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I have a data set that is transformed to stationarity and I'm trying to fit it to an ARIMA model. I found that variance is lowest when the transformed set is differenced to 1, and here are my ACF and PCF plots for that: enter image description here

enter image description here

Running the auto.arima() function on R using the transformed but non-differenced data says I should use an ARIMA(1,1,2) model. I know the difference value of 1 is correct, but I don't understand where the AR(1) and MA(2) models come from. How do you read the ACF and PCF to interpret p=1 and q=2?

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    $\begingroup$ post your data. $\endgroup$ – Tom Reilly Jun 13 '17 at 12:04
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I recommend that whenever you don't understand a command's result, you first look at its help page. The first sentence of ?auto.arima is

Returns best ARIMA model according to either AIC, AICc or BIC value.

Fitting an ARIMA model via an information criterion can yield a different order than the classical approach, and in your case, it does.

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When using forecast::auto.arima, you are fitting a model based on IC. Your model will highly depend on which IC you have chosen.

AIC will usually return models that are more complex as it prefers more parameters compared than to the for example BIC.

However, to obtain a correct model you need to make sure that there is no autocorrelation left in your residuals.

If you go with box-jenkins, you would be fitting models and looking at the residuals correlogram until a model removes the autocorrelation.

Keep in mind that sometimes other effects might be causing non-stationary behavior (eg. long-memory in your series), which might result to overly complicated ARIMA models.

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