Longevity and Mortality Models Andrs M. Villegas white School of - PowerPoint PPT Presentation

Lee-Carter model Australia: male death rates (2014) 0 −2 log death rates −4 −6 −8 −10 0 20 40 60 80 100 age N � β ( i ) x κ t ( i ) β (0) log µ xt = + + α x x γ t − x i =1

Lee-Carter model Australia: male death rates (1921−2014) 0 −2 log death rates −4 −6 −8 −10 0 20 40 60 80 100 age N � β ( i ) x κ t ( i ) β (0) log µ xt = + + α x x γ t − x i =1

Lee-Carter model Australia: male death rates (1921−2014) 0 −2 log death rates −4 −6 −8 −10 0 20 40 60 80 100 age N � β ( i ) x κ t ( i ) β (0) log µ xt = + + α x x γ t − x i =1

Lee-Carter model Australia: male death rates (1921−2014) 0 −2 log death rates −4 −6 −8 −10 0 20 40 60 80 100 age N � β ( i ) x κ t ( i ) β (0) log µ xt = + + α x x γ t − x i =1

Lee-Carter model Australia: male death rates (1921−2014) 0 −2 log death rates −4 −6 −8 −10 0 20 40 60 80 100 age N � β ( i ) x κ t ( i ) β (0) log µ xt = + + α x x γ t − x i =1

Lee-Carter model α x vs. x −2 −4 −6 −8 0 20 40 60 80 100 age ( 1 ) vs. x ( 1 ) vs. t β x κ t 20 0.010 0 −40 0.000 0 20 40 60 80 100 1970 1980 1990 2000 2010 age year

Cairns-Blake-Dowd model EW: death probability male (1961) 0 −1 −2 log qx −3 logit q xt = κ (1) x ) κ (2) t +( x − ¯ −4 t −5 −6 50 60 70 80 90 100 age ( 1 ) vs t ( 2 ) vs t κ t κ t −2.4 0.105 0.100 −2.8 0.095 −3.2 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 year year

Cairns-Blake-Dowd model EW: death probability male (1962) 0 −1 −2 log qx −3 logit q xt = κ (1) x ) κ (2) t +( x − ¯ −4 t −5 −6 50 60 70 80 90 100 age ( 1 ) vs t ( 2 ) vs t κ t κ t −2.4 0.105 0.100 −2.8 0.095 −3.2 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 year year

Cairns-Blake-Dowd model EW: death probability male (1963) 0 −1 −2 log qx −3 logit q xt = κ (1) x ) κ (2) t +( x − ¯ −4 t −5 −6 50 60 70 80 90 100 age ( 1 ) vs t ( 2 ) vs t κ t κ t −2.4 0.105 0.100 −2.8 0.095 −3.2 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 year year

Cairns-Blake-Dowd model EW: death probability male (1964) 0 −1 −2 log qx −3 logit q xt = κ (1) x ) κ (2) t +( x − ¯ −4 t −5 −6 50 60 70 80 90 100 age ( 1 ) vs t ( 2 ) vs t κ t κ t −2.4 0.105 0.100 −2.8 0.095 −3.2 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 year year

Cairns-Blake-Dowd model EW: death probability male (1965) 0 −1 −2 log qx −3 logit q xt = κ (1) x ) κ (2) t +( x − ¯ −4 t −5 −6 50 60 70 80 90 100 age ( 1 ) vs t ( 2 ) vs t κ t κ t −2.4 0.105 0.100 −2.8 0.095 −3.2 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 year year

Cairns-Blake-Dowd model EW: death probability male (1970) 0 −1 −2 log qx −3 logit q xt = κ (1) x ) κ (2) t +( x − ¯ −4 t −5 −6 50 60 70 80 90 100 age ( 1 ) vs t ( 2 ) vs t κ t κ t −2.4 0.105 0.100 −2.8 0.095 −3.2 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 year year

