Title: Modeling Coherent Mortality Forecasts using the Framework of Lee-Carter Model
1Modeling Coherent Mortality Forecasts using the
Framework of Lee-Carter Model
- Presenter Jack C. Yue /National Chengchi
University, Taiwan - Co-author Sharon S. Yang /National Central
University - Yi-Ping Chang /Soochow University
- Yu-Yun Yeh/Cathay Life
Insurance Company - Sept. 26, 2009
2Outline
- Introduction
- The Coherent Mortality Modeling
- Analysis of the Coherent Mortality Modeling
- Mortality Forecasts Coherent Models vs. Single
Population - Conclusions and Discussions
3Introduction
4Motivation
- Men and women in a country or people in nearby
countries share comparable living conditions and
are likely to have similar mortality behaviors. - For example, the correlation coefficients of the
life expectancy for the male and female between
Canada and the U.S. are 0.9922 and is 0.9926,
respectively. - ?The correlation between male and female for
Canada and U.S. are 0.9701 and 0.9666.
5Life expectancy of male and female aged 65 for
Canada and the US
6Coherent Mortality
- Wilson (2001) pointed out a global convergence in
mortality and mentioned that it is improper to
prepare mortality forecasts for individual nation
in isolation from one another. - Li and Lee (2005) mentioned that mortality
patterns in closely related populations are
likely to be similar in some respects and
differences are unlikely to increase in the long
run.
7The Coherent Mortality Modeling
8Lee-Carter Model
- Lee and Carter (1992) proposed the following
mortality model for U.S., -
- where
- ? Central Death Rate of age x, at time t
- ? Intensity of Mortality at time t
(linear!) - ? Average Mortality of age x
- ? Tendency of Mortality change for age x
9Empirical Evidence for LC Model
- LC Model provides fairly accuracy forecasts for
the countries such as the U.S. and Japan. - ?It works well for a single population, one sex
or two sexes combined. - However, using the LC model to forecast two-sex
mortality of a population has been a problem. - ?The values of can be very different for
the male and female. (Divergence problem!)
10Converging Mortality Forecast
- The gaps of life expectancy between developed and
developing countries have been decreasing since
the second half of the 20th century. - Li and Lee (2005) think that a long-term
divergence in life expectancy is unlikely. They
extended the LC model and proposed using the
model for a group of populations with similar
socioeconomic conditions.
11Coherent Mortality Model(Augmented Common Factor
LC Method)
- Li and Lee (2005) modified the original LC model
to multiple populations, assuming that they have
same and . - ?They suggest using the Explanation Ratio
(similar to R2 in regression) to check if
combining populations is appropriate, comparing
to modeling the single population. - ?They found that the proposed model worked well
if combining 15 low mortality countries as a
group.
12Our proposed Study
- In this study, our goal is also to explore the
coherent mortality model. In specific, our study
has different focus - ?Estimation method
- ?Mortality of the elderly
- ?Combining sexes or countries
- ?Goodness-of-fit
-
13Our proposed Study(Estimation Method)
- Unlike the ordinary least squares is used in Li
and Lee (2005), we suggest using the maximum
likelihood estimation (MLE), assuming that the
numbers of deaths follow Poisson distributions.
- ?Due to the nature of nonlinear optimization,
the recursive procedure of Newton method is used.
The initial values of the parameters are obtained
from the singular value decomposition (SVD).
14Our proposed Study(Elderly Data)
- The patterns of mortality improvement are
changing over time and the different age groups
have quite different experiences. - ?For example, the reduction rate of mortality for
the younger group is decreasing, but that for the
elderly is increasing. - ?We will focus on the data of ages 6599
15Our proposed Study(Grouping and Goodness-of-fit)
- We want to know if male and female in a country,
or different countries in a region fits better as
a group. - Li and Lee (2005) did not suggest any tests of
goodness-of-fit. We want to know if there are
any systematic errors which can help to improve
the modeling. We shall look at the residuals of
the coherent model.
16Empirical Analysis of the Coherent Model
17Data and Evaluation Methods
- We use the data of Canada and U.S., in Human
Mortality Database (HMD). - ?Five-year age groups, years 19502005.
