2014 -- 2034 Bayesian Forecasts of Taiwan Population 2014--2034
O
working age population (age 15−64)
2
0
10
20
30
40
50
60
70
Total: 23,383,000 (increased from 21,178,000 in 1994) Female > Male (11,699,000 vs. 11,684,000)
Aging population: proportion of seniors ( 65 and older) =12% (7% in 1994) World’s lowest fertility:
total fertility rate (births per woman) =1.07 (1.76 in 1994)
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Population Forecast - Purposes Policy planning and making for the economy, health and
medical care, social security (pensions), education, infrastructure construction, tax and expenditure,… (government) Future demand for food, water, energy, and other natural
resources; human impacts on the environment and ecology (scientists) Size of future markets (business and industry) Sustainable developments
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Population Forecast – current practice Made regularly by national agencies (;
The United States Census Bureau) Global as well as national population forecasts for all countries
are produced by the United Nations and the World Bank Methodology developed by demographers, economists, and
statisticians Modeling future dynamics based on past trends observed from
historical data
Cohort: age-sex groups
Component: the three processes resulting in population change Birth (fertility) Death (mortality) Migration (immigration & emigration)
Project the population forward in each age-sex subgroup
Both the population size as well as the composition (age distribution)
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Vital Statistics Age-specific mortality rate in a year:
# deaths in an age group in a year mid−year population size in that age group
Age-specific fertility rate in a year:
# births in a year to women of an age group mid−year population size for women in that age group
Total Fertility rate: total number of children a woman would bear during her lifetime if she were to experience the prevailing age-specific fertility rates ( = sum of 1-year age-specific fertility rates)
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× − −
+ − −
= 0, , , 85+, ′, = , () mid-year population sizes at year t by age groups for sex
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= 0, , 14, , , 45, ,0, , 0 ′ ,
birth rates for sex at year t by age groups of women
=
0, 0 0 0 0 1, 0 0 0 83, 0 0 0 0 0 84, 85+,
conditional survival rates at year t by age groups for sex
= 0, , , 85+, ′ , = , ()
net immigration size at year t by age groups for sex
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2(, + , +1,)
, = (1−) 1+0.5 0,
×1 2(, + , +1,)
=1/(1+ the ratio of male to female births) proportion of female births
, : fertility rate at age x and year t
, = 1−0.5 +1,
1+0.5 ,
, : mortality rate at age x and year t for sex k
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Traditional Population Forecasts High, Medium and Low forecasts, based on the assumed levels of future
fertility, mortality and migration Deterministic and of no probability interpretations
Even for very short-term forecasts, the [Low, High] interval usually fails to
contain the actual value (Lee and Tuljapurka 1994 JASA) In “2010 2060 ” (, Sep. 2010):
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23,180 23,200 23,220 23,240 23,260 23,280 23,300 23,320 23,340 23,360 23,380 23,400
2010 2011 2012 2013 2014
th ou
sa nd
decisions (tax increases, benefit cuts, retirement age, energy conservation and carbon reduction) (Lee 2011 Science)
Could be directly assessed via probabilistic rather than
deterministic forecasts Probabilistic forecasts can also be based flexibly on quantiles
of the predictive distribution (Raftery et al. 2012 PNAS) Bayesian approach is a systematic and natural way to
incorporate all known sources of uncertainty in forecasting
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Historical Taiwan Mortality Data Mortality rates for years 1975—2012 by sex and age (1-year)
group mortality rates published by 86 age groups: 0, 1,…, 84, 85+ (85 and above) Armed-shaped profile across ages Declining mortality rates over the last 4 decades
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15
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-9
-8
-7
-6
-5
-4
-3
-2
-1
0
ln, = + + ,
: general age pattern of mortality (constant over time) : general mortality profile over time (constant over age);
unobserved time series process : age-specific coefficient of
, ∼ i.i.d. n (0, 2)
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Lee-Carter’s Proposal: is a random walk with constant drift
= −1 + + , ∼ i.i.d. n (0, 2) linear temporal trend Constraints for identifiability:
∑ =1 = 0, ∑
=1 = 1
