Uncertainty and coherence of mortality projections

Transcription

1 Uncerany and coherence of moraly projecons Søren Fg Jarner and Chresen Dengsøe Absrac hs paper dscusses uncerany and coherence of moraly projecons for sochasc moraly models ha employ a random walk wh drf o descrbe he connued decrease n deah raes over me (he perod effec). he random walk s a popular choce because of s smplcy and because s perceved o assume no srucure on he fuure moraly evoluon. I s rounely used o model he perod effec n, e.g. he model by Lee and Carer (199) and he models consdered n Carns e al. (008). In he paper we sudy how parameer uncerany and srucural uncerany accumulae n random walk ype moraly models, and we show ha snce he random walk does no dsngush beween shor-erm and long-erm uncerany he long-erm forecasng uncerany merely reflecs he varably n shor-erm flucuaons. Nex ncoheren and coheren moraly projecons of groups of counres are consdered. Separae analyses based on random walk ype moraly models end o exaggerae shor-erm dfferences and lead o dvergng projecons, whch seems hghly mplausble n he lgh of hsorc smlares. hs leads o he noon of coherence and convergence of moraly levels. Mos developed counres have experenced a smlar moraly evoluon, due o smlares n soco-economc facors, lfesyle, level of reamen and oher underlyng facors. he hypohess beng ha moraly levels are lkely o connue o evolve n parallel wh emporary counry-specfc devaons. We descrbe he SAIN framework of Jarner and Kryger (008) whch s based on hs hypohess and we use o oban coheren projecons preservng hsorc smlares for a group of developed counres. Keywords: Sochasc moraly modellng, sources of uncerany, srucural uncerany, parameer uncerany, random walk ype moraly models, ncoheren vs. coheren moraly projecons, convergence of moraly levels, SAIN framework. Conac deals: Søren Fg Jarner (sj@ap.dk) and Chresen Dengsøe (cd@ap.dk), he Dansh Labour Marke Supplemenary Penson Fund (AP), Kongens Vænge 8, DK-3400 Hllerød, Denmark 1 Inroducon A key feaure of sochasc moraly modellng s ha s possble o quanfy he uncerany assocaed wh forecased moraly levels. he uncerany s ypcally communcaed n erms of confdence nervals for he quanes of neres, e.g. he underlyng age-specfc deah raes or aggregae measures such as lfe expecancy a brh, or he probably of survval o a ceran age. Ofen one calculaes a hgh probably confdence nerval, e.g. 95%, wh he common nerpreaon ha hs nerval represens he possble values for he gven quany. he fac ha wh some probably he confdence nerval may no conan he rue value s generally gnored. he ably of sochasc moraly models o quany he uncerany probablscally and objecvely s a man advanage compared o sac moraly projecon mehodologes, such as exper judgemen or deermnsc models. Askng dfferen expers or ryng dfferen deermnsc models lead o dfferen subjecve answers. he answers represen a range of plausble values bu hey canno be combned no a sngle probablsc saemen. In hs paper we wll only dscuss sochasc moraly models. 1

2 Many sochasc moraly models rely on a random walk (wh drf), or smlar me-seres model, o descrbe he sochasc evoluon of deah raes. However, he random walk srucure s unable o dsngush beween shor-erm and long-erm uncerany, n he sense ha large shor-erm (annual) flucuaons n moraly levels auomacally accumulae o large long-erm projecon uncerany, whle small shor-erm flucuaons lead o small long-erm uncerany. hs mples ha he projecon uncerany can be very dfferen for wo populaons wh seemngly smlar hsorc moraly mprovemens. Moraly projecons are ypcally performed for one populaon a a me on he bass of he moraly experence of ha populaon alone. Snce mos ndusralzed counres have experenced smlar moraly mprovemens over he las 50 o 60 years one would expec ha separae projecons performed for each counry would lead o smlar resuls. hs, however, s no he case f he projecons are based on randomwalk ype moraly models. he man argumen for usng a random walk o model fuure moraly mprovemens s ha s a smple model ha makes mnmal assumpons abou he srucure of fuure moraly. However, he lack of srucure mples ha random-walk ype moraly models end o exaggerae shor-erm dfferences. hs leads o dvergng projecons for he group of ndusralzed counres, whch seems hghly mplausble n he lgh of hsorc smlares. As he le suggess he purpose of hs paper s wofold. Frs, o show how uncerany eners no projecons based on random-walk ype moraly models. hs wll be done usng he well-known Lee-Carer model nroduced by Lee and Carer (199), bu he conclusons hold rue for many oher moraly models as well. he second purpose s o sugges and llusrae a framework for coheren,.e. preservng hsorc smlares, moraly projecons for groups of relaed populaons. he framework s called SAIN (acronym for Spread Adjused InerNaonal rend) and was nroduced n Jarner and Kryger (008). Sochasc moraly modellng Assume ha we have observed he number of deahs of people of age x n year, D(), and he correspondng exposure, E(),.e. he oal number of years lved by people of age x n year n he populaon under sudy. We have hese daa for a range of ages and a range of years. From hese daa we form he (crude) deah raes m() = D() / E(). Ofen, bu no always, we wll model he age-specfc deah raes for each gender separaely, or for some oher subpopulaon of neres, bu hs wll no be par of he noaon. Consder he followng generc (observaonal) model: m( ) = F( η( ), E( )), (1) where η() s he rue underlyng deah rae and F specfes he dsrbuonal assumpons of he daa gven he rue deah rae and he exposure, ha s, F specfes he error srucure. In sochasc moraly modellng we are prmarly neresed n modellng he underlyng deah rae η, whle we consder he exposures as gven (non-sochasc). he observaonal equaon above s used o esmae parameers n a sochasc model for η, whch s subsequenly used for forecasng he rue deah rae. Generally we do no model fuure exposures and hence we do no explcly model fuure realzed deah raes. Whn he sochasc moraly framework we dsngush beween sysemac and unsysemac varably. Sysemac varably s he varably n m() ha sems from varably n he underlyng rue deah rae η. hs varably s ndependen of he populaon sze. Unsysemac varably s he varably n m() caused by he sochasc naure of deah and he fac ha he populaon s fne. hs varably decreases wh he sze of he populaon.

