Warm Up

Andrew Dickinson

Categorical Variables

Suppose you want to model the relationship between an individual’s pay (\(Pay\)) and their gender (\(Male\)) using the following model:

\[ Pay_i = \beta_0 + \beta_1 Male_i + u_i \]

where \(Male_i\) is a binary variable that equals 1 if the individual is male and 0 otherwise. Which of the following statements is correct?

1. \(\beta_0\) represents the expected pay for males.
   
2. \(\beta_1\) represents the expected difference in pay between males and females.
   
3. \(\beta_0\) represents the expected pay for females.
   
4. \(\beta_1\) represents the expected pay for males.

Solution:

The correct interpretations of the coefficients in the model are:

• \(\beta_0\): This represents the expected pay for females (when \(Male = 0\)).
• \(\beta_1\): This represents the expected difference in pay between males and females.

Therefore, the correct answers are:

Option b: \(\beta_1\) represents the expected difference in pay between males and females.
Option c: \(\beta_0\) represents the expected pay for females.

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Consider the following model attempting to explain a person’s pay (\(Pay\)) as a function of their years of education (\(School\)), gender (\(Male\)), and their interaction:

\[ Pay_i = \beta_0 + \beta_1 School_i + \beta_2 Male_i + \beta_3 (School_i \times Male_i) + u_i \]

For this regression, which of the following statements is closest to being correct?

1. \(\beta_2\) gives the expected difference in pay between males and females, holding years of education constant.
   
2. \(\beta_3\) gives the expected difference in pay for an additional year of schooling for males compared to females.
   
3. \(\beta_1\) gives the expected increase in pay for an additional year of schooling for females.
   
4. \(\beta_0\) gives the expected pay for females with zero years of schooling.

Solution:

The model given is: \[ Pay_i = \beta_0 + \beta_1 School_i + \beta_2 Male_i + \beta_3 (School_i \times Male_i) + u_i \]

We need to interpret the coefficients correctly.

1. \(\beta_2\) gives the expected difference in pay between males and females, holding years of education constant.
2. \(\beta_3\) gives the expected difference in pay for an additional year of schooling for males compared to females.
3. \(\beta_1\) gives the expected increase in pay for an additional year of schooling for females.
4. \(\beta_0\) gives the expected pay for females with zero years of schooling.

Explanation:

• Option 1: Correct. \(\beta_2\) represents the expected difference in pay between males and females, holding years of education constant.
• Option 2: Correct. \(\beta_3\) represents the interaction effect, showing how the pay difference for an additional year of schooling varies between males and females.
• Option 3: Correct. \(\beta_1\) represents the expected increase in pay for an additional year of schooling for females.
• Option 4: Correct. \(\beta_0\) represents the expected pay for females with zero years of schooling.

Therefore, the correct answers are all options because they all provide correct interpretations based on the model.