Cairns-Blake-Dowd model EW: death probability male (1980) 0 −1 −2 log qx −3 logit q xt = κ (1) x ) κ (2) t +( x − ¯ −4 t −5 −6 50 60 70 80 90 100 age ( 1 ) vs t ( 2 ) vs t κ t κ t −2.4 0.105 0.100 −2.8 0.095 −3.2 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 year year

Cairns-Blake-Dowd model EW: death probability male (1990) 0 −1 −2 log qx −3 logit q xt = κ (1) x ) κ (2) t +( x − ¯ −4 t −5 −6 50 60 70 80 90 100 age ( 1 ) vs t ( 2 ) vs t κ t κ t −2.4 0.105 0.100 −2.8 0.095 −3.2 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 year year

Cairns-Blake-Dowd model EW: death probability male (2000) 0 −1 −2 log qx −3 logit q xt = κ (1) x ) κ (2) t +( x − ¯ −4 t −5 −6 50 60 70 80 90 100 age ( 1 ) vs t ( 2 ) vs t κ t κ t −2.4 0.105 0.100 −2.8 0.095 −3.2 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 year year

Cairns-Blake-Dowd model EW: death probability male (2010) 0 −1 −2 log qx −3 logit q xt = κ (1) x ) κ (2) t +( x − ¯ −4 t −5 −6 50 60 70 80 90 100 age ( 1 ) vs t ( 2 ) vs t κ t κ t −2.4 0.105 0.100 −2.8 0.095 −3.2 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 year year

Cohort effects ◮ Statistical evidence � both period and cohort effect have an impact on mortality improvements (Willets 1999, 2004) ◮ Period effects approximate contemporary factors ◮ General health status of the population ◮ Healthcare services available ◮ Critical weather conditions ◮ Cohort effects approximate historical factors ◮ World War II ◮ Diet ◮ Welfare State (in the UK) ◮ Smoking habits

Age-Period-Cohort models: Classic APC model ◮ A widely used structured used in medicine, psychology and demography (Hobcraft et al. (1982), Wilmoth (1990)) log µ xt = α x + κ t + γ t − x ◮ No unique set of parameters resulting in optimal fit due to c = t − x ( α x , κ t , γ t − x ) → ( α x + φ 1 − φ 2 x , κ t + φ 2 t , γ t − x − φ 1 − φ 2 ( t − x )) ( α x , κ t , γ t − x ) → ( α x + c 1 , κ t − c 1 , γ t − x ) ◮ Impose constraints � � � κ t = 0 , γ c = 0 , c γ c = 0 c c t

Age-Period-Cohort models: Identifiability α x vs. x β (1) β (0) x vs. x x vs. x 2 0.02 0.02 tilt 1 0 tilt 2 0.015 0.015 tilt 3 −2 tilt 4 tilt 5 −4 0.01 0.01 −6 −8 0.005 0.005 −10 0 0 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 age (x) age (x) age (x) κ t vs. t ι y vs. y 400 100 200 0 0 −200 −100 −400 −200 1970 1980 1990 2000 1880 1900 1920 1940 1960 1980 2000 calendar year (t) year of birth (y)

Other stochastic mortality models ◮ Lee-Carter extensions ◮ Add cohorts x κ (1) log µ xt = α x + β (1) + β (0) x γ t − x t ◮ CBD extensions ◮ M6: Add cohorts logit q xt = η xt = κ (1) x ) κ (2) + ( x − ¯ + γ t − x t t ◮ M7: Add cohorts and quadratic age effect � � logit q xt = η xt = κ (1) x ) κ (2) κ (3) x ) 2 − ˆ σ 2 +( x − ¯ + ( x − ¯ + γ t − x t t x t ◮ Plat model combines the Lee-Carter and the CBD log µ xt = α x + κ (1) x − x ) κ (2) x − x ) + κ (3) + (¯ + (¯ + γ t − x t t t

