HOUSEHOLDS AND LIVING ARRANGEMENTS
PROJECTIONS AT NATIONAL AND SUBNATIONAL LEVEL
-- An Extended Cohort-component Approach
Yi Zeng
Professor, Duke University and Peking University
1. THE CORE IDEAS OF THE ProFamy
EXTENDED COHORT-COMPONENT METHOD
Core idea 1 : A multi-state accounting model.
☻Unlike most other macrosimulation models
which use the household as the basic unit and
require the non-conventional data on transition
probabilities among household-type statuses,
☻We use individual as the basic unit of analysis
and thus only conventionally available
demographic data are required in ProFamy
model and we forecast households and
population age/sex distributions simultaneously.
Demographic statuses distinguished in our ProFamy model
Status
Sym
Age
X
0,1,2,3,…,W; W is chosen by user
x=0,1,2,3,…,100
Sex
S
1. Female; 2. Male
s=1,2
Race (optional)
R
To be determined by user
r=1,2,3,4
Marital/union status
M
4 or 7 marital status model chosen
by user
m=1,2,3,4,5,6,7
Co-residence with
parent(s)
K
1. With two parents; 2. with one
parent only; 3. Not with parents.
k=1,2,3
Parity
P
p = 0,1,2,…, H; H is chosen by user
p=0,12,3,4,5+
# co-residing children
C
c = 0,1,2,…, H (cp)
c=0,1,2,3,4,5+
Residence (optional)
U
1. Rural; 2. Urban
Not considered
Projection year
t
Single year from t1 to t2, chosen by
user
t1=2000; t2=2050
Definition and codes
U.S. application
Figure 1. Seven marital statuses model
6
 ( p＋1）
p＝0
Core idea 2: an innovative computational strategy in
the periodic demographic accounting process
☻
With needed individual statuses identified, we
would have huge cross-status transition matrices if
adopting conventional computation strategy; e.g., if 7
marital/union statuses, 3 statuses of co-residence
with parents, 6 parity and 6 co-residence statuses
with children are distinguished as what was done in
U.S. applications, one has to estimate a cross-status
transition probabilities matrix with 194,481 elements
at each age of each sex for each race – would require
huge datasets; NOT practical.
 Thus, we adopted an innovative computational
strategy, which was originally proposed by Bongaarts
(1987) and further justified mathematically and
numerically by Zeng (1991)
Figure 2. Computational strategy to calculate changes in
marital/union, co-residence with parents/children, migration and
survival statuses
Changes in marital/union, co-residence with
parents/children, migration and survival statuses
occur in the middle of age interval (x,x+1)
x
X+1
Changes in parity and
maternal statuses
occur in the 1st half of
the single age interval
Changes in parity and
maternal statuses
occur in the 2nd half
of the single age
interval
Core idea 3: A judicious use of stochastic
independence assumptions to face data reality
Also originally suggested by Bongaarts (1987)
and adapted and generalized by Zeng (1987, 1991)
and others.
☻ Statistical basis:
 the real-world mostly allows assumptions of
stochastically independent;
 limited data sources force application of an
independence assumption.
☻ In ProFamy extended cohort-component model,
 marital/union status transitions depend on age, sex,
and race, but independent of other statuses;
 fertility depends on age, race, parity and marital
status, but independent of other statuses;
 mortality depends on age, sex, race and marital
status, but independent of other statuses;
Core idea 4: Use of the harmonic mean to ensures
consistency between the two sexes and between
parents and children in the projection model.
We ensure the consistency between the two
sexes and between parents and children
following the harmonic mean approach,
which satisfies most of the theoretical
requirements and practical considerations
(Pollard, 1977; Schoen, 1981; Keilman, 1985;
Van Imholf and Keilman, 1992; Zeng et al.
1997; 1998).
Core idea 5. Using national model standard schedules and
summary parameters at sub-national level to specify
projected demographic rates of the sub-national region in
future years.
☻The standard schedules formulate the age pattern of
demographic processes. One may take into account anticipated
changes in the age patterns, such as delaying or advancing
marriage and fertility, changes in shape of the curve towards more
spread or more concentrated, through adjusting the parameters
(mean or median, and interquartile range) (Zeng et al., 2000).
