Stochastic Mortality Modelling: Key Drivers and Dependent Residuals

Transcription

1 Sochasic Moraliy Modelling: Key Drivers and Dependen Residuals George Mavros, Andrew J.G. Cairns, George Srefaris, Torsen Kleinow Maxwell Insiue for Mahemaical Sciences, Edinburgh, and Deparmen of Acuarial Mahemaics & Saisics Herio-Wa Universiy, Edinburgh, EH14 4AS, UK. Absrac This aricle proposes an alernaive framework for modelling he sochasic dynamics of moraliy raes. A simple age basis combined wih wo sochasic period facors is used o explain he key moraliy drivers, while he remaining srucure is modelled via a mulivariae auoregressive residuals model. The laer capures he saionary moraliy dynamics and inroduces dependencies beween adjacen age-period cells of he moraliy marix ha, amongs oher hings, can be srucured o capure cohor effecs in a ransparen manner and incorporae across ages correlaions in a naural way. Our approach is compared wih models wih and wihou a univariae cohor process. The age and period relaed laen saes of he moraliy basis are more robus when he residuals surface is modelled via he mulivariae ime-series model, implying ha he process acs indeed independenly of he assumed moraliy basis. Under he Bayesian paradigm, he poserior disribuion of he models is considered in order o explore coherenly he exen of parameer uncerainy. Samples from he poserior predicive disribuion are used o projec moraliy, and an indeph sensiiviy analysis is conduced. The mehodology is easily exendable in muliple ways which give a differen form and degree of significance o he differen componens of moraliy dynamics. Keywords: Sochasic Moraliy Dynamics, Robusness, Parameer uncerainy, Correlaion modelling. Corresponding auhors: grgmavros@gmail.com, A.J.G.Cairns@hw.ac.uk

2 1 Inroducion Sochasic moraliy models allow us o idenify he impac of sysemaic risk on uncerainy in numbers of deahs, and a vas variey of such models has been developed during he las wo decades. Some of he mos influenial ideas underlying he ime-series naure of he field have been se up by Lee and Carer 199), Brouhns e al. 00), Booh e al. 00), Renshaw and Haberman 003, 006), Cairns e al. 006) and Pla 009). In-deph comparaive sudies of various compeiors have been conduced by Cairns e al. 009) and Haberman and Renshaw 011), and cerain models from herein are oday of he mos elaborae academic and indusrial approaches in modelling moraliy raes. All he above, and indeed many more, models are based on he assumpion ha human moraliy dynamics may be broken down ino a mixure of evenually independen deerminisic and sochasic componens which capure age, period and, in some cases, cohor effecs. The number and form of hese ypes of effecs is usually wha disinguishes one model from anoher. Ofen, some of hese effecs are unobservable, and heir laen saes are esimaed hrough opimisaion of an objecive funcion under some assumpion abou he disribuion of deahs. The Poisson and he Binomial models are wo sandard choices. Exrapolaion is hen performed by independen modelling and projecion of he relevan sochasic facors. In his aricle we sugges an alernaive framework, where he exisence of dominan deerminisic and sochasic facors responsible for he main shape of he moraliy surface is also admied, bu a he same ime we invesigae he possibiliy of shor-erm and conemporaneous inerdependencies wihin he daa. Such dependence srucures are modelled, firs, by esimaing he residuals of a simple moraliy srucure joinly wih all oher laen effecs, and second, by srucuring an appropriae mulivariae sochasic model for projecing he newly inroduced residuals process. In conras o he mainsream approach where an addiional univariae cohor process is used o model he saionary par of he moraliy dynamics, our framework is able o accommodae boh serial and cross-age dependence srucures. This urns ou o be sufficien for capuring he underlying dynamics, alhough in a qualiaively disinc manner. The residuals augmenaion builds up a hierarchical srucure of condiionally independen processes, which we hen embed wihin he Bayesian paradigm. By consrucing a Markov chain which has as is saionary disribuion he arge poserior of he model, we sample from i hrough a hybrid Markov chain Mone Carlo MCMC) algorihm. The usage of he full poserior disribuion of he model imposes he correc degree of parameer uncerainy in esimaes and projecions, hence removing he requiremen for any addiional boosrap exercise in he modelling procedure. Our approach shares common elemens wih ha of D Amao e al. 01) in he sense ha ime-series models are employed in boh cases for he residuals marix. However, D Amao e al. 01) work wih he Lee-Carer model, whereas we use an enhanced version of he CBD model. More imporanly, we use a parsimonious mulivariae Markov auoregression insead of he single univariae auoregressive model for each age in he daa in he D Amao e al. 01) approach. Moraliy residuals have also been direcly modelled by Debón e al. 008, 010) under a spaial dependence approach specified by a parameric covariance srucure. In he presen aricle, he main driver of he residuals model is he auoregression, and he covariance of he model eners as an addiional differeniaing sep. 1

3 The srucure of he aricle is as follows. Secion inroduces he noaion and modelling framework, and develops he proposed model. Secion 3 briefly illusraes he fiing procedure and oulines he relaed algorihm for he employed MCMC scheme. In Secion 4 we presen our resuls and examine he sensiiviy of he poserior disribuions of he laen saes and parameers of he models o differen esimaion imeframes. Secion 5 includes comparaive projecions and numerical applicaions of he models, while Secion 6 concludes. Model Descripion In his Secion we develop he proposed framework and specify he paricular modelling assumpions. We assign a paricular parameric residuals model o illusrae he idea which will be examined in he following secions, and briefly presen and commen on he models which will be used for comparaive purposes..1 Noaion and Assumpions Time is assumed o be measured in years, so ha calendar year has he meaning of he ime inerval [, + 1). Dx, ) denoes he number of deahs in year among individuals aged x las birhday on he dae of deah, and Ex, ) denoes he cenral exposure-o-risk for age x during year, which refers o he populaion exposed o he risk of deah. Usually moraliy models describe he cenral deah rae, mx, ), which is presumed consan wihin each cell of he daa, hrough some disribuional assumpion abou deahs, Dx, ). The condiional Poisson assumpion wih mean Ex, )mx, ), independenly in each age-year cell given mx, ) is employed here, where he exposures Ex, ) are considered consans provided in he daa. We denoe by n x and n he number of ages and years, respecively. Hence, he disribuional assumpion ha accompanies all he models in his aricle is saed as follows: ) Dx, ) mx, ) Poisson Ex, )mx, ), 1) where Dx, ) are condiionally independen for x = 1,..., n x, = 1,..., n, given he respecive mx, ) parameers. Under his choice we ensure ha all he models will be consisenly implemened and compared under he same source of idiosyncraic risk. The daa also include n c = n x + n 1 number of cohors across he diagonals of he moraliy able, given as c = x.. Modelling Framework We assume ha he principal moraliy dynamics are given by he following modelling form: M1: log mx, ) ) = α x + κ 1) + κ ) ) x x, ) where α x are age-relaed moraliy level parameers, κ 1) and κ ) are ime varying unobservable saes of he wo sochasic period facors and x is he mean of he ages in he daa. The moraliy surface is developed

