Data Needs and Estimation Procedures

Abstract

This chapter discusses the data needed for household and living arrangements projections at the national or sub-national level employing the ProFamy extended cohort-component model.

Appendices

Appendix 1: Procedures to Ensure the Accuracy of the Base Population for the Projections

1.1 Procedure to Ensure Accurate Total Population Size and Age/Sex Distributions in the Starting Year of the Projection

Define

W(k,m,p,c,x,s,T1) – age(x) and sex(s) specific number of persons with statuses k (co-residence with parents), m (marital/union status), p (parity), c (number of co-residing children) in the starting year (T1) of the projection, derived from the sample dataset.

N(m,x,s,T1) – age-sex-marital/union-status-specific number of persons in the starting year of the projection, based on the 100 % census tabulations.

To ensure accurate total population size and age/sex distributions, W(k, m, p, c, x, s, T1) must be adjusted:

$$ {W}^{\prime}\left(k,m,p,c,x,s,T 1\right)=W\left(k,m,p,c,x,s,T 1\right)\left[N\left(m,x,s,T 1\right)/{\displaystyle \sum_k{\displaystyle \sum_p{\displaystyle \sum_c\mathrm{W}\left(\mathrm{k},\mathrm{m},\mathrm{p},\mathrm{c},\mathrm{x},\mathrm{s},\mathrm{T}1\right)}}}\right] $$

(3.1)

If no age-sex-marital/union-status-specific number of persons based on the 100 % census tabulations are available, but age-sex-specific number of persons (N(x,s,T1)) in the starting year based on the 100 % census tabulations are available, Eq. 3.1 is modified as:

$$ {W}^{\prime}\left(k,m,p,c,x,s,T 1\right)=W\left(k,m,p,c,x,s,T 1\right)\left[N\left(x,s,T 1\right)/{\displaystyle \sum_k{\displaystyle \sum_p{\displaystyle \sum_c{\displaystyle \sum_m\mathrm{W}\left(\mathrm{k},\mathrm{m},\mathrm{p},\mathrm{c},\mathrm{x},\mathrm{s},\mathrm{T}1\right)}}}}\right] $$

(3.2)

1.2 Procedure to Ensure an Accurate Total Number of Households in the Starting Year of the Projection

As described in Sect. 3.1.1 above, we have obtained correct total (100 %) population classified by age, sex and k, m, p, c statuses in the staring year of the projection (W′(k, m, p, c, x, s, T1)). Using our ProFamy model accounting system, we first get a total number of households in the starting year of the projection, which may not be equal to the 100 % census count of the total number of households. For example, the difference between the ProFamy model count and the census count of the total number of households is 1.5–2.0 % using the U.S. 1980 and 1990 census micro data files and the 100 % census tabulations of population age and sex distributions from 1980 and 1990. The reason why there is such a discrepancy is that the sampling fractions of individual persons and household units are not exactly the same. Although the discrepancy is generally small, we need to do some adjustment to ensure an accurate total number of households in the starting year of the projection. We have done this using a simple procedure (Zeng et al. 2006) described below. Note that the following procedure assumes that we do not have census 100 % tabulations of number of households by age of reference persons, which is the usual case.

Define: H1(j) – number of households with size j in the starting year, derived by the ProFamy model count, using both census sample data set and the census 100 % tabulations of the population age-sex (and marital status, if available) distributions.

H2(j) – total number of households with size j in the starting year, based on the 100 % census tabulation.

TH2 – total number of all households in the starting year, based on the 100 % census tabulation.

T(x,s) – age-sex-specific total number of persons including reference and non-reference persons in the starting year, according to the 100 % census tabulation.

W1(x,s,j) – age-sex-specific total number of reference persons of the households with size j in the starting year, according to the ProFamy model count.

NW1(x,s) – age-sex-specific total number of non-reference persons in the starting year, according to ProFamy model count, where $T\left(x,s\right)={\displaystyle \sum_jW1\left(x,s,j\right)}+ NW 1\left(x,s\right)$.

W2(x,s,i) and NW2(x,s) are the adjusted number of reference persons (with household size j) and non-reference persons, respectively.

T(x,s), W1(x,s,j), NW1(x,s), W2(x,s,j) and NW2(x,s) are all 5-year age specific.

• 1. First adjustment to ensure that the household size distribution is consistent with the census 100 % tabulation:

  $$ W{ 2}^{\mathit{\prime}}\left(x,s,j\right)=W 1\left(x,s,j\right)\left(H 2(j)/H 1(j)\right) $$

• 2. Second adjustment to ensure that the total number of all households is consistent with the census 100 % tabulation, while the relative distribution of household size remains unchanged as in step (1):

  $$ W 2\left(x,s,j\right)=W{ 2}^{\mathit{\prime}}\left(x,s,j\right)\Big\{ TH 2/\Big[{\displaystyle \sum_x{\displaystyle \sum_s{\displaystyle \sum_j\mathrm{W}{2}^{\prime}\left(\mathrm{x},\mathrm{s},\mathrm{j}\right)\left]\right\}.}}} $$

• 3. Adjust non-reference persons:

  $$ NW 2\left(x,s\right)= NW 1\left(x,s\right)\left\{\left[T\left(x,s\right)-{\displaystyle \sum_jW2\left(x,s,j\right)\Big]/\Big[}T\left(x,s\right)-W 1\left(x,s\right)\right]\right\} $$

