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So I asked this question and didn't get a response that cleared up my understanding. I came across some material today that provokes my question again in a different context.

In some slides I'm going through, the creator included this diagram:

enter image description here

Which was referred to as "Ballentine Graph".

If we run the regression $\hat y = \hat \beta_0 + \hat \beta_1 X_1$ Then we'll recover the yellow and brown portion of this graph as $\hat \beta_1$. Similarly for $X_2$. If we run the regression $\hat y = \hat \beta_0 + \hat \beta_1 X_1 + \hat \beta_2 X_2$ then $\hat \beta_1$ will now only be the yellow area. Similar results for $X_2$.

My question is, what happens to the brown area when $X_1$ and $X_2$ are both included in the regression? It'd be included in the coefficient recovered if $X_1$ and $X_2$ were included separately but it's not included in either if both regressors are included.

Why does brown disappear from both regressors. Where in mathematical space did it go? To me, it seems like $X_1$ and $X_2$ would have to share it but since this isn't the case, did it go into $\hat \beta_0$? Why?

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The Ballantine diagram is correct. The brown part is lost when regressed on both variables. Just refer to the estimator equations for the coefficients in the two regressor case. There is a minus covariance(X1, X2) term.

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  • $\begingroup$ could you point me in the direction of the derivation of the estimator equation with two regressors? I've only seen bivariate and matrix derivations $\endgroup$ Sep 28, 2020 at 16:00

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