6.2. Consider the linear regression model y = X+e with assumptions (i) to (iv), and let...

 

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6.2. Consider the linear regression model y = X+e with assumptions (i) to (iv), and let the n 1 vector of ones 1 belong to the column space of X. Let n (y - 5) (0 - 3) R = = n (1 (Y T) 1 (i 1)) (y: 1/2 - be the sample multiple correlation coefficient between y and the prediction g. (a) Show that 1 = 1y for = Py, P = X(X'X)-X'. == y (b) Show that R can be written as where P1 - R = y' (P - P)y - [y' (In P)yy'(P P)y] /2 " - (1/n)1n1n. (c) Conclude that the centered coefficient of determination R2 is the square. of R. 6.2. Consider the linear regression model y = X+e with assumptions (i) to (iv), and let the n 1 vector of ones 1 belong to the column space of X. Let n (y - 5) (0 - 3) R = = n (1 (Y T) 1 (i 1)) (y: 1/2 - be the sample multiple correlation coefficient between y and the prediction g. (a) Show that 1 = 1y for = Py, P = X(X'X)-X'. == y (b) Show that R can be written as where P1 - R = y' (P - P)y - [y' (In P)yy'(P P)y] /2 " - (1/n)1n1n. (c) Conclude that the centered coefficient of determination R2 is the square. of R.