4 while he semi-parameric deerminisic age basis consising of he hree vecors α x, 1 uni vecor of lengh n x ) and x x ) ) varies in accordance o he evoluion of he vecor of sochasic facors κ = κ 1), κ ). Equaion ) is ermed M1 in his aricle and is in pars similar o popular models in he lieraure. The righ hand side of he model excluding α x is similar o he CBD model Cairns e al., 006), excep ha i models he logarihmic mx, ) s, while including α x gives a very basic Pla 009) model. Model M1 describes he principal moraliy dynamics in he sense ha i includes he age relaed basis componen and all of he non-saionary sochasic par of he moraliy surface. Alhough ha basis can be exended by a quadraic age erm, and he exisence of a furher sochasic facor, κ 3), can be saisically jusified, we limi ourselves o he above model for simpliciy, and so ha we have a more informaive residual srucure. Even if a hird erm is added in model M1, he age-period ineracions alone are known o be inadequae, a leas for some available daa where prominen saionary variaion appears o exis. Such saionary dynamics come up in he form of diagonal srucures wihin he residual surface, and are modelled by he inclusion of year-of-birh depended parameers represening he cohor, c, each cell of he dynamic life-able belongs o. Model M1 is hen exended such ha: M: log mx, ) ) = α x + κ 1) + κ ) ) x x + γc, 3) where γ c denoe he sochasic cohor effecs ha are responsible for capuring saionary srucures, assuming he same level of persisence for any specific cohor across he whole observaion period. Tha is, for a fixed cohor c, he respecive coefficien γ c will deermine he impac of he cohor effec i suffers, or enjoys, for as long as he dynamics of his cohor are being projeced. Again, M may be viewed as model M6 in Cairns e al. 009) wih he addiion of α x modelling he logarihmic mx, ) s, or a simplified Pla 009) model. The alernaive approach suggesed by his aricle encapsulaes esimaing he residuals marix of model M1, and he laen residual saes are modelled by a mulivariae sochasic process direcly. Modelling he residuals as addiional laen saes allows for inroducing richer inerdependencies wihin he age-year cells of he populaion, compared o he resricive srucure of he cohor effecs. If he assumpion abou he consan conribuion of he cohor parameers o he raes across he diagonals of he moraliy able was really accurae, one would expec under a model ha does no include cohor effecs o observe residual series on he same diagonal having low variabiliy, i.e. he consan cohor effec plus he presumed i.i.d. noise. Examinaion of residual series of he same cohor, under for example M1, reveals persisenly excessive sandard deviaion values, which jusifies he adopion of a more flexible modelling framework. By uilising an appropriae mulivariae residuals model one may assume various srucures across boh dimensions of he moraliy able, while a he same ime capure efficienly he cohor effec. In his case, he augmened modelling equaion becomes: M3: log mx, ) ) = α x + κ 1) + κ ) ) x x + Rx, ), 4) where an addiional residual erm Rx, ) is added for each cell of he daa. Similarly o M, he random vecors across all ages for fixed ime, R := Rx, ), of M3 are assumed o ac independenly of all oher 3

5 sochasic facors of he model. The laen residuals are considered as an addiional conribuion o he moraliy rae explanaory equaion according o a mulivariae sochasic model of dimension n x..3 The Residuals Model We model he residuals marix by a simple firs order vecor auoregressive process VAR). We know a priori ha he dynamics wihin he residuals are principally driven by he diagonal srucures. We also allow for ineracion beween adjacen in age cells of he moraliy marix. Such a relaionship is inroduced via he following residuals regression equaion: Rx, ) = αrx, 1) + βrx 1, 1) + γrx + 1, 1) + Zx, ), 5) where Zx, ) are cell specific error erms and he coefficien β corresponds o he cohor-ype effec. If cohor effecs are primarily driven by lifesyle facors, such as smoking, hen i is likely ha here will be some diffusion beween adjacen cohors, which should hen be capured by facors α and γ. Such a parsimonious srucure imposes flexibiliy in he way cohor effecs impac moraliy raes over ime by allowing he inroducion of addiional serial correlaion wihin he moraliy able. The residuals model may be compacly wrien as: R = AR 1 + Z, 6) ) where Z N nx 0, VZ are i.i.d. vecors of dimension equal o he number of ages in he daa, nx, and covariance marix V Z. The n x n x auoregressive marix A of equaion 5) may be beer wrien as: α α α α 4 α 5 α A = 0 α 4 α 5 α α 4 α 5 α α 4 α 5 α 6 Parameers α 4, α 5 and α 6 correspond o α, β and γ in equaion 5), respecively. α 5, as well as α, carry he cohor informaion of he model. α 6 and α 3 inroduce serial dependence wihin he residuals model. α 1 is he single regression coefficien for he firs age of he daa. A furher source of variabiliy wihin he residuals model comes in he form of he covariance marix V Z. Several choices can be made here. A diagonal srucure implies independence beween he residual saes across he age dimension. We use a single common parameer across he diagonal of V Z for he firs version of model M3 which we label M3a. Furher assumpions, such as consan or auoregressive correlaion, may be imposed by appropriae parameerisaion of V Z. We choose o explore he full underlying srucure by allowing a complee parameerisaion for V Z for he he second version of model M3 which we name M3b. Tha is, we assume a fully parameerised covariance marix which will be easily esimaed wihin he MCMC 4

