Introduction Data And Methods Results

Bayesian forecasting of demographic rates forsmall areas

Junni Zhang

Guanghua School of Management, Peking UniversityJoint Work with John Bryant, Statistics New Zealand

December 21, 2014

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Introduction Data And Methods Results

Section 1

Introduction

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Introduction Data And Methods Results

Demographic Forecasting

Forecasts of the total number of people are not sufficient.Planning for hospitals, bridges, schools, housing, andmuch else besides requires population forecasts for smallareas.Forecasts typically extend many years into the future,because planning for infrastructure requires long timehorizons.

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Introduction Data And Methods Results

Our Application

e.g. counts of “permanent and long-term" departures from NewZealand, region=Kaikoura, time=2012

SexAge Female Male0-4 3 15-9 5 3

10-14 3 315-19 3 120-24 7 525-29 3 830-34 2 135-39 2 440-44 5 145-49 1 550-54 1 055-59 2 160-64 0 065-69 1 070-74 0 275+ 0 0

Cross-classified counts,rates, probabilitiesDimensions: age, sex,region, time

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Introduction Data And Methods Results

Our Application

We produce statistical forecasts for emigration rates for 16 agegroups, 2 sexes, and 73 regions over 25 years.

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Introduction Data And Methods Results

Traditional Demographic Approach

Forecasts are constructed for birth rates, death rates andmigration rates, disaggregated by age and sex.The accounting identity is then repeatedly applied, to giveforecasted population in each period after the base year.

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Introduction Data And Methods Results

Traditional Demographic Approach

Traditionally mathematics, not statisticsData evaluation, rich but informalStruggling with disaggregationUncertainty: ‘low’, ‘medium’, and ‘high’ variants.

When population forecasts are assembled from low,median, and high variants for fertility, mortality andmigration, the results are often counter-intuitive.For instance, combinations of variants that lead to largevariation in population size may lead to small variation inthe ratio of young people to old people.

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Introduction Data And Methods Results

Probabilistic Approaches

Researchers have developed probabalistic approaches toforecasting that combine ideas from demography withideas from the time series literature.

Lee and Carter (1992), Booth (2006), Booth and Tickle(2008), Alkema et al., (2011), Raftery et al. (2012), Bijakand Wisniowski (2010)

However, almost all the research on population hasfocused on national-level projections.

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Introduction Data And Methods Results

Complications with Small Area Forecasts

Increasing prominence of random variation as the databecome more disaggregated.Virtually all geographically-disaggregated data containgaps and breaks due to changes in administrativeboundaries.

Before 2010 there were 73 territorial authorities in NewZealand.During 2010, seven territorial authorities within greaterAuckland were amalgamated into a single unit.

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Introduction Data And Methods Results

Our Approach

We draw on ideas from the literature on small areaestimation, in addition to demography and time seriesstatistics.We develop a Bayesian hierarchical model.

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Introduction Data And Methods Results

Section 2

Data And Methods

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Introduction Data And Methods Results

Direct Estimation: Both Sexes, All Ages

Year

Rat

e

0.000

0.005

0.010

0.015

0.020

1995 2005

New Zealand

1995 2005

Kaikoura

1995 2005

Masterton

1995 2005

Papakura

1995 2005

Rotorua

1995 2005

Manukau

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Introduction Data And Methods Results

Direct Estimation: Female, Age 20-24

Year

Rat

e

0.00

0.05

0.10

0.15

1995 2005

New Zealand

1995 2005

Kaikoura

1995 2005

Masterton

1995 2005

Papakura

1995 2005

Rotorua

1995 2005

Manukau

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Introduction Data And Methods Results

Direct Estimation: Both Sexes, Time 1992 and 2010

Year

Rat

e

0.00

0.02

0.04

0.06

0.08

0.10

0 20 40 60 80

New Zealand

0 20 40 60 80

Kaikoura

0 20 40 60 80

Masterton

0 20 40 60 80

Papakura

0 20 40 60 80

Rotorua

0 20 40 60 80

Manukau

1992 2010

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Introduction Data And Methods Results

Basic Model

For age a, sex s, region r and time t, let xasrt denote populationsize, and let yasrt denote the count of international departures.

yasrtind∼ Poisson(λasrtxasrt)

logλasrt = β0 + βagea + βsex

s + βregr + βtime

t

+ βage:sexas + β

age:regar + β

sex:regsr + β

age:sex:regasr + εasrt

εasrtind∼ N(0, σ2

ε )

