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Given the partitioned regression equation (into $X_1$ and $X_2$), I want to transform $X_1$, say $X^*_1$, such that $X_2$ and $X^*_1$ becomes orthogonal ie. $X_2^T$. $X_1^*$= 0. A matrix can be transformed as $X_1^*$ = $X_1$.p where P is transformation matrix.

If we premultiply $X_1^*$ = $X_1$.p with $X_2^T$ we will get $X_2^T$.$X_1$.p = $X_2^T$.$X_1^*$= 0. How can I find P?

Part (ii) is easy to solve. Parameters of $X_2^T$ and $X_1^*$ will depend on their separate regression with Y.

Question

Greene, Econometrics Analysis, Chapter 3,pg.49

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$\boldsymbol X^*_1$ is the residual of the projection of $\boldsymbol X_1 $ on $\boldsymbol X_2 $. The 'residual maker' matrix is $\boldsymbol I - \boldsymbol X_2 \left(\boldsymbol X_2^T\boldsymbol X_2\right)^{-1}\boldsymbol X_2^T$. Therefore, $$ \boldsymbol X_1^* = \left(\boldsymbol I -\boldsymbol X_2 \left(\boldsymbol X_2^T\boldsymbol X_2\right)^{-1}\boldsymbol X_2^T\right)\boldsymbol X_1.$$

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