Coherent mortality forecasting using functional time series models

1. Coherent mortality forecasting using functional time series models Coherent mortality forecasting 1 Rob J Hyndman Australia: cohort life expectancy at age 50

2. Mortality rates Coherent mortality forecasting 2 0 20 40 60 80 100 −10−8−6−4−20 Australia: male mortality (1921) Age Logdeathrate

3. Mortality rates Coherent mortality forecasting 3 q q q q q q qq qqq qqq q q q q qqqq qq qq qqqqq qqqqqqqq qqq qqq q qqq qqqq qqq qqqqqq qqq qq qq qqq qqqq qqqqqq qqqqqq qqq qqqq q q qq q q 0 20 40 60 80 100 −10−8−6−4−20 Australia: male death rates (1970) Age Logdeathrate

4. Mortality rates Coherent mortality forecasting 3 q q q q qq q q q q qqqq q q q q qqqqqq qqqqqqqqq q qqqq qq qqqqqq qqqq qqq q qq qqq qqqq qqqqqqqqq qqq qqqqqqqqqqqqqq qqqq q q qq qq q q 0 20 40 60 80 100 −10−8−6−4−20 Australia: male death rates (1990) Age Logdeathrate

5. Mortality rates Coherent mortality forecasting 3 0 20 40 60 80 100 −10−8−6−4−20 Australia: male death rates (1921−2009) Age Logdeathrate

6. Mortality rates Coherent mortality forecasting 3 0 20 40 60 80 100 −10−8−6−4−20 Australia: male death rates (1921−2009) Age Logdeathrate

7. Mortality rates Coherent mortality forecasting 3 0 20 40 60 80 100 1234 Australia: mortality sex ratio (1921−2009) Age Sexratioofrates:M/F

8. Outline 1 Functional forecasting 2 Forecasting groups 3 Coherent cohort life expectancy forecasts 4 Conclusions Coherent mortality forecasting 4

9. Outline 1 Functional forecasting 2 Forecasting groups 3 Coherent cohort life expectancy forecasts 4 Conclusions Coherent mortality forecasting Functional forecasting 5

10. Some notation Let yt,x be the observed (smoothed) data in period t at age x, t = 1,...,n. yt,x = ft(x) + σt(x)εt,x ft(x) = µ(x) + K k=1 βt,k φk (x) + et(x) Coherent mortality forecasting Functional forecasting 6 Estimate ft(x) using penalized regression splines. Estimate µ(x) as mean ft(x) across years. Estimate βt,k and φk (x) using functional principal components. εt,x iid ∼ N(0,1) and et(x) iid ∼ N(0,v(x)).

15. Australian male mortality model Coherent mortality forecasting Functional forecasting 7 0 20 40 60 80 −8−7−6−5−4−3−2−1 Age (x) µ(x) 0 20 40 60 80 0.050.100.150.20 Age (x) φ1(x) Year (t) βt1 1920 1960 2000 −505 0 20 40 60 80 −0.15−0.050.050.15 Age (x) φ2(x) Year (t) βt2 1920 1960 2000 −2.0−1.00.01.0 0 20 40 60 80 −0.10.00.10.2 Age (x) φ3(x) Year (t) βt3 1920 1960 2000 −2−101

16. Australian male mortality model Coherent mortality forecasting Functional forecasting 7 1940 1960 1980 2000 020406080100 Residuals Year (t) Age(x)

17. Functional time series model yt,x = ft(x) + σt(x)εt,x ft(x) = µ(x) + K k=1 βt,k φk (x) + et(x) The eigenfunctions φk (x) show the main regions of variation. The scores {βt,k } are uncorrelated by construction. So we can forecast each βt,k using a univariate time series model. Univariate ARIMA models can be used for forecasting. Coherent mortality forecasting Functional forecasting 8

21. Forecasts yt,x = ft(x) + σt(x)εt,x ft(x) = µ(x) + K k=1 βt,k φk (x) + et(x) Coherent mortality forecasting Functional forecasting 9

