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The Distributional Aspects of Social Security and Social Security Reform$

Martin Feldstein and Jeffrey B. Liebman

Print publication date: 2002

Print ISBN-13: 9780226241067

Published to Chicago Scholarship Online: February 2013

DOI: 10.7208/chicago/9780226241890.001.0001

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(p.447) Appendix

(p.447) Appendix

Estimating Life Tables That Reflect Socioeconomic Differences In Mortality

Source:
The Distributional Aspects of Social Security and Social Security Reform
Publisher:
University of Chicago Press

Three of the papers in this volume (Brown, Feldstein-Liebman, and Liebman) make use of a new set of life tables that differentiate mortality experience by sex, race, and education level.1 This appendix describes the methodology used to develop these life tables and presents the estimates themselves.

The underlying data for these life tables come from the National Longitudinal Mortality Survey (NLMS). The NLMS was created by matching individuals who were in the Current Population Survey (CPS) between 1979 and 1985 to death records from the National Death Index (NDI). This match occurred in 1989. The total sample population from the CPS data is 1,046,959, and by 1989 the NDI had death records for 69,385 of the individuals in the sample population. Since the earliest data are from 1979 and the match to mortality records occurred in 1989, each person in the sample was followed for a maximum of ten years (Rogot et al. 1992).

Because the CPS contains detailed demographic information for each sample member, this matched data set can be used to generate mortality estimates that take into account a wide variety of demographic characteristics. However, since the probability of death at a given age is small, especially at younger ages, we limited our categories to ones representing race, (p.448) ethnicity, sex, and education level. We chose these characteristics because, unlike household income or marital status, these are largely predetermined and invariant to change at the time an individual enters the labor force.2,3 In particular, we divided the data into six race-ethnicity-gender categories: non-Hispanic white males, non-Hispanic black males, and Hispanic males (table A.1), and non-Hispanic white females, non-Hispanic black females, and Hispanic females (table A.2). The number of education groupings for which we could produce reliable estimates varied by race and ethnicity. For white males and females, we developed separate life tables for individuals who did not complete high school (“less than high school,” or LTHS), who completed high school but did not complete a four-year college degree (HS+), and who completed four years of college (COL). For black males and females, we did not have a sufficient number of observations to reliably estimate separate life tables for individuals with a college degree. Therefore, the Feldstein-Liebman and Liebman papers (chapters 7 and 1, respectively) used two education subgroups for blacks: individuals who did not complete high school (“LTHS”); and high school graduates, including those who completed college (“HS +”). Brown (chapter 10) used a slightly different approach. For black high school graduates who did not complete college, he estimated a life table that corresponded to that population. Then, for black college graduates he assumed that the ratio of mortality between black college graduates and other blacks was the same as the ratio for whites. The Hispanics samples were not large enough to develop reliable estimates with any variation by education level. Thus, we generate a total of twelve different life tables (six for whites, four for blacks, and two for Hispanics).

The NLMS data file that is available to researchers outside of the U.S. government contains roughly half the observations in the full NLMS. This sample was not sufficient for us to produce precise estimates for all of our groups, so instead we used summary tabulations produced by a Census Bureau employee from the full NLMS that contained nonparametric mortality rates for each age-by-sex-by-ethnicity-by-education group cell from age twenty-five to eighty-four. For example, the nonparametric sample estimate for the mortality rate of sixty-five-year-old college-educated white males is the number of sixty-five-year-old college-educated white males in the NLMS matched to a death record from the NDI before age sixty-six, (p.449)

Table 10A.1 Relative Mortality Rates for Males Aged 25–100 by Race, Ethnicity, and Education: Ratio of Subgroup Male Mortality to General Population Male Mortality

