Neglected heterogeneity and the algebra of least squares

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 grouplevel 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.