Parameters are estimated by the observed (historical) data Future mortality can be predicted based on estimated
parameters and forecasted series
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Properties of Lee-Carter Models Excellent fit to mortality trends in many countries
Parsimonious in the number of parameters used (recall we have 86 time series for each sex group ! ) General time series models can be assumed for the latent process
Most widely applied model for mortality forecasting (Wang and Preston 2009 PNAS) No age-time interaction
Underestimated forecast error (without accounting for estimation
variability)
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Bayesian Approach to Lee-Carter Model (Pedroza 2006 Biostatistics)
Parameters ( = 1,… , ), , , 2 ( = 1,… , ), , 2 are drawn from their joint posterior distribution, with the prior given by either expert opinions or noninformative prior distributions To sample from the joint posterior distribution, an MCMC approach
(Gibbs sampler) is implemented via iteratively sampling from conditional distributions of each of the parameters given assigned values for all the other parameters States ( = 1,… , ) are predicted and updated by Kalman filter,
and sampled in the MCMC by the Kalman smoother
[ | +1,… , , 1,… , ]
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Taiwan Mortality Forecasting
In-sample forecast (using data from 1975 to 2006 as training sample and data from 2007 to 2012 as test sample) Out-of-sample forecast for 2014--2034 (fitting data from 1975--2012) Noninformative prior for the model parameters
3 parallel chains of the Gibbs sampler with over-dispersed initial
values for 2000 iterations Convergence checked by trace plots and the Gelman-Rubin statistic
(Gelman and Rubin 1992 Statistical Science)
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A sample for the posterior distribution of the parameters obtained from the last 1000 iterations in each of the 3 chains
Forecast based on the Lee-Carter model with 3,000 samples of the
parameter MCMC posterior distribution and forecasted states Stratified by sex
Similar results obtained by using a common state vector for male
and female mortality
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2014 2034
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Historical Taiwan Fertility Data Fertility rates data by single-year age group available for ages
of 14, 15,…, 44, 45+ (45 and above) from 1975—2012 Fertility in Taiwan has declined dramatically in the past decades
Total fertility rate in Taiwan has dropped below the replacement
level of 2.1 since 1984, reached a minimum of 0.9 in 2010, and leveled off at around 1.0 since 2011
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Age Patterns of Fertility over Time Age patterns shifted downwards and to the right over time –
declining and postponed fertility
Model for Forecasting Fertility: Two-component State-space Model
ln, = + + + , : general age pattern of fertility (constant over time)
: linear trend over time; unobserved random walk (constant drift)
: nonlinear trend over time; unobserved AR(1) process
, : age-specific slope coefficients for , , ∼ i.i.d. n (0, 2)
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Two-component State Space = −1 + + , ∼ i.i.d. n (0, 2) linear temporal trend
= −1 + + , ∼ i.i.d. n (0, 2) nonlinear temporal trend , are orthogonal:


Taiwan Fertility Forecasting Two-component state space model, with the first component accounting
for the linear (declining) trend, and the second (AR) component for the recent sign of turnaround
Bayesian procedure (Gibbs sampler + Kalman filter + Kalman smoother)
Forecast based on the two-component state space model with 3,000
samples of the parameter values from their posterior distribution and forecasted states from their predictive distribution
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Two latent components of processes (, ) estimated with data from 1975--2012
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0
0.5
1
1.5
2
2.5
3
3.5
III. Migration Forecasting
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Forecast of Migration in Taiwan Migration data (immigration and emigration) are less accurate and
stable Net immigration size for each age-sex-time group obtained
indirectly from the population and mortality data Based on one-component state space model similar to the Lee-
Carter model, with net immigration size in each age-time group as the outcome Bayesian forecasts with the model parameters sampled from the
MCMC posterior distribution (of size 3,000)
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Bayesian Population Forecast Based on the Bayesian forecasts of mortality rates, fertility rates, and
