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 :)