For instance, I have an ARIMA (1,0,1) process (determined by auto.arima function, and analysis of residuals which are white noise). I know how to read an ACF for a strict AR(p) process or a MA(q) process, but how does one infer from the ACF and PACF that we have an ARIMA (p,q) process. Are there any resources out there that can help me with this?

E.g. For my ARIMA (1,0,1) process mentioned above, here is what my ACF and PACF look like:

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How are these characteristic of a ARIMA(1,0,1) process?

Another example. I have what was determined to be a ARIMA(1,0,2) from the b auto.arima function. I have tested for white noise residuals. These are the coeffiecnts: enter image description here enter image description here enter image description here

Again, this ACF and PACF are not characteristic of your sole AR(p) or MA(q) process. It's seems "random". Can you tell me how this is supposed to lead me to determining an ARIMA model (in this case, probably an ARIMA(1,0,2) process ) ?



1 Answer 1


Unfortunately, ACF and PACF plots are only useful in determining ARIMA($p$,$d$,$q$) orders if either $p$ or $q$ are zero. See, e.g., the section on nonseasonal ARIMA in Forecasting: Principles and Practice by Hyndman and Athanasopoulos.

Thus, your best bet might be to investigate your ACF/PACF plots for regularities, add an AR or MA term as suggested by the plot, check ACF/PACF plots of residuals, and iterate until there is no more evidence of structure. This is essentially the approach.

Alternatively, fit multiple models, either exhaustively or greedily, and pick the one with the lowest AIC (or AICc or BIC). This is a more modern approach than Box-Jenkins, and it has arguably superseded the older one, not least because it is more easily automated. (Note that you can't decide on the order of differencing $d$ using information criteria.)


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