Proof

$$ {\displaystyle \sum_x{\displaystyle \sum_s{\displaystyle \sum_jW2\left(x,s,j\right)}}}={\displaystyle \sum_x{\displaystyle \sum_s{\displaystyle \sum_jW{2}^{\prime}\left(x,s,j\right)}}}\left\{ TH 2/\left[{\displaystyle \sum_x{\displaystyle \sum_s{\displaystyle \sum_jW{2}^{\prime}\left(\mathrm{x},\mathrm{s},\mathrm{j}\right)}}}\right]\right\}= TH 2 $$

$$ \begin{array}{l} NW 2\left(x,s\right)= NW 1\left(x,s\right)\left\{\right[T\left(x,s\right)-{\displaystyle \sum_jW2\left(x,s,j\right)\Big]/\Big[}T\left(x,s\right)-W 1\left(x,s\right)\left]\right\} = NW 1\left(x,s\right)\\ {}\Big\{\left[T\left(x,s\right)-{\displaystyle \sum_jW2\left(x,s,j\right)}\right]/\Big[W 1\left(x,s\right)+ NW1\left(x,s\right)-W1\left(x,s\right)\left]\right\} =T\left(x,s\right)-{\displaystyle \sum_jW2\left(x,s,j\right)}\end{array} $$

so, ${\displaystyle \sum_jW2\left(x,s,j\right)}+ NW 2\left(x,s\right)={\displaystyle \sum_jW2\left(x,s,j\right)}+T\left(x,s\right)-{\displaystyle \sum_jW2\left(x,s,j\right)}=T\left(x,s\right)$

Appendix 2: Standardized General Rates of Marriage/Union Formation and Dissolution

The standardized general rate of marriage/union formation and dissolution in the projection year t is defined as the total number of events that would occur if the age-sex-specific rates of occurrence of the events in year t were applied to the most recent census-counted sex-age-marital/union status distribution derived from the census data (Zeng et al. 2006).

Let N _i (x,s,r,T1) denote the number of persons of age x, marital/union status i, race or rural/urban category r, and sex s counted in the most recent census year T1 (i.e., the starting population of our household projection);

m _ij (x,s,r,t), sex-age-status-specific rates of transition from marital/union status i to j in year t $\left(\mathit{\mathsf{i}}\ne \mathit{\mathsf{j}}\right)$.

The m _ij (x,s,r,t) are to be calculated by the ProFamy program, while ensuring the consistency of the two-sex constraints and the projected standardized general rates of marriage/union formation and dissolution, which are defined below, in the year t (see Appendix 4 for details on how to calculate m _ij (x,s,r,t)).

Let GM(r,t) denote the projected race or rural/urban-specific standardized general rate of marriages including first marriage and remarriage for males and females combined.

$$ GM\left(r,t\right)=\frac{{\displaystyle \sum_{x=\alpha}^{\beta }{\displaystyle \sum_{s=1,2}{\displaystyle \sum_i{N}_i\left(x,s,r,T1\right){m}_{i2}\left(x,s,r,t\right)}}}}{{\displaystyle \sum_{x=\alpha}^{\beta }{\displaystyle \sum_{s=1,2}{\displaystyle \sum_i{N}_i\left(x,s,r,T1\right)}}}},i=1,3,4,5,6,7 $$

(3.3)

where α is the lowest age at marriage; β is the higher boundary of the age range in which the general rate of marriage/union formation and dissolution is defined.

Let GD(r,t) denote the projected race- or rural/urban-specific standardized general divorce rate for males and females combined.

$$ GD\left(r,t\right)=\frac{{\displaystyle \sum_{x=\alpha}^{\beta }{\displaystyle \sum_{s=1,2}{N}_2\left(x,s,r,T1\right){m}_{24}\left(x,s,r,t\right)}}}{{\displaystyle \sum_{x=\alpha}^{\beta }{\displaystyle \sum_{s=1,2}{N}_i\left(x,s,r,T1\right)}}}. $$

(3.4)

Let GC(r,t) denote the projected race- or rural/urban-specific standardized general rate of cohabiting of never-married and ever-married males and females combined.

$$ GC\left(r,t\right)=\frac{{\displaystyle \sum_{x=\alpha}^{\beta }{\displaystyle \sum_{s=1,2}\left[{N}_1\left(x,s,r,T1\right){m}_{15}\left(x,s,r,t\right)+{N}_3\left(x,s,r,T\right){m}_{36}\left(x,s,r,t\right){N}_4\left(x,s,r,T 1\right){m}_{47}\left(x,s,r,t\right)\right]}}}{{\displaystyle \sum_{x=\alpha}^{\beta }{\displaystyle \sum_{s=1,2}\left[{N}_1\left(x,s,r,T1\right)+{N}_3\left(x,s,r,T1\right)+{N}_4\left(x,s,r,T1\right)\right]}}} $$

(3.5)

Let GCD(r,t) denote the projected race- or rural/urban-specific standardized general union dissolution rate for males and females combined.

$$ GCD\left(r,t\right)=\frac{{\displaystyle \sum_{x=\alpha}^{\beta }{\displaystyle \sum_{s=1,2}\left[{N}_5\left(x,s,r,T1\right){m}_{51}\left(x,s,r,t\right)+{N}_6\left(x,s,r,T1\right){m}_{63}\left(x,s,r,t\right)+{N}_7\left(x,s,r,T1\right){m}_{74}\left(x,s,r,t\right)\right]}}}{{\displaystyle \sum_{x=\alpha}^{\beta }{\displaystyle \sum_{s=1,2}\left[{N}_5\left(x,s,r,T1\right)+{N}_6\left(x,s,r,T1\right)+{N}_7\left(x,s,r,T1\right)\right]}}} $$

(3.6)

Appendix 3: Procedure to Estimate Proportions of Those Aged 40–44 in Year t Who Do Not Live with Parents and Proportions of Elders Aged x in Year t Living with Adult Child(ren), While Taking into Account the Effects of Large Changes in Fertility

The procedures presented in this Appendix are designed for those populations in which the fertility level has been largely reduced in the past a few decades, implying that the availability of children for old parent(s) to co-reside with (if desired) has been substantially reduced (e.g. the case of China). Although the procedures are applicable to all populations, they may not be necessary for populations such as the U.S. and European countries which did not experience such large reduction in fertility level in recent decades. In that case, one may simply project or assume that the future years’ proportions of those aged 45–49 who do not live with parents and proportions of the elderly living with adult child(ren) will remain constant or by trend extrapolation or expert opinions.

