In This Set of Notes We’re Going to
When I look at a multivariate linear regression model, I try and think of it as a single variable regression model.
Consider the following multivariate linear regression model. Let’s say that we interested in coefficient $\beta$. Economic papers are always reporting the coefficient $\beta$, so it’s important to ask ourselves, what does it capture?
$$ Y_i = \alpha + \beta D_i + \gamma X_i + \varepsilon_i $$
The key to thinking about $\beta$ understand that it’s equivalent to the coefficient $\beta$ in the following single variable regression model.
$$ Y_i = \phi + \beta (D_i - \hat{D_i}) + \eta_i $$
Where $\hat{D}_i$ is the predicted values from regressing $D_i$ on $X_i$. Note, $D_i - \hat{D}_i$ is simple a new variable!
This tells us that $\beta$ captures the linear relationship between the outcome and the part of the treatment variable which cannot be predicted by the controls.