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Regarding median years to retirement and worklife expectancy.

I. Introduction

In the last issue of this Journal (Spring/Summer 1997), Tamorah Hunt, Joyce Pickersgill, and Herbert Rutemiller (HPR) use labor force participation rates for 1992-93 to calculate median years to retirement and worklife expectancies for men and women by educational attainment. They compare their worklife results with those of Shirley Smith (1986) and James Ciecka, Thomas Donley, and Jerry Goldman (CDG) (1995, 1996). This paper contains some general comments about median years to retirement and work-life expectancies and some specific comments on the paper by HPR.

II. Median Years to Retirement and Worklife Expectancies

Consider the hypothetical data in Table 1. Varying portions of a population of 100 people are active between the ages of 20 and 60, and assume that there is no labor force activity beyond age 60. For simplicity, we ignore mortality; therefore, all 100 people survive to age 60. In Panel A of Table 1 the participation rate is 4% for ages 20-49, 2% at age 50, and 1% for ages 51-60. In Panel B the participation rate is 40% for ages 20-49, 20% at age 50, and 10% for ages 51-60. Panel C is identical to Panel B except the participation rate is 20% at age 20. The implied number of people active on a yearly basis and cumulative years of activity are shown in each panel. Finally, median years to retirement (MYR) and worklife expectancies (WLE) at age 20 are shown in each panel. Two types of worklife expectancies are computed: [WLE.sub.g] is the WLE for the general population at age 20 (i.e., relative to all 100 people in the population) and [WLE.sub.a] is the WLE for those who are active at age 20. These hypothetical data are intentionally exaggerated in order to clearly illustrate the general remarks presented below. Although some of these remarks may be obvious, they may be easily forgotten when comparisons are made between MYR and WLE.

[TABULAR DATA 1 NOT REPRODUCIBLE IN ASCII]

To define MYR, let [l.sub.x] and [a.sub.x] denote the number of survivors and the number of actives in a cohort at exact age x. Assume that ax is a monotonically decreasing function, i.e., [a.sub.x] > [a.sub.x+j], j = 1,2,...,m, where [a.sub.x+m] = 0. The median age at retirement, [k.sub.x], is the age at which 1) holds:

1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the number of actives [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] only includes those who were active at age x and all labor force separations are assumed to be final.(1) The MYR is

2) MYR = [k.sub.x]-x..

Remark 1. MYR only depends of the activity ratios ([a.sub.x+j]/[a.sub.x]), j=1,2,...,m.

Although Panels A and B in Table 1 illustrate very different amounts of labor force activity, MYR is 30 years in both panels because the activity ratios are equal at any exact age in both panels. In general, let [Lambda]> 0; and let the initial number of actives be [a'.sub.x] = [[Lambda]a.sub.x] [is less than or equal to] [l.sub.x], where [Lambda] captures the level of labor force activity. Then,

3) [a'.sub.x+j] / [a'.sub.x] = [[Lambda]a.sub.x+j] / [[Lambda]a.sub.x] = [a.sub.x+j] / [a.sub.x]

and in particular

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

implying the same MYR as in 2). This result implies that MYR does not capture the level of labor force activity, but rather focuses exclusively on the activity ratio.

Corollary la. MYR is invariant to labor force participation levels beyond the median point (i.e., in the tail of the distribution).

Suppose that in Panel B of Table 1 participation rates at age 51 and beyond held constant at 19% rather than 10%. This would result in an additional 90 cumulative years of activity and would increase the [WLE.sub.g] from 13.2 years to 14.1 years. MYR, however, would not reflect this change. In general, all information beyond the median point is disregarded in the calculation of MYR. Consequently, many changes in retirement patterns will not be captured by this estimate of retirement behavior.

Remark 2. MYR cannot be meaningfully applied to the general population (i.e., the sum of actives and inactives) for any group.

What does it mean to say that the MYR for the general population in Panel A (i.e., 100 people) is 30 years? At peak participation there are only four people in the labor force, then two people, and then only one person in the labor force. To ascribe a median of 30 years of activity to the entire population makes no sense whatsoever.(2)

Corollary 2a. Comparisons between MYR cannot be meaningfully made across general groups even within the same period of time.