6 algorihm. If we refer o model M3 means ha he assignmen of he srucure for V Z is no relevan. Beyond independence beween he residual saes, version M3a also carries he srong assumpion of homoscedasic errors across all ages of he daa. In conras, all diagonal parameers of V Z for M3b are allowed o vary, so ha we can invesigae he case of a disinc residuals variance parameer for each age of he daa. Marix V Z only accouns for he one-sep ahead variabiliy of he residuals model. Under saionariy, which is ensured if he eigenvalues of marix A belong in he inerval 1, 1), or equivalenly if parameers α 1, α 3 and α 6 are in ha inerval, he long-erm covariance marix, V R, of he VAR model solely depends on A and V Z, and i is given as he soluion of he sysem Lükepohl, 005): V R = AV R A + V Z, 7) which is linear in he enries of V R. The soluion of he above sysem is he consan, long-erm covariance marix of he residuals process. So long as he examined populaion preserves he saionariy requiremen, he conribuion of he residuals model as he projecion horizon exends will be n x -dimensional vecors wih expecaion zero and covariance given by V R..4 Furher Modelling Consideraions The models presened in he previous secions are driven by he same pair of principle sochasic facors, he vecor κ. The componens of κ are usually modelled as correlaed random walks wih consan drifs, so ha hey conribue he non-saionary moraliy dynamics Cairns e al., 006). Hence, for κ of all models we use he following process: κ = δ + κ 1 +, N 0, V ), 8) ) where δ = δ 1, δ is he consan drif vecor and he covariance marix V is parameerised as: V = σ 1 σ 1, σ 1, σ For model M, in addiion o κ we assign a model for he cohor effecs process which is also generally. assumed sochasic. A sandard choice is usually a firs order auoregressive AR1)) process wih drif Cairns e al., 011): γ c = δ γ + α γ γ c 1 + c, c N 0, σ γ). 9) Nex, we commen on he idenifiabiliy consrains for each of he models. A deailed descripion of hese is given in Appendix B. For model M1, equaion ) does no imply a unique se of parameer esimaes. The regression equaion may become idenifiable by imposing he following consrains for he period effecs, κ i), i = 1, : κ i) = 0. 10) Model M is no idenifiable if only he consrains in equaions 10) are used. Addiionally, we impose: γ c = 0, c c c) γ c = 0, c 5 c c) γ c = 0, 11) c

7 where c is he mean of he cohor years of birh. The consrains in equaion 11) ensure γ c will flucuae around zero wih no idenifiable srucure, and hence he choice of model 9) is jusified. For model M3, beyond he consrains of equaion 10) we also use: Rx, ) = 0, x x x x)rx, ) = 0 and α x Rx, ) = 0,. 1) x For his model, in addiion o he wo consrains in equaion 10), we have hree ses of consrains for each calendar year of he daa. This yields 3 n + consrains in oal. We noe here ha under consrains 10) - 1), he likelihood conribuions of he original parameers and he resuling ransformaions are exacly he same deails given in Appendix B). 3 Bayesian Model Fiing Fundamenally, he vas amoun of laen saes and parameers of models M3 moivaes he adopion of he Bayesian paradigm. Under a usual esimaion approach, such as maximum likelihood, each moraliy rae would need o serve for he esimaion of more han one unknown of models M3. Such an esimae would hen be liable o a grea variabiliy which we aemp o balance hrough a more informaive esimaor, such as hose provided using he full poserior disribuion. Furhermore, obaining a poin esimae hrough some opimisaion mehod would also hen require a succeeding, and always essenial, parameer uncerainy assessmen exercise, he mos usual being some sor of parameric boosrap mehod as in Brouhns e al. 005). The unified esimaion and parameer uncerainy assessmen provided by he Bayesian framework, as also described and applied for example by Czado e al. 005) or Pedroza 006), is herefore in our view he mos coheren and comprehensive soluion for he calibraion of models M3. Due o he similar srucure of all he models, he relevan MCMC algorihms are developed in parallel for all of hem. Thus, evenually we compare he framework of residuals modelling, firs, agains he absence of modelling he saionary componen of he moraliy daa, and second, agains modelling wih cohor effecs as in M, by keeping all oher condiions equal. The Bayesian paradigm consiss of combining he likelihood of he model wih prior specificaions for is parameers, so ha he relevan poserior disribuion is derived. The global parameer vecor for all he presened models is denoed by θ. For convenience and o emphasise he dependence on he parameer vecor θ, le also u θ) uθ; x, ) generically denoe he righ-hand-side of equaions ), 3) and 4) for each of M1, M and M3, respecively. 3.1 Likelihood Funcion The Poisson model of Equaion 1) and he bivariae Normal likelihood for κ are common for all models. The wo componens of he likelihood are: l 1 θ) = { Dx, ) uθ; x, ) ) Ex, ) exp uθ; x, )) )}, x, l θ) = 1 { n 1 ) log V ) + n 1 j=1 [ κj+1 κ j δ ) V 1 κj+1 κ j δ )]}. 6

