Non stationarity and forecasting Let's assume we have estimated a linear regression model on a dataset from 2000 to 2017.
The data were stationary.
What happens if the data are no longer stationary in the next years? Do the forecasts stop being valid and should not be taken into account? What should a model developer do in this case?
And a related question: regressing non stationary time series automatically leads to spurious regressions?
 A: Normally, if data were stationary and is with new data not stationary anymore one does not come to the conclusion that it will be so forever in the future. It may be that your time series may get back to its original stationary process, but only after a certain time (lag) that would lead to a model correction factor. The question is, what model/method did you use to forecast?

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*a VAR in levels would become a VECM for example in the above case.

*a LSTM would adapt its weights and wont bother so much about the old things. That process is the same as a normal retraining of a NN. It may be that predictors which were good for prediction in the past are no longer. In that case a NN would use the new weights and wouldnt bother about old data anymore.

If data really is no longer stationary, you should try to wait until you have enough data and try to model a new one ( as stated a NN would atuomatically use new and transformed data). But I believe a stationary process observed over such a period (2000-2017), which were already stationary only got a very strong shock, and it will need more time to get back for it.
In summary: look at the amount of data points that are not statioanry, if they are enough to make a second model than do it. If the situation is unclear, wait and see if the time series gets back to its original state. That would lead to a new method but results and equation will only be correct by a certain factor.
Regressing non stationary time series without prior transformation is not advised. I wouldnt say it leads 'automatically' to a spurious regression as we should have a look at it, even a granger causality could be a spurious regression if both time series has nothing to do with each other: look here at this post: The fallacy of correlating some time series values with specific time points: is there a specific name for it or are there references?, but there are papers dealing with that and they state it is exactly as you say:
https://www.tandfonline.com/doi/abs/10.1080/03610920601041499
