EurostatUNECE Work Session on Demographic Projections LeeCarter Mortality

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 © Jorge Miguel Bravo 1

Agenda 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 © Jorge Miguel Bravo 2

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) © Jorge Miguel Bravo 3

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 what’s the problem? “. . . using extrapolative methods is like driving a car through the rear mirror. . . !” © Jorge Miguel Bravo 4

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 – © Jorge Miguel Bravo produce increasing divergence in life expectancy 5

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 © Jorge Miguel Bravo 6

AP Lee-Carter model © Jorge Miguel Bravo 7

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 © Jorge Miguel Bravo 8

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 © Jorge Miguel Bravo 9

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 © Jorge Miguel Bravo 10

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 loglikelihood function with © Jorge Miguel Bravo 11

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) © Jorge Miguel Bravo 12

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, . . . ) © Jorge Miguel Bravo 13

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

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

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” – … © Jorge Miguel Bravo 16

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