Mortality Model for Multi-Populations: A Semiparametric Comparison Approach Lei Fang Wolfgang Karl Härdle Juhyun Park Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. – Center for Applied Statistics and Economics Humboldt–Universität zu Berlin http://lvb.wiwi.hu-berlin.de http://www.case.hu-berlin.de
Motivation 1-1 Demographic Risk ⊡ Low mortality, low fertility, global aging trend ⊡ Mortality rate is the key to insurance and pension industry Mortality Model for Multi-Populations
Motivation 1-2 Demographic key element: mortality ⊡ Mortality rate: number of death/number of exposure, taken as the log transformation ⊡ Mortality rate: age-specific, male and female, (region-specific) ⊡ Mortality change is more "stable" compared to fertility Note: In following graphs, rates in different years are plotted in rainbow palette so that the earliest years are red and so on. Mortality Model for Multi-Populations
Motivation 1-3 Demographic Risk in Japan 1947 2 0 Log death rate −2 −4 −6 −8 −10 0 20 40 60 80 100 Age Figure 1: Japan female mortality trend: 1947-2012 Mortality Model for Multi-Populations
Motivation 1-4 Demographic Risk in Japan 1947 0.30 0.25 0.20 Fertility rate 0.15 0.10 0.05 0.00 20 30 40 50 Age Figure 2: Japan fertility trend: 1947-2012 Mortality Model for Multi-Populations
Motivation 1-5 Demographic risk in China ⊡ Small sample size: 17 years ⊡ Aging trend is inevitable ⊡ Regional similarities between Japan and China Mortality Model for Multi-Populations
Motivation 1-6 Demographic Risk in China 1994 2 0 Log death rate −2 −4 −6 −8 −10 0 20 40 60 80 Age Figure 3: China female mortality trend: 1994-2010, Japan’s historical fe- male mortality is displayed as grey zone. Mortality Model for Multi-Populations
Motivation 1-7 Demographic Risk in China 1994 2 0 Log death rate −2 −4 −6 −8 −10 0 20 40 60 80 Age Figure 4: China male mortality trend: 1994-2010, Japan’s historical male mortality is displayed as grey zone. Mortality Model for Multi-Populations
Motivation 1-8 Literature Mortality Similarity ⊡ Hanewald (2011): The Lee-Carter mortality index k t correlates significantly with macroeconomic fluctuations in some periods Semiparametric Comparison Model ⊡ Härdle and Marron (1990): Semiparametric comparison of regression curves ⊡ Grith et al. (2013): Shape invariant model Mortality Model for Multi-Populations
Motivation 1-9 Multi-Population Mortality Modeling China ⊡ Is there mortality similarity between China and Japan? ⊡ How can the mortality modeling and forecasting be improved via Japan? Multi-Countries ⊡ How do we generate a multi-population mortality model based on the common shape? Mortality Model for Multi-Populations
Outline 1. Motivation � 2. Classic mortality models 3. Semiparametric comparison model 4. Empirical analysis 5. Reference
Classic mortality models 2-1 Lee-Carter (LC) Method ⊡ A benchmark in demographics: Lee and Carter (1992) ⊡ Idea: use SVD to extract a single time-varying index of mortality/fertility rate level ⊡ Take mortality for analysis: log { y t ( x ) } = a x + b x k t + ε x , t ◮ y t ( x ) observed mortality rate at age x in year t ◮ a x age pattern averaged across years ◮ b x first PC reflecting how fast the mortality changes at each age ◮ k t time-varying index of mortality level ε x , t residual at age x in year t ◮ Mortality Model for Multi-Populations
Classic mortality models 2-2 Hyndman-Ullah (HU) Method ⊡ Variant of LC method: presmooth, orthogonalize, forecast ⊡ Estimate the smooth functions s t ( x ) through the data sets { x , y t ( x ) } for each t : y t ( x ) = s t ( x ) + σ t ( x ) ε t ◮ s t ( x ) smooth function ◮ σ t ( x ) smooth volatility function of y t ( x ) ◮ ε t i.i.d. random error Mortality Model for Multi-Populations
Classic mortality models 2-3 Hyndman-Ullah (HU) Method Use functional principal component analysis (FPCA) K � s t ( x ) = µ ( x ) + β t , k φ k ( x ) + e t ( x ) k = 1 ⊡ µ ( x ) mean of s t ( x ) across years ⊡ φ k ( x ) orthogonal basis functional PCs ⊡ β t , k uncorrelated PC scores ⊡ e t ( x ) is residual function with mean zero Mortality Model for Multi-Populations
Classic mortality models 2-4 Mortality Analysis Figure 5: China’s female mortality decomposition by HU Method: yellow areas represent the 95% confidence intervals for the coefficients forecast. Mortality Model for Multi-Populations