Let’s define the following variables:

• L(42,t) – Proportion of those aged 40–44 (on average aged 42) in year t who do not live with parents;

• S(42,t) – Proportion of those aged 40–44 (on average aged 42) in year t who live with old parents (S(42,t) = 1.0–L(42,t));

• N(x,t) – Proportion of elderly aged x in year t living with adult child(ren) (and the child’s spouse if the child is married);

n ₀(t − x + 25) – Probability of dying of the elderly cohort members aged x in year t before their children reach average age at childbearing; As detailed cohort mortality data are usually not available, we may reasonably assume that n ₀(t − x + 25) is approximately equal to cumulative mortality rate up to average age at childbearing (e.g., age 25) in year t−x + 25;

n ₁(x,t) – Proportion of life-time infecundity of the elderly aged x in year t;

n ₂(x,t) – Proportion of old parents aged x in year t who do not live with adult child among those who have at least one adult child, due to preference of independent living or children’s mobility or other socioeconomic reasons;

n ₃(x,t) – Proportion of old parents aged x in year t who are not able to live with adult child(ren) even if they wish to do so among those who have at least one adult child, due to shortage of children (i.e., child generation size is smaller than parental generation size);

M(t−x + 40) – Proportion of eventually ever-married among adult children of the elderly aged x in year t (assuming the highest age at first marriage is 40; one may adopt a different assumption);

P(t − x + 25) – Male and female combined probability of surviving up to average age at childbearing for the adult children of the elderly aged x in year t; P(t−x + 25) is equal to cumulative survival probability up to average age at childbearing (assuming the average age at childbearing is 25; one may adopt a different assumption) in year t−x + 25.

G(x,t) – Index of offspring resource with respect to potential of co-residence between old parents and adult children; G(x,t) is defined as the sum of half of the average number of married children (as married children may also possibly live with their spouse’s parents if they wish) and the average number of adult children who were never married for whole life, among elderly aged x in year t;

$$ G\left(x,t\right)=0.5M\left(t-x+40\right)\cdot TFR\left(t-x+25\right)\cdot P\left(t-x+25\right)+\left(1-M\left(t-x+40\right)\right)\cdot TFR\left(t-x+25\right)\cdot P\left(t-x+25\right) $$

3.1 Estimation of Proportions of Those Aged 40–44 Who Do Not Live with Parents (L(42,t))

$$ L\left(42,t\right)=1.0-\frac{1-{n}_0\left(t-x+25\right)-{n}_1\left(x,t\right)-{n}_3\left(x,t\right)-\left[1-{n}_0\left(t-x+25\right)-{n}_1\left(x,t\right)-{n}_3\left(x,t\right)\right]\cdot {n}_2\left(x,t\right)}{G\left(x,t\right)}, $$

(3.7)

$$ S\left(42,t\right)=1-L\left(42,t\right); $$

(3.8)

$$ {n}_2\left(x,t\right)=1.0-\frac{S\left(42,t\right)G\left(x,t\right)}{1-{n}_0\left(t-x+25\right)-{n}_1\left(x,t\right)-{n}_3\left(x,t\right)} $$

(3.9)

If G(x, T1) ≥ (1.0 − n ₀ (T1 − x + 25) − n ₁ (x, T1)), n ₃ (x, T1) = 0;

If G(x,t) ≤ (1.0 − n ₀ (t − x + 25) − n ₁ (x,t)),

$$ {n}_3\left(x,t\right)=\left( 1.0-{n}_0\left(t-x+ 25\right)-{n}_1\left(x,t\right)\right)-G\left(x,t\right); $$

(3.10)

Therefore, we only need P(t−x + 25), M(t−x + 40), n ₀ (t−x + 25), n ₁ (x,t), n ₂ (x,t) and TFR(t−x + 25) to estimate L(42,t) and S(42,t), which are needed for family household projection for the countries in which fertility declined substantially in recent decades. P(t−x + 25), M(t−x + 40), n ₀ (t−x + 25), n ₁ (x,t) and TFR(t−x + 25) can be easily estimated from demographic data sources, which is straightforward, but estimation of n ₂(x,t) needs some more discussion. We can estimate the n ₂(x, T1) of the elderly aged x in the census year T1 (i.e., starting year of the projections), based on the observed proportion of those aged 40–44 (on average aged 42) in census year T1 (S(42,T1)) who live with old parents, using the formula (3.9) in either the case (1) or (2) as follows:

• 1. If G(x, T1) ≥ (1.0 − n ₀ (T1 − x + 25) − n ₁ (x, T1)), n ₃ (x, T1) = 0, and

  $$ {n}_2\left(x,T1\right)=1.0-\frac{S\left(42,T1\right)G\left(x,T1\right)}{1-{n}_0\left(T1-x+25\right)-{n}_1\left(x,T1\right)}; $$

• 2. If G(x, T1) ≤ (1.0 − n ₀ (T1 − x + 25) − n ₁ (x, T1)), n ₃ (x, T1) = 1.0 − n ₀ (T1 − x + 25) − n ₁ (x, T1) − G(x, T1), and

  S(x, T1) = 1.0–n ₂ (x, T1) [derived based on replacing n ₃ (x, T1) in Eq. 3.9 by (1.0 − n ₀ (T1 − x + 25) − n ₁ (x, T1) − G(x, T1)).

  $$ {n}_2\left(x,T 1\right)= 1.0-S\left(x,T 1\right), $$

Once we estimated the n ₂ (x, T1) in the census year T1 (i.e., starting year of the projections), we can estimate or project (or assume) the n ₂ (x, T1) in the future years based on trend extrapolation or expert opinions, and then estimate the L(42,t) and S(42,t) in the corresponding future years.

