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Ex: Suppose that after an initil analysis, an ARMA(3,2) model was selected for your data. Adjusted the model, you did not find any significant parameter, but the analysis of the autocorrelation function indicated that the process was not white noise. Give some reason for this to happen.

Well, it is to be expected that even if the model is appropriate the autocorrelations are not "exactly" white noise, but at least they seem to be. As the exercise says that they clearly do not seem to be, then one can conclude that the model is not appropriate.

Regarding the significance of the parameters, I think that if an appropriate model were of a smaller order, then only the higher order parameters would be insignificant and not all of them. What makes me think the following, suppose a correct model was an AR (1) and I set an ARMA (3.2), could it occur from the first autoregressive parameter become non-significant? What I'm trying to say is, could overfitting make significant parameters non-significant?

Perhaps it is the case of using an MA or pure AR of higher orders.

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It could, but also consider that if you don't evaluate outliers, level shifts, changes in trend/level/parameters/variance then you certainly could fit a subpar model.

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