=Paper=
{{Paper
|id=Vol-2018/paper-09
|storemode=property
|title=The Lee-Carter Method for Mortality Forecasting: the Case of the Republic of Bashkortostan
|pdfUrl=https://ceur-ws.org/Vol-2018/paper-09.pdf
|volume=Vol-2018
|authors=Irina Lakman,Denis Popov,Nailya Shamsutdinova
}}
==The Lee-Carter Method for Mortality Forecasting: the Case of the Republic of Bashkortostan==
https://ceur-ws.org/Vol-2018/paper-09.pdf
The Lee–Carter Method for Mortality
Forecasting: the Сase of the Republic
of Bashkortostan?
Irina Lakman1[0000−0001−9876−9202] , Denis Popov2[0000−0002−7698−8858] , and
Nailya Shamsutdinova2[0000−0003−2008−0856]
1
Ufa State Aviation Technical University, Ufa, Russia,
lackmania@mail.ru,
2
Institute for Strategic Studies of the Republic of Bashkortostan, Ufa, Russia
Abstract. The article is devoted to predicting the mortality rates by
age and sex for one of constituent entities of the Russian Federation – the
Republic of Bashkortostan. As initial data, age-specific rates were used
in five-year groups of up to 85 years, published by the Territorial Body
of the Federal State Statistics Service for the Republic of Bashkortostan
(2001–2014) and calculated by the authors (1980–2000). The death rate
was calculated by means of Stata software. For this purpose, the method
was specifically adapted for the Republic of Bashkortostan. However,
correlation of the actual indices of 2015 showed that predicted values
for that year were underestimated. Due to the impact of downward dy-
namic observed nationwide in the 1990s, there are restrictions on using
of methods of extrapolation of death rate for the purpose of population
forecasts and calculation of insurance risks and decision-making in this
field. The outcome for the Republic of Bashkortostan, as for Russia in
whole, indicates that there still remains a crucial task of reducing eco-
nomic losses and losses of human capital as a result of high mortality
of the working-age population, which will bring pressure upon pension
funds.
Keywords: Insurance risks, mortality forecasting rate, death rate, the
Lee–Carter method, ARIMA model
1 Introduction
In the Russian Federation, after a period of high rise of mortality rate during
1990s, the situation changes into upward dynamics since 2004. From this time
onward until 2016, life expectancy at birth increases in figures nationwide totaled
up to 6.6 years.
The dynamic pattern of life expectancy at birth on a global scale has the
tendency to linear growth; however, in Russia this trend has extremely unstable
?
The reported study was funded by Russian Foundation for Basic Research (RFBR)
according to the research project No 17-46-020237 р_а.
character [15]. The increase in life expectancy is one of the main goals of the
socioeconomic and demographic policy of the Russian Federation and its regions.
The Republic of Bashkortostan as one of the large and influential constituent
entity of the Russian Federation, replicates the whole Russia’s situation on the
dynamics of mortality rate. The study of the applicability of prediction methods
at the regional level is now important not only from scientific and practical
viewpoints.
Age-specific mortality rate is the basis for predicting life expectancy, the
future population, its structure, and also of interest to the insurance industry.
Within the framework of human capital research, studying these particular age
groups, where the greatest loss rates are being observed, over time it will be
possible to create a system for loss prevention. Besides, the mortality forecasting
is necessary for model building growth prospects of the insurance and pension
systems, contributing decision-making in the range of acceptable risk.
2 Overview of actual methodological and instrumental
approaches
In the estimation of studies of Sweden, the USA and some other countries,
pension funds are subject to consider longevity risk, which actualizes the search
for new methods for their management [14, 16].
At the other end of the scale, premature mortality decreases the efficiency
of investments into human capital and serves as the reason of economic dam-
age that is currently important for Russia [10]. According to certain estimates,
it amounted to 16.3% of Gross Regional Product (GRP) in the Republic of
Bashkortostan [3]. In order to assess the future prospects and choose priorities
for mortality reduction, it is necessary to resort to modeling its indicators.
For forecasting age-specific mortality index, the Lee–Carter method is widely
used [5,8,9,12,13]. The results attained across the Russian Federation show con-
tinued existence in the future of a big difference between male and female mor-
tality nationwide in comparison with other countries. However, high fluctuations
of mortality in Russia demand the questioning attitude towards the results of
application the Lee–Carter method [10, 11]. However, in the regions of the Rus-
sian Federation, no simulation experiment was performed using the Lee–Carter
model. Coherent forecasts of mortality industrialized countries are justified by
reason of their greater homogeneity [4]. The Russian Federation continues to be
extremely internally heterogeneous [17].
