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the Hyndman-Khandakar algorithm for automatic ARIMA modelling searches the model space and looks only on the AICc as its criteria for finding the best model. Shouldn't it also look at the characteristic roots of the model found?

Isn't it possible that a model fits the data perfectly (as measured by the AICc) but it has characteristic roots which are very close to the unit circle? this would make it numerically unstable and perhaps we could have selected a different model with worse AICc but much better roots?

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The auto.arima() algorithm from the forecast package for R does look at the characteristic roots and will not return a model with near-unit-roots even if it has a low AICc. This is explained at the very end of the link you cite.

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