1

An Extension of the Lee-Carter Model for

Mortality Projections

By

Udi Makov

,Shaul Bar-Lev, Yaser Awad ,

http://maf2008.unive.it/submit.php?id=134

2

Extensions 1:

Bayesian Trend choice

3

The Lee-Carter model :

),,(,

),,(,

2

1

2

,,,

w

iid

tttt

txtxtxxtx

oNwwkk

oFky

.year in age

for ratedeath central theof logarithm the:,

tx

y tx

schedule.mortality theof age across shape general:x

changes. ) (mortality of level

general when the ageat mortality of force

theof logarithm theof tendency theindicates:

t

x

k

x

. at timemortality of level in thetion varia

thedescribest index thaan is:

t

kt

4 4

Estimation Methodology

The methodology is based on the following steps:

1. Fixing appropriate prior distributions for the unknown

parameters

2. Using the Kalman Filter algorithm to estimate the

predicted values of the time series process, including the

variance of the prediction

3. Using the Gibbs sampler to estimate the posterior

distributions of the unknown parameters, including the

parameters of the unobserved time series index, whose values

were estimated by the Kalman Filter algorithm.

5

The Lee Carter model as a state-space model

Equations below are describing the state process model and the

observation model as follow:

equationt measuremen

equation state

,

,1

ttt

ttt

ky

wkk

),0(~ 2IN p

iid

t

),0(~ 2w

iid

t Nw We assumed the process noise is

and the measurement noise is

6

General Presentation of time trend

model

t

wwkk

t

tttttt

21

111

7

where

,,

1

1

,

000

00

0

00

2

1

max

min

t

t

XP

000

00

0

00

In vectorial presentation

wIXkPXk )()()(

),,,( max1minmin kkkk ),,,(

),,,( max1minmin wwww

8

where

))(,(~ 12 UQXNk kT

100

1

010

01

01

)()(

2

2

2

2

PIPIQ

100

1

010

01

01

2

2

2

2

U

9

Extensions

2006) Pedroza, 1992; Carter, & (Lee

0)ARIMA(0,1,0,0,1 : 211 Model

),0(~with , 21 wtttt Nwwkk

),(~ 112

QNk wT

Then the resulting distribution

is:

10

2006) al.,et (Czado trend

linear with 0)ARIMA(1,0,0,0 :2 Model

Then the resulting distribution

is:

),(~ 12 QXNk kT

t

wkk

t

ttttt

21

11

11

growth with smoothing lexponentia Simple

1)ARIMA(0,1,0,1 : 213 Model

Then the resulting distribution

is:

),(~ 112

UQNk wT

11 tttt wwkk

12

ndlinear tre without 2 case As

0)ARIMA(1,0,0,0 : 214 Model

Then the resulting distribution

is:

),(~ 12 QNk wT

ttt wkk 1

13

1)ARIMA(1,0,0 : 215 Model

Then the resulting distribution

is:

),(~ 12 UQNk wT

11 tttt wwkk

14

Model Choice

) et al., nd, (Czadolinear tre

) with ,,ARIMA(,φθ

) ; Pedroza,er, (Lee& Cart

),,ARIMA(

2005

00100

20061992

0100,0,1 21

:2 Model

:1 Model

wth with grosmoothing onential Simple

),,ARIMA(γ,γρ

exp

11001 21 :3 Model

dinear tren without lAs case

),,ARIMA(,γγ

2

0010021 :4 Model

) ,,ARIMA(γ γ 101021 :5 Model

15

Some data consideration We analyze the data for U.S and Ireland male. The

data obtained from the Human Mortality Database .

The data consists of annual age-specific death rates for

years 1959-1999.

The age group for U.S are 0,1-4,5-9,…,105-109,110+.

The age group for Ireland are 0,1-4,5-9,…,105-109.

Therefore; for U.S (P=24, T=6) , for Ireland (P=23,

T=6), that is 1958-1989 grouped by 5 years.

