Demography's dark matter: measuring cohort effects

(May 19, 2014)

My last blog generated quite a bit of interest so I thought I'd write again on cohorts. It's easy to (a) demonstrate the existence of a cohort effect and to (b) fit models with cohort terms, but not so easy to (c) interpret or forecast the fitted cohort coefficients. In this blog I'll fit the following three models:

LC \(\log \mu_{i,j} = \alpha_i + \beta_i \kappa_j\)
LCC \(\log \mu_{i,j} = \alpha_i + \beta_i \kappa_j + \gamma_{j-i}\)
APC \(\log \mu_{i,j} = \alpha_i + \kappa_j + \gamma_{j-i}\)

where \(\mu_{i,j}\) is the force of mortality at age \(i\) in year \(j\). These are the Lee-Carter model (LC), the Lee-Carter model with an added cohort effect (LCC) (a special case of the Renshaw-Haberman model) and the Age-Period-Cohort…

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Tags: cohort effect, APC, Age-Period-Cohort, mortality projections

Forecasting with cohorts for a mature closed portfolio

(Mar 27, 2014)

At a previous seminar I discussed forecasting with the age-period-cohort (APC) model:

$$ \log \mu_{i,j} = \alpha_i + \kappa_j + \gamma_{j-i}$$

where \(\mu_{i,j}\) is the force of mortality at age \(i\) in year \(j\); the parameters \(\alpha_i\) , \(\kappa_j\) and \(\gamma_{j-i}\) are usually thought of as the age, period and cohort parameters respectively. These parameters are not identifiable from the formula above, so we need a set of constraints to fix unique estimates.  Suppose we number the cohorts from 1 (oldest) to \(n_c\) (youngest).  One set of constraints is as follows:

$$\sum \kappa_j = 0,\; \sum \gamma_c = 0,\; \sum c \gamma_c = 0.$$

Forecasting the mortality table now depends on forecasting…

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Tags: APC, Age-Period-Cohort, mortality projections, cohort effect

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