1.7.2.md

Solution to Review Question

by Qiang Gao, updated at May 20, 2017

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Chapter 1 Finite-Sample Properties of OLS

Section 7 Application: Returns to Scale in Electricity Supply

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Review Question 1.7.2 (Change of units)

In Nerlove's data, output is measured in kilowatt hours. If output were measured in megawatt hours, how would the estimated restricted regression change?

Solution

In the restricted regression

$$ \log \left( \frac{ TC_i }{ p_{i3} } \right) = \beta_1 + \beta_2 \log ( Q_i ) + \beta_3 \log \left( \frac{ p_{i1} }{ p_{i3} } \right) + \beta_4 \log \left( \frac{ p_{i2} }{ p_{i3} } \right) + \varepsilon_i, \tag{1.7.6} $$

if output measured in kilowatt hours ($$Q_i$$) is substituted by output measured in megawatt hours ($$ Q'_i \equiv Q_i / 1000 $$),

$$ \begin{align} \log \left( \frac{ TC_i }{ p_{i3} } \right) & = \beta_1 + \beta_2 \log ( 1000 \cdot Q'i ) + \beta_3 \log \left( \frac{ p{i1} }{ p_{i3} } \right) + \beta_4 \log \left( \frac{ p_{i2} }{ p_{i3} } \right) + \varepsilon_i \ & = ( \beta_1 + \beta_2 \log (1000) ) + \beta_2 \log ( Q'i ) + \beta_3 \log \left( \frac{ p{i1} }{ p_{i3} } \right) + \beta_4 \log \left( \frac{ p_{i2} }{ p_{i3} } \right) + \varepsilon_i, \end{align} $$

then the estimated values of slopes ($$ \beta_2 $$, $$ \beta_3 $$ and $$ \beta_4 $$) will not change, but the estimated value of the intercept ($$ \beta_1 $$) will increase by $$ \hat{\beta}_2 \log (1000) $$.

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Copyright ©2017 by Qiang Gao