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For a given ARMA model (order and coefficients are known) we generate simulated data. Model is stationary and invertible. Then using this data, I want to find the best model by trying all combinations of p and q (e.g., p,q < 5) and select model with lowest AIC or BIC. Is it possible that the best candidate model on the basis of AIC or BIC differs from the initial model?

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Yes it's quite possible; indeed with large samples the AIC is nearly certain to choose a different model.

When there's a "true model" in the candidate set of models, the BIC will tend to pick it - at least in larger samples; however, such a situation would be very rare in practice -- when there are a range of very many small effects (a much more common situation in practice) the AIC will often be a better choice.

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