Generalised Age-Period-Cohort stochastic mortality models Recent research has proposed a unifying framework discrete stochastic mortality models ◮ General Age-Period-Cohort model structure Hunt and Blake (2015a) ◮ Generalised (non-)linear model Currie (2016) ◮ R Implementation of GAPC models Villegas et al. (2018)

Generalised Age-Period-Cohort stochastic mortality models 1. Random Component: D xt ∼ Poisson( E c xt µ xt ) or D xt ∼ Binomial( E xt , q xt ) 2. Systematic Component: N � x κ ( i ) β ( i ) β (0) η xt = α x + + x γ t − x t i =1 ◮ Lee-Carter type � β ( i ) x , non-parametric ◮ CBD type � β ( i ) ≡ f ( i ) ( x ), pre-specified parametric function x 3. Link Function: � � D xt �� g = η xt E E xt ◮ log-Poisson: η xt = log µ xt ◮ logit-Binomial: η xt = logit q xt

Generalised Age-Period-Cohort stochastic mortality models 4. Set of parameter constraints: ◮ Need parameters constraints to ensure identifiability 5. Forecasting and simulation ◮ Period indexes: Multivariate random walk with drift  κ (1)  t .   κ t = δ + κ t − 1 + ξ κ . ξ κ t , κ t =  , t ∼ N ( 0 , Σ) , .  κ ( N ) t ◮ Cohort effect: ARIMA( p , q , d ) with drift ∆ d γ c = δ 0 + φ 1 ∆ d γ c − 1 + · · · + φ p ∆ d γ c − p + ǫ c + δ 1 ǫ c − 1 + · · · + δ q ǫ c − q GAPC models can be implemented with the R package StMoMo (http://cran.r-project.org/package=StMoMo)

Other key areas in single population discrete mortality models I ◮ Parameter estimation: ◮ Poisson (Brouhns et al., 2002) ◮ Negative-Binomial (Delwarde et al., 2007a, Li et al. (2009)) ◮ Bayesian and state-space setting (Czado et al., 2005, Pedroza (2006), Kogure et al. (2009), Fung et al. (2016)) ◮ Parameter Smoothing (Delwarde et al., 2007b) and functional data approach (Hyndman and Ullah, 2007) ◮ Bootstrapping and parameter uncertainty ◮ Semiparametric (Brouhns et al., 2005) ◮ Parametric (Koissi et al., 2006) ◮ Comparison of methods (Renshaw and Haberman, 2008) ◮ Modelling of errors: residual dependence (Debón, 2008, Debón et al. (2010), Mavros et al. (2017)) ◮ Identifiability

Other key areas in single population discrete mortality models II ◮ Age-Period Models (Hunt and Blake, 2015b) ◮ APC models (Hunt and Blake, 2015c) ◮ Impact on estimation (Hunt and Villegas, 2015) ◮ Modelling and projecting of period and cohort factors ◮ Optimal calibration period (Booth et al., 2002, Denuit2005) ◮ Regime-switching (Milidonis et al., 2011) ◮ Structural changes (Coelho and Nunes, 2011, van Berkum et al. (2014)) ◮ Selection criteria of most appropriate model ◮ Goodness-of-fit (Cairns et al., 2009, Dowd et al. (2010b)) ◮ Backtesting (Dowd et al., 2010a) ◮ Qualitative properties of forecasts (Cairns et al., 2011b) ◮ Overall performance for England and Wales and the USA (Haberman and Renshaw, 2011).

