Let Y=Z3+ 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 mxm 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 (0) E(Σ) = Σ (f) cov(₁, 3) = 0 1 n-p-1
Let Y=Z3+ 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 mxm 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 (0) E(Σ) = Σ (f) cov(₁, 3) = 0 1 n-p-1
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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
Transcribed Image Text:Let Y=Z3+ € 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 × 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
(e) E(Σ) = Σ
(f) cov(₁,3)=0
1
n-p-1
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