3 As an example consder he followng specalzaon of he generc model: D( ) η ( ) ~ Posson( η( ), E( )) Wh hs specfcaon we have E ( D η ) = Var( D η) = ηe where we have omed he dependence on x and o ease he noaon. Dvdng by he exposure we furher have E( m η) = E( D / E η) = η Var( m η) = Var( D / E μ) = η / E and usng hs we can decompose he varably n m as follows E( η) Var( m) = Var(E( m η )) + E(Var( m η)) = Var( η) + E he varance of η s ndependen of he populaon sze and gves rse o he sysemac varably n m, whle he las erm whch depends on he average rue deah rae and he populaon sze quanfes he unsysemac varably. I s seen ha he unsysemac varably ends o zero as he exposure ends o nfny..1 Sources of sysemac uncerany In hs paper we wll focus on sysemac uncerany,.e. he uncerany ha comes from vewng he underlyng rue deah raes as sochasc. Unsysemac uncerany,.e. he uncerany assocaed wh he acual number of deahs or he realzed deah raes n a populaon when he rue deah rae s known, s generally neglgble n large populaons expec n he very old age groups wh low exposure. here are suaons however where s mporan o consder unsysemac varably, e.g. when prcng moraly dervaves such as survvor swaps wren on specfc (small) populaons. An n-deph dscusson of hs opc s however ousde he scope of he curren paper. Consder a generc sochasc moraly model for he rue deah rae η: η ) = G( ; θ,{ ω } ), () ( s s where G s he model parameerzed by θ, and he ω s form a sequence of sochasc varables represenng he sochasc naure of he deah raes. As he noaon suggess sochasc moraly models wll ofen have he srucure ha deah raes are consdered o evolve hrough me, and hence hey can depend on randomness up o and ncludng he curren me, bu no afer he curren me. In he presen conex we dsngush beween hree sources of uncerany: Srucural uncerany s he uncerany ha relaes o he sochasc naure of he model,.e. he uncerany due o he ω s gven he choce of G and he value of θ. hs uncerany s ofen communcaed by calculang (ponwse) confdence nervals for he deah raes hemselves or for aggregae quanes such as lfe expecancy. 3

4 Parameer uncerany s he uncerany relaed o he value of θ gven he choce of G. Assumng G s correc he uncerany n θ comes from he need o esmae he parameers from (a fne amoun of) daa. he parameer uncerany can be assessed by calculang confdence nervals n a frequens analyss, or he poseror dsrbuon n a Bayesan analyss. Model uncerany s he uncerany relaed o he choce of G. Dfferen choces of G wll lead o dfferen resuls and alhough one may be able o exclude some models on he bass of lack of f here wll be many models ha are conssen wh daa. In he end he specfc choce of model wll be subjecve. he frequens approach o assess model uncerany s o perform a sensvy analyss ryng dfferen models, bu hs does no lead o a probablsc quanfcaon. In a Bayesan framework s n prncple possble o make a probablsc quanfcaon of model uncerany f one s wllng o assgn pror probables on he se of possble models, cf. Carns (000). 3 Random-walk ype moraly models A random walk wh drf μ s a sochasc process of he form X = X 1 + μ + ε where he ε s are ndependen, dencally dsrbued random varables wh mean zero and varance σ. hs process, or relaed me-seres models, les a he core of mos moraly models where s used o descrbe he connued decrease n deah raes over me (he perod effec). I s used n he orgnal Lee-Carer model nroduced by Lee and Carer (199), and also n he vas number of exensons and modfcaons of he orgnal model, e.g. Brouhns e al. (00), Lee and Mller (001), Renshaw and Haberman (006), Renshaw and Haberman (003), de Jong and ckle (006), Curre e al. (004). he random walk also descrbes he perod effec n he class of models consdered n Carns e al. (006) and Carns e al. (008). Usng he ermnology of he prevous secon he srucural uncerany s he uncerany n fuure value of X due o he sochasc evoluon of he process,.e. due o he ε s. he parameer uncerany s he uncerany n he values of he parameers conrollng he drf, μ, and he varance, σ, and orgnaes from he esmaon uncerany of he parameers. Model uncerany s he uncerany assocaed wh choosng a random walk model nsead of any oher model. Normally we assume ha he ε s are normally dsrbued, and hs choce can also be consdered par of he model uncerany. In he followng we wll sudy how he parameer uncerany and srucural uncerany accumulae n random walk-ype moraly models and llusrae he mplcaons for he projecon uncerany of wo dfferen daa ses. he conclusons are vald for all random walk-ype moraly models, bu o focus he dscusson and clarfy he exposon we wll llusrae he pons usng he famlar Lee-Carer model as an example. he Lee-Carer model proposed by Lee and Carer (199) akes he form: log( m ( )) a x + bxk + ε = (3) where he a s and b s are age-dependen ses of parameers conrollng respecvely he moraly level and he response (rae of mprovemen) o he ndex k. he moraly ndex k s a unvarae me-seres capurng he perod effec of moraly mprovemens, and he ε s are measuremen errors he dsrbuon of whch are lef unspecfed n he orgnal paper. In order o ensure denfably he parameers are consraned by x = 1 and k = x b 0 he age-dependen consans (a x ) are esmaed by averages over me of log(m()), whle (b x ) and he me-varyng ndex k are esmaed by sngular value decomposon (SVD) of he marx of he logarhms of deah raes mnus he esmaed a s. 4

