I have been trying to model the value of bitcoin log returns using R and my main objective is to apply an ARCH-GARCH model to the data set and produce some forecast from this. I think it might be applicable to note that this is for academic purposes and apologies if I have misinterpreted any of the concepts explained below.
So, in essence, I have run my dataset and obtained the log returns. I then used an ADF test to confirm stationarity thus I could begin to apply an ARIMA model to the data set. I used the AIC values to determine that the ARIMA(7,0,5) is the best-suited model for the log returns of bitcoin. However, when analysing the residuals for the Arima models the ACF gave evidence for discrete white noise but the QQ plots indicated that the residuals are not normally distributed and are in fact heavy-tailed (possible t-distribution?). This immediately raised some concerns for me as 1. The AIC values which use maximum likelihood function assumes asymptotic normality and thus imply that the model selection is not viable? And 2. R uses ML to determine the model.
Jarque Bera Test
So then my question arises, can I still use a GARCH on the ARIMA(7,0,5) model if the residuals from the ARIMA model are not normally distributed? if it is ok could you explain why? Or am I better off using it on the log data data itself?
I noticed with the GARCH function you can fit a T-distribution given a much better fit on the QQ plot, I was wondering if there is a package within R that lets me fit a distribution to the ARIMA model such as the GARCH function.