8 l 1 follows from he condiional Poisson assumpion in erms of he parameerisaion of he moraliy raes as given in Equaions ), 3) and 4) for M1, M and M3, respecively. l corresponds o he condiional likelihood of he bivariae random walk model for he period effecs, κ, as given by Equaion 8). The likelihood of M also includes he model for γ c as given by equaion 9). The firs sae of he process is assigned he saionary disribuion of he model, so ha: ) δ γ σγ γ 1 {α γ, δ γ, σ γ } N, 1 α γ 1 αγ. 13) Then, he addiional likelihood componen for M becomes: { ) l 3 θ) = 1 σ γ log 1 αγ + n c 1) log σγ) αγ) γ 1 δ ) } n c γ + γ j δ γ α γ γ j 1 ). 1 α γ σ γ For model M3, he likelihood incorporaes he erm of he residuals model of equaion 6). Similarly o above, we assign he saionary disribuion implied by he model for he firs residuals vecor of he daa. In his case he relevan saionary disribuion is given as: j= R 1 {A, V Z } N nx 0, V R ), 14) where he covariance marix V R is given as he soluion of he sysem in equaion 7). componen of he likelihood funcion for M3 is hen given as: The addiional l 4 θ) = 1 { n 1 ) log V Z ) + log V R ) + R 1V 1 R R 1 + n 1 j=1 [ Rj+1 ) ) ]} AR j V 1 Z Rj+1 AR j. Summarising he above discussion, he relevan likelihood funcions, l M1, l M and l M3, for models M1, M and M3, respecively, are given as: M1: l M1 θ) = l 1 θ) + l θ) + C 1, M: l M θ) = l 1 θ) + l θ) + l 3 θ) + C, M3: l M3 θ) = l 1 θ) + l θ) + l 4 θ) + C 3. 15) where C 1, C and C 3 are consans ha do no depend on he parameers of he models. 3. Prior Disribuions The second ingredien of our Bayesian models is obained by he assignmen of prior specificaions for he lowes level free parameers over all layers of hierarchy. The prior disribuion for model M1, which is also a componen of he priors for models M and M3, includes 7

9 wo componens. Firs, he drif, δ, of κ is assigned a bivariae normal prior wih mean δ 0 = 0, 0) and covariance, V 0, he ideniy marix. The prior is cenred close o reasonable values for he parameers; we expec δ 1 o be posiive and δ o be negaive, bu in boh cases considerably close o zero. Hence, he chosen prior is quie diffuse and no very resricive. There is very lile change in our resuls, however, if V 0 was 100 imes he ideniy marix. Marix V is given he Jeffreys non-informaive prior Cairns e al., 006). This choice convenienly leads in a conjugae Inverse-Wishar IW) full condiional poserior for V. The prior disribuion for model M includes addiional componens for he parameers of he model for γ c, as his is given in Equaion 9). For α γ we use a Uniform 1, 1) disribuion, o ensure saionariy of he process and he exisence of he disribuion of Equaion 13). The drif, δ γ, is given a Uniform, ) disribuion. Finally, he variance parameer, σ γ, is given an Inverse-Gamma IG), σ γ IGa, b), where a, b = Tha laer prior is he usual conjugae non-informaive choice for he variance of a normally disribued process, such as he one implied by model 9). The prior disribuion for model M3 includes erms for he elemens of he auoregression marix, A, and he covariance srucure V Z. The diagonal elemens of A, α 1, α 3 and α 6, are assigned Uniform 1, 1) prior disribuions along he same lines of argumenaion as for α γ in M and he relevan commens abou saionariy in Secion. For he remaining elemens of A we use Uniform, ) priors. In he case of a diagonal marix V Z, we assume v IGa, b) a priori, as for σ γ of M. For he fully parameerised covariance marix, V Z, of model M3b, we use again he Jeffreys non-informaive prior, as for V of M1. Le now φ denoe he parameer sub-vecor of θ for which prior disribuions are supplied. For each of M1, } { } } M and M3, he respecive vecors are φ 1 = {δ, V, φ = φ 1, α γ, δ γ, σγ and φ 3 = {φ 1, A, V Z. We summarise he above discussion in he following mahemaical saemens, where he prior p i, i = 1,, 3a, 3b is used for each of our four models. M1: p 1 φ 1 ) = δi 3 log V ) + C 1 i=1 M: p φ ) = p 1 φ 1 ) a + 1) logσγ) b σγ + C M3a: p 3a φ 3 ) = p 1 φ 1 ) a + 1) logv) b v + C 3a M3b: p 3b φ 3 ) = p 1 φ 1 ) n x + 1 log V Z ) + C 3b 16) where C 1, C, C 3a and C 3b are consans which do no depend on he parameers of he models. The use of vague prior disribuions implies ha he consrains described in Secion.4 are necessary in our analysis due o a combinaion of a poenially over-parameerised model and he lack of prior informaion Gelfand and Sahu 1999)). For example, he presence of δ and δ γ in 8) and 9) respecively, for which we use largely non-informaive priors, can lead o idenifiabiliy issues also refleced in MCMC convergence) if consrains are no imposed. This also means ha some of hese consrains, e.g. in 11), could be removed if a model wih δ γ = 0 was considered, or a srong prior wih mean 0 and large precision was assumed. 8

