Model Misspecification When Eliminating a Factor in Age-period-cohort Multiple Classification Models

Abstract

The impossibility of uniquely estimating all of the age, period, and cohort coefficients in age-period-cohort multiple classification (APCMC) models without imposing a constraint on the model is widely recognized. The problem results from a linear dependency in the design matrix, and this dependency involves the linear trends of age effects, period effects, and cohort effects. This article critiques the use of fit statistics to assess the overall importance of the effects of ages, periods, and cohorts in APCMC models. In particular, one proposed strategy to avoid the APCMC model identification problem is to test to see if including only two of the factors in a model (e.g., ages and cohorts) produces a fit that is not significantly different statistically from a model that includes all three factors. If the third factor (in this example periods) does not account for a statistically significant amount of variance, this strategy suggests that one should use the model with only the two factors. This is consistent with model selection approaches. The two-factor model is identified and produces estimates of the individual effects of ages and cohorts. There is, however, a fundamental problem with this approach when used with APCMC models. That problem results from the complete confounding of the linear effects of the three factors.