Mortality improvement rate models Discussion based on: Hunt, A., & Villegas, A. M. (2017). Mortality Improvement Rates: Modeling and Parameter Uncertainty. In Living to 100, Society of Actuaries International Symposium. Retrieved from https://www.soa.org/essays-monographs/ 2017-living-to-100/2017-living-100-monograph-hunt-villegas-paper.pdf

Two parallel approaches to modelling and projecting mortality Mortality Rates Improvement Rates q x , t − 1 , − ln( m x , t q x , t What? x q x , t , µ x , t , m x , t 1 − m x , t − 1 ) Who? Lee and Carter (1992) xxxxxx CMIB (1978) xxxxxx Cairns et al. (2006) xxxxxx xx CMI (2002, 2009, 2016) Brouhns et al. (2005) SOA (1995, 2012) xxxxxxxxxxxxxxx N � β ( i ) x κ ( i ) How? ln m x , t = α x + + γ t − x q x , T + n = q x , T (1 − R x ) n t i =1

Improvement rates in actuarial practice – AA and BB scales × (1 − AA x ) n q x , T + n = q x , T ���� � �� � Reduction factor Base table

Improvement rates in actuarial practice – CMI model Latest CMI projection model uses an APC model on improvement rates (CMI Working paper 90): − ∆ ln m x , t = α x + κ t + γ t − x

Recent academic interest on improvement rate modelling ◮ Mitchell et al. (2013) � ˆ � m x , t +1 ◮ ln = α x + β x κ t + ǫ x , t m x , t ˆ ◮ Benefits of detrending ◮ Estimation with singular value decomposition ◮ Haberman and Renshaw (2012) m x , t − 1 − ˆ ˆ m x , t m x , t ) = η x , t ◮ 0 . 5( ˆ m x , t − 1 + ˆ ◮ Predictor structures η x , t borrowed from mortality rate modelling ◮ Duality between mortality rate and improvement rate modelling ◮ Estimation using Gaussian model with variable dispersion ◮ Haberman and Renshaw (2013), Plat (2011), Danesi et al. (2015), Chuliá et al. (2016), Njenga and Sherris (2011)

Complications with improvement rate modelling Mortality rate Improvement rate 0.06 Improvement rate at age 70 0.10 Mortality rate at age 70 0.04 0.00 −0.10 0.02 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 year year 5e−01 Mortality rate in 2011 (log) Improvement rate in 2011 0.2 5e−03 0.0 1e−04 −0.2 0 20 40 60 80 100 0 20 40 60 80 100 age age ◮ Heteroscedasticity ◮ Patterns are not that clear ◮ Parameter uncertainty ◮ Non-standard distribution

Two alterantive approaches to modelling improvement rates Diagram source: Li et al. (2017) ◮ Route A (Direct) : Direct modelling of improvement rates ◮ Route B (Indirect)) Derive improvement rates from mortality rate model

Route B: Estimation and equivalent mortality rate structure N N � β ( i ) x κ ( i ) � − ∆ ln m x , t = α x + + γ t − x β ( i ) x K ( i ) ln ( m x , t ) = A x − α x t + + Γ t − x t t i =1 i =1 ( 1 ) vs. t ( 1 ) vs. t κ t K t 0.0 0.05 −0.4 κ ( i ) = − ∆ K ( i ) t t −0.8 −0.05 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 year year ⇐ ⇒ γ t − x vs. t−x Γ t − x vs. t−x 0.30 0.00 0.15 x γ c = − ∆Γ c −0.10 0.00 1880 1900 1920 1940 1880 1900 1920 1940 cohort cohort

Route B: Estimation and equivalent mortality rate structure N N � β ( i ) x κ ( i ) � − ∆ ln m x , t = α x + + γ t − x β ( i ) x K ( i ) ln ( m x , t ) = A x − α x t + + Γ t − x t t i =1 i =1 ( 1 ) vs. t ( 1 ) vs. t κ t K t 0.0 0.05 −0.4 κ ( i ) = − ∆ K ( i ) t t −0.8 −0.05 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 year year ⇐ ⇒ γ t − x vs. t−x Γ t − x vs. t−x 0.30 0.00 0.15 x γ c = − ∆Γ c −0.10 0.00 1880 1900 1920 1940 1880 1900 1920 1940 cohort cohort