22. Forecasts yt,x = ft(x) + σt(x)εt,x ft(x) = µ(x) + K k=1 βt,k φk (x) + et(x) where vn+h,k = Var(βn+h,k | β1,k ,...,βn,k ) and y = [y1,x,...,yn,x]. Coherent mortality forecasting Functional forecasting 9 E[yn+h,x | y] = ˆµ(x) + K k=1 ˆβn+h,k ˆφk (x) Var[yn+h,x | y] = ˆσ2 µ (x) + K k=1 vn+h,k ˆφ2 k (x) + σ2 t (x) + v(x)

23. Forecasting the PC scores Coherent mortality forecasting Functional forecasting 10 0 20 40 60 80 −8−7−6−5−4−3−2−1 Age (x) µ(x) 0 20 40 60 80 0.050.100.150.20 Age (x) φ1(x) Year (t) βt1 1920 1980 2040 −20−15−10−505 0 20 40 60 80 −0.15−0.050.050.15 Age (x) φ2(x) Year (t) βt2 1920 1980 2040 −10−8−6−4−202 0 20 40 60 80 −0.10.00.10.2 Age (x) φ3(x) Year (t) βt3 1920 1980 2040 −2−101

24. Forecasts of ft(x) Coherent mortality forecasting Functional forecasting 11 0 20 40 60 80 100 −10−8−6−4−20 Australia: male death rates (1921−2009) Age Logdeathrate

25. Forecasts of ft(x) Coherent mortality forecasting Functional forecasting 11 0 20 40 60 80 100 −10−8−6−4−20 Australia: male death rates (1921−2009) Age Logdeathrate

26. Forecasts of ft(x) Coherent mortality forecasting Functional forecasting 11 0 20 40 60 80 100 −10−8−6−4−20 Australia: male death rates forecasts (2010−2059) Age Logdeathrate

27. Forecasts of ft(x) Coherent mortality forecasting Functional forecasting 11 0 20 40 60 80 100 −10−8−6−4−20 Australia: male death rates forecasts (2010 and 2059) Age Logdeathrate 80% prediction intervals

28. Forecasts of mortality sex ratio Coherent mortality forecasting Functional forecasting 12 0 20 40 60 80 100 01234567 Australia: mortality sex ratio data Age Year

29. Forecasts of mortality sex ratio Coherent mortality forecasting Functional forecasting 12 0 20 40 60 80 100 01234567 Australia: mortality sex ratio data Age Year

30. Forecasts of mortality sex ratio Coherent mortality forecasting Functional forecasting 12 0 20 40 60 80 100 01234567 Australia: mortality sex ratio forecasts Age Year

31. Forecasts of mortality sex ratio Coherent mortality forecasting Functional forecasting 12 0 20 40 60 80 100 01234567 Australia: mortality sex ratio forecasts Age Year Male and female mortality rate forecasts are diverging.

32. Outline 1 Functional forecasting 2 Forecasting groups 3 Coherent cohort life expectancy forecasts 4 Conclusions Coherent mortality forecasting Forecasting groups 13

33. The problem Let ft,j(x) be the smoothed mortality rate for age x in group j in year t. Groups may be males and females. Groups may be states within a country. Expected that groups will behave similarly. Coherent forecasts do not diverge over time. Existing functional models do not impose coherence. Coherent mortality forecasting Forecasting groups 14

39. Forecasting the coefficients yt,x = ft(x) + σt(x)εt,x ft(x) = µ(x) + K k=1 βt,k φk (x) + et(x) We use ARIMA models for each coefficient {β1,j,k ,...,βn,j,k }. The ARIMA models are non-stationary for the first few coefficients (k = 1,2) Non-stationary ARIMA forecasts will diverge. Hence the mortality forecasts are not coherent. Coherent mortality forecasting Forecasting groups 15

42. Male fts model Coherent mortality forecasting Forecasting groups 16 0 20 40 60 80 −8−7−6−5−4−3−2−1 Age µ(x) 0 20 40 60 80 0.050.100.150.20 Age φ1(x) Time βt1 1950 2050 −25−15−505 0 20 40 60 80 −0.15−0.050.050.15 Age φ2(x) Time βt2 1950 2050 −15−10−50 0 20 40 60 80 −0.10.00.10.2 Age φ3(x) Time βt3 1950 2050 −2−101

