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Stochastic population forecasts using FDM
Stochastic populationforecasts using
functional data models
Rob J Hyndman
Department of Econometrics and Business Statistics
Stochastic population forecasts using FDM
Outline
1 Functional time series
2 Current state of Australian populationforecasting
3 Stochastic population forecasting
Stochastic population forecasts using FDM Functional time series
Outline
1 Functional time series
2 Current state of Australian populationforecasting
3 Stochastic population forecasting
Stochastic population forecasts using FDM Functional time series
Mortality rates
Stochastic population forecasts using FDM Functional time series
Fertility rates
Stochastic population forecasts using FDM Functional time series
Some notation
Let yt(xi) be the observed data in period t atage xi, i = 1, . . . ,p, t = 1, . . . ,n.
yt(xi) = st(xi) + σt(xi)εt,i
εt,iiid∼ N(0,1)
st(x) and σt(x) are smooth functions of x.
We need to estimate st(x) from the datafor x1 < x < xp.
We want to forecast whole curve yt(x) fort = n + 1, . . . ,n + h.
Stochastic population forecasts using FDM Functional time series
Some notation
Let yt(xi) be the observed data in period t atage xi, i = 1, . . . ,p, t = 1, . . . ,n.
yt(xi) = st(xi) + σt(xi)εt,i
εt,iiid∼ N(0,1)
st(x) and σt(x) are smooth functions of x.
We need to estimate st(x) from the datafor x1 < x < xp.
We want to forecast whole curve yt(x) fort = n + 1, . . . ,n + h.
Stochastic population forecasts using FDM Functional time series
Some notation
Let yt(xi) be the observed data in period t atage xi, i = 1, . . . ,p, t = 1, . . . ,n.
yt(xi) = st(xi) + σt(xi)εt,i
εt,iiid∼ N(0,1)
st(x) and σt(x) are smooth functions of x.
We need to estimate st(x) from the datafor x1 < x < xp.
We want to forecast whole curve yt(x) fort = n + 1, . . . ,n + h.
Stochastic population forecasts using FDM Functional time series
Some notation
Let yt(xi) be the observed data in period t atage xi, i = 1, . . . ,p, t = 1, . . . ,n.
yt(xi) = st(xi) + σt(xi)εt,i
εt,iiid∼ N(0,1)
st(x) and σt(x) are smooth functions of x.
We need to estimate st(x) from the datafor x1 < x < xp.
We want to forecast whole curve yt(x) fort = n + 1, . . . ,n + h.
Stochastic population forecasts using FDM Functional time series
Functional time series model
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
1 Estimate smooth functions st(x) usingnonparametric regression.
2 Estimate µ(x) as mean st(x) across years.3 Estimate βt,k and φk(x) using functional
principal components.4 Forecast βt,k using time series models.5 Put it all together to get forecasts of yt(x).
Stochastic population forecasts using FDM Functional time series
Functional time series model
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
1 Estimate smooth functions st(x) usingnonparametric regression.
2 Estimate µ(x) as mean st(x) across years.3 Estimate βt,k and φk(x) using functional
principal components.4 Forecast βt,k using time series models.5 Put it all together to get forecasts of yt(x).
Stochastic population forecasts using FDM Functional time series
Functional time series model
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
1 Estimate smooth functions st(x) usingnonparametric regression.
2 Estimate µ(x) as mean st(x) across years.
3 Estimate βt,k and φk(x) using functionalprincipal components.
4 Forecast βt,k using time series models.5 Put it all together to get forecasts of yt(x).
Stochastic population forecasts using FDM Functional time series
Functional time series model
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
1 Estimate smooth functions st(x) usingnonparametric regression.
2 Estimate µ(x) as mean st(x) across years.3 Estimate βt,k and φk(x) using functional
principal components.
4 Forecast βt,k using time series models.5 Put it all together to get forecasts of yt(x).
Stochastic population forecasts using FDM Functional time series
Functional time series model
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
1 Estimate smooth functions st(x) usingnonparametric regression.
2 Estimate µ(x) as mean st(x) across years.3 Estimate βt,k and φk(x) using functional
principal components.4 Forecast βt,k using time series models.
5 Put it all together to get forecasts of yt(x).
Stochastic population forecasts using FDM Functional time series
Functional time series model
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
1 Estimate smooth functions st(x) usingnonparametric regression.
2 Estimate µ(x) as mean st(x) across years.3 Estimate βt,k and φk(x) using functional
principal components.4 Forecast βt,k using time series models.5 Put it all together to get forecasts of yt(x).
Stochastic population forecasts using FDM Functional time series
Functional time series model
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
1 Estimate smooth functions st(x) usingnonparametric regression.
2 Estimate µ(x) as mean st(x) across years.3 Estimate βt,k and φk(x) using functional
principal components.4 Forecast βt,k using time series models.5 Put it all together to get forecasts of yt(x).
Stochastic population forecasts using FDM Functional time series
1. Creating functional time series
0 20 40 60 80 100
−8
−6
−4
−2
0
Australia: male death rates (1921−2003)
Age
Log
deat
h ra
te
Stochastic population forecasts using FDM Functional time series
1. Creating functional time series
0 20 40 60 80 100
−8
−6
−4
−2
0
Australia: male death rates (1921−2003)
Age
Log
deat
h ra
te
Stochastic population forecasts using FDM Functional time series
1. Creating functional time series
Monotonic regression splines
Fit penalized regression spline with a largenumber of knots.
For mortality data, constrain curve to bemonotonic for x > b.
Choosing b = 50 seems to work quite wellin practice for mortality data.
Fit is weighted with weightswt(xi) = σ−2
t (xi) (based on Poisson deaths).
This can be done using a modification ofthe gam function in the mgcv package in R.
Stochastic population forecasts using FDM Functional time series
1. Creating functional time series
Monotonic regression splines
Fit penalized regression spline with a largenumber of knots.
For mortality data, constrain curve to bemonotonic for x > b.
Choosing b = 50 seems to work quite wellin practice for mortality data.
Fit is weighted with weightswt(xi) = σ−2
t (xi) (based on Poisson deaths).
This can be done using a modification ofthe gam function in the mgcv package in R.
Stochastic population forecasts using FDM Functional time series
1. Creating functional time series
Monotonic regression splines
Fit penalized regression spline with a largenumber of knots.
For mortality data, constrain curve to bemonotonic for x > b.
Choosing b = 50 seems to work quite wellin practice for mortality data.
Fit is weighted with weightswt(xi) = σ−2
t (xi) (based on Poisson deaths).
This can be done using a modification ofthe gam function in the mgcv package in R.
Stochastic population forecasts using FDM Functional time series
1. Creating functional time series
Monotonic regression splines
Fit penalized regression spline with a largenumber of knots.
For mortality data, constrain curve to bemonotonic for x > b.
Choosing b = 50 seems to work quite wellin practice for mortality data.
Fit is weighted with weightswt(xi) = σ−2
t (xi) (based on Poisson deaths).
This can be done using a modification ofthe gam function in the mgcv package in R.
Stochastic population forecasts using FDM Functional time series
1. Creating functional time series
Monotonic regression splines
Fit penalized regression spline with a largenumber of knots.
