A Bayesian Approach to Modeling and Projecting Cohort Effects

Abstract

One of the key motivations in the construction of ever more sophisticated mortality models was the realization of the importance of “cohort effects” in the historical data. However, these are often difficult to estimate robustly, due to the identifiability issues present in age/period/cohort mortality models, and exhibit spurious features for the most recent years of birth, for which we have little data. These can cause problems when we project the model into the future. In this study, we show how to ensure that projected mortality rates from the model are independent of the arbitrary identifiability constraints needed to identify the cohort parameters. We then go on to develop a Bayesian approach for projecting the cohort parameters that allows fully for uncertainty in the recent parameters due to the lack of information for these years of birth, which leads to more reasonable projections of mortality rates in future.

Acknowledgments

Material in this article was presented under the title “Projecting Mortality: Identifiability with Trend Changes and Cohort Effects” at the 17th International Congress on Insurance: Mathematics and Economics in July 2013 in Copenhagen, Denmark. We are grateful to participants at this conference, and to Matthias Börger, Frank van Berkum, Andrew Cairns, Pietro Millossovich, and Andrés Villegas for useful discussions regarding this work.

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.

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Notes

1 For simplicity, the dependence of the age functions on ${\theta }^{(i)}$ is supressed in the notation used in the remainder of this article, but not in the model itself.

2 Demographic significance is defined in Hunt and Blake (2020a) as the interpretation of the components of a mortality model in terms of the underlying biological, medical, or socio economic causes of changes in mortality rates that generate them.

3 These are called “invariant transformations” in Hunt and Blake (2020b,d,e) for this reason.

4 Here, X is the number of ages in the data, $\overline{x}=\frac{1}{X}\sum _{x}x$ and ${\sigma }_x=\frac{1}{X}\sum _x{(x-\overline{x})}^2,$ and similarly for $\overline{t},\overline{y},$ etc. Also note that, to aid understanding of these complex relationships, Equations (2)(2) $$\{{\widehat{\alpha }}_x,{\widehat{\kappa }}_t^{(1)},{\widehat{\kappa }}_t^{(2)},{\widehat{\kappa }}_t^{(3)},{\widehat{\gamma }}_y\}=\{{\alpha }_x-a_0,{\kappa }_t^{(1)},{\kappa }_t^{(2)},{\kappa }_t^{(3)},{\gamma }_y+a_0\}$$(2) , (3)(3) $$\begin{array}{cc}\{{\widehat{\alpha }}_x,{\widehat{\kappa }}_t^{(1)},{\widehat{\kappa }}_t^{(2)},{\widehat{\kappa }}_t^{(3)},{\widehat{\gamma }}_y\}=\{{\alpha }_x+a_1(x-\overline{x}),{\kappa }_t^{(1)}-a_1(t-\overline{t}),{\kappa }_t^{(2)},{\kappa }_t^{(3)},& \hspace{1em}{\gamma }_y+a_1(y-\overline{y})\}\end{array}$$(3) , and (4)(4) $$\begin{array}{c}\{{\widehat{\alpha }}_x,{\widehat{\kappa }}_t^{(1)},{\widehat{\kappa }}_t^{(2)},{\widehat{\kappa }}_t^{(3)},{\widehat{\gamma }}_y\}=\{{\alpha }_x-a_2({(x-\overline{x})}^2-{\sigma }_y+{\sigma }_t),{\kappa }^{(1)}-a_2({(t-\overline{t})}^2-{\sigma }_t),\\ \hspace{1em}{\kappa }_t^{(2)}+2a_2(t-\overline{t}),{\kappa }_t^{(3)},{\gamma }_y+a_2({(y-\overline{y})}^2-{\sigma }_y)\}.\end{array}$$(4) do not incorporate the normalization factors required on the age functions in order to ensure that $\sum _x|f^{(i)}(x)|=1\forall i.$ These will need to be included before the model is fitted to data.

5 In principle, if the final time series processes are known in advance or determined by a trial two-step sequential estimation of the model and time series processes, it is possible to fit the model and time series processes to data jointly in a one-step process. This can be done using either maximum likelihood techniques, as in Dowd et al. (2011), or Bayesian Markov chain Monte Carlo techniques, as in Pedroza (2006). However, such techniques are complicated to implement and so are not practical when using sophisticated mortality models or if the model is intended to be used for different data sets for which different time series processes might be appropriate.

6 Note that, for consistency with Figure 4, this uses the same identifiability constraints as are later applied in Section 5.

7 In practice, we take d_x to be given by the fitted mortality rates in the final year of the data. However, the results are relatively insensitive to the choice of d_x as long as these reflect a plausible pattern of deaths from a cohort across different ages.

8 In Hunt and Blake (2020c), this is relaxed and the year of observation is allowed to change to reflect the impact of new observations on the previously estimated cohort parameters.

9 This is a consequence of having a simplified age/cohort structure and setting ${\beta }_x^{(0)}=1,$ discussed in Hunt and Blake (2020a).

10 The model could, theoretically, be extended to allow for more lags and an AR(p) structure via a different choice for Γ_y.

11 Though the distribution for ${\gamma }_y^x$ is not used here, it is necessary when updating the estimates of the cohort parameters for additional data, as done in Hunt and Blake (2020c).

12 Mathematically, this is a consequence of $D_{t-Y}>0.$ More intuitively, it can be seen that ${\sigma }^2{(1-{\rho }^2)}^{-1}$ is the variability of a cohort parameter under the prior distribution from the AR(1) time series without any additional information from the data to refine the parameter estimate.

13 Introduced in Cairns, Blake, and Dowd (2006) 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.”

14 We find log-likelihoods of $-3.09\times {10}^{-4}$ using the estimated parameters and $-3.25\times {10}^{-4}$ using the expectation of the ultimate parameters, which is mainly due to worsening the fit to mortality data at age 0. This may indicate that the most recent fitted cohort parameters attempt to overfit data at this unusual age, rather than capturing genuine lifelong mortality effects.

15 In principle, the other age/period terms in the model could be re-estimated subsequent to finding M(y, t). In practice, however, this was not done in this study.

16 http://www.llma.org/.

17 Note that the bespoke survivor swaps entered into by pension schemes also suffer from these issues as well.

18 Although, historically, annuities were traded contracts and the UK government recently proposed establishing a secondary annuity market, although later withdrew the proposal.