Poisson log-bilinear approach to Lee-Carter modelling and application for Turkish mortality

Yıl 2013, Cilt: 6 Sayı: 1, 14 - 40, 01.03.2013

Selim Demircioğlu Murat Büyükyazıcı

Öz

Poisson log-bilinear approach to Lee-Carter modelling and application for Turkish mortality The aim of this study is to estimate parameters of Lee-Carter model to age-specific death rates by gender in Turkey with both singular value decomposition method and Poisson Log-Bilinear regression approaches and to compare model parameters and death rates which are calculated with these two approaches. For the years between 1937 and 1995, mortality rates, death counts and exposures for both sex have been obtained from the model life table level result of “The Project of Turkish Life and Life Annuity Tables” by using interpolation techniques. Forecasted age-specific death rates by gender in Turkey for 20 years are compared and assessed.

Anahtar Kelimeler

Lee-Carter model, Poisson log-bilinear approach, Model life tables, Death rates, Death counts; Exposures

Poisson log-bilineer yaklaşımıyla Lee-Carter modellemesi ve Türkiye Uygulaması

Öz

Bu çalışmanın amacı, Türkiye yaşa özel ölüm hızlarının Lee-Carter modeli parametrelerini tekil değer ayrışımı yöntemi ile ve bu yönteme seçenek olarak geliştirilen Poisson Log-Bilineer yaklaşımıyla tahmin etmek ve her iki yaklaşımla elde edilen model parametrelerini ve ölüm hızı değerlerini karşılaştırmaktır. Her iki cinsiyet için 1937-1995 yılları arası ölüm hızları, ölüm sayıları ve riske maruz değerler “Türkiye Hayat ve Hayat Annüite Tablolarının Oluşturulması Projesi” kapsamında belirlenmiş olan model hayat tablosu seviyelerine uygun olarak içdeğerleme yöntemi ile elde edilmiştir. Türkiye yaşa özel ölüm hızlarının her iki yaklaşım ile elde edilen 20 yıllık öngörüleri karşılaştırılmış ve az da olsa farklı sonuçlar verdiği görülmüştür.

Anahtar Kelimeler

Lee-Carter modeli, Poisson log-bilineer yaklaşımı, Model hayat tabloları, Ölüm hızları, Ölüm sayıları, Riske maruz değerler

