This article considers the effects of age, period, and cohort in social studies and chronic disease epidemiology through age–period–cohort (APC) analysis. These factors are linearly dependent; thus, the multiple classification model, a regression model that takes these factors as covariates in APC analysis, suffers from an identifiability problem with multiple estimators. A data set of homicide arrest rates is used to illustrate the problem. A smoothing cohort model is proposed that allows flexible structure of the effects for age, period, and cohort and avoids the identifiability problem. Results are provided for the consistency of estimation of model intercept and age effects as the number of periods goes to infinity under a mild bounded cohort condition. This also leads to consistent estimation for period and cohort effects. Analyses of homicide arrest rate and lung cancer mortality rate data demonstrate that the smoothing cohort model yields unique parameter estimation with sensible trend interpretation.
Key Words: consistent estimation • identifiability • intrinsic estimator • semiparametric • singular design matrix • spline smoothing