# American Institute of Mathematical Sciences

February  2021, 4(1): 45-59. doi: 10.3934/mfc.2021001

## The uses and abuses of an age-period-cohort method: On the linear algebra and statistical properties of intrinsic and related estimators

 1 Department of Sociology, The University of British Columbia, Vancouver, BC V6T 1Z1, Canada 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong 3 School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia 4 School of Medicine, University of California, San Francisco, CA 94121, USA 5 Department of Sociology, Social Work, and Anthropology, Utah State University, Logan, UT 84322, USA 6 Department of Sociology, Yale University, New Haven, CT 06511, USA 7 Department of Sociology and Social Science Research Institute, Duke University, Durham, NC 27708, USA

* Corresponding author: Qiang Fu

Received  September 2020 Published  February 2021 Early access  December 2020

Fund Project: This research was partially supported by the Research Grants Council of Hong Kong [Project No. PolyU 15334616] and partially based on class notes provided by Qiang Fu during the course "Age-Period-Cohort Analysis: Principles, Models and Application", given in the Institute for Empirical Social Science Research (IESSR) at Xi'an Jiaotong University (July 2015) and in the School of Public Administration at Zhongnan University of Economics and Law (July 2018)

As a sophisticated and popular age-period-cohort method, the Intrinsic Estimator (IE) and related estimators have evoked intense debate in demography, sociology, epidemiology and statistics. This study aims to provide a more holistic review and critical assessment of the overall methodological significance of the IE and related estimators in age-period-cohort analysis. We derive the statistical properties of the IE from a linear algebraic perspective, provide more precise mathematical proofs relevant to the current debate, and demonstrate the essential, yet overlooked, link between the IE and classical statistical tools that have been employed by scholars for decades. This study offers guidelines for the future use of the IE and related estimators in demographic research. The exposition of the IE and related estimators may help redirect, if not settle, the logic of the debate.

Citation: Qiang Fu, Xin Guo, Sun Young Jeon, Eric N. Reither, Emma Zang, Kenneth C. Land. The uses and abuses of an age-period-cohort method: On the linear algebra and statistical properties of intrinsic and related estimators. Mathematical Foundations of Computing, 2021, 4 (1) : 45-59. doi: 10.3934/mfc.2021001
##### References:

show all references

##### References:
 [1] Yanqing Liu, Jiyuan Tao, Huan Zhang, Xianchao Xiu, Lingchen Kong. Fused LASSO penalized least absolute deviation estimator for high dimensional linear regression. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 97-117. doi: 10.3934/naco.2018006 [2] Wei Li, Yun Teng. Enterprise inefficient investment behavior analysis based on regression analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1015-1025. doi: 10.3934/dcdss.2019069 [3] Jiang Xie, Junfu Xu, Celine Nie, Qing Nie. Machine learning of swimming data via wisdom of crowd and regression analysis. Mathematical Biosciences & Engineering, 2017, 14 (2) : 511-527. doi: 10.3934/mbe.2017031 [4] Bingzheng Li, Zhengzhan Dai. Error analysis on regularized regression based on the Maximum correntropy criterion. Mathematical Foundations of Computing, 2020, 3 (1) : 25-40. doi: 10.3934/mfc.2020003 [5] Shuhua Wang, Zhenlong Chen, Baohuai Sheng. Convergence of online pairwise regression learning with quadratic loss. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4023-4054. doi: 10.3934/cpaa.2020178 [6] Adil Bagirov, Sona Taheri, Soodabeh Asadi. A difference of convex optimization algorithm for piecewise linear regression. Journal of Industrial & Management Optimization, 2019, 15 (2) : 909-932. doi: 10.3934/jimo.2018077 [7] Shaoyong Lai, Qichang Xie. A selection problem for a constrained linear regression model. Journal of Industrial & Management Optimization, 2008, 4 (4) : 757-766. doi: 10.3934/jimo.2008.4.757 [8] Song Wang, Quanxi Shao, Xian Zhou. Knot-optimizing spline networks (KOSNETS) for nonparametric regression. Journal of Industrial & Management Optimization, 2008, 4 (1) : 33-52. doi: 10.3934/jimo.2008.4.33 [9] Baohuai Sheng, Huanxiang Liu, Huimin Wang. Learning rates for the kernel regularized regression with a differentiable strongly convex loss. Communications on Pure & Applied Analysis, 2020, 19 (8) : 3973-4005. doi: 10.3934/cpaa.2020176 [10] Erik Kropat, Gerhard Wilhelm Weber. Fuzzy target-environment networks and fuzzy-regression approaches. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 135-155. doi: 10.3934/naco.2018008 [11] Yang Mi, Kang Zheng, Song Wang. Homography estimation along short videos by recurrent convolutional regression network. Mathematical Foundations of Computing, 2020, 3 (2) : 125-140. doi: 10.3934/mfc.2020014 [12] Qing Xu, Xiaohua (Michael) Xuan. Nonlinear regression without i.i.d. assumption. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 8-. doi: 10.1186/s41546-019-0042-6 [13] Charles Curry, Stephen Marsland, Robert I McLachlan. Principal symmetric space analysis. Journal of Computational Dynamics, 2019, 6 (2) : 251-276. doi: 10.3934/jcd.2019013 [14] Xin Guo, Qiang Fu, Yue Wang, Kenneth C. Land. A numerical method to compute Fisher information for a special case of heterogeneous negative binomial regression. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4179-4189. doi: 10.3934/cpaa.2020187 [15] Lianjun Zhang, Lingchen Kong, Yan Li, Shenglong Zhou. A smoothing iterative method for quantile regression with nonconvex $\ell_p$ penalty. Journal of Industrial & Management Optimization, 2017, 13 (1) : 93-112. doi: 10.3934/jimo.2016006 [16] Yuyuan Ouyang, Trevor Squires. Some worst-case datasets of deterministic first-order methods for solving binary logistic regression. Inverse Problems & Imaging, 2021, 15 (1) : 63-77. doi: 10.3934/ipi.2020047 [17] Lucian Coroianu, Danilo Costarelli, Sorin G. Gal, Gianluca Vinti. Approximation by multivariate max-product Kantorovich-type operators and learning rates of least-squares regularized regression. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4213-4225. doi: 10.3934/cpaa.2020189 [18] Victor Meng Hwee Ong, David J. Nott, Taeryon Choi, Ajay Jasra. Flexible online multivariate regression with variational Bayes and the matrix-variate Dirichlet process. Foundations of Data Science, 2019, 1 (2) : 129-156. doi: 10.3934/fods.2019006 [19] Andrew J. Majda, Yuan Yuan. Fundamental limitations of Ad hoc linear and quadratic multi-level regression models for physical systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1333-1363. doi: 10.3934/dcdsb.2012.17.1333 [20] Yazhe Li, Tony Bellotti, Niall Adams. Issues using logistic regression with class imbalance, with a case study from credit risk modelling. Foundations of Data Science, 2019, 1 (4) : 389-417. doi: 10.3934/fods.2019016

Impact Factor: