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 ...
Review Question 1.1.1 (Change in units in the semi-log form) In the wage equation
$$
\log ( WAGE_i ) = \beta_1 + \beta_2 S_i + \beta_3 TENURE_i + \beta_4 EXPR_i + \varepsilon_i,
\tag{1.1.3}
$$
of Example 1.2, if $$ WAGE $$ is measured in cents rather than in dollars, what difference does it make to the equation?
Let $$ WAGE' $$ denote the $$ WAGE $$ variable measured in cents, that is,
$$
WAGE' = 100 WAGE,
$$
$$
\log ( WAGE' ) = \log (100) + \log ( WAGE ).
$$
Substituting into equation (1.1.3),
$$
\log ( WAGE_i') - \log (100) = \beta_1 + \beta_2 S_i +
\beta_3 TENURE_i + \beta_4 EXPR_i +
\varepsilon_i,
$$
$$
\log (\mathit{WAGE}_i') = \log (100) + \beta_1 + \beta_2 S_i +
\beta_3 \mathit{TENURE}_i + \beta_4 \mathit{EXPR}_i +
\varepsilon_i,
$$
$$
\log ( WAGE_i') = \beta_1' + \beta_2 S_i + \beta_3
TENURE_i + \beta_4 EXPR_i + \varepsilon_i,
$$
where $$ \beta_1' $$ is defined as $$ \beta_1' = \log (100) + \beta_1 $$. So the only difference is $$ \beta_1 $$ is increased by $$ \log (100)$$.
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