1.3.2.md

Solution to Review Question

by Qiang Gao, updated at Apr 23, 2017

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Chapter 1 Finite-Sample Properties of OLS

Section 3 Finite-Sample Properties of OLS

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Review Question 1.3.2 (Example of a linear estimator)

For the consumption function example in Example 1.1, propose a linear and unbiased estimator of $$ \beta_2 $$ that is different from the OLS estimator.

Solution

We propose an estimator $$ \widehat{\beta}_2 = ( CON_2 - CON_1 ) / ( YD_2 - YD_1 ) $$.

1. When $$ YD_1, \ldots, YD_n $$ is known, $$ \widehat{\beta}_2 $$ is a linear combination of $$ CON_1, \ldots, CON_n $$.

2. Because

$$ \begin{align} \mathrm{E} ( \widehat{\beta}_2 \mid YD_1, \ldots, YD_n ) & = \mathrm{E} \left( \left. \frac{ ( \beta_1 + \beta_2 YD_2 + \varepsilon_2) - ( \beta_1 + \beta_2 YD_1 + \varepsilon_1 ) } { YD_2 - YD_1 } \right| YD_1, \ldots, YD_n \right) \ & = \mathrm{E} \left( \left. \frac{ \beta_2 (YD_2 - YD_1) + \varepsilon_2 - \varepsilon_1 } { YD_2 - YD_1 } \right| YD_1, \ldots, YD_n \right) \ & = \beta_2 + 0 - 0 = \beta_2, \end{align} $$

So $$ \widehat{\beta}_2 $$ proposed here is linear and unbiased.

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Copyright ©2017 by Qiang Gao