Can there exist a unit root series that’s Granger-caused, or better predicted with a model other than the AR process we tested using ADF? If a series has a unit root, then it is a function of random white noise. Therefore, it follows a random walk process.
Is it then possible for:

*

*Some other series to Granger-cause the unit root series?


*Another model to predict the unit root series better than the AR process we tested for when conducting the ADF? I.e. a random walk process with drift, trend.
Effectively, are we “done,” as many say, when we find a unit root in a series? I.e. you’re wasting your time trying to find a better model than the AR process we tested using ADF?

The premise is incorrect. A series that has a unit root need not be a random walk series. Take any stationary series that is not white noise and then take its cumulatively sums for each time point. The new series will contain a unit root yet it will not be a random walk.
Now to your questions:

*

*Yes, it is possible for other series to Granger-cause a unit-root series. Take a pair of cointegrated time series that each have a unit root, and you will find that at least one of them causes the other (or perhaps both cause each other). For a detailed exposition, see Giles "Cointegration & Granger Causality".

*In such a case and given a sufficiently long time series (so that model coefficients can be estimated with good precision) a vector error correction model will do a better job forecasting your unit-root series (the one that is Granger-caused by the other) than an AR model would.