5 Snce he dsrbuon of he measuremen errors s unspecfed he orgnal Lee-Carer model s srcly speakng no a fully specfed sascal model. Brouhns e al. (00) consder nsead he model D ) ~ Posson( η ( ) E( )), η( ) = exp( a + b k ) (4) ( x x he nerpreaon s he same as for he orgnal Lee-Carer model, bu phrasng he model as a sandard sascal model allows he use of sandard esmaon mehodology. he auhors propose o use a jon Maxmum Lkelhood Esmaon (MLE) of a x, b x and k. In boh Lee and Carer (199) and Brouhns e al. (00) he dynamcs of he moraly ndex (k ) s esmaed n a second sep as f (k ) were observed raher han esmaed quanes. In prncple jon esmaon of all parameers ncludng hose governng he k-dynamcs s possble by MLE. In a Bayesan framework Czado e al. (005) descrbe how a jon esmaon can be performed usng Markov Chan Mone Carlo (MCMC) echnques. In he followng we wll proceed as f he moraly ndex was observed. 4 Uncerany of moraly projecons he moraly ndex of he Lee-Carer model and all s varans, ncludng he one by Brouhns e al. (00), s ypcally modelled as a random walk wh drf k = 1 + μ + ε (5) k We assume, as s normally done, ha he ε s are ndependen, dencally normally dsrbued N(0,σ ). For ease of noaon we le he daa perod correspond o = 0,,. he sandard parameer esmaes and her dsrbuons are hen gven by 1 ˆ μ = ˆ σ = 1 1 = 1 Δk = 1 ~ N( μ, ( Δk ˆ) μ 1 σ ) ~ σ ( 1) 1 χ 1 (6) where Δk = k k -1 for = 1,,. hese esmaes are he maxmum lkelhood esmaes (excep ha we as cusomary dvde by -1 nsead of o oban a cenral esmae of σ ). By erang he defnng equaon (5) we can express fuure values of he moraly ndex n erms of he las observed value, k, and fuure nnovaons as k + h = k + h + ε ε + h, μ K (7) from whch forecass of he ndex are readly obaned by replacng μ by s esmaed value k + h = k + h ˆ μ + ε 1 + K ε h, (8) and usng he esmaed varance ˆ σ as he varance of he nnovaons. Srcly speakng he k +h n he equaon above represens an esmaed quany o be dsngushed from he rue nde bu o ease he exposon we wll no nroduce addonal noaon o make he dsncon vsble. 5

6 We can now form wo ypes of confdence nervals for he fuure values of he ndex. he frs ype consders only he srucural uncerany, reang he parameers as fxed, whle he oher ype akes he parameer uncerany no accoun also. For fxed parameers we have from (7) E( k Var( k + h + h k k ) = k + hμ ) = Var( ε K + ε + h ) = hσ and snce he (condonal) dsrbuon of k +h s normal we oban a 95%-confdence nerval for k +h by CI 95% ( k + h ) = k + hμˆ ± 1.96 h ˆ σ where we have replaced μ and σ by her esmaed values. akng parameer uncerany of he drf parameer no accoun (reang he varance parameer as fxed) we have from (6) and (8) E( k Var( k + h + h k k ) = k + hμ ) = Var( h ˆ μ + ε K + ε + h ) = h 1 σ + hσ from whch we oban a 95%-confdence nerval for k +h by CI 1 95% ( k h ) = k + h ˆ μ ± 1.96 h ˆ σ + h ˆ σ + (9) In prncple, we can also ake accoun of he parameer uncerany n he varance parameer, bu ha wll generally have only a small mpac compared o he oher unceranes and we do no consder ha here. I follows from (9) ha for small h he uncerany n he ndex wll be domnaed by he srucural uncerany from he nnovaons. However, snce he srucural uncerany ncreases wh he square-roo of h whle he conrbuon from he parameer uncerany ncreases lnearly n h asympocally he uncerany n he drf parameer wll domnae. Ignorng for smplcy all oher sources of uncerany, ncludng he measuremen error, we have from he above and (3) ha he varance of forecased log deah raes s Var(log( m( + h)) = b 1 ( h σ hσ ) x Var( k + h k ) = bx + and we hus oban a 95%-confdence nerval for he deah raes hemselves by 1 ( ax + bx ( k + h ˆ) μ ± 1.96 bx ( h ˆ σ + h ˆ )) CI95% ( m( + h)) = exp σ (10) As seen from he formulas he nnovaon varance, σ, conrols he sze of boh srucural uncerany and parameer uncerany. If σ s large here wll be large flucuaons n he value of he ndex from year o year (shor-erm srucural uncerany), and due o he random walk srucure he varance of he annual flucuaons wll accumulae (lnearly) over me and lead o very large uncerany n he long run (long-erm srucural uncerany). Furhermore, f σ s large he parameer uncerany of he drf parameer wll also be large and furher ncrease he oal uncerany of forecased deah raes. Conversely, f σ s small boh he shor-erm and long-erm srucural uncerany wll be small, and he uncerany n he drf parameer wll also be small. Hence, he fac ha he magnude of he randomness n he model s conrolled by only one parameer and he fac ha he random walk model over me accumulaes all randomness mples ha eher all ypes of uncerany (shor-erm, long-erm and parameer uncerany) are hgh or hey are all low. he maxmum lkelhood esmae of σ depends on he flucuaons (dfferences) from year o year of he moraly nde cf. (6). From a praccal pon of vew hs mples ha f we apply he model o a daa se wh large flucuaons n moraly levels from year o year we wll end o esmae a hgh value of σ (and also a hgh measuremen error), and consequenly boh srucural uncerany and parameer uncerany of 6

