Hey! Help with questions e,f,g,h would be appreciated!

Thanks!

4. Consider the general linear regression model Y = XB + where Y is n x 1, X is nx p and of rank P, B is p x 1, e is n x 1, and e is N(0, oI). n. (a) The hat matrix H is given by H = X(XX)-1X. Show that (I H) is idempotent where I is the n x n identity matrix. (b) Using the least squares method, we minimize RSS = e'e = (Y Xb)'(Y Xb) to obtain b = (XX)-1XY. Show that RSS can also be written as Y'(I H)Y. (c) Obtain an expression for the variance-covariance matrix of the fitted values i, i = 1, 2, in terms of the hat matrix H. (d) e = Y - is the vector of residuals. Are the residuals statistically independent? Justify your answer with an explanation. (e) Show that e = - He. Suppose we denote by hij the (i, j) element of the HAT matrix H. Thus, ei can be written as ei Ei - Lj=1 hijj. What does this equation of e; show? Suppose that we partition X and as X = [X1 : X2] B B1 B2 where X is n x p1, X2 is n x P2, and p1 + P2 = p. B1 is p1 x 1, and B2 is P2 x 1. (f) If the true model is Y = XB + , and we fit the model Y = X1i + u, have we underspecified or overspecified the model? (g) For the case in part (f), Y (XX1)-1X1Y. If the true model is Y = XB + , compute E(b). (h) As a result of model misspecification in part (f), we could obtain an estimator of o2 which is larger than it should be. Does this affect inferences made about the model? Explain. = 4. Consider the general linear regression model Y = XB + where Y is n x 1, X is nx p and of rank P, B is p x 1, e is n x 1, and e is N(0, oI). n. (a) The hat matrix H is given by H = X(XX)-1X. Show that (I H) is idempotent where I is the n x n identity matrix. (b) Using the least squares method, we minimize RSS = e'e = (Y Xb)'(Y Xb) to obtain b = (XX)-1XY. Show that RSS can also be written as Y'(I H)Y. (c) Obtain an expression for the variance-covariance matrix of the fitted values i, i = 1, 2, in terms of the hat matrix H. (d) e = Y - is the vector of residuals. Are the residuals statistically independent? Justify your answer with an explanation. (e) Show that e = - He. Suppose we denote by hij the (i, j) element of the HAT matrix H. Thus, ei can be written as ei Ei - Lj=1 hijj. What does this equation of e; show? Suppose that we partition X and as X = [X1 : X2] B B1 B2 where X is n x p1, X2 is n x P2, and p1 + P2 = p. B1 is p1 x 1, and B2 is P2 x 1. (f) If the true model is Y = XB + , and we fit the model Y = X1i + u, have we underspecified or overspecified the model? (g) For the case in part (f), Y (XX1)-1X1Y. If the true model is Y = XB + , compute E(b). (h) As a result of model misspecification in part (f), we could obtain an estimator of o2 which is larger than it should be. Does this affect inferences made about the model? Explain. =