Skip to main content

Assessing Validity and Application Scope of the Intrinsic Estimator Approach to the Age-Period-Cohort Problem

An Editor's Note to this article was published on 21 November 2013

A Commentary to this article was published on 17 October 2013

A Commentary to this article was published on 17 October 2013

A Commentary to this article was published on 17 October 2013

A Commentary to this article was published on 17 October 2013


In many different fields, social scientists desire to understand temporal variation associated with age, time period, and cohort membership. Among methods proposed to address the identification problem in age-period-cohort analysis, the intrinsic estimator (IE) is reputed to impose few assumptions and to yield good estimates of the independent effects of age, period, and cohort groups. This article assesses the validity and application scope of IE theoretically and illustrates its properties with simulations. It shows that IE implicitly assumes a constraint on the linear age, period, and cohort effects. This constraint not only depends on the number of age, period, and cohort categories but also has nontrivial implications for estimation. Because this assumption is extremely difficult, if not impossible, to verify in empirical research, IE cannot and should not be used to estimate age, period, and cohort effects.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 1


  1. 1.

    One way to characterize the effects of an interval variable like time is to break the effect into two components: linear and nonlinear (curvature or deviations from linearity) trends. It has been known at least since Holford (1983) that the linear components of age, period, and cohort effects cannot be estimated without constraints because they are not identified. In contrast, nonlinear age, period, and cohort trends can be estimated without bias.

  2. 2.

    Yang et al. (2008) refer to this as “CGLIM.”

  3. 3.

    It is important to distinguish data reduction or prediction from coefficient estimation. Because the identification problem does not prevent obtaining a set of solutions with good fit to the data, one can still make good predictions. The PC technique treats such problems as data redundancy and allows obtaining one solution. However, as noted earlier, none of these solutions is the uniquely preferred solution: the solution that APC techniques, including IE, aim to discover. Therefore, providing a solution for the purpose of prediction is not the same as finding a uniquely preferred solution for estimation of separate age, period, and cohort effects.

  4. 4.

    Alternatively, Yang (2008:413) described the computational algorithm of IE as follows: after obtaining r – 1 coefficients in the PC space (w 2, . . . , w r ), “set coefficient w 1 equal to 0 and transform the coefficients vector w = (w 1, . . . , w r )T,” where w 1 corresponds to the null eigenvector direction.

  5. 5.

    Yang et al. (2004, 2008) used \( {\mathbf{b}}_{\mathbf{0}}^{*}=\frac{{\mathbf{b}}_{\mathbf{0}}}{\left\Vert {\mathbf{b}}_{\mathbf{0}}\right\Vert } \), where ‖b 0 ‖ is the length of b 0 , so b * 0 has a length of 1. b 0 is used in this article because it is simply a multiple of b * 0 and is simpler for exposition and computation.

  6. 6.

    The constraint imposed by IE depends on how model (2) is parameterized. If the model is parameterized in terms of orthogonal polynomial contrasts for each of the age, period, and cohort effects, as in Holford (1983), then IE imposes a constraint solely on the linear contrasts of age, period, and cohort effects irrespective of any nonlinear trends that are present. The parameterization used here is more common (e.g., Kupper et al. 1985), and in this parameterization, the constraint on the linear components of the age, period, and cohort effects depends on the nonlinear components when both components are present.

  7. 7.

    Yang and colleagues have used “unbiasedness” in a different sense to mean that the expectation of IE is equal to b 1 , the projection of parameter vector b onto the nonnull space of design matrix X (see, e.g., Yang et al. 2008:1709). This is an important distinction because the true parameter vector b can be very different from its projection b 1 onto the nonnull space, the vector that IE actually estimates. Because APC analysts are usually interested in estimating the true age, period, and cohort effects, the classic concept of unbiasedness is more relevant to APC research than that used by IE’s proponents. Thus, I use “unbiasedness” in its classic sense in my discussion.

  8. 8.