7 forecased deah raes wll be large. Conversely, f we apply he model o a daa se wh a smooh moraly evoluon he esmae of σ wll be small and he forecasng uncerany wll also be small. Fgure 1 llusraes he dfferences n forecasng uncerany n he wo suaons. he fgure shows he evoluon of Dansh and US female moraly for ages 60 and 70 and he confdence nerval (10) for he forecased deah raes. he US moraly evoluon has been raher smooh whle he Dansh moraly evoluon exhbs large varaon from year o year. Consequenly, he forecasng uncerany for he Dansh daa s subsanally larger han for he US daa. However, alhough he Dansh daa flucuaes more n he shor run han he US daa he deah raes n he wo counres sar and end a almos he same level, and s clear ha over he long-run he moraly mprovemens evolve n parallel n Denmark and US. On he bass of he hsorc daa we expec Dansh and US moraly o connue o evolve n parallel and we consder very unlkely ha he moraly levels wll ever devae by so much as he confdence nervals allow. hus he forecasng uncerany whch appears reasonable lookng a each counry n solaon appears unreasonable when compared across counres. In concluson he lack of srucure of he random walk model and he way uncerany accumulaes over me mply ha dfferences n shor-erm varably lead o even larger dfferences n long-erm varably. hs can cause unreasonable large dfferences n forecasng uncerany for populaons wh moraly levels evolvng n parallel. We have used he Lee-Carer model for llusrave purposes as s smple and well-known. However, he problem s a consequence of he underlyng random walk and no he specfc srucure of he Lee-Carer model, and he conclusons apply o all sochasc moraly models ha rely on random walks o model he perod effec. Deah rae 0.% 0.5% 1% % 5% Year Fgure 1: Dansh and US female moraly for ages 60 and 70 wh 95%-confdence nervals for forecass usng separae Lee-Carer models for each counry. he rregular curves are Dansh moraly and he smooher curves are US moraly. 7

8 5 Coherence of moraly projecons Overall mos developed counres have had a smlar moraly evoluon snce around 1950 wh subsanal mprovemens n deah raes and lfe expecancy. here have been counry-specfc devaons from he general paern wh some counres havng a seady declne year afer year whle oher counres have had alernae perods wh large moraly mprovemens and almos sagnaon. In some counres here have even been perods wh slghly ncreasng age-specfc deah raes for some age groups, e.g. Dansh female moraly for age 60 o 70 n he 1990 s, see Jarner e al. (008) for a dealed accoun of he Dansh moraly evoluon. However, he overall pcure remans ha moraly mprovemens hsorcally have occurred n parallel n mos developed counres. I s reasonable o beleve ha he observed smlares n moraly mprovemens are due o smlar developmens n he counres of he underlyng processes governng moraly, e.g. soco-economc facors, lfesyle, nuron, level of reamen ec. Wlson (001) and Wlmoh (1998) provde evdence for convergence n global moraly levels due o convergence of socal and economc facors, see also uljapurkar e al. (000). Gong forward seems lkely ha moraly levels n mos developed counres wll connue o evolve n parallel. Clearly here wll always be counry-specfc dfferences n moraly and n he underlyng facors, and a any pon n me hese dfferences can be subsanal. However, seems unlkely ha dfferences n mprovemen raes can perss n he long run and cause moraly levels o drf apar. he hsorc convergence of moraly levels s generally no preserved n forecass based on separae analyses of he ndvdual counres. Fgure shows he hsorc developmen snce 1950 n female moraly for ages 60 and 70 for he followng 18 ndusralzed counres: Ausra, Ausrala, Belgum, Canada, Fnland, France, Germany, Holland, Denmark, Ialy, Japan, Norway, Porugal, Swzerland, Span, Sweden, Uned Kngdom and Uned Saes. he daa s from he Human Moraly Daabase ( and for each counry consss of age-specfc deah couns and exposures for each year snce 1950 for ages 0 o 100. A Lee-Carer model has been fed for each counry and he forecas for age 70 s shown on he fgure. In he hsorc perod he counres have evolved n parallel wh emporary counry-specfc devaons from he general rend, bu he separae analyses lead o dvergng projecons due o small dfferences beween he counres. he fannng ou of he projecons seems unlkely n vew of he hsorc smlares. hs pon s hghlghed n Fgure 3 whch shows female moraly for age 70 for each counry relave o he average deah rae for all he counres. Hsorcally deah raes have been whn a facor of abou 3/ of he average value (he horzonal doed lnes are ploed a /3 and 3/ respecvely), bu n he projecons 8 ou of he 18 counres fall ousde of hs band n 070. We have used he Lee-Carer o llusrae how separae analyses of seemngly smlar daa ses generally lead o dvergng projecons. However, hs s by no means a problem specfc o he Lee-Carer model. On he conrary, mos moraly models wll when appled o each daa se ndvdually produce dvergng projecons. We call hs phenomenon ncoherence snce he projecons vewed collecvely are clearly a odds wh realy. Alhough mos moraly models are ncoheren n hs sense of he word he class of random walk ype moraly models are parcularly prone o be so due o he accumulaon of all (annual) flucuaons. In essence, random walk ype moraly models exrapolae shor-erm dfferences wh no regard o he absolue level. hs lack of srucure s boh he vrue and he problem of random walk ype moraly models. 8

