i) Consider the regular multiple regression

y = X +

where E() = 0 , Cov() = ²I

Let A(n n) be a known orthogonal matrix. Take

Y = Ay, X = AX, = A

Prove

a) E() = 0 and Cov() = ^2I , I=identity matrix

b) Show that = and ⁿ_i=1 e_i² = ⁿ_i=1 e^*_i²

ii) Let Y = 2 + 3xi + xi² + _i i = 1, 2, . . . , n.

Assume i.i.d. N(0, 2). The following observations are given

(0, 2.1),(1, 6),(1, 5.9),(1.1, 6.1),(2, 7).

(a) Estimate and find a 95% confidence interval for .

(b) Find a 95% prediction interval when x = 2.2.

iii) Regressing on centered data. Let

Yi = + (xi x ) + i = 1, 2, . . . , n.

where i.i.d. N(0, 2).

Find the L.S.E. for and . Compare this model with the regular model (non-centred model) in

which regression is on xi (not on xi x ). Compare the distribution of estimators. Derive a formula

for the prediction interval for new x value.