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Introduction Data And Methods Results

Basic Model

βtimet ∼ a non-stationary polynomial trend model with order p

p = 1:

βtimet = θt ,1 + vt

θt ,1 = θt−1,1 + wt ,1

p = 2:

βtimet = θt ,1 + vt

θt ,1 = θt−1,1 + θt−1,2 + wt ,1

θt ,2 = θt−1,2 + wt ,2

βagea ∼ a non-stationary polynomial trend model with order q

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Introduction Data And Methods Results

Basic Model

βregr = γ>X r + ur , (1)

X r consists of:the logarithm of percent of population born overseas forregion r .the logarithm of percent of population in full-time study forregion r .

urind∼ N(0, σ2

u)

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Introduction Data And Methods Results

Prior Distributions for Other Parameters

βage:sexas

ind∼ N(0, σ2age:sex),

βage:regar

ind∼ N(0, σ2age:reg),

βsex:regsr

ind∼ N(0, σ2sex:reg),

βage:sex:regasr

ind∼ N(0, σ2age:sex:reg).

The regression coefficients and the standard deviations followimproper uniform prior distributions.

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Introduction Data And Methods Results

MCMC

We use a Markov Chain Monte Carlo (MCMC) algorithm todraw the parameters from their posterior distribution.

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Introduction Data And Methods Results

Time Order Terms

time

value

-0.4

-0.2

0.0

0.2

0.4

order1

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013

-0.4

-0.2

0.0

0.2

0.4

order2

We choose p = 1.

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Introduction Data And Methods Results

Age Order Terms

age

value

-2

-1

0

1

2

order1

0-4 5-9 10-1415-1920-2425-2930-3435-3940-4445-4950-5455-5960-6465-6970-74 75+

-2

-1

0

1

2

order2

We choose q = 2.21 / 29

Introduction Data And Methods Results

Missing Values for Region: First Problem

7.5% of records have no regional information at all, eitherbecause the respondent did not provide it, or because theresponse could not be coded.

We address this issue through multiple imputation.

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Introduction Data And Methods Results

Missing Values for Region: First Problem

Statistics New Zealand has information on citizenship thatcannot be released publicly.

“... imputation performed by the data collector (e.g. the CensusBureau) has the important advantage of allowing the use ofinformation available to the data collector but not available to anexternal data analyst. · · · This kind of information, even thoughinaccessible to the user of a public-use file, can often improvethe imputed values.” (Rubin 1987)

∑r

ymisasrtc = yobs

astc

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Introduction Data And Methods Results

Missing Values for Region: Second Problem

From 2010 onwards, if the region is coded as Auckland, there ismissing information on which of the seven original territorial 16authorities within Auckland is associated with the records.

We address this issue through jointly updating these valuesand the parameters within the MCMC algorithm.

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Introduction Data And Methods Results

Section 3

Results

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Introduction Data And Methods Results

Validation Exercise

We pretend that the event of merging seven territorialauthorities within Auckland happened at 2003.The training data include the counts for 73 regions for1991-2002 and the counts for 67 regions, with the countsfor the seven territorial authorities within Auckland merged,for 2003-2005.We predict the emigration counts ypre

asrt for 2006-2010, andobtain their posterior medians, 50% credible intervals and90% credible intervals.We compare the posterior medians and credible intervalswith the observed values of yasrt for 2006-2010.

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Introduction Data And Methods Results

Results of Validation Exercise

The median of yasrt to be predicted equals to 9, and themedian absolute error of using posterior medians of ypre

asrt topredict yasrt is 3.The percentage of yasrt lying inside the 50% credibleintervals is 57.1%.The percentage of yasrt lying inside the 90% credibleintervals is 89.6%.

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Introduction Data And Methods Results

Estimates for Female, Age 20-24

Time

0.05

0.10

0.15

Kaikoura

1995 2000 2005 2010

Gore Masterton

1995 2000 2005 2010

Papakura Rotorua

1995 2000 2005 2010

0.05

0.10

0.15

Manukau

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Introduction Data And Methods Results

Estimates and Prediction for Female, Age 20-24

Time

0.05

0.10

0.15

0.20

Kaikoura

1990 2000 2010 2020 2030 2040

Gore Masterton

1990 2000 2010 2020 2030 2040

Papakura Rotorua

1990 2000 2010 2020 2030 2040

0.05

0.10

0.15

0.20

Manukau

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