10 3.3 Poserior Sampling The preceding assumpions for he models lead o heir respecive poserior disribuions. On he log-scale, he poserior disribuions of he models, πj i θ), i, j) = 1, 1),, ), 3, 3a), 3, 3b), are sums of he respecive likelihood and prior erms. Based on he noaion of he previous secions he poseriors may be wrien as: π i jθ) = l i θ) + p j φ i ). The above log-poserior densiies are complex funcions of he parameers and laen saes of he models. We employ a general mehod based on drawing samples of he model parameers from approximae disribuions which are improved a each sep, known as Markov chain Mone Carlo MCMC) Gelman e al., 003). MCMC sampling converges o he full arge poserior disribuion of he model. The ieraive sampling is based on he full condiional log-poserior disribuions of he model. The condiioning in each case is on he global parameer vecor excep for he parameer, or group of parameers under consideraion, denoed by θ i, where i indicaes he parameers ha are being updaed, while he ohers are fixed a heir laes values. The exac forms of he log-poserior and of he full condiional log-poserior disribuions of he models are given in Appendix A. Generally, he parameers of ineres may be sampled individually, joinly or appropriaely blocked. For models such as hose developed herein, blocking usually leads o grea simplificaions and performance improvemens. The hierarchical srucure of he models unveils several conjugae relaionships, for which he Gibbs sampler is used. In ha case, he full condiional log-poserior has a closed form which is used o sample he parameers a each ieraion. For insances where no analyical resul arises, he Meropolis-Hasings MH) algorihm, or varians of i, is implemened Gamerman and Lopes, 006). For ha mehod an accepance-rejecion calculaion is involved, comparing he proposed sae agains he curren. The full condiional log-poserior is again he compuaional ool required for comparing and choosing beween he wo saes. Nex, we skech ou he poserior sampling procedure for all he models. The similar srucure of he models helps developing he respecive MCMC algorihms joinly. The age relaed moraliy levels, α x, are exponenially conjugae for fixed age x, for all models. The fullcondiional poseriors, which are used o implemen Gibbs seps in he algorihms, are given as follows: expα x ) θ i Gamma Dx, ), Ex, ) exp u θ) )). 17) The full-condiional log-poseriors of he vecor κ of he wo period effecs for fixed ime are mixures of Poisson and bivariae normal densiies, commonly for all models. The differeniaion from one model o anoher depends only on he parameerisaion of he model, u θ). The funcions of he vecors κ 1 and κ n include a single erm coming from he bivariae normal densiy. The period effecs vecors for inermediae years, κ j, include wo such erms. Since no analyical resul is available for heir sampling, we employ MH sampling. Given he curren esimae of he vecor κ we propose a new value by sampling from a bivariae normal disribuion cenred a ha curren value of he chain and appropriaely uned diagonal covariance marix. The echnique is generally known as random walk MH scheme. Alhough more elaborae choices for 9

11 he proposal covariance srucure may be essenial for oher Bayesian models, he random walk scheme works well for our purposes. Transiion, or no, o he proposed sae depends on he corresponding accepance raio. If he j 1 h sae of he algorihm is [κ ] j 1) and he proposed sae of he j h ieraion is κ, hen he required accepance raio is given as: { r = min 1, exp ) ) π κ )} θ i π [κ ] j 1) θ i. Model M includes he cohor effecs, γ c, as addiional laen saes. Their full-condiional log-poseriors are again mixures of Poisson and univariae, in his case, normal erms. There is a single observaion for he boundary cohors of he moraliy able, which increase progressively from boh sides as we move owards inermediae cohors. Due o he assumpion in Equaion 13), he full-condiional log-poserior of he firs sae of he cohor series includes an addiional erm from he saionary disribuion of he process. Similarly o κ, we employ a univariae random walk MH scheme for sampling he laen saes γ c for fixed cohor c. Model M3 includes he residual vecors, R, as addiional laen saes. As for κ, he full-condiional log-poseriors for he vecors a he boundary years of he daa include a single erm from he condiional likelihood of he residuals models, while all inermediae years comprise wo such mulivariae normal erms. Again, for R 1 we include he saionary erm of he mulivariae normal which sems from assumpion in Equaion 14). Sampling he laen residuals vecors is achieved by a mulivariae random walk MH scheme cenred a ha curren value of he chain and appropriaely uned diagonal covariance marix. Having described he sampling procedure for he laen saes for he models, now we deail he mehod for he parameers of he underlying sochasic processes. The random walk model for κ is presen in all models and is parameers, δ and V, are conjugae wih bivariae normal and IW full condiional logposerior disribuions, respecively. As such, he Gibbs sampler is implemened o updae hese parameers. Model M includes he parameers of he cohor process, α γ, δ γ and σ γ. The firs wo have inracable fullcondiional log-poserior, and herefore random walk MH sampling is used o approximae heir poserior disribuions. The full-condiional of σ γ is conjugae such ha: σ γ IG n ) c 1, 1 αγ γ 1 δ ) γ 1 α γ ) n c ) + γj δ γ α γ γ j 1. 18) Model M3 comprises he parameers of he residuals model, A and V Z. Marix A is involved in he summaion included in he l 4 par of he log-likelihood bu moreover, i is also conained in he sysem whose soluion yields he marix V R for he uncondiional poserior erm of R 1. No compac form arises for he funcional specificaion of he full-condiional log-poseriors of hese parameers. Thus, he enries of he auoregressive marix A, α 1,..., α 6, are updaed individually based on normal random walk MH seps cenred a he curren sae of he chain and appropriaely uned proposal variance. A his poin we describe he only differeniaion beween models M3a and M3b, which occurs for he residuals model covariance. j= The covariance marix, V Z, of M3a is assigned a diagonal srucure wih a 10

12 common parameer v. This form implies independence beween he laen residuals across he age dimension for fixed ime,. Parameer v is involved in he saionary covariance V R so ha no precise conjugae relaionship holds, and a normal random walk MH sep is applied o sample for i. On he oher hand, M3b assumes a fully parameerised covariance marix for V Z. Again due o he presence of he saionary covariance marix, V R, he form of he full-condiional log-poserior of V Z is no conjugae. However, i is close o an IW disribuion, which we use as he proposal generaor and se up a MH sampling sep. This process approximaes efficienly he fully parameerised covariance marix V Z which capures he correlaion srucure of he residuals saes across all ages. 4 Empirical Resuls In his secion we presen he resuls of he previous fiing procedures. The Human Moraliy Daabase England and Wales EW) males exposures and deahs sample is used for 50 years, from 1960 o 009, and for 30 ages, 60 o 89 inclusive. Firs, we discuss he fi of he models for he full daa-se. Second, we conduc a sensiiviy analysis for he parameer esimaes, iniially, by fiing he models o he final 30 years of he daa, from 1980 o 009, and hen, by fiing he models o he firs 40 years of daa from 1960 o 1999, hus leaving he final 10 years of he complee daa-se for validaion. The MCMC algorihms are ieraed for 1,050,000 cycles, of which only every 50 h ieraion is kep o yield a poserior sample of size 1,000. The firs 1,000 ieraions are discarded as he burn-in period for all chains. 4.1 Daa Analysis, years Convergence of all models is fas and he applied burn-in period is possibly longer han required. This is more he case for models M1 and M, where he algorihms converge wihin he firs 10 o 0 ieraions. For models M3, some lower level parameers may require up o 00 o 300 runs. The hinning inerval is adequae in he sense ha he individual MCMC chains show no significan auocorrelaion a any lag, excep for some of he elemens of marix A for models M3. Also, he races of some elemens of marix A develop noiceable cross-correlaion. The accepance raes for he MH seps of he algorihms vary beween 10-5% across he laen saes and parameers of he models, and hese values are consisen wih wha is recommended in he relevan lieraure. The majoriy of MCMC races are as expeced, and some of he mos indicaive are displayed in Figures 1 and below. Some furher exhibis for he case of he vecor of laen saes, κ 50, are also given in Appendix C. Firs we commen on he convergence race of he auoregression coefficiens of A for models M3a and M3b. We have excluded he burn-in period for parameers α 4, α 5 and α 6 in order o inspec closely he differences beween he wo poserior disribuions. The disincion beween hese esimaes for he wo models is due o he fully populaed residuals covariance marix of M3b, insead of he diagonal srucure of M3a. The panels of Figure 1 illusrae ha coefficiens α 1, α and α 3 converge o differen values for he wo models. This suggess ha he covariance marix of M3b capures and absorbs significan ineracion 11

