In the causal graph $$X \leftarrow Z \rightarrow M \rightarrow Y$$, $$Z$$ is a good control variable because conditioning on it blocks the backdoor path $$X \leftarrow Z \rightarrow M \rightarrow Y$$. $$Z$$ acts as a confounder since there is a directed path from $$Z$$ to both the treatment $$X$$ and the outcome $$Y$$ (with $$M$$ acting as a mediator). By controlling for $$Z$$, we effectively close this backdoor path and prevent bias.

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

Common Good controls in causal inference

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.

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