3.2 Estimating Proportions of Elderly Living with Adult Child(ren) (N(x,t))

$$ N\left(x,t\right)=1-{n}_1\left(x,t\right)-{n}_3\left(x,t\right)-\left[1-{n}_1\left(x,t\right)-{n}_3\left(x,t\right)\right]\cdot {n}_2\left(x,t\right), $$

(3.11)

We can estimate the N(x,t), using the formula (3.11) in either the case (1) or (2) as follows:

• 1. If G(x,t) ≥ (1.0 − n ₀ (t − x + 25) − n ₁ (x,t)), n ₃ (x,t) = 0, and

  $$ N\left(x,t\right)=1-{n}_1\left(x,t\right)-\left[1-{n}_1\left(x,t\right)\right]\cdot {n}_2\left(x,t\right) $$

• 2. If G(x,t) ≤ (1.0 − n ₀ (t − x + 25) − n ₁ (x,t)), n ₃ (x,t) = 1.0 − n ₀ (t − x + 25) − n ₁ (x,t) − G(x,t), and

  $$ N\left(x,t\right)=G\left(x,t\right)-G\left(x,t\right){n}_2\left(x,t\right) $$

N(x,t) is an average proportion of the old parents aged x who live with an adult child (and the child’s spouse if the child is married), and N(x,t) represents the overall level of co-residence between old parents aged x in year t and their adult children. In the same time, we estimate the sex-age-marital/union status-specific proportions of the elderly living with children as a standard schedule based on the census (or survey) data. Using these standard schedules and the estimated N(x,t) in the future years, we can estimate the sex-age-marital/union status-specific proportions of elderly living with children in the future years.

Appendix 4: Procedure to Calculate Sex-Age-Specific Rates While Ensuring the Consistency of the Two-Sex Constraints and the Projected Standardized General Rates of Marriage/Union Formation and Dissolution

• Input:

• GM(r,t), GD(r,t), GC(r,t), GCD(r,t): projected (or assumed) race- or rural/urban-specific standardized general rates of marriage/union formation and dissolution in the projection year t; r stands for race or rural/urban dimension;

• m ^s _ij (x,s,r): the sex-age-specific standard schedules of o/e rates of transition from marital/union status i to marital/union status j between age x and x + 1.

• Output:

• m _ij (x,s,r,t): the sex-age-specific o/e rates of transition from marital/union status i to marital/union status j(i ≠ j) between age x and x + 1 in the projection year t, that is consistent with the two-sex constraints and the projected standardized general rates of marriage/union formation and dissolution.

• One important conceptual note must be clarified – we adjust the initial standard schedules of age-sex-status-specific o/e rates rather than probabilities of marital/union formation and dissolution to achieve consistency with the projected summary measures and the two-sex constraints. The age-sex-status-specific o/e rate is defined as the number of events that occurred in the age interval divided by the number of person-years lived at risk of experiencing the event. The age-specific rates can be analytically translated to the age-sex-status-specific probabilities using the matrix formula in the context of multiple increment-decrement models (see, e.g., Preston et al. 2001; Schoen 1988; Willekens et al. 1982). This approach could adequately handle the issues of competing risks. Furthermore, adjusting probabilities directly may result in an inadmissible value that is greater than one; adjusting age-specific o/e rates would not yield such an inadmissible probability value, however.

The procedure consists of two steps (refer to: Zeng et al. 2004).

• Step 1. Adjustment to comply with the two-sex constraints, following the harmonic mean approach

We use the harmonic mean approach to ensure two-sex consistency in household projections in monogamous societies. The harmonic mean satisfies most of the theoretical requirements and practical considerations for handling consistency problems in a two-sex model Keilman (1985; Pollard 1977; Schoen 1981).

In order to calculate the number of events that occurred in year t, we need to calculate the mid-year population $\left(\overline{N_i^{\prime }}\left(x,s,r,t\right)\right)$, classified by age, sex, marital/union status and race or rural/urban status, if it is distinguished. The $\overline{N_i^{\prime }}\left(x,s,r,t\right)$ are the averages of the populations at the beginning and the end of the year t and can be considered as an approximation of the person-years lived in status i (i.e., at risk of experiencing the event of transition from status i to j).

Let N _i (x,s,r,t) denote the number of persons of age x, marital/union status i, r status, and sex s at the beginning of year t, which are known through the preceding year’s projection. When t refers to the starting year of the projection, the N _i (x,s,r,t) are derived from the census data. The sex-age-specific rates m _ij (x,s,r,t) and sex-age-specific probabilities P _ij (x,s,r,t) were defined earlier and their relationship can be expressed in the matrix formula (ref. to Willekens et al. 1982). We seek to estimate m _ij (x,s,r,t) through adjusting m _ij (x, s, r, t − 1), which are known through the preceding year’s estimation. When t refers to the starting year of the projection, m _ij (x, s, r, t − 1) are equal to the standard schedules. The estimated m _ij (x,s,r,t) must be consistent with the two-sex constraints of all race groups combined or rural/urban combined and the projected race or rural/urban-specific standardized general rates of the marriage/union formation and dissolution in year t.

$$ {N}_i^{\prime}\left(x+1,s,r,t+1\right)={\displaystyle \sum_j{P}_{ij}^{\prime }}\left(x,s,r,t\right){N}_j\left(x,s,r,t\right). $$

(3.12)

$$ \overline{N_i^{\prime }}\left(x,s,r,t\right)= 0.5\ \left[{N}_i\left(x,s,r,t\right)+{N}_i^{\prime}\left(x+ 1,s,r,t+ 1\right)\right]. $$

(3.13)

Keep in mind for later consideration that $\overline{N_i^{\prime }}\left(x,s,r,t\right)$ (the average of N _i (x,s,r,t) and N′ _i (x + 1, s, r, t + 1)) is only a first approximation, since N′ _i (x + 1, s, r, t + 1) is based on the P′ _ij (x,s,r,t), which are not the final estimates for year t.