For mortality data, constrain curve to bemonotonic for x > b.
Choosing b = 50 seems to work quite wellin practice for mortality data.
Fit is weighted with weightswt(xi) = σ−2
t (xi) (based on Poisson deaths).
This can be done using a modification ofthe gam function in the mgcv package in R.
Stochastic population forecasts using FDM Functional time series
Functional time series model
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
1 Estimate smooth functions st(x) usingnonparametric regression.
2 Estimate µ(x) as mean st(x) across years.3 Estimate βt,k and φk(x) using functional
principal components.4 Forecast βt,k using time series models.5 Put it all together to get forecasts of yt(x).
Stochastic population forecasts using FDM Functional time series
2. Estimate µ(x)
0 20 40 60 80 100
−8
−6
−4
−2
0
Australia: male death rates (1921−2003)
Age
Log
deat
h ra
te
Stochastic population forecasts using FDM Functional time series
2. Estimate µ(x)
0 20 40 60 80 100
−8
−6
−4
−2
0
Australia: male death rates (1921−2003)
Age
Log
deat
h ra
te
Stochastic population forecasts using FDM Functional time series
Functional time series model
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
1 Estimate smooth functions st(x) usingnonparametric regression.
2 Estimate µ(x) as mean st(x) across years.3 Estimate βt,k and φk(x) using
functional principal components.4 Forecast βt,k using time series models.5 Put it all together to get forecasts of yt(x).
Stochastic population forecasts using FDM Functional time series
3. Functional PC
The optimal basis functions
st(x) = µ(x) +K∑
i=0
βt,iφi(x) + et(x).
where et(x) =n−1∑
i=K+1
βt,i φi(x).
For a given K, the basis functions φi(x) whichminimize
MISE =1
n
n∑t=1
∫v2
t (x) dx
are the principal components.
Stochastic population forecasts using FDM Functional time series
3. Functional PC
The optimal basis functions
st(x) = µ(x) +K∑
i=0
βt,iφi(x) + et(x).
where et(x) =n−1∑
i=K+1
βt,i φi(x).
For a given K, the basis functions φi(x) whichminimize
MISE =1
n
n∑t=1
∫v2
t (x) dx
are the principal components.
Stochastic population forecasts using FDM Functional time series
3. Functional PC
(Ramsay and Silverman, 1997,2002).In FDA, each principal component isspecified by a weight function φi(x).
The PC scores for each year are given by
zi,t =
∫φi(x)st(x)dx
The aim is to:
1 Find the weight function φ1(x) that maximizesthe variance of z1,t subject to the constraint∫φ2
i (x)dx = 1.2 Find the weight function φ2(x) that maximizes
the variance of z2,t such that∫φ2
i (x)dx = 1and
∫φ1(x)φ2(x)dx = 0.
3 Find the weight function φ3(x) that . . .
Stochastic population forecasts using FDM Functional time series
3. Functional PC
(Ramsay and Silverman, 1997,2002).In FDA, each principal component isspecified by a weight function φi(x).The PC scores for each year are given by
zi,t =
∫φi(x)st(x)dx
The aim is to:
1 Find the weight function φ1(x) that maximizesthe variance of z1,t subject to the constraint∫φ2
i (x)dx = 1.2 Find the weight function φ2(x) that maximizes
the variance of z2,t such that∫φ2
i (x)dx = 1and
∫φ1(x)φ2(x)dx = 0.
3 Find the weight function φ3(x) that . . .
Stochastic population forecasts using FDM Functional time series
3. Functional PC
(Ramsay and Silverman, 1997,2002).In FDA, each principal component isspecified by a weight function φi(x).The PC scores for each year are given by
zi,t =
∫φi(x)st(x)dx
The aim is to:
1 Find the weight function φ1(x) that maximizesthe variance of z1,t subject to the constraint∫φ2
i (x)dx = 1.2 Find the weight function φ2(x) that maximizes
the variance of z2,t such that∫φ2
i (x)dx = 1and
∫φ1(x)φ2(x)dx = 0.
3 Find the weight function φ3(x) that . . .
Stochastic population forecasts using FDM Functional time series
3. Functional PC
(Ramsay and Silverman, 1997,2002).In FDA, each principal component isspecified by a weight function φi(x).The PC scores for each year are given by
zi,t =
∫φi(x)st(x)dx
The aim is to:1 Find the weight function φ1(x) that maximizes
the variance of z1,t subject to the constraint∫φ2
i (x)dx = 1.
2 Find the weight function φ2(x) that maximizesthe variance of z2,t such that
∫φ2
i (x)dx = 1and
∫φ1(x)φ2(x)dx = 0.
3 Find the weight function φ3(x) that . . .
Stochastic population forecasts using FDM Functional time series
3. Functional PC
(Ramsay and Silverman, 1997,2002).In FDA, each principal component isspecified by a weight function φi(x).The PC scores for each year are given by
zi,t =
∫φi(x)st(x)dx
The aim is to:1 Find the weight function φ1(x) that maximizes
the variance of z1,t subject to the constraint∫φ2
i (x)dx = 1.2 Find the weight function φ2(x) that maximizes
the variance of z2,t such that∫φ2
i (x)dx = 1and
∫φ1(x)φ2(x)dx = 0.
3 Find the weight function φ3(x) that . . .
Stochastic population forecasts using FDM Functional time series
3. Functional PC
(Ramsay and Silverman, 1997,2002).In FDA, each principal component isspecified by a weight function φi(x).The PC scores for each year are given by
zi,t =
∫φi(x)st(x)dx
The aim is to:1 Find the weight function φ1(x) that maximizes
the variance of z1,t subject to the constraint∫φ2
i (x)dx = 1.2 Find the weight function φ2(x) that maximizes
the variance of z2,t such that∫φ2
i (x)dx = 1and
∫φ1(x)φ2(x)dx = 0.
3 Find the weight function φ3(x) that . . .
Stochastic population forecasts using FDM Functional time series
3. Functional PC
0 20 40 60 80 100
−8
−6
−4
−2
Main effects
Age
Mu
0 20 40 60 80 1000.
00.
10.
20.
30.
4
Age
Phi
1
Interaction
Time
Bet
a 1
1950 1970 1990
−2
−1
01
20 20 40 60 80 100
−0.
6−
0.4
−0.
20.
00.
20.
4
Age
Phi
2
Time
Bet
a 2
1950 1970 1990−
0.6
−0.
4−
0.2
0.0
0.2
0.4
Stochastic population forecasts using FDM Functional time series
3. Functional PC
Recap
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
Pure time term excluded as it would make{βt,k} correlated.
We can check if any structure is left in theresiduals εt,x (smoothing problem) andet(x) (modelling problem).
Stochastic population forecasts using FDM Functional time series
3. Functional PC
Recap
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
Pure time term excluded as it would make{βt,k} correlated.
We can check if any structure is left in theresiduals εt,x (smoothing problem) andet(x) (modelling problem).
