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Coherent mortality forecasting using functional time series models
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![Page 1: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/1.jpg)
Coherent mortality
forecasting using functional
time series models
Coherent mortality forecasting 1
Rob J Hyndman
Australia: cohort life expectancy at age 50
Year
Rem
aini
ng li
fe e
xpec
tanc
y
1920 1940 1960 1980 2000 2020 2040 2060
2025
3035
4045
50
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Mortality rates
Coherent mortality forecasting 2
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Mortality rates
Coherent mortality forecasting 3
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0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates (1970)
Age
Log
deat
h ra
te
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Mortality rates
Coherent mortality forecasting 3
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0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates (1990)
Age
Log
deat
h ra
te
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Mortality rates
Coherent mortality forecasting 3
0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates (1921−2009)
Age
Log
deat
h ra
te
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Mortality rates
Coherent mortality forecasting 3
0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates (1921−2009)
Age
Log
deat
h ra
te
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Mortality rates
Coherent mortality forecasting 3
0 20 40 60 80 100
12
34
Australia: mortality sex ratio (1921−2009)
Age
Sex
rat
io o
f rat
es: M
/F
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Outline
1 Functional forecasting
2 Forecasting groups
3 Coherent cohort life expectancy forecasts
4 Conclusions
Coherent mortality forecasting 4
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Outline
1 Functional forecasting
2 Forecasting groups
3 Coherent cohort life expectancy forecasts
4 Conclusions
Coherent mortality forecasting Functional forecasting 5
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Some notation
Let yt ,x be the observed (smoothed) data in period tat 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 principalcomponents.
εt ,xiid∼ N(0,1) and et(x)
iid∼ N(0,v(x)).
![Page 11: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/11.jpg)
Some notation
Let yt ,x be the observed (smoothed) data in period tat 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 principalcomponents.
εt ,xiid∼ N(0,1) and et(x)
iid∼ N(0,v(x)).
![Page 12: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/12.jpg)
Some notation
Let yt ,x be the observed (smoothed) data in period tat 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 principalcomponents.
εt ,xiid∼ N(0,1) and et(x)
iid∼ N(0,v(x)).
![Page 13: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/13.jpg)
Some notation
Let yt ,x be the observed (smoothed) data in period tat 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 principalcomponents.
εt ,xiid∼ N(0,1) and et(x)
iid∼ N(0,v(x)).
![Page 14: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/14.jpg)
Some notation
Let yt ,x be the observed (smoothed) data in period tat 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 principalcomponents.
εt ,xiid∼ N(0,1) and et(x)
iid∼ N(0,v(x)).
![Page 15: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/15.jpg)
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
0.05
0.10
0.15
0.20
Age (x)
φ 1(x
)
Year (t)
β t1
1920 1960 2000
−5
05
0 20 40 60 80
−0.
15−
0.05
0.05
0.15
Age (x)
φ 2(x
)
Year (t)
β t2
1920 1960 2000−2.
0−
1.0
0.0
1.0
0 20 40 60 80
−0.
10.
00.
10.
2
Age (x)
φ 3(x
)
Year (t)
β t3
1920 1960 2000
−2
−1
01
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Australian male mortality model
Coherent mortality forecasting Functional forecasting 7
1940 1960 1980 2000
020
4060
8010
0 Residuals
Year (t)
Age
(x)
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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 mainregions of variation.The scores {βt ,k } are uncorrelated byconstruction. So we can forecast each βt ,kusing a univariate time series model.Univariate ARIMA models can be used forforecasting.
Coherent mortality forecasting Functional forecasting 8
![Page 18: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/18.jpg)
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 mainregions of variation.The scores {βt ,k } are uncorrelated byconstruction. So we can forecast each βt ,kusing a univariate time series model.Univariate ARIMA models can be used forforecasting.
Coherent mortality forecasting Functional forecasting 8
![Page 19: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/19.jpg)
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 mainregions of variation.The scores {βt ,k } are uncorrelated byconstruction. So we can forecast each βt ,kusing a univariate time series model.Univariate ARIMA models can be used forforecasting.
