Eurostat/UNECE Work Session on Demographic Projections Lee-Carter Mortality Projection

© Jorge Miguel Bravo 1

Eurostat/UNECE Work Session on Demographic Projections

Lee-Carter Mortality Projection

with "Limit Life Table"

Jorge Miguel Bravo

University of Évora (Deparment of Economics) and CEFAGE -UE

Lisbon, Portugal, 29th April 2010


1. Introduction and motivation

2. Classical Lee-Carter mortality modelling

3. Lee-Carter Model with "Limit Life Table“

4. Implementation issues

5. Concluding remarks and further research

Agenda


• Mortality forecasting methods currently can be classified into

1. Explanatory methods

Based on structural or causal epidemiological models, analyze the

relationship between age-specific risk factors (e.g., smoking) and

mortality rates

2. Expert-opinion based methods

Involve the use of informed expectations about the future,

alternative low/high scenarios or a targeting approach

3. Extrapolative methods

Assume that future mortality patterns can be estimated by

projecting into the future trends observed in the recent to

medium-term past (e.g., Lee-Carter, APC methods)

Mortality forecasting methods


• Why do we use extrapolative methods? Because...

– of the inherent complexity of the factors affecting

human mortality

– of the current lack of understanding of the intricate

mechanisms governing the aging process

– of the relative stability of the past demographic trends

– they offer a reliable basis for projection

• So what’s the problem?

“... using extrapolative methods is like driving a car through

the rear mirror ...!”

Extrapolative methods


• Since the methods rely on the assumption that future

mortality trends will continue into the future as observed in

the past, they may

– generate biologically implausible scenarios (e.g., null

mortality rates for all ages)

– produce implausible age patterns

Crossover of consecutive mortality rates

Crossover of male/female life expectancy

– produce increasing divergence in life expectancy

Extrapolative methods: limitations


• Age-Period demographic model

• Identification constraints

• Fitting method: OLS by Singular Value Decomposition (SVD)

• Forecasting: Age effects (x and x) are assumed constant + a time

series ARIMA (p,d,q) model for the time component (kt)

• Problem: Asymptotic behavior of the model

Classical AP Lee-Carter model

2

, , ,ln , (0, )x t x x t x t x tm k N

max max

min min

1, 0x t

x tx x t t

k

,ˆ ˆˆˆlim lim exp ˆ 0x t x x tk k

m k


AP Lee-Carter model


• Basic idea (Bravo, 2007)

– There is a “target” life table to which longevity improvements

over time (over a projection horizon) converge

– We explicitly admit that there are (at least in a limited time

horizon) natural limits to longevity improvements

• Rationale

– there is a decline in the physiological parameters associated

with ageing in humans duration of life is limited?

– stylized facts: slowdown in life expectancy at birth increases

observed in many developed countries

AP LC Model with “Limit Life Table”


• Hip. 1: the age-specific forces of mortality are constant within

each rectangle of the Lexis diagram

• Hip. 2: Let denote the instantaneous death rate or

probability of death corresponding to this “target” life table

• Hip. 3: AP LC model is formulated within a Generalized Linear

Model (GLM) framework with a generalized error distribution

age- and period-specific numbers of deaths are independent

realizations from a Poisson distribution with parameters


lim lim,x xq

, , , for 0 , 1x t x t

, , , , ,, x t x t x t x t x tE D E Var D E D


• Age-Period demographic model

with

identification constraints

• GLM model of the response variable Dx,t with logarithmic link and

non-linear parameterized predictor


lim

, ,

ad

x t x x t

max max

min min

1, 0x t

x tx x t t

k

, expad

x t x x tk

lim

, ,log logx t x t x x x tE k


• Fitting method: ML methods with theory-based distributional

assumptions instead of empirical measures (i.e., OLS)

• Parameter estimates are obtained by maximizing the log-likelihood

function

with


,max max

min min

max max

min min

, ,

,

lim, ,

exp( , , ) ln

!

ln( exp ) exp( )

x tdx tx t x t

x x tx x t t x t

x t

x t x x x t x t x x tx x t t

L kd

d k E k c

lim, , , expx t x t x t x x x tE D E k


• Because of the log-bilinear term xkt we cannot use standard

statistical packages that include GLM Poisson regression

• Solution: Use an iterative algorithm for estimating log-bilinear

models developed by Goodman (1979) based on a Newton-

Raphson algorithm Updating-scheme

• Adjust parameter estimates to meet identification constraints

• Forecasting: Age effects (x and x) constant and a time series

ARIMA (p,d,q) model for the time component (kt)


( )

( 1) ( )2 ( )

2

( , , )

ˆˆ ˆ ˆ ˆˆ, , ,( , , )

vx x t

jv vj j x x tv

x x t

j

L k

kL k


• We need a “limit/target” life table as input subjective/informed

assumptions about the future development of a set of important

biological, economic and social variables have to be made

• Alternative approaches

– Use an epidemiological model to define the target life table (TLT)

– Consider the life table of a more advanced population as TLT

– Use the observed gaps between countries and regions

– combination of the lowest mortality rates observed by sex-age groups

– estimates of the lowest achievable cause-specific death rates

– Calibrate some mortality law to express different scenarios on the main

trends in human longevity (e.g., rectangularization survival curve, life

expectancy trends, median, mode, entropy, IQR,...)

Implementation issues


• Duchêne and Wunsch (1988) hypothetical limit life table


1lim

0

, exp

95, 14.40198275, 0.002, 115

x x x

x xl l

q

Idade

l(x)/l(0)

0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0


• 2nd Heligman-Pollard (1980) mortality law


( ) 2exp (ln ln )1

Cx

x Bx x

GHq A D E x F

KGH

Age

ln(mux)

0 20 40 60 80

-10

-8-6

-4-2

Heligman-Pollard2007


• The asymptotic behaviour of the AP LC is unsatisfactory

• We argue that a combination of expert-opinion and extrapolative

methods can be used to forecast mortality rates within the Lee-

Carter framework limit/target life table

• The key implementation is the definition of the “target” life table

• Future research

– Experiment with alternative parameterizations of the GLM

demographic model (e.g., age-specific rates of convergence)

– Consider cohort-specific “targets”

– Consider gender-specific “targets”

– …

Concluding remarks


THANK YOUJORGE MIGUEL BRAVO

([email protected])

Eurostat/UNECE Work Session on Demographic Projections

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Eurostat/UNECE Work Session on Demographic Projections Lee-Carter Mortality Projection