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I have two Ind. V's, $x_1$ and $x_2$. They are slightly correlated with eachother. $x_1$ explains a significant portion of $y$'s variability. Rather than just modeling $y = \beta_0 +\beta_1 x_1 +\beta_2 x_2+\epsilon$ via multiple reg, I want to orthogonalize $x_1$ and $x_2$.

So, first I estimate $x_2= \beta x_1+\epsilon_1$ (I'm intentionally modeling $x_2$ as a function of $x_1$). Then I estimate the model $y=\beta_0 +\beta_1 x_1+ \beta_2 \epsilon_1 +\epsilon$, such that my independent variables $x_1$ and $\epsilon_1$ are uncorrelated.

Is there a formal name for this methodology? I've heard of it before so I believe it is sound but I could be wrong. Basically, I'm trying to test and see if the portion of $x_2$ uncorrelated with $x_1$, e.g. $\epsilon_1$, has a significant effect on $y$ after controlling for $x_1$.

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