I am trying to implement the ARIMAX model. I have a lot of exogenous features and I have no clue which ones are important in predicting my endogenous variable. Does ARIMAX have some sort of native feature selection in that it just places less weight to the variables that aren't that important and more weight to the ones that are? If not, should I be using some sort of feature selection algorithm to determine which variables are important?
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The estimation will weight the less explanatory variables lower than more explanatory variables. However, you'll get massive overfitting and probably some spurious correlation, too. The coefficients will be unreliable and it will probably be a poor forecaster, too. You really want to cut those exogenous variables back to something a bit more parsimonious.
Most people will use the AIC and SIC for model selection. Generally, you want to start with the most general model and then start to drop the parameters that have insignificant t-stats. If you have a few that seem collinear you might want to check the F-stat on the group. At each stage, as you drop variables, make sure that your information criteria are improving. When you get to the point where you can no longer eliminate variables without increased your model selection criterion, stop.
