Cross-classified age-period-cohort models as a constrained estimator

Age-Period-Cohort (APC) models are designed to separate the independent effects of age, time periods, and cohort membership in various outcomes including obesity, smoking, and health status. However, such APC models suffer from an identification problem: there are no valid solutions because of the exact linear dependency among age, period, and cohort. Among methods proposed to address this problem, the cross-classified approach, including Cross-Classified Fixed Effects Models (CCFEM) and Cross-Classified Random Effects Models (CCREM), appears to solve the identification problem and to yield good estimates of the independent effects of age, period, and cohort groups. This paper assesses the validity and application scope of CCFEM and CCREM theoretically and illustrates their properties with simulations. It shows that the cross-classified methods do not automatically solve the identification problem; rather, they address this problem by implicitly imposing multiple constraints on the age, period, and cohort effects. These constraints not only depend on the width of the age, period, and cohort intervals but also have non-trivial implications for estimation. Because these assumptions are extremely difficult, if not impossible, to verify in empirical research, they are qualitatively no different from other constrained estimators' assumptions. The authors conclude that CCFEM and CCREM cannot and should not be used to recover the true age, period, and cohort effects.