The total number of new marriages of persons of sex s (s = 1, 2, referring to females and males, respectively) who were not cohabiting before marriage for all race groups combined or for rural/urban combined in year t (TM(s,t)) is estimated as follows:

$$ TM\left(s,t\right)={\displaystyle \sum_r\Big[{\displaystyle \sum_i{\displaystyle \sum_{x=\alpha}^{\omega}\overline{N_i^{\prime }}\left(x,s,r,t\right){m}_{i2}\left(x,s,r,t-1\right)\Big]}}}, i=1,3,4 $$

where ω is the highest age considered in the family household projection; α is the lowest age at marriage. To meet the two-sex constraint, the sex-age-specific rates of marriage among persons who were not cohabiting before marriage need to be adjusted:

$$ {m}_{i2}^{\prime}\left(x,s,r,t\right)={m}_{i2}\left(x,s,r,t- 1\right)\left[\frac{2 TM\left(1,t\right) TM\left(2,t\right)\Big)}{ TM\left(1,t\right)+ TM\left(2,t\right)}/ TM\left(s,t\right)\right], i=1,3,4 $$

(3.14)

The estimated total number of new divorces of persons of sex s for all race groups combined or for rural/urban combined in year t (TD(s,t)) is

$$ TD\left(s,t\right)={\displaystyle \sum_r\Big[{\displaystyle \sum_{x=\alpha}^{\omega}\overline{N_2^{\prime }}\left(x,s,r,t\right){m}_{24}\left(x,s,r,t-1\right)}}. $$

To meet the two-sex constraint, the sex-age-specific rates of divorce need to be adjusted:

$$ {m}_{24}^{\prime}\left(x,s,r,t\right)={m}_{24}\left(x,s,r,t- 1\right)\left[\frac{2 TD\left(1,t\right) TD\left(2,t\right)\Big)}{ TD\left(1,t\right)+ TD\left(2,t\right)}/ TD\left(s,t\right)\right]. $$

(3.15)

The rates of widowhood depend on spouses’ death rates, which are calculated before the two-sex constraints adjustments, based on the standard mortality schedules and the projected life expectancy at birth in year t. The already projected spouses’ death rates should not be adjusted again; they must be used as a “standard”. Thus, instead of employing the harmonic mean approach, we simply adjust the rates of widowhood to be consistent with the total number of spouses who die in year t. The total number of persons (i.e., spouses) of sex s who died for all race groups combined or for rural/urban combined in year t with an intact marriage before death (TDM(s,t)) based on already projected sex-age-specific death rates is

$$ TDM\left(s,t\right)={\displaystyle \sum_r\Big[{\displaystyle \sum_i{\displaystyle \sum_{x=\alpha}^{\omega}\overline{N_2^{\prime }}\left(x,s,r,t\right){d}_2\left(x,s,r,t\right)}}}\Big] $$

where d ₂(x,s,r,t) is the already projected death rate of married persons of age x and sex s in year t.

The estimated total number of newly widowed persons of sex s for all race groups combined or for rural/urban combined in year t (TW(s,t)) is

$$ TW\left(s,t\right)={\displaystyle \sum_r\Big[{\displaystyle \sum_i{\displaystyle \sum_{x=\alpha}^{\omega}\overline{N_2^{\prime }}\left(x,s,r,t\right){m}_{23}\left(x,s,r,t-1\right)}}}\Big] $$

To meet the two-sex constraint, the sex-age-specific rates of widowhood need to be adjusted using TDM(s,t) as a “standard”:

$$ {m}_{23}^{\prime}\left(x,s,r,t\right)={m}_{23}\left(x,s,r,t- 1\right)\left[\frac{ TDM\left({s}^{-1},t\right)}{ TW\left(s,t\right)}\right]. $$

(3.16)

where “s ⁻¹” indicates the opposite sex of “s”.

The estimated total number of newly cohabiting persons of sex s for all race groups combined or for rural/urban combined in year t (TC(s,t)) is

$$ TC\left(s,t\right)={\displaystyle \sum_r\Big[{\displaystyle \sum_{x=\alpha}^{\omega}\overline{N_1^{\prime }}\left(x,s,r,t\right){m}_{15}\left(x,s,r,t-1\right)}+{\displaystyle \sum_{x=\alpha}^{\omega}\overline{N_3^{\prime }}\left(x,s,r,t\right){m}_{36}\left(x,s,r,t-1\right)}} +{\displaystyle \sum_{x=\alpha}^{\omega}\overline{N_4^{\prime }}\left(x,s,r,t\right){m}_{47}\left(x,s,r,t-1\right)\Big]}. $$

To meet the two-sex constraint, the sex-age-specific rates of cohabiting need to be adjusted:

$$ {m}_{15}^{\prime}\left(x,s,r,t\right)={m}_{15}\left(x,s,r,t- 1\right)\left[\frac{2 TC\left(1,t\right) TC\left(2,t\right)\Big)}{ TC\left(1,t\right)+ TC\left(2,t\right)}/ TC\left(s,t\right)\right]. $$

