Imagine you regressed earnings of individuals on a constant, a binary variable ("Male") which takes on the value 1 for males and is 0 otherwise, and another binary variable ("Female") which takes on the value 1 for females and is 0 otherwise. Because females typically earn less than males, you would expect:

Question

BlackTom AI · Accepted Answer

Including a constant and both Male and Female dummies creates perfect multicollinearity because Male + Female = 1 for every observation.

1) Relevant knowledge points and concepts
- Dummy-variable trap: when you include a full set of mutually exclusive category dummies plus a constant, one regressor is an exact linear combination of the others, producing perfect multicollinearity.
- OLS requires X'X to be invertible; perfect collinearity makes X'X singular, so unique OLS estimates do not exist.
- If you drop one dummy (or the constant), the model is identified and the remaining dummy coefficient is a difference-in-means.

2) Step-by-step reasoning
Start from the specified model:
y = β0 + β1·Male + β2·Female + u.
Because every observation is either male or female, Female = 1 − Male. Substitute:

3) Calculations / algebraic steps
y = β0 + β1·Male + β2·(1 − Male) + u
  = β0 + β2 + (β1 − β2)·Male + u
Let β0* = β0 + β2 and β1* = β1 − β2, then
y = β0* + β1*·Male + u.

This shows only β0* and β1* are identified; infinitely many (β0,β1,β2) triples map to the same (β0*,β1*). Also note 1 − Male − Female = 0 is an exact linear relation among columns of X, so X'X is singular.

4) Why the correct answer is correct
Because of the exact linear dependence (constant = Male + Female), OLS cannot produce unique estimates for β0, β1, and β2 simultaneously. Thus none of the OLS estimators (as uniquely defined values) exist.

5) Why the other options are incorrect
- a: There is no necessary symmetry “same distance from the constant”; coefficients are not identified uniquely so such a claim is meaningless.
- c: A difference-in-means statistic arises when you include one dummy and a constant (or omit the constant and include both dummies appropriately). With both dummies plus a constant, you cannot estimate to obtain that statistic because of collinearity.
- d: Signs are not well-defined because β1 and β2 are not uniquely estimated; you cannot assert one must be positive and the other negative.

6) Conclusion
Perfect multicollinearity occurs, so none of the OLS estimators exist uniquely. This confirms option b.

类似问题

The dummy variable trap is an example of:

Choose ALL CORRECT statement(s) about the imperfect multicollinearity:

The dummy variable trap is an example of:

Choose ALL CORRECT statement(s) about the imperfect multicollinearity:

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