Z is a neutral control here because it :
a. Isn't a confounder.
b. Doesn't block any backdoor paths between X and Y
c. Doesn't open any backdoor paths when controlled.
However, since Z is a cause for X, it may affect the precision of the estimation.

University of Michigan - Ann Arbor

Common Neutral controls in causal inference

Cinelli, Carlos, et al. “A Crash Course in Good and Bad Controls.” Technical Report -UCLA, 21 Mar. 2020.

https://ftp.cs.ucla.edu/pub/stat_ser/r493.pdf

A Crash Course in Good and Bad
Controls

Z does not facilitate a backdoor path between X and Y, and neither is it a confounder. Controlling for Z doesn't create any paths between X and Y either. Thus Z is a neutral control.

Explaining X->Y<-Z

Explaining Z->X->Y

Here Z is a neutral control because it:
a. Isn't a confounder.
b. Doesn't block any backdoor paths between X and Y
c. Doesn't open any backdoor paths when controlled.

However, controlling for Z reduces variations in X, which could affect the precision of the estimation.

Explaining Z<-X->Y

Here Z is a neutral control because it :
a. Isn't a confounder.
b. Doesn't block any backdoor paths between X and Y
c. Doesn't open any backdoor paths when controlled.

Here Z is a cause of W - the mediator. Controlling for Z may increase precision of the estimation.

Explaining        Z
                             |
                            V
                    X->W->Y              

Here Z is a neutral control because it :
a. Isn't a confounder (while W is not controlled)
b. Doesn't block any backdoor paths between X and Y (Unlike W)
c. Doesn't open any backdoor paths when controlled (Unlike W)

Z is an effect of X, so controlling it will decrease variations in X, which may affect the precision of the estimate.

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