White

Black

Hispanic

Age

LTHS

HS+

COL

LTHS

HS+

All

25

1.038637

0.880302

0.719848

3.955698

1.655740

2.734350

26

1.056137

0.881084

0.707339

3.913315

1.673879

2.646963

27

1.073668

0.882054

0.694940

3.868796

1.691346

2.558439

28

1.091155

0.883218

0.682718

3.822210

1.708028

2.469093

29

1.108517

0.884582

0.670741

3.773642

1.723815

2.379255

30

1.125676

0.886149

0.659077

3.723197

1.738603

2.289268

31

1.142551

0.887921

0.647796

3.670994

1.752293

2.199477

32

1.159065

0.889897

0.636961

3.617167

1.764794

2.110232

33

1.175142

0.892075

0.626637

3.561860

1.776026

2.021876

34

1.190709

0.894452

0.616884

3.505232

1.785916

1.934746

35

1.205698

0.897022

0.607757

3.447445

1.794407

1.849165

36

1.220047

0.899778

0.599307

3.388671

1.801450

1.765438

37

1.233700

0.902711

0.591579

3.329082

1.807010

1.683851

38

1.246605

0.905811

0.584612

3.268853

1.811065

1.604665

39

1.258721

0.909067

0.578441

3.208154

1.813603

1.528113

40

1.270009

0.912466

0.573091

3.147154

1.814625

1.454403

41

1.280443

0.915995

0.568585

3.086016

1.814145

1.383712

42

1.290000

0.919640

0.564936

3.024894

1.812183

1.316186

43

1.298666

0.923387

0.562155

2.963932

1.808773

1.251945

44

1.306431

0.927220

0.560244

2.903267

1.803954

1.191078

45

1.313294

0.931124

0.559202

2.843021

1.797774

1.133645

46

1.319259

0.935084

0.559022

2.783308

1.790286

1.079683

47

1.324334

0.939085

0.559694

2.724226

1.781547

1.029204

48

1.328532

0.943113

0.561204

2.665864

1.771621

0.982198

49

1.331870

0.947151

0.563533

2.608297

1.760572

0.938636

50

1.334368

0.951187

0.566664

2.551589

1.748464

0.898471

51

1.336049

0.955206

0.570571

2.495793

1.735367

0.861644

52

1.336936

0.959194

0.575233

2.440950

1.721345

0.828080

53

1.337057

0.963139

0.580622

2.387094

1.706465

0.797696

54

1.336438

0.967026

0.586713

2.334248

1.690792

0.770403

55

1.335107

0.970844

0.593479

2.282425

1.674387

0.746102

56

1.333091

0.974580

0.600891

2.231633

1.657312

0.724694

57

1.330417

0.978223

0.608922

2.181873

1.639623

0.706073

58

1.327113

0.981759

0.617543

2.133139

1.621376

0.690137

59

1.323204

0.985179

0.626728

2.085421

1.602620

0.676780

60

1.318716

0.988471

0.636447

2.038704

1.583406

0.665899

61

1.313671

0.991622

0.646674

1.992969

1.563777

0.657391

62

1.308093

0.994622

0.657382

1.948194

1.543777

0.651157

63

1.302003

0.997459

0.668543

1.904357

1.523443

0.647102

64

1.295420

1.000122

0.680130

1.861430

1.502812

0.645132

65

1.288362

1.002599

0.692117

1.819387

1.481916

0.645157

66

1.280847

1.004878

0.704476

1.778200

1.460786

0.647093

67

1.272891

1.006948

0.717182

1.737841

1.439451

0.650857

68

1.264506

1.008797

0.730207

1.698282

1.417935

0.656370

69

1.255708

1.010413

0.743524

1.659495

1.396264