Suppose Panel A and Panel B refer to two separate demographic groups for the same time period. Except in a very formal arithmetic or algebraic sense, what is the meaning of saying that Group A and Group B have the same MYR? Both panels show MYR = 30; therefore, a forensic analysis of the loss period using MYR would yield the same length of loss for Groups A and B, even though the activity level of one group is ten-fold that of the other. Moreover, suppose the participation rate was 20% at age 49 in Panel B. The implied MYR would be 29 years, and the loss period would be smaller for Group B than for Group A even though Group B's total activity level would be almost ten times as large.

Corollary 2b. Comparisons between MYR cannot be meaningfully made for the same general group across different periods of time.

Suppose Panel A refers to all women in 1979-80, and Panel B refers to all women in 1992-93. What sense would it make to say that there is no change in MYR between 1979-80 and 1992-93 even though women have expanded their labor force activity by a factor of ten?

Remark 3. The concept of MYR, by definition, is confined to the active population only; and, even then, its interpretation is corrupted because of the manner in which it is calculated.

For only the actives, the MYR can reasonably be interpreted as 30 years in both Panel A and Panel B. Of course, there is no guarantee that the same people are active at each age. When calculated from participation rate data, MYR allows accessions and does not require that separations are final. To the extent that people move in and out of the labor force, participation rates are not tracking the activity of the same group of people; and we know little about years to retirement because the actives at any age are not necessarily a subset of those who were initially active. The appropriate definition of median age at retirement for actives at age x is given in 1); but, in applications, the median age at retirement is computed as the value n such that

4) [u.sub.n] + [v.sub.n] / [a.sub.x] = .5

where [u.sub.n] is the number of people who were active at age x who are still active at age n, and [v.sub.n] is the number of people who were inactive at age x who are active at age n. The resulting [MYR.sup.*] is then

5) [MYR.sup.*] = n - x.

Assuming [v.sub.n] > 0, [MYR.sup.*] > MYR.

Participation rates rise until age 34 for men and until age 42 for women (HPR, Appendix A, Tables 1 and 2). Therefore, for x < 34 for men (and x < 42 for women), the activity ratio ([a.sub.x+j] / [a.sub.x]) exceeds one for some j. For example, for 18-year-old men, the activity ratio ([a.sub.x+j] / [a.sub.x]) exceeds one until age 62; for 18-year-old women the ratio exceeds one until age 58. Even after allowing for mortality, years of activity are clearly being contributed by people who were not in the labor force at age 18, but their activity lengthens the measure of the [MYR.sup.*] for those that were active at 18.

The problem caused by labor force accessions and temporary separations does not arise with the increment-decrement model of WLE. In fact, entries and exits lie at the heart of the increment-decrement representation of the labor force in which transition probabilities are applied to a group of initial actives and inactives, thereby allowing separations and accessions to occur over time. However, the potential for inactives to become active later gives rise to calculation anomalies for MYR that contradict its own definition.

Remark 4. There is no method to calculate MYR for those who are currently inactive.

However, WLE can be computed for inactives by using the increment-decrement model of labor force activity.

Remark 5. The median age at retirement for young people can exceed the median age at retirement for older people.

In Panel C the median age at retirement is 51 for a 20-year-old person. However, the median age at retirement for a 21-year-old person is age 50. This is a curious result in that it implies that members of the 20-year-old group are credited with two additional years of activity even though they are only one year younger than members of the 21-year-old group. In other words, MYR ignores information about labor force activity prior to, and after, the median age at retirement.

In general, we have [l.sub.x] = [a.sub.x] + [i.sub.x], where [i.sub.x] is the number of those inactive in the labor force at age x. Calculation problems arise in computing MYR when using either participation rates or transition probabilities because [a.sub.x+i] counts contributions from [i.sub.x] inactives of an earlier age. In addition to the practical problem of computation, there are several theoretical problems stemming from the definition. In order for the median age at retirement to exist, ax must decrease as a function of x; moreover, the nature of the correspondence which associates [a.sub.x] with x is crucial. The definition of MYR essentially requires this correspondence to be inverted, i.e., that an age be associated with a given number of actives. The fact that numbers of actives are influenced by in-migration and out-migration changes the information embedded in the correspondence unless exactly the original cohort is traced throughout. The averaging effect which occurs over

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when computing WLE preserves more information than the MYR process, which relies only on a transformation of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to the age domain.

Remark 6. WLE can be meaningfully applied to the general population and meaningful comparisons can be made across groups and across time.

