On the Structure and Classification of Mortality Models

Abstract

Recently there has been a huge increase in the use of models that examine the structure of mortality rates across the dimensions of age, period and cohort. This article reviews the major developments in the field, provides a holistic analysis of these models, and examines the models’ similarities and differences. Specifically, the article reviews the models that have been proposed to date, investigates the structure of age/period/cohort mortality models, introduces a classification scheme for existing models, and lists the key principles a model user should consider when constructing a new model in this class.

ACKNOWLEDGMENTS

We are grateful to Andrés Villegas, Steven Haberman, Bent Nielsen, Andrew Cairns, Pietro Millossovich, and Ana Debón for many useful discussions on this and related topics which improved this article immeasurably.

DISCLAIMER

This study was performed when Dr. Hunt was a Ph.D. student at Cass Business School, City University of London, and therefore the views expressed within it are held in a personal capacity and do not represent the opinions of Pacific Life Re and should not be read to that effect.

Discussions on this article can be submitted until October 1, 2020. The authors reserve the right to reply to any discussion. Please see the Instructions for Authors found online at http://www.tandfonline.com/uaaj for submission instructions.

Notes

1 Most APC mortality models have only one age/cohort term for the reasons discussed in Section 7. However, some models do incorporate multiple terms—for instance, that proposed in Hatzopoulos and Haberman (2011).

2 Lee and Carter (1992) originally applied the LC model to central mortality rates, $m_{x,t}.$ However, as discussed in Section 4, these are equivalent to the force of mortality, ${\mu }_{x,t},$ under the assumption in Equation (23)(23) $${\mu }_{x,t}={\mu }_{\lfloor x\rfloor ,\lfloor t\rfloor },$$(23) .

3 Where $d_{x,t}$ is the observation of the random death count, $D_{x,t}.$

4 Human Mortality Database (2014).

5 For instance, in Cairns et al. (2006b).

6 One solution to this might be to assume a constant force of mortality over shorter age and period bands, for instance, across months as in Gavrilov and Gavrilova (2011). However, data limitations at high ages tend to make this infeasible.

7 That is, ${\mu }_{x,t}\ge 0$ or $q_{x,t}\in (0,1).$

8 See, for instance, the estimation of models in the CBD family using the LifeMetrics code in Coughlan et al. (2007), where a Poisson distribution of deaths is assumed with a logit link function.

9 These models do not draw a direct link between the use of the logit function and binomial death counts. However, this connection is made explicit in Haberman and Renshaw (2011) and Currie (2014).

10 For this reason, we could alternatively refer to non parametric age functions as “factorial” age functions.

11 For this reason, these age functions could also be called “formulaic.”

12 PCA assumes homogenous, normally distributed residuals and, therefore, is inconsistent with the underlying binomial or Poisson distribution for the death count process. However, the estimates obtained for the parameters using PCA can be used as the starting point for methods such as maximum likelihood, which use the death count process to allow for heterogeneity caused by differences in the underlying exposures.

13 This can be seen with the sharp peak at β₁ in Figure 2.

14 Introduced in Cairns, Blake, and Dowd (2006a) and defined as “a method of reasoning used to establish a causal association (or relationship) between two factors that is consistent with existing medical knowledge.” Note that biological reasonableness is a property of observable quantities such as life expectancies or mortality rates, in contrast to demographic significance, which relates to our interpretation of the terms in a model.

15 We also suffer from the problem that the parameters in the model may not be uniquely identified. This topic and its implications for forecasting are discussed further in Hunt and Blake (2020a, 2020b).

16 See Alai and Sherris (2012) for an example of a model that gives primacy to cohort parameters.

17 For instance, Cairns et al. (2011) raised “the possibility that cohort effects might be partially or completely replaced by well-chosen age and period effects” and see Murphy (2010).

18 See Hunt and Villegas (2015) for a discussion and potential solution for this issue.