Solution to Review Question by Qiang Gao, updated at May 8, 2017 Chapter 1 Finite-Sample Properties of OLS Section 3 Finite-Sample Properties of OLS ... Review Question 1.3.5 Propose an unbiased estimator of $$ \sigma^2 $$ if you had data on $$ \boldsymbol{ \varepsilon } $$. Solution $$ \boldsymbol{ \varepsilon }' \boldsymbol{ \varepsilon } / n $$ is an unbiased estimator of $$ \sigma^2 $$. This is because $$ \begin{align} \mathrm{E} ( \boldsymbol{ \varepsilon }' \boldsymbol{ \varepsilon } / n ) & = \frac{1}{n} \sum_{i=1}^n \mathrm{E} ( \varepsilon_i^2 ) \ & = \frac{1}{n} \sum_{i=1}^n \mathrm{E} [ \mathrm{E} ( \varepsilon_i^2 \mid \mathbf{X} ) ] \ & = \frac{1}{n} \cdot n \sigma^2 \ & = \sigma^2. \end{align} $$ Copyright ©2017 by Qiang Gao