Estimation routes illustration: η x , t = α x ln m(x,t) vs. t Mortality rate at age 40 (log) −6.0 −6.2 −6.4 −6.6 1960 1970 1980 1990 2000 2010 year

Estimation routes illustration: η x , t = α x ln m(x,t) vs. t Mortality rate at age 40 (log) −6.0 −6.2 −6.4 −6.6 1960 1970 1980 1990 2000 2010 year Route 2: “Indirect” ln m x , t = A x − α x t + ǫ x , t x

Estimation routes illustration: η x , t = α x ln m(x,t) vs. t Mortality rate at age 40 (log) −6.0 −6.2 −6.4 −6.6 1960 1970 1980 1990 2000 2010 year Route 2: “Indirect” ln m x , t = A x − α x t + ǫ x , t x

Estimation routes illustration: η x , t = α x ln m(x,t) vs. t Estimated α x Mortality rate at age 40 (log) −6.0 Crude −6.2 −6.4 Fitted −6.6 0.8% 1.0% 1.2% 1.4% 1960 1970 1980 1990 2000 2010 Improvement rate year Route 2: “Indirect” ln m x , t = A x − α x t + ǫ x , t T � ( t − ¯ t )(ln ˆ m x , t − ln ˆ m x , t ) t =0 α x = T � ( t − ¯ t ) 2 t =0

Estimation routes illustration: η x , t = α x ln m(x,t) vs. t Estimated α x Mortality rate at age 40 (log) −6.0 Crude −6.2 −6.4 Fitted −6.6 0.8% 1.0% 1.2% 1.4% 1960 1970 1980 1990 2000 2010 Improvement rate year Route 2: “Indirect” Route 1: “Direct” T ln m x , t = A x − α x t + ǫ x , t α x = 1 � − ∆ ln ˆ m x , t T T t =1 � ( t − ¯ t )(ln ˆ m x , t − ln ˆ m x , t ) α x = ln ˆ m x , 0 − ln ˆ m x , T t =0 α x = T T � ( t − ¯ t ) 2 x t =0

Estimation routes illustration: η x , t = α x ln m(x,t) vs. t Estimated α x Mortality rate at age 40 (log) −6.0 Crude −6.2 −6.4 Fitted −6.6 0.8% 1.0% 1.2% 1.4% 1960 1970 1980 1990 2000 2010 Improvement rate year Route 2: “Indirect” Route 1: “Direct” T ln m x , t = A x − α x t + ǫ x , t α x = 1 � − ∆ ln ˆ m x , t T T t =1 � ( t − ¯ t )(ln ˆ m x , t − ln ˆ m x , t ) α x = ln ˆ m x , 0 − ln ˆ m x , T t =0 α x = T T � ( t − ¯ t ) 2 x t =0

Estimation routes illustration: η x , t = α x ln m(x,t) vs. t Estimated α x Mortality rate at age 40 (log) −6.0 Crude −6.2 −6.4 Fitted −6.6 0.8% 1.0% 1.2% 1.4% 1960 1970 1980 1990 2000 2010 Improvement rate year Route 2: “Indirect” Route 1: “Direct” T ln m x , t = A x − α x t + ǫ x , t α x = 1 � − ∆ ln ˆ m x , t T T t =1 � ( t − ¯ t )(ln ˆ m x , t − ln ˆ m x , t ) α x = ln ˆ m x , 0 − ln ˆ m x , T t =0 α x = T T � ( t − ¯ t ) 2 x t =0

Estimation routes illustration: η x , t = α x ln m(x,t) vs. t Estimated α x Mortality rate at age 40 (log) −6.0 Crude −6.2 −6.4 Fitted −6.6 0.8% 1.0% 1.2% 1.4% 1960 1970 1980 1990 2000 2010 Improvement rate year Route 2: “Indirect” Route 1: “Direct” T ln m x , t = A x − α x t + ǫ x , t α x = 1 � − ∆ ln ˆ m x , t T T t =1 � ( t − ¯ t )(ln ˆ m x , t − ln ˆ m x , t ) α x = ln ˆ m x , 0 − ln ˆ m x , T t =0 α x = T T � ( t − ¯ t ) 2 x t =0

