Is having a high $p$ and $q$ evidence that ARIMA would be a better model than moving average? It is my understanding than a time series having a high $p$ and $q$ would mean that the ARIMA model would be more complex as opposed to having, for example, $p = q = 1$.
My intuition tells me that if a time series supports being modeled with a higher $p$ and $q$ then I should not try using a moving average to predict future values, but a more complex model like ARIMA.
Is there any research related to this? Thanks
 A: The first thing to note here is that a Moving Average (MA) model is a special case of the ARIMA model that occurs when $p=d=0$.  Thus, if you have evidence that $p>0$ or $d>0$ then this constitutes evidence against the MA model.  Any order values above zero constitute a falsification of the MA model, so it is really not terribly important whether you have estimated a large or small (non-zero) order.
Note that if you get a large value of $p$ or $d$ then this might mean that there is a trend in the data that is not easily amenable to the auto-regression and differencing in the ARIMA model, which may mean that the model is not a good model for that data.  However, even if this is the case, since the MA model is a special case of the ARIMA model, that would mean that both models are performing poorly, and it would still not give a reason to prefer a moving average to the full ARIMA model.
A: a high p and a high q probably suggest either bad software or data that is heavily impacted by anthropomorphic effects which need to be identified. If you have such a data set .. post it and I will try and help you . 
