some problem which intuitively seems to work, but where I have some problems to prove. Consider two regression lines fitted by OLS, namely
$\hat{y_i}=b_0 +b_1x_{i1} + b_2x_{i2}$ (1)
and
$\tilde{y_i}=\tilde{b_0} +\tilde{b_1}x_{i1}$ (2)
where $\tilde{y_i}$ is the residual from a regression of $y_i$ on a constant and $x_{i2}$.
Show that
$\mid{\tilde{b_1}\mid}\leq\mid{{b_1}\mid}$
and how you have to change the second equation such that
$\mid{\tilde{b_1}\mid}=\mid{{b_1}\mid}$
How to approach this by using for example the standard normal equations and the resulting properties about the resdiuals? Thanks a lot!