(3.17)

$$ {m}_{36}^{\prime}\left(x,s,r,t\right)={m}_{36}\left(x,s,r,t- 1\right)\left[\frac{2 TC\left(1,t\right) TC\left(2,t\right)\Big)}{ TC\left(1,t\right)+ TC\left(2,t\right)}/ TC\left(s,t\right)\right]. $$

(3.18)

$$ {m}_{47}^{\prime}\left(x,s,r,t\right)={m}_{47}\left(x,s,r,t- 1\right)\left[\frac{2 TC\left(1,t\right) TC\left(2,t\right)\Big)}{ TC\left(1,t\right)+ TC\left(2,t\right)}/ TC\left(s,t\right)\right]. $$

(3.19)

The estimated total number of new marriages of persons of sex s who were cohabiting before marriage for all race groups combined or for rural/urban combined in year t (TCM(s,t)) is

$$ TCM\left(s,t\right)={\displaystyle \sum_r\Big[{\displaystyle \sum_i{\displaystyle \sum_{x=\alpha}^{\omega}\overline{N_i^{\prime }}\left(x,s,r,t\right){m}_{i2}\left(x,s,r,t-1\right)}}}\Big], i=5,6,7. $$

To meet the two-sex constraint, the sex-age-specific rates of marriage of persons who were cohabiting before marriage need to be adjusted:

$$ {m}_{i 2}^{\prime}\left(x,s,r,t\right)={m}_{i 2}\left(x,s,r,t- 1\right)\left[\frac{2\left( TCM\left(1,t\right) TCM\left(2,t\right)\right)}{ TCM\left(1,t\right)+ TCM\left(2,t\right)}/ TCM\left(s,t\right)\right], i=5,6,7. $$

(3.20)

The estimated total number of events of cohabitation union dissolution of persons of sex s for all race groups combined or for rural/urban combined in year t (TCD(s,t)) is

$$ TCD\left(s,t\right)={\displaystyle \sum_r\Big[{\displaystyle \sum_{x=\alpha}^{\omega}\overline{N_5^{\prime }}\left(x,s,r,t\right){m}_{51}\left(x,s,r,t-1\right)}+{\displaystyle \sum_{x=\alpha}^{\omega}\overline{N_6^{\prime }}\left(x,s,r,t\right){m}_{63}\left(x,s,r,t-1\right)}} +{\displaystyle \sum_{x=\alpha}^{\omega}\overline{N_7^{\prime }}\left(x,s,r,t\right){m}_{74}\left(x,s,r,t-1\right)}\Big]. $$

To meet the two-sex constraint, the sex-age-specific o/e rates of cohabitation union dissolution need to be adjusted:

$$ {m}_{51}^{\prime}\left(x,s,r,t\right)={m}_{51}\left(x,s,r,t- 1\right)\left[\frac{2 TCD\left(1,t\right) TCD\left(2,t\right)\Big)}{ TCD\left(1,t\right)+ TCD\left(2,t\right)}/ TCD\left(s,t\right)\right]. $$

(3.21)

$$ {m}_{63}^{\prime}\left(x,s,r,t\right)={m}_{63}\left(x,s,r,t- 1\right)\left[\frac{2\left( TCD\left(1,t\right) TCD\left(2,t\right)\right)}{ TCD\left(1,t\right)+ TCD\left(2,t\right)}/ TCD\left(s,t\right)\right]. $$

(3.22)

$$ {m}_{74}^{\prime}\left(x,s,r,t\right)={m}_{74}\left(x,s,r,t- 1\right)\left[\frac{2\left( TCD\left(1,t\right) TCD\left(2,t\right)\right)}{ TCD\left(1,t\right)+ TCD\left(2,t\right)}/ TCD\left(s,t\right)\right]. $$

(3.23)

The sex-age-specific rates, m ^’ _ij (x,s,t), are adjusted for consistency with the two-sex constraint as described above, but they need to be further adjusted to be consistent with the projected race or rural/urban-specific standardized general rates of marriage/union formation and dissolution in year t, and will be described in Step 2 as follows.

• Step 2. Adjustment for consistency with the projected race- or rural/urban-specific standardized general rates of marriage/union formation and dissolution in year t

To calculate the m _ij (x,s,r,t) while ensuring consistency with the race or rural/urban- specific standardized general rates GM(r,t), GD(r,t), GC(r,t) and GCD(r,t), we first estimate the GM′(r,t), GD′(r,t), GC′(r,t) and GCD′(r,t), based on N _i (x, s, r, T1) derived from the most recent census and the m′ _ij (x,s,r,t), which were consistent with the two-sex constraints and estimated in Step 1 described above, and using the formulas presented in Appendix 2. We then use the same adjustment factor to adjust male and female rates as follows.

$$ {m}_{i2}^{{\prime\prime}}\left(x,s,r,t\right)=\frac{ GM\left(r,t\right)}{G{M}^{\prime}\left(r,t\right)}{m}_{i2}^{\prime}\left(x,s,r,t\right),i=1,3,4,5,6,7. $$

(3.24)

$$ {m}_{24}^{{\prime\prime}}\left(x,s,r,t\right)=\frac{ GD\left(r,t\right)}{G{D}^{\prime}\left(r,t\right)}{m}_{24}^{\prime}\left(x,s,r,t\right). $$

(3.25)

$$ {m}_{15}^{{\prime\prime}}\left(x,s,r,t\right)=\frac{ GC\left(r,t\right)}{G{C}^{\prime}\left(r,t\right)}{m}_{15}^{\prime}\left(x,s,r,t\right). $$