    Yang and colleagues have used empirical data, in which the true effects are unknown, to assess the properties and performance of IE (Yang et al. 2008:1712–1716). However, it is logically impossible to assess the performance of an estimator when the true effects are unknown. If such a cross-model validation of IE for a specific empirical data set were to show that IE yields reasonable estimates, this can only depend on having selected examples that are consistent with the IE’s constraint. Therefore, cross-model comparisons using empirical data are not an appropriate method to validate IE.

  9. 9.

    Although Yang and colleagues correctly pointed out that IE estimates the projection of the “true” effects onto the nonnull space, they compared IE estimates with the “true” parameters, not to the projection (Yang et al. 2008:1718–1722). This is key because the true parameter vector can be very different from its projection onto the nonnull space (the vector that IE actually estimates). That is, what IE actually estimates can be very different from the true APC effects if the true effects do not at least approximately satisfy the LC implicit in IE.

  10. 10.

    Examples include age-period-cohort characteristic models developed by O’Brien (2000) and the mechanism-based approach proposed by Winship and Harding (2008).

  11. 11.

    Examples are the cross-classified random-effects models created by Yang and Land (2006, 2008).


  1. Alwin, D. F. (1991). Family of origin and cohort differences in verbal ability. American Sociological Review, 56, 625–638.

    Article  Google Scholar 

  2. Fabio, A., Leober, R., Balasubramani, G. K., Roth, J., Fu, W., & Farrington, D. P. (2006). Why some generations are more violent than others: Assessment of age, period, and cohort effects. American Journal of Epidemiology, 164, 151–160.

    Article  Google Scholar 

  3. Fu, W. (2000). Ridge estimator in singular design with applications to age-period-cohort analysis of disease rates. Communications in Statistics Theory and Method, 29, 263–278.

    Article  Google Scholar 

  4. Fu, W. J., & Hall, P. (2006). Asymptotic properties of estimators in age-period-cohort analysis. Statistics & Probability Letters, 76, 1925–1929.

    Article  Google Scholar 

  5. Fu, W. J., Land, K. C., & Yang, Y. (2011). On the intrinsic estimator and constrained estimators in age-period-cohort models. Sociological Methods & Research, 40, 453–466.

    Article  Google Scholar 

  6. Glenn, N. D. (2005). Cohort analysis. Thousand Oaks, CA: Sage Publications.

    Google Scholar 

  7. Heckman, J., & Robb, R. (1985). Using longitudinal data to estimate age, period, and cohort effects in earnings equations. In W. M. Mason & S. E. Fienberg (Eds.), Cohort analysis in social research (pp. 137–150). New York: Springer-Verlag.

    Chapter  Google Scholar 

  8. Holford, T. R. (1983). The estimation of age, period and cohort effects for vital rates. Biometrics, 39, 311–324.

    Article  Google Scholar 

  9. Keyes, K., & Miech, R. (2013). Age, period, and cohort effects in heavy episodic drinking in the US from 1985 to 2009. Drug and Alcohol Dependence. Advance online publication. doi:10.1016/j.drugalcdep.2013.01.019

    Google Scholar 

  10. Kupper, L. L., Janis, J., Karmous, A., & Greenberg, B. G. (1985). Statistical age-period-cohort analysis: A review and critique. Journal of Chronic Diseases, 38, 811–830.

    Article  Google Scholar 

  11. Kupper, L. L., Janis, J., Salama, I. A., Yoshizawa, C. N., Greenberg, B. G., & Winsborough, H. H. (1983). Age-period-cohort analysis: An illustration of the problems in assessing interaction in one observation per cell data. Communications in Statistics—Theory and Methods, 12, 2779–2807.

    Google Scholar 

  12. Langley, J., Samaranayaka, A., Davie, J., & Campbell, A. J. (2011). Age, cohort and period effects on hip fracture incidence: Analysis and predictions from New Zealand data 1974–2007. Osteoporosis International, 22, 105–111.

    Article  Google Scholar 

  13. Mason, K. O., Mason, W. H., Winsborough, H. H., & Poole, W. K. (1973). Some methodological issues in cohort analysis of archival data. American Sociological Review, 38, 242–258.

