# The Lee-Carter Family

In a recent paper presented to the Faculty of Actuaries, Stephen Richards and I discussed *model risk* and showed how it can have a material impact on mortality forecasts. Different models have different features, some more desirable than others. This post illustrates a particular problem with the original Lee-Carter model, and shows how it can be combatted via smoothing. The choice of which parameters to smooth in the Lee-Carter model leads to a general family of models, which this note describes.

Let’s start by reminding ourselves of Lee and Carter’s original model (Lee & Carter, 1992), referred to here as LC. We’ll use the CMI data set with claims and central exposed-to-risk arranged in matrices **D** = (*d _{ij}*) and

**E**= (

*e*) respectively; the rows are indexed by age of claim: 40, ... , 89, and the columns by year of claim: 1950, ... , 2005. LC assumed that:

_{ij}log *μ _{ij}* = α

*+*

_{i}*β*

_{i}κ_{j}where *μ _{ij}* is the true, but unknown, force of mortality at age

*i*in year

*j*. The parameters in the equation above are not identifiable so we impose the constraints Σ

*κ*= 0 and Σ

_{j}*κ*= 1; these constraints have no effect on the estimated values of

_{2j}*μ*, nor on any subsequent forecast. LC’s idea is simple yet clever, since it reduces a 2-dimensional forecasting problem to the much simpler 1-dimensional problem of forecasting the time index,

_{ij}**. Estimation is straightforward using maximum likelihood (Brouhns, Denuit & Vermunt, 2002) and Figure 1 shows the resulting parameter estimates, together with the observed and fitted log mortalities for age 70.**

*κ*The idea behind our Lee-Carter family is to exploit the obvious regularities in the parameter estimates seen in Figure 1. This idea is not completely new: Delwarde, Denuit & Eilers (2007) (DDE) observed that irregularities in the estimates of the *β _{i}* can have unfortunate consequences. The upper-right panel of Figure 1 shows a worrying jump between the estimate of

*β*at age 53 (low) and that at age 54 (high). This jump in the values of

_{i}*β*feeds through to the forecasts for ages 53 and 54 which crossover around year 2030, as the upper right panel of Figure 2 shows.

_{i}DDE dealt with this problem by smoothing the *β _{i}*, and not surprisingly, they used the P-spline method of smoothing (Eilers & Marx, 1996); the upper left panel of Figure 2 shows the original LC

**values and their smoothed values. The lower left panel in Figure 2 shows that not only has the crossover problem been solved but the mortality at successive ages progresses in a more regular fashion than in the original LC model. The lower right panel in Figure 2 shows the forecast at age 70 together with the 95% confidence interval.**

*β*But why stop at smoothing just ** β**? The lower left panel in Figure 1 suggests that we could also smooth

**. We also use P-splines to smooth the**

*κ***, and as a bonus, the smoothing process itself also yields forecasts of the**

*κ***values; this has been widely described in the actuarial literature. We refer to this model as the Currie-Richards model (CR). The three models, LC, DDE and CR are the three models used in our paper on longevity risk referred to above.**

*κ*Now that we have got a taste for parameter smoothing we can use a pick ’n’ mix approach. Models which don’t smooth ** κ** forecast by time-series methods; on the other hand, models which do smooth

**use the smoothing process to forecast. Within these two sub-families we have various possibilities.**

*κ*- Time-series sub-family. We already have two members: LC and DDE. If in addition to smoothing
, we smooth*β*, we have a third member, smooth Lee-Carter, and if we replace*α*by a linear function (the Gompertz law) we have what we call Gompertz-Lee-Carter.*α*

- Smooth sub-family with forecasting via the smoothing process instead of a time series. To the existing CR model we can add the models with smooth or linear
.*α*

If you’d like to see some of the theory behind these models and pictures of the output they give, then have a look at the slides from a talk I’ll give at the Actuarial Teachers’ and Researchers’ Conference at Queen’s University, Belfast next month.

## Comments

am presently working on the model for my PhD THESIS in the university,i need to know more on the model

For more details on the model class, take a look at the links in the blog - there's a link to last year's paper at the start, and another link at the end to Iain's slides at a conference. Further conference presentations can be found at http://www.ma.hw.ac.uk/~iain/research/talks.html

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