(p.182) Appendix Three Details of the Chapter 4 Simulations
(p.182) Appendix Three Details of the Chapter 4 Simulations
A. Simulating a Constant Family Size Within Family Type (Table 6)
To implement the simulation reported in table 6, I divide my data into three samples composed of those individuals in each of the three family types. Because family size does not change among persons in single-person family units (which always contain only one individual), I work only with the two samples containing persons in single-headed family types and in married-couple family types. Within each of these family types, I rank all persons by family-size centiles in 1979 and 2007. This means dividing the persons within each family type into one hundred equal-sized groups, ranked by total family size. Of course, there are many persons who live in families of the same size and who need to be allocated among different centile groups. In cases where there are a large number of people with the same family-size level, I randomly assign people into a centile.
Within each 1979 centile, I calculate the mean family size. I then assign the mean family size from the 1979 centile to each person in the equivalent 2007 centile. So, the lowest-ranked centile in 2007 would be assigned the family size of the lowest-ranked centile from 1979; the second-lowest centile in 2007 would be assigned the family size of the second-lowest centile from 1979, and so on. I then recalculate per person (p.183) income in 2007 using 2007 income levels but using the simulated 1979 family size for each individual rather than their actual 2007 family sizes. The simulated family-size distribution of all individuals in 2007 now mimics that of 1979 within each of these family types.
b. Simulating a Constant Share of Persons Across Family Types and then simulating constant shares and constant Family Sizes (Table 7)
I simulate the effect of changes in the relative share of family types by reweighting the 2007 data so that the share of eighteen- to sixty-four-year-olds in each family type is identically equal to its share in 1979. For each person in each of the three family types in the 2007 data, I multiply their person-weight by the ratio of the share of their family type in the eighteen- to sixty-four-year-old population in 1979 divided by the share of their family type in the eighteen- to sixty-four-year-old population in 2007. The result is to increase the weights on persons in married-couple families (which were more populous in 1979) and decrease the weights on persons in single-individual and single-headed family types (which were less populous in 1979). This data is used for the results in column 3 of table 7.
When simulating the effects of constant shares across family types as well as constant family size (column 4 of table 7), I use the reweighted population described in the previous paragraph and adjust it for family-size changes within each family type, using the technique described in section A of this appendix to hold family size constant. This both holds the population weights constant across family types and holds family size constant within family type.
c. Simulating a Constant Distribution of Earnings, Government Income, and other Unearned income (Table 8)
The technique I use here is based on that used in Reed and Cancian (2001). Let me first explain the simulation in part A of table 8. holding the earnings distribution constant at its 1979 levels. I divide the 2007 data (p.184) into three samples, each containing all eighteen- to sixty-four-year-old individuals in each of the three family types. All of my analysis is then done within each sample by family type. For each family-type sample, I rank all persons in the 1979 sample by their family unit’s total earnings and divide them into one thousand equal-sized groups; I refer to each of these as a permillage. Because I do not want to hold family size constant in this simulation, I do this ranking on the basis of each person’s total family earnings (not per person income based on family earnings adjusted for family size). There are often many people with the same total family earnings who need to be assigned to different permillages in order to keep the number of people in each group equal. When there are a number of people tied at the same total family-earnings level, I randomly assign these people to an appropriate permillage. This is the same method I used to break family-size ties in section A of this appendix.
I calculate the mean total family earnings for each permillage in the 1979 distribution. I then divide the 2007 family-type samples into permillages as well. For the persons in each permillage in the 2007 distribution, I assign them mean total family earnings from their equivalent 1979 permillage. Said another way, I rank each person in 2007 from one to one thousand, based on their location in the total family-earnings distribution of their family type, and assign them the total family earnings that someone at an equivalent rank in the 1979 distribution of their family type would have received. This essentially provides a simulated level and distribution of total family earnings for everyone in the 2007 sample that imitates the level and distribution of total family earnings in 1979 by family type.
I simulate total 2007 per person income levels by adding actual family government income and other unearned income for each person to their simulated total family earnings and then adjusting for family size using persons’ actual 2007 family sizes. This provides a measure of 2007 income among all persons, holding the distribution of earnings at its 1979 pattern within each family type while allowing the distribution of government and other unearned income, and family size and family-type composition, to change.
For part B of table 8, I simulate the distribution of total income. I again divide the 1997 and 2007 data into three samples, each containing (p.185) all eighteen- to sixty-four-year-olds within a given family type. Within each family-type sample, I create permillages ordered by total income in 1979 and 2007. Following my treatment of family earnings, I do this ranking on the basis of each person’s total family income (not per person family income adjusted for family size). Within each family-type sample, persons in each 2007 permillage of total income were assigned the total income level of the equivalent permillage in 1979. Hence, my simulation re-creates the 1979 distribution of total income within the 2007 sample for each family type. By working with total income, I take account of all of the correlations between different income components, which change together over time.
Once a simulated 2007 total family income is assigned to each person, I then define simulated per person income by using actual 2007 family size to adjust total family income to per person income for each individual. This allows the distribution of family size to change as it actually does between 1979 and 2007. Hence, I allow family size and family-type composition to change but hold the distribution of income constant.
d. Simulating Constant Income and Constant Family Type Together (Table 9)
The final simulations, shown in table 9, use the simulation techniques described earlier. In part A, I simulate a constant income distribution as described in section C of this appendix. These are the results reported in the first row of part A. Then, using this simulated data, I follow the techniques described in section B of this appendix to hold family size and composition constant in the second row of part A.
Part B reverses these simulations. I first do the simulation described in section B of this appendix to hold family size and composition constant (reported in the second row of part B), and then I use this simulated data to follow the techniques described in section C of this appendix to further hold income distribution and level constant (reported in the first row of part B).
The residuals from both of these two-part simulations are identical and reflect the further effect of the correlation between changes in income and changes in family composition and family size.