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I am trying to fit an ARIMA model to time series data. When I fit the model using auto.arima function in R, ARIMA(2,1,1) model is selected with AIC=6618.16. However, when I played around with the model, I found that ARIMA(6,1,8) gives an AIC=6528.37. Which one should I choose? Here are the ACF and PACF plots after one regular differentiation. ACF PACF

Besides, a portmanteau test (Ljung-Box test) gives the following results for checking the residuals whether are white noise or not. A portmanteau test returns a large p-value, also suggesting that the residuals are white noise (Forecasting: Principles and Practice by Rob J Hyndman and George Athanasopoulos).

Ljung-Box test

data:  Residuals from ARIMA(6,1,8)
Q* = 6.2381, df = 3, p-value = 0.1006

Model df: 14.   Total lags used: 17


Ljung-Box test

data:  Residuals from ARIMA(2,1,1)
Q* = 20.261, df = 7, p-value = 0.005033

Model df: 3.   Total lags used: 10
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If your goal is just to forecast (which, if you are using an ARIMA model then my guess is that's the goal), then out of these two competing models you have quite large evidence to suggest that the additional complexity of the ARIMA(6, 1, 8) model is worth it. Out of these two models, it would be justified to go with the ARIMA(6, 1, 8).

What I would recommend: rerun auto.arima but increase the max.p and max.q arguments to something like 10. There seems to be evidence that your time series has significant autocorrelation over large lags.

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  • $\begingroup$ Thanks for the recommendation. auto.arima chose ARIMA(9,1,1) which has a lower AIC than ARIMA(2,1,1). However, the cross-validation error of ARIMA(2,1,1) was lower than that of ARIMA(9,1,1). Nevertheless, when I checked the residuals by using the Ljung-Box test (using checkresiduals(ARIMA.model) function in R), I realized that the residuals of ARIMA(9,1,1) are white noise, but not ARIMA(2,1,1). This raises another question in my mind. $\endgroup$ Feb 14 at 18:33
  • $\begingroup$ By the way, the AIC value of ARIMA(6,1,8) is much lower than both models. $\endgroup$ Feb 14 at 18:49

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