43. Female fts model Coherent mortality forecasting Forecasting groups 17 0 20 40 60 80 −8−6−4−2 Age µ(x) 0 20 40 60 80 0.050.100.15 Age φ1(x) Time βt1 1950 2050 −30−20−10010 0 20 40 60 80 −0.15−0.050.05 Age φ2(x) Time βt2 1950 2050 −10−505 0 20 40 60 80 −0.4−0.20.0 Age φ3(x) Time βt3 1950 2050 −1.0−0.50.00.51.0

44. Australian mortality forecasts Coherent mortality forecasting Forecasting groups 18 0 20 40 60 80 100 −10−8−6−4−20 (a) Males Age Logdeathrate 0 20 40 60 80 100 −10−8−6−4−20 (b) Females Age Logdeathrate

45. Mortality product and ratios Key idea Model the geometric mean and the mortality ratio instead of the individual rates for each sex separately. pt(x) = ft,M (x)ft,F (x) and rt(x) = ft,M (x) ft,F (x). Product and ratio are approximately independent Ratio should be stationary (for coherence) but product can be non-stationary. Coherent mortality forecasting Forecasting groups 19

49. Product data Coherent mortality forecasting Forecasting groups 20 0 20 40 60 80 100 −8−6−4−20 Australia: product data Age Logofgeometricmeandeathrate

50. Ratio data Coherent mortality forecasting Forecasting groups 21 0 20 40 60 80 100 1234 Australia: mortality sex ratio (1921−2009) Age Sexratioofrates:M/F

51. Model product and ratios pt(x) = ft,M (x)ft,F (x) and rt(x) = ft,M (x) ft,F (x). log[pt(x)] = µp(x) + K k=1 βt,k φk (x) + et(x) log[rt(x)] = µr(x) + L =1 γt, ψ (x) + wt(x). {γt, } restricted to be stationary processes: either ARMA(p,q) or ARFIMA(p,d,q). No restrictions for βt,1,...,βt,K . Forecasts: fn+h|n,M (x) = pn+h|n(x)rn+h|n(x) fn+h|n,F (x) = pn+h|n(x) rn+h|n(x). Coherent mortality forecasting Forecasting groups 22

55. Product model Coherent mortality forecasting Forecasting groups 23 0 20 40 60 80 −8−6−4−2 Age µP(x) 0 20 40 60 80 0.050.100.15 Age φ1(x) Year βt1 1920 1980 2040 −20−10−505 0 20 40 60 80 −0.2−0.10.00.10.2 Age φ2(x) Year βt2 1920 1980 2040 −1.5−0.50.51.0 0 20 40 60 80 −0.20−0.100.000.10 Age φ3(x) Year βt3 1920 1980 2040 −4−2024

56. Ratio model Coherent mortality forecasting Forecasting groups 24 0 20 40 60 80 0.10.20.30.4 Age µR(x) 0 20 40 60 80 −0.10.00.10.2 Age φ1(x) Year βt1 1920 1980 2040 −0.6−0.20.00.20.4 0 20 40 60 80 0.000.100.20 Age φ2(x) Year βt2 1920 1980 2040 −2.0−1.00.01.0 0 20 40 60 80 −0.3−0.10.10.20.3 Age φ3(x) Year βt3 1920 1980 2040 −0.4−0.20.00.20.4

57. Product forecasts Coherent mortality forecasting Forecasting groups 25 0 20 40 60 80 100 −10−8−6−4−2 Age Logofgeometricmeandeathrate

58. Ratio forecasts Coherent mortality forecasting Forecasting groups 26 0 20 40 60 80 100 1234 Age Sexratio:M/F

59. Coherent forecasts Coherent mortality forecasting Forecasting groups 27 0 20 40 60 80 100 −10−8−6−4−2 (a) Males Age Logdeathrate 0 20 40 60 80 100 −10−8−6−4−2 (b) Females Age Logdeathrate

60. Ratio forecasts Coherent mortality forecasting Forecasting groups 28 0 20 40 60 80 100 01234567 Independent forecasts Age Sexratioofrates:M/F 0 20 40 60 80 100 01234567 Coherent forecasts Age Sexratioofrates:M/F

61. Life expectancy forecasts Coherent mortality forecasting Forecasting groups 29 Life expectancy forecasts Year Age 1920 1960 2000 2040 707580859095 1920 1960 2000 2040 707580859095 Life expectancy difference: F−M Year Numberofyears 1960 1980 2000 2020 4567

