Suppose the estimation equation is $$ y=\beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}+\varepsilon $$ where $\varepsilon $ is a disturbance and $x_{1}$ and $x_{2}$ are highly correlated regressors. In a seminar, the presenter showed the results of the following regression $$ y=\gamma _{0}+\gamma _{1}x_{1}+\gamma _{2}\eta +\varepsilon $$ where $\eta $ is the residual of the regression of $x_{1}$ on $x_{2}$
$$ x_{2}=\alpha _{0}+\alpha _{1}x_{1}+\eta $$
Question: What would the advantage of this two-step approach?
