What if your original model suggest p or q beyond 2 I have often found comments that no MA, AR or integration in ARIMA should have a value beyond 2 in social science data. So what do you do if your ARIMA analysis suggest one of these beyond 2? I assume you chose a reduced model, I m not sure how you do this.
 A: There are conventions, rules of thumb, pieces of hard-won experience and whatever you want to call them in many places, in science and elsewhere. If you believe your situation warrants crossing a line that is usually not crossed, you should (a) make double sure you do need to cross that line, and (b) write up a compelling argument for why.
In your particular case, I would suggest first fitting a model with orders restricted to be no larger than 2, then another model with larger potential orders. Then report the AICs. There are a couple of rules of thumb about differences in AIC, e.g., in Burnham & Anderson. If the unresticted model has an AIC that is lower than the AIC of the restricted model by 10, then this is quite an argument. If the unrestricted model improves the AIC by only 0.3, the argument is less convincing. Alternatively, use a holdout sample and see which model yields lower forecast errors. (Are orders relatively stable if you remove observations at the end? If the model is stable, that is another point in its favor.)
Related: Order of ARMA models and Why does default auto.arima stop at (5,2,5)?
A: In my experience studying ARIMA models (51 plus years ) , the phenomena of >2 polynomials either ar/ma or differencing usually suggests/indicates a Gaussian Violation of some sort .
Gaussian violations can be caused by a violation in the expected value or the variance of the errors. Common treatments are Intervention Detection Intervention Analysis Coding in R TSA Package & http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html and power transforms http://stats.stackexchange.com/questions/18844/when-and-why-to-take-the-log-of-a-distribution-of-numbers OR http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html