Coherent mortality forecasting Functional forecasting 8
![Page 20: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/20.jpg)
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 mainregions of variation.The scores {βt ,k } are uncorrelated byconstruction. So we can forecast each βt ,kusing a univariate time series model.Univariate ARIMA models can be used forforecasting.
Coherent mortality forecasting Functional forecasting 8
![Page 21: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/21.jpg)
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
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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] = µ̂(x)+K∑
k=1
β̂n+h ,k φ̂k(x)
Var[yn+h ,x | y] = σ̂2µ (x)+
K∑k=1
vn+h ,k φ̂2k(x)+ σ
2t (x)+ v(x)
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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
0.05
0.10
0.15
0.20
Age (x)
φ 1(x
)
Year (t)
β t1
1920 1980 2040
−20
−15
−10
−5
05
0 20 40 60 80
−0.
15−
0.05
0.05
0.15
Age (x)
φ 2(x
)
Year (t)
β t2
1920 1980 2040
−10
−8
−6
−4
−2
02
0 20 40 60 80
−0.
10.
00.
10.
2
Age (x)
φ 3(x
)
Year (t)
β t3
1920 1980 2040
−2
−1
01
![Page 24: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/24.jpg)
Forecasts of ft(x)
Coherent mortality forecasting Functional forecasting 11
0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates (1921−2009)
Age
Log
deat
h ra
te
![Page 25: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/25.jpg)
Forecasts of ft(x)
Coherent mortality forecasting Functional forecasting 11
0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates (1921−2009)
Age
Log
deat
h ra
te
![Page 26: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/26.jpg)
Forecasts of ft(x)
Coherent mortality forecasting Functional forecasting 11
0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates forecasts (2010−2059)
Age
Log
deat
h ra
te
![Page 27: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/27.jpg)
Forecasts of ft(x)
Coherent mortality forecasting Functional forecasting 11
0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates forecasts (2010 and 2059)
Age
Log
deat
h ra
te
80% prediction intervals
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Forecasts of mortality sex ratio
Coherent mortality forecasting Functional forecasting 12
0 20 40 60 80 100
01
23
45
67
Australia: mortality sex ratio data
Age
Year
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Forecasts of mortality sex ratio
Coherent mortality forecasting Functional forecasting 12
0 20 40 60 80 100
01
23
45
67
Australia: mortality sex ratio data
Age
Year
![Page 30: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/30.jpg)
Forecasts of mortality sex ratio
Coherent mortality forecasting Functional forecasting 12
0 20 40 60 80 100
01
23
45
67
Australia: mortality sex ratio forecasts
Age
Year
![Page 31: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/31.jpg)
Forecasts of mortality sex ratio
Coherent mortality forecasting Functional forecasting 12
0 20 40 60 80 100
01
23
45
67
Australia: mortality sex ratio forecasts
Age
Year
Male and female mortalityrate forecasts arediverging.
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Outline
1 Functional forecasting
2 Forecasting groups
3 Coherent cohort life expectancy forecasts
4 Conclusions
Coherent mortality forecasting Forecasting groups 13
![Page 33: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/33.jpg)
The problem
Let ft ,j(x) be the smoothed mortality ratefor 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 behavesimilarly.Coherent forecasts do not diverge overtime.Existing functional models do notimpose coherence.Coherent mortality forecasting Forecasting groups 14
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The problem
Let ft ,j(x) be the smoothed mortality ratefor 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 behavesimilarly.Coherent forecasts do not diverge overtime.Existing functional models do notimpose coherence.Coherent mortality forecasting Forecasting groups 14
![Page 35: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/35.jpg)
The problem
Let ft ,j(x) be the smoothed mortality ratefor 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 behavesimilarly.Coherent forecasts do not diverge overtime.Existing functional models do notimpose coherence.Coherent mortality forecasting Forecasting groups 14
![Page 36: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/36.jpg)
The problem
Let ft ,j(x) be the smoothed mortality ratefor 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 behavesimilarly.Coherent forecasts do not diverge overtime.Existing functional models do notimpose coherence.Coherent mortality forecasting Forecasting groups 14
![Page 37: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/37.jpg)
The problem
Let ft ,j(x) be the smoothed mortality ratefor 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 behavesimilarly.Coherent forecasts do not diverge overtime.Existing functional models do notimpose coherence.Coherent mortality forecasting Forecasting groups 14
![Page 38: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/38.jpg)
The problem
Let ft ,j(x) be the smoothed mortality ratefor 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 behavesimilarly.Coherent forecasts do not diverge overtime.Existing functional models do notimpose coherence.Coherent mortality forecasting Forecasting groups 14
![Page 39: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/39.jpg)
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 thefirst few coefficients (k = 1,2)Non-stationary ARIMA forecasts will diverge.Hence the mortality forecasts are not coherent.
