Eurostat/UNECE Work Session on Demographic Projections

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

2
Agenda

• Introduction and motivation
• Classical Lee-Carter mortality modelling
• Lee-Carter Model with "Limit Life Table
• Implementation issues
• Concluding remarks and further research

3
Mortality forecasting methods

• 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)

4
Extrapolative 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 whats the problem?
• ... using extrapolative methods is like driving
  a car through the rear mirror ...!

5
Extrapolative methods limitations

• 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

6
Classical AP Lee-Carter model

• 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

7
AP Lee-Carter model
8
AP LC Model with Limit Life Table

• 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

9
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

10
AP LC Model with Limit Life Table

• Age-Period demographic model
• with
• identification constraints
• GLM model of the response variable Dx,t with
  logarithmic link and non-linear parameterized
  predictor

11
AP LC Model with Limit Life Table

• 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

12
AP LC Model with Limit Life Table

• 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)

13
Implementation issues

• 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,...)

14
Implementation issues

• Duchêne and Wunsch (1988) hypothetical limit life
  table

15
Implementation issues

• 2nd Heligman-Pollard (1980) mortality law

16
Concluding remarks

• 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

17

• THANK YOU
• JORGE MIGUEL BRAVO
• (jbravo_at_uevora.pt)

Eurostat/UNECE Work Session on Demographic
Projections