Abstract
In multistate populations, the rates of interstate transfer cannot generally be determined from the size and composition of the populations at the beginning and end of a time interval. With N living states, the population data give only N equations to determine the N 2 possible rates. Here, the QERT (quadratic estimation of rates of transfer) approach is advanced that allows the transfer rates to be estimated when the products of selected pairs of rates can be assumed constant. The solution can be written in closed form and, for N living states, involves no more than N−1 quadratic equations. Compared to the leading alternative approaches, QERT provides very similar numerical estimates, while yielding the underlying behavioral rates, having flexible input requirements, accommodating all structural zeros, and reproducing the exact solution when interstate transfers are strictly hierarchical. The QERT approach is applied to construct labor force life tables for U.S. men and women for 2005–2010. The results show that labor force participation differences between men and women have continued to narrow, and that the QERT approach can generate robust worklife estimates. QERT thus provides new opportunities for demographic analysis in the absence of direct data on behavioral rates.
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Appendix: Constructing Labor Force Life Tables for U.S. Men and Women, 2005–2010
Appendix: Constructing Labor Force Life Tables for U.S. Men and Women, 2005–2010
The 2005 and 2010 U.S. census populations, by gender, were tabulated by age groups \(0{-} 4,{ 5}{-} 9, \ldots ,{ 75}{-} 7 9,{\text{ and 8}}0 +\). At every step, separate calculations, using the same procedures, were done for the male and female data. Because the census populations were not closed to migration, the 2010 population, at all ages over 5, was taken to be the 2005 population survived 5 years by the person-year values in the 2008 U.S. Life Tables. The Bureau of Labor Statistics (BLS) data on the employment status of the civilian noninstitutional population were then applied, age by age, to divide the populations into the statuses In the Labor Force (A) and Not in the Labor Force (N). Since the BLS data had no one in the labor force below age 16, interpolation based on the 2008 U.S. Life Tables was used to allocate persons in the 15–19 year age interval. At every age, the same mortality was assumed to apply to both labor force statuses.
The BLS considers persons to be in the labor force if they are employed or unemployed. Employed persons are those who (i) did any work for pay in the survey reference week, (ii) did unpaid work for 15 or more hours in a family business, or (iii) were temporarily absent from their regular job. Unemployed persons are those without a job who have actively looked for work in the prior 4 weeks and are currently available to work. Persons are classified as not in the labor force if they are neither employed nor unemployed during the reference week (Bureau of Labor Statistics 2014).
Flow Eqs. (13) provide the structure for the life table calculations. First, linear interpolation using 2008 U.S. Life Table values was used to shift the age categories by half an interval, so that the intervals begin at ages \(0,{ 2}. 5,{ 7}. 5, \ldots ,{ 72}. 5,{\text{ and 77}}. 5\). Population (P) values for ages x to x + n were taken to be at age x + n/2. The person-years lived at each age by the age cohort during the 5 year interval from 2005 to 2010 were found from the linear relationship in Eq. (3). The k values were also linearly adjusted to the new age intervals. QERT Eqs. (15) and (16) were then used to determine the age-specific rates of entry to and exit from the labor force.
The life table construction procedures followed Schoen (1988, Chap. 4). Because of the magnitude of the rates at young ages, the exponential (or constant forces) method was used to calculate the state-specific survivorship (ℓ) and person-year (L) values. The equations in Schoen (1988, p. 86) were used to determine the person-years lived, by state, at ages over 77.5. Once the person-year values, by age and state, were found, the number of transfers between states (d ij ) were calculated using the relationship \(d_{ij} = m_{ij} L_{i}\). Summary measures were found in a straightforward manner. The expected number of years in the labor force, \({\text{e}}_{\text{A}} \left( 0 \right)\), is \(\Sigma_{x} L_{A} \left( x \right)/\ell \left( 0 \right)\), i.e., the number of years lived in the labor force by the life table cohort at all ages, divided by the initial size of the life table cohort. The proportion of life lived in state A between ages 17.5 and 67.5 is the sum of the L A values for those ages divided by the sum of the total L values at those ages. The average number of entries to the labor force per person is \(\Sigma_{x} {\text{d}}_{\text{NA}} \left( x \right)/\ell \left( 0 \right)\). The average duration in the labor force per entry is the total number of years lived in the labor force, \(\Sigma_{x} L_{\text{A}}\), divided by \(\Sigma_{x} d_{\text{NA}}\), the total number of labor force entries. Consistency checks verified the flow equations at every age, and confirmed that the total number of entries to each state equaled the total number of exits.
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Schoen, R. Multistate Transfer Rate Estimation from Adjacent Populations. Popul Res Policy Rev 35, 217–240 (2016). https://doi.org/10.1007/s11113-015-9376-7
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DOI: https://doi.org/10.1007/s11113-015-9376-7