# Understanding influence of correlated predictors on target variable

I am working on a problem where I have a "target" variable $$Z$$ that I know for sure is influenced by a "predictor" $$Y$$. I also have a second predictor $$X$$ that is correlated with $$Y$$ (about -.3), and I would like to know whether is has an influence on $$Z$$

If I fit a linear model for $$Z$$ on $$X$$, I get a very significatively non-zero coefficient for the slope, but of course, this tells nothing, as any variable that is correlated with $$Y$$ would do so.

Next thing I tried was to fit a linear model for $$Z$$ on both $$X$$ and $$Y$$. The coefficients for both variables is significant, but, since the variables are correlated, I am not sure about the power of the model to "distinguish" between the effects of each of the predictors.

So I finally went for a third approach: I will build the model in two steps: First, I will fit a model for $$Z$$ on $$Y$$ to eliminate its effects. Then, I'll take its residuals $$\epsilon_i$$ and fit a second linear model for $$\epsilon$$ on $$X$$

Now my question is: Is this a valid method to achieve what I actually want to achieve? Also, should I add more complex terms in the "first-step" model, to eliminate other non-linear effects of $$Y$$ on $$Z$$?

NOTE: When I say "significant", I don't mean one of those $$0.04$$ $$p$$-values that fill mediocre research papers. I mean something more in the roum of $$10^{-10}$$, as sample and effect sizes are both quite big

• Would you please post the data, or a link to the data? – James Phillips Jun 28 at 12:59
• Unfortunatelly, the data belongs to my company, not myself – David Jun 28 at 23:04