1. Demeaned regression. Consider an i.i.d. sample of (yi, xi)' for i=1,2,..., n, where x; is a

k x 1 vector not including a constant. Let 1 = (1, 1,..., 1)' be the nx 1 vector of ones, M₁ be

its annihilator matrix, Y = (y₁,..., yn)' be an n x 1 vector, and X = (X₁,...,xn)' be an n x k

matrix.

(a) Show the following:

(i) M₁Y = Y - 1, where y = n¹Σi=1 yi,

(ii) M₁X = X - 1x', where x = X'1/n = (₁, ...,k)' with j = n¹1 j₁; being the

mean of the j-th regressor.

(b) Let ÿi = Yi - ÿ and X₁ = Xį - X for i = 1,...,n denote the demeaned values. Use the

Frisch-Waugh-Lovell theorem to show that the OLS estimators of 3 in y; = a + x₁ß + €i

and in ÿj; = x;ß + ε; are the same.