I have an ARIMA model, I'm trying to figure out how much an external variable (exogenous covariate) could improve the forecast, so I need to "synthesize" a rate that tell me the usefulness (or impact) of a new external regressor. Does anyone know a good way to do it?
Do this: create a series X, which is a multiple of your target variable say 3 or 5 times or a fraction of your target variable say 10% or 15%. Multiply your target variable values with the selected multiple or fraction but don't multiply all your target values by say a single multiple or fraction value, multiply target values by say numbers in a close range say between 10-13. example below
so Step 1: if my Y (target)= [10,20,30,40..], then I create X as [10*10=100,20*13=160,30*11=330,40*11.5=460...] (created random numbers between 10-13 and multiplied them by target values) Same multiple will create perfect relation between X & Y, and different multiple values in a close range will overcome this to a good extent.
Step 2: To further ensure that X is not a perfect predictor of Y but only reasonably good one and there are other predictors at play like TIME (since you are doing ARIMA), add random noise to X by creating some random numbers and adding them to series X creating a new series say Xnew.
random noise= [25,37,31..]
Xnew= [100+25, 160+37, 330+31..so on]
thus you can now add Xnew to your ARIMAX model.
Xnew is definitely a significant predictor, that you will see in your ARIMAX results.