The Lee–Carter method original modeling contains the following equation:
log mx,t = ax + bx kt + εx,t
Here mx,t – age-specific mortality index for a cohort x in the period of time t
(year), a vector reflects time-mean value effect of influence of age to mortality
index for each cohort x, kt vector reflects an effect of influence of time, average
on age, of mortality index for every period of time t, and the coefficient vector
bx explains the effect of interaction expressing specific sensitivity of mortality
index at age x to changes in time of k [16]. Random errors of the equation are
designated as εx,t .
Coefficient vectors bx and kt are in the original model of the Lee–Carter
method by means of singular value decomposition (SVD) which equates:
A = U SV T
Here U – the orthogonal matrix, V T – the transposed matrix,S – the diagonal
matrix consisting of zero and singular values of matrix A spaced diagonally.
There is a use of experience for finding unknown bx and kt as an alternative to
SVD-analysis of the weighted least spreads method (WLS) [7] and method of
maximum likelihood (ML) [6].
The first base of index construction of ARIMA models (autoregressive inte-
grated moving average) is type definition of process to which time-series belongs.
The approach of J. J. Dolado , T. Jenkinson and S. Sosvilla-Rivero, consistently
applying to the complex hypotheses of the extended test of Dickey–Fuller al-
lows to determine the type of process. Further, if there is a deterministic linear
component in the time series, it is removed. If the time series is an integrable
process in the first or second order, then the procedure for differentiation of the
corresponding order is performed.
At the second stage, the identification procedure of an order of autoregression
and order of process of the moving average is carried out.
At the third stage, the ARIMA model equation coefficients are estimated by
method of least squares and calibrate reversibility of model.
At the third stage, the coefficients of ARIMA model are estimated by the least
squares method and they verify the reversibility of the model, which means, they
test the requirement that the roots of the characteristic process corresponding
to the process lie outside the unit circle.
At the fourth stage, selection of models by using information criteria of
Akaike, Hannah–Quinn is carried out in case the same process can be described
in various equations.
At the fifth stage, quality monitoring of the constructed model is conducted,
screening with the help of specific tests (Jarque–Bera, Durbin–Watson, Breusch–
Godfrey, etc.) so that estimated coefficients of ARIMA model could be unbiased,
well-founded and effective.
At the final stage, forecasting model behavior is estimated, proceeding from
a minimum mean absolute percent error, residual dispersion and odds ratio ac-
cording to Theil inequality (index).
3 Materials and alternatives
For the development of population mortality forecasting by sex-age structure,
customized 5-year age groups of up to 85+ cohorts study has been set up. From
2001 to 2014 sex-age-specific death rates appeared in official publications [1, 2].
From 1980 to 2000 the interval charts were provided by Territorial Body of Fed-
eral State Statistics Service in the Republic of Bashkortostan (Bashkortostan-
stat, Ufa, Russia), calculated by a standard method:
Mx
mx = × 1000 (1)
P̄x
Here mx – age-specific mortality index; – number of the deceased aged x in
a year; P̄x – mid-year population aged x. Change of mortality age distribution
in the Republic of Bashkortostan, as in Russia at large, has irregular nature.
Tremendous losses in the 1990s were suffered by the population at productive age
(employable age). Increase in remaining life expectancy in the country continues
since 2003. However, in this particular region the death rate advances by 2015
in comparison with 1990 in some age groups: among women in the cohort of
25–39 years and among men in the cohort of 25–49 years and 60–74 years old
(Tables 1 and 2).
