Consider the linear regression model given by

Consider the linear regression model discussed in lectures and given by y =X B+ u. (mx1) (rxk)(kel) (mx1) (1) the OLS residual vector, u = y - Xb, may be written as u = My, where M = In - P and P = XXX)-'X. (a) Show that MX= 0 . (xk) ( mck) (b) Show that the OLS residual vector may also be written as u = Mu. (c) Show that u u = u'Mu. (d) Show that Xu= 0 . (karl) (karl) (e) Recall that two vectors, u and v, are said to be orthogonal if u'v = vu = 0. Prove that i is orthogonal to each of the regressors, X1, X2,.....,Xx, in the linear regression model. That is, xju = xu =...= xu = 0, where x, is the ith column of X written as a row vector. Hint: You need to partition X appropriately. (f) i) Prove that if the rank of the matrix X in (1) is less than k, then the matrix XX is not invertible. i) What is the implication of this result for the OLS estimator of the parameter vector B in (1)? Briefly explain