Chapter 12 -'K_t' -Effect of time trend on mortality

[Bill SD]

Hi,
Would someone mind explaining (in the basic Lee Carter model) what is the difference between 'bx' (change in rates at age x due to time trend) and 'kt' (effect of time trend at time t on mortality)? Is it because 'bx' only accounts for past years' mortality data while 'kt' allows for current/future mortality trends?

Confused why:
- need two separate terms and not one (like 'ax' =general shape of mortality at age x)
-if mortality improves with time, then kt reduces as t increases (pg 12 notes)?

Tia

 

[Andrew Martin]

Hello

1.
If we just had one term relating to the time trend, say kt, then this would mean that, according to the model, the change in mortality over time is the same for all ages. So, even if the initial mortality rates are different, the year on year change is the same across all ages. Now this is a plausible model to try and use, however data suggests that there are differences in how mortality rates change over time for different ages.

By adding additional parameters, the bx's, we can make the model more flexible. The change in mortality over time for a specific age is now related to bx * kt. Kt represents the overall trend in mortality over time (so across all ages) and bx represents a scaling factor for that time trend for each specific age.

2.
If the model is log(mx,t) = ax + bx * kt then:

log(mx,t+1) - log(mxt) = bx * (k{t+1} - kt)

So, mortality will improve from year t to year t+1 (fall) if bx * (k{t+1}-kt) < 0.

So if, for example, kt simply decreases over time for all t and bx is positive then yes this corresponds to constantly improving mortality rates.

Hope this helps

Andy