Mediator vs. Moderator variables

I have a co-author who is a psychologist. As we are collaborating on a paper, I found myself unclear about some of the terms, not just in psychology, but also in statistical analysis. One example is the concepts of mediator (mediating variable) and moderator (moderating variable).

As an economist, these two terms are foreign to me. But after googling them, they can be easily explained in terms of econometrics.

Psychologists use this concept of mediating variable to refer to the underlying psychological factor ($x_m$) that can explain why some external factor ($x_e$) lead to a change in a dependent variable ($y$).
Consider a regression of the dependent variable y on the independent variable $x_e$. If we add $x_m$ to the regression significantly change the direct effect of $x_e$, then $x_m$ can be considered as a mediating variable.





The concept of moderator variables, on the other hand, is closely related to interaction terms in econometrics. When we find that the effect of x1 on y depends on the value of x2, then we call x2 a moderator variable, and it has a moderating effect on the relationship between x1 and y. Moderators are usually categorical variables. Back to the above example, we find altruism x1 influences bidding behavior, but the effect is more pronounced for females. Then here gender x2 is a moderating variable that influences the relationship between altruism x1 and bidding in an all-auction y.

In a regression model,  the relationships between y, x1, and x2 can be expressed as the following relationship:
    $y = b_0 + b_1 x_1 + b_2 (x_1 \times x_2) + \epsilon$

The moderating effect of x2 is measured by b2.

P.S.: Biddings in all-pay auctions provide a measure of competitiveness. The above example is essentially saying that altruistic people are less likely to be competitive, but the effect is more significant for female. In other words, we can say that men seem to care less about the opponent in a competitive environment. How do we measure people's altruism? You can use a dictator game in the lab.