Kaynakça

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• Ek- LEM programı çıktısı LEM: log-linear and event history analysis with missing data. Developed by Jeroen Vermunt (c), Tilburg University, The Netherlands. Version 1.0 (September 18, 1997). *** INPUT *** man 2 dim 18 59 lab X T mod {wei(XT),X,spe(T,1a,X,b)} dat deaths.dat sta wei(XT) exposures.dat cri 0.0000001
• ***LOG-LINEAR PARAMETERS *** * TABLE XT [or P(XT)] * effect beta exp(beta) main -4.3893 0.0124 X 1 1.9033 6.7083 2 -0.5082 0.6016 3 -1.7783 0.1689 4 -2.0623 0.1272 5 -1.5917 0.2036 6 -1.2432 0.2885 7 -1.1822 0.3066 8 -1.0529 0.3489 9 -0.8482 0.4282 10 -0.5701 0.5655 11 -0.2520 0.7772 12 0.1115 1.1180 13 0.4841 1.6227 14 0.8856 2.4243 15 1.2842 3.6118 16 1.7046 5.4990 17 2.1425 8.5209 18 2.5733 13.1088 X [spe(T,1a)] 1 1.0000 2 1.4883 3 1.1241 4 1.0626 5 0.9144 6 0.9237 7 0.9756 8 0.9902 9 0.9448 10 0.8508 11 0.6926 12 0.5801 13 0.4495 14 0.3855 15 0.3138 16 0.2552 17 0.2136 18 0.1857
• Ek-EKK yönteminden TDA ile elde edilen Lee-Carter modeli yaş parametreleri – Erkek Yaş Grupları a_x b_x b_x Güncellenmiş -2,48598 0,282076 0,073918819 1 -4,90285 0,434079 0,113751619 5 -6,16954 0,321912 0,084357971 10 -6,45026 0,301421 0,078988184 15 -5,98034 0,259198 0,067923614 20 -5,63136 0,261406 0,068502121 25 -5,57102 0,27725 0,072654001 30 -5,44142 0,281028 0,073644013 35 -5,23646 0,267727 0,070158617 40 -4,95811 0,24033 0,062979173 45 -4,64012 0,195055 0,051114594 50 -4,27643 0,163465 0,04283646 55 -3,90419 0,127152 0,03332055 60 -3,54103 0,125074 0,032775886 65 -3,10402 0,089562 0,02346996 70 -2,68362 0,073299 0,019208285 75 -2,24556 0,061837 0,016204475 80 -1,81466 0,054156 0,014191659
• Ek-EKK yönteminden TDA ile elde edilen Lee-Carter modeli yılparametreleri – Erkek Yıllar k_t k_t Güncellenmiş Yıllar k_t k_t Güncellenmiş 1937 3,173498903 12,11015686 1967 -0,244760378 -0,934012161 1938 3,018566225 11,5189296 1968 -0,32017036 -1,221778674 1939 2,758705373 10,52729363 1969 -0,395477805 -1,509153901 1940 2,534933535 9,673374301 1970 -0,470735182 -1,796338068 2,335389401 8,911908535 1971 -0,545994069 -2,083528 1942 2,152479195 8,21391829 1972 -0,621318496 -2,370968031 1943 1,996795177 7,619823904 1973 -0,69675815 -2,658847772 1944 1,849353189 7,057181329 1974 -0,772367948 -2,947376789 1,713843802 6,540073874 1975 -0,842466112 -3,214873261 1,595727323 6,089338227 1976 -0,907960216 -3,464800515 1947 1,481998938 5,65534766 1977 -0,973679597 -3,715587435 1948 1,371953689 5,235412041 1978 -1,039670915 -3,967412074 1949 1,268144011 4,839271529 1979 -1,105965548 -4,220394169 1,168433347 4,458772963 1980 -1,172591664 -4,474641213 1,071063431 4,087206756 1981 -1,239602586 -4,730356687 1952 0,975714101 3,723351156 1982 -1,307020339 -4,987624639 1953 0,8824384 3,367408583 1983 -1,374892343 -5,246626024 1954 0,793758716 3,029004532 1984 -1,443255534 -5,507501794 0,706457669 2,695861398 1985 -1,511955375 -5,769662229 1956 0,620370731 2,367351335 1986 -1,577180653 -6,018563635 1957 0,535339781 2,042870949 1987 -1,642705221 -6,268607145 1958 0,451226223 1,721891357 1988 -1,70858193 -6,519994431 1959 0,369288012 1,40921295 1989 -1,774835553 -6,772820033 1960 0,29079175 1,109669109 1990 -1,841501142 -7,027217706 1961 0,212961117 0,812665329 1991 -1,874999111 -7,155046853 1962 0,135704175 0,51785077 1992 -1,908616625 -7,283332189 0,058939905 0,224916258 1993 -1,942354713 -7,412077633 1964 -0,017411312 -0,066442032 1994 -1,976223072 -7,541320199 1965 -0,093426973 -0,356519833 1995 -2,010214343 -7,671031803 1966 -0,169182855 -0,645606307
• Ek-EKK yönteminden TDA ile elde edilen Lee-Carter modeli yaş parametreleri – Kadın Yaş Grupları a_x b_x b_x Güncellenmiş -2,70958 0,263891402 0,069365421 1 -4,98237 0,41115775 0,108075255 5 -6,28293 0,34034231 0,089460996 10 -6,53067 0,335908569 0,088295561 15 -6,13288 0,30457293 0,080058802 20 -5,84011 0,287051117 0,07545309 25 -5,68699 0,275966371 0,072539398 30 -5,54092 0,267054437 0,070196844 35 -5,37395 0,244819419 0,064352237 40 -5,17998 0,211120402 0,055494251 45 -4,9281 0,169521423 0,044559712 50 -4,58835 0,151492867 0,039820799 55 -4,23989 0,130732165 0,034363725 60 -3,8086 0,121021064 0,031811105 65 -3,36927 0,096200037 0,025286751 70 -2,88677 0,078727483 0,020693986 75 -2,40592 0,061352419 0,016126847 80 -1,94622 0,053433146 0,01404522
