Question

Let \(d\) be a dummy (binary) variable and let \(z\) be a quantitative variable. Consider the model $$y=\beta_{0}+\delta_{0} d+\beta_{1} z+\delta_{1} d \cdot z+u_{;}$$ this is a general version of a model with an interaction between a dummy variable and a quantitative variable. [An example is in equation \((7.17) .]\) i. Because it changes nothing important, set the error to zero, \(u=0 .\) Then, when \(d=0\) we can write the relationship between \(y\) and \(z\) as the function \(f_{0}(z)=\beta_{0}+\beta_{1} z\). Write the same relationship when \(d=1\), where you should use \(f_{1}(z)\) on the left-hand side to denote the linear function of \(z\). ii. Assuming that \(\delta_{1} \neq 0\) (which means the two lines are not parallel), show that the value of \(z^{*}\) such that \(f_{0}\left(z^{*}\right)=f 1\left(z^{*}\right)\) is \(z^{*}=-\delta_{0} / \delta_{1}\). This is the point at which the two lines intersect [as in Figure 7.2 (b)]. Argue that \(z^{*}\) is positive if and only if \(\delta_{0}\) and \(\delta_{1}\) have opposite signs. iii. Using the data in TWOYEAR, the following equation can be estimated: $$\begin{aligned}\widehat{\log (\text {wage})}=& 2.289-.357 \text { female }+.50 \text { totcoll}+.030 \text { female } \text { -totcoll} \\\&(0.011) \quad (.015) \quad (.003) \quad (.005) \\\n=& 6,763, R^{2}=.202\end{aligned}$$ where all coefficients and standard errors have been rounded to three decimal places. Using this equation, find the value of totcoll such that the predicted values of \(\log (w a g e)\) are the same for men and women. iv. Based on the equation in part (iii), can women realistically get enough years of college so that their earnings catch up to those of men? Explain.

Short answer

Women need about 11.9 years of college for equal predicted wages. It's likely unrealistic due to typical education lengths.

Step-by-step solution

Express f₁(z)

When \( d = 1 \), the relationship becomes \( f_1(z) = \beta_0 + \delta_0 + (\beta_1 + \delta_1) z \). This accounts for the additional effects of \( \delta_0 \) and \( \delta_1 \) when the dummy variable \( d \) is included in the model.

Set Equal Functions for Intersection

To find the point where \( f_0(z) = f_1(z) \), solve the equation \( \beta_0 + \beta_1 z = \beta_0 + \delta_0 + (\beta_1 + \delta_1) z \). This simplifies to \( 0 = \delta_0 + \delta_1 z \).

Solve for z*

From \( \delta_0 + \delta_1 z = 0 \), solve for \( z \) to find \( z^* = -\frac{\delta_0}{\delta_1} \). This is the value at which \( f_0(z) \) and \( f_1(z) \) intersect.

Determine the Sign of z*

Given \( \delta_1 eq 0 \), check the sign of \( z^* = -\frac{\delta_0}{\delta_1} \). \( z^* \) is positive if \( \delta_0 \) and \( \delta_1 \) have opposite signs, since the negative ratio of opposite-signed numbers results in a positive value.

Substitute into Given Equation

Using the equation \( \widehat{\log(\text{wage})} = 2.289 - 0.357 \times \text{female} + 0.50 \times \text{totcoll} + 0.030 \times \text{female} \times \text{totcoll} \), find \( \text{totcoll} \) where predicted \( \log(\text{wage}) \) for men and women are equal. Set \( -0.357 + 0.030 \times \text{totcoll} = 0 \).

Solve for Specific totcoll

From \( -0.357 + 0.030 \times \text{totcoll} = 0 \), calculate \( \text{totcoll} = \frac{0.357}{0.030} \approx 11.9 \). This is the number of college years at which predicted wages are equal for men and women.

Analyze Feasibility

Compare the necessary \( \text{totcoll} \) (11.9 years) to realistic educational achievements. This requires considering typical lengths of higher education programs to determine if women can realistically attain this many years of college to match men's earnings.

Key concepts

Linear regression model

Linear regression is a foundational tool in econometrics, used to understand the relationship between dependent and independent variables. The general form of a linear regression model is expressed as:\[ y = \beta_0 + \beta_1 x + \epsilon \]where:

• \( y \) is the dependent variable, representing the outcome we aim to analyze or predict.
• \( \beta_0 \) is the intercept, indicating the expected value of \( y \) when all other explanatory variables are zero.
• \( \beta_1 \) is the slope coefficient, showing the change in \( y \) for a one-unit change in the independent variable \( x \).
• \( \epsilon \) is the error term, capturing the influence of all other factors not included in the model.

In this context, we're examining how educational attainment may impact wages using the linear regression model framework. By modifying this basic structure, with interactions and dummy variables, we get more nuanced insights of the data.

Dummy variable interaction

A dummy variable is a tool in econometrics that helps model qualitative data in regression analysis. These variables typically take values of 0 or 1, indicating the absence or presence of a certain condition or characteristic. An interaction is a form of combining multiple variables to analyze how they jointly influence the dependent outcome.

In our exercise, the model includes an interaction term of the form \( \delta_1 \cdot d \cdot z \). Here, \( d \) is a dummy variable, and \( z \) is a quantitative variable, interacting to create a combined effect on \( y \).

When \( d = 0 \), the relationship is purely driven by \( \beta_0 + \beta_1 z \). However, when \( d = 1 \), the relationship becomes \( \beta_0 + \delta_0 + (\beta_1 + \delta_1)z \).

• \( \delta_0 \) is the shift in intercept due to \( d \).
• \( \delta_1 \) is the change in slope due to the interaction with \( z \).

This modified regression allows us to capture more complex relationships, determining how the combination of these characteristics influences an outcome like wage.

Quantitative variable analysis

Quantitative variables, such as years of education or total college years, can be measured and expressed numerically. In regression analysis, these variables serve as independent predictors to explain changes in the dependent variable, such as income or wages.

In our scenario, "totcoll" represents the total years of college education and is used as a quantitative variable to ascertain its impact on wages. The coefficient for "totcoll" in the regression equation is positive (0.50), suggesting that as the total years of college increase, so does the log of wages. This reflects a typical finding in econometrics where higher education leads to higher earnings.

Effective quantitative analysis requires careful consideration of variable scaling, the choice of measurement units, and potential standardization to make comparisons between different data series. This helps in drawing valid, actionable signals from the data.

Educational attainment impact

Educational attainment often has a significant impact on earnings, as evidenced by numerous empirical studies in econometrics. A higher number of years in college generally correlates with increased earning potential.

The model given in the exercise emphasizes this relationship by assessing how women's earnings may catch up to men's through educational achievements. Using the regression equation:\[\widehat{\log(\text{wage})} = 2.289 - 0.357 \cdot \text{female} + 0.50 \cdot \text{totcoll} + 0.030 \cdot \text{female} \cdot \text{totcoll}\]We observe that women need approximately 11.9 years of college to match men's predicted log wages.

This analysis highlights not only the role of education in bridging gender gaps in earnings but also the practical constraints of achieving such extensive education. It requires examining the typical duration of educational programs and the feasibility of individuals spending that time in higher education, considering economic and personal factors.