0.663559

70

1.246509

1.011784

0.757105

1.621453

1.374459

0.672350

71

1.236920

1.012898

0.770923

1.584131

1.352542

0.682675

72

1.226954

1.013745

0.784948

1.547505

1.330532

0.694468

73

1.216621

1.014312

0.799153

1.511553

1.308451

0.707664

74

1.205934

1.014589

0.813506

1.476254

1.286316

0.722201

75

1.194905

1.014568

0.827979

1.441593

1.264148

0.738019

76

1.183546

1.014240

0.842540

1.407556

1.241967

0.755056

77

1.171874

1.013599

0.857159

1.374131

1.219795

0.773252

78

1.159905

1.012640

0.871804

1.341313

1.197656

0.792549

79

1.147658

1.011361

0.886445

1.309099

1.175575

0.812886

80

1.135157

1.009765

0.901052

1.277492

1.153581

0.834201

81

1.122428

1.007857

0.915595

1.246500

1.131705

0.856433

82

1.109503

1.005646

0.930048

1.216136

1.109984

0.879519

83

1.096419

1.003150

0.944385

1.186421

1.088458

0.903395

84

1.083219

1.000391

0.958587

1.157381

1.067174

0.927993

85

1.069955

0.997400

0.972637

1.129050

1.046183

0.953247

86

1.056685

0.994215

0.986525

1.101471

1.025545

0.979089

87

1.043478

0.990886

1.000248

1.074694

1.005327

1.005450

88

1.030413

0.987474

1.013813

1.048778

0.985603

1.032261

89

1.017579

0.984053

1.027238

1.023793

0.966457

1.059453

90

1.005079

0.980710

1.040552

0.999819

0.947983

1.086959

91

0.993029

0.977549

1.053801

0.976945

0.930286

1.114714

92

0.981558

0.974689

1.067045

0.955273

0.913482

1.142657

93

0.970812

0.972269

1.080364

0.934917

0.897697

1.170727

94

0.960950

0.970445

1.093856

0.916000

0.883071

1.198870

95

0.952149

0.969394

1.107638

0.898660

0.869755

1.227032

96

0.944598

0.969307

1.121844

0.883042

0.857911

1.255159

97

0.938498

0.970394

1.136623

0.869302

0.847711

1.283189

98

0.934057

0.972871

1.152125

0.857596

0.839330

1.311045

99

0.931479

0.976952

1.168491

0.848078

0.832939

1.338615

100

0.930953

0.982838

1.185832

0.840889

0.828698

1.365729

Note: See text for explanation of abbreviations and makeup of education subgroups.

(p.450) divided by the total number of sixty-five-year-old college-educated white males in the NLMS sample. These tabulations were generously provided to us by Hugh Richards and have been used previously in Richards and Barry (1998).

There are several reasons that we do not use these nonparametric estimates directly. First, the sample sizes for some cells are very small, and therefore some single year of age mortality probabilities have very large standard errors. Consequently, estimates are often not a monotonic function of age for a particular race-gender-education classification, suggesting that some smoothing would be desirable. Second, we needed to construct (p.451)

Table 10A.2 Relative Mortality Rates for Females Aged 25–100 by Race, Ethnicity, and Education: Ratio of Subgroup Female Mortality to General Population Female Mortality

