Causal Inference: Moderation and Mediation I've encountered lots of causal inference terms and jargons (under the Neyman-Rubin potential outcome framework), and I had a question regarding mediator and moderator.
Is it the case that moderation / moderators (interaction terms) necessarily implies unobserved causal mediation?
For example, both gender and race can have impact on income. In this case, if we have a (significant) multiplicative interaction term for gender and race into the regression model (i.e., gender moderates the impact of race, or vice versa), does it implies that there exists unobserved causal mediation (i.e., indirect effects) and we are trying to use this interaction as a proxy for that?
Related Questions:
Causal Inference: Ignorability and Collider
Causal Inference: Selection Bias and Endogeneity
Pearl's Front-door and Back-door

Mediation and moderation are two unrelated concepts that can but do not always occur together.
Mediation occurs when the effect of one variable on another passes through a third variable, e.g., $A \rightarrow M \rightarrow Y$. $M$ is a mediator of the relationship between $A$ and $Y$. $A$ may also have a direct effect on $Y$ or other indirect effects through other mediators.
Moderation occurs when the effect of one variable varies across levels of another variable, e.g., $E[Y^1-Y^0|X=1] \ne E[Y^1-Y^0|X=2]$. $X$ is a moderator of the effect of the treatment on $Y$.
Moderation can occur without mediation when the moderator and the treatment do not cause each other, such as in a randomized experiment where both the treatment and moderator are randomized. There is no mediator because neither variable is caused by the other, but there could be moderation if the effect of the treatment varies across levels of the moderator.