Stochastic population forecasts using FDM Functional time series
3. Functional PC
1950 1960 1970 1980 1990 2000
020
4060
8010
0 Smoothing residuals
Year
Age
Stochastic population forecasts using FDM Functional time series
3. Functional PC
1950 1960 1970 1980 1990 2000
020
4060
8010
0 Modelling residuals
Year
Age
Stochastic population forecasts using FDM Functional time series
Functional time series model
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
1 Estimate smooth functions st(x) usingnonparametric regression.
2 Estimate µ(x) as mean st(x) across years.3 Estimate βt,k and φk(x) using functional
principal components.4 Forecast βt,k using time series models.5 Put it all together to get forecasts of yt(x).
Stochastic population forecasts using FDM Functional time series
4. Forecasting the coefficients
0 20 40 60 80 100
−8
−6
−4
−2
Main effects
Age
Mea
n
0 20 40 60 80 1000.
00.
10.
20.
30.
4
Age
Bas
is fu
nctio
n 1
Interaction
Year
Coe
ffici
ent 1
1960 1980 2000 2020
−6
−4
−2
02
0 20 40 60 80 100
−0.
6−
0.4
−0.
20.
00.
20.
4
Age
Bas
is fu
nctio
n 2
Year
Coe
ffici
ent 2
1960 1980 2000 2020−
0.6
−0.
20.
00.
20.
4
Stochastic population forecasts using FDM Functional time series
4. Forecasting the coefficients
I use exponential smoothing state spacemodels based on Ord, Koehler & Snyder(JASA, 1997) and Hyndman, Koehler,Snyder & Grose (IJF, 2002).
These provide a stochastic framework forexponential smoothing forecasts.The models shown are equivalent to
damped Holt’s methodARIMA(0,1,2)
Univariate models are ok because theseries are uncorrelated.
Stochastic population forecasts using FDM Functional time series
4. Forecasting the coefficients
I use exponential smoothing state spacemodels based on Ord, Koehler & Snyder(JASA, 1997) and Hyndman, Koehler,Snyder & Grose (IJF, 2002).
These provide a stochastic framework forexponential smoothing forecasts.
The models shown are equivalent to
damped Holt’s methodARIMA(0,1,2)
Univariate models are ok because theseries are uncorrelated.
Stochastic population forecasts using FDM Functional time series
4. Forecasting the coefficients
I use exponential smoothing state spacemodels based on Ord, Koehler & Snyder(JASA, 1997) and Hyndman, Koehler,Snyder & Grose (IJF, 2002).
These provide a stochastic framework forexponential smoothing forecasts.The models shown are equivalent to
damped Holt’s methodARIMA(0,1,2)
Univariate models are ok because theseries are uncorrelated.
Stochastic population forecasts using FDM Functional time series
4. Forecasting the coefficients
I use exponential smoothing state spacemodels based on Ord, Koehler & Snyder(JASA, 1997) and Hyndman, Koehler,Snyder & Grose (IJF, 2002).
These provide a stochastic framework forexponential smoothing forecasts.The models shown are equivalent to
damped Holt’s method
ARIMA(0,1,2)
Univariate models are ok because theseries are uncorrelated.
Stochastic population forecasts using FDM Functional time series
4. Forecasting the coefficients
I use exponential smoothing state spacemodels based on Ord, Koehler & Snyder(JASA, 1997) and Hyndman, Koehler,Snyder & Grose (IJF, 2002).
These provide a stochastic framework forexponential smoothing forecasts.The models shown are equivalent to
damped Holt’s methodARIMA(0,1,2)
Univariate models are ok because theseries are uncorrelated.
Stochastic population forecasts using FDM Functional time series
4. Forecasting the coefficients
I use exponential smoothing state spacemodels based on Ord, Koehler & Snyder(JASA, 1997) and Hyndman, Koehler,Snyder & Grose (IJF, 2002).
These provide a stochastic framework forexponential smoothing forecasts.The models shown are equivalent to
damped Holt’s methodARIMA(0,1,2)
Univariate models are ok because theseries are uncorrelated.
Stochastic population forecasts using FDM Functional time series
Functional time series model
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
1 Estimate smooth functions st(x) usingnonparametric regression.
2 Estimate µ(x) as mean st(x) across years.3 Estimate βt,k and φk(x) using functional
principal components.4 Forecast βt,k using time series models.5 Put it all together to get forecasts of yt(x).
Stochastic population forecasts using FDM Functional time series
5. Forecasts of yt(x)
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
Let I = {yt(xi); t = 1, . . . ,n; i = 1, . . . ,p}.
E[yn+h(x) | I,Φ] = µ(x) +K∑
k=1
βn+h|n,k φk(x).
Var[yn+h(x) | I,Φ] =
σ2n+h(x) + σ2
µ(x) +K∑
k=1
vn+h|n,k φ2k(x) + v(x)
where vn+h|n,k = Var(βn+h,k | β1,k, . . . , βn,k).
Stochastic population forecasts using FDM Functional time series
5. Forecasts of yt(x)
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
Let I = {yt(xi); t = 1, . . . ,n; i = 1, . . . ,p}.
E[yn+h(x) | I,Φ] = µ(x) +K∑
k=1
βn+h|n,k φk(x).
Var[yn+h(x) | I,Φ] =
σ2n+h(x) + σ2
µ(x) +K∑
k=1
vn+h|n,k φ2k(x) + v(x)
where vn+h|n,k = Var(βn+h,k | β1,k, . . . , βn,k).
Stochastic population forecasts using FDM Functional time series
5. Forecasts of yt(x)
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
Let I = {yt(xi); t = 1, . . . ,n; i = 1, . . . ,p}.
E[yn+h(x) | I,Φ] = µ(x) +K∑
k=1
βn+h|n,k φk(x).
Var[yn+h(x) | I,Φ] =
σ2n+h(x) + σ2
µ(x) +K∑
k=1
vn+h|n,k φ2k(x) + v(x)
where vn+h|n,k = Var(βn+h,k | β1,k, . . . , βn,k).
Stochastic population forecasts using FDM Functional time series
5. Forecasts of yt(x)
0 20 40 60 80 100
−10
−8
−6
−4
−2
Australia: male death rate forecasts (2004 and 2023)
Age
Log
deat
h ra
te
Stochastic population forecasts using FDM Functional time series
5. Forecasts of yt(x)
0 20 40 60 80 100
−10
−8
−6
−4
−2
Australia: male death rate forecasts (2004 and 2023)
Age
Log
deat
h ra
te
Stochastic population forecasts using FDM Functional time series
5. Forecasts of yt(x)
0 20 40 60 80 100
−10
−8
−6
−4
−2
Australia: male death rate forecasts (2004 and 2023)
Age
Log
deat
h ra
te
Stochastic population forecasts using FDM Functional time series
5. Forecasts of yt(x)
0 20 40 60 80 100
−10
−8
−6
−4
−2
Australia: male death rate forecasts (2004 and 2023)
Age
Log
deat
h ra
te
80% prediction intervals
Stochastic population forecasts using FDM Functional time series
Some referencesErbas, Hyndman and Gertig (2007) Forecastingage-specific breast cancer mortality usingfunctional data models. Statistics in Medicine,26(2), 458–470.