(3.26)

$$ {m}_{36}^{{\prime\prime}}\left(x,s,r,t\right)=\frac{ GC\left(r,t\right)}{G{C}^{\prime}\left(r,t\right)}{m}_{36}^{\prime}\left(x,s,r,t\right). $$

(3.27)

$$ {m}_{47}^{{\prime\prime}}\left(x,s,r,t\right)=\frac{ GC\left(r,t\right)}{G{C}^{\prime}\left(r,t\right)}{m}_{47}^{\prime}\left(x,s,r,t\right). $$

(3.28)

$$ {m}_{51}^{{\prime\prime}}\left(x,s,r,t\right)=\frac{ GC D\left(r,t\right)}{ GC{D}^{\prime}\left(r,t\right)}{m}_{51}^{\prime}\left(x,s,r,t\right). $$

(3.29)

$$ {m}_{63}^{{\prime\prime}}\left(x,s,r,t\right)=\frac{ GC D\left(r,t\right)}{ GC{D}^{\prime}\left(r,t\right)}{m}_{63}^{\prime}\left(x,s,r,t\right). $$

(3.30)

$$ {m}_{74}^{{\prime\prime}}\left(x,s,r,t\right)=\frac{ GC D\left(r,t\right)}{ GC{D}^{\prime }(t)}{m}_{74}^{\prime}\left(x,s,r,t\right). $$

(3.31)

Note that the adjustments described in Step 1 use the mid-year populations $\left(\overline{N_i^{\prime }}\left(x,s,r,t\right)\right)$, which are the preliminarily estimated average of the populations at the beginning and the end of year t and approximations only, since they are not based on the final estimates of the sex-age-specific rates of marriage/union formation and dissolution. Although we use the same adjustment factors for males and females, the rates adjusted in Step 2 may not be exactly consistent with the two-sex constraints mainly because $\overline{N_i^{\prime }}\left(x,s,r,t\right)$ are not the final estimates. We, therefore, need to repeat the adjustment procedure described in Step 1 by using the m″ _ij (x,s,t) estimated in Step 2. More specifically, we calculate N″ _i (x + 1, s, r, t + 1) and $\overline{N_i^{{\prime\prime} }}\left(x,s,r,t\right)$, using the m″ _ij (x,s,t) and employing the formulas 3.12 and 3.13. We then use $\overline{N_i^{{\prime\prime} }}\left(x,s,r,t\right)$ and m″ _ij (x,s,r,t) to replace $\overline{N_i^{\prime }}\left(x,s,r,t\right)$ and m _ij (x, s, r, t − 1) in the formulas in Step 1 to get the new estimates m′″ _ij (x,s,r,t), which satisfy the two-sex constraints. We then use the new estimates of m′″ _ij (x,s,r,t) to calculate the new estimates of race- or rural/urban-specific standardized general rates of marriage/union formation and dissolution: GM″(r,t), GD″(r,t), GC″(r,t), GCD″(r,t). If the absolute values of the relative difference between the new estimates of the standardized general rates and the corresponding projected general rates are all less than a selected criterion (e.g., 0.01 or 0.001), we have achieved our goals for computing the sex-age-specific (and race- or rural/urban-specific if so distinguished) rates of marriage/union formation and dissolution in year t. Otherwise, we need to repeat the adjustment procedures described in Step 2 and Step 1 until the selected criterion is met.

To provide numerical examples, we used the procedure described above in Step 1 and Step 2 with the standardized general rates as summary measures to calculate the time-varying sex-age-specific rates of marriage/union formation and dissolution in the projection years. The standard schedules are based on the estimates of the U.S. sex-age-specific rates of marriage/union formation and dissolution in 1990–1996 (Zeng et al. 2012b). The sex-age-marital/union status distributions of the starting year of the projection were derived from the U.S. 2000 census micro sample data file. We estimated models with seven marital/cohabiting statuses including cohabitation for the four race groups respectively and combined.

The required number of repetitions of Step 1 and Step 2 using standardized general rates as summary measures is between 2 and 4, as indicated in Table 3.4.1.

Table 3.4.1 Number of repetitions of Step 1 and Step 2 in an illustrative example (GM(r,t) decrease by 4 %, GD(r,t) increase by 5 %; GC(r,t) increase by 8 %; GCD(r,t) increase by 6 %)

The results of the illustrative numerical applications listed in Table 3.4.1 demonstrate that the iterative procedures expressed in Steps 1 and 2 are valid for practical applications. Based on the final estimates of m _ij (x,s,r,t) in year t, one can also construct multi-state marital/union status life tables for males and females separately and calculate the detailed sex-specific period life table propensities of transitions from marital/union status i to j (PP _ij (s,r,t)) in the year t. PP _ij (s,r,t) are informative to reflect the gender differentials of the intensities of transitions among various marital/union statuses, which are consistent with the two-sex constraints and the projected standardized general rates of marriage/union formation and dissolution in the projection years.

Appendix 5: Procedure to Estimate General Rates of Marriage/Union Formation and Dissolution at the Starting Year of the Projections

As an illustration of the application, we present procedures to estimate the U.S. race-specific general rates of marriages, divorce, cohabitation, and union dissolution at the state level (Zeng et al. 2013a). The procedures presented here are applicable to other countries and regions; the race dimension (denoted as “r”) may be replaced by a rural–urban dimension, or eliminated if no race and no rural/urban dimension is distinguished in your applications.

5.1 Estimating the U.S. State-Race-Specific General Rates of Marriage and Divorce at the Starting Year of the Projections

Given that we have the published total numbers of marriages and divorces for each of the 50 states and DC for all races combined but not for specific races, we employ the following procedure to estimate the U.S. state-race-specific general rate of marriage (GM(r,T1) ) and divorce ( GD(r,T1) ) in the census year T1 which is the starting year of the projection. The marital/union status codes i and j are defined earlier in Sect. 1 of Chap. 2. To simplify the presentation, we omit the state dimension in all variables and formulas in this Appendix.