The level of activity captured in Panel B is ten times as large as the activity in Panel A, and the corresponding [WLE.sub.g] is ten times larger for Panel B (i.e., 13.2 years versus 1.32 years). In addition, [WLE.sub.g] = 13.0 years in Panel C, a number only slightly smaller than the worklife expectancy in Panel B because there is lees activity contributed by the 20-year-old group in Panel C.

Remark 7. Participation rates can be used to calculate WLE for a general group.

The [WLE.sub.g] calculations in Table 1 are based on participation rates, and their relative magnitudes seem intuitively plausible as discussed in Remark 6. When worklife is computed over all living members of a group (whether active or inactive) at age x, a year of activity supplied by a person at age x + j, j = 1,2,3,...,m should count in [WLE.sub.g] whether, or not, that person was active at age x. For example Figure 1 shows the number of males alive at ages 1874 from an initial population of 100,000 and the number of actives at each age.(3) [WLE.sub.g] at age 18 is simply the area under the labor force curve dividend by the number of survivors at age 18. Figure 1 also shows the corresponding functions for women.

[Figure 1 ILLUSTRATION OMITTED]

Remark 8. Participation rates should not be used to calculate WLE for the active part of a group.

Panel C of Table 1 reveals the basic problem. Many of the years of activity embedded in [WLE.sub.a] are contributed by people who are not part of the labor force at age 20. Therefore, the numerator of [WLE.sub.a] is too large relative to the 20 people in the denominator of the ratio, implying [WLE.sub.a] = 65.0 which is almost twice as large as [WLE.sub.a] = 33.0 in Panel B even though both panels exhibit almost identical cumulative years of activity. How do we fix this problem? Consider 18-year-old men as an example. In its conventional model, the Bureau of Labor Statistics (BLS) (Smith, 1982) used the following procedure: (1) Find the peak participation rate (this is 94.3% at age 34 using the HPR data). (2) Pretend that the labor force equals this maximum participation rate times the number or survivors between ages 18-34. This assumption essentially creates a new labor force curve by extending its peak point to the vertical axis (see Figure 2). (3) The labor force consists of the product of age-specific participation rates and survivors for ages beyond age 34, as shown in Figure 2. (4) Total years of activity, the numerator of the [WLE.sub.a] calculation, is the sum of activity in (2) and (3). (5) The denominator of the [WLE.sub.a] is the product of the maximum participation rate and the number of survivors at age 18. In other words, the BLS solved the problem by inflating both the numerator and denominator of the worklife expectancy calculation to arrive at [WLE.sub.a]. It goes without saying that the magnitude of a ratio is affected in some unpredictable manner when both its numerator and denominator are inflated (Smith, 1982). Figure 2 also illustrates the BLS procedure for estimating worklife in its conventional model for active 18-year-old women.

[Figure 2 ILLUSTRATION OMITTED]

General Conclusions. MYR should not be applied to a general population (Remark 1, Remark 2, and Corollaries 2a and 2b), but WLE can be computed for the general population (Remark 6). When applied to an active population, MYR is corrupted as explained in Remarks 3 and 5. MYR cannot be applied to an inactive population (Remark 4). [WLE.sub.a] for a general population can be based on participation rates (Remark 7). [WLE.sub.a] based on participation rates, for actives is corrupted as explained in Remark 8.

III. Specific Comments About the Paper by Hunt, Pickersgill, and Rutemiller

The age, gender, educational attainment specific participation rates for 1992-93 (HPR, Appendix A, Tables 1 and 2) can be useful to forensic economists. Some uses include direct application in the life-participationemployment model and indirect application to benchmark results from other studies like increment-decrement models of worklife expectancies. Users of the median years to retirement tables should keep in mind that they only apply to active individuals and not to the general population or the inactive population. Moreover, the median years to retirement tables are not of the MYR type defined in 1) and 2) but of the [MYR.sub.*] type defined in 4) and 5). Remarks 2, 3, and 5 apply to the median years to retirement estimates in HPR, Appendix A, Tables 3-5. Remark 1 explains why the [MYR.sub.*] for men and women are very close to each other (e.g., 31.6 years and 31.4 years for 30-year-old men and women, respectively) even though men have much larger participation rates than women. In addition, comparisons between groups (HPR, Appendix A, Tables 4, and 5) and across time (HPR, Appendix B, Tables 1-4) are affected by the issues raised in Remarks 2, 3, and 5.