9 Deah rae 0.% 0.5% 1% % 5% Year Fgure : Hsorc female moraly for ages 60 and 70 for 18 developed counres. he sragh lnes are projeced deah raes for age 70 based on separae Lee- Carer models. In order from op o boom of he projeced value n 070 he counres are: Denmark, Uned Saes, Uned Kngdom, Holland, Canada, Norway, Sweden, Belgum, Porugal, Germany, Ausrala, Ialy, Ausra, Swzerland, Fnland, France, Span and Japan. Deah rae/avg. deah rae Year Fgure 3: Hsorc and projeced female moraly for age 70 for 18 developed counres relave o he average deah rae of he counres. he projecons (sragh lnes) are based on separae Lee-Carer models for each counry. In order from op o boom of he projeced value n 070 he counres are: Denmark, Uned Saes, Uned Kngdom, Holland, Canada, Norway, Sweden, Belgum, Porugal, Germany, Ausrala, Ialy, Ausra, Swzerland, Fnland, France, Span and Japan. 9

10 5.1 SAIN framework he moraly evoluon of mos ndusralzed counres appears o follow he same long-erm rend. he mprovemens occur a dfferen mes n he ndvdual counres and here are dfferences n he varably of he mprovemen raes from year o year beween he counres, bu over he long-run he average mprovemen raes are very smlar and he moraly levels of he counres say whn a farly narrow band of each oher over me. Separae analyses of he counres wll generally no preserve he hsorc, long-erm smlares. In fac, separae analyses end o produce ncoheren,.e. dvergng, projecons. In he followng we wll descrbe a framework for coheren moraly modellng,.e. a framework for producng moraly projecons for a group of counres/populaons whch preserves he common long-erm rend. he framework and a specfc model whn he framework were nroduced n Jarner and Kryger (008) from he perspecve of lnkng a (small) subpopulaon o a larger reference populaon. Relaed deas can be found n L and Lee (005). he framework s dubbed SAIN for Spread Adjused InerNaonal rend and consss of wo pars: 1. A model for he common long-erm rend for he group of populaons under consderaon. Counry-specfc models for he shor o medum-erm devaons of he counry-specfc moraly levels from he common rend (he spread) We assume ha daa are of he form of deah couns and exposures for each counry for a range of ages and a range of years. We le D () denoe he number of deahs n counry for age x n year, and we le E () denoe he correspondng exposure. We do no assume ha he age ranges and years for whch we have daa for he varous counres necessarly concde. Furher, we le D In () and E In () denoe he sum of D () and E () over hose counres for whch we have daa for he gven age and year. We assume ha D In () are ndependen wh D ( ) ~ Posson( ( ) E ( )), In η In In (11) where η In () s he common nernaonal level. he common nernaonal level can be modelled as one pleases. In he applcaon n Jarner and Kryger (008) he common level s modelled by an elaborae fraly model wh he am of producng precse esmaes of fuure mprovemen raes n old-age moraly, whle n he example below we use a Lee-Carer specfcaon of η In (). In he laer case model (11) for he common nernaonal level concdes wh model (4) of Brouhns e al. (00). Havng esmaed he common nernaonal level we consder nex for each counry a model where D () are ndependen wh D ( ) ~ Posson( η ( ) E ( )), η ( ) = ˆ η In ( ) exp( r x ' y ) (1) he naonal moraly levels are gven n erms of spreads o he (esmaed) common nernaonal level. Each spread depends on a counry-specfc mulvarae me-seres ( y ) and a se of fxed, common regressors r x whch deermne he possble shapes of he spread. In Jarner and Kryger (008) and n he example below we use a se of hree regressors whch corresponds o level, slope and curvaure of he spread,.e. r x ' = (1, a1x + a0, b x + b1 x + b0 ) where he consans a1, a0, b, b1 andb 0 are chosen such ha he regressors are normalzed, mean-zero and muually orhogonal. Fxng, raher han esmang, he regressors has he advanage ha we can nerpre he value of he me-seres as he excess moraly n counry compared o he nernaonal level relaed o respecvely he overall level, old versus young age moraly, and old and young age versus mddle-aged moraly. 10

