How to isolate two distinct effects of a unique independent variable in multiple regression analysis?

Question

Salut!

First of all thank you if you're reading this post and I wish you all the best for 2020!

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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

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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?

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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 ($\beta_b \neq e_1$)?

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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 $B_t$)

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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 :)

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Answer 1

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.