16

The distribution of the mortality rates by age groups and time

U.S Male

Smrlo'10 an International Symposium on February 8-11, 2010

17

The distribution of the mortality rates by age groups and time

Ireland Male

Smrlo'10 an International Symposium on February 8-11, 2010

18

Prior distribution

22

11

02

012

2

1

2

2

22

22

0

0,~

),(~),,(~

11int),0(~

),(~),,0(~

),(~),,(~

a

aN

baGammabaGamma

),erval (-to the truncated N

baGammaIN

baGammaIaN

ww

p

xxp

2-w

19

Bayesian Model Choice For models comparison we estimated the Deviance

information criterion – DIC for normal likelihood.

smaller values for “better” models. (Spiegelhalter, et

al. 2002).

where

of valuessimulated theof average sample the

parameters ofnumber effective the

))/(log(2)(

:

2)(

p

yfD

pDDDIC

We used the data of 1958-1989 to estimate the models

parameters and the DIC index.

20 20

Bayesian Model Choice

The DIC indices for all five models using U.S male data are as follows:

DIC[ model 1:ARIMA(0,1,0)]= -149.25

DIC[ model 2:ARIMA(1,0,0)]= -154.26

DIC[ model 3:ARIMA(0,1,1)]= -146.55

DIC[ model 4:ARIMA(1,0,0)]= -108.96

DIC[ model 5:ARIMA(1,0,1)]= -104.96

The DIC indices for all five models using Irish male data are as follows:

DIC[ model 1:ARIMA(0,1,0)]= -130.07

DIC[ model 2: ARIMA(1,0,0)]= -141.77

DIC[ model 3: ARIMA(0,1,1)]= -123.91

DIC[ model 4: ARIMA(1,0,0)]= -88.78

DIC[ model 5: ARIMA(1,0,1)]= -87.35

21

Then we used the data from 1990-1999 to assess the

predictive quality of the models and we compared

between them using also graphical presentation

We used the data of 1958-1989 to estimate the models

parameters.

22 22

Comparison between model 1 and model 2

U.S Male

0.18

0.2

0.22

0.24

0.26

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999

de

ath

ra

te

Forecast Year

AGE group 90-94

observed model 1 model 2

23 23

Comparison between model 1 and model 2

Ireland Male

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999

de

ath

ra

te

Forecast Year

AGE group 90-94

observed model 1 model 2

24

Extensions 2:

Mortality Projections Simultaneous for

Multiple populations

25

26

Exchangeable sequence of random

variables

Formally, an exchangeable sequence of random variables is a finite or infinite sequence

X1, X2, X3, ... of random variables such that for any finite permutation σ of the indices 1, 2, 3, ..., i.e. any permutation σ that leaves all but finitely many indices fixed, the joint probability distribution of the permuted sequence

is the same as the joint probability distribution of the original sequence.

27

iD

iK i i

1 2, , i k

(1) 2

(1)

a (1)

b

(2) 2

(2)

a (2)

b

28

Exchangeability between parameters: s

Suppose we have mortality projection for m countries and we believe the

time trends parameters belong to a joint distribution

Suppose are i.i.d variables, with

),,,( 21 m

0,),,(),/( jbNbj

The joint distribution of is: s

m

j

j bb1

),/(),/(

Suppose and , where assumed

known (maybe chosen to reflect vague prior information)

),(~ 2bbNb ),(~ caIG cabb ,,,2

29 29

REPLACE: Sampling from

TT

kkNkK wnw

2

02

0 ;),,/(

30 30

By the following posterior distributions, for m=2:

TTTTb

T

kkNb

wwwwT

2

1,

2

1,

2

1,

2

1,1,01,*

1 ;),,/(

TTTTb

T

kkNb

wwwwT

2

2,

2

2,

2

2,

2

2,2,02,*

2 ;),,/(

The rest of the posterior distributions should stay the same as before for

each country separately.

