1. In a linear regression model with n. > p. the dependent variable Y : Q + R" is modelled as following the distribution given by Y = $5 + E, E ~ N(0,a2I), where .1: E R\"? is a fixed, known matrix whose columns are assumed to be linearly independent and ,3 6 R\" is a fixed but unknown vector. Similarly, o2 is assumed fixed but unknown. The estimated error (also known as residual) is E = H E, where H = I x(me)'lmT 6 RM\" is known as the hat matrix. Ha: = 0. HT = H. (c) Show that H2 = H. Show that if v E R\" is orthogonal to all column vectors of x. then He = 1). Hence obtain all the eigenvalues of H and their geometric multiplicities (i.e. how many linearly independent eigenvectors there are for each different eigenvalue). [4] (d) Show that the mgf of Z = ETE can be written in the form Mz(s) = KfRn exp (-;- TM'Lu.) du, where K E R is a constant and M 6 RM" is a matrix. Simplify this to obtain an expression for the mgf that does not involve any integrals or determinants. You may use without proof that if the matrix A E Rn\" has eigenvalues A1, . . . ,An then det(A) = L1 Ag. Hint: If A is an eigenvalue of the matrix A, what is the corresponding eigenvalue of I A? [6] (e) Hence obtain an unbiased estimator of 0'2 and provide its sampling distribution. For which matrix a: does your estimator co~coincide with 82 from the lecture? [3]