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bluesilver
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Joined: 12 Sep 2004
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 Posted: Fri, 20 Oct 2006 18:28:27 UTC Post subject: covariance matrix |
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I have the question regarding the calculation of the covariance matrix that is used in multiple linear regression models. Suppose that B is the vector of regression coefficients, X is the matrix of the levels of the regressor variables, and σ^2 is the unknown variance of errors. Then X' is the transpose of X, (X'X)^-1 is the inverse of X'X. Then the covariance matrix is
Cov(B) = σ^2 (X'X)^-1
OK, that is what I see in the textbook. So, my question is: can I use the unbiased estimator of σ^2 (i.e., the residual mean square) instead of σ^2 to calculate the covariance matrix above? If not, please tell me of other methods of calculating the covariance matrix. |
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royhaas
Member of the 'S.O.S. Math' Hall of Fame
Joined: 23 Jun 2003
Posts: 2022
Location: San Antonio,Texas USA
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| Posted: Fri, 20 Oct 2006 19:24:50 UTC Post subject: |
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Yes, use the MSE. Note that in linear regression the X matrix is treated as a constant, so there are no problems. The MSE will have n-p degrees of freedom for a matrix of column rank p.
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Bilbo
Member of the 'S.O.S. Math' Hall of Fame
Joined: 14 Jan 2006
Posts: 690
Location: Yountville, California
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| Posted: Fri, 20 Oct 2006 19:28:04 UTC Post subject: Re: covariance matrix |
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| bluesilver wrote: | I have the question regarding the calculation of the covariance matrix that is used in multiple linear regression models. Suppose that B is the vector of regression coefficients, X is the matrix of the levels of the regressor variables, and σ^2 is the unknown variance of errors. Then X' is the transpose of X, (X'X)^-1 is the inverse of X'X. Then the covariance matrix is
Cov(B) = σ^2 (X'X)^-1
OK, that is what I see in the textbook. So, my question is: can I use the unbiased estimator of σ^2 (i.e., the residual mean square) instead of σ^2 to calculate the covariance matrix above? If not, please tell me of other methods of calculating the covariance matrix. |
Using an estimate of  provides an estimate of the covariance matrix, not the covariance matrix itself. But estimate is all you can do. |
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bluesilver
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Joined: 12 Sep 2004
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| Posted: Fri, 20 Oct 2006 19:28:25 UTC Post subject: |
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| Great. Thank you. I'm just curious - are there also other methods, or there is only one method of calculating the covariance matrix? |
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