Uncertainty of mortality projections
Preparation for ICA 2010 mortality session
Chresten Dengsøe
Uncertainty of mortality projections
Agenda
 Overview
-
Sources of uncertainty
 Uncertainty accumulation under random walk structure
-
Example: Lee-Carter model
-
Short-term and long-term uncertainty
 Incoherent vs. coherent mortality projections
-
Convergence of mortality levels
-
SAINT framework: short-term deviations from long-term trend
 ICA 2010
www.atp.dk
2
Uncertainty of mortality projections
Systematic and unsystematic variability
 Generic mortality model:
m( x, t ) = F (η ( x, t ), ε x ,t )
m(x,t) : realized death rate for age x in year t
η(x,t) : underlying death rate
εx,t
: ”measurement” error
F
: error-structure, e.g. additive or multiplicative
 Systematic and unsystematic variability:
-
variability in η(x,t) is independent of sample size
-
the size of εx,t depends on sample size
www.atp.dk
3
Uncertainty of mortality projections
Sources of uncertainty
Type
Model uncertainty
Lee-Carter, logistic, etc.
Parameter uncertainty
Improvement rates, drift etc.
Structural uncertainty
Stochastic projections
www.atp.dk
Assessment
Sensitivity analysis
Bayesian methods
Confidence intervals
Posterior distributions
Mean forecast with
pointwise confidence bands
4
Uncertainty of mortality projections
Uncertainty accumulation
 Random walk with drift
X t = X t −1 + µ + ε t ,
-
ε t ~ N (0, σ 2 )
used in most mortality models to describe the continued decrease
in death rates over time (the period effect), e.g.
-
the model by Lee and Carter (1992), and its variants
-
the class of models considered in Cairns et al. (2008)
-
parameter uncertainty (uncertainty in μ and σ2)
-
structural uncertainty (stochastic evolution due to εt)
 Instructive to study how parameter uncertainty and
structural uncertainty accumulate in these models
www.atp.dk
5
Uncertainty of mortality projections
Example: Lee-Carter
 The model proposed by Lee and Carter (1992) takes the form
log(m( x, t )) = a x + bx kt + ε x ,t
Systematic
variability
Unsystematic
variability
 To ensure identifiability the parameters are constrained by
∑b
x
x
=1
∑k
t
=0
t
 Estimates of the age-dependent constants (ax) are given by the
averages over time of log(m(x,t)), while (bx) and the time-varying
mortality index kt are obtained by SVD.
www.atp.dk
6
Uncertainty of mortality projections
Refinements
 Brouhns et al. (2002) consider the model
D( x, t ) ~ Poisson ( E ( x, t ) µ ( x, t )), µ ( x, t ) = exp(a x + bx kt )
-
with D the number of deaths, E the exposure and μ the force of mortality
-
introducing a statistical model allows joint MLE of ax, bx and kt
-
the mortality index, (kt), is modelled as in LC
 As in LC, the dynamics of the mortality index is estimated
as if (kt) were observed rather than estimated quantities
-
in principle joint estimation of all parameters – including those governing the
k-dynamics – is possible by MLE
-
Czado et al. (2005) describe a full Bayesian analysis using MCMC
www.atp.dk
7
Uncertainty of mortality projections
Modelling the mortality index
 The index is typically modelled as a random-walk with drift
kt = kt −1 + µ + ε t ,
t = 0,  , T
 Assuming εt i.i.d. N(0,σ2) the estimates and their distributions are
1
µˆ = ∑ ∆kt ~ N ( µ , T −1σ 2 )
T t
σ 2 =
1
 ) 2 ~ σ 2 (T − 1) −1 χ 2
(
∆
−
k
µ
∑ t
T −1
T −1 t
where Δkt = kt-kt-1 for t =1, …,T.
www.atp.dk
8
Uncertainty of mortality projections
Forecasting
 Forecasts are readily obtained from the expression
kT + h = kT + hµˆ + ε T +1 +  + ε T + h , E (kT + h | kT ) = kT + hµˆ
 95%-confidence intervals with and without uncertainty in µ̂
CI 95% (kT + h ) = kT + hµˆ ± 1.96 hσˆ 2 + h 2T −1σˆ 2
CI 95% (kT + h ) = kT + hµˆ ± 1.96 hσˆ 2
 Ignoring other sources of uncertainty these confidence intervals
directly translate into confidence intervals for m(x,t)
www.atp.dk
9
2%
1%
0.2%
0.5%
Death rate
5%
Uncertainty of mortality projections
1930
www.atp.dk
1950
1970
1990
2010
Year
2030
2050
2070
10
Uncertainty of mortality projections
Projection uncertainty under random walk structure
 Variance of forecasted log death rates:
[
Var (log m( x, T + h)) = bˆx2 Var (kT + h ) = bˆx2 hσˆ 2 + T −1h 2σˆ 2
structural
uncertainty
-
]
parameter
uncertainty
assuming for simplicity: Var (aˆ x ) = Var (bˆx ) = Var (σˆ 2 ) = Var (ε x ,t ) = 0
 RW structure implies that the same parameter (σ2) controls
-
uncertainty in annual improvements (short-term structural uncertainty)
-
size of accumulated deviations (long-term structural uncertainty)
-
uncertainty in average improvement (parameter uncertainty)
