Block Constraints in Age–Period–Cohort Models with Unequal-width Intervals

Age–period–cohort (APC) models are designed to estimate the independent effects of age, time periods, and cohort membership. However, APC models suffer from an identification problem: There are no unique estimates of the independent effects that fit the data best because of the exact linear dependency among age, period, and cohort. Among methods proposed to address this problem, using unequal-interval widths for age, period, and cohort categories appears to break the exact linear dependency and thus solve the identification problem. However, this article shows that the identification problem remains in these models; in fact, they just implicitly impose multiple block constraints on the age, period, and cohort effects to achieve identifiability. These constraints depend on an arbitrary choice of widths for the age, period, and cohort intervals and can have nontrivial effects on the estimates. Because these assumptions are extremely difficult, if not impossible, to verify in empirical research, they are qualitatively no different from the assumptions of other constrained estimators. Therefore, the unequal-intervals approach should not be used without an explicit rationale justifying their constraints.