Hyndman and Ullah (2007) Robust forecasting ofmortality and fertility rates: a functional dataapproach. Computational Statistics & DataAnalysis, 51, 4942–4956.Booth, Hyndman, Tickle and De Jong (2006)Lee-Carter mortality forecasting: a multi-countrycomparison of variants and extensions.Demographic Research, 15(9), 289–310.Hyndman (2006) demography: Forecastingmortality and fertility data. R package v0.98.www.robhyndman.info/Rlibrary/demography
Stochastic population forecasts using FDM Functional time series
Some referencesErbas, Hyndman and Gertig (2007) Forecastingage-specific breast cancer mortality usingfunctional data models. Statistics in Medicine,26(2), 458–470.Hyndman and Ullah (2007) Robust forecasting ofmortality and fertility rates: a functional dataapproach. Computational Statistics & DataAnalysis, 51, 4942–4956.
Booth, Hyndman, Tickle and De Jong (2006)Lee-Carter mortality forecasting: a multi-countrycomparison of variants and extensions.Demographic Research, 15(9), 289–310.Hyndman (2006) demography: Forecastingmortality and fertility data. R package v0.98.www.robhyndman.info/Rlibrary/demography
Stochastic population forecasts using FDM Functional time series
Some referencesErbas, Hyndman and Gertig (2007) Forecastingage-specific breast cancer mortality usingfunctional data models. Statistics in Medicine,26(2), 458–470.Hyndman and Ullah (2007) Robust forecasting ofmortality and fertility rates: a functional dataapproach. Computational Statistics & DataAnalysis, 51, 4942–4956.Booth, Hyndman, Tickle and De Jong (2006)Lee-Carter mortality forecasting: a multi-countrycomparison of variants and extensions.Demographic Research, 15(9), 289–310.
Hyndman (2006) demography: Forecastingmortality and fertility data. R package v0.98.www.robhyndman.info/Rlibrary/demography
Stochastic population forecasts using FDM Functional time series
Some referencesErbas, Hyndman and Gertig (2007) Forecastingage-specific breast cancer mortality usingfunctional data models. Statistics in Medicine,26(2), 458–470.Hyndman and Ullah (2007) Robust forecasting ofmortality and fertility rates: a functional dataapproach. Computational Statistics & DataAnalysis, 51, 4942–4956.Booth, Hyndman, Tickle and De Jong (2006)Lee-Carter mortality forecasting: a multi-countrycomparison of variants and extensions.Demographic Research, 15(9), 289–310.Hyndman (2006) demography: Forecastingmortality and fertility data. R package v0.98.www.robhyndman.info/Rlibrary/demography
Stochastic population forecasts using FDM Current state of Australian population forecasting
Outline
1 Functional time series
2 Current state of Australian populationforecasting
3 Stochastic population forecasting
Stochastic population forecasts using FDM Current state of Australian population forecasting
ABS population projections
The Australian Bureau of Statistics providepopulation “projections”.“The projections are not intended as predictions orforecasts, but are illustrations of growth and change inthe population that would occur if assumptions madeabout future demographic trends were to prevail overthe projection period.While the assumptions are formulated on the basis of anassessment of past demographic trends, both inAustralia and overseas, there is no certainty that any ofthe assumptions will be realised. In addition, noassessment has been made of changes innon-demographic conditions.”ABS 3222.0 - Population Projections, Australia, 2004 to 2101
Stochastic population forecasts using FDM Current state of Australian population forecasting
ABS population projections
The ABS provides three projection scenarioslabelled “High”, “Medium” and “Low”.
Based on assumed mortality, fertility andmigration rates
No objectivity.
No dynamic changes in rates allowed
No variation allowed across ages.
No probabilistic basis.
Not prediction intervals.
Most users use the “Medium” projection,but it is unrelated to the mean, median ormode of the future distribution.
Stochastic population forecasts using FDM Current state of Australian population forecasting
ABS population projections
The ABS provides three projection scenarioslabelled “High”, “Medium” and “Low”.
Based on assumed mortality, fertility andmigration rates
No objectivity.
No dynamic changes in rates allowed
No variation allowed across ages.
No probabilistic basis.
Not prediction intervals.
Most users use the “Medium” projection,but it is unrelated to the mean, median ormode of the future distribution.
Stochastic population forecasts using FDM Current state of Australian population forecasting
ABS population projections
The ABS provides three projection scenarioslabelled “High”, “Medium” and “Low”.
Based on assumed mortality, fertility andmigration rates
No objectivity.
No dynamic changes in rates allowed
No variation allowed across ages.
No probabilistic basis.
Not prediction intervals.
Most users use the “Medium” projection,but it is unrelated to the mean, median ormode of the future distribution.
Stochastic population forecasts using FDM Current state of Australian population forecasting
ABS population projections
The ABS provides three projection scenarioslabelled “High”, “Medium” and “Low”.
Based on assumed mortality, fertility andmigration rates
No objectivity.
No dynamic changes in rates allowed
No variation allowed across ages.
No probabilistic basis.
Not prediction intervals.
Most users use the “Medium” projection,but it is unrelated to the mean, median ormode of the future distribution.
Stochastic population forecasts using FDM Current state of Australian population forecasting
ABS population projections
The ABS provides three projection scenarioslabelled “High”, “Medium” and “Low”.
Based on assumed mortality, fertility andmigration rates
No objectivity.
No dynamic changes in rates allowed
No variation allowed across ages.
No probabilistic basis.
Not prediction intervals.
Most users use the “Medium” projection,but it is unrelated to the mean, median ormode of the future distribution.
Stochastic population forecasts using FDM Current state of Australian population forecasting
ABS population projections
The ABS provides three projection scenarioslabelled “High”, “Medium” and “Low”.
Based on assumed mortality, fertility andmigration rates
No objectivity.
No dynamic changes in rates allowed
No variation allowed across ages.
No probabilistic basis.
Not prediction intervals.
Most users use the “Medium” projection,but it is unrelated to the mean, median ormode of the future distribution.
Stochastic population forecasts using FDM Current state of Australian population forecasting
ABS population projections
The ABS provides three projection scenarioslabelled “High”, “Medium” and “Low”.
Based on assumed mortality, fertility andmigration rates
No objectivity.
No dynamic changes in rates allowed
No variation allowed across ages.
No probabilistic basis.
Not prediction intervals.
Most users use the “Medium” projection,but it is unrelated to the mean, median ormode of the future distribution.
Stochastic population forecasts using FDM Current state of Australian population forecasting
ABS population projections
Australian total population
Year
Mill
ions
1920 1940 1960 1980 2000 2020 2040
510
1520
2530
A
B
C
What do these projections mean?
Stochastic population forecasts using FDM Current state of Australian population forecasting
ABS population projections
Australian total population
Year
Mill
ions
1920 1940 1960 1980 2000 2020 2040
510
1520
2530
A
B
C
What do these projections mean?
Stochastic population forecasts using FDM Stochastic population forecasting
Outline
1 Functional time series
2 Current state of Australian populationforecasting
3 Stochastic population forecasting
Stochastic population forecasts using FDM Stochastic population forecasting
Stochastic population forecasts
Forecasts represent median of future distribution.
Percentiles allow information about uncertainty
Prediction intervals with specified probabilitycoverage for population size and all derivedvariables (total fertility rate, life expectancy,old-age dependencies, etc.)