Let: N _i (x,s,r,T1) denote the number of persons of age x, race r, marital/union status i and sex s counted in the census year T1 in the state;

M _ij (x,s,r), the national model standard schedules of the race-sex-age-specific o/e rates of transition from marital/cohabiting status i to j(i ≠ j), where i and j represent the seven marital/union statuses.

m _ij (x,s,r,T1), the estimated race-sex-age-specific o/e rates of transition from marital/union status i to j in the census year T1(i ≠ j) in the state; and TM(T1), the published all-races-combined total number of marriages (newly married couples) including first marriages and remarriages that occurred in the state in the census year T1.

We assume that the race-sex-age-specific o/e rates of first marriage and remarriage in the state in the census year are proportional to the corresponding national model standard schedule rates,

m _i2(x, s, r, T1) = γ(T1)M _i2(x,s,r); i ≠ 2 (The subscript 2 represents currently married status); where the γ (T1) is estimated as:

$$ \gamma \left(T 1\right)=\frac{2 TM(T1)}{{\displaystyle \sum_{x=\alpha}^{\beta }{\displaystyle \sum_{s=1,2}{\displaystyle \sum_r{\displaystyle \sum_i{N}_i\left(x,s,r,T1\right){m}_{i2}\left(x,s,r\right)}}}}}, i\ne 2 $$

(3.32)

where α (usually taken as 15) and β are the low and the upper boundary of the age range in which the events of marriage/union formation and dissolution occur.

We then use the estimated m _i2 (x,s,r,T1) and N _i (x,s,r,T1) to calculate the estimated race-specific GM (r,T1) in year T1 for the state, employing the formula 3.3 in Appendix 2.

Let TD(T1) denote the published all-races-combined total number of couples who divorced in the state in the census year T1. We assume that the race-sex-age-specific o/e rates of divorce in the state in the census year are proportional to the corresponding model standard schedule rates of divorce, namely, m ₂₄ (x,s,r,T1) = δ (T1) M ₂₄ (x,s,r); (The subscript 2 and 4 represent currently married and divorced) where the δ (T1) is estimated as:

$$ \delta \left(T 1\right)=\frac{2 TD(T1)}{{\displaystyle \sum_{x=\alpha}^{\beta }{\displaystyle \sum_{s=1,2}{\displaystyle \sum_r{\displaystyle \sum_i{N}_2\left(x,s,r,T1\right){M}_{24}\left(x,s,r\right)}}}}}, $$

(3.33)

We then use the estimated m ₂₄ (x,s,r,T1) and N ₂ (x,s,r,T1) to calculate the estimated race-specific GD (r,T1) in year T1 for the state, employing the formula 3.4 in Appendix 2.

5.2 Estimating the U.S. Race-Specific General Rates of Cohabitation and Union Dissolution at the State Level

Because we do not have published data on the total numbers of cohabitation formation and dissolution events at the state level, we cannot estimate the U.S. race-specific general rate of cohabitating (GC(r,T1)) and union dissolution (GCD(r,T1)) at the state level directly; thus, we need to employ an indirect estimation approach by iterative proportional fitting. First, we use the previous census data as the base population and the race-specific model standard schedules and the other estimated demographic parameters as input to project the household distributions (race-specific) from the previous census year to the most recent census year (T1), which is the starting year of the projection. Through such projections, we obtain the projected all-races-combined proportions of households with a cohabiting couple among all households in the most recent census year, denoted as PC. Second, we then compare PC with the observed all-races-combined proportion of the households with a cohabiting couple among all households observed in the most recent census, denoted as CC. If the PC is higher (or lower) than CC by a margin of a pre-determined criterion (e.g., 1 %), we proportionally adjust the race-sex-age-specific rates of cohabitation union formation and dissolution by the following formulas:

$$ {m}_{15}\left(x,s,r,T 1\right)={M}_{15}\left(x,s,r\right)\ \left( 2- PC/ CC\right); $$

(3.34)

$$ {m}_{36}\left(x,s,r,T 1\right)={M}_{36}\left(x,s,r\right)\ \left( 2- PC/ CC\right); $$

(3.35)

$$ {m}_{47}\left(x,s,r,T 1\right)={M}_{47}\left(x,s,r\right)\ \left( 2- PC/ CC\right); $$

(3.36)

$$ {m}_{51}\left(x,s,r,T 1\right)={M}_{51}\left(x,s,r\right)\ \left( PC/ CC\right); $$

(3.37)

$$ {m}_{63}\left(x,s,r,T 1\right)={M}_{63}\left(x,s,r\right)\ \left( PC/ CC\right); $$

(3.38)

$$ {m}_{74}\left(x,s,r,T 1\right)={M}_{74}\left(x,s,r\right)\ \left( PC/ CC\right); $$

(3.39)

We then use the above-adjusted rates and the other data to redo the projection from the previous census year to the starting year T1, and calculate the new projected PC and compare it with CC. If the new PC is still higher (or lower) than CC by a margin of a pre-determined criterion, we repeat this iterative proportional fitting procedure until the projected PC and the observed CC are reasonably close to each other (e.g., with a relative difference between −1 % and +1 %), and we then estimate the GC(r,T1) and GCD(r,T1), employing the formulas 3.5 and 3.6 in Appendix 2

Given the cohabitation data constraints at the state level, this procedure produces reasonably good estimates of the GC(r,T1) and GCD(r,T1), as shown by the results of the validation testing projections from 1990 to 2000 for each of the 50 states and DC presented in Chap. 4.