Discussion
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:
\(Wage_{i}\) = wage of the employee \(i\)
\(Pass_{i}\) =
1, if the employee \(i\) passes in the annual appraisal
0, if the employee \(i\) fails in the annual appraisal
\(Fail_{i}\) =
1, if the employee \(i\) fails in the annual appraisal
0, if the employee \(i\) passes in the annual appraisal
Based on the above information, which one of the following regression models cannot be estimated?
Answer: A
The model that cannot be estimated is \(Wage_{i} = \alpha + \beta Pass_{i} + \gamma Fail_{i} + u_{1i}\).
Every employee either passes or fails, so \(Pass_{i} + Fail_{i} = 1\) for all \(i\). That sum is exactly the column of ones that carries the intercept \(\alpha\). Including an intercept and both dummies therefore creates perfect multicollinearity, the dummy variable trap: the regressor matrix is not of full rank and least squares cannot separate \(\alpha\), \(\beta\) and \(\gamma\).
The other three are estimable. With both dummies and no intercept, \(\beta\) and \(\gamma\) are simply the mean wages of those who pass and those who fail. With an intercept and one dummy, the omitted category becomes the base group: \(\alpha\) is its mean wage and the dummy's coefficient is the difference between the groups.
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