The probability of future events can beestimated.
Economic planning is better based onprediction intervals than point forecasts.
Stochastic models allow true policy analysis.
Stochastic population forecasts using FDM Stochastic population forecasting
Stochastic population forecasts
Forecasts represent median of future distribution.
Percentiles allow information about uncertainty
Prediction intervals with specified probabilitycoverage for population size and all derivedvariables (total fertility rate, life expectancy,old-age dependencies, etc.)
The probability of future events can beestimated.
Economic planning is better based onprediction intervals than point forecasts.
Stochastic models allow true policy analysis.
Stochastic population forecasts using FDM Stochastic population forecasting
Stochastic population forecasts
Forecasts represent median of future distribution.
Percentiles allow information about uncertainty
Prediction intervals with specified probabilitycoverage for population size and all derivedvariables (total fertility rate, life expectancy,old-age dependencies, etc.)
The probability of future events can beestimated.
Economic planning is better based onprediction intervals than point forecasts.
Stochastic models allow true policy analysis.
Stochastic population forecasts using FDM Stochastic population forecasting
Stochastic population forecasts
Forecasts represent median of future distribution.
Percentiles allow information about uncertainty
Prediction intervals with specified probabilitycoverage for population size and all derivedvariables (total fertility rate, life expectancy,old-age dependencies, etc.)
The probability of future events can beestimated.
Economic planning is better based onprediction intervals than point forecasts.
Stochastic models allow true policy analysis.
Stochastic population forecasts using FDM Stochastic population forecasting
Stochastic population forecasts
Forecasts represent median of future distribution.
Percentiles allow information about uncertainty
Prediction intervals with specified probabilitycoverage for population size and all derivedvariables (total fertility rate, life expectancy,old-age dependencies, etc.)
The probability of future events can beestimated.
Economic planning is better based onprediction intervals than point forecasts.
Stochastic models allow true policy analysis.
Stochastic population forecasts using FDM Stochastic population forecasting
Stochastic population forecasts
Forecasts represent median of future distribution.
Percentiles allow information about uncertainty
Prediction intervals with specified probabilitycoverage for population size and all derivedvariables (total fertility rate, life expectancy,old-age dependencies, etc.)
The probability of future events can beestimated.
Economic planning is better based onprediction intervals than point forecasts.
Stochastic models allow true policy analysis.
Stochastic population forecasts using FDM Stochastic population forecasting
Demographic growth-balance equation
Demographic growth-balance equationPt+1(x + 1) = Pt(x)− Dt(x, x + 1) + Gt(x, x + 1)
Pt+1(0) = Bt − Dt(B,0) + Gt(B,0)
x = 0,1,2, . . . .
Pt(x) = population of age x at 1 January, year tBt = births in calendar year t
Dt(x, x + 1) = deaths in calendar year t of persons aged x atthe beginning of year t
Dt(B,0) = infant deaths in calendar year tGt(x, x + 1) = net migrants in calendar year t of persons
aged x at the beginning of year tGt(B,0) = net migrants of infants born in calendar year t
Stochastic population forecasts using FDM Stochastic population forecasting
Demographic growth-balance equation
Demographic growth-balance equationPt+1(x + 1) = Pt(x)− Dt(x, x + 1) + Gt(x, x + 1)
Pt+1(0) = Bt − Dt(B,0) + Gt(B,0)
x = 0,1,2, . . . .
Pt(x) = population of age x at 1 January, year tBt = births in calendar year t
Dt(x, x + 1) = deaths in calendar year t of persons aged x atthe beginning of year t
Dt(B,0) = infant deaths in calendar year tGt(x, x + 1) = net migrants in calendar year t of persons
aged x at the beginning of year tGt(B,0) = net migrants of infants born in calendar year t
Stochastic population forecasts using FDM Stochastic population forecasting
Key ideas
Build a stochastic functional model foreach of mortality, fertility and net migration.
Treat all observed data as functional (i.e.,smooth curves of age) rather than discretevalues.Use the models to simulate future samplepaths of all components giving the entireage distribution at every year into the future.Compute future births, deaths, net migrantsand populations from simulated rates.Combine the results to get age-specificstochastic population forecasts.
Stochastic population forecasts using FDM Stochastic population forecasting
Key ideas
Build a stochastic functional model foreach of mortality, fertility and net migration.Treat all observed data as functional (i.e.,smooth curves of age) rather than discretevalues.
Use the models to simulate future samplepaths of all components giving the entireage distribution at every year into the future.Compute future births, deaths, net migrantsand populations from simulated rates.Combine the results to get age-specificstochastic population forecasts.
Stochastic population forecasts using FDM Stochastic population forecasting
Key ideas
Build a stochastic functional model foreach of mortality, fertility and net migration.Treat all observed data as functional (i.e.,smooth curves of age) rather than discretevalues.Use the models to simulate future samplepaths of all components giving the entireage distribution at every year into the future.
Compute future births, deaths, net migrantsand populations from simulated rates.Combine the results to get age-specificstochastic population forecasts.
Stochastic population forecasts using FDM Stochastic population forecasting
Key ideas
Build a stochastic functional model foreach of mortality, fertility and net migration.Treat all observed data as functional (i.e.,smooth curves of age) rather than discretevalues.Use the models to simulate future samplepaths of all components giving the entireage distribution at every year into the future.Compute future births, deaths, net migrantsand populations from simulated rates.
Combine the results to get age-specificstochastic population forecasts.
Stochastic population forecasts using FDM Stochastic population forecasting
Key ideas
Build a stochastic functional model foreach of mortality, fertility and net migration.Treat all observed data as functional (i.e.,smooth curves of age) rather than discretevalues.Use the models to simulate future samplepaths of all components giving the entireage distribution at every year into the future.Compute future births, deaths, net migrantsand populations from simulated rates.Combine the results to get age-specificstochastic population forecasts.
Stochastic population forecasts using FDM Stochastic population forecasting
The available data
In most countries, the following data areavailable:Pt(x) = population of age x at 1 January, year tEt(x) = population of age x at 30 June, year tBt(x) = births in calendar year t to females of age xDt(x) = deaths in calendar year t of persons of age x
From these, we can estimate:
mt(x) = Dt(x)/Et(x) = central death rate incalendar year t;ft(x) = Bt(x)/EF
t (x) = fertility rate forfemales of age x in calendar year t.
Stochastic population forecasts using FDM Stochastic population forecasting
The available data
In most countries, the following data areavailable:Pt(x) = population of age x at 1 January, year tEt(x) = population of age x at 30 June, year tBt(x) = births in calendar year t to females of age xDt(x) = deaths in calendar year t of persons of age x
From these, we can estimate:
mt(x) = Dt(x)/Et(x) = central death rate incalendar year t;
ft(x) = Bt(x)/EFt (x) = fertility rate for
females of age x in calendar year t.
Stochastic population forecasts using FDM Stochastic population forecasting
The available data
In most countries, the following data areavailable:Pt(x) = population of age x at 1 January, year tEt(x) = population of age x at 30 June, year tBt(x) = births in calendar year t to females of age xDt(x) = deaths in calendar year t of persons of age x
From these, we can estimate:
mt(x) = Dt(x)/Et(x) = central death rate incalendar year t;ft(x) = Bt(x)/EF
t (x) = fertility rate forfemales of age x in calendar year t.