☻The summary parameters, e.g, TFR, General rates of marriage
and divorce, etc., can be used to “tune” the household and
population projections up or down for demographic scenarios.
☻However, Data for estimating race-sex-age-specific standard
schedules of the demographic rates for household projection
may not be available at the sub-national level.
-- The core ideas 2,3,4 are not detailed here due to time
constrains
☻The age-race-sex-specific standard schedules at the
national level can be employed as model standard schedules
for projections at the sub-national level.
☻This is similar to the widely practiced application of model life
tables (e.g., Coale, Demeny, and Vaughn, 1983; U.N., 1982), the
Brass logit relational life table model (e.g. Murray, 2003), the Brass
Relational Gompertz Fertility Model (Brass, 1974), and other
parameterized models (e.g. Coale and Trussell, 1974; Rogers, 1986)
in population projections and estimations.
☻Numerous studies have demonstrated that parameterized
models consisting of a model standard schedule and a few
summary parameters offer an efficient and realistic way to
project or estimate demographic age-sex-specific rates.
☻The demographic summary parameters are most crucial
for determining changes in level and age pattern of the agespecific rates, as long as the model standard schedules
reveal the general age patterns.
(Brass, 1978; Booth, 1984; Paget and Timaeus, 1994; Zeng
et al., 1994)
2. A Comparison between the ProFamy Extended Cohort
Component Model and Still-Widely-Used Headship Rate Method
(1) Linkage with demographic rates
 Headship Rate: cannot link to demographic events,
extremely hard to incorporate demographic
assumptions of fertility, mortality, marriage/union
formation and dissolution etc. (Mason and Racelis
1992; Spicer et al., 1992)
 The ProFamy model: Use demographic rates from
conventional sources as input; closely link projected
households with demographic rates and summary
measures on marriage/union formation and
dissolution, fertility and mortality etc.
The ProFamy model household,
elderly living arrangement and
population projection: using
demographic rates as input
Headship-rate household projection:
cross-sectional extrapolation of the
age-specific headship-rate, without
linkage to demographic rate
(2) Information produced and their
adequacy for planning
Headship Rate: little information on household types and no household
sizes projection, inadequate for planning purposes (Bell & Cooper, 1990),
especially most households consumptions (e.g. home vehicles, housing,
energy use…) largely depends on household size.
Households types projected by headship rates methods
(Bureau of the Census, 1996)
Code
Household type
Household size
1
Married couple household
Not available
2
Female-headed household,no spouse
Not available
3
Male-headed household,,no spouse
Not available
4
Female non-family household
Not available
5
Male non-family household
Not available
The ProFamy model needs conventionally available data and projects much
more detailed information on households and living arrangements
Type code
Household types
Household sizes
One generation households
1-6
One person only by sex and marital status
1
7-12
One person & other/non-relative by sex and marital
status of the person
2,3,4,5,or 6+
13-14
One married couple only; One cohabiting couple only
2
15-16
One married couple & other/non-relative; One cohabiting
couple & other/non-relative
3,4,5,6,or 7+
Two-generation households
17-18
Married couple & children; Cohabiting couple & children
3,4,5,6,7,8,or 9+
19-24
Single-parent & children by sex and marital status of the
single parent
2,3,4,5,6,7,8,or
9+
Three-generation households
25-28
Married (or cohabiting) couple with children and 1 or 2
grandparents
4,5,6,7,8,or 9+
29-40
Sex-marital status-specific single-parent & children & 1
or 2 grandparents
3,4,5,6,7,8,or 9+
3. Data needed for household forecasting at
national and sub-national levels
(1) Base population
Contents of the data
A census micro data file for the state, with
a few needed variables of sex, age, race
(optional), marital/union status,
relationship to the householder, and
whether living in a private or institutional
household.
Main data resources
(US applications)
Census 5% micro data
or more recent and
cumulative American
Community Survey
(ACS) data files and the
If a sample data set is used, 100%
published online 100%
tabulations of age-sex distributions of the census or ACS crossentire population and those living in group tabulations.
quarters, derived from the census data
must be provided.