62. Coherent forecasts for J groups pt(x) = [ft,1(x)ft,2(x)···ft,J (x)]1/J and rt,j (x) = ft,j (x) pt(x), log[pt(x)] = µp(x) + K k=1 βt,k φk (x) + et(x) log[rt,j (x)] = µr,j (x) + L l=1 γt,l,j ψl,j (x) + wt,j (x). Coherent mortality forecasting Forecasting groups 30 pt(x) and all rt,j (x) are approximately independent. Ratios satisfy constraint rt,1(x)rt,2(x)···rt,J (x) = 1. log[ft,j (x)] = log[pt(x)rt,j (x)]

67. Coherent forecasts for J groups µj (x) = µp(x) + µr,j (x) is group mean zt,j (x) = et(x) + wt,j (x) is error term. {γt, } restricted to be stationary processes: either ARMA(p,q) or ARFIMA(p,d,q). No restrictions for βt,1,...,βt,K . Coherent mortality forecasting Forecasting groups 31 log[ft,j (x)] = log[pt(x)rt,j (x)] = log[pt(x)] + log[rt,j ] = µj (x) + K k=1 βt,k φk (x) + L =1 γt, ,j ψ ,j (x) + zt,j (x)

72. Li-Lee method Li & Lee (Demography, 2005) method is a special case of our approach. ft,j (x) = µj (x) + βtφ(x) + γt,j ψj (x) + et,j (x) where f is unsmoothed log mortality rate, βt is a random walk with drift and γt,j is AR(1) process. No smoothing. Only one basis function for each part, Random walk with drift very limiting. AR(1) very limiting. The γt,j coeﬃcients will be highly correlated with each other, and so independent models are not appropriate. Coherent mortality forecasting Forecasting groups 32

77. Outline 1 Functional forecasting 2 Forecasting groups 3 Coherent cohort life expectancy forecasts 4 Conclusions Coherent mortality forecasting Coherent cohort life expectancy forecasts 33

78. Life expectancy calculation Using standard life table calculations: For x = 0,1,...,ω − 1: qx = mx/(1 + (1 − ax)mx) x+1 = x(1 − qx) Lx = x[1 − qx(1 − ax)] Tx = Lx + Lx+1 + ··· + Lω−1 + Lω+ ex = Tx/Lx where ax = 0.5 for x ≥ 1 and a0 taken from Coale et al (1983). qω+ = 1, Lω+ = lx/mx, and Tω+ = Lω+. Period life expectancy: let mx = mx,t for some year t. Cohort life expectancy: let mx = mx,t+x for birth cohort in year t. Coherent mortality forecasting Coherent cohort life expectancy forecasts 34

82. Cohort life expectancy Because we can forecast mx,t we can estimate the mortality rates for each birth cohort (using actual values when they are available). We can simulate future mx,t in order to estimate the uncertainty associated with ex. Coherent mortality forecasting Coherent cohort life expectancy forecasts 35

83. Cohort life expectancy Because we can forecast mx,t we can estimate the mortality rates for each birth cohort (using actual values when they are available). We can simulate future mx,t in order to estimate the uncertainty associated with ex. Coherent mortality forecasting Coherent cohort life expectancy forecasts 35

84. Cohort life expectancy Coherent mortality forecasting Coherent cohort life expectancy forecasts 36 Age 0 20 40 60 80 Year 1950 2000 2050 2100 2150 Logdeathrate −10 −5

85. Simulate future mortality rates pt(x) = ft,M (x)ft,F (x) and rt(x) = ft,M (x) ft,F (x). log[pt(x)] = µp(x) + K k=1 βt,k φk (x) + et(x) log[rt(x)] = µr(x) + L =1 γt, ψ (x) + wt(x). {γt, } and {βt,k } simulated. {et(x)} and {wt(x)} bootstrapped. Generate many future sample paths for ft,M (x) and ft,F (x) to estimate uncertainty in ex. Coherent mortality forecasting Coherent cohort life expectancy forecasts 37

89. Cohort life expectancy Coherent mortality forecasting Coherent cohort life expectancy forecasts 38 Australia: cohort life expectancy at age 50 Year Remaininglifeexpectancy 1920 1940 1960 1980 2000 2020 2040 2060 20253035404550

