Neglected heterogeneity and the algebra of least squares

Abstract

This paper explores an algebraic relationship between two types of coefficients for a regression with several predictors and an additive binary group variable. In a general regression, the regression coefficients are allowed to be group-specific, the restricted regression imposes constant coefficients. The key result is that the restricted coefficients imposing homogeneity are not necessarily a convex average of the unrestricted coefficients obtained from the more general regression. In the context of treatment effect estimation with several treatment arms and group-level controls, this means that the estimated effect of a specific treatment can be non-zero, and statistically significant, even if the estimated unrestricted effects are zero in each group.