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I have plotted an ACF plot and found all its lagged variables exceed the confidence interval range. I have tested again with Ljung box test on its residuals and most of the lagged variables have a significant test statistics to reject null hypothesis (i.e. p k = 0). In the end result, I get an AR(1) model given that the PACF test statistic is significant at lag 1.

What I confused is that why we don't include MA terms here although ACF test statistics are mostly significant. It contradict with what I thought ONLY DON'T include MA terms if the ACF test statistics of the lagged variables are NOT significant (or does not exceed confidence interval).

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I think you should consider the time frequency of your data. It looks like there is a seasonality every 6 lags, judging from the extrema at t-6, t-12, t-18 etc. Now I think about it, you probably have monthly data and there is some seasonal effect every half year. Like a big spike in ice cream sales in the summer and a big negative effect of ice cream sales in the winter, something like that. You should incorporate this seasonality in your model first, and do your ACF/PACF analysis after to see what structure is left.

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