How to isolate two distinct effects of a unique independent variable in multiple regression analysis? Salut!
First of all thank you if you're reading this post and I wish you all the best for 2020!

Edit 1: All variables are continuous variables (no dummy or categorical)
Edit 2: An illustration of the problem has been added at the end of the post

Problem
My question can be illustrated with a simple regression as follow :
\begin{equation}
Y_t = \beta_a*A_t + \varepsilon_t \;\;\;\;\;\;\; (1)
\end{equation}
Assume that the effect of variable $A_t$ on $Y_t$ have two (and only two) specific channels, meaning that $A_t$ have two distinct effects (let's call them $e_1$ and $e_2$ for the sake of simplicity) on $Y_t$. 
In equation (1), we can only conclude about the overall effect of $A$ on $Y$; where $\beta_a$ would represent both the sum of the effects $e_1 + e_2$, right?

Now assume we can characterize one of the channel using another variable, say $B_{t}$. This variable $B_t$ allows us to control for the effect $e_1$, but also capture other effects. Including it in Equation (1), we obtain :
\begin{equation}
Y_t = \beta_a*A_t + \beta_b*B_t + \varepsilon_t \;\;\;\;\;\;\; (2)
\end{equation}
When we estimate both Equation $(1)$ and $(2)$, we find two different $\hat{\beta_a}$. But, can we draw any conclusion from there about $e_1$ and $e_2$?
My intuition is that:


*

*In Equation $(1)$: $\hat{\beta_a}$ accounts for $e_1 + e_2$

*In Equation $(2)$: $\hat{\beta_a}$ accounts only for $e_2$ since we control for $e_1$ with the variable $B_t$
Is my intuition correct? If yes, is there a way I can estimate/measure $e_1$ from there ($\beta_b \neq e_1$)? 

Potential solution:
Would integrating an interaction term of $A$ and $B$ be a way to capture $e_1$ (as follow)?
\begin{equation}
Y_t = \beta_a*A_t + \beta_b*B_t + \beta_{ab}*A_tB_t + \varepsilon_t \;\;\;\;\;\;\; (3)
\end{equation}
In which:


*

*$\hat{\beta_a}$ : would account for $e_2$

*$\hat{\beta_b}$ : would account for the general effect of $B_t$

*$\hat{\beta_{ab}}$ : would account for $e_1$ (which would change based on the level of $B_t$)



Any help would be much appreciated. I hope the problem is clear enough, don't hesitate to ask if you need further information/specification. Thank you :)


 A: Multiple regression model always capture effect et ceteris paribus for each variable, meaning when every other variables are held constant. I think this property is enough to justify my first intuition, that is:


*

*In Equation $(1)$: $\hat{\beta_a}$ accounts for $e_1 + e_2$

*In Equation $(2)$: $\hat{\beta_a}$ accounts only for $e_2$; $\hat{\beta_b}$ accounts for the general effect of $B_t$
$e_1$ is not captured anymore in equation $(2)$, but rather goes is in the error term $\varepsilon_t$, am I right ?
Thus, adding an interaction term, see equation $(3)$, would rather leads :


*

*$\hat{\beta_a}$: accounts for $e_2$

*$\hat{\beta_b}$: accounts for the general effect of $B_t$

*$\hat{\beta_{ab}}$: accounts for the interaction between $e_2$ and $B_t$
I can argue without the measure of $e_1$, but I'm still very curious if there is a way to measure it?
Since we know the value of ($e_1$ + $e_2$) and the value of $e_2$ only, I guess there is some sort of method, maybe I've learned it before but can't remember. Any help would be welcome please. 
