1.4.2.md

Solution to Review Question

by Qiang Gao, updated at May 10, 2017

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Chapter 1 Finite-Sample Properties of OLS

Section 4 Hypothesis Testing under Normality

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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 $$ \bar{y} $$.

Because

$$ SE( b_k ) \equiv \sqrt{ s^2 \cdot \left( ( \mathbf{X}' \mathbf{X} )^{-1} \right)_{kk} }, $$

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 $$ \bar{y} $$.

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Copyright ©2017 by Qiang Gao