Short-term and long-term uncertainty cannot be distinguished
www.atp.dk
11
Uncertainty of mortality projections
Long-term uncertainty reflects short-term deviations
DK 95%-CI including drift uncertainty
US data
US 95%-CI including drift uncertainty
1%
2%
DK data
0.2%
0.5%
Death rate
5%
DK and US female mortality for ages 60 and 70
Similar mortality history in DK and US,
but projection uncertainty much higher in
DK due to higher short-term variability
1930
www.atp.dk
1950
1970
1990
2010
Year
2030
2050
2070
12
Uncertainty of mortality projections
Incoherent mortality projections
 Convergence of mortality levels
-
Similar mortality evolution in most developed countries, due to
similarities in socio-economic factors, lifestyle, level of treatment etc.
-
Mortality levels are likely to continue to evolve in parallel
with temporary country-specific deviations
 Incoherent mortality projections
-
Separate analyses exaggerate short-term differences
and lead to diverging projections
-
Seems highly implausible in the light of historic similarities
-
Example*: Austria, Australia, Belgium, Canada, Finland, France, Germany,
Holland, Denmark, Italy, Japan, Norway, Portugal, Switzerland, Spain, Sweden,
United Kingdom, United States
www.atp.dk
*Data from Human Mortality Database 13
Uncertainty of mortality projections
Divergent Lee-Carter projections from separate analyses
5%
International female mortality for ages 60 and 70
Denmark
US
UK
2%
Canada
Norway
Sweden
Belgium
1%
Death rate
Holland
Portugal
Germany
0.5%
Australia
Italy
Austria
Switzerland
0.2%
Finland
France
Spain
Japan
1950
www.atp.dk
1970
1990
2010
Year
2030
2050
2070
14
Uncertainty of mortality projections
2.5
2.0
1.5
0.5
1.0
Death rate/avg. death rate
Historic similarities are not preserved
1950
www.atp.dk
1970
1990
2010
Year
2030
2050
2070
15
Uncertainty of mortality projections
Large variation in projected mortality and levels of uncertainty
0.5%
0.2%
www.atp.dk
US
UK
SWE
SPA
SCH
POR
NOR
JAP
ITA
DK
HOL
GER
FRA
FIN
CAN
BEL
AUS
AU
0.05% 0.1%
Death rate
1%
2%
Mean forecast and 95%-CI of female mortality for age 70 in 2070
16
Uncertainty of mortality projections
Short-term deviations from long-term trend
 All countries appear to follow the same long-term trend
-
But improvements occur at different times in the individual countries
-
Variation in annual improvement rates differ between countries
 Separate analyses
-
Diverging projections with non-overlapping confidence intervals
-
Unreasonable variation in forecasting uncertainty
-
inability of RW to distingush between short- and long-term uncertainty
 Coherent mortality projections
-
Model common long-term trend
-
Allow country-specific short-term deviations from trend
www.atp.dk
17
Uncertainty of mortality projections
SAINT (Spread Adjusted InterNational Trend) framework
 Common international trend
-
Model of choice for μInt(x,t) estimated from pooled international data
 Model for country i:
log µi ( x, t ) = log µ Int ( x, t ) + yti ' rx
-
Multivariate, stationary time series model for yti
-
-
controls length and magnitude of deviations (spread)
The spread, log μi(x,t) - log μInt(x,t), is parameterized by rx
-
e.g. level, slope and curvature
 The framework is described in Jarner and Kryger (2008)
www.atp.dk
18
Uncertainty of mortality projections
Example
 Lee-Carter specification of μInt(x,t)
 Three-dimensional vector autoregressive model for yti
yt = Ayt −1 + b + ε t , ε t ~ N 3 (0, Ω)
 In this case the forecast variance takes the form
Var (log µi ( x, T + h)) = bˆx (hσˆ 2 + h 2 Var ( µˆ )) + Var ( yTi + h ' rx )
→ ∞ for h → ∞
→ ci < ∞ for h → ∞
-
First term is the Lee-Carter variance of the common long-term trend
-
Second term is the (bounded) variance of the country-specific spread
www.atp.dk
19
Uncertainty of mortality projections
Coherent projections and levels of uncertainty
2%
1%
0.2%
0.5%
Death rate
5%
International female mortality for ages 60 and 70
1950
www.atp.dk
1970
1990
2010
Year
2030
2050
2070
20
Uncertainty of mortality projections
ICA 2010 – uncertainty themes
 Sources of uncertainty
-
systematic, unsystematic, model, parameter and structural uncertainty
 Methods for assessing forecasting ability and uncertainty
-
”biological reasonableness”, bootstrap, fancharts etc.
 Model performance – backtesting techniques
 Uncertainty accumulation
 Coherent mortality projections
 Model selection
-
much effort is spent making the right analysis of the wrong models!
www.atp.dk
21
Uncertainty of mortality projections
References