In the section of their paper entitled "Worklife Expectancy Calculated Using Labor Force Participation Rates and Transition Probabilities," HPR compare their WLE indirect corrected estimates for actives to the WLE estimates generated by Smith (1986). They conclude that their technique yields results that are very close to Smith's. However, HPR never use independent BLS data for 1979-80 to generate their WLE estimates. Rather, they apply their techniques to activity levels generated by 1979-80 transition probabilities used in Smith's increment-decrement model. At a later point in their paper, HPR apply their technique to the period 1992-93; but they do something different in this application. They actually use independently supplied BLS participation rates to generate WLE estimates. They then compare their WLE estimates with those generated by CDG (1995, 1996). Their corrected estimates are not as close as they are in the 1979-80 period. This should not be surprising because the indirect WLE estimates for 1979-80 are based on transition probabilities, but the indirect WLE estimates for 1992-93 are based on independent BLS participation rates. The indirect WLE estimates for 1992-93 generally exceed the WLE estimates by CDG. Moreover, differences are larger for actives and for less educated groups (whose members probably have higher turnover rates than general populations). Remark 8 explains the problems encountered when worklife expectancies for actives are based on participation rates--when both the numerator and denominator of a ratio are inflated, the effect on the ratio is unknown.

HPR's corrected estimates assume the following (consider an 18-year-old as an example): (1) Assume that the participation rate is 100% at ages 18, 19, and 20. (2) Pretend that the labor force equals 100% times the number of survivors between ages 18-20. (3) The labor force consists of the product of age-specific participation rates and survivors beyond age 20. (4) Total years of activity, the numerator of the [WLE.sub.a] calculation, is the sum of activity in (2) and (3). (5) The denominator of the [WLE.sub.a] is the number of survivors at age 18. In other words, HPR inflate both the numerator and denominator of the worklife expectancy calculation, but in a manner different than the BLS, to arrive at [WLE.sub.a] (see Figure 3). The HPR procedure seems to be less extreme than the BLS procedure, but it still suffers from the similar fundamental problems, viz., it creates labor force activity for three years that does not exist, it counts activity after three years that is contributed by people who were not initially active, and it assumes an initially active population that is too large. It follows that this approach affects the WLE ratio in an unpredictable manner (Smith, 1982).

[Figure 3 ILLUSTRATION OMITTED]

None of the foregoing problems exists in the increment-decrement formulation of [WLE.sub.a] because transition probabilities are allowed to act on a group of arbitrary size that is assumed to be active. No activity is ever supplied by anyone who was not an initial member of the group.

A significant part of HPR's paper focuses on the merits of basing median years to retirement and worklife expectancy estimates on participation rates as opposed to transition probabilities. HPR favor the use of participation rates. In summarizing they say:

1. For estimating median years to retirement, participation rates

yield exactly the same estimates as transition probabilities;

and participation rates are simpler to obtain and less subject

to bias.

2. For estimating worklife expectancies for a general population,

the situation is the same as the previous statement about median

years to retirement.

3. For estimating worklife expectancies for an active

population, transition probabilities are required for a direct

calculation; but "corrected estimates" based on participation

rates yield reasonably close estimates.

We do not believe that the first statement is demonstrated in the paper. If the statement refers to the general population, the concept of median years to retirement has little meaning (Remark 2) and the statement itself has little meaning.(4) Likewise, the statement has little meaning if median years to retirement refers to the inactive population (Remark 4). If the statement refers to an active population, then Remarks 3 and 5 come into play for median years to retirement estimates based on participation rates. Remark 3 is especially important because it shows that median years to retirement estimates based on participation rates are upward biased. Transition probabilities, when applied to an initially active population, are not subject to the problems discussed in Remarks 3 and 5.

In regard to the second statement, it is the case that worklife expectancies for the general population can be computed from participation rate data. Comparisons (in Appendix C of the HPR paper) show participation-rate based worklife expectancies exceeding transition-probability based worklife expectancies for 1992-93. The differences in the two types of estimates are about 4.4% for all men and 2.6% for all women at age 18. By age 30, the percentage difference is approximately 3% for all men and 2% for all women. This seems to be a common phenomenon as shown in Table 2 for 1970 and 1977 data (BLS, 1976 and Smith, 1982), i.e., worklife expectancies based on participation rates in a "conventional model" are consistently larger than those based on transition rates in the increment-decrement model.
Table 2
Comparison of Worklife Expectancies for Men Computed with
Participation Rates and Transition Probabilities

