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3 Let B be the OLS estimate from the regression of y on X. Let A be a (k + 1) × (k + 1) nonsingular matrix and define z, = x,A, t = 1, ..., n. Therefore, z, is 1 × (k + 1) and is a nonsingular linear com- bination of x,. Let Z be then x (k + 1) matrix with rows z,. Let B denote the OLS estimate from a regression of y on Z. (i) Show that ß = AB. (ii) Let ŷ, be the fitted values from the original regression and let ŷ, be the fitted values from regress- ing y on Z. Show that y, = sions compare? (iii) Show that the estimated variance matrix for ß is ô'A-(X'X)-'A-", where ô2 is the usual vari- ance estimate from regressing y on X. (iv) Let the B; be the OLS estimates from regressing y, on 1, x,1, ..., Xk, and let the B; be the OLS es- timates from the regression of y, on 1, a,x,1,..., ax, where a; + 0, j = 1, ..., k. Use the results from part (i) to find the relationship between the B; and the B,. (v) î., for all t = 1, 2, .., n. How do the residuals from the two regres- = se(B,)/la,l. Assuming the setup of part (iv), use part (iii) to show that se(B;) (vi) Assuming the setup of part (iv), show that the absolute values of the t statistics for B; and B; are identical.

3 Let B be the OLS estimate from the regression of y on X. Let A be a (k + 1) × (k + 1) nonsingular matrix and define z, = x,A, t = 1, ..., n. Therefore, z, is 1 × (k + 1) and is a nonsingular linear com- bination of x,. Let Z be then x (k + 1) matrix with rows z,. Let B denote the OLS estimate from a regression of y on Z. (i) Show that ß = AB. (ii) Let ŷ, be the fitted values from the original regression and let ŷ, be the fitted values from regress- ing y on Z. Show that y, = sions compare? (iii) Show that the estimated variance matrix for ß is ô'A-(X'X)-'A-", where ô2 is the usual vari- ance estimate from regressing y on X. (iv) Let the B; be the OLS estimates from regressing y, on 1, x,1, ..., Xk, and let the B; be the OLS es- timates from the regression of y, on 1, a,x,1,..., ax, where a; + 0, j = 1, ..., k. Use the results from part (i) to find the relationship between the B; and the B,. (v) î., for all t = 1, 2, .., n. How do the residuals from the two regres- = se(B,)/la,l. Assuming the setup of part (iv), use part (iii) to show that se(B;) (vi) Assuming the setup of part (iv), show that the absolute values of the t statistics for B; and B; are identical.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 23EQ
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3 Let B be the OLS estimate from the regression of y on X. Let A be a (k + 1) × (k + 1) nonsingular matrix and define z, = x,A, t = 1, ..., n. Therefore, z, is 1 × (k + 1) and is a nonsingular linear com- bination of x,. Let Z be then x (k + 1) matrix with rows z,. Let B denote the OLS estimate from a regression of y on Z. (i) Show that B = A-'B. (ii) Let ŷ, be the fitted values from the original regression and let ŷ, be the fitted values from regress- ing y on Z. Show that y, sions compare? (iii) Show that the estimated variance matrix for ß is ô'A-(X'X)-'A-", where &² is the usual vari- ance estimate from regressing y on X. (iv) Let the B; be the OLS estimates from regressing y, on 1, x,1, ..., Xk, and let the B; be the OLS es- timates from the regression of y, on 1, a,x,1,..., ax, where a; + 0, j = 1, ..., k. Use the results from part (i) to find the relationship between the B; and the B;. (v) î, for all t = 1, 2, .., n. How do the residuals from the two regres- = se(B,)/la,l. Assuming the setup of part (iv), use part (iii) to show that se(B;) (vi) Assuming the setup of part (iv), show that the absolute values of the t statistics for B; and B; are identical.
Transcribed Image Text:3 Let B be the OLS estimate from the regression of y on X. Let A be a (k + 1) × (k + 1) nonsingular matrix and define z, = x,A, t = 1, ..., n. Therefore, z, is 1 × (k + 1) and is a nonsingular linear com- bination of x,. Let Z be then x (k + 1) matrix with rows z,. Let B denote the OLS estimate from a regression of y on Z. (i) Show that B = A-'B. (ii) Let ŷ, be the fitted values from the original regression and let ŷ, be the fitted values from regress- ing y on Z. Show that y, sions compare? (iii) Show that the estimated variance matrix for ß is ô'A-(X'X)-'A-", where &² is the usual vari- ance estimate from regressing y on X. (iv) Let the B; be the OLS estimates from regressing y, on 1, x,1, ..., Xk, and let the B; be the OLS es- timates from the regression of y, on 1, a,x,1,..., ax, where a; + 0, j = 1, ..., k. Use the results from part (i) to find the relationship between the B; and the B;. (v) î, for all t = 1, 2, .., n. How do the residuals from the two regres- = se(B,)/la,l. Assuming the setup of part (iv), use part (iii) to show that se(B;) (vi) Assuming the setup of part (iv), show that the absolute values of the t statistics for B; and B; are identical.
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