White

Black

Hispanic

Age

LTHS

HS+

COL

LTHS

HS+

All

25

1.131830

1.042855

0.864270

2.649466

0.711158

1.185041

26

1.138879

1.033564

0.853033

2.681900

0.759445

1.166252

27

1.146161

1.024175

0.841630

2.713179

0.808499

1.146999

23

1.153650

1.014734

0.830117

2.743030

0.858107

1.127352

29

1.161315

1.005292

0.818552

2.771177

0.908037

1.107389

30

1.169120

0.995900

0.807000

2.797344

0.958041

1.087195

31

1.177026

0.986615

0.795528

2.821258

1.007857

1.066864

32

1.184991

0.977492

0.784209

2.842658

1.057211

1.046494

33

1.192969

0.968588

0.773113

2.861297

1.105824

1.026188

34

1.200911

0.959962

0.762316

2.876947

1.153415

1.006053

35

1.208766

0.951668

0.751889

2.889405

1.199706

0.986194

36

1.216484

0.943759

0.741903

2.898496

1.244427

0.966718

37

1.224014

0.936286

0.732426

2.904078

1.287320

0.947730

38

1.231305

0.929294

0.723523

2.906042

1.328144

0.929328

39

1.238307

0.922825

0.715251

2.904318

1.366680

0.911606

40

1.244973

0.916912

0.707664

2.898870

1.402733

0.894650

41

1.251260

0.911586

0.700807

2.889702

1.436135

0.878538

42

1.257127

0.906868

0.694717

2.876854

1.466749

0.863336

43

1.262538

0.902776

0.689426

2.860399

1.494467

0.849103

44

1.267460

0.899317

0.684956

2.840443

1.519212

0.835883

45

1.271865

0.896495

0.681321

2.817122

1.540938

0.823713

46

1.275732

0.894306

0.678527

2.790594

1.559631

0.812615

47

1.279043

0.892741

0.676576

2.761040

1.575300

0.802604

48

1.281783

0.891785

0.675459

2.728658

1.587985

0.793680

49

1.283944

0.891419

0.675163

2.693658

1.597746

0.785839

50

1.285521

0.891619

0.675669

2.656258

1.604664

0.779064

51

1.286514

0.892359

0.676953

2.616684

1.608839

0.773333

52

1.286924

0.893609

0.678987

2.575160

1.610383

0.768617

53

1.286758

0.895336

0.681741

2.531909

1.609422

0.764879

54

1.286022

0.897507

0.685181

2.487152

1.606090

0.762081

55

1.284729

0.900087

0.689271

2.441099

1.600527

0.760178

56

1.282888

0.903041

0.693974

2.393957

1.592875

0.759124

57

1.280515

0.906332

0.699253

2.345918

1.583281

0.758872

58

1.277623

0.909925

0.705068

2.297165

1.571888

0.759371

59

1.274228

0.913786

0.711383

2.247871

1.558841

0.760572

60

1.270345

0.917878

0.718159

2.198195

1.544278

0.762424

61

1.265990

0.922170

0.725359

2.148283

1.528335

0.764878

62

1.261180

0.926627

0.732947

2.098270

1.511143

0.767885

63

1.255929

0.931220

0.740887

2.048281

1.492826

0.771397

64

1.250253

0.935917

0.749147

1.998427

1.473503

0.775369

65

1.244169

0.940690

0.757692

1.948808

1.453286

0.779756

66

1.237690

0.945511

0.766492

1.899516

1.432281

0.784515

67

1.230832

0.950354

0.775515

1.850632

1.410588

0.789605

68

1.223608

0.955194

0.784734

1.802229

1.388299

0.794988

69

1.216033

0.960008

0.794121

1.754371

1.365503

0.800626

70

1.208121

0.964774

0.803648

1.707115

1.342281

0.806484

71

1.199884

0.969470

0.813292

1.660513

1.318708

0.812530

72

1.191338

0.974077

0.823027

1.614610

1.294857

0.818731

73

1.182495

0.978578

0.832833

1.569445

1.270794

0.825059

74

1.173370

0.982955

0.842686

1.525054

1.246581

0.831488

75

1.163978

0.987194

0.852568

1.481470

1.222277

0.837990

76

1.154333

0.991282

0.862458

1.438720

1.197938

0.844545

77

1.144453

0.995206

0.872340

1.396832

1.173615

0.851130

78

1.134354

0.998956

0.882198

1.355828

1.149360

0.857728

79

1.124055

1.002525

0.892016

1.315733

1.125221

0.864320

80

1.113577

1.005906

0.901782

1.276566

1.101244

0.870894

81

1.102942

1.009098

0.911485

1.238350

1.077476

0.877438

82

1.092175

1.012098

0.921115

1.201104

1.053961

0.883943

83

1.081303

1.014909

0.930667

1.164851

1.030744

0.890404

84

1.070359

1.017537

0.940135

1.129611

1.007872

0.896818

85

1.059375

1.019993

0.949520

1.095409

0.985389

0.903188

86

1.048391

1.022291

0.958823

1.062268

0.963344

0.909521

87

1.037452

1.024452

0.968053

1.030215

0.941787

0.915827

88

1.026606

1.026501

0.977222

0.999281

0.920770

0.922125

89

1.015911

1.028472

0.986350

0.969499

0.900349

0.928440

90

1.005432

1.030411

0.995464

0.940908

0.880586

0.934805

91

0.995245

1.032370

1.004602

0.913552

0.861547

0.941266

92

0.985438

1.034420

1.013814

0.887483

0.843308

0.947881

93

0.976114

1.036646

1.023168

0.862764

0.825954

0.954725

94

0.967396

1.039158

1.032753

0.839468

0.809585

0.961894

95

0.959433

1.042093

1.042685

0.817688

0.794317

0.969514

96

0.952405

1.045627

1.053121

0.797536

0.780291

0.977746

97

0.946537

1.049985

1.064264

0.779154

0.767676

0.986801

98

0.942108

1.055461

1.076390

0.762723

0.756685

0.996955

99

0.939474

1.062435

1.089863

0.748476

0.747586

1.008572

100

0.939091

1.071408

1.105174

0.736714

0.740721

1.022134

Note: See text for explanation of abbreviations and makeup of education subgroups.

