A regressor that is the sum of two others

Question

You regress firm returns on debt, equity, and assets, where by construction assets = debt + equity. All three go in with an intercept, and the regression will not estimate. Why does including the total plus both components break OLS, and what do you remove?

Accepted Answer

An exact adding-up relationship. By construction, 	exttt{assets} = 	exttt{debt} + 	exttt{equity}, so the assets column is an exact linear combination of the other two. That is perfect collinearity: the three columns are linearly dependent, the design matrix is singular, and there is no unique least-squares fit. No high-correlation subtlety here, it is an exact identity. Why it is unidentified. Because assets carries no information beyond debt + equity, the model cannot decide how much credit each column deserves; infinitely many coefficient triples give the same fit. The data simply does n…

A regressor that is the sum of two others

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