An Extension of the Lee-Carter Model for Mortality Projections...3 The Lee-Carter model : , ( , ),,...
Transcript of An Extension of the Lee-Carter Model for Mortality Projections...3 The Lee-Carter model : , ( , ),,...
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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
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Extensions 1:
Bayesian Trend choice
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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
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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.
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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
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General Presentation of time trend
model
t
wwkk
t
tttttt
21
111
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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
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where
))(,(~ 12 UQXNk kT
100
1
010
01
01
)()(
2
2
2
2
PIPIQ
100
1
010
01
01
2
2
2
2
U
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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:
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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
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growth with smoothing lexponentia Simple
1)ARIMA(0,1,0,1 : 213 Model
Then the resulting distribution
is:
),(~ 112
UQNk wT
11 tttt wwkk
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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
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1)ARIMA(1,0,0 : 215 Model
Then the resulting distribution
is:
),(~ 12 UQNk wT
11 tttt wwkk
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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
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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.
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The distribution of the mortality rates by age groups and time
U.S Male
Smrlo'10 an International Symposium on February 8-11, 2010
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The distribution of the mortality rates by age groups and time
Ireland Male
Smrlo'10 an International Symposium on February 8-11, 2010
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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
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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.
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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
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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.
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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
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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
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Extensions 2:
Mortality Projections Simultaneous for
Multiple populations
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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.
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iD
iK i i
1 2, , i k
(1) 2
(1)
a (1)
b
(2) 2
(2)
a (2)
b
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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
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REPLACE: Sampling from
TT
kkNkK wnw
2
02
0 ;),,/(
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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%
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Thank you