Solution to Review Question by Qiang Gao, updated at Mar 13, 2017
Chapter 1 Finite-Sample Properties of OLS Section 1 The Classical Linear Regression Model ...
Review Question 1.1.6 (An exercise in conditional and unconditional expectations) Show that Assumptions 1.2 and 1.4 imply
$$
\begin{align}
\mathrm{Var} ( \varepsilon_i ) & = \sigma^2
&&
(i = 1, 2, \ldots, n)
\\
\text{and} \qquad
\mathrm{Cov} ( \varepsilon_i, \varepsilon_j ) & = 0
&&
(i \neq j; i, j = 1, 2, \ldots, n). \tag{$\ast$}
\end{align}
$$
Strict exogeneity implies $$ \mathrm{E} (\varepsilon_i) = 0 $$. So $$ (\ast) $$ is equivalent to
$$
\begin{align}
\mathrm{E} ( \varepsilon_i^2 ) & = \sigma^2
&&
(i = 1, 2, \ldots, n)
\\
\text{and} \qquad
\mathrm{E} ( \varepsilon_i \varepsilon_j ) & = 0
&&
(i \neq j; i, j = 1, 2, \ldots, n).
\end{align}
$$
(1) For $$i = 1, 2, \ldots, n$$ ,
$$
\begin{align}
\mathrm{E} (\varepsilon_i^2)
& =
\mathrm{E} [\mathrm{E} (\varepsilon_i^2 \mid \mathbf{X})]
&&
\text{(law of total expectations)}
\ & =
\sigma^2.
&&
\text{(Assumption 1.4)}
\end{align}
$$
(2) For $$i \neq j; i, j = 1, 2, \ldots, n$$ ,
$$
\begin{align}
\mathrm{E} (\varepsilon_i \varepsilon_j)
& =
\mathrm{E} [ \mathrm{E} (\varepsilon_i \varepsilon_j \mid \mathbf{X}) ]
&&
\text{(law of total expectations)}
\ & = 0.
&&
\text{(Assumption 1.4)}
\end{align}
$$
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