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In a regression what is the matrix $(X'X)^{-1}$? What do its interior elements represent?

amoeba
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michael guay
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  • I do believe that $(X'X)^{-1}$ is the covariance-variance matrix where each diagonal is the variance of $B_{i}$ and the other elements are co variances between different $\beta$'s. Unfortunately, I do not use Stata so I can't help you with your other question but maybe someone more informed can tel you :) Welcome to the site! – nicefella Mar 22 '14 at 17:34
  • Yes, that is the covariance matrix. The other one X * X' is called the Gram matrix and is useful when finding the dual solution. – Dave31415 Mar 22 '14 at 18:24
  • Try `matrix list e(V)` right after you have run the `reg` function. – Penguin_Knight Mar 22 '14 at 19:30
  • Duplicate of this question? http://stats.stackexchange.com/q/73463/3277 – ttnphns Mar 23 '14 at 01:42
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    There is also a discussion of the role of $(X'X)^{-1}$ in regression here: http://blog.stata.com/2011/03/03/understanding-matrices-intuitively-part-1/ – Maarten Buis Mar 23 '14 at 12:23
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    I stumbled across this question just after providing an answer to it at http://stats.stackexchange.com/questions/104704. To summarize, although this matrix is *not* the covariance matrix of the estimates of the regression coefficients, a scalar multiple of it is the covariance matrix. Because that multiple is usually not known, it typically is estimated (from the variance of the residuals). The square roots of the diagonal elements are routinely reported in regression output (as the "standard errors" of the coefficients). – whuber Jun 25 '14 at 22:08
  • The linear equation for multiple input/output system, the vector equation for a single input / output pair (x, y) can be written as y=Ax. For multiple samples, (X, Y) it can be more convent to write Y = X A'. This can be solved for A with X' Y = X' X A then (X' X)^(-1) X' Y = A. The term (X' X)^(-1) X is the input, and can be processed using singular value decomposition to determine how many states the input represents, which tells the rank of A, which may be unknown. So, without measuring any output information, the rank can be determined from the information contained in inputs only. – user3969377 Nov 01 '14 at 18:46

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