The same variable in two units: percent and decimal

Question

Building a feature table, you include a stock's prior return twice by mistake: once as a decimal (ret_dec, e.g. 0.03) and once as a percentage (ret_pct, e.g. 3.0). Both go into the regression. It fails to estimate.

Why does this break OLS even though there are no dummy variables?

Accepted Answer

Perfect collinearity is not only about dummies. The two columns satisfy an exact linear relationship: 	exttt{ret\pct} = 100 	imes 	exttt{ret\dec}. One column is a perfect rescaling of the other, so they are perfectly collinear. The design matrix is singular and OLS has no unique solution, exactly as in the dummy variable trap. The lesson: perfect collinearity is any exact linear dependence among the columns, whether it comes from dummies summing to one, a unit duplication, or a constructed variable. Why the coefficients would be meaningless anyway. Even if software forced a solution, the s…

The same variable in two units: percent and decimal

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