11 We frs esmae he me seres from (1) based on he counry-specfc daa and he esmaed value of he common nernaonal rend ˆ η In ( ). Noe, snce he regressors are fxed he maxmum lkelhood esmae of y depends only on daa for year. Second, we esmae a mulvarae, saonary me-seres model for ( y ). he me-seres model conrols he lengh and magnude of he counry-specfc devaons from he common level. he assumed saonary ensures ha he devaons are effecvely bounded such ha n he long run moraly levels n each counry wll evolve n parallel wh he common rend whle allowng subsanal shor o medum erm devaons from he rend. In he example below we use a hreedmensonal vecor auoregressve (VAR) model for ( y ): y = A y + b + ε ε ~ N (0, Ω ) (13) 1, 3 In Jarner and Kryger (008) s argued ha here should be no mean erm n model, bu for llusrave purposes we nclude a mean erm n he model n he example n hs paper. Forecass are easly obaned by separaely forecasng he common nernaonal rend and each counry-specfc spread usng (13), and hen combnng he wo forecass o produce a forecas for he naonal moraly levels usng he relaon n (1). Assumng ndependence of he nernaonal rend and he naonal spreads he forecas varance akes he form: Var (logη ( x, + h)) = Var(logη ( + h)) + r ' Var( y ) r In where s he las observaon year and h s he forecas horzon. he frs erm n he varance decomposon s he varance of he forecased nernaonal level. hs erm s common for all counres and wll ypcally, e.g. f we use a Lee-Carer model for he nernaonal level, end o nfny as h ends o nfny. he second erm comes from he counry-specfc spread and wll ncrease o a fne asympoc value as h ends o nfny due o saonary of he me-seres. Hence asympocally he uncerany of he common rend wll domnae and he long-run forecasng uncerany wll herefore be (almos) he same for all counres. Confdence nervals for he naonal moraly levels can be obaned by CI 95% x ( r ' yˆ ± 1.96 Var(log ( h)) ) ( η ( + h)) = ˆ η ( + h)exp + η In x h + x x h where ˆ η (, ) and In + y + h ˆ denoe forecased values. As an example of he proposed framework we have appled o he group of 18 developed counres consdered n Secon 5. We used a Lee-Carer model for he common nernaonal level and separae VARmodels of he form (13) for he counry-specfc spreads. Due o daa problems was no possble o esmae he me-seres for all years for Fnland. Furher, he esmaed VAR-model for Japan, Porugal and Span was no saonary. In a more dealed analyss boh problems could be remeded by changng he esmaon rounes and/or ryng dfferen me-seres models. However, for he purpose of hs example we have smply excluded he four counres. Fgure 4 shows he resuls for he remanng 14 counres: Ausra, Ausrala, Belgum, Canada, France, Germany, Holland, Denmark, Ialy, Norway, Swzerland, Sweden, Uned Kngdom and Uned Saes. As seen he mean forecass of he moraly levels evolve n parallel and converge o he common level offse by he mean erm of he VAR-model. Noe ha convergence of he mean forecass does no mply ha he acual deah raes wll convergence. he model s sochasc and each counry has a spread ha develops ndependenly of he oher spreads, hence he realzed deah raes wll never concde. he doed lnes are 95%-confdence nervals and he funnel hey shape essenally defnes he regon where all deah raes are expeced o le n he fuure. In conras o he forecas based on separae Lee-Carer models n Fgure, he SAIN forecas closely reflecs he hsorc smlares of he counres. 11

12 Deah rae 0.% 0.5% 1% % 5% Year Fgure 4: Hsorc female moraly for ages 60 and 70 for a group of developed counres, and projeced deah raes for age 70. he projecons are based on a common Lee-Carer model and separae me-seres models for he spreads. he doed lnes are 95%-confdence nervals for he forecased values. 6 Conclusons We have used he erm random walk ype moraly models for sochasc moraly models ha use a random walk wh drf o model he connued decrease n moraly over me,.e. he perod effec. hs class conans mos of he moraly models commonly used n pracse, e.g. he model of Lee and Carer (199) and he class of models consdered n Carns e al. (008). he consequences of he random walk srucure for forecasng uncerany and for he forecased levels are summarsed below: Uncerany: In random walk ype moraly models he nnovaon varance s esmaed from he annual flucuaons n moraly levels. Snce he nnovaon varance also deermnes he accumulaed (srucural) uncerany and he rend (parameer) uncerany hs can lead o large dfferences n forecasng uncerany for smlar populaons. Level: In random walk ype moraly models he rend s esmaed from he accumulaed annual dfferences. he model does no dsngush beween an underlyng rend and devaons from he rend whch mples ha forecass for populaons wh (small) hsorc dfferences wll dverge. Insead of usng separae (random walk ype) moraly models we nroduced he SAIN framework for obanng coheren projecons for groups of populaons. In hs framework we model a common rend and separae, saonary devaons from he rend. Regardng forecasng uncerany and he forecased levels he SAIN srucure mples Uncerany: Populaons wh dfferences n he sze of annual flucuaons wll have dfferen shorerm (spread) uncerany. However, snce he rend uncerany s he same and devaons are essenally bounded he populaons wll have smlar long-erm forecasng uncerany. 1

