1.1.4.md

Solution to Review Question

by Qiang Gao, updated at Mar 13, 2017

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Chapter 1 Finite-Sample Properties of OLS

Section 1 The Classical Linear Regression Model

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Review Question 1.1.4 (Normally distributed ramdom sample)

Consider a random sample on consumption and disposable income, $$ ( CON_i, YD_i ) $$, $$ ( i = 1, 2, \ldots, n ) $$. Suppose the joint distribution of $$ ( CON_i, YD_i ) $$ (which is the same across $$ i $$ because of the random sample assumption) is normal. Clearly, Assumption 1.3 is satisfied; the rank of $$ \mathbf{X} $$ would be less than $$ K $$ only by pure accident. Show that the other assumptions, Assumptions 1.1, 1.2, and 1.4, are satisfied. Hint: if two random variables, $$ y $$ and $$ x $$, are jointly normally distributed, then the conditional expectation is linear in $$ x $$, i.e.,

$$ \mathrm{E} ( y \mid x ) = \beta_1 + \beta_2 x, $$

and the conditional variance, $$ \mathrm{Var} ( y \mid x ) $$, does not depend on $$ x $$. Here, the fact that the distribution is the same across $$ i $$ is important; if the distribution differed across $$ i $$, $$ \beta_1 $$ and $$ \beta_2 $$ could vary across $$ i $$.

Solution (with flaw)

(1) We define the error term as

$$ \varepsilon_i = CON_i - \beta_1 - \beta_2 YD_i, $$

then Assumption 1.1 is satisfied.

(2) Because

$$ \begin{align} \mathrm{E} (\varepsilon_i \mid \mathbf{X}) & = \mathrm{E} ( CON_i - \beta_1 - \beta_2 YD_i \mid \mathbf{X} ) && \text{(definition of $\varepsilon_i$)} \ & = \mathrm{E} ( CON_i \mid YD_i ) - \beta_1 - \beta_2 YD_i && \text{(linearity of conditional expectations)} \ & = \beta_1 + \beta_2 \mathit{YD}_i - \beta_1 - \beta_2 \mathit{YD}_i && \text{(hint)} \ & = 0, \end{align} $$

Assumption 1.2 holds.

(3) To prove Assumption 1.4,

$$ \begin{align} \mathrm{E} ( \varepsilon_i^2 \mid \mathbf{X} ) & = \mathrm{Var} ( \varepsilon_i \mid \mathbf{X} ) + \mathrm{E} ( \varepsilon_i \mid \mathbf{X} )^2 && \text{(definition of $\mathrm{Var} (\cdot)$)} \ & = \mathrm{Var} ( \varepsilon_i \mid \mathbf{X} ) && \text{(Assumption 1.2)} \ & = \mathrm{Var} ( CON_i - \beta_1 - \beta_2 YD_i \mid YD_i ) && \text{(definition of $\varepsilon_i$)} \ & = \mathrm{Var} ( CON_i \mid YD_i ) && \text{($\mathrm{Var} (ax + b) = a^2 \mathrm{Var} (x)$ )} \ & = \sigma^2 > 0, && \text{(hint)} \end{align} $$

and

$$ \begin{align} \mathrm{E} ( \varepsilon_i \varepsilon_j \mid \mathbf{X} ) & = \mathrm{E} ( \varepsilon_i \mid \mathbf{x}_i ) \mathrm{E} ( \varepsilon_j \mid \mathbf{x}_j ) && \text{(Review Question 1.1.2)} \ & = 0. && \text{(Assumption 1.2)} \end{align} $$

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Copyright ©2017 by Qiang Gao