1. © 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, 29 th April 2010

2. © Jorge Miguel Bravo 2 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

3. © Jorge Miguel Bravo 3 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

4. © Jorge Miguel Bravo 4 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

5. © Jorge Miguel Bravo 5 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

6. © Jorge Miguel Bravo 6 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 (k t ) Problem: Asymptotic behavior of the model Classical AP Lee-Carter model

7. © Jorge Miguel Bravo 7 AP Lee-Carter model

8. © Jorge Miguel Bravo 8 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”

9. © Jorge Miguel Bravo 9 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 AP LC Model with “Limit Life Table”

10. © Jorge Miguel Bravo 10 Age-Period demographic model with identification constraints GLM model of the response variable D x,t with logarithmic link and non-linear parameterized predictor AP LC Model with “Limit Life Table”

11. © Jorge Miguel Bravo 11 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 AP LC Model with “Limit Life Table”

12. © Jorge Miguel Bravo 12 Because of the log-bilinear term  x k t 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 (k t ) AP LC Model with “Limit Life Table”

13. © Jorge Miguel Bravo 13 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

14. © Jorge Miguel Bravo 14 Duchêne and Wunsch (1988) hypothetical limit life table Implementation issues

15. © Jorge Miguel Bravo 15 2 nd Heligman-Pollard (1980) mortality law Implementation issues

16. © Jorge Miguel Bravo 16 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

17. © Jorge Miguel Bravo 17 THANK YOU JORGE MIGUEL BRAVO (jbravo@uevora.pt) Eurostat/UNECE Work Session on Demographic Projections