Where *2,

2

1,,01,0

1 },,,,,,,,{ mwwmm kkKK

Similarly; we assumed Exchangeability on other parameters

of the model: Alpha and Beta

31

Validation

• We used the data from U.S and Ireland male of

1958-1989 to estimate the models parameters

• Then we used the data from 1990-1999 to

assess the predictive quality of the models and

we compared between them using also

graphical presentation

32

Exchangeability w.r.t Ө

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0

theta

0

1

2

3

theta_Ex

theta_Usa_Ex

theta_Ir_Ex

theta_Usa

theta_Ir

Figure 1: Posterior dis. of theta : USA and Ireland

33

Exchangeability w.r.t α

-1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0

alpha

0

2

4

6

8 alpha.Ex

alpha.Ir

alpha.Ir.Ex

alpha.Usa.Ex

alpha.Usa

Figure 3: Posterior dis. of alpha, age-group 105-109: Usa and Ireland

34

Exchangeability w.r.t β

-1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

beta

0

4

8

12

beta.ex

beta.Usa

beta.Usa.ex

beta.Ir.Ex

beta.Ir

Figure 5: Posterior dis. of beta, age-group 105-109: Usa and Ireland

35

Impact on α & β

age alpha_Ir alpha_Ir_Ex beta_ir beta_ir_Ex alpha_Usa alpha_Usa_Ex beta_Usa beta_Usa_Ex

0 -4.032 -4.034 0.313 0.314 -3.991 -3.995 0.145 0.151

1-4 -7.141 -7.140 0.200 0.199 -7.121 -7.124 0.102 0.103

5-9 -7.822 -7.816 0.152 0.152 -7.807 -7.801 0.108 0.109

10-14 -7.969 -7.961 0.079 0.079 -7.730 -7.736 0.076 0.078

70-74 -2.778 -2.787 0.011 0.007 -2.917 -2.911 0.047 0.045

75-79 -2.352 -2.357 0.007 0.005 -2.544 -2.533 0.043 0.039

80-84 -1.922 -1.930 0.009 0.012 -2.128 -2.121 0.032 0.031

85-89 -1.492 -1.507 0.022 0.022 -1.730 -1.721 0.028 0.028

90-94 -1.156 -1.163 0.010 0.014 -1.371 -1.361 0.021 0.023

95-99 -0.735 -0.748 0.008 0.006 -1.063 -1.041 0.008 0.007

100-104 -0.644 -0.661 -0.063 -0.062 -0.937 -0.931 -0.029 -0.030

105-109 -0.740 -0.749 -0.168 -0.173 -1.032 -1.019 -0.063 -0.064

110+ -0.683 -0.704 -0.112 -0.114 -1.258 -1.231 -0.060 -0.062

36

Impact on α & β

age beta_ir beta_ir_Ex alpha_Usa alpha_Usa_Ex

105-109 -0.168 -0.173 -1.032 -1.019

37

Impact on k(t)

year=t k(t)-Usa k(t)-Usa-Ex k(t)-Ir k(t)-Ir-Ex

60-64 0.698 0.683 1.259 1.277

65-69 1.315 1.303 0.979 0.973

70-74 1.173 1.161 0.432 0.425

75-79 0.119 0.116 -0.067 -0.074

80-84 -1.145 -1.134 -0.835 -0.833

85-89 -2.160 -2.130 -1.768 -1.768

theta -0.204 -0.187 -0.189 -0.194

38

Impact on Ө

k(t)-Usa k(t)-Usa-Ex k(t)-Ir k(t)-Ir-Ex

theta -0.204 -0.187 -0.189 -0.194

39

Impact on RMSE

Mortality at age 105

0.2755 USA

0.2705 USA Ex

1.2750 Ireland

1.0401 Ireland Ex

Improvement

1.8%

Improvement

18.4%

40

Thank you