An alternative procedure for calculating the least-squares coefficient B 1 is as follows: 1. Regress Y on X 2 through X k , obtaining residuals E Y/2 ... k . 2. Regress X 1 on X 2 through X k ,...

An alternative procedure for calculating the least-squares coefficient B₁
is as follows:

1. Regress Y on X₂
through X_k
, obtaining residuals E_{Y/2
... k}
.

2. Regress X₁
on X₂
through X_k
, obtaining residuals E_{1/2
... k}
.

3. Regress the residuals E_{Y/2 ... k}
on the residuals E_{1/2 ... k}
. The slope for this simple regression is the multiple-regression slope for X₁, that is, B₁.

 (a) Apply this procedure to the multiple regression of prestige on education, income, and percentage of women in the Canadian occupational prestige data, confirming that the coefficient for education is properly recovered.

(b) The intercept for the simple regression in Step 3 is 0. Why is this the case?

(c) In light of this procedure, is it reasonable to describe B₁
as the ‘‘effect of X₁
on Y when the influence of X_{2; ...}
; X_k
is removed from both X₁
and Y’’?

(d) The procedure in this problem reduces the multiple regression to a series of simple regressions (in Step 3). Can you see any practical application for this procedure? (See the discussion of added-variable plots in Section 11.6.1.)