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What if the AIC/BIC is lower (negatively speaking) with the model including just the constant with respect to other ARMA versions?

I don't think because k=1 it is lower by construction.

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Lower AIC/BIC suggests the constant-only model should be preferred to the other competing models. This signals that the extra variance introduced by estimating extra parameters is greater than the reduction in squared bias due to the extra parameters.

No, AIC/BIC of a constant-only model is not lower than AIC/BIC of competing models by construction.

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