Question

For a sample of employees participating in an annual appraisal, the effect of the participants’ performance on their wages is studied. Information are collected, and variables of interest are defined below:

• A. \(Wage_i=\alpha+\beta Pass_i+\gamma Fail_i+u_{1i}\)
• B. \(Wage_i=\beta Pass_i+\gamma Fail_i+u_{2i}\)
• C. \(Wage_i=\alpha+\gamma Fail_i+u_{3i}\)
• D. \(Wage_i=\alpha+\beta Pass_i+u_{4i}\)

Hint:

With an intercept, always omit one dummy variable category to avoid the dummy variable trap.

Answer 1

The Correct Option is A

Step 1: Understand the dummy variables.
There are two categories: employees who pass and employees who fail.
For every employee, exactly one of the two events must occur.
So,
\[ Pass_i+Fail_i=1 \]

Step 2: Understand the dummy variable trap.
If a regression model includes an intercept and all dummy variables for all categories, then perfect multicollinearity occurs.
This is called the dummy variable trap.

Step 3: Analyze option (A).
Option (A) includes intercept \(\alpha\), \(Pass_i\), and \(Fail_i\).
Since
\[ Pass_i+Fail_i=1 \] and the intercept column is also equal to \(1\), one variable is an exact linear combination of the others.
So, this model cannot be estimated.

Step 4: Analyze option (B).
Option (B) includes both \(Pass_i\) and \(Fail_i\), but no intercept.
Without intercept, the model can be estimated because there is no perfect dummy variable trap.

Step 5: Analyze option (C).
Option (C) includes intercept and only \(Fail_i\).
This is estimable because one category, pass, acts as the base category.

Step 6: Analyze option (D).
Option (D) includes intercept and only \(Pass_i\).
This is also estimable because one category, fail, acts as the base category.

Step 7: Final conclusion.
Therefore, the regression model that cannot be estimated is
\[ \boxed{Wage_i=\alpha+\beta Pass_i+\gamma Fail_i+u_{1i}} \]
Hence, the correct answer is option (A).