Coherent mortality forecasting Forecasting groups 15
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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 thefirst few coefficients (k = 1,2)Non-stationary ARIMA forecasts will diverge.Hence the mortality forecasts are not coherent.
Coherent mortality forecasting Forecasting groups 15
![Page 41: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/41.jpg)
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 thefirst few coefficients (k = 1,2)Non-stationary ARIMA forecasts will diverge.Hence the mortality forecasts are not coherent.
Coherent mortality forecasting Forecasting groups 15
![Page 42: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/42.jpg)
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
0.05
0.10
0.15
0.20
Age
φ 1(x
)
Time
β t1
1950 2050
−25
−15
−5
05
0 20 40 60 80
−0.
15−
0.05
0.05
0.15
Age
φ 2(x
)
Time
β t2
1950 2050
−15
−10
−5
0
0 20 40 60 80
−0.
10.
00.
10.
2
Age
φ 3(x
)
Time
β t3
1950 2050
−2
−1
01
![Page 43: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/43.jpg)
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
0.05
0.10
0.15
Age
φ 1(x
)
Time
β t1
1950 2050
−30
−20
−10
010
0 20 40 60 80
−0.
15−
0.05
0.05
Age
φ 2(x
)
Time
β t2
1950 2050
−10
−5
05
0 20 40 60 80
−0.
4−
0.2
0.0
Age
φ 3(x
)
Time
β t3
1950 2050
−1.
0−
0.5
0.0
0.5
1.0
![Page 44: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/44.jpg)
Australian mortality forecasts
Coherent mortality forecasting Forecasting groups 18
0 20 40 60 80 100
−10
−8
−6
−4
−2
0(a) Males
Age
Log
deat
h ra
te
0 20 40 60 80 100
−10
−8
−6
−4
−2
0
(b) Females
Age
Log
deat
h ra
te
![Page 45: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/45.jpg)
Mortality product and ratios
Key idea
Model the geometric mean and the mortality ratioinstead of the individual rates for each sexseparately.
pt(x) =√ft ,M(x)ft ,F(x) and rt(x) =
√ft ,M(x)
/ft ,F(x).
Product and ratio are approximatelyindependent
Ratio should be stationary (for coherence) butproduct can be non-stationary.Coherent mortality forecasting Forecasting groups 19
![Page 46: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/46.jpg)
Mortality product and ratios
Key idea
Model the geometric mean and the mortality ratioinstead of the individual rates for each sexseparately.
pt(x) =√ft ,M(x)ft ,F(x) and rt(x) =
√ft ,M(x)
/ft ,F(x).
Product and ratio are approximatelyindependent
Ratio should be stationary (for coherence) butproduct can be non-stationary.Coherent mortality forecasting Forecasting groups 19
![Page 47: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/47.jpg)
Mortality product and ratios
Key idea
Model the geometric mean and the mortality ratioinstead of the individual rates for each sexseparately.
pt(x) =√ft ,M(x)ft ,F(x) and rt(x) =
√ft ,M(x)
/ft ,F(x).
Product and ratio are approximatelyindependent
Ratio should be stationary (for coherence) butproduct can be non-stationary.Coherent mortality forecasting Forecasting groups 19
![Page 48: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/48.jpg)
Mortality product and ratios
Key idea
Model the geometric mean and the mortality ratioinstead of the individual rates for each sexseparately.
pt(x) =√ft ,M(x)ft ,F(x) and rt(x) =
√ft ,M(x)
/ft ,F(x).
