My time series is very leptokurtic and non-normal, which is of course highly common for time series data. However, I don't exactly understand why that is not a problem for ARIMA modeling and GARCH modeling? I know that stationarity is the most important condition, but why is the non-normality of the series not an issue?
Why should it be a problem? Note that modelling assumptions are typically made about the distribution of residuals or the conditional distribution of the dependent variable.
I have just seen some research papers do the Jarque-Bera test on the series, discussing that the distribution of time series was non-normal. I understand that it does not pose any issues for time series modeling so then I was slightly confused about why normality tests would even be done? Just to describe the time series?
Normality tests could be used for descriptive purposes. Or they could be applied on residuals or standardized residuals from some models where the estimation of parameters may rely on an assumption of normality.