    Article  Google Scholar 

  14. Miech, R., Koester, S., & Dorsey-Holliman, B. (2011). Increasing US mortality due to accidental poisoning: The role of the baby boom cohort. Addiction, 106, 806–815.

    Article  Google Scholar 

  15. O’Brien, R. M. (2000). Age period cohort characteristic models. Social Science Research, 29, 123–139.

    Article  Google Scholar 

  16. O’Brien, R. M. (2011a). Constrained estimators and age-period-cohort models. Sociological Methods & Research, 40, 419–452.

    Article  Google Scholar 

  17. O’Brien, R. M. (2011b). Intrinsic estimators as constrained estimators in age-period-cohort accounting models. Sociological Methods & Research, 40, 467–470.

    Article  Google Scholar 

  18. Preston, S. H., & Wang, H. (2006). Sex mortality differences in the United States: The role of cohort smoking patterns. Demography, 43, 631–646.

    Article  Google Scholar 

  19. Rodgers, W. L. (1982a). Estimable functions of age, period, and cohort effects. American Sociological Review, 47, 774–787.

    Article  Google Scholar 

  20. Rodgers, W. L. (1982b). Reply to comment by Smith, Mason, and Fienberg. American Sociological Review, 47, 793–796.

    Article  Google Scholar 

  21. Ryder, N. B. (1965). The cohort as a concept in the study of social change. American Sociological Review, 30, 843–861.

    Article  Google Scholar 

  22. Schwadel, P. (2011). Age, period, and cohort effects on religious activities and beliefs. Social Science Research, 40, 181–192.

    Article  Google Scholar 

  23. Winkler, R., & Warnke, K. (2013). The future of hunting: An age-period-cohort analysis of deer hunter decline. Population Environment, 34, 460–480.

    Article  Google Scholar 

  24. Winship, C., & Harding, D. J. (2008). A mechanism-based approach to the identification of age-period-cohort models. Sociological Methods & Research, 36, 362–401.

    Article  Google Scholar 

  25. Yang, Y. (2008). Trends in U.S. adult chronic disease mortality, 1960–1999: Age, period, and cohort variations. Demography, 45, 387–416.

    Article  Google Scholar 

  26. Yang, Y., Fu, W. J., & Land, K. C. (2004). A methodological comparison of age-period-cohort models: The intrinsic estimator and conventional generalized linear models. Sociological Methodology, 34, 75–110.

    Article  Google Scholar 

  27. Yang, Y., & Land, K. C. (2006). A mixed models approach to the age-period-cohort analysis of repeated cross-section surveys, with an application to data on trends in verbal test scores. Sociological Methodology, 36, 75–97.

    Article  Google Scholar 

  28. Yang, Y., & Land, K. C. (2008). Age–period–cohort analysis of repeated cross-section surveys: Fixed or random effects? Sociological Methods & Research, 36, 297–326.

    Article  Google Scholar 

  29. Yang, Y., Schulhofer-Wohl, S., Fu, W. J., & Land, K. C. (2008). The intrinsic estimator for age-period-cohort analysis: What it is and how to use it. American Journal of Sociology, 113, 1697–1736.

    Article  Google Scholar 

Download references


I am grateful to James Hodges, John Robert Warren, Robert O’Brien, Christopher Winship, Daniel Powers, Carolyn Liebler, Samir Soneji, Ann Meier, Ian Ross Macmillan, Caren Arbeit, Julia Drew, Catherine Fitch, Julian Wolfson, and Wenjie Liao for their helpful comments. I also thank the Maryland Population Research Center for support. A version of this article was presented at the 2012 meeting of the Population Association of America, San Francisco, CA.

Author information



Corresponding author

Correspondence to Liying Luo.

Electronic Supplementary Material

Below is the link to the electronic supplementary material.

Fig. S1

(PDF 204 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Luo, L. Assessing Validity and Application Scope of the Intrinsic Estimator Approach to the Age-Period-Cohort Problem. Demography 50, 1945–1967 (2013).

Download citation


  • Age-period-cohort analysis
  • Intrinsic estimator
  • Identification problem
  • Constrained estimator
  • Linear constraint