Introduction to Statistical Modeling with SAS/STAT Software

Estimating the Error Variance

The least squares principle does not provide for a parameter estimator for $\sigma ^2$. The usual approach is to use a method-of-moments estimator that is based on the sum of squared residuals. If the model is correct, then the mean square for error, defined to be $\mr{SSR}$ divided by its degrees of freedom,

\begin{align*} \widehat{\sigma }^2 & = \frac{1}{n-\mr{rank}(\bX )} \left(\bY -\bX \widehat{\bbeta }\right)’ \left(\bY -\bX \widehat{\bbeta }\right) \\ & = \mr{SSR}/(n-\mr{rank}(\bX )) \end{align*}

is an unbiased estimator of $\sigma ^2$.