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I have run an ARIMA model on univariate time series data. I have the below statistical results with the lag differencing at 3. I am not sure which of the model to select to forecast. Any help to choose the order out of the below 2 would be much appreciated.

Note: dataframe d.ratio is with 3rd lag differencing.

Ordering: AR (1), diff (3), MA (1)

Ordering: AR (2), diff (3), MA (3)

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    $\begingroup$ Why don't you use forecast::auto.arima()? $\endgroup$ – Stephan Kolassa Aug 19 '20 at 11:11
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    $\begingroup$ What is "3rd lag differencing"? There's no differencing in your printout, but either a third difference or a seasonal difference with data with frequency 3 would be unusual. Can you show the ACF and PACF of the original data? Also, to more directly answer the question: you need to do out-of-sample testing of forecast performance, you can't decide based on those printouts. $\endgroup$ – Chris Haug Aug 19 '20 at 12:16
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    $\begingroup$ If you are trying to generate forecasts for practical purposes, then as @StephanKolassa said, why don't you use auto.arima()? If you want to do the model selection on your own for learning purposes, then you might want to use the AIC or the BIC as your model selection criteria. As Chris Haug mentions, out of sample performance is the gold standard, but when you don't have enough data, AIC or BIC are your best bet. Notice that you captured the AIC in your first screen shot, but not your second. Also, ideally model selection shouldn't happen between just two models. Try some more parameters. $\endgroup$ – Skander H. Aug 19 '20 at 13:01
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Personally I usually prefer to go for the simpler model i.e. the model with fewer parameters. I would then rather go for the ARIMA(1,0,1) therefore. This is of course given that the residuals are white noise (if you are following the Box & Jenkins procedure).
You do this by using Box.test(residuals(model),lag=15,type=c("Ljung-Box))

But as @StepahanKolassa mentioned, you can use auto.arima in the forecast library. Just remember to give your undifferenced data to auto.arima since it will determine the order of differencing for you. More info on the forecast package here.

If you want to know more on choosing the order of ARIMA models by the Box & Jenkins approach, you might find this post helpful.

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