                               Panel A

                     [WLE.sub.g] in 1970 Computed with

          Participation         Transition
Age          Rates             Probabilities       Difference

16          41.4 Years           38.7 Years         2.7 Years
18          40.6                 38.1               2.5
20          39.4                 37.3               2.1
25          35.6                 34.4               1.2
30          31.2                 30.6                .6
35          25.8                 24.7               1.1
40          21.3                 20.3               1.0

                               Panel B

                     [WLE.sub.g] in 1977 Computed with

          Participation         Transition
Age          Rates             Probabilities       Difference

16          40.8 Years           38.5 Years         2.3 Years
18          39.9                 37.8               2.1
20          38.6                 36.8               1.8
25          34.7                 33.4               1.3
30          30.3                 29.2               1.1
35
40




Source: BLS (1976, Table A-1) and Smith (1982, Tables 3, A-2, and B-2)

It is possible that a matched household sample produces biased estimates of transition probabilities. The active-to-active transition probability may be understated for young people who are active but move to another residence (in the same general area) or relocate for better job opportunities. To the extent that these people are lost from the matched sample, the active-to-active transition probability is underestimated and the increment-decrement model picks up too little activity. The inactive-to-active transition probability for young people may be underestimated because people enter the labor force and relocate as they complete their education. During retirement years, the active-to-inactive probability may be understated when people relocate because of retirement, and the increment-decrement model may record too much activity for this age group. The theoretical possibility of bias notwithstanding, Perrachi and Welch (1995) investigated possible biases from matched samples and concluded that no such bias exists.

Table 3 shows participation rates, active-to-active transition probabilities, and inactive-to-active transition probabilities from the matched sample for men and women at various ages (CDG, 1995). Exact-age participation rates, computed from HPR, also are shown; these participation rates are two-year averages which reflect that people are on average one-half year older than their reported age. Inspection of Table 3 reveals that the male matched-sample participation rates from CDG are close to the exact-age participation rates from HPR, the average absolute difference being only .013 and the average difference being only .005, with the CDG participation rates being slightly larger on average. Female participation rates from CDG's matched sample also closely agree with exact-age participation rates from HPR. The average absolute difference and average difference are .014 and .008, respectively. Therefore, there does not seem to be any matched-sample bias; or, if there is bias, it affects the HPR participation rates as well.
Table 3
Transition Probabilities and Participation Rates (P.R.) for
Men and Women at Various Exact Ages in 1992-93

                       Men

        CDG        Probability         Probability        Exact-Age
Age     P.R.     Active-to-Active   Inactive-to-Active    HPR P.R.

19      65.0%         81.6%               39.4%             65.7%
20      69.6          83.4                40.3              72.9
25      90.3          93.5                46.2              90.6
30      93.6          96.9                39.4              94.1
35      95.4          97.8                31.0              94.2
40      94.6          97.6                26.4              93.7
45      94.2          97.3                21.5              93.1
50      91.0          96.4                17.1              90.5
55      86.6          92.7                12.1              84.8
60      72.8          84.2                 7.3              70.9
65      37.2          75.3                 4.3              36.1
70      20.3          72.5                 2.6              18.6

                      Women

        CDG        Probability         Probability        Exact-Age
Age     P.R.     Active to Active   Inactive to Active    HPR P.R.

19      59.8%         80.6%               37.3%             59.8%
20      66.0          81.5                35.1              64.7
25      76.0          88.4                30.8              74.7
30      76.0          90.0                25.7              73.4
35      75.6          91.8                22.5              74.6
40      78.4          93.2                21.4              77.8
45      79.6          93.8                17.5              78.0
50      73.7          93.1                12.3              72.9
55      62.7          90.9                 9.0              64.6
60      52.0          83.4                 5.2              48.4
65      22.9          72.3                 2.9              23.7
70      10.1          65.4                 1.5              11.0




Source: CDG (1995, data used to compute worklife expectancies) and HPR (the participation rate at exact age x is the average of the rates reported by HPR for average ages x-1 and x).

When making comparisons between the estimates of WLE based on participation and transition probabilities the following should be kept in mind. Participation rates represent a snapshot of labor market activity, whereas transition probabilities are the result of a dynamic process and capture, at least to some degree, the duration of the labor force experience. Consequently, participation rates will increasingly overstate the amount of labor force attachment as labor market turnover increases. Given that transition probabilities are predicated on state changes they are not as susceptible to such changes.

