StMoMo : An R Package for St ochastic Mo rtality Mo delling Andrés M. Villegas , Vladimir Kaishev, Pietro Millossovich Cass Business School, City University London 7 September 2015, Lyon Eleventh International Longevity Risk and Capital Markets Solutions Conference
Agenda ◮ Motivation and Literature Review ◮ Generalised Age-Period-Cohort mortality models ◮ StMoMo package ◮ Conclusions
StMoMo : St ochastic Mo rtality Mo delling Who is MoMo?
StMoMo : St ochastic Mo rtality Mo delling Who is MoMo? x
StMoMo : St ochastic Mo rtality Mo delling Who is MoMo? x
StMoMo : St ochastic Mo rtality Mo delling Who is MoMo? x
Advances in mortality modelling ◮ Lee-Carter model (Lee and Carter 1992) ◮ Add more bilinear age-period components (Renshaw and Haberman 2003) ◮ Add a cohort effect (Renshaw and Haberman 2006) ◮ Two factor CBD model (Cairns, Blake, and Dowd 2006) ◮ Add cohort effect, quadratic age term (Cairns et al. 2009) ◮ Combine with features of the Lee-Carter (Plat 2009) ◮ Many more models proposed in the literature (e.g. Aro and Pennanen (2011), O’Hare and Li (2012), Börger, Fleischer, and Kuksin (2013), Alai and Sherris (2014))
Mortality modelling in R ◮ Demography (Hyndman 2014) ◮ Lee-Carter model and several of its variants ◮ ilc (Butt, Haberman, and Shang 2014) ◮ Lee-Carter with cohorts and Lee-Carter under a Poisson framework ◮ Lifemetrics ( http://www.macs.hw.ac.uk/~andrewc/lifemetrics/ ) ◮ CBD and extensions ◮ Lee-Carter with cohorts and Lee-Carter under a Poisson framework
Limitation of existing R packages ◮ Not easily expandable to include new models ◮ Limited forecasting and simulation capabilities ◮ Limited tools for goodness-of-fit analysis ◮ Do not allow for parameter uncertainty
Limitation of existing R packages ◮ Not easily expandable to include new models ◮ Limited forecasting and simulation capabilities ◮ Limited tools for goodness-of-fit analysis ◮ Do not allow for parameter uncertainty ◮ StMoMo seeks to overcome these limitations
StMoMo : An R package for St ochastic Mo rtality Mo delling ◮ On CRAN: http://cran.r-project.org/web/packages/StMoMo/ ◮ Development version on Github: https://github.com/amvillegas/StMoMo ◮ To install the stable version on R CRAN: install.packages ("StMoMo") ◮ To load within R: library (StMoMo)
Overview of the structure of StMoMo
Generalised Age-Period-Cohort stochastic mortality models StMoMo is based on the unifying framework of the family of Generalised Age-Period-Cohort stochastic mortality models ◮ General Age-Period-Cohort model structure (Hunt and Blake 2015) ◮ Generalised (non-)linear model (Currie 2014)
General Age-Period-Cohort model structure EW: male death rates (1961) 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
General Age-Period-Cohort model structure EW: male death rates (1965) 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
General Age-Period-Cohort model structure EW: male death rates (1970) 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
General Age-Period-Cohort model structure EW: male death rates (1975) 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
General Age-Period-Cohort model structure EW: male death rates (1980) 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
General Age-Period-Cohort model structure EW: male death rates (1985) 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
General Age-Period-Cohort model structure EW: male death rates (1990) 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
General Age-Period-Cohort model structure EW: male death rates (1995) 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
General Age-Period-Cohort model structure EW: male death rates (2000) 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
General Age-Period-Cohort model structure EW: male death rates (2005) 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
General Age-Period-Cohort model structure EW: male death rates (2010) 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
General Age-Period-Cohort model structure EW: male death rates (1961−2011) 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
General Age-Period-Cohort model structure EW: male death rates (1961−2011) 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
General Age-Period-Cohort model structure EW: male death rates (1961−2011) 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
General Age-Period-Cohort model structure EW: male death rates (1961−2011) 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
General Age-Period-Cohort model structure EW: male death rates (1961−2011) 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
General Age-Period-Cohort model structure EW: male death rates (1961−2011) 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
General Age-Period-Cohort model structure EW: male death rates (1961−2011) 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
Generalised Age-Period-Cohort stochastic mortality models 1. Random Component : D xt ∼ Poisson ( E c D xt ∼ Binomial ( E 0 xt µ xt ) or xt , q xt )
Generalised Age-Period-Cohort stochastic mortality models 1. Random Component : D xt ∼ Poisson ( E c D xt ∼ Binomial ( E 0 xt µ xt ) or 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
Generalised Age-Period-Cohort stochastic mortality models 1. Random Component : D xt ∼ Poisson ( E c D xt ∼ Binomial ( E 0 xt µ xt ) or 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 : ◮ Most mortality models are only identifiable up to a transformation ◮ Need parameters constraints to ensure identifiability ◮ Constraint function v mapping an arbitrary vector of parameters � � , κ (1) t , ..., κ ( N ) α x , β (1) x , ..., β ( N ) , β (0) θ := x , γ t − x t x into a vector of transformed parameters � � v ( θ ) = ˜ α x , ˜ x , ..., ˜ κ (1) κ ( N ) , ˜ β (1) β ( N ) β (0) θ = ˜ , ˜ t , ..., ˜ x , ˜ γ t − x t x satisfying the model constraints with no effect on the predictor η xt (i.e. θ and ˜ θ result in the same η xt )
GAPC stochastic mortality models with StMoMo GAPC model are constructed using the function StMoMo (link, staticAgeFun, periodAgeFun, cohortAgeFun, constFun)
GAPC stochastic mortality models with StMoMo GAPC model are constructed using the function StMoMo (link, staticAgeFun, periodAgeFun, cohortAgeFun, constFun) ◮ link : defines the link and random component . i =1 β ( i ) x κ ( i ) β (0) � N = + + η xt α x x γ t − x t
GAPC stochastic mortality models with StMoMo GAPC model are constructed using the function StMoMo (link, staticAgeFun, periodAgeFun, cohortAgeFun, constFun) ◮ link : defines the link and random component . ◮ The predictor is defined via: i =1 β ( i ) x κ ( i ) β (0) � N = + + η xt α x x γ t − x t
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