1.4.7.md

Solution to Review Question

by Qiang Gao, updated at May 11, 2017

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Chapter 1 Finite-Sample Properties of OLS

Section 4 Hypothesis Testing under Normality

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Review Question 1.4.7 (Variance of $$s^2$$)

Show that, under Assumptions 1.1—1.5,

$$ \mathrm{Var} ( s^2 \mid \mathbf{X} ) = \frac{ 2 \sigma^4 }{ n - K }. $$

Hint: If a random variable is distributed as $$ \chi^2 (m) $$, then its mean is $$m$$ and variance $$2m$$.

Solution

Because

$$ s^2 \equiv \frac{ \mathbf{e}' \mathbf{e} }{ n - K } = \frac{ \sigma^2 }{ n - K } \left( \frac{ \boldsymbol{\varepsilon} }{ \sigma } \right)' \mathbf{M} \left( \frac{ \boldsymbol{\varepsilon} }{ \sigma } \right), $$

by property of variance,

$$ \mathrm{Var} ( s^2 \mid \mathbf{X} ) = \frac{ \sigma^4 }{ (n - K)^2 } \mathrm{Var} ( q \mid \mathbf{X} ), \tag{1} $$

where

$$ q \equiv \left( \frac{ \boldsymbol{\varepsilon} }{ \sigma } \right)' \mathbf{M} \left( \frac{ \boldsymbol{\varepsilon} }{ \sigma } \right) $$

defined in page 36 in text is distributed as $$ q \mid \mathbf{X} \sim \chi^2 (n - K)$$. So $$ \mathrm{Var} ( q \mid \mathbf{X} ) = 2(n - K) $$, and substituting into (1)

$$ \mathrm{Var} ( s^2 \mid \mathbf{X} ) = \frac{ 2 \sigma^4 }{ n - K }. $$

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Copyright ©2017 by Qiang Gao