Table 1. The age-specific male death rate from 1990 to 2015
Kohort 1990 1995 2000 2005 2010 2015
0–1 18.3 21.5 17.5 14.4 7.7 8.0
1–4 1.4 1.2 1.1 0.8 0.6 0.3
5–9 0.7 0.8 0.6 0.5 0.3 0.3
10–14 0.5 0.8 0.5 0.5 0.4 0.3
15–19 1.5 2.3 2.5 1.6 1.3 1.0
20–24 2.8 5.0 5.5 3.9 3.2 2.4
25–29 3.2 6.0 6.0 5.8 5.8 3.5
30–34 3.8 7.1 7.3 7.4 7.7 6.4
35–39 4.9 9.3 8.5 9.3 8.2 9.1
40–44 6.8 11.4 11.1 12.9 9.7 10.0
45–49 10.0 15.6 14.7 16.0 13.0 13.1
50–54 13.9 21.6 19.8 22.2 17.5 16.6
55–59 20.1 26.4 28.3 29.9 23.8 23.0
60–64 28.9 37.1 38.4 42.1 35.9 33.4
65–69 41.0 51.7 52.9 53.2 49.1 43.1
70–74 60.5 69.9 71.3 74.9 66.8 62.0
75–79 90.3 101.5 94.9 104.2 97.0 87.2
80–84 135.0 149.4 132.1 142.8 136.6 125.0
85+ 213.8 248.5 238.2 235.3 204.3 193.8
4 Experimental setup
After receiving coefficients bx and kt by means of singular value decomposition
executed using Stata software, parameterization of ARIMA models for kt series
Table 2. The age-specific female death rate from 1990 to 2015
Kohort 1990 1995 2000 2005 2010 2015
0–1 13.9 15.1 12.0 9.4 6.6 6.8
1–4 1.0 1.0 0.8 0.7 0.5 0.3
5–9 0.4 0.4 0.4 0.3 0.2 0.2
10–14 0.3 0.4 0.3 0.2 0.3 0.3
15–19 0.8 0.9 0.7 0.8 0.7 0.5
20–24 1.0 1.0 1.1 0.9 0.8 0.8
25–29 0.8 1.3 1.3 1.4 1.5 1.3
30–34 1.0 1.4 1.5 1.9 2.1 2.2
35–39 1.4 2.1 2.2 2.6 2.3 2.9
40–44 2.1 3.4 2.9 3.5 3.2 3.2
45–49 3.8 5.1 4.2 4.8 4.2 4.2
50–54 5.3 7.3 6.9 7.0 5.3 5.7
55–59 8.2 9.8 9.8 11.0 9.2 7.4
60–64 12.4 15.7 14.7 15.8 13.4 11.1
65–69 19.4 23.5 23.0 22.7 20.8 17.0
70–74 32.8 35.7 37.3 38.5 32.0 27.9
75–79 53.1 63.3 59.2 62.6 56.2 47.6
80–84 88.4 105.5 101.6 100.7 92.9 84.2
85+ 181.5 203.6 198.6 214.6 184.2 171.4
by sex distribution was carried out. In Fig. 1 and 2, kt time series are shown
calculated for men and women respectively.
Fig. 1. Male death rate (kt )
Fig. 2. Female death rate (kt )
As a result, the best model for estimation of kt on male death rate is ARIMA
(2,1,2) model:
∆kt = 0.023 · ∆kt−1 − 0.724∆kt−2 − 0.214 · εt−1 − 0.04 · εt−2 + 0.01 + εt
The best model for estimation of kt on female death rate is the ARIMA
(1,2,2) model:
∆2 kt = −0.381 · ∆2 kt−1 − 0.862 · εt−1 − 0.871 · εt−2 − 0.005 + εt
Both models were checked for lack of residual autocorrelation by Ljung–Box
Q-test and for normality of their distribution by Jarque–Bera test.
The Lee–Carter models designed for other countries have kt process as random-
walk process in terms of DS (I (1)) with a constant [6]. For the Republic of
Bashkortostan kt process serves as a framework for predictive model of the Lee–
Carter method, presents the actual process of DS (I (2)) with a constant and
AR component (AR (1)) for women and DS (I (1)) with a constant and AR
component (AR (2)) for men. This indicates clearly the difference of mortality
dynamics nature in the regions of Russia and in other countries.
Within the scope of ARIMA (2,1,2) and (1,2,2) models, forecasts for kt for
the period from 2015 to 2030 has been generated.
On the basis of the ARIMA (2,1,2) and (1,2,2) models, forecasts for kt for
the period from 2015 to 2030 were constructed (Tables 3 and 4).
5 Conclusion
The simulation observations point out persistent growth of mortality rate for
men in all five-year cohorts from 25 to 74 years and for women from 25 to 49
years. Noticeable reduction index follows in both genders in group of 85+ years.