Estimation routes illustration: η x , t = α x ln m(x,t) vs. t Estimated α x Mortality rate at age 40 (log) −6.0 Crude −6.2 −6.4 Fitted −6.6 0.8% 1.0% 1.2% 1.4% 1960 1970 1980 1990 2000 2010 Improvement rate year Route 2: “Indirect” Route 1: “Direct” T ln m x , t = A x − α x t + ǫ x , t α x = 1 � − ∆ ln ˆ m x , t T T t =1 � ( t − ¯ t )(ln ˆ m x , t − ln ˆ m x , t ) α x = ln ˆ m x , 0 − ln ˆ m x , T t =0 α x = T T � ( t − ¯ t ) 2 x t =0

Estimation routes illustration: η x , t = α x ln m(x,t) vs. t Estimated α x Mortality rate at age 40 (log) −6.0 Crude −6.2 −6.4 Fitted −6.6 0.8% 1.0% 1.2% 1.4% 1960 1970 1980 1990 2000 2010 Improvement rate year Route 2: “Indirect” Route 1: “Direct” T ln m x , t = A x − α x t + ǫ x , t α x = 1 � − ∆ ln ˆ m x , t T T t =1 � ( t − ¯ t )(ln ˆ m x , t − ln ˆ m x , t ) α x = ln ˆ m x , 0 − ln ˆ m x , T t =0 α x = T T � ( t − ¯ t ) 2 x t =0

Estimation routes illustration: η x , t = α x ln m(x,t) vs. t Estimated α x Mortality rate at age 40 (log) −6.0 Crude −6.2 −6.4 Fitted −6.6 0.8% 1.0% 1.2% 1.4% 1960 1970 1980 1990 2000 2010 Improvement rate year Route 2: “Indirect” Route 1: “Direct” T ln m x , t = A x − α x t + ǫ x , t α x = 1 � − ∆ ln ˆ m x , t T T t =1 � ( t − ¯ t )(ln ˆ m x , t − ln ˆ m x , t ) α x = ln ˆ m x , 0 − ln ˆ m x , T t =0 α x = T T � ( t − ¯ t ) 2 x t =0

Estimation routes illustration: η x , t = α x ln m(x,t) vs. t Estimated α x Mortality rate at age 40 (log) −6.0 Crude −6.2 −6.4 Fitted −6.6 0.8% 1.0% 1.2% 1.4% 1960 1970 1980 1990 2000 2010 Improvement rate year Route 2: “Indirect” Route 1: “Direct” T ln m x , t = A x − α x t + ǫ x , t α x = 1 � − ∆ ln ˆ m x , t T T t =1 � ( t − ¯ t )(ln ˆ m x , t − ln ˆ m x , t ) α x = ln ˆ m x , 0 − ln ˆ m x , T t =0 α x = T T � ( t − ¯ t ) 2 x t =0

Parameter estimates: LC and CBD Black lines: ‘’Indirect” approach, Red lines: ”Direct” approach LC ( 1 ) vs. x ( 1 ) vs. t β x κ t 0.06 2.0 0.04 1.5 1.0 0.02 0.5 −0.02 −0.5 20 30 40 50 60 70 80 90 1970 1980 1990 2000 2010 age year CBD ( 1 ) vs. t ( 2 ) vs. t κ t κ t 0.002 0.04 0.02 0.000 0.00 −0.02 −0.002 1970 1980 1990 2000 2010 1970 1980 1990 2000 2010 year year

Key message ◮ Improvement rates are an intuitive and natural way to interpret mortality data ◮ Compelling reasons for formulating and communicating projection models in terms of improvement rates ◮ Important differences between approached to fitting improvement rate models ◮ Considerable parameter uncertainty ◮ Implication for projections and robustness ◮ Compelling reasons for estimating projection models in terms of mortality rates

