Neglected heterogeneity, Simpson's paradox, and the anatomy of least squares

This paper explores an algebraic relationship between two types of coefficients for a regression with several predictors and a group structure. 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 are not necessarily a convex average of the group-specific coefficients. In the context of regression with two independent variables and two groups, I show that the coefficient of a regressor estimated from pooled data can be negative, even though the separately estimated coefficients are positive in each group, providing an additional example of Simpson's paradox.