Moderated regression: Why do we calculate a *product* term between the predictors? Moderated regression analyses are often used in social sciences to assess the interaction between two or more predictors/covariates. 
Typically, with two predictor variables, the following model is applied:
$Y = β_0 + β_1*X + β_2*M + β_3*XM + e$
Notice that the test of moderation is operationalized by the product term $XM$ (the multiplication between independent variable $X$ and moderator variable $M$). My very fundamental question is: why do we actually calculate a product term between $X$ and $M$? Why not, for instance, the absolute difference $|M-X|$ or just the sum $X + M$? 
Interestingly, Kenny alludes to this issue here http://davidakenny.net/cm/moderation.htm by saying: "As will be seen, the test of moderation is not always operationalized by the product term XM" but no further explanation is given. A formal illustration or proof would be enlightening, I guess/hope.
 A: You won't find a formal proof for using multiplicative moderator. You can support this approach by other means. For instance, look at the Taylor-MacLaurin expansion of a function $f(X,M)$:
$$f(X,M)=f(0,0)+\frac{\partial f(0,0)}{\partial T} T+\frac{\partial f(0,0)}{\partial M} M+\frac{\partial^2 f(0,0)}{\partial T\partial M} TM
+\frac{\partial^2 f(0,0)}{2\partial T^2} T^2
+\frac{\partial^2 f(0,0)}{2\partial M^2} M^2\dots$$
If you plug a function of this form $f(X,M)=\beta_0+\beta_XX+\beta_MM+\beta_{XM}XM$ into the Taylor equation, you get this:
$$f(X,M)=\beta_0+\beta_XX +\beta_MM +\beta_{XM}XM$$
So, the rationale here is that this particular multiplicative form of the moderation is basically a second order Taylor approximation of a generic moderation relationship $f(X,M)$
UPDATE:
if you include quadratic terms, as @whuber suggested then this will happen:
$$g(X,M)=b_0+b_XX +b_MM +b_{XM}XM+b_{X2}X^2 +b_{M2}M^2$$
plug this into Taylor:
$$g(X,M)=b_0+b_XX +b_MM +b_{XM}XM  +b_{X2}X^2 +b_{M2}M^2$$
This shows that our new model $g(X,M)$ with quadratic terms corresponds to a full second order Taylor approximation, unlike the original moderation model $f(X,M)$. 
A: A "moderator" affects the regression coefficients of $Y$ against $X$: they might change as values of the moderator change.  Thus, in full generality, the simple regression model of moderation is
$$\mathbb{E}(Y) = \alpha(M) + \beta(M)X$$
where $\alpha$ and $\beta$ are functions of the moderator $M$ rather than constants unaffected by values of $M$.
In the same spirit in which regression is founded on a linear approximation of the relationship between $X$ and $Y$, we may hope that both $\alpha$ and $\beta$ are--at least approximately--linear functions of $M$ throughout the range of values of $M$ in the data:
$$\eqalign{
\mathbb{E}(Y) &= \alpha_0 + \alpha_1 M + O(M^2) + (\beta_0 + \beta_1 M + O(M^2))X \\
&= \alpha_0 + \beta_0 X + \alpha_1 M + \beta_1 MX + O(M^2) + O(M^2)X.
}$$
Dropping the nonlinear ("big-O") terms, in the hope they are too small to matter, gives the multiplicative (bilinear) interaction model
$$\mathbb{E}(Y) = \alpha_0 + \beta_0 X + \alpha_1 M + \beta_1 MX.\tag{1}$$
This derivation suggests an interesting interpretation of the coefficients:  $\alpha_1$ is the rate at which $M$ changes the intercept while $\beta_1$ is the rate at which $M$ changes the slope.  ($\alpha_0$ and $\beta_0$ are the slope and intercept when $M$ is (formally) set to zero.)  $\beta_1$ is the coefficient of the "product term" $MX$.  It answers the question in this way: 

We model the moderation with a product term $MX$ when we expect the moderator $M$ will (approximately, on average) have a linear relationship with the slope of $Y$ vs $X$.


Of interest is that this derivation points the way towards a natural extension of the model, which might suggest ways to check goodness of fit.  If you are not concerned with nonlinearity in $X$--you either know or assume that model $(1)$ is accurate--then you would want to extend the model to accommodate the terms that were dropped:
$$
\mathbb{E}(Y) = \alpha_0 + \beta_0 X + \alpha_1 M + \beta_1 MX + \alpha_2M^2 + \beta_2 M^2X.
$$
Testing the hypothesis $\alpha_2=\beta_2=0$ evaluates the goodness of fit.  Estimating $\alpha_2$ and $\beta_2$ could indicate in what way model $(1)$ might need to be extended: to incorporate nonlinearity in $M$ (when $\alpha_2 \ne 0$) or a more complicated moderating relationship (when $\beta_2 \ne 0$) or possibly both.  (Note that this test would not be suggested by a power series expansion of a generic function $f(X,M)$.)

Finally, if you were to discover that the interaction coefficient $\beta_1$ were not significantly different from zero, but that the fit is nonlinear (as evidenced by a significant value of $\beta_2$), then you would conclude (a) there is moderation but (b) it is not modeled by an $MX$ term, but instead by some higher-order terms beginning with $M^2X$.  This might be the kind of phenomenon to which Kenny was referring.
A: If you use the sum of predictors to model their interaction, your equation would be:
$$
\begin{eqnarray}
Y &=& \beta_0 + \beta_1X + \beta_2M + \beta_3(X + M) + e\\
  &=& \beta_0 + \beta_1X + \beta_2M + \beta_3X + \beta_3M + e\\
  &=& \beta_0 + (\beta_1 + \beta_3)X + (\beta_2 + \beta_3)M + e \\
  &=& \beta_0 + \beta_1'X + \beta_2'M + e
\end{eqnarray}
$$
where $\beta_1'=\beta_1+\beta_3$ and $\beta_2'=\beta_2+\beta_3$. Therefore, your model would have no interaction at all. Clearly, this is not the case with product.
Recall the definition of the absolute value:
$$
|X-M| = \begin{cases}
          X-M, & X \geq M\\
          M-X, & X < M
        \end{cases}     
$$
Although you can reduce the model $\beta_0 + \beta_1X + \beta_2M + \beta_3|X-M| + e$ to the one with only $X$ and $M$ terms, using the def. of $|X-M|$, the absolute value is a "specialized form of moderation that is unlikely to be realistic in many situations", as pointed out in the comment below.
