Linear regression - Should I keep the original variable when using its shifted versions

Question

I am using linear regression for modelling the effect of investment in different media channels on sales. Let's say spending on Google Ads effects the sales after two weeks. Now, I have three versions of the variable: the original, shifted one week, and shifted two weeks. The last one correlates the most with the dependent variable. Which version(s) should I include in my model? Why? Should I do this at all?

Answer 1

This is called regression with lagged variables. It could well be that different time lags of the same predictor variable all are important predictors, so you could do this. Cross-validate your models to see which works best!

There are similar questions, see Inclusion of lagged dependent variable in regression