Semiparametric comparison model 3-1 Mortality trends comparison ⊡ Time-varying indicator k t derived from Lee-Carter model presents similar pattern. 150 100 50 Kt 0 −50 −100 1950 1960 1970 1980 1990 2000 2010 Time Figure 6: China mortality trend (short curves) vs. Japan mortality trend (long curves): female, male. Mortality Model for Multi-Populations
Semiparametric comparison model 3-2 Semiparametric comparison model two-country case Take China and Japan for example ⊡ Use k t derived from LC model log { y t ( x ) } = a x + b x k t + ε x , t , (1) ⊡ Infer China’s mortality trend via Japan’s trend � t − θ 2 � k c ( t ) = θ 1 k j + θ 4 , (2) θ 3 k c ( t ) is the time-varying indicator for China ◮ k j ( t ) is the time-varying indicator for Japan ◮ θ = ( θ 1 , θ 2 , θ 3 , θ 4 ) ⊤ are shape deviation parameters ◮ Mortality Model for Multi-Populations
Semiparametric comparison model 3-3 Model estimation ⊡ Estimation procedure � 2 � � u − θ 2 � � k c ( u ) − θ 1 ˆ ˆ min k j − θ 4 w ( u ) du , (3) θ 3 θ t c k c ( t ) and ˆ ˆ ◮ k j ( t ) are the nonparametric estimates of the original time-varying indicators, t c is the China data’s time interval ◮ the comparison region satisfies the condition � w ( u ) = 1 [ a , b ] { ( u − θ 2 ) /θ 3 } , tj where tj is the time interval of Japan’s mortality data, a ≥ inf ( tj ) and b ≤ sup ( tj ) . Mortality Model for Multi-Populations
Semiparametric comparison model 3-4 Algorithm ⊡ Iterate based on the scheme (3) ⊡ Set up the prior estimates θ 0 = ( θ 0 4 ) ⊤ and 1 , θ 0 2 , θ 0 3 , θ 0 the nonparametric estimates of ˆ k c ( t ) and ˆ k j ( t ) ⊡ Update ( θ 1 , θ 2 , θ 3 , θ 4 ) ⊤ ⊡ Reach convergence Mortality Model for Multi-Populations
Semiparametric comparison model 3-5 Semiparametric comparison model multi-country case ⊡ k i ( t ) is a derived time-varying mortality indicator for country i , with i ∈ { 1 , ..., n } , n = 36 stands for 36 countries. ⊡ The curves can be represented in the form � t − θ i 2 � k i ( t ) = θ i 1 k 0 + θ i 4 , (4) θ i 3 k i ( t ) is the time-varying indicator for country i ◮ k 0 ( t ) is a reference curve, understood as common trend ◮ θ = ( θ i 1 , θ i 2 , θ i 3 , θ i 4 ) ⊤ are shape deviation parameters ◮ Mortality Model for Multi-Populations
Semiparametric comparison model 3-6 Estimation of Common Trend ⊡ Synchronization k i ( θ i 3 t + θ i 2 ) = θ i 1 k 0 ( t ) + θ i 4 , (5) ⊡ Identification conditions (normalize) N N � � T − 1 θ i 1 = T − 1 θ i 3 = 1 , (6) i = 1 i = 1 N N � � T − 1 θ i 2 = T − 1 θ i 4 = 0 (7) i = 1 i = 1 ⊡ Common trend curve N k 0 ( t ) = T − 1 � k t ( θ i 3 t + θ i 2 ) (8) i = 1 Mortality Model for Multi-Populations
Semiparametric comparison model 3-7 Initial Value and Algorithm ⊡ Choose a group of countries with bigger sample size and set their average curve k 0 ( t ) as initial reference curve ⊡ Repeat the iteration of two-country case and generate initial θ 0 for the other countries ⊡ Get the common trend based on formula (8) ⊡ Iterate based on the above procedures ⊡ Update ( θ 1 , θ 2 , θ 3 , θ 4 ) ⊤ ⊡ Reach convergence Mortality Model for Multi-Populations
Empirical results 4-1 Demographic Data ⊡ China Mortality: age-specific (0,90+), male and female Years: 1994-2010 Data Source: China Statistical Year Book ⊡ The other 35 countries Mortality: age-specific (0,110+), male and female Extracted ages: (0,90) Years: it differs from 14 years (Chile) to 261 years (Sweden) Data Source: Human Mortality Database Mortality Model for Multi-Populations
Empirical results 4-2 Mortality trends comparison ⊡ Intuitive comparison: time delay between China and Japan female mortality trend. 150 100 50 Kt 0 −50 −100 1950 1960 1970 1980 1990 2000 2010 Time Figure 7: Japan trend, Japan smoothed trend, China trend and China smoothed trends of no-delay, 20-, 23- and 25- year delay respectively. Mortality Model for Multi-Populations
Empirical results 4-3 Understanding θ θ = ( θ 1 , θ 2 , θ 3 , θ 4 ) ⊤ = ( 1 , θ 2 , 1 , θ 4 ) ⊤ ⊡ θ 1 is the general trend adjustment, possibly selected as 1. ⊡ θ 2 is the time-delay parameter ⊡ θ 3 is the time acceleration parameter, possibly selected as 1. ⊡ θ 4 is the vertical shift parameter 150 150 100 100 50 50 Kt Kt 0 0 −50 −50 −100 −100 1950 1960 1970 1980 1990 2000 2010 1950 1960 1970 1980 1990 2000 2010 Time Time Figure 8: Time delay θ 2 = 23 Figure 9: Vertical shift θ 4 = − 85 Mortality Model for Multi-Populations
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