Product and ratio are approximatelyindependent
Ratio should be stationary (for coherence) butproduct can be non-stationary.Coherent mortality forecasting Forecasting groups 19
![Page 49: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/49.jpg)
Product data
Coherent mortality forecasting Forecasting groups 20
0 20 40 60 80 100
−8
−6
−4
−2
0Australia: product data
Age
Log
of g
eom
etric
mea
n de
ath
rate
![Page 50: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/50.jpg)
Ratio data
Coherent mortality forecasting Forecasting groups 21
0 20 40 60 80 100
12
34
Australia: mortality sex ratio (1921−2009)
Age
Sex
rat
io o
f rat
es: M
/F
![Page 51: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/51.jpg)
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 ,kφk(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
![Page 52: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/52.jpg)
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 ,kφk(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
![Page 53: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/53.jpg)
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 ,kφk(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
![Page 54: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/54.jpg)
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 ,kφk(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
![Page 55: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/55.jpg)
Product model
Coherent mortality forecasting Forecasting groups 23
0 20 40 60 80
−8
−6
−4
−2
Age
µ P(x
)
0 20 40 60 80
0.05
0.10
0.15
Age
φ 1(x
)
Year
β t1
1920 1980 2040
−20
−10
−5
05
0 20 40 60 80−0.
2−
0.1
0.0
0.1
0.2
Age
φ 2(x
)
Year
β t2
1920 1980 2040
−1.
5−
0.5
0.5
1.0
0 20 40 60 80
−0.
20−
0.10
0.00
0.10
Age
φ 3(x
)
Year
β t3
1920 1980 2040
−4
−2
02
4
![Page 56: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/56.jpg)
Ratio model
Coherent mortality forecasting Forecasting groups 24
0 20 40 60 80
0.1
0.2
0.3
0.4
Age
µ R(x
)
0 20 40 60 80
−0.
10.
00.
10.
2
Age
φ 1(x
)
Year
β t1
1920 1980 2040
−0.
6−
0.2
0.0
0.2
0.4
0 20 40 60 80
0.00
0.10
0.20
Age
φ 2(x
)
Year
β t2
1920 1980 2040−2.
0−
1.0
0.0
1.0
0 20 40 60 80
−0.
3−
0.1
0.1
0.2
0.3
Age
φ 3(x
)
Year
β t3
1920 1980 2040
−0.
4−
0.2
0.0
0.2
0.4
![Page 57: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/57.jpg)
Product forecasts
Coherent mortality forecasting Forecasting groups 25
0 20 40 60 80 100
−10
−8
−6
−4
−2
Age
Log
of g
eom
etric
mea
n de
ath
rate
![Page 58: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/58.jpg)
Ratio forecasts
Coherent mortality forecasting Forecasting groups 26
0 20 40 60 80 100
12
34
Age
Sex
rat
io: M
/F
![Page 59: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/59.jpg)
Coherent forecasts
Coherent mortality forecasting Forecasting groups 27
0 20 40 60 80 100
−10
−8
−6
−4
−2
(a) Males
Age
Log
deat
h ra
te
0 20 40 60 80 100
−10
−8
−6
−4
−2
(b) Females
Age
Log
deat
h ra
te
![Page 60: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/60.jpg)
Ratio forecasts
Coherent mortality forecasting Forecasting groups 28
0 20 40 60 80 100
01
23
45
67
Independent forecasts
Age
Sex
rat
io o
f rat
es: M
/F
0 20 40 60 80 100
01
23
45
67
Coherent forecasts
Age
Sex
rat
io o
f rat
es: M
/F
![Page 61: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/61.jpg)
Life expectancy forecasts
Coherent mortality forecasting Forecasting groups 29
Life expectancy forecasts
Year
Age
1920 1960 2000 2040
7075
8085
9095
1920 1960 2000 2040
7075
8085
9095Life expectancy difference: F−M
Year
Num
ber
of y
ears
1960 1980 2000 2020
45
67
![Page 62: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/62.jpg)
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 ,kφk(x)+ et(x)
log[rt ,j(x)] = µr ,j(x)+L∑
l=1
γt ,l ,jψl ,j(x)+wt ,j(x).
Coherent mortality forecasting Forecasting groups 30
pt(x) and all rt ,j(x)are approximatelyindependent.