(2)-I Model standard schedules at national level (can be
used for households projections at sub-national level)
Contents of the data
(a) Age-race-sex-specific death rates (maritalstatus specific, if possible).
Main data resources
Census Bureau’s
estimates, Schoen and
Standish (2001)
(b) Age-race-sex-specific o/e rates of
marriage/union formation and dissolution
Pooled NSFH, NSFG,
CPS, SIPP data sets, see
(c) Age-race-parity-specific o/e rates of marital Zeng and Land et al.
(2006).
and non-marital fertility
(d) Age-race-sex-specific net rates of leaving
the parental home, estimated based on two
The 1990, and 2000
adjacent census micro data files and the intracensuses micro data files
cohort iterative method (Coale1984; 1985;
Stupp 1988; Zeng, Coale et al., 1994).
(e) Age-sex-specific rates of international
emigration and immigration.
Census 5% micro data or
ACS data files
(2)-I I Model standard schedules at sub-national level
(f) Race-sex-age-specific rates of domestic in-migration
and out-migration for each state
Census 5% micro data,
ACS data files
(3) Demographic summary measures for the nation and sub-national
regions
(a) Race-specific general rates of marriage and general
rates of divorce
(b) Race-specific general rates of cohabiting and general
rates of union dissolution
(c) Race-specific Total Fertility Rates (TFR) by parity
(d) Race-sex-specific Life expectancies at birth
(e) Race-sex-specific total numbers of male and female
migrants
(f) Race-sex-specific mean age at first marriage and births
Based on, census
micro data, vital
statistics and pooled
survey data sets
Based on estimates
released by the Census
Bureau and the
National Center for
Health Statistics
4. Validation of the extended cohort-component method for
household forecasting at sub-national level
Zeng and Land et al. (2006) and Zeng et al. (2008) did
validation tests of households projections for US and China
at national level from 1990 to 2000, and then compared to
the 2000 census observations.
We do TWO sets of validation tests of household forecasts
from 1990 to 2000 for each of the 50 states and DC fo USA,
all using the national model standard schedules.
(1) Using the 1990 census data as base population and the
summary measures estimated based on data before 1991,
and compare the projected and the census-observed in
2000.
(2) Using the 1990 census data as base population and
summary measures estimated based on data in 1990s, and
compares the projected and the census-observed in 2000.
Figure 3a. Distributions of the absolute percent errors (APE)
of forecasts from 1990 to 2000, 6 main indices of
households for each of the 50 states and DC, in total 306
pairs of comparisons between ProFamy forecasted and
census observations in 2000
(A) based on data before 1991
(B) including data in 1990s
APE5.0-9.99
(0.6%)
APE >=10.0
(6.7%)
APE >=10.0
(0.0%)
APE3.0-4.99
(11.5%)
APE5.0-9.99
(12.9%)
APE <1.00
(29.1%)
APE3.0-4.99
(17.4%)
APE <1.00
(45.9%)
APE 1.0-2.99
(42.0%)
APE 1.0-2.99
(33.9%)
Figure 3b. Distributions of the absolute percent errors (APE)
of forecasts from 1990 to 2000, 6 main indices of population
for each of the 50 states and DC, in total 306 pairs of
comparisons between ProFamy forecasted and census
observations in 2000
(C) based on data before 1991
APE5.0-9.99
(9.5%)
APE3.0-4.99
(16.5%)
APE >=10.0
(0.8%)
APE <1.00
(29.7%)
APE 1.0-2.99
(43.4%)
(D) including data in 1990s
APE5.0-9.99
(7.3%)
APE >=10.0
(4.2%)
APE3.0-4.99
(14.3%)
APE 1.0-2.99
(39.8%)
APE <1.00
(34.5%)
Table 2a. The Mean Absolute Percent Error, Mean Algebraic
Percent Error and Median Absolute Percent Error of the main
indices of household projection between the ProFamy projections
from 1990 to 2000 and the Census observations in 2000 for each
of the 50 states and DC
Table 2b. The Mean Absolute Percent Error, Mean Algebraic
Percent Error and Median Absolute Percent Error of the main
indices of population projection between the ProFamy projections
from 1990 to 2000 and the Census observations in 2000 for each
of the 50 states and DC
The discrepancies are within a very reasonable range, and the
ProFamy extended cohort component approach is validated at
sub-national level.