(p.452) out-of-sample estimates of age-specific mortality rates, because mortality estimates for people above age eighty-four are required for our analysis. Third, this methodology yields data for a period mortality table describing the mortality experience of people alive during the NLMS. In contrast, our research projects required cohort life tables representative of people born in a given year.

To smooth the data, we estimated a nonlinear model for age-specific mortality separately for each group. With the proper choice of three parameters, the Gompertz-Makeham survival function can be applied from (p.453) age twenty-five almost to the end of life (Jordan 1991). The Gompertz-Makeham formula is usually written as

AppendixEstimating Life Tables That Reflect Socioeconomic Differences In Mortality

where x is age, lx is the number of people in the population alive at age x, and lo is the number of people in the population alive at birth. The parameters to be estimated are c, g, and s. Note that if lo is normalized to 1, then lx is the cumulative probability of survival at age x, and we can define qx, the mortality rate at age x, as

AppendixEstimating Life Tables That Reflect Socioeconomic Differences In Mortality

Then, rearranging the Gompertz-Makeham formula and solving for qx yields the equation that we estimate using nonlinear least squares:

AppendixEstimating Life Tables That Reflect Socioeconomic Differences In Mortality

We estimate the parameters in this equation separately for each of our twelve groups described above. By substituting the nonlinear least squares estimates of c, g, and s into the equation above, fitted estimates of mortality rates for a particular group at age x are formed. This approach guarantees that the fitted mortality rates are a monotonic function of age. Figure A.1 shows the fitted mortality rates and original nonparametric estimates for two of our twelve groups. The fitted values track the original data quite closely.4

As we mentioned above, the estimates from the NLMS are period estimates. Our basic approach to producing cohort life tables is to assume that the ratio of age-specific mortality in our disaggregated groups to more aggregated age-specific mortality rates stays constant over time. Then, to generate disaggregated cohort life tables, we can simply apply a ratio of period mortality rates from our data to the more aggregated cohort life tables published by the Social Security Administration or Census Bureau. For example, the cohort age-specific mortality estimates for white female college graduates could be constructed as

AppendixEstimating Life Tables That Reflect Socioeconomic Differences In Mortality

(p.454)

AppendixEstimating Life Tables That Reflect Socioeconomic Differences In Mortality

Fig. A.1 Comparison of fitted and nonparametric estimates of mortality rates

(p.455) The assumption that relative mortality rates stay constant over time is clearly a strong one, though there is some evidence, for example, that differences in mortality rates between socioeconomic groups were not shrinking in the late twentieth century and may actually have been widening (Pappas et al. 1993; Preston and Taubman 1994). Moreover, even if relative mortality rates among demographic subgroups remain constant, changes in the share of the population within each demographic group could alter the relationship between a subgroup's mortality ratio and the aggregate ratio.

Because there has been such significant change in the educational attainment of Americans over the past century, we make one further adjustment to our data. Specifically, we produce aggregate mortality tables from our data weighting the various subgroups to represent the distribution of education rates (within sex-by-race cells) for thirty- to thirty-four-year-olds in the March 1999 CPS. Ratios of subgroup mortality to these aggregates will therefore be appropriate for use with aggregate cohort life tables for cohorts born in the late 1960s and for other cohorts with a similar distribution of educational attainment.5

Figure A.2 illustrates the impact of this adjustment for white males. The dashed line displays our unadjusted fitted estimates, which lie almost exactly on top of the published Vital Statistics period life table for white males in 1989–91. The dark line shows our estimate of aggregate white male mortality under the counterfactual assumption that the educational shares of the population had remained constant over time at rates like those for recent cohorts. Because recent cohorts have a larger share of individuals in the high-education (and therefore low-mortality) groups than the older cohorts in the NLMS did, this reweighted aggregate estimate shows lower mortality rates than the estimates based on the actual period data do.

