Dear colleagues

I'm thinking of a method to isolate the effect of a particular variable (X) on say Y and then use that it in an entirely different regression and another dependent variable.

For example, assume I have the following regressions:


yi​=β0​+β1​xi​+β2zi​+ϵi​ (1)


and let's say that the negative effect of x on y is established (just for the sake of argument), what is the best way to isolate the reduction of Y that is attributed to x and use it in another regression (2)?


ji​=β0​+β1​(reduction of Y attributed to x in regression 1)i​+β2pi​+ϵi​ (2)


I thought about running the regression with and without x and then compare the residuals but not entirely convinced.Or should I save β1​ and use it?

Any Ideas?


Thank you