Is ARIMA-GARCH nested within ARIMA? I wanted to compare ARIMA(1,1,1)-GARCH(1,1) and ARIMA(1,1,1) model forecasts with a Diebold-Mariano test, but I know that it cannot be used for nested models. Is ARIMA-GARCH technically nested within ARIMA and thus I cannot use the test? I'm just unsure on how to interpret whether these models are nested or not? Thanks in advance for your input! :)

ARIMA-GARCH is not nested within ARIMA, but ARIMA is nested within ARIMA-GARCH. This is because you can obtain ARIMA from ARIMA-GARCH by simplifying the latter model's conditional variance equation to a constant.
Regardless of that, the concern about nested models in the Diebold-Mariano test is only relevant in a narrow field of applications. It matters if you are interested in finding out (1) which of the two models might have generated the data or (2) which model might be the best predictor given perfect estimation precision of the model's parameters. For (1) you can do better than use a Diebold-Mariano test; compare BICs of the two models estimated on the full sample instead. Meanwhile, (2) considers an unrealistic setting that is irrelevant in practice. For most other applications, nestedness is a non-issue. See Diebold "Comparing Predictive Accuracy, Twenty Years Later: A Personal Perspective on the Use and Abuse of Diebold–Mariano Tests" (2015) (and the comments, and the rejoinder) for more details.
