# Causal model: Controlling for moderating effect of confounder (rather than the confounder itself)

My linear regression model can be described by the following causal graph:

I want to explain the sign of the causal effect of $$X$$ on $$Y$$, but unfortunately cannot measure $$Z$$ directly. Thus, I can only block the backdoor via $$Z$$ as good a possible. Obviously, I control for $$A$$. But I would also like to control for the moderator $$B$$.

Ideally, I would like to control for the moderating effect of $$B$$ directly with the "sub-regression"

$$X=\gamma_0 + \gamma_1 Z + \gamma_2 B +\gamma_3 BZ+\zeta .$$

But as I don't care about the direction, could I just rearrange this equality as follows? $$Z=\delta_0+\delta_1 \frac{B}{1+B}+\delta_1\frac{X}{1+B}+\eta$$

($$\zeta$$ and $$\eta$$ are the error terms.) If so, my full regression model would be $$Y=\beta_0+\beta_1X+\beta_2A+\beta_3 \frac{B}{1+B}+\beta_4\frac{X}{1+B}+\varepsilon$$

Is this the correct regression model? Should it be different (e.g. just $$\beta_3 B+\beta_4 BX$$)? Or do I try something which is impossible?

• This is not a causal graph. Do you think that B influences X, but not Y? Then B would not moderate the effect of X on Y. For that to occur, B would also need to have an open path to Y not running through X. – Julian Schuessler Feb 9 at 10:08
• Thank you Julian for giving this question some thought. Isn't the path through Z (for which I don't control fully through A) the open path to Y for B? – Tom Pape Feb 9 at 17:21