Solution to Review Question by Qiang Gao, updated at Mar 20, 2017
Chapter 1 Finite-Sample Properties of OLS Section 1 The Classical Linear Regression Model ...
Review Question 1.1.2 (Conditional cross-moment of error terms) Prove the last equality in (1.1.15),
$$
\mathrm{E} (\varepsilon_i \varepsilon_j \mid \mathbf{X}) \mathrm{E} (\varepsilon_i \mid \mathbf{x}_i)
\mathrm{E} (\varepsilon_j \mid \mathbf{x}_j)
\qquad
\text{(for $i \neq j$ )}.
\tag{1.1.15}
$$
$$
\begin{align}
\mathrm{E} ( \varepsilon_i \varepsilon_j \mid \mathbf{X} )
& =
\mathrm{E} [ \mathrm{E} ( \varepsilon_i \varepsilon_j \mid \mathbf{X}, \varepsilon_j ) \mid \mathbf{X} ]
&&
\text{(law of iterated expectations)}
\\
& = \mathrm{E} [ \varepsilon_j \mathrm{E} ( \varepsilon_i \mid \mathbf{X}, \varepsilon_j ) \mid \mathbf{X} ]
&&
\text{($\varepsilon_j$ is constant under condition)}
\\
& = \mathrm{E} [ \varepsilon_j \mathrm{E} ( \varepsilon_i \mid \mathbf{x}_i ) \mid \mathbf{X} ]
&&
\text{(random sample)} \\
& = \mathrm{E} ( \varepsilon_i \mid \mathbf{x}_i ) \mathrm{E} ( \varepsilon_j \mid \mathbf{X})
&&
\text{($\mathbf{x}_i$ is known conditional on $\mathbf{X}$)} \\
& = \mathrm{E} ( \varepsilon_i \mid \mathbf{x}_i ) \mathrm{E} ( \varepsilon_j \mid \mathbf{x}_j)
&&
\text{(random sample)}
\end{align}
$$
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