- The log-likelihood, mean absolute percentage
error (MAPE), - and AIC (BIC) are used to evaluate the model
fit.
18Parameter estimates of coherent LC and original
LC models (US Male 6599)
19Parameter estimates of coherent LC and original
LC models (US Female 6599)
20Empirical Results of Parameters
- For the cases of U.S. 6599 (years 1950-2000),
the estimated results of and from
combining countries and those from combining
genders look similar, unlike those from the
single population. - ?It seems that the coherent model produce similar
estimates, no matter if the group variable is sex
or country. - ?The intercepts are almost identical.
21Log-likelihood for Different Coherent Groups
Country Gender Coherent group All Ages 029 3064 6599
USA Male Country -10887.0 -2989.2 -3392.1 -3243.2
USA Male Gender -1210.03 -2292.5 -3476.7 -6019.0
USA Female Country -9251.7 -2102.4 -2632.7 -4015.5
USA Female Gender -12048.0 -1784.2 -3314.4 -6462.2
Canada Male Country -13027.0 -3526.5 -3157.6 -3825.2
Canada Male Gender -16122.0 -2940.8 -3096.1 -9123.2
Canada Female Country -9756.1 -2477.6 -2559.6 -4265.1
Canada Female Gender -16678.0 -2121.8 -3223.8 -10071.8
Note The numbers in red are preferred.
22Mortality Forecasts Coherent Models vs. Single
Population
23Forecasting Mortality
- We use the prediction error to evaluate the
coherent model and the single LC model. - ?The data of years 1950-2000 are used to reach
parameter estimates, and the data of years
2001-2005 are treated as unknown. Also, we
focus on the elderly, ages 65-99. - ?Unlike using the R2 or log-likelihood, we can
use the predicted MAPE to evaluate the models,
without adjusting degree of freedom.
24Forecast MAPE for Higher Ages (65-99) Coherent
Groups vs. Single Population
Country Gender MAPE
USA Male Coherent Country Group 1.18
USA Male Coherent Gender Group 2.86
USA Male Single Population 1.90
USA Female Coherent Country Group 1.03
USA Female Coherent Gender Group 5.42
USA Female Single Population 0.89
Canada Male Coherent Country Group 2.86
Canada Male Coherent Gender Group 2.90
Canada Male Single Population 5.50
Canada Female Coherent Country Group 2.16
Canada Female Coherent Gender Group 1.36
Canada Female Single Population 3.69
25Forecasting Mortality (conti.)
- The predicted MAPEs are all very small,
indicating the LC-type models are a good
candidate to model the elderly mortality rates. - ?On average, combining countries has smaller
predicted MAPE than combining gender, similar to
the result in estimation. - ?Combining countries almost dominates the single
population, except for the case of U.S. female.
26Forecasting Mortality (conti.)
- In addition to the predicted MAPE, we also
compare the variances of predicted mortality
rates for the coherent model to those for the
single population model. - ?As expected, since the coherent model use more
samples (i.e., combining country or gender), the
variances and the prediction intervals are
smaller.
2795 Confidence Interval of Simulated Mortality
(U.S. Male Aged 65). Coherent Country Model
2895 Confidence Interval of Simulated Mortality
(U.S. Male Aged 65). Single Mortality Model
2995 Confidence Interval of Simulated Mortality
(U.S. Male Aged 65, 75, 85).
Age 85
Age 75
Age 65
3095 Confidence Interval of Simulated Annuity
Value (U.S. Male Aged 65) Coherent Country Model
3195 Confidence Interval of Simulated Annuity
Value (U.S. Male Aged 65) Single Mortality Model
32Conclusion
33Conclusion
- The coherent model is to treat populations with
similar socioeconomic conditions as a group. - ?We found that the data of Canada and U.S. do
support the coherent model, and it seems that
treating Canada and U.S. as a group is more
appropriate (than combining gender). - ?We provide an alternative approach in this
study, in addition to the R2 measure.
34Discussions
- How do we choose a group of populations with
similar conditions? - ?What measures can we use? (trial-and-error?)
- ?Should we check if and are consistent?
- There are problems in goodness-of-fit.
- ?Does it matter?
- ?Should we select the age groups?
- Are the parameters fixed?
- ?If not, what can we do?
35Thank you!!