Chronological age is the time elapsed between birth (cohort) and observation (period), so for data arrayed by age × time there is no way to identify simultaneously the linear elements of age, period and cohort, absent some restriction on the model. The association of the data structure with a set of challenges and strictures from statistics and mathematics has had the unfortunate consequence of causing us to overlook what the basic identity A ≡ P – C is telling us about how we should be thinking about age, period and cohort as explanatory concepts. For period and cohort – historical time – why would you want to imagine linear trends in both?
This chapter advances no means of identification that does not already exist, but does highlight some commonalities, since just-identifying restrictions on linear terms, including bounding solutions (e.g., monotonicity) turn out to be isomorphic with the equality constraint on neighboring effect coefficients that kicked off the social science literature. It is on conceptual grounds, therefore, that I suggest a preference among them: primary identification via a zero linear trend, usually in period. Trends that are purportedly the combination of offsetting period and cohort effects may be represented parsimoniously by one or the other. Trends in historical time may be of less interest than the oscillations around this trend, especially with respect to cohorts. The theory of age effects is usually developmental and longitudinal, hence within cohorts, not cross-sectional. Age–cohort models are thus the canonical framework for the study of age-differentiated outcomes over time. Period-specific events give rise to effects that can just as usefully be interpreted in terms of differences between cohorts. From the standpoint of extrapolation, the nonlinear terms associated with cohort have a value that the nonlinear terms associated with period do not.