Original equation is as below: $$y = X_1\beta_1 + X_2\beta_2 + u $$ I have to compare the estimator $\hat\beta_2$ and residuals $\hat u$ with the below different regression models:
$$y = X_2\beta_2 + v$$ $$P_xy = X_1\beta_1 + X_2\beta_2 + v$$ $${M_{x_1}}y = X_2\beta_2 + v$$ $${M_{x_1}}y = X_1\beta_1 + {M_{x_1}}X_2\beta_2 +v$$
Here, $P_x = X(X'X)^{-1}X'$ is the projection matrix and $M_x = I - P_x$ is the annihilator matrix. The subscription of these matrices indicate which subspace of regressor it is projecting or annihilating (i.e. $M_{x_1} = I - P_{x_1} = I - X_1({X_1}{'}{X_1})^{-1}X_1'$). Also, the residual $v$ is to differenciating from the original residuals $u$, pointing out the possibility that the residuals can be different in these regression models.
This is one of Davidson's questions that I want to work on and understand. I know that the first three equations will result in different estimators and residuals, but I was wondering how I can show it mathematically to prove that they are not identical.
For the last equation, would it be the same estimator by FWL theorem? Could anyone help me with this question?