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 x m covariance matrix of any row of € 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 σij. Define the sample covariance matrix of the residuals by = Σ = Prove 1 n-p-1 (c) E() = 0 (d) cov(i, k) = Oik (I-H) -d'ê

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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 € 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
(c) E(ê) = 0
(d) cov(ei, k) = Oik (I-H)
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 € 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 (c) E(ê) = 0 (d) cov(ei, k) = Oik (I-H)
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