90. Complete code Coherent mortality forecasting Coherent cohort life expectancy forecasts 39 library(demography) # Read data aus <- hmd.mx("AUS","username","password","Australia") # Smooth data aus.sm <- smooth.demogdata(aus) #Fit model aus.pr <- coherentfdm(aus.sm) # Forecast aus.pr.fc <- forecast(aus.pr, h=100) # Compute life expectancies e50.m.aus.fc <- flife.expectancy(aus.pr.fc, series="male", age=50, PI=TRUE, nsim=1000, type="cohort") e50.f.aus.fc <- flife.expectancy(aus.pr.fc, series="female", age=50, PI=TRUE, nsim=1000, type="cohort")

91. Forecast accuracy evaluation Coherent mortality forecasting Coherent cohort life expectancy forecasts 40 Australian female cohort e50 Year Remainingyears 1920 1940 1960 1980 2000 2628303234363840

92. Forecast accuracy evaluation Coherent mortality forecasting Coherent cohort life expectancy forecasts 41 Australian female cohort e50: Data to 1955 Year Remainingyears 1920 1940 1960 1980 2000 2628303234363840

93. Forecast accuracy evaluation Compute age 50 remaining cohort life expectancy with a rolling forecast origin beginning in 1921. Compare against actual cohort life expectancy where available. Compute 80% prediction interval actual coverage. Coherent mortality forecasting Coherent cohort life expectancy forecasts 42

96. Forecast accuracy evaluation Coherent mortality forecasting Coherent cohort life expectancy forecasts 43 5 10 15 20 25 0.00.20.40.60.81.01.2 Mean Absolute Forecast Errors Forecast horizon Years 1 2 3 4 Male Female

97. Forecast accuracy evaluation Coherent mortality forecasting Coherent cohort life expectancy forecasts 43 5 10 15 20 25 020406080100 80% prediction interval coverage Forecast horizon Percentagecoverage 1 2 3 4

98. Outline 1 Functional forecasting 2 Forecasting groups 3 Coherent cohort life expectancy forecasts 4 Conclusions Coherent mortality forecasting Conclusions 44

99. Some conclusions New, automatic, flexible method for coherent forecasting of groups of functional time series. Suitable for age-specific mortality. Based on geometric means and ratios, so interpretable results. More general and flexible than existing methods. Easy to compute prediction intervals for any computable statistics. Coherent mortality forecasting Conclusions 45

104. Selected references Hyndman, Ullah (2007). “Robust forecasting of mortality and fertility rates: A functional data approach”. Computational Statistics and Data Analysis 51(10), 4942–4956 Hyndman, Shang (2009). “Forecasting functional time series (with discussion)”. Journal of the Korean Statistical Society 38(3), 199–221 Hyndman, Booth, Yasmeen (2013). “Coherent mortality forecasting: the product-ratio method with functional time series models”. Demography 50(1), 261–283 Booth, Hyndman, Tickle (2013). “Prospective Life Tables”. Computational Actuarial Science, with R. ed. by Charpentier. Chapman & Hall/CRC, 323–348 Hyndman (2013). demography: Forecasting mortality, fertility, migration and population data. v1.16. cran.r-project.org/package=demography Coherent mortality forecasting Conclusions 46

105. Selected references Hyndman, Ullah (2007). “Robust forecasting of mortality and fertility rates: A functional data approach”. Computational Statistics and Data Analysis 51(10), 4942–4956 Hyndman, Shang (2009). “Forecasting functional time series (with discussion)”. Journal of the Korean Statistical Society 38(3), 199–221 Hyndman, Booth, Yasmeen (2013). “Coherent mortality forecasting: the product-ratio method with functional time series models”. Demography 50(1), 261–283 Booth, Hyndman, Tickle (2013). “Prospective Life Tables”. Computational Actuarial Science, with R. ed. by Charpentier. Chapman & Hall/CRC, 323–348 Hyndman (2013). demography: Forecasting mortality, fertility, migration and population data. v1.16. cran.r-project.org/package=demography Coherent mortality forecasting Conclusions 46 ¯ Papers and R code: robjhyndman.com ¯ Email: Rob.Hyndman@monash.edu

Coherent mortality forecasting using functional time series models

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