Four quarterly seasonal dummies plus a constant

Question

To model seasonality in quarterly sales you build four dummies, Q1, Q2, Q3, Q4, and include all four plus an intercept. The estimation fails. Explain the failure and give the two standard fixes.

Accepted Answer

The trap again. Each observation falls in exactly one quarter, so Q1 + Q2 + Q3 + Q4 = 1 always. The intercept is a column of 1s, identical to that sum, so the four seasonal dummies and the intercept are perfectly collinear. The design matrix cannot be inverted and there is no unique fit. Fix 1: drop one quarter. Keep the intercept and, say, Q2, Q3, Q4, dropping Q1. Now Q1 is the baseline season, the intercept is the average Q1 level, and each remaining coefficient is that quarter's difference from Q1. Fix 2: drop the intercept. Keep all four dummies and no constant. Each coefficient is then t…

Four quarterly seasonal dummies plus a constant

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