Ratios satisfy constraintrt ,1(x)rt ,2(x) · · ·rt ,J(x) = 1.
log[ft ,j(x)] = log[pt(x)rt ,j(x)]
![Page 63: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/63.jpg)
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 ,kφk(x)+ et(x)
log[rt ,j(x)] = µr ,j(x)+L∑
l=1
γt ,l ,jψl ,j(x)+wt ,j(x).
Coherent mortality forecasting Forecasting groups 30
pt(x) and all rt ,j(x)are approximatelyindependent.
Ratios satisfy constraintrt ,1(x)rt ,2(x) · · ·rt ,J(x) = 1.
log[ft ,j(x)] = log[pt(x)rt ,j(x)]
![Page 64: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/64.jpg)
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 ,kφk(x)+ et(x)
log[rt ,j(x)] = µr ,j(x)+L∑
l=1
γt ,l ,jψl ,j(x)+wt ,j(x).
Coherent mortality forecasting Forecasting groups 30
pt(x) and all rt ,j(x)are approximatelyindependent.
Ratios satisfy constraintrt ,1(x)rt ,2(x) · · ·rt ,J(x) = 1.
log[ft ,j(x)] = log[pt(x)rt ,j(x)]
![Page 65: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/65.jpg)
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 ,kφk(x)+ et(x)
log[rt ,j(x)] = µr ,j(x)+L∑
l=1
γt ,l ,jψl ,j(x)+wt ,j(x).
Coherent mortality forecasting Forecasting groups 30
pt(x) and all rt ,j(x)are approximatelyindependent.
Ratios satisfy constraintrt ,1(x)rt ,2(x) · · ·rt ,J(x) = 1.
log[ft ,j(x)] = log[pt(x)rt ,j(x)]
![Page 66: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/66.jpg)
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 ,kφk(x)+ et(x)
log[rt ,j(x)] = µr ,j(x)+L∑
l=1
γt ,l ,jψl ,j(x)+wt ,j(x).
Coherent mortality forecasting Forecasting groups 30
pt(x) and all rt ,j(x)are approximatelyindependent.
Ratios satisfy constraintrt ,1(x)rt ,2(x) · · ·rt ,J(x) = 1.
log[ft ,j(x)] = log[pt(x)rt ,j(x)]
![Page 67: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/67.jpg)
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 ,kφk(x)+L∑`=1
γt ,`,jψ`,j(x)+ zt ,j(x)
![Page 68: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/68.jpg)
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 ,kφk(x)+L∑`=1
γt ,`,jψ`,j(x)+ zt ,j(x)
![Page 69: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/69.jpg)
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 ,kφk(x)+L∑`=1
γt ,`,jψ`,j(x)+ zt ,j(x)
![Page 70: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/70.jpg)
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 ,kφk(x)+L∑`=1
γt ,`,jψ`,j(x)+ zt ,j(x)
![Page 71: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/71.jpg)
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 ,kφk(x)+L∑`=1
γt ,`,jψ`,j(x)+ zt ,j(x)
![Page 72: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/72.jpg)
Li-Lee method
Li & Lee (Demography, 2005) method is a specialcase of our approach.
ft ,j(x) = µj(x)+ βtφ(x)+γt ,jψj(x)+ et ,j(x)
where f is unsmoothed log mortality rate, βt is arandom 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 correlatedwith each other, and so independent models arenot appropriate.Coherent mortality forecasting Forecasting groups 32
![Page 73: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/73.jpg)
Li-Lee method
Li & Lee (Demography, 2005) method is a specialcase of our approach.
ft ,j(x) = µj(x)+ βtφ(x)+γt ,jψj(x)+ et ,j(x)
where f is unsmoothed log mortality rate, βt is arandom 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 correlatedwith each other, and so independent models arenot appropriate.Coherent mortality forecasting Forecasting groups 32
![Page 74: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/74.jpg)
Li-Lee method
Li & Lee (Demography, 2005) method is a specialcase of our approach.
ft ,j(x) = µj(x)+ βtφ(x)+γt ,jψj(x)+ et ,j(x)
where f is unsmoothed log mortality rate, βt is arandom 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 correlatedwith each other, and so independent models arenot appropriate.Coherent mortality forecasting Forecasting groups 32
![Page 75: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/75.jpg)
Li-Lee method
Li & Lee (Demography, 2005) method is a specialcase of our approach.
