Constraints in Random Effects Age-Period-Cohort Models

Liying Luo, James S. Hodges

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


Random effects (RE) models have been widely used to study the contextual effects of structures such as neighborhoods or schools. The RE approach has recently been applied to age-period-cohort (APC) models that are unidentified because the predictors are exactly linearly dependent. However, research has not fully explained how the RE specification identifies these otherwise unidentified APC models. We address this challenge by first making explicit that RE-APC models have greater—not less—rank deficiency than the traditional fixed-effects model, followed by two empirical examples. We then provide intuition and a mathematical proof to explain that for APC models with one RE, treating one effect as an RE is equivalent to constraining the estimates of that effect’s linear component and the random intercept to be zero. For APC models with two REs, the effective constraints implied by the model depend on the true (i.e., in the data-generating mechanism) nonlinear components of the effects that are modeled as REs, so that the estimated linear components of the REs are determined by the true nonlinear components of those effects. In conclusion, RE-APC models impose arbitrary although highly obscure constraints and thus do not differ qualitatively from other constrained APC estimators.

Original languageEnglish (US)
Pages (from-to)276-317
Number of pages42
JournalSociological Methodology
Issue number1
StatePublished - Aug 1 2020

Bibliographical note

Funding Information:
The project benefited from support provided by the Population Research Institute, the Pennsylvania State University. We thank Wayne Osgood and anonymous reviewers for their helpful comments.


  • age-period-cohort model
  • identification problem
  • linear and nonlinear effects
  • linear dependency
  • random effects models


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