4. Consider the general linear regression model Y = X+ where Y is n x 1,...

 

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4. Consider the general linear regression model Y = X+ where Y is n x 1, X is n x p and of rank P, 3 is p x 1, e is n x 1, and e is N(0, 21). (a) The hat matrix H is given by H = X(X'X) X'. 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= (X'X) X'Y. Show that RSS can also be written as Y'(I - H)Y. (e) Obtain an expression for the variance-covariance matrix of the fitted values , i = 1, 2, ..., n, 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 h, the (i, j) element of the HAT matrix H. Thus, e, can be written as e, e- hijes. What does this equation of e, show? Suppose that we partition X and as x=[X1 X2] 3-[31] where X1 is n x P1, X2 is n x p2, and p + P2 = p. 3 is p x 1, and 32 is p2 x 1. (f) If the true model is Y = X+e, and we fit the model Y = X + u, have we underspecified or overspecified the model? (g) For the case in part (f), b = (X'X)-'X'Y. If the true model is Y = X3 + e, compute E(bi). (h) As a result of model misspecification in part (f), we could obtain an estimator of o which is larger than it should be. Does this affect inferences made about the model? Explain. 1378 x 0 4. Consider the general linear regression model Y = X+ where Y is n x 1, X is n x p and of rank P, 3 is p x 1, e is n x 1, and e is N(0, 21). (a) The hat matrix H is given by H = X(X'X) X'. 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= (X'X) X'Y. Show that RSS can also be written as Y'(I - H)Y. (e) Obtain an expression for the variance-covariance matrix of the fitted values , i = 1, 2, ..., n, 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 h, the (i, j) element of the HAT matrix H. Thus, e, can be written as e, e- hijes. What does this equation of e, show? Suppose that we partition X and as x=[X1 X2] 3-[31] where X1 is n x P1, X2 is n x p2, and p + P2 = p. 3 is p x 1, and 32 is p2 x 1. (f) If the true model is Y = X+e, and we fit the model Y = X + u, have we underspecified or overspecified the model? (g) For the case in part (f), b = (X'X)-'X'Y. If the true model is Y = X3 + e, compute E(bi). (h) As a result of model misspecification in part (f), we could obtain an estimator of o which is larger than it should be. Does this affect inferences made about the model? Explain. 1378 x 0