However, the ProFamy approach needs substantially more data than
does the classic headship-rate method.
Is it still worthwhile to employ the new ProFamy approach rather than
the classic headship-rate method, if the users only simply needs
the projections of the home-based consumption demands, such as
numbers of housing units by number of bedrooms, but do not care
about the details of the household characteristics and the statuses
of the reference persons, such as marital/union status, coresidence status with parents and children, etc.?
To answer this question, we project from 1990 to 2000 housing demands by #
of bedrooms for each of 50 states and DC, employing headship-rate model
and ProFamy approach using data before 1990. By comparing the
projected and the census-observed # of housing units by # of bedrooms in
2000, we estimated/compared the forecasts errors, by the headship-rate
method and the ProFamy approach.
Table 3. Forecast errors of Mean Algebraic Percent Error (MALPE),
Mean Absolute Percent Error (MAPE) and Median Absolute
Percent Error (MEDAPE) of housing demands projections from
1990 to 2000 (compared to the 2000 census observations),
Comparisons between the ProFamy cohort-component approach
and the constant headship-rates
The constant headship-rate did much worse than ProFamy in housing
demand forecasting, but one may argue that we could have headship
rates changing… So, we did another test below:
Table 4. Forecast errors of Mean Algebraic Percent Error (MALPE),
Mean Absolute Percent Error (MAPE) and Median Absolute Percent
Error (MEDAPE) of housing demands projections from 1990 to 2000
(compared to the 2000 census observations), Comparisons between
the ProFamy cohort-component approach and the adjusted changing
headship-rates, both approaches resulted in the same projected total
number of households as observed in the 2000 census. The
headship-rate still did substantially worse.
-- The changing headship-rate model still did substantially
worse. Why? The censuses data shown, as compared to 1990,
the 1, 2, 3, 45, and 6+ persons households in 2000 increased
by 20.6, 16.9, 9.2, 9.3 and 15.1 percent, respectively. American
households with 12 persons (which more likely need 0-1
bedroom) and 6+ persons (which more likely need 4-bedrooms)
increase substantially faster than the 3- and 45 person
households (which more likely need 23 bedrooms).
Thus, the headship-rate method, which cannot forecast
households by size, resulted in substantially more serious
forecast errors in projecting the demands of housing units by
number of bedrooms, as compared to the ProFamy approach
whose forecasts do include detailed households size
information.
5. A summary of findings of households projections from 1990
to 2000 for the 50 states and DC
(1) the average household size would decrease considerably in
almost all states up to 2020 or so, especially in the states with
higher degree of population aging, and remain relatively stable
afterwards;
(2) % of one-person households would increase substantially
in all states.
(3) Husband-wife households would decrease moderately and
cohabiting-couple households would increase substantially, up
to 2020 or so, and remain relatively stable afterwards;
(4) Directions of changes in percent of single-parent
households among the two-generation households are
diversified, increase moderately in some states but decrease
moderately or remain unchanged in the other states.
(5) Percent of households with at least one elder aged 65+ of
the total number of households, percent of elderly aged 65+
living alone and percent of oldest-old aged 80+ living alone
will increase substantially and pervasively in all states.
6. Conclusion remarks:
ProFamy extended cohort-component model does
substantially better than the still-widely-used
classic headship rates method in households
projections.
In addition to the academic research, the ProFamy
method/software can be used for home-based
consumption and services needs/costs projections.
For example, ProFamy method/software was
employed for the U.S. households energy
consumption projections in Dalton et al. (2008),
for the U.S. housing projections at national and
sub-national levels in Smith et al. (2008; 2012), for
Austrian and the U.S. home-based vehicles
consumption projections in Prskawetz et al. (2004)
and Feng et al. (2011).
Thank You！