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Wilmoth, John. 1989. "Variation in Vital Rates by Age, Period, and Cohort." PSC Research Report No. 89-141. April 1989.
The analysis of age-specific vital rates is studied in this paper, with special attention given to the problem of decomposing an array of rates into factors related to age, period, and cohort. A complete, symmetric decomposition of the data array into age, period, and cohort components is not attempted. Instead, a choice is made to focus on the age and period dimensions and to derive an initial description of the matrix structure with regard to changes only in those two directions. This two-dimensional description is then augmented by a consideration of residual patterns which seem clearly linked to cohorts.
The empirical section of the paper describes in great detail the structure of an array of mortality rates for French males ages 0-89 over the years 1946-1981. Apart from an error term, the array can be effectively decomposed into three parts: an additive portion showing the shape of an average age pattern of mortality over the period with an adjustment in level for each year; a pair of multiplicative terms depicting the slow transformation of this age pattern over time; and a constant diagonal term representing the level of excess mortality for cohorts over the period of observation. This description results from fitting models of the form, where k = j - i and i, j, and k are indices referring to ages, periods, and cohorts, respectively. The model is built up piece by piece in an exploratory analysis, then justified formally within a least-squares framework.
The motivation for considering such a model is offered both in terms of substantive issues and in light of the general topic of age-period-cohort analysis. The method was developed in order to isolate cohorts whose mortality experience over the postwar period has been unusually high or low. These results may be important for understanding the interplay between processes of debilitation and selection in the development of cohort mortality patterns. The choice to apply a model which is asymmetric in age, period, and cohort is justified by a detailed discussion of the problems of identification in models involving perfectly collinear independent variables. An important conclusion is that traditional modeling approaches which treat age, period, and cohort in a symmetric fashion are fundamentally flawed.