Partial correlation (see Exercise 5.8): (a) Illustrate how the partial correlation r Y1/2 can be represented using geometric vectors. Draw the vectors y(; x( 1, and x( 2, and define e1 {E (1)2 } and e...

Partial correlation (see Exercise 5.8):

(a) Illustrate how the partial correlation r_Y1/2
can be represented using geometric vectors. Draw the vectors y(; x( 1, and x( 2, and define e1 {E₍₁₎₂} and e_Y
{E_(Y)2} (where i is the subscript for observations).

(b) (Use the vector diagram in part (a) to show that the incremental F-test for the hypothesis H₀: β₁
= 0 can be written as

Recalling part (b) of Exercise 5.8, why is this result intuitively plausible?