Solutions to Review Question

by Qiang Gao, updated at Mar 13, 2017

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?

Solution

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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