Empirical Assessments and a Comparison with the Headship Rate Method

Abstract

One useful way to validate a projection model and computer program is to project between two past dates for which the observations are known, and then compare the observed data with the projected data. We assessed the accuracy of the ProFamy method and program by projecting: (1) U.S. households by race from 1990 to 2000 (Zeng et al. 2006), (2) Chinese households by rural and urban areas from 1990 to 2000 (Zeng et al. 2008), and (3) Chinese households by rural and urban areas and Eastern, Middle, and Western regions from 2000 to 2010 (Zeng et al. 2013b).

Appendix 1: Procedures to Project Housing Demands Based on Household Projections Employing the Headship Rate Method or the ProFamy Approach

The projections are for each of the 50 U.S. states and DC, but we omit the state dimension in all variables and formulas in this Appendix to simplify the presentation.

1.1 Housing Demand Projections by the Constant Headship Rate Method

The projection conducted by the U.S. Census Bureau (1996), which is a typical well-done household projection using the headship rate method, projected age-sex-household type-specific headship rates in the future. The five household types listed in Table 4.6 and ten age groups were distinguished in the U.S. Census Bureau (1996) projection.

Let Hr(x,s,h) denote the age-sex-household type-specific headship rates estimated based on the 1990 census data, where x and s refer to age and sex of the household head; h refers to the household type.

P2000(x,s), number of persons aged x with sex s in year 2000, projected by the conventional cohort-component method for population projection by age and sex which is also part of the ProFamy model output.

HH(x,s,h), the number of households by age/sex of the household head and household type in 2000 projected by the headship rate method.

b(x,s,h,i), the age-sex-household type-specific proportions of households with i bedroom(s) (i =1, 2, 3, and 4, referring to 0–1, 2, 3, 4+ bedrooms), estimated using the 1990 census data; ${\displaystyle \sum_ib\left(x,s,h,i\right)=1.0}$

HU(x,s,h,i), the number of housing units by number of bedrooms and by age/sex of the household head and household type in 2000 projected using the headship rate method.

The estimators for projecting HH(x,s,h) and HU(x,s,h,i) are:

$$ HH\left(x,s,h\right)= Hr\left(x,s,h\right)\ P 2000\left(x,s\right) $$

and

$$ HU\left(x,s,h,i\right)= HH\left(x,s,h\right)\ b\left(x,s,h,i\right) $$

1.2 Housing Demand Projections by the ProFamy Extended Cohort-Component Approach

Let EH(x,s,m,z) denote the age-sex-household type-size-specific number of households in 2000, projected employing the ProFamy approach based on data before 1991; where x: 5-year age group; s: sex; m: household type (marital/union status of the reference person); z: household size – 1, 2, 3, 4, 5, 6+ persons.

eb(x,s,m,z,i), the age-sex-household type-size specific proportions of households with i bedroom(s) (i =1, 2, 3, and 4, referring to 0–1, 2, 3, 4+ bedrooms, respectively), estimated using the 1990 census data; ${\displaystyle \sum_i eb\left(x,s,m,z,i\right)=1.0}$,

EHU(x,s,m,z,i), the age-sex-household type-size specific number of housing units by number of bedrooms in 2000 projected employing the ProFamy approach.

The estimator for projecting EHU(x,s,,m,z,i) is:

$$ EHU\left(x,s,m,z,,i\right)= EH\left(x,s,m,z\right)\ eb\left(x,s,m,z,i\right) $$

1.3 Adjusted Housing Demand Projections by Changing Headship Rates

Instead of assuming constant headship rates, time-varying headship rates in future years may be projected by regression or other trend extrapolation methods based on time series data. For example, the U.S. Census Bureau (1996) performed time series regression models to project time-varying age-sex-household type-specific headship rates in future years. However, even if numbers of households are assumed to be correctly projected using the changing headship rates based on regression or other trend extrapolation method, it is possible that the headship rates still may result in biased projections of household consumption demands, which largely depend on household size (Myers et al. 2002), because the classic headship rate method excludes household size (U.S. Census Bureau 1996). To test this hypothesis, we conducted another assessment in which the changing headship rates are assumed to produce the same number of households as those observed in the 2000 census in each of the 50 U.S. states and DC. More specifically, we proportionally adjust the age-sex-household type-specific headship rates observed in 1990 by multiplying the ratio of the 2000-census observed total number of households to the total number of households in 2000 projected by the constant headship rates; we use these adjusted and changing age-sex-household type-specific headship rates to project the number of households (whose sum is equal to the census-observed total number of households), and then to project the housing units by number of bedrooms in 2000.

Let TH2000 denote total number of households counted in the 2000 census of the state.

The estimator to project age-sex-household type-specific number of housing units by number of bedrooms in 2000, based on adjusted age-sex-household type-specific headship rates, which result in the projected total number of households being equal to the total number of households observed in the 2000 census, is:

$$ HU'\left(x,s,h,i\right)= HH\left(x,s,h\right)\kern0.5em \frac{ TH2000}{{\displaystyle \sum_x{\displaystyle \sum_s{\displaystyle \sum_h HH\left(x,s,h\right)}}}}\kern0.5em b\left(x,s,h,i\right) $$

1.4 Adjusted Housing Demand Projections by the ProFamy Extended Cohort-Component Approach

To ensure comparability, we proportionally adjusted the age-sex-household type-size-specific number of households projected by the ProFamy approach to result in the same projected total number of households as the 2000-census observations; we then produced the adjusted housing unit projections. The estimator to project age-sex-household type-size-specific number of housing units by number of bedroom in 2000, based on the adjusted age-sex-household type-size-specific projected number of households, whose sum is equal to the total number of households observed in the 2000 census, is:

$$ EHU'\left(x,s,m,z,i\right)= EH\left(x,s,m,z\right)\kern0.5em \frac{ TH2000}{{\displaystyle \sum_x{\displaystyle \sum_s{\displaystyle \sum_m{\displaystyle \sum_z EH\left(x,s,m,z\right)}}}}}\kern0.5em eb\left(x,s,m,z,i\right) $$