My linear regression model can be described by the following causal graph: enter image description here

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?

  • $\begingroup$ 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. $\endgroup$ – Julian Schuessler Feb 9 at 10:08
  • $\begingroup$ 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? $\endgroup$ – Tom Pape Feb 9 at 17:21

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