ft ,j(x) = µj(x)+ βtφ(x)+γt ,jψj(x)+ et ,j(x)
where f is unsmoothed log mortality rate, βt is arandom 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 correlatedwith each other, and so independent models arenot appropriate.Coherent mortality forecasting Forecasting groups 32
![Page 76: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/76.jpg)
Li-Lee method
Li & Lee (Demography, 2005) method is a specialcase of our approach.
ft ,j(x) = µj(x)+ βtφ(x)+γt ,jψj(x)+ et ,j(x)
where f is unsmoothed log mortality rate, βt is arandom 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 correlatedwith each other, and so independent models arenot appropriate.Coherent mortality forecasting Forecasting groups 32
![Page 77: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/77.jpg)
Outline
1 Functional forecasting
2 Forecasting groups
3 Coherent cohort life expectancy forecasts
4 Conclusions
Coherent mortality forecasting Coherent cohort life expectancy forecasts 33
![Page 78: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/78.jpg)
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 someyear t .Cohort life expectancy: let mx =mx ,t+x for birthcohort in year t .Coherent mortality forecasting Coherent cohort life expectancy forecasts 34
![Page 79: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/79.jpg)
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 someyear t .Cohort life expectancy: let mx =mx ,t+x for birthcohort in year t .Coherent mortality forecasting Coherent cohort life expectancy forecasts 34
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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 someyear t .Cohort life expectancy: let mx =mx ,t+x for birthcohort in year t .Coherent mortality forecasting Coherent cohort life expectancy forecasts 34
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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 someyear t .Cohort life expectancy: let mx =mx ,t+x for birthcohort in year t .Coherent mortality forecasting Coherent cohort life expectancy forecasts 34
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Cohort life expectancy
Because we can forecast mx ,t we canestimate the mortality rates for eachbirth cohort (using actual values whenthey are available).We can simulate future mx ,t in order toestimate the uncertainty associatedwith ex .
Coherent mortality forecasting Coherent cohort life expectancy forecasts 35
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Cohort life expectancy
Because we can forecast mx ,t we canestimate the mortality rates for eachbirth cohort (using actual values whenthey are available).We can simulate future mx ,t in order toestimate the uncertainty associatedwith ex .
Coherent mortality forecasting Coherent cohort life expectancy forecasts 35
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Cohort life expectancy
Coherent mortality forecasting Coherent cohort life expectancy forecasts 36
Age
0
20
40
60
80Year
1950
2000
2050
2100
2150
Log death rate
−10
−5
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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 ,kφk(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
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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 ,kφk(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
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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 ,kφk(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
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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 ,kφk(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
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Cohort life expectancy
Coherent mortality forecasting Coherent cohort life expectancy forecasts 38
Australia: cohort life expectancy at age 50
Year
Rem
aini
ng li
fe e
xpec
tanc
y
1920 1940 1960 1980 2000 2020 2040 2060
2025
3035
4045
50
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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")
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Forecast accuracy evaluation
Coherent mortality forecasting Coherent cohort life expectancy forecasts 40
Australian female cohort e50
Year
Rem
aini
ng y
ears
1920 1940 1960 1980 2000
2628
3032
3436
3840
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Forecast accuracy evaluation
Coherent mortality forecasting Coherent cohort life expectancy forecasts 41
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Forecast accuracy evaluation
Compute age 50 remaining cohort lifeexpectancy with a rolling forecast originbeginning in 1921.Compare against actual cohort lifeexpectancy where available.Compute 80% prediction interval actualcoverage.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 42
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Forecast accuracy evaluation
Compute age 50 remaining cohort lifeexpectancy with a rolling forecast originbeginning in 1921.Compare against actual cohort lifeexpectancy where available.Compute 80% prediction interval actualcoverage.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 42
