commented Apr 20, 2021 at 7:28