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

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.

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.

Explaining X->Z->W<-U->Y

University of Michigan - Ann Arbor

A variable is neutral to be controlled or adjusted for in regression analysis for causal inference if controlling it does not affect the estimated causal relation between the predictor and the outcome of interest. However, controlling it may affect the precision of the estimation.

Neutral control 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

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