H. Booth, J. Maindonald and L. Smith (2002). Applying Lee-Carter under conditions of varying mortality decline. Population Studies 56, 325-336

N. Brouhns, M. Denuit, J.K. Vermunt (2002). A Poisson log-bilinear regression approach to the construction of projected lifetables. IME 31, 373-393

A. Cairns (2000). A discussion of parameter and model uncertainty in insurance. IME 27, 313-330

A. Cairns, D. Blake, K. Dowd, G. Coughlan, D. Epstein and M. Khalaf-Allah (2008). Mortality density forecasts: An analysis of six stochastic
mortality models. Pensions Institute Discussion Paper PI-0801

A. Cairns, D. Blake, K. Dowd, G. Coughlan, D. Epstein and A. Ong (2007). A quantitative comparison of stochastic mortality models using data from
England & Wales and the United States. Pensions Institute Discussion Paper PI-0701

C. Czado, A. Delwarde and M. Denuit (2005). Bayesian Poisson log-bilinear mortality projections. IME 36, 260-284

S. Jarner and E. Kryger (2008). Modelling adult mortality in small populations: The SAINT model. Submitted for publication

S. Jarner, E. Kryger and C. Dengsøe (2008). The evolution of death rates and life expectancy in Denmark. SAJ 108, 147-173

M.-C. Koissi, A. Shapiro and G. Högnäs (2006). Evaluating and extending the Lee-Carter model for mortality forecasting: Bootstrap confidence
intervals. IME 38, 1-20

P. de Jong and L. Tickle (2006). Extending Lee-Carter mortality forecasting. Mathematical Population Studies 13, 1-18

R. Lee and L. Carter (1992). Modeling and forecasting of U.S. mortality. JASA 87, 659-675

R. Lee and T. Miller (2001). Evaluating the performance of the Lee-Carter method for forecasting mortality. Demography 38, 537-549

N. Li and R. Lee (2005). Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method. Demography 42, 575-594

S. Tuljapurkar, N. Li and C. Boe (2000). A universal pattern of mortality decline in the G7 countries. Nature 405, 789-792

C. Wilson (2001). On the scale of global demographic convergence 1950-2000. Population and Development Review 27, 155-171
www.atp.dk
22