Table A.1 contains the estimated ratios of subgroup age-specific mortality rates to aggregate mortality rates for males. Table A.2 contains the identical data for females.

One last issue requires discussion. Table A.2 indicates that mortality estimates for college-educated white males above age eighty-seven are higher than mortality rates for white males with less education at similar ages. A similar phenomenon appears at age ninety among white females. Because our summary tabulations from the NLMS do not contain data for ages above 84, the fitted estimates for ages 85 to 100 are heavily dependent on the relatively noisy raw data at slightly younger ages. Hence, it is possible that this crossover in the mortality estimates is simply the product (p.456)

AppendixEstimating Life Tables That Reflect Socioeconomic Differences In Mortality

Fig. A.2 The impact of adjusting mortality rates for changes in education shares

(p.457) of measurement error in the NLMS data combined with the functional form assumptions implicit in the Gompertz-Makeham formula. However, it is also possible that the crossover is a real phenomenon. In particular, if it is assumed that there exists a maximum age of survival, then at some very high age we would necessarily observe a crossover.6

References

Deaton, Angus, and Cristina Paxson. 2001. Mortality, education, income and inequality among American cohorts. In Themes in the economics of aging, ed. David A. Wise, 129–70. Chicago: University of Chicago Press.

Jordan, C. W. 1991. Life contingencies. 2d. Chicago: Society of Actuaries.

Pappas, Gregory, Susan Queen, Wilbur Hadden, and Gail Fisher. 1993. The increasing disparity in mortality between socioeconomic groups in the United States: 1960 and 1986. New England Journal of Medicine 329 (2): 103–9.

Preston, Samuel H., Irma T. Elo, Ira Rosenwaike, and Mark Hill. 1996. African-American mortality at older ages: Results of a matching study. Demography 33 (2): 193–209.

Preston, Samuel, and Paul Taubman. 1994. Socioeconomic differences in adult mortality and health status. In Demography of aging, ed. Linda G. Martin and Samuel H. Preston, 279–318. Washington, D.C.: National Academy Press.

Richards, Hugh, and Ronald Barry. 1998. U.S. life tables for 1990 by sex, race, and education. Journal of Forensic Economics 11 (1): 9–26.

Rogot, Eugene, et al. 1992. A mortality study of 1.3 million persons by demographic, social and economic factors: 1979–1985 follow-up. Bethesda, Md.: National Institutes of Health and National Heart, Lung, and Blood Institute. (p.458)

Notes:

Jeffrey R. Brown is an assistant professor of public policy at the John F. Kennedy School of Government, Harvard University, and a faculty research fellow at the National Bureau of Economic Research. Jeffrey B. Liebman is associate professor of public policy at the John F. Kennedy School of Government, Harvard University, and a faculty research fellow at the National Bureau of Economic Research. Joshua Pollet is a graduate student in the department of economics at Harvard University and a research assistant at the National Bureau of Economic Research.

(1.) Because we have continued to refine these mortality tables over time, the three papers use slightly different versions of these mortality tables.

(2.) In addition to the demographic variables, the NLMS includes a CPS-based measure of annual family income. While recent research by Deaton and Paxson (2001) has shown that income has an impact on mortality even when race and education are controlled for, we chose not to incorporate income into our analysis. It is difficult to establish causality when examining correlations between contemporaneous income and mortality, because of the possibility that negative health shocks affect both measures.

(3.) Some individuals do, of course, obtain additional education after age twenty-five, the initial age used in our analysis.

(4.) This is true for the other ten groups as well.

(5.) We used the late 1960s birth cohorts in creating our weights because they were the youngest for whom we could safely assume that nearly all members of the cohort had completed their educations.

(6.) See Preston et al. (1996) for a discussion of the racial crossover in mortality.