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Forecast accuracy evaluation
Compute age 50 remaining cohort lifeexpectancy with a rolling forecast originbeginning in 1921.Compare against actual cohort lifeexpectancy where available.Compute 80% prediction interval actualcoverage.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 42
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Forecast accuracy evaluation
Coherent mortality forecasting Coherent cohort life expectancy forecasts 43
5 10 15 20 25
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Mean Absolute Forecast Errors
Forecast horizon
Year
s
1 2 3 4
Male
Female
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Forecast accuracy evaluation
Coherent mortality forecasting Coherent cohort life expectancy forecasts 43
5 10 15 20 25
020
4060
8010
080% prediction interval coverage
Forecast horizon
Per
cent
age
cove
rage
1 2 3 4
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Outline
1 Functional forecasting
2 Forecasting groups
3 Coherent cohort life expectancy forecasts
4 Conclusions
Coherent mortality forecasting Conclusions 44
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Some conclusions
New, automatic, flexible method forcoherent forecasting of groups offunctional time series.Suitable for age-specific mortality.Based on geometric means and ratios,so interpretable results.More general and flexible than existingmethods.Easy to compute prediction intervals forany computable statistics.Coherent mortality forecasting Conclusions 45
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Some conclusions
New, automatic, flexible method forcoherent forecasting of groups offunctional time series.Suitable for age-specific mortality.Based on geometric means and ratios,so interpretable results.More general and flexible than existingmethods.Easy to compute prediction intervals forany computable statistics.Coherent mortality forecasting Conclusions 45
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Some conclusions
New, automatic, flexible method forcoherent forecasting of groups offunctional time series.Suitable for age-specific mortality.Based on geometric means and ratios,so interpretable results.More general and flexible than existingmethods.Easy to compute prediction intervals forany computable statistics.Coherent mortality forecasting Conclusions 45
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Some conclusions
New, automatic, flexible method forcoherent forecasting of groups offunctional time series.Suitable for age-specific mortality.Based on geometric means and ratios,so interpretable results.More general and flexible than existingmethods.Easy to compute prediction intervals forany computable statistics.Coherent mortality forecasting Conclusions 45
![Page 103: Coherent mortality forecasting using functional time series models](https://reader033.fdocuments.net/reader033/viewer/2022052321/55506301b4c905c0448b5344/html5/thumbnails/103.jpg)
Some conclusions
New, automatic, flexible method forcoherent forecasting of groups offunctional time series.Suitable for age-specific mortality.Based on geometric means and ratios,so interpretable results.More general and flexible than existingmethods.Easy to compute prediction intervals forany computable statistics.Coherent mortality forecasting Conclusions 45
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Selected referencesHyndman, Ullah (2007). “Robust forecasting of mortalityand fertility rates: A functional data approach”.Computational Statistics and Data Analysis 51(10),4942–4956Hyndman, Shang (2009). “Forecasting functional timeseries (with discussion)”. Journal of the KoreanStatistical Society 38(3), 199–221Hyndman, Booth, Yasmeen (2013). “Coherent mortalityforecasting: the product-ratio method with functionaltime series models”. Demography 50(1), 261–283Booth, Hyndman, Tickle (2013). “Prospective Life Tables”.Computational Actuarial Science, with R. ed. by
Charpentier. Chapman & Hall/CRC, 323–348Hyndman (2013). demography: Forecasting mortality,fertility, migration and population data. v1.16.cran.r-project.org/package=demography
Coherent mortality forecasting Conclusions 46
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Selected referencesHyndman, Ullah (2007). “Robust forecasting of mortalityand fertility rates: A functional data approach”.Computational Statistics and Data Analysis 51(10),4942–4956Hyndman, Shang (2009). “Forecasting functional timeseries (with discussion)”. Journal of the KoreanStatistical Society 38(3), 199–221Hyndman, Booth, Yasmeen (2013). “Coherent mortalityforecasting: the product-ratio method with functionaltime series models”. Demography 50(1), 261–283Booth, Hyndman, Tickle (2013). “Prospective Life Tables”.Computational Actuarial Science, with R. ed. by
Charpentier. Chapman & Hall/CRC, 323–348Hyndman (2013). demography: Forecasting mortality,fertility, migration and population data. v1.16.cran.r-project.org/package=demography
Coherent mortality forecasting Conclusions 46
å Papers and R code:robjhyndman.com
å Email: [email protected]