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Our discussion on page 200 of the variance-covariance matrix for the ordinary least squares (OLS) estimators (Equation 8.33b) neglicted to mention how to estimate sigma^2_e, the residual variance.
Equation 8.33b gives the variance-covariance matrix for the vector b of OLS estimators as
Var(b) = (X^{T}X)^{-1} sigma^2_e
We can estimate sigma^2_e from the residual sums of squares,
RSS = (y - Xb)^{T}(y - Xb)
If the model estimates p parameters, then the estimate of sigma^2_e is simply RSS/(N-p) where N is the number of data points. Thus,
Var(b) = (X^{T}X)^{-1} (y - Xb)^{T}(y - Xb) /(N-p)
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Bruce Walsh. jbwalsh@u.arizona.edu . Comments welcome.