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Consider a multiple regression model XB + €, (n x 1) (n x p)(p x 1) (n x 1) where n = the number of obsevations, k = the number of regressors, and p k+ 1, with E(e) = 0 and Cov(e) = o²In. 1. Under the Gauss-Markov assumption, show that the LSE for 3 is given by 3 = (X'X)-'X'y by using matrix algebra and calculus. 2. Prove (1) E(§) = B (2) Cov(B) = (X'X)-? 3. Show îy = B'Xy = ÿ'ŷ = y'Hy (Hint) ў 3 Xв, В- (X'X)-1X'у. and H %3D X(X'X)-1X". 4. Show that (1) SST = (y. - 9)² = y/y – nj = y' ( I, - J) y (2) SSR = (yi- 9)² = B'X'y – ny² = y' ( H - i=1 (3) SSE =(y: – ŷ.)² = y'y – BX'y = y' (In – H) y i=1 1 5. Show that the residual vector, e, can be written as e = (In - H)y, where H is a hat matrix. Furthermore, show both H and (I- H) are idempotent matrices. 6. Find E(e) and Cov(e). What is the Variance of e;? What is the Variance of Cov(e;, e;)?

Consider a multiple regression model XB + €, (n x 1) (n x p)(p x 1) (n x 1) where n = the number of obsevations, k = the number of regressors, and p k+ 1, with E(e) = 0 and Cov(e) = o²In. 1. Under the Gauss-Markov assumption, show that the LSE for 3 is given by 3 = (X'X)-'X'y by using matrix algebra and calculus. 2. Prove (1) E(§) = B (2) Cov(B) = (X'X)-? 3. Show îy = B'Xy = ÿ'ŷ = y'Hy (Hint) ў 3 Xв, В- (X'X)-1X'у. and H %3D X(X'X)-1X". 4. Show that (1) SST = (y. - 9)² = y/y – nj = y' ( I, - J) y (2) SSR = (yi- 9)² = B'X'y – ny² = y' ( H - i=1 (3) SSE =(y: – ŷ.)² = y'y – BX'y = y' (In – H) y i=1 1 5. Show that the residual vector, e, can be written as e = (In - H)y, where H is a hat matrix. Furthermore, show both H and (I- H) are idempotent matrices. 6. Find E(e) and Cov(e). What is the Variance of e;? What is the Variance of Cov(e;, e;)?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Regression analysis
Consider a multiple regression model = XB + €, (n x 1) (n x p)(p x 1) (n x 1) where n = the number of obsevations, k = the number of regressors, and p = k + 1, with E(e) = 0 and Cov(e) = 0²I„. 1. Under the Gauss-Markov assumption, show that the LSE for 3 is given by B = (X'X)-'X'y by using matrix algebra and calculus. 2. Prove (1) Ε(β)-β' (2) Cov(8) = (X'X)-'g² 3. Show ปีข = 3 Xy = ปีบิ = y Hบ (Hint) ŷ = XB, B = (X'X)-'X'y, and H = X(X'X)-!X'. 4. Show that (1) SST = >( - 9)? = y'y – nữ = y' (2) SSR = ( – n)° = B'X'y – nj = y' (H - J)y i=1 (3) SSE = (- ŷ;)² = y'y – B'X'y = y' (In – H) y i=1 1 5. Show that the residual vector, e, can be written as e = (In – H)y, where H is a hat matrix. Furthermore, show both H and (I,- H) are idempotent matrices. 6. Find E(e) and Cov(e). What is the Variance of e;? What is the Variance of Cov(e;, e;)?
Transcribed Image Text:Consider a multiple regression model = XB + €, (n x 1) (n x p)(p x 1) (n x 1) where n = the number of obsevations, k = the number of regressors, and p = k + 1, with E(e) = 0 and Cov(e) = 0²I„. 1. Under the Gauss-Markov assumption, show that the LSE for 3 is given by B = (X'X)-'X'y by using matrix algebra and calculus. 2. Prove (1) Ε(β)-β' (2) Cov(8) = (X'X)-'g² 3. Show ปีข = 3 Xy = ปีบิ = y Hบ (Hint) ŷ = XB, B = (X'X)-'X'y, and H = X(X'X)-!X'. 4. Show that (1) SST = >( - 9)? = y'y – nữ = y' (2) SSR = ( – n)° = B'X'y – nj = y' (H - J)y i=1 (3) SSE = (- ŷ;)² = y'y – B'X'y = y' (In – H) y i=1 1 5. Show that the residual vector, e, can be written as e = (In – H)y, where H is a hat matrix. Furthermore, show both H and (I,- H) are idempotent matrices. 6. Find E(e) and Cov(e). What is the Variance of e;? What is the Variance of Cov(e;, e;)?
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