Table 3. Forecasted death rates for male population from 2015 to 2025
Kohort 2015 2016 2019 2022 2025
0–1 9.7 9.5 8.8 8.2 7.6
1–4 0.3 0.3 0.2 0.2 0.1
5–9 0.3 0.3 0.3 0.3 0.3
10–14 0.4 0.4 0.4 0.4 0.4
15–19 1.7 1.7 1.7 1.8 1.8
20–24 4.0 4.0 4.1 4.2 4.3
25–29 6.1 6.2 6.4 6.7 6.9
30–34 8.6 8.8 9.3 9.8 10.4
35–39 10.0 10.1 10.6 11.1 11.6
40–44 12.4 12.6 13.0 13.5 14.1
45–49 15.7 15.9 16.4 16.9 17.4
50–54 21.4 21.6 22.2 22.8 23.4
55–59 28.7 29.0 29.7 30.5 31.2
60–64 40.5 40.8 41.8 42.9 43.9
65–69 52.5 52.8 53.6 54.5 55.4
70–74 71.5 71.7 72.4 73.1 73.8
75–79 100.3 100.5 101.0 101.5 102.0
80–84 137.7 137.6 137.2 136.8 136.4
85+ 206.7 204.8 199.6 194.4 189.4
Table 4. Forecasted death rates for female population from 2015 to 2025
Kohort 2015 2016 2019 2022 2025
0–1 5.3 4.9 3.9 3.1 2.4
1–4 0.6 0.6 0.5 0.5 0.5
5–9 0.2 0.1 0.1 0.1 0.1
10–14 0.2 0.2 0.1 0.1 0.1
15–19 0.6 0.6 0.5 0.5 0.5
20–24 0.9 0.9 0.9 0.9 0.9
25–29 1.6 1.6 1.8 2.0 2.2
30–34 2.7 2.8 3.3 4.0 4.8
35–39 3.0 3.1 3.4 3.8 4.3
40–44 3.5 3.6 3.8 4.0 4.2
45–49 4.5 4.5 4.6 4.6 4.7
50–54 6.1 6.1 6.1 6.1 6.1
55–59 9.6 9.6 9.7 9.8 9.9
60–64 13.7 13.7 13.7 13.7 13.7
65–69 20.3 20.3 20.1 20.0 19.8
70–74 31.1 30.6 29.3 27.9 26.5
75–79 56.3 56.3 56.1 55.9 55.7
80–84 93.1 92.9 92.1 91.4 90.5
85+ 179.9 178.5 174.3 169.8 165.1
In spite of the fact that positive changes were outlined in separate groups
of active working-age in recent years, the Lee–Carter model lets us see increase
in practically all cohorts. In case if trends of continued existence in the last
34 years remain, death rate only in children and advanced ages may decrease.
The projected values of age-specific mortality rates received by means of the
Lee–Carter model application indicate preserving of considerable inequality of
cohorts. The use of the Lee–Carter model for mortality forecasting makes it
possible to concentrate attention on specific problematic age groups, refraction
of situation in which allows avoiding evolvement of the negative scenario.
The medium-term projected perspective indicates continuance of male su-
permortality problem. However, the circumstances with female mortality rate
are not so positive either.
The results by applying the Lee–Carter model should be considered par-
ticularly in terms of Russia, its mortality factor is needed to be explained ex-
ceptionally. They have an effect of nonlinear dynamics of death rate due to
dramatic discontinuity during compound crisis in the 1990s. It is necessary to
mention that, in the first place, this break was observed among the population
at employable age. The common trend of infant and child mortality, declining
throughout 1990s made an impact on the results of modeling which indicated
continuation of this trend. It is noteworthy that the cohort of 30–34 years hap-
pens to be the highest possible death rates for both sexes. In accordance with
Russian research, the greatest contribution falls in the lost years of potential life
in some regions of Russia [10]. We have the opportunity to correlate the received
forecast results with the actual results for 2015. Predicted indices turned out to
be less optimistic.
There exists dozens of ways to consider the non-linearity of the perspective
dynamics of mortality, besides all of them contribute to reasonably accurate pre-
dictions of mortality and, respectively, the remaining life expectancy [13]. A lack
of the Lee–Carter model has been noted in the form of constant-rate of reducing
mortality, which leads to overestimation of the future level of mortality, espe-
cially in older age groups [4]. In our case, the Lee–Carter model pointed to an
underestimation of reduction of mortality which is related to the inconsistent
fluctuation of death rate. Due to the impact of downward dynamic observed na-
tionwide in the 1990s, there are restrictions on using methods of extrapolation
of death rate for the purpose of population forecasts and calculation of insur-
ance risks and decision-making in this field. The outcome for the Republic of
Bashkortostan, as for Russia in whole, indicates that there still remains a cru-
cial task of reducing economic losses and losses of human capital as a result of
high mortality of the working-age population, which will bring pressure upon
pension funds.
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