13 from he auoregression for he firs wo ages of he daa. A paricular poin can be made for he behaviour of α1. The single variance parameer v of M3a, and he absence of correlaion wih residuals of oher years of he daa, leads α1 o be very close o one, due o he imposed consrain. The implicaion will be ha residuals R60, ) will be deermined by a very srong auoregressive relaionship. On he conrary, α1 of M3b is cenred o significanly lower values and is poserior disribuion is more volaile han α1 of M3a. The poserior spread of all oher parameers of marix A is greaer for model M3a, implying ha he fully populaed covariance marix, VZ, reduces he variabiliy of he auoregressive parameers. AR coefficien, α, EW daa AR coefficien, α1, EW daa Ieraion Ieraion AR coefficien, α3, EW daa. AR coefficien, α4, EW daa Ieraion Ieraion AR coefficien, α5, EW daa. AR coefficien, α6, EW daa Ieraion Ieraion Figure 1: MCMC races for elemens of auoregressive marix A. Black race is for model M3a and grey race is for model M3b. Nex we invesigae he behaviour of he MCMC chains for some of he laen residual saes and for he residuals covariance marix elemens. Figure compares he MCMC races of R65, 009), R75, 009) and R85, 009) beween models M3a and M3b. 1

14 Residual sae, R65,009), EW daa. Residual sae, R75,009), EW daa Ieraion Ieraion Residual sae, R85,009), EW daa. Elemens of marix V Z, EW daa Ieraion Ieraion Sd. Residuals, M1 Sd. Residuals, M Age Age Year Year Sd. Residuals, M3a Sd. Residuals, M3b Age Age Year Year Figure : Top and lef of nd from op: Comparaive MCMC races for models M3a and M3b in black and grey colour, respecively. nd from op: Righ frame compares he race of he single variance parameer v of marix V Z of M3a agains ha of he variance parameer for age 65 of marix V Z of M3b. Lower wo panels: Sandardised residuals plos of all models, indicaed in grey and black squares for posiive and negaive values, respecively. 13

15 These hree cases display he disincive behaviour of he residuals across ages. The laen residual sae for age 75 is clearly negaive, whereas for ages 65 and 85 i is posiive. In all hree cases he residual sae compensaes for wha is no capured by he moraliy basis of model M1, resuling in a beer fi. The boom righ panel in he second row of Figure includes he plo of he MCMC race of he variance parameer v, he common variance parameer across all ages, of marix V Z for M3a, along wih he race of v 65 of V Z, he variance of he residuals series for age 65, for M3b. The poserior disribuion of v of M3a has much smaller variance compared o v 65 of M3b, and his is generally he case across all ages. We compare he goodness of fi of he models by inspecing he bi-dimensional sandardised residuals plos. The lower wo panels of Figure display grey-black plos, where grey are posiive and black are negaive sandardised residuals values for all models. The plo of M1 indicaes an obvious remaining srucure no capured by he model. The plo of M is improved bu clusers of posiive or negaive sandardised residuals can sill be observed. Boh models M3 depic fairly random plos across boh dimensions. A sligh improvemen in ha respec could resul from he addiion of a furher age-period ineracion erm in he moraliy basis of M1. Figure 3 plos he poserior medians of he laen age and period effecs of all models for he hree differen ime-frame esimaions. In his subsecion, we focus on he poins corresponding o he full daa ), and commen on he disincive behaviour of he laen saes esimaes beween he four models. Commens regarding he sensiiviy of he laen saes o he esimaion ime-frame are reserved for he nex subsecion. The age dependen moraliy basis, α x, appears idenical for all models. The key drivers of he moraliy dynamics as capured by he vecor of period effecs, κ, signify wo main poins. Firs, he esimaes of he period effec vecors are no affeced by he correlaion srucure of model M3b, and are pracically idenical beween M3a and M3b. Second, he κ s of models M3 are much closer o hose of M1. This is in conras o he respecive behaviour of M, where he cohor process disors he κ esimaes, especially for κ ). Small discrepancies beween he κ ) s of M1 and of models M3 may be noiced for he very final years in he daa. The rend of he period effecs esimaes also deermine he parameers, δ and V, based on which he random walk model for κ will be projeced. Close inspecion of he corresponding densiies in Figure 4 shows ha alhough he il of he period effecs of M, he only observable difference occurs a he ails of he poserior disribuions of parameers σ and σ 1, while he drifs remain almos unchanged. Neverheless, he addiion of he cohor process, γ, appears o affec he κ series, and models M3 seem o capure he saionary sochasic componen of he moraliy dynamics in a properly independen manner. Furher, we look ino how he esimaed residuals model is described by is parameers. The model is primarily deermined by he auoregressive marix, A, and differences in he auoregression coefficiens of models M3a and M3b are solely due o he full correlaion srucure of M3b in conras o he single-parameer diagonal covariance marix of M3a. Figure 5 shows he sampled poserior densiies of he elemens of A for models M3. As above, we focus in he full daa esimaed densiies in black and reserve commens regarding he sensiiviy o he esimaion ime-frame for he nex subsecion. As also commened earlier, he mos sriking differeniaion occurs for coefficien α 1, which in he presence of he full covariance marix of M3b 14

