I am trying to prove the following two statements as they relate to partial correlation in regression: 1. $r^2_{y*x_m*(x_1,...,x_{m-1})}=\frac{RSS_m-RSS}{RSS_m}$
- $RSS_m-RSS=(y^T(I-H_m)x_m)^2/(x^T(I-H_m)x_m)$
Here $RSS_m=y^T(I-H_m)y$ is for the reduced model and $RSS$ is for the full model. $H_m$ is the hat matrix for the reduced model. I am not very good with matrices so I have no idea what to do. Could anyone help me?
For the first one I was thinking to start by setting $r^2_{y*x_m*(x_1,...,x_{m-1})}=\frac{(y^T(I-H_m)x_m)^2}{(y^T(I-H_m)y)(x^T(I-H_m)x_m)}$ but I don't know where to go from here. I can see $RSS_m$ in the denominator there but I don't get how they all go together.