For a linear regression model given as Y = XB+ where Y is an n-dimensional response...

 

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For a linear regression model given as Y = XB+ where Y is an n-dimensional response vector, X is an n xp predictor matrix, and is an n- dimensional random error matrix with E(e) = 0 and Var(e) = Io2. For the least square estimator b = (X'X)-1X'Y and the fitted value = Xb = X(X'X)-X'Y, answer the following questions: a. Prove that Var(b) = 0 (X'X)-1. b. Prove that the hat matrix H = X(X'X)-1X' is idempotent and symmetric. c. Prove that (Y - 1)' (Y - 1) = ( 1)'( 1) + (Y )' (Y ) where Y = Y; and 1 is an n-dimensional vector whose elements are all 1's. (Hint: Use the fact that 1'(-) 0 without a proof.) Li=1 For a linear regression model given as Y = XB+ where Y is an n-dimensional response vector, X is an n xp predictor matrix, and is an n- dimensional random error matrix with E(e) = 0 and Var(e) = Io2. For the least square estimator b = (X'X)-1X'Y and the fitted value = Xb = X(X'X)-X'Y, answer the following questions: a. Prove that Var(b) = 0 (X'X)-1. b. Prove that the hat matrix H = X(X'X)-1X' is idempotent and symmetric. c. Prove that (Y - 1)' (Y - 1) = ( 1)'( 1) + (Y )' (Y ) where Y = Y; and 1 is an n-dimensional vector whose elements are all 1's. (Hint: Use the fact that 1'(-) 0 without a proof.) Li=1