16 is significanly shifed o lower values. Beyond he qualiaively differen behaviour of he R60, ) series, α x, EW daa M1 M M3a M3b Age κ 1), EW daa M1 M M3a M3b Year M1 M M3a M3b κ ), EW daa Year Figure 3: Poserior medians of age dependen moraliy basis, α x, and period effecs, κ, for models M1, M, M3a and M3b. 15

17 Drif, δ 1, EW daa. Drif, δ, EW daa. Densiy M1 M M3a M3b Densiy M1 M M3a M3b e+00 5e 04 1e 03 Covariance marix, σ 1, EW daa. Covariance marix, σ, EW daa. Densiy M1 M M3a M3b Densiy M1 M M3a M3b e+00 1e 06 e 06 3e 06 4e 06 Covariance marix, σ 1, EW daa. Densiy M1 M M3a M3b 0e+00 1e 05 e 05 3e 05 4e 05 5e 05 6e 05 Figure 4: Poserior densiies of he bivariae random walk model parameers for he vecor κ for models M1, M, M3a and M3b. 16

18 AR coefficien, α 1, EW daa. AR coefficien, α, EW daa. Densiy A B C M3a M3b C A B C A B Densiy B A C A B C M3a M3b A B C AR coefficien, α 3, EW daa. AR coefficien, α 4, EW daa. Densiy M3a M3b A B C B C A B A C Densiy M3a M3b A B C C C A A B B AR coefficien, α 5, EW daa. AR coefficien, α 6, EW daa. Densiy M3a M3b A B C C A A C B B Densiy M3a M3b A B C A B C B C A Figure 5: Poserior densiies for he elemens of he auoregression marix A for models M3a and M3b. 17

19 a furher implicaion of ha shif is ha he residuals model of M3b is clearly saionary, whereas M3a srongly relies on he resricion of α 1 being less han 1 wihin he algorihm. Deviaion is also observed for he α 3 coefficien of he second age of he daa, which under M3b is clearly cenred around a zero. Coefficiens α 4, α 5 and α 6 describe he residuals behaviour for he majoriy of ages. The main difference beween he poserior disribuions of hese parameers under models M3a and M3b appears for α 6, which in he presence of he full covariance marix of M3b is cenred around zero. Combined wih he similar shif of α 3, we conclude ha he covariance marix of M3b absorbs significance from he ime dimension of he moraliy surface and embodies he dependency in he age dimension. Figure 6 examines he poserior esimaes of he covariance marix V Z for models M3. The op panel shows he elemen-wise poserior median of V Z of M3 scaled o illusrae he correlaion for he full daa-se esimaion. The graph shows mosly posiive residual correlaion for adjacen ages, which ypically decreases uniformly. This is an addiional dependence srucure across he ages in he daa which will incorporae furher variabiliy in he forecass beyond he cohor effec which is mainly capured by coefficiens α and α 5 of marix A. The variance parameers of M3b across ages are compared he o common variance parameer of M3a in he lower righ panel of Figure 6, where he median esimaes are shown. The esimae for he firs age in he daa-se is excluded since i is significanly grea and disors he illusraion. We observe ha for he majoriy of ages he residuals variance parameers of M3b are lower han he single parameer of M3a. This was expeced due o he effec of he covariance elemens of M3b. 4. Sensiiviy Analysis Here we refer o he plos of he previous secion and examine he robusness of he parameers wih respec o he esimaion ime-frame. In Figure 3 we examine he impac of he ime-frame on he poserior medians of he laen age and period effecs. For each differen esimaion ime-frame he age-relaed basis, α x, is idenical for all models. The period effecs κ 1) perfecly mach for he esimaion across all models, whereas wih he daa sligh deviaions occur in some cases under M, bu less inense han hose observed for he full daa. Similar behaviour is shown for period effecs κ ), where he esimaes for he ime-frame for M are closer o hose of he oher hree models. In Figure 4 we assess he impac of he calibraion ime-frame on he parameer esimaes of he random walk model for κ. The greaes differences wihin he drif vecor occur for δ 1, which is shifed significanly lower and higher, compared o he full daa medians, for he and esimaions, respecively. The second componen, δ, appears o remain cenred around a sable region in all hree cases, wih an obvious shif o higher values under he ime-frame Also, noiceable are he shifs in he densiies across he four models for ha ime-frame, mos possibly due o he inclusion of only 30 years. The poserior disribuions of he hree parameers of he covariance marix of he random walk model share similar behaviours. Firs, all hree esimaed densiies are significanly lower for he smaller ime-frame, Second, all hree esimaed densiies for he 18

20 ime-frame develop a heavier righ ail Age Age M3a, residuals variance, v, EW daa. M3a and M3b, residuals variances vs age Densiy 0e+00 1e+05 e+05 3e+05 4e e 05 e 05 3e 05 4e 05 5e 05 6e 05 7e 05 e-05 4e-05 6e-05 8e-05 1e Age Figure 6: Poserior summaries of covariance srucures for models M3a and M3b. Top: Implied correlaion marix from he poserior medians of he covariance marix V Z of model M3b for he full EW daa-se. Boom lef: Poserior densiy of parameer v of he covariance marix V Z of model M3a for all hree simulaions. Boom righ: Poserior medians of diagonal elemens of V Z of M3b agains he consan parameer v of M3a for he full EW daa-se. 19