migration, using the cohort-component formula = the proportion of female births=1/(1+1.08) 1.08 : the average sex ratio (M:F) at birth in Taiwan for 2010--2014 ( > 1.05 the historical worldwide value) the 95% probability (prediction) intervals, or other quantiles of forecasts,
can be naturally derived from the Bayesian procedure (MCMC samples) Using 2006 (in-sample forecast) or 2012 (out-of-sample forecast)
population as a baseline population
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In-sample Forecasts of Taiwan Population (in thousands) Bias= ( posterior mean − real )
2007 2008 2009 2010 2011 2012 Male 97.5% 11,617 11,654 11,690 11,728 11,763 11,796
mean 11,607 11,633 11,655 11,673 11,686 11,696 2.5% 11,597 11,613 11,621 11,620 11,613 11,597 bias 7 15 23 37 46 36
Female 97.5% 11,322 11,393 11,465 11,538 11,608 11,678 mean 11,312 11,374 11,433 11,488 11,539 11,586 2.5% 11,302 11,356 11,403 11,443 11,475 11,499 bias -5 -6 -14 -17 -14 -25
Total 97.5% 22,937 23,042 23,144 23,247 23,344 23,433 mean 22,919 23,007 23,088 23,161 23,225 23,281 2.5% 22,902 22,975 23,036 23,082 23,113 23,134 bias 2 9 9 20 32 11
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11,550
11,600
11,650
11,700
11,750
11,800
11,850
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11,250
11,300
11,350
11,400
11,450
11,500
11,550
11,600
11,650
11,700
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22,800
22,900
23,000
23,100
23,200
23,300
23,400
23,500
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Observed vs. In-sample Forecasted Age Profiles for 2012
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2013 2014 2034 Male 97.5% 11,685 11,703 11,780
mean 11,675 11,684 11,046 2.5% 11,665 11,665 10,369 mean − actual -4 1 ??
Female 97.5% 11,671 11,726 12,847 mean 11,661 11,705 11,748 2.5% 11,651 11,685 10,688 bias -4 6 ??
Total 97.5% 23,353 23,423 24,300 mean 23,335 23,390 22,793 2.5% 23,318 23,357 21,409 mean − actual -8 7 ??
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Sex Ratio (M/F) over Age Groups
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Old Age Dependence Ratio (PADR) Forecast
OADR= population of ages 65+ population of ages 15−64 = # older dependents
working age population (%)
Young Age Dependence Ratio (YADR) Forecast
YADR= population of ages 0−14 population of ages 15−64 = # younger dependents
working age population
working age population (%)
Take-home Message -- Technical:
The Lee-Carter Model fits the Taiwan mortality data well and provides accurate in-sample forecasts The newly proposed two-component state-space model with
orthogonal linear and non-linear trend components fits and forecasts the Taiwan fertility data well Bayesian prediction interval (distribution) accounts for various
sources of uncertainty simultaneously and coherently
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Take-home Message -- General:
After 20 years, Taiwan population is expected to: reduce overall by > 150 thousand (with 75% probability)
but the changes are heterogeneous across age groups:
have > one in every four aged 65 and over (with 95% probability)
have > doubled old age dependency ratio (with 95% probability)
“Demographic dividend” is coming to an end: dependency ratio starts
rise now Difficult and painful choices/decision for government and people to make
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24-50% 31-43% 8-25% 99-148% 73-235%
Thank You !!
Bayesian Forecasts of Taiwan Population2014--2034
Dependence Ratio () # dependents (age 0−14, 65+) working age population (age 15−64)
Taiwan Population –At present:
Vital Statistics
Cohort-component Equation:
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Lee-Carter’s Proposal:
Taiwan Mortality Forecasting
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II. Fertility Forecasting
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Two-component State Space
Taiwan Fertility Forecasting
Two latent components of processes ( , ) estimated with data from 1975--2012
In-sample Prediction: 2012
III. Migration Forecasting
Historical Taiwan Migration (Net Immigration) Data
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IV. Population Forecasting
Bayesian Population Forecast
In-sample Forecast of Taiwan Population: Male
In-sample Forecast of Taiwan Population: Female
In-sample Forecast of Taiwan Population: Total
Observed vs. In-sample Forecasted Age Profiles for 2012
Out-of-sample Forecasts of Taiwan Population(in thousands)
Taiwan Population Forecasts:2014--2034
Age Profiles (Population Pyramids)2034 vs. 2012
Median Age
Working Age (15 - 64) Population Proportion
Old Age Dependence Ratio (PADR) Forecast
Young Age Dependence Ratio (YADR) Forecast
Dependence Ratio (DR) Forecast