(Partitioned Fit) In the previous chapter, we note in (3.16) that

$$\hat{\beta_1} = [X_1'(I - P_x)X_1]^{-1}X_1'(I - P_x)y$$

is the solution to

$$\underset{\beta_1}{min} \ (y - X_1\beta_1)'(I - P_x)(y - X_1\beta_1)$$

if X is full-(column) rank.

(a) Exercise 4.6 does not imply this. Why not?

(b) Show that it is true nevertheless. (HINT: Recall that $I - P_x$ is symmetric and idempotent.)