Multipopulation (discrete) mortality models Discussion based on: Villegas, A. M., Haberman, S., Kaishev, V. K., & Millossovich, P. (2017). A Comparative Study of Two-Population Models for the Assessment of Basis Risk in Longevity Hedges. ASTIN Bulletin, 47(03), 631–679. https://doi.org/10.1017/asb.2017.18

Universe of Multipopulation Models Relative Lee-Carter + Cohorts Augmented Common Factor xt = α x + β (1) log m i x κ t + γ t − x + + Cohorts x + β (2) log m i xt = α i α i x κ i x + β x κ t + t Multipopulation GLM � N j =1 β ( j , i ) κ ( j , i ) + β (0 , i ) γ i Villegas and Haberman (2014) x t x t − x Hatzopoulos and Haberman (2013) , Yang et al. (2016) Ahmadi and Li (2014) Co-integrated Lee-Carter log m i xt = α i x + β i x κ i Augmented Common Factor t Relative P-Splines Carter and Lee (1992) , x + β x κ t + β ( i ) log m i xt = α i x κ i Biatat and Currie (2010) t Li and Hardy (2011) , Li and Lee (2005) , Li and Hardy (2011) Plat + Lee-Carter Yang and Wang (2013) Hyndman et al. (2013) , Li (2012) Wan and Bertschi (2015) Stratified Lee-Carter xt = α x + α i + β x κ t log m i Lee-Carter + Butt and Haberman (2009) , Extensions of VAR/VECM Debón et al. (2011) log m i xt = α i x + β x κ i the Lee-Carter t Other Zhou et al. (2014) models Common Factor log m i xt = α i x + β x κ t Common Age Effect Carter and Lee (1992) , x κ ( j , i ) log m i xt = α i x + � j β j Li and Lee (2005) , t Li and Hardy (2011) Kleinow (2015) Saint model Jarner and Kryger (2011a) Three-way Lee-Carter Bayesian log m i xt = α i x + β x λ i κ t two-population APC Russolillo et al. (2011) log m i xt = α i x + κ i t + γ i t − x Gravity model - Cairns et al. (2011a) Two-population APC log m i xt = α i x + κ i t + γ i t − x Joint- κ Dowd et al. (2011) Plat Relative model log m i xt = α i x + β i x κ t Plat (2009a) Carter and Lee (1992) , Li and Hardy (2011) , Wilmoth and Valkonen (2001) , Two-population CBD (M5) Extensions of xt = κ ( i , 1) x ) κ ( i , 2) Delwarde et al. (2006) logit q i + ( x − ¯ t t the CBD model Li et al. (2015) Two-population M6 Two-population M7 � κ ( i , 3) xt = κ ( i , 1) x ) κ ( i , 2) logit q i + γ i logit q i xt = κ ( i , 1) x ) κ ( i , 2) � ( x − ¯ x ) 2 − ˆ σ 2 + γ i + ( x − ¯ + ( x − ¯ + t t t − x t t x t t − x Li et al. (2015) Li et al. (2015)

Coherent forecasts for multiple populations ◮ See for example Li and Lee (2005), Hyndman et al. (2013) Source: Hyndman et al. (2013)

Quantifying socio-economic differences in mortality ◮ See for example Villegas and Haberman (2014), Cairns et al. (2016) Source: Villegas and Haberman (2014) log m i xt = α x + α i x + β x λ i κ t

Assessing longevity basis risk ◮ See for example Li and Hardy (2011), Villegas et al. (2017), Li et al. (2018) Source: Villegas et al. (2017)

Borrowing strength from a larger population ◮ Derive the trend from the larger population and model the spread (ratio) between the larger and the smaller population (Jarner and Kryger, 2011b; van Berkum et al., 2017) Source: van Berkum et al. (2017)