• Ek-EKK yönteminden TDA ile elde edilen Lee-Carter modeli yılparametreleri – Kadın Yıllar k_t k_t Güncellenmiş Yıllar k_t k_t Güncellenmiş 1937 3,504803615 13,33355329 1967 -0,252908987 -0,962158176 1938 3,342774814 12,71713654 1968 -0,336748564 -1,281114553 1939 3,071843395 11,68641445 1969 -0,420695625 -1,600479841 1940 2,840746811 10,80723862 1970 -0,504814506 -1,920498794 2,642590424 10,05337934 1971 -0,589169359 -2,241415469 1942 2,460775333 9,361688311 1972 -0,673839209 -2,563530513 1943 2,301576992 8,756039667 1973 -0,758887975 -2,887087087 1944 2,150451648 8,18110365 1974 -0,844386456 -3,21235454 2,006773669 7,634500132 1975 -0,927374265 -3,528070482 1,870836223 7,117344427 1976 -1,008260386 -3,835790838 1947 1,739260876 6,61678374 1977 -1,089591507 -4,14520413 1948 1,611271491 6,129865365 1978 -1,171443382 -4,456598564 1949 1,491524348 5,674303489 1979 -1,253873534 -4,770192975 1,376722471 5,23755521 1980 -1,336936284 -5,086194022 1,264116014 4,80915911 1981 -1,420718395 -5,404931777 1952 1,153347424 4,387754932 1982 -1,505268845 -5,726592575 1953 1,044759228 3,974645764 1983 -1,590669886 -6,051489335 1954 0,943695012 3,590160568 1984 -1,676992441 -6,379891866 0,843843122 3,2102875 1985 -1,763993244 -6,710874706 1956 0,745018423 2,834322245 1986 -1,845545555 -7,021129488 1957 0,647040687 2,461579143 1987 -1,927879398 -7,334357502 1958 0,549749782 2,091449 1988 -2,011081231 -7,650887669 1959 0,455044618 1,73115596 1989 -2,095204327 -7,970922659 1960 0,36485819 1,388053843 1990 -2,180317114 -8,294722792 1961 0,27512695 1,046683424 1991 -2,223266357 -8,458117403 1962 0,185741942 0,706630202 1992 -2,266494213 -8,62257196 0,096605365 0,3675221 1993 -2,310005683 -8,788105485 1964 0,007619587 0,028987693 1994 -2,353817138 -8,954780267 1965 -0,081312077 -0,309340847 1995 -2,397923067 -9,122575333 1966 -0,169099446 -0,643316067
• Ek- EÇO yöntemi ile elde edilen Lee-Carter modeli a parametresi değerleri – Erkek 0,1115
• Ek- EÇO yöntemi ile elde edilen Lee-Carter modeli b parametresi değerleri – Erkek Yaş Grupları b_x b_x Güncellenmiş 1,0000 0,074903562 1 1,4883 0,111478971 5 1,1241 0,084199094 10 1,0626 0,079592525 15 0,9144 0,068491817 20 0,9237 0,06918842 25 0,9756 0,073075915 30 0,9902 0,074169507 35 0,9448 0,070768885 40 0,8508 0,06372795 45 0,6926 0,051878207 50 0,5801 0,043451556 55 0,4495 0,033669151 60 0,3855 0,028875323 65 0,3138 0,023504738 70 0,2552 0,019115389 75 0,2136 0,015999401 80 0,1857 0,013909591 sum: 13,3505 1,0000
• Ek- EÇO yöntemi ile elde edilen Lee-Carter modeli k parametresi değerleri - Erkek Yıllar Güncellenmiş k_t k_t Güncellenmiş 1937 0,9053 1967 -0,076 -1,01330295 1938 0,8621 1968 -0,097 -1,2923284 1939 0,7885 1969 -0,117 -1,5673487 1940 0,7253 9,68311765 1970 -0,138 -1,83836385 1941 0,6666 8,8994433 1971 -0,158 -2,1067089 1942 0,6142 8,1998771 1972 -0,178 -2,37238385 1943 0,5682 7,5857541 1973 -0,197 -2,6353887 1944 0,5256 7,0170228 1974 -0,217 -2,89572345 1945 0,4868 6,4990234 1975 -0,236 -3,1560582 1946 0,4521 1976 -0,256 -3,41639295 1947 0,4193 5,59786465 1977 -0,275 -3,67539265 1948 0,3881 1978 -0,295 -3,93172225 1949 0,3577 1979 -0,314 -4,18538175 1950 0,3284 4,3843042 1980 -0,333 -4,43904125 1951 0,3001 1981 -0,351 -4,6886956 1952 0,2729 1982 -0,37 -4,93834995 1953 0,2464 3,2895632 1983 -0,388 -5,1853342 1954 0,2203 1984 -0,407 -5,4309834 1955 0,1949 1985 -0,425 -5,67529755 1956 0,1702 2,2722551 1986 -0,443 -5,91827665 1957 0,1461 1987 -0,461 -6,15858565 1958 0,1225 1988 -0,479 -6,3975596 1959 0,0993 1,32570465 1989 -0,497 -6,63653355 1960 0,0761 1990 -0,515 -6,87417245 1961 0,0534 0,7129167 1991 -0,524 -6,99165685 1962 0,031 0,4138655 1992 -0,533 -7,10914125 1963 0,0091 0,12148955 1993 -0,541 -7,2279607 1964 -0,0125 1994 -0,55 -7,3454451 1965 -0,0337 1995* -0,559 -7,4602594 1966 -0,0548 -0,7316074 13,351