Do the p values associated with ARIMA coefficients have any significance attached to them particularly when they are small?
To be precise, can it happen that for an ARIMA(2,0,0) model the lag 2 coefficient is significant, particularly at 5%, as indicated by the p value, and yet ARIMA(1,0,0) has a lower AIC than that of ARIMA(2,0,0) leading to the conclusion that ARIMA(1,0,0) is the better model?
My understanding based on Why applying model selection using AIC gives me non-significant p-values for the variables is that if a coefficient has a p-value greater than 5% it might still be a part of the model as determined by AIC. What if a coefficient has p-value less than 5%? Can it still NOT be a part of the final model as determined by AIC or by any other criterion?
The third last paragraph in https://robjhyndman.com/hyndsight/tests2/ says that a variable significant by p value might still not be a part of the final model. But the predictor that is being spoken about is said to have a high variance. Are predictors not supposed to be fixed? I did not quite understand this part. Also, the coefficient of the predictor is said to be small and significant for large sample size and so not adding much information given the high variance of the predictor. What if the coefficient is large and significant? What then?