Stochastic population forecasts using functional data models · Stochastic population forecasts...

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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

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