Age–Period–Cohort Models and the Perpendicular Solution

Abstract Separating the effects of ages, periods, and cohorts is a classic problem not only in epidemiology but also in demography and the social sciences in general. Frost provides a classic example in epidemiology that I use as an empirical example. In the classic age–period–cohort (APC) approach a single constraint is used to eliminate the linear dependency and identify the model. Among the infinite number of possible choices for a constraint, a particular constraint has been discovered and rediscovered in multiple guises. This constraint produces a solution to the APC model that is perpendicular to the null vector. Among these guises are the minimum norm solution, the Moore–Penrose solution, the principle components solution, the intrinsic estimator, the singular value decomposition solution, the partial least squares solution, and the maximum entropy solution. The results based on the perpendicular constraint provide a solution that has a smaller variance than any other constrained solution. This would be an important advantage, if there were some reason to believe that the perpendicular solutions were an unbiased estimate of the parameters that generated the dependent variable. To help explicate the common features of these solutions, this paper carefully examines the algebraic, geometric, and statistical characteristics of the perpendicular solution.