I computed an algorithm to find out the best ARMA (p,q) model via minimisation of the AIC. It turned out ARMA(5,5) is the best one with AIC=-2693.12.
However, the inverse roots of the AR and MA characteristic polynomial are the following:
Many of them looks very close to the unitary circle. That makes me feel like I'm in presence of a near non-stationarity & invertibility series (isn't it?). However ACF and PACF show the model as great for capturing autocorrelation in the residuals.
If I use auto.arima() in $R$ to find the best p,q instead, it turns out the best model is a simple AR(1). The AIC worst off to -2687.08.
By looking on internet I figured out that auto.arima() looks yes at AIC (if you specify so) but also at "numerical stability" in returning the "optimal" orders p,q.
What fools me is:
- What does it mean? What are the implication for a statistical analysis?
,and consequently:
- Which order should I use? Should I trade some AIC "points" in exchange of much numerical stability proposed by
auto.arima()? - Would the previous answer change in case the scope of my ARMA model is forecasting or testing?
Here the dput() of my dataset for replicability.
Best Answer
I took your 1488 daily values from this economic time series and found an adequate/sufficient model. There are a number of outliers ( only a few shown here ) and the best forecast is a simple constant.
. The Actual/Fit and Forecast are here 
. The residual ACF suggesting sufficiency is here 
.
Very far from correct but directionally ok is https://people.duke.edu/~rnau/arimrule.htm detailing the way an ARIMA model is formed (IF and only if there are no deterministic structure ( such as pulses) in the data. The acf of the original series is here 
. Your "problem" is that you think auto.arima is a model identification tool … not so much ! .. It can be under very rare circumstances such as if the following 6 characteristics are true for your data .1 ) no pulses in the the data 2) no step/level shifts in the data …. 3) no seasonal pulses in the data …. 4) no deterministic time trends in the data … 5) parameters are constant over time …..6 )error variance is constant over time . Unfortunately your data doesn't match the required profile .