Stochastic population forecasts using FDM Stochastic population forecasting
Australia’s start-of-year population
0 20 40 60 80 100
050
000
1000
0015
0000
Australia: female population (1921−2004)
Age
Pop
ulat
ion
Stochastic population forecasts using FDM Stochastic population forecasting
Mortality rates
0 20 40 60 80 100
−8
−6
−4
−2
0
Australia: male death rates (1921−2003)
Age
Log
deat
h ra
te
Stochastic population forecasts using FDM Stochastic population forecasting
Fertility rates
15 20 25 30 35 40 45 50
050
100
150
200
250
Australia fertility rates (1921−2003)
Age
Fer
tility
rat
e
Stochastic population forecasts using FDM Stochastic population forecasting
Net migration
We need to estimate migration data based ondifference in population numbers afteradjusting for births and deaths.
Demographic growth-balance equationGt(x, x + 1) = Pt+1(x + 1)− Pt(x) + Dt(x, x + 1)
Gt(B,0) = Pt+1(0) − Bt + Dt(B,0)
x = 0,1,2, . . . .
Note: “net migration” numbers also includeerrors associated with all estimates. i.e., a“residual”.
Stochastic population forecasts using FDM Stochastic population forecasting
Net migration
We need to estimate migration data based ondifference in population numbers afteradjusting for births and deaths.
Demographic growth-balance equationGt(x, x + 1) = Pt+1(x + 1)− Pt(x) + Dt(x, x + 1)
Gt(B,0) = Pt+1(0) − Bt + Dt(B,0)
x = 0,1,2, . . . .
Note: “net migration” numbers also includeerrors associated with all estimates. i.e., a“residual”.
Stochastic population forecasts using FDM Stochastic population forecasting
Net migration
We need to estimate migration data based ondifference in population numbers afteradjusting for births and deaths.
Demographic growth-balance equationGt(x, x + 1) = Pt+1(x + 1)− Pt(x) + Dt(x, x + 1)
Gt(B,0) = Pt+1(0) − Bt + Dt(B,0)
x = 0,1,2, . . . .
Note: “net migration” numbers also includeerrors associated with all estimates. i.e., a“residual”.
Stochastic population forecasts using FDM Stochastic population forecasting
Net migration
0 20 40 60 80 100
−40
00−
2000
020
0040
00
Australia: male net migration (1922−2003)
Age
Net
mig
ratio
n
Stochastic population forecasts using FDM Stochastic population forecasting
Net migration
0 20 40 60 80 100
−40
00−
2000
020
0040
00
Australia: female net migration (1922−2003)
Age
Net
mig
ratio
n
Stochastic population forecasts using FDM Stochastic population forecasting
Stochastic population forecastsComponent models
Data: age/sex-specific mortality rates,fertility rates and net migration.
Models: Five functional time series modelsfor mortality (M/F), fertility and netmigration (M/F) assuming independencebetween components.For each component:
Stochastic population forecasts using FDM Stochastic population forecasting
Stochastic population forecastsComponent models
Data: age/sex-specific mortality rates,fertility rates and net migration.Models: Five functional time series modelsfor mortality (M/F), fertility and netmigration (M/F) assuming independencebetween components.
For each component:
Stochastic population forecasts using FDM Stochastic population forecasting
Stochastic population forecastsComponent models
Data: age/sex-specific mortality rates,fertility rates and net migration.Models: Five functional time series modelsfor mortality (M/F), fertility and netmigration (M/F) assuming independencebetween components.For each component:
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
Stochastic population forecasts using FDM Stochastic population forecasting
Functional time series
Let gλ(u) =
{log(u) λ = 0;xλ−1λ λ > 0.
Mortality rates:yt(xi) = g0(mt(xi)) where mt(xi) =empirical mortality rate at age xi.
Fertility rates:yt(xi) = g0.45(pt(xi)) where pt(xi) =empirical fertility rate at age xi.
Net migration:yt(xi) = empirical net migration at age xi.
Stochastic population forecasts using FDM Stochastic population forecasting
Functional time series
Let gλ(u) =
{log(u) λ = 0;xλ−1λ λ > 0.
Mortality rates:yt(xi) = g0(mt(xi)) where mt(xi) =empirical mortality rate at age xi.
Fertility rates:yt(xi) = g0.45(pt(xi)) where pt(xi) =empirical fertility rate at age xi.
Net migration:yt(xi) = empirical net migration at age xi.
Stochastic population forecasts using FDM Stochastic population forecasting
Functional time series
Let gλ(u) =
{log(u) λ = 0;xλ−1λ λ > 0.
Mortality rates:yt(xi) = g0(mt(xi)) where mt(xi) =empirical mortality rate at age xi.
Fertility rates:yt(xi) = g0.45(pt(xi)) where pt(xi) =empirical fertility rate at age xi.
Net migration:yt(xi) = empirical net migration at age xi.
Stochastic population forecasts using FDM Stochastic population forecasting
Mortality: female
0 20 40 60 80 100
−10
−8
−6
−4
−2
Australia: female death rates (1950−2003)
Age
Log
deat
h ra
te
Stochastic population forecasts using FDM Stochastic population forecasting
Mortality: female
0 20 40 60 80 100
−8
−6
−4
−2
Main effects
Age
Mu
0 20 40 60 80 1000.
10.
20.
30.
4
Age
Phi
1
Interaction
Time
Bet
a 1
1950 1970 1990
−2
−1
01
20 20 40 60 80 100
−0.
4−
0.2
0.0
0.2
Age
Phi
2
Time
Bet
a 2
1950 1970 1990−
0.4
−0.
20.
00.
10.
2
Stochastic population forecasts using FDM Stochastic population forecasting
Mortality: female
0 20 40 60 80 100
−10
−8
−6
−4
−2
Australia: forecast female log death rates (2004, 2023)
Age
Log
deat
h ra
te
Stochastic population forecasts using FDM Stochastic population forecasting
Mortality: female
0 20 40 60 80 100
−10
−8
−6
−4
−2
Australia: forecast female log death rates (2004, 2023)
Age
Log
deat
h ra
te
●
●
●●●
●
●
●●●
●
●●●
●●
●●●●●●●●
●●●●●●
●●●
●●●●●●●
●●●●●●●●●●●●●●
●●●
●●●●●●●
●●●●●
●●●●●●●●●●●●●
●●●
●●●
●●●●●●●●●
●●●●
Stochastic population forecasts using FDM Stochastic population forecasting
Fertility
15 20 25 30 35 40 45 50
050
100
150
200
250
Australia fertility rates (1921−2003)
Age
Fer
tility
rat
e
Stochastic population forecasts using FDM Stochastic population forecasting
Fertility
15 20 25 30 35 40 45 50
05
1015
20
Main effects
Age
Mu
15 20 25 30 35 40 45 500.
00.
40.
81.
2
Age
Phi
1
Interaction
Time
Bet
a 1
1920 1940 1960 1980 2000
−4
−2
02
415 20 25 30 35 40 45 50
−0.
