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ACF and PACF for monthly average temperature time series:

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The very first thing you can tell based on your ACF is that your time series is blatantly, obviously seasonal. Which is not overly surprising for monthly temperature data.

So you should take seasonal differences and then examine (P)ACF plots for the seasonally differenced data.

Better yet, use information criteria instead of the outdated Box-Jenkins approach. I recommend Forecasting: Principles and Practice, 2nd ed., by Athanasopoulos & Hyndman as an excellent forecasting textbook.

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I totally agree that outdated Box-Jenkins approaches that ignore outliers, level shifts ,local time trends , seasonal pulses , changes in parameters over time , deterministic changes in error variance should be avoided i.e. the approach of auto.arima which determines the best of a set of simple old-fashioned/outdated/simple arima models.

To review , auto.arima in a brute force list-based procedure that tries a fixed set of models and selects the calculated AIC based upon estimated parameters. The AIC should be calculated from residuals using models that control for intervention administration, otherwise the intervention effects are taken to be Gaussian noise, underestimating the actual model's autoregressive effect and thus miscalculates the model parameters which leads directly to an incorrect error sum of squares and ultimately an incorrect AIC. Most SE responders do not point out this assumption when they promote the free auto.arima tool which I think is a serious error of omission.

Modern/Correct/Advanced ARIMA time series analysis is conducted by identifying a starting model and then iterating to refine the initially suggested model as detailed in If I am convinced that a series is mostly trend+season, what is it I should check about the residuals? will be of help here.

If you wish please post your data I will illustrate this for you and the possible failings/omissions of auto.arima.

As an example of a very bad model identification using auto.arima see https://www.omicsonline.org/open-access/an-implementation-of-the-mycielski-algorithm-as-a-predictor-in-r-2090-4541-1000195.php?aid=65324 .

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