13 Level: he long-erm (mean) forecass for all he populaons consdered wll converge o he same long-erm rend offse by a facor reflecng hsorc dfferences beween he populaons,.e. all longerm forecass wll evolve n parallel. We have dscussed he noon of coheren projecons for groups of counres and ha s also he ermnology used n he SAIN framework. However, he concep of coherence s also of relevance n oher conexs where we wan o assure ha forecass for relaed populaons do no dverge, e.g. forecass for females and males n he same populaon, and he SAIN framework may be useful n hese suaons also. References N. Brouhns, M. Denu, J.K. Vermun (00). A Posson log-blnear regresson approach o he consrucon of projeced lfeables. IME 31, A. Carns (000). A dscusson of parameer and model uncerany n nsurance. IME 7, A. Carns, D. Blake and K. Dowd (006). A wo-facor model for sochasc moraly wh parameer uncerany: heory and calbraon. Journal of Rsk and Insurance 73, A. Carns, D. Blake, K. Dowd, G. Coughlan, D. Epsen and M. Khalaf-Allah (008). Moraly densy forecass: An analyss of sx sochasc moraly models. Pensons Insue Dscusson Paper PI I.D. Curre, M. Durban, P.H.C. Elers (004). Smoohng and forecasng moraly raes. Sascal Modellng 4, C. Czado, A. Delwarde and M. Denu (005). Bayesan Posson log-blnear moraly projecons. IME 36, S. Jarner and E. Kryger (008). Modellng adul moraly n small populaons: he SAIN model. Pensons Insue Dscusson Paper PI-090 (hp:// S. Jarner, E. Kryger and C. Dengsøe (008). he evoluon of deah raes and lfe expecancy n Denmark. SAJ 108, P. de Jong and L. ckle (006). Exendng Lee-Carer moraly forecasng. Mahemacal Populaon Sudes 13, R. Lee and L. Carer (199). Modelng and forecasng of U.S. moraly. JASA 87, R. Lee and. Mller (001). Evaluang he performance of he Lee-Carer mehod for forecasng moraly. Demography 38, N. L and R. Lee (005). Coheren moraly forecass for a group of populaons: An exenson of he Lee- Carer mehod. Demography 4, A.E. Renshaw and S. Haberman (003). Lee-Carer moraly forecasng wh age-specfc enhancemen. IME 33, A.E. Renshaw and S. Haberman (006). A cohor-based exenson o he Lee-Carer model for moraly reducon facors. IME 38, S. uljapurkar, N. L and C. Boe (000). A unversal paern of moraly declne n he G7 counres. Naure 405, J.R. Wlmoh (1998). Is he pace of Japanese moraly declne convergng oward nernaonal rends? Populaon and Developmen Revew 4, C. Wlson (001). On he scale of global demographc convergence Populaon and Developmen Revew 7,

14 Uncerany and coherence of moraly projecons ICA, Cape own, 7-1 March 010 Søren Fg Jarner Chresen Dengsøe

15 Agenda Sochasc moraly modellng Sources of uncerany Random-walk ype models Uncerany accumulaon Convergence of moraly levels Coheren projecons

16 Sochasc moraly modellng Daa for age x n year D() : Deah coun E() : Exposure Deah rae m() = D()/E() Generc moraly model m( ) F( ( ), E( )) Deah rae 0.3% 0.5% 1% % m(60,) fng η(60,) exrapolaon η-pah η-pah mean forecas η-pah η() : rue deah rae F : error-srucure Year

17 Sysemac and unsysemac varably wo ypes of varably n m() Varably n η() sysemac varably, ndependen of sample sze Measuremen error unsysemac varably, depends on sample sze Example D( ) ( ) ~ Posson( ( ) E( )) E( D ) Var( D ) E E( m ) E( D / E ) Var( m ) Var( D / E ) / E Var( m) Var(E( m )) E(Var( m )) sysemac varably unsysemac varably E( ) Var( ) E

18 Varably n η Generally ( ) G( ;,{ s} s ) θ : parameers G : model srucure ω s : randomness Parameer uncerany uncerany n θ (magnude of ω s) Deah rae 0.3% 0.5% 1% % m(60,) fng η(60,) exrapolaon η-pah cond. mean η-pah mean η-pah cond. mean Srucural uncerany unerany due o ω s Year

19 Sources of sysemac uncerany ype Assessmen Model uncerany Lee-Carer, logsc, ec. Sensvy analyss Bayesan mehods Parameer uncerany Improvemen raes, drf ec. Confdence nervals Poseror dsrbuons Srucural uncerany Sochasc projecons Mean forecas wh ponwse confdence bands

20 Uncerany accumulaon Random walk wh drf X, ~ N(0, ) X 1 buldng block used n mos moraly models o descrbe he connued decrease n deah raes over me (he perod effec), e.g. he model by Lee and Carer (199), and s varans he class of models consdered n Carns e al. (008) parameer uncerany (uncerany n μ and σ ) srucural uncerany (sochasc evoluon due o ε ) Insrucve o sudy how parameer uncerany and srucural uncerany accumulae n hese models

21 Example: Lee-Carer he model proposed by Lee and Carer (199) akes he form log( m ( )) ax bxk Sysemac varably Unsysemac varably o ensure denfably he parameers are consraned by x b 1 k 0 x Esmaes of he age-dependen consans (a x ) are gven by he averages over me of log(m()), whle (b x ) and he me-varyng moraly ndex k are obaned by SVD.

22 Refnemens Brouhns e al. (00) consder he model D( ) ~ Posson( E( ) ( )), ( ) exp( ax bxk ) μ s he force of moraly nroducng a sascal model allows jon MLE of a x, b x and k he moraly nde (k ), s modelled as n LC As n LC, he dynamcs of he moraly ndex s esmaed as f (k ) were observed raher han esmaed quanes n prncple jon esmaon of all parameers ncludng hose governng he k-dynamcs s possble by MLE Czado e al. (005) descrbe a full Bayesan analyss usng MCMC

23 Modellng he moraly ndex he ndex s ypcally modelled as a random-walk wh drf k, k 1 0,, Assumng ε..d. N(0,σ ) he esmaes and her dsrbuons are ˆ 1 k ~ N(, 1 ) 1 1 ( k ) ~ ( 1) 1 1 where Δk = k -k -1 for =1,,.

