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I was having the same doubts with NBEATSx, but I solved it with a simple experiment. I literally gave a slightly modified version of the prediction (yhat) as exogenous feature, and the model was able to capture it. So maybe try that just to check that it can do that.

Like you purposefully introduced data leakage to see if the model can use it to improve? I'll give it a try.

Precisely! My logic was - If exogenous variables are truly informative for the model, then it should be able to capture the leak and reproduce the results.