Deciding p and q for an ARMA model based on ACF plot Without just obviously trying different combinations of p and q with a grid search, there should be a more intuitive method that some people mention but others don't. To clarify my understanding, how exactly do you determine the p and q of an ARMA model based on the ACF plot of the stationary process? For example, I have an acf below that I have to determine the parameters for (after differencing a process by lag 12 to remove annual seasonality).
I know that q can be decided by counting the number of peaks until termination but there is some convolution with the addition of the AR model. At the same time, I threw this into auto.arima to see what it gives and the best model is either ARMA(1, 1) or ARMA(2, 2). This doesn't make as much sense to me because there's some kind of sinusoid pattern that's relatively annual still, which I believe should result in an ARMA(12, 0) or something.
Thanks for any resources I should look at!

 A: It can be difficult to find the order of an ARMA process by the ACF alone, since the AR and MA components will behave in ”opposite” ways. If you visualise the correlation with a PACF and EACF as well, and read up on what they do I think things will be clearer.
A: As awend has stated, you will need to look at the PACF in conjunction with the ACF. You're right in saying that you should count the number of lags (you've said peak, but lag is a better word in my opinion) until termination. However, you also need to pay attention to how the lags are terminating. Are they cutting off or are they tailing off? Answering that question can often be a matter of judgement.
Generally, an MA(q) process will cut off at lag q on the ACF and tail off on the PACF. It's the reverse for an AR(p). An AR(p) will cut off at lag p on the PACF and tail off on the ACF. If both the ACF and PACF are tailing, this suggests an ARMA process.
The above gets more complicated when you introduce seasonality. You've said that you've seasonally differenced the data, so I'm wondering whether you've considered a SARIMA model instead of just ARIMA. Also, if your data is indeed stationary, then the ACF is definitely suggesting some seasonality. But you will also need to check the PACF as well to confirm whether seasonal MA or seasonal AR (or seasonal ARMA). For seasonality, you still need to check whether the lags are tailing off or cutting off, however, you also need to look where it is doing so. A pure SMA will cut off at every S on the ACF and tail off at every S on the PACF. For example, you've suggested S=12, so you will look for whether the ACF is cutting off at lag=12,24,36,etc, on the ACF. Given your plots, it looks tailing off to me. It's the reverse for the pure SAR. To confirm whether you've got a seasonal AR or seasonal MA, you'll need to look at the PACF as well.
The above does not hold strictly true, especially for mixed SARMAs, but should hold as a good guide. You've stated that you want a more intuitive approach than the grid search, but I will stress that playing around with different orders of the model is extremely beneficial once you have a starting point.
Hope that helps.