24 Forecasng Forecass are readly obaned from he expresson 95%-confdence nervals wh and whou uncerany n Ignorng oher sources of uncerany hese confdence nervals drecly ranslae no confdence nervals for m() ˆ ) E(, ˆ 1 h k k k h k k h h h 95% 1 95% ˆ 1.96 ˆ ) ( ˆ ˆ 1.96 ˆ ) ( h h k k CI h h h k k CI h h ˆ

25 Uncerany n forecass wh and whou uncerany n drf US female moraly for ages 60 and 70 Deah rae 0.% 0.5% 1% % 5% Daa Mean forecas 95%-CI whou drf uncerany 95%-CI wh drf uncerany Year

26 Projecon uncerany under random walk srucure Varance of forecased log deah raes: Var(log m( h)) bˆ x Var( k h ) bˆ x h ˆ 1 h ˆ srucural uncerany parameer uncerany assumng for smplcy: ˆ Var ( aˆ x ) Var( b x ) Var( ˆ ) Var( ) 0 RW srucure mples ha he same parameer (σ ) conrols uncerany n annual mprovemens (shor-erm srucural uncerany) sze of accumulaed devaons (long-erm srucural uncerany) uncerany n average mprovemen (parameer uncerany)

27 Long-erm uncerany reflecs shor-erm devaons DK and US female moraly for ages 60 and 70 Deah rae 0.% 0.5% 1% % 5% DK daa US daa Smlar moraly hsory n DK and US, bu projecon uncerany much hgher n DK due o hgher shor-erm varably DK 95%-CI ncludng drf uncerany US 95%-CI ncludng drf uncerany Year

28 Incoheren moraly projecons Convergence of moraly levels Smlar moraly evoluon n mos developed counres, due o smlares n soco-economc facors, lfesyle, level of reamen ec. Moraly levels are lkely o connue o evolve n parallel wh emporary counry-specfc devaons Incoheren moraly projecons Separae analyses exaggerae shor-erm dfferences and lead o dvergng projecons Seems hghly mplausble n he lgh of hsorc smlares Example * : Ausra, Ausrala, Belgum, Canada, Fnland, France, Germany, Holland, Denmark, Ialy, Japan, Norway, Porugal, Swzerland, Span, Sweden, Uned Kngdom, Uned Saes *Daa from Human Moraly Daabase

29 Dvergen Lee-Carer projecons from separae analyses Inernaonal female moraly for ages 60 and 70 Deah rae 0.% 0.5% 1% % 5% Denmark US UK Holland Canada Norway Sweden Belgum Porugal Germany Ausrala Ialy Ausra Swzerland Fnland France Span Japan Year

30 Hsorc smlares are no preserved Female moraly for age 70 relave o nernaonal average Deah rae/avg. deah rae Denmark US UK Holland Canada Norway Sweden Belgum Porugal Germany Ausrala Ialy Ausra Swzerland Fnland France Span Japan Year

31 Deah rae AU AUS BEL CAN FIN FRA GER HOL DK IA JAP NOR POR SCH SPA SWE UK 0.05% 0.1% 0.% 0.5% 1% % US Large varaon n projeced moraly and levels of uncerany Mean forecas and 95%-CI of female moraly for age 70 n 070

32 Shor-erm devaons from long-erm rend All counres appear o follow he same long-erm rend Bu mprovemens occur a dfferen mes n he ndvdual counres Varaon n annual mprovemen raes dffer beween counres Separae analyses Dvergng projecons wh non-overlappng confdence nervals Unreasonable varaon n forecasng uncerany nably of RW o dsngush beween shor- and long-erm uncerany Coheren moraly projecons Model common long-erm rend Allow counry-specfc shor-erm devaons from rend

33 SAIN (Spread Adjused InerNaonal rend) framework Common nernaonal rend Model of choce for μ In () esmaed from pooled nernaonal daa Model for counry : log ( x, ) log In ( ) y ' r x Mulvarae, saonary me seres model for y conrols lengh and magnude of devaons (spread) he spread, log μ () - log μ In (), s parameerzed by r x e.g. level, slope and curvaure for example, r x = (1, x-a, (x-a) ) he SAIN-model s descrbed n Jarner and Kryger (008)

34 Example Lee-Carer specfcaon of μ In () hree-dmensonal vecor auoregressve model for y y A y 1 b, ~ N3 (0, ) In hs case he forecas varance akes he form Var(log ( h)) bˆ x ( h ˆ h Var( ˆ)) Var( y h ' r x ) for h c for h Frs erm s he Lee-Carer varance of he common long-erm rend Second erm s he (bounded) varance of he counry-specfc spread

35 Coheren projecons and levels of uncerany Inernaonal female moraly for ages 60 and 70 Deah rae 0.% 0.5% 1% % 5% Year

36 Random walk ype moraly models Uncerany Large annual flucuaons Large accumulaed uncerany (srucural) Large rend uncerany (parameer) Unreasonably large dfferences n projecon uncerany Level Shor-erm dfferences Dvergng projecons A odds wh expeced moraly convergence

37 Coheren moraly modellng Uncerany Large annual flucuaons Large, bu bounded, spread uncerany No mpac on rend uncerany Smlar long-erm uncerany, dfferen shor-erm uncerany Level Shor-erm dfferences Dfferen levels relave o common rend Long-erm (mean) forecass evolve n parrallel