LINEAR MODELS

QUESTION 2 [22] Consider a general linear model y = X3 + e, where e is distributed with mean 0 and variance o-I and y is distributed with mean X3 and variance o'I. The normal equations for X X not of full rank is X X30 = X y. For any generalised inverse G of X X, corresponding solution is given by 30=GX y. (a) Find the expected value and variance of 30 (6) (b) Find the residual sum of squares, SSE. (5) (c) Show that the an unbised estimator of o is given by a2 = SSE N-r(X) . (4) (d) Show that 30 and & are independent and SSE is distributed Chi-Square with N - r degrees of freedm where r = rank(X). (7)