21 A final observaion for all hree parameers of he covariance marix concerns he sabiliy of he esimaion across models for he ime-frame, where all densiies appear o be closer one o anoher compared o some insances of he oher wo cases. Figure 5 focuses on he robusness of he elemens of marix A of he residuals model. I can be seen ha parameers α, α 3 and α 6 of M3b do no change subsanially under he differen ime-frames. On he conrary, α 4 and α 5 of M3b appear o shif by changing he esimaion ime-frame, his being more inense for α 5 which reflecs he impac of he cohor effec. Also, he esimaes of he elemens of A of M3a seem o be sysemaically impaced by he change in he esimaion ime-frame. Finally, in he lower lef panel of Figure 6 we compare he change in he esimaed poserior densiy of he single variance parameer v of he diagonal covariance marix V Z of model M3a by changing he esimaion ime-frame. For he smaller daa-ses his resuls in poseriors cenred around significanly lower values wih reduced poserior variance compared o he densiy from he esimaion of he full ime-frame. 5 Applicaions In his secion we presen applicaions of he moraliy models. Firs we look a he forecasing properies of he models, and we conduc a brief back-esing exercise. Furher, we calculae sandard moraliy risk merics in order o quanify he differences beween he forecass of he moraliy models. Finally, we examine he difference in he forecass when he saionary moraliy dynamics are modelled by a cohor process and when modelled by he residuals. 5.1 Forecasing Properies Firs we examine he forecass of he models under esimaion based on he full daa, We have a join poserior sample of size 0,000 from which we sample wihou replacemen 10,000 imes and forecas he models for 10 years ahead. The full se of hisorical observaions and associaed model forecass for he firs 50 years of he log-raes for ages 65, 75 and 85 are shown in Figure 7. The forecass of M1 conain only non-saionary dynamics. If he saionary moraliy dynamics are modelled via he cohor process, γ c, he graph of M shows increasingly volaile forecass for as long as esimaed saes are incorporaed in he forecass and unil he cohor process converges. This is paricularly eviden for he higher ages, 75 and 85, of he graphs. Model M3a seems capable of capuring he saionary moraliy dynamics in a similar manner o M since he same rend is generaed by he forecass for all hree ages. However, he observed curvaure is creaed by he Markov mulivariae auoregressive relaionship of he residuals model raher han he volaile cohor esimaes of M. Hence, our framework capures he acual moraliy dynamics insead of jus incorporaing volaile laen effecs ha improve he fi of he model o he given daa-se. Moreover, he evoluion of he projeced raes under M3a is much smooher compared o M. The graph for M3b shows ha he smooh projecions of M3a are due o he sronger auoregression implied by coefficien α 1 of he residuals model of M3a. 0

22 Model M1 Model M Age 85 Age 85 Log rae Age 75 Age 65 Log rae Age 75 Age Year Year Model M3a Model M3b Age 85 Age 85 Log rae Age 75 Age 65 Log rae Age 75 Age Year Year Figure 7: Hisorical observaions and associaed forecass of log-raes for ages 65, 75 and 85 for 50 years ahead for all models. Fan chars illusrae 50% o 95% confidence bands a 5% sep. 1

23 Model M1 Model M Log rae Age 85 Age 75 Age Log rae Age 85 Age 75 Age Year Year Model M3a Model M3b Log rae Age 85 Age 75 Age Log rae Age 85 Age 75 Age Year Year Figure 8: Hisorical observaions and associaed forecass of log-raes for ages 65, 75 and 85 for 60 years ahead for all models. The graphs show he forecass under all hree esimaion ime-frames.

24 Figure 8 plos he hisorical observaions and he projeced medians of he four models under all hree esimaion ime-frames, his ime for half he forecased period, up unil 070. The graphs allow close examinaion of he qualiaive properies of he forecass, and in paricular he smoohness of model M3a in conras o he jagged oucomes of models M and M3b. A furher poin o noe is he difference beween he forecased medians depending on he esimaion ime-frame. If he final 30 years of he daa-se are only employed o deduce he parameers of he underlying sochasic models, fuure raes appear significanly lower compared o he forecass under he full available ime-frame. The principal reason for his behaviour is lower esimae of he drif componen, δ 1, under he runcaed daa esimaion. Finally, we backes he moraliy models by comparing he projeced raes for years agains he observed. In all cases, he models fail o capure he realised raes, and he discrepancies are generally greaer as he examined age increases. 5. Moraliy Risk Merics In his secion we calculae moraliy indices and merics in order o compare he behaviour of he models and quanify heir differences. A sandard moraliy relaed financial measure is he price of an annuiy. Based on he forecass of he previous secion, Table 1 shows he mean and sandard deviaion of he value of a 5-year erm immediae uni annuiy paid in arrears issued o a life aged 65 in 010, he firs forecased calendar year, under a consan ineres rae of 4%. We have calculaed he summary saisics of he annuiy based on he fi of he models for wo of he esimaion ime-frames of he previous secion. We do expec he annuiy disribuion o be higher under esimaion ime-frame , since he forecased moraliy raes are considerably lower in ha case. As he resuls show, his urns o be indeed he case for all our models. Addiionally, we see ha he sandard deviaion of he annuiy disribuion is significanly lower for he 30 years daa-se. Focusing on he behaviour across models for fixed esimaion window, model M1 yields he highes mean values and model M he lowes. Models M3 resul in mean values ha are close, and he expeced annuiy value under M3b urns ou o be greaer for boh esimaion ime-frames. Finally, model M1 reurns he lowes annuiy sandard deviaion since i does no ake ino accoun he saionary moraliy dynamics. Nex we examine he forecasing impac of modelling he saionary moraliy dynamics under he hree models. In Figure 9 we plo he difference in he median forecass beween M1 and each of he hree models which include a saionary dynamics componen, for ages 65, 75 and 85. Essenially, we plo he raio: sx, ) Mj = mx, )Mj mx, ) M1 mx, ) M1, 19) where mx, ) Mj is he median moraliy rae forecas for model Mj = M, M3a, M3b. sx, ) Mj can be seen as he median forecased spread of each model if M1 is he basis. 3