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
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Review Question 1.1.3 (Combining linearity and strict exogeneity)
Show that Assumptions 1.1 and 1.2 imply
$$
\mathrm{E} ( y_i \mid \mathbf{X} ) = \mathbf{x}_i' \boldsymbol{\beta}
\qquad
\text{($i = 1, 2, \ldots, n$)}
\tag{1.1.20}
$$
Conversely, show that this assumption implies that there exist error terms that satisfy those two assumptions.
(1) Firstly, we prove Assumption 1.1 and 1.2 imply (1.1.20).
$$
\begin{align}
\mathrm{E} ( y_i \mid \mathbf{X} )
& = \mathrm{E} ( \mathbf{x}_i' \boldsymbol{\beta} + \varepsilon_i \mid \mathbf{X} )
&&
\text{(Assumption 1.1)}
\ & =
\mathrm{E} ( \mathbf{x}_i' \boldsymbol{\beta} \mid \mathbf{X} ) + \mathrm{E} ( \varepsilon_i \mid \mathbf{X} )
&&
\text{(linearity of conditional expectatioins)}
\ & =
\mathrm{E} ( \mathbf{x}_i' \boldsymbol{\beta} \mid \mathbf{X} )
&&
\text{(Assumption 1.2)}
\& =
\mathbf{x}_i' \boldsymbol{\beta}
&&
\text{($\mathbf{x}_i$ is known conditional on $\mathbf{X}$, $\boldsymbol{\beta}$ is constant)}
\end{align}
$$
(2) Conversely, we prove (1.1.20) implies there exist error terms that satisfy Assumption 1.1 and 1.2.
We define the error term as
$$
\varepsilon_i = y_i - \mathbf{x}_i' \boldsymbol{\beta},
$$
then Assumption 1.1 is satisfied. To prove assumption 1.2, notice that
$$
\begin{align}
\mathrm{E} (\varepsilon_i \mid \mathbf{X})
& =
\mathrm{E} ( y_i - \mathbf{x}_i' \boldsymbol{\beta} \mid \mathbf{X})
&&
\text{(definition of $\varepsilon_i$)}
\ & =
\mathrm{E} ( y_i \mid \mathbf{X} ) - \mathrm{E} ( \mathbf{x}_i' \boldsymbol{\beta} \mid \mathbf{X} )
&&
\text{(linearity of conditional expectations)}
\ & =
\mathbf{x}_i' \boldsymbol{\beta} - \mathrm{E} ( \mathbf{x}_i' \boldsymbol{\beta} \mid \mathbf{X} )
&&
\text{(1.1.20)}
\ & =
\mathbf{x}_i' \boldsymbol{\beta} - \mathbf{x}_i' \boldsymbol{\beta}
&&
\text{($\mathbf{x}_i$ is known conditional on $\mathbf{X}$, $\boldsymbol{\beta}$ is constant)}
\ & = 0.
\end{align}
$$
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