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: See the two groups.
Every worker either passes or fails, so the two dummies always add up as
\[ Pass_i+Fail_i=1 \]

Step 2: Recall the dummy trap.
The intercept term is also a column of ones. If a model keeps the intercept and both dummies, then one column is an exact copy of a mix of the others. This perfect overlap is the dummy variable trap, and the model cannot be estimated.

Step 3: Look at option A.
It has the intercept, $Pass_i$ and $Fail_i$ all together. Since $Pass_i+Fail_i$ equals the intercept column of ones, this is the trap. So A cannot be estimated.

Step 4: Check the safe ones.
Option B drops the intercept, so both dummies are fine. Options C and D keep the intercept but use only one dummy, leaving the other group as the base. All three are estimable.

Step 5: Conclude.
\[ \boxed{Wage_i=\alpha+\beta Pass_i+\gamma Fail_i+u_{1i}} \]