50.
00.
51.
0
Age
Phi
2
Time
Bet
a 2
1920 1940 1960 1980 2000−
3−
2−
10
12
3
Stochastic population forecasts using FDM Stochastic population forecasting
Fertility
15 20 25 30 35 40 45 50
050
100
150
200
250
Australia: forecast fertility rates (2004, 2023)
Age
Fer
tility
rat
e
Stochastic population forecasts using FDM Stochastic population forecasting
Fertility
15 20 25 30 35 40 45 50
050
100
150
200
250
Australia: forecast fertility rates (2004, 2023)
Age
Fer
tility
rat
e
●●
●●
●●
●●
●
●
●
●●
●● ● ●
●●
●
●
●
●
●
●
●●
●● ● ● ● ● ● ●
Stochastic population forecasts using FDM Stochastic population forecasting
Migration: male
0 20 40 60 80 100−20
00−
1000
010
0020
0030
00 Australia: male net migration (1973−2003)
Age
Net
mig
ratio
n
Stochastic population forecasts using FDM Stochastic population forecasting
Migration: male
0 20 40 60 80 100
020
040
060
080
0
Main effects
Age
Mu
0 20 40 60 80 1000.
00.
10.
20.
30.
4
Age
Phi
1
Interaction
Time
Bet
a 1
1975 1985 1995
−10
000
1000
2000
0 20 40 60 80 100
−0.
6−
0.2
0.2
0.4
Age
Phi
2
Time
Bet
a 2
1975 1985 1995−
1500
−50
00
500
Stochastic population forecasts using FDM Stochastic population forecasting
Migration: male
0 20 40 60 80 100
−20
00−
1000
010
0020
0030
00
Australia: male net migration (1973−2003)
Age
Net
mig
ratio
n
Stochastic population forecasts using FDM Stochastic population forecasting
Migration: male
0 20 40 60 80 100
−20
00−
1000
010
0020
0030
00
Male net migration
Age
Num
ber
peop
le
●
●●
●●●●●●●●●
●●●●
●
●
●
●●●
●●●
●
●●●
●●
●
●●●●
●●●●●
●●●
●●●●
●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●
●●●●●●●●●●●●●●●●●
Stochastic population forecasts using FDM Stochastic population forecasting
Migration: female
0 20 40 60 80 100
−20
00−
1000
010
0020
00
Australia: female net migration (1973−2003)
Age
Net
mig
ratio
n
Stochastic population forecasts using FDM Stochastic population forecasting
Migration: female
0 20 40 60 80 100
020
060
010
00
Main effects
Age
Mu
0 20 40 60 80 1000.
00.
10.
20.
30.
4
Age
Phi
1
Interaction
Time
Bet
a 1
1975 1985 1995
−15
00−
500
500
1500
0 20 40 60 80 100
−0.
6−
0.2
0.2
0.6
Age
Phi
2
Time
Bet
a 2
1975 1985 1995−
1500
−50
00
500
Stochastic population forecasts using FDM Stochastic population forecasting
Migration: female
0 20 40 60 80 100
−20
00−
1000
010
0020
0030
00
Australia: female net migration (1973−2003)
Age
Net
mig
ratio
n
Stochastic population forecasts using FDM Stochastic population forecasting
Migration: female
0 20 40 60 80 100
−20
00−
1000
010
0020
0030
00
Female net migration
Age
Num
ber
peop
le
●
●●●●●●
●●●●●●●●
●●
●
●
●
●●
●●
●●
●
●●●●
●
●●●●●
●●●●●
●●●●●●
●●●
●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●
●●
●●●●●●
●●●●●●●●●●●
Stochastic population forecasts using FDM Stochastic population forecasting
Simulation
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
For each of mFt (x), mM
t (x), ft(x), GFt (x, x + 1), and
GMt (x, x + 1):
Generate random sample paths of βt,k fort = n + 1, . . . ,n + h conditional on β1,k, . . . , βn,k.Generate random values for et(x) and εt,x.
Use simulated rates to generate Bt(x), DFt (x, x + 1),
DMt (x, x + 1) for t = n + 1, . . . ,n + h, assuming
deaths and births are Poisson.
Stochastic population forecasts using FDM Stochastic population forecasting
Simulation
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
For each of mFt (x), mM
t (x), ft(x), GFt (x, x + 1), and
GMt (x, x + 1):
Generate random sample paths of βt,k fort = n + 1, . . . ,n + h conditional on β1,k, . . . , βn,k.
Generate random values for et(x) and εt,x.
Use simulated rates to generate Bt(x), DFt (x, x + 1),
DMt (x, x + 1) for t = n + 1, . . . ,n + h, assuming
deaths and births are Poisson.
Stochastic population forecasts using FDM Stochastic population forecasting
Simulation
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
For each of mFt (x), mM
t (x), ft(x), GFt (x, x + 1), and
GMt (x, x + 1):
Generate random sample paths of βt,k fort = n + 1, . . . ,n + h conditional on β1,k, . . . , βn,k.Generate random values for et(x) and εt,x.
Use simulated rates to generate Bt(x), DFt (x, x + 1),
DMt (x, x + 1) for t = n + 1, . . . ,n + h, assuming
deaths and births are Poisson.
Stochastic population forecasts using FDM Stochastic population forecasting
Simulation
yt(x) = st(x) + σt(x)εt,x
st(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
For each of mFt (x), mM
t (x), ft(x), GFt (x, x + 1), and
GMt (x, x + 1):
Generate random sample paths of βt,k fort = n + 1, . . . ,n + h conditional on β1,k, . . . , βn,k.Generate random values for et(x) and εt,x.
Use simulated rates to generate Bt(x), DFt (x, x + 1),
DMt (x, x + 1) for t = n + 1, . . . ,n + h, assuming
deaths and births are Poisson.
Stochastic population forecasts using FDM Stochastic population forecasting
SimulationDemographic growth-balance equation used toget population sample paths.
Demographic growth-balance equationPt+1(x + 1) = Pt(x)− Dt(x, x + 1) + Gt(x, x + 1)
Pt+1(0) = Bt − Dt(B,0) + Gt(B,0)
x = 0,1,2, . . . .
10000 sample paths of population Pt(x), deathsDt(x) and births Bt(x) generated fort = 2004, . . . ,2023 and x = 0,1,2, . . . ,.
This allows the computation of the empiricalforecast distribution of any demographic quantitythat is based on births, deaths and populationnumbers.
Stochastic population forecasts using FDM Stochastic population forecasting
SimulationDemographic growth-balance equation used toget population sample paths.
Demographic growth-balance equationPt+1(x + 1) = Pt(x)− Dt(x, x + 1) + Gt(x, x + 1)
Pt+1(0) = Bt − Dt(B,0) + Gt(B,0)
x = 0,1,2, . . . .
10000 sample paths of population Pt(x), deathsDt(x) and births Bt(x) generated fort = 2004, . . . ,2023 and x = 0,1,2, . . . ,.
This allows the computation of the empiricalforecast distribution of any demographic quantitythat is based on births, deaths and populationnumbers.
Stochastic population forecasts using FDM Stochastic population forecasting
SimulationDemographic growth-balance equation used toget population sample paths.
