Constrained Estimators and Age-Period-Cohort Models

If a researcher wants to estimate the individual age, period, and cohort coefficients in an age-period-cohort (APC) model, the method of choice is constrained regression, which includes the intrinsic estimator (IE) recently introduced by Yang and colleagues. To better understand these constrained models, the author shows algebraically how each constraint is associated with a specific generalized inverse that is associated with a particular solution vector that (when the model is just identified under the constraint) produces the least square solution to the APC model. The author then discusses the geometry of constrained estimators in terms of solutions being orthogonal to constraints, solutions to various constraints all lying on a line single line in multidimensional space, the distance on that line between various solutions, and the crucial role of the null vector. This provides insight into what characteristics all constrained estimators share and what is unique about the IE. The first part of the article focuses on constrained estimators in general (including the IE), and the latter part compares and contrasts the properties of traditionally constrained APC estimators and the IE. The author concludes with some cautions and suggestions for researchers using and interpreting constrained estimators.