Inverse AR/MA roots and near non-stationarity/invertibility 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.
 A: 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 .
