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Review Question 1.4.2 (Computation of test statistics)
Verify that $$ SE( b_k ) $$ as well as $$ \mathbf{b} $$, $$ SSR $$, $$ s^2 $$, and $$ R^2 $$ can be calculated from the following sample averages: $$ \mathbf{S}{ \mathbf{xx} } $$, $$ \mathbf{s}{ \mathbf{xy} } $$, $$ \mathbf{y}' \mathbf{y} / n $$, and $$ \bar{y} $$.
Solution
In review question 1.2.9, it has been shown that $$ \mathbf{b} $$, $$ SSR $$, $$ s^2 $$, and $$ R^2 $$ can be calculated from sample averages $$ \mathbf{S}{ \mathbf{xx} } $$, $$ \mathbf{s}{ \mathbf{xy} } $$, $$ \mathbf{y}' \mathbf{y} / n $$, and $$ \bar{y} $$.
by definition that $$ \mathbf{S}{\mathbf{xx}} = \frac{1}{n} \mathbf{X}' \mathbf{X} $$, it is obvious that $$ SE( b_k ) $$ can be calculated from sample averages of $$ \mathbf{S}{ \mathbf{xx} } $$, $$ \mathbf{s}_{ \mathbf{xy} } $$, $$ \mathbf{y}' \mathbf{y} / n $$, and $$ \bar{y} $$.