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

Abstract

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.

Mixed models, linear dependency, and identification in age‐period‐cohort models

“…This paper is not the first to suggest that the mixed model representation of the APC model is identified with no other constraints than those inherent in the statistical model itself . In addition to this observation: (i) I show clearly how the mixed‐model approach achieves model identification by constraining the solution through the shrinkage associated with random effects.…”

“…It must maximize the likelihood function by jointly minimizing the squared deviations of the observed values of the dependent variable from the predicted values and the variation of the random effects. As noted above by Luo and Hodges [: 23] ‘. .…”

“…does not incur the identification problem because the three effects are not assumed to be linear and additive at the same level of analysis’ [: 191]. In a recent paper, Luo and Hodges note that in published applications of the HAPC model researchers use unequal age, period, and cohort intervals (different widths in years) and that this procedure adds multiple constraints to the model (see also Feinberg and Mason ; Holford ; and Osmond and Gardner . If one codes cohorts in 7‐year grouping and periods in 5‐year groupings, the linear dependency in the APC model is broken, because an observation's age and cohort does not determine the period of the observation.…”

“…This is implied by those noting that the solutions shift depending on the factor that is treated as random and that when two of the factors are modeled as fixed effects and one is modeled as random effects, the random effects tend to have a near zero trend across time of . Luo and Hodges [: 23] note this constraint in specific terms ‘. .…”

An Assessment and Extension of the Mechanism-Based Approach to the Identification of Age-Period-Cohort Models

“…In all estimation models, A , P , and C are treated as categorical variables through dummy coding (10 dummy variables for age, 4 dummy variables for period, and 14 dummy variables for cohort because cohort categories were forced to overlap in order to maintain the linear identity, C = P – A ). By generating cohort in this way, we maintain the linear dependency among age, period, and cohort and therefore follow the equal interval width definition (Luo and Hodges 2016). …”