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Let Y=Z3+ e denote the standard form of the multivariate regression model where Y is a matrix of n observations on m dependent variables, Z is the nx (r + 1) design matrix, 3 is the matrix of regression coefficients and e denotes the error matrix. Let Σ denote the m x m covariance matrix of any row of e and assume the rows are independent. The least square estimate for 3 is 3 = (Z'Z)-¹ZY. The projection matrix is H = Z(Z'Z)-¹Z'. Let the ith column of 3 be denoted by 3, and the i, jth element of Σ by dij. Define the sample covariance matrix of the residuals by Σ = Prove 1 n-p-1 (g) Z'(I-H) = 0 (h) Qê= (I-11')ê = ê.

Let Y=Z3+ e denote the standard form of the multivariate regression model where Y is a matrix of n observations on m dependent variables, Z is the nx (r + 1) design matrix, 3 is the matrix of regression coefficients and e denotes the error matrix. Let Σ denote the m x m covariance matrix of any row of e and assume the rows are independent. The least square estimate for 3 is 3 = (Z'Z)-¹ZY. The projection matrix is H = Z(Z'Z)-¹Z'. Let the ith column of 3 be denoted by 3, and the i, jth element of Σ by dij. Define the sample covariance matrix of the residuals by Σ = Prove 1 n-p-1 (g) Z'(I-H) = 0 (h) Qê= (I-11')ê = ê.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CR: Review Exercises
Problem 62CR
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Let Y Z3+ e denote the standard form of the multivariate regression model where Y is a matrix of n observations on m dependent variables, Z is the nx (r + 1) design matrix, 3 is the matrix of regression coefficients and denotes the error matrix. Let Σ denote the m x m covariance matrix of any row of e and assume the rows are independent. The least square estimate for 3 is 3 = (Z'Z)-¹Z'Y. The projection matrix is H = Z(Z'Z)-¹Z'. Let the ith column of 3 be denoted by 3, and the i, jth element of Σ by oij. Define the sample covariance matrix of the residuals by Prove Σ = 1 n-p-1 (g) Z'(I-H) = 0 (h) Qê= (I-11')ê = ê.
Transcribed Image Text:Let Y Z3+ e denote the standard form of the multivariate regression model where Y is a matrix of n observations on m dependent variables, Z is the nx (r + 1) design matrix, 3 is the matrix of regression coefficients and denotes the error matrix. Let Σ denote the m x m covariance matrix of any row of e and assume the rows are independent. The least square estimate for 3 is 3 = (Z'Z)-¹Z'Y. The projection matrix is H = Z(Z'Z)-¹Z'. Let the ith column of 3 be denoted by 3, and the i, jth element of Σ by oij. Define the sample covariance matrix of the residuals by Prove Σ = 1 n-p-1 (g) Z'(I-H) = 0 (h) Qê= (I-11')ê = ê.
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