Consider the augmented regression given in (8.5) y = X +di+u where is a scalar and di = 1 for the i-th observation and

Consider the augmented regression given in (8.5) y = Xβ

∗+diϕ+u where ϕ is a scalar and di = 1 for the i-th observation and 0 otherwise. Using the Frisch-Waugh Lovell Theorem given in section 7.3, verify that

(a) β

∗

= (X

(i)X(i))−1X

(i)y(i) = β(i).

(b) ϕ = (d

i

¯ PXdi)−1d

i

¯ PXy = ei/(1 − hii) where ¯ PX = I − PX.

(c) Residual Sum of Squares from (8.5) = (Residual Sum of Squares with di deleted) − e2i

/(1−

hii).

(d) Assuming Normality of u, show that the t-statistic for testing ϕ = 0 is t = ϕ/s.e.(ϕ) = e∗

i as given in (8.3).