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Bayesian Ridge Estimation of Age-Period-Cohort Models

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Dynamic Demographic Analysis

Part of the book series: The Springer Series on Demographic Methods and Population Analysis ((PSDE,volume 39))

Abstract

Age-Period-Cohort (APC) analysis offers a framework to study trends in the three temporal dimensions underlying age by period tables. However, the perfect linear relationship among age, period, and cohort leads to a well-known identification issue due perfect colinearity from the identity Cohort = Period − Age. A number of methods have been proposed to deal with this identification issue, e.g., the intrinsic estimator (IE), which may be viewed as a limiting form of ridge regression. Bayesian regression offers an alternative approach to modeling tabular age, period, cohort data. This study views the ridge estimator from a Bayesian perspective by introducing prior distributions for the ridge parameters, which permits these parameters to be estimated jointly with the substantive parameters rather than being assigned (and fixed) a-priori. Results show that a Bayesian ridge model with a common prior for the ridge parameter yields estimated age, period, and cohort effects similar to those based on the intrinsic estimator and to those based on a conventional ridge estimator with a shrinkage penalty obtained from cross-validation. The performance of Bayesian models with distinctive priors for the ridge parameters of age, period, and cohort effects is, however, affected by the choice of prior distributions. Further investigation of the influence of the choice of prior distributions is therefore warranted.

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Notes

  1. 1.

    Suzuki (2012) provides insight into the nature, conceptualization, and meaning of the underlying temporal dimensions.

  2. 2.

    Identification may be less an issue using a Bayesian approach where inference is carried out using simulation, as opposed to the traditional numerical methods using least squares.

  3. 3.

    This implies that last category effects are: \( {\alpha}_a=-{\displaystyle \sum_{i=1}^{a-1}{\alpha}_i} \), \( {\beta}_p=-{\displaystyle \sum_{j=1}^{p-1}}{\beta}_j \), and \( {\gamma}_{a+p-1}=-{\displaystyle \sum_{k=1}^{a+p-2}}{\gamma}_k \). The Bayesian approach adopted here makes it straightforward to monitor these quantities along with the other parameters of interest.

  4. 4.

    Variance components in Bayesian models are typically parameterized in terms of precision, i.e., \( {\sigma}^{-2} \) rather than variance σ 2.

  5. 5.

    Keyes et al. (2010) provide an application to APC modeling and a useful comparison with other methods.

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Authors and Affiliations

  1. Department of Sociology and Population Research Center, University of Texas at Austin, 305 E. 23d Street (G-1800), Austin, TX, 78712-1699, USA

    Minle Xu & Daniel A. Powers

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  1. Minle Xu
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  2. Daniel A. Powers
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Correspondence to Minle Xu .

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Editors and Affiliations

  1. Population Research Institute, Pennsylvania State University Population Research Institute, University Park, Pennsylvania, USA

    Robert Schoen

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Xu, M., Powers, D.A. (2016). Bayesian Ridge Estimation of Age-Period-Cohort Models. In: Schoen, R. (eds) Dynamic Demographic Analysis. The Springer Series on Demographic Methods and Population Analysis, vol 39. Springer, Cham. https://doi.org/10.1007/978-3-319-26603-9_17

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  • DOI: https://doi.org/10.1007/978-3-319-26603-9_17

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