Both a male and a female dummy, plus an intercept

Question

In a wage regression you code gender with two dummies, Male (1 if male) and Female (1 if female), and include both along with an intercept. The regression will not estimate.

Why not, and how do you fix it?

Accepted Answer

The redundancy. Every person is either male or female in this coding, so Male + Female = 1 for every observation. The intercept column is also all 1s, so it equals Male + Female exactly. That is perfect collinearity: one column is an exact linear combination of the others, the design matrix is singular, and there is no unique solution. The fix. Use a single dummy. Keep, say, Female and the intercept and drop Male. Then the intercept is the average for males (the baseline), and the Female coefficient is the male-to-female difference. Equivalently, drop the intercept and keep both dummies, givi…

Both a male and a female dummy, plus an intercept

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