## Abstract

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.

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## Notes

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

- 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.
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.
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.
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.
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.
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.
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.
- 11.

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## Acknowledgments

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.

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Luo, L. Assessing Validity and Application Scope of the Intrinsic Estimator Approach to the Age-Period-Cohort Problem.
*Demography* **50, **1945–1967 (2013). https://doi.org/10.1007/s13524-013-0243-z

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### Keywords

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