Z is a good control here because it blocks the backdoor path X-Z-M-Y.

Z is the confounder since there is a path from Z to X and Y (with M as a mediator). Thus by controlling for Z we effectively negate the backdoor path.

Explaining X<-Z->M->Y

Z is a good control here because it blocks the backdoor path X-U-Z-Y.

U is the confounder along this path. Z is a mediator along the U->Z->Y path. Thus by controlling for Z we effectively negate the backdoor path and resolve the confounding effect of U.

Explaining X->U->Z->Y

Z is a good control here because it blocks the backdoor path X-Z-U-Y.

U is the confounder along this path. Z is a mediator along the U->Z->X path. Thus by controlling for Z we effectively negate the backdoor path.

Good Control X<-U->Z->Y

U is the confounder here since there is a path to both X and Y (via mediators Z and M) from U. However, the hollow circle shows us that U cannot be controlled for. 
If we control Z however, we break the path from U to Y via Z and M. Thus U is no longer a confounder and we can break the back door path. 
Z is thus a good control.

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

U is the confounder here since there is a path to both X(via mediator Z) and Y (via mediator M) from U. However, the hollow circle shows us that U cannot be controlled for. 
If we control Z however, we break the path from U to X via Z. Thus U is no longer a confounder and we can break the back door path. 
Z is thus a good control.

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

Common causes of X and Y or any mediator between X and Y confound the causal effect of X on Y.

University of Michigan - Ann Arbor

1- Bad controls
2- Good controls (also known as “confounders” or “deconfounders”)
3- Neutral controls

Types of control variables 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

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

Bad Controls are those variables when added to the regression equation will impose bias or other disparities between the real effect size to be estimated, and the regression coefficient that represents it. They are defined on the contrary to good controls (i.e., confounders or deconfounders), which should be adjusted (controlled) in a regression model to improve the accuracy of estimating the desired parameters. Bad and Good Controls are necessary to be considered when deciding which variables to control for in regression analysis for causal inference.

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