Demographic growth-balance equationPt+1(x + 1) = Pt(x)− Dt(x, x + 1) + Gt(x, x + 1)
Pt+1(0) = Bt − Dt(B,0) + Gt(B,0)
x = 0,1,2, . . . .
10000 sample paths of population Pt(x), deathsDt(x) and births Bt(x) generated fort = 2004, . . . ,2023 and x = 0,1,2, . . . ,.
This allows the computation of the empiricalforecast distribution of any demographic quantitythat is based on births, deaths and populationnumbers.
Stochastic population forecasts using FDM Stochastic population forecasting
Forecasts of life expectancy at age 0Forecast female life expectancy
Year
Age
1960 1980 2000 2020
7580
85
Forecast male life expectancy
Year
Age
1960 1980 2000 2020
7075
8085
Stochastic population forecasts using FDM Stochastic population forecasting
Forecasts of TFRForecast Total Fertility Rate
Year
TF
R
1920 1940 1960 1980 2000 2020
1500
2000
2500
3000
3500
Stochastic population forecasts using FDM Stochastic population forecasting
Population forecastsForecast population: 2023
Population ('000)
Age
Male Female
150 100 50 0 50 100 150
010
3050
7090
010
3050
7090
Age
Forecast population pyramid for 2023, along with 80%prediction intervals. Dashed: actual population pyramid for2003.
Stochastic population forecasts using FDM Stochastic population forecasting
Population forecasts
● ● ●●
●●
●●
●●
●●
●●
●●
●●
●●
●●
●●
●●
●●
●
1980 1990 2000 2010 2020
78
910
1112
13Total females
Year
Pop
ulat
ion
(mill
ions
)
Twenty-year forecasts of total population along with 80% and95% prediction intervals. Dashed lines show the ABS (2003)projections, series A, B and C.
Stochastic population forecasts using FDM Stochastic population forecasting
Population forecasts
● ● ● ● ●●
●●
●●
●●
●●
●●
●●
●●
●●
●●
●●
●●
●
1980 1990 2000 2010 2020
78
910
1112
13Total males
Year
Pop
ulat
ion
(mill
ions
)
Twenty-year forecasts of total population along with 80% and95% prediction intervals. Dashed lines show the ABS (2003)projections, series A, B and C.
Stochastic population forecasts using FDM Stochastic population forecasting
Old-age dependency ratioOld−age dependency ratio forecasts
Year
ratio
1920 1940 1960 1980 2000 2020
0.10
0.15
0.20
0.25
0.30
Stochastic population forecasts using FDM Stochastic population forecasting
Advantages of stochasticsimulation approach
Functional data analysis provides a way offorecasting age-specific mortality, fertility andnet migration.
Stochastic age-specific cohort-componentsimulation provides a way of forecasting manydemographic quantities with predictionintervals.
No need to select combinations of assumedrates.
True prediction intervals with specified coveragefor population and all derived variables (TFR, lifeexpectancy, old-age dependencies, etc.)
Stochastic population forecasts using FDM Stochastic population forecasting
Advantages of stochasticsimulation approach
Functional data analysis provides a way offorecasting age-specific mortality, fertility andnet migration.
Stochastic age-specific cohort-componentsimulation provides a way of forecasting manydemographic quantities with predictionintervals.
No need to select combinations of assumedrates.
True prediction intervals with specified coveragefor population and all derived variables (TFR, lifeexpectancy, old-age dependencies, etc.)
Stochastic population forecasts using FDM Stochastic population forecasting
Advantages of stochasticsimulation approach
Functional data analysis provides a way offorecasting age-specific mortality, fertility andnet migration.
Stochastic age-specific cohort-componentsimulation provides a way of forecasting manydemographic quantities with predictionintervals.
No need to select combinations of assumedrates.
True prediction intervals with specified coveragefor population and all derived variables (TFR, lifeexpectancy, old-age dependencies, etc.)
Stochastic population forecasts using FDM Stochastic population forecasting
Advantages of stochasticsimulation approach
Functional data analysis provides a way offorecasting age-specific mortality, fertility andnet migration.
Stochastic age-specific cohort-componentsimulation provides a way of forecasting manydemographic quantities with predictionintervals.
No need to select combinations of assumedrates.
True prediction intervals with specified coveragefor population and all derived variables (TFR, lifeexpectancy, old-age dependencies, etc.)
Stochastic population forecasts using FDM Stochastic population forecasting
Extensions
We intend extending this to Australian states,and to other countries.
We intend extending the individual models to:
allow dependencies between sexesallow cohort effectsallow interaction between fertility andmigration?
Software and papers:Hyndman and Booth (2006). Working paper:“Stochastic population forecasts usingfunctional data models for mortality, fertilityand migration”.
www.robhyndman.info
Stochastic population forecasts using FDM Stochastic population forecasting
Extensions
We intend extending this to Australian states,and to other countries.We intend extending the individual models to:
allow dependencies between sexesallow cohort effectsallow interaction between fertility andmigration?
Software and papers:Hyndman and Booth (2006). Working paper:“Stochastic population forecasts usingfunctional data models for mortality, fertilityand migration”.
www.robhyndman.info
Stochastic population forecasts using FDM Stochastic population forecasting
Extensions
We intend extending this to Australian states,and to other countries.We intend extending the individual models to:
allow dependencies between sexes
allow cohort effectsallow interaction between fertility andmigration?
Software and papers:Hyndman and Booth (2006). Working paper:“Stochastic population forecasts usingfunctional data models for mortality, fertilityand migration”.
www.robhyndman.info
Stochastic population forecasts using FDM Stochastic population forecasting
Extensions
We intend extending this to Australian states,and to other countries.We intend extending the individual models to:
allow dependencies between sexesallow cohort effects
allow interaction between fertility andmigration?
Software and papers:Hyndman and Booth (2006). Working paper:“Stochastic population forecasts usingfunctional data models for mortality, fertilityand migration”.
www.robhyndman.info
Stochastic population forecasts using FDM Stochastic population forecasting
Extensions
We intend extending this to Australian states,and to other countries.We intend extending the individual models to:
allow dependencies between sexesallow cohort effectsallow interaction between fertility andmigration?
Software and papers:Hyndman and Booth (2006). Working paper:“Stochastic population forecasts usingfunctional data models for mortality, fertilityand migration”.
www.robhyndman.info
Stochastic population forecasts using FDM Stochastic population forecasting
Extensions
We intend extending this to Australian states,and to other countries.We intend extending the individual models to:
allow dependencies between sexesallow cohort effectsallow interaction between fertility andmigration?
Software and papers:Hyndman and Booth (2006). Working paper:“Stochastic population forecasts usingfunctional data models for mortality, fertilityand migration”.
www.robhyndman.info
Stochastic population forecasts using FDM Stochastic population forecasting
Extensions
We intend extending this to Australian states,and to other countries.We intend extending the individual models to:
allow dependencies between sexesallow cohort effectsallow interaction between fertility andmigration?
Software and papers:Hyndman and Booth (2006). Working paper:“Stochastic population forecasts usingfunctional data models for mortality, fertilityand migration”.
www.robhyndman.info