At a more practical level, worklife expectancy for the general population have little application in the world of forensic economics where conditional estimates (based on current labor force status) are the norm. Indeed, one of the shortcomings of the increment-decrement WLE model is that it includes demonstrated labor force activity for only one period in the past. If anything, more accurate estimates of WLE would be predicated on an extended history of observed labor force participation. The use of participation rates rather than transition probabilities in the calculation of WLE will tend to decrease the accuracy of WLE rather than increase them.

Regarding HPR's third statement, Remark 8 identifies the difficulty encountered, and the fix that is applied, when worklife expectancies for an active population are based on participation rates. The corrected estimates may be close to those based on transition rates, but the standard for judging the accuracy of the corrected estimates is the set of estimates that are based on transition probabilities.

(1) That is, MYR does not allow entry or re-entry; only permanent separations are allowed to occur. Separations can occur because of death and can be voluntary or involuntary for survivors. If separations were not final, there may be more than one age at which 1) holds, and the MYR in 2) would not be unique. Other anomalies could also occur if separations were not final. For example, suppose 100 people are active at age 20; and ignore mortality for simplicity. Consider two situations. Situation A: Assume 49 people withdraw from the labor force between ages 2145, but the remaining 61 stay active between (yes 21-45. Between ages 46-70, assume that the 49 previous inactives become active, but 49 of the previous 51 actives become inactive. Assume that everyone leaves the labor force at age 71. The median age at retirement is age 71. Now, consider Situation B: Assume 49 people withdraw from the labor force between ages 21-46 Just as in Situation A, but these separations are final. Two additional final withdrawals occur at age 46. The remaining 49 people stay active to age 70, and they leave the labor force at age 71. The median age at retirement is 46. In both Situation A and Situation B, approximately half of the population is active between ages 21-70. However, median age at retirement is 25 years greater in Situation A because of reentry into the labor force.

(2) In fact, for the general population of 100 people in Panel A, the number of actives already is well below 50% at age 20.

(3) Survivors are from Vital Statistics of the United States, 1991, (1995). The labor force is the product of the number of survivors and age-specific participation rates (HPR, 1996).

(4) HPR may indeed be referring to the general population because they use the notation [A.sub.k] to denote the number of actives at average age [k.sub.x] without restricting the people who contribute to [A.sub.k] to have been active at average age x. In fact, the discussion preceding the definition of median years to retirement seems to explicitly allow inactives at average age x to contribute to [A.sub.k]. This gives the impression that HPR's apply the concept of median years to retirement to the general population. However, at another point in their paper, HPR do refer to estimating MYR "...for the active population...."

References

Ciecka, James, Thomas Donley, and Jerry Goldman, "A Markov Process Model of Worklife Expectancies Based on Labor Market Activity in 1992-93," Journal of Legal Economics, 1995, 5(3), 17-41.

-----, "Errata to A Markov Process Model of Worklife Expectancies Based on Labor Market Activity in 1992-93," Journal of Legal Economics, 1996, 6(1), 8185.

Hunt, Tamorah, Joyce Pickersgill, and Herbert Rutemiller, "Median Years to Retirement and Worklife Expectancy," Journal of Forensic Economics, 1997, 10(2), 171-205.

Perrachi, Franco, and Finis Welch, "How Representative Are Matched Cross-Sections? Evidence from the Current Population Survey," Journal of Econometrics, 1995, 68, 153-79.

Smith, Shirley J., Tables of Working Life: The Increment-Decrement Model, Bulletin 2135, US Department of Labor, Bureau of Labor Statistics, February, 1982.

-----, Worklife Estimates: The Effects of Race and Education, Bulletin 2254, U.S. Department of Labor, Bureau of Labor Statistics, February, 1986.

U.S. Department of Health and Human Services, National Center for Health Statistics, Vital Statistics of the United States, 1991, Washington: US Government Printing Office, 1995.

U.S. Department of Labor, Bureau of Labor Statistics, Length of Working Life for Men and Women, 1970, Special Labor Force Report 187, 1976.

James, Ciecka, Thomas Donley and Jerry Goldman(*)

(*) DePaul University, Chicago, IL. Ciecka and Donley, Department of Economics; Goldman, Department of Mathematical Sciences.
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Author:Ciecka, James; Donley, Thomas; Goldman, Jerry
Publication:Journal of Forensic Economics
Date:Sep 22, 1997
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