The dummy variable trap: when collinearity is perfect

Question

You regress returns on an intercept plus four dummy variables, one for each calendar quarter (Q1, Q2, Q3, Q4). The software drops a column or errors out.

What is happening, how is it different from the near-collinearity case, and what is the fix?

Accepted Answer

The failure. The four quarter dummies sum to exactly 1 in every row (each day is in exactly one quarter), and that is identical to the intercept column of all 1s. So the design matrix X has an exact linear dependence among its columns: perfect collinearity. Then X'X is singular (non-invertible), and the OLS formula \hat\beta = (X'X)^{-1}X'y has no unique solution, infinitely many coefficient vectors fit identically. This is the "dummy variable trap." How it differs from near-collinearity. With merely high correlation (say 0.95), X'X is still invertible, so OLS returns a unique estimate, just …

The dummy variable trap: when collinearity is perfect

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