Consider the augmented regression y = X + PXDp + u, where Dp is an n p matrix of dummy variables for

Consider the augmented regression y = Xβ

∗ + ¯ PXDpϕ∗ + u, where Dp is an n × p matrix of dummy variables for the p suspected observations. Note that ¯ PXDp rather than Dp appear in this equation. Compare with (8.6). Let ep = D

pe, then E(ep) = 0, var(ep) = σ2D

p

¯ PXDp. Verify that

(a) β

∗

= (XX)−1Xy = βOLS and

(b) ϕ

∗ = (D

p

¯ PXDp)−1D

p

¯ PXy = (D

p

¯ PXDp)−1D

pe = (D

p

¯ PXDp)−1ep.

(c) Residual Sum of Squares = (Residual Sum of Squares with Dp deleted) − e

p(D

p

¯ PX)D−1 p ep.

Using the Frisch-Waugh Lovell Theorem show this residual sum of squares is the same as that for (8.6).

(d) Assuming normality of u, verify (8.7) and (8.9).

(e) Repeat this exercise for problem 4 with ¯ PXdi replacing di. What do you conclude?