Linear Models

QUESTION 4 [23 Marks] 4.1 Consider a general linear model y : X + e, where e is distributed with mean 0 and variance 0'21, and y is distributed with mean X8 and variance 0'21. The normal equations for XIX not of full rank is XIX? : X'y. For any generalised inverse G of XX, the corresponding solution is given by 3 : GX'y. 4.1.1 Find the expected value and variance of 3. 4.1.2 Find the residual sum of squares, SSE. SSE Nr(X) ' 4.1.3 Show that the an unbiased estimator of 02 is given by 32 : 4.1.4 Show that E and glare independent, and that E is distributed ChiSquare with (N 3") degrees of U2 freedom where r : rank(X)