HOUSEHOLDS AND LIVING ARRANGEMENTS PROJECTIONS AT NATIONAL AND SUB- NATIONAL 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 Definition and codes U.S. application AgeX0,1,2,3, …,W; W is chosen by userx=0,1,2,3, …,100 SexS1. Female; 2. Males=1,2 Race (optional)RTo be determined by userr=1,2,3,4 Marital/union statusM4 or 7 marital status model chosen by user m=1,2,3,4,5,6,7 Co-residence with parent(s) K1. With two parents; 2. with one parent only; 3. Not with parents. k=1,2,3 ParityPp = 0,1,2, …, H; H is chosen by userp=0,12,3,4,5+ # co-residing childrenCc = 0,1,2, …, H (cp)c=0,1,2,3,4,5+ Residence (optional)U1. Rural; 2. UrbanNot considered Projection yeartSingle year from t 1 to t 2, chosen by user t 1 =2000; t 2 =2050

Figure 1. Seven marital statuses model

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) xX+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).

☻ 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). 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 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 age- specific 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) CodeHousehold typeHousehold size 1Married couple householdNot available 2Female-headed household,no spouseNot available 3Male-headed household,,no spouseNot available 4Female non-family householdNot available 5Male non-family householdNot available

The ProFamy model needs conventionally available data and projects much more detailed information on households and living arrangements Type codeHousehold typesHousehold sizes One generation households 1-6One person only by sex and marital status1 7-12One person & other/non-relative by sex and marital status of the person 2,3,4,5,or One married couple only; One cohabiting couple only One married couple & other/non-relative; One cohabiting couple & other/non-relative 3,4,5,6,or 7+ Two-generation households Married couple & children; Cohabiting couple & children 3,4,5,6,7,8,or Single-parent & children by sex and marital status of the single parent 2,3,4,5,6,7,8,or 9+ Three-generation households 25-28Married (or cohabiting) couple with children and 1 or 2 grandparents 4,5,6,7,8,or 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 Main data resources (US applications) 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. If a sample data set is used, 100% tabulations of age-sex distributions of the entire population and those living in group quarters, derived from the census data must be provided. Census 5% micro data or more recent and cumulative American Community Survey (ACS) data files and the published online 100% census or ACS cross- tabulations.

Contents of the dataMain data resources (a) Age-race-sex-specific death rates (marital- status specific, if possible). 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 Zeng and Land et al. (2006). (c) Age-race-parity-specific o/e rates of marital and non-marital fertility (d) Age-race-sex-specific net rates of leaving the parental home, estimated based on two adjacent census micro data files and the intra- cohort iterative method (Coale1984; 1985; Stupp 1988; Zeng, Coale et al., 1994). The 1990, and 2000 censuses micro data files (e) Age-sex-specific rates of international emigration and immigration. Census 5% micro data or ACS data files (2)-I Model standard schedules at national level (can be used for households projections at sub-national level)

(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 Based on, census micro data, vital statistics and pooled survey data sets (b) Race-specific general rates of cohabiting and general rates of union dissolution (c) Race-specific Total Fertility Rates (TFR) by parityBased on estimates released by the Census Bureau and the National Center for Health Statistics (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

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 (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

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 (D) including data in 1990s

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, co- residence 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 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).

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